**Meet the editor**

Ivan Padron was born in Camajuani, Cuba in 1970. He received his bachelor degree in Physics from the University of Havana in 1993, and his Ph. D in Applied Physics from New Jersey Institute of Technology in 2010. His research has been directed towards diaphragm based Fabry-Perot interferometric sensors. His critical thinking and enterprising personality where acknowledged in

two U.S. patents and one patent application, and offered him the opportunity of publish his work in varies peer-reviewed scientific journals, participates in numerous international conferences and to serve as a peer reviewer of the IEEE Sensor Journal. He is a member of the American Physical Society and IEEE Professional Association. His research interests include optical sensors and the application of physics to life sciences and medicine.

Contents

**Preface IX** 

Cheng-Chih Hsu

**Part 1 Interferometry Methods and Research 1** 

Ali Reza Bahrampour, Sara Tofighi, Marzieh Bathaee and Farnaz Farman

Chapter 3 **One-Shot Phase-Shifting Interferometry** 

and Cruz Meneses-Fabián

Chapter 4 **Phosphor-Based White Light Emitting** 

Chapter 5 **Similariton-Based Spectral Interferometry** 

Eneas N. Morel and Jorge R. Torga

Tae Hyun Baek and Myung Soo Kim

**by Amplitude Modulation 171** 

Cruz Meneses-Fabian and Uriel Rivera-Ortega

Chapter 7 **Speckle Interferometry for Displacement** 

Chapter 8 **Phase-Shifting Interferometry** 

**with Phase-Gratings and Modulation** 

Chapter 1 **Optical Fiber Interferometers and Their Applications 3** 

Chapter 2 **The Applications of the Heterodyne Interferoemetry 31** 

**of Polarization Using N≥4 Interferograms 65**  Gustavo Rodríguez Zurita, Noel-Ivan Toto-Arellano

Wee Keat Chong, Xiang Li and Yeng Chai Soh

**Diode (LED) for Vertical Scanning Interferometry (VSI) 81** 

**for Signal Analysis on Femtosecond Time Scale 99**  Levon Mouradian, Aram Zeytunyan and Garegin Yesayan

Chapter 6 **Spectral Low Coherence Interferometry: A Complete Analysis of the Detection System and the Signal Processing 125** 

**Measurement and Hybrid Stress Analysis 149** 

## Contents

## **Preface XIII**

## **Part 1 Interferometry Methods and Research 1**


X Contents


Contents VII

Chapter 20 **Application of Interferometry to Analysis**

Akos Ledeczi and Miklos Maroti

Veronica Makarova and Valery Kulichikhin

Chapter 21 **Interferometry in Wireless Sensor Networks 437** Sandor Szilvasi, Peter Volgyesi, Janos Sallai,

**of Polymer-Polymer and Polymer-Solvent Interactions 393**

Akira Kimachi

Chapter 20 **Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 393**  Veronica Makarova and Valery Kulichikhin

## Chapter 21 **Interferometry in Wireless Sensor Networks 437**  Sandor Szilvasi, Peter Volgyesi, Janos Sallai, Akos Ledeczi and Miklos Maroti

VI Contents

Chapter 9 *N***-Shots 2***N***-Phase-Steps Binary Grating Interferometry 195** Cruz Meneses-Fabian, Gustavo Rodriguez-Zurita

**Scattering Measurements with Suppressed Influence**

Masanori Ota, Shinsuke Udagawa, Tatsuro Inage and Kazuo Maeno

**for Measurement of Static and Dynamic Phase Objects 263**

**of Liposomes Diffusion Through Artificial Membranes 281** 

**of Phase Gratings and Applications in Simultaneous PSI 297**

**for Optical Transmission Line Using Interferometry 315** 

Noel-Ivan Toto-Arellano, David-Ignacio Serrano-García, Amalia Martínez-García and Gustavo Rodríguez-Zurita

Michał Arabski, Sławomir Wąsik, Zuzanna Drulis-Kawa, Hubert Grześkiewicz, Jerzy Gubernator and Wiesław Kaca

Chapter 15 **Experimental π Phase-Shifts Observed in the Fourier Spectra** 

Noel-Ivan Toto-Arellano, Gustavo Rodríguez-Zurita, Amalia Martínez-García, David-Ignacio Serrano-García

Takeshi Tsujimura, Koichi Yoshida and Kuniaki Tanaka

**of Photoresist Thin Films Using Interferometry 359** Fotis Kossivas, Charalabos Doumanidis and Andreas Kyprianou

**Interferometry with Correlation Image Sensor 377** 

and María-Graciela Hernández-Orduña

**States by Light Wave Scattering 339**

Xiaoji Zhou and Xuguang Yue

**Modulated Low Coherence Interferometry 209** 

and Noel-Ivan Toto-Arellano

Chapter 10 **Path Length Resolved Dynamic Light** 

Chapter 11 **Interferometric Measurement** 

**of Optical Properties Using Phase** 

**in Shock Tube Experiments 209**

**Part 2 Recent Interferometry Applications 245** 

Chapter 13 **Simultaneous Phase Shifting Shearing Interferometry** 

Chapter 12 **Interferometry for Fusion Devices 247** Luis Esteban and Miguel Sánchez

Chapter 14 **Laser Interferometric Determination** 

Chapter 16 **Length Measurement** 

Chapter 17 **High Order Momentum** 

Chapter 18 **Thickness Measurement** 

Chapter 19 **Real-Time Heterodyne** 

Akira Kimachi

Babu Varghese and Wiendelt Steenbergen

Preface

This book compiles recent studies on interferometry and its applications in science and technology. It is intended as an up-to-date reference of theoretical and experimental

The book is divided in two sections. The first one is an overview of different interferometry techniques and their general applications. Chapter 1 is concentrated on the classical field interferometry. This chapter describes and explains the structures of different type of fiber interferometers, and the standard methods for signal recovering. Chapter 2 is an overview of the heterodyne interferometry and its applications. It includes some of recent development techniques and a summary of the advantages and disadvantages of the heterodyne interferometer. Chapter 3 describes the Phase-shifting interferometry through an experimental set-up for a polarizing two-window phase-grating common-path interferometer. The system is able to obtain *n* = (*N*+1) interferograms with only one shot. Chapter 4 introduces vertical scanning interferometry and shows that the use of phosphor-based LED on vertical scanning interferometry affects the repeatability and accuracy of vertical scanning interferometry, especially repeatability. However, the undesired effects of phosphor-based white LED can be removed by applying a constraint on the input to existing reconstruction algorithm. In Chapter 5 is developed and implemented a similariton based self-referencing method of spectral interferometry for the complete characterization of femtosecond signal. The method is based on the similariton generation from the part of signal and its use as a reference for the interference with the signal in the spectrometer. Chapter 6 offers an overview of Low coherence interferometry including the Basis of Low Coherence Interferometry and Experimental configurations, covering aspects of the spectral domain low coherence Interferometry such as the interference signal, detection system, signal processing and parameter limits. Chapter 7 presents a reliable hybrid method for characterizing stress around the circular hole in a tensile-loaded steel plate. The method utilized only few micro-scale x-displacement data measured by speckle interferometry, in conjunction with phase shifting method using Fourier transform to calculate stress components and eventually stress concentration at an angle of 90 degrees. Chapter 8 is discusses the phenomenon of optical interference for two waves elliptically polarized, and demonstrates with a numerical analysis and a computer simulation

aspects of interferometry and their applications.

## Preface

The book is divided in two sections. The first one is an overview of different interferometry techniques and their general applications. Chapter 1 is concentrated on the classical field interferometry. This chapter describes and explains the structures of different type of fiber interferometers, and the standard methods for signal recovering. Chapter 2 is an overview of the heterodyne interferometry and its applications. It includes some of recent development techniques and a summary of the advantages and disadvantages of the heterodyne interferometer. Chapter 3 describes the Phase-shifting interferometry through an experimental set-up for a polarizing two-window phase-grating common-path interferometer. The system is able to obtain *n* = (*N*+1) interferograms with only one shot. Chapter 4 introduces vertical scanning interferometry and shows that the use of phosphor-based LED on vertical scanning interferometry affects the repeatability and accuracy of vertical scanning interferometry, especially repeatability. However, the undesired effects of phosphor-based white LED can be removed by applying a constraint on the input to existing reconstruction algorithm. In Chapter 5 is developed and implemented a similariton based self-referencing method of spectral interferometry for the complete characterization of femtosecond signal. The method is based on the similariton generation from the part of signal and its use as a reference for the interference with the signal in the spectrometer. Chapter 6 offers an overview of Low coherence interferometry including the Basis of Low Coherence Interferometry and Experimental configurations, covering aspects of the spectral domain low coherence Interferometry such as the interference signal, detection system, signal processing and parameter limits. Chapter 7 presents a reliable hybrid method for characterizing stress around the circular hole in a tensile-loaded steel plate. The method utilized only few micro-scale x-displacement data measured by speckle interferometry, in conjunction with phase shifting method using Fourier transform to calculate stress components and eventually stress concentration at an angle of 90 degrees. Chapter 8 is discusses the phenomenon of optical interference for two waves elliptically polarized, and demonstrates with a numerical analysis and a computer simulation

This book compiles recent studies on interferometry and its applications in science and technology. It is intended as an up-to-date reference of theoretical and experimental aspects of interferometry and their applications.

the viability of a new method of phase-shifting based on the amplitude variation of two fields considered as the reference beams in a scheme of a three beam interferometer. In Chapter 9, based on the interference of two waves monochromatic and coherences, the phase shifting interferometry is studied and a method to reduce the number of captures needed is proposed on the basis of grating interferometry and modulation of linear polarization. Chapter 10 introduces a new bio-optical method "Path length resolved optical Doppler perfusion monitoring," to determine path length distributions of multiple scattered light in static and dynamic turbid media using phase modulated coherence gated interferometry. Also, is presented the first path length resolved Doppler measurements of multiply scattered light from human skin. Chapter 11 describes two topics related to interferometric measurement in shock tube experiment. Laser Interferometry Computed Tomography technique is applied to the measurement of high-speed, unsteady and 3D flow field induced by discharging shock waves, and Laser Differential Interferometer is applied to velocity and density measurement in micro-scale shock tube.

Preface XI

USA

**Dr Ivan Padron**

New Jersey Institute of Technology,

The book has been made possible as the outcome of the outstanding work done by all authors and by the professional assistance of Publishing Process Manager, Ms. Petra

Nenadic, during all phases of editing.

The second section is devoted to more specific interferometry applications including in chapter12 interferometry for magnetic fusion plasmas, followed by Measurement of Static and Dynamic Phase Objects using Simultaneous Phase Shifting Shearing Interferometry in chapter 13. Chapter 14 offers a biological application of laser interferometry in the Analysis of liposome diffusion. In chapter 15, theoretical and experimental evidences of π phase-shifts in the Fourier spectra of phase gratings and phase grids is presented, which are of considerable relevance when gratings or grids are used for interferometric applications, specifically in Phase shifting interferometry. Chapter 16 proposes a new switching method for in-service optical transmission lines that transfer live optical signals. The method uses optical fibers, instead of using electric apparatus to control the transmission speed. Chapter 17 introduces a novel method for testing large aspherical surfaces by subaperture stitching interferometry. Chapter 18 shows two methods of getting high order momentum states by resonant superradiant scattering and by a sequence of pulsed standing waves, corresponding to traveling wave scattering and standing wave scattering, respectively. Chapter 19 describes the thickness measurement of photoresist thin films using interferometry techniques. Chapter 20 presets a real-time 2-D HI technique based on the use of the correlation image sensor as a two-dimensional array of pixels, each of which simultaneously demodulate the amplitude and phase of incident heterodyne beams at an ordinary frame rate. Chapter 21 is an application of interferometry to analysis of polymer-polymer and polymer-solvent interactions. Chapter 21 describes the baseline approach of radio interferometry based localization and details the many developments that occurred since its introduction; including the mathematical foundation of the localization method and a method for tracking mobile nodes in Wireless Sensor Networks utilizing the phenomena of Doppler effects and radio interferometry.

The book has been made possible as the outcome of the outstanding work done by all authors and by the professional assistance of Publishing Process Manager, Ms. Petra Nenadic, during all phases of editing.

X Preface

tube.

interferometry.

the viability of a new method of phase-shifting based on the amplitude variation of two fields considered as the reference beams in a scheme of a three beam interferometer. In Chapter 9, based on the interference of two waves monochromatic and coherences, the phase shifting interferometry is studied and a method to reduce the number of captures needed is proposed on the basis of grating interferometry and modulation of linear polarization. Chapter 10 introduces a new bio-optical method "Path length resolved optical Doppler perfusion monitoring," to determine path length distributions of multiple scattered light in static and dynamic turbid media using phase modulated coherence gated interferometry. Also, is presented the first path length resolved Doppler measurements of multiply scattered light from human skin. Chapter 11 describes two topics related to interferometric measurement in shock tube experiment. Laser Interferometry Computed Tomography technique is applied to the measurement of high-speed, unsteady and 3D flow field induced by discharging shock waves, and Laser Differential Interferometer is applied to velocity and density measurement in micro-scale shock

The second section is devoted to more specific interferometry applications including in chapter12 interferometry for magnetic fusion plasmas, followed by Measurement of Static and Dynamic Phase Objects using Simultaneous Phase Shifting Shearing Interferometry in chapter 13. Chapter 14 offers a biological application of laser interferometry in the Analysis of liposome diffusion. In chapter 15, theoretical and experimental evidences of π phase-shifts in the Fourier spectra of phase gratings and phase grids is presented, which are of considerable relevance when gratings or grids are used for interferometric applications, specifically in Phase shifting interferometry. Chapter 16 proposes a new switching method for in-service optical transmission lines that transfer live optical signals. The method uses optical fibers, instead of using electric apparatus to control the transmission speed. Chapter 17 introduces a novel method for testing large aspherical surfaces by subaperture stitching interferometry. Chapter 18 shows two methods of getting high order momentum states by resonant superradiant scattering and by a sequence of pulsed standing waves, corresponding to traveling wave scattering and standing wave scattering, respectively. Chapter 19 describes the thickness measurement of photoresist thin films using interferometry techniques. Chapter 20 presets a real-time 2-D HI technique based on the use of the correlation image sensor as a two-dimensional array of pixels, each of which simultaneously demodulate the amplitude and phase of incident heterodyne beams at an ordinary frame rate. Chapter 21 is an application of interferometry to analysis of polymer-polymer and polymer-solvent interactions. Chapter 21 describes the baseline approach of radio interferometry based localization and details the many developments that occurred since its introduction; including the mathematical foundation of the localization method and a method for tracking mobile nodes in Wireless Sensor Networks utilizing the phenomena of Doppler effects and radio

**Dr Ivan Padron** New Jersey Institute of Technology, USA

**Part 1** 

**Interferometry Methods and Research** 

## **Part 1**

**Interferometry Methods and Research** 

**1** 

*Iran* 

**Optical Fiber Interferometers** 

Ali Reza Bahrampour, Sara Tofighi, Marzieh Bathaee and Farnaz Farman

Interference as a wave characteristic of the electromagnetic wave has many applications in science, technology and medicine (Grattan & Meggit, 1997; Wang et al., 2011). The fringe visibility of the first order interference experiments such as the famous double slit Young experiment and Michelson interferometer, is determined by the first order correlation function (Gerry & Knight, 2005). The first order interference is also called the field interference. In Hanbury-Brown and Twiss (HBT) experiment, fringes are due to the intensity interference and visibility is determined by the second order correlation function. (Brown & Twiss, 1956; Scully & Zubairy 2001). In quantum optics, nonlinear Lithography and quantum Lithography, interferometry based on higher order correlation function is of prime importance (Bentley & Boyd, 2004; Boto et al., 2000). However in all of these interferometries, the fringe pattern depends on the optical path difference (OPD) and feature of light source. This chapter is concentrated on the classical field interferometry. The fringe existence is a characteristic of

The phenomenon of interference of light is used in many high precision measuring systems and sensors. The optical path can be controlled by optical waveguides and optical fibers. The use of optical fibers allows making such devices extremely compact and economic.

Among the lots of advantages of optical fibers is their ability to reduce the effects of wave front distortion by the atmospheric turbulence and compact beam-splitter and combiner. These abilities made optical fiber as a suitable medium for transportation of light in long baseline interferometers which are used for gravitational wave detection, intruder sensor, structural health monitoring and long length leak detection systems (Sacharov, 2001; Cahill, 2007; Cahill & Stokes, 2008; Jia et al., 2008; Mishra & Soni, 2011, Bahrampour et al., 2012).

Other advantages that make optical fibers become useful elements in sensing technologies are high elongation sensitivity, fast response to internal or external defects such as temperature and tension, electromagnetic noise disturbance immunity, less power

In this chapter the different structures of optical fibers which are important in fiber interferometry are taken into consideration. The structures of different types of fiber interferometers are described. The sensitivity of coherent light optical fiber interferometers is compared with those of the incoherent and white light optical fiber interferometers. The

consumption and potential for large scale multiplexing (Higuera & Miguel, 2002).

spatial or temporal coherences between the two light beams.

**1. Introduction** 

**and Their Applications** 

*Sharif University of Technology* 

## **Optical Fiber Interferometers and Their Applications**

## Ali Reza Bahrampour, Sara Tofighi, Marzieh Bathaee and Farnaz Farman *Sharif University of Technology Iran*

## **1. Introduction**

Interference as a wave characteristic of the electromagnetic wave has many applications in science, technology and medicine (Grattan & Meggit, 1997; Wang et al., 2011). The fringe visibility of the first order interference experiments such as the famous double slit Young experiment and Michelson interferometer, is determined by the first order correlation function (Gerry & Knight, 2005). The first order interference is also called the field interference. In Hanbury-Brown and Twiss (HBT) experiment, fringes are due to the intensity interference and visibility is determined by the second order correlation function. (Brown & Twiss, 1956; Scully & Zubairy 2001). In quantum optics, nonlinear Lithography and quantum Lithography, interferometry based on higher order correlation function is of prime importance (Bentley & Boyd, 2004; Boto et al., 2000). However in all of these interferometries, the fringe pattern depends on the optical path difference (OPD) and feature of light source. This chapter is concentrated on the classical field interferometry. The fringe existence is a characteristic of spatial or temporal coherences between the two light beams.

The phenomenon of interference of light is used in many high precision measuring systems and sensors. The optical path can be controlled by optical waveguides and optical fibers. The use of optical fibers allows making such devices extremely compact and economic.

Among the lots of advantages of optical fibers is their ability to reduce the effects of wave front distortion by the atmospheric turbulence and compact beam-splitter and combiner. These abilities made optical fiber as a suitable medium for transportation of light in long baseline interferometers which are used for gravitational wave detection, intruder sensor, structural health monitoring and long length leak detection systems (Sacharov, 2001; Cahill, 2007; Cahill & Stokes, 2008; Jia et al., 2008; Mishra & Soni, 2011, Bahrampour et al., 2012).

Other advantages that make optical fibers become useful elements in sensing technologies are high elongation sensitivity, fast response to internal or external defects such as temperature and tension, electromagnetic noise disturbance immunity, less power consumption and potential for large scale multiplexing (Higuera & Miguel, 2002).

In this chapter the different structures of optical fibers which are important in fiber interferometry are taken into consideration. The structures of different types of fiber interferometers are described. The sensitivity of coherent light optical fiber interferometers is compared with those of the incoherent and white light optical fiber interferometers. The

Optical Fiber Interferometers and Their Applications 5

polarization aligned with one of the principal axes of the birefrigent fiber, the light propagates without any disturbance in its polarization state. The birefringence parameter of the fiber is defined by the difference between the two refractive indices corresponding to the two principal axes ���� � ��, where �� and �� are the refractive indices of the slow and fast axis respectively. Sometimes birefringence is defined in terms of the fiber beat length �� �� �⁄ that is defined as the length of fiber over which the phase difference between the fast and slow waves becomes ��. The beat length should be smaller than the perturbation periods introduced in the drawing process as well as the physical bends and twists. Consequently short beat length fibers preserve the polarization direction. This kind of fibers are called Polarization Maintained optical Fibers and denoted by PMF. Several

Fig. 1. Cross section of (a) elliptical core fiber (b) elliptical jacket fiber (c) side tunnel fiber (d)

Light propagation in standard optical fibers and PMFs is based on the total internal reflection effect. Bragg diffraction effect can also be employed to confine the light in the core of fiber with periodic structure in the cladding. The micro structured fiber which is also called photonic crystal fiber (PCF) as shown in Fig. 2, consists of numerous air holes within a silica host. Usually the air holes are in a periodic arrangement around silica or a hollow core. The silica core PCF is called holey fiber, high delta or cobweb fiber while hollow core is named photonic band gap fiber (PBGF). The simplest structure of the holey fiber is a regular hexagonal lattice of small holes with a defect in the center such that the hole in the center is missed. In the holey fibers, guiding mechanism is also based on the total internal reflection. Air holes in the cladding area cause an effective lowering of the average refractive index (Poli, et al., 2007). In hollow-core fibers, field confinement in the air core is based on the

Optical fibers are suitable transmission lines for several applications such as high capacity long-haul communication networks and long length optical interferometry. In many applications such as integrated circuits, the transmission length is less than one millimeter.

types of PMFs are shown in Fig. 1 (Okamoto, 2006).

Fig. 2. Cross section of holey fiber (left) and hollow fiber (right).

PANDA fiber (e) Bow-tie fiber.

**2.3 Photonic crystal fibers** 

band gap effect.

**2.4 Slab optical waveguide** 

standard methods for signal recovering are explained. A brief discussion on the noise sources appears in this chapter. Due to the immunity of the optical fibers to the lightening and electromagnetic noise, optical fibers are suitable sensors for transient measurement in harsh environments such as current measurement in high voltage transformers (Grattan & Meggit, 1999). The optical fiber hydrophone systems are based on elasto-optic effect in optical fiber coil, which is installed in one arm of an optical fiber interferometer (OFI) (Freitas, 2011). The optical fiber interferometers can be employed as biochemical sensors (Gopel et al., 1991). The cooperation of optical fiber interferometry and Plasmon can improve the sensitivity of biosensors to one molecule detection system (De Vos et al., 2009). The mechanical quantities such as pressure, velocity, acceleration and displacement can be measured by optical fiber interferometers (Shizhuo et al., 2008). Among a lot of applications of optical fiber interferometers, only some applications such as linear and nonlinear photonic circuits and distributed optical fiber sensors are mentioned in this chapter.

## **2. Optical fibers structures**

## **2.1 Standard fibers**

An optical fiber is a cylindrical structure that transports electromagnetic waves in the infrared or visible bands of electromagnetic spectrum. In practice optical fibers are highly flexible and transparent dielectric material. The optical fiber consists of three different layers. Core is the central region which is surrounded by the cladding. These two layers are protected by protective jacket. The core refractive index can be uniform or graded while the cladding index is typically uniform. For light guiding, it is necessary that the core index be greater than the cladding index. Most of the light energy propagates in the core and only a small fraction travels in the cladding. The cladding radius is so large that the jacket has no effect on the light propagation in the optical fiber structure.

Depending on the dimensionless frequency � � ������� � � ��� � )� �⁄ ⁄� where � is the core radius, � is the wavelength of the light in free space, *nco* and *nclad* are the core and clad refractive indices respectively, optical fibers are divided into multimode (���) and single mode fibers (�� � � � �� ), where �� is cutoff frequency (Agrawal, 2007).

The optical fibers whose core and cladding have very nearly the same refractive index are named weakly guiding fibers. The corresponding eigen value equation is simpler than the exact fiber characteristic equation. The notation ����� introduces the weakly guiding modes. The fundamental mode �����is denoted by �����(Okamoto, 2006). The normalized propagation constant versus the dimensionless frequency is called the dispersion curve.

Depending on the coupling and optical fiber physical parameters, bounded, radiation and evanescent modes can exist in an optical fiber. The total incident power can be transported by the bounded and radiation modes while evanescent modes store power near the excitation source (Snyder, 1983).

## **2.2 Polarization maintained optical fibers**

Birefringent optical fibers are those fibers that display two distinct refractive indices depending on the polarization direction of the light entering into them. The two principal axes of the birefringent fibers are named the fast and slow axis. For a light beam whose

standard methods for signal recovering are explained. A brief discussion on the noise sources appears in this chapter. Due to the immunity of the optical fibers to the lightening and electromagnetic noise, optical fibers are suitable sensors for transient measurement in harsh environments such as current measurement in high voltage transformers (Grattan & Meggit, 1999). The optical fiber hydrophone systems are based on elasto-optic effect in optical fiber coil, which is installed in one arm of an optical fiber interferometer (OFI) (Freitas, 2011). The optical fiber interferometers can be employed as biochemical sensors (Gopel et al., 1991). The cooperation of optical fiber interferometry and Plasmon can improve the sensitivity of biosensors to one molecule detection system (De Vos et al., 2009). The mechanical quantities such as pressure, velocity, acceleration and displacement can be measured by optical fiber interferometers (Shizhuo et al., 2008). Among a lot of applications of optical fiber interferometers, only some applications such as linear and nonlinear

photonic circuits and distributed optical fiber sensors are mentioned in this chapter.

effect on the light propagation in the optical fiber structure.

Depending on the dimensionless frequency � � �������

mode fibers (�� � � � �� ), where �� is cutoff frequency (Agrawal, 2007).

An optical fiber is a cylindrical structure that transports electromagnetic waves in the infrared or visible bands of electromagnetic spectrum. In practice optical fibers are highly flexible and transparent dielectric material. The optical fiber consists of three different layers. Core is the central region which is surrounded by the cladding. These two layers are protected by protective jacket. The core refractive index can be uniform or graded while the cladding index is typically uniform. For light guiding, it is necessary that the core index be greater than the cladding index. Most of the light energy propagates in the core and only a small fraction travels in the cladding. The cladding radius is so large that the jacket has no

radius, � is the wavelength of the light in free space, *nco* and *nclad* are the core and clad refractive indices respectively, optical fibers are divided into multimode (���) and single

The optical fibers whose core and cladding have very nearly the same refractive index are named weakly guiding fibers. The corresponding eigen value equation is simpler than the exact fiber characteristic equation. The notation ����� introduces the weakly guiding modes. The fundamental mode �����is denoted by �����(Okamoto, 2006). The normalized propagation constant versus the dimensionless frequency is called the dispersion curve.

Depending on the coupling and optical fiber physical parameters, bounded, radiation and evanescent modes can exist in an optical fiber. The total incident power can be transported by the bounded and radiation modes while evanescent modes store power near the

Birefringent optical fibers are those fibers that display two distinct refractive indices depending on the polarization direction of the light entering into them. The two principal axes of the birefringent fibers are named the fast and slow axis. For a light beam whose

� � ���

� )� �⁄ ⁄� where � is the core

**2. Optical fibers structures** 

excitation source (Snyder, 1983).

**2.2 Polarization maintained optical fibers** 

**2.1 Standard fibers** 

polarization aligned with one of the principal axes of the birefrigent fiber, the light propagates without any disturbance in its polarization state. The birefringence parameter of the fiber is defined by the difference between the two refractive indices corresponding to the two principal axes ���� � ��, where �� and �� are the refractive indices of the slow and fast axis respectively. Sometimes birefringence is defined in terms of the fiber beat length �� �� �⁄ that is defined as the length of fiber over which the phase difference between the fast and slow waves becomes ��. The beat length should be smaller than the perturbation periods introduced in the drawing process as well as the physical bends and twists. Consequently short beat length fibers preserve the polarization direction. This kind of fibers are called Polarization Maintained optical Fibers and denoted by PMF. Several types of PMFs are shown in Fig. 1 (Okamoto, 2006).

Fig. 1. Cross section of (a) elliptical core fiber (b) elliptical jacket fiber (c) side tunnel fiber (d) PANDA fiber (e) Bow-tie fiber.

## **2.3 Photonic crystal fibers**

Light propagation in standard optical fibers and PMFs is based on the total internal reflection effect. Bragg diffraction effect can also be employed to confine the light in the core of fiber with periodic structure in the cladding. The micro structured fiber which is also called photonic crystal fiber (PCF) as shown in Fig. 2, consists of numerous air holes within a silica host. Usually the air holes are in a periodic arrangement around silica or a hollow core. The silica core PCF is called holey fiber, high delta or cobweb fiber while hollow core is named photonic band gap fiber (PBGF). The simplest structure of the holey fiber is a regular hexagonal lattice of small holes with a defect in the center such that the hole in the center is missed. In the holey fibers, guiding mechanism is also based on the total internal reflection. Air holes in the cladding area cause an effective lowering of the average refractive index (Poli, et al., 2007). In hollow-core fibers, field confinement in the air core is based on the band gap effect.

Fig. 2. Cross section of holey fiber (left) and hollow fiber (right).

## **2.4 Slab optical waveguide**

Optical fibers are suitable transmission lines for several applications such as high capacity long-haul communication networks and long length optical interferometry. In many applications such as integrated circuits, the transmission length is less than one millimeter.

Optical Fiber Interferometers and Their Applications 7

interferometer (OFI), the position of minimum can be shifted to maximum position if the optical path difference varies by odd number of half wavelengths. At least two optical paths are necessary for an interfererometery experiment. These optical paths can be in one optical fiber with two or more different optical fiber modes. Each of modes defines one optical path for the interferometer such as the Sagnac interferometer where the optical paths are defined by the clockwise and counter clockwise modes. The optical paths can be defined by separate optical fibers such as Mach-Zehnder OFI. There are many interferometer configurations that have been realized with the optical fiber. To see the principle of their operation, the detail of some interferometers such as Sagnac, birefringence OFI, Mach-Zehnder, Michelson, Moiré

The configuration of a Sagnac optical fiber is illustrated by Fig. 3. The optical source is a single mode stabilized coherent semi-conductor or Erbium doped optical fiber laser. The laser output beam is assumed to be well collimated with uniform phase. The laser beam enters the lossless 3dB optical fiber coupler (OFC). At the OFC the injected light splits into two parts with equal intensity that each of them travels around single mode optical fiber coil

Due to this specific configuration, fiber Sagnac interferometer has been used for rotation sensing primarily. In a non-rotating Sgnac interferometer, the clockwise (CW) and counter clockwise (CCW) modes are in phase while for a rotating Sagnac configuration due to the rotating velocity, the optical path of one of the modes is shorten and the other one is lengthen. The Sagnac effect causes the interference spectrum depends on the angular frequency of the setup (Sagnac, 1913). Analysis can be based on the Doppler frequency difference between the CW and CCW modes. The detector output frequency is the beating frequency of CW and CCW modes. When rotational axis is oriented along the optical fiber coil axis, the phase difference of CW and CCW modes is �� � ����� �� ⁄ (Burns, 1993; Vali & Shorthill, 1976), where � is the free space optical wavelength, ��*′* is the area of Sagnac coil, N is the number of the coil turn and�� is the angular velocity. The sensitivity is the ratio of the phase difference to the angular velocity � � ���� �� ⁄ , which is increased by increasing the coil radius, total fiber length and laser frequency. Optical fiber loss and packaging

Sagnac fiber interferometers can also be employed for sensing nonreciprocal and timevarying phenomena. So they become applicable tools for detection current, acoustic wave, strain and temperature. The optical gyroscope based on sagnac interferometer is commercially available (Bohnert et al., 2002; Lin et al., 2004; Starodumov et al., 1997; Dong &

in opposite directions. The output of Sagnac coil is guided toward a single detector.

and Fabry-Perot interferometer are presented.

Fig. 3. A schematic diagram of Sagnac fiber interferometer.

criteria limit the total fiber length and coil radius respectively.

Tam, 2007; Fu et al., 2010)

**3.1 Sagnac optical fiber interferometer** 

Optical waveguides are proper for such applications. A dielectric waveguide consists of a dielectric with refractive index n1 which is deposited on a substrate with refractive index n2. The refractive index of the medium above the layer n1 is indicated by n3. To achieve true guiding modes, it is necessary that n1 be larger than n2 and n3. The propagating modes of the slab waveguides are TE and TM modes. The mathematical mode analysis of slab waveguides can be found in any standard text book (Adams, 1981). Depending on the propagation constant �, modes of narrow dielectric strip waveguide are also classified into the bounded, radiation and evanescent modes (Adams, 1981; Snyder, 1983).

## **2.5 Fiber Bragg gratings**

The modes of optical fibers and waveguides are propagated without coupling to each other in the absence of any perturbation. Coupling to the desired modes can be controlled by changing the amplitude and the phase of the perturbation in the optical fiber. The coupled mode theory (CMT) can be found in the standard text books (Huang, 1984). If the refractive index of the core varies periodically, due to the Bragg diffraction effect, scattering from different periods can be constructive for some frequencies and destructive for the other ones. Depending on the period length, the periodic structures are classified as either long period grating (LPG) or fiber Bragg grating (FBG). The period of the LPG is of the order of micrometer while in the FBG, it is of the order of nanometer. The operation of the LPG is on the basis of coupling the fundamental core mode to higher order co-propagating cladding modes. The coupling wavelength is obtained by the linear momentum conservation law or the phase matching equation � � ��� � ��)Λ, where �� and �� are the core and cladding mode propagation constants respectively and Λ is the period of the LPG (Kashyap, 1999).

FBG can be employed as a frequency selective reflector or a polarization selective rotator. In the reflector state, the forward modes are coupled to the backward modes. While in the polarization rotator, a mode with a definite polarization is coupled to another mode with different polarization. In the frequency selective reflector, coupling to the backward modes occurs in a narrow range of wavelengths around the wavelength for which Bragg condition is satisfied � � �����Λ, where ���� is the effective refractive index of the core. Bandwidth of FBG is typically below 1nm and depends on the amount of refractive index variation and the length of FBG. The governing equations of the FBG can be obtained from the conservation of energy and momentum (Kashyap, 1999; Chen, 2006).

Depending on the application of FBG, the period of the structure can vary in a definite way or randomly along the optical fiber core. This structure is named chirped FBG which has many applications in optical networks and sensors (Kashyap, 1999; Rao, 1997).

## **3. Basic optical fiber interferometer configurations**

Interferometry is based on the superimposing of two or more light beams to measure the phase difference between them. Interferometer utilizes two light beams with the same frequency. Typically an incident light beam of interferometer is split into two or more parts and then recombine together to create an interference pattern. The integer number of wavelength for the optical path difference between the two paths corresponds to constructive points and odd number of half wavelengths corresponds to destructive points of the interference pattern. So in the output optical spectrum of the optical fiber

Optical waveguides are proper for such applications. A dielectric waveguide consists of a dielectric with refractive index n1 which is deposited on a substrate with refractive index n2. The refractive index of the medium above the layer n1 is indicated by n3. To achieve true guiding modes, it is necessary that n1 be larger than n2 and n3. The propagating modes of the slab waveguides are TE and TM modes. The mathematical mode analysis of slab waveguides can be found in any standard text book (Adams, 1981). Depending on the propagation constant �, modes of narrow dielectric strip waveguide are also classified into

The modes of optical fibers and waveguides are propagated without coupling to each other in the absence of any perturbation. Coupling to the desired modes can be controlled by changing the amplitude and the phase of the perturbation in the optical fiber. The coupled mode theory (CMT) can be found in the standard text books (Huang, 1984). If the refractive index of the core varies periodically, due to the Bragg diffraction effect, scattering from different periods can be constructive for some frequencies and destructive for the other ones. Depending on the period length, the periodic structures are classified as either long period grating (LPG) or fiber Bragg grating (FBG). The period of the LPG is of the order of micrometer while in the FBG, it is of the order of nanometer. The operation of the LPG is on the basis of coupling the fundamental core mode to higher order co-propagating cladding modes. The coupling wavelength is obtained by the linear momentum conservation law or the phase matching equation � � ��� � ��)Λ, where �� and �� are the core and cladding mode propagation constants respectively and Λ is the period of the LPG (Kashyap, 1999). FBG can be employed as a frequency selective reflector or a polarization selective rotator. In the reflector state, the forward modes are coupled to the backward modes. While in the polarization rotator, a mode with a definite polarization is coupled to another mode with different polarization. In the frequency selective reflector, coupling to the backward modes occurs in a narrow range of wavelengths around the wavelength for which Bragg condition is satisfied � � �����Λ, where ���� is the effective refractive index of the core. Bandwidth of FBG is typically below 1nm and depends on the amount of refractive index variation and the length of FBG. The governing equations of the FBG can be obtained from the

the bounded, radiation and evanescent modes (Adams, 1981; Snyder, 1983).

conservation of energy and momentum (Kashyap, 1999; Chen, 2006).

**3. Basic optical fiber interferometer configurations** 

many applications in optical networks and sensors (Kashyap, 1999; Rao, 1997).

Depending on the application of FBG, the period of the structure can vary in a definite way or randomly along the optical fiber core. This structure is named chirped FBG which has

Interferometry is based on the superimposing of two or more light beams to measure the phase difference between them. Interferometer utilizes two light beams with the same frequency. Typically an incident light beam of interferometer is split into two or more parts and then recombine together to create an interference pattern. The integer number of wavelength for the optical path difference between the two paths corresponds to constructive points and odd number of half wavelengths corresponds to destructive points of the interference pattern. So in the output optical spectrum of the optical fiber

**2.5 Fiber Bragg gratings** 

interferometer (OFI), the position of minimum can be shifted to maximum position if the optical path difference varies by odd number of half wavelengths. At least two optical paths are necessary for an interfererometery experiment. These optical paths can be in one optical fiber with two or more different optical fiber modes. Each of modes defines one optical path for the interferometer such as the Sagnac interferometer where the optical paths are defined by the clockwise and counter clockwise modes. The optical paths can be defined by separate optical fibers such as Mach-Zehnder OFI. There are many interferometer configurations that have been realized with the optical fiber. To see the principle of their operation, the detail of some interferometers such as Sagnac, birefringence OFI, Mach-Zehnder, Michelson, Moiré and Fabry-Perot interferometer are presented.

## **3.1 Sagnac optical fiber interferometer**

The configuration of a Sagnac optical fiber is illustrated by Fig. 3. The optical source is a single mode stabilized coherent semi-conductor or Erbium doped optical fiber laser. The laser output beam is assumed to be well collimated with uniform phase. The laser beam enters the lossless 3dB optical fiber coupler (OFC). At the OFC the injected light splits into two parts with equal intensity that each of them travels around single mode optical fiber coil in opposite directions. The output of Sagnac coil is guided toward a single detector.

Fig. 3. A schematic diagram of Sagnac fiber interferometer.

Due to this specific configuration, fiber Sagnac interferometer has been used for rotation sensing primarily. In a non-rotating Sgnac interferometer, the clockwise (CW) and counter clockwise (CCW) modes are in phase while for a rotating Sagnac configuration due to the rotating velocity, the optical path of one of the modes is shorten and the other one is lengthen. The Sagnac effect causes the interference spectrum depends on the angular frequency of the setup (Sagnac, 1913). Analysis can be based on the Doppler frequency difference between the CW and CCW modes. The detector output frequency is the beating frequency of CW and CCW modes. When rotational axis is oriented along the optical fiber coil axis, the phase difference of CW and CCW modes is �� � ����� �� ⁄ (Burns, 1993; Vali & Shorthill, 1976), where � is the free space optical wavelength, ��*′* is the area of Sagnac coil, N is the number of the coil turn and�� is the angular velocity. The sensitivity is the ratio of the phase difference to the angular velocity � � ���� �� ⁄ , which is increased by increasing the coil radius, total fiber length and laser frequency. Optical fiber loss and packaging criteria limit the total fiber length and coil radius respectively.

Sagnac fiber interferometers can also be employed for sensing nonreciprocal and timevarying phenomena. So they become applicable tools for detection current, acoustic wave, strain and temperature. The optical gyroscope based on sagnac interferometer is commercially available (Bohnert et al., 2002; Lin et al., 2004; Starodumov et al., 1997; Dong & Tam, 2007; Fu et al., 2010)

$$I\_n = \frac{I\_0}{q} \left[ 3 + 2\cos(\varphi\_{12} + \theta\_n) + 2\cos(\varphi\_{23} + \theta\_n) + 2\cos(\varphi\_{31} + \theta\_n) \right]; \ n = 1, 2, 3\tag{1}$$

$$\begin{aligned} I\_n &= \frac{l\_0}{9} [a\_1^2 + a\_2^2 + a\_3^2 + 2a\_1 a\_2 \cos(\varphi\_{12} + \theta\_n) \\ &+ 2a\_2 a\_3 \cos(\varphi\_{23} + \theta\_n) + 2 \, a\_1 a\_3 \cos(\varphi\_{31} + \theta\_n)]; \quad n = 1, 2, 3 \end{aligned} \tag{2}$$

$$
\begin{pmatrix} E\_{out}^{(1)} \\ E\_{out}^{(2)} \end{pmatrix} = \begin{pmatrix} Y^{11} & Y^{12} \\ Y^{21} & Y^{22} \end{pmatrix} \begin{pmatrix} E\_{in}^{(1)} \\ E\_{in}^{(2)} \end{pmatrix} \tag{3}
$$

$$E\_{out}^{(1)} = [Y^{11} + Y^{12}Y^{21}P(1 - Y^{22}P)^{-1}]E\_{in}^{(1)}\tag{4}$$

$$I\_j = I\_0 \left| Y\_{j1}^{11} + (Y^{12}Y^{21}P(1 - Y^{22}P)^{-1})\_{j1} \right|^2, \qquad j = 1, \cdots, N \tag{5}$$

$$E = \Sigma \overrightarrow{E\_I} e^{-l[\widehat{\Xi}(\mathbf{x}a\_f + \mathbf{y}b\_f) + \varphi\_f]} \quad + \text{c.c.} \tag{6}$$

$$I = \sum\_{l=1}^{N} I\_l + \Sigma\_{l \neq j} \sqrt{l\_l I\_j} \cos(\theta\_l - \theta\_j) \cos\left\{\frac{k}{D} \left[ (a\_l - a\_j)\mathbf{x} + (b\_l - b\_j)\mathbf{y} \right] - \varphi\_{lj} \right\},\tag{7}$$

Optical Fiber Interferometers and Their Applications 13

second mirror of IFFPI shown in Fig. 11 (a) is an internal mirror which can be produced by splicing of polished fibers or by polished coated fibers. Both mirrors of the IFFPI shown in Fig. 11 (b) are internal fiber mirrors while those are used in the IFFPI presented in Fig. 11 (c) are FBG reflectors. Depending on the application of IFFPI, one of the configurations

In Fig. 12 (a) the air-gap cavity is bounded by the end of a polished fiber and a diaphragm mirror. The cavity length is of the order of several microns and can be increased by convex mirror diaphragm. In another configuration presented in Fig. 12 (b) a thin film of transparent solid material is coated on the end of the fiber. The air-gap cavity between two polished fiber surfaces, where the fibers are aligned in a hollow tube is another configuration of EFFPI (Fig. 12 (c)). The structure shown in Fig. 12 (d) is called the in-line fiber etalon (ILFE). The ILFE is constructed of a hollow-core fiber spliced between two single mode fibers. The diffraction loss causes to limit the practical length of EFFPI to a few

The interferometric techniques are known as the precise method for measuring physical quantities that can induce the optical path difference (OPD) in the interferometers. The

presented in Fig. 11 can be used.

Fig. 11. Schematic configurations of three IFFPI.

Fig. 12. Schematic configurations of four EFFPI.

hundred of microns (Shizhuo, 2008).

**3.7 White light fiber interferometry** 

Four different EFFPI configurations are shown in Fig. 12.

Fig. 9. Two arrangements of optical fiber Moiré interferometer and their interference patterns. a) The polarizations of the three fibers are in the same direction b) fibers 2 and 3 with vertical and horizontal polarization and fiber 3 with angle 45*°* .

## **3.6 Optical fiber Fabry-Perot interferometer**

A Fabry-Perot (FP) consists of two optically parallel reflectors with reflectance ��(�) and ��(�) separated by a cavity of length *L*. Reflectors can be mirrors, interface of two dielectrics or fiber Bragg gratings. The cavity may be an optical fiber or any other optical medium.

Two different optical fiber Fabry-Perot interferometers are shown in Fig. 10.

Fig. 10. (a) Fabry-perot based on the light transmission (b) Fabry-perot based on the light reflection.

One is based on the light transmission through a Fabry-Perot, while the other is based on the reflection. Due to multiple reflections, the reflected and transmitted spectrums are functions of cavity length, medium index of refraction and mirrors reflectivity. Because of energy conservation law, the transmitted spectrum is opposite to the reflected spectrum.

Optical fiber Fabry-Perots are classified as intrinsic and extrinsic types. In the intrinsic fiber FP interferometer (IFFPI), the two mirrors are separated by a single mode fiber, while in the extrinsic fiber FP interferometer (EFFPI), the two mirrors are separated by an air gap or by some solid material other than fiber. In both IFFPI and EFFPI, light from emitter to the FP and from FP to the detector are transmitted by a single mode fiber. Fig. 11 shows schematic configurations of three IFFPI. One end of the fiber shown in Fig. 11 (a) is polished as a mirror. For higher reflection the polished end is coated with switchable dielectric layers. The

Fig. 9. Two arrangements of optical fiber Moiré interferometer and their interference patterns. a) The polarizations of the three fibers are in the same direction b) fibers 2 and 3

A Fabry-Perot (FP) consists of two optically parallel reflectors with reflectance ��(�) and ��(�) separated by a cavity of length *L*. Reflectors can be mirrors, interface of two dielectrics or fiber Bragg gratings. The cavity may be an optical fiber or any other optical medium.

Fig. 10. (a) Fabry-perot based on the light transmission (b) Fabry-perot based on the light

conservation law, the transmitted spectrum is opposite to the reflected spectrum.

One is based on the light transmission through a Fabry-Perot, while the other is based on the reflection. Due to multiple reflections, the reflected and transmitted spectrums are functions of cavity length, medium index of refraction and mirrors reflectivity. Because of energy

Optical fiber Fabry-Perots are classified as intrinsic and extrinsic types. In the intrinsic fiber FP interferometer (IFFPI), the two mirrors are separated by a single mode fiber, while in the extrinsic fiber FP interferometer (EFFPI), the two mirrors are separated by an air gap or by some solid material other than fiber. In both IFFPI and EFFPI, light from emitter to the FP and from FP to the detector are transmitted by a single mode fiber. Fig. 11 shows schematic configurations of three IFFPI. One end of the fiber shown in Fig. 11 (a) is polished as a mirror. For higher reflection the polished end is coated with switchable dielectric layers. The

.

with vertical and horizontal polarization and fiber 3 with angle 45*°*

Two different optical fiber Fabry-Perot interferometers are shown in Fig. 10.

**3.6 Optical fiber Fabry-Perot interferometer** 

reflection.

second mirror of IFFPI shown in Fig. 11 (a) is an internal mirror which can be produced by splicing of polished fibers or by polished coated fibers. Both mirrors of the IFFPI shown in Fig. 11 (b) are internal fiber mirrors while those are used in the IFFPI presented in Fig. 11 (c) are FBG reflectors. Depending on the application of IFFPI, one of the configurations presented in Fig. 11 can be used.

Fig. 11. Schematic configurations of three IFFPI.

Four different EFFPI configurations are shown in Fig. 12.

Fig. 12. Schematic configurations of four EFFPI.

In Fig. 12 (a) the air-gap cavity is bounded by the end of a polished fiber and a diaphragm mirror. The cavity length is of the order of several microns and can be increased by convex mirror diaphragm. In another configuration presented in Fig. 12 (b) a thin film of transparent solid material is coated on the end of the fiber. The air-gap cavity between two polished fiber surfaces, where the fibers are aligned in a hollow tube is another configuration of EFFPI (Fig. 12 (c)). The structure shown in Fig. 12 (d) is called the in-line fiber etalon (ILFE). The ILFE is constructed of a hollow-core fiber spliced between two single mode fibers. The diffraction loss causes to limit the practical length of EFFPI to a few hundred of microns (Shizhuo, 2008).

## **3.7 White light fiber interferometry**

The interferometric techniques are known as the precise method for measuring physical quantities that can induce the optical path difference (OPD) in the interferometers. The

Optical Fiber Interferometers and Their Applications 15

standard optical fiber interferometer and their combinations (Yuan, 2002; Mercado et al., 2001;). As an example Fig. 13 shows a white light fiber optic Michelson interferometer working in the spatial domain. The LED light is coupled to the two path of Michelson interferometer through a 2×2 OFC without insertion loss. The reflected beams recombine on the PIN detector of the WLI. The scanning mirror is adjusted for maximum output

Fig. 13. (a) A schematic configuration of WLI Michelson interferometer (b) Input LED

(b) (a) M1

Detector

As shown in Fig. 13(c) for OPD less than the source coherence length, the white-light fringe pattern is produced. The position of the highest amplitude corresponds to the exactly zero optical path difference between two beams. After some mathematical manipulations for LED parameters presented in (Yuan, 1997), the normalized interference fringe pattern is calculated and result is presented in Fig. 14. The results of three peaks LED are compared with those of a normal LED to see how the multi wavelength white light source increases the precision of the central fringe position measurement relative to the single white light

OFC

LED �

Movable mirror M2

Fig. 14. The spectrum distribution of the light source (up) and their normalized interference

The phase difference between two coherent light beams is detected by interferometric methods, which are most sensitive techniques for optical path difference measurement. The OPD variations have sufficiently low frequency components. So when they are converted to

fringe pattern (down) (a) Three peaks LED (b) Normal LED.

**4. Signal recovering methods** 

corresponding to the position of central fringe.

spectrum (c) Interference fringe pattern.

(c)

source.

coherent length of the narrow band sources such as Lasers are greater than the optical path length difference in the interferometers. Due to the periodic nature of the interferogram fringes, the interferometric measurement suffers from an integer multiple ofʹߨphase ambiguity. Hence interferometers driven by narrow line-width Lasers do not produce absolute data unless extra complexity is added to the interferometer. By employing the short coherent light that is illuminated from wide-band light sources, the phase ambiguity is eliminated. In wide bandwidth interferometer the fringes of the interferogram are narrowly located in the zero path length difference region (Flourney et al., 1972). So the phase difference can be determined without the phase ambiguity by measuring the fringe peak or the envelop peak of the interferogram. This type of interferometry is named as white light or low coherence interferometry. In white light interferometry (WLI) corresponding to each wavelength a separate fringe system is produced. The electric field at any point of observation is the sum of electric fields of these individual patterns. In a WLI which is adjusted such that the optical path difference is zero at the center of the field of view, the electric field of different wavelengths exhibits the maximum at the center point. The fringes of different wavelengths will no longer coincide as moving away from the center of the pattern. The fringe pattern is a sequence of colors whose saturation decreases rapidly. The central bright white light fringe can be used to adjust the WLI.

The light sources such as fluorescent lamp, SLDs, LEDs, Laser diodes near threshold, optically pumped Erbium-doped fibers and tungsten lamps, can be used in the WLI. The spectral width of SLD and LED is between 20 and 100 nm. It is expected that at the operating wavelength (1.3 µm) of these types of light sources, the coherent length is between 17 and 85 µm. Because of the wave-train damping, the Doppler effect, disturbances by neighbor atoms, noises and mode mixing effects, the practical coherent length is less than those are predicted previously.

In the WLI, one of the two arms is used as the measurement arm and the other one as the reference arm. The length of the reference arm can be controlled by different methods such as moving mirrors or Piezoelectric (PZT) devices. Generally the operation of WLI is based on the balancing the two arms of the interferometer and compensating the OPD in the measurement arm. Therefore the desired measurement can be achieved.

As the OPD between the two paths of a WLI is varied, the intensity of interference fringe drops from a maximum to a minimum value. The maximum intensity corresponds to the central white bright fringe. Measurement of the position of the central fringe in the WLI is of prime importance. Because the distance between the central fringe and its adjacent side fringes is too small and the presence of noise, the determination of the central fringe position is inaccurate, so there are some ambiguities in the central fringe identification. This problem can be solved by employing a combinational source of two or three multimode Laser diodes with different wavelengths.

White light fiber interferometers (WLFI) can be designed on different topologies of single or multi-mode fiber interferometers. Each of the single mode and multi-mode fibers has their own advantages and disadvantages. For example usually white light single mode fiber interferometer provides stable and large signal to noise ratio while in the interferometers based on multi-mode fiber, cheaper optical components are employed (Song et al., 2001; Manojlovi et al., 2010). Generally there are several WLI topologies corresponding to the

coherent length of the narrow band sources such as Lasers are greater than the optical path length difference in the interferometers. Due to the periodic nature of the interferogram fringes, the interferometric measurement suffers from an integer multiple ofʹߨphase ambiguity. Hence interferometers driven by narrow line-width Lasers do not produce absolute data unless extra complexity is added to the interferometer. By employing the short coherent light that is illuminated from wide-band light sources, the phase ambiguity is eliminated. In wide bandwidth interferometer the fringes of the interferogram are narrowly located in the zero path length difference region (Flourney et al., 1972). So the phase difference can be determined without the phase ambiguity by measuring the fringe peak or the envelop peak of the interferogram. This type of interferometry is named as white light or low coherence interferometry. In white light interferometry (WLI) corresponding to each wavelength a separate fringe system is produced. The electric field at any point of observation is the sum of electric fields of these individual patterns. In a WLI which is adjusted such that the optical path difference is zero at the center of the field of view, the electric field of different wavelengths exhibits the maximum at the center point. The fringes of different wavelengths will no longer coincide as moving away from the center of the pattern. The fringe pattern is a sequence of colors whose saturation decreases rapidly. The

The light sources such as fluorescent lamp, SLDs, LEDs, Laser diodes near threshold, optically pumped Erbium-doped fibers and tungsten lamps, can be used in the WLI. The spectral width of SLD and LED is between 20 and 100 nm. It is expected that at the operating wavelength (1.3 µm) of these types of light sources, the coherent length is between 17 and 85 µm. Because of the wave-train damping, the Doppler effect, disturbances by neighbor atoms, noises and mode mixing effects, the practical coherent length is less than

In the WLI, one of the two arms is used as the measurement arm and the other one as the reference arm. The length of the reference arm can be controlled by different methods such as moving mirrors or Piezoelectric (PZT) devices. Generally the operation of WLI is based on the balancing the two arms of the interferometer and compensating the OPD in the

As the OPD between the two paths of a WLI is varied, the intensity of interference fringe drops from a maximum to a minimum value. The maximum intensity corresponds to the central white bright fringe. Measurement of the position of the central fringe in the WLI is of prime importance. Because the distance between the central fringe and its adjacent side fringes is too small and the presence of noise, the determination of the central fringe position is inaccurate, so there are some ambiguities in the central fringe identification. This problem can be solved by employing a combinational source of two or three multimode Laser diodes

White light fiber interferometers (WLFI) can be designed on different topologies of single or multi-mode fiber interferometers. Each of the single mode and multi-mode fibers has their own advantages and disadvantages. For example usually white light single mode fiber interferometer provides stable and large signal to noise ratio while in the interferometers based on multi-mode fiber, cheaper optical components are employed (Song et al., 2001; Manojlovi et al., 2010). Generally there are several WLI topologies corresponding to the

central bright white light fringe can be used to adjust the WLI.

measurement arm. Therefore the desired measurement can be achieved.

those are predicted previously.

with different wavelengths.

standard optical fiber interferometer and their combinations (Yuan, 2002; Mercado et al., 2001;). As an example Fig. 13 shows a white light fiber optic Michelson interferometer working in the spatial domain. The LED light is coupled to the two path of Michelson interferometer through a 2×2 OFC without insertion loss. The reflected beams recombine on the PIN detector of the WLI. The scanning mirror is adjusted for maximum output corresponding to the position of central fringe.

Fig. 13. (a) A schematic configuration of WLI Michelson interferometer (b) Input LED spectrum (c) Interference fringe pattern.

As shown in Fig. 13(c) for OPD less than the source coherence length, the white-light fringe pattern is produced. The position of the highest amplitude corresponds to the exactly zero optical path difference between two beams. After some mathematical manipulations for LED parameters presented in (Yuan, 1997), the normalized interference fringe pattern is calculated and result is presented in Fig. 14. The results of three peaks LED are compared with those of a normal LED to see how the multi wavelength white light source increases the precision of the central fringe position measurement relative to the single white light source.

Fig. 14. The spectrum distribution of the light source (up) and their normalized interference fringe pattern (down) (a) Three peaks LED (b) Normal LED.

## **4. Signal recovering methods**

The phase difference between two coherent light beams is detected by interferometric methods, which are most sensitive techniques for optical path difference measurement. The OPD variations have sufficiently low frequency components. So when they are converted to

Optical Fiber Interferometers and Their Applications 17

methods are based on the use of orthogonal components without using heterodyne methods. OFIs usually have two outputs. A 2×2 OFC is employed to combine the two path beams of the interferometer and form the interference pattern. By energy conservation law it

output is dark, all energy must be presented in the other output and vice versa. So no orthogonal components can be found in the outputs. The orthogonal components are produced by the heterodyne methods. The output coupler can be modified such that the orthogonal components directly exist in the outputs. As an example a 3×3 coupler can be employed as the output coupler of the interferometer to create output with orthogonal components without employing the heterodyne detection method (Choma et al., 2003).

Calculation of signal to noise ratio strongly depends on the OFI topology. In principle the noise sources belong to the light source, optical fibers, detector, electronic circuits and environment (Bottacchi, 2008; Tucker & Baney, 2001). Moreover any random process in each stage of interferometry: signal generation, transmission and detection can be considered as a

The laser generation is the result of the quantum interaction of electromagnetic wave and matter. The spontaneous and stimulated emissions are quantum effects and are the noise sources in the phase and amplitude of laser output (Linde, 1986; Clark, 1999; Tsuchida, 1998). On the other hand the interaction of light with universal modes of surrounding reservoir through the mirror coupling and stimulated emission in active medium bath are also noise sources for the laser output (Scully & Zubairy, 2001). The cavity filtering and feedback can reduce the laser noise significantly (Sanders et al., 1992; Cliché et al., 2007). The phase and amplitude of laser noise cause to increase the bandwidth of the laser light. In single mode lasers by proper design of optical cavities, the bandwidth can be reduced to several kilohertz, which gives several tens of kilometers for coherent length. The mode competition and cross saturation effects are new noise sources in multimode lasers that can

Rayleigh scattering, Mie scattering, core cladding interface scattering, Brillouin scattering, absorption and amplification parts in the optical fiber, are the main noise sources in OFI arms and transmission parts. Some parts of the scattered light are trapped in the guided region and travel in both direction of the fiber, contribute to the phase and amplitude noises. Other parts are scattered out of the optical fiber and affect the amplitude noise only. Except the Brillouin and Raman scatterings, all other effects are linear and do not change the light frequency. Both the Brillouin and Raman scattering have two different components Stokes and anti-Stokes frequencies. The Stokes and anti-Stokes Brillouin shifts are due to the lightacoustic phonon interaction and are about ±25 GHz, while the Stokes and anti-Stokes Raman shift correspond to the optical phonon-photon interaction and are of the order of 13 THz (Agrawal, 2007). Beating between Stokes, anti-Stokes and direct beam can occur, but such a high beating frequencies cannot be observed at the output response of any realistic detector and are eliminated intrinsically by the low pass filter detector. The Brillouin and Raman scattering loss can be considered as a source of amplitude noise. The Rayleigh, Brillouin and Raman scattering are symmetrically distributed with respect to the forward

out of phase from one another. When one

is easy to show that the two outputs are 180°

**5. Noise sources in optical fiber interferometers** 

be employed in wide band fiber interferometry.

noise source.

the light intensity on the observation plan, they can easily be detected by photo diodes, photo diode matrix or charge coupled devices (CCD). The detector output is proportional to ���(�), where � is the phase difference. In the presence of cosine and the absence of sinusoidal signal an ambiguity exists in the phase recovering. If the phase amplitude falls outside the (0 to 2π) range, in addition to sine and cosine values we must track the history of phase angle variation to know in which quadrant it precisely lie. Numerous methods have been devised for recovering the path length difference from the output signals of OFI. The methods based on the production of new frequencies are called heterodyne detection otherwise are named homodyne methods. The main three methods are briefly described in following.

## **4.1 Phase generated carrier (PGC) homodyne detection**

In this method the interferometer Laser source is driven by combination of direct and sinusoidal current. The Laser output power and wavelength are modulated by the Laser current variation. In the presence of path length difference, the change of the wavelengths indicates itself as a change in the output phase. The current frequency can be observed in the received phase in the output. Each of the current frequency and its harmonics carries in their sidebands a copy of the created phase modulated signal. Two of these copies are chosen by band pass filters. Proper control of the amplitude of the sinusoidal current and filters configuration guaranty that the chosen copies have the same amplitude. The filters outputs are used as the inputs of an electronic mixer. One of the outputs of the mixer is proportional to sine, while the other one is proportional to the cosine of the interesting signal. Sometimes the method is also called Pseudo-Heterodyne Detection (PHD) (Jackson et al., 1982).To produce the phase shift instead of Laser frequency modulation it is possible to create phase shift with a cylindrical Piezoelectric, which is wrapped around one arm of the interferometer and is derived with a sinusoidal voltage (Hoeling et al., 2001). This case is called synthetic heterodyne method (Strauss, 1994).

## **4.2 Fringe-rate methods**

When large phase shift is produced in an interferometer, two new methods which are called fringe-counting and fringe-rate demodulation become feasible (Barone et al., 1994; Crooker & Garrett, 1987). These methods are based on the transitions of interferometric outputs across some central value. In the fringe counting method, on a suitable period of time the transitions are counted digitally. The instantaneous frequency is determined by the ratio of the counting number to the counting time. Because in practice one must wait a short time to obtain at least one count, it is impossible to obtain an instantaneous count. However the phase can be obtained by integrating the instantaneous frequency. In the fringe-rate method, the transitions are used as inputs of a frequency to voltage converter circuit (FVC). To obtain the phase difference, the output of FVC circuit is integrated. There is no transition for weak signal. The minimum detectable signal is of the order of π radian.

## **4.3 Homodyne method**

The operation range of the synthetic heterodyne method is limited above to π radian, while the fringe-counting and fringe-rate techniques are limited from below to π radian (Dorrer et al., 2001). A number of homodyne techniques are employed to bridge this region. All these

the light intensity on the observation plan, they can easily be detected by photo diodes, photo diode matrix or charge coupled devices (CCD). The detector output is proportional to ���(�), where � is the phase difference. In the presence of cosine and the absence of sinusoidal signal an ambiguity exists in the phase recovering. If the phase amplitude falls outside the (0 to 2π) range, in addition to sine and cosine values we must track the history of phase angle variation to know in which quadrant it precisely lie. Numerous methods have been devised for recovering the path length difference from the output signals of OFI. The methods based on the production of new frequencies are called heterodyne detection otherwise are named homodyne methods. The main three methods are briefly described in

In this method the interferometer Laser source is driven by combination of direct and sinusoidal current. The Laser output power and wavelength are modulated by the Laser current variation. In the presence of path length difference, the change of the wavelengths indicates itself as a change in the output phase. The current frequency can be observed in the received phase in the output. Each of the current frequency and its harmonics carries in their sidebands a copy of the created phase modulated signal. Two of these copies are chosen by band pass filters. Proper control of the amplitude of the sinusoidal current and filters configuration guaranty that the chosen copies have the same amplitude. The filters outputs are used as the inputs of an electronic mixer. One of the outputs of the mixer is proportional to sine, while the other one is proportional to the cosine of the interesting signal. Sometimes the method is also called Pseudo-Heterodyne Detection (PHD) (Jackson et al., 1982).To produce the phase shift instead of Laser frequency modulation it is possible to create phase shift with a cylindrical Piezoelectric, which is wrapped around one arm of the interferometer and is derived with a sinusoidal voltage (Hoeling et al., 2001). This case is

When large phase shift is produced in an interferometer, two new methods which are called fringe-counting and fringe-rate demodulation become feasible (Barone et al., 1994; Crooker & Garrett, 1987). These methods are based on the transitions of interferometric outputs across some central value. In the fringe counting method, on a suitable period of time the transitions are counted digitally. The instantaneous frequency is determined by the ratio of the counting number to the counting time. Because in practice one must wait a short time to obtain at least one count, it is impossible to obtain an instantaneous count. However the phase can be obtained by integrating the instantaneous frequency. In the fringe-rate method, the transitions are used as inputs of a frequency to voltage converter circuit (FVC). To obtain the phase difference, the output of FVC circuit is integrated. There is no transition for weak

The operation range of the synthetic heterodyne method is limited above to π radian, while the fringe-counting and fringe-rate techniques are limited from below to π radian (Dorrer et al., 2001). A number of homodyne techniques are employed to bridge this region. All these

**4.1 Phase generated carrier (PGC) homodyne detection** 

called synthetic heterodyne method (Strauss, 1994).

signal. The minimum detectable signal is of the order of π radian.

**4.2 Fringe-rate methods** 

**4.3 Homodyne method** 

following.

methods are based on the use of orthogonal components without using heterodyne methods. OFIs usually have two outputs. A 2×2 OFC is employed to combine the two path beams of the interferometer and form the interference pattern. By energy conservation law it is easy to show that the two outputs are 180° out of phase from one another. When one output is dark, all energy must be presented in the other output and vice versa. So no orthogonal components can be found in the outputs. The orthogonal components are produced by the heterodyne methods. The output coupler can be modified such that the orthogonal components directly exist in the outputs. As an example a 3×3 coupler can be employed as the output coupler of the interferometer to create output with orthogonal components without employing the heterodyne detection method (Choma et al., 2003).

## **5. Noise sources in optical fiber interferometers**

Calculation of signal to noise ratio strongly depends on the OFI topology. In principle the noise sources belong to the light source, optical fibers, detector, electronic circuits and environment (Bottacchi, 2008; Tucker & Baney, 2001). Moreover any random process in each stage of interferometry: signal generation, transmission and detection can be considered as a noise source.

The laser generation is the result of the quantum interaction of electromagnetic wave and matter. The spontaneous and stimulated emissions are quantum effects and are the noise sources in the phase and amplitude of laser output (Linde, 1986; Clark, 1999; Tsuchida, 1998). On the other hand the interaction of light with universal modes of surrounding reservoir through the mirror coupling and stimulated emission in active medium bath are also noise sources for the laser output (Scully & Zubairy, 2001). The cavity filtering and feedback can reduce the laser noise significantly (Sanders et al., 1992; Cliché et al., 2007). The phase and amplitude of laser noise cause to increase the bandwidth of the laser light. In single mode lasers by proper design of optical cavities, the bandwidth can be reduced to several kilohertz, which gives several tens of kilometers for coherent length. The mode competition and cross saturation effects are new noise sources in multimode lasers that can be employed in wide band fiber interferometry.

Rayleigh scattering, Mie scattering, core cladding interface scattering, Brillouin scattering, absorption and amplification parts in the optical fiber, are the main noise sources in OFI arms and transmission parts. Some parts of the scattered light are trapped in the guided region and travel in both direction of the fiber, contribute to the phase and amplitude noises. Other parts are scattered out of the optical fiber and affect the amplitude noise only. Except the Brillouin and Raman scatterings, all other effects are linear and do not change the light frequency. Both the Brillouin and Raman scattering have two different components Stokes and anti-Stokes frequencies. The Stokes and anti-Stokes Brillouin shifts are due to the lightacoustic phonon interaction and are about ±25 GHz, while the Stokes and anti-Stokes Raman shift correspond to the optical phonon-photon interaction and are of the order of 13 THz (Agrawal, 2007). Beating between Stokes, anti-Stokes and direct beam can occur, but such a high beating frequencies cannot be observed at the output response of any realistic detector and are eliminated intrinsically by the low pass filter detector. The Brillouin and Raman scattering loss can be considered as a source of amplitude noise. The Rayleigh, Brillouin and Raman scattering are symmetrically distributed with respect to the forward

Optical Fiber Interferometers and Their Applications 19

properties make long length and short length waveguide and fiber interferometer sensors, suitable in novel applications such as oil and gas pipeline monitoring, temperature distribution measurement in the depth of ocean and intruder sensors. Among the wide range of applications of waveguides and fiber interferometers, only a few applications in the optical communication networks and special types of fiber and waveguide interferometric

The key devices in optical DWDM communication networks are re-amplifying, re-shaping and re-timing (3R-regenerator) systems. In re-shaping and re-timing circuits the nonlinear networks such as clipper, clampers, switching and flip-flops are of prime importance. While in add-drop filters, the linear filters such as tune and notch filters have an important role. On the basis of a nonlinear Mach-Zehnder interferometer, the structure of an all optical

An optical fiber with high nonlinear Kerr effect such as Chalcogenide glasses is employed in one of the Mach-Zehnder interferometer arms. So in the presence of a suitable light intensity at the input of optical fiber coupler 1 (OFC 1), the change of refractive index ( ���� � �� ) causes a �-phase shift in the upper arm of the interferometer relative to the lower arm. It is assumed that in the absence of the input, the interferometer arms are balanced and the outputs � and �̅ are in the constructive and destructive conditions respectively. The 0 and 1 digital states are represented by destructive and constructive output ports. In the presence of the OFC 1 input, the output � changes to 0 and the �̅ switches to 1. This interferometer is an optical logic inverter. The structure shown in Fig. 15 is also used in quantum nondemolition experiments (Gerry & Knight, 2005). For small input intensities Fig. 15 acts as an intensity modulator circuit and �̅ output is approximately proportional to the input intensity. By varying input intensity, the output varies from its maximum value to zero, i.e.

The inverter of Fig. 16 is designed on the basis of optical waveguides to avoid the high

Fig. 16. A schematic of an inverter based on an optical waveguide and micro-ring resonator.

Fig. 15. A schematic of an all optical inverter. OFC is an optical 2×2 coupler.

this circuit operates as a light controlled variable attenuator.

length nonlinear optical fibers in the inverter design.

OC is 2×2 optical coupler.

sensors are mentioned in this section.

inverter is shown in Fig. 15.

**6.1 Applications in optical fiber networks** 

and backward direction while those of Mie and core-cladding interference scattering are mainly in the forward direction.

Mode coupling is another source of noise in multimode fiber interferometer. In such an interferometer the mode coupling noise must be taken into account. Absorption and amplification correspond to the interaction of light with reservoirs so according to quantum Langevin equation there are some noises in the output (Scully & Zubairy, 2001). Generally Avalanch photo diode (APD), PIN diode, charge coupled device (CCD) and photo multiplier (PM) are used as the electronic detector of the OFIs. Dark current noise, shot noise, background noise, thermal noise and flicker noise are common in all of the optical detectors. The generation and recombination of electron hole are a stochastic process in semiconductor detectors and are the noise sources of such detectors. The avalanche effect, the basis of operation of APDs is a random process and causes noise generation in avalanche photo diode. The same effect on the anodes of PM can be a noise source in PM detectors. The amplifier noise which is consistent of the shot noise, Johnson noise, burst noise and flicker noise of different solid state electronic elements of the amplifier is the final intrinsic noise of OFI.

The fiber parameters can be affected by the environmental physical variations such as mechanical vibration, acoustic agitation, pressure, tension and thermal variations. In a controlled way this effects can be used to make the optical fibers as a sensor for these physical quantities, while in OFIs are noise sources. As an example the population of Stokes and anti-Stokes photons are functions of the fiber temperature and can be used to design a high precision temperature sensor for water, oil and gas leak detection systems (Harris et al., 2010; Chelliah et al., 2010). The Stokes and anti-Stokes parameters of Brillouin scattering are functions of fiber strain and fiber temperature. This effect is used to measure the strain and temperature simultaneously for structural health monitoring systems (Güemes, 2006; Bahrampour & Maasoumi, 2010). The optical fiber sensitivity to mechanical variation and acoustic waves are employed for various applications such as acoustic, vibration and ultrasonic detectors for under water sensor systems. However the output signal is affected by all the noise sources and the aim is to denoise signal by the signal processing methods. Depending on the signal, one of the denoising methods such as Fourier regularized deconvolution (ForD) and Fourier wavelet regularized deconvolution (ForWaRD) method can be employed (Bahrampour & Askari, 2006; Bahrampour et al., 2012). The wavelet deconvolution method generally use to denoise transient signals. The short time Fourier method is employed to denoise the music-like signals of a fiber intruder detector based on the birefrigent fiber interferometer (Bahrampour et al., 2012).

## **6. Applications of optical fiber interferometers**

Optical fiber interferometers as a precise measuring interferometer or sensitive tools have many applications in all branches of science and technology (Shizhuo et al., 2008). The OFIs can be employed to design the optical components for the inline signal processing, such as band pass filters in optical communication networks. The same topologies can be easily fabricated by the light waveguides in the integrated circuits by means of photolithographic process for application in optical transmitters and receivers. Because of high sensitivity of the interferometer, the linear and nonlinear properties of optical fiber can be detected. These

and backward direction while those of Mie and core-cladding interference scattering are

Mode coupling is another source of noise in multimode fiber interferometer. In such an interferometer the mode coupling noise must be taken into account. Absorption and amplification correspond to the interaction of light with reservoirs so according to quantum Langevin equation there are some noises in the output (Scully & Zubairy, 2001). Generally Avalanch photo diode (APD), PIN diode, charge coupled device (CCD) and photo multiplier (PM) are used as the electronic detector of the OFIs. Dark current noise, shot noise, background noise, thermal noise and flicker noise are common in all of the optical detectors. The generation and recombination of electron hole are a stochastic process in semiconductor detectors and are the noise sources of such detectors. The avalanche effect, the basis of operation of APDs is a random process and causes noise generation in avalanche photo diode. The same effect on the anodes of PM can be a noise source in PM detectors. The amplifier noise which is consistent of the shot noise, Johnson noise, burst noise and flicker noise of different solid state electronic elements of the amplifier is the final intrinsic

The fiber parameters can be affected by the environmental physical variations such as mechanical vibration, acoustic agitation, pressure, tension and thermal variations. In a controlled way this effects can be used to make the optical fibers as a sensor for these physical quantities, while in OFIs are noise sources. As an example the population of Stokes and anti-Stokes photons are functions of the fiber temperature and can be used to design a high precision temperature sensor for water, oil and gas leak detection systems (Harris et al., 2010; Chelliah et al., 2010). The Stokes and anti-Stokes parameters of Brillouin scattering are functions of fiber strain and fiber temperature. This effect is used to measure the strain and temperature simultaneously for structural health monitoring systems (Güemes, 2006; Bahrampour & Maasoumi, 2010). The optical fiber sensitivity to mechanical variation and acoustic waves are employed for various applications such as acoustic, vibration and ultrasonic detectors for under water sensor systems. However the output signal is affected by all the noise sources and the aim is to denoise signal by the signal processing methods. Depending on the signal, one of the denoising methods such as Fourier regularized deconvolution (ForD) and Fourier wavelet regularized deconvolution (ForWaRD) method can be employed (Bahrampour & Askari, 2006; Bahrampour et al., 2012). The wavelet deconvolution method generally use to denoise transient signals. The short time Fourier method is employed to denoise the music-like signals of a fiber intruder detector based on the birefrigent fiber interferometer

Optical fiber interferometers as a precise measuring interferometer or sensitive tools have many applications in all branches of science and technology (Shizhuo et al., 2008). The OFIs can be employed to design the optical components for the inline signal processing, such as band pass filters in optical communication networks. The same topologies can be easily fabricated by the light waveguides in the integrated circuits by means of photolithographic process for application in optical transmitters and receivers. Because of high sensitivity of the interferometer, the linear and nonlinear properties of optical fiber can be detected. These

mainly in the forward direction.

noise of OFI.

(Bahrampour et al., 2012).

**6. Applications of optical fiber interferometers** 

properties make long length and short length waveguide and fiber interferometer sensors, suitable in novel applications such as oil and gas pipeline monitoring, temperature distribution measurement in the depth of ocean and intruder sensors. Among the wide range of applications of waveguides and fiber interferometers, only a few applications in the optical communication networks and special types of fiber and waveguide interferometric sensors are mentioned in this section.

## **6.1 Applications in optical fiber networks**

The key devices in optical DWDM communication networks are re-amplifying, re-shaping and re-timing (3R-regenerator) systems. In re-shaping and re-timing circuits the nonlinear networks such as clipper, clampers, switching and flip-flops are of prime importance. While in add-drop filters, the linear filters such as tune and notch filters have an important role. On the basis of a nonlinear Mach-Zehnder interferometer, the structure of an all optical inverter is shown in Fig. 15.

Fig. 15. A schematic of an all optical inverter. OFC is an optical 2×2 coupler.

An optical fiber with high nonlinear Kerr effect such as Chalcogenide glasses is employed in one of the Mach-Zehnder interferometer arms. So in the presence of a suitable light intensity at the input of optical fiber coupler 1 (OFC 1), the change of refractive index ( ���� � �� ) causes a �-phase shift in the upper arm of the interferometer relative to the lower arm. It is assumed that in the absence of the input, the interferometer arms are balanced and the outputs � and �̅ are in the constructive and destructive conditions respectively. The 0 and 1 digital states are represented by destructive and constructive output ports. In the presence of the OFC 1 input, the output � changes to 0 and the �̅ switches to 1. This interferometer is an optical logic inverter. The structure shown in Fig. 15 is also used in quantum nondemolition experiments (Gerry & Knight, 2005). For small input intensities Fig. 15 acts as an intensity modulator circuit and �̅ output is approximately proportional to the input intensity. By varying input intensity, the output varies from its maximum value to zero, i.e. this circuit operates as a light controlled variable attenuator.

The inverter of Fig. 16 is designed on the basis of optical waveguides to avoid the high length nonlinear optical fibers in the inverter design.

Fig. 16. A schematic of an inverter based on an optical waveguide and micro-ring resonator. OC is 2×2 optical coupler.

Optical Fiber Interferometers and Their Applications 21

In addition to the logic gates and nonlinear circuits, the fiber interferometers can be used to design linear circuits such as different types of optical filters. In many applications such as selection of a narrow spectrum from a broad band spectrum, band pass filter is of prime importance. Due to the Bragg diffraction effect, each FBG fiber can be used as a notch or band stop filter. As shown in Fig. 19, an optical coupler is employed to detect the reflected spectrum of FBG. In this filter, the input power splits into two parts by the OFC. The light reflected by a FBG is again equally split between the ports 1 and 2. Hence only 25% of the

To eliminate the insertion-loss of the band pass filter several interferometric methods was proposed (Kashyap, 1999). On the basis of Michelson, Mach-Zehnder and Fabry-Perot interferometers three different design of band pass filters are presented in Fig. 20(a-c)

FBG F-P cavity FBG

Amplitude

Frequency

OFC FBG <sup>1</sup>

Fig. 20. Interferometric design of band pass filter (a) Michelson (b) Mach-Zehnder (c) Fabry-

FBG2 (c)

OFC2

In both arms of Michelson interferometer as shown in Fig. 20(a) a FBG is employed. The arms of Michelson interferometer are designed in such a way that the reflected light from FBG 2 arrives at the input port of OFC, has � out of phase with respect to the light reflected from FBG 1. In such a condition light from FBG 1 and FBG 2 interfere constructively at the output port 2, so that 100% of the light at the Bragg wavelength appears at the output port of band pass filter (Kashyap, 1999). The band pass spectrum can be designed by the profile of chirped spectrum. The dual grating Mach-Zehnder interferometer band pass filter as shown in Fig. 20 (b) is designed for the application in add-drop filters. The principle of operation is the same as that is demonstrated in Michelson band pass filter. Here ''UV trimming'' is used to balance the interferometer after the gratings are written. ''UV trimming'' relies on photo induced change in the refractive index to adjust the optical path difference. The simplest band pass filter is an inline Fabry-Perot interferometer. In distributed feedback (DFB) lasers, two FBG can be employed instead of mirrors. As shown in Fig. 20(c) a single � �⁄ phase-shifted FBG has a sharp Lorentzian line shape band pass in the middle of band stop. The broader transmission band width is obtained by cascading several structure (Haus & Lai, 1992). Number of band pass peaks that they appear within

the band stop increases by increasing the gap between the two grating sections.

light is in the output port 2 of the band pass filter (Kashyap, 1999).

2

Fig. 19. A schematic of band pass filter.

OFC

OFC1

FBG1

FBG1 FBG2 UV UV

respectively(Kashyap, 1999).

1 2

> 1 2

(b)

(a)

Perot.

As shown in Fig. 16, the Kerr cell that is shown in Fig. 15 is replaced by a high dispersive nonlinear element such as micro-ring or microsphere. The phase difference between the input and output of micro-ring can be changed by the resonance frequency of the micro-ring which is controllable due to the cross-Kerr effect. The upper input to the micro-ring causes to change the micro-ring refractive index and therefore the resonance frequency is changed. So the phase difference between the input and output of the upper arm of the interferometer is changed. On the basis of optical fiber and optical waveguide interferometers in combination with ultrahigh nonlinear optical elements (UHNO), such as semiconductor optical amplifiers (SOA), different high frequency optical classical logic gates are designed and demonstrated. Also quantum interferometers such as Hong-Ou-Mandel interferometer are employed to design quantum gates (Hong et al., 1987; Olindo et al., 2006).

The bi-stability effect is the basis of the clipper and flip-flop circuits. Most bi-stability designs include both a cavity with nonlinear medium and a feedback (Bahrampour et al., 2008a, 2008b, 2008c). A novel OFI with common mode compensation is proposed by Backman (Backman, 1989). The Backman interferometer consists of a Mach-Zehnder interferometer with one nonlinear path and re-circulating delay line as shown in Fig. 17. The output intensity |����| � versus the input intensity |���| � in the steady state has the bi-stability behavior.

Fig. 17. A schematic of Backman interferometer. NL is nonlinear line and DL is delay line.

Flip-flops are building blocks of the sequential logic circuits such as time recovering circuits. As usual a reset-set (RS) flip-flop can be designed on the basis of regenerative feedback in the two inverter circuit. Fig.18 shows a RS flip-flop based on the two Mach-Zehnder interferometer inverters. The optical fiber couplers OFC 1 and OFC 2 are 3 × 3 couplers and OFC 3 and OFC 4 are 2 × 2 couplers. The bias light inserts to the upper and lower Mach-Zehnder interferometers (MZI 1, MZI 2) by a 2×2 coupler. R and S are the reset and set trigger inputs. � � �̅ and � � �� are the outputs of MZI 1 and MZI 2 respectively. Due to the energy conservation law, �̅ and �� are the logic complement of � and � outputs. In the absence of set and reset �=�=1. The output complement of each inverter is connected to the control input of the other inverter. This network of Fig.18 has two stable (� = 0� �� = 1) and (� = 1� �� = 0) states. In the presence of trigger signal at S or R inputs, this system can switch between these two stable states.

Fig. 18. A schematic of an optical Flip-Flop by combination of two Mach-Zehnder Interferometers.

As shown in Fig. 16, the Kerr cell that is shown in Fig. 15 is replaced by a high dispersive nonlinear element such as micro-ring or microsphere. The phase difference between the input and output of micro-ring can be changed by the resonance frequency of the micro-ring which is controllable due to the cross-Kerr effect. The upper input to the micro-ring causes to change the micro-ring refractive index and therefore the resonance frequency is changed. So the phase difference between the input and output of the upper arm of the interferometer is changed. On the basis of optical fiber and optical waveguide interferometers in combination with ultrahigh nonlinear optical elements (UHNO), such as semiconductor optical amplifiers (SOA), different high frequency optical classical logic gates are designed and demonstrated. Also quantum interferometers such as Hong-Ou-Mandel interferometer

The bi-stability effect is the basis of the clipper and flip-flop circuits. Most bi-stability designs include both a cavity with nonlinear medium and a feedback (Bahrampour et al., 2008a, 2008b, 2008c). A novel OFI with common mode compensation is proposed by Backman (Backman, 1989). The Backman interferometer consists of a Mach-Zehnder interferometer with one nonlinear path and re-circulating delay line as shown in Fig. 17. The output intensity |����|

Fig. 17. A schematic of Backman interferometer. NL is nonlinear line and DL is delay line.

Fig. 18. A schematic of an optical Flip-Flop by combination of two Mach-Zehnder

Flip-flops are building blocks of the sequential logic circuits such as time recovering circuits. As usual a reset-set (RS) flip-flop can be designed on the basis of regenerative feedback in the two inverter circuit. Fig.18 shows a RS flip-flop based on the two Mach-Zehnder interferometer inverters. The optical fiber couplers OFC 1 and OFC 2 are 3 × 3 couplers and OFC 3 and OFC 4 are 2 × 2 couplers. The bias light inserts to the upper and lower Mach-Zehnder interferometers (MZI 1, MZI 2) by a 2×2 coupler. R and S are the reset and set trigger inputs. � � �̅ and � � �� are the outputs of MZI 1 and MZI 2 respectively. Due to the energy conservation law, �̅ and �� are the logic complement of � and � outputs. In the absence of set and reset �=�=1. The output complement of each inverter is connected to the control input of the other inverter. This network of Fig.18 has two stable (� = 0� �� = 1) and (� = 1� �� = 0) states. In the presence of trigger signal at S or R inputs, this system can

� in the steady state has the bi-stability behavior.

�

are employed to design quantum gates (Hong et al., 1987; Olindo et al., 2006).

versus the input intensity |���|

switch between these two stable states.

Interferometers.

In addition to the logic gates and nonlinear circuits, the fiber interferometers can be used to design linear circuits such as different types of optical filters. In many applications such as selection of a narrow spectrum from a broad band spectrum, band pass filter is of prime importance. Due to the Bragg diffraction effect, each FBG fiber can be used as a notch or band stop filter. As shown in Fig. 19, an optical coupler is employed to detect the reflected spectrum of FBG. In this filter, the input power splits into two parts by the OFC. The light reflected by a FBG is again equally split between the ports 1 and 2. Hence only 25% of the light is in the output port 2 of the band pass filter (Kashyap, 1999).

Fig. 19. A schematic of band pass filter.

To eliminate the insertion-loss of the band pass filter several interferometric methods was proposed (Kashyap, 1999). On the basis of Michelson, Mach-Zehnder and Fabry-Perot interferometers three different design of band pass filters are presented in Fig. 20(a-c) respectively(Kashyap, 1999).

Fig. 20. Interferometric design of band pass filter (a) Michelson (b) Mach-Zehnder (c) Fabry-Perot.

In both arms of Michelson interferometer as shown in Fig. 20(a) a FBG is employed. The arms of Michelson interferometer are designed in such a way that the reflected light from FBG 2 arrives at the input port of OFC, has � out of phase with respect to the light reflected from FBG 1. In such a condition light from FBG 1 and FBG 2 interfere constructively at the output port 2, so that 100% of the light at the Bragg wavelength appears at the output port of band pass filter (Kashyap, 1999). The band pass spectrum can be designed by the profile of chirped spectrum. The dual grating Mach-Zehnder interferometer band pass filter as shown in Fig. 20 (b) is designed for the application in add-drop filters. The principle of operation is the same as that is demonstrated in Michelson band pass filter. Here ''UV trimming'' is used to balance the interferometer after the gratings are written. ''UV trimming'' relies on photo induced change in the refractive index to adjust the optical path difference. The simplest band pass filter is an inline Fabry-Perot interferometer. In distributed feedback (DFB) lasers, two FBG can be employed instead of mirrors. As shown in Fig. 20(c) a single � �⁄ phase-shifted FBG has a sharp Lorentzian line shape band pass in the middle of band stop. The broader transmission band width is obtained by cascading several structure (Haus & Lai, 1992). Number of band pass peaks that they appear within the band stop increases by increasing the gap between the two grating sections.

Optical Fiber Interferometers and Their Applications 23

Fig. 22. A schematic of a birefringent fiber intruder detection system. FM is the frequency modulator, LD is laser diod, x-po is a x-polarizer, FPBS is a fiber polarization beam splitter, FPR is a fiber polarization rotator, APD is an avalanche photo-diode detector, ALPF is an active electronic low pass filter, A/D is an analog to digital convertor, Com. a computer system for signal processing and denoising. (b) Ramp input to the FM system (Bahrampour

distributed fiber sensors is optical time domain reflectometer (OTDR) which is based on the monitoring the Rayleigh back scattering along the fiber. On the basis of Raman and Brilliouin scattering the OTDR is developed to the Raman time optical domain reflectometer (ROTDR) and Brillouin optical time domain reflectometer (BOTDR) respectively. The OTDR, ROTDR and BOTDR optical fiber sensors have applications in structural health monitoring (Glisic & Inaudi, 2008). In the absence of any intrusion, the �OTDR signal is saved in an electronic memory and it is compared with the �OTDR output continuously (Juarez et al., 2005). Due to the elasto-optic effect, in the presence of an intrusion, the fiber refractive index and hence the phase of the back scattered signal changes. This phase

IV ) Phase modulated sensors: Variation of the optical length of optical fiber causes a phase shift of the light beam��� � ��(��� � ���)⁄��, where �� is the free space light wavelength and n is the fiber refractive index. Phase shifts usually are measured by interferometric methods. Refractive index and fiber length can vary due to the characteristic of various measurands and therefore the cross sensitivity occurs. To avoid the cross sensitivity, special design of jacketing is necessary. The material of the jacketing is chosen such that to improve the effect of desired measurand and attenuates the others. A schematic of an optical fiber intrusion detector system is presented in Fig. 23. The light laser source through an optical circulator and a 50:50 coupler is connected to a Faraday rotating mirror (FRM). The intrusion distance (��) is the length between the intrusion point and FRM. The parts 3 and 4 of the coupler are connected with a fiber of length �� to form a delay loop. The returned light through the port 1 of 50:50 coupler and circulator is transported to the detector . The intrusion point is determined after signal processing. There are four different paths in the

changes can be measured at the sensor output (Righini et al., 2009).

system for transmission of light from source to detector:

Path IV: 1→4→�� → � → � → ��� → � → � → �� →4→1

Path I: 1 → � � ��� � � → � → �� →4→1 Path II: 1→4→�� → � → � → ��� → � → 1

Path III: 1 → � → ��� → � → 1

et al, 2012).

## **6.2 Some applications in optical fiber sensors**

The traveling wave in the dielectric medium of optical fibers and waveguides can be perturbed by their environment. This is the basic idea of the optical fiber sensors (OFS). The interaction of quantity of interest (which is called the measurand), with the optical fiber produces a modulation in the parameters of propagating light beam within the fiber. Generally there are four beam parameters for measurand modulation:

I) Intensity modulation: the intensity modulated fiber sensors are simplest and low cost fiber sensors for measuring the position, pressure and vibration in medical and industrial applications (Polygerinos et al., 2011, Jayanthkumar et al., 2006). Fig. 21 shows a distributed oil leak detection system based on intensity modulated sensor (Carrillo, 2002). The leaked oil causes to expand the polymer around the optical fiber. Due to the wrapped strain-less steel wire, the fiber bending loss increases and the oil leakage position can be measured by a commercial optical time domain reflectometer (OTDR) (Righini et al., 2009).

Fig. 21. Distributed Oil leak detection system.

II) Wavelength modulation: The measurands such as temperature and strain can be modulated on the resonance frequency of an inline Fabry-Perot or Bragg wavelength of an inline LPG. An optical or chemical transducers joint at the end of a fiber can be used as the wavelength modulator. Interaction of the measurand with transducer causes to change the spectral properties of transducer. The measurement of the optical spectrum of the transducer through the optical fiber makes possibility to monitor measurand status (Righini et al., 2009).

III) Polarization modulation: In birefringent optical fibers the two fundamental modes propagate with slightly different phase velocities. On the basis of high birefringent fibers several methods for current and magnetic field measurement are designed and manufactured. Fig. 22 shows the principle of operation of an intrusion sensor based on the birefringent optical fiber (Bahrampour et al., 2012).

An x-polarized ramped frequency modulated laser is injected to the birefringent fiber sensor. At the cross point of the intrusion and the fiber sensor, energy from the x-polarized mode is converted to the y-polarized mode. Due to different velocities of the x- and ypolarized modes a beating frequency is observed at the output of the detector. The intrusion position can be obtained from the output beating frequency.

The optical fiber and waveguide sensors that have been investigated and proposed for science, industrial, military, biochemical, biomedical, environment, automotive, avionic and geophysical applications are countless. One of the basic characteristics of the optical fiber sensors is their ability for long length distributed sensing. One of the most popular

The traveling wave in the dielectric medium of optical fibers and waveguides can be perturbed by their environment. This is the basic idea of the optical fiber sensors (OFS). The interaction of quantity of interest (which is called the measurand), with the optical fiber produces a modulation in the parameters of propagating light beam within the fiber.

I) Intensity modulation: the intensity modulated fiber sensors are simplest and low cost fiber sensors for measuring the position, pressure and vibration in medical and industrial applications (Polygerinos et al., 2011, Jayanthkumar et al., 2006). Fig. 21 shows a distributed oil leak detection system based on intensity modulated sensor (Carrillo, 2002). The leaked oil causes to expand the polymer around the optical fiber. Due to the wrapped strain-less steel wire, the fiber bending loss increases and the oil leakage position can be measured by a

Undisturbed fiber Disturbed fiber

Wrapped strain-less steel wire

II) Wavelength modulation: The measurands such as temperature and strain can be modulated on the resonance frequency of an inline Fabry-Perot or Bragg wavelength of an inline LPG. An optical or chemical transducers joint at the end of a fiber can be used as the wavelength modulator. Interaction of the measurand with transducer causes to change the spectral properties of transducer. The measurement of the optical spectrum of the transducer through the optical fiber makes possibility to monitor measurand status (Righini

Swollen polymer

III) Polarization modulation: In birefringent optical fibers the two fundamental modes propagate with slightly different phase velocities. On the basis of high birefringent fibers several methods for current and magnetic field measurement are designed and manufactured. Fig. 22 shows the principle of operation of an intrusion sensor based on the

An x-polarized ramped frequency modulated laser is injected to the birefringent fiber sensor. At the cross point of the intrusion and the fiber sensor, energy from the x-polarized mode is converted to the y-polarized mode. Due to different velocities of the x- and ypolarized modes a beating frequency is observed at the output of the detector. The intrusion

The optical fiber and waveguide sensors that have been investigated and proposed for science, industrial, military, biochemical, biomedical, environment, automotive, avionic and geophysical applications are countless. One of the basic characteristics of the optical fiber sensors is their ability for long length distributed sensing. One of the most popular

**6.2 Some applications in optical fiber sensors** 

Fig. 21. Distributed Oil leak detection system.

birefringent optical fiber (Bahrampour et al., 2012).

position can be obtained from the output beating frequency.

et al., 2009).

Generally there are four beam parameters for measurand modulation:

commercial optical time domain reflectometer (OTDR) (Righini et al., 2009).

(a) (b)

Dry polymer

Fig. 22. A schematic of a birefringent fiber intruder detection system. FM is the frequency modulator, LD is laser diod, x-po is a x-polarizer, FPBS is a fiber polarization beam splitter, FPR is a fiber polarization rotator, APD is an avalanche photo-diode detector, ALPF is an active electronic low pass filter, A/D is an analog to digital convertor, Com. a computer system for signal processing and denoising. (b) Ramp input to the FM system (Bahrampour et al, 2012).

distributed fiber sensors is optical time domain reflectometer (OTDR) which is based on the monitoring the Rayleigh back scattering along the fiber. On the basis of Raman and Brilliouin scattering the OTDR is developed to the Raman time optical domain reflectometer (ROTDR) and Brillouin optical time domain reflectometer (BOTDR) respectively. The OTDR, ROTDR and BOTDR optical fiber sensors have applications in structural health monitoring (Glisic & Inaudi, 2008). In the absence of any intrusion, the �OTDR signal is saved in an electronic memory and it is compared with the �OTDR output continuously (Juarez et al., 2005). Due to the elasto-optic effect, in the presence of an intrusion, the fiber refractive index and hence the phase of the back scattered signal changes. This phase changes can be measured at the sensor output (Righini et al., 2009).

IV ) Phase modulated sensors: Variation of the optical length of optical fiber causes a phase shift of the light beam��� � ��(��� � ���)⁄��, where �� is the free space light wavelength and n is the fiber refractive index. Phase shifts usually are measured by interferometric methods. Refractive index and fiber length can vary due to the characteristic of various measurands and therefore the cross sensitivity occurs. To avoid the cross sensitivity, special design of jacketing is necessary. The material of the jacketing is chosen such that to improve the effect of desired measurand and attenuates the others. A schematic of an optical fiber intrusion detector system is presented in Fig. 23. The light laser source through an optical circulator and a 50:50 coupler is connected to a Faraday rotating mirror (FRM). The intrusion distance (��) is the length between the intrusion point and FRM. The parts 3 and 4 of the coupler are connected with a fiber of length �� to form a delay loop. The returned light through the port 1 of 50:50 coupler and circulator is transported to the detector . The intrusion point is determined after signal processing. There are four different paths in the system for transmission of light from source to detector:

Path I: 1 → � � ��� � � → � → �� →4→1 Path II: 1→4→�� → � → � → ��� → � → 1 Path III: 1 → � → ��� → � → 1 Path IV: 1→4→�� → � → � → ��� → � → � → �� →4→1

Optical Fiber Interferometers and Their Applications 25

To minimize the effects of other noises, whole setup including optical elements and beam path are kept in ultra-high vacuum (10�� − 10�������) and the optical elements are suspended on top of the seismic isolation system. A highly stabilized laser and active control system for adjusting cavity length are employed in these devices. So the long

Fig. 24 shows another configuration for GW detection that uses optical fiber in the arms of Michelson interferometer. GW optical fiber interferometric detector is very small, cheap and

simple to build and operate (Cahill, 2007; Sacharov, 2001; Cahill & Stokes, 2008).

Fig. 24. A schematic of gravitational wave optical fiber interferometric detector.

Backman, A.B. (1989) *Journal of Lightwave Technology*, Vol. 7

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(ForWaRD) for lidar systems based on TEA CO2 laser. Optics Communications,

**7. Reference** 

0471279692

pp. 240-249

Diego

2456

Vol. 257, No. 1, pp. 97-111.

baseline Laser interferometer GW detectors are high cost projects.

Fig. 23. A schematic of an optical fiber intrusion detector system.

All path differences except the optical path difference of the path I and II are so large, so they have no effect on the interference signal. The electric field at the photo detector is the superposition of the electric field corresponding to the four different paths. The time dependence of the strength of an intrusion at the detector and the instant � is �� � ���������, where *Δφ0* and *ωs* are the induced amplitude and frequency by intruder respectively. The light intensity of interference of electric field of path I and path II on the photo detector is easily obtained.

$$I\_{ln} = 2E\_0^2 \cos\left\{4\Delta\varphi\_0 \sin\frac{\pi\_2\omega\_s}{2} \cos\frac{\pi\_1\omega\_s}{2} \cos\omega\_s(t - t\_0)\right\},\tag{8}$$

where �� � 2���/�, �� � 2���/� and �� � (�� � ��)/2. The relative amplitude of the frequency components of the detector outputs are functions of the intrusion distances and can be obtained by the Fourier transform method (Jia et al., 2008).

According to the general relativity, gravitational waves (GW) are produced when the curvature of spacetime disturbed by accelerating mass. The ripples in the curvature of spacetime propagate at the speed of light. A GW causes a tiny time dependent quadruple change of strain in the plane transverse to the wave's propagation direction. The space is stretched in one direction while is shrunk along its perpendicular direction. The strength of GW '*h'* is expressed by the dimensionless strain �� �⁄ . Due to the quadruple nature of Michelson interferometer, it is suitable device for gravitational wave detection. The interference pattern is linear measure of the strain. The amplitudes of GWs radiated from astrophysical sources at Earth are typically of the order of 10��� or smaller. Detection of such weak strains needs high sensitive devices. For increasing the sensitivity of the interferometer, the lengths of the interferometer arms were increased to the order of kilometers and multi-path cell or Fabry-Perot optical cavity was used in each arm. So the light can be stored for a time comparable to the time scale of GW signal. Long base line gravitational wave detectors such as LIGO, VIRGO, GEO and TAMA are now operational. In the presence of GW, the change in the length of arms is very small. Hence many noises such as seismic noise, thermal noise and quantum noise limit the sensitivity of interferometer. Quantum noise is the fundamental and unavoidable noise in new generation of these interferometers and is due to the light-interferometer interaction. So the sensitivity of laser GW detector depends on the quantum state of light. It was shown that depending on the parameters of interferometer such as arm's lengths, frequency of laser and mass of mirrors, the optimum quantum state for the dark port is vacuum squeezed state with specific squeezing factor. By employing this optimum quantum state in the dark port, the quantum noise and optimum laser power reduce one order of magnitude relative to the conventional interferometers (Tofighi et al., 2010).

To minimize the effects of other noises, whole setup including optical elements and beam path are kept in ultra-high vacuum (10�� − 10�������) and the optical elements are suspended on top of the seismic isolation system. A highly stabilized laser and active control system for adjusting cavity length are employed in these devices. So the long baseline Laser interferometer GW detectors are high cost projects.

Fig. 24 shows another configuration for GW detection that uses optical fiber in the arms of Michelson interferometer. GW optical fiber interferometric detector is very small, cheap and simple to build and operate (Cahill, 2007; Sacharov, 2001; Cahill & Stokes, 2008).

Fig. 24. A schematic of gravitational wave optical fiber interferometric detector.

## **7. Reference**

24 Interferometry – Research and Applications in Science and Technology

1 2 4 3

Ld

OFC Lx

All path differences except the optical path difference of the path I and II are so large, so they have no effect on the interference signal. The electric field at the photo detector is the superposition of the electric field corresponding to the four different paths. The time dependence of the strength of an intrusion at the detector and the instant � is �� � ���������, where *Δφ0* and *ωs* are the induced amplitude and frequency by intruder respectively. The light intensity of interference of electric field of path I and path II on the

�

where �� � 2���/�, �� � 2���/� and �� � (�� � ��)/2. The relative amplitude of the frequency components of the detector outputs are functions of the intrusion distances and

According to the general relativity, gravitational waves (GW) are produced when the curvature of spacetime disturbed by accelerating mass. The ripples in the curvature of spacetime propagate at the speed of light. A GW causes a tiny time dependent quadruple change of strain in the plane transverse to the wave's propagation direction. The space is stretched in one direction while is shrunk along its perpendicular direction. The strength of GW '*h'* is expressed by the dimensionless strain �� �⁄ . Due to the quadruple nature of Michelson interferometer, it is suitable device for gravitational wave detection. The interference pattern is linear measure of the strain. The amplitudes of GWs radiated from astrophysical sources at Earth are typically of the order of 10��� or smaller. Detection of such weak strains needs high sensitive devices. For increasing the sensitivity of the interferometer, the lengths of the interferometer arms were increased to the order of kilometers and multi-path cell or Fabry-Perot optical cavity was used in each arm. So the light can be stored for a time comparable to the time scale of GW signal. Long base line gravitational wave detectors such as LIGO, VIRGO, GEO and TAMA are now operational. In the presence of GW, the change in the length of arms is very small. Hence many noises such as seismic noise, thermal noise and quantum noise limit the sensitivity of interferometer. Quantum noise is the fundamental and unavoidable noise in new generation of these interferometers and is due to the light-interferometer interaction. So the sensitivity of laser GW detector depends on the quantum state of light. It was shown that depending on the parameters of interferometer such as arm's lengths, frequency of laser and mass of mirrors, the optimum quantum state for the dark port is vacuum squeezed state with specific squeezing factor. By employing this optimum quantum state in the dark port, the quantum noise and optimum laser power reduce one order of magnitude relative to the

��� ���� �

�����(����)�� (8)

FRM

Fig. 23. A schematic of an optical fiber intrusion detector system.

Circulator

Detector

���� �������� ����

can be obtained by the Fourier transform method (Jia et al., 2008).

photo detector is easily obtained.

��� � 2��

Laser

conventional interferometers (Tofighi et al., 2010).


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**2** 

**The Applications of** 

Cheng-Chih Hsu

*Taiwan* 

*Yuan-Tung Road, Chung-Li,* 

**the Heterodyne Interferoemetry** 

*Department of Photonics Engineering, Yuan Ze University,* 

Optical interferometry is widely used in many precision measurements such as displacement[1, 2], vibration[3, 4], surface roughness[5, 6], and optical properties[7-14] of the object. For example, holographic interferomter [1-3] can be used to measure the surface topography of the rigid object. The emulsion side of the photographic plate faces the object and is illuminated by a plane wave at normal incidence. Therefore, the reflection type hologram is recorded the interference signals between the incident wave and scattered wave from the object within the emulsion layer. Then the hologram is reconstructed with laser light and the information of object surface can be obtained. The Speckle interferometry [2-4] can be used to measure the motion of the rough surface. To compare the two exposure specklegrams, then the phase difference related to the surface movement can be obtained. Abbe refractometer [7, 8] is an easy method to determine the refractive index of the material based on the total internal reflection (TIR). That means the refractive index of the testing sample will be limit by the hemisphere prism installed in the refractometer. The ellipsometer [9-12] is widely used to measure the thickness and refractive index of film or bulk materials. Typically, the optical components of ellipsometer included polarizer, compensator, sample, and analyzer. Hence, there were many different types of ellipsometer for refractive index and thickness measurement of the sample. Most popular type is rotating polarizer and analyzer ellipsometer which can be divided into rotating polarizer type and rotating analyzer type. Both of them are analysis of the ellipsometric angles (, ) which determined directly from the adjustable angular settings of the optical components. The accuracy of the ellipsometric measurement are typically within the range 0.01 and 0.05 in

Compare to previous method, the heterodyne interferometry give much more flexibility of different kinds of the measurement purposes with suitable optical configuration. In this chapter, I will review the heterodyne interferometry and focus on the applications of this kind of interferometer. First of all, I will briefly introduce the history and applications of heterodyne interferometry that will be discussed in this chapter. Before I mention the applications of the heterodyne interferometry, I would like to describe several types of heterodyne interferometry. Then I would like to describe the precision positioning with optical interferometer and focus on the heterodyne grating interferometer. After that, I will

**1. Introduction** 

(, ) [13, 14].


## **The Applications of the Heterodyne Interferoemetry**

Cheng-Chih Hsu

*Department of Photonics Engineering, Yuan Ze University, Yuan-Tung Road, Chung-Li, Taiwan* 

## **1. Introduction**

30 Interferometry – Research and Applications in Science and Technology

Wang,C.; Trivedi, S.; Kutcher,S.; Rodriguez, P.; Jin,F.; Swaminathan, V. ; Nagaraj, S.;

Weihs, G.; Reck, M.; Weinfurter, H. & Zeilinger, A. (1996). All-fiber three-path Mach– Zehnder interferometer. *OPTICS LETTERS*, Vol. 21, No. 4, pp. 302-304 Yuan, L. (1997). White-light interferometric fiber-optic strain sensor from three-peak-

Yuan, L. (2002). Multiplexed, White-Light Interferometric Fiber-Optic Sensor Matrix with a Long-Cavity, Fabry-Perot Resonator. *Applied Optics*, Vol. 41, No. 22, pp. 4460-4466 Yuan, L.; Liu, Y. & Sun, W. (2005). Fiber optic Moiré interferometric profilometry.

*Proceedings of SPIE*, Vol. 5633, ISBN 9780819455888

6250

Quoraishee, S. & Prasad. N.S. (2011). Non-Contact Human Cardiac Activity Monitoring Using a High Sensitivity Pulsed Laser Vibrometer. *Proceedings of Conference on Lasers and Electro-Optics (CLEO)* , ISBN: 978-1-4577-1223-4, Baltimore

wavelength broadband LED source. *APPLIED OPTICS*, Vol. 36, No. 25 pp. 6246-

Optical interferometry is widely used in many precision measurements such as displacement[1, 2], vibration[3, 4], surface roughness[5, 6], and optical properties[7-14] of the object. For example, holographic interferomter [1-3] can be used to measure the surface topography of the rigid object. The emulsion side of the photographic plate faces the object and is illuminated by a plane wave at normal incidence. Therefore, the reflection type hologram is recorded the interference signals between the incident wave and scattered wave from the object within the emulsion layer. Then the hologram is reconstructed with laser light and the information of object surface can be obtained. The Speckle interferometry [2-4] can be used to measure the motion of the rough surface. To compare the two exposure specklegrams, then the phase difference related to the surface movement can be obtained. Abbe refractometer [7, 8] is an easy method to determine the refractive index of the material based on the total internal reflection (TIR). That means the refractive index of the testing sample will be limit by the hemisphere prism installed in the refractometer. The ellipsometer [9-12] is widely used to measure the thickness and refractive index of film or bulk materials. Typically, the optical components of ellipsometer included polarizer, compensator, sample, and analyzer. Hence, there were many different types of ellipsometer for refractive index and thickness measurement of the sample. Most popular type is rotating polarizer and analyzer ellipsometer which can be divided into rotating polarizer type and rotating analyzer type. Both of them are analysis of the ellipsometric angles (, ) which determined directly from the adjustable angular settings of the optical components. The accuracy of the ellipsometric measurement are typically within the range 0.01 and 0.05 in (, ) [13, 14].

Compare to previous method, the heterodyne interferometry give much more flexibility of different kinds of the measurement purposes with suitable optical configuration. In this chapter, I will review the heterodyne interferometry and focus on the applications of this kind of interferometer. First of all, I will briefly introduce the history and applications of heterodyne interferometry that will be discussed in this chapter. Before I mention the applications of the heterodyne interferometry, I would like to describe several types of heterodyne interferometry. Then I would like to describe the precision positioning with optical interferometer and focus on the heterodyne grating interferometer. After that, I will

The Applications of the Heterodyne Interferoemetry 33

Fig. 1. The first heterodyne interferometer constructed by Zeeman laser [15].

Fig. 2. The heterodyne light source constructed with moving grating [17].

Fig. 3. The heterodyne light source constructed with rotation radial grating [17].

review some refractometer using heterodyne interferometer. In this section I would like to quick look some useful methods for measuring the refractive index and thickness of bulk material or thin film structure. In addition, the measurement of the optic axis and birefringence of the birefringent crystal will also be discussed in this section. The final application of the heterodyne interferometry that I would like to talk about is the concentration measurement. In this section, I will roughly classify the method into two categories. One is fiber type sensor; another is a non-fiber type sensor. And I will discuss the surface plasmon resonance (SPR) sensor in fiber-type and non-fiber type sensors. Finally, I would like to give the short conclusion, which summarized the advantages and disadvantages of the heterodyne interferometer.

## **2. Heterodyne interferometry**

This section will introduce the development history of the heterodyne interferometry and describe the fundamental theory and basic optical configuration of the heterodyne interferometer.

## **2.1 History of Heterodyne light source developement**

Hewlett Packard Company (HP) developed the first commercial heterodyne interferometer for precision positioning since 1966. Until now, HP systems have widely used in industry, scientific research, and education. J. A. Dahlquist, D. G. Peterson, and W. Culshaw [15] demonstrated an optical interferometer, which used Zeeman laser properties in 1966. They had the application of an axial magnetic field and resulted in the frequency difference between the right hand and left hand circular polarization states of the He-Ne laser. Because of these two polarization states are affected by equally thermal drift and mechanism vibration of the laser, the frequency difference are extremely stable. Therefore, this light source with different frequency is so called the heterodyne light source. Figure 1 showed that the first heterodyne interferometer which constructed with Zeeman laser. As you can see, the frequency shift coming from the moving mirror will be carried with 2. Then these two lights with different frequency will be interference at 45 and the distance-varying phase can be detected.

There are many methods can construct the optical frequency shift such as rotation or moving grating method [16, 17], accousto-optical modulator (AOM) [18, 19], electro-optical modulator (EOM) [20, 21], and modulating two slightly different wavelengths of laser diodes [22]. Suzuki and Hioki [16] proposed the idea of moving grating method for constructing the heterodyne light source in 1967. As the grating moves along y-axis with the

velocity *v*, the frequency shift will be introduced into the 1 order diffracted beam with . *v a*

By suitable arrangement of the optical configuration, either one of these frequency shifted signals can be selected and to form the heterodyne light source. W. H. Stevenson [17] proposed the rotation radial grating to form the heterodyne light source in which he showed that the frequency shift were linear increased with the rotation rate of the radial grating up to 6k rpm. And the maximum frequency shift in this case was 500 kHz.

An acousto-optic modulator (AOM) uses the acousto-optic effect to diffract and shift the frequency of the light [18, 19]. The piezoelectric transducer attaches to the quartz and the

review some refractometer using heterodyne interferometer. In this section I would like to quick look some useful methods for measuring the refractive index and thickness of bulk material or thin film structure. In addition, the measurement of the optic axis and birefringence of the birefringent crystal will also be discussed in this section. The final application of the heterodyne interferometry that I would like to talk about is the concentration measurement. In this section, I will roughly classify the method into two categories. One is fiber type sensor; another is a non-fiber type sensor. And I will discuss the surface plasmon resonance (SPR) sensor in fiber-type and non-fiber type sensors. Finally, I would like to give the short conclusion, which summarized the advantages and

This section will introduce the development history of the heterodyne interferometry and describe the fundamental theory and basic optical configuration of the heterodyne

Hewlett Packard Company (HP) developed the first commercial heterodyne interferometer for precision positioning since 1966. Until now, HP systems have widely used in industry, scientific research, and education. J. A. Dahlquist, D. G. Peterson, and W. Culshaw [15] demonstrated an optical interferometer, which used Zeeman laser properties in 1966. They had the application of an axial magnetic field and resulted in the frequency difference between the right hand and left hand circular polarization states of the He-Ne laser. Because of these two polarization states are affected by equally thermal drift and mechanism vibration of the laser, the frequency difference are extremely stable. Therefore, this light source with different frequency is so called the heterodyne light source. Figure 1 showed that the first heterodyne interferometer which constructed with Zeeman laser. As you can

two lights with different frequency will be interference at 45 and the distance-varying

There are many methods can construct the optical frequency shift such as rotation or moving grating method [16, 17], accousto-optical modulator (AOM) [18, 19], electro-optical modulator (EOM) [20, 21], and modulating two slightly different wavelengths of laser diodes [22]. Suzuki and Hioki [16] proposed the idea of moving grating method for constructing the heterodyne light source in 1967. As the grating moves along y-axis with the velocity *v*, the frequency shift will be introduced into the 1 order diffracted beam with . By suitable arrangement of the optical configuration, either one of these frequency shifted signals can be selected and to form the heterodyne light source. W. H. Stevenson [17] proposed the rotation radial grating to form the heterodyne light source in which he showed that the frequency shift were linear increased with the rotation rate of the radial grating up

An acousto-optic modulator (AOM) uses the acousto-optic effect to diffract and shift the frequency of the light [18, 19]. The piezoelectric transducer attaches to the quartz and the

2. Then these

 *v a*

see, the frequency shift coming from the moving mirror will be carried with

to 6k rpm. And the maximum frequency shift in this case was 500 kHz.

disadvantages of the heterodyne interferometer.

**2.1 History of Heterodyne light source developement** 

**2. Heterodyne interferometry** 

interferometer.

phase can be detected.

Fig. 1. The first heterodyne interferometer constructed by Zeeman laser [15].

Fig. 2. The heterodyne light source constructed with moving grating [17].

Fig. 3. The heterodyne light source constructed with rotation radial grating [17].

$$\text{EO}\{45^{\circ}, \Gamma\} = \begin{pmatrix} \cos 45^{\circ} & -\sin 45^{\circ} \\ \sin 45^{\circ} & \cos 45^{\circ} \end{pmatrix} \begin{pmatrix} e^{\frac{\Gamma}{2}} & 0 \\ 0 & e^{-\frac{\Gamma}{2}} \end{pmatrix} \begin{pmatrix} \cos 45^{\circ} & \sin 45^{\circ} \\ -\sin 45^{\circ} & \cos 45^{\circ} \end{pmatrix} = \begin{pmatrix} \cos \frac{\pi}{2} & \sin \frac{\pi}{2} \\ \sin \frac{\pi}{2} & \cos \frac{\pi}{2} \end{pmatrix} \tag{1}$$

$$\text{EO(}\omega t\text{)} = \begin{pmatrix} e^{-t\text{mr}\frac{l\omega t}{2}} & 0\\ 0 & e^{t\text{mr}\frac{-l\omega t}{2}} \end{pmatrix} = \begin{pmatrix} e^{t\frac{\omega t}{2}} & 0\\ 0 & e^{-t\frac{\omega t}{2}} \end{pmatrix} \tag{2}$$

$$E = EO(\omega t) \cdot E\_{ln} = \begin{pmatrix} e^{\frac{\omega t}{2}} & 0 \\ 0 & e^{-\frac{\omega t}{2}} \end{pmatrix} \cdot \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} e^{i\omega\_0 t} = \frac{1}{\sqrt{2}} \begin{pmatrix} e^{\frac{\omega t}{2}} \\ e^{-\frac{\omega t}{2}} \end{pmatrix} e^{i\omega\_0 t} \tag{3}$$

$$\begin{aligned} E' &= Q(0^\infty) \cdot EO(\omega t) \cdot E\_{ln} \\ &= \begin{pmatrix} 1 \ 0 \\ 0 \ \dot{l} \end{pmatrix} \begin{pmatrix} \cos(\omega t/2) & l \sin(\omega t/2) \\ l \sin(\omega t/2) & \cos(\omega t/2) \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} \cos(\omega t/2) \\ -\sin(\omega t/2) \end{pmatrix} \\ &= \frac{1}{n} \begin{pmatrix} 1 \\ l \end{pmatrix} e^{\frac{l\omega t}{2}} + \frac{1}{n} \begin{pmatrix} 1 \\ -1 \end{pmatrix} e^{-\frac{l\omega t}{2}} . \end{aligned} \tag{4}$$

$$I\_r = I'[1 + \cos(\omega t)],\tag{5}$$

$$E\_t = AN\_t(a) \cdot W \cdot E\_{ln} = \frac{1}{2} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} e^{\frac{l\phi}{z}} & 0 \\ 0 & e^{-\frac{l\phi}{z}} \end{pmatrix} \begin{pmatrix} \cos\frac{\omega t}{z} \\ -\sin\frac{\omega t}{z} \end{pmatrix}$$

$$= [\cos a \cos \frac{\omega t}{2} e^{i\frac{\phi}{z}} - \sin a \sin \frac{\omega t}{2} e^{-i\frac{\phi}{z}}] \cdot \begin{pmatrix} \cos a \\ \sin a \end{pmatrix} \tag{6}$$

$$I\_t = |E\_t|^2 = \frac{1}{2} \left[ 1 + 2\sqrt{A^2 + B^2} \cos(\omega t + \phi) \right] \tag{7}$$

$$A = \frac{1}{2} \left( \cos^2 \alpha - \sin^2 \alpha \right),\tag{8a}$$

$$B = \cos \mathfrak{a} \sin \mathfrak{a} \cos \mathfrak{p},\tag{8b}$$

$$\phi = \tan^{-1}(\frac{B}{A}) = \tan^{-1}\frac{2\cos a \sin a \cos \phi}{\left(\cos^2 a - \sin^2 a\right)'}\tag{8c}$$

$$E\_t = AN(\alpha) \cdot S \cdot E'$$

$$= \begin{pmatrix} \cos^2 \alpha & \sin \alpha \cos \alpha\\ \sin \alpha \cos \alpha & \sin^2 \alpha \end{pmatrix} \begin{pmatrix} r\_p & 0\\ 0 & r\_s \end{pmatrix} \begin{pmatrix} \cos \frac{\omega t}{2} \\ -\sin \frac{\omega t}{2} \end{pmatrix}$$

$$= \left( r\_p \cos \alpha \cos \frac{\omega t}{2} - r\_s \sin \alpha \sin \frac{\omega t}{2} \right) \begin{pmatrix} \cos \alpha\\ \sin \alpha \end{pmatrix} \tag{9}$$

$$I\_t = |E\_t|^2 = I\_0[1 + \cos(\omega t + \phi)] \tag{10}$$

$$I\_0 = \frac{(r\_p^{-2}\cos^2a + r\_s^{-2}\sin^2a)}{2} \tag{11a}$$

$$\phi = \tan^{-1}\left(\frac{2\sin a \cos a \cdot r\_p r\_s}{r\_p^{-2} \cos^2 a + r\_s^{-2} \sin^2 a}\right) \tag{11b}$$

$$r\_p = \frac{n\_2 \cos \theta - n\_1 \cos \theta\_t}{n\_2 \cos \theta + n\_1 \cos \theta\_t} \tag{12a}$$

$$\tau\_{\rm s} = \frac{n\_1 \cos \theta - n\_2 \cos \theta\_t}{n\_1 \cos \theta + n\_2 \cos \theta\_t} \,\tag{12b}$$

$$r\_p(\beta) = \frac{r\_{1p} + r\_{2p} \exp(l\beta)}{1 + r\_{1p}r\_{2p} \exp(l\beta)},\tag{13a}$$

$$r\_{\mathbf{s}}(\beta) = \frac{r\_{1\mathbf{s}} + r\_{2\mathbf{s}} \exp(i\beta)}{1 + r\_{1\mathbf{s}}r\_{2\mathbf{s}} \exp(i\beta)}\tag{13b}$$

$$\beta = \frac{2\pi n\_2 d \cos(\theta\_t)}{\lambda}. \tag{13c}$$

$$E\_t = \begin{pmatrix} \frac{e^{\frac{l\omega t}{2} - lk(2d\_f)}}{e^{\frac{l\omega t}{2} - lk(2d\_s)}} \end{pmatrix} e^{l\omega\_0 t} \tag{14}$$

$$I\_t = \frac{1}{2} \left\{ 1 + \cos \left[ \omega t - \frac{4\pi}{\lambda} (d\_p - d\_s) \right] \right\}. \tag{15}$$

The Applications of the Heterodyne Interferoemetry 41

C. C. Hsu [29] proposed the grating heterodyne interferometry (GHI) to measure the inplane displacement. The schematic diagram of this method is shown in figure 10. The diffracted grating has mounted on the motorized stage and four diffracted lights will diffract and propagate in the x-z and y-z planes which are for measuring the displacement in x- and y- directions respectively. Based on the Jones calculation, the E-field of the 1

diffracted beams can be expressed

D*<sup>x</sup>*<sup>2</sup>

Fig. 10. 2-D displacement measurement system with GHI [29].

2-D diffraction grating

Dr

PBS1

D*<sup>y</sup>*<sup>1</sup>

AN4

AN5

Dy2

AN3

AN1 P1

Lens

P4

ೣ

݀௫ቁήܧு ൌ ቀ݈݅݇௫േଵ േ ݅ ଶగ

*x*-*y* stage

P2

Heterodyne light source



P3

BS

AN2

PBS2

D*<sup>x</sup>*<sup>1</sup>

݀௬൰ήܧு ൌ ൬݈݅݇௬േଵ േ ݅ ଶగ

where *g*x and *g*y are the grating pitch in x- and y- directions, *d*x and *d*y are the displacement along the x- and y- directions respectively. To consider x-direction displacement measurement, the 1st order diffracted lights will be collected by a lens L and propagate into two paths: (1) prism P2 polarization beam splitter PBS2 analyzer AN2 (45) detector

ೣ

݀௫ቁ ቀ ഘȀమ

݀௬൰ ቀ ഘȀమ

షഘȀమቁ, (x-direction) (16a)

+1

*y* 

*x* 

+1

షഘȀమቁ, (y-direction) (16b)

ܧ௫േଵ ן ቀ݈݅݇௫േଵ േ ݅ ଶగ

ܧ௬േଵ ן ൬݈݅݇௬േଵ േ ݅ ଶగ

and

If the mirror M1 is moved with time, the phase difference ସగௗ ఒ will be changed and the displacement variation can be measured by comparing the testing signal and reference signal with lock-in amplifier.

Fig. 8. The optical configuration of the non-common path displacement measurement.

## **3. Accurate positioning with heterodyne interferometer**

Nano-scale positioning devices have become a significant requirement in scientific instruments used for nanotechnology applications. These devices can be applied to nanohandling, nanomanipulation, and nanofabrication. In addition, they are an essential part of the scanning probe microscopy (SPM) and widely used in many research fields. The precision positioning devices consist of three principle parts, which are the rolling component, the driving system and the position sensor. Piezoelectric actuator is the most popular method for driving system and commercial products have been on the market for a few decades. Therefore, the piezoelectric actuator and the position sensor will play the role of the positioning and the feedback control of the rolling element. To achieve the high resolution positioning, the sensing methods of position sensor become more important and have attracted great attention over the past two decades. In this section, we will introduce a few of typical precision positioning methods [24-30] which used heterodyne interferometry.

displacement variation can be measured by comparing the testing signal and reference

BS

AN

*<sup>I</sup>*<sup>t</sup> *<sup>I</sup>*<sup>r</sup>

Dt

Fig. 8. The optical configuration of the non-common path displacement measurement.

Nano-scale positioning devices have become a significant requirement in scientific instruments used for nanotechnology applications. These devices can be applied to nanohandling, nanomanipulation, and nanofabrication. In addition, they are an essential part of the scanning probe microscopy (SPM) and widely used in many research fields. The precision positioning devices consist of three principle parts, which are the rolling component, the driving system and the position sensor. Piezoelectric actuator is the most popular method for driving system and commercial products have been on the market for a few decades. Therefore, the piezoelectric actuator and the position sensor will play the role of the positioning and the feedback control of the rolling element. To achieve the high resolution positioning, the sensing methods of position sensor become more important and have attracted great attention over the past two decades. In this section, we will introduce a few of typical precision positioning methods [24-30] which used heterodyne interferometry.

PC

**3. Accurate positioning with heterodyne interferometer** 

Lock-in Amplifier ఒ will be changed and the

M1

*d*p

OSC

*<sup>d</sup>*<sup>s</sup> PBS

M2

If the mirror M1 is moved with time, the phase difference ସగௗ

signal with lock-in amplifier.

Heterodyne light source

Function Generator C. C. Hsu [29] proposed the grating heterodyne interferometry (GHI) to measure the inplane displacement. The schematic diagram of this method is shown in figure 10. The diffracted grating has mounted on the motorized stage and four diffracted lights will diffract and propagate in the x-z and y-z planes which are for measuring the displacement in x- and y- directions respectively. Based on the Jones calculation, the E-field of the 1 diffracted beams can be expressed

Fig. 10. 2-D displacement measurement system with GHI [29].

$$E\_{\mathbf{x}\pm 1} \propto \exp\left(ilk\_{\mathbf{x}\pm 1} \pm l \frac{2\pi}{\mathbf{g}\_{\mathbf{x}}} d\_{\mathbf{x}}\right) \cdot E\_H = \exp\left(ilk\_{\mathbf{x}\pm 1} \pm l \frac{2\pi}{\mathbf{g}\_{\mathbf{x}}} d\_{\mathbf{x}}\right) \begin{pmatrix} e^{i\omega t/2} \\ e^{-i\omega t/2} \end{pmatrix} \text{ (x-direction)} \quad \text{(16a)}$$

and

$$E\_{\mathbf{y}\pm 1} \propto \exp\left(ikl\_{\mathbf{y}\pm 1} \pm l \frac{2\pi}{\mathbf{g}\_{\mathbf{y}}} d\_{\mathbf{y}}\right) \cdot E\_H = \exp\left(ikl\_{\mathbf{y}\pm 1} \pm l \frac{2\pi}{\mathbf{g}\_{\mathbf{y}}} d\_{\mathbf{y}}\right) \begin{pmatrix} e^{i\omega t/2} \\ e^{-i\omega t/2} \end{pmatrix} \text{ (y-direction)}\tag{16b}$$

where *g*x and *g*y are the grating pitch in x- and y- directions, *d*x and *d*y are the displacement along the x- and y- directions respectively. To consider x-direction displacement measurement, the 1st order diffracted lights will be collected by a lens L and propagate into two paths: (1) prism P2 polarization beam splitter PBS2 analyzer AN2 (45) detector

$$I\_{\mathbf{x}1} \propto \left| E\_{+\mathbf{1}\mathbf{s},\mathbf{x}} + E\_{+\mathbf{1}\mathbf{p},\mathbf{x}} \right|^2 = \frac{1}{2} \left| \mathbf{1} + \cos \left( \omega t + k \{ l\_{-\mathbf{1},\mathbf{x}} - l\_{+\mathbf{1},\mathbf{x}} \} - \frac{4\pi}{\varrho\_{\mathbf{x}}} d\_{\mathbf{x}} \right) \right|, \tag{17a}$$

$$I\_{\mathbf{x}2} \propto \left| E\_{-\mathbf{1}\mathbf{s},\mathbf{x}} + E\_{+\mathbf{1}\mathbf{p},\mathbf{x}} \right|^2 = \frac{1}{2} \left| \mathbf{1} + \cos \left( \omega t + k \left( l\_{+\mathbf{1},\mathbf{x}} - l\_{-\mathbf{1},\mathbf{x}} \right) - \frac{4\pi}{\mathbf{g}\_{\mathbf{x}}} d\_{\mathbf{x}} \right) \right|, \tag{17b}$$

$$I\_{\rm y1} \propto \left| E\_{+1 \rm s, \mathcal{y}} + E\_{-1 \rm p, \mathcal{y}} \right|^2 = \frac{1}{2} \left| 1 + \cos \left( \omega t + k \{ l\_{-1, \mathcal{y}} - l\_{+1, \mathcal{y}} \} - \frac{4 \pi}{\mathcal{g}\_{\mathcal{Y}}} d\_{\mathcal{Y}} \right) \right|, \tag{18a}$$

$$I\_{y2} \ll \left| E\_{-1s,y} + E\_{+1p,y} \right|^2 = \frac{1}{2} \left| 1 + \cos \left( \omega t + k \left( l\_{+1,y} - l\_{-1,y} \right) - \frac{4\pi}{\mathfrak{g}\_{\mathcal{Y}}} d\_{\mathcal{Y}} \right) \right|. \tag{18b}$$

$$d\phi\_l = \frac{8\pi}{\mathfrak{g}\_l} d\_l + 2kl\_l \quad \text{( $l=x,y$ )},\tag{19}$$

$$E\_B = E\_C = f\{180^\circ\} \cdot E\_H = \begin{pmatrix} e^{-i\omega t/2} \\ e^{i\omega t/2} \end{pmatrix}, E\_A = E\_D = f\{0^\circ\} \cdot E\_H = \begin{pmatrix} e^{i\omega t/2} \\ e^{-i\omega t/2} \end{pmatrix}. \tag{20}$$

$$I\_1 = 1 + \cos(\omega t - \phi\_{x1}),\tag{21a}$$

$$I\_2 = 1 + \cos(\omega t - \phi\_{\rm Y1}).\tag{21b}$$

$$I\_3 = 1 + \cos(\omega t),\tag{2}$$

The Applications of the Heterodyne Interferoemetry 45

angle between the moving direction and the grating, which can be alignment by mounting 2D grating on the rotation stage. In their case, the larger difference was about 12 *μ*m in the ydirection and the smallest difference was about 29 nm in the x- direction for a displacement of 1 mm. Based on the error analysis, if the phase resolution (0.001) of the lock-in amplifier is considered, the corresponding displacement resolutions of the differential and single type interferometers are estimated to be 9 pm and 4.5 pm for a grating pitch of 3.2 m, respectively. If only high frequency noise is considered, the measurement resolution of the differential and single type QCOP interferometers can be estimated to be 1.41 nm and 2.52

Fig. 13. Displacement measurement of the quadrangular motion with the QCOP method

Optical constants of the materials such as refractive index, birefringence, optical activity, and thickness are significant parameters in material science. There are many methods [9-12, 31-40] can determine those factors, most popular method is ellipsometer [9-12]. Recently, these factors can be obtained by heterodyne interferometer. In this section, we will review some novel methods [31-40] for optical constants measurement with heterodyne

C. C. Hsu [32] proposed a novel method for determine the refractive index of the bulk materials with normal incident circular heterodyne interferometer (NICHI) and the schematic diagram is shown in figure 14. The circular heterodyne light source was incident into the modified Twyman-Green interferometer, in which the testing signal reflected from the sample can be interfered and carried by the circular heterodyne light beams. Based on

**4. Optical constants measurement with heterodyne interferometer** 

Jones calculation, the interference signal measured by D can be written as

nm.

[30].

interferometer.

Fig. 12. The schematic of the single type and differential type QCOP heterodyne grating interferometer.

These three signals *I*1, *I*2, and *I*<sup>3</sup> are sent into the lock-in amplifier and the phase differences *x* = *x*1 (between *I*1 and *I*3) and *y* = *<sup>y</sup>*1 (between *I*2 and *I*3) are given by

$$d\phi\_l = \frac{2\pi}{\mathfrak{g}\_l} d\_l \quad \text{ ( $l=x,y$ )},\tag{23}$$

where *di* is the displacement in x- and y- directions; *gi* is the grating pitch of 2-D grating in xand y- directions. It is obvious that the 2-D displacement can be obtained as the phase difference and grating pitch of the 2-D grating are given. In the differential type QCOP method, two polarization beam splitters (PBSs) are used to separate the two overlapping beams into four parts. Therefore, the interference signals detected by D4, D5, D6, and D7 can be written as

$$I\_4 = 1 + \cos(\omega t - \phi\_{\ge 1}) \text{ and } I\_5 = 1 + \cos(\omega t + \phi\_{\ge 1}); \quad \text{(for x-direction)}\tag{24a}$$

$$I\_6 = 1 + \cos(\omega t - \phi\_{y1}) \text{ and } I\_7 = 1 + \cos(\omega t + \phi\_{y1}). \quad \text{(for y-direction)}\tag{24b}$$

These two pairs signal are sent into the multi-channel lock-in amplifier, the phase differences *<sup>x</sup>* = *x*1 – (–*x*1) (between *I*4 and *I*5) and *<sup>y</sup>* = *y*1 – (–*<sup>y</sup>*1) (between *I*6 and *I*7) are 4 times of *i*.

Figure 13 shows a top view of the experimental results in the XY section and the XY stepper moves with a displacement of 1 mm. It is clear that the slight difference between the results measured by the laser encoder and QCOP method. The difference is coming from a tiny

Fig. 12. The schematic of the single type and differential type QCOP heterodyne grating

*y* = 

> �� <sup>=</sup> �� ��

*x*1) (between *I*4 and *I*5) and

These three signals *I*1, *I*2, and *I*<sup>3</sup> are sent into the lock-in amplifier and the phase differences

where *di* is the displacement in x- and y- directions; *gi* is the grating pitch of 2-D grating in xand y- directions. It is obvious that the 2-D displacement can be obtained as the phase difference and grating pitch of the 2-D grating are given. In the differential type QCOP method, two polarization beam splitters (PBSs) are used to separate the two overlapping beams into four parts. Therefore, the interference signals detected by D4, D5, D6, and D7 can

These two pairs signal are sent into the multi-channel lock-in amplifier, the phase

Figure 13 shows a top view of the experimental results in the XY section and the XY stepper moves with a displacement of 1 mm. It is clear that the slight difference between the results measured by the laser encoder and QCOP method. The difference is coming from a tiny

*<sup>y</sup>*1 (between *I*2 and *I*3) are given by

�� = � � ������ � ���) and �� = � � ������ � ���); (for x- direction) (24a)

�� = � � ������ � ���) and �� = � � ������ � ���). (for y- direction) (24b)

*<sup>y</sup>* = *y*1 – (–

��������� = �� �), (23)

*<sup>y</sup>*1) (between *I*6 and *I*7) are 4

interferometer.

be written as

differences

times of

*<sup>x</sup>* = *x*1 – (–

*i*.

*x*1 (between *I*1 and *I*3) and

*x* =  angle between the moving direction and the grating, which can be alignment by mounting 2D grating on the rotation stage. In their case, the larger difference was about 12 *μ*m in the ydirection and the smallest difference was about 29 nm in the x- direction for a displacement of 1 mm. Based on the error analysis, if the phase resolution (0.001) of the lock-in amplifier is considered, the corresponding displacement resolutions of the differential and single type interferometers are estimated to be 9 pm and 4.5 pm for a grating pitch of 3.2 m, respectively. If only high frequency noise is considered, the measurement resolution of the differential and single type QCOP interferometers can be estimated to be 1.41 nm and 2.52 nm.

Fig. 13. Displacement measurement of the quadrangular motion with the QCOP method [30].

## **4. Optical constants measurement with heterodyne interferometer**

Optical constants of the materials such as refractive index, birefringence, optical activity, and thickness are significant parameters in material science. There are many methods [9-12, 31-40] can determine those factors, most popular method is ellipsometer [9-12]. Recently, these factors can be obtained by heterodyne interferometer. In this section, we will review some novel methods [31-40] for optical constants measurement with heterodyne interferometer.

C. C. Hsu [32] proposed a novel method for determine the refractive index of the bulk materials with normal incident circular heterodyne interferometer (NICHI) and the schematic diagram is shown in figure 14. The circular heterodyne light source was incident into the modified Twyman-Green interferometer, in which the testing signal reflected from the sample can be interfered and carried by the circular heterodyne light beams. Based on Jones calculation, the interference signal measured by D can be written as

$$I\_t = |E\_1 + E\_2|^2 = I\_0[1 + \chi \cos(\omega t + \phi)],\tag{25}$$

$$I\_0 = \left[\frac{1}{2}(\frac{n-1}{n+1})^2 + 2r\_m^{\prime 2} + 2(\frac{n-1}{n+1})r\_m \sin 2a \sin(\phi\_{d1} - \phi\_{d2})\right]. \tag{26a}$$

$$\chi = \frac{\frac{\sqrt{A^2 + B^2}}{\binom{1}{2}\binom{n-1}{n+1}^2 + 2r\_m^2 + 2\binom{n-1}{n+1}r\_m \sin 2\alpha \sin(\phi\_{d1} - \phi\_{d2})}}{\binom{1}{2}\binom{n-1}{n+1}\binom{n-1}{n+1}\binom{n-1}{n+1}\binom{n-1}{n+1}\binom{n-1}{n+1}\binom{n-1}{n+1}\binom{n-1}{n+1}\binom{n-1}{n+1}}\tag{26b}$$

$$\phi = \tan^{-1}(\frac{B}{A}),\tag{26c}$$

$$\mathbf{A} = \frac{1}{2} \left[ \left( \frac{n-1}{n+1} \right)^2 - 4r\_m{}^2 \right] \cdot \left( \cos^2 \alpha - \sin^2 \alpha \right),\tag{27a}$$

$$\mathbf{B} = \frac{1}{2} \mathbf{I} \left( \frac{n-1}{n+1} \right)^2 - 4r\_m{}^2 \mathbf{I} \cdot \sin 2\alpha - 2 \frac{n-1}{n+1} r\_m,\tag{27b}$$

$$
\Delta \ln n = \frac{1}{\frac{d\phi}{dn}} |\Delta \phi| = [\frac{ac - b}{cd}] |\Delta \phi|, \tag{28}
$$

$$a = \frac{2[-\frac{4r\_m}{(1+n)^2} + \frac{2(n-1)}{(1+n)^3}\sin 2a]}{[\left(\frac{n-1}{1+n}\right)^2 - 4r\_m \, ^2][\cos^2 a - \sin^2 a]},\tag{29a}$$

$$b = 2\frac{4(n-1)}{(1+n)^3} \left\{ \frac{-2(n-1)r\_m}{1+n} + \frac{1}{2} \left[ \left(\frac{n-1}{n+1}\right)^2 - 4r\_m{}^2 \right] \cdot \sin 2a \right\},\tag{29b}$$

$$\mathbf{c} = \mathbb{I}\left(\frac{n-1}{n+1}\right)^2 - 4r\_m^{-2}\mathbb{I}^2\langle\cos^2 a - \sin^2 a\rangle,\tag{29c}$$

$$d = 1 + \frac{4\left[\frac{-2(n-1)r\_m}{n+1} + \frac{1}{2}\left(\frac{n-1}{n+1}\right)^2 - 4r\_m z^2\right] \sin 2a}{[\left(\frac{n-1}{n+1}\right)^2 - 4r\_m z^2]^2 [\cos^2 a - \sin^2 a]^2}. \tag{29d}$$

$$I\_1 = I\_0 + \chi\_1 \cos(2\pi f t + \phi\_1)$$

$$= \frac{1}{2} \left\{ r^2 + r\_m^{-2} - 2rr\_m \cos \left[ 2\pi f t + \frac{\pi}{2} - (\phi\_{d1} - \phi\_{d2} + \phi\_r) \right] \right\} \tag{30}$$

$$I\_2 = \left[r r\_m \sin(\phi\_{d1} - \phi\_{d2} + \phi\_r)\right] \cos(2\pi f t) + \left|\frac{1}{2}(r\_m^{-2} - r^2)\right| \cos(2\pi f t) + \mathcal{O},\tag{31a}$$

$$\phi\_2 = \cot^{-1} \left[ \frac{2rr\_m \sin(\phi\_{d1} - \phi\_{d2} + \phi\_r)}{(r\_m^{-2} - r^2)} \right]. \tag{31b}$$

$$n = \frac{\cot \phi\_2 - r\_m \cos \phi\_1 + r\_m \sqrt{\cos^2 \phi\_1 + \cot^2 \phi\_2}}{\cot \phi\_2 + r\_m \cos \phi\_1 - r\_m \sqrt{\cos^2 \phi\_1 + \cot^2 \phi\_2}}. \tag{32}$$

$$I(t) = \sqrt{{A\_0}^2 + {B\_0}^2} \cos(2\pi f t + \varphi) + \mathcal{C}\_0,\tag{33a}$$

$$\varphi = \tan^{-1} \left( \frac{-B\_0}{A\_0} \right). \tag{33b}$$

$$E\_t = \begin{pmatrix} \cos^2 \infty & \sin \arg \cos \alpha \\ \sin \alpha \cos \alpha & \sin^2 \alpha \end{pmatrix} \begin{pmatrix} r\_p & 0 \\ 0 & r\_s \end{pmatrix} \begin{pmatrix} \cos \frac{\omega t}{2} \\ -\sin \frac{\omega t}{2} \end{pmatrix} = \begin{pmatrix} r\_p \cos^2 a \cos \frac{\omega t}{2} - r\_s \sin a \cos a \sin \frac{\omega t}{2} \\ r\_p \sin a \cos a \cos \frac{\omega t}{2} - r\_s \sin^2 a \sin \frac{\omega t}{2} \end{pmatrix}. \tag{34}$$

$$I\_t = |E\_t|^2 = I\_0[1 + \frac{\sqrt{A^2 + B^2}}{I\_0} \cos(\omega t + \phi)],\tag{35}$$

$$I\_0 = \frac{1}{2} \zeta \left| r\_p \right|^2 \cos^2 a + |r\_s|^2 \sin^2 a \rangle,\tag{36a}$$

$$A = \frac{1}{2} \zeta \left| \mathbf{r}\_p \right|^2 \cos^2 a - |\mathbf{r}\_s|^2 \sin^2 a \rangle,\tag{36b}$$

$$B = \frac{1}{2} \left( \mathfrak{r}\_p \mathfrak{r}\_s \,^\* + \mathfrak{r}\_s \mathfrak{r}\_p \,^\* \right) \sin \mathfrak{a} \cos \mathfrak{a},\tag{36c}$$

$$\phi = \tan^{-1}(\frac{B}{A}) = \tan^{-1}[\frac{(r\_p r\_s^\* + r\_s r\_p^\*)\sin a \cos a}{(|r\_p|^2 \cos^2 a - |r\_s|^2 \sin^2 a)}].\tag{36d}$$

$$E\_t = AN(\mathcal{J}) \cdot \begin{bmatrix} ^{lsp} & r\_{ps} \\ r\_{sp} & r\_{ss} \end{bmatrix} \cdot E\_l$$

$$= \left( \left( r\_{pp} \cos \mathcal{J} + r\_{sp} \sin \mathcal{J} \right) \cos \frac{\omega t}{2} - \left( r\_{ps} \cos \mathcal{J} + r\_{ss} \sin \mathcal{J} \right) \sin \frac{\omega t}{2} \right) \begin{pmatrix} \cos \mathcal{J} \\ \sin \mathcal{J} \end{pmatrix} \tag{37}$$

$$|I\_t = |E\_t|^2 = \frac{(r\_{pp}\cos\beta + r\_{sp}\sin\beta)^2 + (r\_{ps}\cos\beta + r\_{ss}\sin\beta)^2}{2} [1 + \cos(\omega t + \phi)],\tag{38a}$$

$$\phi = \tan^{-1}\left(\frac{2(r\_{pp}\cos\beta + r\_{sp}\sin\beta)(r\_{ps}\cos\beta + r\_{ss}\sin\beta)}{\left(r\_{pp}\cos\beta + r\_{sp}\sin\beta\right)^2 + (r\_{ps}\cos\beta + r\_{ss}\sin\beta)^2}\right). \tag{38b}$$

$$\phi = \tan^{-1} \left( \frac{2r\_{pp}r\_{ps}}{r\_{pp}^{-2} - r\_{ps}^{-2}} \right). \tag{39}$$

$$\Phi = \tan^{-1} \left( \frac{\sin 2\beta \cdot r\_{pp} r\_{ps}}{r\_{pp} \, ^2 \cos^2 \beta - r\_{\infty} \, ^2 \sin^2 \beta} \right). \tag{40}$$

$$\mathbf{E} = \mathbf{A}(0) \cdot \mathbf{S}(\theta) \cdot Q\_1(45) \cdot EO(90) \cdot P(45) \cdot E\_{ln}$$

$$= \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} \frac{1-l}{2} & \frac{1+l}{2} \\ \frac{1+l}{2} & \frac{1-l}{2} \end{bmatrix} \begin{bmatrix} e^{-l\omega t/2} & 0 \\ 0 & e^{-\omega t/2} \end{bmatrix} \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{bmatrix} \begin{bmatrix} 0 \\ E\_0 e^{w\_0 t} \end{bmatrix},\tag{41}$$

$$I\_t = I\_{dc} \left[ 1 - \sin(\omega t - 2\theta) \right],\tag{42}$$

$$C = \frac{100\theta}{L\{\theta\}}\tag{43}$$

The Applications of the Heterodyne Interferoemetry 55

developed novel methods for measuring the solution concentration. Because of the advantages of the optical method such as high sensitivity, high resolution, non-contact, and quick response, optical measurement method become more popular in past few decades. And these methods can roughly divide into fiber type sensor [41, 44-45, 49] and non-fiber type sensor [42-43, 46-48, 50]. In this section, we will review some recent development in both type sensors for measuring concentration of the specific chemical compound [41-50]. M. H. Chiu [41] developed a D-shape fiber sensor with SPR property and integrated with heterodyne interferometer which could detect variation in the alcohol concentration of 2%. The optical setup was shown in figure 22. The heterodyne light source was guided into the sensor and suffered the attenuate total reflection (ATR) at the sensing region. Because of the refractive index of the sample will be varied as the concentration changed. And induce the phase difference between the p- and s- polarizations. To measure the phase difference can be

Figure 22 shows that the testing signal detected by photodetector and sent into the

where *I*0 and *V* are the average intensity and visibility; *L* and *h* are the sensing length and

incident angle at the interface between fiber core and metal film. Based on the Fresnel

medium. Figure 23 shows that the results measured by the D-shape fiber sensor for different

*s*) is the phase difference between p- and s- polarizations;

�� ��� ��

*<sup>s</sup>*) from the amplitude reflection coefficient under ATR

*<sup>s</sup>*) is the function of the refractive index of the sensing

���, (44)

*<sup>i</sup>* is the

�(�) � �� �� � � ��� ��� � ��(�����)

obtained the concentration variation of the sample.

Fig. 22. The scheme of the D-shape fiber sensor [41].

core diameter; (

*p*-

equation, one can derive the (

concentration of the alcohol.

condition and it is obvious that (

phasemeter. Therefore, the interference signal can be written as

*p*-

> *p*-

Fig. 20. Schematic diagram of circular heterodyne interferometer for measuring the optical rotation angle in a chiral medium [40].

Fig. 21. Optical rotation angle of glucose solution with concentration variation [40].

## **5. Concentration measurement with heterodyne interferometer**

The concentration of solution is an important factor in food, chemical and biochemical industrial, especially in health care and disease prevention. For example, the blood glucose concentration is related to the diabetes. To control the blood glucose concentration within the normal level is critical issue to the diabetic daily care. Therefore, many researchers

Fig. 20. Schematic diagram of circular heterodyne interferometer for measuring the optical

Fig. 21. Optical rotation angle of glucose solution with concentration variation [40].

The concentration of solution is an important factor in food, chemical and biochemical industrial, especially in health care and disease prevention. For example, the blood glucose concentration is related to the diabetes. To control the blood glucose concentration within the normal level is critical issue to the diabetic daily care. Therefore, many researchers

**5. Concentration measurement with heterodyne interferometer** 

rotation angle in a chiral medium [40].

developed novel methods for measuring the solution concentration. Because of the advantages of the optical method such as high sensitivity, high resolution, non-contact, and quick response, optical measurement method become more popular in past few decades. And these methods can roughly divide into fiber type sensor [41, 44-45, 49] and non-fiber type sensor [42-43, 46-48, 50]. In this section, we will review some recent development in both type sensors for measuring concentration of the specific chemical compound [41-50].

M. H. Chiu [41] developed a D-shape fiber sensor with SPR property and integrated with heterodyne interferometer which could detect variation in the alcohol concentration of 2%. The optical setup was shown in figure 22. The heterodyne light source was guided into the sensor and suffered the attenuate total reflection (ATR) at the sensing region. Because of the refractive index of the sample will be varied as the concentration changed. And induce the phase difference between the p- and s- polarizations. To measure the phase difference can be obtained the concentration variation of the sample.

Fig. 22. The scheme of the D-shape fiber sensor [41].

Figure 22 shows that the testing signal detected by photodetector and sent into the phasemeter. Therefore, the interference signal can be written as

$$I(t) = I\_0 \left\{ 1 + V \cos \left[ \omega t + \left( \frac{L(\phi\_p - \phi\_s)}{2h \tan \theta\_l} \right) \right] \right\} \tag{44}$$

where *I*0 and *V* are the average intensity and visibility; *L* and *h* are the sensing length and core diameter; (*p*-*s*) is the phase difference between p- and s- polarizations; *<sup>i</sup>* is the incident angle at the interface between fiber core and metal film. Based on the Fresnel equation, one can derive the (*p*-*<sup>s</sup>*) from the amplitude reflection coefficient under ATR condition and it is obvious that (*p*-*<sup>s</sup>*) is the function of the refractive index of the sensing medium. Figure 23 shows that the results measured by the D-shape fiber sensor for different concentration of the alcohol.

The Applications of the Heterodyne Interferoemetry 57

As the heterodyne light source enters the sensing part, the light beam undergoes total internal reflection (TIR) and the phase difference between the p- and s- polarization states

where *n*1 and *n*2 are the refractive indices of the immobilized GOx and the testing solution.

*<sup>t</sup>* and *m* are the incident angle and the number of TIRs that occur at the interface between the GOx and the testing solution. After dripping the testing sample onto the sensor, the phase will vary as the glucose reacts with the GOx to be converted into gluconic acid and

It means that the refractive index (*n*2) will change and consequently the phase will change. Besides, the refractive index *n*2 is a function of the concentration of the testing sample. Different concentration of the solution has different refractive index. Therefore, one can determine the concentration variation by measuring the phase variation. In their methods, the phase difference can be carried in the heterodyne interference signal and written as

�� = ��[1 + cos(��) + ��]. To deserve to be mentioned, they found that the pH property between the testing sample and sensor is critical issue for rapid measurement. Figure 25 shows that the response time and response efficiency of the fiber sensor. It is clear that the response time for measuring glucose solution was shorter than those for serum measurement. And the response efficiency for measuring glucose solution was faster than those for serum measurement at

Based on their results, this fiber sensor has good linearity of the calibration curve for glucose solution and serum sample. And they showed the best resolutions were 0.1 and 0.136 mg/dl

One of the non-fiber type sensors is SPR (surface Plasmon resonance) sensor which has been applied in field such as pharmaceutical development and life sciences. And SPR provides ultra high sensitivity for detecting tiny refractive index (RI) changes or other quantities which can be converted into an equivalent RI. The heterodyne interferometer detects the SPR phase by using a Zeeman laser or optical modulator, such as an acousto-optic modulator or electro-optic modulator and has been reported in the literature. Heterodyne phase detection techniques offer the high measurement performance high sensitivity and high resolution in real-time. J. Y. Lee [43] proposed wavelength-modulation circular heterodyne interferometer (WMCHI) with SPR sensor for measuring the different

The SPR sensor had the Kretschmann configuration consists of a BK7 prism coated with a 50 nm gold film and integrated with micro-fluid channel which used to inject the testing

concentration of alcohol. The diagram of the WMCHI is shown in figure 26.

∙ tan�� �

����� ���(�� �� )�

��� ��∙��� ��

��� gluconic acid + H�O�. (46)

�, (45)

� ��� ��

���

�� = ����� <sup>=</sup> �

hydrogen peroxide. The chemical reaction can be formulated as follows:

Glucose + O�

for glucose solution and serum based sample, respectively.

can be written as

different GOx concentration.

Fig. 23. The experiment result of different concentration of alcohol measured by D-shape fiber sensor [41].

In Chiu's results, they can observe the concentration variation 2 %, in which the corresponding refractive index variation is about 0.0009. Based on error analysis, their method can be reached 210-6 refractive index unit.

Recently, T. Q. Lin [44] and C. C. Hsu [45] developed a fiber sensor which immobilized glucose oxidase (GOx) on the fiber core for measuring the glucose concentration in serum and phosphate buffer solution (PBS). Their measurement method integrated the fiber sensor and heterodyne interferometer which showed in figure 24.

Fig. 24. Schematic diagram of the measurement system and preliminary test of the glucose fiber sensor [44].

6 %

8 %

10 %

4 %

Fig. 23. The experiment result of different concentration of alcohol measured by D-shape

0 %

2 %

In Chiu's results, they can observe the concentration variation 2 %, in which the corresponding refractive index variation is about 0.0009. Based on error analysis, their

Recently, T. Q. Lin [44] and C. C. Hsu [45] developed a fiber sensor which immobilized glucose oxidase (GOx) on the fiber core for measuring the glucose concentration in serum and phosphate buffer solution (PBS). Their measurement method integrated the fiber sensor

Fig. 24. Schematic diagram of the measurement system and preliminary test of the glucose

fiber sensor [41].

fiber sensor [44].

method can be reached 210-6 refractive index unit.

and heterodyne interferometer which showed in figure 24.

As the heterodyne light source enters the sensing part, the light beam undergoes total internal reflection (TIR) and the phase difference between the p- and s- polarization states can be written as

$$\phi\_t = m\phi\_{TIR} = \frac{L}{h\tan\theta\_t} \cdot \tan^{-1}\left(\frac{\sqrt{\sin^2\theta\_t - (\frac{n\_2}{n\_1})^2}}{\tan\theta\_t \cdot \sin\theta\_t}\right) \tag{45}$$

where *n*1 and *n*2 are the refractive indices of the immobilized GOx and the testing solution. *<sup>t</sup>* and *m* are the incident angle and the number of TIRs that occur at the interface between the GOx and the testing solution. After dripping the testing sample onto the sensor, the phase will vary as the glucose reacts with the GOx to be converted into gluconic acid and hydrogen peroxide. The chemical reaction can be formulated as follows:

$$\text{Glucose} + \text{O}\_2 \xrightarrow{\text{GOx}} \text{gluconic acid} + \text{H}\_2\text{O}\_2. \tag{46}$$

It means that the refractive index (*n*2) will change and consequently the phase will change. Besides, the refractive index *n*2 is a function of the concentration of the testing sample. Different concentration of the solution has different refractive index. Therefore, one can determine the concentration variation by measuring the phase variation. In their methods, the phase difference can be carried in the heterodyne interference signal and written as

$$I\_t = I\_0[1 + \cos(at) + \phi\_t].$$

To deserve to be mentioned, they found that the pH property between the testing sample and sensor is critical issue for rapid measurement. Figure 25 shows that the response time and response efficiency of the fiber sensor. It is clear that the response time for measuring glucose solution was shorter than those for serum measurement. And the response efficiency for measuring glucose solution was faster than those for serum measurement at different GOx concentration.

Based on their results, this fiber sensor has good linearity of the calibration curve for glucose solution and serum sample. And they showed the best resolutions were 0.1 and 0.136 mg/dl for glucose solution and serum based sample, respectively.

One of the non-fiber type sensors is SPR (surface Plasmon resonance) sensor which has been applied in field such as pharmaceutical development and life sciences. And SPR provides ultra high sensitivity for detecting tiny refractive index (RI) changes or other quantities which can be converted into an equivalent RI. The heterodyne interferometer detects the SPR phase by using a Zeeman laser or optical modulator, such as an acousto-optic modulator or electro-optic modulator and has been reported in the literature. Heterodyne phase detection techniques offer the high measurement performance high sensitivity and high resolution in real-time. J. Y. Lee [43] proposed wavelength-modulation circular heterodyne interferometer (WMCHI) with SPR sensor for measuring the different concentration of alcohol. The diagram of the WMCHI is shown in figure 26.

The SPR sensor had the Kretschmann configuration consists of a BK7 prism coated with a 50 nm gold film and integrated with micro-fluid channel which used to inject the testing

$$E\_1 = P(45^\circ) \cdot f\_{SPR} \cdot f\_Q(45^\circ) \cdot E\_h$$

$$= \frac{1}{2} \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} |r\_p|e^{i\phi p} & 0 \\ 0 & |r\_s|e^{i\phi s} \end{bmatrix} \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & i \\ i & 1 \end{bmatrix} \begin{bmatrix} e^{i(\phi p - \omega t)/2} \\ e^{-i(\phi p\_0 - \omega t)/2} \end{bmatrix}$$

$$= \frac{1}{2\sqrt{2}} \left[ \left( |r\_p|e^{i\phi p} + i|r\_s|e^{i\phi s} \right) e^{\frac{i(\phi p - \omega t)}{2}} + \left( i|r\_p|e^{i\phi p} + |r\_s|e^{i\phi s} \right) e^{-\frac{\ell(\phi p - \omega t)}{2}} \right] \begin{bmatrix} 1 \\ 1 \end{bmatrix} \right] \tag{47a}$$

$$\mathbf{E}\_2 = P(-4\mathbf{\hat{s}}^\circ) \cdot f\_{SPR} \cdot f\_Q(4\mathbf{\hat{s}}^\circ) \cdot \mathbf{E}\_h$$

$$\mathbf{E} = \frac{1}{2\sqrt{2}} \left[ \left( |r\_p|e^{i\phi p} + i|r\_s|e^{i\phi s} \right) e^{\frac{l(\phi p - \omega t)}{2}} + \left( i|r\_p|e^{i\phi p} + |r\_s|e^{i\phi s} \right) e^{-\frac{l(\phi p - \omega t)}{2}} \right] \begin{bmatrix} 1 \\ -1 \end{bmatrix} \tag{47b}$$

$$\Phi = \left(\Phi\_0 + \tan^{-1}\frac{B}{A}\right) - \left(\Phi\_0 - \tan^{-1}\frac{B}{A}\right) = 2\tan^{-1}\left(\frac{\left|r\_p\right|^2 - \left|r\_s\right|^2}{2\left|r\_p\right|\left|r\_s\right|\cos\phi}\right). \tag{48}$$

$$I\_{p1+p2} \text{(\Delta\omega t)} = A\_{p1} A\_{p2} \cos(\Delta\omega t + \phi\_n), \tag{49a}$$

$$I\_{\rm s1+s2}(\Delta\omega t) = A\_{\rm s1}A\_{\rm s2}\cos(\Delta\omega t + \phi\_{\rm s}),\tag{49b}$$

The Applications of the Heterodyne Interferoemetry 61

(a) (b)

In this chapter, we reviewed some recent development or state of the art techniques. It shows that the heterodyne interferometry is a mature technique and can be applied to many different aspects. For example, the diffraction grating heterodyne interferometry (DGHI) provided nanometer resolution for precision positioning which can be integrated with motorized stage. Full-field circular heterodyne interferometry (FFCHI) can be used to investigate the twodimensional optical properties, such as refractive index and birefrigence of testing sample. For this point of view, the heterodyne interferometry can be a refractometer with high accuracy. To integrate with optical sensor, the heterodyne interferometry can be used to diagnose the concentration of the body fluid such as blood glucose and glycerin or the protein interaction between body-antibody. Therefore, the heterodyne interferometry is a powerful, flexible,

[1] L. C. De Backer, "In-plane displacement measurement by speckle interferometry," Non-

Fig. 29. The measurement results of the PSPR method [42].

(c)

integrable, and reliable technique for precision metrology.

destructive Testing 8, pp. 177-180 (1975).

**6. Conclusion** 

**7. References** 

Fig. 27. The relationship between and and the incident angle [43].

respectively. *P* and *<sup>S</sup>* are the phase differences of the reflected P and P and the reflected S and S waves respectively. In equation (49), *P* and *<sup>S</sup>* are equal to 0 and these two interference signals will remain at maximum intensity under the SPR proceeded.

Fig. 28. The schematic of the amplitude sensitivity PSPR [42].

Based on this method, Chou demonstrated three different testing samples with concentration variation which were sucrose, glycerin-water solution, and rabbit anti-mouse IgG. In figure 29, the best of these sample are 810-8, 7.610-7, and 210-9, respectively.

Fig. 27. The relationship between

×103

2

0






Phase difference (deg)

and S waves respectively. In equation (49),

*P* and 

respectively.

Fig. 28. The schematic of the amplitude sensitivity PSPR [42].

interference signals will remain at maximum intensity under the SPR proceeded.

and and the incident angle [43].

Incident angle (deg)

65 65.5 66 66.5 67 67.5 68

*P* and

Based on this method, Chou demonstrated three different testing samples with concentration variation which were sucrose, glycerin-water solution, and rabbit anti-mouse IgG. In figure 29, the best of these sample are 810-8, 7.610-7, and 210-9, respectively.

*<sup>S</sup>* are the phase differences of the reflected P and P and the reflected S

*r*

*<sup>S</sup>* are equal to 0 and these two

Fig. 29. The measurement results of the PSPR method [42].

## **6. Conclusion**

In this chapter, we reviewed some recent development or state of the art techniques. It shows that the heterodyne interferometry is a mature technique and can be applied to many different aspects. For example, the diffraction grating heterodyne interferometry (DGHI) provided nanometer resolution for precision positioning which can be integrated with motorized stage. Full-field circular heterodyne interferometry (FFCHI) can be used to investigate the twodimensional optical properties, such as refractive index and birefrigence of testing sample. For this point of view, the heterodyne interferometry can be a refractometer with high accuracy. To integrate with optical sensor, the heterodyne interferometry can be used to diagnose the concentration of the body fluid such as blood glucose and glycerin or the protein interaction between body-antibody. Therefore, the heterodyne interferometry is a powerful, flexible, integrable, and reliable technique for precision metrology.

## **7. References**

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The Applications of the Heterodyne Interferoemetry 63

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**3** 

*México* 

**One-Shot Phase-Shifting Interferometry with** 

**Using N≥4 Interferograms** 

*Benemérita Universidad Autónoma de Puebla,* 

and Cruz Meneses-Fabián

Gustavo Rodríguez Zurita, Noel-Ivan Toto-Arellano

**Phase-Gratings and Modulation of Polarization** 

Phase-shifting interferometry requires (PSI) of several interferograms of the same optical field with similar characteristics but shifted by certain phase values to retrieve the optical phase. This task has been usually performed by stages with great success and requires of a series of sequential shots [Meneses et al., 2006a]. However, time-varying phase distributions are excluded from this schema. Several efforts for single-shot phase-shifting interferometry have been tested successfully [Novak et al.,2005 ;Rodriguez et al., 2008a], but some of them require of non-standard components and they need to be modified in some important respects in order to get more than four interferograms. Two-windows grating interferometry, on the other hand, has been proved to be an attractive technique because of its mechanical stability as a common-path interferometer [Arrizón and De La Llave 2004]. Moreover, gratings can be used as convenient phase modulators because they introduce phase shifts through lateral displacements. In this regard, phase gratings offer more multiplexing capabilities than absorption gratings (more useful diffraction orders because higher diffraction efficiencies can be achieved). Furthermore, with two phase gratings with their vector gratings at 90° (grids) there appear even more useful diffraction orders[Toto et al., 2008]. Modulation of polarization can be independently applied to each diffraction order to introduce a desired phase-shift in each interference pattern instead of using lateral translations. These properties combine to enable phase-shifting interferometric systems that require of only a single-shot, thus enabling phase inspection of moving subjects. Also, more than four interferograms can be acquired that way. A simple interferogram processing enables the use of interference fringes with different fringe modulations and intensities. In this chapter, the basic properties of two-windows phase grating interferometry (TWPGI) and modulation of polarization is reviewed on the basis of the far-field diffraction properties of phase gratings and grids. Phase shifts in the diffraction orders can be used as an advantage because they simplify the needed polarization filter distributions. It is finally remarked, that these interferometers are compatible with interference fringes exhibiting spatial frequencies of relative low values and, therefore, no great loss of resolution is related with several interferograms when simultaneously using the same image field of the camera. To extract optical phase distributions which evolve in time, the capture of the *n* shifted

**1. Introduction** 


## **One-Shot Phase-Shifting Interferometry with Phase-Gratings and Modulation of Polarization Using N≥4 Interferograms**

Gustavo Rodríguez Zurita, Noel-Ivan Toto-Arellano and Cruz Meneses-Fabián *Benemérita Universidad Autónoma de Puebla, México* 

## **1. Introduction**

64 Interferometry – Research and Applications in Science and Technology

[38] Y. L. Chen, H. C. Hsieh, W. T. Wu, W. Y. Chang, and D. C. Su, "Alternative method for

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27566 (2010).

(2008).

1~100608-3 (2011).

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linear birefringent retardation plate," Optics Communications 133, pp. 11-16 (1997).

modulated circular heterodyne interferometer for measuring the rotation angle in a

Plasmon resonance technology and heterodyne interferometry," Optics Letters 30,

surface plasmon resonance and wavelength modulated heterodyne

immobilized GOD technique for rapid measurement," Optics Express 18 pp. 27560-

measuring glucose concentration in serum," Chinese Optics Letters, 9 pp. 100608-

index using a laser interferometer," Optics and Laser in Engineering 46, pp. 666-670

sensor for liquids and solids using dielectric multilayer films deposited on optical

Phase-shifting interferometry requires (PSI) of several interferograms of the same optical field with similar characteristics but shifted by certain phase values to retrieve the optical phase. This task has been usually performed by stages with great success and requires of a series of sequential shots [Meneses et al., 2006a]. However, time-varying phase distributions are excluded from this schema. Several efforts for single-shot phase-shifting interferometry have been tested successfully [Novak et al.,2005 ;Rodriguez et al., 2008a], but some of them require of non-standard components and they need to be modified in some important respects in order to get more than four interferograms. Two-windows grating interferometry, on the other hand, has been proved to be an attractive technique because of its mechanical stability as a common-path interferometer [Arrizón and De La Llave 2004]. Moreover, gratings can be used as convenient phase modulators because they introduce phase shifts through lateral displacements. In this regard, phase gratings offer more multiplexing capabilities than absorption gratings (more useful diffraction orders because higher diffraction efficiencies can be achieved). Furthermore, with two phase gratings with their vector gratings at 90° (grids) there appear even more useful diffraction orders[Toto et al., 2008]. Modulation of polarization can be independently applied to each diffraction order to introduce a desired phase-shift in each interference pattern instead of using lateral translations. These properties combine to enable phase-shifting interferometric systems that require of only a single-shot, thus enabling phase inspection of moving subjects. Also, more than four interferograms can be acquired that way. A simple interferogram processing enables the use of interference fringes with different fringe modulations and intensities. In this chapter, the basic properties of two-windows phase grating interferometry (TWPGI) and modulation of polarization is reviewed on the basis of the far-field diffraction properties of phase gratings and grids. Phase shifts in the diffraction orders can be used as an advantage because they simplify the needed polarization filter distributions. It is finally remarked, that these interferometers are compatible with interference fringes exhibiting spatial frequencies of relative low values and, therefore, no great loss of resolution is related with several interferograms when simultaneously using the same image field of the camera. To extract optical phase distributions which evolve in time, the capture of the *n* shifted

One-Shot Phase-Shifting Interferometry

rulings of the second grating along the

  *q r*

**3.2 Two-window phase-grating interferometry: fringe modulation** 

*q r*

transform of the phase grid becomes

a lattice with a period given by X0.

through an example.

**3.1 Phase grids** 

written as

 

with Phase-Gratings and Modulation of Polarization Using N≥4 Interferograms 67

Each polarizing filter transmission axis is adjusted at different angle ψ*i*, so as to obtain the desired phase shift *ξi* for each pair of orders. For a 90° phase-shift ξ*i* between interfering fields, the polarization angles *ψi* in each diffraction order must be 0°, 45°, 90° and 135° for the case of ideal quarter-wave retardation (= 90°). In the next sections, some particularities arising from the optical components available for our set-up are discussed. Among these, the calculation of *ψi* for the case of a non exact quarter-wave retardation is considered

Object and image planes are described by (*x,y*) coordinates. A periodic phase-only

frequency coordinates scaled to the wavelength and the focal length *f*. In the plane (*u,v*), the period of G is denoted by *d* (the same in both axis directions) and thus, its spatial frequency by *d*. Two neighboring diffraction orders have a distance of X0=*f*/*d* in the image plane. Then, uX0. Taking the rulings of one grating along the direction and the

<sup>0</sup> <sup>0</sup> 2 sin 2 2 sin 2 <sup>2</sup> <sup>2</sup> , 2 2 *g x g y iA X iA X i qX i rX*

where the frequencies along each axes directions are taken of the same value The Fourier

0 0 , 22 ,

which consists of point-like diffraction orders distributed in the image plane on the nodes of

Phase grating interferometry is based on a phase grating placed as the pupil of a 4f Fourier optical system [Rodriguez et al., 2008a;Thomas and Wyant ,1976]. The use of two windows at the object plane in conjunction with phase grating interferometry allows interference between the optical fields associated to each window with higher diffraction efficiency [Arrizon and Sanchez, 2004; Ramijan, 1978]. Such a system performs as a common path interferometer (Fig. 1). When using birefringent plates which do not perform exactly as quarter-wave plates for the wavelength employed, the polarization angles of the linear

To calculate the phase shifts induced in a more general polarization states by linear

polarizing filters to obtain 90° phase-shifts must change[Rodriguez et al., 2008a].

polarizers, consider two fields whose Jones vectors are described respectively by

 

*q gr g*

*Gxy J A J A x qX y rX* 

*G e e J Ae J Ae*

 

direction, the resulting centered phase grid can be

*q g r g*

 

*q r*

(2)

(1)

/*f* are the

**3. Interference patterns with polarizing filters and retarding plates** 

transmittance G() is placed in the frequency plane (*u,v*). Then =*u*/*f* and =

 

interferograms with one shot is desirable. Some approaches to perform this task have been already demonstrated [Wyant, 2004; Rodriguez et al., 2009], although only for *n* = 4 to our knowledge. Among these systems, the one using two windows in the object plane of a 4*f* system with a phase-grating in the Fourier plane and modulation of polarization (TWPGI) is a very simple possibility [Rodriguez 2008a]. In this communication, the capability of TWPGI to capture more than four interferograms in one shot is demonstrated with the introduction of a phase grid in place of the grating. To test TWPGI for more than four interferograms, the case of *n*=(*N*+1) interferograms has been chosen. This method reduces errors in phase calculations when noisy interferograms are involved [Malacara D, 1998]. Experimental results for *n* = 5, 7, 9 interferograms are shown.

## **2. Experimental setup**

A The Fig. 1 shows the arrangement of an ideal one-shot phase-shifting grating interferometer incorporating modulation of polarization. A combination of a quarter-wave plate Q and a linear polarizing filter P generates linearly polarized light at an appropriate azimuth angle (45°) entering the interferometer. Two quarter-wave plates (QL and QR) with their orthogonal fast axes are placed in front of the two windows of the common-path interferometer so as to generate left and right circularly polarized light as the corresponding beam leaves each window, see Fig. 1(a). A phase grating is placed at the system´s Fourier plane as the pupil. In the image plane, Fig. 1(b) superimposition of diffraction orders result, causing replicated images to interfere.

Fig. 1. (a) One-shot phase-shifting grating interferometer with modulation of polarization. A, B: windows. PBS: Polarizing Beam splitter; *M*i: Mirrors; G(µ,ν): Phase grid. (b) *ψi*: polarizing angles (with *i* = 1,2,3,4 to obtain phase-shifts *ξi* =0°, 90°, 180°, 270° respectively). Translation of coordinates around the order position: *xq = x − qF0* and *yr = y − rF0. '*= 1.519 *rad.*

The phase shifting ξ*i* , *i*= 1…4, results after placing a linear polarizer to each one of the interference patterns generated on each diffracting orders in the exit plane (P1, P2, P3, P4). Each polarizing filter transmission axis is adjusted at different angle ψ*i*, so as to obtain the desired phase shift *ξi* for each pair of orders. For a 90° phase-shift ξ*i* between interfering fields, the polarization angles *ψi* in each diffraction order must be 0°, 45°, 90° and 135° for the case of ideal quarter-wave retardation (= 90°). In the next sections, some particularities arising from the optical components available for our set-up are discussed. Among these, the calculation of *ψi* for the case of a non exact quarter-wave retardation is considered through an example.

## **3. Interference patterns with polarizing filters and retarding plates**

## **3.1 Phase grids**

66 Interferometry – Research and Applications in Science and Technology

interferograms with one shot is desirable. Some approaches to perform this task have been already demonstrated [Wyant, 2004; Rodriguez et al., 2009], although only for *n* = 4 to our knowledge. Among these systems, the one using two windows in the object plane of a 4*f* system with a phase-grating in the Fourier plane and modulation of polarization (TWPGI) is a very simple possibility [Rodriguez 2008a]. In this communication, the capability of TWPGI to capture more than four interferograms in one shot is demonstrated with the introduction of a phase grid in place of the grating. To test TWPGI for more than four interferograms, the case of *n*=(*N*+1) interferograms has been chosen. This method reduces errors in phase calculations when noisy interferograms are involved [Malacara D, 1998]. Experimental

A The Fig. 1 shows the arrangement of an ideal one-shot phase-shifting grating interferometer incorporating modulation of polarization. A combination of a quarter-wave plate Q and a linear polarizing filter P generates linearly polarized light at an appropriate azimuth angle (45°) entering the interferometer. Two quarter-wave plates (QL and QR) with their orthogonal fast axes are placed in front of the two windows of the common-path interferometer so as to generate left and right circularly polarized light as the corresponding beam leaves each window, see Fig. 1(a). A phase grating is placed at the system´s Fourier plane as the pupil. In the image plane, Fig. 1(b) superimposition of diffraction orders result,

Fig. 1. (a) One-shot phase-shifting grating interferometer with modulation of polarization. A, B: windows. PBS: Polarizing Beam splitter; *M*i: Mirrors; G(µ,ν): Phase grid. (b) *ψi*: polarizing angles (with *i* = 1,2,3,4 to obtain phase-shifts *ξi* =0°, 90°, 180°, 270° respectively). Translation of coordinates around the order position: *xq = x − qF0* and *yr = y − rF0. '*= 1.519

The phase shifting ξ*i* , *i*= 1…4, results after placing a linear polarizer to each one of the interference patterns generated on each diffracting orders in the exit plane (P1, P2, P3, P4).

results for *n* = 5, 7, 9 interferograms are shown.

causing replicated images to interfere.

**2. Experimental setup** 

*rad.*

Object and image planes are described by (*x,y*) coordinates. A periodic phase-only transmittance G() is placed in the frequency plane (*u,v*). Then =*u*/*f* and =/*f* are the frequency coordinates scaled to the wavelength and the focal length *f*. In the plane (*u,v*), the period of G is denoted by *d* (the same in both axis directions) and thus, its spatial frequency by *d*. Two neighboring diffraction orders have a distance of X0=*f*/*d* in the image plane. Then, uX0. Taking the rulings of one grating along the direction and the rulings of the second grating along the direction, the resulting centered phase grid can be written as

$$G\left(\mu,\xi\_{\mathfrak{r}}\right) = e^{i2\pi A\_{\mathfrak{r}}\sin\left[2\pi \cdot X\_{\mathfrak{r}}\mu\right]}e^{i2\pi A\_{\mathfrak{r}}\sin\left[2\pi \cdot X\_{\mathfrak{r}}\nu\right]} = \sum\_{q=-\infty}^{\infty}J\_q\left(2\pi A\_{\mathfrak{r}}\right)e^{i2\pi qX\_{\mathfrak{r}}\mu}\sum\_{r=-\infty}^{\infty}J\_r\left(2\pi A\_{\mathfrak{r}}\right)e^{i2\pi \cdot rX\_{\mathfrak{r}}\nu} \tag{1}$$

where the frequencies along each axes directions are taken of the same value The Fourier transform of the phase grid becomes

$$\tilde{G}\left(\mathbf{x},\mathbf{y}\right) = \sum\_{q=-\alpha}^{q=\alpha} \sum\_{r=-\alpha}^{r=\alpha} f\_q\left(2\pi A\_g\right) f\_r\left(2\pi A\_g\right) \delta\left(\mathbf{x} - qX\_0, \mathbf{y} - rX\_0\right) \tag{2}$$

which consists of point-like diffraction orders distributed in the image plane on the nodes of a lattice with a period given by X0.

## **3.2 Two-window phase-grating interferometry: fringe modulation**

Phase grating interferometry is based on a phase grating placed as the pupil of a 4f Fourier optical system [Rodriguez et al., 2008a;Thomas and Wyant ,1976]. The use of two windows at the object plane in conjunction with phase grating interferometry allows interference between the optical fields associated to each window with higher diffraction efficiency [Arrizon and Sanchez, 2004; Ramijan, 1978]. Such a system performs as a common path interferometer (Fig. 1). When using birefringent plates which do not perform exactly as quarter-wave plates for the wavelength employed, the polarization angles of the linear polarizing filters to obtain 90° phase-shifts must change[Rodriguez et al., 2008a].

To calculate the phase shifts induced in a more general polarization states by linear polarizers, consider two fields whose Jones vectors are described respectively by

One-Shot Phase-Shifting Interferometry

 

with

and also [4]

Plots of

 

( , ') 

Fig. 2. (a) Phase shift

several values of '

( , ') 

. as a function of

retardation and experimental retardation. (b) Amplitude *A*( , ')

for several values of '

  . Insert: '

as a function of

for

for ideal

*L qr R q r*

*LJ*

> 

*A*( , ') 1 sin(2 ) cos( ')

 and *A*( , ') 

with Phase-Gratings and Modulation of Polarization Using N≥4 Interferograms 69

2

1 1 1 1

*A JJ J J JJJ J x y*

 

( , ') 2 cos ( , ') ( ', ')

*L L J JJ* 

 , sin( ') ( , ') cot(2 ) cos( ') 1 tan( ) cos( ') *ArcTan*

are shown in Fig. 2 for several values of '

, ' *<sup>L</sup>*

 

*R R J JJ* 

, (9)

. (10)

 

> 

.

, (8)

*qr q r qrq r*

1 1

*J JJ J J J i x y*

' ' exp ( ', ')

2 2

cos sin sin cos

 , ' *<sup>L</sup>*

$$
\vec{J}\_L(\mathbf{x}, \mathbf{y}) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ e^{i\alpha^\*} \end{pmatrix} \quad \vec{J}\_R(\mathbf{x}, \mathbf{y}) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ e^{-i\alpha^\*} \end{pmatrix} \tag{3}
$$

These vectors represent the polarization states of two beams emerging from a retarding plate with phase retardation ±'. Each beam enters the plate with linear polarization at ±45° with respect to the plate fast axis. Due to their orientations, the electric fields of the beams rotate in opposite directions, thereby the indices *L* and *R*. A convenient window pair for a grating interferometer implies an amplitude transmittance given by

$$
\vec{t}\_0(\mathbf{x}, y) = \vec{f}\_L \cdot w(\mathbf{x} - \frac{\mathbf{x}\_0}{2}, y - \frac{y\_0}{2}) + \vec{f}\_R w'(\mathbf{x} + \frac{\mathbf{x}\_0}{2}, y + \frac{y\_0}{2}) \,. \tag{4}
$$

and x0 and y0 give the mutual separations between the centers of each window along the coordinate axis. One rectangular aperture can be written as w(x,y)= rect[x/a]rect[y/b] whereas the second one, as w´(x,y)= w(x,y) exp{ i(x,y)}, a relative phase between the windows being described with the function . a represents the side length of each window. The image *t x <sup>i</sup>* ,*y* formed by the system consists basically of replications of each window at distances X0, that is, the convolution of *t x* <sup>0</sup> ,*y* with the point spread function of the system, defined by the inverse Fourier transform of

$$
\tilde{\Gamma}\_2(\mu, \nu) = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \mathbf{G}\_2(\mu, \nu) \,. \tag{5}
$$

This results into the following

$$\begin{split} \vec{t}\_f(\mathbf{x}, \mathbf{y}) &= \vec{t}\_2(\mathbf{x}, \mathbf{y}) \ast \mathfrak{T}^{-1} \left\{ \tilde{\Gamma}\_2 \left( \mu, \nu \right) \right\} \\ &= \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \cdot \vec{t}\_2(\mathbf{x}, \mathbf{y}) \ast \mathfrak{T}^{-1} \left\{ \tilde{\mathcal{G}}\_2 \left( \mu, \nu \right) \right\}. \end{split} \tag{6}$$

Assuming *y*0=*x*0, by invoking the condition of matching first-neighboring orders, X0=x0, *q*'=*q*+1 and *r*'=*r*+1, and the image is then basically described by

$$\begin{aligned} \overline{f}\_0(\mathbf{x}, y) \* \tilde{\mathbf{G}}(\mathbf{x}, y) &= \\ \overline{f}\_L \sum\_{q, r}^\infty I\_q I\_r \cdot w(\mathbf{x} - (q + \mathbf{\dot{J}}\_2') \mathbf{x}\_0, y - (r + \mathbf{\dot{J}}\_2') \mathbf{x}\_0) \\ + \overline{f}\_R \sum\_{q', r'}^\infty I\_q \cdot I\_{r'} \cdot w'(\mathbf{x} - (q + \mathbf{\dot{J}}\_2') \mathbf{x}\_0, y - (r + \mathbf{\dot{J}}\_2') \mathbf{x}\_0) &= \\ \sum\_{q = -\infty}^\infty \sum\_{r = -\infty}^\infty \left\{ \overline{f}\_L I\_q I\_r + \overline{f}\_R I\_{q+1} I\_{r+1} \cdot \exp\left[ i\phi \left( \mathbf{x} - (q + \mathbf{\dot{J}}\_2') \mathbf{x}\_0, y - (r + \mathbf{\dot{J}}\_2') \mathbf{x}\_0 \right) \right] \right\} \end{aligned} \tag{7}$$

where some inessential constants are dropped. By selecting the diffraction term of order *qr*, after placing a linear polarizing filter with transmission axis at the angle *ψ*, *LJ* , its irradiance results proportional to

$$\begin{aligned} & \left\| \overline{I}\_{L} \, ^{\prime} I\_{q} I\_{r} + \overline{I}\_{R} \, ^{\prime} I\_{q+1} I\_{r+1} \cdot \exp\left[ i \varphi(\mathbf{x}^{\prime}, \mathbf{y}^{\prime}) \right] \right\|^{2} = \\ & A(\mathbf{y}^{\prime}, \mathbf{a}^{\prime}) \cdot \left[ \left( I\_{q} I\_{r} \right)^{2} + \left( I\_{q+1} I\_{r+1} \right)^{2} + 2 I\_{q} I\_{r} I\_{q+1} I\_{r+1} \cdot \cos \left[ \xi^{\prime}(\mathbf{y}^{\prime}, \mathbf{a}^{\prime}) - \varphi(\mathbf{x}^{\prime}, \mathbf{y}^{\prime}) \right] \right]^{\prime} \end{aligned} \tag{8}$$

with

68 Interferometry – Research and Applications in Science and Technology

1 1

 

*e* 

, (3)

(,) <sup>2</sup> *<sup>R</sup> <sup>i</sup> J xy*

0 0 0 0

 

1 2 2

. (5)

 

1 1

, its irradiance

. (6)

(7)

*x x <sup>y</sup> <sup>y</sup> t xy J wx y J w x y* , (4)

'

'

These vectors represent the polarization states of two beams emerging from a retarding plate with phase retardation ±'. Each beam enters the plate with linear polarization at ±45° with respect to the plate fast axis. Due to their orientations, the electric fields of the beams rotate in opposite directions, thereby the indices *L* and *R*. A convenient window pair for a

0( , ) ( , ) ´( , ) 22 22 *L R*

and x0 and y0 give the mutual separations between the centers of each window along the coordinate axis. One rectangular aperture can be written as w(x,y)= rect[x/a]rect[y/b] whereas the second one, as w´(x,y)= w(x,y) exp{ i(x,y)}, a relative phase between the windows being described with the function . a represents the side length of each window. The image *t x <sup>i</sup>* ,*y* formed by the system consists basically of replications of each window at distances X0, that is, the convolution of *t x* <sup>0</sup> ,*y* with the point spread function of the

2 2 <sup>1</sup> 1 0 , (,) <sup>2</sup> 0 1

 

(,) , . <sup>2</sup> 0 1

*JJJ J J J i x q x y r x* 

where some inessential constants are dropped. By selecting the diffraction term of order *qr*,

1 1 2 2 0 0

exp ( ) , ( )

*t xy G*

*G*

1 2 2

Assuming *y*0=*x*0, by invoking the condition of matching first-neighboring orders, X0=x0,

1 1 2 2 0 0

'( ( ' ) , ( ' ) )

( ( ), ( ))

1 1 ' ' 2 2 <sup>0</sup> <sup>0</sup>

(,) (,) , 1 1 0

*J JJ wx q x y r x*

*J JJ wx q x y r x*

after placing a linear polarizing filter with transmission axis at the angle *ψ*, *LJ*

 

*<sup>f</sup> t xy t xy*

*q*'=*q*+1 and *r*'=*r*+1, and the image is then basically described by

*Lqr Rq r*

,

*L qr q r*

(,) (,)

*t xy Gxy*

', '

*R qr q r*

1 1

 

*e* 

(,) <sup>2</sup> *<sup>L</sup> <sup>i</sup> J xy*

grating interferometer implies an amplitude transmittance given by

system, defined by the inverse Fourier transform of

This results into the following

0

*q r*

results proportional to

$$
\vec{J}\_{\varphi}^{L} = \begin{pmatrix} \cos \psi & -\sin \psi \\ \sin \psi & \cos \psi \end{pmatrix}, \ \vec{J}\_{L}{}^{\prime} = \mathbf{J}\_{\varphi}^{L} \vec{J}\_{L}{}^{\prime}, \ \vec{J}\_{R}{}^{\prime} = \mathbf{J}\_{\varphi}^{L} \vec{J}\_{R}{}^{\prime} \tag{9}
$$

and also [4]

$$A(\nu, \alpha') = 1 + \sin(2\nu) \cdot \cos(\alpha') \cdot \xi(\nu, \alpha') = \operatorname{ArcTan} \left[ \frac{\sin(\alpha')}{\frac{\cot(2\nu)}{1 + \tan(\nu) \cdot \cos(\alpha')} + \cos(\alpha')} \right] \cdot \tag{10}$$

Plots of ( , ') and *A*( , ') are shown in Fig. 2 for several values of ' .

Fig. 2. (a) Phase shift ( , ') as a function of for several values of ' . Insert: ' for ideal retardation and experimental retardation. (b) Amplitude *A*( , ') as a function of for several values of ' .

One-Shot Phase-Shifting Interferometry

next sections taking only four of 16.

Fig. 4. Experimental patters for a phase-grid.

with Phase-Gratings and Modulation of Polarization Using N≥4 Interferograms 71

For the case of the diffractions orders belonging to a phase-grid constructed with two crossed gratings of equal frequency, the corresponding interference patterns are shown in Fig.4. Each grating gives patterns as in Fig. 4(a) when placed alone in the system of Fig. 1 with no plate retarders neither linear polarizing filters. The whole figure is a composite image because patterns of higher order have lower intensities. The fringe modulation signs are in agreement with the conclusions derived from Fig. 2. The relative phase values of the 16 patterns within the square (drawn with dotted lines in the patterns of Fig. 4) employing

Any grating displacement on its plane only introduces a constant phase term in Eqs. (6) and (8) which, in turn, only shifts each interference pattern by the same amount independent of the diffraction order [Arrizon and Sanchez 2004; Meneses et al., 2006b]. Modulation of polarization employed to attain the needed shifts in each interferogram is described in the

**4. Experimental testing of the phase-shifts in phase grids** 

the method from Kreis, 1986 can be seen in Table 1.

Thus, an interference pattern between fields associated to each window must appear within each replicated window. It is shifted by an amount induced by polarization. For the case of exact quarter-wave retardation, *A*(,/2)=1 and (,/2)=2. Otherwise, these quantities must be evaluated with Eqs. (8). The fringe modulation mqr of each pattern would be of the form

$$m\_{qr} = \frac{2\,\mathrm{J}\_q\mathrm{J}\_{q-1}\mathrm{J}\_r\mathrm{J}\_{r-1}}{\mathrm{J}\_q^2\mathrm{J}\_r^2 + \mathrm{J}\_{q-1}^2\mathrm{J}\_{r-1}^2} \tag{11}$$

The Fourier spectrum of the grid in our tests behaves as sketched in Fig. 3, where two equal phase gratings are shown with their respective +4th diffraction order assumed negative [Fig. 3(a)]. Thus, the -4th diffraction order results also negative. A phase grid is formed with the gratings at 90° and the resulting Fourier spectrum forms a rectangular reticule [Fig. 3(b)]. Due to the phase difference between orders, there are orders pointing out toward the reader (circles) or away (crosses). Because the window are displaced, two Fourier spectra become shifted from the origin diagonally and in opposite directions [Fig. 3(c)]. Similar rows and columns are encircled within the dotted lines. Under our matching condition, the order *qr* superimposes with the order (*q*-1)(*r*-1). Thus, some orders are in phase (dots with dots or crosses with crosses, but only one symbol depicted) and others out of phase (dot with cross). Then, only one symbol means positive contrast, while both symbols mean contrast reversal [Fig. 3(c)].

Fig. 3. a) One dimensional spectra of identical phase gratings to be crossed in order to construct a grid. b) Corresponding image plane for the phase grid. c) Two shifted Fourier spectra are superimposed according to the windows displacement A-B.

Thus, an interference pattern between fields associated to each window must appear within each replicated window. It is shifted by an amount induced by polarization. For the case of

must be evaluated with Eqs. (8). The fringe modulation mqr of each pattern would be of the

2 *qq rr*

*qr q r JJ JJ*

*JJ J J* 

The Fourier spectrum of the grid in our tests behaves as sketched in Fig. 3, where two equal phase gratings are shown with their respective +4th diffraction order assumed negative [Fig. 3(a)]. Thus, the -4th diffraction order results also negative. A phase grid is formed with the gratings at 90° and the resulting Fourier spectrum forms a rectangular reticule [Fig. 3(b)]. Due to the phase difference between orders, there are orders pointing out toward the reader (circles) or away (crosses). Because the window are displaced, two Fourier spectra become shifted from the origin diagonally and in opposite directions [Fig. 3(c)]. Similar rows and columns are encircled within the dotted lines. Under our matching condition, the order *qr* superimposes with the order (*q*-1)(*r*-1). Thus, some orders are in phase (dots with dots or crosses with crosses, but only one symbol depicted) and others out of phase (dot with cross). Then, only one symbol means positive contrast, while both symbols mean contrast reversal

(,/2)=2

1 1 22 2 2

1 1

. Otherwise, these quantities

(11)

/2)=1 and

Fig. 3. a) One dimensional spectra of identical phase gratings to be crossed in order to construct a grid. b) Corresponding image plane for the phase grid. c) Two shifted Fourier

spectra are superimposed according to the windows displacement A-B.

,

*qr*

*m*

exact quarter-wave retardation, *A*(

form

[Fig. 3(c)].

## **4. Experimental testing of the phase-shifts in phase grids**

For the case of the diffractions orders belonging to a phase-grid constructed with two crossed gratings of equal frequency, the corresponding interference patterns are shown in Fig.4. Each grating gives patterns as in Fig. 4(a) when placed alone in the system of Fig. 1 with no plate retarders neither linear polarizing filters. The whole figure is a composite image because patterns of higher order have lower intensities. The fringe modulation signs are in agreement with the conclusions derived from Fig. 2. The relative phase values of the 16 patterns within the square (drawn with dotted lines in the patterns of Fig. 4) employing the method from Kreis, 1986 can be seen in Table 1.

Any grating displacement on its plane only introduces a constant phase term in Eqs. (6) and (8) which, in turn, only shifts each interference pattern by the same amount independent of the diffraction order [Arrizon and Sanchez 2004; Meneses et al., 2006b]. Modulation of polarization employed to attain the needed shifts in each interferogram is described in the next sections taking only four of 16.

Fig. 4. Experimental patters for a phase-grid.

One-Shot Phase-Shifting Interferometry

**5.1 Case of four interferograms** 

taken at a different

2 1*J* , <sup>2</sup> 2*J* , <sup>2</sup> 3*J* and

> 

resulting in similar conclusions.

( , /2) 2 

where

Fig. 5. Polarizing filters array for 90° phase stepping.

such that 12 3 4

 

**5.2 Case of five, seven, and nine interferograms** 

cover two patterns with 180° phase shift in between.

 and *A*( , /2) 1 

retarders, as is well known. Dependence of

with Phase-Gratings and Modulation of Polarization Using N≥4 Interferograms 73

For phase-shifting interferometry with four patterns, four irradiances can be used, each one

tan

 

2 4*J*

> 

angle. The relative phase can be calculated as [Schwieder, 1983]

(12)

on the coordinates of the centered point has

(13)

are the intensity measurements with the values of

 ( ) 0, ( ) /2, ( ) , ( ) 3 /2 respectively. Note that

, so a good choice for the retarders is quarter-wave

2 2 1 3 2 2 2 4

*J J J J*

> 

been simplified to *x*,*y*. The same fringe modulation *mq* results as in Eq. (8). Therefore, the discussion about fringe modulation given in previous sections is retained when introducing the modulation of polarization. Such polarization modulation can be made also for grids,

To demonstrate the use of the several interferograms obtained to extract phase under the conditions as described above, we choose the symmetrical N+1 phase steps algorithms for data processing in the cases N= 4, 6, 8. The phase for N shifts is given by [Malacara, 1998]:

1

*i i N*

> *i i*

*N*

<sup>1</sup> sin 2

*N*

*N*

*<sup>i</sup> <sup>I</sup>*

1

*x y <sup>i</sup> <sup>I</sup>*

where N+1 is the number of interferograms. The Fig. 6 shows the polarizing filters employed. For the case of five interferograms, only three linear polarizing filters have to be placed. The transmission axes of the filter pairs Pn can be the same for each as long as they

tan , <sup>1</sup> cos 2


Table 1. Phase shifts of the 16 patterns within the dotted square of Fig. 3, as measured by the method from Kreis 2986.

## **5. Phase-grid interference patterns with modulation of polarization**

Incorporating modulation of polarization, a TWPGI can be used for dynamic interferometry. This system is able to obtain four interferograms 90° phase-apart with only one shot. Phase evolving in time can then be calculated and displayed on the basis of phase-shifting techniques with four interferograms. The system performs as previous proposals to attain four interferograms with a single shot [Barrientos et al., 1999; Novak et al., 2005]. In the following sections, a variant of a TWPGI able to capture N≥4 interferograms in one shot is described. It consists of the set-up shown in Fig. 1. The system uses a grid as a beam splitter in a way that resembles the well-known double-frequency shearing interferometer as proposed by Wyant,2004, but our proposal differs from it not only because of its modulation of polarization, the use of a single frequency and the use of two windows, but also in the phase steps our system introduces. Besides, ours is not a shearing interferometer of any type.

The Fig. 1 shows the arrangement of a one-shot phase-shifting grid interferometer including modulation of polarization with retarders for the windows and linear polarizers on the image plane. The system generates several diffraction orders of similar irradiances in the average but not equal fringe modulations, as expected. Each interferogram image was scaled to the same values of grey levels (from 0 to 255). Previous reports show that a simplification for the polarizing filters array can be attained when using the phase shifts of to obtain values of of 0, and , due to the -shifts, only two linear polarizing filters have to be placed (instead of four filters, without the -shifts). The transmission axes of the filter pairs P1, P3 and P2, P4 can be the same for each as long as they cover two patterns 180° phase apart (Fig. 4). The needed values of have to be of and with ideal quarter-wave retarders. But considering the retarders at disposal, it can be shown with Eq. (10) that can be of and °. They are sketched in Fig. 4. The square enclosing the 16 windows replicas in the same figure is to be compared with the similar square of Fig. 5 (dotted lines).

Fig. 5. Polarizing filters array for 90° phase stepping.

## **5.1 Case of four interferograms**

72 Interferometry – Research and Applications in Science and Technology

Table 1. Phase shifts of the 16 patterns within the dotted square of Fig. 3, as measured by the

Incorporating modulation of polarization, a TWPGI can be used for dynamic interferometry. This system is able to obtain four interferograms 90° phase-apart with only one shot. Phase evolving in time can then be calculated and displayed on the basis of phase-shifting techniques with four interferograms. The system performs as previous proposals to attain four interferograms with a single shot [Barrientos et al., 1999; Novak et al., 2005]. In the following sections, a variant of a TWPGI able to capture N≥4 interferograms in one shot is described. It consists of the set-up shown in Fig. 1. The system uses a grid as a beam splitter in a way that resembles the well-known double-frequency shearing interferometer as proposed by Wyant,2004, but our proposal differs from it not only because of its modulation of polarization, the use of a single frequency and the use of two windows, but also in the phase steps our system introduces. Besides, ours is not a shearing interferometer of any

The Fig. 1 shows the arrangement of a one-shot phase-shifting grid interferometer including modulation of polarization with retarders for the windows and linear polarizers on the image plane. The system generates several diffraction orders of similar irradiances in the average but not equal fringe modulations, as expected. Each interferogram image was scaled to the same values of grey levels (from 0 to 255). Previous reports show that a simplification for the polarizing filters array can be attained when using the phase shifts of to obtain values of of 0, and , due to the -shifts, only two linear polarizing filters have to be placed (instead of four filters, without the -shifts). The transmission axes of the filter pairs P1, P3 and P2, P4 can be the same for each as long as they cover two patterns 180° phase apart (Fig. 4). The needed values of have to be of and with ideal quarter-wave retarders. But considering the retarders at disposal, it can be shown with Eq. (10) that can be of and °. They are sketched in Fig. 4. The square enclosing the 16 windows replicas in the same figure is to be compared with the similar square of Fig.

**5. Phase-grid interference patterns with modulation of polarization** 

method from Kreis 2986.

type.

5 (dotted lines).

For phase-shifting interferometry with four patterns, four irradiances can be used, each one taken at a different angle. The relative phase can be calculated as [Schwieder, 1983]

$$\tan \phi = \frac{\left\| \vec{J}\_1 \right\|^2 - \left\| \vec{J}\_3 \right\|^2}{\left\| \vec{J}\_2 \right\|^2 - \left\| \vec{J}\_4 \right\|^2} \tag{12}$$

where 2 1*J* , <sup>2</sup> 2*J* , <sup>2</sup> 3*J* and 2 4*J* are the intensity measurements with the values of such that 12 3 4 ( ) 0, ( ) /2, ( ) , ( ) 3 /2 respectively. Note that ( , /2) 2 and *A*( , /2) 1 , so a good choice for the retarders is quarter-wave retarders, as is well known. Dependence of on the coordinates of the centered point has been simplified to *x*,*y*. The same fringe modulation *mq* results as in Eq. (8). Therefore, the discussion about fringe modulation given in previous sections is retained when introducing the modulation of polarization. Such polarization modulation can be made also for grids, resulting in similar conclusions.

## **5.2 Case of five, seven, and nine interferograms**

To demonstrate the use of the several interferograms obtained to extract phase under the conditions as described above, we choose the symmetrical N+1 phase steps algorithms for data processing in the cases N= 4, 6, 8. The phase for N shifts is given by [Malacara, 1998]:

$$\tan\phi(x,y) = \frac{\sum\_{i}^{N+1} I\_i \sin\left(2\pi \frac{i-1}{N}\right)}{\sum\_{i}^{N+1} I\_i \cos\left(2\pi \frac{i-1}{N}\right)}\tag{13}$$

where N+1 is the number of interferograms. The Fig. 6 shows the polarizing filters employed. For the case of five interferograms, only three linear polarizing filters have to be placed. The transmission axes of the filter pairs Pn can be the same for each as long as they cover two patterns with 180° phase shift in between.

One-Shot Phase-Shifting Interferometry

factor : 0.405 rad.

with Phase-Gratings and Modulation of Polarization Using N≥4 Interferograms 75

Fig. 7. Upper row: phase dot. Four 90° phase-shifted interferograms and unwrapped phase. Lower row: phase step. Four 90° phase-shifted interferograms and unwrapped phase.

Fig. 8. Unwrapped calculated phases along typical raster lines of each object of Fig. 5. Scale

Fig. 6. Polarizing filters array for five, seven and nine interferograms. a) 12 interference patterns detected with a polarizing filter at =35°covering all of them. Some -shifts can be recognized when reversal contrasts are present. b) N=4, symmetrical five. c) N= 6, symmetrical seven. d) N= 8, symmetrical nine.

## **6. Experimental results**

Two objects for testing are a phase disk and a phase step. When each object was placed separately in one of the windows using the TWPGI with the polarizers array, the interferograms of Fig. 7 were obtained. For each object, the four interferograms are shown together with the calculated unwrapped phase. However, more than four interferograms could be used, whether for N-steps phase-shifting interferometry [Shwieder et al., 1983] or for averaging images with the same shift. Examples, some typical raster lines for each unwrapped phase are shown in Fig. 8(in arbitrary phase units).

Fig. 6. Polarizing filters array for five, seven and nine interferograms. a) 12 interference

Two objects for testing are a phase disk and a phase step. When each object was placed separately in one of the windows using the TWPGI with the polarizers array, the interferograms of Fig. 7 were obtained. For each object, the four interferograms are shown together with the calculated unwrapped phase. However, more than four interferograms could be used, whether for N-steps phase-shifting interferometry [Shwieder et al., 1983] or for averaging images with the same shift. Examples, some typical raster lines for each

recognized when reversal contrasts are present. b) N=4, symmetrical five. c) N= 6,

=35°covering all of them. Some -shifts can be

patterns detected with a polarizing filter at

**6. Experimental results** 

symmetrical seven. d) N= 8, symmetrical nine.

unwrapped phase are shown in Fig. 8(in arbitrary phase units).

Fig. 7. Upper row: phase dot. Four 90° phase-shifted interferograms and unwrapped phase. Lower row: phase step. Four 90° phase-shifted interferograms and unwrapped phase.

Fig. 8. Unwrapped calculated phases along typical raster lines of each object of Fig. 5. Scale factor : 0.405 rad.

One-Shot Phase-Shifting Interferometry

with Phase-Gratings and Modulation of Polarization Using N≥4 Interferograms 77

Fig. 10. Typical frames from an unwrapped phase from interferograms of oil flowing.

Fig. 9. Flow of oil drops on glass. Phase-shifted interferograms and unwrapped phases. Upper two rows: two examples of five 90° phase-shifts. Center rows: seven 60° phase-shifts. Lower rows: nine 45° phase-shifts. Reference square for scale dimensions.

Fig. 9. Flow of oil drops on glass. Phase-shifted interferograms and unwrapped phases. Upper two rows: two examples of five 90° phase-shifts. Center rows: seven 60° phase-shifts.

Lower rows: nine 45° phase-shifts. Reference square for scale dimensions.

Fig. 10. Typical frames from an unwrapped phase from interferograms of oil flowing.

One-Shot Phase-Shifting Interferometry

38, pp. 2069-2074

*Opt.*, Vol. 44, pp. 6861-6868.

*Optical Testing.*

13-17.

7806-7817.

Vol. 22, pp. 3421-3432.

*Opt. Eng*., Vol. 15, pp. 477.

**8. Acknowledgements** 

grant 102137/43055.

**9. References** 

with Phase-Gratings and Modulation of Polarization Using N≥4 Interferograms 79

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Robledo-Sánchez C.,(2008). One-Shot Phase-Shifting Phase-Grating Interferometry with Modulation of Polarization: case of four interferograms, *Opt. Express*, Vol. 16,

(2008). One-shot phase-shifting interferometry: five, seven, and nine

(2009).Adjustable lateral-shear single-shot phase-shifting interferometry for moving

Wave-Front Measuring Interferometry: some systematic error sources, *Appl. Opt.*,

Phase shifts in the Fourier spectra of phase gratings and phase grids: an application

Dimensional Periodic Filters, *Opt. Lett*. Vol. 29, pp. 141-143.

Ronchi Ruling, *J. Opt. Soc. Am. A*, Vol. 23, pp. 298-305.

Transform Method, *J. Opt. Soc. Am. A,* Vol. 3, pp. 847-855.

interferograms, *Opt Letters*, Vol. 33, pp. 2788-2790.

phase distributions, *Meas. Sci. Technol.* Vol. 20, pp. 115902.

Considering the retarders at disposal, according to Eq. (10) it can be shown that *<sup>n</sup>* can be of 1 0 , <sup>2</sup> 46.577 and 3 92.989 , each step being of 90°. For the case of symmetrical seven, each step is of 60°, so it can be shown that *<sup>n</sup>* are 1 0 , 2 30.800 , 3 62.330 and 4 92.989 . For symmetrical nine, 1 0 , 2 92.989 , 3 22.975 , 4 46.577 and 5 157.903 . In this case, each step has to be 45°. The corresponding results are shown in Fig. 9, where the object was an oil drop running down over a microscope slide. Each interferogram was subject to rescaling and normalization and then, a filtering process prior to phase calculation through Eq. (13).

## **6.1 Moving distributions**

Immersion oil was applied to a glass microscope slide and allowed to flow under the effect of gravity by tilting the slide slightly. The slide was put in front of one of the object windows of the system of Fig. 1. Fig. 10 shows the resulting unwrapped phase evolution of oil flow (Case N=4).

## **7. Final remarks**

The experimental set-up for a polarizing two-window phase-grating common-path interferometer has been described. This system is able to obtain four interferograms 90° phase-apart in only one shot. Therefore, it is suitable to carry out phase extraction using phase shifting techniques. Phase evolving in time can then be calculated and displayed. The system is considerably simpler than previous proposals to attain four interferograms with only one shot. In its present form, it is, however, best suited to relative small objects which do not introduce polarization changes. Because it works with interferograms placed relatively far from the optical axes, experimental results suggest that some method has to be introduced to compensate mainly for distortion, among other off-axis aberrations. This compensation could be optical (as a better design of the optical imaging system) or digital (fringe distortion compensation by inverse transformation). In the experiments, the use of is described retarding plates which are not quarter-wave plates. Although they can perform well enough in principle, it seems better to use quarter-wave plates because no additional variations of the interferogram amplitude arise. Also in this case, a simpler polarization filter array can be used taking advantage of the diffraction properties of a phase grating. Some special phase gratings design could optimize the interferometric system described.

This system is able to obtain *n* = (*N*+1) interferograms with only one shot ( 16 *n* ). Tests with 2 / *N* phase-shifts were presented, but other approaches using different phase-shifts could be attained using linear polarizers with their transmission axes at the proper angle before detection. The phase shifts of due to the grid spectra allows the use of a number of polarizing filters which is less than the number of interferograms, simplifying the filter array. Other configurations for the window positions which are different as the one reported in this communication can also be possible. The accuracy in measurements is the one typical of phase-shifting. Some trade-offs appear while placing several images over the same detector field, but for low frequencies interferograms (with respect to the inverse of the pixel spacing) the influence of these factors seems to be rather small if noticeable. The interferometer could be used for objects with no changes of polarization.

## **8. Acknowledgements**

Partial support from Benemérita Universidad Autónoma de Puebla (BUAP), project: 154984 (CONACYT-BUAP) is also acknowledged. Author N.-I. Toto-Arellano expresses sincere appreciation to Luisa, Miguel and Damian for the support provided, and to CONACYT for grant 102137/43055.

## **9. References**

78 Interferometry – Research and Applications in Science and Technology

corresponding results are shown in Fig. 9, where the object was an oil drop running down over a microscope slide. Each interferogram was subject to rescaling and normalization and

Immersion oil was applied to a glass microscope slide and allowed to flow under the effect of gravity by tilting the slide slightly. The slide was put in front of one of the object windows of the system of Fig. 1. Fig. 10 shows the resulting unwrapped phase evolution of

The experimental set-up for a polarizing two-window phase-grating common-path interferometer has been described. This system is able to obtain four interferograms 90° phase-apart in only one shot. Therefore, it is suitable to carry out phase extraction using phase shifting techniques. Phase evolving in time can then be calculated and displayed. The system is considerably simpler than previous proposals to attain four interferograms with only one shot. In its present form, it is, however, best suited to relative small objects which do not introduce polarization changes. Because it works with interferograms placed relatively far from the optical axes, experimental results suggest that some method has to be introduced to compensate mainly for distortion, among other off-axis aberrations. This compensation could be optical (as a better design of the optical imaging system) or digital (fringe distortion compensation by inverse transformation). In the experiments, the use of is described retarding plates which are not quarter-wave plates. Although they can perform well enough in principle, it seems better to use quarter-wave plates because no additional variations of the interferogram amplitude arise. Also in this case, a simpler polarization filter array can be used taking advantage of the diffraction properties of a phase grating. Some special phase gratings design could optimize the

This system is able to obtain *n* = (*N*+1) interferograms with only one shot ( 16 *n* ). Tests

interferometer could be used for objects with no changes of polarization.

 *N* phase-shifts were presented, but other approaches using different phase-shifts could be attained using linear polarizers with their transmission axes at the proper angle before detection. The phase shifts of due to the grid spectra allows the use of a number of polarizing filters which is less than the number of interferograms, simplifying the filter array. Other configurations for the window positions which are different as the one reported in this communication can also be possible. The accuracy in measurements is the one typical of phase-shifting. Some trade-offs appear while placing several images over the same detector field, but for low frequencies interferograms (with respect to the inverse of the pixel spacing) the influence of these factors seems to be rather small if noticeable. The

92.989 . For symmetrical nine, 1

92.989 , each step being of 90°. For the case of

157.903 . In this case, each step has to be 45°. The

*<sup>n</sup>* can be

92.989 ,

 *<sup>n</sup>* are 1 0 ,

 0 , 2 

Considering the retarders at disposal, according to Eq. (10) it can be shown that

then, a filtering process prior to phase calculation through Eq. (13).

symmetrical seven, each step is of 60°, so it can be shown that

of 1 

2 

3 

 0 , <sup>2</sup> 

30.800 , 3

22.975 , 4

**6.1 Moving distributions** 

oil flow (Case N=4).

**7. Final remarks** 

interferometric system described.

with 2 / 

46.577 and 3

62.330 and 4

46.577 and 5


**4** 

*Singapore* 

**Phosphor-Based White Light Emitting Diode** 

Wee Keat Chong1, Xiang Li1 and Yeng Chai Soh2 *1Singapore Institute of Manufacturing Technology, A\*STAR* 

*2Nanyang Technological University* 

**(LED) for Vertical Scanning Interferometry (VSI)** 

Vertical scanning interferometry (VSI) is an established optical method for surface profile measurement by analyzing a series of interference patterns of low coherence light with known optical path difference among them. As white light is commonly used as low coherence light source, vertical scanning interferometry is also known as white light

Vertical scanning interferometry is most commonly used as surface profilometer which is considered as an enabling and supporting technology to other fields such as surface finishing, machining and material science. It is a non-contact three-dimensional surface measurement technique that provides accuracy up to nanometer level and measurement

Fig. 1 graphically illustrates the schematic diagram of vertical scanning interferometry in Michelson interferometer configuration; the light beam from the light source is split into two: one to reference surface and one to measurement surface, then these light beams reflect and interfere with each other. Interference pattern occurs when the optical path difference (OPD) between these two light beams is small, within the coherence length of the light source. The interference pattern is known as interferogram (as shown in Fig. 2), it is recorded by area-based photo-sensitive sensor such as CCD camera. Correlogram is the function of intensity response of each pixel against optical path difference, and it is further processed for height profile measurement. Fig. 3 graphically illustrates correlogram and

Apart from the light source, the hardware of vertical scanning interferometry has not changed much in the past one decade. As phosphor-based white light emitting diode (LED) promises greater power, longer lifetime, low heat dissipation and compactness, it is

The conventional light source for white light has a very broad and smooth spectrum, for example Fig. 4 shows the spectrum of Quartz Tungsten Halogen Lamps (model no 6315 from NewPort). However due to the spectral response of photo detector, the effective spectrum of conventional white light is considered as single Gaussian function in visible light spectrum. On the other hand, phosphor-based white LED consists of single color LED

**1. Introduction**

interferometry (WLI).

range up to a few hundred micrometers.

coherence peak function of vertical scanning interferometry.

replacing the conventional light source in vertical scanning interferometry.

for one shot phase-shifting interferometry, Opt. Express, Vol. 16, pp. 19330- 19341

Wyant J.C. (2004). Vibration insensitive interferometric optical testing, in Frontiers in Optics, OSA Technical Digest, OTuB2.

## **Phosphor-Based White Light Emitting Diode (LED) for Vertical Scanning Interferometry (VSI)**

Wee Keat Chong1, Xiang Li1 and Yeng Chai Soh2 *1Singapore Institute of Manufacturing Technology, A\*STAR 2Nanyang Technological University Singapore* 

## **1. Introduction**

80 Interferometry – Research and Applications in Science and Technology

Wyant J.C. (2004). Vibration insensitive interferometric optical testing, in Frontiers in Optics,

19341

OSA Technical Digest, OTuB2.

for one shot phase-shifting interferometry, Opt. Express, Vol. 16, pp. 19330-

Vertical scanning interferometry (VSI) is an established optical method for surface profile measurement by analyzing a series of interference patterns of low coherence light with known optical path difference among them. As white light is commonly used as low coherence light source, vertical scanning interferometry is also known as white light interferometry (WLI).

Vertical scanning interferometry is most commonly used as surface profilometer which is considered as an enabling and supporting technology to other fields such as surface finishing, machining and material science. It is a non-contact three-dimensional surface measurement technique that provides accuracy up to nanometer level and measurement range up to a few hundred micrometers.

Fig. 1 graphically illustrates the schematic diagram of vertical scanning interferometry in Michelson interferometer configuration; the light beam from the light source is split into two: one to reference surface and one to measurement surface, then these light beams reflect and interfere with each other. Interference pattern occurs when the optical path difference (OPD) between these two light beams is small, within the coherence length of the light source. The interference pattern is known as interferogram (as shown in Fig. 2), it is recorded by area-based photo-sensitive sensor such as CCD camera. Correlogram is the function of intensity response of each pixel against optical path difference, and it is further processed for height profile measurement. Fig. 3 graphically illustrates correlogram and coherence peak function of vertical scanning interferometry.

Apart from the light source, the hardware of vertical scanning interferometry has not changed much in the past one decade. As phosphor-based white light emitting diode (LED) promises greater power, longer lifetime, low heat dissipation and compactness, it is replacing the conventional light source in vertical scanning interferometry.

The conventional light source for white light has a very broad and smooth spectrum, for example Fig. 4 shows the spectrum of Quartz Tungsten Halogen Lamps (model no 6315 from NewPort). However due to the spectral response of photo detector, the effective spectrum of conventional white light is considered as single Gaussian function in visible light spectrum. On the other hand, phosphor-based white LED consists of single color LED

Phosphor-Based White Light Emitting Diode (LED) for Vertical Scanning Interferometry (VSI) 83

Fig. 3. Example of correlogram and corresponding fringe contrast function (also known as

(normally blue) and phosphor of different color (normally yellow) to produce white light, so there are two peaks in its spectrum. Fig. 5 compares the effective intensity spectrum of conventional white light and phosphor-based white LED, the major difference between

Fig. 4. Spectral irradiance at 0.5 m from the 6315 1000W QTH Lamp3 (by NewPort).

based white LED.

As most prior works (Guo, Zhao, & Chen, 2007; Gurov, Ermolaeva, & Zakharov, 2004; Mingzhou, Chenggen, Cho Jui, Ivan, & Shihua, 2005; Pavli?ek & Soubusta, 2004) assume the use of conventional white light, the effects of phosphor-based white LED on vertical scanning interferometer is the focus on this chapter. Other than that, this chapter also covers a computationally efficient signal modelling method for vertical scanning interferometry and method to improve performance of vertical scanning interferometry with phosphor-

coherence peak function) in vertical scanning interferometry.

these two light sources is the number of peaks in spectral domain.

Fig. 1. Schematic diagram of vertical scanning interferometry in Michelson interferometer configuration.

Fig. 2. Example of interferogram of vertical scanning interferometry: Fringe on a tilted flat surface captured using a CCD camera.

Fig. 1. Schematic diagram of vertical scanning interferometry in Michelson interferometer

Fig. 2. Example of interferogram of vertical scanning interferometry: Fringe on a tilted flat

surface captured using a CCD camera.

configuration.

Fig. 3. Example of correlogram and corresponding fringe contrast function (also known as coherence peak function) in vertical scanning interferometry.

(normally blue) and phosphor of different color (normally yellow) to produce white light, so there are two peaks in its spectrum. Fig. 5 compares the effective intensity spectrum of conventional white light and phosphor-based white LED, the major difference between these two light sources is the number of peaks in spectral domain.

Fig. 4. Spectral irradiance at 0.5 m from the 6315 1000W QTH Lamp3 (by NewPort).

As most prior works (Guo, Zhao, & Chen, 2007; Gurov, Ermolaeva, & Zakharov, 2004; Mingzhou, Chenggen, Cho Jui, Ivan, & Shihua, 2005; Pavli?ek & Soubusta, 2004) assume the use of conventional white light, the effects of phosphor-based white LED on vertical scanning interferometer is the focus on this chapter. Other than that, this chapter also covers a computationally efficient signal modelling method for vertical scanning interferometry and method to improve performance of vertical scanning interferometry with phosphorbased white LED.

Phosphor-Based White Light Emitting Diode (LED) for Vertical Scanning Interferometry (VSI) 85

model parameters and physical setting of the system, for example, the term *F(k)* in Equation (1) is equal to the intensity spectrum which can be measured by optical spectrum analyzer. The drawback of this generalized model is intensive computational load: as the intensity spectrum of light source is either difficult to be expressed in symbolic form or modelled as Gaussian, Equation (1) has to be solved by numerical integration which is a computationally intensive and time consuming process. Therefore there is a simplified version of the

2

2

This simplified model is based on assumptions that the numerical aperture of objective is small and the intensity spectrum of light, *F(k)* in Equation (1), is single Gaussian function. The advantage of such simplification is computational efficiency, and the drawback is poor correlation between the model parameters and theoretical parameters, for example, the parameter *σ* in Equation (2) does not have physical meaning (such as the spectrum of light source). As such, the parameters of the simplified model have to be determined empirically, and it is impossible to simulate the correlogram based on specified spectrum and/or

To reduce computational load, de Groot and Lega (2004) proposed simplification in frequency domain: Equation (1) is first transformed into frequency domain, followed by simplification, applying numerical integration and lastly inverse Fourier transform back to its original domain. This approach is 200 times faster than direct numerical integration of the generalized model, but it is still a time consuming process due to (1) the use of Fourier

Before looking into the computationally efficient signal modelling, let's look into the cause of intensive computational load of using Equation (1) – numerical integration. Numerical integration is a process calculating the approximated value of a definite integral which can

*b a fa fb b a f x dx fa k*

1

1

 (3)

*n n*

and inverse Fourier transform and (2) it still requires numerical integration process.

( ) () () ( ) <sup>2</sup> *b n*

*a k*

( ) 2( ) ( ) ( ) ( )exp cos *dc amplitude*

*zz zz Iz I z I z*

0 0

*m*

(2)

generalized model and it can be expressed as follows:

*z* is defocus position (related to optical path difference),

*z0* is related to the profile of sample surface, *Idc* is constant value and not related to interference,

*Iamplitude* is amplitude of interference signal, *λm* is equivalent wavelength of light, *σ* is related to coherent length of light and

**2.1 Computationally efficient signal modeling** 

where

*Ø* is phase offset.

numerical aperture value.

be expressed as following:

where

Fig. 5. Comparing effective intensity spectrum of conventional white light and phosphorbased white LED (LXHL-LW6C by LumiLEDs).

## **2. Theory**

Besides optical path difference, the correlogram is affected by the following factors: (1) spectrum of light source (2) numerical aperture of objective (3) optical transfer function of imaging system (4) reflectivity of reference and sample surface (5) phase change at reference and sample surface. The intensity response can be formulated as following:

$$I\_{\text{interference}}(z) = C\_1 \int\_{bandwidth} \int\_0^{\theta\_0} \{k^2 \times \cos[2k(z - z\_0)\cos\theta + \phi] \} $$

$$\times \sin\theta \cos\theta d\theta \text{[}F(k)dk\tag{1}$$

where

*C1* is a constant,

*z* is a independent variable which corresponds to height change by piezoelectric positioner, *z0* is the height which corresponds to surface profile,

*k* is angular wave number (*k=2π/λ*),

*sin θ0* is numerical aperture (NA) of objective lens,

*Ø* is the phase offset and *F(k)* is the intensity spectrum of light source.

Detail on derivation and modeling of the intensity response can be found in literature by Chim and Kino (1990), Kino and Chim (1990), de Groot and Lega (2004) and Sheppard and Larkin (1995).

Equation (1) is a generalized model that can simulate the effects of changing spectrum of light source and numerical aperture of objective lens. There is strong correlation between model parameters and physical setting of the system, for example, the term *F(k)* in Equation (1) is equal to the intensity spectrum which can be measured by optical spectrum analyzer. The drawback of this generalized model is intensive computational load: as the intensity spectrum of light source is either difficult to be expressed in symbolic form or modelled as Gaussian, Equation (1) has to be solved by numerical integration which is a computationally intensive and time consuming process. Therefore there is a simplified version of the generalized model and it can be expressed as follows:

$$I(z) = I\_{dc}(z) + I\_{\text{supidual}}(z) \exp\left(\frac{-(z - z\_0)^2}{\sigma^2}\right) \cos\left(\frac{2\pi(z - z\_0)}{\lambda\_m} + \phi\right) \tag{2}$$

where

84 Interferometry – Research and Applications in Science and Technology

Fig. 5. Comparing effective intensity spectrum of conventional white light and phosphor-

Besides optical path difference, the correlogram is affected by the following factors: (1) spectrum of light source (2) numerical aperture of objective (3) optical transfer function of imaging system (4) reflectivity of reference and sample surface (5) phase change at reference

> 0 2 interference 1 <sup>0</sup> <sup>0</sup> ( ) { cos[2 ( )cos ]

> > sin cos } ( )

 

*z* is a independent variable which corresponds to height change by piezoelectric positioner,

Detail on derivation and modeling of the intensity response can be found in literature by Chim and Kino (1990), Kino and Chim (1990), de Groot and Lega (2004) and Sheppard and

Equation (1) is a generalized model that can simulate the effects of changing spectrum of light source and numerical aperture of objective lens. There is strong correlation between

 

*d F k dk* (1)

and sample surface. The intensity response can be formulated as following:

*bandwidth I z C k kz z* 

based white LED (LXHL-LW6C by LumiLEDs).

*z0* is the height which corresponds to surface profile,

*Ø* is the phase offset and *F(k)* is the intensity spectrum of light source.

*sin θ0* is numerical aperture (NA) of objective lens,

*k* is angular wave number (*k=2π/λ*),

**2. Theory** 

where

*C1* is a constant,

Larkin (1995).

*z* is defocus position (related to optical path difference), *z0* is related to the profile of sample surface, *Idc* is constant value and not related to interference, *Iamplitude* is amplitude of interference signal, *λm* is equivalent wavelength of light, *σ* is related to coherent length of light and *Ø* is phase offset.

This simplified model is based on assumptions that the numerical aperture of objective is small and the intensity spectrum of light, *F(k)* in Equation (1), is single Gaussian function. The advantage of such simplification is computational efficiency, and the drawback is poor correlation between the model parameters and theoretical parameters, for example, the parameter *σ* in Equation (2) does not have physical meaning (such as the spectrum of light source). As such, the parameters of the simplified model have to be determined empirically, and it is impossible to simulate the correlogram based on specified spectrum and/or numerical aperture value.

To reduce computational load, de Groot and Lega (2004) proposed simplification in frequency domain: Equation (1) is first transformed into frequency domain, followed by simplification, applying numerical integration and lastly inverse Fourier transform back to its original domain. This approach is 200 times faster than direct numerical integration of the generalized model, but it is still a time consuming process due to (1) the use of Fourier and inverse Fourier transform and (2) it still requires numerical integration process.

## **2.1 Computationally efficient signal modeling**

Before looking into the computationally efficient signal modelling, let's look into the cause of intensive computational load of using Equation (1) – numerical integration. Numerical integration is a process calculating the approximated value of a definite integral which can be expressed as following:

$$\int\_{a}^{b} f(\mathbf{x})d\mathbf{x} \approx \frac{(b-a)}{n} \left( \frac{f(a) + f(b)}{2} - \sum\_{k=1}^{n-1} f\left(a + k\frac{b-a}{n}\right) \right) \tag{3}$$

where

*n* is the number of interval

The accuracy of numerical integration process is proportional to the parameter *n* in Equation (3). However, as *n* goes up, the computational load increases.

There are two cases which numerical integration is required: (1) the integrand may be known for certain region only and/or (2) the anti-derivative of the integrand does not exist. For signal simulation of vertical scanning interferometry, the intensity spectrum, *F(k)* in Equation (1), it is either sampled by spectrum analyzer or modelled as Gaussian (which is not an explicit integral). As such, the generalized model, i.e. Equation (1), has to be solved by numerical integration which is computationally intensive.

Chong et al. (2010a) proposed to remove the numerical integration process by representing a single Gaussian function as a sum of two piecewise cosine functions, which can be expressed as following:

$$a\exp\left(\frac{-\left(\mathbf{x}-\mathbf{x}\_m\right)^2}{\sigma^2}\right) = \begin{cases} a\_1 \cos\left(\frac{2\pi(\mathbf{x}-\mathbf{x}\_m)}{b\_1}\right) + a\_2 \cos\left(\frac{2\pi(\mathbf{x}-\mathbf{x}\_m)}{b\_2}\right) & \text{for } (\mathbf{x}\_m \cdot \mathbf{c}) \le \mathbf{x} \le (\mathbf{x}\_m + \mathbf{c})\\ 0 & \text{else} \end{cases} \tag{4}$$

The transformation from single Gaussian (with parameters of a, xm and σ ) to a sum of two piecewise cosine functions (with parameters of a1,a2,b1,b2,xm and c) is modeled as a linear transformation and solved by trust region approach (for generating data) and linear regression, followed by minimizing error of fitting with respect to *Crange*. As a result, the unknown in Equation (4) can now be expressed as following:

$$\begin{cases} a\_1 = 0.7888a \\ a\_2 = 0.2049a \\ \chi\_m = \chi\_m \\ b\_1 = 7.6777\sigma - 0.0078 \\ b\_2 = -2.4769\sigma + 0.0024 \\ c = 0.852 \min(|b\_1|, |b\_2|) \end{cases} \tag{5}$$

Phosphor-Based White Light Emitting Diode (LED) for Vertical Scanning Interferometry (VSI) 87

0 0 0 0 0 0 0 0

*C g zz k b g zz k b g zz k b g zz k b*

( 2 2 2) 2 2 2 2 cos sin ( ) <sup>4</sup>

*<sup>D</sup> b k k bUk b k k bUk bU bU k <sup>b</sup> bU*

*m m*

*b k k bUk b k k bUk bU k <sup>b</sup> bU b b*

2 22 2 22 2( ) cos sin ( )

 

Chong et al.'s model (2010a) reduces the computational time by 256800 times compared to conventional direct numerical integration on Equation (1), and it is 2784 times faster than de

Phosphor-based white LED consists of single color LED (normally blue) and phosphor of different color (normally yellow) to produce white light, so there are two peaks in its

In general, the angular wave number of blue and yellow light are 14.37 rad/nm (438nm) and 11.25 rad/nm (558nm) respectively, the spread of blue and yellow light in spectrum domain are 0.4941 rad/nm and 1.439 rad/nm; these values vary slightly across

Compared to conventional white light (with reference to Fig. 5), the intensity spectrum of

*f k BYratio e e*

*kblue* and *kyellow* indicate the peak angular wave number of blue and yellow light

*blue* and *σyellow* indicate the spread of blue and yellow light in spectrum domain

*m m*

*b b*

2 2 00 00

[2 cos sin 2 cos 8 44

*<sup>b</sup> C k <sup>z</sup> <sup>z</sup> <sup>k</sup> <sup>z</sup> <sup>z</sup>*

cos 2 cos cos ]

 

spectrum and the intensity spectrum can be expressed as follows:

( )

p*/λ*)

phosphor-based white LED is significantly different.

0 0

*zz z z*

*k k*

 

cos 2

*<sup>m</sup> <sup>k</sup>*

*ul*

*a*

2 2 sin

Groot and Lega (2004) 's approach.

**2.2 Phosphor-based white LED** 

*k* is angular wave number (=*2*

manufacturers/model.

18 4 4 cos

*D zz z z*

*bk k <sup>b</sup>*

2 2 0 0

*m*

 

*ll*

*k*

0

*gzz kb*

*U zz*

where

σ

*where*

0

Let ( , , , , )

( , , , , ) ( , ,0, , ) ( , , , , ) ( , ,0, , )

*ul ul ll ll*

*b b*

 

*k zz dk*

 

 

2 2 <sup>2</sup> <sup>2</sup> ( ) sin ( )

*bUk b k k bUk <sup>m</sup> bU bU*

<sup>2</sup> <sup>2</sup>

(7)

*yellow blue blue yellow k k k k*

 

  (6)

2

2

 

 

 

Next, Equation (1) is derived to elementary form as follows:

$$\begin{split} I\_{\text{inferference}}(\mathbf{z}) &= \mathbb{C} \int\_{\text{bandrwidth}} k^2 F(k) \left\{ \cos[2k(z - z\_0)\cos\theta + \theta] \sin\theta \cos\theta d\theta d\theta dk \\ &= \mathbb{C} \int\_{\text{bandrwidth}} k^2 F(k) \left[ -\frac{2kz\cos\theta - 2kz\_0\cos\theta)\sin(2kz\cos\theta - \theta)}{4k^2 z^2 - 8k^2 z\_0 z + 4k^2 z\_0^2} \right]\_0^{\theta\_0} dk \\ &= \mathbb{C} \int\_{\text{ds}} \frac{k^2 - k\_{\text{m}}}{8z\_0 z - 4z^2 - 4z\_0^2} [(2kz\cos\theta\_0 - 2kz\_0\cos\theta\_0)\sin(2kz\cos\theta\_0 - \theta) - 2kz\_0\cos\theta\_0] dk}{8z\_0 z - 4z^2 - 4z\_0^2} \end{split}$$
 
$$\begin{split} &= \mathbb{C} \int\_{k\_l} \frac{k^2 - k\_{\text{m}}}{8z\_0 z - 4z^2 - 4z\_0^2} [(2kz\cos\theta\_0 - 2kz\_0\cos\theta\_0)\sin(2kz\cos\theta\_0 + \theta) - 2kz\_0\cos\theta\_0] dk} \\ &= \frac{2kz\_0\cos\theta\_0 + \phi}{(2kz - 2kz\_0)\sin(2kz - 2kz\_0 + \phi) - \cos(2kz - 2kz\_0 + \phi)] dk} \end{split}$$

$$=\mathbb{C}\int\_{k\_{\rm{ul}}}^{k\_{\rm{ul}}} \frac{a\cos\left(2\pi\frac{k-k\_{m}}{b}\right)}{8z\_{0}z-4z^{2}-4z\_{0}^{2}}[2k\cos\theta\_{0}(z-z\_{0})\sin\left(2k\cos\theta\_{0}(z-z\_{0})+\phi\right)+$$

$$\cos\left(2k\cos\theta\_{0}(z-z\_{0})+\phi\right)-\cos\phi\,\mathrm{d}k}\tag{6}$$

$$=\mathbb{C}\left[\left(g(z,z\_{0},\theta\_{0},k\_{\rm{ul}},b)-g(z,z\_{0},0,k\_{\rm{ul}},b)\right)-\left(g(z,z\_{0},\theta\_{0},k\_{\rm{ul}},b)-g(z,z\_{0},0,k\_{\rm{ul}},b)\right)\right]$$

 0 2 2 Let ( , , , , ) ( 2 2 2) 2 2 2 2 cos sin ( ) <sup>4</sup> 2 22 2 22 2( ) cos sin ( ) 2 2 sin *m m m m m gzz kb <sup>D</sup> b k k bUk b k k bUk bU bU k <sup>b</sup> bU b b b k k bUk b k k bUk bU k <sup>b</sup> bU b b bk k <sup>b</sup>* 2 2 0 0 2 2 <sup>2</sup> <sup>2</sup> ( ) sin ( ) 18 4 4 *bUk b k k bUk <sup>m</sup> bU bU b b where D zz z z* 

$$\boldsymbol{\omega} \boldsymbol{I} = \left(\boldsymbol{z} - \boldsymbol{z}\_0\right) \cos \theta$$

The accuracy of numerical integration process is proportional to the parameter *n* in Equation

There are two cases which numerical integration is required: (1) the integrand may be known for certain region only and/or (2) the anti-derivative of the integrand does not exist. For signal simulation of vertical scanning interferometry, the intensity spectrum, *F(k)* in Equation (1), it is either sampled by spectrum analyzer or modelled as Gaussian (which is not an explicit integral). As such, the generalized model, i.e. Equation (1), has to be solved

Chong et al. (2010a) proposed to remove the numerical integration process by representing a single Gaussian function as a sum of two piecewise cosine functions, which can be

> *m m <sup>m</sup> m m x x x x x x aa x c x x c*

The transformation from single Gaussian (with parameters of a, xm and σ ) to a sum of two piecewise cosine functions (with parameters of a1,a2,b1,b2,xm and c) is modeled as a linear transformation and solved by trust region approach (for generating data) and linear regression, followed by minimizing error of fitting with respect to *Crange*. As a result, the

 

1 2

22 2 22 0 0

 

 

*kz kzz kz*

00 0

*kz kz kz kz kz kz dk*

2 2 sin(2 2 ) cos(2 2 )]

 

2 cos ) cos(2 cos 2 cos ) ( ) 48 4

2( ) 2( ) cos + cos for ( - ) ( )

(4)

(5)

 

 

  0

0

0 else

(3). However, as *n* goes up, the computational load increases.

by numerical integration which is computationally intensive.

*a b b* 

unknown in Equation (4) can now be expressed as following:

Next, Equation (1) is derived to elementary form as follows:

2 interference 0

c

os 2

*<sup>m</sup> <sup>k</sup>*

*a C*

*ul*

*ll*

*k*

*bandwidth*

*bandwidth*

1 2 2 1 2

> 1 2

  0.7888 0.2049

*a a a a*

7.6777 0.0078 2.4769 0.0024 0.852min( , )

*c bb*

0 2 0 0

0 0 0 00

2 cos ) cos(2 cos 2 cos )

[ 2 cos 2 cos sin(2 cos 8 44

*<sup>b</sup> kz kz kz zz z z*

*kz kz kz*

2 2 0 00 0

 

 

*kz kz kz kz kz kz C kFk dk*

2 cos 2 cos sin(2 cos

 

1 2

0

0 0

*k k*

0

*m m*

( ) ( ) cos[2 ( )cos ]sin cos

*I z C kFk kz z d dk*

*x x b b*

*n* is the number of interval

expressed as following:

exp

<sup>2</sup>

Chong et al.'s model (2010a) reduces the computational time by 256800 times compared to conventional direct numerical integration on Equation (1), and it is 2784 times faster than de Groot and Lega (2004) 's approach.

## **2.2 Phosphor-based white LED**

Phosphor-based white LED consists of single color LED (normally blue) and phosphor of different color (normally yellow) to produce white light, so there are two peaks in its spectrum and the intensity spectrum can be expressed as follows:

$$f(k) = BYratio \times e^{-\left(\frac{k - k\_{blue}}{\sigma\_{hhw}}\right)^2} + e^{-\left(\frac{k - k\_{yellow}}{\sigma\_{yellow}}\right)^2} \tag{7}$$

where

*k* is angular wave number (=*2*p*/λ*) *kblue* and *kyellow* indicate the peak angular wave number of blue and yellow light σ*blue* and *σyellow* indicate the spread of blue and yellow light in spectrum domain

In general, the angular wave number of blue and yellow light are 14.37 rad/nm (438nm) and 11.25 rad/nm (558nm) respectively, the spread of blue and yellow light in spectrum domain are 0.4941 rad/nm and 1.439 rad/nm; these values vary slightly across manufacturers/model.

Compared to conventional white light (with reference to Fig. 5), the intensity spectrum of phosphor-based white LED is significantly different.

Phosphor-Based White Light Emitting Diode (LED) for Vertical Scanning Interferometry (VSI) 89

Based on the intensity spectrum of LxHL-LW6C and numerical aperture of 0.4, the corresponding correlogram is simulated using the computational efficient signal modelling by Chong et al and shown in Fig. 7 (b). The distinctive feature highlighted in Fig. 7 (b) is the

result of having two peaks in the spectrum of phosphor-based white LED.

Fig. 7. Comparing spectrum and correlogram of phosphor-based white LED and

simulated correlogram based on spectrum of (c).

(as shown in Fig. 7(c) and (d)).

LW6C was used.

conventional white light: (a) spectrum of phosphor-based LED, LXHL-LW6C; (b) simulated correlogram based on spectrum of (a); (c) effective spectrum of conventional white light; (d)

As shown in Fig. 7, the correlogram of vertical scanning interferometry using phosphorbased white LED is significantly different from that using light source of Gaussian spectrum

Next, an experimental verification is conducted to verify the simulation result. The experiment configuration is as follows: a 40x Nikon mirau-based interferometric objective with numerical aperture of 0.4 and phosphor-based white LED from LumiLEDs LXHL-

**3.1 Effects on correlogram** 

## **3. Effects of phosphor-based white LED on vertical scanning interferometry**

As mentioned earlier, the intensity spectrum of phosphor-based white LED is significantly different from conventional lighting which most prior arts adopted. In this section, the effects of phosphor-based white LED on vertical scanning interferometry in correlogram and reconstructed surface profile level are investigated by simulation, followed by experimental verification.

Fig. 6. Intensity spectrum of phosphor-based LED, LXHL-LW6C by LumiLEDs and comparison between its presentations by three Gaussian functions and Chong et al.'s method.

To study the effects of phosphor-based white LED, a commercial off the shelf phosphorbased white LED, LXHL-LW6C by LumiLEDs is selected, and the numerical aperture of objective is assumed to 0.4 which is a typical value for 20x objectives. This configuration is adopted for both simulation and experimental verification.

To simulate the correlogram of LXHL-LW6C by LumiLED, the intensity spectrum (as shown in Fig. 6) is first fitted to three Gaussian functions by non-linear least square fitting method; next each Gaussian terms is replaced with a sum of two piecewise cosine functions according Equation (4) and Equation (5); by Equation (6), contribution of each cosine term is calculated; lastly, the resultant intensity response is the sum of contribution by six cosine terms.

Fig. 6 shows that Chong et al.'s method represents the intensity spectrum of LXHL-LW6C by LumiLED well compared to representation by three Gaussian functions and the original intensity spectrum.

## **3.1 Effects on correlogram**

88 Interferometry – Research and Applications in Science and Technology

**3. Effects of phosphor-based white LED on vertical scanning interferometry**  As mentioned earlier, the intensity spectrum of phosphor-based white LED is significantly different from conventional lighting which most prior arts adopted. In this section, the effects of phosphor-based white LED on vertical scanning interferometry in correlogram and reconstructed surface profile level are investigated by simulation, followed by

Fig. 6. Intensity spectrum of phosphor-based LED, LXHL-LW6C by LumiLEDs and comparison between its presentations by three Gaussian functions and Chong et al.'s

adopted for both simulation and experimental verification.

To study the effects of phosphor-based white LED, a commercial off the shelf phosphorbased white LED, LXHL-LW6C by LumiLEDs is selected, and the numerical aperture of objective is assumed to 0.4 which is a typical value for 20x objectives. This configuration is

To simulate the correlogram of LXHL-LW6C by LumiLED, the intensity spectrum (as shown in Fig. 6) is first fitted to three Gaussian functions by non-linear least square fitting method; next each Gaussian terms is replaced with a sum of two piecewise cosine functions according Equation (4) and Equation (5); by Equation (6), contribution of each cosine term is calculated; lastly, the resultant intensity response is the sum of contribution by six cosine

Fig. 6 shows that Chong et al.'s method represents the intensity spectrum of LXHL-LW6C by LumiLED well compared to representation by three Gaussian functions and the original

experimental verification.

method.

terms.

intensity spectrum.

Based on the intensity spectrum of LxHL-LW6C and numerical aperture of 0.4, the corresponding correlogram is simulated using the computational efficient signal modelling by Chong et al and shown in Fig. 7 (b). The distinctive feature highlighted in Fig. 7 (b) is the result of having two peaks in the spectrum of phosphor-based white LED.

Fig. 7. Comparing spectrum and correlogram of phosphor-based white LED and conventional white light: (a) spectrum of phosphor-based LED, LXHL-LW6C; (b) simulated correlogram based on spectrum of (a); (c) effective spectrum of conventional white light; (d) simulated correlogram based on spectrum of (c).

As shown in Fig. 7, the correlogram of vertical scanning interferometry using phosphorbased white LED is significantly different from that using light source of Gaussian spectrum (as shown in Fig. 7(c) and (d)).

Next, an experimental verification is conducted to verify the simulation result. The experiment configuration is as follows: a 40x Nikon mirau-based interferometric objective with numerical aperture of 0.4 and phosphor-based white LED from LumiLEDs LXHL-LW6C was used.

Phosphor-Based White Light Emitting Diode (LED) for Vertical Scanning Interferometry (VSI) 91

respectively. Among these three algorithms, only FDA is phase-based approach; the other two are fringe contrast based approach. Among these two methods, the major difference is that Gaussian fitting method by Mingzhou et al assumes the fringe contrast function is a

Mingzhou et al.'s method recovers height by first finding the envelope of the correlogram, followed by Gaussian fitting. Ai and Novak proposed that the centroid of the correlogram corresponds to the maximum of fringe. As phase-based approach, Groot and Deck's method breaks down the white light into multiple single wavelength components and applies phase

Simulation is used for investigating the effects of phosphor based white LED on reconstructed height profile. For comparison purpose, two sets of data are simulated: one is based on light with Gaussian spectrum; another one is based on phosphor-based white LED, LXHL-LW6C by LumiLEDs. A line profile of 1um step height is selected, the line profile consists of 256 surface points and each surface point has a corresponding intensity response. The sampling interval of the intensity response is 50nm, and each intensity response is corrupted by Gaussian white noise (zero mean, variance of 0.05). Next we reconstructed these two sets of data using these three construction algorithms mentioned earlier, and the reconstructed profiles are shown in Fig. 9 and Fig. 10. The repeatability (in term of standard

Gaussian signal while the centroid approach does not.

signal analysis similar to phase shifting interferometry.

analysis by Groot and Deck

domain analysis by Groot and Deck

deviation) and accuracy of reconstructed profiles are further analyzed.

Fig. 9. For phosphor-based LED set, 1um step height reconstructed using (a) Gaussian Fitting by Mingzhou et al. (b) Centroid approach by Ai and Novak (c) Frequency domain

Fig. 10. For light source of Gaussian spectrum, 1um step height reconstructed using (a) Gaussian Fitting by Mingzhou et al. (b) Centroid approach by Ai and Novak (c) Frequency

Fig. 8. Graphical comparison between computationally efficient signal modeling by Chong et al., computationally intensive model and experimental data.

Fig. 8 graphically compares the experimental data with simulation data, it shows that the distinctive feature due to the use of phosphor-based white LED (highlighted in Fig. 7(b) ) is consistent with the experimental data.

By experiment and simulation, it is shown that the intensity response (also known as correlogram) of vertical scanning interferometry using phosphor-based white LED significantly different from light source of Gaussian spectrum. The fringe contrast function (envelope function of correlogram) is not longer a single Gaussian function (Chong, Li, & Wijesoma, 2010b).

## **3.2 Effects of reconstructed height profile**

As correlogram alone does not provide information on height, a reconstruction algorithm is required to transform correlogram into height information. There are two categories for surface height profile reconstruction algorithm: (1) fringe contrast based approach and (2) phase-based approach. Fringe contrast based approach finds the maximum of fringe contrast function which corresponds to the height profile; while phase-based approach transforms intensity response into frequency domain, followed by phase signal analysis.

This section investigates the effects of phosphor-based LED on three reconstruction algorithms proposed by Gaussian fitting method (Mingzhou et al., 2005), Centroid approach (Ai & Novak, n.d.), and Frequency domain analysis (FDA) (P. J. de Groot & Deck, 1994)

Fig. 8. Graphical comparison between computationally efficient signal modeling by Chong

Fig. 8 graphically compares the experimental data with simulation data, it shows that the distinctive feature due to the use of phosphor-based white LED (highlighted in Fig. 7(b) ) is

By experiment and simulation, it is shown that the intensity response (also known as correlogram) of vertical scanning interferometry using phosphor-based white LED significantly different from light source of Gaussian spectrum. The fringe contrast function (envelope function of correlogram) is not longer a single Gaussian function (Chong, Li, &

As correlogram alone does not provide information on height, a reconstruction algorithm is required to transform correlogram into height information. There are two categories for surface height profile reconstruction algorithm: (1) fringe contrast based approach and (2) phase-based approach. Fringe contrast based approach finds the maximum of fringe contrast function which corresponds to the height profile; while phase-based approach transforms intensity response into frequency domain, followed by phase signal analysis.

This section investigates the effects of phosphor-based LED on three reconstruction algorithms proposed by Gaussian fitting method (Mingzhou et al., 2005), Centroid approach (Ai & Novak, n.d.), and Frequency domain analysis (FDA) (P. J. de Groot & Deck, 1994)

et al., computationally intensive model and experimental data.

consistent with the experimental data.

**3.2 Effects of reconstructed height profile** 

Wijesoma, 2010b).

respectively. Among these three algorithms, only FDA is phase-based approach; the other two are fringe contrast based approach. Among these two methods, the major difference is that Gaussian fitting method by Mingzhou et al assumes the fringe contrast function is a Gaussian signal while the centroid approach does not.

Mingzhou et al.'s method recovers height by first finding the envelope of the correlogram, followed by Gaussian fitting. Ai and Novak proposed that the centroid of the correlogram corresponds to the maximum of fringe. As phase-based approach, Groot and Deck's method breaks down the white light into multiple single wavelength components and applies phase signal analysis similar to phase shifting interferometry.

Simulation is used for investigating the effects of phosphor based white LED on reconstructed height profile. For comparison purpose, two sets of data are simulated: one is based on light with Gaussian spectrum; another one is based on phosphor-based white LED, LXHL-LW6C by LumiLEDs. A line profile of 1um step height is selected, the line profile consists of 256 surface points and each surface point has a corresponding intensity response. The sampling interval of the intensity response is 50nm, and each intensity response is corrupted by Gaussian white noise (zero mean, variance of 0.05). Next we reconstructed these two sets of data using these three construction algorithms mentioned earlier, and the reconstructed profiles are shown in Fig. 9 and Fig. 10. The repeatability (in term of standard deviation) and accuracy of reconstructed profiles are further analyzed.

Fig. 9. For phosphor-based LED set, 1um step height reconstructed using (a) Gaussian Fitting by Mingzhou et al. (b) Centroid approach by Ai and Novak (c) Frequency domain analysis by Groot and Deck

Fig. 10. For light source of Gaussian spectrum, 1um step height reconstructed using (a) Gaussian Fitting by Mingzhou et al. (b) Centroid approach by Ai and Novak (c) Frequency domain analysis by Groot and Deck

Phosphor-Based White Light Emitting Diode (LED) for Vertical Scanning Interferometry (VSI) 93

Fig. 12. Comparing the measurement repeatability of different algorithms between different light sources: measurement repeatability is inversely proportional to the standard deviation

As identified earlier, phosphor-based white LED degrades the performance of two reconstruction algorithms (Gaussian fitting method and Frequency domain analysis approach) as it breaks the assumption adopted by these reconstruction algorithms. These undesired effects of phosphor-based white LED can be avoided by either redesigning reconstruction algorithm that does not assume the distribution of intensity spectrum or setting constraint on the input to existing reconstruction algorithm such that the assumption is valid. Redesigning a reconstruction algorithm is beyond the scope of this chapter, so a constraint is applied to make assumption required by Gaussian fitting and frequency

For Gaussian fitting method, the two valleys (distinctive features highlighted in Fig. 7 (b)) make the fringe contrast function has 3 peaks and can not be modelled as single Gaussian function. However as shown in Fig. 13, the envelope of correlogram between two valleys (distinctive features highlighted in Fig. 7 (b)) can be reasonably modelled as single Gaussian, setting a constraint selecting only these data for single Gaussian fitting would fulfil the assumption of the original Gaussian fitting approach. So, instead of fitting the whole correlogram to a Gaussian, Gaussian fitting approach is modified such that it fit subset of

For frequency domain analysis approach, the required assumption is that there should be only one peak in the spatial frequency ranges from 20rad/um to 30 rad/um which

**4. Modification and improvement for phosphor-based white LED on VSI** 

of perfectly flat surface, ideal value is zero.

domain analysis approach valid.

correlogram to a Gaussian.

Fig. 11. Comparing measurement accuracy of different algorithms between different light sources, ideal value is 1um.

Fig. 11 shows the comparison of accuracy among different algorithms using different lighting. As the accuracy of individual algorithm is highly dependent on the complexity of the algorithm, it is beyond the scope of this investigation. However it is noticeable that the accuracy of all three algorithms decreases as the light is switched to phosphor-based white LED.

Fig. 12 shows the comparison of measurement repeatability among different algorithms using different lighting, and it is clear that the use phosphor-based LED decreases the repeatability of measurement (standard deviation is inversely proportional to repeatability) at different scale: Gaussian fitting method by Mingzhou et al. suffers most, followed by frequency domain analysis by Groot and Deck, lastly the centroid approach by Ai and Novak. This observation can be explained as follows:


Fig. 11. Comparing measurement accuracy of different algorithms between different light

Fig. 11 shows the comparison of accuracy among different algorithms using different lighting. As the accuracy of individual algorithm is highly dependent on the complexity of the algorithm, it is beyond the scope of this investigation. However it is noticeable that the accuracy of all three algorithms decreases as the light is switched to phosphor-based white

Fig. 12 shows the comparison of measurement repeatability among different algorithms using different lighting, and it is clear that the use phosphor-based LED decreases the repeatability of measurement (standard deviation is inversely proportional to repeatability) at different scale: Gaussian fitting method by Mingzhou et al. suffers most, followed by frequency domain analysis by Groot and Deck, lastly the centroid approach by Ai and

 For the centroid approach by Ai and Novak: This method does not make assumption on the fringe contrast function, so the change in fringe contrast function has relatively little

 For Gaussian fitting by Mingzhou et al.: This method assumes that the fringe contrast function is a Gaussian function, so it suffers the worst repeatability. As the assumption on the fringe contrast function is not valid, the fitting process is unable to produce good

 For the frequency domain analysis by Groot and Deck: Although this method processes the correlogram in the frequency domain only, the change in fringe contrast function does affect the amount and the quality of information selected for frequency domain

analysis. This effect leads to improvement discussed in next session.

sources, ideal value is 1um.

Novak. This observation can be explained as follows:

effect on its reconstruction.

result.

LED.

Fig. 12. Comparing the measurement repeatability of different algorithms between different light sources: measurement repeatability is inversely proportional to the standard deviation of perfectly flat surface, ideal value is zero.

## **4. Modification and improvement for phosphor-based white LED on VSI**

As identified earlier, phosphor-based white LED degrades the performance of two reconstruction algorithms (Gaussian fitting method and Frequency domain analysis approach) as it breaks the assumption adopted by these reconstruction algorithms. These undesired effects of phosphor-based white LED can be avoided by either redesigning reconstruction algorithm that does not assume the distribution of intensity spectrum or setting constraint on the input to existing reconstruction algorithm such that the assumption is valid. Redesigning a reconstruction algorithm is beyond the scope of this chapter, so a constraint is applied to make assumption required by Gaussian fitting and frequency domain analysis approach valid.

For Gaussian fitting method, the two valleys (distinctive features highlighted in Fig. 7 (b)) make the fringe contrast function has 3 peaks and can not be modelled as single Gaussian function. However as shown in Fig. 13, the envelope of correlogram between two valleys (distinctive features highlighted in Fig. 7 (b)) can be reasonably modelled as single Gaussian, setting a constraint selecting only these data for single Gaussian fitting would fulfil the assumption of the original Gaussian fitting approach. So, instead of fitting the whole correlogram to a Gaussian, Gaussian fitting approach is modified such that it fit subset of correlogram to a Gaussian.

For frequency domain analysis approach, the required assumption is that there should be only one peak in the spatial frequency ranges from 20rad/um to 30 rad/um which

Phosphor-Based White Light Emitting Diode (LED) for Vertical Scanning Interferometry (VSI) 95

Fig. 14. Illustrating the effects of selecting (a) 64 data (b) 32 data around the maximum of correlogram in spatial frequency domain. According to FDA approach, there should be only 1 peak (in the region around 20 to 30 rad/um), this assumption is met in (b) which 32 data

Fig. 15 shows that the proposed modification on Gaussian fitting approach and frequency domain analysis works, the constraint on the input to reconstruction algorithms improves

Fig. 15. Simulation verification on the proposed modification: Comparing standard deviation of measuring perfectly flat surface reproduced by proposed modification and

As an experimental verification, the configuration is identical to the earlier simulation but measured a 10um80nm standard step height. 256 correlograms are collected at sampling

around peak are selected.

original.

the measurement repeatability.

correspond to wavelength of 628nm and 419nm. However for phosphor-based white LED, there are two peaks in the frequency ranges of interest (as shown in Fig. 14(a)), so we applied a constraint on the input for frequency domain analysis such that it would not have two peaks in spatial frequency domain analysis. As the distinctive features are the result of phosphor-based LED, we applied only data between these two features for frequency domain analysis. Fig. 14 (b) confirms that the assumption of frequency domain analysis is met by reducing the amount of data for frequency domain analysis, there is only one distinct peak in spatial frequency domain.

Fig. 13. Fitting the fringe contrast function with single Gaussian: Selecting a subset of data (such as -0.5um<=defocus position<= 0.5um) would lead to a good fit compared to selecting all data.

correspond to wavelength of 628nm and 419nm. However for phosphor-based white LED, there are two peaks in the frequency ranges of interest (as shown in Fig. 14(a)), so we applied a constraint on the input for frequency domain analysis such that it would not have two peaks in spatial frequency domain analysis. As the distinctive features are the result of phosphor-based LED, we applied only data between these two features for frequency domain analysis. Fig. 14 (b) confirms that the assumption of frequency domain analysis is met by reducing the amount of data for frequency domain analysis, there is only one distinct

> envelopew vs. x Fitting all data

Fitting only a subset of data


defocus position(um)

Fig. 13. Fitting the fringe contrast function with single Gaussian: Selecting a subset of data (such as -0.5um<=defocus position<= 0.5um) would lead to a good fit compared to selecting

peak in spatial frequency domain.

0.2

all data.

0.3

0.4

0.5

0.6

Normalized signal amplitude

0.7

0.8

0.9

1

Fig. 14. Illustrating the effects of selecting (a) 64 data (b) 32 data around the maximum of correlogram in spatial frequency domain. According to FDA approach, there should be only 1 peak (in the region around 20 to 30 rad/um), this assumption is met in (b) which 32 data around peak are selected.

Fig. 15 shows that the proposed modification on Gaussian fitting approach and frequency domain analysis works, the constraint on the input to reconstruction algorithms improves the measurement repeatability.

Fig. 15. Simulation verification on the proposed modification: Comparing standard deviation of measuring perfectly flat surface reproduced by proposed modification and original.

As an experimental verification, the configuration is identical to the earlier simulation but measured a 10um80nm standard step height. 256 correlograms are collected at sampling

Phosphor-Based White Light Emitting Diode (LED) for Vertical Scanning Interferometry (VSI) 97

reconstructed height profile varies depending on the assumption adopted by individual reconstruction algorithm. However the undesired effects of phosphor-based white LED can be removed by applying a constraint on the input to existing reconstruction

I would like to thanks my supervisors, Prof Soh Yeng Chai, Dr Li Xiang Leon and Dr Sardha WIJESOMA, whose help, stimulating suggestion and encouragement helped me in my

I would also like to thanks my colleagues from Singapore Institute of Manufacturing Technology (SIMTech), especially Dr Zhang Ying for their support, guidance and valuable

Ai, C., & Novak, E. K. (n.d.). Centroid approach for estimating modulation peak in broad-

Chim, S. S. C., & Kino, G. S. (1990). Correlation microscope. *Optics Letters*, *15*(10), 579-581.

Chong, W. K., Li, X., & Wijesoma, S. (2010a). Computationally efficient signal modeling for

Chong, W. K., Li, X., & Wijesoma, S. (2010b). Effects of phosphor-based LEDs on vertical

de Groot, P., & Colonna de Lega, X. (2004). Signal Modeling for Low-Coherence Height-

Groot, P. J. de, & Deck, L. L. (1994). Surface profiling by frequency-domain analysis of white

Guo, H., Zhao, Z., & Chen, M. (2007). Efficient iterative algorithm for phase-shifting

Gurov, I., Ermolaeva, E., & Zakharov, A. (2004). Analysis of low-coherence interference

Kino, Gordon S., & Chim, S. S. C. (1990). Mirau correlation microscope. *Applied Optics*,

Mingzhou, L., Chenggen, Q., Cho Jui, T., Ivan, R., & Shihua, W. (2005). Measurement of

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transparent coating thickness by the use of white light interferometry. *Third* 

algorithm.

research.

advice

**7. References** 

**6. Acknowledgment**

interval of 50nm, the repeatability of measuring optically flat surface was used to measure the performance of modified Gaussian fitting, modified frequency domain analysis, original Gaussian fitting and original frequency domain analysis.

Fig. 16. Experimental verification on the proposed modification: Comparing standard deviation of measuring perfectly flat surface reproduced by proposed modification and original.

Fig. 16 shows that the repeatability of both modified reconstruction algorithm are better than original one, the result agrees with simulation result, but it is controversial to common norm that more data leads to better result.

The result shows that for surface reconstruction of vertical scanning interferometry, it is more important to fulfil the assumption of reconstruction algorithm rather than fitting as many data as possible. The modification of inputting a subset of correlogram (which is around the maximum) improves the performance of reconstruction algorithm for phosphorbased LED, and it is applicable to reconstruction algorithm of both fringe contrast based and phase-based approach.

## **5. Conclusion**

In this chapter, it is shown that the use of phosphor-based LED on vertical scanning interferometry affect the repeatability and accuracy of vertical scanning interferometry, especially repeatability. The effect on the correlogram is inevitable, as the fringe contrast function can not longer be modelled as single Gaussian function. The effect on the reconstructed height profile varies depending on the assumption adopted by individual reconstruction algorithm. However the undesired effects of phosphor-based white LED can be removed by applying a constraint on the input to existing reconstruction algorithm.

## **6. Acknowledgment**

96 Interferometry – Research and Applications in Science and Technology

interval of 50nm, the repeatability of measuring optically flat surface was used to measure the performance of modified Gaussian fitting, modified frequency domain analysis, original

Fig. 16. Experimental verification on the proposed modification: Comparing standard deviation of measuring perfectly flat surface reproduced by proposed modification and

Fig. 16 shows that the repeatability of both modified reconstruction algorithm are better than original one, the result agrees with simulation result, but it is controversial to common

The result shows that for surface reconstruction of vertical scanning interferometry, it is more important to fulfil the assumption of reconstruction algorithm rather than fitting as many data as possible. The modification of inputting a subset of correlogram (which is around the maximum) improves the performance of reconstruction algorithm for phosphorbased LED, and it is applicable to reconstruction algorithm of both fringe contrast based and

In this chapter, it is shown that the use of phosphor-based LED on vertical scanning interferometry affect the repeatability and accuracy of vertical scanning interferometry, especially repeatability. The effect on the correlogram is inevitable, as the fringe contrast function can not longer be modelled as single Gaussian function. The effect on the

Gaussian fitting and original frequency domain analysis.

original.

phase-based approach.

**5. Conclusion** 

norm that more data leads to better result.

I would like to thanks my supervisors, Prof Soh Yeng Chai, Dr Li Xiang Leon and Dr Sardha WIJESOMA, whose help, stimulating suggestion and encouragement helped me in my research.

I would also like to thanks my colleagues from Singapore Institute of Manufacturing Technology (SIMTech), especially Dr Zhang Ying for their support, guidance and valuable advice

## **7. References**


**5** 

**Similariton-Based Spectral Interferometry** 

Levon Mouradian, Aram Zeytunyan and Garegin Yesayan

*Ultrafast Optics Laboratory, Faculty of Physics,* 

*Yerevan State University,* 

*Armenia* 

**for Signal Analysis on Femtosecond Time Scale** 

The *signal analysis problem on the femtosecond time scale* employs the powerful arsenal of contemporary optics, involving the methods of nonlinear and adaptive optics, Fourier optics and holography, spectral interferometry, etc. The nonlinear-optical techniques of FROG and its modifications [1], the most popular and commercialized, provide accurate and complete determination of the temporal amplitude and phase by recording high-resolution spectrograms, which are further decoded by means of iterative phase-retrieval procedures. The approach of *spectral interferometry (SI)* [2] and its developments to the methods of SPIDER [3], SPIRIT [4], and SORBETS [5] have the advantage of non-iterative phase retrieval. The recently developed and effectively applied MIIPS technique [6] operates with spectral phase measurement through its adaptive compensation up to transform-limited pulse shaping by using feedback from the SHG process. All these methods are based on the spectral phase determination, spectrum measurement, and reconstruction of a temporal pulse. The pulse direct measurement is possible by the transfer of temporal information to the space or frequency domain, or to the time domain with a lager - measurable scale. The time-to-space mapping through Fourier holography, implemented in real time by using a special multiple-quantum-well photorefractive device [7] or second-harmonic-generation crystal [8], provides a temporal resolution given by the duration of reference pulse. In an alternative approach, the 103 up-conversion time microscope demonstrates a 300 fs resolution [9]. In the time-to-frequency conversion approach, the methods of optical frequency inter-modulation by sum-frequency generation [10,11] and electro-optical modulation [12–13] are limited to the picosecond domain. The technique of optical chirpedpulse gating [14] demonstrates sub-picosecond resolution. The method of pulse spectrotemporal imaging through temporal lensing is more promising, having as a principal limit of resolution the ~1 fs nonlinear response time of silica [15–20]. Its recent modifications, implemented in the silicon chip [21] and similariton-induced parabolic temporal lens [22], provide accurate, high-resolution direct measurement of a pulse in the spectrometer as in

Many modern scientific and technological problems, such as revealing the character and peculiarities of nonlinear-dispersive similariton [23], studies of the nonlinear-dispersive regime of spectral compression and shaping of transform-limited rectangular pulses [24],

**1. Introduction** 

the femtosecond optical oscilloscope.

*International Conference on Experimental Mechanics and Third Conference of the Asian Committee on Experimental Mechanics* (Vol. 5852, pp. 401-406). SPIE.


## **Similariton-Based Spectral Interferometry for Signal Analysis on Femtosecond Time Scale**

Levon Mouradian, Aram Zeytunyan and Garegin Yesayan *Ultrafast Optics Laboratory, Faculty of Physics, Yerevan State University, Armenia* 

## **1. Introduction**

98 Interferometry – Research and Applications in Science and Technology

Pavliček, P., & Soubusta, J. (2004). Measurement of the Influence of Dispersion on

Sheppard, C. J. R., & Larkin, K. G. (1995). Effect of numerical aperture on interference fringe spacing. *Applied Optics*, *34*(22), 4731-4734. doi:10.1364/AO.34.004731

*Committee on Experimental Mechanics* (Vol. 5852, pp. 401-406). SPIE.

doi:10.1364/AO.43.000766

*International Conference on Experimental Mechanics and Third Conference of the Asian* 

White-Light Interferometry. *Applied Optics*, *43*(4), 766-770.

The *signal analysis problem on the femtosecond time scale* employs the powerful arsenal of contemporary optics, involving the methods of nonlinear and adaptive optics, Fourier optics and holography, spectral interferometry, etc. The nonlinear-optical techniques of FROG and its modifications [1], the most popular and commercialized, provide accurate and complete determination of the temporal amplitude and phase by recording high-resolution spectrograms, which are further decoded by means of iterative phase-retrieval procedures. The approach of *spectral interferometry (SI)* [2] and its developments to the methods of SPIDER [3], SPIRIT [4], and SORBETS [5] have the advantage of non-iterative phase retrieval. The recently developed and effectively applied MIIPS technique [6] operates with spectral phase measurement through its adaptive compensation up to transform-limited pulse shaping by using feedback from the SHG process. All these methods are based on the spectral phase determination, spectrum measurement, and reconstruction of a temporal pulse. The pulse direct measurement is possible by the transfer of temporal information to the space or frequency domain, or to the time domain with a lager - measurable scale. The time-to-space mapping through Fourier holography, implemented in real time by using a special multiple-quantum-well photorefractive device [7] or second-harmonic-generation crystal [8], provides a temporal resolution given by the duration of reference pulse. In an alternative approach, the 103 up-conversion time microscope demonstrates a 300 fs resolution [9]. In the time-to-frequency conversion approach, the methods of optical frequency inter-modulation by sum-frequency generation [10,11] and electro-optical modulation [12–13] are limited to the picosecond domain. The technique of optical chirpedpulse gating [14] demonstrates sub-picosecond resolution. The method of pulse spectrotemporal imaging through temporal lensing is more promising, having as a principal limit of resolution the ~1 fs nonlinear response time of silica [15–20]. Its recent modifications, implemented in the silicon chip [21] and similariton-induced parabolic temporal lens [22], provide accurate, high-resolution direct measurement of a pulse in the spectrometer as in the femtosecond optical oscilloscope.

Many modern scientific and technological problems, such as revealing the character and peculiarities of nonlinear-dispersive similariton [23], studies of the nonlinear-dispersive regime of spectral compression and shaping of transform-limited rectangular pulses [24],

Similariton-Based Spectral Interferometry for Signal Analysis on Femtosecond Time Scale 101

the femtosecond oscilloscope, based on the similariton-induced parabolic lens, is given by the bandwidth of similariton [22], and the application range for similariton-based SI [43] is as large as broadband the similariton-reference is. The pulse compression ratio is also as high as the spectral broadening factor is [40,41]. Experimentally, our SI study permitted the complete characterization of the nonlinear-dispersive similariton of up to 5 THz bandwidths [36,37]. For broadband similaritons (of up to 50-THz bandwidths), we applied the chirp measurement technique through spectral compression and frequency tuning in the sum-

In our method of *similariton-based SI*, the part of signal is injected into a fiber to generate the nonlinear-dispersive similariton-reference. The residual part of the signal, passing an optical time delay, is coupled with the similariton in a spectrometer. The spectral fringe pattern, on the background of the signal and similariton spectra, completely covers the signal spectrum, and the whole phase information becomes available for any signal. The known spectral phase of the similariton-reference allows to retrieve the signal spectral phase, and by measuring also the signal spectrum, to reconstruct the complex temporal amplitude of the signal through Fourier transformation. Thus, the method of similariton-based SI joins the advantages of both the classic SI [2] and spectral shearing interferometry [3-5], combining the simplicity of the principle and configuration with the self-referencing performance [46, 47]. Experimentally examining the similariton-based SI, we carried out the comparative measurements with a prototype of the femtosecond oscilloscope (FO) based on the pulse spectrotemporal imaging in the similariton-induced temporal lens. The resolution of such a similariton-based FO is given by the transfer function of the similariton's spectrum [22], and FO with a similariton-reference of the bandwidth of 50 THz bandwidths provides the direct measurement of temporal pulse in a spectrometer of 7 fs temporal resolution. In the comparative experiments, we demonstrate and study the methods of similariton-based SI and spectrotemporal imaging as two applications of similariton. The reference-based methods become self-referencing by the use of similariton. The results of the measurements carried out by similariton-based SI and FO are in quantitative accordance. The similaritonbased spectrotemporal imaging has the advantage of direct pulse measurement leading to the development of a femtosecond optical oscilloscope, but it does not give phase information without additional interferometric measurement. The similariton-based SI

provides the complete (amplitude and phase) characterization of femtosecond signal.

similariton-referencing methods of SI and temporal lensing.

Concluding the chapter introduction, our studies on the generation of nonlinear-dispersive similaritons of up to 50 THz bandwidths, and their experimental characterization by means of the methods of SI and chirp measurement through the technique of frequency tuning in spectral compression process are presented below. Our studies state that only fiber dispersion determines the phase of broadband nonlinear-dispersive similariton. Afterwards, our demonstration and study of the similariton-based SI for femtosecond signal complete characterization and the comparative experiments with FO based on the similariton-induced temporal lens are presented, carried out together with theoretical check and autocorrelation measurements, evidencing the quantitative accordance and high precision of both the

Outlining this chapter, after a brief review of modern methods of signal analysis on the femtosecond time scale, we present our studies of the nature and peculiarities of similariton generated in passive fiber carried out by classic SI, then – research of broadband similariton

frequency generation process [22,44,45].

characterization of prism-lens dispersive delay line [25], etc, however, along with the amplitude information, demand also the phase information, possible through additional interferometric measurements, motivating the urgency of SI methods for the complete characterization of femtosecond signal. The *classic method of SI* is based on the interference of the signal and reference beams spectrally dispersed in a spectrometer, with the spectral fringe pattern caused by the difference of spectral phases [2]. The known spectral phase of the reference permits to retrieve the spectral phase of signal, and, together with the spectrum measurement, to recover the complex temporal amplitude of the signal through Fourier transformation. The setup of classic SI is rather simple, and the measurement is accurate as any interferometric one, but its application range is restricted by the bandwidth of the reference. The SI characterization of a signal that has undergone a nonlinear interaction with medium requires a special broadband reference to fully cover the broadened signal spectrum. To avoid this restriction, the self-referencing methods of spectral shearing interferometry are developed [3-5]. This improvement promotes the SImethods to the class of the most popular and commercialized methods of accurate measurements on the femtosecond time scale, making them compatible with the FROG techniques [1], at the expense of a complicated optical arrangement. Our developed method of similariton-based SI, along with its self-referencing performance, keeps the simplicity of the principle and configuration of the classic SI [2].

*Similaritons*, pulses with the distinctive property of self-similar propagation, recently attract the attention of researchers, due to fundamental interest and prospective applications in ultrafast optics and photonics, particularly for high-power pulse amplification, optical telecommunications, ultrafast all-optical signal processing, etc [26,27]. The self-similar propagation of the high-power pulse with parabolic temporal, spectral, and phase profiles was predicted theoretically in the 90's [28]. In practice, the generation of such parabolic similaritons is possible in active fibers, such as rare-earth-doped fiber amplifiers [29-31] and Raman fiber amplifiers [32], as well as in the laser resonator [33]. The generation of parabolic similariton has been also proposed in a tapered fiber with decreasing normal dispersion, using either passive dispersion-decreasing fiber [34] or a hybrid configuration with Raman amplification [35]. Another type of similariton is generated in a conventional uniform and passive (without gain) fiber under the combined impacts of Kerr-nonlinearity and dispersion [23]. In contrast to the parabolic similariton with parabolic amplitude and phase profiles, this nonlinear-dispersive similariton has only parabolic phase but maintains its temporal (and spectral) shape during the propagation, as well. Our SI studies of this type of pulses [36,37] show the linearity of their chirp (parabolic phase), with a slope given only by the fiber dispersion. This property leads to the spectrotemporal similarity and selfspectrotemporal imaging of nonlinear-dispersive similariton, with accuracy given both by spectral broadening and pulse stretching. The relation of such a similariton with the flat-top "rectangular" pulse, shaped in the near field of nonlinear-dispersive self-interaction, leads to its important peculiarity: the bandwidth of nonlinear-dispersive similariton is given by the input pulse power, with slightly varying coefficients given by the input pulse shape [23, 38,39]. Both the parabolic similariton of active fiber and nonlinear-dispersive similariton of passive fiber are of interest for applications in ultrafast optics, especially for pulse compression [40,41] and shaping [42], similariton-referencing temporal lensing and spectrotemporal imaging [23], and SI [43]. The applications to ultrafast optics, however, demand the generation and study of broadband similariton. Particularly, the resolution of

characterization of prism-lens dispersive delay line [25], etc, however, along with the amplitude information, demand also the phase information, possible through additional interferometric measurements, motivating the urgency of SI methods for the complete characterization of femtosecond signal. The *classic method of SI* is based on the interference of the signal and reference beams spectrally dispersed in a spectrometer, with the spectral fringe pattern caused by the difference of spectral phases [2]. The known spectral phase of the reference permits to retrieve the spectral phase of signal, and, together with the spectrum measurement, to recover the complex temporal amplitude of the signal through Fourier transformation. The setup of classic SI is rather simple, and the measurement is accurate as any interferometric one, but its application range is restricted by the bandwidth of the reference. The SI characterization of a signal that has undergone a nonlinear interaction with medium requires a special broadband reference to fully cover the broadened signal spectrum. To avoid this restriction, the self-referencing methods of spectral shearing interferometry are developed [3-5]. This improvement promotes the SImethods to the class of the most popular and commercialized methods of accurate measurements on the femtosecond time scale, making them compatible with the FROG techniques [1], at the expense of a complicated optical arrangement. Our developed method of similariton-based SI, along with its self-referencing performance, keeps the simplicity of

*Similaritons*, pulses with the distinctive property of self-similar propagation, recently attract the attention of researchers, due to fundamental interest and prospective applications in ultrafast optics and photonics, particularly for high-power pulse amplification, optical telecommunications, ultrafast all-optical signal processing, etc [26,27]. The self-similar propagation of the high-power pulse with parabolic temporal, spectral, and phase profiles was predicted theoretically in the 90's [28]. In practice, the generation of such parabolic similaritons is possible in active fibers, such as rare-earth-doped fiber amplifiers [29-31] and Raman fiber amplifiers [32], as well as in the laser resonator [33]. The generation of parabolic similariton has been also proposed in a tapered fiber with decreasing normal dispersion, using either passive dispersion-decreasing fiber [34] or a hybrid configuration with Raman amplification [35]. Another type of similariton is generated in a conventional uniform and passive (without gain) fiber under the combined impacts of Kerr-nonlinearity and dispersion [23]. In contrast to the parabolic similariton with parabolic amplitude and phase profiles, this nonlinear-dispersive similariton has only parabolic phase but maintains its temporal (and spectral) shape during the propagation, as well. Our SI studies of this type of pulses [36,37] show the linearity of their chirp (parabolic phase), with a slope given only by the fiber dispersion. This property leads to the spectrotemporal similarity and selfspectrotemporal imaging of nonlinear-dispersive similariton, with accuracy given both by spectral broadening and pulse stretching. The relation of such a similariton with the flat-top "rectangular" pulse, shaped in the near field of nonlinear-dispersive self-interaction, leads to its important peculiarity: the bandwidth of nonlinear-dispersive similariton is given by the input pulse power, with slightly varying coefficients given by the input pulse shape [23, 38,39]. Both the parabolic similariton of active fiber and nonlinear-dispersive similariton of passive fiber are of interest for applications in ultrafast optics, especially for pulse compression [40,41] and shaping [42], similariton-referencing temporal lensing and spectrotemporal imaging [23], and SI [43]. The applications to ultrafast optics, however, demand the generation and study of broadband similariton. Particularly, the resolution of

the principle and configuration of the classic SI [2].

the femtosecond oscilloscope, based on the similariton-induced parabolic lens, is given by the bandwidth of similariton [22], and the application range for similariton-based SI [43] is as large as broadband the similariton-reference is. The pulse compression ratio is also as high as the spectral broadening factor is [40,41]. Experimentally, our SI study permitted the complete characterization of the nonlinear-dispersive similariton of up to 5 THz bandwidths [36,37]. For broadband similaritons (of up to 50-THz bandwidths), we applied the chirp measurement technique through spectral compression and frequency tuning in the sumfrequency generation process [22,44,45].

In our method of *similariton-based SI*, the part of signal is injected into a fiber to generate the nonlinear-dispersive similariton-reference. The residual part of the signal, passing an optical time delay, is coupled with the similariton in a spectrometer. The spectral fringe pattern, on the background of the signal and similariton spectra, completely covers the signal spectrum, and the whole phase information becomes available for any signal. The known spectral phase of the similariton-reference allows to retrieve the signal spectral phase, and by measuring also the signal spectrum, to reconstruct the complex temporal amplitude of the signal through Fourier transformation. Thus, the method of similariton-based SI joins the advantages of both the classic SI [2] and spectral shearing interferometry [3-5], combining the simplicity of the principle and configuration with the self-referencing performance [46, 47]. Experimentally examining the similariton-based SI, we carried out the comparative measurements with a prototype of the femtosecond oscilloscope (FO) based on the pulse spectrotemporal imaging in the similariton-induced temporal lens. The resolution of such a similariton-based FO is given by the transfer function of the similariton's spectrum [22], and FO with a similariton-reference of the bandwidth of 50 THz bandwidths provides the direct measurement of temporal pulse in a spectrometer of 7 fs temporal resolution. In the comparative experiments, we demonstrate and study the methods of similariton-based SI and spectrotemporal imaging as two applications of similariton. The reference-based methods become self-referencing by the use of similariton. The results of the measurements carried out by similariton-based SI and FO are in quantitative accordance. The similaritonbased spectrotemporal imaging has the advantage of direct pulse measurement leading to the development of a femtosecond optical oscilloscope, but it does not give phase information without additional interferometric measurement. The similariton-based SI provides the complete (amplitude and phase) characterization of femtosecond signal.

Concluding the chapter introduction, our studies on the generation of nonlinear-dispersive similaritons of up to 50 THz bandwidths, and their experimental characterization by means of the methods of SI and chirp measurement through the technique of frequency tuning in spectral compression process are presented below. Our studies state that only fiber dispersion determines the phase of broadband nonlinear-dispersive similariton. Afterwards, our demonstration and study of the similariton-based SI for femtosecond signal complete characterization and the comparative experiments with FO based on the similariton-induced temporal lens are presented, carried out together with theoretical check and autocorrelation measurements, evidencing the quantitative accordance and high precision of both the similariton-referencing methods of SI and temporal lensing.

Outlining this chapter, after a brief review of modern methods of signal analysis on the femtosecond time scale, we present our studies of the nature and peculiarities of similariton generated in passive fiber carried out by classic SI, then – research of broadband similariton

Similariton-Based Spectral Interferometry for Signal Analysis on Femtosecond Time Scale 103

average power *p* ~ 100 mW at a 76 MHz repetition rate, the nonlinear interaction length

allows us to roughly split the impacts of nonlinear self-interaction and dispersive deformation of the pulse, assuming that first we have pure nonlinear self-phase modulation of the pulse and spectral broadening, and afterwards pure dispersive stretching and dispersive-spectronic propagation. For the phase of output pulse, we have <sup>2</sup>

*NL* of [ ( ~ , )] *NL <sup>t</sup> FT A f L t*

propagation step of nonlinear self-interaction. Assuming it parabolic at the central energycarrying part of the pulse at the propagation distances of ~ *LNL* , we have the phase

have for the nonlinear, dispersive, and overall chirp slopes at the output, respectively:

fs, *p* ~ 100 mW average power at a 76 MHz repetition rate, *f* ~ 1 m of fiber), we will have

Thus, for the femtosecond pulse nonlinear-dispersive self-interaction at *f* ~ 1 m of fiber, we

nonlinear-dispersive spectron-similariton, that practically the fiber dispersion determines

<sup>2</sup> ( , ) | ( , )| exp( /2) *Aft Af Ct iCt* 

Another interesting issue is the relation of nonlinear-dispersive similariton with the rectangular pulses, shaped due to the pulse nonlinear-dispersive self-interaction at the fiber lengths ~ 2 *D NL f LL* . For such nonlinear-dispersive rectangular pulses, the temporal

respectively, since the pulse optimal compression ratio is 0 / *<sup>c</sup> t t L L D NL* / /2 *L f <sup>D</sup>* / ,

dispersive self-interaction in fiber, the chirp slope becomes equal to the one of pulse

nonlinear-dispersive rectangular pulses can be considered as an earlier step of the shaping

Summarizing our rough analysis, we can expect the spectronic nature of nonlineardispersive similariton of passive fiber, its spectrotemporal similarity and imaging with the

To check the terms and conclusions of the rough analytical discussion above, a quantitative analysis of the process is carried out through *numerical modeling* based on the complete wave pattern. The mathematical description of the pulse nonlinear-dispersive self-interaction in

, with *CNL*

<sup>0</sup> /*<sup>t</sup>* <sup>2</sup> 

*<sup>t</sup> CCt NL*

broadening and dispersive pulse stretching factors ( *b*

*C C CC C* (1 / ) *NL* , with the accuracy of *C C*/ *NL* ~1% .

stretching and spectral broadening are up to 0 *t t* 2 and

 

*n I* is much shorter than the dispersive one: *LNL* ~ 1 cm << *LD* ~ 10 cm. This

, for spectral broadening *<sup>b</sup>* ~ 10 and pulse stretching *<sup>s</sup>* ~ 10 ( <sup>0</sup> *<sup>t</sup>* ~ 100

(,) *z t* <sup>2</sup> *Ct C C* (1 / )/2 *NL* . Considering the nonlinear spectral

 /

<sup>0</sup> /*s* , and *C C CC* (1 / ) *NL*

~1% precision. Considering the key peculiarity of the

. Thus, in this case the chirp slope obtains the value

*<sup>f</sup>* , starting from the fiber lengths *<sup>f</sup>* <sup>~</sup> <sup>2</sup> *L LD NL* , and

, and with the scaling coefficient of the chirp slope <sup>1</sup>

*LD <sup>f</sup> <sup>t</sup> <sup>f</sup>* . Therefore, during the pulse nonlinear-

. (2)

*NL* " . For the overall output phase

 <sup>0</sup> 2 ( /) 

*L f <sup>D</sup>* ,

<sup>2</sup> *C* ( ) *f* ,

0 and *s* / <sup>0</sup> *t t* ), we

, come from the pulse initial

*<sup>D</sup>*( , ) /2 *f t Ct* ,

<sup>1</sup> *C sb* [1 ( ) ] .

*LNL* <sup>1</sup> 0 20 ( ) 

*C t NL* / <sup>0</sup>

Since <sup>1</sup> *C C sb* / () *NL*

 

and an additional phase term

*D NL* , we have

have spectron of ~1/ *sb* = <sup>2</sup> *C* /

and 0 / *<sup>c</sup> t t* <sup>0</sup> 2 /

accuracy ~1/ *sb* = <sup>2</sup> *C* /

given by dispersion only.

*C* 

dispersive propagation <sup>1</sup>

of nonlinear-dispersive similariton.

<sup>2</sup> *C* ( ) 

 / *t* <sup>1</sup> 0 02 (2 / )/(2 ) ( )

*NL*( ,) *f t* 1 2 2 2

 

2 <sup>0</sup> 

*NL* / 2| ( / ) / 2 *C*

*b* , *C*

the chirp slope, we can describe it following way:

in view of problems of femtosecond signal analysis-synthesis, and finally – comparative studies on the development of novel method of self-referencing similariton-based SI.

## **2. Spectral interferometric study of nonlinear-dispersive similariton**

In this section, our spectral interferometric studies for the complete characterization of the similariton generated in passive fiber due to the combined impacts of nonlinearity and dispersion are described. Studies of the generation of nonlinear-dispersive similariton of passive fiber, its distinctive properties, especially its origin, nature and relation with the spectron and rectangular pulses, and the temporal, spectral and phase features in view of potential applications are presented. The nonlinear-spectronic character of such a similariton, with the key specificity of linear chirping, leads to important applications to the signal analysis - synthesis problems in ultrafast optics.

The outline of this section is the following: first, a rough analytical discussion and numerical studies are given to reveal the features of nonlinear-dispersive similariton, afterwards, experiments for the spectral interferometric characterization of nonlinear-dispersive similariton are presented to demonstrate the peculiarities of similariton predicted by the theory, then the technique for the measurement of similariton chirp by the use of spectrometer and autocorrelator is described, and finally, the study of the bandwidth (and duration) rule of nonlinear-dispersive similariton is presented.

## **2.1 Similariton self-shaped (generated) in passive fiber**

In a *rough analysis* of nonlinear-dispersive similariton, first we consider the pulse propagation through a pure dispersive medium. In the far field of dispersion a *spectron* pulse is shaped [23], which repeats its spectral profile, in the temporal analogy of the Fraunhofer zone diffraction, and therefore propagates self-similarly. Mathematically, the solution of the dispersion equation for temporal amplitude *Aft* ( ,) 1 2 <sup>2</sup> *FT A i f* [ (0, )exp( / 2)] at the propagation distance *<sup>f</sup> LD* obtains the form

$$A(f,t) \propto \tilde{A}(0,o) \exp(i\beta\_2 f o^2 / 2)\Big|\_{o=Ct} = \tilde{A}(0,o)|\_{o=Ct} \exp(iCt^2/2) \,. \tag{1}$$

In Eq. (1), *A*( ) [ ( )] *FT A t* is the complex spectral amplitude, *FT* – the operator of Fourier transformation, *C d dt* / 1 1 0 2 [ "( )] ( ) *<sup>f</sup>* – chirp slope; *LD* 2 1 2 0 ( ) – dispersive length, 0 "( ) – second derivative of the dispersion-induced spectral phase ( ) at the central frequency 0 , 2 – coefficient of second order dispersion, and 0 – input spectral bandwidth. The condition of temporal Fraunhofer zone, meaning enough large pulse stretching *s* / *<sup>o</sup> t t* <sup>2</sup> / 1 *<sup>o</sup> C* , gives the 1 /*s* <sup>2</sup> *C* /*<sup>o</sup>* precision of the spectron's spectrotemporal similarity *A*( ,) *f t* | (0, )| *A Ct* | ( , )| *A f Ct* . For a 100 fs pulse propagating in a standard single-mode fiber *LD* is of ~10 cm, and at the output of 1-m fiber we will have pulse stretching of *s f* /*LD* ~ 10, and spectrotemporal similarity of spectron of the 1 /*s* ~10% precision.

For a nonlinear-dispersively propagating pulse, the nonlinear self-interaction broadens the spectrum and increases the impact of dispersion, leading to a higher-precision spectronsimilariton shaping. Quantitatively, for 100-fs pulse radiation in a single mode fiber with

in view of problems of femtosecond signal analysis-synthesis, and finally – comparative

In this section, our spectral interferometric studies for the complete characterization of the similariton generated in passive fiber due to the combined impacts of nonlinearity and dispersion are described. Studies of the generation of nonlinear-dispersive similariton of passive fiber, its distinctive properties, especially its origin, nature and relation with the spectron and rectangular pulses, and the temporal, spectral and phase features in view of potential applications are presented. The nonlinear-spectronic character of such a similariton, with the key specificity of linear chirping, leads to important applications to the

The outline of this section is the following: first, a rough analytical discussion and numerical studies are given to reveal the features of nonlinear-dispersive similariton, afterwards, experiments for the spectral interferometric characterization of nonlinear-dispersive similariton are presented to demonstrate the peculiarities of similariton predicted by the theory, then the technique for the measurement of similariton chirp by the use of spectrometer and autocorrelator is described, and finally, the study of the bandwidth (and

In a *rough analysis* of nonlinear-dispersive similariton, first we consider the pulse propagation through a pure dispersive medium. In the far field of dispersion a *spectron* pulse is shaped [23], which repeats its spectral profile, in the temporal analogy of the Fraunhofer zone diffraction, and therefore propagates self-similarly. Mathematically, the solution of the dispersion equation for temporal amplitude *Aft* ( ,)

at the propagation distance *<sup>f</sup> LD* obtains the form

 

spectral bandwidth. The condition of temporal Fraunhofer zone, meaning enough large

pulse propagating in a standard single-mode fiber *LD* is of ~10 cm, and at the output of 1-m fiber we will have pulse stretching of *s f* /*LD* ~ 10, and spectrotemporal similarity of

For a nonlinear-dispersively propagating pulse, the nonlinear self-interaction broadens the spectrum and increases the impact of dispersion, leading to a higher-precision spectronsimilariton shaping. Quantitatively, for 100-fs pulse radiation in a single mode fiber with

<sup>2</sup> (0, )| exp( / 2) *<sup>A</sup> Ct iCt*

*FT A t* is the complex spectral amplitude, *FT* – the operator of Fourier

*<sup>f</sup>* – chirp slope; *LD*

– second derivative of the dispersion-induced spectral phase

2 – coefficient of second order dispersion, and

 *Ct* | ( , )| *A f*

*<sup>o</sup> C* , gives the 1 /*s* <sup>2</sup> *C* /

 

. (1)

 

2 1 2 0 ( ) –

*<sup>o</sup>* precision of the

*Ct* . For a 100 fs

( ) 

0 – input

<sup>2</sup> (0, )exp( / 2) *Ct A if*

 / 1 1 0 2 [ "( )] ( )

 

studies on the development of novel method of self-referencing similariton-based SI.

**2. Spectral interferometric study of nonlinear-dispersive similariton** 

signal analysis - synthesis problems in ultrafast optics.

duration) rule of nonlinear-dispersive similariton is presented.

**2.1 Similariton self-shaped (generated) in passive fiber** 

*Aft* ( ,) <sup>2</sup>

pulse stretching *s* / *<sup>o</sup> t t* <sup>2</sup> / 1

0 , 

spectron's spectrotemporal similarity *A*( ,) *f t* | (0, )| *A*

1 2 <sup>2</sup> *FT A i f* [ (0, )exp( / 2)] 

In Eq. (1), *A*( ) [ ( )] 

dispersive length, 0

at the central frequency

transformation, *C d dt*

 

 "( ) 

spectron of the 1 /*s* ~10% precision.

average power *p* ~ 100 mW at a 76 MHz repetition rate, the nonlinear interaction length *LNL* <sup>1</sup> 0 20 ( ) *n I* is much shorter than the dispersive one: *LNL* ~ 1 cm << *LD* ~ 10 cm. This allows us to roughly split the impacts of nonlinear self-interaction and dispersive deformation of the pulse, assuming that first we have pure nonlinear self-phase modulation of the pulse and spectral broadening, and afterwards pure dispersive stretching and dispersive-spectronic propagation. For the phase of output pulse, we have <sup>2</sup> *<sup>D</sup>*( , ) /2 *f t Ct* , and an additional phase term *NL* of [ ( ~ , )] *NL <sup>t</sup> FT A f L t* , come from the pulse initial propagation step of nonlinear self-interaction. Assuming it parabolic at the central energycarrying part of the pulse at the propagation distances of ~ *LNL* , we have the phase *NL*( ,) *f t* 1 2 2 2 *NL* / 2| ( / ) / 2 *C <sup>t</sup> CCt NL* , with *CNL NL* " . For the overall output phase *D NL* , we have (,) *z t* <sup>2</sup> *Ct C C* (1 / )/2 *NL* . Considering the nonlinear spectral broadening and dispersive pulse stretching factors ( *b* /0 and *s* / <sup>0</sup> *t t* ), we have for the nonlinear, dispersive, and overall chirp slopes at the output, respectively: *C t NL* / <sup>0</sup> 2 <sup>0</sup> *b* , *C* <sup>0</sup> /*<sup>t</sup>* <sup>2</sup> <sup>0</sup> /*s* , and *C C CC* (1 / ) *NL* <sup>1</sup> *C sb* [1 ( ) ] . Since <sup>1</sup> *C C sb* / () *NL* , for spectral broadening *<sup>b</sup>* ~ 10 and pulse stretching *<sup>s</sup>* ~ 10 ( <sup>0</sup> *<sup>t</sup>* ~ 100 fs, *p* ~ 100 mW average power at a 76 MHz repetition rate, *f* ~ 1 m of fiber), we will have *C C CC C* (1 / ) *NL* , with the accuracy of *C C*/ *NL* ~1% .

Thus, for the femtosecond pulse nonlinear-dispersive self-interaction at *f* ~ 1 m of fiber, we have spectron of ~1/ *sb* = <sup>2</sup> *C* / ~1% precision. Considering the key peculiarity of the nonlinear-dispersive spectron-similariton, that practically the fiber dispersion determines the chirp slope, we can describe it following way:

$$A(f,t) \propto |\bar{A}(f,o)|\_{o=Cr} \exp(iCt^2/2) \,. \tag{2}$$

Another interesting issue is the relation of nonlinear-dispersive similariton with the rectangular pulses, shaped due to the pulse nonlinear-dispersive self-interaction at the fiber lengths ~ 2 *D NL f LL* . For such nonlinear-dispersive rectangular pulses, the temporal stretching and spectral broadening are up to 0 *t t* 2 and <sup>0</sup> 2 ( /) *L f <sup>D</sup>* , respectively, since the pulse optimal compression ratio is 0 / *<sup>c</sup> t t L L D NL* / /2 *L f <sup>D</sup>* / , and 0 / *<sup>c</sup> t t* <sup>0</sup> 2 / . Thus, in this case the chirp slope obtains the value *C* / *t* <sup>1</sup> 0 02 (2 / )/(2 ) ( ) *LD <sup>f</sup> <sup>t</sup> <sup>f</sup>* . Therefore, during the pulse nonlineardispersive self-interaction in fiber, the chirp slope becomes equal to the one of pulse dispersive propagation <sup>1</sup> <sup>2</sup> *C* ( ) *<sup>f</sup>* , starting from the fiber lengths *<sup>f</sup>* <sup>~</sup> <sup>2</sup> *L LD NL* , and nonlinear-dispersive rectangular pulses can be considered as an earlier step of the shaping of nonlinear-dispersive similariton.

Summarizing our rough analysis, we can expect the spectronic nature of nonlineardispersive similariton of passive fiber, its spectrotemporal similarity and imaging with the accuracy ~1/ *sb* = <sup>2</sup> *C* / , and with the scaling coefficient of the chirp slope <sup>1</sup> <sup>2</sup> *C* ( ) *f* , given by dispersion only.

To check the terms and conclusions of the rough analytical discussion above, a quantitative analysis of the process is carried out through *numerical modeling* based on the complete wave pattern. The mathematical description of the pulse nonlinear-dispersive self-interaction in

Similariton-Based Spectral Interferometry for Signal Analysis on Femtosecond Time Scale 105

Fig. 2. Shaping and evolution of nonlinear-dispersive similariton in the fiber ( *R* = 30). From

Our simulations show that even in case of pulses with complex initial forms the output pulse has nearly parabolic form at its central energy-carrying part. The irregularities of the temporal and spectral profiles are forced out to the edges during the nonlinear-dispersive self-interaction, and pulse and spectrum become more and more parabolic. However, the pulse dispersive stretching decreases its peak intensity, finally minimizing the impact of nonlinear self-phase modulation. The spectrum does not change any more and the further

It is important that our simulations for the nonlinear-dispersive similariton at a given fiber length and different power values confirm also the prediction of the analytical discussion: the output chirp slope is practically independent of the input pulse intensity, and depends only on the fiber length. The increase of input pulse power leads to the spectral broadening and temporal stretching of pulse, keeping the chirp coefficient unchanged: the chirp slope is the same in all cases, even in case of pulse pure dispersive propagation. It allows extracting the full information on the nonlinear-dispersive similariton having the spectrum and fiber length. This statement is checked for sufficiently powerful pulses, when the character of pulse self-interaction is nonlinear-dispersive (but not pure dispersive), and the result is the

We studied also the chirp of nonlinear-dispersive similariton versus the chirp of input pulse: the chirp slope of similariton is practically constant, when the pulse intensity is high enough. In case of the dispersive propagation, the induced chirp simply imposes on the initial chirp according to Eq. (1). In case of nonlinear-dispersive propagation, the chirp "forgets" about the initial chirp according to analytic discussion above: it becomes

left to right: *f L* / *<sup>D</sup>* = 1; 2; 3; 4. From top to bottom: pulse, chirp, and spectrum.

alteration of the pulse shape has a dispersive character only.

same: the chirp slope is independent of the power.

independent of the input chirp according to Eq. (2).

fiber is based on the standard nonlinear Schrödinger equation (NLSE) with the terms of Kerr nonlinearity and second-order dispersion, adequate to the pulse durations of 50 fs [23]. The split-step Fourier method is applied to solve the NLSE. In simulations, the pulse propagation distance is expressed in dispersive lengths *LD* ( *LD* ~ 10 cm for 100-fs input pulses); the power of radiation in fiber is given by the nonlinear parameter *<sup>R</sup> L L D NL* / 2 1 2 0 0 20 ( ) *n I* <sup>~</sup> <sup>0</sup>*I* ( *<sup>p</sup>* = 100 mW average power of a 100-fs pulse radiation at a 76 MHz repetition rate in a standard single-mode fiber corresponds to *R* = 6; *p* 100uW is adequate to the pulse pure dispersive propagation of *R* = 0). The dimensionless running time *t* and centralized frequency are normalized to the input pulse duration <sup>0</sup> *t* and bandwidth 0 1 / *t* , respectively.

Fig. 1 and 2 show the dynamics of similariton shaping: pulse (top row), chirp (middle row) and spectrum (bottom row) are shown during the pulse propagation in fiber. Fig. 1 illustrates the first step of nonlinear-dispersive self-interaction when, typically, rectangular pulses are shaped, and Fig. 2 shows the step of similariton shaping. The spectral broadening and decreasing of the pulse peak power lead to the "activation" of dispersion; the pulse obtains a linear chirp (parabolic phase), and the self-spectrotemporal similarity of nonlineardispersive similariton takes place (Fig. 2).

Fig. 1. Shaping of rectangular pulses ( *R* = 30). From left to right: pulse evolution in fiber for *f* /*LD* = 0.1; 0.2; 0.3; and 0.4. From top to bottom: pulse, chirp, and spectrum.

fiber is based on the standard nonlinear Schrödinger equation (NLSE) with the terms of Kerr nonlinearity and second-order dispersion, adequate to the pulse durations of 50 fs [23]. The split-step Fourier method is applied to solve the NLSE. In simulations, the pulse propagation distance is expressed in dispersive lengths *LD* ( *LD* ~ 10 cm for 100-fs input pulses); the power of radiation in fiber is given by the nonlinear parameter

at a 76 MHz repetition rate in a standard single-mode fiber corresponds to *R* = 6; *p* 100uW is adequate to the pulse pure dispersive propagation of *R* = 0). The dimensionless running time *t* and centralized frequency are normalized to the input pulse duration

Fig. 1 and 2 show the dynamics of similariton shaping: pulse (top row), chirp (middle row) and spectrum (bottom row) are shown during the pulse propagation in fiber. Fig. 1 illustrates the first step of nonlinear-dispersive self-interaction when, typically, rectangular pulses are shaped, and Fig. 2 shows the step of similariton shaping. The spectral broadening and decreasing of the pulse peak power lead to the "activation" of dispersion; the pulse obtains a linear chirp (parabolic phase), and the self-spectrotemporal similarity of nonlinear-

Fig. 1. Shaping of rectangular pulses ( *R* = 30). From left to right: pulse evolution in fiber for

*f* /*LD* = 0.1; 0.2; 0.3; and 0.4. From top to bottom: pulse, chirp, and spectrum.

*n I* <sup>~</sup> <sup>0</sup>*I* ( *<sup>p</sup>* = 100 mW average power of a 100-fs pulse radiation

*<sup>R</sup> L L D NL* / 2 1

<sup>0</sup> *t* and bandwidth 0

2 0 0 20 ( ) 

dispersive similariton takes place (Fig. 2).

 

1 / *t* , respectively.

Fig. 2. Shaping and evolution of nonlinear-dispersive similariton in the fiber ( *R* = 30). From left to right: *f L* / *<sup>D</sup>* = 1; 2; 3; 4. From top to bottom: pulse, chirp, and spectrum.

Our simulations show that even in case of pulses with complex initial forms the output pulse has nearly parabolic form at its central energy-carrying part. The irregularities of the temporal and spectral profiles are forced out to the edges during the nonlinear-dispersive self-interaction, and pulse and spectrum become more and more parabolic. However, the pulse dispersive stretching decreases its peak intensity, finally minimizing the impact of nonlinear self-phase modulation. The spectrum does not change any more and the further alteration of the pulse shape has a dispersive character only.

It is important that our simulations for the nonlinear-dispersive similariton at a given fiber length and different power values confirm also the prediction of the analytical discussion: the output chirp slope is practically independent of the input pulse intensity, and depends only on the fiber length. The increase of input pulse power leads to the spectral broadening and temporal stretching of pulse, keeping the chirp coefficient unchanged: the chirp slope is the same in all cases, even in case of pulse pure dispersive propagation. It allows extracting the full information on the nonlinear-dispersive similariton having the spectrum and fiber length. This statement is checked for sufficiently powerful pulses, when the character of pulse self-interaction is nonlinear-dispersive (but not pure dispersive), and the result is the same: the chirp slope is independent of the power.

We studied also the chirp of nonlinear-dispersive similariton versus the chirp of input pulse: the chirp slope of similariton is practically constant, when the pulse intensity is high enough. In case of the dispersive propagation, the induced chirp simply imposes on the initial chirp according to Eq. (1). In case of nonlinear-dispersive propagation, the chirp "forgets" about the initial chirp according to analytic discussion above: it becomes independent of the input chirp according to Eq. (2).

Similariton-Based Spectral Interferometry for Signal Analysis on Femtosecond Time Scale 107

increases with the decreasing of the chirp slope, which is approximately equal to *C* <sup>1</sup>

(a)

9 m.

Fig. 4 and 5 illustrate the typical behavior of nonlinear-dispersive similariton in case of *f* =

(b) (c)

(d)

Fig. 4. (a-d) Schematic of the experiment given by the spectrograms of the relevant steps; (e) spectral phases of nonlinear-dispersive similaritons generated from input single- (thin black line), double- (red ) and distant double-peak (blue ) pulses in comparison with the one

frequency (rad/fs)

(e)

Having the spectral phase and spectral profile, the temporal profile of nonlinear-dispersive similariton is retrieved. Fig. 5 shows the spectral and temporal profiles of the similaritons with the spectral phases of Fig. 4(e). The black curves are the spectra and the gray-dotted curves are the pulses. They coincide with each other, that is, takes place the self-

for pure dispersively propagated single-peak pulse (thick yellow line).

.

## **2.2 Spectral interferometric characterization of nonlinear-dispersive similariton**

We carried out experimental studies to check-confirm the predictions of our rough discussion and numerical analysis above. We applied the classic method of SI [2] to completely characterize and study the generation process and peculiarities of nonlineardispersive similariton of a passive fiber.

Fig. 3 schematically illustrates our experiment. Using a Mach-Zender interferometer, we split the input radiation of a standard Coherent Verdi V10-Mira 900F femtosecond laser system into two parts.

Fig. 3. Schematic of the experiment. Laser – Coherent Verdi V10-Mira 900F femtosecond laser system, M – mirrors, P – prisms, L – lens, OSA – optical spectral analyzer, and SMF – single-mode optical fiber.

The low-power pulse serves as a reference. For the high-power pulse, first we filter its spectrum of the bandwidth = 11 nm down to the value = 2 nm. We use standard polarization-preserving fibers Newport F-SPF @820 nm and ThorLabs HP @ 780 nm of different lengths – 1 m, 9 m, and 36 m. The spectra of the pulses at the output of fiber are broadened, however, the spectrum of the reference pulse covers them completely. This allows measuring the spectral phase of similariton within the whole range of its spectrum. The spectral interferometric fringe pattern is recorded by an optical spectrum analyzer (OSA Ando 6315), and the spectral phase is retrieved. Having the spectrum and retrieved spectral phase, the temporal profile of the similariton is reconstructed by Fourier transformation.

The performance of the experiment is given schematically in Fig. 4 by means of the spectrograms of relevant steps. Fig. 4(a) shows the spectrum of the laser pulse, (b) is the spectrum of spectrally filtered and shaped pulse, (c) is the spectrum of nonlinear-dispersive similariton and (d) is the SI fringe pattern. Fig. 4(e) shows the measured spectral phases of the similaritons generated from different input pulses. The spectral phases are parabolic ( <sup>2</sup> /2 ) and their coefficients have nearly the same values in all cases of dispersive and nonlinear-dispersive propagations: = 0.32 ps2 for the pure dispersive propagation of single-peak pulse, and 0.33 ps2, 0.328 ps2, 0.35 ps2 for the nonlinear-dispersive propagations of single-, double- and distant double-peak pulses, respectively. The parabolic phase (linear chirp) leads to the self-spectrotemporal imaging of similariton. The accuracy of imaging

We carried out experimental studies to check-confirm the predictions of our rough discussion and numerical analysis above. We applied the classic method of SI [2] to completely characterize and study the generation process and peculiarities of nonlinear-

Fig. 3 schematically illustrates our experiment. Using a Mach-Zender interferometer, we split the input radiation of a standard Coherent Verdi V10-Mira 900F femtosecond laser

Fig. 3. Schematic of the experiment. Laser – Coherent Verdi V10-Mira 900F femtosecond laser system, M – mirrors, P – prisms, L – lens, OSA – optical spectral analyzer, and SMF –

Laser

The low-power pulse serves as a reference. For the high-power pulse, first we filter its

polarization-preserving fibers Newport F-SPF @820 nm and ThorLabs HP @ 780 nm of different lengths – 1 m, 9 m, and 36 m. The spectra of the pulses at the output of fiber are broadened, however, the spectrum of the reference pulse covers them completely. This allows measuring the spectral phase of similariton within the whole range of its spectrum. The spectral interferometric fringe pattern is recorded by an optical spectrum analyzer (OSA Ando 6315), and the spectral phase is retrieved. Having the spectrum and retrieved spectral phase, the temporal profile of the similariton is reconstructed by Fourier transformation.

The performance of the experiment is given schematically in Fig. 4 by means of the spectrograms of relevant steps. Fig. 4(a) shows the spectrum of the laser pulse, (b) is the spectrum of spectrally filtered and shaped pulse, (c) is the spectrum of nonlinear-dispersive similariton and (d) is the SI fringe pattern. Fig. 4(e) shows the measured spectral phases of the similaritons generated from different input pulses. The spectral phases are parabolic

single-peak pulse, and 0.33 ps2, 0.328 ps2, 0.35 ps2 for the nonlinear-dispersive propagations of single-, double- and distant double-peak pulses, respectively. The parabolic phase (linear chirp) leads to the self-spectrotemporal imaging of similariton. The accuracy of imaging

= 11 nm down to the value

L

have nearly the same values in all cases of dispersive

= 0.32 ps2 for the pure dispersive propagation of

= 2 nm. We use standard

**2.2 Spectral interferometric characterization of nonlinear-dispersive similariton** 

dispersive similariton of a passive fiber.

system into two parts.

single-mode optical fiber.

( <sup>2</sup> 

spectrum of the bandwidth

/2 ) and their coefficients

and nonlinear-dispersive propagations:

increases with the decreasing of the chirp slope, which is approximately equal to *C* <sup>1</sup> . Fig. 4 and 5 illustrate the typical behavior of nonlinear-dispersive similariton in case of *f* = 9 m.

Fig. 4. (a-d) Schematic of the experiment given by the spectrograms of the relevant steps; (e) spectral phases of nonlinear-dispersive similaritons generated from input single- (thin black line), double- (red ) and distant double-peak (blue ) pulses in comparison with the one for pure dispersively propagated single-peak pulse (thick yellow line).

Having the spectral phase and spectral profile, the temporal profile of nonlinear-dispersive similariton is retrieved. Fig. 5 shows the spectral and temporal profiles of the similaritons with the spectral phases of Fig. 4(e). The black curves are the spectra and the gray-dotted curves are the pulses. They coincide with each other, that is, takes place the self-

Similariton-Based Spectral Interferometry for Signal Analysis on Femtosecond Time Scale 109

similariton chirp studies by a simpler way of the spectrum and autocorrelation track measurements, and afterwards calculation of the spectrum autocorrelation. The comparison of the measured and calculated autocorrelations extracts the chirp slope. A good accordance between the chirp slope values measured by this and SI methods occurs. This simple method permits checking easily the results on similariton chirp numerical study. The experimental results, in agreement with the theory, show that the chirp slope of nonlineardispersive similariton is practically independent of the input pulse phase modulation in a

Taking into account the relation between nonlinear-dispersive similariton and rectangular pulse discussed, it seems reasonable to expect that the *bandwidth of similariton* is equal to the one for rectangular pulse. To determine the bandwidth (and afterwards the duration) of nonlinear-dispersive similariton, the relation for the pulse optimal compression can be used [23]. This gives the following relation for the spectral broadening of nonlinear-dispersive

*L L D NL* / / / *in* <sup>0</sup> *kW t*

power of pulse radiation with a repetition rate *v* , and *k* <sup>1</sup>

given by the fiber parameters ( *n*2 – coefficient of the Kerr nonlinearity, 0 0

the input pulse power, *in t* – input pulse duration, *W pv* / – pulse energy, *p* – average

is checked numerically (a) and experimentally (b). The confirmation of the truthfulness of

*k P* , / <sup>2</sup> *t Ck*

For comparison, the spectral bandwidth of the similariton generated in a fiber amplifier is

Fig. 7. (a) Simulation: spectral broadening versus *R*; (b) Experiment: similariton bandwidth

 versus 1/2 (/ ) *in p t* . Blue lines correspond to the transform-limited pulse of 100 fs duration, red and cyan – to 140 fs pulse, green and black – to 225 fs pulse, magenta and yellow – to 320 fs pulse. Red, green and magenta are related to pulses stretched in a medium with normal dispersion, and cyan, black, and yellow – to pulses stretched in a medium with

2 – group-velocity dispersion coefficient, *S* – fiber mode area). This thesis (Fig. 7)

 

, where *W* and *P* are the input pulse energy

/ <sup>0</sup> *k P*

20 2 *n S* 

bandwidth and *t* duration

*f P* . (3)

 2 / 

, where *P* is

– wave

( ) is a coefficient

wide range of <sup>2</sup>

number,

 

 0 0 

similariton: / *<sup>o</sup> b R* 

of nonlinear-dispersive similariton:

( ) *z* 2 1/3

anomalous dispersion.

from -3 to +3.

the brief discussion above gives the following rule for the

 

0 2 <sup>2</sup> [( ) /(2 )] exp( / 3) *g n PW S gz*

 

and power, and *g* is the gain coefficient [31].

spectrotemporal imaging of nonlinear-dispersive similariton. A good spectrotemporal similarity is seen in case of input single-peak pulse (a). For the input double-peak pulse of (b), the matching between the spectral and temporal profiles of similaritons is qualitative only. To obtain a quantitative agreement, one must use a longer fiber, increasing the coefficient: the spectral and temporal profiles of similaritons practically coincide for the thick orange line of (b), showing the temporal profile of similariton for the increased coefficient.

Fig. 5. Self-imaging of nonlinear-dispersive similaritons generated from input (a) single- , and (b) double-peak pulses. Black solid curves show the spectra of nonlinear-dispersively propagated pulses (the black dotted one of (a) stands for pure dispersively propagated pulse), gray dotted lines show the retrieved temporal profiles. The thick orange curve of (b) shows the temporal profile of similariton for 4increased coefficient of spectral phase.

To show the relation between the nonlinear-dispersive similariton and the rectangular pulse of fiber, spectral interferometric measurements are carried out using a short fiber ( *f* = 1 m). Fig. 6 illustrates the shaping of a rectangular pulse in a nonlinear-dispersive fiber.

Fig. 6. (a) Spectrum of nonlinear-dispersive rectangular pulse with the relevant spectral phase (black solid line) and fitted parabola (pink). The blue line is a high-order polynomial fit. (b) Temporal profile of the rectangular pulse (blue) in comparison with the pulse retrieved by the fitted parabolic spectral phase (pink).

Here the black curves are the spectrum and spectral phase measured, the blue curve is a high-order polynomial fit and the pink is a fitted parabola. The measured spectral phase has a parabolic shape only at the central energy-carrying part of spectrum. Deviation from the parabola at the wings leads to the shaping of a rectangular pulse shown in Fig. 6(b) with the blue curve. Even in this case the chirp slope at the central energy-carrying part of the pulse/spectrum is also determined only by the fiber length ( = 0.0465 ps2). The pink curve of Fig. 6(b) is the retrieved pulse by the fitted parabolic spectral phase.

The complete and precise SI study confirmed the principal description of nonlineardispersive similariton by Eq. (2), leading to its self-spectrotemporal similarity and imaging by the scaling coefficient of the chirp slope <sup>1</sup> <sup>2</sup> *C z* ( ) . This allows carrying out the

spectrotemporal imaging of nonlinear-dispersive similariton. A good spectrotemporal similarity is seen in case of input single-peak pulse (a). For the input double-peak pulse of (b), the matching between the spectral and temporal profiles of similaritons is qualitative only. To obtain a quantitative agreement, one must use a longer fiber, increasing the

coefficient: the spectral and temporal profiles of similaritons practically coincide for the thick orange line of (b), showing the temporal profile of similariton for the increased

Fig. 5. Self-imaging of nonlinear-dispersive similaritons generated from input (a) single- , and (b) double-peak pulses. Black solid curves show the spectra of nonlinear-dispersively propagated pulses (the black dotted one of (a) stands for pure dispersively propagated pulse), gray dotted lines show the retrieved temporal profiles. The thick orange curve of (b)

frequency (rad/fs) frequency (rad/fs)

To show the relation between the nonlinear-dispersive similariton and the rectangular pulse of fiber, spectral interferometric measurements are carried out using a short fiber ( *f* = 1 m).

Fig. 6 illustrates the shaping of a rectangular pulse in a nonlinear-dispersive fiber.

Fig. 6. (a) Spectrum of nonlinear-dispersive rectangular pulse with the relevant spectral phase (black solid line) and fitted parabola (pink). The blue line is a high-order polynomial fit. (b) Temporal profile of the rectangular pulse (blue) in comparison with the pulse

Here the black curves are the spectrum and spectral phase measured, the blue curve is a high-order polynomial fit and the pink is a fitted parabola. The measured spectral phase has a parabolic shape only at the central energy-carrying part of spectrum. Deviation from the parabola at the wings leads to the shaping of a rectangular pulse shown in Fig. 6(b) with the blue curve. Even in this case the chirp slope at the central energy-carrying part of the

The complete and precise SI study confirmed the principal description of nonlineardispersive similariton by Eq. (2), leading to its self-spectrotemporal similarity and imaging

> <sup>2</sup> *C z* ( )

increased coefficient of spectral phase.

(a) (b)

= 0.0465 ps2). The pink curve

. This allows carrying out the

shows the temporal profile of similariton for 4

retrieved by the fitted parabolic spectral phase (pink).

pulse/spectrum is also determined only by the fiber length (

by the scaling coefficient of the chirp slope <sup>1</sup>

of Fig. 6(b) is the retrieved pulse by the fitted parabolic spectral phase.

coefficient.

similariton chirp studies by a simpler way of the spectrum and autocorrelation track measurements, and afterwards calculation of the spectrum autocorrelation. The comparison of the measured and calculated autocorrelations extracts the chirp slope. A good accordance between the chirp slope values measured by this and SI methods occurs. This simple method permits checking easily the results on similariton chirp numerical study. The experimental results, in agreement with the theory, show that the chirp slope of nonlineardispersive similariton is practically independent of the input pulse phase modulation in a wide range of <sup>2</sup> 0 0 from -3 to +3.

Taking into account the relation between nonlinear-dispersive similariton and rectangular pulse discussed, it seems reasonable to expect that the *bandwidth of similariton* is equal to the one for rectangular pulse. To determine the bandwidth (and afterwards the duration) of nonlinear-dispersive similariton, the relation for the pulse optimal compression can be used [23]. This gives the following relation for the spectral broadening of nonlinear-dispersive similariton: / *<sup>o</sup> b R L L D NL* / / / *in* <sup>0</sup> *kW t* / <sup>0</sup> *k P* , where *P* is the input pulse power, *in t* – input pulse duration, *W pv* / – pulse energy, *p* – average power of pulse radiation with a repetition rate *v* , and *k* <sup>1</sup> 20 2 *n S* ( ) is a coefficient given by the fiber parameters ( *n*2 – coefficient of the Kerr nonlinearity, 0 0 2 / – wave number, 2 – group-velocity dispersion coefficient, *S* – fiber mode area). This thesis (Fig. 7) is checked numerically (a) and experimentally (b). The confirmation of the truthfulness of the brief discussion above gives the following rule for the bandwidth and *t* duration of nonlinear-dispersive similariton:

$$
\Delta o = k \sqrt{P} \quad , \quad \Delta t = \Delta o \;/\; \mathcal{C} = k \beta\_2 f \sqrt{P} \;. \tag{3}
$$

For comparison, the spectral bandwidth of the similariton generated in a fiber amplifier is ( ) *z* 2 1/3 0 2 <sup>2</sup> [( ) /(2 )] exp( / 3) *g n PW S gz* , where *W* and *P* are the input pulse energy and power, and *g* is the gain coefficient [31].

Fig. 7. (a) Simulation: spectral broadening versus *R*; (b) Experiment: similariton bandwidth versus 1/2 (/ ) *in p t* . Blue lines correspond to the transform-limited pulse of 100 fs duration, red and cyan – to 140 fs pulse, green and black – to 225 fs pulse, magenta and yellow – to 320 fs pulse. Red, green and magenta are related to pulses stretched in a medium with normal dispersion, and cyan, black, and yellow – to pulses stretched in a medium with anomalous dispersion.

Similariton-Based Spectral Interferometry for Signal Analysis on Femtosecond Time Scale 111

50 THz bandwidth. It is conditioned by the physical pattern of the process: the nonlinear self-interaction of powerful pulses leads to large spectral broadening (with the factor of *b* ~10 just at the first ~1 cm of propagation in fiber, in our case), substantially increasing the impact of dispersion (with the factors of <sup>2</sup> *b* ~100 for SOD, and <sup>3</sup> *b* ~1000 for TOD), resulting in the pulse stretching and peak intensity decreasing (with the factor of <sup>2</sup> *b* ~100), and thus essentially decreasing and blocking out the high-order nonlinear effects. The impact of highorder nonlinear effects can be significant in case of high-power pulse propagation in lowdispersion fibers, e.g., for photonic crystal fibers with all-normal flattened dispersion, where a longer and more efficient nonlinear self-interaction results in the generation of octave

The numerical analysis shows that the fiber TOD is expressed additively in the

( ,) *f t* ( , )| *Ct f* 

simulations for the SOD and TOD coefficients of fused silica (

( ,) *f t* parabolic spectral and temporal phase profiles due to its small impact as compared to the SOD, and its value is the same for the nonlinear-dispersive similariton and the

<sup>3</sup> <sup>3</sup>

27.44 fs3/mm). Thus, the chirp measurement of broadband similariton becomes urgent, since it gives the TOD of fiber and permits to generalize the description of Eq. (2'). The SI study permitted the complete characterization of nonlinear-dispersive similariton of up to 5 THz bandwidths [23]. For broadband similaritons (of up to 50-THz bandwidths), the chirp measurement technique through spectral compression and frequency tuning in the sum-

In the *experiment*, a broadband nonlinear-dispersive similariton of 50-THz FWHMbandwidth is generated in a piece of passive fiber and the chirp measurement is carried out through frequency tuning in the SFG-spectral compression process [44-47]. Fig. 8 shows the schematic of the experimental setup. The Coherent Verdi V10 + Mira 900F femtosecond laser system is used, with the following parameters of radiation: 100 fs pulse duration, 76 MHz repetition rate, 1.6 W average power, 800 nm central wavelength. Beam-splitter (BS) splits the laser radiation into high- and low-power parts (80%+20%). The high-power pulse (100 fs FWHM-duration and pulse energy of up to 7 nJ, corresponding to 70 kW peak power) is injected into a standard single-mode fiber (1.65 m Newport F-SPF PP@820 nm) by means of a 10 microscope objective and a broadband nonlinear-dispersive similariton is generated.

Fig. 9(a) shows the spectrum of 107-nm (50 THz) FWHM-bandwidth similariton, recorded by the optical spectrum analyzer Ando 6315 (OSA). This spectral profile represents the spectrotemporal image of the generated similariton in the approximation of its linear chirp. The asymmetry in this spectrotemporal profile evidences the impact of TOD. Although the initial pulse asymmetry can also cause the spectral asymmetry in the near field of dispersion, typically for the picosecond-scale experiments, the impact of the possible

 

*f*

3 is the TOD coefficient. The precision of Eq. (4) is of ~4%, according to the

6 *Ct*

 <sup>3</sup> <sup>3</sup>

3 2 <sup>2</sup> 6

*t f* 

, (4)

<sup>2</sup> = 36.11 fs2/mm and

<sup>3</sup> =

(,) *f* and

spanning supercontinuum [48].

spectron pulse of pure dispersive propagation:

frequency generation process is applied [44,45].

where

This revealed property of nonlinear-dispersive similariton of Eq. (3) can be used for the measurement of femtosecond pulse duration, alternatively to the autocorrelation technique.

*Concluding,* our spectral interferometric studies demonstrate the following properties of nonlinear-dispersive similariton generated in a passive fiber:


## **3. Broadband similariton for femtosecond signal analysis and synthesis**

The applications to ultrafast optics demand the generation and study of *broadband similariton*  [43,44]. Particularly, the pulse compression ratio is as high as the spectral broadening factor is [41], the resolution of the femtosecond oscilloscope, based on the similariton-induced parabolic lens is given by the bandwidth of similariton [22], and for the similariton-based SI, the application range is as large as broadband the similariton-reference is [43,46].

In this section, our studies on the generation and characterization of a broadband, 50-THz bandwidth nonlinear-dispersive similariton are presented with the objective to reveal its distinctive properties in view of its applications to the signal synthesis and analysis problems on the femtosecond time scale. According to the spectral-interferometric characterization of nonlinear-dispersive similariton of the bandwidths of 5 THz ( ~10 nm at ~ 800 nm), it is described by Eq. (2), or for its slowly varying amplitude *Aft* ( ,) and phase ( , ) *<sup>f</sup> f t* we have:

$$A(f,t) \propto \tilde{A}^\*(f,\alpha)|\_{\alpha=\odot \quad \text{'}},\\\phi\_f(\mathbf{t}) = -\phi\_f(\alpha)\Big|\_{\alpha=\odot \quad \text{'}} = \beta\_2 f \alpha^2 \Big/ 2\Big|\_{\alpha=\odot \quad \text{'}} = \mathbf{C}t^2 \Big/ 2\Big.\tag{2'}$$

For comparison, the spectron pulse has the same dispersion-induced phase [Eq. (1)]. The applications of similariton demand to check and generalize this key peculiarity for broadband pulses.

First, the *numerical modeling* is carried out to have the complete physical pattern and reveal the distinctive peculiarities of the generation and propagation of broadband similariton. The mathematical description, based on the generalized nonlinear Schrödinger equation, considers the high-order terms of third-order dispersion (TOD), shock wave (selfsteepening) and delayed nonlinear response (related to the Raman gain), together with the principal terms of self-phase modulation and second order dispersion (SOD) [44,47]. The split-step Fourier method is used in the procedures of numerical solution of the equation in simulations for 100-fs pulses of a standard laser with up to 500 mW of average power at a 76 MHz repetition rate (70 kW of peak power) in a few meters of standard single-mode fiber (losses are negligible). Simulations in these conditions show that the high-order nonlinear factors of shock wave and delayed nonlinear response do not impact on the process under study; however, the impact of TOD is expressed in the generated broadband similaritons of

This revealed property of nonlinear-dispersive similariton of Eq. (3) can be used for the measurement of femtosecond pulse duration, alternatively to the autocorrelation technique. *Concluding,* our spectral interferometric studies demonstrate the following properties of

linear chirp, with a slope given only by the fiber dispersion and independent of the

property of spectrotemporal similarity / self-spectrotemporal imaging, with accuracy

The applications to ultrafast optics demand the generation and study of *broadband similariton*  [43,44]. Particularly, the pulse compression ratio is as high as the spectral broadening factor is [41], the resolution of the femtosecond oscilloscope, based on the similariton-induced parabolic lens is given by the bandwidth of similariton [22], and for the similariton-based SI,

In this section, our studies on the generation and characterization of a broadband, 50-THz bandwidth nonlinear-dispersive similariton are presented with the objective to reveal its distinctive properties in view of its applications to the signal synthesis and analysis problems on the femtosecond time scale. According to the spectral-interferometric

800 nm), it is described by Eq. (2), or for its slowly varying amplitude *Aft* ( ,) and

 <sup>2</sup> / 2 *Ct <sup>f</sup>* 

  ~10 nm

<sup>2</sup> *Ct* / 2 . (2')

**3. Broadband similariton for femtosecond signal analysis and synthesis** 

the application range is as large as broadband the similariton-reference is [43,46].

characterization of nonlinear-dispersive similariton of the bandwidths of 5 THz (

 <sup>2</sup>

For comparison, the spectron pulse has the same dispersion-induced phase [Eq. (1)]. The applications of similariton demand to check and generalize this key peculiarity for

First, the *numerical modeling* is carried out to have the complete physical pattern and reveal the distinctive peculiarities of the generation and propagation of broadband similariton. The mathematical description, based on the generalized nonlinear Schrödinger equation, considers the high-order terms of third-order dispersion (TOD), shock wave (selfsteepening) and delayed nonlinear response (related to the Raman gain), together with the principal terms of self-phase modulation and second order dispersion (SOD) [44,47]. The split-step Fourier method is used in the procedures of numerical solution of the equation in simulations for 100-fs pulses of a standard laser with up to 500 mW of average power at a 76 MHz repetition rate (70 kW of peak power) in a few meters of standard single-mode fiber (losses are negligible). Simulations in these conditions show that the high-order nonlinear factors of shock wave and delayed nonlinear response do not impact on the process under study; however, the impact of TOD is expressed in the generated broadband similaritons of

*Ct* , () ( ) *f f Ct t*

nonlinear-dispersive similariton generated in a passive fiber:

relation with the rectangular pulse of nonlinear-dispersive fiber;

determined by spectral broadening and pulse stretching together; only initial pulse power determines the spectral bandwidth of similariton.

amplitude, chirp and power of the input pulse;

at ~ 

phase ( , ) *<sup>f</sup>* 

broadband pulses.

*f t* we have:

( , ) \* ( , )| *Aft A f*

  50 THz bandwidth. It is conditioned by the physical pattern of the process: the nonlinear self-interaction of powerful pulses leads to large spectral broadening (with the factor of *b* ~10 just at the first ~1 cm of propagation in fiber, in our case), substantially increasing the impact of dispersion (with the factors of <sup>2</sup> *b* ~100 for SOD, and <sup>3</sup> *b* ~1000 for TOD), resulting in the pulse stretching and peak intensity decreasing (with the factor of <sup>2</sup> *b* ~100), and thus essentially decreasing and blocking out the high-order nonlinear effects. The impact of highorder nonlinear effects can be significant in case of high-power pulse propagation in lowdispersion fibers, e.g., for photonic crystal fibers with all-normal flattened dispersion, where a longer and more efficient nonlinear self-interaction results in the generation of octave spanning supercontinuum [48].

The numerical analysis shows that the fiber TOD is expressed additively in the (,) *f* and ( ,) *f t* parabolic spectral and temporal phase profiles due to its small impact as compared to the SOD, and its value is the same for the nonlinear-dispersive similariton and the spectron pulse of pure dispersive propagation:

$$
\Delta\phi(f,t) \approx \left.\Delta\phi(f,\alpha)\right|\_{\alpha=\text{Cr}} = \left.-\frac{\beta\_3 f}{6} \alpha^3\right|\_{\alpha=\text{Cr}} \approx \left.-\frac{\beta\_3}{6\beta\_2^3 f^2} t^3\right|,\tag{4}
$$

where 3 is the TOD coefficient. The precision of Eq. (4) is of ~4%, according to the simulations for the SOD and TOD coefficients of fused silica ( <sup>2</sup> = 36.11 fs2/mm and <sup>3</sup> = 27.44 fs3/mm). Thus, the chirp measurement of broadband similariton becomes urgent, since it gives the TOD of fiber and permits to generalize the description of Eq. (2'). The SI study permitted the complete characterization of nonlinear-dispersive similariton of up to 5 THz bandwidths [23]. For broadband similaritons (of up to 50-THz bandwidths), the chirp measurement technique through spectral compression and frequency tuning in the sumfrequency generation process is applied [44,45].

In the *experiment*, a broadband nonlinear-dispersive similariton of 50-THz FWHMbandwidth is generated in a piece of passive fiber and the chirp measurement is carried out through frequency tuning in the SFG-spectral compression process [44-47]. Fig. 8 shows the schematic of the experimental setup. The Coherent Verdi V10 + Mira 900F femtosecond laser system is used, with the following parameters of radiation: 100 fs pulse duration, 76 MHz repetition rate, 1.6 W average power, 800 nm central wavelength. Beam-splitter (BS) splits the laser radiation into high- and low-power parts (80%+20%). The high-power pulse (100 fs FWHM-duration and pulse energy of up to 7 nJ, corresponding to 70 kW peak power) is injected into a standard single-mode fiber (1.65 m Newport F-SPF PP@820 nm) by means of a 10 microscope objective and a broadband nonlinear-dispersive similariton is generated.

Fig. 9(a) shows the spectrum of 107-nm (50 THz) FWHM-bandwidth similariton, recorded by the optical spectrum analyzer Ando 6315 (OSA). This spectral profile represents the spectrotemporal image of the generated similariton in the approximation of its linear chirp. The asymmetry in this spectrotemporal profile evidences the impact of TOD. Although the initial pulse asymmetry can also cause the spectral asymmetry in the near field of dispersion, typically for the picosecond-scale experiments, the impact of the possible

Similariton-Based Spectral Interferometry for Signal Analysis on Femtosecond Time Scale 113

characterizes the generated broadband similariton. Its projections represent the temporal

spectrotemporal similarity of broadband similariton described by Eqs. (2') and (4). In general, through the Fourier transformation of complex temporal amplitude, given by the measured temporal pulse and chirp, the spectral complex amplitude and afterwards the spectral phase could be retrieved. For our case of the broadband similariton, the Eq. (2')

, as well as with the derivative of spectral phase ( )

Fig. 9. (a) Measured spectrum of broadband similariton, and (b) 3D frequency tuning patterns with spectral compression (sharp peaks) and without it (thick lines) for the chirp measurement: the 40 nm (75 THz) frequency tuning at 400 nm is adequate to the 160 nm spectral range of similariton at 800 nm for the 32 ps range of temporal delay between SFG-

separately (inset). The circles in Fig. 10 are the measured experimental points; the dashed

curve obtained this way. It is described by the polynomial ( )

. The quadratic component of ( )

permits to have the spectral phase information by a simple scaling

*<sup>f</sup>* ( )/ *t C* for the derivative of spectral phase.

, and the curve

 *f* 

( )*t* connected with the

, according to the

*Ct* , and we have

 *f*   *f* = –

is shown

and spectral profiles of the intensity *I t*( ) and *I*( )

(a)

(b)

chirp ( ) ( ) *<sup>f</sup> t t* 

interacting pulses.

15.06 103 fs3 <sup>2</sup>

Fig. 10 shows the ( )

 *f* 

– 81.13 103 fs2

and solid curves are for the linear and parabolic fits, respectively.

( ) *<sup>f</sup>*

asymmetry of the initial laser pulse on the spectrotemporal shape of similariton becomes insignificant, according to our simulations.

For the characterization of broadband similariton, its intensity profile can be measured by means of the cross-correlation technique, using the SFG-interaction of similariton with the laser pulse (as a reference). A spectral detection of the SFG-signal will give also information on the chirp of similariton, modifying the cross-correlation technique to the cross-correlation frequency-resolved optical gating (XFROG) [49,50]. Our additional modification is the use of a dispersively chirped reference pulse, which provides a spectrally compressed SFG-signal in a wider spectral range, and thus, more efficient measurement [22,44]. Experimentally, the low-power pulse is directed into the dispersive delay line with anomalous dispersion (Dline; conventional prism compressor consisting of a 3.5-m separated SF11 prism pair with a reverse mirror) and stretch it 22 times, resulting in the pulse autocorrelation duration of 3.1 ps. Then, using a lens, we direct the similariton and the dispersively stretched pulse to the nonlinear -barium borate crystal (BBO, type 1 – ooe, 800 nm operating wavelength) to have SFG-spectral compression. The SFG-interaction of up- and down-chirped pulses results in the chirp cancellation and spectral compression, and a temporal delay between these two pulses leads to the frequency shift of the SFG-signal, according to the concept of the temporal lens [22]. In the experiment, the temporal delay is provided by shifting the reverse mirror of the D-line and recording the relevant SFG-compressed spectra by OSA. Measurements with D-line and without it are carried out, replacing the D-line with a simple temporal delay (TD), to compare the techniques of SFG-spectral compression and XFROG. Fig. 9(b) shows the relevant 3D frequency tuning patterns for the chirp measurement.

Fig. 8. Schematic of experimental setup: laser – Coherent Verdi V10 + Mira 900F femtosecond laser system, BS – beam splitter, M – mirrors, D-line – dispersive delay line (conventional prism compressor consisting of a SF11 prism pair with a reverse mirror), TD – temporal delay, fiber – Newport F-SPF PP@820 nm, SFG – BBO crystal for SFG, OSA – optical spectrum analyzer.

The technique of SFG-spectral compression, as compared to XFROG, is more efficient, providing sharper spectral signal in a wider spectral (and temporal) range. The temporal delay between the SFG-interacting pulses in the range of 16 ps results in a 20 nm wavelength shift for the 22 times SFG-spectrally compressed signal (down to 0.12 nm at 400 nm central wavelength), corresponding to the chirp measurement of similariton in the span range of 160 nm (75 THz) at 800 nm central wavelength. This 3D pattern completely

asymmetry of the initial laser pulse on the spectrotemporal shape of similariton becomes

For the characterization of broadband similariton, its intensity profile can be measured by means of the cross-correlation technique, using the SFG-interaction of similariton with the laser pulse (as a reference). A spectral detection of the SFG-signal will give also information on the chirp of similariton, modifying the cross-correlation technique to the cross-correlation frequency-resolved optical gating (XFROG) [49,50]. Our additional modification is the use of a dispersively chirped reference pulse, which provides a spectrally compressed SFG-signal in a wider spectral range, and thus, more efficient measurement [22,44]. Experimentally, the low-power pulse is directed into the dispersive delay line with anomalous dispersion (Dline; conventional prism compressor consisting of a 3.5-m separated SF11 prism pair with a reverse mirror) and stretch it 22 times, resulting in the pulse autocorrelation duration of 3.1 ps. Then, using a lens, we direct the similariton and the dispersively stretched pulse to the


have SFG-spectral compression. The SFG-interaction of up- and down-chirped pulses results in the chirp cancellation and spectral compression, and a temporal delay between these two pulses leads to the frequency shift of the SFG-signal, according to the concept of the temporal lens [22]. In the experiment, the temporal delay is provided by shifting the reverse mirror of the D-line and recording the relevant SFG-compressed spectra by OSA. Measurements with D-line and without it are carried out, replacing the D-line with a simple temporal delay (TD), to compare the techniques of SFG-spectral compression and XFROG. Fig. 9(b) shows the relevant 3D frequency tuning patterns for the chirp measurement.

Fig. 8. Schematic of experimental setup: laser – Coherent Verdi V10 + Mira 900F

femtosecond laser system, BS – beam splitter, M – mirrors, D-line – dispersive delay line (conventional prism compressor consisting of a SF11 prism pair with a reverse mirror), TD – temporal delay, fiber – Newport F-SPF PP@820 nm, SFG – BBO crystal for SFG, OSA –

The technique of SFG-spectral compression, as compared to XFROG, is more efficient, providing sharper spectral signal in a wider spectral (and temporal) range. The temporal delay between the SFG-interacting pulses in the range of 16 ps results in a 20 nm wavelength shift for the 22 times SFG-spectrally compressed signal (down to 0.12 nm at 400 nm central wavelength), corresponding to the chirp measurement of similariton in the span range of 160 nm (75 THz) at 800 nm central wavelength. This 3D pattern completely

insignificant, according to our simulations.

nonlinear

optical spectrum analyzer.

characterizes the generated broadband similariton. Its projections represent the temporal and spectral profiles of the intensity *I t*( ) and *I*( ) , and the curve ( )*t* connected with the chirp ( ) ( ) *<sup>f</sup> t t* , as well as with the derivative of spectral phase ( ) *f* , according to the spectrotemporal similarity of broadband similariton described by Eqs. (2') and (4). In general, through the Fourier transformation of complex temporal amplitude, given by the measured temporal pulse and chirp, the spectral complex amplitude and afterwards the spectral phase could be retrieved. For our case of the broadband similariton, the Eq. (2') permits to have the spectral phase information by a simple scaling *Ct* , and we have ( ) *<sup>f</sup> <sup>f</sup>* ( )/ *t C* for the derivative of spectral phase.

Fig. 9. (a) Measured spectrum of broadband similariton, and (b) 3D frequency tuning patterns with spectral compression (sharp peaks) and without it (thick lines) for the chirp measurement: the 40 nm (75 THz) frequency tuning at 400 nm is adequate to the 160 nm spectral range of similariton at 800 nm for the 32 ps range of temporal delay between SFGinteracting pulses.

Fig. 10 shows the ( ) *f* curve obtained this way. It is described by the polynomial ( ) *f* = – 15.06 103 fs3 <sup>2</sup> – 81.13 103 fs2 . The quadratic component of ( ) *f* is shown separately (inset). The circles in Fig. 10 are the measured experimental points; the dashed and solid curves are for the linear and parabolic fits, respectively.

Similariton-Based Spectral Interferometry for Signal Analysis on Femtosecond Time Scale 115

present its demonstration in comparison with the method of pulse spectrotemporal imaging in the similariton-induced temporal lens. Application of similariton to the reference-based methods upgrades them up to the self-referencing ones, substantially improving their performance due to the enlarged application range along with the simplicity of the principle

Our experimental study of nonlinear-dispersive similariton, described in sec. 2.2, as a remarkable example for the demonstration-application of SI-approach, clearly shows that the classic SI provides accurate measurement with a rather simple setup, but its application range is restricted by the bandwidth of the reference. As mentioned above, the SI characterization of a signal that has undergone a nonlinear interaction with medium requires a special broadband reference to fully cover the broadened signal spectrum. To avoid this restriction, the self-referencing methods of spectral shearing interferometry, such as SPIDER [3] and SPIRIT [4], are developed. This improvement promotes the SI to the class of the most popular and commercialized methods of accurate measurements on the femtosecond time scale, making it compatible with the FROG [1] and MIIPS [6] techniques, at the expense of a more complicated optical arrangement. Our proposed method of similariton-based SI, along with its self-referencing performance, keeps the simplicity of the

Describing the principle of *similariton-based SI*, for its implementation the setup of Fig. 8 is used, removing the nonlinear BBO crystal and D-line. Splitting the signal beam, its part is injected into a fiber to generate the nonlinear-dispersive similariton-reference, with the

> | ( , )|exp[ ( , )] *Af i f*

similariton in a spectrometer with an appropriate time delay. The spectral fringe pattern

whole phase information becomes available, for any signal. The known spectral phase of the

also the signal spectrum, to reconstruct the complex temporal amplitude *A t* (0, ) of the signal through Fourier transformation. Thus, the method of similariton-based SI joins the advantages of both the classic SI [2] and spectral shearing interferometry [3-5], combining the simplicity of the principle and configuration with the self-referencing performance. Examining the similariton-based SI, we compare its measurements with the ones carried out by a prototype of the femtosecond oscilloscope (FO) based on the pulse spectrotemporal imaging in the similariton-induced temporal lens in the SFG process. Our comparative study, involving also theoretical and autocorrelation check, along with the demonstration and study of the similariton-based SI, serves also for the inspection of the prototype of similariton-based FO, the measurements of which previously were compared with the

The method of SFG-*spectrotemporal imaging* for direct femtosecond scale measurements is based on the conversion of temporal information to the spectral domain in a similaritoninduced parabolic temporal lens [22,46,47]. The setup of the similariton-based SI is modified to FO by replacing the temporal delay (TD) with a dispersive delay line (D-line) and placing

 

 

 | (0, )|exp[ (0, )] *A i* 

*f* , on the background of the signal and

 

(0, ) 

. The residual part of the signal

 <sup>2</sup> | (0, )| *A* 

, is coupled with the

, and the

, and by measuring

similariton spectra, completely covers the signal spectrum *S*(0, )

similariton-reference allows to retrieve the signal spectral phase

and configuration.

principle and configuration of the classic SI.

of a complex spectral amplitude *A*(0, )

2| (0, )|| ( , )|cos[ (0, ) ( , )] *A Af*

complex spectral amplitude *A f* (,)

autocorrelation only [22].

( ) *SSI* 

Concluding, we generated nonlinear-dispersive similariton of 50 THz bandwidth, and carried out its complete characterization through the chirp measurement, using the technique of frequency tuning in the SFG-spectral compression. Our studies state that only fiber dispersion determines the phase (chirp) of broadband nonlinear-dispersive similariton. The fiber TOD results in the same additional phase for broadband nonlinear-dispersive similariton and spectron. The ~1% accuracy of the linear fit for the chirp of the 50-THz bandwidth similariton gives the range of applications for aberration-free similariton-based spectrotemporal imaging [22] and spectral interferometry [45-47]. The described approach to the generation and characterization of broadband similariton can be helpful also for its applications in pulse compression [40,41] and CARS microscopy [51]. For these applications, the low value of TOD can impact significantly and it should be considered more carefully.

Fig. 10. Broadband similariton's chirp ( ) *<sup>f</sup> t* and derivative of spectral phase ( ) *f* with its quadratic component, separately (inset). The ~1% difference from the linear fit in the range of ~50 THz gives the range of applications for the similariton-based methods of signal characterization (Section 4).

## **4. Similariton based self-referencing spectral interferometry for femtosecond pulse characterization**

Our studies on broadband similariton serve as a basis for development of a novel method of similariton-based of SI for the femtosecond pulse complete characterization. Below we

Concluding, we generated nonlinear-dispersive similariton of 50 THz bandwidth, and carried out its complete characterization through the chirp measurement, using the technique of frequency tuning in the SFG-spectral compression. Our studies state that only fiber dispersion determines the phase (chirp) of broadband nonlinear-dispersive similariton. The fiber TOD results in the same additional phase for broadband nonlinear-dispersive similariton and spectron. The ~1% accuracy of the linear fit for the chirp of the 50-THz bandwidth similariton gives the range of applications for aberration-free similariton-based spectrotemporal imaging [22] and spectral interferometry [45-47]. The described approach to the generation and characterization of broadband similariton can be helpful also for its applications in pulse compression [40,41] and CARS microscopy [51]. For these applications, the low value of TOD can impact significantly and it should be considered more carefully.

Fig. 10. Broadband similariton's chirp ( ) *<sup>f</sup>*

characterization (Section 4).

**pulse characterization** 

quadratic component, separately (inset). The ~1% difference from the linear fit in the range of ~50 THz gives the range of applications for the similariton-based methods of signal

**4. Similariton based self-referencing spectral interferometry for femtosecond** 

Our studies on broadband similariton serve as a basis for development of a novel method of similariton-based of SI for the femtosecond pulse complete characterization. Below we

*t* and derivative of spectral phase ( )

 *f* 

with its

present its demonstration in comparison with the method of pulse spectrotemporal imaging in the similariton-induced temporal lens. Application of similariton to the reference-based methods upgrades them up to the self-referencing ones, substantially improving their performance due to the enlarged application range along with the simplicity of the principle and configuration.

Our experimental study of nonlinear-dispersive similariton, described in sec. 2.2, as a remarkable example for the demonstration-application of SI-approach, clearly shows that the classic SI provides accurate measurement with a rather simple setup, but its application range is restricted by the bandwidth of the reference. As mentioned above, the SI characterization of a signal that has undergone a nonlinear interaction with medium requires a special broadband reference to fully cover the broadened signal spectrum. To avoid this restriction, the self-referencing methods of spectral shearing interferometry, such as SPIDER [3] and SPIRIT [4], are developed. This improvement promotes the SI to the class of the most popular and commercialized methods of accurate measurements on the femtosecond time scale, making it compatible with the FROG [1] and MIIPS [6] techniques, at the expense of a more complicated optical arrangement. Our proposed method of similariton-based SI, along with its self-referencing performance, keeps the simplicity of the principle and configuration of the classic SI.

Describing the principle of *similariton-based SI*, for its implementation the setup of Fig. 8 is used, removing the nonlinear BBO crystal and D-line. Splitting the signal beam, its part is injected into a fiber to generate the nonlinear-dispersive similariton-reference, with the complex spectral amplitude *A f* (,) | ( , )|exp[ ( , )] *Af i f* . The residual part of the signal of a complex spectral amplitude *A*(0, ) | (0, )|exp[ (0, )] *A i* , is coupled with the similariton in a spectrometer with an appropriate time delay. The spectral fringe pattern ( ) *SSI* 2| (0, )|| ( , )|cos[ (0, ) ( , )] *A Af f* , on the background of the signal and similariton spectra, completely covers the signal spectrum *S*(0, ) <sup>2</sup> | (0, )| *A* , and the whole phase information becomes available, for any signal. The known spectral phase of the similariton-reference allows to retrieve the signal spectral phase (0, ) , and by measuring also the signal spectrum, to reconstruct the complex temporal amplitude *A t* (0, ) of the signal through Fourier transformation. Thus, the method of similariton-based SI joins the advantages of both the classic SI [2] and spectral shearing interferometry [3-5], combining the simplicity of the principle and configuration with the self-referencing performance. Examining the similariton-based SI, we compare its measurements with the ones carried out by a prototype of the femtosecond oscilloscope (FO) based on the pulse spectrotemporal imaging in the similariton-induced temporal lens in the SFG process. Our comparative study, involving also theoretical and autocorrelation check, along with the demonstration and study of the similariton-based SI, serves also for the inspection of the prototype of similariton-based FO, the measurements of which previously were compared with the autocorrelation only [22].

The method of SFG-*spectrotemporal imaging* for direct femtosecond scale measurements is based on the conversion of temporal information to the spectral domain in a similaritoninduced parabolic temporal lens [22,46,47]. The setup of the similariton-based SI is modified to FO by replacing the temporal delay (TD) with a dispersive delay line (D-line) and placing

Similariton-Based Spectral Interferometry for Signal Analysis on Femtosecond Time Scale 117

Fig. 11. Similariton-based SI for pulses dispersively stretched and chirped in SF11 glasses of different thickness: (a) retrieved spectral phases with the measured spectrum (dotted line), and (b) autocorrelation functions of SI-reconstructed temporal pulses (solid) in comparison

at the wavelength 850 nm, according to the data provided by the fiber manufacturer). Then different multi-peak signal pulses are shaped inserting thin glass plates in parts of the beam. The beam parts passed through the plates obtain time delay with respect to the freepropagated part. The movement of the plates along the vertical axis adjusts the power proportion among the peaks. The thicknesses of the plates give the time delay between the peaks; e.g. a 0.12 mm thick glass plate gives a 200 fs delay, if assuming the refractive index of the plate equal to 1.5. Using double- and triple- peak signal pulses, we carry out SI-

The application range and limitations of similariton-based SI are conditioned by the parameters of input radiation and fiber, which are necessary to generate the similaritonreference with parabolic phase. Fig. 12 and 13 illustrate the experiment for the typical regime preceding the similariton shaping: results for double- and triple-peak signal pulses are shown. As Fig. 12 illustrates, having the spectrum (b, black) and SI-retrieved spectral phase (b, blue solid line) of pulse, its temporal profile (c, blue solid line) is reconstructed through Fourier transformation. To check the precision of our measurements through similariton-based SI, we calculate the autocorrelation of reconstructed pulse (d, blue solid line) and compare it with the intensity autocorrelation measured at the input of the system (d, black). The spectral shape of nonlinear-dispersive similariton (a) ensures the fulfillment of necessary conditions for the parabolicity of the spectral phase of similariton (according to [23]). The structure of the similariton spectrum (a), strange at first glance, is typical for short lengths of nonlinear-dispersive interaction, and is observed also for parabolic similaritons generated in fiber amplifiers [52]. The blue solid and red dashed curves in Fig. 12

= 1.995 104 fs2 (5% difference), respectively. Fig. 13 shows the analogue procedures

Finally, we compare the measurements of the similariton-based SI and FO, together with a theoretical check. The double-peak signal pulses are shaped with the spectral domain amplitude- and phase-modulation given by the peaks' temporal distance *T* and their

<sup>2</sup> = 40 fs2/mm ( *D*

*At T* ( ) corresponds to the complex

= 2.1 104 fs2

= –103 ps/nm/km

with autocorrelation traces measured (dashed).

*f* with the values *f* = 49 cm and

measurements and compare the results with measured autocorrelation tracks.

correspond to the pulse reconstruction with the spectral phase coefficients

. The temporal amplitude *A t*( ) <sup>0</sup> *A t*( ) <sup>0</sup>

for a pulse with more complex sub-structure.

expression 2 

and 

proportion

a nonlinear crystal for SFG at the system output (returning to the initial configuration of Fig. 8). In the spectral domain, the dispersive delay works as a parabolic phase modulator, and the signal (0, ) *A* passed through is described as ( , ) *A d* <sup>2</sup> (0, )exp( / 2) *A i <sup>d</sup>* , with the given coefficient <sup>1</sup> *d d <sup>C</sup>* . In the fiber arm, we have a nonlinear-dispersive similariton with the known parameters as in the case of similariton-based SI. In both arms of the setup, we have practically linearly chirped pulses, and the temporal and spectral complex amplitudes repeat each other in the temporal Fraunhofer zone, i.e. spectron pulses are formed [47]: *Adt* ( ,) *A d* (, ) , and *Aft* ( ,) *A f* (,) with *C t <sup>d</sup>*, *<sup>f</sup>* . Under the conditions of the opposite and same value chirps *C CC f d* , and constant similariton spectrum throughout the signal spectrum, the output temporal SFG-signal repeats the input spectral amplitude: ( ) *A t SFG Adt A f t* ( ,) ( ,) *<sup>A</sup>*(0, ) . Accordingly, the output spectral and input temporal amplitudes repeat each other ( ) *A SFG A t* (0, ) , and the output SFG-spectrum displays directly the input temporal pulse: ( ) *SSFG* <sup>2</sup> | ( )| *A SFG* <sup>2</sup> | (0, )| *A t I t* (0, ) , with the scale *Ct* . The resolution of such a similariton-based FO is given by the transfer function of the similariton's spectrum [22,47], and FO with a similariton-reference of the bandwidth of a few tens of nanometers provides the direct measurement of temporal pulse in a spectrometer, exceeding the resolution of the achievement of silicon-chip-based ultrafast optical oscilloscope [21] by an order of magnitude.

In the *experiment*, different amplitude- and phase-modulated pulses at the setup input are shaped and the signal radiation is split by a beam-splitter (80% + 20%). The low-power part is directed to the TD or D-line (SF 11 prism pair with the reverse mirror) for similaritonbased SI and spectrotemporal imaging, respectively. In the second path, the high-power pulse (with average power of up to 500 mW) is injected into a standard single-mode fiber (1.65 m Newport F-SPF PP@820 nm) by a microscope objective (10 ) to generate broadband nonlinear-dispersive similaritons. For the SI-measurements, these two pulses are coupled directly into the OSA and the SI fringe pattern and signal spectrum are registered. To retrieve the spectral phase, the Fourier-transform algorithm of the fringe-pattern analysis are used [44-47]. For the FO-measurements, a BBO crystal at the input of OSA is placed, and the SFG-spectrotemporal image is registered directly. The similariton-based SI and FO measurements are carried out together with the autocorrelation check by a standard APE PulseCheck autocorrelator.

First, the similariton-based SI for the laser pulses stretched and chirped in SF11 glasses of different thickness is tested, comparing the results with the autocorrelation measurements (Fig. 11). For the dispersion-induced parabolic spectral phases of the stretched pulses, the coefficients of the dispersion-induced parabolic spectral phases [Fig. 11(a)] are the following: "( ) =1.94 10-3, 4.94 10-3, 6.34 10-3, and 10.78 10-3 ps2 for the 0, 2, 3, and 5-cm glasses, respectively. The SI-reconstructed pulses, correspondingly, have durations of 108, 197, 252, and 365 fs, in a good accordance with the autocorrelation durations of 156, 298, 369 and 539 fs of the measurements shown in Fig. 11(b).

Afterwards, measurements for multi-peak pulses are carried out together with the autocorrelation check. The SI calibrating measurement of the coefficient for similariton, using the known laser pulse as a reference, gives the value = 2.1 104 fs2, in accordance with the

a nonlinear crystal for SFG at the system output (returning to the initial configuration of Fig. 8). In the spectral domain, the dispersive delay works as a parabolic phase modulator, and

the known parameters as in the case of similariton-based SI. In both arms of the setup, we have practically linearly chirped pulses, and the temporal and spectral complex amplitudes repeat each other in the temporal Fraunhofer zone, i.e. spectron pulses are formed [47]:

with

*SFG* 

function of the similariton's spectrum [22,47], and FO with a similariton-reference of the bandwidth of a few tens of nanometers provides the direct measurement of temporal pulse in a spectrometer, exceeding the resolution of the achievement of silicon-chip-based ultrafast

In the *experiment*, different amplitude- and phase-modulated pulses at the setup input are shaped and the signal radiation is split by a beam-splitter (80% + 20%). The low-power part is directed to the TD or D-line (SF 11 prism pair with the reverse mirror) for similaritonbased SI and spectrotemporal imaging, respectively. In the second path, the high-power pulse (with average power of up to 500 mW) is injected into a standard single-mode fiber (1.65 m Newport F-SPF PP@820 nm) by a microscope objective (10 ) to generate broadband nonlinear-dispersive similaritons. For the SI-measurements, these two pulses are coupled directly into the OSA and the SI fringe pattern and signal spectrum are registered. To retrieve the spectral phase, the Fourier-transform algorithm of the fringe-pattern analysis are used [44-47]. For the FO-measurements, a BBO crystal at the input of OSA is placed, and the SFG-spectrotemporal image is registered directly. The similariton-based SI and FO measurements are carried out together with the autocorrelation check by a standard APE

First, the similariton-based SI for the laser pulses stretched and chirped in SF11 glasses of different thickness is tested, comparing the results with the autocorrelation measurements (Fig. 11). For the dispersion-induced parabolic spectral phases of the stretched pulses, the coefficients of the dispersion-induced parabolic spectral phases [Fig. 11(a)] are the following:

Afterwards, measurements for multi-peak pulses are carried out together with the auto-

"( ) =1.94 10-3, 4.94 10-3, 6.34 10-3, and 10.78 10-3 ps2 for the 0, 2, 3, and 5-cm glasses, respectively. The SI-reconstructed pulses, correspondingly, have durations of 108, 197, 252, and 365 fs, in a good accordance with the autocorrelation durations of 156, 298, 369

coefficient for similariton, using

= 2.1 104 fs2, in accordance with the

opposite and same value chirps *C CC f d* , and constant similariton spectrum throughout the signal spectrum, the output temporal SFG-signal repeats the input spectral

*Ct* . The resolution of such a similariton-based FO is given by the transfer

 <sup>2</sup> | ( )| *A SFG* 

*d d <sup>C</sup>* . In the fiber arm, we have a nonlinear-dispersive similariton with

 <sup>2</sup> (0, )exp( / 2) *A i* 

 *<sup>d</sup>* , with the

*C t <sup>d</sup>*, *<sup>f</sup>* . Under the conditions of the

. Accordingly, the output spectral and input

*A t* (0, ) , and the output SFG-spectrum

<sup>2</sup> | (0, )| *A t I t* (0, ) , with

passed through is described as ( , ) *A d*

(,) 

the signal (0, ) *A*

*Adt* ( ,) *A d*

the scale

, and *Aft* ( ,) *A f*

amplitude: ( ) *A t SFG Adt A f t* ( ,) ( ,) *<sup>A</sup>*(0, )

temporal amplitudes repeat each other ( ) *A*

optical oscilloscope [21] by an order of magnitude.

and 539 fs of the measurements shown in Fig. 11(b).

correlation check. The SI calibrating measurement of the

the known laser pulse as a reference, gives the value

displays directly the input temporal pulse: ( ) *SSFG*

given coefficient <sup>1</sup>

(, ) 

PulseCheck autocorrelator.

 

Fig. 11. Similariton-based SI for pulses dispersively stretched and chirped in SF11 glasses of different thickness: (a) retrieved spectral phases with the measured spectrum (dotted line), and (b) autocorrelation functions of SI-reconstructed temporal pulses (solid) in comparison with autocorrelation traces measured (dashed).

expression 2 *f* with the values *f* = 49 cm and <sup>2</sup> = 40 fs2/mm ( *D*= –103 ps/nm/km at the wavelength 850 nm, according to the data provided by the fiber manufacturer). Then different multi-peak signal pulses are shaped inserting thin glass plates in parts of the beam. The beam parts passed through the plates obtain time delay with respect to the freepropagated part. The movement of the plates along the vertical axis adjusts the power proportion among the peaks. The thicknesses of the plates give the time delay between the peaks; e.g. a 0.12 mm thick glass plate gives a 200 fs delay, if assuming the refractive index of the plate equal to 1.5. Using double- and triple- peak signal pulses, we carry out SImeasurements and compare the results with measured autocorrelation tracks.

The application range and limitations of similariton-based SI are conditioned by the parameters of input radiation and fiber, which are necessary to generate the similaritonreference with parabolic phase. Fig. 12 and 13 illustrate the experiment for the typical regime preceding the similariton shaping: results for double- and triple-peak signal pulses are shown. As Fig. 12 illustrates, having the spectrum (b, black) and SI-retrieved spectral phase (b, blue solid line) of pulse, its temporal profile (c, blue solid line) is reconstructed through Fourier transformation. To check the precision of our measurements through similariton-based SI, we calculate the autocorrelation of reconstructed pulse (d, blue solid line) and compare it with the intensity autocorrelation measured at the input of the system (d, black). The spectral shape of nonlinear-dispersive similariton (a) ensures the fulfillment of necessary conditions for the parabolicity of the spectral phase of similariton (according to [23]). The structure of the similariton spectrum (a), strange at first glance, is typical for short lengths of nonlinear-dispersive interaction, and is observed also for parabolic similaritons generated in fiber amplifiers [52]. The blue solid and red dashed curves in Fig. 12 correspond to the pulse reconstruction with the spectral phase coefficients = 2.1 104 fs2 and = 1.995 104 fs2 (5% difference), respectively. Fig. 13 shows the analogue procedures for a pulse with more complex sub-structure.

Finally, we compare the measurements of the similariton-based SI and FO, together with a theoretical check. The double-peak signal pulses are shaped with the spectral domain amplitude- and phase-modulation given by the peaks' temporal distance *T* and their proportion . The temporal amplitude *A t*( ) <sup>0</sup> *A t*( ) <sup>0</sup> *At T* ( ) corresponds to the complex

Similariton-Based Spectral Interferometry for Signal Analysis on Femtosecond Time Scale 119

 

double-peak pulses, the laser beam is expanded and a thin glass plate in its part is placed, as described above. The thickness of the plate gives the time delay between the peaks; e.g. a 0.12 mm thick glass plate gives a 200 fs delay, according to autocorrelation check. The similariton-based SI and FO are comparatively experimented using the double-peak signal pulse: the SI-reconstructed pulses are compared with spectrotemporal images of

, with the

  ( ) 

*T T* phase-modulation. To shape such

 <sup>2</sup> 1 2 cos( ) 

 *T*

 = 0 *A i* ( ) ( )exp[ ( )] 

 <sup>1</sup> arctan[(sin ) /( cos )] 

Fig. 14. Comparison of similariton-based SI and FO for a double-peak signal pulse:

curves are for the theory, similariton-based SI and FO, respectively.

(a)measured spectrum, (b) retrieved spectral phase, and (c) pulse. Dashed, solid and dotted

Fig. 14 illustrates this experiment by the results for a double-peak signal pulse: the quantitative accordance of the measured spectrum (a) and retrieved spectral phase (b) with the theoretical curves (dashed) leads to an accurate pulse reconstruction through similariton-based SI (c, solid). An accurate spectrotemporal imaging (c, dotted) is ensured by the similariton of the bandwidths of 40 nm. The differences between these independent SI- and FO-measurements and theoretical curve are hardly seen, evidencing both the accuracy of the mentioned measurements and the potential of similariton-based methods.

spectral amplitude ( ) *A*

( ) 

amplitude- and

the signal.

Fig. 12. Reconstruction of double-peak pulse (shaped by means of a 130- m thick glass) through similariton-based SI in comparison with autocorrelation measurement: (a) spectrum of nonlinear-dispersive similariton; (b) retrieved spectral phase and measured spectrum; (c) reconstructed pulse temporal profile; and (d) autocorrelation tracks. Blue solid and red dashed curves are for = 2.1 104 fs2 and = 1.995 104 fs2 (5% difference), respectively, and the black one in (d) is the measured autocorrelation track.

Fig. 13. Reconstruction of three-peak pulse (shaped by means of 145 and 120-um thick glasses) through similariton-based SI in comparison with autocorrelation: (a) measured spectrum; (b) retrieved spectral phases; (c) reconstructed pulse temporal profiles and autocorrelation tracks (inset). Blue dotted, red dashed and green solid curves are for = 2.1 104 fs2 , 1.995 104 fs2, and 1.89 104 fs2, respectively, and the black one of the inset is the measured autocorrelation.

*I* (au)

autocorrelation (au)

Fig. 12. Reconstruction of double-peak pulse (shaped by means of a 130-

1


(inset). Blue dotted, red dashed and green solid curves are for

0

intensity (au)

and red dashed curves are for

spectrum

2.32 2.34 2.36 2.38

frequency (rad/fs)

spectral phase

frequency (rad/fs)

2.32 2.34 2.36 2.38

(b)

(a)

0

2

phase (rad)

4

6

0

intensity (au)

1

*I* (au)

through similariton-based SI in comparison with autocorrelation measurement: (a) spectrum of nonlinear-dispersive similariton; (b) retrieved spectral phase and measured spectrum; (c) reconstructed pulse temporal profile; and (d) autocorrelation tracks. Blue solid

= 2.1 104 fs2 and

145 um 120 um

pulse shaping (c)

Fig. 13. Reconstruction of three-peak pulse (shaped by means of 145 and 120-um thick glasses) through similariton-based SI in comparison with autocorrelation: (a) measured spectrum; (b) retrieved spectral phases; (c) reconstructed pulse temporal profiles and autocorrelation tracks

and 1.89 104 fs2, respectively, and the black one of the inset is the measured autocorrelation.


time (fs)

= 2.1 104 fs2 , 1.995 104 fs2,

respectively, and the black one in (d) is the measured autocorrelation track.


delay (fs)

500

(rad/fs) 2

(rad)

= 1.995 104 fs2 (5% difference),

pulse

2.32 2.34 2.36

(a) (b)

(c) (d)

0 0


m thick glass)

spectral amplitude ( ) *A* = 0 *A i* ( ) ( )exp[ ( )] , with the ( ) <sup>2</sup> 1 2 cos( ) *T* amplitude- and ( ) <sup>1</sup> arctan[(sin ) /( cos )] *T T* phase-modulation. To shape such double-peak pulses, the laser beam is expanded and a thin glass plate in its part is placed, as described above. The thickness of the plate gives the time delay between the peaks; e.g. a 0.12 mm thick glass plate gives a 200 fs delay, according to autocorrelation check. The similariton-based SI and FO are comparatively experimented using the double-peak signal pulse: the SI-reconstructed pulses are compared with spectrotemporal images of the signal.

Fig. 14. Comparison of similariton-based SI and FO for a double-peak signal pulse: (a)measured spectrum, (b) retrieved spectral phase, and (c) pulse. Dashed, solid and dotted curves are for the theory, similariton-based SI and FO, respectively.

Fig. 14 illustrates this experiment by the results for a double-peak signal pulse: the quantitative accordance of the measured spectrum (a) and retrieved spectral phase (b) with the theoretical curves (dashed) leads to an accurate pulse reconstruction through similariton-based SI (c, solid). An accurate spectrotemporal imaging (c, dotted) is ensured by the similariton of the bandwidths of 40 nm. The differences between these independent SI- and FO-measurements and theoretical curve are hardly seen, evidencing both the accuracy of the mentioned measurements and the potential of similariton-based methods.

Similariton-Based Spectral Interferometry for Signal Analysis on Femtosecond Time Scale 121

similariton-induced temporal lens. Our comparative study, carried out together with theoretical check and autocorrelation measurements, evidences the quantitative accordance and high precision of both the similariton-referencing methods of spectral interferometry and spectrotemporal imaging for accurate femtosecond-scale temporal measurements. The similariton-based spectrotemporal imaging has the advantage of direct pulse measurement leading to the development of a femtosecond optical oscilloscope, but it does not give the phase information without additional interferometric measurement. The novel method of similariton-based spectral interferometry, with a rather simple setup and self-referencing performance, provides the complete (amplitude and phase), high-resolution characterization

The work was carried out within the framework of the 978027 project of Science for Peace Programme of North Atlantic Treaty Organization, collaborative programme project IE007 of Centre National de la Recherche Scientifique (CNRS), France – State Committee of Science (SCS), Armenia, and ANSEF grant # PS-opt-2903. A. Zeytunyan also acknowledges SCS, National Foundation of Science and Advanced Technology (NFSAT), and Civilian Research and Development Foundation (CDRF) for financial support in the framework of the Early

[1] D.J.Kane, R.Trebino "Single-shot measurement of the intensity and phase of an arbitrary

S. Akturk, M. Kimmel, P. O'Shea, R. Trebino "Measuring spatial chirp in ultrashort pulses using singleshot frequency-resolved optical gating" Opt. Express 11, 68–78

[2] J. Piasecki, B. Colombeau, M. Vampouille, C. Froehly, J.A. Arnaud, "Nouvelle méthode

[3] C. Iaconis and I. A. Walmsley, "Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses," Opt. Lett. 23, 792–794 (1998). [4] V.Messager, F.Louradour, C.Froehly, A.Barthélémy "Coherent measurement of short

F. Reynaud, F. Salin, and A. Barthélémy, "Measurement of phase shifts introduced by nonlinear optical phenomena on subpicosecond pulses," Opt. Lett. 14, 275–277

ultrashort pulse by using frequency-resolved optical gating" Opt.Lett.18, 823–825

de mesure de la réponse impulsionnelle des fibres optiques," Appl. Opt. 19, 3749

laser pulses based on spectral interferometry resolved in time" Opt.Lett.28, 743–745

M. Lelek, F. Louradour, A. Barthélémy, C. Froehly, T. Mansourian, L. Mouradian, J.-P. Chambaret, G. Chériaux, B. Mercier, "Two-dimensional spectral shearing interferometry resolved in time for ultrashort optical pulse characterization," J. Opt.

Career Support Program (grant numbers A-16 and ECSP-09-50).

of femtosecond signal.

**6. Acknowledgments** 

**7. References** 

(1993).

(2003).

(1980).

(1989).

(2003).

Soc. Am. B 25, A17–A24 (2008).

Obviously, the demonstrated methods of femtosecond signal characterization can be implemented also by the use of "standard" parabolic similaritons generated in active or dispersion decreasing fibers. In a recent progress in the generation of parabolic broadband similaritons, bandwidths of up to 11 THz (40 nm at 1050 nm central wavelength) are achieved [52]. However, the use of the nonlinear-dispersive similariton generated in a piece of standard passive fiber currently is more beneficial, providing larger bandwidths and thus larger application ranges with technically simpler experimental arrangement.

Thus, the methods of similariton-based SI and spectrotemporal imaging are experimentally demonstrated as two applications of similariton. The reference-based methods become selfreferencing by the use of similariton. The described comparative study, involving also theoretical check and autocorrelation measurements, ensures the quantitative accordance and high precision of both the similariton-referencing methods. While the method of similariton-based spectrotemporal imaging has the advantage of direct pulse measurement, and thus leads to the development of a femtosecond optical oscilloscope, it does not give phase information. The method of similariton-based SI provides the complete (amplitude and phase) characterization of femtosecond signal. The method of similariton-based SI provides the complete (amplitude and phase) characterization of femtosecond signal.

## **5. Conclusion**

Our spectral interferometric studies demonstrate the spectronic nature and distinctive properties of nonlinear-dispersive similariton, of up to 5-THz bandwidth, generated in a passive fiber. The key property of nonlinear-dispersive similariton of having a parabolic phase (linear chirp), given by the fiber dispersion only, leads to its spectrotemporal similarity and thus to its self-spectrotemporal imaging, with the accuracy given by spectral broadening and pulse stretching together.

Generating similaritons of 50-THz bandwidth, we carry out their complete characterization through the chirp measurement, using the technique of frequency tuning in the process of spectral compression by sum-frequency generation. The studies permit to generalize the description of nonlinear-dispersive similaritons, verifying that only fiber dispersion determines the phase (chirp) of such broadband similaritons. The third order dispersion of fiber results in the same additional phase for broadband nonlinear-dispersive similariton and spectron. The ~1% accuracy of the linear fit for the chirp of the 50-THz bandwidth similariton gives the range of applications for aberration-free similariton-based spectrotemporal imaging and spectral interferometry. The described approach to the generation and characterization of broadband similariton can be helpful also for its applications in pulse compression and CARS microscopy.

We develop and implement a similariton based self-referencing method of spectral interferometry for the complete characterization of femtosecond signal. The method is based on the similariton generation from the part of signal and its use as a reference for the interference with the signal in the spectrometer. Therefore, the method of similariton-based spectral interferometry combines the advantage of the simple principle and configuration with the self-referencing performance. We experiment the similariton-based method of spectral interferometry in comparison with the measurements carried out with the prototype of femtosecond oscilloscope based on the spectrotemporal imaging in a similariton-induced temporal lens. Our comparative study, carried out together with theoretical check and autocorrelation measurements, evidences the quantitative accordance and high precision of both the similariton-referencing methods of spectral interferometry and spectrotemporal imaging for accurate femtosecond-scale temporal measurements. The similariton-based spectrotemporal imaging has the advantage of direct pulse measurement leading to the development of a femtosecond optical oscilloscope, but it does not give the phase information without additional interferometric measurement. The novel method of similariton-based spectral interferometry, with a rather simple setup and self-referencing performance, provides the complete (amplitude and phase), high-resolution characterization of femtosecond signal.

## **6. Acknowledgments**

120 Interferometry – Research and Applications in Science and Technology

Obviously, the demonstrated methods of femtosecond signal characterization can be implemented also by the use of "standard" parabolic similaritons generated in active or dispersion decreasing fibers. In a recent progress in the generation of parabolic broadband similaritons, bandwidths of up to 11 THz (40 nm at 1050 nm central wavelength) are achieved [52]. However, the use of the nonlinear-dispersive similariton generated in a piece of standard passive fiber currently is more beneficial, providing larger bandwidths and thus

Thus, the methods of similariton-based SI and spectrotemporal imaging are experimentally demonstrated as two applications of similariton. The reference-based methods become selfreferencing by the use of similariton. The described comparative study, involving also theoretical check and autocorrelation measurements, ensures the quantitative accordance and high precision of both the similariton-referencing methods. While the method of similariton-based spectrotemporal imaging has the advantage of direct pulse measurement, and thus leads to the development of a femtosecond optical oscilloscope, it does not give phase information. The method of similariton-based SI provides the complete (amplitude and phase) characterization of femtosecond signal. The method of similariton-based SI provides the complete (amplitude and phase) characterization of femtosecond signal.

Our spectral interferometric studies demonstrate the spectronic nature and distinctive properties of nonlinear-dispersive similariton, of up to 5-THz bandwidth, generated in a passive fiber. The key property of nonlinear-dispersive similariton of having a parabolic phase (linear chirp), given by the fiber dispersion only, leads to its spectrotemporal similarity and thus to its self-spectrotemporal imaging, with the accuracy given by spectral

Generating similaritons of 50-THz bandwidth, we carry out their complete characterization through the chirp measurement, using the technique of frequency tuning in the process of spectral compression by sum-frequency generation. The studies permit to generalize the description of nonlinear-dispersive similaritons, verifying that only fiber dispersion determines the phase (chirp) of such broadband similaritons. The third order dispersion of fiber results in the same additional phase for broadband nonlinear-dispersive similariton and spectron. The ~1% accuracy of the linear fit for the chirp of the 50-THz bandwidth similariton gives the range of applications for aberration-free similariton-based spectrotemporal imaging and spectral interferometry. The described approach to the generation and characterization of broadband similariton can be helpful also for its

We develop and implement a similariton based self-referencing method of spectral interferometry for the complete characterization of femtosecond signal. The method is based on the similariton generation from the part of signal and its use as a reference for the interference with the signal in the spectrometer. Therefore, the method of similariton-based spectral interferometry combines the advantage of the simple principle and configuration with the self-referencing performance. We experiment the similariton-based method of spectral interferometry in comparison with the measurements carried out with the prototype of femtosecond oscilloscope based on the spectrotemporal imaging in a

larger application ranges with technically simpler experimental arrangement.

**5. Conclusion** 

broadening and pulse stretching together.

applications in pulse compression and CARS microscopy.

The work was carried out within the framework of the 978027 project of Science for Peace Programme of North Atlantic Treaty Organization, collaborative programme project IE007 of Centre National de la Recherche Scientifique (CNRS), France – State Committee of Science (SCS), Armenia, and ANSEF grant # PS-opt-2903. A. Zeytunyan also acknowledges SCS, National Foundation of Science and Advanced Technology (NFSAT), and Civilian Research and Development Foundation (CDRF) for financial support in the framework of the Early Career Support Program (grant numbers A-16 and ECSP-09-50).

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Similariton-Based Spectral Interferometry for Signal Analysis on Femtosecond Time Scale 123

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**6** 

**Spectral Low Coherence Interferometry:** 

**and the Signal Processing** 

*Universidad Tecnológica Nacional, Facultad Regional Delta* 

Eneas N. Morel and Jorge R. Torga

*Campana, Buenos Aires* 

*Argentina* 

**A Complete Analysis of the Detection System** 

Low coherence interferometry (LCI) is an optical technique in which light is used as an instrument to obtain high resolution optical images in a great diversity of materials. It is possible to find in the literature, a variety of configurations with different names, based on the same principle: low coherence reflectometry (OLCR), optical coherence tomography (OCT), white light interferometry (WLI), are only some examples of this. The main idea of this technique is to measure the echo time delay of backscattered light in the sample through the characterization of the interference intensity obtained when the light coming from the sample and the light reflected in a reference surface overlap. When a low coherence source is used, the interference signal is temporally and spatially localized, so it is possible to use this property to obtain distance values or parameters related to the time of flight of the light,

LCI has been proposed and studied since the beginning of the optical science. However, for the last 20 years, there has been a dramatically increase in its applications, mainly due to the development of optical coherence tomography and the evolution of new light sources. The first works of in- surface and optical material characterization was reported in 1960. In 80´s some applications in fibre optics characterization (Takada et al, 1987), topography surfaces and internal structures in transparent media (Youngquist et al, 1987) were proposed with a LCI set-up. It is generally accepted that first biological application was reported by (Fercher, 1988). After these first works, optical coherent tomography (OCT) became a powerful technique for medical diagnosis, (Huang, 1991), in which images of the human retina and coronary artery were obtained. Since then, OCT has evolved, and nowadays it is a wellestablished technique for ophthalmic diagnosis and other biological tissues (Brezinski, 2006). New developments in light sources, fibre optic elements, detectors and processing techniques allow a dramatic increase in resolution and speed of image acquisition. These advances transform this technique in a powerful three-dimensional visualization method which has a wide diversity of applications. Over the last ten years its growth has been explosive; proof of this is the increasing number of publications, patents and companies involved in this subject (www.octnews.org). There is no doubt that this evolution has been

**1. Introduction** 

reflected in different sections of a sample.


## **Spectral Low Coherence Interferometry: A Complete Analysis of the Detection System and the Signal Processing**

Eneas N. Morel and Jorge R. Torga *Universidad Tecnológica Nacional, Facultad Regional Delta Campana, Buenos Aires Argentina* 

## **1. Introduction**

124 Interferometry – Research and Applications in Science and Technology

[37] A.S.Zeytunyan, K.A.Palanjyan, G.L.Yesayan, L.Kh.Mouradian, "Spectral-

[38] C.Finot, B.Kibler, L.Provost, S.Wabnitz, "Beneficial impact of wave-breaking for

[39] A.S.Zeytunyan, H.R.Madatyan, G.L.Yesayan, L.Kh.Mouradian "Diagnostics of femto-

[40] V.I. Kruglov, D. Méchin, J.D. Harvey, "High compression of similariton pulses under the influence of higher-order effects," J. Opt. Soc. Am. B 24, 833–838 (2007). [41] K.Palanjyan, A.Muradyan, A.Zeytunyan, G.Yesayan, L.Mouradian, "Pulse

[42] C. Finot, G. Millot "Synthesis of optical pulses by use of similaritons" Opt. Express 12,

[43] A.Zeytunyan, G.Yesayan, L.Mouradian, F.Louradour, A.Barthélémy, "Applications of

[45] A.Zeytunyan, A.Muradyan, G.Yesayan, L.Mouradian, F.Louradour, A.Barthélémy

[46] A.Zeytunyan, A.Muradyan, G.Yesayan, L.Mouradian, F.Louradour, A.Barthelemy,

[47] A.Zeytunyan, A.Muradyan, G.Yesayan, L.Mouradian, F.Louradour, A.Barthélémy

[48] A.M. Heidt, "Pulse preserving flat-top supercontinuum generation in all-normal dispersion photonic crystal fibers," J. Opt. Soc. Am. B 27, 550-559 (2010). [49] D.T. Reid, P. Loza-Alvarez, C.T.A. Brown, T. Beddard, W. Sibbett, "Amplitude and

[50] J. Dudley, X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O'Shea, R. Trebino, S. Coen, R.

[51] A.F. Pegoraro, A. Ridsdale, D.J. Moffatt, Y. Jia, J.P. Pezacki, A. Stolow, "Optimally

[52] W.H. Renninger, A. Chong, F.W. Wise, "Self-similar pulse evolution in an all-normal-

femtosecond pulses" Opt. Commun. v. 284, pp. 3742–3747 (2011).

frequency-resolved optical gating," Opt. Lett. 25, 1478-1480 (2000).

experiments," Opt. Express 10, 1215-1221 (2002).

dispersion laser," Phys. Rev. A 82, 021805(R) (2010).

Opt. Express 17, 2984-2996 (2009).

imaging," in *Frontiers in Optics*, OSA Tech. Digest, paper FWI5 (2009). [44] A.Zeytunyan, A. Muradyan, G. Yesayan, L. Mouradian, "Broadband similariton" Laser

64–69 (2010).

Am. B 25, 1938–1948 (2008).

SPIE 7998, 79980N (2010).

Physics 20, 1729–1732 (2010).

Digest, paper NME46 (2010).

Italy, paper Mo.2.E.5 (2010).

5104–5109 (2004).

Contemp. Phys. 45, 169–171 (2010).

interferometric study of nonlinear-spectronic similariton:" J. Contemp. Phys. 45,

coherent continuum formation in normally dispersive nonlinear fibers," J. Opt. Soc.

second laser pulses based on generation of nonlinear-dispersive similariton" J.

compression down to 17 femtoseconds by generating broadband similariton," Proc.

similariton in ultrafast optics: spectral interferometry and spectrotemporal

"Measuring of Broadband Similariton Chirp" Nonlinear Photonics OSA Tech.

Similariton for Femtosecond Optics Proc. ECOC 2010 (19-23 Sep. 2010), Torino,

"Generation of broadband similaritons for complete characterization of

phase measurement of mid-infrared femtosecond pulses by using cross-correlation

Windeler, "Cross-correlation frequency resolved optical gating analysis of broadband continuum generation in photonic crystal fiber: simulations and

chirped multimodal CARS microscopy based on a single Ti:sapphire oscillator,"

Low coherence interferometry (LCI) is an optical technique in which light is used as an instrument to obtain high resolution optical images in a great diversity of materials. It is possible to find in the literature, a variety of configurations with different names, based on the same principle: low coherence reflectometry (OLCR), optical coherence tomography (OCT), white light interferometry (WLI), are only some examples of this. The main idea of this technique is to measure the echo time delay of backscattered light in the sample through the characterization of the interference intensity obtained when the light coming from the sample and the light reflected in a reference surface overlap. When a low coherence source is used, the interference signal is temporally and spatially localized, so it is possible to use this property to obtain distance values or parameters related to the time of flight of the light, reflected in different sections of a sample.

LCI has been proposed and studied since the beginning of the optical science. However, for the last 20 years, there has been a dramatically increase in its applications, mainly due to the development of optical coherence tomography and the evolution of new light sources. The first works of in- surface and optical material characterization was reported in 1960. In 80´s some applications in fibre optics characterization (Takada et al, 1987), topography surfaces and internal structures in transparent media (Youngquist et al, 1987) were proposed with a LCI set-up. It is generally accepted that first biological application was reported by (Fercher, 1988). After these first works, optical coherent tomography (OCT) became a powerful technique for medical diagnosis, (Huang, 1991), in which images of the human retina and coronary artery were obtained. Since then, OCT has evolved, and nowadays it is a wellestablished technique for ophthalmic diagnosis and other biological tissues (Brezinski, 2006).

New developments in light sources, fibre optic elements, detectors and processing techniques allow a dramatic increase in resolution and speed of image acquisition. These advances transform this technique in a powerful three-dimensional visualization method which has a wide diversity of applications. Over the last ten years its growth has been explosive; proof of this is the increasing number of publications, patents and companies involved in this subject (www.octnews.org). There is no doubt that this evolution has been

Spectral Low Coherence Interferometry:

In figure 4 it is shown a tomography of a finger skin.

**1.1 Basis of low coherence Interferometry** 

shown in figure 5 below.

Fig. 5. Michelson interferometer set-up.

Fig. 4. Human finger pad. Photos used with permission of Thorlabs®.

A Complete Analysis of the Detection System and the Signal Processing 127

Fig. 3. 3D optical profiling of an M2 metal screw. Photos used with permission of Thorlabs®.

Interference can be considered as the wave pattern that is obtained after the superposition of two or more waves. In low coherence interferometry, superposition is obtained with a broadband light source, and the wave pattern in this situation is considerably different from

In interferometry techniques the interference signal is usually obtained from the superposition between light backscattered from a reference arm and a sample arm as it is

that obtained with a standard monochromatic source. (Born, M. & Wolf, E., 1999).

the result of the numerous successful OCT applications in the medical field, especially in ophthalmology (Drexler et al, 2008). Over the last years its application in material characterization and non-destructive techniques is another field that is growing fast (Goode, 2009). Micro structures and MEMS characterization, surface topography, structural parameters in semitransparent materials, analysis and visualization of structural vibrations, are only a few examples (Bruce et al, 1991), (Wiesauer et al, 2005), (Wiesner et al, 2010).

Nowadays, there are several companies that offer commercial systems based on this tecnique. Polytec®, Thorlabs®, Carl Zeiss® are only some examples. Figure 1 show an image obtained with the Stratus from Carl Zeiss®. This equipment is capable of exploring the eye at a speed of several thousand lines per second.

Fig. 1. a) Stratus 3000 Carl Zeiss®. b) OCT image of retina, taken with Stratus 3000 (Michael P Kelly, Duke Eye Imaging, Duke University Eye Center, Durham, NC). Photos used with permission of Carl Zeiss®.

Another example is shown in figure 2; TSM-1200 TopMap® µ.Lab from Polytec® that can be used to acquire high-resolution topographical maps of functional surfaces and microstructures.

Fig. 2. TMS-1200 TopMap µ.Lab. Photos used with permission of Polytec®.

High-speed 3D OCT imaging can provide comprehensive data that combines the advantages of optical coherence tomography and microscopy in a single system. Shown below are some 3D image data sets of two samples; in figures 4 it is shown how this technique can obtain the surface topography of a screw.

the result of the numerous successful OCT applications in the medical field, especially in ophthalmology (Drexler et al, 2008). Over the last years its application in material characterization and non-destructive techniques is another field that is growing fast (Goode, 2009). Micro structures and MEMS characterization, surface topography, structural parameters in semitransparent materials, analysis and visualization of structural vibrations, are only a few examples (Bruce et al, 1991), (Wiesauer et al, 2005), (Wiesner et al, 2010).

Nowadays, there are several companies that offer commercial systems based on this tecnique. Polytec®, Thorlabs®, Carl Zeiss® are only some examples. Figure 1 show an image obtained with the Stratus from Carl Zeiss®. This equipment is capable of exploring the eye at

Fig. 1. a) Stratus 3000 Carl Zeiss®. b) OCT image of retina, taken with Stratus 3000 (Michael P Kelly, Duke Eye Imaging, Duke University Eye Center, Durham, NC). Photos used with

Another example is shown in figure 2; TSM-1200 TopMap® µ.Lab from Polytec® that can be used to acquire high-resolution topographical maps of functional surfaces and

Fig. 2. TMS-1200 TopMap µ.Lab. Photos used with permission of Polytec®.

technique can obtain the surface topography of a screw.

High-speed 3D OCT imaging can provide comprehensive data that combines the advantages of optical coherence tomography and microscopy in a single system. Shown below are some 3D image data sets of two samples; in figures 4 it is shown how this

a speed of several thousand lines per second.

a b

permission of Carl Zeiss®.

microstructures.

Fig. 3. 3D optical profiling of an M2 metal screw. Photos used with permission of Thorlabs®.

In figure 4 it is shown a tomography of a finger skin.

Fig. 4. Human finger pad. Photos used with permission of Thorlabs®.

## **1.1 Basis of low coherence Interferometry**

Interference can be considered as the wave pattern that is obtained after the superposition of two or more waves. In low coherence interferometry, superposition is obtained with a broadband light source, and the wave pattern in this situation is considerably different from that obtained with a standard monochromatic source. (Born, M. & Wolf, E., 1999).

In interferometry techniques the interference signal is usually obtained from the superposition between light backscattered from a reference arm and a sample arm as it is shown in figure 5 below.

Fig. 5. Michelson interferometer set-up.

Spectral Low Coherence Interferometry:

**1.2 Experimental configurations** 

systems are expensive comparatively.

particular characteristics. They are commonly known as: 1. Time domain low coherence interferometry (TDLCI) 2. Spectral domain low coherence interferometry (SDLCI) 3. Sweep source domain low coherence interferometry (SSDLCI)

performed.

within the coherence length of the light source ( figure 6-b).

interferometer, the sample with its holder and the detection system.

A Complete Analysis of the Detection System and the Signal Processing 129

When the light source is monochromatic, the total intensity shows the same variation (figure 6-a). Using a low coherence source, interference is only observed when the path length is

We will show a generic system of low coherence interferometry with a brief description of the experimental set-up and the function of each component. The general configuration can be separated in four well differentiated sections: the broadband light source, the

Figure 7 shows the basic set-up used in most of the low coherence applications. The initial system is the broadband light source [1], that is a critical factor since an appropriate selection in its characteristics is crucial in the final quality of the images obtained (Drexler et al, 2008). Nowadays there is an important offer of broadband sources with different center

Diode systems are probably the most utilized light sources, as the superluminiscent LED`s (SLD). They have the advantage to be easy to hand and operate; also, their spatial dimension is reduced, their price is usually lower than other systems, and they are appropriated for nolaboratory applications. Amplified spontaneous emission (ASE) sources are an interesting alternative as they offer a broad bandwidth in the infrared region. Finally there is a group based on ultra-short laser pulses and supercontinuum light sources which is an attractive option as it offers high power, broad bandwidth and good quality in spatial mode. These

The output beam from the source is directed to the interferometer. There is a great variety of configurations proposed in the literature, but in most of these works a Michelson type is used (Schmitt et al, 1999). A typical configuration is shown in figure 7. The output beam is split in the beamsplitter [4]. One of the beams goes to the reference surface [2]-usually a mirror- and the other beam goes to the sample [3]. After the reflection on each surface, both beams are sent to the detector [5]. The superposition generates the interference signal.

With this signal it is obtained a measurement of the OPD between both arms. In most cases the light beam is focused on the sample so that the measurement process is performed point to point on the desired region. For opaque samples, the light is reflected only on the surface, so a topography measurement is obtained. For transparent or semi-transparent media, light is reflected from sub-surface structures in the sample, so a tomography measurement is

Usually the sample-holder or the light beam can be displaced in a 2-D lateral movement (axis x and y in the figure 7). This way the sample can be inspected in the region of interest. Although there are many configurations proposed based on the general configuration described before, it is possible to make a division into three main groups, each one with its

wavelengths, power, and beam quality (stability, noise, single transverse mode, etc.).

The total intensity at the detector at a time t, is obtained as the superposition of *er(t) = Re(Er(t))* and *es(t) = Re(Es(t+τ))*, the electric field from the reference and from the sample respectively. *Re(E(t))*, indicates the real part of the complex expression of the electric field, *E(t)*. τ is the difference in time of flight of light in each arm and it is a consequence of the different optical path length. The symbol < > indicates temporal average, K is a constant factor.

$$I = \mathcal{K} \left\langle \left| E\_r + E\_s \right|^2 \right\rangle = I\_r + I\_s + 2\mathcal{K} \langle \text{Re}(E\_r(t+\tau)E\_s(t)) \rangle \tag{1}$$

The reflectivity in the interface at the end of the reference and the sample arm are considered by the coefficient *R*, which is assumed constant. If *Io* is the intensity at the output of the light source, the intensity of each arm after the reflections is: *Ir = RrIo* and *Is = RsIo*. Then, equation 1 can be written:

$$I = R\_r I\_o + R\_s I\_o + 2\sqrt{R\_s R\_r} \operatorname{Re}(\Gamma(\tau)) \tag{2}$$

Delay (a.u.) (b)

In the last term in equation 2, usually named the interference term, the complex coherence function *Г(τ) = K< Eo(t) Eo(t+τ)>* is defined (Goodman, 1984); that is the correlation of the electric field. This function is closely related to the concept of coherence time *-T -* a measure of the delay between two beams which is necessary to blur the interference term. A useful way to define this concept is (Goodman, 1984):

2

Fig. 6. Total intensity (I), as a function of time delay(τ) for a coherent source a) and for a low coherence source b).

Delay (a.u.) (a)

It is possible to illustrate this concept -close to the basic idea of low coherence interferometry- considering the total intensity *I* as a function of τ, the time delay. To do this, we perform an axial scan of the reference sample. In the simplified interferometer scheme of figure 5 this can be done with a movement of the reference surface with a constant velocity (axis z figure 5).

I (a.u.)

When the light source is monochromatic, the total intensity shows the same variation (figure 6-a). Using a low coherence source, interference is only observed when the path length is within the coherence length of the light source ( figure 6-b).

## **1.2 Experimental configurations**

128 Interferometry – Research and Applications in Science and Technology

The total intensity at the detector at a time t, is obtained as the superposition of *er(t) = Re(Er(t))* and *es(t) = Re(Es(t+τ))*, the electric field from the reference and from the sample respectively. *Re(E(t))*, indicates the real part of the complex expression of the electric field, *E(t)*. τ is the difference in time of flight of light in each arm and it is a consequence of the different optical

The reflectivity in the interface at the end of the reference and the sample arm are considered by the coefficient *R*, which is assumed constant. If *Io* is the intensity at the output of the light source, the intensity of each arm after the reflections is: *Ir = RrIo* and *Is = RsIo*.

In the last term in equation 2, usually named the interference term, the complex coherence function *Г(τ) = K< Eo(t) Eo(t+τ)>* is defined (Goodman, 1984); that is the correlation of the electric field. This function is closely related to the concept of coherence time *-T -* a measure of the delay between two beams which is necessary to blur the interference term. A useful

> 2 <sup>1</sup> ( )

 

(0) *T d* 

2



0

I (a.u.)

Fig. 6. Total intensity (I), as a function of time delay(τ) for a coherent source a) and for a low

It is possible to illustrate this concept -close to the basic idea of low coherence interferometry- considering the total intensity *I* as a function of τ, the time delay. To do this, we perform an axial scan of the reference sample. In the simplified interferometer scheme of figure 5 this can be done with a movement of the reference surface with a constant velocity

0.5

1

<sup>2</sup> 2 Re( ( ) ( )) *r s rs r s I KE E I I K Et E t* (1)

2 Re( ( )) *ro so s r I RI RI RR* (2)

(3)

0 0.2 0.4 0.6 0.8 1

T

Delay (a.u.) (b)

path length. The symbol < > indicates temporal average, K is a constant factor.

Then, equation 1 can be written:

way to define this concept is (Goodman, 1984):

0 0.2 0.4 0.6 0.8 1

Delay (a.u.) (a)


coherence source b).

(axis z figure 5).


0

I (a.u.)

0.5

1

We will show a generic system of low coherence interferometry with a brief description of the experimental set-up and the function of each component. The general configuration can be separated in four well differentiated sections: the broadband light source, the interferometer, the sample with its holder and the detection system.

Figure 7 shows the basic set-up used in most of the low coherence applications. The initial system is the broadband light source [1], that is a critical factor since an appropriate selection in its characteristics is crucial in the final quality of the images obtained (Drexler et al, 2008). Nowadays there is an important offer of broadband sources with different center wavelengths, power, and beam quality (stability, noise, single transverse mode, etc.).

Diode systems are probably the most utilized light sources, as the superluminiscent LED`s (SLD). They have the advantage to be easy to hand and operate; also, their spatial dimension is reduced, their price is usually lower than other systems, and they are appropriated for nolaboratory applications. Amplified spontaneous emission (ASE) sources are an interesting alternative as they offer a broad bandwidth in the infrared region. Finally there is a group based on ultra-short laser pulses and supercontinuum light sources which is an attractive option as it offers high power, broad bandwidth and good quality in spatial mode. These systems are expensive comparatively.

The output beam from the source is directed to the interferometer. There is a great variety of configurations proposed in the literature, but in most of these works a Michelson type is used (Schmitt et al, 1999). A typical configuration is shown in figure 7. The output beam is split in the beamsplitter [4]. One of the beams goes to the reference surface [2]-usually a mirror- and the other beam goes to the sample [3]. After the reflection on each surface, both beams are sent to the detector [5]. The superposition generates the interference signal.

With this signal it is obtained a measurement of the OPD between both arms. In most cases the light beam is focused on the sample so that the measurement process is performed point to point on the desired region. For opaque samples, the light is reflected only on the surface, so a topography measurement is obtained. For transparent or semi-transparent media, light is reflected from sub-surface structures in the sample, so a tomography measurement is performed.

Usually the sample-holder or the light beam can be displaced in a 2-D lateral movement (axis x and y in the figure 7). This way the sample can be inspected in the region of interest.

Although there are many configurations proposed based on the general configuration described before, it is possible to make a division into three main groups, each one with its particular characteristics. They are commonly known as:


Spectral Low Coherence Interferometry:

each pixel, has been already presented (Vakhtin et al, 2003).

total intensity is obtained as a function of the wavelength *(*

**2.2 The interference signal and the detection system** 

sample arm . The expression for the total intensity is then:

path length in each arm, both measured from the beamsplitter.

*Rr1Io (k)* and *Is1(k) = R1sIo (k)*, as mentioned in equation 5, so:

figure 8), were *k=2* 

A Complete Analysis of the Detection System and the Signal Processing 131

between the sample and the reference is obtained from the analysis of the superposition of the spectrum of the light source reflected in each arm of the interferometer. This spectrum is commonly measured with a spectrometer. Most of the detection systems employ linear arrays as a sensor element. Some works with 2-D CCD systems and individual processing of

Figure 8 below illustrates the basic experimental set-up assuming a simple situation in which there are only two reflections, one at the end of each of the arms of the interferometers. The interference signal is measured with the spectrometer sensor where the

Fig. 8. A typical spectral doamin low coherence interferometry experimental set-up.

It is assumed that *Io(k)* encodes the power spectral dependence of the light source ([1] in

superposition of the spectrum of the light coming from the two reflections. We call *Ir1(k)* to the intensity coming from the reference arm and *Is1(k*) to the intensity coming from the

The first two terms are known as DC intensities in the literature (Drexler et al, 2008). The last term in equation 4 is the interference component, and it includes the OPD dependence that, in this simple situation, is given by the value ΔXr1s1 = xr1-xs1, where xr1 and xs1 are the total

To simplify the expression, we use *β2 = Rs1/Rr1*, the reflectivity coefficients ratio, and it is assumed that these coefficients have no dependence on the wavenumber, that is: *Ir1(k) =* 

The figure 9 shows an example of an image obtained in a 2D-CCD sensor at the end of the spectrometer (see set-up picture). The image corresponds to the total intensity described

<sup>2</sup>

<sup>1</sup> 1 1 ( ) ( ) 1 2 cos *r o r s Ik R I k k x* (5)

*/λ,* is the wavenumber. The interference fringes surges from the

*Ik I k I k I kI k k x* ( ) ( ) ( ) 2 ( ) ( ) cos *r s* 1 1 *r s* 1 1 *r s* 1 1 (4)

.

In TDLCI mode the total interference is obtained with a one element detector( [5] in figure 7), usually a photodiode, and the total intensity is registered while the optical path of the reference arm changes from a maximum to a minimum value that is predefined for each setup (displacement in the z direction – figure 7). While the reference arm is moving the sample is maintained in the same position (Drexler et al, 2008).

In SDLCI mode the detection [5] is performed with a spectrometer. In this scheme the interference signal is obtained with the superposition of the spectral intensity of both arms. (Drexler et al, 2008).

Fig. 7. A typical low coherence interferometry experimental set-up.

Based on the same idea, in the SSDLCI configuration, a tuneable laser and a one element photodetector are used. The light source is swept in wavelength as a function of time, so the spectral components are not encoded by spatial separation, they are encoded in time. After a complete swept we obtain the spectrum one by one wavelength (Choma et al 2003), and consequently the interference signal.

## **2. Spectral domain low coherence interferometry**

## **2.1 Introduction**

Spectral domain low coherence interferometry (SDLCI) -also known as Fourier domain low coherence interferometry (FDLCI)- is, as mentioned before, one of the configurations commonly used in low coherence interferometry. The first result using this technique (Fercher et al, 1995) was for ophthalmological measurements and some years later (Hasler et al, 1998) in dermatological applications. After these initial works, it began to be shown as a competitive technique with the time domain method (TD-OCT), a well established method for optical coherence tomography (OCT) applications at that time. Since then, SDLCI has shown several advantages which have favoured its development and the high level of acceptance that it has nowadays.

These advantages can be summarized in the lack of need for a fast mechanical scanning mechanism (Drexler et al, 2008) that brings the simplicity of no moving parts in the reference arm of the interferometer, and the superior sensitivity of the detection (Leitgeb et al, 2003). The SDLI typical configuration is shown in figure 8. The light source and the interferometer scheme follow the same characteristics described before, but the detection system is a distinct point in this technique. The main idea is that the optical path difference

In TDLCI mode the total interference is obtained with a one element detector( [5] in figure 7), usually a photodiode, and the total intensity is registered while the optical path of the reference arm changes from a maximum to a minimum value that is predefined for each setup (displacement in the z direction – figure 7). While the reference arm is moving the

In SDLCI mode the detection [5] is performed with a spectrometer. In this scheme the interference signal is obtained with the superposition of the spectral intensity of both arms.

Based on the same idea, in the SSDLCI configuration, a tuneable laser and a one element photodetector are used. The light source is swept in wavelength as a function of time, so the spectral components are not encoded by spatial separation, they are encoded in time. After a complete swept we obtain the spectrum one by one wavelength (Choma et al 2003), and

Spectral domain low coherence interferometry (SDLCI) -also known as Fourier domain low coherence interferometry (FDLCI)- is, as mentioned before, one of the configurations commonly used in low coherence interferometry. The first result using this technique (Fercher et al, 1995) was for ophthalmological measurements and some years later (Hasler et al, 1998) in dermatological applications. After these initial works, it began to be shown as a competitive technique with the time domain method (TD-OCT), a well established method for optical coherence tomography (OCT) applications at that time. Since then, SDLCI has shown several advantages which have favoured its development and the high level of

These advantages can be summarized in the lack of need for a fast mechanical scanning mechanism (Drexler et al, 2008) that brings the simplicity of no moving parts in the reference arm of the interferometer, and the superior sensitivity of the detection (Leitgeb et al, 2003). The SDLI typical configuration is shown in figure 8. The light source and the interferometer scheme follow the same characteristics described before, but the detection system is a distinct point in this technique. The main idea is that the optical path difference

sample is maintained in the same position (Drexler et al, 2008).

Fig. 7. A typical low coherence interferometry experimental set-up.

**2. Spectral domain low coherence interferometry** 

consequently the interference signal.

acceptance that it has nowadays.

**2.1 Introduction** 

(Drexler et al, 2008).

between the sample and the reference is obtained from the analysis of the superposition of the spectrum of the light source reflected in each arm of the interferometer. This spectrum is commonly measured with a spectrometer. Most of the detection systems employ linear arrays as a sensor element. Some works with 2-D CCD systems and individual processing of each pixel, has been already presented (Vakhtin et al, 2003).

Figure 8 below illustrates the basic experimental set-up assuming a simple situation in which there are only two reflections, one at the end of each of the arms of the interferometers. The interference signal is measured with the spectrometer sensor where the total intensity is obtained as a function of the wavelength *(*.

Fig. 8. A typical spectral doamin low coherence interferometry experimental set-up.

## **2.2 The interference signal and the detection system**

It is assumed that *Io(k)* encodes the power spectral dependence of the light source ([1] in figure 8), were *k=2 /λ,* is the wavenumber. The interference fringes surges from the superposition of the spectrum of the light coming from the two reflections. We call *Ir1(k)* to the intensity coming from the reference arm and *Is1(k*) to the intensity coming from the sample arm . The expression for the total intensity is then:

$$I(k) = I\_{r1}(k) + I\_{s1}(k) + 2\sqrt{I\_{r1}(k)I\_{s1}(k)}\cos\left(k\Delta x\_{r1s1}\right) \tag{4}$$

The first two terms are known as DC intensities in the literature (Drexler et al, 2008). The last term in equation 4 is the interference component, and it includes the OPD dependence that, in this simple situation, is given by the value ΔXr1s1 = xr1-xs1, where xr1 and xs1 are the total path length in each arm, both measured from the beamsplitter.

To simplify the expression, we use *β2 = Rs1/Rr1*, the reflectivity coefficients ratio, and it is assumed that these coefficients have no dependence on the wavenumber, that is: *Ir1(k) = Rr1Io (k)* and *Is1(k) = R1sIo (k)*, as mentioned in equation 5, so:

$$I(k) = R\_{r1} I\_o(k) \left( 1 + \beta^2 + 2\beta \cos\left(k \Delta x\_{r1s1}\right) \right) \tag{5}$$

The figure 9 shows an example of an image obtained in a 2D-CCD sensor at the end of the spectrometer (see set-up picture). The image corresponds to the total intensity described

Spectral Low Coherence Interferometry:

A Complete Analysis of the Detection System and the Signal Processing 133

and xs1s2 = 2.n.d . Where *xs1, xs2, xr1* and 2.nd*,* are the total optical path distances

The usual way to obtain the OPD values from the total intensity data is to apply the Fourier transform. We call *FI(x)* to the Fourier Transform of *I(k)*, where x is the conjugate variable of

As an example it is shown the expression for *FI(x)* in the situation mentioned before (equation 4), where the interference signal is obtained from only two reflections (r1 and s1).

According to the Wiener–Kinchin theorem (Goodman, 1984), the first term in the convolution is the coherence function *Г(x)* and the second term yields two delta functions

1 1 1 1 1 1 1 1

 

*x R R RR x x x x* (9)

( )( ) 2 ( ) ( )

*r s s r r s r s*

Figure 11-a and 11-b illustrate the interference signals in two situations that correspond to different deep modulation *(β = 1, β = 0.5)*. Figures 11-c and 11-d show the absolute value of the corresponding Fourier transform for each signal. This graph shows three "interference peaks". One is centred in the origin of the *x* axis and the "lateral peaks", denoted as P and Q

*FI x F I k F R R R R x k* ( ) ( ( )) 0 1 *r s rs* 1 1 2 cos( ) 1 1*r s*<sup>1</sup> (8)

*xr1s2 = xr1- (xs1+2. n.d)* 

Fig. 10. Experimental set-up for thickness measurement of a slab.

corresponding to each interface indicated in figure 10.

the wavenumber *k*.

We denote the convolution as ( ).

located at *±Δxr1s1* (Papoulis, 1962).

in the figures, are centred in *± Δxr1s1*.

Being α a constant factor.

Therefore, the last expression can be written as:

() .()

*FI x x*

So:

So in this situation the OPD values in equation 7 are: *Δxr1s1 = xr1- xs1 ,* 

before in equation 4. In this particular example, the source is a superluminescent diode (λo = 840 nm, Δλ = 20nm) and R1s = R1r 0.08.

Fig. 9. a) Interferences fringes obtained with a 2-D spectrometer sensor b) Intensity as a function of k obtained from a row of pixel from the image a).

Figure 9-b represents total intensity, obtained from a line of pixels of the image shown in figure 9-a. The modulation in the curve is produced by the interference term over the Gaussian spectrum of the light source.

In a general situation in which it is assumed that there are several reflections in the reference arm and in the sample arm, the total intensity expression can be written as:

$$I(k) = \sum\_{i=1}^{N} I\_i(k) + \sum\_{i,j=1}^{N} 2\sqrt{I\_i(k)I\_j(k)}\cos\left(\Delta x\_{\vec{\eta}}\right) \tag{6}$$

N represents the total number of reflectors in the sample and in the reference. The sub index i or j identifies the region where the reflection is produced in the reference (r1, r2, r3, …) or in the sample arm (s1, s2, s3, …). To illustrate this point, we show a typical application in which the reference arm is a mirror (1r) and the sample is a slab in air. The slab has two interfaces (1s and 2s), an inner group index *n*, and a thickness *d*.

$$\begin{aligned} I(k) &= I\_{r1}(k) + I\_{s1}(k) + I\_{s2}(k) \\ &+ 2\sqrt{I\_{r1}(k)I\_{s1}(k)}\cos(\Delta x\_{r1s1}) \\ &+ 2\sqrt{I\_{r1}(k)I\_{s2}(k)}\cos(\Delta x\_{r1s2}) \\ &+ 2\sqrt{I\_{s1}(k)I\_{s2}(k)}\cos(\Delta x\_{s1s2}) \end{aligned} \tag{7}$$

The first three terms are the DC terms mentioned before. The second and the third term correspond to the cross correlation between the reference and each of the sample surfaces; the forth term is the correlation component that surges from the reflections on both interfaces of the sample. (see figure 10).

before in equation 4. In this particular example, the source is a superluminescent diode (λo =

0

50

100

I (a.u.)

Fig. 9. a) Interferences fringes obtained with a 2-D spectrometer sensor b) Intensity as a

Figure 9-b represents total intensity, obtained from a line of pixels of the image shown in figure 9-a. The modulation in the curve is produced by the interference term over the

In a general situation in which it is assumed that there are several reflections in the reference

*i i j ij*

( ) ( ) 2 ( ) ( ) cos

*Ik I k I kI k x*

N represents the total number of reflectors in the sample and in the reference. The sub index i or j identifies the region where the reflection is produced in the reference (r1, r2, r3, …) or in the sample arm (s1, s2, s3, …). To illustrate this point, we show a typical application in which the reference arm is a mirror (1r) and the sample is a slab in air. The slab has two

112

*rss*

2 ( ) ( ) cos 2 ( ) ( ) cos 2 ( ) ( ) cos

 

The first three terms are the DC terms mentioned before. The second and the third term correspond to the cross correlation between the reference and each of the sample surfaces; the forth term is the correlation component that surges from the reflections on both

() () () ()

*Ik I k I k I k*

arm and in the sample arm, the total intensity expression can be written as:

*N N*

*i ij*

interfaces (1s and 2s), an inner group index *n*, and a thickness *d*.

1 ,1

150

200

250

0 100 200 300 400 500 600

Pixel (b)

(7)

(6)

 

1 1 1 1 1 2 1 2 1 2 1 2

*r s r s r s r s s s s s*

*I kI k x I kI k x I kI k x*

840 nm, Δλ = 20nm) and R1s = R1r 0.08.

Interference Fringe (a)

Gaussian spectrum of the light source.

interfaces of the sample. (see figure 10).

function of k obtained from a row of pixel from the image a).

Fig. 10. Experimental set-up for thickness measurement of a slab.

So in this situation the OPD values in equation 7 are: *Δxr1s1 = xr1- xs1 , xr1s2 = xr1- (xs1+2. n.d)*  and xs1s2 = 2.n.d . Where *xs1, xs2, xr1* and 2.nd*,* are the total optical path distances corresponding to each interface indicated in figure 10.

The usual way to obtain the OPD values from the total intensity data is to apply the Fourier transform. We call *FI(x)* to the Fourier Transform of *I(k)*, where x is the conjugate variable of the wavenumber *k*.

As an example it is shown the expression for *FI(x)* in the situation mentioned before (equation 4), where the interference signal is obtained from only two reflections (r1 and s1). So:

$$FI(\mathbf{x}) = F(I\_0(k)) \otimes F\left(R\_{r1} + R\_{s1} + 2\sqrt{R\_{r1}R\_{s1}}\cos(\Delta \mathbf{x}\_{r1s1}k)\right) \tag{8}$$

We denote the convolution as ( ).

According to the Wiener–Kinchin theorem (Goodman, 1984), the first term in the convolution is the coherence function *Г(x)* and the second term yields two delta functions located at *±Δxr1s1* (Papoulis, 1962).

Therefore, the last expression can be written as:

$$\begin{aligned} FI(\mathbf{x}) &= \alpha \Gamma(\mathbf{x}) \otimes \\ &\left[ \mathcal{S}(\mathbf{x}) (R\_{r1} + R\_{z1}) + 2\sqrt{R\_{z1} R\_{r1}} \left( \mathcal{S}(\mathbf{x} - \Delta \mathbf{x}\_{r1\varepsilon}) + \mathcal{S}(\mathbf{x} + \Delta \mathbf{x}\_{r1\varepsilon}) \right) \right] \end{aligned} \tag{9}$$

Being α a constant factor.

Figure 11-a and 11-b illustrate the interference signals in two situations that correspond to different deep modulation *(β = 1, β = 0.5)*. Figures 11-c and 11-d show the absolute value of the corresponding Fourier transform for each signal. This graph shows three "interference peaks". One is centred in the origin of the *x* axis and the "lateral peaks", denoted as P and Q in the figures, are centred in *± Δxr1s1*.

Spectral Low Coherence Interferometry:

**2.3 Signal detection and parameter limits** 

spectral range of the spectrometer *(*

function as follows:

*z*

the spot size.

written as:

A Complete Analysis of the Detection System and the Signal Processing 135

In an ideal situation, the intensity function *I(k)* would be obtained with a "perfect" spectrometer. In a real situation, it is important to evaluate the spectrometer parameters in order to optimize the measurements and evaluate possible artifacts, errors or limitations introduced for the "real" set-up. We assume a detector model based on a grating spectrometer with a linear array sensor. There are some important issues to be considered in

a. The spectrometer is used for separating the different wavelengths of the light source spectrum and to focus each of them on the sensor that is considered as a linear array of Np pixels. After its calibration it is possible to define a *k* axis in the array direction. The

value assigned to the first pixel and *kNp* to the last one. We also define the *"Range"*

( ) *<sup>a</sup>*

*ka* indicates the central wavelength in the spectrometer. Along this work we assume

b. The dimensions of the spot of each wavelength focused on the sensor is taken into account by the point spread function (*Psf(k)*) (Hu et al, 2007). The width, the shape and the position of the center of this spot depends on the spectrometer characteristics such as the entrance slit width, diffraction grating and focal length and eventually on the spatial profile of the light source in the spectrometer entrance (Dorrer et al, 2000). We

0.

The " ideal" detected signal is then modified by the convolution with this function:

´ . ( ´) ´ ( ) *rI k I k Psf k k dk I Psf k*

c. The value obtained in a particular pixel "i" is the total intensity integrated over a width *(Δkpix)*, which is defined by the pixel dimensions, the grating dispersion and the focus length of the spectrometer (Leitgeb et al 2003; Hu et al 2007). This average value can be

( )( ). ( ) ( ) *r i <sup>i</sup> I k I Psf k Pix k ki dk I Psf Pix k*

*<sup>k</sup> Psf k psf e*

*k k Range k <sup>k</sup>*

*a*

2 2 ( ) 4ln(2)

*i psf k k*

*ka))* is given by the interval *[k1, kNp]* where *k1* is the

(10)

(11)

(12)

(13)

*kpsf)* is a measurement of

this evaluation, more details can be found in (Hu et al, 2007; Jeon et al, 2011).

a *rect* function defined as 0 if │z│>1/2 and 1 if │z│<1/2.

assume a Gaussian shape (Hu et al, 1997) where its FWHM *(*

Fig. 11. a) and b) shows the total interference signal as a function of wavenumber k, for two different relations. c) and d) shows its corresponding Fourier transform as a function of x (the conjugate variable of k) .

Figure 12 shows an example of how the position of the interference peak changes according to the increment of OPD between both arms. The peaks in order zero no contain OPD information.

Fig. 12. Interference peak vs OPD increments.

## **2.3 Signal detection and parameter limits**

134 Interferometry – Research and Applications in Science and Technology

Fig. 11. a) and b) shows the total interference signal as a function of wavenumber k, for two different relations. c) and d) shows its corresponding Fourier transform as a function of x

Figure 12 shows an example of how the position of the interference peak changes according to the increment of OPD between both arms. The peaks in order zero no contain OPD

(the conjugate variable of k) .

Fig. 12. Interference peak vs OPD increments.

information.

In an ideal situation, the intensity function *I(k)* would be obtained with a "perfect" spectrometer. In a real situation, it is important to evaluate the spectrometer parameters in order to optimize the measurements and evaluate possible artifacts, errors or limitations introduced for the "real" set-up. We assume a detector model based on a grating spectrometer with a linear array sensor. There are some important issues to be considered in this evaluation, more details can be found in (Hu et al, 2007; Jeon et al, 2011).

a. The spectrometer is used for separating the different wavelengths of the light source spectrum and to focus each of them on the sensor that is considered as a linear array of Np pixels. After its calibration it is possible to define a *k* axis in the array direction. The spectral range of the spectrometer *(ka))* is given by the interval *[k1, kNp]* where *k1* is the value assigned to the first pixel and *kNp* to the last one. We also define the *"Range"* function as follows:

$$Range(k) = \prod \left( \frac{k - k\_a}{\Delta k\_a} \right) \tag{10}$$

*ka* indicates the central wavelength in the spectrometer. Along this work we assume *z*a *rect* function defined as 0 if │z│>1/2 and 1 if │z│<1/2.

b. The dimensions of the spot of each wavelength focused on the sensor is taken into account by the point spread function (*Psf(k)*) (Hu et al, 2007). The width, the shape and the position of the center of this spot depends on the spectrometer characteristics such as the entrance slit width, diffraction grating and focal length and eventually on the spatial profile of the light source in the spectrometer entrance (Dorrer et al, 2000). We assume a Gaussian shape (Hu et al, 1997) where its FWHM *(kpsf)* is a measurement of the spot size.

$$\begin{array}{c} \text{1.} \text{Pf}\left(k\right) = \text{psf}\_0.e \end{array} \tag{11}$$

The " ideal" detected signal is then modified by the convolution with this function:

$$I\_r\left(k\right) = \bigcap\_{-\phi}^{\phi} I\left(k'\right).Psf\left(k-k'\right)dk' = I \otimes Psf\left(k\right) \tag{12}$$

c. The value obtained in a particular pixel "i" is the total intensity integrated over a width *(Δkpix)*, which is defined by the pixel dimensions, the grating dispersion and the focus length of the spectrometer (Leitgeb et al 2003; Hu et al 2007). This average value can be written as:

$$I\_r(k\_i) = \bigcap\_{-\infty}^{\infty} (I \otimes Psf)(k). \text{Pix}(k - ki) \, dk = I \otimes Psf \otimes \text{Pix}(k\_i) \tag{13}$$

Spectral Low Coherence Interferometry:

available for signal processing avoiding aliasing.

psf(x) pixel(x)

Interference Peak

0 1 2 3 4 5

x (mm) (b)

xpsf=4.ln(2)/kpsf (Bajraszewski et al, 2008), (Leitgeb et al, 2003).

it is shown these terms for a typical situation.

0

*<sup>k</sup> FI x FI c x*

*n*

sin

That can be written as:

0.5

0

0.5

FI (a.u.)

Zoom of figure a).

**2.4 Signal processing** 

1

FI (a.u.)

1

A Complete Analysis of the Detection System and the Signal Processing 137

This expression is easily obtained from equation 18 and some mathematics relations (Papoulis, 1969). *FIo* is a constant factor that also absorbs all the pure phase terms that result from the Fourier transform. The new delta comb has a period x=2kpix*,* the interval

The expression in brackets in equation 19, shows the Fourier transform of the "ideal." intensity and the modifications introduced by the Fourier transform of the *psf* and the *pixel* function.

*x nx*

2 *a*

*pix*

2 2

0 0.5 1 1.5 2 2.5 3 3.5

x (mm) (c)

x 10-3

(19)

sin .exp ( ( )) <sup>2</sup> 16ln(2)

0 0.2 0.4

FI (a.u.)

x 10-3

Fig. 14. a) Interference peak and the psf and pixel Fourier Transform functions. b) and c)

This modification means a sensitivity fall-off when the OPD value increase. In the figure 14

As it can be appreciated from the figure 14, the shape of both functions (the *sinc* and *psf* Fourier transform) generate a decrease in the visibility amplitude of the interference peak that goes to zero when x increase. So, there are two parameters that limit the dynamic range of the technique. One is: xpix=2/kp, the first zero of the *sinc* function. The other is the FWHM of the Gaussian function obtained as the Fourier transform of the *psf*:

In the last section we showed some considerations in the interference image, imposed by the real characteristics of the detector. The SDLCI detection process ends with an array of Np

<sup>0</sup> 0.005 0.01 0.015 0.02 0.025 0.03 <sup>0</sup>

x (mm) (a)

*<sup>k</sup> k x psf c x FIk*

The convolution takes into account the process described before. A first approach to this function can be to assume a rectangular shape where its width is: *Δkpix = Δka / Np* (Hu et al, 2007) (Wang et al, 2008). Here it is assumed by simplicity that the fill factor is 100%.

$$\Pr(k - k\_i) = \Pi(\frac{k - k\_i}{\Delta k\_{pix}}) \tag{14}$$

More complex dependence with *k* can be thought, considering additional effects such as crosstalk or fill factor (Jeon et al, 2011).

d. The measurement of the total intensity *I(k)* obtained from the detector is a discrete collection of *Np* values. Each value corresponds to the average intensity measured in one pixel, for this reason, it is convenient to define a sampling comb of delta functions as (Dorrer et al, 2000):

$$\mathbf{C}\left(k\right) = \sum\_{i=-\nu}^{+\nu} \mathbb{S}\left(k - k\_i\right) \tag{15}$$

Where *ki* is the value assigned to the pixel with index *i* (Wang et al,2006); (Hu et al, 2007).

The following figure shows a schematic of the spectrometer sensor and a detail of how two wavelengths (*i* and *j*) have been focused on it. The picture also shows how the *"pix"* and *"psf"* function combine to define the average value in pixel j.

Fig. 13. Schematic of the spectrometer sensor array and the "psf" and "pixel" functions.

Then, the final expression for the signal intensity is:

$$I(k) = \text{Range}(k) (\text{C} \otimes p\text{sf} \otimes p\text{ixel} \otimes I) (k) \tag{16}$$

$$I(k) = \prod \left( k - k\_a \right) \sum\_{i = -\infty}^{\infty} \delta \left( k - k\_i \right) \left( p \text{ixel} \otimes psf \otimes I(k) \right) \tag{17}$$

Under these considerations the A scan signal *FI(x)* is obtained applying the Fourier transform of equation 17. Here the *x* variable is the conjugated of *k.*

$$\begin{aligned} FI\_4(\mathbf{x}) &= F(\prod (k - k\_a)) \\ &\otimes F\left(\sum\_{i = -\infty}^{\infty} \delta(k - k\_i)\right) \\ &\otimes \{ F(p \text{pixel}(k)) F(p \text{sf}(k)) F(I(k)) \} \end{aligned} \tag{18}$$

That can be written as:

136 Interferometry – Research and Applications in Science and Technology

The convolution takes into account the process described before. A first approach to this function can be to assume a rectangular shape where its width is: *Δkpix = Δka / Np* (Hu et al, 2007) (Wang et al, 2008). Here it is assumed by simplicity that the fill factor is 100%.

> ( )*<sup>i</sup> i*

More complex dependence with *k* can be thought, considering additional effects such as

d. The measurement of the total intensity *I(k)* obtained from the detector is a discrete collection of *Np* values. Each value corresponds to the average intensity measured in one pixel, for this reason, it is convenient to define a sampling comb of delta functions

> *i Ck k k*

Fig. 13. Schematic of the spectrometer sensor array and the "psf" and "pixel" functions.

*a i*

Under these considerations the A scan signal *FI(x)* is obtained applying the Fourier

*a*

*F kk*

 

*F pixel k F psf k FIk*

*i*

() () ()

*I k k k k k pixel psf I k*

*i*

*FI x F k k*

*i*

transform of equation 17. Here the *x* variable is the conjugated of *k.*

4

*"pix"* and *"psf"* function combine to define the average value in pixel j.

Where *ki* is the value assigned to the pixel with index *i* (Wang et al,2006); (Hu et al, 2007). The following figure shows a schematic of the spectrometer sensor and a detail of how

*k k Pix k k*

crosstalk or fill factor (Jeon et al, 2011).

*i* and 

Then, the final expression for the signal intensity is:

*<sup>i</sup>*

as (Dorrer et al, 2000):

two wavelengths (

*pix*

(14)

(15)

*j*) have been focused on it. The picture also shows how the

*I k Ran gek C* ( )( *psf pixel I k* ) (16)

(17)

(18)

*k*

$$\begin{aligned} FI(\mathbf{x}) &= FI\_0 \sin c \left(\frac{\Delta k\_a}{2}\mathbf{x}\right) \\ &\quad \otimes \sum\_{n=-\infty}^{\infty} \delta \left(\mathbf{x} - n\Delta \mathbf{x}\right) \\ &\quad \otimes \left[ \text{sinc} \left(\frac{\Delta k\_p \text{pix}}{2} \mathbf{x}\right) \exp\left(-\frac{\Delta k^2}{16\ln(2)} \mathbf{x}^T\right) \mathbf{f}(I(k)) \right] \end{aligned} \tag{19}$$

This expression is easily obtained from equation 18 and some mathematics relations (Papoulis, 1969). *FIo* is a constant factor that also absorbs all the pure phase terms that result from the Fourier transform. The new delta comb has a period x=2kpix*,* the interval available for signal processing avoiding aliasing.

The expression in brackets in equation 19, shows the Fourier transform of the "ideal." intensity and the modifications introduced by the Fourier transform of the *psf* and the *pixel* function.

Fig. 14. a) Interference peak and the psf and pixel Fourier Transform functions. b) and c) Zoom of figure a).

This modification means a sensitivity fall-off when the OPD value increase. In the figure 14 it is shown these terms for a typical situation.

As it can be appreciated from the figure 14, the shape of both functions (the *sinc* and *psf* Fourier transform) generate a decrease in the visibility amplitude of the interference peak that goes to zero when x increase. So, there are two parameters that limit the dynamic range of the technique. One is: xpix=2/kp, the first zero of the *sinc* function. The other is the FWHM of the Gaussian function obtained as the Fourier transform of the *psf*: xpsf=4.ln(2)/kpsf (Bajraszewski et al, 2008), (Leitgeb et al, 2003).

## **2.4 Signal processing**

In the last section we showed some considerations in the interference image, imposed by the real characteristics of the detector. The SDLCI detection process ends with an array of Np

Spectral Low Coherence Interferometry:

OPD measurement.

0

1

0

0.5

Amplitude (Norm.)

consuming.

0.5

Amplitude (Norm.)

1

Gaussian).

A Complete Analysis of the Detection System and the Signal Processing 139

Another point that can be avoided or smoothed by signal processing is the edge effects in the Fourier transform of the *Range* function proposed in equation 10, to take into account the spectrometer range. To modify the shape of the rectangular function the windowing technique can be used (Oppenheim et al, 2000). With this tool it is possible to minimize the effects that result in spectral leakage in the *FI(x)* signal, and to increase the resolution in the

There are many types of windows available in the literature. Figure 16 shows common functions (Rectangular, Bartlett, Hanning, Hamming, Blackman, Blackman-Harris and



Amplitude (dB)

A spectral analysis of the different window functions is recommended to determine the best to use. In some cases it is possible to apply more than one window simultaneously. To increment the resolution some techniques can be used (Boaz Porat, 1997). One of the most popular is known as Zero Padding (ZP) (Dorrer et al, 2000; Yun et al, 2003). In this technique the original array *I[k]* is extended with zeros, then, the length of the new array changes to *Mp> Np*. As a consequence, the sampling interval can be reduced by a *Np/Mp* factor with the disadvantages that mean computing longer vectors and more time

to ZP but in a localized region of the x axis is the Chirp Fourier Transform (CFT). This

*F q Ine q q q q*

0

/ , 0 1

(22)


0

0 50 100 150

*xsamp* in a similar way

Rectangular Triangular Hann Hamming Blackman Blackman-Harris Gaussian

Samples

0 100 200 300 400 500 600

0 10 20 30 40 50

Samples

Samples (a)

Fig. 16. Typical windows functions and b) the corresponding FFT curves. c)

An alternative technique that allows reducing the sampling interval

If a sequence I[n] with *0 ≤ n ≤ Np-1* is used, we can write the DFT:

0

*Np*

*n*

predefines the sampling interval and the region of interest on the x axis.

 1

*j q*

data points that contain the information of the depth profile of the sample. This process usually continues with the analysis of the Digital Fast Fourier Transform (FFT) of this vector. The classical algorithm is shown in equation 20, where Np is the I[n] length.

$$FI\left[q\right] = \sum\_{n=0}^{N\_p - 1} I\left[n\right] \mathcal{W}\_{N\_p}^{qn} \quad \prime \quad \mathcal{W}\_{N\_p}^{qn} = e^{-j\left(\frac{2\pi}{N\_p}\right)} \tag{20}$$

In figure 15 a typical example is shown. The interference signal is registered in a 640-point array (figure 15-a). Its conventional FFT is shown in figure 15-b. The interference spectrum has been obtained in a wavelength range from 780 to 880 nm. The modulation corresponds to a situation in which there is only one surface (*r1* and *s1*) in both arms of the interferometer, as presented in equation 4.

Fig. 15. a) Interference signal and b) Interference peak.

The OPD value xr1s1 in this scheme is the abscissa coordinate of the maximum point of the lateral peak in the Fourier transform curve (figure 15-b). With the conventional Cooley-Tukey FFT algorithm the sampling interval is:

$$
\Delta \mathbf{x}\_{samp} = \frac{2\pi}{\mathcal{N}\_p \Delta k\_{p\text{ix}}} \tag{21}
$$

Where Np and Δkpix are the parameters previously defined. In this step, spectral calibration is a critical process. First, a conversion of the measured spectrographs from λ-space to kspace is needed. As the spectra obtained by the spectrometer are not necessarily evenly spaced, a posterior resample to be uniformly spaced in k-axis is required. This implies a careful calibration of each pixel of the sensor. There are several works in which this point is presented (Dorrer et al, 2000; Hu et al, 2007) and different experimental and post process arrangements are proposed (Bajraszewski et all, 2008; Jeon et al 2011).

data points that contain the information of the depth profile of the sample. This process usually continues with the analysis of the Digital Fast Fourier Transform (FFT) of this

<sup>2</sup> <sup>1</sup>

*p p*

*qn qn N*

*p*

 

(20)

/

*N N*

*N j*

In figure 15 a typical example is shown. The interference signal is registered in a 640-point array (figure 15-a). Its conventional FFT is shown in figure 15-b. The interference spectrum has been obtained in a wavelength range from 780 to 880 nm. The modulation corresponds to a situation in which there is only one surface (*r1* and *s1*) in both arms of the

<sup>780</sup> <sup>790</sup> <sup>800</sup> <sup>810</sup> <sup>820</sup> <sup>830</sup> <sup>840</sup> <sup>850</sup> <sup>860</sup> <sup>870</sup> <sup>880</sup> <sup>0</sup>

 (a)

<sup>0</sup> 0.05 0.1 0.15 0.2 0.25 0.3 <sup>0</sup>

x (mm) (b)

The OPD value xr1s1 in this scheme is the abscissa coordinate of the maximum point of the lateral peak in the Fourier transform curve (figure 15-b). With the conventional Cooley-

*samp*

Where Np and Δkpix are the parameters previously defined. In this step, spectral calibration is a critical process. First, a conversion of the measured spectrographs from λ-space to kspace is needed. As the spectra obtained by the spectrometer are not necessarily evenly spaced, a posterior resample to be uniformly spaced in k-axis is required. This implies a careful calibration of each pixel of the sensor. There are several works in which this point is presented (Dorrer et al, 2000; Hu et al, 2007) and different experimental and post process

*x*

arrangements are proposed (Bajraszewski et all, 2008; Jeon et al 2011).

2

*N k*

*p pix*

(21)

vector. The classical algorithm is shown in equation 20, where Np is the I[n] length.

*FI q I n W W e*

*n*

interferometer, as presented in equation 4.

Fig. 15. a) Interference signal and b) Interference peak.

Tukey FFT algorithm the sampling interval is:

FI(x) (a.u.)

Intensity I(k) (a.u.)

*p*

0

Another point that can be avoided or smoothed by signal processing is the edge effects in the Fourier transform of the *Range* function proposed in equation 10, to take into account the spectrometer range. To modify the shape of the rectangular function the windowing technique can be used (Oppenheim et al, 2000). With this tool it is possible to minimize the effects that result in spectral leakage in the *FI(x)* signal, and to increase the resolution in the OPD measurement.

There are many types of windows available in the literature. Figure 16 shows common functions (Rectangular, Bartlett, Hanning, Hamming, Blackman, Blackman-Harris and Gaussian).

Fig. 16. Typical windows functions and b) the corresponding FFT curves. c)

A spectral analysis of the different window functions is recommended to determine the best to use. In some cases it is possible to apply more than one window simultaneously.

To increment the resolution some techniques can be used (Boaz Porat, 1997). One of the most popular is known as Zero Padding (ZP) (Dorrer et al, 2000; Yun et al, 2003). In this technique the original array *I[k]* is extended with zeros, then, the length of the new array changes to *Mp> Np*. As a consequence, the sampling interval can be reduced by a *Np/Mp* factor with the disadvantages that mean computing longer vectors and more time consuming.

An alternative technique that allows reducing the sampling interval *xsamp* in a similar way to ZP but in a localized region of the x axis is the Chirp Fourier Transform (CFT). This predefines the sampling interval and the region of interest on the x axis.

If a sequence I[n] with *0 ≤ n ≤ Np-1* is used, we can write the DFT:

$$F\left(\boldsymbol{\theta}[\boldsymbol{q}]\right) = \sum\_{n=0}^{N\_p - 1} I\left[n\right] e^{-j\boldsymbol{\theta}[\boldsymbol{q}]} \quad \text{/} \quad \boldsymbol{\theta}[\boldsymbol{q}] = \boldsymbol{\theta}\_0 + q\Delta\boldsymbol{\theta}\_\prime \ 0 \le \boldsymbol{q} \le q - 1 \tag{22}$$

Spectral Low Coherence Interferometry:

displacement.

500 1000

Fig. 20. Schematic of the sample.

Peak Amplitude (a.u.)

Peak Position (a.u.)

A Complete Analysis of the Detection System and the Signal Processing 141

At first a calibration process is shown. As it has been mentioned before, it is necessary to link the interferometer signal with the OPD absolute values. A simple way to do this is to move the reference surface through controlled displacements (OPD) or steps, and simultaneously register the abscise coordinate of the center of the FTI peak that corresponds to each step. Figure 18-a shows the peak abscise coordinate (pixel number) versus displacement. Figure 18-b shows the peak amplitude versus the corresponding

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>0</sup>

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>0</sup>

Fig. 19. a) OPD in pixel vs Displacement and b) Amplitud variation of interference peak.

explained in section 2.3. For this particular set-up the *psf* function is predominant.

shows no-real OPD values that can be misunderstood (figure 19).

the figure 20 (2 mm on the gauge surface and 2mm on the reference plane).

The amplitude fall-off shows the consequence of the "real parameters" of the detector

When the OPD value is close to the limit imposed by Nyquist, aliasing effects can affect the interference signal (Dorrer et al, 2000). If the Nyquist limit is transposed, the measurement

In the next example the technique is used to obtain the profile of a metal sample. The sample is a gauge (class 2-norm ISO 3650, model M7T the C.E. Johansson Inc.); its nominal thickness is 1100 ± 0.45μm; this gauge was placed on a second gauge which was used as a reference plane, as shown in figure 20. A 4- mm length profile was obtained on the region indicated in

Displacement (mm) (a)

Displacement (mm) (b)

Where *Δθ* is the sampling resolution desired, *θ0* the initial *x* value and *K* the number of additional points. Therefore, through *θ0, Δθ* and *Np*, you can delimit the frequency range in which you want to work. The operation number required is approximately M+N.

In figure 17, the original signal FI[x], FIZP[x] and FICFT[x] are shown, the sequence I[k] has 640 points but in the FI[x] only six point defined the interference peak (figure 17-a). To increment the resolution in a factor of 5, we used ZP and CFT. The resolution level is equal in both cases but FIZP[x] has 3200 points and FICFT[x] has only 81 points as showed in figure 17-b and 17-c.

Fig. 17. (a) FI[x], (b) FIZP[x] and (c) FICFT[x].

## **3. Experimental results**

In this section we show results obtained from an experimental configuration similar to that presented in section 2.1. The idea is to show that with a relatively simple set-up, it is possible to apply this technique in a variety of applications with interesting results. The light source is a superluminescent infrared diode, centered at 840nm, with a 20- nm bandwidth and a 5-mw output power. The interferometer is a Michelson type, in air , and the detector is a spectrometer Ocean Optics model 4000. With this configuration we get a 2-mm dynamic range and the axial resolution is lower than 10 microns.

Fig. 18. a) Calibration curve and b) Amplitude variation of interference peak.

Where *Δθ* is the sampling resolution desired, *θ0* the initial *x* value and *K* the number of additional points. Therefore, through *θ0, Δθ* and *Np*, you can delimit the frequency range in

In figure 17, the original signal FI[x], FIZP[x] and FICFT[x] are shown, the sequence I[k] has 640 points but in the FI[x] only six point defined the interference peak (figure 17-a). To increment the resolution in a factor of 5, we used ZP and CFT. The resolution level is equal in both cases but FIZP[x] has 3200 points and FICFT[x] has only 81 points as showed in figure

<sup>0</sup> 0.05 0.1 0.15 0.2 0.25 0.3 <sup>0</sup>

<sup>0</sup> 0.05 0.1 0.15 0.2 0.25 0.3 <sup>0</sup>

<sup>0</sup> 0.05 0.1 0.15 0.2 0.25 0.3 <sup>0</sup>

x (mm) (c)

In this section we show results obtained from an experimental configuration similar to that presented in section 2.1. The idea is to show that with a relatively simple set-up, it is possible to apply this technique in a variety of applications with interesting results. The light source is a superluminescent infrared diode, centered at 840nm, with a 20- nm bandwidth and a 5-mw output power. The interferometer is a Michelson type, in air , and the detector is a spectrometer Ocean Optics model 4000. With this configuration we get a 2-mm dynamic



Fig. 18. a) Calibration curve and b) Amplitude variation of interference peak.

Displacement (mm) (b)

Displacement (mm) (a)

x (mm) (b)

x (mm) (a)

which you want to work. The operation number required is approximately M+N.

17-b and 17-c.

FIST (a.u.)

FIZP (a.u.)

FICFT (a.u.)

5000

5000

5000

**3. Experimental results** 

1000 2000

Peak Amplitude (a.u.)

Peak Position (a.u.)

Fig. 17. (a) FI[x], (b) FIZP[x] and (c) FICFT[x].

range and the axial resolution is lower than 10 microns.

At first a calibration process is shown. As it has been mentioned before, it is necessary to link the interferometer signal with the OPD absolute values. A simple way to do this is to move the reference surface through controlled displacements (OPD) or steps, and simultaneously register the abscise coordinate of the center of the FTI peak that corresponds to each step. Figure 18-a shows the peak abscise coordinate (pixel number) versus displacement. Figure 18-b shows the peak amplitude versus the corresponding displacement.

Fig. 19. a) OPD in pixel vs Displacement and b) Amplitud variation of interference peak.

The amplitude fall-off shows the consequence of the "real parameters" of the detector explained in section 2.3. For this particular set-up the *psf* function is predominant.

When the OPD value is close to the limit imposed by Nyquist, aliasing effects can affect the interference signal (Dorrer et al, 2000). If the Nyquist limit is transposed, the measurement shows no-real OPD values that can be misunderstood (figure 19).

In the next example the technique is used to obtain the profile of a metal sample. The sample is a gauge (class 2-norm ISO 3650, model M7T the C.E. Johansson Inc.); its nominal thickness is 1100 ± 0.45μm; this gauge was placed on a second gauge which was used as a reference plane, as shown in figure 20. A 4- mm length profile was obtained on the region indicated in the figure 20 (2 mm on the gauge surface and 2mm on the reference plane).

Fig. 20. Schematic of the sample.

Spectral Low Coherence Interferometry:

and D1 is the OPD between both samples faces.

Fig. 23. Sagnac-Michelson interferometer.

The expressions for the different OPD are defined as follow:

A Complete Analysis of the Detection System and the Signal Processing 143

With this set-up it is possible to obtain measurement of two sample's face simultaneously employing a third reference surface, which is an advantage over other techniques as the interferometric gauge block, (Decker & Pekelsky, 1997). This is very useful for thickness measurements in opaque samples or where the refraction index is unknown. In figure 23 it is shown a basic Sagnac-Michelson interferometer set-up (Morel & Torga, 2009). This is a ring interferometer type were M1 is the reference mirror; M2 and M3 are the mirrors

BS1 and BS2 are two beam splitters. The light source is a superluminescent diode and the detector system ends in a spectrometer. The reference arm ends in mirror E1. After BS2, the two beams are directed to each of the faces of the sample; after their reflections in the sample face and reflection E1, we obtain three interference signals which let us know the

In order to identify the origin of each of the interference signals employed in the measurements, we define Dr as the optical path difference (OPD) between the reference arm and the beam in the ring interferometer when the sample has been removed; D2 and D3 represent the OPD between the reference arm and the reflections in each of the sample faces,

*D L L L L L d LR <sup>r</sup>* 12 345 2 2 (23)

*D LLL LLL* 1 413 523 2 2 (24)

The last example is to show a different configuration in the interferometer system.

employed to send the light to each face of the sample (S in the figure 1).

sample thickness. We measure the interference signals in the detector system.

Fig. 21. a) Profile obtained by SDLCI by a sequence of points measured on the sample surface. b) Dispersion in the axial distance measurement.

The profile is obtained by measuring the axial distance in a sequence of points on the sample surface (figure 21-a). Each point is measured after a lateral displacement by steps of 50 microns. The value of gauge thickness obtained from this measurement is: 1100 ± 0.71 μm. Figure 21-b shows the dispersion in each point measured, which is much lower than the coherence length of the source, except on the borders of the sample step.

In the following example an application on 3D- surface measurement is shown. These measurements are critical to the successful manufacturing of precision parts. Components and structures ranging from submillimeter to centimeter size can be found in many fields including the automotive, aerospace, semiconductor and data storage industries.

Fig. 22. b)- c) and d) are 3D scan results of a sector of the sample shown in a).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

STEP (mm) (b)

The profile is obtained by measuring the axial distance in a sequence of points on the sample surface (figure 21-a). Each point is measured after a lateral displacement by steps of 50 microns. The value of gauge thickness obtained from this measurement is: 1100 ± 0.71 μm. Figure 21-b shows the dispersion in each point measured, which is much lower than the

In the following example an application on 3D- surface measurement is shown. These measurements are critical to the successful manufacturing of precision parts. Components and structures ranging from submillimeter to centimeter size can be found in many fields

> 0 10 20

 5 10

60

300

Z (mm)

30

330


120

(c) X (mm)

90

270

 vs Module (e)

180 0

210

150

240

Y (mm)

Fig. 21. a) Profile obtained by SDLCI by a sequence of points measured on the sample

coherence length of the source, except on the borders of the sample step.

including the automotive, aerospace, semiconductor and data storage industries.

STEP (mm) (a)

0 0.5 1

0

surface. b) Dispersion in the axial distance measurement.

Iron Test Tube (a)

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> -10

Displacement (mm) (d)

Fig. 22. b)- c) and d) are 3D scan results of a sector of the sample shown in a).

Diameter (mm)

0.01

DISTANCES (mm)

0.02

DISTANCES (mm)

The last example is to show a different configuration in the interferometer system.

With this set-up it is possible to obtain measurement of two sample's face simultaneously employing a third reference surface, which is an advantage over other techniques as the interferometric gauge block, (Decker & Pekelsky, 1997). This is very useful for thickness measurements in opaque samples or where the refraction index is unknown. In figure 23 it is shown a basic Sagnac-Michelson interferometer set-up (Morel & Torga, 2009). This is a ring interferometer type were M1 is the reference mirror; M2 and M3 are the mirrors employed to send the light to each face of the sample (S in the figure 1).

BS1 and BS2 are two beam splitters. The light source is a superluminescent diode and the detector system ends in a spectrometer. The reference arm ends in mirror E1. After BS2, the two beams are directed to each of the faces of the sample; after their reflections in the sample face and reflection E1, we obtain three interference signals which let us know the sample thickness. We measure the interference signals in the detector system.

In order to identify the origin of each of the interference signals employed in the measurements, we define Dr as the optical path difference (OPD) between the reference arm and the beam in the ring interferometer when the sample has been removed; D2 and D3 represent the OPD between the reference arm and the reflections in each of the sample faces, and D1 is the OPD between both samples faces.

Fig. 23. Sagnac-Michelson interferometer.

The expressions for the different OPD are defined as follow:

$$D\_r = \left(L\_1 + L\_2 + 2L\_3 + L\_4 + L\_5 + d\right) - \mathcal{D}\left(LR\right) \tag{23}$$

$$D\_1 = \mathcal{Z}\left(L\_4 + L\_1 + L\_3\right) - \mathcal{Z}\left(L\_5 + L\_2 + L\_3\right) \tag{24}$$

Spectral Low Coherence Interferometry:

<sup>0</sup> <sup>5</sup> <sup>10</sup> 0.05

STEP (mm) (a)

improve the information obtained from the images.

think is nowadays largely unexplored.

**5. Acknowledgment** 

support and images design.

Argentina.

0.1 0.15 0.2 0.25

1.04 1.06 1.08 1.1 1.12

**4. Conclusions** 

THICKNESS (mm)

D1

(A.U.)

A Complete Analysis of the Detection System and the Signal Processing 145

<sup>0</sup> <sup>5</sup> <sup>10</sup> 0.32

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> 1.02

Fig. 25. Surface profiles and thickness obtained from the sample shown in figure 24.

In this chapter we present a brief introduction to the theory of low coherence interferometry and the main configurations used nowadays. Then, we focus in giving a detailed analysis of the technique "Spectral domain low coherence interferometry" that offer interesting advantages over alternative methods. We give a description of the limitations and experimental design considerations to take account in the detection of the interference intensity and in the signal processing. We also show how this knowledge is important to

At the end we show some experimental results with the intention of giving examples of the enormous potential of this technique, specially in the non- destructive tests area, where we

E. Morel and J. Torga are professors at the Universidad Tecnológica Nacional and are members of the Consejo Nacional de Investigaciones Científicas y Técnicas de la República

Thanks to Hernán Miranda for his collaboration and special thanks to Pamela Morel for her

STEP (mm) (d)

STEP (mm) (b)

<sup>0</sup> <sup>5</sup> <sup>10</sup> 0.45

STEP (mm) (c)

0.5 0.55

D3

**DARK ZONE**

(A.U.)

0.34 0.36 0.38 0.4

D2

(A.U.)

$$D\_2 = \mathcal{Z}\left(L\_4 + L\_1 + L\_3\right) - \mathcal{Z}\left(LR\right) \tag{25}$$

$$D\_3 = \mathcal{Z}\left(L\_5 + L\_2 + L\_3\right) - \mathcal{Z}\left(LR\right) \tag{26}$$

We showed that it is possible to get the sample thickness (d) with the following relations:

$$d = \left| \frac{\left(D\_2 + D\_3\right) - 2D\_r}{2} \right| \tag{27}$$

As D2 and D3 are the OPD between the sample and a reference plane, a lateral displacement of the sample let us obtain the surface topography of each of the faces. From the same curves it is possible to improve the alignment of the sample with the reference plane (in our set-up the mirror E1).

An example of a typical measurement obtained with this set-up is shown in figure 25. The nominal thickness of each gauge is 1.1 mm and 1.05 mm. Both are positioned so that one side has an exposed step and the other side forms a planar surface. The total lateral displacement in the experiment is about 10 mm. We called e1 to the step between the two gauges; e3 and e2 are the thickness of each gauge.

Fig. 24. Schematic of the sample.

In figure 25.a it is shown the profile obtained from the measurement of D1. Figures 25.b and 25.c show the profiles of both faces of the sample (D2 and D3). From these results it is possible to obtain the height of the step (e1 in figure 24) and the thickness of both gauges (e3 and e2 in the figure 24). The slopes of these curves let us to obtain the relative alignment of each of the sample faces with a reference plane and the relative alignment between both gauges.

The values obtained for the gauges thickness are 1091.7 µm and 1047.46 µm, with an average dispersion of 1.32 µm, in very good agreement with the expected value.

There is a dark zone in the curves that appears when the sample presents abrupt changes in surface topography, so the light is scattered in high angles and the collected light intensity is under the detection sensitivity. The optic to focus light on the sample can be selected to minimize dark zone usually lowering the dynamical range of the interferometer.

This method enables simultaneous measurements of physical thickness and refractive group index without any prior knowledge on samples as showed in (Park et al, 2011).

Fig. 25. Surface profiles and thickness obtained from the sample shown in figure 24.

## **4. Conclusions**

144 Interferometry – Research and Applications in Science and Technology

We showed that it is possible to get the sample thickness (d) with the following relations:

 2 3 2 2 *DD Dr <sup>d</sup>*

As D2 and D3 are the OPD between the sample and a reference plane, a lateral displacement of the sample let us obtain the surface topography of each of the faces. From the same curves it is possible to improve the alignment of the sample with the reference plane (in our set-up

An example of a typical measurement obtained with this set-up is shown in figure 25. The nominal thickness of each gauge is 1.1 mm and 1.05 mm. Both are positioned so that one side has an exposed step and the other side forms a planar surface. The total lateral displacement in the experiment is about 10 mm. We called e1 to the step between the two

In figure 25.a it is shown the profile obtained from the measurement of D1. Figures 25.b and 25.c show the profiles of both faces of the sample (D2 and D3). From these results it is possible to obtain the height of the step (e1 in figure 24) and the thickness of both gauges (e3 and e2 in the figure 24). The slopes of these curves let us to obtain the relative alignment of each of the sample faces with a reference plane and the relative alignment between both

The values obtained for the gauges thickness are 1091.7 µm and 1047.46 µm, with an average

There is a dark zone in the curves that appears when the sample presents abrupt changes in surface topography, so the light is scattered in high angles and the collected light intensity is under the detection sensitivity. The optic to focus light on the sample can be selected to

This method enables simultaneous measurements of physical thickness and refractive group

dispersion of 1.32 µm, in very good agreement with the expected value.

minimize dark zone usually lowering the dynamical range of the interferometer.

index without any prior knowledge on samples as showed in (Park et al, 2011).

the mirror E1).

gauges; e3 and e2 are the thickness of each gauge.

Fig. 24. Schematic of the sample.

gauges.

*D L L L LR* 2 413 2 2 (25)

*D L L L LR* 3 523 2 2 (26)

(27)

In this chapter we present a brief introduction to the theory of low coherence interferometry and the main configurations used nowadays. Then, we focus in giving a detailed analysis of the technique "Spectral domain low coherence interferometry" that offer interesting advantages over alternative methods. We give a description of the limitations and experimental design considerations to take account in the detection of the interference intensity and in the signal processing. We also show how this knowledge is important to improve the information obtained from the images.

At the end we show some experimental results with the intention of giving examples of the enormous potential of this technique, specially in the non- destructive tests area, where we think is nowadays largely unexplored.

## **5. Acknowledgment**

E. Morel and J. Torga are professors at the Universidad Tecnológica Nacional and are members of the Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina.

Thanks to Hernán Miranda for his collaboration and special thanks to Pamela Morel for her support and images design.

Spectral Low Coherence Interferometry:

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A Complete Analysis of the Detection System and the Signal Processing 147

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coherence tomography for process control of laser micromachining"; Review of

vivo human retinal imaging by Fourier domain optical coherence tomography", J.

ophthalmic imaging by ultrafast spectral interferometry" Proc. SPIE 4956, 4956-11

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Born, M. & Wolf, E. (2002), Cambridge University Press, ISBN 13 978-0-521-64222-4, London,

Bruce L. Danielson and C. Y. Boisrobert, "Absolute optical ranging using low coherence interferometry," Appl. Opt. 30, 2975-2979 (1991). doi:10.1364/AO.30.002975. Choma, M,; Sarunic, M.; Yang, C.; and Izatt, J. "Sensitivity advantage of swept source and

Decker JE, Pekelsky JR. "Gauge block calibration by optical interferometry at the national

Dorrer, C.; Belabas, N.; Likforman, J.; and Joffre, M.; "Spectral resolution and sampling

Drexler, W. & Fujimoto, J. "Optical Coherence Tomography: Technology and Applications",

Fercher, A.; Hitzenberger, C.; Kamp, G.; El-Zaiat, S.; "Measurement of intraocular distances

Fujimoto, J. (2002). Optical Coherence Tomography: Introduction. In: Bouma B E & Tearney

Goode, B. (2009). Optical Coherence Tomography: OCT aims for industrial application,

Häusler, G. and Lindner, M.; "Coherence radar and spectral radar – new tools for dermatological diagnosis" J. Biomed. Opt. 3, 21-31(1998). doi:10.1117/1.429899. Huang, D.; Swanson, EA .; Lin, CP.; Schuman, JS .; Stinson, WG .; Chang, W .; Hee, MR .;

Hu, Z.; Pan, Y. & Rollins, A.; "Analytical model of spectrometer-based two-beam spectral interferometry," Appl. Opt. 46, 8499-8505 (2007). doi:10.1364/AO.46.008499. Jeon, M.; Kim, J.; Jung, U.; Lee, C.; Jung, W. and Boppart, S.A.; "Full-range k-domain

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Kowalczyk A., "Improved spectral optical coherence tomography using optical frequency comb" Opt. Express 16, 4163-4176 (2008). doi:10.1364/OE.16.004163. Boaz, P.; "A course in digital signal processing" John Wiley & Sons, 1997. ISBN: 978-0-471-

Fourier domain optical coherence tomography," Opt. Express 11, 2183-2189 (2003).

research council of Canada". Measurement Science Conference Pasadena,

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by backscattering spectral interferometry," Opt.Commun. 117, 43-48,(1995).

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Flotte, T.; Gregory, K.; Puliafito, CA. and Fuyimoto, J.; "Optical coherence tomography", Science, 22 November 1991:Vol. 254. DOI: 10.1126/

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DOI: 10.1016/0030-4018(95)00119-S.

Laser Focus World, Vol 45 (9).


**7** 

**Speckle Interferometry for Displacement** 

**Measurement and Hybrid Stress Analysis** 

*2Department of Electronic Engineering, Kunsan National University,* 

*1School of Mechanical and Automotive Engineering, Kunsan National University,* 

**1.1 Application of ESPI to measurement of out-of-plane displacement in a spot welded** 

Laser speckle interferometry can be used to detect the locations of stress concentration of an object and deformation over a whole area to be measured through the shape information in fringe patterns (Cloud, 1998; Petzing & Tyrer, 1998). ESPI can obtain interferometric fringes by a subtraction process ( before after *II I* | | ) of the image data. Small displacement can be measured from such interferometric fringes corresponding to the order of the laser

Several researches about in-plane displacement and vibration properties analyzed by ESPI have been reported and the application of ESPI is increasing (Rastogi, 2001). In this paper, the out-of-plane displacements of a partially spot welded canti-levered plate and of a normal canti-levered plate are measured and compared by application of 4-step phase

Figure 1-1 shows the arrangement of the ESPI optical system for measuring out-of-plane displacement. The interferometric fringe patterns caused by phase differences between the specimen and the reference plane with a PZT (piezoelectric transducer) must be analyzed because the interferometric fringe patterns contain information about out-of-plane displacement of the specimen. The phase difference is made by an optical path difference. The out-of-plane displacement, *w* , caused by the optical path difference is as follows:

> <sup>4</sup> *<sup>w</sup>*

(1-1)

shifting method to the analysis of fringe patterns in ESPI (Baek et al., 2002).

**1.2.1 ESPI for measurement of out-of-plane displacement** 

**1. Introduction** 

**canti-levered plate** 

wavelength.

**1.2 Optics of ESPI** 

Tae Hyun Baek1 and Myung Soo Kim2

*Daehangno, Gunsan City,* 

*The Republic of Korea* 

*Jeonbuk,* 


## **Speckle Interferometry for Displacement Measurement and Hybrid Stress Analysis**

Tae Hyun Baek1 and Myung Soo Kim2

*1School of Mechanical and Automotive Engineering, Kunsan National University, 2Department of Electronic Engineering, Kunsan National University, Daehangno, Gunsan City, Jeonbuk, The Republic of Korea* 

## **1. Introduction**

148 Interferometry – Research and Applications in Science and Technology

Youngquist, R; Sally Carr, & Davies D. (1987 "Performance of fourier domain vs. time

Yun, S.; Tearney, G.; Bouma, B.; Park, B. and Boer, J.; "High-speed spectral-domain optical

doi:10.1364/OE.11.000889.

doi:10.1364/OE.11.003598.

domain optical coherence tomography," Opt. Express 11, 889-894 (2003).

coherence tomography at 1.3 μm wavelength," Opt. Express 11, 3598-3604 (2003).

## **1.1 Application of ESPI to measurement of out-of-plane displacement in a spot welded canti-levered plate**

Laser speckle interferometry can be used to detect the locations of stress concentration of an object and deformation over a whole area to be measured through the shape information in fringe patterns (Cloud, 1998; Petzing & Tyrer, 1998). ESPI can obtain interferometric fringes by a subtraction process ( before after *II I* | | ) of the image data. Small displacement can be measured from such interferometric fringes corresponding to the order of the laser wavelength.

Several researches about in-plane displacement and vibration properties analyzed by ESPI have been reported and the application of ESPI is increasing (Rastogi, 2001). In this paper, the out-of-plane displacements of a partially spot welded canti-levered plate and of a normal canti-levered plate are measured and compared by application of 4-step phase shifting method to the analysis of fringe patterns in ESPI (Baek et al., 2002).

## **1.2 Optics of ESPI**

## **1.2.1 ESPI for measurement of out-of-plane displacement**

Figure 1-1 shows the arrangement of the ESPI optical system for measuring out-of-plane displacement. The interferometric fringe patterns caused by phase differences between the specimen and the reference plane with a PZT (piezoelectric transducer) must be analyzed because the interferometric fringe patterns contain information about out-of-plane displacement of the specimen. The phase difference is made by an optical path difference. The out-of-plane displacement, *w* , caused by the optical path difference is as follows:

$$
\omega = \frac{\mathcal{L}}{4\pi} \phi
\tag{1-1}
$$

Speckle Interferometry for Displacement Measurement and Hybrid Stress Analysis 151

The speckle pattern contains lots of noise and the noise must be filtered out before the phase unwrapping process. The Gaussian blur process in a commercial image processing software

The specimen used in this experiment is a 2 mm-thick canti-lever made of steel plate (*E*=200

(a) Before weld (b) After weld

Figure 1-3 shows the alignment of the optical components in which the optical set-up for Twyman-Green interferometry is used to measure the out-of-plane displacement. The phase shifting method is used to advance the precision degree of the speckle pattern in the measurement of out-of-plane displacement, and the phase shifting is performed by the PZT

The expression for the deflection curve for a canti-lever beam subjected to a concentrated

6

*EI*

3 2 3

(1-4)

*<sup>P</sup> <sup>x</sup> Lx*

In Eq. (1-4), *E* is the Young's elastic modulus, *I* is the moment of inertia, *L* is the distance from the fixed support to the loading point, and *x* is the distance from the fixed support to

Fig. 1-2. Dimensions of the specimen used for measurement of out-of-plane displacement.

(a) Normal canti-levered plate, (b) Spot welded canti-levered plate.

that is controlled with a personal computer.

load *P* , as shown in Fig. 1-2 (a), is

any arbitrary point.

 0.3 ). The shape and size of the specimen is shown in Fig. 1-2. Fig. 1-2 (a) is the normal canti-levered plate that is not spot-welded, and Fig. 1-2 (b) is the canti-levered plate

package (Adobe Photoshop, Version 5.5) is used to eliminate the speckle noise.

**1.3 Experiment** 

GPa, 

**1.3.1 Specimen and experimental set-up** 

that is spot-welded on the rear side.

In Eq. (1-1), *w* is z-directional (out-of-plane) displacement, is the wavelength of the laser, and represents the phase difference between before and after out-of-plane displacement.

Fig. 1-1. Block diagram of ESPI optical system for measuring out-of-plane displacement.

## **1.2.2 4-step phase shifting method in ESPI**

The 4-step phase shifting method used in this paper is employed to move the reference plane with the PZT by / 2 radians in each step and to obtain four fringe patterns with relative phase difference. The phase map is obtained from the four fringe patterns using the arc tangent function. The light intensity of the fringe pattern in ESPI, *<sup>i</sup> I* , is expressed as follows:

$$I\_i = I\_0 \left\{ 1 + m(\mathbf{x}, \mathbf{y}) \cos[\phi(\mathbf{x}, \mathbf{y}) + \alpha] \right\} \tag{1-2}$$

In the above equation, (, ) *<sup>i</sup> I xy* is the measured light intensity, <sup>0</sup> *I* is the average intensity, *mxy* (, ) is the contrast, (, ) *x y* is the phase difference, and is the phase introduced by PZT. The phase map, where the magnitude and sign of displacement can be known, can be obtained as follow.

$$\phi(x,y) = \tan^{-1}\left[\frac{I\_4 - I\_2}{I\_1 - I\_3}\right] \tag{1-3}$$

In Eq. (1-3), 1*I* , <sup>2</sup> *I* , <sup>3</sup> *I* and <sup>4</sup> *I* are the light intensities at 0 , / 2 , , 3 /2 radians, respectively. Eq. (1-3) uses the four fringe patterns with different phases, so that this method is called the 4-step phase shifting method. The phase map obtained by this method has the phase between radians and radians due to the property of the arc tangent function. Thus, the phase map has the discontinuous phase at every 2 radians, but this discontinuity can be eliminated by use of the phase unwrapping process and the continuous displacement of the specimen can be obtained (Ghiglia & Pritt, 1998).

The speckle pattern contains lots of noise and the noise must be filtered out before the phase unwrapping process. The Gaussian blur process in a commercial image processing software package (Adobe Photoshop, Version 5.5) is used to eliminate the speckle noise.

## **1.3 Experiment**

150 Interferometry – Research and Applications in Science and Technology

Fig. 1-1. Block diagram of ESPI optical system for measuring out-of-plane displacement.

The 4-step phase shifting method used in this paper is employed to move the reference

relative phase difference. The phase map is obtained from the four fringe patterns using the arc tangent function. The light intensity of the fringe pattern in ESPI, *<sup>i</sup> I* , is expressed as

*I I mxy xy <sup>i</sup>* <sup>0</sup> 1 ( , ) cos[ ( , ) ]

In the above equation, (, ) *<sup>i</sup> I xy* is the measured light intensity, <sup>0</sup> *I* is the average intensity,

PZT. The phase map, where the magnitude and sign of displacement can be known, can be

<sup>1</sup> 4 2 ( , ) tan

respectively. Eq. (1-3) uses the four fringe patterns with different phases, so that this method is called the 4-step phase shifting method. The phase map obtained by this method has the

discontinuity can be eliminated by use of the phase unwrapping process and the continuous

*x y*

Thus, the phase map has the discontinuous phase at every 2

displacement of the specimen can be obtained (Ghiglia & Pritt, 1998).

In Eq. (1-3), 1*I* , <sup>2</sup> *I* , <sup>3</sup> *I* and <sup>4</sup> *I* are the light intensities at

radians and

(, ) *x y* is the phase difference, and

1 3

*I I*

*I I*

/ 2 radians in each step and to obtain four fringe patterns with

 0 , / 2 , , 3 /2 

radians due to the property of the arc tangent function.

(1-2)

(1-3)

is the phase introduced by

radians,

radians, but this

represents the phase difference between before and after out-of-plane displacement.

is the wavelength of the laser,

In Eq. (1-1), *w* is z-directional (out-of-plane) displacement,

**1.2.2 4-step phase shifting method in ESPI** 

plane with the PZT by

*mxy* (, ) is the contrast,

obtained as follow.

phase between

follows:

and 

## **1.3.1 Specimen and experimental set-up**

The specimen used in this experiment is a 2 mm-thick canti-lever made of steel plate (*E*=200 GPa, 0.3 ). The shape and size of the specimen is shown in Fig. 1-2. Fig. 1-2 (a) is the normal canti-levered plate that is not spot-welded, and Fig. 1-2 (b) is the canti-levered plate that is spot-welded on the rear side.

Fig. 1-2. Dimensions of the specimen used for measurement of out-of-plane displacement. (a) Normal canti-levered plate, (b) Spot welded canti-levered plate.

Figure 1-3 shows the alignment of the optical components in which the optical set-up for Twyman-Green interferometry is used to measure the out-of-plane displacement. The phase shifting method is used to advance the precision degree of the speckle pattern in the measurement of out-of-plane displacement, and the phase shifting is performed by the PZT that is controlled with a personal computer.

The expression for the deflection curve for a canti-lever beam subjected to a concentrated load *P* , as shown in Fig. 1-2 (a), is

$$\mathcal{S} = \frac{P}{6EI} \left(\mathbf{x}^3 - 3L\mathbf{x}^2\right) \tag{14}$$

In Eq. (1-4), *E* is the Young's elastic modulus, *I* is the moment of inertia, *L* is the distance from the fixed support to the loading point, and *x* is the distance from the fixed support to any arbitrary point.

Speckle Interferometry for Displacement Measurement and Hybrid Stress Analysis 153

Fig. 1-5. Speckle fringe patterns of a spot welded canti-levered plate at difference phases for

(a) (b)

Gaussian Blur Original Image

150 200 250 300 350 400 450 500 Pixel Distance

(a) (b)

Fig. 1-8. Wrapped phase images of (a) normal and (b) spot welded canti-lever plate.

Fig. 1-6. (a) Original image and (b) Gaussian blurred image obtained from Fig. 1-4.

 0, /2, , 3 /2 

  radians.

0

Fig. 1-7. Comparison of light intensity along line A-A of Figs. 1-6 (a) and (b).

10

20

30

Light Intensity

40

50

60

Fig. 1-3 Optical systems for ESPI measurement of out-of-plane displacement.

## **1.3.2 Results of experiment**

Figure 1-4 shows the speckle fringe patterns of a normal canti-lever at difference phases for 0, / 2, , 3 / 2 . Figure 1-5 shows the speckle fringe patterns of a spot welded cantilevered plate at difference phases for 0, / 2, , 3 / 2 . Those fringe patterns are obtained through a subtraction of before-displacement and after-displacement results of a specimen. The different phases in Figs. 1-4 and 1-5 are created by the PZT which controls the phases of fringe pattern by 0, / 2, , 3 / 2 , as mentioned before.

As seen in Figs. 1-4 and 1-5, the change of fringe patterns is shown near the welded area of the spot welded specimen. To eliminate the noise of the high frequency component in the speckle, the Gaussian blur process is applied to the original image obtained in the experiment. Figures 1-6 (a) and (b) are the original image and the Gaussian blurred image, respectively. When the Gaussian blurred image is compared with the original image in Fig. 1-6, it is clear that the high frequency noise is eliminated.

Figure 1-7 shows the light intensities along the line A-A in Fig. 1-6 (a) and (b). There is no doubt that the high frequency noise is eliminated in the Gaussian blurred image. Figure 1-8 is the phase map calculated by Eq. (1-3) in which four Gaussian blurred images are used.

Fig. 1-4. Speckle fringe patterns of a normal canti-levered plate at difference phases for 0, / 2, , 3 / 2 radians.

Fig. 1-3 Optical systems for ESPI measurement of out-of-plane displacement.

Figure 1-4 shows the speckle fringe patterns of a normal canti-lever at difference phases for

obtained through a subtraction of before-displacement and after-displacement results of a specimen. The different phases in Figs. 1-4 and 1-5 are created by the PZT which controls the

As seen in Figs. 1-4 and 1-5, the change of fringe patterns is shown near the welded area of the spot welded specimen. To eliminate the noise of the high frequency component in the speckle, the Gaussian blur process is applied to the original image obtained in the experiment. Figures 1-6 (a) and (b) are the original image and the Gaussian blurred image, respectively. When the Gaussian blurred image is compared with the original image in Fig.

Figure 1-7 shows the light intensities along the line A-A in Fig. 1-6 (a) and (b). There is no doubt that the high frequency noise is eliminated in the Gaussian blurred image. Figure 1-8 is the phase map calculated by Eq. (1-3) in which four Gaussian blurred images are used.

Fig. 1-4. Speckle fringe patterns of a normal canti-levered plate at difference phases for

0, / 2, , 3 / 2 . Figure 1-5 shows the speckle fringe patterns of a spot welded canti-

 

0, / 2, , 3 / 2 , as mentioned before.

0, / 2, , 3 / 2 . Those fringe patterns are

**1.3.2 Results of experiment** 

phases of fringe pattern by

levered plate at difference phases for

1-6, it is clear that the high frequency noise is eliminated.

radians.

 

 0, / 2, , 3 / 2 

 

Fig. 1-5. Speckle fringe patterns of a spot welded canti-levered plate at difference phases for 0, /2, , 3 /2 radians.

Fig. 1-6. (a) Original image and (b) Gaussian blurred image obtained from Fig. 1-4.

Fig. 1-7. Comparison of light intensity along line A-A of Figs. 1-6 (a) and (b).

Fig. 1-8. Wrapped phase images of (a) normal and (b) spot welded canti-lever plate.

Speckle Interferometry for Displacement Measurement and Hybrid Stress Analysis 155

(a) (b)

*m* , 2.134

*m* , 3.219

*m* , 5.432

*m* . That is, it is proved that the physical out-of-

(a) Normal canti-lever (b) Spot welded canti-lever

Figure 1-12 (b) shows the displacement distribution obtained from the spot welded canti-

10mm, 20mm, 30mm from the fixed area of the normal canti-levered plate. Therefore, the displacement for the same load decreases as the canti-levered plate is reinforced by spotwelding, and the spot welded area that is not visible can be easily detected by use of speckle

The 4-step phase shifting method applied to an ESPI experiment has been used for the measurement of out-of-plane displacement in the normal canti-levered plate and the spot welded canti-levered plate. The measured displacement of the normal canti-levered plate

plane displacement can be directly measured and precise measurement with a nanometer resolution is possible. Also, a welded area that is not visible from the surface can be detected and a small out-of-plane displacement in the welded area can be measured. The distribution

*m* , 1.183

Fig. 1-12. Displacement distribution obtained from theory and from the phase shifting

mm, 30 mm from the fixed area of the spot welded canti-levered plate, respectively.

Displacement(m)

Fig. 1-11. 3-D view of unwrapped phase image of (a) Fig. 11(a) and (b) Fig. 11(b).

0 5 10 15 20 25 30 35 Distance(mm)

method along line A-A of Fig. 1-8 (a) and (b).

As a reference, the other displacements, 1.006

levered plate. The displacements, 0.582

**1.4 Conclusions and discussions** 

agreed to the theoretical value within 0.076

0

interferometry.

1

1.5

Displacement(m)

0.5

2

3

Theory Experiment

2.5

Welding Region

0 5 10 15 20 25 30 35 Distance(mm)

*m* , are measured at 10 mm, 20

*m* , are estimated at

Theory w/o spot weld Experiment

Fig. 1-9. (a) Wrapped and unwrapped phase distributions along line A-A of Fig. 1-8 (a) and along line A-A of Fig. 1-8 (b).

Fig. 1-10. Unwrapped phase maps of Fig. 1-8 (a) and (b).

When Figs. 1-8 (a) and (b) are compared, the phase map of the normal canti-levered plate is uniform, but that of the spot welded canti-levered plate shows a phase reversal at the welded area. Figure 1-9 (a) shows wrapped and unwrapped phase distributions along line A-A of Fig. 1-8 (a) of the normal canti-levered plate. Figure 1-9 (b) shows wrapped and unwrapped phase distributions along line A-A of Fig. 1-8 (b) of the spot welded cantilevered plate. Figure 1-10 (a) and (b) are the unwrapped phase maps of Fig. 1-8 (a) and (b). Figure 1-11 is 3-D view of the unwrapped phase image of Fig. 1-10. It is clearly seen in Figs. 1-11 (a) and (b) that continuous displacement occurred in the normal canti-levered plate but the displacement at the spot welded area was hump-shaped in the spot welded cantilevered plate.

Figure 1-12 (a) shows the displacement distribution obtained from the theory and from the phase shifting method along line A-A of Fig. 1-8. It shows that the result of ESPI is almost the same as that of the theoretical calculation for the normal canti-levered plate which is not spot welded. The maximum error of 0.076 *m* occurs at approximately 7.9 mm from the fixed area of the canti-levered plate. However, in general, the measured displacement by the ESPI experiment is quite close to the theoretically expected displacement. Thus, it is proved that the physical out-of-plane displacement can be directly measured by the ESPI method.

0

400

800

Light Intensity

(a) (b)

Fig. 1-9. (a) Wrapped and unwrapped phase distributions along line A-A of Fig. 1-8 (a) and

(a) (b)

When Figs. 1-8 (a) and (b) are compared, the phase map of the normal canti-levered plate is uniform, but that of the spot welded canti-levered plate shows a phase reversal at the welded area. Figure 1-9 (a) shows wrapped and unwrapped phase distributions along line A-A of Fig. 1-8 (a) of the normal canti-levered plate. Figure 1-9 (b) shows wrapped and unwrapped phase distributions along line A-A of Fig. 1-8 (b) of the spot welded cantilevered plate. Figure 1-10 (a) and (b) are the unwrapped phase maps of Fig. 1-8 (a) and (b). Figure 1-11 is 3-D view of the unwrapped phase image of Fig. 1-10. It is clearly seen in Figs. 1-11 (a) and (b) that continuous displacement occurred in the normal canti-levered plate but the displacement at the spot welded area was hump-shaped in the spot welded canti-

Figure 1-12 (a) shows the displacement distribution obtained from the theory and from the phase shifting method along line A-A of Fig. 1-8. It shows that the result of ESPI is almost the same as that of the theoretical calculation for the normal canti-levered plate which is not

fixed area of the canti-levered plate. However, in general, the measured displacement by the ESPI experiment is quite close to the theoretically expected displacement. Thus, it is proved that the physical out-of-plane displacement can be directly measured by the ESPI method.

1200

1600

150 200 250 300 350 400 450 500 550 Pixel Distance

*m* occurs at approximately 7.9 mm from the

Unwrapped Phase Wrapped Phase

150 200 250 300 350 400 450 500 Pixel Distance

Fig. 1-10. Unwrapped phase maps of Fig. 1-8 (a) and (b).

spot welded. The maximum error of 0.076

Unwrapped Phase Wrapped Phase

0

levered plate.

along line A-A of Fig. 1-8 (b).

500

1000

1500

Light Intensity

2000

2500

Fig. 1-11. 3-D view of unwrapped phase image of (a) Fig. 11(a) and (b) Fig. 11(b).

Fig. 1-12. Displacement distribution obtained from theory and from the phase shifting method along line A-A of Fig. 1-8 (a) and (b).

Figure 1-12 (b) shows the displacement distribution obtained from the spot welded cantilevered plate. The displacements, 0.582 *m* , 1.183 *m* , 2.134 *m* , are measured at 10 mm, 20 mm, 30 mm from the fixed area of the spot welded canti-levered plate, respectively.

As a reference, the other displacements, 1.006 *m* , 3.219 *m* , 5.432 *m* , are estimated at 10mm, 20mm, 30mm from the fixed area of the normal canti-levered plate. Therefore, the displacement for the same load decreases as the canti-levered plate is reinforced by spotwelding, and the spot welded area that is not visible can be easily detected by use of speckle interferometry.

## **1.4 Conclusions and discussions**

The 4-step phase shifting method applied to an ESPI experiment has been used for the measurement of out-of-plane displacement in the normal canti-levered plate and the spot welded canti-levered plate. The measured displacement of the normal canti-levered plate agreed to the theoretical value within 0.076 *m* . That is, it is proved that the physical out-ofplane displacement can be directly measured and precise measurement with a nanometer resolution is possible. Also, a welded area that is not visible from the surface can be detected and a small out-of-plane displacement in the welded area can be measured. The distribution

Speckle Interferometry for Displacement Measurement and Hybrid Stress Analysis 157

unwrapping algorithm of the wrapped phase (Ghiglia & Pritt, 1998). Then, the in-plane

(, ) 4 sin( / 2) *<sup>u</sup> u x <sup>y</sup>* 

Fig. 2-1. Schematic diagram of speckle interferometry with phase-shifting method for in-

The specimens used in this experiment are a rectangular steel plate with a through-thickness circular hole at the center and a rectangular steel plate with a partly through-thickness circular hole at the center on rear side as in Fig. 2-2. Both of the rectangular steel plates have the same size that is 147 mm x 27.7 mm with thickness of 1.2 mm. The diameter of throughthickness circular hole is 12 mm. Also the diameter of partly through-thickness circular hole is 12 mm but it is uniformly cut by 0.8 mm on the rear side. Figure 2-3 is the picture of optical experiment system. Ar laser with wavelength of 515 nm is used in the optical

(a) (b) Fig. 2-2. Dimensions of specimens used in-plane displacement (unit : mm) (a) rectangular steel plate with through-thickness circular hole (b) rectangular steel plate with partly

At first, optical experiment is performed with the steel plate with through-thickness circular hole at the center. Four fringe patterns, 1*I* , <sup>2</sup> *I* , <sup>3</sup> *I* , and <sup>4</sup> *I* , are taken in the experiment and they are stored in PC. However, the four fringe patterns have lots of speckle noises, so that

 (2-3)

is the angle between two incident lights onto the

*x y* is obtained with

*x y* as follows (Rastogi, 2001) ;

(, ) *x y* in Eq. (2-2) is so-called wrapped phase. Unwrapped phase (, ) *<sup>u</sup>*

 

displacement of specimen, *u* , is obtained from the phase (, ) *<sup>u</sup>*

plane displacement measurement (

through-thickness circular hole.

**2.3 Optical experiment** 

specimen).

experiment.

of displacement shows a slight non-linearity in the period of fringe, and therefore, further work is necessary for more precise measurement. Also, noise due to speckle has to be eliminated before the phase shifting method is applied. The Gaussian blur process is used to eliminate the noise in this work.

## **2. Measurement for in-plane displacement of tensile plates with throughthickness circular hole and partly through-thickness circular hole by use of speckle interferometry**

## **2.1 Introduction**

Speckle interferometry is an optical technique to measure displacement of a specimen by a coherent light from a laser. There are several phase extraction methods in speckle interferometry to measure the displacement, and phase shifting method (PSM) is one of the methods (Creath, 1988). In PSM, a known phase produced by piezoelectric transducer (PZT) is added into optical beam and resultant interference patterns are processed to get information about displacement of a specimen (Kim & Baek, 2006). In this paper, in-plane displacement of a steel plate with a partly through-thickness circular hole and a steel plate of a through-thickness circular hole is measured by simple optical system of speckle interferometry with PSM. Especially, the circular hole of steel plate with a partly through-thickness circular hole is not visible because the circular hole is cut on the rear side of the plate. This means that one cannot see any deformation or defect of the specimen.

## **2.2 Phase shifting method in speckle interferometry**

Figure 2-1 is an optical system to measure in-plane displacement of specimens by use of PSM in speckle interferometry. A specimen is placed in a loading device and tensile load is applied to the specimen by the loading device in order to make in-plane displacement on the specimen. When two optical beams from a laser illuminate a specimen in speckle interferometry as shown Fig. 2-1, the optical beams make interference fringe patterns. The interference fringe patterns, *<sup>i</sup> I* , can be represented by Eq. (2-1).

$$I\_i = I\_0 + I\_c \cos\left[\phi(\mathbf{x}, \mathbf{y}) + \alpha\_i\right] \tag{2-1}$$

where <sup>0</sup> *I* is the average intensity of fringe pattern and *<sup>c</sup> I* is the contrast of fringe pattern.

A known phase, *<sup>i</sup>* , is added into one of the optical beams through controlling PZT. (, ) *x y* is the phase of fringe pattern caused by the in-plane displacement of a specimen which tensile load is applied to. When 0, / 2 , , and 3 /2 radians are used for *<sup>i</sup>* , four fringe patterns are taken through CCD camera and stored in PC. When 1*I* , <sup>2</sup> *I* , <sup>3</sup> *I* , and <sup>4</sup> *I* , represent the fringe patterns for the known phases of 0, / 2 , , and 3 /2 radians, respectively, the phase, (, ) *x y* , is calculated by use of Eq. (2-2).

$$\phi(\mathbf{x}, \mathbf{y}) = \tan^{-1} \left[ \frac{I\_4 - I\_2}{I\_1 - I\_3} \right] \tag{2-2}$$

(, ) *x y* in Eq. (2-2) is so-called wrapped phase. Unwrapped phase (, ) *<sup>u</sup> x y* is obtained with unwrapping algorithm of the wrapped phase (Ghiglia & Pritt, 1998). Then, the in-plane displacement of specimen, *u* , is obtained from the phase (, ) *<sup>u</sup> x y* as follows (Rastogi, 2001) ;

> (, ) 4 sin( / 2) *<sup>u</sup> u x <sup>y</sup>*

  (2-3)

Fig. 2-1. Schematic diagram of speckle interferometry with phase-shifting method for inplane displacement measurement ( is the angle between two incident lights onto the specimen).

## **2.3 Optical experiment**

156 Interferometry – Research and Applications in Science and Technology

of displacement shows a slight non-linearity in the period of fringe, and therefore, further work is necessary for more precise measurement. Also, noise due to speckle has to be eliminated before the phase shifting method is applied. The Gaussian blur process is used to

Speckle interferometry is an optical technique to measure displacement of a specimen by a coherent light from a laser. There are several phase extraction methods in speckle interferometry to measure the displacement, and phase shifting method (PSM) is one of the methods (Creath, 1988). In PSM, a known phase produced by piezoelectric transducer (PZT) is added into optical beam and resultant interference patterns are processed to get information about displacement of a specimen (Kim & Baek, 2006). In this paper, in-plane displacement of a steel plate with a partly through-thickness circular hole and a steel plate of a through-thickness circular hole is measured by simple optical system of speckle interferometry with PSM. Especially, the circular hole of steel plate with a partly through-thickness circular hole is not visible because the circular hole is cut on the rear side of the plate. This means that one cannot see any

Figure 2-1 is an optical system to measure in-plane displacement of specimens by use of PSM in speckle interferometry. A specimen is placed in a loading device and tensile load is applied to the specimen by the loading device in order to make in-plane displacement on the specimen. When two optical beams from a laser illuminate a specimen in speckle interferometry as shown Fig. 2-1, the optical beams make interference fringe patterns. The

> <sup>0</sup> cos ( , ) *ic i I I I xy*

where <sup>0</sup> *I* is the average intensity of fringe pattern and *<sup>c</sup> I* is the contrast of fringe pattern.

 / 2 , 

(, ) *x y* is the phase of fringe pattern caused by the in-plane displacement of a specimen

fringe patterns are taken through CCD camera and stored in PC. When 1*I* , <sup>2</sup> *I* , <sup>3</sup> *I* , and <sup>4</sup> *I* ,

(, ) *x y* , is calculated by use of Eq. (2-2).

( , ) tan *I I x y I I*

*<sup>i</sup>* , is added into one of the optical beams through controlling PZT.

 / 2 , 

 , and 3 /2 

1 4 2 1 3

(2-1)

radians are used for

 , and 3 /2 

(2-2)

*<sup>i</sup>* , four

radians,

**2. Measurement for in-plane displacement of tensile plates with throughthickness circular hole and partly through-thickness circular hole by use of** 

eliminate the noise in this work.

**speckle interferometry** 

deformation or defect of the specimen.

which tensile load is applied to. When 0,

**2.2 Phase shifting method in speckle interferometry** 

interference fringe patterns, *<sup>i</sup> I* , can be represented by Eq. (2-1).

represent the fringe patterns for the known phases of 0,

**2.1 Introduction** 

A known phase,

respectively, the phase,

The specimens used in this experiment are a rectangular steel plate with a through-thickness circular hole at the center and a rectangular steel plate with a partly through-thickness circular hole at the center on rear side as in Fig. 2-2. Both of the rectangular steel plates have the same size that is 147 mm x 27.7 mm with thickness of 1.2 mm. The diameter of throughthickness circular hole is 12 mm. Also the diameter of partly through-thickness circular hole is 12 mm but it is uniformly cut by 0.8 mm on the rear side. Figure 2-3 is the picture of optical experiment system. Ar laser with wavelength of 515 nm is used in the optical experiment.

Fig. 2-2. Dimensions of specimens used in-plane displacement (unit : mm) (a) rectangular steel plate with through-thickness circular hole (b) rectangular steel plate with partly through-thickness circular hole.

At first, optical experiment is performed with the steel plate with through-thickness circular hole at the center. Four fringe patterns, 1*I* , <sup>2</sup> *I* , <sup>3</sup> *I* , and <sup>4</sup> *I* , are taken in the experiment and they are stored in PC. However, the four fringe patterns have lots of speckle noises, so that

Speckle Interferometry for Displacement Measurement and Hybrid Stress Analysis 159

(a) (b)

Fig. 2-5. Wrapped phase maps (a) steel plate with a through-thickness circular hole, (b) steel

 (a) A-A (b) B-B (c) C-C Fig. 2-6. Displacement distribution along A-A, B-B, and C-C line of Fig. 2-5 (a).

 (a) A-A (b) B-B (c) C-C Fig. 2-7. Displacement distribution along A-A, B-B, and C-C line of Fig. 2-5 (b).

same as the physical properties of structural steel that are *E*=200 GPa and

of the specimens is plotted in Figs. 2-6 and 2-7 by use of Eq. (2-3).

in Fig. 2-5 (a) and (b). Quantitative data are acquired along the lines, A-A, B-B, and C-C in

For comparative purpose, the two specimens of Fig. 2-2 are analyzed by ANSYS. Figures 2-8 (a) and (b) are the models for ANSYS. The physical properties used for the analysis are the

ANSYS discretization for the rectangular steel plate with a through-thickness circular hole at the center used 8 node quadrilateral elements. The ANSYS discretization for the rectangular

(, ) *x y* , are obtained for the two specimens and shown

0.3 . The

(, ) *x y* , along the lines are unwrapped and the in-plane displacement

plate with a partly through-thickness circular hole.

Using Eq. (2-2), the wrapped phases,

Fig. 2-5. The phases,

they are processed by use of an image processing algorithm. In this work, Gaussian blur algorithm that is available commercially in Adobe Photoshop is used to process the fringe patterns. The processed four fringe patterns are shown in Fig. 2-4. The same experimental procedures are used to get four fringe patterns for the steel plate with partly throughthickness circular hole at the center on rear side.

Fig. 2-3. Optical setup for measurement of in-plane displacement.

Fig. 2-4. Fringe patterns of specimen : (a) Steel plate with a through-thickness circular hole processed by Gaussian Blur (b) Steel plate with a partly through-thickness circular hole processed by Gaussian Blur.

they are processed by use of an image processing algorithm. In this work, Gaussian blur algorithm that is available commercially in Adobe Photoshop is used to process the fringe patterns. The processed four fringe patterns are shown in Fig. 2-4. The same experimental procedures are used to get four fringe patterns for the steel plate with partly through-

thickness circular hole at the center on rear side.

Fig. 2-3. Optical setup for measurement of in-plane displacement.

(a) (b)

processed by Gaussian Blur.

Fig. 2-4. Fringe patterns of specimen : (a) Steel plate with a through-thickness circular hole processed by Gaussian Blur (b) Steel plate with a partly through-thickness circular hole

Fig. 2-5. Wrapped phase maps (a) steel plate with a through-thickness circular hole, (b) steel plate with a partly through-thickness circular hole.

Fig. 2-6. Displacement distribution along A-A, B-B, and C-C line of Fig. 2-5 (a).

Fig. 2-7. Displacement distribution along A-A, B-B, and C-C line of Fig. 2-5 (b).

Using Eq. (2-2), the wrapped phases, (, ) *x y* , are obtained for the two specimens and shown in Fig. 2-5 (a) and (b). Quantitative data are acquired along the lines, A-A, B-B, and C-C in Fig. 2-5. The phases, (, ) *x y* , along the lines are unwrapped and the in-plane displacement of the specimens is plotted in Figs. 2-6 and 2-7 by use of Eq. (2-3).

For comparative purpose, the two specimens of Fig. 2-2 are analyzed by ANSYS. Figures 2-8 (a) and (b) are the models for ANSYS. The physical properties used for the analysis are the same as the physical properties of structural steel that are *E*=200 GPa and 0.3 . The ANSYS discretization for the rectangular steel plate with a through-thickness circular hole at the center used 8 node quadrilateral elements. The ANSYS discretization for the rectangular

Speckle Interferometry for Displacement Measurement and Hybrid Stress Analysis 161

geometric discontinuity. In conjunction with our previous studies (Baek et al., 2000; Baek & Rowlands, 1999; Baek & Kim, 2005; Baek et al., 2006), the hybrid method employing the least-squares method integrated with Laurent series representation of the stress function was used to estimate reliable edge data around the circular hole in a tensile-loaded plate from a relatively few measured x-displacement data away from the boundary. Speckle interferometry has been explored and integrated with other methods for the optical measurement of in-plane and out-of-plane displacement in a material (Schwider, 1989; Steinzig & Takahashi, 2006). Different from the previous works, this study utilized considerably a few number of in-plane micro-scale x-displacement only measured by

The present technique employs fairly general expressions for the stress functions, and traction-free conditions which are satisfied at the geometric discontinuity using conformal mapping and analytical continuation. The approach is illustrated using the x-displacement

In the absence of body forces and rigid body motion, the stresses and displacements under plane and rectilinear orthotropy can be written as (Gerhardt, 1984; Rhee & Rowlands, 2002).

> 11 2 2 11 2 2

> > and <sup>2</sup> ( )

*k k*

 *cB cC*

(3-2)

(3-1)

are related to each other by the

 

2Re ( ) ( ) 2Re ( ) ( ) .

 

conformal mapping and analytic continuation. For a traction-free physical boundary, the two functions within sub-region of Fig. 3-1 can be written as Laurent expansions,

( ) and ( ) 1 1 2 22 *m m k k <sup>k</sup>*

The coefficients of Eq. (3-2) are *kk k c a ib* where *<sup>k</sup> a* and *<sup>k</sup> b* are real numbers. In addition to satisfying the traction-free conditions on the hole boundary , the stresses and displacements of Eqs. (3-1) and (3-2) associated with these stress functions (1) and (2)

 

*up p vq q*

*k m k m*

Fig. 3-1. Mapping of holes from the physical z-plane into the -plane.

speckle interferometry using PSM/FT for hybrid stress analysis.

**3.2 Theoretical background** 

The two complex stress functions <sup>1</sup> ( )

satisfy equilibrium and compatibility.

respectively (Gerhardt, 1984; Baek & Rowlands, 2001)

**3.2.1 Basic equations** 

as input data obtained from phase-shifting method in speckle interferometry.

steel plate with a partly through-thickness circular hole at the center used 10 node tetrahedral elements.

Fig. 2-8. ANSYS discretization. (a) Steel plate with a circular hole in Fig. 2 (a). (b) Steel plate with partly through-thickness circular hole in Fig. 2 (b).

The results of the two specimens from ANYSYS are plotted also in Figs. 2-6 and 2-7. As shown in Figs. 2-6 and 2-7, the results of the optical experiments for speckle interferometry with 4-step phase shifting method agree with those of ANSYS.

## **2.4 Conclusions and discussions**

In-plane displacements of two specimens are measured by simple optical system of speckle interferometry using 4-step phase shifting method; one is the rectangular steel plate with a through-thickness circular hole and the other is the rectangular steel plate with a partly through-thickness circular hole. The circular hole of the steel plate with a partly throughthickness circular hole is not visible because the circular hole is cut on the rear side of the plate, so that one cannot see any deformation or defect of the specimen. The fringe patterns acquired by optical experiment of speckle interferometry are processed by image processing algorithm of Gaussian blur in Adobe Photoshop and the in-plane displacements of the two specimens are obtained by the processed fringe patterns. Also the in-plane displacements of the two specimens are calculated by use of ANSYS.

The results of optical experiments are quite comparable to those of calculation with ANSYS. Based on the optical experiments, speckle interferometry can be applied to easily detect or measure defect or deformation in a specimen that is not visible.

## **3. A hybrid stress measurement using only x-displacements by Phase Shifting Method with Fourier Transform (PSM/FT) in laser speckle interferometry and least squares method**

## **3.1 Introduction**

Stress raisers have been one of the main concerns when it comes to design analysis. Due to the complexities associated with it, numerous and continuing investigation are done to develop techniques to accurately measure stress concentration around the geometric boundaries. Several methods can be found in the literature ranging from FEM, the use of photoelastic-data, hybrid method and other various numerical and experimental procedures (Lekhnitskii; Tsai & Cheron, 1968; Kobayshi, 1993; Dally & Riley, 1991; Pilkey's, 2008).

In this paper, we present stress concentration measurement method using only xcomponent displacement data of selected points along straight lines away from the geometric discontinuity. In conjunction with our previous studies (Baek et al., 2000; Baek & Rowlands, 1999; Baek & Kim, 2005; Baek et al., 2006), the hybrid method employing the least-squares method integrated with Laurent series representation of the stress function was used to estimate reliable edge data around the circular hole in a tensile-loaded plate from a relatively few measured x-displacement data away from the boundary. Speckle interferometry has been explored and integrated with other methods for the optical measurement of in-plane and out-of-plane displacement in a material (Schwider, 1989; Steinzig & Takahashi, 2006). Different from the previous works, this study utilized considerably a few number of in-plane micro-scale x-displacement only measured by speckle interferometry using PSM/FT for hybrid stress analysis.

The present technique employs fairly general expressions for the stress functions, and traction-free conditions which are satisfied at the geometric discontinuity using conformal mapping and analytical continuation. The approach is illustrated using the x-displacement as input data obtained from phase-shifting method in speckle interferometry.

## **3.2 Theoretical background**

## **3.2.1 Basic equations**

160 Interferometry – Research and Applications in Science and Technology

steel plate with a partly through-thickness circular hole at the center used 10 node

Fig. 2-8. ANSYS discretization. (a) Steel plate with a circular hole in Fig. 2 (a). (b) Steel plate

The results of the two specimens from ANYSYS are plotted also in Figs. 2-6 and 2-7. As shown in Figs. 2-6 and 2-7, the results of the optical experiments for speckle interferometry

In-plane displacements of two specimens are measured by simple optical system of speckle interferometry using 4-step phase shifting method; one is the rectangular steel plate with a through-thickness circular hole and the other is the rectangular steel plate with a partly through-thickness circular hole. The circular hole of the steel plate with a partly throughthickness circular hole is not visible because the circular hole is cut on the rear side of the plate, so that one cannot see any deformation or defect of the specimen. The fringe patterns acquired by optical experiment of speckle interferometry are processed by image processing algorithm of Gaussian blur in Adobe Photoshop and the in-plane displacements of the two specimens are obtained by the processed fringe patterns. Also the in-plane displacements of

The results of optical experiments are quite comparable to those of calculation with ANSYS. Based on the optical experiments, speckle interferometry can be applied to easily detect or

Stress raisers have been one of the main concerns when it comes to design analysis. Due to the complexities associated with it, numerous and continuing investigation are done to develop techniques to accurately measure stress concentration around the geometric boundaries. Several methods can be found in the literature ranging from FEM, the use of photoelastic-data, hybrid method and other various numerical and experimental procedures (Lekhnitskii; Tsai & Cheron, 1968; Kobayshi, 1993; Dally & Riley, 1991; Pilkey's, 2008).

In this paper, we present stress concentration measurement method using only xcomponent displacement data of selected points along straight lines away from the

**3. A hybrid stress measurement using only x-displacements by Phase Shifting Method with Fourier Transform (PSM/FT) in laser speckle** 

(a) (b)

with 4-step phase shifting method agree with those of ANSYS.

with partly through-thickness circular hole in Fig. 2 (b).

the two specimens are calculated by use of ANSYS.

**interferometry and least squares method** 

**3.1 Introduction** 

measure defect or deformation in a specimen that is not visible.

**2.4 Conclusions and discussions** 

tetrahedral elements.

In the absence of body forces and rigid body motion, the stresses and displacements under plane and rectilinear orthotropy can be written as (Gerhardt, 1984; Rhee & Rowlands, 2002).

$$\begin{split} \mu &= 2\operatorname{Re}\Big[p\_1\Phi(\zeta\_1) + p\_2\Psi(\zeta\_2)\Big] \\ \nu &= 2\operatorname{Re}\Big[q\_1\Phi(\zeta\_1) + q\_2\Psi(\zeta\_2)\Big]. \end{split} \tag{3-1}$$

The two complex stress functions <sup>1</sup> ( ) and <sup>2</sup> ( ) are related to each other by the conformal mapping and analytic continuation. For a traction-free physical boundary, the two functions within sub-region of Fig. 3-1 can be written as Laurent expansions, respectively (Gerhardt, 1984; Baek & Rowlands, 2001)

$$\Phi(\zeta\_1') = \sum\_{k=-m}^{m} \zeta\_1^{'k} \quad \text{and} \quad \Psi(\zeta\_2') = \sum\_{k=-m}^{m} \left(\overline{c\_k} B \zeta\_2^k + c\_k C \zeta\_2^k\right) \tag{3-2}$$

The coefficients of Eq. (3-2) are *kk k c a ib* where *<sup>k</sup> a* and *<sup>k</sup> b* are real numbers. In addition to satisfying the traction-free conditions on the hole boundary , the stresses and displacements of Eqs. (3-1) and (3-2) associated with these stress functions (1) and (2) satisfy equilibrium and compatibility.

Fig. 3-1. Mapping of holes from the physical z-plane into the -plane.

Speckle Interferometry for Displacement Measurement and Hybrid Stress Analysis 163

Knowing {*d*} at various locations in Eq. (3-8) allows the best values of unknown coefficients {*c*} in a least square sense (Sanford, 1980). For *n* measured x-displacements and *m* terms, the coefficients {*c*} of Eqs. (3-8) were obtained from Eq. (3-10) by the least squares method

> 

*m n m n nm m n c U U Ud*

 

1 12 2 1 12 2 1 *T T*

In speckle interferometry using phase-shifting method with Fourier transformation (PSM/FT) (Morimoto & Fujisawa, 1994), interference fringe pattern is obtained by subtracting the pattern from the pre-loading and post-loading conditions of the specimen.

*I xy Axy xy B xy* , : ( , )cos ( , ) ,

where *A*(*x,y*) is the amplitude of the brightness in the pattern and *B*(*x,y*) is the average brightness. Piezoelectric transducer (PZT) can control an optical path length. is the known phase which is added into one of the two optical beams by controlling the PZT. The known

A phase-shifting method using Fourier transform had been well-developed for the measurement of in-plane displacement by speckle interferometry (Morimoto & Fujisawa,

can be calculated as

 , <sup>0</sup> , : cos , *<sup>j</sup> <sup>j</sup> x y F x y A x y B e d Ae*

Speckle interferometry experiment was performed with setup as shown in Fig. 3-2 to acquire the needed data. In Fig. 3-2, LA is a laser, PA is a pin-hole assembly, CL is a collimating lens, BS is a non-polarizing beam splitter, MR1 and MR2 are mirrors. SL is a specimen installed in a tensile loading device, CCD is a CCD camera, and PC is a personal computer. PC controls the

 

, tan Re , :

*x y F xy*

 

is the -directional Fourier transform of Eq. (3-11). In Eqs. (3-12) and (3-13),

radians at equal intervals.

1 0

Im , :

*F xy*

0

 

(3-13)

 

1

 

(3-11)

(3-10)

(*x,y*) is the phase of fringe

(3-12)

<sup>0</sup> is a

, the displacement of a specimen can

 

pattern caused by the in-plane displacement of a specimen.

fundamental frequency. Using the calculated phase,

movement of PZT through the control board CNT ( Kim et al., 2005).

**3.2.3 In-plane displacement by speckle interferometry** 

The intensity of the fringe pattern is calculated as

phase covers the region from 0 to 2

1994; Kim et al., 2005). The phase

where

be obtained.

**3.3 Optical experiments** 

expressed as

The primes denote differentiation with respect to the argument. Complex material parameters *µl* (*l* = 1, 2) are the two distinct roots of the characteristic Eq. (3-3) for an orthotropic material under plane stress (Lekhnitskii, Tsai & Cheron, 1968; Gerhardt, 1984)

$$(S\_{11}\mu^4 + (\mathcal{D}S\_{12} + S\_{66})\mu^2 + S\_{22} = 0\tag{3-3}$$

where *Sij* are the elastic compliances. The material properties *pl* and *ql* are defined as

$$\begin{aligned} p\_1 &= S\_{11}\mu\_1^2 + S\_{12}, & p\_2 &= S\_{11}\mu\_2^2 + S\_{12} \\ q\_1 &= S\_{12}\mu\_1 + \frac{S\_{22}}{\mu\_1}, & q\_2 &= S\_{12}\mu\_2 + \frac{S\_{22}}{\mu\_2} \end{aligned} \tag{3-4}$$

The inverse of the mapping function namely <sup>1</sup> , maps the geometry of interest from the physical z-plane into the -plane *( l l* ). For orthotropic materials, the conformal transformation from the unit circle in the -plane to the hole in the z-plane of radius R is shown in Fig. 3-1 and is given by

$$z\_{\parallel} = \alpha\_{l}(\mathcal{L}\_{l}) = \frac{R}{2} \left[ (1 - i\mu\_{l})\mathcal{L}\_{l} + \frac{1 - i\mu\_{l}}{\mathcal{L}\_{l}} \right] \tag{3-5}$$

where *i* 1 . The inverse of (3-5) is

$$\alpha o^{-1}(z\_l) = \zeta\_l = \frac{z\_l \pm \sqrt{z\_l^2 - R^2(1 + \mu\_l^2)}}{R(1 - i\mu\_l)}\tag{3-6}$$

The branch of the square root of Eq. (3-6) is chosen so that *<sup>l</sup>* 1 (*l* = 1, 2). Complex quantities *B* and *C* in Eq. (3-2) depend on material properties defined as

$$B = \frac{\overline{\mu}\_2 - \overline{\mu}\_1}{\mu\_2 - \overline{\mu}\_2}, \quad C = \frac{\overline{\mu}\_2 - \mu\_1}{\mu\_2 - \overline{\mu}\_2} \tag{3-7}$$

### **3.2.2 Least-squares method**

Combining Eqs. (3-2), (3-3) and (3-5) gives the following expressions for the displacements through regions 1 and 2 of Fig. 3-1. In matrix form,

$$\{d\} = [U] \{\mathbf{c}\} \tag{3-8}$$

where <sup>T</sup> {} {, } *u u* , and <sup>T</sup> {} { , } *k k c ab* . [ ] *U* is a rectangular coefficient matrix whose size depends on the number of terms k of the power series expansions of Eq. (3-1) and given by

$$U(1,j) = 2\operatorname{Re}\left\{p\_1\zeta\_1^k + p\_2(C\zeta\_2^k + B\zeta\_2^{-k})\right\} \tag{3-9a}$$

$$U(\mathcal{D}, j) = 2\operatorname{Re}\left\{q\_1\zeta\_1^k + q\_2(C\zeta\_2^{\prime k} + B\zeta\_2^{\prime - k})\right\} \tag{3-9b}$$

In Eqs. (3-9a) and (3-9b), *j km* 2( ) 1 if *k* 0 and *j km* 2( ) 1 if *k* 0.

Knowing {*d*} at various locations in Eq. (3-8) allows the best values of unknown coefficients {*c*} in a least square sense (Sanford, 1980). For *n* measured x-displacements and *m* terms, the coefficients {*c*} of Eqs. (3-8) were obtained from Eq. (3-10) by the least squares method expressed as

$$\begin{Bmatrix} \mathbf{c} \end{Bmatrix} = \left( \begin{bmatrix} U \\ \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} U \\ \end{bmatrix} \begin{bmatrix} U \\ \end{bmatrix} \right)^{-1} \begin{bmatrix} U \\ \end{bmatrix}^{\mathrm{T}} \begin{Bmatrix} U \\ \end{Bmatrix}^{\mathrm{T}} \begin{Bmatrix} d \\ \end{Bmatrix} \tag{3-10}$$

## **3.2.3 In-plane displacement by speckle interferometry**

In speckle interferometry using phase-shifting method with Fourier transformation (PSM/FT) (Morimoto & Fujisawa, 1994), interference fringe pattern is obtained by subtracting the pattern from the pre-loading and post-loading conditions of the specimen. The intensity of the fringe pattern is calculated as

$$I(\mathbf{x}, \mathbf{y}; \alpha) = A(\mathbf{x}, \mathbf{y}) \cos \left[ \phi(\mathbf{x}, \mathbf{y}) - \alpha \right] + B(\mathbf{x}, \mathbf{y}) \tag{3-11}$$

where *A*(*x,y*) is the amplitude of the brightness in the pattern and *B*(*x,y*) is the average brightness. Piezoelectric transducer (PZT) can control an optical path length. is the known phase which is added into one of the two optical beams by controlling the PZT. The known phase covers the region from 0 to 2 radians at equal intervals. (*x,y*) is the phase of fringe pattern caused by the in-plane displacement of a specimen.

A phase-shifting method using Fourier transform had been well-developed for the measurement of in-plane displacement by speckle interferometry (Morimoto & Fujisawa, 1994; Kim et al., 2005). The phase can be calculated as

$$\phi(\mathbf{x}, \mathbf{y}) = -\tan^{-1}\left(\frac{\operatorname{Im}\left[F\_a(\mathbf{x}, \mathbf{y}; a\_0)\right]}{\operatorname{Re}\left[F\_a(\mathbf{x}, \mathbf{y}; a\_0)\right]}\right) \tag{3-12}$$

where

162 Interferometry – Research and Applications in Science and Technology

The primes denote differentiation with respect to the argument. Complex material parameters *µl* (*l* = 1, 2) are the two distinct roots of the characteristic Eq. (3-3) for an orthotropic material under plane stress (Lekhnitskii, Tsai & Cheron, 1968; Gerhardt, 1984)

> 

2 2 1 11 1 12 2 11 2 12

22 22

 

 

*l*

*l*

(3-6)

(3-7)

{ } [ ]{ } *d Uc* (3-8)

(3-9a)

(3-9b)

(3-4)

, maps the geometry of interest from the

(3-5)

*<sup>l</sup>* 1 (*l* = 1, 2). Complex

). For orthotropic materials, the conformal


1 2

(2 ) 0 (3-3)

4 2 <sup>11</sup> 12 66 <sup>22</sup> *S SS S*

1 12 1 2 12 2

*l ll l l*

 ,

*pS S pS S S S qS qS*

,

 namely <sup>1</sup> 

<sup>1</sup> () 1 <sup>2</sup>

<sup>1</sup> <sup>1</sup> (1 ) ( ) (1 )

21 21 22 22

Combining Eqs. (3-2), (3-3) and (3-5) gives the following expressions for the displacements

depends on the number of terms k of the power series expansions of Eq. (3-1) and given by

(1, ) 2Re ( ) 11 2 2 2 *k kk U j p pC B* 

(2, ) 2Re ( ) 11 2 2 2 *k kk U j q qC B* 

In Eqs. (3-9a) and (3-9b), *j km* 2( ) 1 if *k* 0 and *j km* 2( ) 1 if *k* 0.

 

 

, and <sup>T</sup> {} { , } *k k c ab* . [ ] *U* is a rectangular coefficient matrix whose size

 

 

*z zR <sup>z</sup> R i*

*B C* , 

> 

 

22 2

*l*

*l l*

*<sup>R</sup> <sup>i</sup> z i*

where *Sij* are the elastic compliances. The material properties *pl* and *ql* are defined as


*l l*

quantities *B* and *C* in Eq. (3-2) depend on material properties defined as

 

through regions 1 and 2 of Fig. 3-1. In matrix form,

The branch of the square root of Eq. (3-6) is chosen so that

The inverse of the mapping function

shown in Fig. 3-1 and is given by

where *i* 1 . The inverse of (3-5) is

**3.2.2 Least-squares method** 

where <sup>T</sup> {} {, } *u u*

transformation from the unit circle in the

physical z-plane into the

$$F\_a(\mathbf{x}, \mathbf{y}; a\boldsymbol{\alpha}\_0) = \prod\_{-\mathbf{r}}^{\pi} \left[ A \cos \left( \phi(\mathbf{x}, \mathbf{y}) - \boldsymbol{\alpha} \right) + B \right] e^{-j\boldsymbol{\alpha}} da = \pi A e^{-j\phi(\mathbf{x}, \boldsymbol{\gamma})} \tag{3-13}$$

is the -directional Fourier transform of Eq. (3-11). In Eqs. (3-12) and (3-13), <sup>0</sup> is a fundamental frequency. Using the calculated phase, , the displacement of a specimen can be obtained.

## **3.3 Optical experiments**

Speckle interferometry experiment was performed with setup as shown in Fig. 3-2 to acquire the needed data. In Fig. 3-2, LA is a laser, PA is a pin-hole assembly, CL is a collimating lens, BS is a non-polarizing beam splitter, MR1 and MR2 are mirrors. SL is a specimen installed in a tensile loading device, CCD is a CCD camera, and PC is a personal computer. PC controls the movement of PZT through the control board CNT ( Kim et al., 2005).

Speckle Interferometry for Displacement Measurement and Hybrid Stress Analysis 165

Size of each fringe pattern is 640 x 480 with 8 bits brightness. 16 fringe patterns out of the 32

Fig. 3-4. Picture of optical experiment system for speckle interferometry by PSM/FT. (① He-Ne CW laser, ②pin-hole assembly, ③ collimating lens, ④ PZT control stage, ⑤ beam

The accuracy of the least-squares method for calculating displacement components {,}*<sup>T</sup> u*

as presented in the preceding section was related with the input data acquired through experiment by phase-shifting method in speckle interferometry (Baek & Rudolphi, 2010).

Percent errors for each data was calculated and shown in Table 3-1. As can be observed, the level of precision of the hybrid method is too close with the original input data as indicated by the % Errors. Thus, the method is significantly accurate and reliable for calculating such

The simplicity of the test specimen being isotropic, containing symmetrically-shaped discontinuity facilitates the ease of reliable verification. Finite element anlysis was done to establish a benchmark for comparison purposes. Figure 3-7 shows the ABAQUS discretization of the quarter steel plate. In the vicinity of the circular hole, the ABAQUS model of Fig. 3-7 utilizes elements on the edge of the hole as small as <sup>o</sup> 1.5 by 0.013*r*, where *r* is the radius of the circular hole. Tangential stress around the quarter hole was determined and compared as shown in Fig. 3-8. Different values of m term in the complex stress functions was tested to see its effect, it came out that at point of high stress concentration

 90 in Fig. 3-8). Well-comparable results were attained at decreasing value of m with the best value equal to 1 – accurate by less than one percent error. Figure 3-9 reveals the preceding observation and shows normalized tangential stress at the edge of the quarter

hole in a steel plate with *m* 1 in the complex stress function.

splitter, ⑥ mirror, ⑦ CCD camera, ⑧ mirror, ⑨ specimen in loading device).

parameter.

( <sup>o</sup> 

/8 radian for the rectangular

fringe patterns are shown as examples at every interval of

steel plate containing a hole in Fig. 3-5 (Kim et al., 2005).

Figure 3-3 shows the steel plate specimen (E=200 GPa, 0.3 ) which was subjected to tensile load during the procedure. The test specimen of Fig. 3 had been used for experiment for the development and application of phase-shifting method in speckle interferometry. The accuracy and reliability of the said in-plane displacement measuring method had been established (Baek et al., 2008). For this reason, it is a useful tool for the hybrid stress analysis presented herein (Baek & Rowlands, 1999; Baek & Rowlands, 2001). In this study, phaseshifting method using Fourier transform was utilized.

Fig. 3-2. Schematic diagram of speckle interferometry experiment for x-displacement data acquisition.

Figure 3-4 shows the picture of optical experiment system for speckle interferometry by PSM/FT. The in-plane x-displacement of the specimen, u, along the longitudinal direction is obtained through

$$
\mu = \frac{\lambda}{4\pi\sin\theta} \phi\_u \tag{3-14}
$$

where u is the longitudinal displacement, is the wavelength of light from a laser and *u* is the phase obtained from longitudinal displacement.

Fig. 3-3. Finite-width uni-axially tensile-loaded steel plate.

The laser used in the experiment is He-Ne laser and is 633 nm. The angle between incident light and vertical line to the specimen, as shown in Fig. 3-2, is around <sup>o</sup> 26 . The specimen is installed in a loading device that applies tensile load to the specimen in order to make in-plane displacement. Fringe patterns of *Ixy* (, : ) in Eq. (3-11) are taken through CCD camera in Fig. 3-2. The fringe patterns consist of 32 patterns which are sequentially phase-shifted by PZT stage and are saved in PC. Phase shifting at each step is /16 radian.

tensile load during the procedure. The test specimen of Fig. 3 had been used for experiment for the development and application of phase-shifting method in speckle interferometry. The accuracy and reliability of the said in-plane displacement measuring method had been established (Baek et al., 2008). For this reason, it is a useful tool for the hybrid stress analysis presented herein (Baek & Rowlands, 1999; Baek & Rowlands, 2001). In this study, phase-

Fig. 3-2. Schematic diagram of speckle interferometry experiment for x-displacement data

Figure 3-4 shows the picture of optical experiment system for speckle interferometry by PSM/FT. The in-plane x-displacement of the specimen, u, along the longitudinal direction is

> 4 sin *<sup>u</sup> <sup>u</sup>*

(3-14)

is the wavelength of light from a laser and

is 633 nm. The angle between

in Eq. (3-11) are taken through

/16 radian.

as shown in Fig. 3-2, is around <sup>o</sup> 26 . The

*u*

specimen is installed in a loading device that applies tensile load to the specimen in order to

CCD camera in Fig. 3-2. The fringe patterns consist of 32 patterns which are sequentially

phase-shifted by PZT stage and are saved in PC. Phase shifting at each step is

0.3 ) which was subjected to

Figure 3-3 shows the steel plate specimen (E=200 GPa,

shifting method using Fourier transform was utilized.

acquisition.

obtained through

where u is the longitudinal displacement,

is the phase obtained from longitudinal displacement.

Fig. 3-3. Finite-width uni-axially tensile-loaded steel plate.

The laser used in the experiment is He-Ne laser and

make in-plane displacement. Fringe patterns of *Ixy* (, : )

incident light and vertical line to the specimen,

Size of each fringe pattern is 640 x 480 with 8 bits brightness. 16 fringe patterns out of the 32 fringe patterns are shown as examples at every interval of /8 radian for the rectangular steel plate containing a hole in Fig. 3-5 (Kim et al., 2005).

Fig. 3-4. Picture of optical experiment system for speckle interferometry by PSM/FT. (① He-Ne CW laser, ②pin-hole assembly, ③ collimating lens, ④ PZT control stage, ⑤ beam splitter, ⑥ mirror, ⑦ CCD camera, ⑧ mirror, ⑨ specimen in loading device).

The accuracy of the least-squares method for calculating displacement components {,}*<sup>T</sup> u* as presented in the preceding section was related with the input data acquired through experiment by phase-shifting method in speckle interferometry (Baek & Rudolphi, 2010).

Percent errors for each data was calculated and shown in Table 3-1. As can be observed, the level of precision of the hybrid method is too close with the original input data as indicated by the % Errors. Thus, the method is significantly accurate and reliable for calculating such parameter.

The simplicity of the test specimen being isotropic, containing symmetrically-shaped discontinuity facilitates the ease of reliable verification. Finite element anlysis was done to establish a benchmark for comparison purposes. Figure 3-7 shows the ABAQUS discretization of the quarter steel plate. In the vicinity of the circular hole, the ABAQUS model of Fig. 3-7 utilizes elements on the edge of the hole as small as <sup>o</sup> 1.5 by 0.013*r*, where *r* is the radius of the circular hole. Tangential stress around the quarter hole was determined and compared as shown in Fig. 3-8. Different values of m term in the complex stress functions was tested to see its effect, it came out that at point of high stress concentration ( <sup>o</sup> 90 in Fig. 3-8). Well-comparable results were attained at decreasing value of m with the best value equal to 1 – accurate by less than one percent error. Figure 3-9 reveals the preceding observation and shows normalized tangential stress at the edge of the quarter hole in a steel plate with *m* 1 in the complex stress function.

Speckle Interferometry for Displacement Measurement and Hybrid Stress Analysis 167

1 0 0 - 2 2.6032 2.5791 -0.926 3 2.8628 2.8397 -0.807 4 4.3754 4.3678 -0.174 5 5.5334 5.5499 0.298 6 6.8069 6.8555 0.714 7 7.3259 7.3421 0.221 8 7.8783 7.8483 -0.381 9 8.0719 8.0536 -0.227

Table 3-1. Comparison between input and calculated x-displacement for different values of

0 10 20 30 40 50 60 70 80 90

Angle, 

Fig. 3-8. Normalized tangential stress at the edge of the quarter hole in a steel plate with

*m* ) Calculated: x-disp. (

*m* ) % Error\*

Fig. 3-7. ABAQUS discretization of the quarter steel plate (*t* = 1.15mm).

Hybrid, m=1 Hybrid, m=3 Hybrid, m=5 ABAQUS

different values of m term in the complex stress function.

Data Point Input: x-disp. (

\*Note: Calculated Input %Error 100(%) Input 

*m* 1 of the complex stress functions.



0

1

  2

3

4

Fig. 3-5. 16 fringe patterns of rectangular steel plate containing a circular hole at every interval of /8 radian.

Fig. 3-6. Wrapped phase of steel plate with circular hole. Lines AB and BC are xdisplacement extraction regions.


\*Note: Calculated Input %Error 100(%) Input 

166 Interferometry – Research and Applications in Science and Technology

α 0 α

8/4 α

8/8 α

α 

α 

α 8/12 α

/8 radian.

interval of

 

displacement extraction regions.

8/13 α

Fig. 3-6. Wrapped phase of steel plate with circular hole. Lines AB and BC are x-

Fig. 3-5. 16 fringe patterns of rectangular steel plate containing a circular hole at every

/8 α

8/5 α

8/9 α

8/10 α

8/14 α

/82 α

/86 α

/83

8/7

8/11

8/15

Table 3-1. Comparison between input and calculated x-displacement for different values of *m* 1 of the complex stress functions.

Fig. 3-7. ABAQUS discretization of the quarter steel plate (*t* = 1.15mm).

Fig. 3-8. Normalized tangential stress at the edge of the quarter hole in a steel plate with different values of m term in the complex stress function.

Speckle Interferometry for Displacement Measurement and Hybrid Stress Analysis 169

Rastogi, P. (Editor) (2001). *Digital Speckle Pattern Interferometry and Related Techniques,* John

Baek, T.; Kim, M., Na, E. & Koh, S. (2003). Application of ESPI to Measurement of Out-of-

Ghiglia, D. & Pritt, M. (1998). *Two-Dimensional Phase Unwrapping*, John Wiley & Sons, Inc.,

Creath, K. (1988). Phase-measurement interferometry techniques, *Progress in Optics*, Vol.26,

Kim, M. & Baek, T. (2006). Measurement for in-plane displacement of tensile plates with

Kobayshi, A. (Editor) (1993). *Handbook on Experimental Mechanics*, 2nd Revised Edition, Society for Experimental Mechanics, VCH Publishers, Inc., ISBN 978-1560816409, Dally, J. & Riley, W. (1991). Experimental Stress Analysis, 3rd Edition, McGraw-Hill, Inc.,

Pilkey, W. & Pilkey, D. (2008). *Peterson's Stress Concentration Factors*, John Wiley & Sons, Inc.,

Baek, T.; Kim, M., Rhee, J. & Rowlands, R. (2000). Hybrid Stress Analysis of Perforated

Baek, T. & Rowlands, R. (1999). Experimental Determination of Stress Concentrations in

Baek, T. & Kim, M. (2005). Computer Simulation of Photoelastic Fringe Patterns for Stress Analysis, *Lecture Notes in Computer Science*, Vol.3398, pp. 214-221, ISSN 0302-9743 Baek, T. Koh, S. & Park, T. (2006). An Improved Hybrid Full-field Stress Analysis of

Schwider, J. (1989). Phase shifting Interferometry: reference phase error reduction, Applied

Steinzig, M. & Takahashi, T. (2006). Residual Stress Measurement Using the Hole Drilling

Experimental Techniques, Vol. 27, Issue 6, pp. 59-63, ISSN 0732-8818 Gerhardt, T. (1984). A Hybrid/Finite Element Approach for Stress Analysis of Notched Anisotropic Material, *ASME Journal of Applied Mechanics*, Vol.51, No.4, pp. 804-810 Rhee, J. & Rowlands, R (2002). Moiré-Numerical Hybrid Analysis of Cracks in Orthotropic Media, *Experimental Mechanics*, Vo.42, No.3, pp. 311-317, ISSN 1741-2765 Baek, T. & Rowlands, R. (2001). Hybrid Stress Analysis of Perforated Composites using Strain Gages, *Experimental Mechanics*, Vol.41, No.2, pp. 194-202, ISSN 1741-2765 Sanford, R. (1980). Application of the Least-Squares Method to Photoelastic Analysis,

Experimental Mechanics, Vol.20, No.6, pp. 192-197, ISSN 1741-2765

Tensile Plates using Multiplied and Sharpened Photoelastic Data and Complex-Variable Techniques, *JSME International Journal*, *Series A: Solid Mechanics and* 

Orthotropic Composites, *Journal of Strain Analysis*, Vol.34, No.2, pp. 69-81, ISSN

Circularly Perforated Plate by Photoelasticity and Finite Element Analysis, *Key* 

Method and Laser Speckle Interferometry, Part IV: Measurement Accuracy,

plane Displacement in a Spot Welded Canti-levered Plate, *International Journal of the* 

partly through-thickness circular hole and through-thickness circular hole by use of speckle interferometry, *Key Engineering Materials*, Vols.321-323, pp. 77-80, ISSN

Wiley and Sons, Ltd., ISBN 978-0471490524, New York, USA

*Korean Society of Precision Engineering*, Vol.4, No.5, pp. 41-46

Lekhnitskii, S.; Tsai, S. & Cheron, T. (1968). *Anisotropic Plates*, Gordon and Breach

*Material Engineering*, Vol.43, No.4, pp. 327-333, ISSN 0914-8809

*Engineering Materials*, Vol.326-328, pp. 1209-1212, ISSN 1162-9795

ISBN 978-0471249351, New York, USA

ISBN 978-0470048245, New Jersey, USA

Optics, Vol.28, Issue 18, pp. 3889-3892

pp. 349-393, ISSN 0079-6638

1162-9795

0309-3247

New York, USA

Fig. 3-9. Normalized tangential stress at the edge of the quarter hole in a steel plate with *m* 1 in the complex stress function.

## **3.4 Conclusions and discussions**

A reliable hybrid method for characterizing stress around the circular hole in a tensileloaded steel plate is presented. The method utilized only few micro-scale x-displacement data measured by a well-established optical technique, speckle interferometry, in conjunction with phase shifting method using Fourier transform to calculate stress components and eventually stress concentration at <sup>o</sup> 90 . The use of few input data may reduce experiment time and relatively increase data processing speed.

Different values of m term in the complex stress functions were tested to see its effect. In the comparison between input x-displacement data and calculated data the hybrid method is effective with an error below 1% in all values of *m*. On the other hand, it came out that at point of high stress concentration ( <sup>o</sup> 90 ), well-comparable results were attained at decreasing value of *m*. The best value of m is consistently known to be equal to 1 of which results are accurate by less than one percent error. Results showed that the method is accurate and reliable as compared with the widely-used FEM software, ABAQUS.

## **4. Acknowledgement**

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number: 2010-0021248)

## **5. References**

Cloud, G. (1998). *Optical Methods of Engineering Analysis*, Cambridge University Press, ISBN 978-0521636421, Cambridge, United Kingdom

Petzing, J. & Tyrer, J. (1998). Recent Developments and Applications in Electronic Speckle Pattern Interferometry, *Journal of Strain Analysis*, Vol.33, No.2, pp. 153-169, ISSN 0309-3247

0 10 20 30 40 50 60 70 80 90

Angle, 

Fig. 3-9. Normalized tangential stress at the edge of the quarter hole in a steel plate with

A reliable hybrid method for characterizing stress around the circular hole in a tensileloaded steel plate is presented. The method utilized only few micro-scale x-displacement data measured by a well-established optical technique, speckle interferometry, in conjunction with phase shifting method using Fourier transform to calculate stress

Different values of m term in the complex stress functions were tested to see its effect. In the comparison between input x-displacement data and calculated data the hybrid method is effective with an error below 1% in all values of *m*. On the other hand, it came out that at

decreasing value of *m*. The best value of m is consistently known to be equal to 1 of which results are accurate by less than one percent error. Results showed that the method is

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and

Cloud, G. (1998). *Optical Methods of Engineering Analysis*, Cambridge University Press, ISBN

Petzing, J. & Tyrer, J. (1998). Recent Developments and Applications in Electronic Speckle

Pattern Interferometry, *Journal of Strain Analysis*, Vol.33, No.2, pp. 153-169, ISSN

90 . The use of few input data may

90 ), well-comparable results were attained at


*m* 1 in the complex stress function.

**3.4 Conclusions and discussions** 


0

1

  2

3

Hybrid, m=1 ABAQUS

components and eventually stress concentration at <sup>o</sup>

point of high stress concentration ( <sup>o</sup>

Technology (Grant number: 2010-0021248)

978-0521636421, Cambridge, United Kingdom

**4. Acknowledgement** 

0309-3247

**5. References** 

reduce experiment time and relatively increase data processing speed.

accurate and reliable as compared with the widely-used FEM software, ABAQUS.

4


**8** 

*México* 

**Phase-Shifting Interferometry** 

Cruz Meneses-Fabian and Uriel Rivera-Ortega

In optics, the superposition of two or more light beams at any point over space can produce the apparition of interference fringes. When these fringes are applied to resolve a problem in industry or they are related with some property of an investigation matter of interest in some area of physics, chemistry, biology, etc., the evaluation of them is a very necessary task. One of the most used methods for phase extraction, as a result of fringes evaluation, is based on a phase change between the interference beams by a known value, while their amplitudes are keeping constant. It is called phase-shifting interferometry, phase-sampling interferometry, or phase-stepping interferometry, which are abbreviated by "PSI" (Schwider, 1990). In this technique a set of *N* interferograms changed in phase are created, which are represented by a set of *N* equations, where each equation has three unknowns called as background light, modulation light and the object phase. These spatial unknowns are considered constant during the application of the PSI technique. Then a 3 *N* system is formed and therefore it can be resolved when 3 *N* . Many methods to introduce a constant phase have been proposed as for example by changing the optical frequency, wavelength, index of refraction, distance, optical path, for instance; but also with some properties or effect of the light such as the polarization, diffraction, Zeeman effect, Doppler effect, for instance (Schwider, 1990; Malacara, 2007). In this chapter, our major interest aims to propose a new method to generate a phase change in the interferogram based on the amplitude

modulation of the electric field (Meneses-Fabian and Rivera-Ortega, 2011).

wavefront whose shape is to be measured, this beam is known as the probe beam.

Interferometry uses the superposition principle of electromagnetic waves when certain conditions of coherence are achieved to extract information about them. If light from a source is divided into two to be superposed again at any point in space, the intensity in the superposition area varies from maxima (when two waves crests reach the same point simultaneously ) to minima (when a wave trough and a crest reach the same point) ; having by this what is known as an interference pattern. Interferometry uses interferometers which are the instruments that use the interference of light to make precise measurements of surfaces, thicknesses, surface roughness, optical power, material homogeneity, distances and so on based on wavefront deformations with a high accuracy of the order of a fraction of the wavelength through interference patterns. In a two wave interferometer one wave is typically a flat wavefront known as the reference beam and the other is a distorter

**1. Introduction** 

**by Amplitude Modulation** 

*Benemérita Universidad Autónoma de Puebla, Facultad de Ciencias Físico-Matemáticas, Puebla,* 


## **Phase-Shifting Interferometry by Amplitude Modulation**

Cruz Meneses-Fabian and Uriel Rivera-Ortega *Benemérita Universidad Autónoma de Puebla, Facultad de Ciencias Físico-Matemáticas, Puebla, México* 

## **1. Introduction**

170 Interferometry – Research and Applications in Science and Technology

Morimoto, Y. & Fujisawa, M. (1994). Fringe Pattern Analysis by a Phase-Shifting Method using Fourier Transform, *Optical Engineering*, Vol.33, No.11, pp. 3709-3714 Kim, M. ; Baek, T., Morimoto, Y., & Fujigaki, M. (2005). Application of Phase-shifting

Baek, T.; Chung, T. & Panganiban, H. (2008). Full-Field Stress Determination Around

*Journal of Solid Mechanics and Material Engineering*, Vol.2, No.6, pp. 756-762 Baek, T. & Rowlands, R. (1999). Experimental Determination of Stress Concentrations in

Baek, T. & Rudolphi, T. (2010). A Hybrid Stress Measurement Using only x-displacement by

Vol.25, No.3, pp. 171-177

Engineering, pp. 49-54, ISSN 0217-9849

0309-3247

Method using Fourier Transform to Measurement of In-plane Displacement by Speckle Interferometry, *Journal of the Korean Society for Nondestructive Testing*,

Circular Discontinuity in a Tensile-Loaded Plate using x-displacements Only,

Orthotropic Composites, *Journal of Strain Analysis*, Vol.34, No.2, pp. 69-81, ISSN

Phase Shifting Method with Fourier Transform (PSM/FT) in Laser Speckle Interferometry and Least Squares Method, *International Journal of Precision Engineering and Manufacturing*, Vol.11, No.1, Korean Society for Precision

> In optics, the superposition of two or more light beams at any point over space can produce the apparition of interference fringes. When these fringes are applied to resolve a problem in industry or they are related with some property of an investigation matter of interest in some area of physics, chemistry, biology, etc., the evaluation of them is a very necessary task. One of the most used methods for phase extraction, as a result of fringes evaluation, is based on a phase change between the interference beams by a known value, while their amplitudes are keeping constant. It is called phase-shifting interferometry, phase-sampling interferometry, or phase-stepping interferometry, which are abbreviated by "PSI" (Schwider, 1990). In this technique a set of *N* interferograms changed in phase are created, which are represented by a set of *N* equations, where each equation has three unknowns called as background light, modulation light and the object phase. These spatial unknowns are considered constant during the application of the PSI technique. Then a 3 *N* system is formed and therefore it can be resolved when 3 *N* . Many methods to introduce a constant phase have been proposed as for example by changing the optical frequency, wavelength, index of refraction, distance, optical path, for instance; but also with some properties or effect of the light such as the polarization, diffraction, Zeeman effect, Doppler effect, for instance (Schwider, 1990; Malacara, 2007). In this chapter, our major interest aims to propose a new method to generate a phase change in the interferogram based on the amplitude modulation of the electric field (Meneses-Fabian and Rivera-Ortega, 2011).

> Interferometry uses the superposition principle of electromagnetic waves when certain conditions of coherence are achieved to extract information about them. If light from a source is divided into two to be superposed again at any point in space, the intensity in the superposition area varies from maxima (when two waves crests reach the same point simultaneously ) to minima (when a wave trough and a crest reach the same point) ; having by this what is known as an interference pattern. Interferometry uses interferometers which are the instruments that use the interference of light to make precise measurements of surfaces, thicknesses, surface roughness, optical power, material homogeneity, distances and so on based on wavefront deformations with a high accuracy of the order of a fraction of the wavelength through interference patterns. In a two wave interferometer one wave is typically a flat wavefront known as the reference beam and the other is a distorter wavefront whose shape is to be measured, this beam is known as the probe beam.

Phase-Shifting Interferometry by Amplitude Modulation 173

After the waves have travelled along two separated arms and they have accumulated phase

For the particular case of two waves ( 2 *n* ), and using the Eq. (3) for a vector treatment, we

2 1

It can be seen in Eq. (4) that there are three terms, by doing the math they can be expressed

<sup>2</sup> \* 2 2 ( )( ) *k k k k i i i i k k k kx k E Ee e E Ee e E E <sup>y</sup> kx ky kx ky*

where the values for *k* 2,1 ; by doing this the first two terms of Eq. (4) will be obtained, the

<sup>1</sup> <sup>1</sup> <sup>22</sup> ( () ) <sup>2</sup> <sup>1</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup>

cos(

*eeEEeeEE <sup>i</sup>*

Re2 2cos2 )cos( <sup>2</sup> <sup>21</sup> <sup>21</sup>

(8) that the interference of two elliptically polarized waves can also be generated by the addition of two interference patterns, one of them with components in *x* direction

> sincos

*baI*

*yxx* cos2 *<sup>y</sup>*

*bbbbb* ;

modulation light *b* depend on the components *<sup>x</sup> a* , *<sup>y</sup> a* , *<sup>x</sup> b* , *<sup>y</sup> b* and also on the phase

where *a* is known as the background light, *b* as the modulation light and

*yx aaa* ; <sup>22</sup> <sup>2</sup>

will also generate a change in *a* and *b* .

*CBA*

*i*

2 2

**E EE i j i j** , (5)

*I <sup>T</sup>* **EEEEE** Re2 **EE** ; (4)

*x y*

*<sup>x</sup> <sup>y</sup>* **jiEE ji** , (6)

*EE yy*

 *xx yy babaI* , (8)

<sup>1</sup>*yy EEa <sup>y</sup>* , *<sup>x</sup> EEb* <sup>21</sup> *xx* 2 y 1 2 2 *y yy b EE* . It can be observed in Eq.

. Note that a phase shifting

tan

 

 

)cos( , (9)

*yy ba* .

<sup>12</sup> . By taking Eqs. (5-7) a general expression for the interference

 <sup>2</sup> \* 1

> 

, (7)

)cos( if <sup>22</sup> *BAC* and

. (10)

as the background *a* and the

cos sin

*yx y bb b*

indicates an

by varying

delays, they recombine again by means of a beam splitter giving as a result a field **E***<sup>T</sup>* .

2 21

> 

can find that the corresponding irradiance is

third term of this equation is obtained by

therefore the resulting interference term is

 

2

On the other hand, it is known that

<sup>1</sup>*xx EEa <sup>x</sup>* , <sup>2</sup>

of two waves is obtained

2

where <sup>2</sup>

\* 1

\*

) cos

2

<sup>1</sup> **EE** *EE xx*

2

)cos( *xx ba* and the other with components in *y* )) ( cos(

*AB* ; by applying this identity to Eq. (8) is obtained

additional phase shifting, which can be expressed by

difference between the amplitude of the waves

It is worth noting from Eq. (10) that both the phase shifting

as follows

where <sup>12</sup> y

tan

*<sup>x</sup> b* , *<sup>y</sup> b* and

2

There are several well studied and know methods to generate phase-shifts which will be briefly discussed, however the present method of PSI based on the amplitude modulation into two beams named reference beams in a scheme of a three beam interferometry will be amply discussed; this discussion will be done for a particular case where the phase difference between the reference beam is conditioned to be 2 ; and for a general case where the phase difference between the reference beam should be within the range of 0,2 but .

We can represent *n* optical perturbations in a complex form with elliptical polarization and traveling on z direction as follows:

$$\mathbf{E}\_k = (\mathbf{i}E\_{kx} + \mathbf{j}E\_{ky}e^{i\delta\_k})e^{i\phi} \tag{1}$$

where *k* goes from 1 , *n k* is the phase of each wave and *<sup>k</sup>* is the relative phase difference between the component of each wave (by simplicity the temporal and spatial dependencies have been omitted). In the particular case that *<sup>k</sup> m* where *m* is an integer number, the wave will be linearly polarized; on the other hand, if *kx EE ky* and *<sup>k</sup> m* 212 the wave will be circularly polarized.

The phenomenon of interference can occur when two or more waves overlap in space. Mathematically the resulting wave is the vector addition,

$$\mathbf{E}\_{\tau} = \sum\_{k=1}^{n} \mathbf{E}\_{k} \cdot \tag{2}$$

When the field is observed by a detector, the result is the average of the field energy by area unit during the integration time of the detector, that is, the irradiance, which can be demonstrated that is proportional to the squared module of the amplitude. However, it is usually accepted the approximation

> 2 2 1 *n T k k I E* **<sup>E</sup>** . (3)

An interferometer is an instrument used to generate wave light interference to measure with high accuracy small deformations of the wave front. The general scheme of a two wave interferometer can be observed in Figure 1, where the electromagnetic wave **E** is typically divided in two coherent parts that is, in a wave **E**1 and **E**<sup>2</sup> , where **E**<sup>1</sup> is the reference wave and **E**2 is the probe wave.

Fig. 1. Scheme of a two wave interferometer.

There are several well studied and know methods to generate phase-shifts which will be briefly discussed, however the present method of PSI based on the amplitude modulation into two beams named reference beams in a scheme of a three beam interferometry will be amply discussed; this discussion will be done for a particular case where the phase

where the phase difference between the reference beam should be within the range of

We can represent *n* optical perturbations in a complex form with elliptical polarization and

*<sup>k</sup> kx ky eeEE*

difference between the component of each wave (by simplicity the temporal and spatial

number, the wave will be linearly polarized; on the other hand, if *kx EE ky* and

The phenomenon of interference can occur when two or more waves overlap in space.

 *n k T k* 1

When the field is observed by a detector, the result is the average of the field energy by area unit during the integration time of the detector, that is, the irradiance, which can be demonstrated that is proportional to the squared module of the amplitude. However, it is

2

An interferometer is an instrument used to generate wave light interference to measure with high accuracy small deformations of the wave front. The general scheme of a two wave interferometer can be observed in Figure 1, where the electromagnetic wave **E** is typically divided in two coherent parts that is, in a wave **E**1 and **E**<sup>2</sup> , where **E**<sup>1</sup> is the reference wave

2

1 *n T k k I E* 

*k* is the phase of each wave and

*<sup>k</sup> <sup>k</sup> ii*

( **jiE** ) (1)

 *<sup>k</sup> m*

**EE** . (2)

**<sup>E</sup>** . (3)

2 ; and for a general case

*<sup>k</sup>* is the relative phase

where *m* is an integer

difference between the reference beam is conditioned to be

dependencies have been omitted). In the particular case that

212 the wave will be circularly polarized.

Mathematically the resulting wave is the vector addition,

 0,2

*<sup>k</sup> m*

 but .

traveling on z direction as follows:

where *k* goes from 1 , *n*

usually accepted the approximation

and **E**2 is the probe wave.

Fig. 1. Scheme of a two wave interferometer.

After the waves have travelled along two separated arms and they have accumulated phase delays, they recombine again by means of a beam splitter giving as a result a field **E***<sup>T</sup>* .

For the particular case of two waves ( 2 *n* ), and using the Eq. (3) for a vector treatment, we can find that the corresponding irradiance is

$$I = \left| \mathbf{E}\_r \right|^2 = \left| \mathbf{E}\_1 + \mathbf{E}\_2 \right|^2 = \left| \mathbf{E}\_1 \right|^2 + \left| \mathbf{E}\_2 \right|^2 + 2 \operatorname{Re} \left\langle \mathbf{E}\_1 \cdot \mathbf{E}\_2 \right\rangle;\tag{4}$$

It can be seen in Eq. (4) that there are three terms, by doing the math they can be expressed as follows

$$\left|\mathbf{E}\_{k}\right|^{2} = \mathbf{E}\_{k} \cdot \mathbf{E}\_{k} \stackrel{\*}{=} \left[ (\mathbf{i}E\_{kx} + \mathbf{j}E\_{ky}e^{-i\delta\_{k}})e^{i\phi\_{k}} \right] \cdot \left[ (\mathbf{i}E\_{kx} + \mathbf{j}E\_{ky}e^{i\delta\_{k}})e^{-i\phi\_{k}} \right] = E\_{kx} \stackrel{2}{\rightarrow} + E\_{ky} \stackrel{2}{\rightarrow} \tag{5}$$

where the values for *k* 2,1 ; by doing this the first two terms of Eq. (4) will be obtained, the third term of this equation is obtained by

$$\mathbf{E}\_1 \, ^\ast \cdot \mathbf{E}\_2 = \left[ (\mathbf{i}E\_{1x} + \mathbf{j}E\_{1y}e^{-i\delta\_1})e^{-\phi\_1} \right] \cdot \left[ (\mathbf{i}E\_{2x} + \mathbf{j}E\_{2y}e^{i\delta\_1})e^{\phi\_1} \right] \,\tag{6}$$

therefore the resulting interference term is

$$\mathbf{2}\operatorname{Re}\left|\mathbf{E}\_{1}\right.^{\circ}\cdot\mathbf{E}\_{2}\left|=2E\_{1x}E\_{2x}\cos\phi+2E\_{1y}E\_{2y}\cos(\phi+\delta)\right.\tag{7}$$

where <sup>12</sup> y <sup>12</sup> . By taking Eqs. (5-7) a general expression for the interference of two waves is obtained

$$I = a\_x + b\_x \cos \phi + a\_y + b\_y \cos(\phi + \delta), \tag{8}$$

where <sup>2</sup> 2 2 <sup>1</sup>*xx EEa <sup>x</sup>* , <sup>2</sup> 2 2 <sup>1</sup>*yy EEa <sup>y</sup>* , *<sup>x</sup> EEb* <sup>21</sup> *xx* 2 y 1 2 2 *y yy b EE* . It can be observed in Eq. (8) that the interference of two elliptically polarized waves can also be generated by the addition of two interference patterns, one of them with components in *x* direction )cos( *xx ba* and the other with components in *y* )) ( cos( *yy ba* .

On the other hand, it is known that sincos *CBA* )cos( if <sup>22</sup> *BAC* and tan*AB* ; by applying this identity to Eq. (8) is obtained

$$I = a + b\cos(\phi + \psi) \, , \tag{9}$$

where *a* is known as the background light, *b* as the modulation light and indicates an additional phase shifting, which can be expressed by

$$a = a\_x + a\_y; \; \; b^2 = b\_x^2 + 2b\_x b\_y \cos \delta + b\_y^2; \; \tan \psi = \frac{b\_y \sin \delta}{b\_x + b\_y \cos \delta}. \tag{10}$$

It is worth noting from Eq. (10) that both the phase shifting as the background *a* and the modulation light *b* depend on the components *<sup>x</sup> a* , *<sup>y</sup> a* , *<sup>x</sup> b* , *<sup>y</sup> b* and also on the phase difference between the amplitude of the waves . Note that a phase shifting by varying *<sup>x</sup> b* , *<sup>y</sup> b* and will also generate a change in *a* and *b* .

Phase-Shifting Interferometry by Amplitude Modulation 175

Fig. 2. Scheme of the most commonly used interferometers: a) Twyman-Green, b) Mach-

Another method to generate phase-shifting is by means of inserting a glass plate in the light

angles formed by the normal and the light beams outside and inside the plate respectively. An special requirement is that the plate must be placed in a collimated light beam to avoid

respecting to the optical axis hence )coscos)(( '

*nnt*

 

0 is generated when the plate is tilted an

, where *t* is the

 and ' are the

. The angles

Zehnder, c) Fizeau.

angle

aberrations.

**2.1.4 Tilting a glass plate** 

beam (Wyant y Shagam 1978). The phase-shift

thickness of the plate, *n* is the refraction index and *k* 2

## **2. Phase-shifting interferometry**

The first studies in the phase shifting techniques can be found in the work of Carré 1966, but it really started with Crane 1969, Moore 1973 , Brunning *et al*. 1974 and some others. These techniques have also been applied in speckle patter interferometry (Nakadate and Saito 1985; Creath 1985; Robinson and William 1986) and also to holographic interferometry (Nakadate *et al*. 1986; Stetson and Brohinski 1988).

In phase shifting interferometry, a reference wave front is moved along its propagation direction respecting to the probe wave front changing with this the phase difference between them. It is possible to determine the phase of the probe wave by measuring the irradiance changes corresponding to each phase shifting.

## **2.1 Some phase-shifting methods**

There are many experimental ways to generate phase shifting, the most common are:

## **2.1.1 Moving a mirror**

This method is based on the change in the optical path of a beam by means of moving a mirror that is in the beam trajectory. This movement can be made by using a piezoelectric transducer (Soobitsky, 1988; Hayes, 1989). The phase-shifting is given by *dco*))(2( , where *dco* is the optical path difference. As an illustrative example if the mirror is translated a distance of 8 and due to the beam travels two times the same trajectory, the value of the phase-shifting is 2 . Examples of interferometers with phase-shifting generated by a piezoelectric are: Twyman-Green, Mach-Zehnder, Fizeau, which are represented in Figure 2. The first two make a phase-shifting by means of moving a mirror that is placed in the reference beam. In the Fizeau interferometer the phase-shifting is made by the translations of the reference or probe object.

## **2.1.2 Rotating a phase plate**

A phase-shifting can also be generated by means of rotating a retarder phase plate (Crane 1969; Okoomian 1969; Bryngdahl 1972; Sommargren 1975; Shagam y Wyant 1978; Hu 1983, Zhi 1983; Kothiyal and Delisle 1984, 1985; Salbut y Patorski 1990). As a particular case, if a circularly polarized wave passes through a half wave retarder plate rotated 45° the direction of rotation will be inverted, thus a phase-shifting of 2 will be present.

## **2.1.3 Displacing a diffraction grating**

This method is based on the perpendicular displacement respecting to the light beam of a diffraction grating (Suzuki y Hioki 1967; Stevenson 1970; Bryngdahl 1976; Srinivasan *et al.*1985, Meneses *et al*. 2009). If the diffraction grating is moved a small distance *y* , the changes in the phase are given by )2( *ydmy* where *d* is the period of the grating and *m* is the diffraction order. As an example, if the grating of Figure 3 is perpendicularly moved respecting to the optical axis, a phase-shifting will be generated.

The first studies in the phase shifting techniques can be found in the work of Carré 1966, but it really started with Crane 1969, Moore 1973 , Brunning *et al*. 1974 and some others. These techniques have also been applied in speckle patter interferometry (Nakadate and Saito 1985; Creath 1985; Robinson and William 1986) and also to holographic interferometry

In phase shifting interferometry, a reference wave front is moved along its propagation direction respecting to the probe wave front changing with this the phase difference between them. It is possible to determine the phase of the probe wave by measuring the

There are many experimental ways to generate phase shifting, the most common

This method is based on the change in the optical path of a beam by means of moving a mirror that is in the beam trajectory. This movement can be made by using a piezoelectric

where *dco* is the optical path difference. As an illustrative example if the mirror is

generated by a piezoelectric are: Twyman-Green, Mach-Zehnder, Fizeau, which are represented in Figure 2. The first two make a phase-shifting by means of moving a mirror that is placed in the reference beam. In the Fizeau interferometer the phase-shifting is made

A phase-shifting can also be generated by means of rotating a retarder phase plate (Crane 1969; Okoomian 1969; Bryngdahl 1972; Sommargren 1975; Shagam y Wyant 1978; Hu 1983, Zhi 1983; Kothiyal and Delisle 1984, 1985; Salbut y Patorski 1990). As a particular case, if a circularly polarized wave passes through a half wave retarder plate rotated 45° the direction

This method is based on the perpendicular displacement respecting to the light beam of a diffraction grating (Suzuki y Hioki 1967; Stevenson 1970; Bryngdahl 1976; Srinivasan *et al.*1985, Meneses *et al*. 2009). If the diffraction grating is moved a small distance *y* , the

*m* is the diffraction order. As an example, if the grating of Figure 3 is perpendicularly

moved respecting to the optical axis, a phase-shifting will be generated.

8 and due to the beam travels two times the same trajectory,

2 . Examples of interferometers with phase-shifting

2 will be present.

)2( *ydmy* where *d* is the period of the grating and

 *dco*))(2( ,

transducer (Soobitsky, 1988; Hayes, 1989). The phase-shifting is given by

 

**2. Phase-shifting interferometry** 

**2.1 Some phase-shifting methods** 

**2.1.1 Moving a mirror** 

translated a distance of

the value of the phase-shifting is

**2.1.2 Rotating a phase plate** 

**2.1.3 Displacing a diffraction grating** 

changes in the phase are given by

are:

(Nakadate *et al*. 1986; Stetson and Brohinski 1988).

 

by the translations of the reference or probe object.

of rotation will be inverted, thus a phase-shifting of

irradiance changes corresponding to each phase shifting.

Fig. 2. Scheme of the most commonly used interferometers: a) Twyman-Green, b) Mach-Zehnder, c) Fizeau.

## **2.1.4 Tilting a glass plate**

Another method to generate phase-shifting is by means of inserting a glass plate in the light beam (Wyant y Shagam 1978). The phase-shift 0 is generated when the plate is tilted an angle respecting to the optical axis hence )coscos)(( ' *nnt* , where *t* is the thickness of the plate, *n* is the refraction index and *k* 2 . The angles and ' are the angles formed by the normal and the light beams outside and inside the plate respectively. An special requirement is that the plate must be placed in a collimated light beam to avoid aberrations.

Phase-Shifting Interferometry by Amplitude Modulation 177

d) Fig. 3. Scheme of interferometers where the method of phase-shifting by polarizations is

Other method to generate phase-shift is by means of changing the frequency of the laser source, there are two ways to do this; one of them is by illuminating the interferometer with a Zeeman laser. The laser frequency is divided in two orthogonally polarized output frequencies by means of a magnetic field (Burgwald and Kruger 1970). The other method is by using an unbalanced interferometer, that is an interferometer with a very long optical path difference and using a laser diode with its frequency controlled by electrical current as proposed by Ishii *et al*. (1991) and subsequently studied by Onodera e Ishii (1996). This method is based on the fact that the phase difference in an interferometer is proportional to

In summary, the known techniques for phase-shifting that have been mentioned are applied in two beams interferometers; however none of these techniques use a variation on the

sampling a certain number of interference pattern which differ from each other due to

cos( ) *<sup>n</sup>* <sup>0</sup>*<sup>n</sup> baI* 

> *<sup>n</sup> C*

; *bC* sin

0 is made for *N* steps, then *N* intensity values *<sup>n</sup> I* will

*on* cos <sup>0</sup> sin , (20)

. (19)

. (21)

by

The phase-shifting interferometry is based on the reconstruction of the phase

the product of the optical path difference and the temporal frequency.

amplitude fields, that is why this option will be discussed in this chapter.

*<sup>n</sup> BAI*

*aA* ; *bB* cos

used.

**2.1.6 Changing the laser source** 

**2.2 Phase extraction methods** 

be measured, (where ,...,1 *Nn* )

 /2 *Nn* . Eq. (19) can be rewritten as follows

<sup>0</sup> . If a shift of

different values of

where <sup>0</sup>*<sup>n</sup>*

where

## **2.1.5 Rotating a polarizer**

If in Eq. (1) *k* 2,1 , *yxyx EEEEE* <sup>02211</sup> , 1 2 , 2 2 then we have two circularly polarized waves with the same amplitude with opposite rotations, that is

$$\mathbf{E}\_1 = (\mathbf{i} + e^{i\frac{\theta}{2}}\mathbf{j})E\_0 e^{i\phi} \,\,\,\,\,\,\,\,\tag{11}$$

$$\mathbf{E}\_2 = (\mathbf{i} + e^{-i\frac{\pi}{2}}\mathbf{j})E\_0 e^{i\phi\_i} \,. \tag{12}$$

Expressing the Eqs. (11-12) with the notation for the Jones vector

$$E\_1 = E\_0 \binom{1}{i} e^{i\phi\_i} \tag{13}$$

$$E\_z = E\_0 \begin{pmatrix} 1 \\ -i \end{pmatrix} e^{i\phi\_i} \tag{14}$$

if we interfere those waves, then there will not be an interference term so no fringes will be observed, however if those waves are passed through a polarizer at an angle and they interfere then

$$E\_1 \stackrel{\cdot}{=} \begin{bmatrix} \cos^2 a & \sin a \cos a \\ \sin a \cos a & \sin^2 a \end{bmatrix} \stackrel{\cdot}{i} \stackrel{\cdot}{l}\_0 e^{i\phi\_i} = E\_0 \begin{bmatrix} \cos a \\ \sin a \end{bmatrix} e^{i(a+\phi\_i)} \tag{15}$$

$$E\_2 = \begin{bmatrix} \cos^2 a & \sin a \cos a \\ \sin a \cos a & \sin^2 a \end{bmatrix} \begin{bmatrix} 1 \\ -i \end{bmatrix} E\_0 e^{i\phi\_i} = E\_0 \begin{bmatrix} \cos a \\ \sin a \end{bmatrix} e^{-i(a-\phi\_i)} \tag{16}$$

$$I = 2E\_o \, ^2 \left[ 1 + \cos(\phi - 2a) \right] \,\tag{17}$$

where

$$
\begin{bmatrix}
\cos^2 a & \sin a \cos a \\
\sin a \cos a & \sin^2 a
\end{bmatrix}
\tag{18}
$$

is the Jones matrix that represents a linear polarizer at an angle . It is observed in Eq. (17) that there is a phase–shift in the interference pattern which is twice the angle at which the polarizer is placed.

An application of phase-shifting generated by polarization is by using the scheme of Figure 3, which is based in a common path interferometer consisting of two windows in the input plane. In this interferometer is possible to have four interference patters in one shot, each pattern shifted by 90°. To do this a binary grating is used to generate the interference between the diffraction orders of two circularly polarized beams with opposite rotations; a linear polarizer at an angle is placed in front of each diffraction order thus, the phaseshifting is obtained (Rodriguez-Zurita *et al*. 2008).

circularly polarized waves with the same amplitude with opposite rotations, that is

<sup>0</sup> <sup>2</sup> <sup>1</sup> )(

<sup>0</sup> <sup>2</sup> <sup>2</sup> )(

**jiE** *eEe*

01

02

observed, however if those waves are passed through a polarizer at an angle

*EE*

*<sup>i</sup> <sup>i</sup> eEeE <sup>i</sup>*

*<sup>i</sup> <sup>i</sup> eEeEi*

)2cos(12 <sup>2</sup> *EI* <sup>0</sup>

that there is a phase–shift in the interference pattern which is twice the angle at which the

An application of phase-shifting generated by polarization is by using the scheme of Figure 3, which is based in a common path interferometer consisting of two windows in the input plane. In this interferometer is possible to have four interference patters in one shot, each pattern shifted by 90°. To do this a binary grating is used to generate the interference between the diffraction orders of two circularly polarized beams with opposite rotations; a

cossin sin cos cossin

2

if we interfere those waves, then there will not be an interference term so no fringes will be

*<sup>i</sup> <sup>i</sup>*

*<sup>i</sup> <sup>i</sup>*

1 1

2

*ie i*

2 0 0

2 0 0

 

 

1 cos

1 cos

 

2

*E* , (16)

*E* , (15)

 

*i e i EE* 

1

1

2

2 , 2

**jiE** *eEe* , (11)

. (12)

, (13)

; (14)

)(

)(

, (17)

, (18)

. It is observed in Eq. (17)

1 1 sin

2 2 sin

is placed in front of each diffraction order thus, the phase-

 

 

2 then we have two

and they

**2.1.5 Rotating a polarizer** 

interfere then

where

polarizer is placed.

linear polarizer at an angle

If in Eq. (1) *k* 2,1 , *yxyx EEEEE* <sup>02211</sup> , 1

Expressing the Eqs. (11-12) with the notation for the Jones vector

2

2

cossin sin cos cossin

cossin sin cos cossin

 

is the Jones matrix that represents a linear polarizer at an angle

shifting is obtained (Rodriguez-Zurita *et al*. 2008).

'

1

'

2

Fig. 3. Scheme of interferometers where the method of phase-shifting by polarizations is used.

## **2.1.6 Changing the laser source**

Other method to generate phase-shift is by means of changing the frequency of the laser source, there are two ways to do this; one of them is by illuminating the interferometer with a Zeeman laser. The laser frequency is divided in two orthogonally polarized output frequencies by means of a magnetic field (Burgwald and Kruger 1970). The other method is by using an unbalanced interferometer, that is an interferometer with a very long optical path difference and using a laser diode with its frequency controlled by electrical current as proposed by Ishii *et al*. (1991) and subsequently studied by Onodera e Ishii (1996). This method is based on the fact that the phase difference in an interferometer is proportional to the product of the optical path difference and the temporal frequency.

In summary, the known techniques for phase-shifting that have been mentioned are applied in two beams interferometers; however none of these techniques use a variation on the amplitude fields, that is why this option will be discussed in this chapter.

## **2.2 Phase extraction methods**

The phase-shifting interferometry is based on the reconstruction of the phase by sampling a certain number of interference pattern which differ from each other due to different values of <sup>0</sup> . If a shift of 0 is made for *N* steps, then *N* intensity values *<sup>n</sup> I* will be measured, (where ,...,1 *Nn* )

$$I\_n = a + b\cos(\phi + \varphi\_{0n})\,. \tag{19}$$

where <sup>0</sup>*<sup>n</sup>* /2 *Nn* .

Eq. (19) can be rewritten as follows

$$I\_u = A + B\cos\varphi \nu\_{\alpha u} + C\sin\varphi \nu\_{\alpha u'} \tag{20}$$

where

$$A = a \; ; \quad B = b \cos \phi \; ; \quad \text{C} = -b \sin \phi \; . \tag{21}$$

Phase-Shifting Interferometry by Amplitude Modulation 179

 

The deformations of the wavefront in an interferogram can also be calculated by a method that uses the Fourier transform. This method was originally proposed Takeda *et al.* (1982) using the Fourier transform in one dimension along an scanned line. Later Macy (1983) extended the Takeda's method in two dimensions by adding the information of multiple scanned lines and obtaining slices of the phase in two dimensions. Bone *et al* (1986) extended

Once the interferogram is obtained, its Fourier transform is calculated, by doing this an image in the Fourier space as in Figure 4a) is obtained, then one lateral spectrum is taken and filtered by means of a layer so that all the irradiance values outside the layer will be multiplied by cero, after that this spectrum is translated to the origin and its Fourier

To describe mathematically this process, the general equation for the irradiance is taken (Eq.

*i*

*i*

, (34)

),(),(),(),( *eyxceyxcyxayxI* \* , (33)

, where

we have ),( )21(),( *yxi beyxc*

, therefore

0 is known as the

2 3 cos <sup>4</sup> *baI ba* 

sin

 

31 <sup>1</sup> <sup>24</sup> tan *II II*

, (31)

, (30)

. (32)

 *ba* cos)cos(

<sup>3</sup> *baI*

the Macy's work by applying the Fourier transform in two dimensions.

transform is obtained giving by this the resulting wavefront under test.

and the phase of the form *x*<sup>0</sup> 2

\* *caI* <sup>0</sup> *c* 

*yxc*

where ~*a* , ~*c* y ~\* *c* are complex Fourier amplitudes. By means of a digital filtering one lateral spectrum is isolated by using a filtering window and then translated to the origin by doing

> 

The calculated phase will present discontinuities because it is obtained by using the inverse tangent function Eq. (23). Because the inverse tangent is a multivalued function, the solution

) ),( <sup>~</sup> ),( <sup>~</sup> ),( <sup>~</sup> ,( <sup>~</sup> <sup>0</sup>

),(Im tan),( <sup>1</sup>

.

*yxc yx*

(9)) expressing the cosine as a complex exponential

Applying the Fourier transform to Eq. (33)

Obtaining the inverse Fourier transform of *c*(,)

the resulting phase is ),(Re

<sup>0</sup> , as shown in Figure 4b).

**2.4 Unwrapping the phase** 

being ),( )21(),( *yxi beyxc*

spatial-carrier frequency

0 

and the phase at each point is

**2.2.3 The Fourier transform method** 

It can be shown that based on a least-squares fit that B and C meet the next equations in an analytical form

$$B = \frac{2}{N} \sum\_{n=1}^{N} I\_n \cos \psi\_{0n} \; ; \; C = \frac{2}{N} \sum\_{n=1}^{N} I\_n \sin \psi\_{0n} \; . \tag{22}$$

A combination of Eq. (21) and Eqs. (22) can give us the basic equation of the Phase Sampling Interferometry (PSI)

$$\phi = \tan^{-1} \frac{-C}{B} = \tan^{-1} \frac{\sum I\_n \sin \nu\_{0n}}{\sum I\_n \cos \nu\_{0n}} \,. \tag{23}$$

In general, a minimum of three samples are needed to know the phase because there are three unknowns in the general interference equation Eq. (9): *a* , *b* and . However a better accuracy can be guaranteed with more than three shifts.

### **2.2.1 Three steps technique**

Since we need a minimum of three interferograms to reconstruct the wavefront, the phase can be calculated with a phase-shift of 2 per exposition. The three intensity measurements can be expressed as (Wyant, Koliopoulus, Bhushan and George 1984)

$$I\_1 = a + b\cos\left(\phi + \frac{1}{4}\pi\right),\tag{24}$$

$$I\_2 = a + b\cos\left(\phi + \frac{3}{4}\pi\right) \tag{25}$$

$$I\_{\rm s} = a + b \cos \left( \phi + \frac{5}{4} \pi \right),\tag{26}$$

the phase at each point is

$$\phi = \tan^{-1} \left( \frac{I\_3 - I\_2}{I\_1 - I\_2} \right) \cdot \tag{27}$$

## **2.2.2 Four steps technique**

A common algorithm used to calculate the phase is the four steps method (Wyant 1982). In this case the four intensity measurements can be expressed as

$$I\_1 = a + b\cos\phi\_{\prime} \tag{28}$$

$$I\_z = a + b\cos\left(\phi + \frac{1}{2}\pi\right) = a - b\sin\phi\_\text{\,s} \tag{29}$$

<sup>3</sup> *baI ba* cos)cos( , (30)

$$I\_4 = a + b\cos\left(\phi + \frac{3}{2}\pi\right) = a + b\sin\phi\,\tag{31}$$

and the phase at each point is

178 Interferometry – Research and Applications in Science and Technology

It can be shown that based on a least-squares fit that B and C meet the next equations in an

; 0

A combination of Eq. (21) and Eqs. (22) can give us the basic equation of the Phase Sampling

*C I B I*

Since we need a minimum of three interferograms to reconstruct the wavefront, the phase

measurements can be expressed as (Wyant, Koliopoulus, Bhushan and George 1984)

1 1 0

1 <sup>2</sup> sin *N*

sin

cos *n n n n*

 

 

 

4 5

> 

 

A common algorithm used to calculate the phase is the four steps method (Wyant 1982). In

<sup>1</sup> *baI* cos

2 1 cos <sup>2</sup> *baI ba* 

this case the four intensity measurements can be expressed as

21 <sup>1</sup> <sup>23</sup> tan *II II*

sin

4 3

4 1 *n C I N*

*n n*

0

. (22)

. (23)

2 per exposition. The three intensity

cos <sup>1</sup> *baI* , (24)

cos <sup>2</sup> *baI* , (25)

cos <sup>3</sup> *baI* , (26)

. (27)

, (29)

, (28)

because there are

. However a better

0

cos

*n n*

tan tan

1

In general, a minimum of three samples are needed to know the phase

three unknowns in the general interference equation Eq. (9): *a* , *b* and

*N*

*n B I N*

2

accuracy can be guaranteed with more than three shifts.

can be calculated with a phase-shift of

analytical form

Interferometry (PSI)

**2.2.1 Three steps technique** 

the phase at each point is

**2.2.2 Four steps technique** 

$$\phi = \tan^{-1} \left( \frac{I\_4 - I\_2}{I\_1 - I\_3} \right) \,\, . \tag{32}$$

## **2.2.3 The Fourier transform method**

The deformations of the wavefront in an interferogram can also be calculated by a method that uses the Fourier transform. This method was originally proposed Takeda *et al.* (1982) using the Fourier transform in one dimension along an scanned line. Later Macy (1983) extended the Takeda's method in two dimensions by adding the information of multiple scanned lines and obtaining slices of the phase in two dimensions. Bone *et al* (1986) extended the Macy's work by applying the Fourier transform in two dimensions.

Once the interferogram is obtained, its Fourier transform is calculated, by doing this an image in the Fourier space as in Figure 4a) is obtained, then one lateral spectrum is taken and filtered by means of a layer so that all the irradiance values outside the layer will be multiplied by cero, after that this spectrum is translated to the origin and its Fourier transform is obtained giving by this the resulting wavefront under test.

To describe mathematically this process, the general equation for the irradiance is taken (Eq. (9)) expressing the cosine as a complex exponential

$$I(\mathbf{x}, \mathbf{y}) = a(\mathbf{x}, \mathbf{y}) + c(\mathbf{x}, \mathbf{y})e^{-i\boldsymbol{\nu}} + c^\dagger(\mathbf{x}, \mathbf{y})e^{i\boldsymbol{\nu}},\tag{33}$$

being ),( )21(),( *yxi beyxc* and the phase of the form *x*<sup>0</sup> 2 , where 0 is known as the spatial-carrier frequency

Applying the Fourier transform to Eq. (33)

$$\widetilde{\mathbb{T}}(\mu,\nu) = \mathfrak{A}(\mu,\nu) + \mathfrak{E}(\mu + \mu\_0,\nu) + \mathfrak{E}^\*(\mu - \mu\_0,\nu) \,. \tag{34}$$

where ~*a* , ~*c* y ~\* *c* are complex Fourier amplitudes. By means of a digital filtering one lateral spectrum is isolated by using a filtering window and then translated to the origin by doing 0 <sup>0</sup> , as shown in Figure 4b).

Obtaining the inverse Fourier transform of *c*(,) we have ),( )21(),( *yxi beyxc* , therefore the resulting phase is ),(Re ),(Im tan),( <sup>1</sup> *yxc yxc yx* .

### **2.4 Unwrapping the phase**

The calculated phase will present discontinuities because it is obtained by using the inverse tangent function Eq. (23). Because the inverse tangent is a multivalued function, the solution

Phase-Shifting Interferometry by Amplitude Modulation 181

As it was described in the previous section, many proposed techniques for phase-shifting interferometry (PSI) are based on the interference of two waves where just one interference term is present and the phase shift is done with a constant phase difference between them. In the present chapter a new method for phase-shifting based on the amplitude variation of the field in a scheme of a three beam interferometer is widely discussed. In that interferometer, two beams will be considered as the reference beam and the other beam as the probe beam. The expression for the irradiance due to the interference of those beams will

have three interference terms, however due to a constant phase difference of

introduced between the reference beams, one interference term will be canceled and the two remaining will be put in quadrature. Because of this and applying some trigonometric identities to the resulting pattern it is possible to model it mathematically as a two wave interferometer where the interference term will contain an additional phase that depends on the amplitude variations of the reference beam. Since the phase-shift depends on the variation of the amplitude of the fields, the visibility may not remain constant, however it will be shown that if the amplitudes are seen as an ordered pair over an arc segment in the first quadrant of a circle whose radius is the square root of the addition of the squared

reference beams, to overcome this difficulty it is necessary an analysis for a general case where the phase difference between the reference beams is arbitrary; it will be shown that despite that the conditions for the particular case are not obtained it is still possible to generate phase-shifting by means of the amplitude variations of the fields and to keep a

Let's have three waves interfering at any point in space, which are linearly polarized at the

)exp( *nn <sup>n</sup> iAE*

contains the wavefront. According to Eq. (2) the interference of these three waves at any point in space is the addition of the three fields, being the irradiance according to Eq. (3)

in which three interference terms, the background and the modulation light that is given by the addition of the intensities of each wave are present. Let's consider the first and third wave as the reference wave, and the second wave as the probe wave. By simplicity the next

<sup>21</sup> <sup>21</sup> <sup>31</sup> <sup>31</sup> <sup>32</sup> <sup>32</sup> )cos(2)cos(2)cos(2 <sup>2</sup>

2 is

2 between the

*<sup>n</sup>* is the phase that

, (38)

, (36)

*EEEI* <sup>221</sup> , (37)

*AA*

**3. PSI by amplitude modulation** 

amplitudes it is possible to keep a constant visibility.

same plane and traveling in the *z* direction

<sup>2</sup>

3

2 2

<sup>1</sup> *AAAAAI*

**3.1 Ideal case** 

doing the math in Eq. (37)

2

conditions will be chosen

But it could be difficult to get experimentally a phase difference of

with 3 *n* ,2,1 , being *An* the amplitude considered as nonnegative real,

*AA*

constant visibility, those amplitudes must be seen as ordered pairs over ellipses.

Fig. 4. a) Intensity distribution obtained with the FT of the interference pattern, b) spectrum translated to the origin.

for is a saw tooth function (Figure 5a), where the discontinuities occur every time changes by 2 . If increases, the slope of the function is positive and vice versa if the phase decreases. The final step in the process of measuring the fringe pattern is to unwrap the phase along a line or path counting the discontinuities at 2 and adding 2 each time the angle of the phase jumps from 2 to cero and subtracting 2 if the angle changes from cero to 2 . Figure 5b) shows the dates from Figure 5a) after the unwrapping. The key of a trustable unwrapping algorithm is its capacity of detecting the discontinuities with high accuracy

Fig. 5. a) Discontinuities of a wrapped phase, b) unwrapped phase. The basic principle of the phase unwrapping is to "integrate" the wrapped phase (in units of 2 ) along a data line. The phase gradient is calculated for each pixel

$$
\Delta \phi = \phi\_n - \phi\_{n-1} \tag{35}
$$

where *n* is the pixel number. If exceeds a certain threshold like , then a discontinuity is assumed. This phase discontinuity is fixed by adding or subtracting 2 depending on the sign of , Itoh (1982). The most common principle used to fix those phase discontinuities is based on the fact that the phase difference among any pair of points measured by integrating the phase along a path between these points is independent from the chosen path, provided they don't pass through a phase discontinuity.

## **3. PSI by amplitude modulation**

180 Interferometry – Research and Applications in Science and Technology

Fig. 4. a) Intensity distribution obtained with the FT of the interference pattern, b) spectrum

is a saw tooth function (Figure 5a), where the discontinuities occur every time

phase decreases. The final step in the process of measuring the fringe pattern is to unwrap

trustable unwrapping algorithm is its capacity of detecting the discontinuities with high

Fig. 5. a) Discontinuities of a wrapped phase, b) unwrapped phase. The basic principle of the phase unwrapping is to "integrate" the wrapped phase (in units of 2 ) along a data

> 

is assumed. This phase discontinuity is fixed by adding or subtracting 2 depending on the

is based on the fact that the phase difference among any pair of points measured by integrating the phase along a path between these points is independent from the chosen

exceeds a certain threshold like

, Itoh (1982). The most common principle used to fix those phase discontinuities

path, provided they don't pass through a phase discontinuity.

the phase along a line or path counting the discontinuities at 2 and adding

increases, the slope of the function is positive and vice versa if the

 to cero and subtracting 2 if the angle changes from 

 2 each time 

5a) after the unwrapping. The key of a

, then a discontinuity

*nn* <sup>1</sup> , (35)

translated to the origin.

 cero to 2 . Figure 5b) shows the dates from Figure

line. The phase gradient is calculated for each pixel

where *n* is the pixel number. If

the angle of the phase jumps from 2

changes by 2 . If

accuracy

sign of

for  As it was described in the previous section, many proposed techniques for phase-shifting interferometry (PSI) are based on the interference of two waves where just one interference term is present and the phase shift is done with a constant phase difference between them.

In the present chapter a new method for phase-shifting based on the amplitude variation of the field in a scheme of a three beam interferometer is widely discussed. In that interferometer, two beams will be considered as the reference beam and the other beam as the probe beam. The expression for the irradiance due to the interference of those beams will have three interference terms, however due to a constant phase difference of 2 is introduced between the reference beams, one interference term will be canceled and the two remaining will be put in quadrature. Because of this and applying some trigonometric identities to the resulting pattern it is possible to model it mathematically as a two wave interferometer where the interference term will contain an additional phase that depends on the amplitude variations of the reference beam. Since the phase-shift depends on the variation of the amplitude of the fields, the visibility may not remain constant, however it will be shown that if the amplitudes are seen as an ordered pair over an arc segment in the first quadrant of a circle whose radius is the square root of the addition of the squared amplitudes it is possible to keep a constant visibility.

But it could be difficult to get experimentally a phase difference of 2 between the reference beams, to overcome this difficulty it is necessary an analysis for a general case where the phase difference between the reference beams is arbitrary; it will be shown that despite that the conditions for the particular case are not obtained it is still possible to generate phase-shifting by means of the amplitude variations of the fields and to keep a constant visibility, those amplitudes must be seen as ordered pairs over ellipses.

## **3.1 Ideal case**

Let's have three waves interfering at any point in space, which are linearly polarized at the same plane and traveling in the *z* direction

$$E\_u = A\_n \exp(i\phi\_u)\_\prime \tag{36}$$

with 3 *n* ,2,1 , being *An* the amplitude considered as nonnegative real, *<sup>n</sup>* is the phase that contains the wavefront. According to Eq. (2) the interference of these three waves at any point in space is the addition of the three fields, being the irradiance according to Eq. (3)

$$I = \left| E\_1 + E\_2 + E\_2 \right|^2,\tag{37}$$

doing the math in Eq. (37)

$$I = A\_1^{\ 2} + A\_2^{\ 2} + A\_3^{\ 2} + 2A\_1 A\_2 \cos(\phi\_1 - \phi\_2) + 2A\_1 A\_3 \cos(\phi\_1 - \phi\_3) + 2A\_2 A\_3 \cos(\phi\_2 - \phi\_3) \, \prime \tag{38}$$

in which three interference terms, the background and the modulation light that is given by the addition of the intensities of each wave are present. Let's consider the first and third wave as the reference wave, and the second wave as the probe wave. By simplicity the next conditions will be chosen

$$
\phi\_1 = 0 \; ; \; \phi\_2 = \phi \quad \text{and} \; \; \phi\_3 = \pi/2 \; . \tag{39}
$$

Phase-Shifting Interferometry by Amplitude Modulation 183

Fig. 6. Amplitudes of first and third wave in a reference system. Any point in the arc keeps

wave could be achieved by means of a retarder plate of a quarter wavelenght. The amplitude variations can be done by using neutral density filters or by using the diffraction orders generated by a grating (for example a Ronchi ruling), where each order is attenuated

To prove the viability of the present proposal, we have carried out a numerical simulation in

<sup>2</sup> *AA <sup>r</sup>* 1 ; 1,0, *AA* <sup>31</sup> ; <sup>22</sup>

such a form guarantees that the interference pattern will have a maximum contrast.

*AE* <sup>11</sup> ; )exp( <sup>2</sup> *iE*

*I*

Figure 7a) shows the values of the amplitudes needed for a phase-shifting of three steps *N* 3 . The left graphic indicates three points over the arc )0,1( , )22,22( , )1,0( which

remains constant. Figure 7b) shows the interferograms modeled by Eq. (49) for the phase steps indicated in Figure 7a). A bar diagram above each interferogram indicates the amplitude levels of the waves needed for each phase-shift. In a very similar way Figure 8

; <sup>3</sup> *AA <sup>r</sup>* sin

. (46)

2 between the first and third

*yx* , (47)

<sup>33</sup> *iAE* , (48)

2,4,0 respectively, while *Ar*

)cos(22 . (49)

is varied as required for PSI.

In this case the amplitudes are in quadrature, *A*1 and *A*3 are given by

which for simplicity, the following considerations have been assumed

In a possible experimental setup the phase difference of

<sup>1</sup> *AA <sup>r</sup>* cos

*Ar* constant, and

in accord with the sinc function.

being the interference patter

Therefore, the three fields could take the form

are the amplitude variations to obtain the phase steps

show the phase-shifting for the case of four steps 4 *N* .

The waves *E*1 and *E*3 will be chosen as homogeneous no tilted plane waves with a phase difference between them of 2 , is the phase of the object contained in the second wave. Substituting Eq. (39) into Eq. (38) we have

$$I = A\_1^{\ \ 2} + A\_2^{\ \ 2} + A\_3^{\ \ 2} + 2A\_1 A\_2 \cos \phi + 2A\_2 A\_3 \sin \phi \ . \tag{40}$$

In this equation it can be observed that one of the three interference terms has been cancelled and the two remaining are now in quadrature. Regrouping the above equation

$$I = A\_1^{\ 2} + A\_2^{\ 2} + A\_3^{\ 2} + 2A\_2 \{ A\_1 \cos \phi + A\_3 \sin \phi \} \, \tag{41}$$

that can be rewritten as

$$I = A\_1^{\;\;\;2} + A\_2^{\;\;2} + A\_3^{\;\;2} + 2A\_2 \sqrt{A\_1^{\;\;2} + A\_3^{\;\;2}} \left( \frac{A\_1}{\sqrt{A\_1^{\;\;2} + A\_3^{\;\;2}}} \cos \phi + \frac{A\_3}{\sqrt{A\_1^{\;\;2} + A\_3^{\;\;2}}} \sin \phi \right), \tag{42}$$

$$I = A\_r^{(2)} + A\_2^{(2)} + 2A\_r A\_2 \{ \cos \phi \cos \psi + \sin \phi \sin \psi \} \, , \tag{43}$$

where <sup>2</sup> 3 2 <sup>11</sup> cos *AAA* and <sup>2</sup> 3 2 <sup>13</sup> sin*AAA* . Eq. (43) can be expressed as

$$I = A\_r^{\ 2} + A\_2^{\ 2} + 2A\_r A\_2 \cos(\phi - \psi) \tag{44}$$

which as it was indicated in Eq. (9) is the expression for a fringe pattern due to the interference of two waves, being

$$\left|A\_{\text{v}}\right|^{2} = \left|A\_{\text{v}}\right|^{2} + \left|A\_{\text{v}}\right|^{2} \quad \tan \psi = \frac{A\_{\text{v}}}{A\_{\text{v}}} \tag{45}$$

where *Ar* is the reference amplitude and is an additional phase, both of them depending on the amplitude variations of the first wave *A*1 and the third wave *A*<sup>3</sup> . It can be observed from this relationship that it is possible to generate a phase-shifting with the variations of those amplitudes; however because this also affects *Ar* , there may be a change in the visibility of the fringes, so it can be thought that it would not be possible to apply the PSI techniques for the phase extraction (because one important condition to apply the PSI phase extraction techniques is that the visibility remains constant). One way to keep *Ar* constant is to consider the amplitudes of waves one and three as an ordered pair ),( *AA* <sup>31</sup> which must be contained over an arc segment of radius *Ar* in the first quadrant (because the amplitudes are considered positive), hence *Ar* will be within the range 2,0 , as can be seen in Figure 6. However it is possible to generate a negative amplitude modulation if the phase difference in the reference waves is , having negative real amplitudes, by this the range of will be 0 and the phase extraction techniques 2, in PSI could be applied without any modification.

The waves *E*1 and *E*3 will be chosen as homogeneous no tilted plane waves with a phase

21

In this equation it can be observed that one of the three interference terms has been cancelled and the two remaining are now in quadrature. Regrouping the above equation

 and 2 3 

<sup>32</sup> sin2cos2

<sup>1</sup> *AAAAAI AA* . (40)

*AAA* . Eq. (43) can be expressed as

<sup>2</sup>

. (39)

, (41)

, (43)

, (45)

is an additional phase, both of them depending

, having negative real amplitudes, by this the range of

sin

*AA A*

2 1 3  

 2 3

, (44)

2,0 , as can be seen in Figure

is the phase of the object contained in the second wave.

0 <sup>1</sup> ; 2 

3

3

<sup>1</sup> *AAAAAAI*

 

3

<sup>13</sup> sin

*<sup>r</sup> rAAAAI*

which as it was indicated in Eq. (9) is the expression for a fringe pattern due to the

2 3

on the amplitude variations of the first wave *A*1 and the third wave *A*<sup>3</sup> . It can be observed from this relationship that it is possible to generate a phase-shifting with the variations of those amplitudes; however because this also affects *Ar* , there may be a change in the visibility of the fringes, so it can be thought that it would not be possible to apply the PSI techniques for the phase extraction (because one important condition to apply the PSI phase extraction techniques is that the visibility remains constant). One way to keep *Ar* constant is to consider the amplitudes of waves one and three as an ordered pair ),( *AA* <sup>31</sup> which must be contained over an arc segment of radius *Ar* in the first quadrant (because the amplitudes

6. However it is possible to generate a negative amplitude modulation if the phase

in PSI could be applied without any

; 1 <sup>3</sup> tan *A A*

2

2 1

*AA <sup>A</sup> AAAAAAI* , (42)

2

2 1 2 1

2 3

3

2 2

) <sup>12</sup> cos(2 <sup>3</sup> sin <sup>2</sup>

2

2 3

3

2

2 2

2

2 1

where <sup>2</sup>

<sup>11</sup> cos

interference of two waves, being

where *Ar* is the reference amplitude and

difference in the reference waves is

modification.

2 2

2 cos

 ) <sup>2</sup> sinsincos(cos2 <sup>2</sup> 2

*AAA* and <sup>2</sup>

2

2 *<sup>r</sup> AAA*

<sup>2</sup> )cos(2 <sup>2</sup>

are considered positive), hence *Ar* will be within the range

will be 0 and the phase extraction techniques 2,

2 12

*<sup>r</sup> rAAAAI*

 2 , 

2

difference between them of

that can be rewritten as

Substituting Eq. (39) into Eq. (38) we have

Fig. 6. Amplitudes of first and third wave in a reference system. Any point in the arc keeps *Ar* constant, and is varied as required for PSI.

In this case the amplitudes are in quadrature, *A*1 and *A*3 are given by

$$A\_1 = A\_r \cos \psi \; ; A\_3 = A\_r \sin \psi \; . \tag{46}$$

In a possible experimental setup the phase difference of 2 between the first and third wave could be achieved by means of a retarder plate of a quarter wavelenght. The amplitude variations can be done by using neutral density filters or by using the diffraction orders generated by a grating (for example a Ronchi ruling), where each order is attenuated in accord with the sinc function.

To prove the viability of the present proposal, we have carried out a numerical simulation in which for simplicity, the following considerations have been assumed

$$A\_2 = A\_r = \mathbf{1};\ A\_1, A\_3 \in \left[0, \mathbf{1}\right];\ \phi = \mathbf{x}^2 + \mathbf{y}^2,\tag{47}$$

such a form guarantees that the interference pattern will have a maximum contrast. Therefore, the three fields could take the form

$$E\_1 = A\_1 \; \text{'} \; E\_2 = \exp(\text{i}\,\phi) \; E\_3 = \text{i}A\_3 \; \text{'} \tag{48}$$

being the interference patter

$$I = 2 + 2\cos(\phi - \psi) \,. \tag{49}$$

Figure 7a) shows the values of the amplitudes needed for a phase-shifting of three steps *N* 3 . The left graphic indicates three points over the arc )0,1( , )22,22( , )1,0( which are the amplitude variations to obtain the phase steps 2,4,0 respectively, while *Ar* remains constant. Figure 7b) shows the interferograms modeled by Eq. (49) for the phase steps indicated in Figure 7a). A bar diagram above each interferogram indicates the amplitude levels of the waves needed for each phase-shift. In a very similar way Figure 8 show the phase-shifting for the case of four steps 4 *N* .

Phase-Shifting Interferometry by Amplitude Modulation 185

Fig. 8. Amplitude variations for a PSI of four steps: a) a point over the arc yields a phaseshift while the reference amplitude is kept constant; b) interferograms shifted in phase by

Fig. 9. Interference patterns with different visibility a) Phase-shifted interferograms due to the amplitudes shown in b); b) points over arcs and straight lines that give phase-shift

the amplitude variations indicated in a).

keeping in some cases a constant visibility.

Fig. 7. Amplitude variations for a PSI of three steps: a) a point on the arc yields a phase-shift while the reference amplitude is kept constant; b) interferogram shifted in phase by the amplitude variations indicated in a).

The columns of Figure 9a) show simulated interference pattern for different phase-shifts, which are obtained when the amplitude values *A*1 and *A*3 are over straight lines that form an angle respecting to the axis *A*<sup>1</sup> , Figure 10b); by doing this, the visibility of the patterns remains constant, but to get a maximum visibility the value of the amplitude *A*2 must be equal to the value of *Ar* , which can be observed in the first row of Figure 9a), where the values of *A*1 and *A*3 are over an arc segment of radius *Ar* 1 , Figure 9b). If we have different values of but its corresponding amplitudes are not over the same arc segment, the interference patterns will not have a constant visibility, what can be seen in Figure 9a) if for each value of we take a different row (indicated by different symbols).

Fig. 7. Amplitude variations for a PSI of three steps: a) a point on the arc yields a phase-shift while the reference amplitude is kept constant; b) interferogram shifted in phase by the

The columns of Figure 9a) show simulated interference pattern for different phase-shifts, which are obtained when the amplitude values *A*1 and *A*3 are over straight lines that form

remains constant, but to get a maximum visibility the value of the amplitude *A*2 must be equal to the value of *Ar* , which can be observed in the first row of Figure 9a), where the values of *A*1 and *A*3 are over an arc segment of radius *Ar* 1 , Figure 9b). If we have

the interference patterns will not have a constant visibility, what can be seen in Figure 9a) if

we take a different row (indicated by different symbols).

respecting to the axis *A*<sup>1</sup> , Figure 10b); by doing this, the visibility of the patterns

but its corresponding amplitudes are not over the same arc segment,

amplitude variations indicated in a).

an angle

different values of

for each value of

Fig. 8. Amplitude variations for a PSI of four steps: a) a point over the arc yields a phaseshift while the reference amplitude is kept constant; b) interferograms shifted in phase by the amplitude variations indicated in a).

Fig. 9. Interference patterns with different visibility a) Phase-shifted interferograms due to the amplitudes shown in b); b) points over arcs and straight lines that give phase-shift keeping in some cases a constant visibility.

Phase-Shifting Interferometry by Amplitude Modulation 187

<sup>2</sup> *<sup>r</sup> AAAAI <sup>r</sup>*

sin tan

shift; this can be achieved if the amplitude *Ar* in la Eq. (59) remains constant while

2 2 3

2 2

*a x*

<sup>2</sup> 2tan <sup>22</sup> *ba*

*A A*

*<sup>r</sup> <sup>r</sup> Ar*

2 2

sin 

2

Being in our case *ba* , so Eq. (61) can be written as in Eq.(62), where

 

2 3

*<sup>r</sup> AAAAA* cos2

 *AA*

To apply the PSI techniques, the visibility in Eq. (58) must remain constant for each phase-

with the changes of *A*1 and *A*<sup>3</sup> . To define the behavior of the amplitudes which satisfy the

cos2

cos

<sup>2</sup> sin 

*ab*

is the inclination angle of the ellipse, *a* is the maximum value of the ellipse over

*b y* 2

; *Ar*

It has been deduced the general expression of a fringe pattern of two waves where *Ar* is

*<sup>A</sup>*<sup>3</sup> <sup>2</sup> sin sin

31 2

2 .

31 2 3 2 cos

> 2 2

2

*ab xy*

depend on the amplitude variations of *A*1 and *A*3 as well as

2

, (62)

, (63)

*Ar <sup>a</sup>* ; (64)

2 in Eq. (60) a family of ellipses inclined to an

2 31

cossin2sin

*<sup>A</sup>* , (61)

*AA*

is an additional phase, both given by

*<sup>A</sup>* , (60)

2 2

, (57)

. (58)

; (59)

varies

*Ar AA* <sup>31</sup> <sup>2</sup> cos

> 2 1

cos

equivalent to the reference amplitude and

it can be seen that *Ar* and

the phase difference between the first and third wave

condition to keep *Ar* constant, Eq. (59) will be rewritten as

2 2 1

*A*

which can take the form of the equation of an ellipse

the *x* axis and *b* is the maximum value over the *y* axis.

2

<sup>2</sup> sin sin

substituting the values of *a* , *b* and

will be obtained

such that

where 

angle 

 ) <sup>2</sup> cos(2 <sup>1</sup> 22

2

where

## **3.2 General case**

As demonstrated in the previous section, it is possible to have a new method of PSI by means of the field amplitude variations based on the scheme of a three beam interferometer modeled as a two beam interferometer, where the reference beam and a constant phase term (used to generate the phase-shift) were given in function of the two reference beams. Due to a phase difference of 2 between the reference beams one of the interference terms was canceled, leaving the two remaining in quadrature.

However under experimental conditions it is not always possible to obtain a phase difference of 2 between the reference beams. Despite of this it will be shown that it is still possible to generate phase-shift by means of the amplitude variation of the fields, where now to keep a constant visibility the amplitudes must be seen as ordered pairs over an ellipse instead of a circle, extending with this the range of the phase-shifting until ,0 instead of 2,0 .

Let's consider again three linearly polarized waves at the same plane traveling on *z* direction as shown in Eq. (36), whose irradiance can be expressed as in Eq. (37)

$$I = A\_1^{\ 2} + A\_2^{\ 2} + A\_3^{\ 2} + 2A\_1 A\_2 \cos(\phi\_1 - \phi\_2) + 2A\_1 A\_3 \cos(\phi\_1 - \phi\_3) + 2A\_2 A\_3 \cos(\phi\_2 - \phi\_3) \ , \tag{50}$$

where the intensities of the three waves are present in the background light and also in the three interference terms. For the general case, the phases that will be considered are

$$
\phi\_1 \neq 0 \; ; \; \phi\_2 = \phi\_1 + \Delta\phi\_1 + \phi \; ; \; \text{and} \; \; \phi\_3 = \phi\_1 + \Delta\phi\_2 \; ; \tag{51}
$$

<sup>1</sup> is a constant phase difference between the first and second wave, 2 is a constant phase difference between the first and third wave; therefore between the second and third wave will exist a phase difference of <sup>2</sup> <sup>1</sup> . In summary, in the absence of a phase object it will be considered a phase difference between each pair of waves. It is important to notice that when <sup>1</sup> 0 and 2 2 the general case studied in section 3.1 will be obtained. Substituting Eq. (51) into Eq. (50) we have

$$I = A\_1^{\;\;\;2} + A\_2^{\;\;\;2} + A\_3^{\;\;\;2} + 2A\_1A\_2\cos(\phi + \Delta\phi\_1) + 2A\_1A\_3\cos(\Delta\phi\_2) + 2A\_2A\_3\cos(\phi + \Delta\phi\_1 - \Delta\phi\_2) \,,\tag{52}$$

$$I = A\_1^{\ 2} + A\_2^{\ 2} + A\_3^{\ 2} + 2A\_1 A\_3 \cos \Delta \phi\_2 + 2A\_2 \left[ A\_1 \cos(\phi + \Delta \phi\_1) + A\_3 \cos(\phi + \Delta \phi\_1 - \Delta \phi\_2) \right]\_{\ }\tag{53}$$

$$I = A\_1^{\ 2} + A\_2^{\ 2} + A\_3^{\ 2} + 2A\_1 A\_3 \cos \Delta \phi\_2 + 2A\_2 \left[ A\_1 \cos(\phi + \Delta \phi\_1) + \Delta \phi\_2 \right] + \dotsb$$

$$+ A\_3 \cos \Delta \phi\_2 \cos(\phi + \Delta \phi\_1) + A\_3 \sin \Delta \phi\_2 \sin(\phi + \Delta \phi\_1) \Big|\_{\phi} \tag{54}$$

(2cos2 )cos <sup>31</sup> <sup>2</sup> <sup>312</sup> <sup>2</sup> 2 3 2 2 2 <sup>1</sup> *AAAAAI AAA* cos( )sin(sin) *A*<sup>31</sup> <sup>2</sup> <sup>1</sup> , (55)

$$I = A\_{z}^{\ 2} + A\_{r}^{\ 2} + \Im A\_{z} A\_{r} \left[ \cos \psi \cos(\phi + \Lambda \phi\_{1}) + \sin \psi \sin(\phi + \Lambda \phi\_{1}) \right],\tag{56}$$

where

186 Interferometry – Research and Applications in Science and Technology

As demonstrated in the previous section, it is possible to have a new method of PSI by means of the field amplitude variations based on the scheme of a three beam interferometer modeled as a two beam interferometer, where the reference beam and a constant phase term (used to generate the phase-shift) were given in function of the two reference beams. Due to

However under experimental conditions it is not always possible to obtain a phase

possible to generate phase-shift by means of the amplitude variation of the fields, where now to keep a constant visibility the amplitudes must be seen as ordered pairs over an ellipse instead of a circle, extending with this the range of the phase-shifting until ,0

Let's consider again three linearly polarized waves at the same plane traveling on *z*

where the intensities of the three waves are present in the background light and also in the

phase difference between the first and third wave; therefore between the second and third

it will be considered a phase difference between each pair of waves. It is important to notice

<sup>31</sup> <sup>2</sup> <sup>12</sup> cos(2cos2 <sup>1</sup> ) <sup>2</sup>

*A A* 3 2 13 2 1 cos cos( ) sin sin( )

*A*<sup>31</sup> <sup>2</sup>

<sup>2</sup> *<sup>r</sup> AAAAI <sup>r</sup>*

*AA*

; and

direction as shown in Eq. (36), whose irradiance can be expressed as in Eq. (37)

*AA*

three interference terms. For the general case, the phases that will be considered are

<sup>1</sup> 0 ; 112 

<sup>1</sup> is a constant phase difference between the first and second wave,

<sup>2</sup> 

*AA*

> *AA*

3

2 3

2 2

2 2

<sup>1</sup> *AAAAAI*

<sup>1</sup> *AAAAAI*

 cos( )sin(sin) 

2 between the reference beams one of the interference terms was

2 between the reference beams. Despite of this it will be shown that it is still

<sup>21</sup> <sup>21</sup> <sup>31</sup> <sup>31</sup> <sup>32</sup> <sup>32</sup> )cos(2)cos(2)cos(2 <sup>2</sup>

2 the general case studied in section 3.1 will be obtained.

cos(2 cos(2)cos(2) ) <sup>21</sup> <sup>1</sup> <sup>31</sup> <sup>2</sup> <sup>32</sup> <sup>21</sup>

cos(2cos2 cos() ) <sup>31</sup> <sup>2</sup> <sup>12</sup> <sup>31</sup> <sup>21</sup>

 

(2cos2 )cos <sup>31</sup> <sup>2</sup> <sup>312</sup> <sup>2</sup>

cos(cos2 )sin(sin) <sup>2</sup> <sup>1</sup> <sup>1</sup>

 

*AAA*

*AA*

<sup>1</sup> . In summary, in the absence of a phase object

*AA*

*A*

<sup>213</sup> , (51)

, (52)

, (54)

, (53)

<sup>1</sup> , (55)

, (56)

, (50)

2 is a constant

**3.2 General case** 

a phase difference of

2,0 .

2

2

2

difference of

instead of

that when

3

2 

2 2

wave will exist a phase difference of

<sup>1</sup> 0 and

2 2

<sup>1</sup> *AAAAAI*

2 2

<sup>1</sup> *AAAAAI*

Substituting Eq. (51) into Eq. (50) we have

2 3

> 2 3

> > 2

2

22

<sup>1</sup> *AAAAAI*

canceled, leaving the two remaining in quadrature.

$$\cos\varphi = \frac{A\_1 + A\_3 \cos\Delta\phi\_2}{A\_r}; \sin\varphi = \frac{A\_3 \sin\Delta\phi\_2}{A\_r} \tag{57}$$

$$I = A\_2 \, ^2 + A\_r \, ^2 + 2A\_2 A\_r \cos(\phi + \Lambda \phi\_l - \psi) \,. \tag{58}$$

It has been deduced the general expression of a fringe pattern of two waves where *Ar* is equivalent to the reference amplitude and is an additional phase, both given by

$$\left|A\_r\right|^2 = A\_1^{-2} + A\_3^{-2} + 2A\_1A\_3\cos\Delta\phi\_2;\tag{59}$$

$$\tan \psi = \frac{A\_3 \sin \Delta \phi\_2}{A\_1 + A\_3 \cos \Delta \phi\_2} \tag{60}$$

it can be seen that *Ar* and depend on the amplitude variations of *A*1 and *A*3 as well as the phase difference between the first and third wave 2 .

To apply the PSI techniques, the visibility in Eq. (58) must remain constant for each phaseshift; this can be achieved if the amplitude *Ar* in la Eq. (59) remains constant while varies with the changes of *A*1 and *A*<sup>3</sup> . To define the behavior of the amplitudes which satisfy the condition to keep *Ar* constant, Eq. (59) will be rewritten as

$$\sin^2 \Delta \phi\_2 = \frac{A\_1^{1^2}}{A\_r^2} \sin^2 \Delta \phi\_2 + \frac{A\_3^{1^2}}{A\_r^2} \sin^2 \Delta \phi\_2 + 2 \frac{A\_1 A\_3}{A\_r^2} \sin^2 \Delta \phi\_2 \cos \Delta \phi\_2,\tag{61}$$

which can take the form of the equation of an ellipse

$$
\sin^2 \alpha = \frac{\chi^2}{a^2} + \frac{y^2}{b^2} - 2\frac{\chi y}{ab}\cos \alpha \,\,\,\,\,\tag{62}
$$

such that

$$\tan 2\nu = \frac{2ab}{a^2 - b^2} \cos a \,, \tag{63}$$

where is the inclination angle of the ellipse, *a* is the maximum value of the ellipse over the *x* axis and *b* is the maximum value over the *y* axis.

Being in our case *ba* , so Eq. (61) can be written as in Eq.(62), where

$$a = \frac{A\_r}{\sin \Delta \phi\_2};$$

substituting the values of *a* , *b* and 2 in Eq. (60) a family of ellipses inclined to an angle will be obtained

Phase-Shifting Interferometry by Amplitude Modulation 189

shifting. When a certain phase-shift is needed, the amplitudes *A*1 and *A*3 will be given by Eq. (67) which comply the conditions for a constant visibility and they could be seen as points over the arc of an ellipse at the first quadrant as it is shown for several values of

A numerical simulation will be shown in order to prove the viability of the proposal. For

*I*

such a form that the interference pattern will have a maximum contrast, where the

2 <sup>3</sup> tan

 3 1 cos *A*<sup>1</sup> ; sin

Figure 12a) shows the ellipse at the first quadrant for this particular case, which is inclined

12b), and these are indicated with points over the arc corresponding to the ellipse in Figure

reference waves it is still possible to generate phase-shifting with the proposed method, besides that it will not be necessary to use an optical device (as phase retarders) to generate

<sup>2</sup> . The variations of the amplitudes can be done by using neutral filters or also by using the diffraction orders produced by a grating as for example a Ronchi ruling, where each

4 with an ellipticity of *e* 3 . Figure 12b) shows the amplitudes corresponding to the

12 with 8 *k* ,...,1,0 were generated by the amplitudes shown in Figure

sin

32,0 .

Figure 12 shows the interferograms for 9 *N* , being the phase step of

This analysis has the advantage that even if there is a phase difference of

diffraction order is attenuated according to the sinc function.

2 3

2 

; )32exp( <sup>3</sup> *iE*

are given by

31

31 3

*AA A* 

 32 ; <sup>22</sup> 

)cos(22 , (71)

<sup>2</sup> 1 <sup>1</sup> *AAAA* , (72)

 3

. (73)

<sup>2</sup> *<sup>A</sup>*<sup>3</sup> . (74)

 12 ; the

2 between the

2 determines directly the range of the phase-

*yx* , (69)

, (70)

2

It can be observed form Eq. (68) that

simplicity, the next considerations have been taken

therefore the three fields can be expressed as

the interference pattern can be written as

amplitudes *A*<sup>1</sup> , *A*3 and the additional phase

Substituting Eq. (69) in Eq. (67) is gotten

phase shift within the range

 *<sup>k</sup> k* <sup>2</sup> *AA <sup>r</sup>* 1 ;

 <sup>11</sup> 0 ;

*AE* <sup>11</sup> ; )exp( <sup>2</sup> *iE*

in Figure 10.

to 

phase-steps

12a).

$$\tan 2\omega = -\frac{\cos \Delta \phi\_2}{0} = \begin{cases} \text{index} & \Delta \phi\_2 = \pi/2 + n\pi \\ -\infty & \Delta \phi\_2 \in [0, \pi/2) \cup (3\pi/2, 2\pi] \\ +\infty & \Delta \phi\_2 \in (\pi/2, 3\pi/2) \end{cases} \tag{65}$$

therefore can take the next values

$$\boldsymbol{\omega} = \begin{bmatrix} \text{indet}, & \Delta \phi\_2 = \pi f 2 + m \pi \\ -\pi/4, & \Delta \phi\_2 \in \left[0, \pi/2\right) \cup \left(3\pi/2, 2\pi\right] \\ \pi/4, & \Delta \phi\_2 \in \left(\pi/2, 3\pi/2\right) \end{bmatrix} . \tag{66}$$

The amplitudes in Eq. (61) can take the next parametric form

$$A\_1 = \frac{A\_r}{\sin \Delta \phi\_2} \sin(\Delta \phi\_2 - \nu) \text{ ; } A\_3 = \frac{A\_r}{\sin \Delta \phi\_2} \sin \nu \text{ .} \tag{67}$$

whose parameter is the phase-shift within a valid range of

$$\psi \in \begin{cases} \left[ 0, \Lambda \phi\_2 \right] & \Lambda \phi\_2 \in \left( 0, \pi \right) \\ \left[ \Lambda \phi\_2 - \pi, \pi \right] & \Lambda \phi\_2 \in \left( \pi, 2\pi \right) \end{cases} \tag{68}$$

that is obtained considering that the amplitudes *A*1 and *A*3 are positive, therefore they must be in the first quadrant and that 2 *m* being *m* an integer number. However it is possible to have a negative modulation in the amplitude if a phase difference of in the reference waves is properly implemented, thus it is possible to have real negative amplitudes, hence will be within the range of 2,0 , and the known PSI techniques could be applied without modifications.

Fig. 10. Amplitudes of the wave one and three put in a reference system for several values of <sup>2</sup> . a) family of ellipses; any point on any ellipse keeps *Ar* constant while is varied as it is required for PSI, b) amplitudes in a parametric form respecting to corresponding to the ellipses in a).

sin( ) sin <sup>2</sup> 2

*Ar <sup>A</sup>* ;

 

2 2 2

 )2,(,, ,,0 ),0(

2 2 2 2 

that is obtained considering that the amplitudes *A*1 and *A*3 are positive, therefore they

reference waves is properly implemented, thus it is possible to have real negative

Fig. 10. Amplitudes of the wave one and three put in a reference system for several values of

<sup>2</sup> . a) family of ellipses; any point on any ellipse keeps *Ar* constant while

it is required for PSI, b) amplitudes in a parametric form respecting to

possible to have a negative modulation in the amplitude if a phase difference of

will be within the range of 2,0

 

,indet 2

 

 

 

The amplitudes in Eq. (61) can take the next parametric form

1

must be in the first quadrant and that 2 *m*

could be applied without modifications.

whose parameter is the phase-shift

cos 2tan

can take the next values

therefore

amplitudes, hence

the ellipses in a).

0

2

2 2 2

within a valid range of

 

 ,4 23,2 2,232,0 ,4-

 23,2 2,232,0

*m*

 

indet 2

<sup>3</sup> *Ar <sup>A</sup>* , (67)

being *m* an integer number. However it is

, and the known PSI techniques

is varied as

corresponding to

in the

sin

, (68)

sin <sup>2</sup>

, (65)

. (66)

*n*

It can be observed form Eq. (68) that 2 determines directly the range of the phaseshifting. When a certain phase-shift is needed, the amplitudes *A*1 and *A*3 will be given by Eq. (67) which comply the conditions for a constant visibility and they could be seen as points over the arc of an ellipse at the first quadrant as it is shown for several values of 2 in Figure 10.

A numerical simulation will be shown in order to prove the viability of the proposal. For simplicity, the next considerations have been taken

$$A\_2 = A\_r = \mathbf{1} \; ; \; \Delta \phi\_1 = \phi\_1 = 0 \; ; \; \Delta \phi\_2 = 2\pi f \mathbf{\mathcal{S}} \; ; \; \phi = \mathbf{x}^2 + \mathbf{y}^2 \; , \tag{69}$$

therefore the three fields can be expressed as

$$E\_1 = A\_1; \; E\_2 = \exp(i\phi); \; E\_3 = \exp(i\,2\pi/3) \; , \tag{70}$$

the interference pattern can be written as

$$I = 2 + 2\cos(\phi - \psi) \, , \tag{71}$$

such a form that the interference pattern will have a maximum contrast, where the amplitudes *A*<sup>1</sup> , *A*3 and the additional phase are given by

$$\mathbf{1} = \mathbf{A\_1}^2 + \mathbf{A\_3}^2 - \mathbf{A\_1}\mathbf{A\_3} \tag{72}$$

$$
\tan \psi = \frac{\sqrt{3}A\_3}{2A\_1 - A\_3} \,. \tag{73}
$$

Substituting Eq. (69) in Eq. (67) is gotten

$$A\_1 = \cos\psi + \frac{1}{\sqrt{3}}\sin\psi \; ; \; A\_3 = \frac{2}{\sqrt{3}}\sin\psi \; . \tag{74}$$

Figure 12a) shows the ellipse at the first quadrant for this particular case, which is inclined to 4 with an ellipticity of *e* 3 . Figure 12b) shows the amplitudes corresponding to the phase shift within the range 32,0 .

Figure 12 shows the interferograms for 9 *N* , being the phase step of 12 ; the phase-steps *<sup>k</sup> k* 12 with 8 *k* ,...,1,0 were generated by the amplitudes shown in Figure 12b), and these are indicated with points over the arc corresponding to the ellipse in Figure 12a).

This analysis has the advantage that even if there is a phase difference of 2 between the reference waves it is still possible to generate phase-shifting with the proposed method, besides that it will not be necessary to use an optical device (as phase retarders) to generate <sup>2</sup> . The variations of the amplitudes can be done by using neutral filters or also by using the diffraction orders produced by a grating as for example a Ronchi ruling, where each diffraction order is attenuated according to the sinc function.

Phase-Shifting Interferometry by Amplitude Modulation 191

In this chapter have been discussed the phenomenon of optical interference for two waves elliptically polarized, the phase shifting interferometry and the commonly used methods to generate that shifting, as well as the methods for phase extraction. It has been demonstrated with a numerical analysis and a computer simulation the viability of a new method of phase-shifting based on the amplitude variation of two fields considered as the reference beams in a scheme of a three beam interferometer, for which two cases were

A particular case was considered when the phase difference between the reference waves is

 2 , hence one of the three interference terms is cancelled while the two remaining are put in quadrature. To get a constant visibility, the amplitude of the reference waves must be over an arc segment in the first quadrant of a circle whose radius is *Ar* . Due to the amplitudes are considered to be real positive, the phase-shifting will be within the range of

2,0 . But, theoretically this range could be extended until 0 if the amplitudes are 2,

modulated moreover in their negative part, what can be done by an appropriate phase

despite of this the phase-shifting by amplitude modulation is also possible, and it includes

during the PSI application, the amplitudes must be over an ellipse instead of a

phase-stepping range when the amplitudes are modulated no-negative and can also reach a

This work was partially supported by Vicerrectoría de Investigación y Estudios de Posgrado of Benemérita Universidad Autónoma de Puebla under grant MEFC and by Programa de

Bone D. J., Bachor H.-A., and Sandeman R. J., (1986). Fringe-Pattern Analysis Using a 2-D

Bruning J. H., Herriott D. R., Gallagher J. E., Rosenfield D.P., White A. D., and Brangaccio

Bryngdahl O., (1972). Polarization Type Interference Fringe Shifter*, J. Opt. Soc. Am*., Vol. 62,

D. J., (1974). Digital Wavefront Measuring Interferometer for Testing Surfaces and

 4 if 23,2 <sup>2</sup> 

, however when the negative part is taken in account, the range can

In a more general case, the phase differences

considered to be arbitrary and within the range 2,0

radians.

Mejoramiento del Profesorado (México) under grant PROMEP/103.5/09/4544.

Fourier . Transform, *Appl. Opt*., Vol. 25, pp. 1653-1660 Born M. and Wolf E.; Cambridge University Press (1993). *Principles of Optics*

Lenses, *Appl. Opt.,* Vol. 3, pp. 2693-2703

2 between the reference waves was

 or 2,232,0 <sup>2</sup> ,

<sup>2</sup> . In this study it was shown that

2 . Besides in order to keep a constant visibility

2 directly influences in both the ellipticity and the

**4. Conclusions** 

analyzed:

range until of

**6. References** 

reach until

change in the reference waves by

the particular case given when 2

circumference, which is inclined at

respectively, also it was found that

 ,0 

pp. 462-464 .

 2,0 .

**5. Acknowledgment** 

 

Fig. 11. Amplitude variations for a PSI of 9 steps when <sup>2</sup> 32 : a) a point over the arc yields a phase-shifting while the reference amplitude is kept constant; b) amplitudes ),( *AA* 31 for the phase steps indicated in a).

Fig. 12. Interferograms shifted in phase by the amplitude variations indicated in Figure 11. The phase steps are shown from left to right and from top to bottom.

## **4. Conclusions**

190 Interferometry – Research and Applications in Science and Technology

yields a phase-shifting while the reference amplitude is kept constant; b) amplitudes

Fig. 12. Interferograms shifted in phase by the amplitude variations indicated in Figure 11.

The phase steps are shown from left to right and from top to bottom.

<sup>2</sup> 

Fig. 11. Amplitude variations for a PSI of 9 steps when

),( *AA* 31 for the phase steps indicated in a).

32 : a) a point over the arc

In this chapter have been discussed the phenomenon of optical interference for two waves elliptically polarized, the phase shifting interferometry and the commonly used methods to generate that shifting, as well as the methods for phase extraction. It has been demonstrated with a numerical analysis and a computer simulation the viability of a new method of phase-shifting based on the amplitude variation of two fields considered as the reference beams in a scheme of a three beam interferometer, for which two cases were analyzed:

A particular case was considered when the phase difference between the reference waves is 2 , hence one of the three interference terms is cancelled while the two remaining are put in quadrature. To get a constant visibility, the amplitude of the reference waves must be over an arc segment in the first quadrant of a circle whose radius is *Ar* . Due to the amplitudes are considered to be real positive, the phase-shifting will be within the range of 2,0 . But, theoretically this range could be extended until 0 if the amplitudes are 2, modulated moreover in their negative part, what can be done by an appropriate phase change in the reference waves by radians.

In a more general case, the phase differences 2 between the reference waves was considered to be arbitrary and within the range 2,0 <sup>2</sup> . In this study it was shown that despite of this the phase-shifting by amplitude modulation is also possible, and it includes the particular case given when 2 2 . Besides in order to keep a constant visibility during the PSI application, the amplitudes must be over an ellipse instead of a circumference, which is inclined at 4 if 23,2 <sup>2</sup> or 2,232,0 <sup>2</sup> , respectively, also it was found that 2 directly influences in both the ellipticity and the phase-stepping range when the amplitudes are modulated no-negative and can also reach a range until of ,0 , however when the negative part is taken in account, the range can reach until 2,0 .

## **5. Acknowledgment**

This work was partially supported by Vicerrectoría de Investigación y Estudios de Posgrado of Benemérita Universidad Autónoma de Puebla under grant MEFC and by Programa de Mejoramiento del Profesorado (México) under grant PROMEP/103.5/09/4544.

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a Single Phase Step Technique, *Appl. Opt*., Vol. 27, pp. 4362-4364

amplitude modulation, *Opt. Lett.* Vol. 36, pp. 2417-2419

Itoh K., (1982). Analysis of Phase Unwrapping Algorithm, *Appl. Opt*., Vol. 21, pp. 2470 Kothiyal M. P. and Delisle C., (1984). Optical Frequency Shifter for Heterodyne Interferometry Using Counterrotating Wave Plates, *Opt. Lett*., Vol 9, 319-321 Macy William. W., Jr., (1983). Two-Dimensional Fringe Pattern Analysis, *Appl. Opt.,* Vol. 22,

Crane R., (1969). Interference Phase Measurement, *Appl. Opt*., Vol. 8, pp. 538-542

Theory and Error Analysis, *Appl. Opt*., Vol. 22, pp. 2052-2056

Vibration, *Eur. Southern Obs. Conf Proc*., Vol. 1, pp. 561-567

Tunable Laser .Diode, *Opt Lett.,* Vol. 12, pp. 233-235

(Marcel Dekker, New York, 1998), pp. 247-248.

Arbitrary Unknown Phase Steps for Diffraction Objects, *Opt Lett.,* Vol. 29 pp. 183-

Rotational .. Symmetry, *Opt. Commun*., Vol. 17, pp. 43

*Hewlett Packard J*., Vol. 21, pp. 14

Gasvik K. J.; John Wiley & Sons Ltd (1996). *Optical Metrology*

185

*Optical Testing* 

pp. 3898-3901

3068

Vol. 24, pp. 2172-2180

Malacara D.; Wiley (2007). *Optical Shop Testing*

Hech E.; Addison-Wesley ( 1972). *Optics*


**9** 

*México* 

*N***-Shots 2***N***-Phase-Steps Binary** 

Cruz Meneses-Fabian, Gustavo Rodriguez-Zurita

In physical optics, the interference effect consists in superposing two or more optical fields in a region of space, which mathematically results in the vector sum of them, it is certain because of in physical optics the superposition principle is valid. So, when this sum is observed with some optical detector such as a human eye or a CCD camera, the irradiance of the total field is obtained, and it can be understood as the sum of the irradiance from each individual field, known as background light, plus an interference additional term per each pair of fields, which consists of a cosine of the phase difference between the two waves and of a factor given mainly by the product of waves amplitudes known as a modulation light. The total effect shows brilliant and obscure zones known as interference fringes, also called as a fringe pattern, or an interferogram (Born & Wolf, 1993). This effect was first reported by Thomas Young in 1801 (Young, 1804; Shamos, 1959), more late it also was observed by Newton, Fizeau, Michelson, etc. (Hecht, 2002). They designed many optical arranges now known as interferometers in order to interfere two o more waves trying to meet the optimal conditions to have a maximum quality in the fringes. Many studies have demonstrated that the fringe quality, (better known as visibility of interference pattern), as well as the shape and the number of fringes is depending of each parameter in the optical field such as the amplitude, the polarization state, the wavelength, the frequency, the coherence degree, the phase and because of it so many applications among physics and other sciences like biology, medicine, astronomy, etc., or also in engineering have been extensively made (Kreis, 2005). However, an intermediate step between the fringe pattern and the direct application in some topic of interest as the evaluation and processing of this pattern must be realized. In this regarding, many proposal have been amply discussed, for example in interferometry of two waves when the phase difference is the variable to be calculated, one of the techniques more widely used consists of performing consecutively shifts of constant phase between the waves that interfere. Then, for each phase-step a new interferogram is gotten and therefore for *N* steps, *N* interferograms are obtained. Mathematically, a *N*x3 system of equations is formed since the with phase-step the object phase, the background light, and the modulation of light are considered unknown with respect to position and constant with respect to time, is known as phase-shifting interferometry (PSI) (Creath, 1993; Malacara, 2007; Schwider, 1990). The phase shift is introduced by a shifting device, which can be done

**1. Introduction** 

**Grating Interferometry** 

and Noel-Ivan Toto-Arellano *Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla* 


## *N***-Shots 2***N***-Phase-Steps Binary Grating Interferometry**

Cruz Meneses-Fabian, Gustavo Rodriguez-Zurita and Noel-Ivan Toto-Arellano *Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla México* 

## **1. Introduction**

194 Interferometry – Research and Applications in Science and Technology

Wizinowich P. L., (1990). Phase Shifting Interferometry in the Presence of Vibration: A New

Wyant J. C. and Shagam R. N., (1978). Use of Electronic Phase Measurement Techniques in

Meng X.F., Cai L.Z., Wang Y.R., Yang X.L., Xu X.F., Dong G.Y., Shen X.X., Cheng X.C.,

(2008). Wavefront reconstruction by two-step generalized phase-shifting

Algorithm and System, *Appl. Opt*., Vol. 29, pp. 3271-3279

interferometry, *Opt. Commun.,* Vol. 281*,* pp. 5701-5705 *.* 

Optical Testing, *Proc ICO-11, Madrid*, pp. 659-662

In physical optics, the interference effect consists in superposing two or more optical fields in a region of space, which mathematically results in the vector sum of them, it is certain because of in physical optics the superposition principle is valid. So, when this sum is observed with some optical detector such as a human eye or a CCD camera, the irradiance of the total field is obtained, and it can be understood as the sum of the irradiance from each individual field, known as background light, plus an interference additional term per each pair of fields, which consists of a cosine of the phase difference between the two waves and of a factor given mainly by the product of waves amplitudes known as a modulation light. The total effect shows brilliant and obscure zones known as interference fringes, also called as a fringe pattern, or an interferogram (Born & Wolf, 1993). This effect was first reported by Thomas Young in 1801 (Young, 1804; Shamos, 1959), more late it also was observed by Newton, Fizeau, Michelson, etc. (Hecht, 2002). They designed many optical arranges now known as interferometers in order to interfere two o more waves trying to meet the optimal conditions to have a maximum quality in the fringes. Many studies have demonstrated that the fringe quality, (better known as visibility of interference pattern), as well as the shape and the number of fringes is depending of each parameter in the optical field such as the amplitude, the polarization state, the wavelength, the frequency, the coherence degree, the phase and because of it so many applications among physics and other sciences like biology, medicine, astronomy, etc., or also in engineering have been extensively made (Kreis, 2005). However, an intermediate step between the fringe pattern and the direct application in some topic of interest as the evaluation and processing of this pattern must be realized. In this regarding, many proposal have been amply discussed, for example in interferometry of two waves when the phase difference is the variable to be calculated, one of the techniques more widely used consists of performing consecutively shifts of constant phase between the waves that interfere. Then, for each phase-step a new interferogram is gotten and therefore for *N* steps, *N* interferograms are obtained. Mathematically, a *N*x3 system of equations is formed since the with phase-step the object phase, the background light, and the modulation of light are considered unknown with respect to position and constant with respect to time, is known as phase-shifting interferometry (PSI) (Creath, 1993; Malacara, 2007; Schwider, 1990). The phase shift is introduced by a shifting device, which can be done

*N*-Shots 2*N*-Phase-Steps Binary Grating Interferometry 197

system of equations (Malacara, et. al., 1998; Creath, 1993; Millerd & Brock, 2003). Because

demonstrated to be very useful (Schreiber & Bruning, 2007;), especially for well contrasted interferograms (Schwider, 1990). To obtain phase-shifted interferograms, a number of procedures have been demonstrated but many of them needs of *N* shots to capture all of those interferograms. Thus, a simplification is desirable in order to reduce the time of capture. Single-shot interferometers capturing all needed interferograms simultaneously are good examples of this effort (Barrientos-Garcia, et. al., 1999; Novak, 2005). On the other side, phase shifts can be induced by mechanical shifts of a proper element, as a piezoelectric stack (Bruning, et. al., 1974) or a grating (Schwider, 1990). Modulation of polarization is another useful technique (Creath, 1993). In grating interferometry, a grating can be transversally

for example. But in order to obtain several values, the same number of displacements is required (Hariharan, et. al., 1987; Hu, et. al., 2008; Novák, et. al., 2008). Besides, when using gratings as phase shifters, the grating displacement must be carried out with sufficient precision. The higher the grating frequency, the smaller the grating displacement required. Thus, the use of high frequency rulings could compromise the precision of the phase shift. In this chapter, a method to reduce the number of captures from 2*N* to *N* is proposed by means of common-path grating interferometry (Meneses-Fabian, et. al., 2006) in conjunction with crossed linear polarization filters for modulation of polarization (Nomura, et. al., 2006) and grating displacements. A common-path phase-shifting interferometer can be constructed with two-window in the object plane of a 4*f* Fourier-transform system and a grating as its pupil (Arrizon & De-La-Llave, 2004). One-shot phase-shifting interferometers have already been proposed with phase-gratings and elliptical polarization (Rodriguez-Zurita, et. al., 2008a; Kreis, 1986). However, a more common Ronchi grating can be used for the case of 4 *N* because the diffraction efficiencies for diffraction orders 1 are sufficiently good enough to display adequate interferograms. Interference of firstneighboring orders can be obtained in the image plane when proper matching conditions are fulfilled and the phase shifting can be performed with grating displacement driven by an actuator (Meneses-Fabian, et. al., 2006). Because several diffraction orders are to be found in the image plane, some additional shifts can be induced by use of linear polarization instead of circular or elliptical polarization. The calculation of the values of the shifts induced by a grating displacement can be carried out with a method developed by Thomas Kreis (Kreis, 1986), thus alleviating the need of a more detailed calibration. This method is particularly useful for the case of 4 *N* , where the proposed simplification reduces the number of required grating displacements to only one. This particular value does not need to be known beforehand because it can be calculated directly by the above mentioned method of Kreis to find a solution for the optical phase. Then, the number of shots required to capture the four interferograms would result in only two. We restricted ourselves to the

the case of 4 *N* interferograms and 3 *N* 1 equal shifts 90

 *<sup>k</sup>*<sup>1</sup> *<sup>k</sup>*<sup>1</sup> 

*<sup>k</sup> k* 0*N* 1 in order to form a solvable

<sup>0</sup> 0 , it is possible to use the

, has been

*<sup>k</sup>* . Among several possibilities,

90 (Meneses-Fabian, et. al., 2006),

 1 2 3 

*<sup>i</sup>* not necessarily of the same value. Experimental results

phase displacements by certain phase amounts

corresponding phase shifts, each denoted by

displaced by a quarter of a period to obtain shifts of

case of 4 *N* precisely but with

are presented.

one of these phase amounts can be taken as reference, say

with Zeeman effect shifters, acousto-optical modulators, rotating polarizers or translating gratings, among other possibilities. Spatial techniques also have been introduced and widely studied. These consist of introducing a spatial phase variation into interferogram (Takeda, et. al., 1982). Typically, this variation is a linear function, but when the fringes forms closed loops this carrier is not appropriate, and to overcome this shortage a quadratic phase has been proposal (Malacara, et. al., 1998). Others methods for phase extraction have also been proposed (Moore & Mendoza-Santoyo, 1995; Peng, et. al., 1995).

Based on the interference of two monochromatic and coherent waves, the present chapter speaks about the phase shifting interferometry. The principal idea consists of obtaining two interferograms shifted in phase by 180° in a single-shot. By performing an arbitrary phasestep, another two interferograms shifted 180° in phase are captured in a second shot. This way, four interferograms result shifted in phase in two shots with two phase steps, considering the first phase step equal to zero. The arbitrary phase-step is considered to be between 0° and 180° and it will be measured under the concept of generalized phase-shifting interferometry (GPSI) (Xu, et. al., 2008). Therefore, in general with *N*-phase-steps 2*N* interferograms will be captured in *N* shots only. Fringe patterns are obtained from an interferometer build using a 4*f* optical correlator of double Fourier transform, where, at the input plane two apertures are considered. One of them is crossed by a reference beam and the other is considered as a probe window where a phase object is placed. These windows are considered as an input transmittance function. In the Fourier plane, a binary grating (Ronchi ruling) with certain period is placed as a spatial filter function. Then with the appropriate conditions of the wavelength, the grating period, and the focal length of the lenses, at the image plane the interference of the fields in each window is achieved and replicated around diffraction order. Then, by using orthogonal linear polarization in the windows, it is possible to demonstrate that a phase shifting of 180° can be obtained by observing the superposition with adequate transmittance angle of another linear polarizer. An arbitrary phase-shifting is later obtained with a grating displacement.

In summary, a method to reduce the number of captures needed in phase-shifting interferometry is proposed on the basis of grating interferometry and modulation of linear polarization. In this chapter, the case of four interferograms is considered. A common-path interferometer is used with two windows in the object plane and a Ronchi grating as the pupil, thus forming several replicated images of each window over the image plane. The replicated images, under proper matching conditions, superimpose in such a way so that they produce interference patterns. Orders 0 and +1 and -1 and 0 form useful patterns to extract the optical phase differences associated to the windows. A phase of is introduced between these orders using linear polarizing filters placed in the windows and also in the replicated windows, so two -shifted patterns can be captured in one shot. An unknown translation is then applied to the grating in order to produce another shift in the each pattern. A second and final shot captures these last patterns. The actual grating displacement and the phase shift can be determined according to the method proposed by (Kreis, 1986) before applying proper phase-shifting techniques to finally calculate the phase difference distribution between windows. Along this chapter a theoretical model is amply discussed and it is verified with both a numeral simulation and experimental results.

## **2. Theoretical analysis**

Phase-shifting interferometry retrieves phase distributions from a certain number *N* of interferograms (Creath, 1993; Malacara, 2007). Each interferogram *<sup>k</sup> I* must result from

with Zeeman effect shifters, acousto-optical modulators, rotating polarizers or translating gratings, among other possibilities. Spatial techniques also have been introduced and widely studied. These consist of introducing a spatial phase variation into interferogram (Takeda, et. al., 1982). Typically, this variation is a linear function, but when the fringes forms closed loops this carrier is not appropriate, and to overcome this shortage a quadratic phase has been proposal (Malacara, et. al., 1998). Others methods for phase extraction have also been

Based on the interference of two monochromatic and coherent waves, the present chapter speaks about the phase shifting interferometry. The principal idea consists of obtaining two interferograms shifted in phase by 180° in a single-shot. By performing an arbitrary phasestep, another two interferograms shifted 180° in phase are captured in a second shot. This way, four interferograms result shifted in phase in two shots with two phase steps, considering the first phase step equal to zero. The arbitrary phase-step is considered to be between 0° and 180° and it will be measured under the concept of generalized phase-shifting interferometry (GPSI) (Xu, et. al., 2008). Therefore, in general with *N*-phase-steps 2*N* interferograms will be captured in *N* shots only. Fringe patterns are obtained from an interferometer build using a 4*f* optical correlator of double Fourier transform, where, at the input plane two apertures are considered. One of them is crossed by a reference beam and the other is considered as a probe window where a phase object is placed. These windows are considered as an input transmittance function. In the Fourier plane, a binary grating (Ronchi ruling) with certain period is placed as a spatial filter function. Then with the appropriate conditions of the wavelength, the grating period, and the focal length of the lenses, at the image plane the interference of the fields in each window is achieved and replicated around diffraction order. Then, by using orthogonal linear polarization in the windows, it is possible to demonstrate that a phase shifting of 180° can be obtained by observing the superposition with adequate transmittance angle of another linear polarizer. An arbitrary phase-shifting is later obtained with a grating displacement.

In summary, a method to reduce the number of captures needed in phase-shifting interferometry is proposed on the basis of grating interferometry and modulation of linear polarization. In this chapter, the case of four interferograms is considered. A common-path interferometer is used with two windows in the object plane and a Ronchi grating as the pupil, thus forming several replicated images of each window over the image plane. The replicated images, under proper matching conditions, superimpose in such a way so that they produce interference patterns. Orders 0 and +1 and -1 and 0 form useful patterns to

between these orders using linear polarizing filters placed in the windows and also in the

translation is then applied to the grating in order to produce another shift in the each pattern. A second and final shot captures these last patterns. The actual grating displacement and the phase shift can be determined according to the method proposed by (Kreis, 1986) before applying proper phase-shifting techniques to finally calculate the phase difference distribution between windows. Along this chapter a theoretical model is amply

Phase-shifting interferometry retrieves phase distributions from a certain number *N* of interferograms (Creath, 1993; Malacara, 2007). Each interferogram *<sup>k</sup> I* must result from

discussed and it is verified with both a numeral simulation and experimental results.


is introduced

extract the optical phase differences associated to the windows. A phase of

replicated windows, so two

**2. Theoretical analysis** 

proposed (Moore & Mendoza-Santoyo, 1995; Peng, et. al., 1995).

phase displacements by certain phase amounts *<sup>k</sup> k* 0*N* 1 in order to form a solvable system of equations (Malacara, et. al., 1998; Creath, 1993; Millerd & Brock, 2003). Because one of these phase amounts can be taken as reference, say <sup>0</sup> 0 , it is possible to use the corresponding phase shifts, each denoted by *<sup>k</sup>*<sup>1</sup> *<sup>k</sup>*<sup>1</sup> *<sup>k</sup>* . Among several possibilities, the case of 4 *N* interferograms and 3 *N* 1 equal shifts 90 1 2 3 , has been demonstrated to be very useful (Schreiber & Bruning, 2007;), especially for well contrasted interferograms (Schwider, 1990). To obtain phase-shifted interferograms, a number of procedures have been demonstrated but many of them needs of *N* shots to capture all of those interferograms. Thus, a simplification is desirable in order to reduce the time of capture. Single-shot interferometers capturing all needed interferograms simultaneously are good examples of this effort (Barrientos-Garcia, et. al., 1999; Novak, 2005). On the other side, phase shifts can be induced by mechanical shifts of a proper element, as a piezoelectric stack (Bruning, et. al., 1974) or a grating (Schwider, 1990). Modulation of polarization is another useful technique (Creath, 1993). In grating interferometry, a grating can be transversally displaced by a quarter of a period to obtain shifts of 90 (Meneses-Fabian, et. al., 2006), for example. But in order to obtain several values, the same number of displacements is required (Hariharan, et. al., 1987; Hu, et. al., 2008; Novák, et. al., 2008). Besides, when using gratings as phase shifters, the grating displacement must be carried out with sufficient precision. The higher the grating frequency, the smaller the grating displacement required. Thus, the use of high frequency rulings could compromise the precision of the phase shift.

In this chapter, a method to reduce the number of captures from 2*N* to *N* is proposed by means of common-path grating interferometry (Meneses-Fabian, et. al., 2006) in conjunction with crossed linear polarization filters for modulation of polarization (Nomura, et. al., 2006) and grating displacements. A common-path phase-shifting interferometer can be constructed with two-window in the object plane of a 4*f* Fourier-transform system and a grating as its pupil (Arrizon & De-La-Llave, 2004). One-shot phase-shifting interferometers have already been proposed with phase-gratings and elliptical polarization (Rodriguez-Zurita, et. al., 2008a; Kreis, 1986). However, a more common Ronchi grating can be used for the case of 4 *N* because the diffraction efficiencies for diffraction orders 1 are sufficiently good enough to display adequate interferograms. Interference of firstneighboring orders can be obtained in the image plane when proper matching conditions are fulfilled and the phase shifting can be performed with grating displacement driven by an actuator (Meneses-Fabian, et. al., 2006). Because several diffraction orders are to be found in the image plane, some additional shifts can be induced by use of linear polarization instead of circular or elliptical polarization. The calculation of the values of the shifts induced by a grating displacement can be carried out with a method developed by Thomas Kreis (Kreis, 1986), thus alleviating the need of a more detailed calibration. This method is particularly useful for the case of 4 *N* , where the proposed simplification reduces the number of required grating displacements to only one. This particular value does not need to be known beforehand because it can be calculated directly by the above mentioned method of Kreis to find a solution for the optical phase. Then, the number of shots required to capture the four interferograms would result in only two. We restricted ourselves to the case of 4 *N* precisely but with *<sup>i</sup>* not necessarily of the same value. Experimental results are presented.

*N*-Shots 2*N*-Phase-Steps Binary Grating Interferometry 199

 <sup>0</sup> , *<sup>d</sup> w n G rect n* 

*k k u f* /

bright-band width and grating period, respectively. The symbol means convolution.

*<sup>O</sup>*(,) (,) , *xy xy G*

with <sup>1</sup> the inverse Fourier-transform operation assumed to be performed by the second transforming lens as a convention taken in this work in accordance with an inversion in the

1 11 (,) , , exp , 2 22 *<sup>O</sup> n A A B <sup>B</sup>*

where *w x <sup>A</sup>* , ,, *y w x y A x y* and *w x <sup>B</sup>* , ,, *y w x yB x y* have been defined and

has been substituted with

matching condition). According to Eq. (4), the amplitude in the image plane consists of a

window *w x nN x y* 0 0 12 , (as observed through a linear polarizer with transmission

row of copies of the entrance transmittance, each copy separated by 1 *d d*

*<sup>n</sup>* with respect to the horizontal) can be described as

2

In particular, for the case 0 *N* 1 and knowing that the irradiance results proportional to the square modulus of the field amplitude, <sup>2</sup> (,) (,) *<sup>w</sup> I x <sup>y</sup>* **<sup>t</sup>** *<sup>x</sup> <sup>y</sup>* , the corresponding interference

<sup>0</sup> ( , ) , , cos ( , ) 2 *n n*

*<sup>u</sup> Ixy a xy b xy xy*

**<sup>J</sup>** ;

 

*<sup>w</sup>*( , ) , cos *xy CAxy C Bxy <sup>n</sup>* 

sin cos sin *L n nn*

 *n B* **J J J** , and

*nn n*

cos sin cos

 

 **J J** , sin *<sup>L</sup>* 

2

first neighbours. By adjusting the distance 0 *x* between windows such that *N x* 0 0

 for <sup>1</sup> *w d* 2 

*n nn xy C w x x y w x x y i x x y N NN*

In the image plane, the amplitude distribution can be written as the convolution between

 

<sup>1</sup>

 **t t** , (3)

0 00

*n nN* <sup>0</sup> , sin exp ,

**t J** . (5)

, and assuming that 0 *x* equals some multiple

*d d f u* , (diffraction orders

*f u* from the

*<sup>d</sup>* and

 

is satisfied, the field amplitude (,) *<sup>w</sup>***t** *x y* of

 *<sup>n</sup> i xy*

> cos sin

*d*

*u*

 

 

*n n*

**J** . (6)

, (7)

 

0 00

**t J <sup>J</sup>** , (4)

 

, (2)

 

*u f v f* , where (,) *u v* are

with 0, , *k wd* as a label for displacement,

 

image coordinates. Using Eq. (2), the convolution results in

integer *N*0 of the period, in other words, *xN N* 00 0

the actual spatial coordinates, and

*n*

axis at angle

 *n A* **J** 

where cos *<sup>L</sup>* 

pattern is given by

 

<sup>1</sup> , *G*

 <sup>1</sup> <sup>2</sup> <sup>0</sup> sinc 2 exp 2 *C n i nu u n d*

*<sup>n</sup> C xn n d*

also assuring that the inequality *w d a fu*

with the spatial frequency coordinates are given by (,) / , /

the amplitude of the object and the impulse response of the system, i.e.,

Fig. 1. Experimental setup. Pj , j = 0…4: linear polarizers. A, B: rectangular windows in object plane. *G* , : Ronchi grating ( , the spatial frequencies escalated by *f* ). DC(x): actuator. f: focal length. Side views of windows (lower left) and polarizers with *<sup>j</sup>* in image plane (lower right) are also sketched.

## **2.1 Theoretical background**

Fig.1 shows the experimental setup. It comprises a 4*f* Fourier-transform system under monochromatic illumination at wavelength . The transforming lenses have a focal length of *f*. Linear polarizer *P*0 have its transmission axis at 45° and the linear polarizers *P*1 and *P*<sup>2</sup> , over windows A and B, have its transmission axis at 0° and 90° respectively. The object plane (input plane) consists of two similar rectangular windows A and B, each of sides *wa* and *wb* . The windows centers are separated by the distance 0 *x* . In general, an amplitude distribution of the form *A x* ,*y* can be considered in the window A as a reference wave, while *Bxy i xy* , exp , can represent the amplitude distribution in the window B (a test object, for instance). Then, the input transmittance can be expressed by.

$$\mathbf{f}(\mathbf{x},\mathbf{y}) = w(\mathbf{x} + \frac{1}{2}\mathbf{x}\_0, \mathbf{y}) A \left(\mathbf{x} + \frac{1}{2}\mathbf{x}\_0, \mathbf{y}\right) \mathbf{I}\_A + w(\mathbf{x} - \frac{1}{2}\mathbf{x}\_0, \mathbf{y}) \mathbf{B}\left(\mathbf{x} - \frac{1}{2}\mathbf{x}\_0, \mathbf{y}\right) \exp\left[i\boldsymbol{\phi}\left(\mathbf{x} - \frac{1}{2}\mathbf{x}\_0, \mathbf{y}\right)\right] \mathbf{I}\_B \tag{1}$$

where 1 0 *<sup>A</sup>* **<sup>J</sup>** , <sup>0</sup> 1 *<sup>B</sup>* **J** are Jones vectors corresponding to orthogonal linear polarization states and the window function is written as *w x*(,) / / *y rect x a rect w w y b* , where *rect x a* / *<sup>w</sup>* is the rectangle function on *x* direction of width *wa* and *recty* /*bw* is the rectangle function on *y* direction of width *wb* .

A binary absorptive grating *G*(,) with spatial period *ud* and bright-band width *uw* is placed in the frequency plane (Fourier plane, Fig.1). An actuator (DC) can translate the grating longitudinally through a given distance *u*<sup>0</sup> . The grating (a Ronchi grating) can be written.

Fig. 1. Experimental setup. Pj , j = 0…4: linear polarizers. A, B: rectangular windows in

Fig.1 shows the experimental setup. It comprises a 4*f* Fourier-transform system under

of *f*. Linear polarizer *P*0 have its transmission axis at 45° and the linear polarizers *P*1 and *P*<sup>2</sup> , over windows A and B, have its transmission axis at 0° and 90° respectively. The object plane (input plane) consists of two similar rectangular windows A and B, each of sides *wa* and *wb* . The windows centers are separated by the distance 0 *x* . In general, an amplitude distribution of the form *A x* ,*y* can be considered in the window A as a reference wave,

can represent the amplitude distribution in the window B (a test

**J** are Jones vectors corresponding to orthogonal linear polarization

with spatial period *ud* and bright-band width *uw* is

0 0 00 0 1 1 11 1 ( , ) ( , ) , ( , ) , exp , 2 2 22 2 *<sup>A</sup> <sup>B</sup> xy wx x yA x x y wx x yB x x y i x x y*

**t J <sup>J</sup>** (1)

states and the window function is written as *w x*(,) / / *y rect x a rect w w y b* , where *rect x a* / *<sup>w</sup>* is the rectangle function on *x* direction of width *wa* and *recty* /*bw* is the

placed in the frequency plane (Fourier plane, Fig.1). An actuator (DC) can translate the grating longitudinally through a given distance *u*<sup>0</sup> . The grating (a Ronchi grating) can be written.

the spatial frequencies escalated by

. The transforming lenses have a focal length

 *<sup>j</sup>* in

*f* ). DC(x):

,

actuator. f: focal length. Side views of windows (lower left) and polarizers with

object, for instance). Then, the input transmittance can be expressed by.

 

object plane. *G*

,

**2.1 Theoretical background** 

while *Bxy i xy* , exp ,

1 0 *<sup>A</sup>* **<sup>J</sup>** , <sup>0</sup>

where

image plane (lower right) are also sketched.

monochromatic illumination at wavelength

1 *<sup>B</sup>* 

rectangle function on *y* direction of width *wb* .

A binary absorptive grating *G*(,)

: Ronchi grating (

$$\text{G}\left(\mu,\nu\right) = \text{rect}\left(\frac{\mu-\mu\_0}{\mu\_w}\right) \otimes \sum\_{n=-\infty}^{\infty} \delta\left(\mu-n\mu\_d\right),\tag{2}$$

with the spatial frequency coordinates are given by (,) / , / *u f v f* , where (,) *u v* are the actual spatial coordinates, and *k k u f* / with 0, , *k wd* as a label for displacement, bright-band width and grating period, respectively. The symbol means convolution.

In the image plane, the amplitude distribution can be written as the convolution between the amplitude of the object and the impulse response of the system, i.e.,

$$\mathbf{t}\_O(\mathbf{x}, y) = \mathbf{t}(\mathbf{x}, y) \otimes \mathfrak{T}^{-1} \left\{ \mathbf{G}(\mu, \nu) \right\},\tag{3}$$

with <sup>1</sup> the inverse Fourier-transform operation assumed to be performed by the second transforming lens as a convention taken in this work in accordance with an inversion in the image coordinates. Using Eq. (2), the convolution results in

$$\mathbf{t}\_{O}(\mathbf{x},\mathbf{y}) = \sum\_{n=-\infty}^{\infty} \mathbf{C}\_{n} \left\{ w\_{A} \left[ \mathbf{x} - \left( \frac{n}{N\_{0}} - \frac{1}{2} \right) \mathbf{x}\_{0}, \mathbf{y} \right] \mathbf{J}\_{A} + w\_{B} \left[ \mathbf{x} - \left( \frac{n}{N\_{0}} + \frac{1}{2} \right) \mathbf{x}\_{0}, \mathbf{y} \right] \exp\left[ i\phi \left( \mathbf{x} - \left( \frac{n}{N\_{0}} + \frac{1}{2} \right) \mathbf{x}\_{0}, \mathbf{y} \right) \right] \mathbf{J}\_{B} \right\} \tag{4}$$

where *w x <sup>A</sup>* , ,, *y w x y A x y* and *w x <sup>B</sup>* , ,, *y w x yB x y* have been defined and <sup>1</sup> , *G <sup>n</sup> C xn n d* has been substituted with <sup>1</sup> <sup>2</sup> <sup>0</sup> sinc 2 exp 2 *C n i nu u n d* for <sup>1</sup> *w d* 2 , and assuming that 0 *x* equals some multiple integer *N*0 of the period, in other words, *xN N* 00 0 *d d f u* , (diffraction orders matching condition). According to Eq. (4), the amplitude in the image plane consists of a row of copies of the entrance transmittance, each copy separated by 1 *d d f u* from the first neighbours. By adjusting the distance 0 *x* between windows such that *N x* 0 0 *<sup>d</sup>* and also assuring that the inequality *w d a fu* is satisfied, the field amplitude (,) *<sup>w</sup>***t** *x y* of window *w x nN x y* 0 0 12 , (as observed through a linear polarizer with transmission axis at angle *<sup>n</sup>* with respect to the horizontal) can be described as

$$\mathbf{t}\_w(\mathbf{x}, y) = \left\{ \mathbf{C}\_n A(\mathbf{x}, y) \cos(\psi\_n) + \mathbf{C}\_{n - N\_0} B(\mathbf{x}, y) \sin(\psi\_n) \exp\left[i\phi(\mathbf{x}, y)\right] \right\} \mathbf{J}\_\psi \,. \tag{5}$$

where cos *<sup>L</sup> n A* **J J J** , sin *<sup>L</sup> n B* **J J J** , and

$$\mathbf{J}\_{\boldsymbol{\nu}}^{L} = \begin{pmatrix} \cos^{2}(\boldsymbol{\nu}\_{n}) & \sin(\boldsymbol{\nu}\_{n})\cos(\boldsymbol{\nu}\_{n})\\ \sin(\boldsymbol{\nu}\_{n})\cos(\boldsymbol{\nu}\_{n}) & \sin^{2}(\boldsymbol{\nu}\_{n}) \end{pmatrix}; \qquad \mathbf{J}\_{\boldsymbol{\nu}} = \begin{pmatrix} \cos(\boldsymbol{\nu}\_{n})\\ \sin(\boldsymbol{\nu}\_{n}) \end{pmatrix}. \tag{6}$$

In particular, for the case 0 *N* 1 and knowing that the irradiance results proportional to the square modulus of the field amplitude, <sup>2</sup> (,) (,) *<sup>w</sup> I x <sup>y</sup>* **<sup>t</sup>** *<sup>x</sup> <sup>y</sup>* , the corresponding interference pattern is given by

$$I(\mathbf{x}, y) = a\_n(\mathbf{x}, y) + b\_n(\mathbf{x}, y) \cos \left| \phi(\mathbf{x}, y) - 2\pi \frac{\mu\_0}{\mu\_d} \right|,\tag{7}$$

*N*-Shots 2*N*-Phase-Steps Binary Grating Interferometry 201

Comparing Eq. (9) and Eq. (10) is easy to note that *a x* <sup>0</sup> ,*y* and *a x* <sup>1</sup> ,*y* in general are

2 2 22 0 0 11 2 2

 

> 

1

45 .

1

2 , or well could be at 90° each other, when 0 1

1 over the window *wx x* <sup>0</sup> 2 ,*y* could be at complementary

, (11)

, (12)

, (13)

, (14)

 

, (15)

, (16)

 

 2 is totally

in Eq. (16) would be only coincident with Eq. (14) for

 2 . For

0 over the window *wx x* <sup>0</sup> 2 ,*y* and the

11 11 cos sin cos sin 4 4

> 2 2 0 1 cos cos 1

> > 0 1 cos sin

> > > 2 2

For the modulation light, in order to do *b x* <sup>0</sup> ,*y* equal to *b x* <sup>1</sup> ,*y* or *b x* <sup>1</sup> ,*y* , from Eq.

0 1

1

<sup>1</sup> 2 

 2 2 22 0 0 11 2 2 11 11 cos , sin , cos , sin , 4 4 *A B AB*

*I xy I xy I xy I xy*

; and 2 2

, (17)

2 0 1 1 1 sin sin

4

, (18)

sin 2 sin 2 

0

45 as it was written in the above example, but the case 0 1

*case 2*: when *I x A B* , , *y I x y* , from Eq. (9a) and Eq. (10a) it must be complied

coincident with Eqs. (14), so it can be used to apply to PSI technique.

 

2 2 0 1 2

1 1 cos cos

 0

45 , or well 0

*case 1*: when *I x A B* , , *y I x y* , then from Eq. (9a) and Eq. (10a) must be complied

different and only they are equals in two cases:

which means that the polarizer placed at angle

 

45 , then 0

 0 1 

(9b) and Eq. (10b) it must be complied

which conduce to

and finally can be established

polarizer placed at angle

whereof can be deduced

a particular case of

as a particular case, we can suppose

4

1 

example: if 1

angles, when 0 1 

obtaining so

where *a x <sup>n</sup>* ,*y* is known as background light, *b x <sup>n</sup>* ,*y* is known as a modulation light and they are given by

$$a\_n(\mathbf{x}, \mathbf{y}) = \frac{1}{4} \text{sinc}^2\left(\frac{1}{2}n\right) \cos^2\nu\_n I\_A(\mathbf{x}, \mathbf{y}) + \frac{1}{4} \text{sinc}^2\left(\frac{1}{2}(n-1)\right) \sin^2\nu\_n I\_B(\mathbf{x}, \mathbf{y})\,\,,\tag{8a}$$

$$h\_n(\mathbf{x}, y) = \frac{1}{4} \text{sinc}\left(\frac{1}{2}n\right) \text{sinc}\left[\frac{1}{2}(n-1)\right] \text{sinc}\left(2\psi\_n\right) \sqrt{I\_A(\mathbf{x}, y)I\_B(\mathbf{x}, y)}\right],\tag{8b}$$

with <sup>2</sup> , , *AI x <sup>y</sup> A x <sup>y</sup>* and <sup>2</sup> , , *BI x <sup>y</sup> B x <sup>y</sup>* . The pattern in Eq. (7) results to be modulated by the functions sin and sinc . Note that *a x <sup>n</sup>* ,*y* and *b x <sup>n</sup>* ,*y* would be independent of position if the illumination is uniform in each window at the input plane. Otherwise, *a x <sup>n</sup>* ,*y* and *b x <sup>n</sup>* ,*y* can be smooth functions of the position and, in such a case they must give rise to corresponding spectra of a given extension, however small.

In the practice, 0,1 *n* are two cases of interest, for 0 *n* the window of observation would be *wx x* <sup>0</sup> 2 ,*y* and Eq. (8) could be reduced to

$$a\_0(\mathbf{x}, y) = \frac{1}{4} \cos^2 \varphi\_0 I\_A(\mathbf{x}, y) + \frac{1}{\pi^2} \sin^2 \varphi\_0 I\_s(\mathbf{x}, y),\tag{9a}$$

$$h\_o(\mathbf{x}, y) = \frac{1}{2\pi} \sin(2\varphi\_o) \sqrt{I\_A(\mathbf{x}, y) I\_s(\mathbf{x}, y)}\,\mathrm{},\tag{9b}$$

Fig. 2. Numerical simulation of phase shifting interferometry for four unequal phase-steps, which are obtained with two camera shots, that is with 2 *N* , then 4 2*N* interferograms changed in phase are obtained: (a) phase function considered in this simulation and (b) for interferograms changed in phase by 0, , , .

and for 1 *n* the window of observation would be *wx x* <sup>0</sup> 2 ,*y* and Eq. (8) could be reduced to

$$a\_1(\mathbf{x}, \mathbf{y}) = \frac{1}{\pi^2} \cos^2 \varphi\_1 I\_A(\mathbf{x}, \mathbf{y}) + \frac{1}{4} \sin^2 \varphi\_1 I\_B(\mathbf{x}, \mathbf{y}) \, \, \, \, \, \tag{10a}$$

$$h\_1(\mathbf{x}, \mathbf{y}) = \frac{1}{2\pi} \text{sinc}(2\boldsymbol{\nu}\_1) \sqrt{I\_A(\mathbf{x}, \mathbf{y}) I\_B(\mathbf{x}, \mathbf{y})} \,. \tag{10b}$$

Comparing Eq. (9) and Eq. (10) is easy to note that *a x* <sup>0</sup> ,*y* and *a x* <sup>1</sup> ,*y* in general are different and only they are equals in two cases:

*case 1*: when *I x A B* , , *y I x y* , then from Eq. (9a) and Eq. (10a) must be complied

$$\frac{1}{4}\cos^2\psi\_0 + \frac{1}{\pi^2}\sin^2\psi\_0 = \frac{1}{\pi^2}\cos^2\psi\_1 + \frac{1}{4}\sin^2\psi\_{1'}\tag{11}$$

which conduce to

$$
\cos^2 \psi\_0 + \cos^2 \psi\_1 = 1 \,\,\,\,\,\tag{12}
$$

obtaining so

200 Interferometry – Research and Applications in Science and Technology

where *a x <sup>n</sup>* ,*y* is known as background light, *b x <sup>n</sup>* ,*y* is known as a modulation light and

1 1 2 2 1 1 2 2 , sinc cos , sinc 1 sin , 4 2 4 2 *<sup>n</sup> n A n B a x <sup>y</sup> n Ix*

11 1 , sinc sinc 1 sin 2 , , 42 2 *<sup>n</sup> nA B b x <sup>y</sup> n n*

with <sup>2</sup> , , *AI x <sup>y</sup> A x <sup>y</sup>* and <sup>2</sup> , , *BI x <sup>y</sup> B x <sup>y</sup>* . The pattern in Eq. (7) results to be modulated by the functions sin and sinc . Note that *a x <sup>n</sup>* ,*y* and *b x <sup>n</sup>* ,*y* would be independent of position if the illumination is uniform in each window at the input plane. Otherwise, *a x <sup>n</sup>* ,*y* and *b x <sup>n</sup>* ,*y* can be smooth functions of the position and, in such a case

In the practice, 0,1 *n* are two cases of interest, for 0 *n* the window of observation would

*<sup>a</sup> <sup>x</sup> <sup>y</sup> <sup>I</sup> <sup>x</sup> <sup>y</sup> <sup>I</sup> <sup>x</sup> <sup>y</sup> <sup>A</sup> <sup>B</sup>* sin , <sup>1</sup>

*<sup>b</sup> <sup>x</sup> <sup>y</sup> <sup>I</sup> <sup>x</sup> <sup>y</sup> <sup>I</sup> <sup>x</sup> <sup>y</sup> <sup>A</sup> <sup>B</sup>* sin <sup>2</sup> , , <sup>2</sup>

Fig. 2. Numerical simulation of phase shifting interferometry for four unequal phase-steps, which are obtained with two camera shots, that is with 2 *N* , then 4 2*N* interferograms changed in phase are obtained: (a) phase function considered in this simulation and (b) for

> .

and for 1 *n* the window of observation would be *wx x* <sup>0</sup> 2 ,*y* and Eq. (8) could be

 2 2 11 1 2 1 1 , cos , sin , <sup>4</sup> *A B a xy* 

1 1 <sup>1</sup> , sin 2 , , <sup>2</sup> *A B b xy* 

 *I xy I xy*

*I xyI xy*

, (10a)

. (10b)

0

0 2

they must give rise to corresponding spectra of a given extension, however small.

cos , <sup>4</sup> <sup>1</sup> , <sup>0</sup>

 

2

<sup>1</sup> , <sup>0</sup>

be *wx x* <sup>0</sup> 2 ,*y* and Eq. (8) could be reduced to

0

interferograms changed in phase by 0, , ,

reduced to

 

*y n Ix y*

*I x y I x y*

, (9a)

, (9b)

, (8a)

, (8b)

2

they are given by

$$
\cos \varphi\_0 = \pm \sin \varphi\_{1\prime} \tag{13}
$$

and finally can be established

$$\nu\_0 = \begin{cases} \pi/2 - \nu\_1 \\ \pi/2 + \nu\_1 \end{cases} \tag{14}$$

which means that the polarizer placed at angle 0 over the window *wx x* <sup>0</sup> 2 ,*y* and the polarizer placed at angle 1 over the window *wx x* <sup>0</sup> 2 ,*y* could be at complementary angles, when 0 1 2 , or well could be at 90° each other, when 0 1 2 . For example: if 1 45 , then 0 45 , or well 0 45 .

For the modulation light, in order to do *b x* <sup>0</sup> ,*y* equal to *b x* <sup>1</sup> ,*y* or *b x* <sup>1</sup> ,*y* , from Eq. (9b) and Eq. (10b) it must be complied

$$
\pm \sin(2\psi\_0) = \pm \sin(2\psi\_1) \,, \tag{15}
$$

whereof can be deduced

$$\nu\_0 = \begin{cases} \pm \nu\_1 \\ \pm \pi/2 \mp \nu\_1 \end{cases} \tag{16}$$

a particular case of 0 1 in Eq. (16) would be only coincident with Eq. (14) for 1 45 as it was written in the above example, but the case 0 1 2 is totally coincident with Eqs. (14), so it can be used to apply to PSI technique.

*case 2*: when *I x A B* , , *y I x y* , from Eq. (9a) and Eq. (10a) it must be complied

$$\frac{1}{4}\cos^2\varphi\_0 I\_A(\mathbf{x},\mathbf{y}) + \frac{1}{\pi^2}\sin^2\varphi\_0 I\_B(\mathbf{x},\mathbf{y}) = \frac{1}{\pi^2}\cos^2\varphi\_1 I\_A(\mathbf{x},\mathbf{y}) + \frac{1}{4}\sin^2\varphi\_1 I\_B(\mathbf{x},\mathbf{y})\,'\tag{17}$$

as a particular case, we can suppose

$$\frac{1}{4}\cos^2\psi\_0 = \frac{1}{\pi^2}\cos^2\psi\_1; \text{and } \frac{1}{\pi^2}\sin^2\psi\_0 = \frac{1}{4}\sin^2\psi\_1. \tag{18}$$

*N*-Shots 2*N*-Phase-Steps Binary Grating Interferometry 203

obtained using only an unknown grating shift and, thus, two camera shots. The desired

23 01 0 1

*<sup>w</sup>* denotes the wrapped phase to be unwrapped further. From Eq. (24), for the case

, the well-known formula for four shifts can be obtained (Schwider, 1990). Eq.

 from the same captured interferograms, the procedure suggested by (Kreis, 1986) can be applied. This procedure is based on the Fourier transform analysis of fringes and a variant of it is proposed in the following sections to conceal it with the desired phase extraction.

to keep constant the visibility in the interference pattern as it was made in *case 1*, this manner the same structure of Eq. (23) could be obtained and the solution for the wrapped phase is obtained with the same Eq. (24). Note that the *case 2* is more general, so also the

( , ) ( , ) ( , ) ( , ) cos ( , ) arctan ( , ) ( , ) sin *<sup>w</sup> I xy I xy I xy I xy x y I xy I xy*

 0, , , 

(a) Patters indicated in Eq. (25), (b) Fourier transform from Eq. (25a) and spectrum filtered, (c) wrapped phases, (d) unwrapped phases, (e) phase difference of the unwrapped phases in

Fig. 4. Numerical simulations of phase extraction: (a) wrapped phase obtained with Eq. (24), (b) unwrapped phase, and (c) phase difference between the calculated phase in (b) and the

57.51 and 1

   

, (24)

32.49 in order

can be

to be useful. In order to calculate

by using the modified Kreis method:

interferograms with phase displacements of 01 2 3

phase distribution can be calculated from

(25) requires, of course, the knowledge of the value

For the case 2, the polarizer angle must be placed to 0

Fig. 3. Numerical simulations for determination of

proposal phase in Fig. (2.a) as a measurement of the error.

57.51 is also valid for the case 1.

(b1) and (b2), (f) a data line from (e), (g) subsample of the data line in (f).

where 

of 90 

polarizer angle

whereof leads to plan a system of equations, where <sup>2</sup> <sup>0</sup> cos and <sup>2</sup> <sup>1</sup> cos would be the unknowns,

$$\begin{aligned} \pi^2 \cos^2 \psi\_0 - 4 \cos^2 \psi\_1 &= 0\\ -4 \cos^2 \psi\_0 + \pi^2 \cos^2 \psi\_1 &= \pi^2 - 4 \end{aligned} \tag{19}$$

being possible to found

$$\cos^2 \psi\_0 = \frac{4}{\pi^2 + 4} \text{; and } \cos^2 \psi\_1 = \frac{\pi^2}{\pi^2 + 4} \text{.}\tag{20}$$

from Eq. (20) the following relation can be deduced

$$\cos^2 \psi\_0 = \sin^2 \psi\_1 \tag{21}$$

then, 0 1 2 , which are complementary angles, as in the *case 1* illustrated with Eq. (14). Finally, taking in account Eq. (20) the values of the polarizer angles can be computed,

$$
\varphi\_0 = \pm 57.51^\circ \text{; and } \varphi\_1 = \pm 32.49^\circ \text{.} \tag{22}
$$

these values for transmission angles of the polarizers also must comply *b x* 0 1 , , *y b x y* or *b x* 0 1 , , *y b x y* due to they are a particular case of Eq. (16).

In order to achieve the PSI technique, selecting *P*3 at angle 0 45 for 0 *n* and *P*4 at angle 1 45 for 1 *n* with no grating displacement, 0 *u* 0 , with idea to implemented *case 1* [Meneses-Fabian, et. al., 2009], this results *a x* 0 1 , ,, *y a x y a x y* and also *b x* 0 1 , ,, *y b x y b x y* , and of this manner two complementary patterns are first obtained at the image plane within replication regions given by *wx x* <sup>0</sup> 2 ,*y* and *wx x* <sup>0</sup> 2 ,*y* . The possible changing in the fringe modulations was narrowly reduced after a normalization procedure, since the interferograms can be converted in digital patterns with the same modulation. Such patterns can be written as

$$I\_0(\mathbf{x}, y) = a(\mathbf{x}, y) + b(\mathbf{x}, y) \cos[\phi(\mathbf{x}, y)],\tag{23a}$$

$$I\_1(\mathbf{x}, y) = a(\mathbf{x}, y) - b(\mathbf{x}, y) \cos[\phi(\mathbf{x}, y)]. \tag{23b}$$

They correspond to patterns with a phase shift of 1 . Secondly, by performing an arbitrary translation of value 0 0 /2 *u ud* , the introduced phase shift is less than radians. Then, another two interferograms result. Each one can be expressed as follows

$$I\_2(\mathbf{x}, y) = a(\mathbf{x}, y) + b(\mathbf{x}, y) \cos \left[ \phi(\mathbf{x}, y) - \Delta \phi \right],\tag{23c}$$

$$I\_3(\mathbf{x}, y) = a(\mathbf{x}, y) - b(\mathbf{x}, y) \cos \left[ \phi(\mathbf{x}, y) - \Delta \phi \right]. \tag{23d}$$

The corresponding phase shift for them is <sup>3</sup> . Considering patterns 1*I* and 2*I* they differ by a phase shift of <sup>2</sup> , where 0 2 / *u ud* . With this procedure, four

0 1 2 22 2 0 1 cos 4cos 0 4cos cos 4

; and

2 2 0 1 cos sin 

(14). Finally, taking in account Eq. (20) the values of the polarizer angles can be computed,

these values for transmission angles of the polarizers also must comply *b x* 0 1 , , *y b x y* or

*I x* 0( , ) ( , ) ( , )cos ( , ) *y a x y b x y*

*I x* 1( , ) ( , ) ( , )cos ( , ) *y a x y b x y*

arbitrary translation of value 0 0 /2 *u ud* , the introduced phase shift is less than

radians. Then, another two interferograms result. Each one can be expressed as follows

*I xy axy bxy xy* 2( , ) ( , ) ( , )cos ( , )

*I xy axy bxy xy* 3( , ) ( , ) ( , )cos ( , )

 <sup>3</sup> 

 , where 0 

1 

 2 / 

57.51 ; and 1

 

 

2 <sup>1</sup> <sup>2</sup> cos

 45 for 1 *n* with no grating displacement, 0 *u* 0 , with idea to implemented *case 1* [Meneses-Fabian, et. al., 2009], this results *a x* 0 1 , ,, *y a x y a x y* and also *b x* 0 1 , ,, *y b x y b x y* , and of this manner two complementary patterns are first obtained at the image plane within replication regions given by *wx x* <sup>0</sup> 2 ,*y* and *wx x* <sup>0</sup> 2 ,*y* . The possible changing in the fringe modulations was narrowly reduced after a normalization procedure, since the interferograms can be converted in digital patterns

2 , which are complementary angles, as in the *case 1* illustrated with Eq.

22 2

4

0 2 4

 <sup>0</sup> cos 

, (19)

2

4 

 and <sup>2</sup> <sup>1</sup> cos 

would be the

, (20)

, (21)

32.49 . (22)

45 for 0 *n* and *P*4 at

*x y* , (23a)

*x y* . (23b)

. Secondly, by performing an

, (23c)

. (23d)

. Considering patterns 1*I* and 2*I* they

*u ud* . With this procedure, four

whereof leads to plan a system of equations, where <sup>2</sup>

2

0 

*b x* 0 1 , , *y b x y* due to they are a particular case of Eq. (16). In order to achieve the PSI technique, selecting *P*3 at angle 0

with the same modulation. Such patterns can be written as

They correspond to patterns with a phase shift of

The corresponding phase shift for them is

<sup>2</sup> 

differ by a phase shift of

cos

from Eq. (20) the following relation can be deduced

unknowns,

being possible to found

then, 0 1 

angle 1 

  interferograms with phase displacements of 01 2 3 0, , , can be obtained using only an unknown grating shift and, thus, two camera shots. The desired phase distribution can be calculated from

$$\phi\_w(\mathbf{x}, y) = \arctan\left| \frac{I\_2(\mathbf{x}, y) - I\_3(\mathbf{x}, y) - \left[I\_0(\mathbf{x}, y) - I\_1(\mathbf{x}, y)\right] \cos(\Delta\phi)}{\left[I\_0(\mathbf{x}, y) - I\_1(\mathbf{x}, y)\right] \sin(\Delta\phi)}\right|, \tag{24}$$

where *<sup>w</sup>* denotes the wrapped phase to be unwrapped further. From Eq. (24), for the case of 90 , the well-known formula for four shifts can be obtained (Schwider, 1990). Eq. (25) requires, of course, the knowledge of the value to be useful. In order to calculate from the same captured interferograms, the procedure suggested by (Kreis, 1986) can be applied. This procedure is based on the Fourier transform analysis of fringes and a variant of it is proposed in the following sections to conceal it with the desired phase extraction.

For the case 2, the polarizer angle must be placed to 0 57.51 and 1 32.49 in order to keep constant the visibility in the interference pattern as it was made in *case 1*, this manner the same structure of Eq. (23) could be obtained and the solution for the wrapped phase is obtained with the same Eq. (24). Note that the *case 2* is more general, so also the polarizer angle 57.51 is also valid for the case 1.

Fig. 3. Numerical simulations for determination of by using the modified Kreis method: (a) Patters indicated in Eq. (25), (b) Fourier transform from Eq. (25a) and spectrum filtered, (c) wrapped phases, (d) unwrapped phases, (e) phase difference of the unwrapped phases in (b1) and (b2), (f) a data line from (e), (g) subsample of the data line in (f).

Fig. 4. Numerical simulations of phase extraction: (a) wrapped phase obtained with Eq. (24), (b) unwrapped phase, and (c) phase difference between the calculated phase in (b) and the proposal phase in Fig. (2.a) as a measurement of the error.

*N*-Shots 2*N*-Phase-Steps Binary Grating Interferometry 205

Fig. 6. Test objects. (a) Interference patterns (b) Unwrapped phase. Upper row: phase dot,

Kreis. The used filter is a unit step, so the suppression of the left half of the spectrum is

wrapped phase from the inverse Fourier transform of the already filtered 1 *g xy* (,) , which is shown in Fig. 3-c1. Fig. 3-c2 shows the same procedure as applied to Fig. 3-2a (i.e., 2 *g xy* (,) ). At this stage, the phases of the inverse Fourier transform of each filtered interference pattern <sup>1</sup> *g xy* (,) and 2 *g xy* (,) are obtained. Therefore, these phases are wrapped, modified phases whose respective numerical integration results in two unwrapped, modified phases (Figs. 3 d1 and 3-d2). These unwrapped phases result in monotonous functions which are different

interference pattern points, so an average over some range can be sufficient to calculate

with a good approximation. Fig. 3-f shows a line of Fig. 3-e1, and Fig. 3-g shows a section of the Fig. 3-f, where the used region to measure phase difference is indicated with an elliptic

' 0.39273364, where the bar means average, so its difference with respect to the initial

is of the order of <sup>5</sup> 3.4556 10 rads. Taken

*<sup>u</sup>* , the phase calculated from

algorithm [Malacara, et. al., 1998]. The results of these stages are shown in Fig. 4.

, but their difference in each point (,) *x y* gives the modified phase

(Figs. 3-e). This modified phase difference has to be constant for all

. The resulting value over the entire lower region of Fig. 3-e2 was of

*w* can be determined with Eq. (24) and the desired phase distribution

radians. Lower

, while the red line shows the modified

*<sup>w</sup>* with standard unwrapping

, the wrapped

' as

consists of extracting the

50.007 0.874 radians. Central row: phase step, 25.726 0.449

radians.

achieved. The following stage of the procedure to find

trace. The dots show the ideal phase difference

row: still oil, 60.7 1.059 

from the desired phase

phase difference '

induced phase

phase distribution

can be identified with

 

difference '

### **2.2 Determination of**

Subtraction of Eq. (23b) from Eq. (23a) and Eq. (23d) from Eq. (23c) gives

$$\log\_1(\mathbf{x}, y) = I\_0(\mathbf{x}, y) - I\_1(\mathbf{x}, y) = 2b(\mathbf{x}, y)\cos[\phi(\mathbf{x}, y)] \,\_{1'} \tag{25a}$$

$$\log\_2(\mathbf{x}, y) = I\_2(\mathbf{x}, y) - I\_3(\mathbf{x}, y) = 2b(\mathbf{x}, y)\cos\left[\phi(\mathbf{x}, y) - \Delta\phi\right],\tag{25b}$$

procedure which eliminates *a*(*x*, *y*) . In Eq. (25), some dependence on position has been considered to include effects such as non uniform illumination, non linear detection or imperfections in the optical components. These subtractions avoid to apply the Fourier transformation and the spatial filtering usually performed for the same purpose of eliminating *a*(*x*, *y*) (Kreis, 1986). It is remarked that the Fourier transform procedure to eliminate *a*(*x*, *y*) introduces an error due to the fact that, in general, the spectra from *a*(*x*, *y*) and *b x* , cos , *y x y* can be found mixed one with each other over the Fourier plane. Therefore, filtering out the *a*(*x*, *y*) -spectrum around the zero frequency excludes also low frequencies from *b x* , cos , *y x y* and, as a consequence, there is a corresponding loss of information related to *x*,*y* and . To calculate , the method introduced by (Kreis, 1986) is employed (an alternative can be seen in (Meng, et. al., 2008). An advantage of the variant that is proposed in this work consists of the elimination of *a*(*x*, *y*) by subtracting two patterns. This way, there is no loss of information due to frequency suppression in the Fourier plane, as is the case of the method as proposed by (Kreis, 1990). The proposed technique is illustrated in the example of the Fig.2. In addition, the technique is valid even when the phase function is more complex. The 3-D plot at the left is the phase distribution (,) *x y* , while the other plots are phase-shifted interferograms calculated from (,) *x y* . Interferograms 0 *I* and 1 *I* are mutually shifted by radians, as well as 2 *I* and 3 *I* , but between 0 *I* and 2 *I* , and 1 *I* and 3 *I* is the same arbitrary phase of 7 0.39269908 radians. This situation illustrates the kind of results that can be obtained with the setup of Fig.1. Then, the problem is to find from the four interference patterns assuming that its value is not known. The solution is illustrated in Fig.3. The plots included are shown as an array of rows (seven letters) and columns (two numbers).

According to Eqs. (25.a) and (25.b), Figs.3-a1 and 3-a2 show the subtractions ( , ) <sup>1</sup> *g x y* and ( , ) <sup>2</sup> *g x y* respectively, where the four irradiances *<sup>k</sup> I* were taken from the same interferograms of Fig. 2. Next, Fig. 3-b1 shows the Fourier spectrum of 3-a1 only ( ( , ) <sup>1</sup> *g x y* ), while Fig. 3-b2 depicts its resulting filtered spectrum in accordance with the method of

Fig. 5. Tilted wavefront for testing. (a) Interference patterns. (b) Unwrapped phase. Phase shift measured according to the modified Kreis method: 14.4 0.2513274 radians.

*g*<sup>101</sup> ( , ) ( , ) ( , ) 2 , cos ( , ) *x y I x y I x y b x y*

procedure which eliminates *a*(*x*, *y*) . In Eq. (25), some dependence on position has been considered to include effects such as non uniform illumination, non linear detection or imperfections in the optical components. These subtractions avoid to apply the Fourier transformation and the spatial filtering usually performed for the same purpose of eliminating *a*(*x*, *y*) (Kreis, 1986). It is remarked that the Fourier transform procedure to eliminate *a*(*x*, *y*) introduces an error due to the fact that, in general, the spectra from *a*(*x*, *y*)

Therefore, filtering out the *a*(*x*, *y*) -spectrum around the zero frequency excludes also low

. To calculate

1986) is employed (an alternative can be seen in (Meng, et. al., 2008). An advantage of the variant that is proposed in this work consists of the elimination of *a*(*x*, *y*) by subtracting two patterns. This way, there is no loss of information due to frequency suppression in the Fourier plane, as is the case of the method as proposed by (Kreis, 1990). The proposed technique is illustrated in the example of the Fig.2. In addition, the technique is valid even when the phase function is more complex. The 3-D plot at the left is the phase distribution

(,) *x y* , while the other plots are phase-shifted interferograms calculated from

radians. This situation illustrates the kind of results that can be obtained with the setup of

value is not known. The solution is illustrated in Fig.3. The plots included are shown as an

According to Eqs. (25.a) and (25.b), Figs.3-a1 and 3-a2 show the subtractions ( , ) <sup>1</sup> *g x y* and ( , ) <sup>2</sup> *g x y* respectively, where the four irradiances *<sup>k</sup> I* were taken from the same interferograms of Fig. 2. Next, Fig. 3-b1 shows the Fourier spectrum of 3-a1 only ( ( , ) <sup>1</sup> *g x y* ), while Fig. 3-b2 depicts its resulting filtered spectrum in accordance with the method of

Fig. 5. Tilted wavefront for testing. (a) Interference patterns. (b) Unwrapped phase. Phase

shift measured according to the modified Kreis method: 14.4 0.2513274

between 0 *I* and 2 *I* , and 1 *I* and 3 *I* is the same arbitrary phase of

*g xy I xy I xy bxy xy* <sup>223</sup> ( , ) ( , ) ( , ) 2 , cos ( , )

*x y* and, as a consequence, there is a corresponding loss of

*x y* can be found mixed one with each other over the Fourier plane.

*x y* , (25a)

, (25b)

(,) *x y* .

7 0.39269908

, the method introduced by (Kreis,

radians, as well as 2 *I* and 3 *I* , but

 

radians.

from the four interference patterns assuming that its

**2.2 Determination of** 

and *b x* , cos , *y*

information related to

frequencies from *b x* , cos , *y*

Fig.1. Then, the problem is to find

*x*,*y* and

Interferograms 0 *I* and 1 *I* are mutually shifted by

array of rows (seven letters) and columns (two numbers).

Subtraction of Eq. (23b) from Eq. (23a) and Eq. (23d) from Eq. (23c) gives

Fig. 6. Test objects. (a) Interference patterns (b) Unwrapped phase. Upper row: phase dot, 50.007 0.874 radians. Central row: phase step, 25.726 0.449 radians. Lower row: still oil, 60.7 1.059 radians.

Kreis. The used filter is a unit step, so the suppression of the left half of the spectrum is achieved. The following stage of the procedure to find consists of extracting the wrapped phase from the inverse Fourier transform of the already filtered 1 *g xy* (,) , which is shown in Fig. 3-c1. Fig. 3-c2 shows the same procedure as applied to Fig. 3-2a (i.e., 2 *g xy* (,) ). At this stage, the phases of the inverse Fourier transform of each filtered interference pattern <sup>1</sup> *g xy* (,) and 2 *g xy* (,) are obtained. Therefore, these phases are wrapped, modified phases whose respective numerical integration results in two unwrapped, modified phases (Figs. 3 d1 and 3-d2). These unwrapped phases result in monotonous functions which are different from the desired phase , but their difference in each point (,) *x y* gives the modified phase difference ' (Figs. 3-e). This modified phase difference has to be constant for all interference pattern points, so an average over some range can be sufficient to calculate with a good approximation. Fig. 3-f shows a line of Fig. 3-e1, and Fig. 3-g shows a section of the Fig. 3-f, where the used region to measure phase difference is indicated with an elliptic trace. The dots show the ideal phase difference , while the red line shows the modified phase difference ' . The resulting value over the entire lower region of Fig. 3-e2 was of ' 0.39273364, where the bar means average, so its difference with respect to the initial induced phase is of the order of <sup>5</sup> 3.4556 10 rads. Taken ' as , the wrapped phase distribution *w* can be determined with Eq. (24) and the desired phase distribution can be identified with *<sup>u</sup>* , the phase calculated from *<sup>w</sup>* with standard unwrapping algorithm [Malacara, et. al., 1998]. The results of these stages are shown in Fig. 4.

*N*-Shots 2*N*-Phase-Steps Binary Grating Interferometry 207

being smaller than a quarter of period, but this condition is possible to verify by observing that the fringes do not shift enough to adopt a complementary fringe modulation. Because in this proposed method the phase shifts are achieved either by modulation of polarization or by grating displacement, not all of the shifts result necessarily of the same value. The requirements of the arrangement are not very restrictive because it uses only very basic optical components, such as linear polarizers at angles of 0°, 45 , or 90°. A Ronchi grating from 500 to 1500 lines per inch can be used. As remarked before, the higher the grating frequency, the larger the distance between windows and more space to place samples becomes available, but the grating displacement has to be smaller. This last feature does not represent an impediment in this method because the actuator responsible for the displacement does not need of a calibration within a certain range, neither a precise displacement at a given prescribed value. Only one unknown displacement is needed and its value can be calculated each time it is employed. As for the fringe modulation, it is close to unity when using the 0 and ±1 diffraction orders for a typical Ronchi grating. Also, the two used patterns have the same fringe modulation because, as long as the two diffraction orders which superpose are of equal value, the involved amplitudes are the same. These features to extract static phase distributions make this proposal competitive as compared to the existing ones. Although in this chapter was discussed the PSI method for four steps, this proposal can be extended for *N* 3 , with which 2*N* interferograms changed in phase

This work was partially supported from VIEP-BUAP under grant MEFC and PROMEP under

Arrizon, V. & De-La-Llave, D. (2004) Common-path interferometry with one-dimensional

Barrientos-García B.; Moore A. J.; Perez-Lopez C.; Wang L. & Tshudi T. (1999). Transient

Bruning, J. H.; Herriott, D. R.; Gallagher, J. E.; Rosenfeld, D. P.; White, A. D. & Brangaccio,

Creath K., "Phase-measurement interferometry techniques," Progress in Optics, Vol. XXVI,

Hariharan, P.; Oreb, B. F.; Eiju, T. (1987) Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm, *Applied Optics*, Vol. 26, 2504-2506

Kreis, T. (1986). Digital holographic interference-phase measurement using the Fouriertransform method, *Journal of the Optical Society of America A*, Vol. 3, 847-855. Kreis, T.; WILEY-VCH Verlag GmbH & Co. KGaA (2005). *Handbook of Holographic* 

Deformation Measurement with Electronic Speckle Pattern Interferometry by Use of a Holographic Optical Element for Spatial Phase Stepping, *Applied Optics*. Vol.

D. J. (1974) Digital wavefront measurement interferometer for testing optical

would be obtained with *N* camera shots.

grant PROMEP/103.5/09/4544) are greatly appreciated.

38, 5944-5947, ISSN 1559-128X

Hech E.; Addison-Wesley (1972). *Optics*

periodic filters, *Optics Letters*, Vol. 29, 141-143

Born, M. and Wolf, E.; Cambridge University Press (1993). *Principles of Optics* 

surfaces and lenses, *Applied Optics*, Vol. 13, pp. 2693-2703.

E. Wolf, ed., (Elsevier Science, 1993), pp. 349-393.

*interferometry, optical and digital methods*

**5. Acknowledgment** 

**6. References** 

Fig. 7. Typical slice profiles of the phase dot and the phase step along arbitrary directions.

## **3. Experimental results**

The experimental setup follows closely the sketch of Fig.1, with *f* 479 , *mm* a laser He-Ne emitting at 632.8 , *nm* with a linear polarization at 45° by using *P*<sup>0</sup> . 10 *wa mm* , 13 *wb mm* , and 0 *x mm* 12.45 , so the condition *<sup>w</sup>* <sup>0</sup> *a x* is fulfilled. The period of the Ronchi grating was 25.4 *u m <sup>d</sup>* . The grating was mounted on an actuator (Newport CMA-25CC). Note that the diffraction-order matching condition *x N* 0 0 *f ud* for 0 *N* 1 is satisfied. The CCD camera (COHU 4815) is adjusted to capture the images of two interference patterns ( *wx x* <sup>0</sup> 2 ,*y* and *wx x* <sup>0</sup> 2 ,*y* ) simultaneously. The fringe modulations of this interferogram pair are mutually complementary. By one actuator displacement, another pair of complementary interferograms can be obtained. This is shown in Fig. 5, where the calculated value of the introduced phase was of 14.4 0.2513274 radians. The unwrapped phase is also shown. Each captured interferogram was subject to the same processing so as to get images with gray levels ranging from 0 to 255 before the use of Eq. (8). The sets of interferograms from three more objects are shown in Fig. 6. Two of these objects were prepared by evaporating magnesium fluoride (MgF2) on glass substrates (a phase dot, upper row, and a phase step, center row). The third object was oil deposited on a glass plate (bottom row). Each object was placed separately in one of the windows using the interferometer of Fig.1. Each set of four interferograms were obtained with a grating displacement between two of them as described. En each example, the phase-shifts induced by polarization can be visually identified from the complementary modulation contrasts of each pair. The calculated unwrapped phase for each object is shown in the rightmost column. Two typical profiles for some unwrapped phase are shown in Fig. (7). The resulting unwrapping phases do not display discontinuities, as is the case when the wrong is taken in Eq. (8).

## **4. Conclusion**

The proposed phase-shifting interferometer is able to capture four useful interferograms with only one grating displacement and two shots. This grating displacement only requires

being smaller than a quarter of period, but this condition is possible to verify by observing that the fringes do not shift enough to adopt a complementary fringe modulation. Because in this proposed method the phase shifts are achieved either by modulation of polarization or by grating displacement, not all of the shifts result necessarily of the same value. The requirements of the arrangement are not very restrictive because it uses only very basic optical components, such as linear polarizers at angles of 0°, 45 , or 90°. A Ronchi grating from 500 to 1500 lines per inch can be used. As remarked before, the higher the grating frequency, the larger the distance between windows and more space to place samples becomes available, but the grating displacement has to be smaller. This last feature does not represent an impediment in this method because the actuator responsible for the displacement does not need of a calibration within a certain range, neither a precise displacement at a given prescribed value. Only one unknown displacement is needed and its value can be calculated each time it is employed. As for the fringe modulation, it is close to unity when using the 0 and ±1 diffraction orders for a typical Ronchi grating. Also, the two used patterns have the same fringe modulation because, as long as the two diffraction orders which superpose are of equal value, the involved amplitudes are the same. These features to extract static phase distributions make this proposal competitive as compared to the existing ones. Although in this chapter was discussed the PSI method for four steps, this proposal can be extended for *N* 3 , with which 2*N* interferograms changed in phase would be obtained with *N* camera shots.

## **5. Acknowledgment**

This work was partially supported from VIEP-BUAP under grant MEFC and PROMEP under grant PROMEP/103.5/09/4544) are greatly appreciated.

## **6. References**

206 Interferometry – Research and Applications in Science and Technology

Fig. 7. Typical slice profiles of the phase dot and the phase step along arbitrary directions.

The experimental setup follows closely the sketch of Fig.1, with *f* 479 , *mm* a laser He-Ne

13 *wb mm* , and 0 *x mm* 12.45 , so the condition *<sup>w</sup>* <sup>0</sup> *a x* is fulfilled. The period of the Ronchi

The CCD camera (COHU 4815) is adjusted to capture the images of two interference patterns ( *wx x* <sup>0</sup> 2 ,*y* and *wx x* <sup>0</sup> 2 ,*y* ) simultaneously. The fringe modulations of this interferogram pair are mutually complementary. By one actuator displacement, another pair of complementary interferograms can be obtained. This is shown in Fig. 5, where the

unwrapped phase is also shown. Each captured interferogram was subject to the same processing so as to get images with gray levels ranging from 0 to 255 before the use of Eq. (8). The sets of interferograms from three more objects are shown in Fig. 6. Two of these objects were prepared by evaporating magnesium fluoride (MgF2) on glass substrates (a phase dot, upper row, and a phase step, center row). The third object was oil deposited on a glass plate (bottom row). Each object was placed separately in one of the windows using the interferometer of Fig.1. Each set of four interferograms were obtained with a grating displacement between two of them as described. En each example, the phase-shifts induced by polarization can be visually identified from the complementary modulation contrasts of each pair. The calculated unwrapped phase for each object is shown in the rightmost column. Two typical profiles for some unwrapped phase are shown in Fig. (7). The resulting unwrapping

The proposed phase-shifting interferometer is able to capture four useful interferograms with only one grating displacement and two shots. This grating displacement only requires

632.8 , *nm* with a linear polarization at 45° by using *P*<sup>0</sup> . 10 *wa mm* ,

. The grating was mounted on an actuator (Newport CMA-25CC).

*f ud* for 0 *N* 1 is satisfied.

14.4 0.2513274 radians. The

is taken in Eq. (8).

**3. Experimental results** 

Note that the diffraction-order matching condition *x N* 0 0

calculated value of the introduced phase was of

phases do not display discontinuities, as is the case when the wrong

grating was 25.4 *u m <sup>d</sup>*

emitting at

**4. Conclusion** 


Born, M. and Wolf, E.; Cambridge University Press (1993). *Principles of Optics* 


**10** 

*The Netherlands* 

**Path Length Resolved Dynamic Light Scattering** 

**Measurements with Suppressed Influence of** 

**Optical Properties Using Phase Modulated** 

*Biomedical Photonic Imaging Group, MIRA Institute for Biomedical Technology* 

In optical Doppler measurements, the path length of the light is unknown. This complicates the noninvasive diagnosis of tissue with light. For example, in laser Doppler blood flowmetry, the coherent light delivered into the tissue interacts with static as well as moving scatterers, e.g. red blood cells and it records values averaged over different and basically unknown path lengths. One of the important limitations of this technique is the dependence of the perfusion signal on the optical properties of the tissue, i.e. absorption coefficient, scattering coefficient and anisotropy factor. These dependences result from the varying optical path length of detected photons; the longer the optical path length the greater is the probability for Doppler scattering events to occur, thus yielding an overestimation of the

If red blood cells in vascular blood can be regarded as independent scatterers, the average number of collisions between photons and red blood cells which is used to determine the

Where Σsc is the scattering cross section of RBCs, [RBC] is the number of moving RBCs in 1 mm3 of tissue and *L* is the mean path length of the detected light. For a homogeneous tissue of fixed concentration and blood volume, the value of *m* is proportional to the average path length of the detected light. The average path lengths will be different for different tissue types due to the changes in tissue optical properties in terms of absorption and scattering and thus laser Doppler flowmetry provides only a relative measure of the perfusion level. Therefore, development of techniques for monitoring Doppler shifts with path length information would result in more-quantitative and more reliable tissue perfusion

To facilitate quantitative path length resolved dynamic light scattering measurements with suppressed influence of optical properties we have developed an improved method based

blood perfusion, compared to the short path length situation [1].

concentration of blood cells moving in the tissue is given by

**1. Introduction**

information.

**Low Coherence Interferometry** 

*and Technical Medicine, University of Twente, Enschede,* 

( )\*[ ]\* *m rbc RBC L sc* (1)

Babu Varghese and Wiendelt Steenbergen


## **Path Length Resolved Dynamic Light Scattering Measurements with Suppressed Influence of Optical Properties Using Phase Modulated Low Coherence Interferometry**

Babu Varghese and Wiendelt Steenbergen

*Biomedical Photonic Imaging Group, MIRA Institute for Biomedical Technology and Technical Medicine, University of Twente, Enschede, The Netherlands* 

## **1. Introduction**

208 Interferometry – Research and Applications in Science and Technology

Malacara, D.; Servín, M.; & Malacara, Z.; Marcel Dekker, New York (1998). *Interferogram* 

Meneses-Fabian, C.; Rodriguez-Zurita, G.; Vazquez-Castillo, J. F.; Robledo-Sanchez C. &

Meneses-Fabian, C.; Rodriguez-Zurita, G.; Encarnacion-Gutierrez, Ma. C. & Toto-Arellano

Meng, X. F.; Cai, L. Z.; Wang, Y. R.; Yang, X. L.; Xu, X. F.; Dong, G. Y.; Shen, X. X. & Cheng,

Millerd J. E. & Brock N. J. (2003). Methods and apparatus for splitting, imaging, and measuring wavefronts in interferometry, U. S. Patent 20030053071A1. Moore, A. J. & Mendoza-Santoyo, F., (1995). Phase demodulation in the space domain without a fringe carrier, *Optical Laser Engineering*, Vol. 23, pp. 319–330. Nomura,T.; Murata, S.; Nitanai, E. & Numata, T. (2006). Phase-shifting digital holography

Novák, M.; Millerd, J.; Brock, N.; North-Morris, M.; Hayes, J. & Wyant, J. (2005). Analysis of

Novák, J.; Novák, P. & Mikš, A. (2008) Multi-step phase-shifting algorithms insensitive to linear phase shift errors, *Optics Communications*, Vol. 281, 5302–5309 Peng, X.; Zhou, S. M. & Gao Z. (1995). An automatic demodulation technique for a non-

Rodriguez-Zurita, G.; Meneses-Fabian, C.; Toto-Arellano, N. I.; Vazquez-Castillo, J. &

Rodriguez-Zurita, G.; Toto-Arellano, N. I.; Meneses-Fabian, C. & Vazquez Castillo, J. (2008b)

Schreiber, H. & Bruning, J.H., (2007). Phase shifting interferometry, Chapter 14, in: *Optical* 

Schwider, J. "Advanced Evaluation Techniques in Interferometry," Progress in Optics. Vol.

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Takeda M., Ina H., and Kobayashi S., (1982). Fourier-Transform Method of Fringe-Pattern

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*Shop Testing*, D. Malacara Ed., Wiley & Sons, New York, 547-655.

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*Society of America A*, Vol. 72, pp. 156-160

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with a phase difference between orthogonal polarizations, *Applied Optics*, Vol. 45

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X. C. (2008). *Optics Communications*, Vol. 281 5701–5705.

In optical Doppler measurements, the path length of the light is unknown. This complicates the noninvasive diagnosis of tissue with light. For example, in laser Doppler blood flowmetry, the coherent light delivered into the tissue interacts with static as well as moving scatterers, e.g. red blood cells and it records values averaged over different and basically unknown path lengths. One of the important limitations of this technique is the dependence of the perfusion signal on the optical properties of the tissue, i.e. absorption coefficient, scattering coefficient and anisotropy factor. These dependences result from the varying optical path length of detected photons; the longer the optical path length the greater is the probability for Doppler scattering events to occur, thus yielding an overestimation of the blood perfusion, compared to the short path length situation [1].

If red blood cells in vascular blood can be regarded as independent scatterers, the average number of collisions between photons and red blood cells which is used to determine the concentration of blood cells moving in the tissue is given by

$$
\overline{m} = \Sigma\_{sc} (r\hbar c)^{\*} [RBC]^{\*} L \tag{1}
$$

Where Σsc is the scattering cross section of RBCs, [RBC] is the number of moving RBCs in 1 mm3 of tissue and *L* is the mean path length of the detected light. For a homogeneous tissue of fixed concentration and blood volume, the value of *m* is proportional to the average path length of the detected light. The average path lengths will be different for different tissue types due to the changes in tissue optical properties in terms of absorption and scattering and thus laser Doppler flowmetry provides only a relative measure of the perfusion level. Therefore, development of techniques for monitoring Doppler shifts with path length information would result in more-quantitative and more reliable tissue perfusion information.

To facilitate quantitative path length resolved dynamic light scattering measurements with suppressed influence of optical properties we have developed an improved method based

Path Length Resolved Dynamic Light Scattering Measurements

modulating the phase in the reference path.

**3. Phase modulated low coherence Mach-Zehnder interferometry**

In our research, we have developed a new bio-optical technique for path length resolved laser Doppler perfusion monitoring, by combining the principles of coherence gated interferometry and laser Doppler blood flowmetry [2-3]. The method is based on a phase modulated fiber optic low coherence Mach-Zehnder interferometer, in which the limited temporal coherence acts as a band pass filter in selecting the photons that have traveled a specific path length. We use a fiber-optic Mach–Zehnder interferometer (Fig.1) with a superluminescent diode (*λc*=832nm, *ΔλFWHM* =17 nm, *LC* =18 μm) as the light source. A single mode fiber-optic coupler with a splitting ratio of 90:10 is used to create a reference arm (10%) and a sample arm (90%). Single mode fibers (mode field diameter=5.3 mm, NA=0.14) are used for illumination, while multimode graded-index fibers (core diameter =100 mm, NA=0.29) are used for detection, providing a large detection window. The path length of the reference arm is varied by reflection of the light in a translatable retroreflector and the position of the retroreflector is adjusted to yield an optical path length equal to the optical

with Suppressed Influence of Optical Properties Using Phase Modulated Low... 211

were compared with the predictions of the dynamic light scattering (DLS) and diffusive wave spectroscopy (DWS) theories in the single scattered and diffusion regimes, respectively. They showed the dependence of detection of multiply scattered light on the geometry of the detection optics and on the anisotropy of the scattering [8]. Even though these studies modeled the single scattering and the diffusive regimes of light fluctuations, they did not model the transition between the two regimes. With a free beam Michelson interferometer, Wax et al. applied path-length resolved DLS spectroscopy and a theoretical model was developed to predict this transition regime across the full range of path lengths from single scattering through diffusive transport [9]. By comparing the trends in the measured power spectra for various-sized microspheres with a theoretical treatment that decomposes the total power spectrum by the number of scattering events, they correlated the detection of multiply scattered light with scattering anisotropy. We reported the development of fiber-optic Mach-Zehnder interferometer for path length resolved measurements with two spatially separated fibers for illumination and detection, as used in conventional laser Doppler perfusion monitors [10]. Low coherence interferometry with phase modulation of the reference beam has been adopted by Ishii et al. in single scattering spectroscopy to analyze the characteristics of extremely dense colloidal suspensions [11]. Doppler optical coherence tomography based on low-coherence single-mode fiber optic Michelson interferometry has been proposed for path length resolved measurements adopting on axis back reflection and confocal detection of singly scattered photons [12]. Here two embodiments were reported due to the possibility of performing interferometric measurements either in the time domain or in the Fourier domain. In time domain OCT the path length of the reference arm is varied in time. In frequency domain OCT the broadband interference is acquired with spectrally separated detectors either by encoding the optical frequency in time with a spectrally scanning source (Swept source OCT) or with a dispersive detector, like a grating and a linear detector array (Spectral Domain or Fourier Domain OCT) [13-14]. In these techniques, optical path length distributions can be obtained about photons which where Doppler shifted by the medium. The photons that has been scattered by static structures can also be made to contribute to the interferometric signal by

on phase modulated low coherence interferometry [2-3]. In this chapter, we aim at describing the state-of-art of this novel interferometric technique and we show that we can measure dynamic properties of particles, independent of the optical properties of the surrounding tissue matrices. Furthermore, we demonstrate the feasibility of phase modulated low coherence interferometry in measuring *in vivo* optical path lengths and path length resolved Doppler shifts.

## **2. Review of coherence domain path length resolved approaches in laser Doppler flowmetry**

To obtain path length distributions with widths of a few millimeters, several successful approaches based on low coherence interferometric methods were reported. In low coherence interferometry, a user-positioned coherence gate selects the light that has traveled a known optical path length in the medium to interfere with reference light. Dougherty et al. presented a new approach based on coherence modulation of semiconductor lasers using the variable coherence properties of the semiconductor laser [4]. In this technique, they exploited variations in effective coherence length properties of certain types of laser diodes by regulating the input drive current to these devices. For a long coherence length, all photons interfere, while for a short coherence length only photons with almost the same path length will interfere. This will relatively suppress the deep photons, since the (few) deep photons will only interfere with the few deep photons but not with the (many) shallow photons. However, these methods still give no control over the optical path length traveled by the detected light. McKinney et al. [5] and Haberland et al. [6] used a wavelength modulated continuous wave source as a variable-coherence source for measuring path length distributions. The frequency of the modulation used was much faster than the integration time of the detection and the authors demonstrated that the speckle contrast ratio measured in that way was linked to the photons path-length distribution [5]. Haberland et al. [6] used such a wavelength modulated source to demonstrate that chirp optical coherence tomography (OCT) can be an alternative to short coherence tomography with the advantage of a simplified optical set-up. However they reduced their investigations to unscattered light. Later, Tualle et al. [7] reported the development of a low cost interferometric set-up to record the scattered light by the use of a wavelength modulated continuous wave source. The principle of this technique relies on the facts that the shape of the time-resolved signal corresponds to the path-length distribution of the scattered light inside the turbid medium and the path-length differences can be measured using an interferometer. They showed that the study of the speckle pattern fluctuations within the modulation period can provide much more information, and that this information can be used to completely reconstruct the scattered light path-length distribution, or equivalently to perform time-resolved measurements. However, the slow rate that they used for the wavelength modulation limited their experiments to static scattering media.

With a fiber-optic low coherence Michelson interferometer, Bizheva et al. demonstrated that particle dynamics of highly scattering media can be imaged and quantified in the single scattering regime with dynamic low coherence tomography (LCI) by examining the intensity fluctuations of the backscattered light and extracting information from the photocurrent power spectrum [8]. Later, they showed that dynamic LCI permits pathlength-resolved measurements of particle dynamics in highly scattering media with the ability to separate singly scattered, multiply scattered, and diffusive light and the results

on phase modulated low coherence interferometry [2-3]. In this chapter, we aim at describing the state-of-art of this novel interferometric technique and we show that we can measure dynamic properties of particles, independent of the optical properties of the surrounding tissue matrices. Furthermore, we demonstrate the feasibility of phase modulated low coherence interferometry in measuring *in vivo* optical path lengths and path

**2. Review of coherence domain path length resolved approaches in laser** 

wavelength modulation limited their experiments to static scattering media.

With a fiber-optic low coherence Michelson interferometer, Bizheva et al. demonstrated that particle dynamics of highly scattering media can be imaged and quantified in the single scattering regime with dynamic low coherence tomography (LCI) by examining the intensity fluctuations of the backscattered light and extracting information from the photocurrent power spectrum [8]. Later, they showed that dynamic LCI permits pathlength-resolved measurements of particle dynamics in highly scattering media with the ability to separate singly scattered, multiply scattered, and diffusive light and the results

To obtain path length distributions with widths of a few millimeters, several successful approaches based on low coherence interferometric methods were reported. In low coherence interferometry, a user-positioned coherence gate selects the light that has traveled a known optical path length in the medium to interfere with reference light. Dougherty et al. presented a new approach based on coherence modulation of semiconductor lasers using the variable coherence properties of the semiconductor laser [4]. In this technique, they exploited variations in effective coherence length properties of certain types of laser diodes by regulating the input drive current to these devices. For a long coherence length, all photons interfere, while for a short coherence length only photons with almost the same path length will interfere. This will relatively suppress the deep photons, since the (few) deep photons will only interfere with the few deep photons but not with the (many) shallow photons. However, these methods still give no control over the optical path length traveled by the detected light. McKinney et al. [5] and Haberland et al. [6] used a wavelength modulated continuous wave source as a variable-coherence source for measuring path length distributions. The frequency of the modulation used was much faster than the integration time of the detection and the authors demonstrated that the speckle contrast ratio measured in that way was linked to the photons path-length distribution [5]. Haberland et al. [6] used such a wavelength modulated source to demonstrate that chirp optical coherence tomography (OCT) can be an alternative to short coherence tomography with the advantage of a simplified optical set-up. However they reduced their investigations to unscattered light. Later, Tualle et al. [7] reported the development of a low cost interferometric set-up to record the scattered light by the use of a wavelength modulated continuous wave source. The principle of this technique relies on the facts that the shape of the time-resolved signal corresponds to the path-length distribution of the scattered light inside the turbid medium and the path-length differences can be measured using an interferometer. They showed that the study of the speckle pattern fluctuations within the modulation period can provide much more information, and that this information can be used to completely reconstruct the scattered light path-length distribution, or equivalently to perform time-resolved measurements. However, the slow rate that they used for the

length resolved Doppler shifts.

**Doppler flowmetry** 

were compared with the predictions of the dynamic light scattering (DLS) and diffusive wave spectroscopy (DWS) theories in the single scattered and diffusion regimes, respectively. They showed the dependence of detection of multiply scattered light on the geometry of the detection optics and on the anisotropy of the scattering [8]. Even though these studies modeled the single scattering and the diffusive regimes of light fluctuations, they did not model the transition between the two regimes. With a free beam Michelson interferometer, Wax et al. applied path-length resolved DLS spectroscopy and a theoretical model was developed to predict this transition regime across the full range of path lengths from single scattering through diffusive transport [9]. By comparing the trends in the measured power spectra for various-sized microspheres with a theoretical treatment that decomposes the total power spectrum by the number of scattering events, they correlated the detection of multiply scattered light with scattering anisotropy. We reported the development of fiber-optic Mach-Zehnder interferometer for path length resolved measurements with two spatially separated fibers for illumination and detection, as used in conventional laser Doppler perfusion monitors [10]. Low coherence interferometry with phase modulation of the reference beam has been adopted by Ishii et al. in single scattering spectroscopy to analyze the characteristics of extremely dense colloidal suspensions [11]. Doppler optical coherence tomography based on low-coherence single-mode fiber optic Michelson interferometry has been proposed for path length resolved measurements adopting on axis back reflection and confocal detection of singly scattered photons [12]. Here two embodiments were reported due to the possibility of performing interferometric measurements either in the time domain or in the Fourier domain. In time domain OCT the path length of the reference arm is varied in time. In frequency domain OCT the broadband interference is acquired with spectrally separated detectors either by encoding the optical frequency in time with a spectrally scanning source (Swept source OCT) or with a dispersive detector, like a grating and a linear detector array (Spectral Domain or Fourier Domain OCT) [13-14]. In these techniques, optical path length distributions can be obtained about photons which where Doppler shifted by the medium. The photons that has been scattered by static structures can also be made to contribute to the interferometric signal by modulating the phase in the reference path.

## **3. Phase modulated low coherence Mach-Zehnder interferometry**

In our research, we have developed a new bio-optical technique for path length resolved laser Doppler perfusion monitoring, by combining the principles of coherence gated interferometry and laser Doppler blood flowmetry [2-3]. The method is based on a phase modulated fiber optic low coherence Mach-Zehnder interferometer, in which the limited temporal coherence acts as a band pass filter in selecting the photons that have traveled a specific path length. We use a fiber-optic Mach–Zehnder interferometer (Fig.1) with a superluminescent diode (*λc*=832nm, *ΔλFWHM* =17 nm, *LC* =18 μm) as the light source. A single mode fiber-optic coupler with a splitting ratio of 90:10 is used to create a reference arm (10%) and a sample arm (90%). Single mode fibers (mode field diameter=5.3 mm, NA=0.14) are used for illumination, while multimode graded-index fibers (core diameter =100 mm, NA=0.29) are used for detection, providing a large detection window. The path length of the reference arm is varied by reflection of the light in a translatable retroreflector and the position of the retroreflector is adjusted to yield an optical path length equal to the optical

Path Length Resolved Dynamic Light Scattering Measurements

*<sup>m</sup>* . Hence the shape of the peak around

mutually interfere to generate beats at frequency

waves S1 and S2 with Doppler shifts

*OM1=OM2=1/2.OR.tan*

*<sup>m</sup>* +

path lengths in the sample.

*Di* and

SPIE).

with Suppressed Influence of Optical Properties Using Phase Modulated Low... 213

their sum phasor M is perpendicular to R. The amplitude of M1 and M2 is

called 'heterodyne'. In practice from a turbid sample waves are obtained with a distribution of Doppler shifts, leading to a similar distribution of spectral components centered around

The component of the reference light represented by the average phasor (OR) that still is at the original light source frequency will also interfere with the scattered light from the sample and thus, apart from the peak around the phase modulation frequency, a heterodyne component will also occur at low frequencies. Finally, the sample phasors S1 and S2, will

'homodyne'. Hence, the spectrum at low frequencies is a mixture of homodyne interference of light remitted by the sample, and heterodyne interference between sample light and unmodulated reference light, while the spectral component around the phase modulation frequency is the pure Doppler shift distribution. The power spectrum measured for widely different optical path lengths in the sample and the reference arm, contains the ordinary homodyne signal due to the mutual interference of scattered light over almost equal optical

Fig. 2. Phasor diagram for the interfering fields (top) and the resulting power spectrum (bottom). (Figure reprinted from Fig. 1 in Ref. [2] Copyright (2008) with permission from

positive frequencies show up in the power spectrum interference peaks are expected at

*D*1 and

*/2* to achieve the desired phase modulation angle. The sample

*Di* (*i=1,2)*. This interference of sample light with reference light will be

*<sup>D</sup>*<sup>1</sup> -

*<sup>D</sup>*<sup>2</sup> interfere with both M1 and M2.

*<sup>m</sup>* corresponds to the Doppler shift distribution.

*<sup>D</sup>*<sup>2</sup> , a component that we call

Since only

*m* -

path length of a certain part of the photons in the sample arm. The reference beam is polarized using a linear polarizer and the phase is sinusoidally modulated at 6 kHz using an electro optic broadband phase modulator with a peak optical phase shift of 2.04 radians applied to the modulator. The AC photocurrent is measured with a 12 bit analogue to digital converter sampling at 40 kHz. The coherence length of the light source, and the intermodal dispersion in the detection fiber, define the path-length resolution of the measurement.

Fig. 1. Schematic of the fiber optic Mach–Zehnder interferometer. (Figure reprinted from Fig. 2 in Ref. [2] Copyright (2008) with permission from SPIE).

In phase modulated low coherence interferometry, for sufficiently small phase modulation angles, the power spectrum measured when the path length difference between the reference light and the scattered light is within the coherence length of light source, is a mixture of homodyne interference of light remitted by the sample, and heterodyne interference between sample light and unmodulated reference light, and a heterodyne spectral component around the phase modulation frequency resulting from the interference between sample light and modulated reference light. Path length resolved optical properties of the media are measured from the heterodyne peak appearing at the modulation frequency. This can be understood from the phasor description of the interfering fields, as shown in fig. 2. Here the reference wave and two Doppler shifted sample waves are represented by phasors R, S1 and S2, respectively. The angle is the peak phase deviation due to phase modulation in the reference light, which is depicted by a sinusoidal oscillation of the reference phasor between the extreme phasors R1 and R2 on either side of the average phasor R. For small values of , the oscillation of the reference phasor between R1 and R2 is equivalent to the summation of the average reference phasor R, and two phasors M1 and M2 of equal length that rotate in opposite directions with a constant angular speed equal to the phase modulation frequency *<sup>m</sup>* . The initial phase of M1 and M2 should be chosen such that

path length of a certain part of the photons in the sample arm. The reference beam is polarized using a linear polarizer and the phase is sinusoidally modulated at 6 kHz using an electro optic broadband phase modulator with a peak optical phase shift of 2.04 radians applied to the modulator. The AC photocurrent is measured with a 12 bit analogue to digital converter sampling at 40 kHz. The coherence length of the light source, and the intermodal dispersion in the detection fiber, define the path-length resolution of the measurement.

Fig. 1. Schematic of the fiber optic Mach–Zehnder interferometer. (Figure reprinted from

In phase modulated low coherence interferometry, for sufficiently small phase modulation angles, the power spectrum measured when the path length difference between the reference light and the scattered light is within the coherence length of light source, is a mixture of homodyne interference of light remitted by the sample, and heterodyne interference between sample light and unmodulated reference light, and a heterodyne spectral component around the phase modulation frequency resulting from the interference between sample light and modulated reference light. Path length resolved optical properties of the media are measured from the heterodyne peak appearing at the modulation frequency. This can be understood from the phasor description of the interfering fields, as shown in fig. 2. Here the reference wave and two Doppler shifted sample waves are

due to phase modulation in the reference light, which is depicted by a sinusoidal oscillation of the reference phasor between the extreme phasors R1 and R2 on either side of the average

equivalent to the summation of the average reference phasor R, and two phasors M1 and M2 of equal length that rotate in opposite directions with a constant angular speed equal to the

, the oscillation of the reference phasor between R1 and R2 is

*<sup>m</sup>* . The initial phase of M1 and M2 should be chosen such that

is the peak phase deviation

Fig. 2 in Ref. [2] Copyright (2008) with permission from SPIE).

represented by phasors R, S1 and S2, respectively. The angle

phasor R. For small values of

phase modulation frequency

their sum phasor M is perpendicular to R. The amplitude of M1 and M2 is *OM1=OM2=1/2.OR.tan /2* to achieve the desired phase modulation angle. The sample waves S1 and S2 with Doppler shifts *D*1 and *<sup>D</sup>*<sup>2</sup> interfere with both M1 and M2. Since only positive frequencies show up in the power spectrum interference peaks are expected at *m* - *Di* and *<sup>m</sup>* +*Di* (*i=1,2)*. This interference of sample light with reference light will be called 'heterodyne'. In practice from a turbid sample waves are obtained with a distribution of Doppler shifts, leading to a similar distribution of spectral components centered around *<sup>m</sup>* . Hence the shape of the peak around *<sup>m</sup>* corresponds to the Doppler shift distribution. The component of the reference light represented by the average phasor (OR) that still is at the original light source frequency will also interfere with the scattered light from the sample and thus, apart from the peak around the phase modulation frequency, a heterodyne component will also occur at low frequencies. Finally, the sample phasors S1 and S2, will mutually interfere to generate beats at frequency *<sup>D</sup>*<sup>1</sup> -*<sup>D</sup>*<sup>2</sup> , a component that we call 'homodyne'. Hence, the spectrum at low frequencies is a mixture of homodyne interference of light remitted by the sample, and heterodyne interference between sample light and unmodulated reference light, while the spectral component around the phase modulation frequency is the pure Doppler shift distribution. The power spectrum measured for widely different optical path lengths in the sample and the reference arm, contains the ordinary homodyne signal due to the mutual interference of scattered light over almost equal optical path lengths in the sample.

Fig. 2. Phasor diagram for the interfering fields (top) and the resulting power spectrum (bottom). (Figure reprinted from Fig. 1 in Ref. [2] Copyright (2008) with permission from SPIE).

Path Length Resolved Dynamic Light Scattering Measurements

with Suppressed Influence of Optical Properties Using Phase Modulated Low... 215

Fig. 3. Power spectra measured for water suspension of Polystyrene microspheres for two

**suppressed influence of optical absorption properties of surrounding tissue** 

To study the effect of absorption on Doppler shift, measurements were performed on three samples with identical scattering properties but increasing absorption levels [2]. The media were an aqueous suspension of 25% of Intralipid 20% [17] and the same suspensions with

The estimations of path length distributions of photons in the aqueous Intralipid suspension

*<sup>a</sup>* = 0.001 mm-1) and for identical suspensions with different absorption levels (0.50 mm-1 and 0.85 mm-1) are shown in fig.4. The estimation of the optical path length distribution is obtained for increasing absorption levels. The minimum path length is the same for all absorptions and is related to the fiber distance of 500 micrometer. At this path length all the distributions start to increase independent of the absorption. As the photons with longer path length have a greater probability to be absorbed in an increasingly absorbing medium, M0 decreases with the absorption. Hence the path length distribution narrows and shows a decrease in the average intensity as the absorption coefficient increases. Lambert-Beer's law can describe the effect of absorption on the path length distribution. According to the law of Lambert–Beer, the light intensity *I0* in an absorbing medium decays exponentially as *I(L)= I0* 

different peak optical phase shifts ( 0.51 and 2.04 radians), with the position of the retroreflector corresponding to an optical path length difference of 1.3 mm. (Figure reprinted from Fig. 1 in Ref. [16] Copyright (2008) with permission from Elsevier B.V.).

**4. Path length resolved dynamic light scattering measurements with** 

absorption coefficients of 0.50 mm-1 and 0.85 mm-1.

**matrices** 

(

The fundamental output quantity of a laser Doppler perfusion monitor is the first moment of the power spectrum *P()* of the detector signal; in general, the ith moment is being defined as

$$M\_i = \bigcap\_a^b P(o)a^i d o$$

Here *a* and *b* are device dependent low and high cut-off frequencies. With *i*=0, a quantity is obtained which is proportional to the concentration of moving red blood cells, while i=1 describes red blood cell flux, which is the product of concentration and the root mean square of the red cell velocity, at least for low blood concentrations [15]. In our instrument, for large phase modulation angles ( =2.04 radians) the power spectra contain interference peaks at both the phase modulation frequency and higher harmonics (Fig.3). Optical path length distributions are obtained by adding the areas of all interference peaks (after subtraction of the background noise, and within a bandwidth of 2 kHz around all center frequencies) in the power spectrum [16]. The area of the Doppler broadened peak, excluding the statically scattered light contribution at the interference peaks, forms an estimation of the amount of Doppler shifted light at that specific optical path length. The average Doppler shift corresponding to the Doppler shifted light is calculated from the weighted first moments (M1/M0) of the heterodyne peak at the modulation frequency, after correction for the sample signal and for the reference arm noise (in a bandwidth of 50 Hz-2 kHz close to the phase modulation frequency and its higher harmonics, indicated by *a* and *b* in Eq.2)

$$\mathcal{M}\_i = \sum\_{j=1}^{3} \int\_{j o\_m + a}^{j o\_m + b} P(o)(o - j o\_m)^i da o \tag{3}$$

To determine the parameters path length resolved, we measured the power spectra with the Mach-Zehnder low coherence setup. First, the background noise from the power spectrum around the modulation frequency (*m* = 6 kHz) is subtracted. The calculated *M0(0,b,m)* of the broadened interference peak (until b=2 kHz from the phase modulation peak) is proportional to the total number of detected photons for that given (by the reference arm) path length. The full width at half maximum (FWHM) of the interference signal in a statically scattering medium has a value *<sup>s</sup>* (s = 50-60 Hz in our system) whereas in the case of dynamic media a Doppler broadened spectral peak around the phase modulation frequency is formed [2]. The area of the Doppler broadened peak, excluding the statically scattered light contribution at the interference peaks, forms an estimation of the amount of the Doppler shifted light at that specific optical path length. For a given optical path length, the fraction of Doppler shifted photons *fD* is then given by *fD= M0(s,b,m)/ M0(0,b,m).* Here we regard the Doppler fraction *fD* as a measure of the concentration of particles moving in the static matrix. The average speed of the moving particles is represented by the average Doppler shift of the Doppler shifted fraction of the detected light, which in terms of equation (1) is <*ωD>=M1(s,b,m)/M0(s,b,m)* with the frequency *<sup>D</sup>* in Hz.

The fundamental output quantity of a laser Doppler perfusion monitor is the first moment

( ) *b*

Here *a* and *b* are device dependent low and high cut-off frequencies. With *i*=0, a quantity is obtained which is proportional to the concentration of moving red blood cells, while i=1 describes red blood cell flux, which is the product of concentration and the root mean square of the red cell velocity, at least for low blood concentrations [15]. In our instrument,

peaks at both the phase modulation frequency and higher harmonics (Fig.3). Optical path length distributions are obtained by adding the areas of all interference peaks (after subtraction of the background noise, and within a bandwidth of 2 kHz around all center frequencies) in the power spectrum [16]. The area of the Doppler broadened peak, excluding the statically scattered light contribution at the interference peaks, forms an estimation of the amount of Doppler shifted light at that specific optical path length. The average Doppler shift corresponding to the Doppler shifted light is calculated from the weighted first moments (M1/M0) of the heterodyne peak at the modulation frequency, after correction for the sample signal and for the reference arm noise (in a bandwidth of 50 Hz-2 kHz close to the phase modulation frequency and its higher harmonics, indicated by *a* and *b* in Eq.2)

( )( )

*i*

 

(3)

*m* = 6 kHz) is subtracted. The calculated *M0(0,b,*

*<sup>s</sup>* (s = 50-60 Hz in our system) whereas in the case

*s,b,*

*<sup>D</sup>* in Hz.

*m)/ M0(0,b,*

*m).* Here

*m)* of

*i a M P d* 

3

*m*

*j b*

*i m*

*M P jd*

To determine the parameters path length resolved, we measured the power spectra with the Mach-Zehnder low coherence setup. First, the background noise from the power spectrum

the broadened interference peak (until b=2 kHz from the phase modulation peak) is proportional to the total number of detected photons for that given (by the reference arm) path length. The full width at half maximum (FWHM) of the interference signal in a

of dynamic media a Doppler broadened spectral peak around the phase modulation frequency is formed [2]. The area of the Doppler broadened peak, excluding the statically scattered light contribution at the interference peaks, forms an estimation of the amount of the Doppler shifted light at that specific optical path length. For a given optical path length,

we regard the Doppler fraction *fD* as a measure of the concentration of particles moving in the static matrix. The average speed of the moving particles is represented by the average Doppler shift of the Doppler shifted fraction of the detected light, which in terms of

*m)* with the frequency

*m*

*j j a*

1

the fraction of Doppler shifted photons *fD* is then given by *fD= M0(*

*s,b,m)/M0(s,b,* *i*

*)* of the detector signal; in general, the ith moment is being

(2)

=2.04 radians) the power spectra contain interference

of the power spectrum *P(*

for large phase modulation angles (

around the modulation frequency (

equation (1) is <*ωD>=M1(*

statically scattering medium has a value

defined as

Fig. 3. Power spectra measured for water suspension of Polystyrene microspheres for two different peak optical phase shifts ( 0.51 and 2.04 radians), with the position of the retroreflector corresponding to an optical path length difference of 1.3 mm. (Figure reprinted from Fig. 1 in Ref. [16] Copyright (2008) with permission from Elsevier B.V.).

## **4. Path length resolved dynamic light scattering measurements with suppressed influence of optical absorption properties of surrounding tissue matrices**

To study the effect of absorption on Doppler shift, measurements were performed on three samples with identical scattering properties but increasing absorption levels [2]. The media were an aqueous suspension of 25% of Intralipid 20% [17] and the same suspensions with absorption coefficients of 0.50 mm-1 and 0.85 mm-1.

The estimations of path length distributions of photons in the aqueous Intralipid suspension (*<sup>a</sup>* = 0.001 mm-1) and for identical suspensions with different absorption levels (0.50 mm-1 and 0.85 mm-1) are shown in fig.4. The estimation of the optical path length distribution is obtained for increasing absorption levels. The minimum path length is the same for all absorptions and is related to the fiber distance of 500 micrometer. At this path length all the distributions start to increase independent of the absorption. As the photons with longer path length have a greater probability to be absorbed in an increasingly absorbing medium, M0 decreases with the absorption. Hence the path length distribution narrows and shows a decrease in the average intensity as the absorption coefficient increases. Lambert-Beer's law can describe the effect of absorption on the path length distribution. According to the law of Lambert–Beer, the light intensity *I0* in an absorbing medium decays exponentially as *I(L)= I0* 

Path Length Resolved Dynamic Light Scattering Measurements

experimental data and the calculated values (for n = 1.33,

length distributions have been correctly measured.

by the exponential decay function *exp(-L.*

0

200

400

600

Average Doppler shift (Hz)

SPIE).

particles are embedded.

800

1000

1200

with Suppressed Influence of Optical Properties Using Phase Modulated Low... 217

original Intralipid and the same suspensions with high absorption coefficients are mutually related by Lambert-Beer's law. The path length distributions of original Intralipid multiplied

figure 4 in linear and logarithmic scales. There is a good agreement between the

basis of Lambert-Beer's law up to an optical path length of 4.5 mm, which proves that path

 Intralipid, 0.001 mm-1 Intralipid+ink, 0.50 mm-1 Intralipid+ink, 0.85 mm-1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Optical path length (mm)

Fig. 5. The average Doppler shift extracted from the phase modulation peak, as a function of the optical path length for an aqueous Intralipid suspension with different absorption coefficients. (Figure reprinted from Fig. 5 in Ref. [2] Copyright (2008) with permission from

The average Doppler shift, measured from the width of Doppler broadened phase modulation interference peaks is represented in fig.5 as a function of the optical path length. The average Doppler shift increases with the optical path length, which can be expected from the increase in the number of scattering events with the optical path length. For a given

mm-1 the Doppler broadening of path length resolved heterodyne spectra is shown to be independent of the absorption level, for a given optical path length. Therefore, our results indicate that for absorption levels realistic for tissue, our method enables Doppler measurements independent of the absorption level of the medium in which the moving

medium with a constant scattering coefficient but absorption coefficients

*a/n)* and the experimental data are shown in

*<sup>a</sup>* = 0.50 and 0.85 mm-1) on the

*<sup>a</sup>* = 0.001 and 0.50

Fig. 4. Optical path length distributions estimated from the zero order moment of the phase modulation peak for an aqueous Intralipid suspension (a = 0.001 mm-1) and for identical suspensions with two different absorption coefficients (0.50 mm-1 and 0.85 mm-1), but equal reduced scattering coefficient (linear and logarithmic scales). The lines result from the application of Lambert-Beer's law on the experimental dataset for zero absorption. (Figure reprinted from Fig. 4 in Ref. [2] Copyright (2008) with permission from SPIE).

*exp(-L. a/n)* with *L* the optical path length*,* and *<sup>a</sup>* and *n* the absorption coefficient and the refractive index of the medium. To validate that the results shown in figure 4 represent the true optical path length distributions, we verify whether the path length distribution of the

*a a*

 IL,No absorption IL, =0.5 mm-1 IL, =0.85 mm-1

> *a*

)

)

Lambert-Beer's( =0.5mm-1

Lambert-Beer's( =0.85mm-1

*a*


012345678

Optical path length (mm)

refractive index of the medium. To validate that the results shown in figure 4 represent the true optical path length distributions, we verify whether the path length distribution of the

*<sup>a</sup>* and *n* the absorption coefficient and the

Fig. 4. Optical path length distributions estimated from the zero order moment of the phase modulation peak for an aqueous Intralipid suspension (a = 0.001 mm-1) and for identical suspensions with two different absorption coefficients (0.50 mm-1 and 0.85 mm-1), but equal reduced scattering coefficient (linear and logarithmic scales). The lines result from the application of Lambert-Beer's law on the experimental dataset for zero absorption. (Figure

reprinted from Fig. 4 in Ref. [2] Copyright (2008) with permission from SPIE).

*a/n)* with *L* the optical path length*,* and

0.0

10-10

10-9

M0(a.u)

*exp(-L.*

10-8

1.0x10-8

2.0x10-8

M0(a.u)

3.0x10-8

4.0x10-8

5.0x10-8

original Intralipid and the same suspensions with high absorption coefficients are mutually related by Lambert-Beer's law. The path length distributions of original Intralipid multiplied by the exponential decay function *exp(-L. a/n)* and the experimental data are shown in figure 4 in linear and logarithmic scales. There is a good agreement between the experimental data and the calculated values (for n = 1.33, *<sup>a</sup>* = 0.50 and 0.85 mm-1) on the basis of Lambert-Beer's law up to an optical path length of 4.5 mm, which proves that path length distributions have been correctly measured.

Fig. 5. The average Doppler shift extracted from the phase modulation peak, as a function of the optical path length for an aqueous Intralipid suspension with different absorption coefficients. (Figure reprinted from Fig. 5 in Ref. [2] Copyright (2008) with permission from SPIE).

The average Doppler shift, measured from the width of Doppler broadened phase modulation interference peaks is represented in fig.5 as a function of the optical path length. The average Doppler shift increases with the optical path length, which can be expected from the increase in the number of scattering events with the optical path length. For a given medium with a constant scattering coefficient but absorption coefficients *<sup>a</sup>* = 0.001 and 0.50 mm-1 the Doppler broadening of path length resolved heterodyne spectra is shown to be independent of the absorption level, for a given optical path length. Therefore, our results indicate that for absorption levels realistic for tissue, our method enables Doppler measurements independent of the absorption level of the medium in which the moving particles are embedded.

Path Length Resolved Dynamic Light Scattering Measurements

0.4

Copyright (2010) with permission from OSA).

decreases with increasing scattering of the static material.

sensitive to the influence of the scattering properties of the static medium.

0.6

Doppler fraction of photons

0.8

1.0

with Suppressed Influence of Optical Properties Using Phase Modulated Low... 219

'

'

'

,static=1.4 mm-1)

,static=0.8 mm-1)

,static=0.4 mm-1)

Expt, MC (s

Expt, MC (s

Expt, MC(s

 1-exp(-0.67\*OPL/1.33) 1-exp(-0.54\*OPL/1.33)

012345

Fig. 6. The fraction of Doppler shifted photons as a function of optical path length, as a result of experiments (open markers), Monte Carlo simulations (filled markers), and experimental decay models, one with a theoretical decay rate (dashed line), and one with the best fit to the experimental results (thick line). (Figure reprinted from Fig. 2 in Ref. [19]

length as expected theoretically and experimentally [3]. However, <*ωD>* also shows a different behaviour for optical path lengths smaller than 2mm. The average Doppler shift

The overall dependence on the static matrix optical properties on the Doppler shift is small, as depicted in figure 7. In the case of a higher scattering coefficient (s'=1.4mm-1), for optical path lengths between 2.5 and 3.5 mm, the Doppler broadening is lower in comparison with those obtained for the lower scattering levels. We may express the overall dependence of the measured concentration, represented by *fD*, and the particle velocity, represented by M1/M0, by their average value. This yields average Doppler fractions *<fD>* of 0.692, 0.685 and 0.694, and average Doppler shifts <M1/M0>, of 447.7, 447.9 and 442.2 Hz, for s'=0.4, 0.8 and 1.4 mm-1, respectively. Measurements at a single optical path length may be more suitable in practice. For single path lengths, figures 6 and 7 feature maximum variations of 10% for both fD and M1/M0. These results clearly illustrate that the average Doppler shift measured with the low coherence interferometer, averaged over all optical path lengths, is much less

Optical path length (mm)

## **5. Path length resolved dynamic light scattering measurements with suppressed influence of scattering properties of surrounding tissue matrices**

To study the effect of scattering of surrounding tissue matrices on Doppler shift, mixed static-dynamic scattering phantoms were prepared with aqueous suspensions of polystyrene microspheres of 4.7 m and 0.20 m respectively [19]. Three scattering phantoms with the same concentration of particles 0.20 m (g=0.18, s'=0.55 mm-1, a=0.001 mm-1) were prepared and two scattering levels of the static medium were realized (s'=1.4, 0.8, 0.4 mm-1). For estimating the flux of particles moving inside static matrices in absolute terms, we related the outcomes of our measurements to the optical and dynamical properties of the dynamic part of the medium. Here we focus on the concentration of moving particles, which may be retrieved from models which relate the measured Doppler fraction *fD* to the contribution of the dynamic part of the medium to the total scattering coefficient of the entire medium. We will consider a simple exponential decay model and compare it with the gold standard provided by the Monte Carlo simulation technique. In the exponential decay model we assume that the fraction of *unshifted* light decays exponentially with the traveled optical path length *lopt*. Consequently, the fraction of Doppler shifted photons will be given by *fD=1-exp(-s,dynℓopt/n),* with *s,dyn* the scattering coefficient of the ensemble of moving particles. Monte Carlo simulations were performed with the algorithm and software as described by De Mul [18]. The single mode fiber used in the experiment was modeled as a point source. Photon detection was performed in a ring with inner and outer radius of 0.25 and 0.35 mm (in agreement with the core diameter and position of the real detection fiber), concentric to the light beam for illumination. The simulated numerical apertures for illumination and detection were identical to the experimental values. The three mixed static-dynamic phantoms were exactly mimicked, with the scattering phase functions being calculated using Mie's theory. Photons which were scattered by the 0.20 m particles were given a Doppler label. For each medium and each path length, 20000 photons were detected.

Figure 6 shows the fraction of Doppler shifted photons *fD* as a function of the optical path length, for the three media. As expected, the measured Doppler fraction increases with the optical path length and the confounding influence of the surrounding static matrices is suppressed. Furthermore, fig. 6 shows the results of Monte Carlo simulations and for the exponential decay model *fD=1-exp(-s,dynℓopt/n)*. The models in general predict higher values of the Doppler fraction than the experimental values. Furthermore, the experimental and Monte Carlo results show biphasic behaviour, with a different trend for optical path lengths below and larger than 2 mm. For a given optical path length, *fD* is independent of the influence of static matrices, in particular for optical path lengths larger than 2 mm. Furthermore *fD* increases with optical path length with a trend that can be depicted by the simple exponential decay model. However, the theoretically predicted Doppler fractions are higher than the experimental values. Nevertheless, if in this model we define scattering coefficient s,dyn as a fitting parameter, it appears that the exponential decay model properly fits the observations for *ℓopt>*2mm, with s,dyn=0.55 mm-1. For *ℓopt>*2mm, the predicted Doppler fractions by Monte Carlo simulations are similar for the three media and are in good agreement with the experimental results. Figure 7 shows the average Doppler shift generated by the moving particles <*ωD>=M1(s,b,m)/M0(s,b,m)* as a function of optical path length. For optical path lengths larger than 2 mm, <*ωD>* increases linearly with optical path

**suppressed influence of scattering properties of surrounding tissue matrices**  To study the effect of scattering of surrounding tissue matrices on Doppler shift, mixed static-dynamic scattering phantoms were prepared with aqueous suspensions of polystyrene microspheres of 4.7 m and 0.20 m respectively [19]. Three scattering phantoms with the same concentration of particles 0.20 m (g=0.18, s'=0.55 mm-1, a=0.001 mm-1) were prepared and two scattering levels of the static medium were realized (s'=1.4, 0.8, 0.4 mm-1). For estimating the flux of particles moving inside static matrices in absolute terms, we related the outcomes of our measurements to the optical and dynamical properties of the dynamic part of the medium. Here we focus on the concentration of moving particles, which may be retrieved from models which relate the measured Doppler fraction *fD* to the contribution of the dynamic part of the medium to the total scattering coefficient of the entire medium. We will consider a simple exponential decay model and compare it with the gold standard provided by the Monte Carlo simulation technique. In the exponential decay model we assume that the fraction of *unshifted* light decays exponentially with the traveled optical path length *lopt*. Consequently, the fraction of Doppler shifted

**5. Path length resolved dynamic light scattering measurements with** 

*s,dynℓopt/n),* with

ensemble of moving particles. Monte Carlo simulations were performed with the algorithm and software as described by De Mul [18]. The single mode fiber used in the experiment was modeled as a point source. Photon detection was performed in a ring with inner and outer radius of 0.25 and 0.35 mm (in agreement with the core diameter and position of the real detection fiber), concentric to the light beam for illumination. The simulated numerical apertures for illumination and detection were identical to the experimental values. The three mixed static-dynamic phantoms were exactly mimicked, with the scattering phase functions being calculated using Mie's theory. Photons which were scattered by the 0.20 m particles were given a Doppler label. For each medium and each path length, 20000 photons

Figure 6 shows the fraction of Doppler shifted photons *fD* as a function of the optical path length, for the three media. As expected, the measured Doppler fraction increases with the optical path length and the confounding influence of the surrounding static matrices is suppressed. Furthermore, fig. 6 shows the results of Monte Carlo simulations and for the

of the Doppler fraction than the experimental values. Furthermore, the experimental and Monte Carlo results show biphasic behaviour, with a different trend for optical path lengths below and larger than 2 mm. For a given optical path length, *fD* is independent of the influence of static matrices, in particular for optical path lengths larger than 2 mm. Furthermore *fD* increases with optical path length with a trend that can be depicted by the simple exponential decay model. However, the theoretically predicted Doppler fractions are higher than the experimental values. Nevertheless, if in this model we define scattering coefficient s,dyn as a fitting parameter, it appears that the exponential decay model properly fits the observations for *ℓopt>*2mm, with s,dyn=0.55 mm-1. For *ℓopt>*2mm, the predicted Doppler fractions by Monte Carlo simulations are similar for the three media and are in good agreement with the experimental results. Figure 7 shows the average Doppler shift

> *s,b,m)/M0(s,b,*

length. For optical path lengths larger than 2 mm, <*ωD>* increases linearly with optical path

*s,dyn* the scattering coefficient of the

*m)* as a function of optical path

*s,dynℓopt/n)*. The models in general predict higher values

photons will be given by *fD=1-exp(-*

exponential decay model *fD=1-exp(-*

generated by the moving particles <*ωD>=M1(*

were detected.

Fig. 6. The fraction of Doppler shifted photons as a function of optical path length, as a result of experiments (open markers), Monte Carlo simulations (filled markers), and experimental decay models, one with a theoretical decay rate (dashed line), and one with the best fit to the experimental results (thick line). (Figure reprinted from Fig. 2 in Ref. [19] Copyright (2010) with permission from OSA).

length as expected theoretically and experimentally [3]. However, <*ωD>* also shows a different behaviour for optical path lengths smaller than 2mm. The average Doppler shift decreases with increasing scattering of the static material.

The overall dependence on the static matrix optical properties on the Doppler shift is small, as depicted in figure 7. In the case of a higher scattering coefficient (s'=1.4mm-1), for optical path lengths between 2.5 and 3.5 mm, the Doppler broadening is lower in comparison with those obtained for the lower scattering levels. We may express the overall dependence of the measured concentration, represented by *fD*, and the particle velocity, represented by M1/M0, by their average value. This yields average Doppler fractions *<fD>* of 0.692, 0.685 and 0.694, and average Doppler shifts <M1/M0>, of 447.7, 447.9 and 442.2 Hz, for s'=0.4, 0.8 and 1.4 mm-1, respectively. Measurements at a single optical path length may be more suitable in practice. For single path lengths, figures 6 and 7 feature maximum variations of 10% for both fD and M1/M0. These results clearly illustrate that the average Doppler shift measured with the low coherence interferometer, averaged over all optical path lengths, is much less sensitive to the influence of the scattering properties of the static medium.

Path Length Resolved Dynamic Light Scattering Measurements

0.0

Copyright (2008) with permission from SPIE).

1.0x10-7

2.0x10-7

M0(a.u)

3.0x10-7

with Suppressed Influence of Optical Properties Using Phase Modulated Low... 221

Doppler+Non-shifted

 Doppler Non-shifted


Fig. 8. Intensity of Doppler-shifted, nonshifted photons and the average Doppler shift as a function of optical path length measured in skin (Figure reprinted from Fig. 1 in Ref. [8]

Here we have presented optical path length distributions and path length resolved Doppler shifts of multiply scattered light, extracted from the spectral peak that was generated by phase modulation of the reference arm in a low coherence Mach Zehnder interferometer. As such, these data can also be obtained without modulation, but then we only can obtain information about photons which where Doppler shifted by the medium. Hence, phase modulation will enable us to measure path length distributions of static, and mixed static and dynamic media. A second advantage of using phase modulation is that the information can be shifted to higher frequencies, where often the noise level is lower and its spectrum is more flat than for low frequencies [2]. The path length resolved perfusion measurements presented here may overcome the inherent limitation of conventional LDPM that restrict its clinical usefulness, where the perfusion signal depends on an unknown photon path length. This will enable to correctly interpret or counter-act the inter-and intra-individual variations in the LDF readings introduced by the variance in tissue optical properties. This approach enables to discriminate between

Optical path length (mm)

0

100

200

M1/M0 (Hz)

300

 M1 /M0

Fig. 7. The weighted Doppler shift measured as a function of optical path length in the medium. (Figure reprinted from Fig. 3 in Ref. [20] Copyright (2010) with permission from OSA).

## **6. Path length resolved optical Doppler perfusion monitoring**

To assess the feasibility of the technique for path length resolved optical Doppler perfusion monitoring, measurements were performed on the skin of the dorsal side of the right forearm of a healthy human volunteer (Skin type- Type II) in the sitting position [20]. A probe holder (PH 08) was attached to the skin with a double-sided adhesive tape. The subject rested approximately 10 minutes prior to the measurements. Skin sites were avoided with visible large superficial blood vessels, hair and pigment variations.

The intensity of Doppler shifted and nonshifted photons measured in skin as a function of optical path length are shown in Fig. 8. The fraction of Doppler shifted photons and nonshifted photons averaged over the entire optical path length measured from the respective areas of the optical path lengths are 22 and 78%, respectively. As shown in Fig. 8, the weighted first moment *M1/M0* of the Doppler shifted light, which represents the average Doppler shift, increased with the optical path length due to the greater probability of interaction of photons with moving scatterers for large optical path lengths. Further *in vivo* studies were performed to measure the variations in perfusion to external stimuli, inter-and intra-individual variations in optical path lengths and path length resolved Doppler shifts, and to compare these results with the perfusion signal measured with a conventional LDPM [21].

' *s* ' *s* ' 

 =1.4 mm *<sup>s</sup>* -1 =0.8 mm-1 =0.4 mm-1

0123456

To assess the feasibility of the technique for path length resolved optical Doppler perfusion monitoring, measurements were performed on the skin of the dorsal side of the right forearm of a healthy human volunteer (Skin type- Type II) in the sitting position [20]. A probe holder (PH 08) was attached to the skin with a double-sided adhesive tape. The subject rested approximately 10 minutes prior to the measurements. Skin sites were avoided

The intensity of Doppler shifted and nonshifted photons measured in skin as a function of optical path length are shown in Fig. 8. The fraction of Doppler shifted photons and nonshifted photons averaged over the entire optical path length measured from the respective areas of the optical path lengths are 22 and 78%, respectively. As shown in Fig. 8, the weighted first moment *M1/M0* of the Doppler shifted light, which represents the average Doppler shift, increased with the optical path length due to the greater probability of interaction of photons with moving scatterers for large optical path lengths. Further *in vivo* studies were performed to measure the variations in perfusion to external stimuli, inter-and intra-individual variations in optical path lengths and path length resolved Doppler shifts, and to compare these results with the perfusion signal measured with a conventional LDPM

Fig. 7. The weighted Doppler shift measured as a function of optical path length in the medium. (Figure reprinted from Fig. 3 in Ref. [20] Copyright (2010) with permission from

**6. Path length resolved optical Doppler perfusion monitoring** 

with visible large superficial blood vessels, hair and pigment variations.

Optical path length (mm)

300

400

500

Average Doppler shift (Hz)

OSA).

[21].

600

700

Fig. 8. Intensity of Doppler-shifted, nonshifted photons and the average Doppler shift as a function of optical path length measured in skin (Figure reprinted from Fig. 1 in Ref. [8] Copyright (2008) with permission from SPIE).

Here we have presented optical path length distributions and path length resolved Doppler shifts of multiply scattered light, extracted from the spectral peak that was generated by phase modulation of the reference arm in a low coherence Mach Zehnder interferometer. As such, these data can also be obtained without modulation, but then we only can obtain information about photons which where Doppler shifted by the medium. Hence, phase modulation will enable us to measure path length distributions of static, and mixed static and dynamic media. A second advantage of using phase modulation is that the information can be shifted to higher frequencies, where often the noise level is lower and its spectrum is more flat than for low frequencies [2]. The path length resolved perfusion measurements presented here may overcome the inherent limitation of conventional LDPM that restrict its clinical usefulness, where the perfusion signal depends on an unknown photon path length. This will enable to correctly interpret or counter-act the inter-and intra-individual variations in the LDF readings introduced by the variance in tissue optical properties. This approach enables to discriminate between

Path Length Resolved Dynamic Light Scattering Measurements

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Handbook of Coherent Domain Optical Methods, Biomedical Diagnostics, Environment and Material Science, Tuchin, Valery V. (Ed.), 2004, Kluwer

particle flux in a static matrix with suppressed influence of optical properties, using

the Doppler-shifted photons resulting from interaction with the moving red blood cells and the non-shifted light scattered only by the surrounding static tissue matrices [20]. Another important feature of this approach is the tunable depth resolved perfusion information that can be achieved. By changing the optical path length in the reference arm, the photons migrated deeper into the tissue can be made to interfere with the reference light and thus enable to discriminate between the perfusion signal from superficial and deeper layers of tissue. Determination of superficial burn depth may be an important application of our technique [19]. However, further developments and fundamental research are required in developing this into a tool that is suitable for use in a clinical environment, with acceptable measurement times and suitable patient interfaces.

## **7. Conclusions**

To summarize, we have developed a new bio-optical method "Path length resolved optical Doppler perfusion monitoring," to determine path length distributions of multiple scattered light in static and dynamic turbid media using phase modulated coherence gated interferometry. We have shown that path length-resolved dynamic light scattering can measure the dynamic properties of a medium independent of its optical absorption properties, at least when absorption levels are applied in the range found for biological tissues. Furthermore, we showed that our method enables optical Doppler or dynamic light scattering measurements of dynamic media embedded in a static medium, with suppressed dependence of the effect of the scattering coefficient of the static matrix in which the moving particles are embedded. Also, we have presented the first path length resolved Doppler measurements of multiply scattered light from human skin. The results presented here show that this approach has potential applications in discriminating between statically and dynamically scattered light in the perfusion signal. In general, path length resolved dynamic light scattering, of which the basic technique is presented in this work, may overcome the influence of photon path lengths on the measured perfusion signal in laser Doppler techniques and makes it possible to perform depth resolved perfusion measurements with suppression of the confounding influence of optical properties in the tissue matrix.

## **8. References**


the Doppler-shifted photons resulting from interaction with the moving red blood cells and the non-shifted light scattered only by the surrounding static tissue matrices [20]. Another important feature of this approach is the tunable depth resolved perfusion information that can be achieved. By changing the optical path length in the reference arm, the photons migrated deeper into the tissue can be made to interfere with the reference light and thus enable to discriminate between the perfusion signal from superficial and deeper layers of tissue. Determination of superficial burn depth may be an important application of our technique [19]. However, further developments and fundamental research are required in developing this into a tool that is suitable for use in a clinical environment, with acceptable measurement times and suitable patient interfaces.

To summarize, we have developed a new bio-optical method "Path length resolved optical Doppler perfusion monitoring," to determine path length distributions of multiple scattered light in static and dynamic turbid media using phase modulated coherence gated interferometry. We have shown that path length-resolved dynamic light scattering can measure the dynamic properties of a medium independent of its optical absorption properties, at least when absorption levels are applied in the range found for biological tissues. Furthermore, we showed that our method enables optical Doppler or dynamic light scattering measurements of dynamic media embedded in a static medium, with suppressed dependence of the effect of the scattering coefficient of the static matrix in which the moving particles are embedded. Also, we have presented the first path length resolved Doppler measurements of multiply scattered light from human skin. The results presented here show that this approach has potential applications in discriminating between statically and dynamically scattered light in the perfusion signal. In general, path length resolved dynamic light scattering, of which the basic technique is presented in this work, may overcome the influence of photon path lengths on the measured perfusion signal in laser Doppler techniques and makes it possible to perform depth resolved perfusion measurements with suppression of the confounding influence of optical

[1] P. Shepherd, and P. Å. Öberg, Laser-Doppler Blood Flowmetry (Kluwer Academic,

[2] B. Varghese, V. Rajan, T. G. Van Leeuwen, and W. Steenbergen, "Path-length-resolved

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discrimination," Rev. Sci. Instrum. 63, 3220–3221(1992).

measurements of multiple scattered photons in static and dynamic turbid media using phase-modulated low-coherence interferometry," J. Biomed. Opt. 12(2),

Doppler broadening and optical path lengths of multiply scattered light by phase modulated low coherence interferometry," Opt. Express, 15 (15), 9157-9165 (2007). [4] G. Dougherty, "A laser Doppler flowmeter using variable coherence to effect depth

**7. Conclusions** 

properties in the tissue matrix.

Boston, 1990).

024020 (2007).

**8. References** 


**11** 

*Japan* 

**Interferometric Measurement** 

Masanori Ota, Shinsuke Udagawa, Tatsuro Inage and Kazuo Maeno

This chapter describes applications of interferometry to the shock tube experiments. The first topic is Laser Interferometric Computed Tomography (LICT) technique to realize the three-dimensional (3D) density measurement of high-speed and unsteady flow field behind shock waves discharging from nozzles. The second topic is measurement of propagating shock wave in micro-scale shock tube by interferometic approach. Micro-scale shock tube is being researched in several fields of science recently and micro-scale shock wave has possibilities of applications for various fields - medical, engineering, …etc. Clarifying the characteristics of micro-scale shock tube to generate the micro-scale shock wave is very

The purpose of this investigation is to develop Laser Interferometric Computed Tomography (LICT) technique to observe high-speed, unsteady and three-dimensional (3-D) flow field that includes shock wave, and to clarify 3-D flow phenomena induced by shock waves. In our previous study, 3-D complex flow discharged from a square nozzle and a pair of circular nozzles was measured by LICT tchnique (Maeno et al., 2005; Ota et al., 2005; Honma et al., 2003a, 2003b). The shock Mach number at the exits of the nozzles were both higher and lower than 2.0. As a result, various phenomena of 3-D flow field were clarified by several imaging technique such as pseudo-color images, pseudo-schlieren images by pseudo-schlieren technique, 3-D isopycnic images, etc. Three-dimensional and complex flow phenomena behind shock wave were elucidated precisely and reported (Honma et al., 2003a, Maeno et al., 2005), therefore this chapter reports measurement results mainly.

Diaphragmless shock tube is employed to produce a shock wave with good reproducibility. Figure 1 illustrates a schematic diagram of LICT experimental apparatus and observation system. The observation system consists of a CCD camera, a Mach-Zehnder interferometer, a pulsed nitrogen laser, a delay/pulse generator, an oscilloscope, and a personal computer. The shock wave is generated by a diaphragmless shock tube driver in the low-pressure tube of 3.1 meters in length, and its inner cross section is 40 mm x 40 mm square. A rotating plug

**2. Laser Interferometric Computed Tomography (LICT) technique** 

**1. Introduction** 

important step for the application.

**2.1 Experimental apparatus** 

**in Shock Tube Experiments** 

*Graduate School of Engineering, Chiba University* 


## **Interferometric Measurement in Shock Tube Experiments**

Masanori Ota, Shinsuke Udagawa, Tatsuro Inage and Kazuo Maeno *Graduate School of Engineering, Chiba University Japan* 

## **1. Introduction**

224 Interferometry – Research and Applications in Science and Technology

[20] B. Varghese, V. Rajan, T. G. van Leeuwen and W. Steenbergen, "Path length resolved optical Doppler perfusion monitoring," J. Biomed. Opt. Lett., 12(6):060508 (2007). [21] B. Varghese, V. Rajan, T. G. van Leeuwen and W. Steenbergen, "In vivo optical path

in Surgery and Medicine, 42(9), 692-700(2010).

lengths and path length resolved doppler shifts of multiply scattered light," Lasers

This chapter describes applications of interferometry to the shock tube experiments. The first topic is Laser Interferometric Computed Tomography (LICT) technique to realize the three-dimensional (3D) density measurement of high-speed and unsteady flow field behind shock waves discharging from nozzles. The second topic is measurement of propagating shock wave in micro-scale shock tube by interferometic approach. Micro-scale shock tube is being researched in several fields of science recently and micro-scale shock wave has possibilities of applications for various fields - medical, engineering, …etc. Clarifying the characteristics of micro-scale shock tube to generate the micro-scale shock wave is very important step for the application.

## **2. Laser Interferometric Computed Tomography (LICT) technique**

The purpose of this investigation is to develop Laser Interferometric Computed Tomography (LICT) technique to observe high-speed, unsteady and three-dimensional (3-D) flow field that includes shock wave, and to clarify 3-D flow phenomena induced by shock waves. In our previous study, 3-D complex flow discharged from a square nozzle and a pair of circular nozzles was measured by LICT tchnique (Maeno et al., 2005; Ota et al., 2005; Honma et al., 2003a, 2003b). The shock Mach number at the exits of the nozzles were both higher and lower than 2.0. As a result, various phenomena of 3-D flow field were clarified by several imaging technique such as pseudo-color images, pseudo-schlieren images by pseudo-schlieren technique, 3-D isopycnic images, etc. Three-dimensional and complex flow phenomena behind shock wave were elucidated precisely and reported (Honma et al., 2003a, Maeno et al., 2005), therefore this chapter reports measurement results mainly.

## **2.1 Experimental apparatus**

Diaphragmless shock tube is employed to produce a shock wave with good reproducibility. Figure 1 illustrates a schematic diagram of LICT experimental apparatus and observation system. The observation system consists of a CCD camera, a Mach-Zehnder interferometer, a pulsed nitrogen laser, a delay/pulse generator, an oscilloscope, and a personal computer.

The shock wave is generated by a diaphragmless shock tube driver in the low-pressure tube of 3.1 meters in length, and its inner cross section is 40 mm x 40 mm square. A rotating plug

Interferometric Measurement in Shock Tube Experiments 227

Fig. 3. Layout of rotating plug.

Fig. 4. Mach-Zehnder interferometer.

2.50.

**2.2 Projection data for LICT measurement** 

Figure 5 shows a finite-fringe interferogram at rotation angle

displacement of fringe pattern, *∆h* is interval of fringe pattern,

with Mach-Zehnder interferometer illustrated in Fig. 4. To obtain higher accuracy, we have taken interferogram of the lower half part of whole flow field, taking benefit of the two-axis symmetrical characteristics of the flow field. Thick blanked line indicates the central axis of rotating plug, and thin blanked line indicates the central axis of circular nozzles. The distance between these two central axes is 4mm. In this figure *zs* is a frontal position of the primary shock wave, *D* is a diameter of circular nozzle (4mm), and *zs /D* is the normalized frontal position of the primary shock wave. In this paper we discuss the case data of *zs*/*D* =

In LICT measurement, three-dimensional density distribution is reconstructed from multidirectional projection data. Projection data are calculated from Eq. 1, where *∆H* is

observation light, and *K* is Gladstone-Dale index. *∆H* and *∆h* in the right hand side in Eq. 1 are calculated from finite-fringe interferogram as shown in Fig. 5. Firstly, we calculate *∆h* at A section of no flow area. Secondly, we calculate *∆H* from displacement of fringe pattern

= 90º taken by CCD camera

is wavelength of

is installed at the end of the low-pressure tube. The duct is open to the low-pressure test section.

To obtain the 3-D image of flow field, we need multidirectional projection data for a reproducible flow. A set of experiments has been performed for several rotation angles at the combination of fixed initial gas conditions for the high-pressure chamber and the lowpressure tube. Figure 2 shows coordinate system of rotating plug relative to the light pass *s*. We define *x* and *y* axes as shown in Fig.2, where these axes rotate with rotating plug. The *z* is central axis of rotating plug and is perpendicular to *x* and *y* axes. Rotation angle can be controlled from outside the shock tube with introduced rotation driving equipment. The experiment is performed for 19 rotation angles between 0 degree and 90 degrees at fivedegree intervals while the light path *s* is fixed, taking benefit of the two-axis symmetrical characteristics of the flow field. The 3-D density distribution is reconstructed from a set of projection data for the same *Mi* and *zs*. The Mach number of the incident shock wave *Mi* is calculated by pressure jump across the shock wave at the pressure transducer installed at 61mm ahead of the inlet of the rotating plug as shown in Fig. 3. In this paper *Mi* is fixed to 2.0. Mach-Zehnder interferometer is conventional one as shown in Fig. 4

Fig. 1. Schematic diagram of experimental apparatus.

Fig. 2. Coordinate system of the rotating plug.

Fig. 3. Layout of rotating plug.

is installed at the end of the low-pressure tube. The duct is open to the low-pressure test

To obtain the 3-D image of flow field, we need multidirectional projection data for a reproducible flow. A set of experiments has been performed for several rotation angles at the combination of fixed initial gas conditions for the high-pressure chamber and the lowpressure tube. Figure 2 shows coordinate system of rotating plug relative to the light pass *s*. We define *x* and *y* axes as shown in Fig.2, where these axes rotate with rotating plug. The *z*

controlled from outside the shock tube with introduced rotation driving equipment. The experiment is performed for 19 rotation angles between 0 degree and 90 degrees at fivedegree intervals while the light path *s* is fixed, taking benefit of the two-axis symmetrical characteristics of the flow field. The 3-D density distribution is reconstructed from a set of projection data for the same *Mi* and *zs*. The Mach number of the incident shock wave *Mi* is calculated by pressure jump across the shock wave at the pressure transducer installed at 61mm ahead of the inlet of the rotating plug as shown in Fig. 3. In this paper *Mi* is fixed to

can be

is central axis of rotating plug and is perpendicular to *x* and *y* axes. Rotation angle

2.0. Mach-Zehnder interferometer is conventional one as shown in Fig. 4

Fig. 1. Schematic diagram of experimental apparatus.

Fig. 2. Coordinate system of the rotating plug.

section.

Fig. 4. Mach-Zehnder interferometer.

## **2.2 Projection data for LICT measurement**

Figure 5 shows a finite-fringe interferogram at rotation angle = 90º taken by CCD camera with Mach-Zehnder interferometer illustrated in Fig. 4. To obtain higher accuracy, we have taken interferogram of the lower half part of whole flow field, taking benefit of the two-axis symmetrical characteristics of the flow field. Thick blanked line indicates the central axis of rotating plug, and thin blanked line indicates the central axis of circular nozzles. The distance between these two central axes is 4mm. In this figure *zs* is a frontal position of the primary shock wave, *D* is a diameter of circular nozzle (4mm), and *zs /D* is the normalized frontal position of the primary shock wave. In this paper we discuss the case data of *zs*/*D* = 2.50.

In LICT measurement, three-dimensional density distribution is reconstructed from multidirectional projection data. Projection data are calculated from Eq. 1, where *∆H* is displacement of fringe pattern, *∆h* is interval of fringe pattern, is wavelength of observation light, and *K* is Gladstone-Dale index. *∆H* and *∆h* in the right hand side in Eq. 1 are calculated from finite-fringe interferogram as shown in Fig. 5. Firstly, we calculate *∆h* at A section of no flow area. Secondly, we calculate *∆H* from displacement of fringe pattern

Interferometric Measurement in Shock Tube Experiments 229

Figure 7 shows the pseudo-color images of normalized density distribution and the pseudo-

section. In pseudo-color image the density level is shown in visible-light color spectrum. The black line with white blank indicates the rotating plug with circular nozzles, where white blank corresponds to a pair of circular nozzles. The experimental condition is *Mi* = 2.0, *zs/D* = 2.50 as mentioned above. The smoothing filter is applied to the reconstructed 3-D density distribution with FBP algorithm before the calculation of the density gradient, for reducing the noise in the resultant pseudo-schlieren image. In pseudo-color image two vortex rings around the discharged flow from two circular open cylinders are illustrated. The primary shock wave (PSW), secondary shock wave (SSW), contact surface (CS1, CS2), and

Pseudo-color image of normalized density distribution at *x-y* cross-section is shown in Fig. 8. Position of the *x-y* cross section is indicated with normalized distance *z/D* where *z* is the distance between the rotating plug's wall and the *x-y* cross section. In these images cross sectional shape of PSW, TSW and two vortex rings around the discharged flow is exhibited. Figure 9 shows the pseudo-schlieren images at *x-y* cross-sections. In these images PSW, the shock-vortex interaction between TSW and vortex, shock-shock interaction between TSW

 /

0 at *y-z* cross-

schlieren images indicating the magnitude of the density gradient

transmitted shock wave (TSW) are exhibited clearly in pseudo-schlieren image.

Fig. 7. Pseudo-color (left) and pseudo-schlieren (right) image at *y-z* cross-section.

**2.3 Results and discussion** 

and PSW are exhibited clearly.

comparing with A section. We obtain the integrated value of density change along the light pass from Eq. 1. Figure 6 shows the calculated projection data at B section of rotation angle = 90º. The horizontal axis indicates the vertical position of the finite-fringe image shown in Fig. 5, the origin of the coordinates is located in left top position of this image. The vertical axis of Fig.6 indicates the calculated integrated value of density change and its value is normalized with the initial density ( 0) at A section. We repeat this process for all projection angles at one cross section to obtain the multidirectional projection data at one cross section. Then 2-D density distribution is reconstructed from these projection data with appropriate reconstruction algorithm. Finally, 3-D density distribution is obtained as collection of reconstructed 2-D density distribution.

$$\int\_{0}^{d} \left\{ \rho(\mathbf{x}, y, z) - \rho\_{0} \right\} ds = \frac{\Delta H}{\Delta \mathbf{l}} \frac{\lambda}{K} \tag{1}$$

Fig. 5. Finite-fringe interferogram at rotation angle = 90º.

Fig. 6. Calculated projection data at B section in Fig. 5.

## **2.3 Results and discussion**

228 Interferometry – Research and Applications in Science and Technology

comparing with A section. We obtain the integrated value of density change along the light pass from Eq. 1. Figure 6 shows the calculated projection data at B section of rotation angle

= 90º. The horizontal axis indicates the vertical position of the finite-fringe image shown in Fig. 5, the origin of the coordinates is located in left top position of this image. The vertical axis of Fig.6 indicates the calculated integrated value of density change and its value is

angles at one cross section to obtain the multidirectional projection data at one cross section. Then 2-D density distribution is reconstructed from these projection data with appropriate reconstruction algorithm. Finally, 3-D density distribution is obtained as collection of

<sup>0</sup> <sup>0</sup> , , *<sup>d</sup> <sup>H</sup> x y z ds h K*

= 90º.

0 100 200 300 400 500

Distance (Pixel)

 

0) at A section. We repeat this process for all projection

(1)

normalized with the initial density (

reconstructed 2-D density distribution.

Fig. 5. Finite-fringe interferogram at rotation angle


Fig. 6. Calculated projection data at B section in Fig. 5.




Integrated value of density change (/

 

)

0

20

40

60

Central axis (*z*)

80

Figure 7 shows the pseudo-color images of normalized density distribution and the pseudoschlieren images indicating the magnitude of the density gradient /0 at *y-z* crosssection. In pseudo-color image the density level is shown in visible-light color spectrum. The black line with white blank indicates the rotating plug with circular nozzles, where white blank corresponds to a pair of circular nozzles. The experimental condition is *Mi* = 2.0, *zs/D* = 2.50 as mentioned above. The smoothing filter is applied to the reconstructed 3-D density distribution with FBP algorithm before the calculation of the density gradient, for reducing the noise in the resultant pseudo-schlieren image. In pseudo-color image two vortex rings around the discharged flow from two circular open cylinders are illustrated. The primary shock wave (PSW), secondary shock wave (SSW), contact surface (CS1, CS2), and transmitted shock wave (TSW) are exhibited clearly in pseudo-schlieren image.

Pseudo-color image of normalized density distribution at *x-y* cross-section is shown in Fig. 8. Position of the *x-y* cross section is indicated with normalized distance *z/D* where *z* is the distance between the rotating plug's wall and the *x-y* cross section. In these images cross sectional shape of PSW, TSW and two vortex rings around the discharged flow is exhibited. Figure 9 shows the pseudo-schlieren images at *x-y* cross-sections. In these images PSW, the shock-vortex interaction between TSW and vortex, shock-shock interaction between TSW and PSW are exhibited clearly.

Fig. 7. Pseudo-color (left) and pseudo-schlieren (right) image at *y-z* cross-section.

Interferometric Measurement in Shock Tube Experiments 231

Three-dimensional observation of flow field around object will be important to apply CT measurement to more general case. Therefore LICT technique has been applied to the flow field around a circular cylinder as shown in Fig. 10. Diameter of a cylinder is 4mm, length 10mm and it was installed at intervals of 8mm from a cylindrical nozzle. In this case the problem is reconstruction has to be done from incomplete projection data. The object in the observation area blocks off the observation light for interferometry as shown in Fig. 11, the calculated projection data also contain the blank part which corresponds to the position of the circular cylinder as indicated in Fig. 12. This section describes the reconstruction of flow

**3. LICT Measurement of flow field around the object** 

field around object with ART algorithm and results.

Fig. 11. Finite-fringe interferogram at rotation angle

= 90º.

Fig. 10. Coordinate system of rotating plug and finite-fringe interferogram.

Fig. 8. Pseudo-color images of density distribution at *x-y* cross-sections.

Fig. 9. Pseudo-schlieren images of density gradient ( /<sup>0</sup> ) at *x-y* cross-sections.

Fig. 8. Pseudo-color images of density distribution at *x-y* cross-sections.

Fig. 9. Pseudo-schlieren images of density gradient (

 /

<sup>0</sup> ) at *x-y* cross-sections.

## **3. LICT Measurement of flow field around the object**

Three-dimensional observation of flow field around object will be important to apply CT measurement to more general case. Therefore LICT technique has been applied to the flow field around a circular cylinder as shown in Fig. 10. Diameter of a cylinder is 4mm, length 10mm and it was installed at intervals of 8mm from a cylindrical nozzle. In this case the problem is reconstruction has to be done from incomplete projection data. The object in the observation area blocks off the observation light for interferometry as shown in Fig. 11, the calculated projection data also contain the blank part which corresponds to the position of the circular cylinder as indicated in Fig. 12. This section describes the reconstruction of flow field around object with ART algorithm and results.

Fig. 10. Coordinate system of rotating plug and finite-fringe interferogram.

Fig. 11. Finite-fringe interferogram at rotation angle = 90º.

Interferometric Measurement in Shock Tube Experiments 233

Larger projection number will better to obtain higher accuracy, however it will cause inefficiency of experiments and data processing. Figure 13 and 14 show that ART is effective for reconstruction from incomplete projection data when LICT measurement is applied to the measurement of flow field around a circular cylinder. In this chapter blanked part in projection data where a circular cylinder is captured in projection plane does not contribute to reconstruction. Projection number is 19 rotation angles from 0º to 90º at 5º intervals as

Fig. 13. Comparison between FBP and ART at *y-z* cross-section.

Fig. 14. Comparison between FBP and ART at *x-y* cross-section.

previous case.

Fig. 12. Calculated projection data at A-A' section in Fig. 11.

## **3.1 Reconstruction algorithm**

In previous section, 3-D density distribution was reconstructed by FBP (Filtered Back Projection) with Shepp and Logan type filter (Shepp & Logan, 1974). FBP is the most common technique for the tomographic reconstruction. In this chapter the density distribution of unsteady flow field around a circular cylinder is reconstructed by ART (Algebraic Reconstruction Technique). This technique is one of the iterative reconstruction method and consists of assuming that the cross section consists of an array of unknowns, and then setting up algebraic equations for the unknowns in terms of the measured projection data (Kak & Slaney, 1988). ART is much simpler than FBP method which is the transform-based method and we have used in our previous study. For FBP a large number of projections is required for higher accuracy in reconstructed image, in the situation where it is not possible to obtain these projections the reconstructed image is suffer from many streaky noise. The reconstruction from incomplete projection data is more amenable to solution by ART. Figure 13 shows comparison between FBP and ART reconstruction. Left figure is pseudo-color image at the same plane illustrated in Fig. 7 and right figure is pseudo-schlieren image. Upper half image is reconstructed by FBP and lower half is by ART. In pseudo-color image, reconstructed density distribution by ART is clearer than FBP result. The smoothing filter is applied to the reconstructed density distribution before the calculation of density gradient for FBP result as mentioned above. On the other hand, no smoothing filter is applied for ART result however density gradient is captured distinctly. Figure 14 is comparison between FBP and ART illustrated by pseudo-color and pseudoschlieren image at *x-y* cross section (normalized position *z/D* = 0.74). In pseudo-color image some radial noises from the center can be seen in FBP result (upper), however these noises are reduced in ART result (lower) drastically. Though no smoothing filter is applied, flow phenomena are clearly seen with ART result in pseudo-shclieren image. In our experiments density distribution is reconstructed from 19 projections that is based on our experiences.

In previous section, 3-D density distribution was reconstructed by FBP (Filtered Back Projection) with Shepp and Logan type filter (Shepp & Logan, 1974). FBP is the most common technique for the tomographic reconstruction. In this chapter the density distribution of unsteady flow field around a circular cylinder is reconstructed by ART (Algebraic Reconstruction Technique). This technique is one of the iterative reconstruction method and consists of assuming that the cross section consists of an array of unknowns, and then setting up algebraic equations for the unknowns in terms of the measured projection data (Kak & Slaney, 1988). ART is much simpler than FBP method which is the transform-based method and we have used in our previous study. For FBP a large number of projections is required for higher accuracy in reconstructed image, in the situation where it is not possible to obtain these projections the reconstructed image is suffer from many streaky noise. The reconstruction from incomplete projection data is more amenable to solution by ART. Figure 13 shows comparison between FBP and ART reconstruction. Left figure is pseudo-color image at the same plane illustrated in Fig. 7 and right figure is pseudo-schlieren image. Upper half image is reconstructed by FBP and lower half is by ART. In pseudo-color image, reconstructed density distribution by ART is clearer than FBP result. The smoothing filter is applied to the reconstructed density distribution before the calculation of density gradient for FBP result as mentioned above. On the other hand, no smoothing filter is applied for ART result however density gradient is captured distinctly. Figure 14 is comparison between FBP and ART illustrated by pseudo-color and pseudoschlieren image at *x-y* cross section (normalized position *z/D* = 0.74). In pseudo-color image some radial noises from the center can be seen in FBP result (upper), however these noises are reduced in ART result (lower) drastically. Though no smoothing filter is applied, flow phenomena are clearly seen with ART result in pseudo-shclieren image. In our experiments density distribution is reconstructed from 19 projections that is based on our experiences.

Fig. 12. Calculated projection data at A-A' section in Fig. 11.

**3.1 Reconstruction algorithm** 

Larger projection number will better to obtain higher accuracy, however it will cause inefficiency of experiments and data processing. Figure 13 and 14 show that ART is effective for reconstruction from incomplete projection data when LICT measurement is applied to the measurement of flow field around a circular cylinder. In this chapter blanked part in projection data where a circular cylinder is captured in projection plane does not contribute to reconstruction. Projection number is 19 rotation angles from 0º to 90º at 5º intervals as previous case.

Fig. 13. Comparison between FBP and ART at *y-z* cross-section.

Fig. 14. Comparison between FBP and ART at *x-y* cross-section.

Interferometric Measurement in Shock Tube Experiments 235

Fig. 16. Pseudo-color images of density distribution at *x-y* cross-sections.

Fig. 17. Pseudo-schlieren images of density gradient (

 / 

<sup>0</sup> ) at *x-y* cross-sections.

## **3.2 Results and discussion**

The resultant image in *y-z* cross section is illustrated in Fig. 15. The left is pseudo-color image of normalized density distribution and right is pseudo-schlieren image. The vertical thick line with white blank indicates rotating plug's wall and white blank indicates a cylindrical nozzle. The position of a circular cylinder is indicated with two horizontal blanked lines. The vortex around discharging flow from a cylindrical nozzle is identified in pseudo-color image and primary shock wave (PSW), secondary shock wave (SSW), contact surface (CS) and reflected shock wave (RSW) from circular cylinder is clearly seen in pseudo-schlieren image.

Figure 16 illustrates pseudo-color images of normalized density distribution in *x-y* cross section. Six cross sections (position A∼F) that are parallel to rotating plug's wall are indicated. The normalized position of cross section (*z/D*) is shown at upper side of each image. The position of a circular cylinder is indicated with blanked circle. In position A∼C, the cross sectional shape of vortex around the discharging flow from a cylindrical nozzle is captured. The reflected shock wave (RSW) from circular cylinder is seen in position D∼F. Four slanting noise from a circular cylinder is appeared in position E and F, this is influence of reconstruction from incomplete projection data.

Fig. 15. Pseudo-color (left) and pseudo-schlieren (right) image at *y-z* cross-section.

The resultant image in *y-z* cross section is illustrated in Fig. 15. The left is pseudo-color image of normalized density distribution and right is pseudo-schlieren image. The vertical thick line with white blank indicates rotating plug's wall and white blank indicates a cylindrical nozzle. The position of a circular cylinder is indicated with two horizontal blanked lines. The vortex around discharging flow from a cylindrical nozzle is identified in pseudo-color image and primary shock wave (PSW), secondary shock wave (SSW), contact surface (CS) and reflected shock wave (RSW) from circular cylinder is clearly seen in

Figure 16 illustrates pseudo-color images of normalized density distribution in *x-y* cross section. Six cross sections (position A∼F) that are parallel to rotating plug's wall are indicated. The normalized position of cross section (*z/D*) is shown at upper side of each image. The position of a circular cylinder is indicated with blanked circle. In position A∼C, the cross sectional shape of vortex around the discharging flow from a cylindrical nozzle is captured. The reflected shock wave (RSW) from circular cylinder is seen in position D∼F. Four slanting noise from a circular cylinder is appeared in position E and F, this is influence

Fig. 15. Pseudo-color (left) and pseudo-schlieren (right) image at *y-z* cross-section.

**3.2 Results and discussion** 

pseudo-schlieren image.

of reconstruction from incomplete projection data.

Fig. 16. Pseudo-color images of density distribution at *x-y* cross-sections.

Fig. 17. Pseudo-schlieren images of density gradient ( / <sup>0</sup> ) at *x-y* cross-sections.

Interferometric Measurement in Shock Tube Experiments 237

pressure. The buffer chamber and the region behind the main piston are separated by the movement of a sub piston as shown in the left side of Fig. 19. Subsequently, the main piston moves to the left side by that the driver gas flows into the region behind the main piston. The high-pressure chamber and the shock tube are also separated by movement of the main piston. Test gas is introduced from the test gas inlet after the introduction of the driver gas. The buffer chamber maintains a low-pressure state, less than 0.1 kPa. The driver gas pressures *p*4 is 0.9 MPa and the test gas pressure *p*1 is maintained at atmospheric pressure. The sub piston moves to the right side by controlling the electromagnetic valve, and then the main piston rapidly moves to the left side by the connection of the buffer chamber and the region behind the main piston. The shock tube and the high-pressure chamber are

connected by rapid movement of a main piston as shown in the right side of Fig. 19.

Fig. 19. Schematic drawing of the operating procedure of the diaphragmless driver section.

Generally the direct measurement of shock wave, by using pressure transducer is very difficult especially in the shock wave propagating in small diameter tube, caused by the difference of the representative scales between the tube and the transducer. Thus,

Figure 20 shows the laser differential interferometer used in this experiment. The laser differential interfeormeter is the polarization phase difference interferometer by using three Wollaston prisms, developed by Smeets (Smeets, 1972; Smeets, 1977). The bright and dark,

contactless measurement by using laser interferometry is very important and useful.

observed in the interferometric fringe, is measured by photo detectors.

Fig. 20. Schematic drawing of the laser differential interferometer.

piston movement driver gas flow

piston movement driver gas flow test gas flow

**4.2 Measurement system** 

**4.2.1 Laser differential interferometer** 

## **4. Velocity measurement of shock wave in micro-scale shock tube by laser differential interferometer**

Recently, the micro-shock waves have attracted attention of researchers in several fields of science. The shock wave propagating in 5.3mm inner diameter tube was measured using pressure transducers by Brouillette. As a consequence, it is experimentally clarified that the shear stress and the heat transfer between a test gas and a wall lead to significant deviations from the normal theory, especially in a small diameter shock tube (Brouillette, 2003). However, it is predicted that the pressure transducer becomes disturbance in case of using smaller tube from the difference of the representative scales between the tube and the transducer. Thus, it is very important to establish the contactless measurement method for the shock wave propagating in small diameter tubes. In this study, we measured the velocities of shock wave and the density ratios across the shock wave, generated by originally developed diaphragmless driver section, propagating in 2 and 3 mm inner diameter tubes by using laser differential interferometer.

## **4.1 Diaghragmless driver section with two pistons**

Figure 18 shows a schematic drawing of the diahpragmless driver section, we developed; it consists of a main piston instead of a diaphragm, a sub piston, a buffer section, and a highpressure section.

Fig. 18. Schematic drawing of the diaphragmless driver section with two pistons.

First, the buffer chamber and the region behind the sub piston are evacuated by a rotary pump, then the sub piston moves to the right side by the pressure difference. The pressure difference occurs between the high-pressure section and the region behind the main piston by the connection between the buffer chamber and the region behind the main piston after the sub piston movement. After movement of the pistons, the high-pressure section and the shock tube are evacuated by a rotary pump. The driver gas is introduced from the driver gas inlet after evacuation, and then the sub piston moves to the left side by the inlet gas

Recently, the micro-shock waves have attracted attention of researchers in several fields of science. The shock wave propagating in 5.3mm inner diameter tube was measured using pressure transducers by Brouillette. As a consequence, it is experimentally clarified that the shear stress and the heat transfer between a test gas and a wall lead to significant deviations from the normal theory, especially in a small diameter shock tube (Brouillette, 2003). However, it is predicted that the pressure transducer becomes disturbance in case of using smaller tube from the difference of the representative scales between the tube and the transducer. Thus, it is very important to establish the contactless measurement method for the shock wave propagating in small diameter tubes. In this study, we measured the velocities of shock wave and the density ratios across the shock wave, generated by originally developed diaphragmless driver section, propagating in 2 and 3 mm inner

Figure 18 shows a schematic drawing of the diahpragmless driver section, we developed; it consists of a main piston instead of a diaphragm, a sub piston, a buffer section, and a high-

optical window

to rotary pump

driver gas inlet

buffer chamber

sub piston

main piston

high pressure section

to rotary pump test gas inlet

First, the buffer chamber and the region behind the sub piston are evacuated by a rotary pump, then the sub piston moves to the right side by the pressure difference. The pressure difference occurs between the high-pressure section and the region behind the main piston by the connection between the buffer chamber and the region behind the main piston after the sub piston movement. After movement of the pistons, the high-pressure section and the shock tube are evacuated by a rotary pump. The driver gas is introduced from the driver gas inlet after evacuation, and then the sub piston moves to the left side by the inlet gas

e.m. valve

optical window

Fig. 18. Schematic drawing of the diaphragmless driver section with two pistons.

**4. Velocity measurement of shock wave in micro-scale shock tube by laser** 

**differential interferometer** 

pressure section.

to rotary pump

diameter tubes by using laser differential interferometer.

**4.1 Diaghragmless driver section with two pistons** 

shock tube

pressure. The buffer chamber and the region behind the main piston are separated by the movement of a sub piston as shown in the left side of Fig. 19. Subsequently, the main piston moves to the left side by that the driver gas flows into the region behind the main piston. The high-pressure chamber and the shock tube are also separated by movement of the main piston. Test gas is introduced from the test gas inlet after the introduction of the driver gas. The buffer chamber maintains a low-pressure state, less than 0.1 kPa. The driver gas pressures *p*4 is 0.9 MPa and the test gas pressure *p*1 is maintained at atmospheric pressure. The sub piston moves to the right side by controlling the electromagnetic valve, and then the main piston rapidly moves to the left side by the connection of the buffer chamber and the region behind the main piston. The shock tube and the high-pressure chamber are connected by rapid movement of a main piston as shown in the right side of Fig. 19.

Fig. 19. Schematic drawing of the operating procedure of the diaphragmless driver section.

## **4.2 Measurement system**

Generally the direct measurement of shock wave, by using pressure transducer is very difficult especially in the shock wave propagating in small diameter tube, caused by the difference of the representative scales between the tube and the transducer. Thus, contactless measurement by using laser interferometry is very important and useful.

## **4.2.1 Laser differential interferometer**

Figure 20 shows the laser differential interferometer used in this experiment. The laser differential interfeormeter is the polarization phase difference interferometer by using three Wollaston prisms, developed by Smeets (Smeets, 1972; Smeets, 1977). The bright and dark, observed in the interferometric fringe, is measured by photo detectors.

Fig. 20. Schematic drawing of the laser differential interferometer.

Interferometric Measurement in Shock Tube Experiments 239

The voltage signal *V* is stored to an oscilloscope. Here, k is the constant defined from the

 0 

as 2 in Fig. 22. Here, the refractive index of the medium is expressed as *n* and the length of the medium is obtained as the inner diameter of the shock tube *d*, the optical path difference

Here, *n*2 and *n*1 are the refractive index of the gas behind the shock wave and the gas in front of the shock wave, respectively. The relation between the refractive index *n* and the density

n-1=K

0

0 and *K* are the density at the normal condition and the non-dimensional Gladstone-Dale constant, respectively. The relation between the voltage signal *V* and the density

> <sup>0</sup> 2 1 0

Here, the voltage signal *V* can be linearly-approximated to *V* at *V*=0 neighborhood as

 

2 dK V=V cos - 

is the gain of the amplifier. Figure 22 shows the relation between the voltage signals

<sup>2</sup> V= V -V = kA cos =V cos

and the optical path difference between both beams. The gain of the amplifier

1 is obtained from Eq. 5, 6 and 7 as follows:

Fig. 22. The interfering signals from the laser differential interferometer.

 -*V*

 

= n -n d 2 1 (6)

(8)

(5)

 

  is amplified by the

is denoted

(7)

characteristics of the photo detector. The voltage signal *V* = *V*

 

is expressed by Gladstone-Dale's formula as follows:

between both beams is obtained as follows:

differential amplifier circuit.

Here, 

Here, 

difference

2-

follows, as shown in Fig. 22:

The laser beam from He-Ne laser head is separated into two parallel and orthogonal polarizing components after passing through the first Wollaston prism and a convex lens. The two-polarized and parallel beams are converged on the second Wollaston prism by a second convex lens after passing through the shock tube. Where, there is no interferometric fringe from the combined beams caused by the discrepancy between the polarization planes of both beams. Here, the shock wave propagates in the shock tube in a direction from the bottom to top in Fig. 20, and it is considered that the shock wave arrives between two beams. The density in front of the shock wave and the density behind the shock wave are 1 and 2, respectively. Two beams passing through the same optical path as light vectors are *Ey* and *Ez*, respectively as shown in Fig. 21.

Fig. 21. Schematic drawing of the optical vectors

The light vectors *Ey* and *Ez* are denoted as follows:

$$\mathbf{E\_y = A\_y \cos\left(\frac{2\pi}{\lambda}\mathbf{x}\right)} \mathbf{\hat{E\_z = A\_z \cos\left\{\frac{2\pi}{\lambda}(\mathbf{x} + \Delta\phi)\right\}}} \tag{2}$$

Here *Ay* and *Az* are the constants for a laser light intensity. *x*, , and are the distance of the traveling direction of laser, the wavelength of laser, and the optical path difference between both beams, respectively. The optical path difference is occurred by the density difference across the shock wave. The third Wollaston prism should be settled as rotated 45 degrees for each polarization planes. The light vector *Ey* is separated into *Ey* and *Ey*, respectively. Similarly, The light vector *Ez* is separated into *Ez* and *Ez*, respectively. Here, and are the axes inclined 45 degree from *y* and *z* axes. After passing through the second Wollaston prism, the agreement of polarization plane causes interference at and directions, respectively. The intensities of interfering lights for and directions are denoted as follows:

$$\mathbf{I}\_{\eta} = \left\langle \left| \mathbf{E}\_{\mathbf{y},\eta} + \mathbf{E}\_{x\eta} \right|^{2} \right\rangle = \mathbf{A}^{2} \cos^{2} \left( \frac{\pi}{\mathcal{L}} \Delta \phi \right), \quad \mathbf{I}\_{\zeta} = \left\langle \left| \mathbf{E}\_{\mathbf{y},\zeta} + \mathbf{E}\_{x\zeta} \right|^{2} \right\rangle = \mathbf{A}^{2} \sin^{2} \left( \frac{\pi}{\mathcal{L}} \Delta \phi \right) \tag{3}$$

Here, *Ay* = *Az* = *A*. The intensities of two polarized interfering beams *I* and *I* are converted into the voltage signals *V* and *V*by the photo detectors as follows:

$$\mathbf{V}\_{\eta} = \mathbf{k} \mathbf{I}\_{\eta} = \mathbf{k} \mathbf{A}^{2} \cos^{2} \left(\frac{\pi}{\mathcal{A}} \Delta \phi\right), \quad \mathbf{V}\_{\zeta} = \mathbf{k} \mathbf{I}\_{\zeta} = \mathbf{k} \mathbf{A}^{2} \sin^{2} \left(\frac{\pi}{\mathcal{A}} \Delta \phi\right) \tag{4}$$

The laser beam from He-Ne laser head is separated into two parallel and orthogonal polarizing components after passing through the first Wollaston prism and a convex lens. The two-polarized and parallel beams are converged on the second Wollaston prism by a second convex lens after passing through the shock tube. Where, there is no interferometric fringe from the combined beams caused by the discrepancy between the polarization planes of both beams. Here, the shock wave propagates in the shock tube in a direction from the bottom to top in Fig. 20, and it is considered that the shock wave arrives between two beams. The density in front of the shock wave and the density behind the shock wave are

2, respectively. Two beams passing through the same optical path as light vectors are

y y z z 2 2 E =A cos x , E =A cos x+

Here *Ay* and *Az* are the constants for a laser light intensity. *x*, , and are the distance of the traveling direction of laser, the wavelength of laser, and the optical path difference between both beams, respectively. The optical path difference is occurred by the density difference across the shock wave. The third Wollaston prism should be settled as rotated 45 degrees for

> and *Ez*

2 2 2 2 2 2

by the photo detectors as follows:

 

y z y z I = E +E =A cos , I = E +E =A sin

2 2 2 2 V =kI =kA cos , V =kI =kA sin

axes inclined 45 degree from *y* and *z* axes. After passing through the second Wollaston

 

 

 and 

  
> and *Ey*

> > and

> > >

 

 and *I*

, respectively. Here,

(3)

 

(4)

  , respectively.

directions,

are converted

 and are the

directions are denoted as

(2)

each polarization planes. The light vector *Ey* is separated into *Ey*

prism, the agreement of polarization plane causes interference at

Here, *Ay* = *Az* = *A*. The intensities of two polarized interfering beams *I*

and 

follows:

into the voltage signals *V*

 

*Ey* and *Ez*, respectively as shown in Fig. 21.

Fig. 21. Schematic drawing of the optical vectors

The light vectors *Ey* and *Ez* are denoted as follows:

Similarly, The light vector *Ez* is separated into *Ez*

respectively. The intensities of interfering lights for

 and *V*

  1 The voltage signal *V* is stored to an oscilloscope. Here, k is the constant defined from the characteristics of the photo detector. The voltage signal *V* = *V* -*V* is amplified by the differential amplifier circuit.

$$\mathbf{V} = \alpha \left( \mathbf{V}\_{\eta} \cdot \mathbf{V}\_{\zeta} \right) = \alpha \mathbf{k} \mathbf{A}^2 \cos \left( \frac{\pi}{\mathcal{A}} \Delta \phi \right) = \mathbf{V}\_0 \cos \left( \frac{\pi}{\mathcal{A}} \Delta \phi \right) \tag{5}$$

Here, is the gain of the amplifier. Figure 22 shows the relation between the voltage signals and the optical path difference between both beams. The gain of the amplifier is denoted as 2 in Fig. 22. Here, the refractive index of the medium is expressed as *n* and the length of the medium is obtained as the inner diameter of the shock tube *d*, the optical path difference between both beams is obtained as follows:

$$
\Delta \phi = (\mathbf{n}\_2 \cdot \mathbf{n}\_1) \mathbf{d} \tag{6}
$$

Here, *n*2 and *n*1 are the refractive index of the gas behind the shock wave and the gas in front of the shock wave, respectively. The relation between the refractive index *n* and the density is expressed by Gladstone-Dale's formula as follows:

$$\text{n-1=K} \frac{\rho}{\rho\_0} \tag{7}$$

Here, 0 and *K* are the density at the normal condition and the non-dimensional Gladstone-Dale constant, respectively. The relation between the voltage signal *V* and the density difference 2-1 is obtained from Eq. 5, 6 and 7 as follows:

$$\mathbf{V} = \mathbf{V}\_0 \cos\left\{\frac{2\pi \mathbf{d} \mathbf{K}}{\lambda \rho\_0} (\rho\_2 \cdot \rho\_1)\right\} \tag{8}$$

Here, the voltage signal *V* can be linearly-approximated to *V* at *V*=0 neighborhood as follows, as shown in Fig. 22:

Fig. 22. The interfering signals from the laser differential interferometer.

$$\pm \Delta \mathbf{V} = 2\mathbf{V}\_0 \frac{\pi d \mathbf{K}}{\lambda \rho\_0} (\rho\_2 \cdot \rho\_1) \tag{9}$$

Interferometric Measurement in Shock Tube Experiments 241

Figure 24 shows the time variation of typical signal trace from the interferometer. The horizontal and vertical axes indicate the time and the voltage. The signal rising and decay in Fig. 24 show that the shock wave passes through the first beam and the second beam of the interferometer. The signal is obtained at *ls*=800mm in *d*=2mm tube. The Mach number of

<sup>s</sup> <sup>s</sup>

between the signal rising and decay as shown in Fig. 24, and *a*1 is the sound velocity of the driven gas. The Mach number is obtained as 1.53 in this case by calculating from Eq. 12.

> **2/1**

shock wave and the Mach number. The horizontal and the vertical axes indicate the Mach

 

2 1 s

1 1 s +1 M <sup>=</sup> -1 M +2

Fig. 24. The interfering signals from the laser differential interferometer.

2/

<sup>l</sup> M = a 

1

*V* in Fig. 24. Figure 25 shows the relation between the density ratio across the

2

2

2/

(12)

1 can be calculated from Eq. 11 by using

is the time difference

1. The black line shows the

(13)

**4.3.1 Shock wave detection by the interferometer** 

**4.3.2 Density ratio across the shock wave** 

The density ratio across the shock wave

theoretical value obtained from the following;

measured

shock wave *Ms* can be calculated by using the following equation;

Here, *ls* is the distance of the two beams of the interferometer,

number *Ms* and the density ratio across the shock wave

Additionally, equation 9 can be rewritten by considering that the value required in this experiment is the absolute value of the voltage signal *V* for the density difference 2-1 as follows:

$$\left| \pm \Delta \mathbf{V} \right| = \Delta \mathbf{V} = 2 \mathbf{V}\_0 \frac{\pi d \mathbf{K}}{\lambda \rho\_0} (\rho\_2 \cdot \rho\_1) \tag{10}$$

Moreover, the above Eq. 10 can be deformed by using density ratio across the shock wave 2/1 as follows:

$$\frac{\rho\_2}{\rho\_1} = 1 + \frac{\Delta \mathbf{V}}{2\mathbf{V}\_0} = \frac{\lambda}{\pi d \mathbf{K}} \frac{\rho\_0}{\rho\_1} \tag{11}$$

## **4.2.2 Shock wave measurement system**

Figure 23 shows the shock wave measurement system used in this experiment. The glass tube, connected to the high pressure section, has the length *l* = 1000 mm. The position of laser differential interferometer *ls* is changed from 200 to 800 mm. The inner diameter of the tube *d* is used as 2 and 3 mm in this experiment. The pressure transducer is settled at the end of the tube to detect the reflected shock wave. The signals obtained from the interferometer and the pressure transducer are stored to an oscilloscope. The initial pressure ratio *p*4/*p*1 maintains constant as 9, in driven pressure *p*1 = 0.1 MPa. The driver and driven gases are helium and air, respectively.

Fig. 23. The interfering signals from the laser differential interferometer.

## **4.3 Results and discussion**

The interferometric signal of the shock wave, generated by using the diaphragmless driver section as mentioned above, propagating in the small diameter tubes is obtained from the laser differential interferometer.

Additionally, equation 9 can be rewritten by considering that the value required in this

dK V = V=2V - 

Moreover, the above Eq. 10 can be deformed by using density ratio across the shock wave

2 0 101 <sup>V</sup> 1 =

Figure 23 shows the shock wave measurement system used in this experiment. The glass tube, connected to the high pressure section, has the length *l* = 1000 mm. The position of laser differential interferometer *ls* is changed from 200 to 800 mm. The inner diameter of the tube *d* is used as 2 and 3 mm in this experiment. The pressure transducer is settled at the end of the tube to detect the reflected shock wave. The signals obtained from the interferometer and the pressure transducer are stored to an oscilloscope. The initial pressure ratio *p*4/*p*1 maintains constant as 9, in driven pressure *p*1 = 0.1 MPa. The driver and driven

Fig. 23. The interfering signals from the laser differential interferometer.

The interferometric signal of the shock wave, generated by using the diaphragmless driver section as mentioned above, propagating in the small diameter tubes is obtained from the

2V dK

 

 

experiment is the absolute value of the voltage signal *V* for the density difference

follows:

1 as follows:

**4.2.2 Shock wave measurement system** 

gases are helium and air, respectively.

**4.3 Results and discussion** 

laser differential interferometer.

2/

0 21 0 dK V=2V - 

 

0 21 0

 

(9)

(10)

(11)

2-1 as

## **4.3.1 Shock wave detection by the interferometer**

Figure 24 shows the time variation of typical signal trace from the interferometer. The horizontal and vertical axes indicate the time and the voltage. The signal rising and decay in Fig. 24 show that the shock wave passes through the first beam and the second beam of the interferometer. The signal is obtained at *ls*=800mm in *d*=2mm tube. The Mach number of shock wave *Ms* can be calculated by using the following equation;

$$\mathbf{M}\_s = \frac{\Delta \mathbf{l}\_s}{\Delta \boldsymbol{\tau} \cdot \mathbf{a}\_1} \tag{12}$$

Here, *ls* is the distance of the two beams of the interferometer, is the time difference between the signal rising and decay as shown in Fig. 24, and *a*1 is the sound velocity of the driven gas. The Mach number is obtained as 1.53 in this case by calculating from Eq. 12.

### **4.3.2 Density ratio across the shock wave 2/1**

The density ratio across the shock wave 2/1 can be calculated from Eq. 11 by using measured *V* in Fig. 24. Figure 25 shows the relation between the density ratio across the shock wave and the Mach number. The horizontal and the vertical axes indicate the Mach number *Ms* and the density ratio across the shock wave 2/1. The black line shows the theoretical value obtained from the following;

$$\frac{\rho\_2}{\rho\_1} = \frac{(\gamma\_1 + 1)\mathbf{M}\_s^{\;2}}{(\gamma\_1 \cdot 1)\mathbf{M}\_s^{\;2} + 2} \tag{13}$$

Fig. 24. The interfering signals from the laser differential interferometer.

Interferometric Measurement in Shock Tube Experiments 243

some improvements in reconstruction will be necessary to clarify complex flow

For velocity measurement of shock wave in micro-scale shock tube, we measured the velocities of the shock wave and the density ratios across the shock wave propagating in the small diameter tubes, generated by the diaphragmless driver section, by using laser differential interferometer. From our results the following points can be concluded. Firstly, the relation between the Mach number and the density ratio across the shock wave is obtained. The measured density ratio is lower than the theory. It is considered that the Gladstone-Dale constant at the normal condition is used for the calculation, nevertheless the temperature behind the shock wave becomes higher. Additionally, the influence of the inner diameter of the shock tube *d* can be seen slightly. Secondly the detailed experiments are required to estimate the influence of the shock wave attenuation by using longer

Authors greatly appreciate Hiroki Honma of Professor Emeritus of Chiba University and Professor Walter Garen of Hochschule Emden/Leer, University of Applied Sciences for their

Brouillette, M. (2003). Shock waves at microscales, *Shock waves,* Vol.13, No.1, (April 2003),

Honma, H., Ishihara, M., Yoshimura, T., Maeno, K. and Morioka, T. (2003a), Interferometric

vortices discharged from open ends, *Shock Waves* (2003a), 13, pp.179-190. Honma, H., Maeno, K., Morioka, T. and Kaneta, T. (2003b), Some topics on three-

Maeno, K., Kaneta, T., Yoshimura, T., Morioka, T. and Honma, H. (2005), Pseudo-schlieren

vortices discharged from open ends, *Shock Waves*, Vol.14, No.4, pp.239-249. Ota, M., Koga, T. and Maeno, K. (2005), Interferometric CT Measurement and Novel

Ota M., Koga, T. and Maeno, K. (2006), Laser interferometric CT measurement of the

*Journal of Applied Physics*, Vol. 44, No. 42 pp. L1293-1294.

*Trans. Nucl, Sci.*, Ns-21, pp.21-43.

CT measurement of three-dimensional flow phenomena on shock waves and

dimensional features of shock waves and vortices discharged from open ends, *Proc. of the 5th International Workshop on Shock/Vortex Interaction* Hosted by National Cheng Kung University, The Kaohsiung Grand Hotel Kaohsiung, October 27-31. Kak A. C., Slaney M. (1988), Principles of Computerized Tomographic Imaging, *IEEE*, New

CT measurement of three-dimensional flow phenomena on shock waves and

Expression Method of Discharged Flow Field with Unsteady Shock Waves, *Japanese* 

unsteady super sonic shock-vortex flow field dischargeing from two parallel and cylindrical nozzles, *Measurement Science and Technology*, Vol. 17, pp. 2066-2071. Shepp, A. L. and Logan, F. B. (1974), The Fourier Reconstruction of a Head Section, *IEEE* 

phenomena.

shock tube.

**6. Acknowledgment** 

**7. References** 

York.

valuable advice and discussion.

pp. 3-12, DOI 10.1007/s00193-003-0191-4.

Fig. 25. Relation between 2/1 and *Ms*.

The measured density ratio is lower than the theory. It is considered that the Gladston-Dale constant at the normal condition is used for the calculation, nevertheless the temperature behind the shock wave becomes higher. Additionally, the influence of the inner diameter of the shock tube *d* can be seen slightly.

## **5. Conclusion**

This chapter described two topics related to interferometric measurement in shock tube experiment. LICT technique was applied to the measurement of high-speed, unsteady and 3D flow field induced by discharging shock waves, and laser differential interferometer was applied to velocity and density measurement in micro-scale shock tube.

For LICT measurement, 3D flow phenomena behind discharging shock wave from two parallel and circular nozzles were captured in detail. Various phenomena were clarified shock-shock interaction, shock-vortex interaction, SSW, CS1, CS2, …etc. LICT measurement was also applied to flow field around a circular cylinder to aim at extending the tomographic measurement to more general case. Density distribution was reconstructed from incomplete projection data containing blanked part by ART algorithm. Three-dimensional flow phenomena can be captured from resultant images. However some improvements in reconstruction will be necessary to clarify complex flow phenomena.

For velocity measurement of shock wave in micro-scale shock tube, we measured the velocities of the shock wave and the density ratios across the shock wave propagating in the small diameter tubes, generated by the diaphragmless driver section, by using laser differential interferometer. From our results the following points can be concluded. Firstly, the relation between the Mach number and the density ratio across the shock wave is obtained. The measured density ratio is lower than the theory. It is considered that the Gladstone-Dale constant at the normal condition is used for the calculation, nevertheless the temperature behind the shock wave becomes higher. Additionally, the influence of the inner diameter of the shock tube *d* can be seen slightly. Secondly the detailed experiments are required to estimate the influence of the shock wave attenuation by using longer shock tube.

## **6. Acknowledgment**

Authors greatly appreciate Hiroki Honma of Professor Emeritus of Chiba University and Professor Walter Garen of Hochschule Emden/Leer, University of Applied Sciences for their valuable advice and discussion.

## **7. References**

242 Interferometry – Research and Applications in Science and Technology

1.0 1.2 1.4 1.6 1.8 2.0

Mach number *Ms*

The measured density ratio is lower than the theory. It is considered that the Gladston-Dale constant at the normal condition is used for the calculation, nevertheless the temperature behind the shock wave becomes higher. Additionally, the influence of the inner diameter of

This chapter described two topics related to interferometric measurement in shock tube experiment. LICT technique was applied to the measurement of high-speed, unsteady and 3D flow field induced by discharging shock waves, and laser differential interferometer was

For LICT measurement, 3D flow phenomena behind discharging shock wave from two parallel and circular nozzles were captured in detail. Various phenomena were clarified shock-shock interaction, shock-vortex interaction, SSW, CS1, CS2, …etc. LICT measurement was also applied to flow field around a circular cylinder to aim at extending the tomographic measurement to more general case. Density distribution was reconstructed from incomplete projection data containing blanked part by ART algorithm. Three-dimensional flow phenomena can be captured from resultant images. However

1.0

Fig. 25. Relation between

**5. Conclusion** 

the shock tube *d* can be seen slightly.

2/

1 and *Ms*.

applied to velocity and density measurement in micro-scale shock tube.

1.5

Density ratio

2/1

2.0

2.5

Theory

*d*=2mm *d*=3mm


**Part 2** 

**Recent Interferometry Applications** 


## **Part 2**

**Recent Interferometry Applications** 

244 Interferometry – Research and Applications in Science and Technology

Smeets, G. (1972). Laser interferometer for high sensitivity measurements on transient phase

Smeets, G. (1977). Flow diagnostics by laser interferometry, *IEEE transactions on Aerospace* 

(March 1972), pp. 186-190, DOI 10.1109/TAES.1972.309488.

10.1109/TAES.1977.308441.

objects, *IEEE transactions on Aerospace and Electronic Systems,* Vol.AES-8, No.2,

*and Electronic Systems,* Vol.AES-13, No.2, (March 1977), pp. 82-89, DOI

**0**

**12**

*Spain*

**Interferometry for Fusion Devices**

Interferometers are optical arrangements capable of measuring changes in the optical paths by measuring the phase differences. They allow to perform a wide range of measurements, including the measurement of the line-electron density of the fusion plasmas Innocente & Tudisco (2006); Innocente et al. (1997); Joffrin & Riva (2003); Kornejew et al. (2006); Sanchez

> *Lp* 0

The interferometric measurements are obtained through the observation of the interference

**E**(**r**, *t*) = **E**(**r**)ei(**k**·**r**−!t+`)

where *λ* is the wavelength in the medium, and **e***<sup>k</sup>* is a unitary vector in the direction of the

In general, interferometers use at least two beams, probing and reference, following different optical paths, *L*<sup>1</sup> and *L*2, which finally interfere in an square law detector1. The optical paths

*path*(*t*)

where the integration is taken along a geometric path with a fixed origin (beam splitter) and

Considering that these waves have angular frequencies *ω*<sup>1</sup> and *ω*2, and that they are plane, monochromatic, and coherent (from the same source), in this case the following

*u*<sup>1</sup> = *u*01e<sup>i</sup>(k01·L1(t)−!1t+`01 )

*L* = 

**E**(**r**) is the amplitude vector, *ω* is the angular frequency, and **k** is the wave number vector:

*<sup>k</sup>* <sup>=</sup> <sup>2</sup>*<sup>π</sup>*

where *ne* is the electron density and *Lp* is the integration path in the plasma.

between two or more beams, generally obtained by splitting an incident beam. The solution to the electromagnetic free space wave equation can be expressed as:

**1. Introduction**

propagation.

are given by:

end (detector), *N* is the refractive index.

<sup>1</sup> An square law detector measures intensities

approximation to the electric fields is considered:

et al. (2008); Zeeland et al. (2006):

Luis Esteban and Miguel Sánchez *CIEMAT-EURATOM Association*

*ne*d*l*, (1)

*<sup>λ</sup>* · **<sup>e</sup>***k*, (3)

*N*d*l*, (4)

, (5)

, (2)

## **Interferometry for Fusion Devices**

Luis Esteban and Miguel Sánchez *CIEMAT-EURATOM Association Spain*

## **1. Introduction**

Interferometers are optical arrangements capable of measuring changes in the optical paths by measuring the phase differences. They allow to perform a wide range of measurements, including the measurement of the line-electron density of the fusion plasmas Innocente & Tudisco (2006); Innocente et al. (1997); Joffrin & Riva (2003); Kornejew et al. (2006); Sanchez et al. (2008); Zeeland et al. (2006):

$$\int\_{0}^{L\_p} n\_\varepsilon \mathrm{d}l \,\,\tag{1}$$

where *ne* is the electron density and *Lp* is the integration path in the plasma.

The interferometric measurements are obtained through the observation of the interference between two or more beams, generally obtained by splitting an incident beam.

The solution to the electromagnetic free space wave equation can be expressed as:

$$\mathbf{E}(\mathbf{r},t) = \mathbf{E}(\mathbf{r})\mathbf{e}^{\mathbf{i}(\mathbf{k}\cdot\mathbf{r}-\mathbb{I}t+\overset{\circ}{\ })},\tag{2}$$

**E**(**r**) is the amplitude vector, *ω* is the angular frequency, and **k** is the wave number vector:

$$k = \frac{2\pi}{\lambda} \cdot \mathbf{e}\_{k\prime} \tag{3}$$

where *λ* is the wavelength in the medium, and **e***<sup>k</sup>* is a unitary vector in the direction of the propagation.

In general, interferometers use at least two beams, probing and reference, following different optical paths, *L*<sup>1</sup> and *L*2, which finally interfere in an square law detector1. The optical paths are given by:

$$L = \int\_{path(t)} \mathrm{Ndd}\_{\prime} \tag{4}$$

where the integration is taken along a geometric path with a fixed origin (beam splitter) and end (detector), *N* is the refractive index.

Considering that these waves have angular frequencies *ω*<sup>1</sup> and *ω*2, and that they are plane, monochromatic, and coherent (from the same source), in this case the following approximation to the electric fields is considered:

$$\mu\_1 = \mu\_{01} \mathbf{e}^{\mathbf{i}(\mathbf{k}\_{01} \cdot \mathbf{L}\_1(\mathbf{t}) - \mathbf{l}\_1 \mathbf{t} + \mathbf{l}\_{01})},\tag{5}$$

<sup>1</sup> An square law detector measures intensities

$$\mu\_2 = \mu\_{02} \mathbf{e}^{\mathbf{i}(\mathbf{k}\_{02} \cdot \mathbf{L}\_2(\mathbf{t}) - \mathbf{l}\_2 \mathbf{t} + \mathbf{l}\_{02})}.\tag{6}$$

At the detector, the interference signal has the form:

$$I = (\mu\_{01} + \mu\_{02})^2 = I\_1 + I\_2 + 2\sqrt{I\_1 I\_2} \cos\left((\omega\_1 - \omega\_2)t + k\_0 \cdot (L\_1(t) - L\_2(t))\right),\tag{7}$$

where *I* is the intensity modulated at frequency Ω = *ω*<sup>1</sup> − *ω*2, and *I*<sup>1</sup> and *I*<sup>2</sup> are the respective intensities of the two waves. Since *k*<sup>01</sup> � *k*02, the average between the two values *k*<sup>0</sup> = (*k*<sup>01</sup> + *k*02)/2, has been considered in Eq. 7. Using *k*0, the effective wavelength can be expressed as:

$$
\lambda\_{eff} = \frac{2\lambda\_1 \lambda\_2}{\lambda\_1 + \lambda\_2}.\tag{8}
$$

Fig. 1. a) A Michelson interferometer; the light is divided into two beams in the beam-splitter B1; Both beams are reflected back at mirrors M1 and M2 and recombined in the output. b) A Mach-Zehnder interferometer. The incoming beam is divided in beam splitter B1 into reference and measuring beams, which are reflected at mirrors M1 and M2, and finally

Interferometry for Fusion Devices 249

The expression to compute the line integrated electron density can be deduced from the Appleton-Hartree relation for a cold plasma. This expression would be valid for the TJ-II and the W7-X plasmas. However, for plasmas with an electron temperature of 20 eV, such as those expected at the ITER tokamak, a relativistic correction must be included in the line integrated electron density. In this Section the main expressions, for cold plasmas, Subsection 2.1 and for weakly relativistic plasmas, Subsection 2.2, are summarized. A detailed deduction

For the measurement of the electron density, the relation between this quantity and the

Adopting the cold plasma approximation, the refractive index of the plasma is given by the

<sup>1</sup>

*X* = *ω*<sup>2</sup>

*<sup>N</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>ω</sup>*<sup>2</sup>

being *ωp* the plasma frequency, and *ω* and the frequency of the radiation. The refractive index

*p <sup>ω</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *ne*

<sup>2</sup>*Y*<sup>2</sup> sin2 *<sup>θ</sup>*

2

*nc*

+ (<sup>1</sup> − *<sup>X</sup>*)2*Y*<sup>2</sup> cos<sup>2</sup> *<sup>θ</sup>*

*<sup>p</sup>*/*ω*2, (12)

, (13)

*<sup>N</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>X</sup>*, (11)

. (10)

can be found in Hartfuss et al. (1997); Hojo & Kawahata (2009); Hutchinson (1990).

recombined at beam splitter B2.

**2. Propagation of electromagnetic waves in plasmas**

**2.1 Propagation of electromagnetic waves in cold plasmas**

changes in the refractive index of the plasma must be found.

<sup>1</sup> <sup>−</sup> *<sup>X</sup>* <sup>−</sup> <sup>1</sup>

where:

becomes:

Appleton-Hartree relation, Hartfuss et al. (1997); Hutchinson (1990):

*<sup>N</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>X</sup>*(<sup>1</sup> <sup>−</sup> *<sup>X</sup>*)

<sup>2</sup>*Y*<sup>2</sup> sin2 *<sup>θ</sup>* <sup>±</sup>

If the magnetic field is negligible, *Y* → 0, the refractive index becomes:

The alternate current from the detector is:

$$i = i\_0 \cos\left(\Omega t + \phi(t)\right),\tag{9}$$

the phase *φ*(*t*) corresponds to the optical path length differences.

Usually, both angular frequencies are equal (i.e., Ω = 0), in this case, the interferometer is called homodyne. If Δ*ω* �= 0 (the measuring and reference beams operate at slightly different frequencies), the interferometer is called heterodyne. The frequencies are typically shifted using Acousto-Optic-Modulators (AOM) for the infrared Ohtsuka & Sasaki (1974). However, other techniques can be adopted for this purpose, in agreement with the used wavelengths, for instance rotating gratings in the case of far-infrared wavelengths.

The phase *φ*(*t*) carries the information related to the difference in the optical path-lengths between the two beams, due to the mechanical movements or changes in the refractive index. The initial phases of the probing and the reference beams, *θ*<sup>01</sup> and *θ*02, are integration constants, which for convenience are considered zero.

With respect to the optical setup, two configurations that are commonly used for plasma refractive index measurements are the Michelson and the Mach-Zehnder schemes. However, in practice, the setups used in fusion are a mixture of these two optical configurations. Both are two-beam interferometers; in the case of the Michelson configuration, the incoming and outgoing beams, travel in the same direction and in opposite senses, while in the Mach-Zehnder one the propagation follows only one sense. The difference between both is shown in Figure 1.

This Chapter is structured as follows. In Section 2 the relation between the line-integrated electron density and the phase shift suffered by the waves that cross the plasma is deduced for a cold plasma model. In Subsection 2.2 extends the cold plasma model for weakly relativistic plasmas, such as those expected in the ITER tokamak. In Section 3 the possible wavelengths to be used in these type of diagnostics are discussed. Section 4 outlines the main characteristics of the middle IR interferometers at TJ-II and at W7-X. Finally the basic expressions of Gaussian beam propagation are developed in Section 6. In addition, several phase measuring algorithms are listed, and the interpolation phase measuring algorithm is described in detail. The main conclusions of the chapter are summarized in Section 8.

2 Will-be-set-by-IN-TECH

*u*<sup>2</sup> = *u*02e<sup>i</sup>(k02·L2(t)−!2t+`02 )

where *I* is the intensity modulated at frequency Ω = *ω*<sup>1</sup> − *ω*2, and *I*<sup>1</sup> and *I*<sup>2</sup> are the respective intensities of the two waves. Since *k*<sup>01</sup> � *k*02, the average between the two values *k*<sup>0</sup> = (*k*<sup>01</sup> + *k*02)/2, has been considered in Eq. 7. Using *k*0, the effective wavelength can be expressed as:

*<sup>λ</sup>eff* <sup>=</sup> <sup>2</sup>*λ*1*λ*<sup>2</sup>

Usually, both angular frequencies are equal (i.e., Ω = 0), in this case, the interferometer is called homodyne. If Δ*ω* �= 0 (the measuring and reference beams operate at slightly different frequencies), the interferometer is called heterodyne. The frequencies are typically shifted using Acousto-Optic-Modulators (AOM) for the infrared Ohtsuka & Sasaki (1974). However, other techniques can be adopted for this purpose, in agreement with the used wavelengths,

The phase *φ*(*t*) carries the information related to the difference in the optical path-lengths between the two beams, due to the mechanical movements or changes in the refractive index. The initial phases of the probing and the reference beams, *θ*<sup>01</sup> and *θ*02, are integration

With respect to the optical setup, two configurations that are commonly used for plasma refractive index measurements are the Michelson and the Mach-Zehnder schemes. However, in practice, the setups used in fusion are a mixture of these two optical configurations. Both are two-beam interferometers; in the case of the Michelson configuration, the incoming and outgoing beams, travel in the same direction and in opposite senses, while in the Mach-Zehnder one the propagation follows only one sense. The difference between both is

This Chapter is structured as follows. In Section 2 the relation between the line-integrated electron density and the phase shift suffered by the waves that cross the plasma is deduced for a cold plasma model. In Subsection 2.2 extends the cold plasma model for weakly relativistic plasmas, such as those expected in the ITER tokamak. In Section 3 the possible wavelengths to be used in these type of diagnostics are discussed. Section 4 outlines the main characteristics of the middle IR interferometers at TJ-II and at W7-X. Finally the basic expressions of Gaussian beam propagation are developed in Section 6. In addition, several phase measuring algorithms are listed, and the interpolation phase measuring algorithm is

described in detail. The main conclusions of the chapter are summarized in Section 8.

*λ*<sup>1</sup> + *λ*<sup>2</sup>

At the detector, the interference signal has the form:

*I* = (*u*<sup>01</sup> + *u*02)<sup>2</sup> = *I*<sup>1</sup> + *I*<sup>2</sup> + 2

The alternate current from the detector is:

the phase *φ*(*t*) corresponds to the optical path length differences.

for instance rotating gratings in the case of far-infrared wavelengths.

constants, which for convenience are considered zero.

shown in Figure 1.

. (6)

. (8)

*<sup>I</sup>*<sup>1</sup> *<sup>I</sup>*<sup>2</sup> cos ((*ω*<sup>1</sup> <sup>−</sup> *<sup>ω</sup>*2)*<sup>t</sup>* <sup>+</sup> *<sup>k</sup>*<sup>0</sup> · (*L*1(*t*) <sup>−</sup> *<sup>L</sup>*2(*t*))), (7)

*i* = *i*<sup>0</sup> cos (Ω*t* + *φ*(*t*)), (9)

Fig. 1. a) A Michelson interferometer; the light is divided into two beams in the beam-splitter B1; Both beams are reflected back at mirrors M1 and M2 and recombined in the output. b) A Mach-Zehnder interferometer. The incoming beam is divided in beam splitter B1 into reference and measuring beams, which are reflected at mirrors M1 and M2, and finally recombined at beam splitter B2.

### **2. Propagation of electromagnetic waves in plasmas**

The expression to compute the line integrated electron density can be deduced from the Appleton-Hartree relation for a cold plasma. This expression would be valid for the TJ-II and the W7-X plasmas. However, for plasmas with an electron temperature of 20 eV, such as those expected at the ITER tokamak, a relativistic correction must be included in the line integrated electron density. In this Section the main expressions, for cold plasmas, Subsection 2.1 and for weakly relativistic plasmas, Subsection 2.2, are summarized. A detailed deduction can be found in Hartfuss et al. (1997); Hojo & Kawahata (2009); Hutchinson (1990).

### **2.1 Propagation of electromagnetic waves in cold plasmas**

For the measurement of the electron density, the relation between this quantity and the changes in the refractive index of the plasma must be found.

Adopting the cold plasma approximation, the refractive index of the plasma is given by the Appleton-Hartree relation, Hartfuss et al. (1997); Hutchinson (1990):

$$N^2 = 1 - \frac{X(1 - X)}{1 - X - \frac{1}{2}Y^2 \sin^2 \theta \pm \sqrt{\left[\left(\frac{1}{2}Y^2 \sin^2 \theta\right)^2 + (1 - X)^2 Y^2 \cos^2 \theta\right]}}.\tag{10}$$

If the magnetic field is negligible, *Y* → 0, the refractive index becomes:

$$N^2 = 1 - X\_\prime \tag{11}$$

where:

$$X = \omega\_p^2 / \omega^2,\tag{12}$$

being *ωp* the plasma frequency, and *ω* and the frequency of the radiation. The refractive index becomes:

$$N^2 = 1 - \frac{\omega\_p^2}{\omega^2} = 1 - \frac{n\_c}{n\_c} \,\tag{13}$$

*ne* is the electron density and *nco* = *ω*2me*�*0/*e*<sup>2</sup> is the cutoff plasma density2.

The phase shift suffered by an electromagnetic wave is given by Hutchinson (1990):

$$
\phi = \int\_{\text{plasma}} k(z) \,\text{d}z.\tag{14}
$$

where a relativistic correction *β*, has been introduced:

if *n* � *nco* the phase difference can be finally written as:

also due to the mechanical vibrations and thermal drifts.

approximated by its Taylor expansion as:

**3. Wavelength choice**

the electron mass respectively.

to cancel mechanical vibrations

*<sup>β</sup>* <sup>=</sup> <sup>3</sup>*K*2(*ρ*) *ρ*<sup>2</sup> <sup>∞</sup>

<sup>e</sup>−*ργ* � <sup>e</sup>−*ρ*e−*<sup>ρ</sup> <sup>p</sup>*<sup>2</sup>

and the correction factor *β* can be simplified as Hojo & Kawahata (2009):

<sup>Δ</sup>*<sup>φ</sup>* <sup>=</sup> *<sup>ω</sup>* 2*cn*

<sup>0</sup> (*p*4/*γ*2)e−*ργ*d*<sup>p</sup>*

*<sup>p</sup>* <sup>=</sup> <sup>|</sup>**p**|/(*mec*) is the normalized momentum, *<sup>γ</sup>* <sup>=</sup> (<sup>1</sup> <sup>+</sup> *<sup>p</sup>*2), *<sup>ρ</sup>* <sup>=</sup> <sup>m</sup>*ec*2/*Te* and *<sup>K</sup>*2(*ρ*) is the modified Bessel function. Considering a weakly relativistic plasma (*γ* < *c*), e−*ργ* can be

Interferometry for Fusion Devices 251

2 

> 5 2*ρ*

 *z*<sup>2</sup> *z*1

2*π*Δ*l*

me*�*<sup>0</sup> *npeak*

*β* � 1 +

2*ρ* 5 + 2*ρ*

In practice, these phase differences will not only occur due to plasma density variations, but

Mechanical vibrations, in one-color arrangements are one of the limiting factors for the wavelength choice. As it can be seen in the former equation, these effects become particularly important for low wavelengths. However, they can be suppressed using two-color setups. The upper limit is imposed by the cutoff density, *nCO*, since when the probing wavelength is long enough it will not propagate through the plasma (evanescent modes). The probing wavelength must satisfy the next relation to ensure that it will propagate through the plasma:

*φ* = *φ<sup>p</sup>* +

*λ* <

2*π*c e

where *npeak* is the maximum electron plasma density and *e* and m*<sup>e</sup>* are the electron charge and

In interferometers operating in the middle infrared range, mechanical vibrations are higher than variations caused by plasma density fluctuations. This circumstance forces to use a second interferometer sharing the same optical path as the first one, so this contribution can be canceled in the final compensated signal3. These type of arrangements receive the name

<sup>3</sup> The compensated signal is obtained after subtracting the optical paths of both interferometers in order

<sup>1</sup> <sup>+</sup> *<sup>ρ</sup> <sup>p</sup>*<sup>4</sup> 8

, (20)

, (21)

, (22)

*n*(*z*)d*z*. (23)

*<sup>λ</sup>* . (24)

, (25)

In the particular case of interferometry, the measuring beam will travel through the plasma and also through vacuum. The vacuum contribution to the k vector can be removed comparing the phases of both measuring and reference beams Hartfuss et al. (1997); Hutchinson (1990):

$$
\phi = \int\_{plasma} (N(z) - 1) \frac{\omega}{c} dz. \tag{15}
$$

*ω* is the radiation frequency and *c* is the speed of light.

Substituting Eq. 13 in Eq. 15, the phase shift becomes:

$$\phi = \frac{\omega}{c} \int\_{plasma} \left\{ \sqrt{\left( 1 - \frac{n\_\ell(z)}{n\_\ell} \right)} - 1 \right\} \,\mathrm{d}z. \tag{16}$$

If *ne* � *nco*, the refractive index, *N* can be approximated by the first two terms of its Taylor expansion:

$$N \simeq 1 - \frac{1}{2} \frac{n\_{\mathcal{C}}}{n\_{\mathcal{C}}} \tag{17}$$

the phase difference, *φ* becomes:

$$
\phi = \text{r}\_{\text{e}} \lambda \int\_{\text{plasma}} n\_{\text{e}}(z) \text{d}z. \tag{18}
$$

This integral receives the name o line integrated electron density, LID and for simplicity is denoted as *ne*(*z*)d*z* Hartfuss et al. (1997). The LID measurement is computed after measuring the phase differences of the output signals, Δ*φ* these phase differences are calculated around the carrying frequency Ω. The path across the plasma receives the name of chord length.

### **2.2 Corrections for weakly relativistic plasmas**

In ITER, it is expected an electron temperature in the order of 20 keV, which implies that the cold plasma approximation is no longer valid. A correction factor *β*, must be included for the computation of the refractive index, Hojo & Kawahata (2009). Under these conditions, considering the dispersion relation for the ordinary mode, the refractive index would now become:

$$N = \sqrt{1 - \frac{1}{\beta} \frac{\omega\_p^2}{\omega^2}},\tag{19}$$

<sup>2</sup> The cutoff density establishes the minimum frequency of the waves that will propagate through the plasma. Below this value the propagation modes are evanescent

4 Will-be-set-by-IN-TECH

*plasma*

In the particular case of interferometry, the measuring beam will travel through the plasma and also through vacuum. The vacuum contribution to the k vector can be removed comparing the phases of both measuring and reference beams Hartfuss et al. (1997);

(*N*(*z*) − 1)

<sup>1</sup> <sup>−</sup> *ne*(*z*) *nc*

*ω c*

> − 1

*k*(*z*)d*z*. (14)

d*z*. (15)

d*z*. (16)

, (17)

*ne*(*z*)d*z*. (18)

*<sup>ω</sup>*<sup>2</sup> , (19)

*ne* is the electron density and *nco* = *ω*2me*�*0/*e*<sup>2</sup> is the cutoff plasma density2.

*φ* = 

*φ* = 

*plasma*

*φ* = re*λ*

*ω* is the radiation frequency and *c* is the speed of light. Substituting Eq. 13 in Eq. 15, the phase shift becomes:

> *<sup>φ</sup>* <sup>=</sup> *<sup>ω</sup> c*

Hutchinson (1990):

expansion:

chord length.

become:

the phase difference, *φ* becomes:

**2.2 Corrections for weakly relativistic plasmas**

The phase shift suffered by an electromagnetic wave is given by Hutchinson (1990):

*plasma*

If *ne* � *nco*, the refractive index, *N* can be approximated by the first two terms of its Taylor

2 *ne nc*

*<sup>N</sup>* � <sup>1</sup> <sup>−</sup> <sup>1</sup>

plasma

This integral receives the name o line integrated electron density, LID and for simplicity is denoted as *ne*(*z*)d*z* Hartfuss et al. (1997). The LID measurement is computed after measuring the phase differences of the output signals, Δ*φ* these phase differences are calculated around the carrying frequency Ω. The path across the plasma receives the name of

In ITER, it is expected an electron temperature in the order of 20 keV, which implies that the cold plasma approximation is no longer valid. A correction factor *β*, must be included for the computation of the refractive index, Hojo & Kawahata (2009). Under these conditions, considering the dispersion relation for the ordinary mode, the refractive index would now

*N* =

plasma. Below this value the propagation modes are evanescent

 <sup>1</sup> <sup>−</sup> <sup>1</sup> *β ω*2 *p*

<sup>2</sup> The cutoff density establishes the minimum frequency of the waves that will propagate through the

where a relativistic correction *β*, has been introduced:

$$\beta = \frac{3K\_2(\rho)}{\rho^2 \int\_0^\infty (p^4/\gamma^2) \mathbf{e}^{-\rho \gamma} \mathbf{d}p'} \tag{20}$$

*<sup>p</sup>* <sup>=</sup> <sup>|</sup>**p**|/(*mec*) is the normalized momentum, *<sup>γ</sup>* <sup>=</sup> (<sup>1</sup> <sup>+</sup> *<sup>p</sup>*2), *<sup>ρ</sup>* <sup>=</sup> <sup>m</sup>*ec*2/*Te* and *<sup>K</sup>*2(*ρ*) is the modified Bessel function. Considering a weakly relativistic plasma (*γ* < *c*), e−*ργ* can be approximated by its Taylor expansion as:

$$\mathbf{e}^{-\rho\gamma} \simeq \mathbf{e}^{-\rho} \mathbf{e}^{-\rho \frac{p^2}{2}} \left(1 + \rho \frac{p^4}{8}\right) \,. \tag{21}$$

and the correction factor *β* can be simplified as Hojo & Kawahata (2009):

$$
\beta \simeq 1 + \frac{5}{2\rho'} \tag{22}
$$

if *n* � *nco* the phase difference can be finally written as:

$$
\Delta\phi = \frac{\omega}{2cn} \frac{2\rho}{5+2\rho} \int\_{z\_1}^{z\_2} n(z)dz. \tag{23}
$$

## **3. Wavelength choice**

In practice, these phase differences will not only occur due to plasma density variations, but also due to the mechanical vibrations and thermal drifts.

$$
\phi = \phi\_p + \frac{2\pi\Delta l}{\lambda}.\tag{24}
$$

Mechanical vibrations, in one-color arrangements are one of the limiting factors for the wavelength choice. As it can be seen in the former equation, these effects become particularly important for low wavelengths. However, they can be suppressed using two-color setups. The upper limit is imposed by the cutoff density, *nCO*, since when the probing wavelength is long enough it will not propagate through the plasma (evanescent modes). The probing wavelength must satisfy the next relation to ensure that it will propagate through the plasma:

$$
\lambda < \frac{2\pi \text{c}}{\text{e}} \sqrt{\frac{\text{m}\_{\text{e}} \varepsilon\_0}{n\_{peak}}} \,\text{}\tag{25}
$$

where *npeak* is the maximum electron plasma density and *e* and m*<sup>e</sup>* are the electron charge and the electron mass respectively.

In interferometers operating in the middle infrared range, mechanical vibrations are higher than variations caused by plasma density fluctuations. This circumstance forces to use a second interferometer sharing the same optical path as the first one, so this contribution can be canceled in the final compensated signal3. These type of arrangements receive the name

<sup>3</sup> The compensated signal is obtained after subtracting the optical paths of both interferometers in order to cancel mechanical vibrations

Fig. 2. Cutoff density versus wavelength from the middle infrared range to the *μ*wave range. With shorter wavelengths higher densities can be measured. However, the vibration problem becomes more significant and forces to use two color arrangement.

of two-color interferometers. From Eq. 18 the optical path length difference from one of the interferometers is calculated as:

$$
\lambda\_1 \Delta \phi\_1 = 2 \pi r\_\text{e} \lambda\_1^2 \int\_{\text{plasma}} n\_\text{e}(l) \text{dl} + \lambda\_1 k\_{01} \int\_{\text{out}} \text{Ndl}\_\text{\textdegree} \tag{26}
$$

The second term of the equation takes into account the optical path length outside the plasma. The final line integrated density is computed subtracting the optical path-length differences between both interferometers if the second term is canceled4, Hartfuss et al. (1997); Hutchinson (1990):

$$\int n\_{\varepsilon} \mathrm{d}l = \frac{\Delta \phi\_1 \lambda\_1 - \Delta \phi\_2 \lambda\_2}{2 \pi p \cdot \mathbf{r}\_{\varepsilon} (\lambda\_1^2 - \lambda\_2^2)},\tag{27}$$

Incident Beam

Interferometry for Fusion Devices 253

Fig. 3. Refraction suffered by a beam of light when crossing a non-uniform plasma.

*<sup>θ</sup>* <sup>=</sup> *<sup>λ</sup>*2*re* 2*π* 

The deflection has a quadratic dependence with the wavelength, so for low wavelengths this effect is to be lower and for the range of interest in this chapter, 1.064 − 10.591 *μ*m, it is negligible. For example, if a parabolic density profile is considered, the maximum deflexion

> *<sup>θ</sup>* <sup>=</sup> *npeakλ*2e2 4*π*2c2*�*<sup>0</sup>

where *npeak* is the maximum electron density reached. Considering a peak density of 1020 m−<sup>3</sup> the deflexion suffered by a 10.591 *μ*m beam would be of 10−<sup>5</sup> rad, which is several orders of magnitude below the typical alignment errors. However, for larger wavelengths, for instance in the far IR region this effect can be significant and should be taken into account. In Table **??** a summary of different wavelengths used in interferometers operating in TJ-II, W7-X, DIII-D, JET, LHD, Tore Supra and ITER fusion machines is shown Braithwaite et al. (1989); Brower et al. (2006); Gil et al. (2001); Kawahata et al. (1997); Kornejew et al. (2006); Zeeland et al.

TJ II is a medium scale flexible Heliac type stellarator Alejaldre (2005) installed in the Laboratorio Nacional de Fusión of CIEMAT, Spain for the study of helical axis plasmas in a wide range of parameters Alejaldre (1989). TJ-II device has a major radius of 1.5 m, a minor of 0.28 m and the magnetic field generated in the center of the plasma is of 1 T. The plasma is heated using an ECRH<sup>6</sup> system based on two gyrotrons working at 53.2 GHz and delivering

Substituting Eq. 17 into the former equation, the deflection angle becomes:

**4. Overview of the TJ-II and the W7-X interferometers**

<sup>6</sup> Electron Cyclotron Resonance Heating

angle would be:

(2006).

Refracted Beam

∇*ne*d*l*. (29)

, (30)

*p* is an integer that corresponds to the number of times that the beams cross the plasma, r*e* is the classical electron radius, *λ*<sup>1</sup> and *λ*<sup>2</sup> are the long and short wavelengths and Δ*φ*<sup>1</sup> and Δ*φ*<sup>2</sup> are the phase differences between the probing and reference signals of the two interferometers. Calculating the phase differences and hence the optical path length differences *λ*1Δ*φ*<sup>1</sup> and *λ*2Δ*φ*2, the line integrated density is obtained from Eq. 27. An appropriate phase measuring algorithm is needed for this aim. Estimating the plasma chord-length (path-length for the line-integral), Lchord, the average density is evaluated as < *ne* >= *nee*d*l*/*L*chord. The plasma chord-length depends on the plasma configuration, and is estimated using the spatial profile measured by the Thomson Scattering system, Herranz et al. (2008; 2000).

Another limiting factor, is that in practice plasmas are not uniform. A density gradient normally exists in the radial direction of the plasma, this will cause a deflection of the output beam with respect to the incident one, Figure 3. If the deflected angle, *θ* is large enough a miss alignment may occur and by the interference signal can be lost at the detector.

In general, the deflection angle is given by Hutchinson (1990):

$$
\theta = \frac{\lambda}{2\pi} \frac{\mathrm{d}\phi}{\mathrm{d}y} = \frac{\mathrm{d}}{\mathrm{d}y} \int \mathrm{Ndl}.\tag{28}
$$

<sup>4</sup> The non-cancelation of the second term results in systematic error in the final electron density

6 Will-be-set-by-IN-TECH

Infrared Far Infrared μwave

10−6 10−5 10−4 10−3 10−2 <sup>1015</sup>

Fig. 2. Cutoff density versus wavelength from the middle infrared range to the *μ*wave range. With shorter wavelengths higher densities can be measured. However, the vibration problem

of two-color interferometers. From Eq. 18 the optical path length difference from one of the

The second term of the equation takes into account the optical path length outside the plasma. The final line integrated density is computed subtracting the optical path-length differences between both interferometers if the second term is canceled4, Hartfuss et al. (1997);

> *ne*d*<sup>l</sup>* <sup>=</sup> <sup>Δ</sup>*φ*1*λ*<sup>1</sup> <sup>−</sup> <sup>Δ</sup>*φ*2*λ*<sup>2</sup> <sup>2</sup>*π<sup>p</sup>* · <sup>r</sup>*e*(*λ*<sup>2</sup>

*p* is an integer that corresponds to the number of times that the beams cross the plasma, r*e* is the classical electron radius, *λ*<sup>1</sup> and *λ*<sup>2</sup> are the long and short wavelengths and Δ*φ*<sup>1</sup> and Δ*φ*<sup>2</sup> are the phase differences between the probing and reference signals of the two interferometers. Calculating the phase differences and hence the optical path length differences *λ*1Δ*φ*<sup>1</sup> and *λ*2Δ*φ*2, the line integrated density is obtained from Eq. 27. An appropriate phase measuring algorithm is needed for this aim. Estimating the plasma chord-length (path-length for the line-integral), Lchord, the average density is evaluated as < *ne* >= *nee*d*l*/*L*chord. The plasma chord-length depends on the plasma configuration, and is estimated using the spatial profile

Another limiting factor, is that in practice plasmas are not uniform. A density gradient normally exists in the radial direction of the plasma, this will cause a deflection of the output beam with respect to the incident one, Figure 3. If the deflected angle, *θ* is large enough a miss

*ne*(*l*)d*l* + *λ*1*k*<sup>01</sup>

<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*<sup>2</sup> 2)  out

*N*d*l*, (26)

, (27)

*N*d*l*. (28)

becomes more significant and forces to use two color arrangement.

measured by the Thomson Scattering system, Herranz et al. (2008; 2000).

In general, the deflection angle is given by Hutchinson (1990):

alignment may occur and by the interference signal can be lost at the detector.

*<sup>θ</sup>* <sup>=</sup> *<sup>λ</sup>* 2*π* d*φ* <sup>d</sup>*<sup>y</sup>* <sup>=</sup> <sup>d</sup> d*y* 

<sup>4</sup> The non-cancelation of the second term results in systematic error in the final electron density

1 

plasma

*λ*1Δ*φ*<sup>1</sup> = 2*π*re*λ*<sup>2</sup>

λ=15μm λ=1mm

λ [m]

1020

interferometers is calculated as:

Hutchinson (1990):

Cutoff density [m−3]

1025

1030

Fig. 3. Refraction suffered by a beam of light when crossing a non-uniform plasma.

Substituting Eq. 17 into the former equation, the deflection angle becomes:

$$
\theta = \frac{\lambda^2 r\_\varepsilon}{2\pi} \int \nabla n\_\varepsilon \mathbf{d}l.\tag{29}
$$

The deflection has a quadratic dependence with the wavelength, so for low wavelengths this effect is to be lower and for the range of interest in this chapter, 1.064 − 10.591 *μ*m, it is negligible. For example, if a parabolic density profile is considered, the maximum deflexion angle would be:

$$\theta = \frac{n\_{peak}\lambda^2 \mathbf{e}^2}{4\pi^2 \mathbf{c}^2 \epsilon\_0},\tag{30}$$

where *npeak* is the maximum electron density reached. Considering a peak density of 1020 m−<sup>3</sup> the deflexion suffered by a 10.591 *μ*m beam would be of 10−<sup>5</sup> rad, which is several orders of magnitude below the typical alignment errors. However, for larger wavelengths, for instance in the far IR region this effect can be significant and should be taken into account. In Table **??** a summary of different wavelengths used in interferometers operating in TJ-II, W7-X, DIII-D, JET, LHD, Tore Supra and ITER fusion machines is shown Braithwaite et al. (1989); Brower et al. (2006); Gil et al. (2001); Kawahata et al. (1997); Kornejew et al. (2006); Zeeland et al. (2006).

## **4. Overview of the TJ-II and the W7-X interferometers**

TJ II is a medium scale flexible Heliac type stellarator Alejaldre (2005) installed in the Laboratorio Nacional de Fusión of CIEMAT, Spain for the study of helical axis plasmas in a wide range of parameters Alejaldre (1989). TJ-II device has a major radius of 1.5 m, a minor of 0.28 m and the magnetic field generated in the center of the plasma is of 1 T. The plasma is heated using an ECRH<sup>6</sup> system based on two gyrotrons working at 53.2 GHz and delivering

<sup>6</sup> Electron Cyclotron Resonance Heating

Fig. 5. Detail of the W7-X Stellarator.

has been implemented in FPGA type devices.

**5. Detection schemes at TJ-II and W7-X**

devices.

minimizing the thermo-optical effect in the vessel windows Sánchez & Sánchez (2005). The configuration CO2-CO permits the use of one single detector for both wavelengths, however a correct frequency diplexion strategy must be fulfilled. There only exists one interferometer

Interferometry for Fusion Devices 255

Both the IR interferometer at TJ-II and the test prototype currently installed at IPP-Greifswald use an hybrid Michelson Mach-Zhender configuration, with two colors to compensate mechanical vibrations and heterodyne modulation. For the shake of simplicity in Figure 6 a two color Mach-Zhender arrangement operating as a fusion diagnostic is illustrated. One arm of the interferometer crosses the plasma and the other arm that travels the same optical path is used as a reference. In the probing arm (the one that crosses the plasma) a frequency displacement *ω*<sup>1</sup> is introduced using Acousto-optical modulation. When the probing beam reaches an square law detector it interferes with the reference beam and an output signal is generated. The frequency of this signal is that of the displacement introduced, *ω*1. The second interferometer is used to cancel mechanical vibrations and shares the same optical path as the first one. The final line-integrated density is obtained by subtracting the measures of both

In the TJ-II infrared interferometer always important signal processing tasks have been carried out by traditional analog systems. The speed and accuracy needed for this application suggests to replace these type of systems by digital electronics. Taking into account the frequencies handled the only possible solution is the use of FPGAs. All the signal processing

In the TJ-II Stellarator the configuration CO2 [10.591 *μ*m] − Nd : YAG [1.064 *μ*m] forces to use two detectors, Figure 7. The two interference signals are sampled using two ADCs and the phases are compared with the phases of two reference signals extracted directly from the

fusion diagnostic with a similar detection scheme Innocente & Tudisco (2006).


Table 1. Optical configurations for interferometers operating in different fusion devices

Fig. 4. Aerial and lateral view of the TJ-II device it has four periods, 32 toroidal coils, 2 poloidal coils and 2 vertical coils.

each up to 200 kW of power to the plasma, and NBI7. The maximum electronic densities reached are in the order of 8 · 1019m−3. To measure in this density range an heterodyne interferometer operating in the middle IR range is installed.

In IPP Greifswald the W7-X stellarator is currently in an advanced stage of construction. W7-X will be the biggest stellarator in the world. The major radius of W7-X will be of 5.5 m and the minor of 0.53 m with a maximum magnetic field in the axis of 3 T. W7-X will have ECRH heating systems in continuous mode (30 MW of power will be reached using this procedure), NBI and ICRH<sup>8</sup> (pulsed). W7-X device is an evolution of the W7-AS, being the first fusion device of the stellarator type with fully optimized, modular super-conducting coils aiming on high performance plasma under steady state conditions.

A key diagnostic that is being developed is a one channel two color heterodyne IR interferometer dedicated for density control and to control the Thomson scattering system Kornejew et al. (2006). It is planned to build it using CO2 (10.591 *μ*m) for the long wavelength and CO (5.295 *μ*m) for the short one. The choice of the short wavelength has the purpose of

<sup>7</sup> Neutral Beam Injection

<sup>8</sup> Ion Cyclotron Resonance Heating

8 Will-be-set-by-IN-TECH

**Fusion Device** *λ*<sup>1</sup> *λ*<sup>2</sup> Peak density TJ-II CO2 <sup>−</sup> 10.6 *<sup>μ</sup>*m NdYAG <sup>−</sup> 1.064 *<sup>μ</sup>*<sup>m</sup> <sup>∼</sup> <sup>8</sup> · 1019 <sup>m</sup>−<sup>3</sup> W7-X<sup>5</sup> CO2 <sup>−</sup> 10.6 *<sup>μ</sup>*m CO <sup>−</sup> 5.295 *<sup>μ</sup>*<sup>m</sup> <sup>&</sup>gt; 1020 <sup>m</sup>−<sup>3</sup> DIII-D CO2 <sup>−</sup> 10.6 *<sup>μ</sup>*m HeNe <sup>−</sup> 0.633 *<sup>μ</sup>*<sup>m</sup> <sup>&</sup>gt; 1020 <sup>m</sup>−<sup>3</sup> JET DCN <sup>−</sup> <sup>195</sup> *<sup>μ</sup>*m CH3OH <sup>−</sup> 118.8 *<sup>μ</sup>*<sup>m</sup> <sup>&</sup>gt; 1020 <sup>m</sup>−<sup>3</sup> LHD CH3OH <sup>−</sup> 118.8 *<sup>μ</sup>*m - <sup>&</sup>gt; 1020 <sup>m</sup>−<sup>3</sup> TORE SUPRA DCN <sup>−</sup> <sup>195</sup> *<sup>μ</sup>*m CH3OH <sup>−</sup> 118.8 *<sup>μ</sup>*<sup>m</sup> <sup>&</sup>gt; 1020 <sup>m</sup>−<sup>3</sup> ITER CO2 <sup>−</sup> 10.6 *<sup>μ</sup>*m CO <sup>−</sup> 5.295 *<sup>μ</sup>*<sup>m</sup> <sup>&</sup>gt; 1022 <sup>m</sup>−<sup>3</sup>

Table 1. Optical configurations for interferometers operating in different fusion devices

Fig. 4. Aerial and lateral view of the TJ-II device it has four periods, 32 toroidal coils, 2

each up to 200 kW of power to the plasma, and NBI7. The maximum electronic densities reached are in the order of 8 · 1019m−3. To measure in this density range an heterodyne

In IPP Greifswald the W7-X stellarator is currently in an advanced stage of construction. W7-X will be the biggest stellarator in the world. The major radius of W7-X will be of 5.5 m and the minor of 0.53 m with a maximum magnetic field in the axis of 3 T. W7-X will have ECRH heating systems in continuous mode (30 MW of power will be reached using this procedure), NBI and ICRH<sup>8</sup> (pulsed). W7-X device is an evolution of the W7-AS, being the first fusion device of the stellarator type with fully optimized, modular super-conducting coils aiming on

A key diagnostic that is being developed is a one channel two color heterodyne IR interferometer dedicated for density control and to control the Thomson scattering system Kornejew et al. (2006). It is planned to build it using CO2 (10.591 *μ*m) for the long wavelength and CO (5.295 *μ*m) for the short one. The choice of the short wavelength has the purpose of

poloidal coils and 2 vertical coils.

<sup>7</sup> Neutral Beam Injection

<sup>8</sup> Ion Cyclotron Resonance Heating

interferometer operating in the middle IR range is installed.

high performance plasma under steady state conditions.

Fig. 5. Detail of the W7-X Stellarator.

minimizing the thermo-optical effect in the vessel windows Sánchez & Sánchez (2005). The configuration CO2-CO permits the use of one single detector for both wavelengths, however a correct frequency diplexion strategy must be fulfilled. There only exists one interferometer fusion diagnostic with a similar detection scheme Innocente & Tudisco (2006).

Both the IR interferometer at TJ-II and the test prototype currently installed at IPP-Greifswald use an hybrid Michelson Mach-Zhender configuration, with two colors to compensate mechanical vibrations and heterodyne modulation. For the shake of simplicity in Figure 6 a two color Mach-Zhender arrangement operating as a fusion diagnostic is illustrated. One arm of the interferometer crosses the plasma and the other arm that travels the same optical path is used as a reference. In the probing arm (the one that crosses the plasma) a frequency displacement *ω*<sup>1</sup> is introduced using Acousto-optical modulation. When the probing beam reaches an square law detector it interferes with the reference beam and an output signal is generated. The frequency of this signal is that of the displacement introduced, *ω*1. The second interferometer is used to cancel mechanical vibrations and shares the same optical path as the first one. The final line-integrated density is obtained by subtracting the measures of both devices.

In the TJ-II infrared interferometer always important signal processing tasks have been carried out by traditional analog systems. The speed and accuracy needed for this application suggests to replace these type of systems by digital electronics. Taking into account the frequencies handled the only possible solution is the use of FPGAs. All the signal processing has been implemented in FPGA type devices.

## **5. Detection schemes at TJ-II and W7-X**

In the TJ-II Stellarator the configuration CO2 [10.591 *μ*m] − Nd : YAG [1.064 *μ*m] forces to use two detectors, Figure 7. The two interference signals are sampled using two ADCs and the phases are compared with the phases of two reference signals extracted directly from the

DETECTOR ADC

needed to isolate the correspondent signals.

filters Kornejew et al. (2006); Sanchez et al. (2008).

**6. Beam propagation**

The beam intensity is given by:

the best.

Fig. 8. Detection scheme based in two detectors, a beam splitter and two optical filters are

Interferometry for Fusion Devices 257

PVM-2TE-10.6 0.8 · 108 1.0 · 108 PCI-2TE-10.6 2.0 · 109 1.8 · 109 PVI-2TE-5 5.0 · 1011 <sup>−</sup>

Detector Detectivity at 5.296 [*μ*m] Detectivity at 10.591 [*μ*m]

Table 2. Detectivity for several Vigo Systems detectors at the wavelengths 5.296 *μ*m and 10.591 *μ*m. PV stands for photo-voltaic and PC for photo-conductive. The final choice has been the PCI-2TE-10.6.

detectivity<sup>9</sup> of these detectors is shown, being the photoconductive solution (PCI-2TE-10.6)

The line-electron density measurement at W7-X will require a precision of 1018 m−2, as it is described in Esteban, Sánchez, Sánchez, Kornejew, Hirsch, López, Fernández & Nieto-Taladriz (2010) the electronic filters needed to split both frequency components (CO2 and CO interference signals), must have an attenuation in the rejection band of 43 dB. This task has been solved using digital filters Esteban, Sánchez, Sánchez, Kornejew, Hirsch, López, Fernández & Nieto-Taladriz (2010) implemented in hardware as a better alternative than the analog diplexer used in the Frascati Tokamak Innocente & Tudisco (2006). The intermediate modulating frequencies are of 25 MHz for the CO and of 40 MHz for the CO2 and have been chosen in such a way that the useful bandwidth of the interferometer is not limited by the

The predominant propagation modes of the lasers used are the TEM00 the beams can be considered to be gaussian. The propagation of gaussian beams is well described in the classic literature Kogelnik & Li (1966); Self (1983), in this section the basic relations are summarized.

*πω*<sup>2</sup> <sup>e</sup>−2( *<sup>r</sup>*

<sup>9</sup> The detectivity is defined as the signal-to-noise ratio (SNR) at the detector output normalized to a 1 W

*ω* ) 2

. (31)

*<sup>I</sup>*(*r*, *<sup>z</sup>*) = <sup>2</sup>

radiant power, a 1 cm<sup>2</sup> detector optical area and a 1 HZ bandwidth

FILTER1

Processing System

FILTER2

Fig. 6. Two color infrared heterodyne Mach-Zehnder interferometer employing a classical two detector scheme.

Fig. 7. Detection scheme based in two detectors, a beam splitter and two optical filters are needed to isolate the correspondent signals.

oscillator that drives the AOMs. In principle no electronic input split preprocessing filter is needed, however, in practice these are useful to increase the signal-to-noise ratio (SNR) of the input interference signals.

The combinations, CO2 [10.591 *μ*m] − CO [5.295 *μ*m] and CO2 [12.1 *μ*m] − CO2 [9 *μ*m] are the ones planned to be used in the W7-X Stellarator and in the ITER Tokamak, respectively. In both cases the wavelengths can be detected by a single detector, Figure 8. Provided that the signals can be split using electronic filters the number of optical components can be significantly reduced. Taking into account the port restriction access in W7-X this would be a great advantage.

In the particular case of the W7-X interferometer, several detectors from Vigo system show the same response for both CO2 [10.591 *μ*m] and CO [5.295 *μ*m] wavelengths. In Table **??** the 10 Will-be-set-by-IN-TECH

PLASMA

Fig. 6. Two color infrared heterodyne Mach-Zehnder interferometer employing a classical

Fig. 7. Detection scheme based in two detectors, a beam splitter and two optical filters are

oscillator that drives the AOMs. In principle no electronic input split preprocessing filter is needed, however, in practice these are useful to increase the signal-to-noise ratio (SNR) of the

The combinations, CO2 [10.591 *μ*m] − CO [5.295 *μ*m] and CO2 [12.1 *μ*m] − CO2 [9 *μ*m] are the ones planned to be used in the W7-X Stellarator and in the ITER Tokamak, respectively. In both cases the wavelengths can be detected by a single detector, Figure 8. Provided that the signals can be split using electronic filters the number of optical components can be significantly reduced. Taking into account the port restriction access in W7-X this would be a

In the particular case of the W7-X interferometer, several detectors from Vigo system show the same response for both CO2 [10.591 *μ*m] and CO [5.295 *μ*m] wavelengths. In Table **??** the

DETECTOR1 ADC1

OSCILLATOR

DETECTOR2

needed to isolate the correspondent signals.

BS

FILTER2

FILTER1

input interference signals.

great advantage.

two detector scheme.

AOM2

AOM1

DETECTOR

Processing System

DETECTOR

ADC1

Fig. 8. Detection scheme based in two detectors, a beam splitter and two optical filters are needed to isolate the correspondent signals.


Table 2. Detectivity for several Vigo Systems detectors at the wavelengths 5.296 *μ*m and 10.591 *μ*m. PV stands for photo-voltaic and PC for photo-conductive. The final choice has been the PCI-2TE-10.6.

detectivity<sup>9</sup> of these detectors is shown, being the photoconductive solution (PCI-2TE-10.6) the best.

The line-electron density measurement at W7-X will require a precision of 1018 m−2, as it is described in Esteban, Sánchez, Sánchez, Kornejew, Hirsch, López, Fernández & Nieto-Taladriz (2010) the electronic filters needed to split both frequency components (CO2 and CO interference signals), must have an attenuation in the rejection band of 43 dB. This task has been solved using digital filters Esteban, Sánchez, Sánchez, Kornejew, Hirsch, López, Fernández & Nieto-Taladriz (2010) implemented in hardware as a better alternative than the analog diplexer used in the Frascati Tokamak Innocente & Tudisco (2006). The intermediate modulating frequencies are of 25 MHz for the CO and of 40 MHz for the CO2 and have been chosen in such a way that the useful bandwidth of the interferometer is not limited by the filters Kornejew et al. (2006); Sanchez et al. (2008).

### **6. Beam propagation**

The predominant propagation modes of the lasers used are the TEM00 the beams can be considered to be gaussian. The propagation of gaussian beams is well described in the classic literature Kogelnik & Li (1966); Self (1983), in this section the basic relations are summarized.

The beam intensity is given by:

$$I(r,z) = \frac{2}{\pi\omega^2} \mathbf{e}^{-2\left(\frac{r}{\omega}\right)^2}.\tag{31}$$

<sup>9</sup> The detectivity is defined as the signal-to-noise ratio (SNR) at the detector output normalized to a 1 W radiant power, a 1 cm<sup>2</sup> detector optical area and a 1 HZ bandwidth

Laser Power [W] *θ* [mrad] *ω*<sup>0</sup> [mm] *M*<sup>2</sup> Nd : YAG − 1.064 *μ*m 0.3 4 0.2 < 1.2 CO2 − 10.591 *μ*m 8 5.5 1.5 < 1.2 Table 3. Propagation characteristics of the Nd:YAG and CO2 beams, the curvature radius of both beams must be the same for equal axial positions. Both optical paths must share the

Interferometry for Fusion Devices 259

The power received by the detector is calculated integrating the radiation intensity, Eq. 31 over the detector area. If the detectors are located at a position z, such that the condition,

> *z zR*

Considering that both reference and probing beams reach the detector with intensities, *I*<sup>1</sup> and

the AC component of the interference, <sup>√</sup>*I*<sup>1</sup> *<sup>I</sup>*<sup>2</sup> has been plotted together with the reference and probing intensities as it can be seen in Figure **??** the intensity of the interference is maximum when both probing and measuring intensities are equal, *I*<sup>1</sup> = *I*2. This condition can be met by

As a final remark it should be notices that the beam waists of both colors must be located at the same geometrical positions this issue is specially important in multichannel interferometers.

from the experience gain at TJ-II it can be said that the first is likely to be very sensitive to mechanical movements, in any case it should be ensures a high pointing stability of the beams.

The line-electron density computation, indicated in Eq. 27 implies the use of an appropriate phase measuring algorithm. Several solutions have been developed in this sense. Some procedures are based on counting the cycles of a high speed clock in synchronism with the zero crossings of the probing signal (2004). These approaches, typically require a clock frequency higher than the frequency of the probing signals and are likely to suffer fringe jumps and synchronism problems if they are not carefully designed. Another method suitable for measuring phase shifts lower than one period of the input signal consist in decomposing

In this sense, three approaches can be considered to obtain the different chords:

<sup>=</sup> *<sup>λ</sup><sup>z</sup> πω*0

*<sup>λ</sup>*2*z*<sup>2</sup> <sup>e</sup>−2( *<sup>r</sup>πω*<sup>0</sup> *<sup>λ</sup><sup>z</sup>* )

2

, (36)

. (37)

*<sup>I</sup>*<sup>1</sup> *<sup>I</sup>*<sup>2</sup> cos (2*πν* <sup>+</sup> <sup>Δ</sup>*φ*), (38)

*ω* � *ω*<sup>0</sup>

*<sup>I</sup>*(*r*, *<sup>z</sup>*) = <sup>2</sup>*πω*<sup>0</sup>

*z* � *z*<sup>0</sup> is met, then the beam radius *ω*(*z*) can be approximated by:

*I*<sup>2</sup> and knowing that the total interference intensity at one detector is:

*I* = *I*<sup>1</sup> + *I*<sup>2</sup> + 2

same optical axis.

is the same at the detectors.

and the intensity, for one beam becomes:

adjusting properly the AOMs.

2. Using independent modules,

1. Expanded beam,

3. Using beam splitters,

**7. Phase measurement**

Fig. 9. Schematic layout of the one channel infrared interferometer of TJ-II, The green and red lines correspond to the CO2 and Nd:YAG beams. The black line after the first beam combiner (BC1) represents the path for both reference beams while the blue line represents the path for the probing beams. Both interference signals are detected in a photoconductor detector (PC), CO2 and in an Avalanche Photo detector (APD), Nd:YAG. M12 is a periscope that launches both probing beams into the plasma, they cross a ZnSe window, the plasma and are reflected back to the periscope. The beam expander located at the output of the CO2 sets both beam waists, Nd:YAG and CO2 in the same position.

being, *r* the radius and *ω* the beam radius:

$$
\omega(z) = \omega\_0 \left[ 1 + \left(\frac{z}{z\_R}\right)^2 \right],
\tag{32}
$$

*ω*<sup>0</sup> is the beam waist and *zR* is the Rayleigh range that separates the near and far fields:

$$z\_R = \frac{\pi \omega\_0^2}{\lambda}.\tag{33}$$

The have angle beam divergence is given by:

$$
\theta = M^2 \frac{\lambda}{\pi \omega\_0},
\tag{34}
$$

where *M*<sup>2</sup> is the beam quality factor or the beam propagation factor.

In Figure 9 the schematic layout of the TJ-II infrared interferometer is illustrated and in In Figure **??** an unfolded virtual representation has been adopted for simplicity, in which the gaussian, Nd:YAG and CO2 beams, travel in one direction and equal distances as they do in the interferometer, in this particular case 15 m until the beams reach the detectors.

Table **??** summarizes the main propagation characteristics of both lasers. At this point the Nd : YAG beam has a diameter of 5 cm and the CO2 of 10 cm, the beam waists of both beams are located in the same positions so the curvature radius:

$$R(z) = z \left[ 1 + \left( \frac{z\_R}{z} \right)^2 \right],\tag{35}$$

12 Will-be-set-by-IN-TECH

Fig. 9. Schematic layout of the one channel infrared interferometer of TJ-II, The green and red lines correspond to the CO2 and Nd:YAG beams. The black line after the first beam combiner (BC1) represents the path for both reference beams while the blue line represents the path for the probing beams. Both interference signals are detected in a photoconductor detector (PC), CO2 and in an Avalanche Photo detector (APD), Nd:YAG. M12 is a periscope that launches both probing beams into the plasma, they cross a ZnSe window, the plasma and are reflected back to the periscope. The beam expander located at the output of the CO2 sets both beam

*ω*(*z*) = *ω*<sup>0</sup>

where *M*<sup>2</sup> is the beam quality factor or the beam propagation factor.

are located in the same positions so the curvature radius:

 1 + *z zR*

*ω*<sup>0</sup> is the beam waist and *zR* is the Rayleigh range that separates the near and far fields:

*zR* <sup>=</sup> *πω*<sup>2</sup> 0

*<sup>θ</sup>* <sup>=</sup> *<sup>M</sup>*<sup>2</sup> *<sup>λ</sup>*

In Figure 9 the schematic layout of the TJ-II infrared interferometer is illustrated and in In Figure **??** an unfolded virtual representation has been adopted for simplicity, in which the gaussian, Nd:YAG and CO2 beams, travel in one direction and equal distances as they do in

Table **??** summarizes the main propagation characteristics of both lasers. At this point the Nd : YAG beam has a diameter of 5 cm and the CO2 of 10 cm, the beam waists of both beams

> 1 + *zR z* 2

the interferometer, in this particular case 15 m until the beams reach the detectors.

*R*(*z*) = *z*

*πω*0

<sup>2</sup> 

, (32)

*<sup>λ</sup>* . (33)

, (34)

, (35)

waists, Nd:YAG and CO2 in the same position.

being, *r* the radius and *ω* the beam radius:

The have angle beam divergence is given by:


Table 3. Propagation characteristics of the Nd:YAG and CO2 beams, the curvature radius of both beams must be the same for equal axial positions. Both optical paths must share the same optical axis.

is the same at the detectors.

The power received by the detector is calculated integrating the radiation intensity, Eq. 31 over the detector area. If the detectors are located at a position z, such that the condition, *z* � *z*<sup>0</sup> is met, then the beam radius *ω*(*z*) can be approximated by:

$$
\omega \simeq \omega\_0 \frac{z}{z\_{\mathbb{R}}} = \frac{\lambda z}{\pi \omega\_0} \,\,\,\,\tag{36}
$$

and the intensity, for one beam becomes:

$$I(r,z) = \frac{2\pi\omega\upsilon\_0}{\lambda^2 z^2} e^{-2\left(\frac{r\pi\omega\_0}{\lambda z}\right)^2}.\tag{37}$$

Considering that both reference and probing beams reach the detector with intensities, *I*<sup>1</sup> and *I*<sup>2</sup> and knowing that the total interference intensity at one detector is:

$$I = I\_1 + I\_2 + 2\sqrt{I\_1 I\_2} \cos\left(2\pi\nu + \Delta\phi\right),\tag{38}$$

the AC component of the interference, <sup>√</sup>*I*<sup>1</sup> *<sup>I</sup>*<sup>2</sup> has been plotted together with the reference and probing intensities as it can be seen in Figure **??** the intensity of the interference is maximum when both probing and measuring intensities are equal, *I*<sup>1</sup> = *I*2. This condition can be met by adjusting properly the AOMs.

As a final remark it should be notices that the beam waists of both colors must be located at the same geometrical positions this issue is specially important in multichannel interferometers. In this sense, three approaches can be considered to obtain the different chords:


from the experience gain at TJ-II it can be said that the first is likely to be very sensitive to mechanical movements, in any case it should be ensures a high pointing stability of the beams.

## **7. Phase measurement**

The line-electron density computation, indicated in Eq. 27 implies the use of an appropriate phase measuring algorithm. Several solutions have been developed in this sense. Some procedures are based on counting the cycles of a high speed clock in synchronism with the zero crossings of the probing signal (2004). These approaches, typically require a clock frequency higher than the frequency of the probing signals and are likely to suffer fringe jumps and synchronism problems if they are not carefully designed. Another method suitable for measuring phase shifts lower than one period of the input signal consist in decomposing

Fig. 10. Ilustration of the parameters involved in the interpolation algorithm. In this example just the negative edge zero crossings are considered.

the signals in its In-Phase & Quadrature components (IQ scheme). As the phase shifts in IR-interferometer operating as fusion diagnostic are of several periods the use of this technique requires of an additional fringe counting mechanism. Pulse width modulation based schemes are also used for phase measurements in interferometry Choi et al. (2005); Mlynek et al. (2010) Another method based on Analog-to-Digital conversion, ADC, zero crossing detection and linear interpolation Zubarev & Khilchenko (2003) was used in the TJ-II interferometer with good results Sanchez et al. (2004) this has been the deciding factor to implement this phase measuring algorithm in the real time measuring FPGA-based system. Below the IQ scheme and the interpolation algorithm are explained in detail.

### **7.1 Interpolation algorithm**

During several TJ-II campaigns an interpolation algorithm has been used to compute the phase differences of the interferometric signals Sanchez et al. (2004). This algorithm counts both the zero crossings of the signals n, and the periods of time between zero crossings, Figure 10. Counting the number of samples in a period and adding it to the fraction of sample correspondent to each zero crossing, the phase at each sampling time can be obtained easily. The sample time fractions are computed interpolating linearly two consecutive samples around a zero cross, each value is given by Sanchez et al. (2004); Zubarev & Khilchenko (2003):

$$\pi\_{\mathfrak{n}} = \frac{|S\_{\bar{j}}|}{|S\_{\bar{j}}| + |S\_{\bar{j}+1}|} \Delta t\_{\prime} \tag{39}$$

<sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> 10−4

Interferometry for Fusion Devices 261

Fig. 11. Interpolation error according to the number of samples per period of the signals, as it can be seen in the figure with less than four samples per period the error increases abruptly.

Using the same procedure, *tn*+<sup>1</sup> can be calculated. The calculation of the phase value

Where *tn*+<sup>1</sup> − *tn* is the current period. in Figure 11 the error introduced by the algorithm for

Precisions of 1/400 of CO2 fringe have been achieved with this algorithm . This precision is equivalent to an error in the line integrated density of <sup>±</sup>1017m−3. The algorithm was implemented in an off-line processing system and has been routinely working since autumn of 2003. Currently the FPGA-based processing is installed in the TJ-II IR-interferometer, allowing the readout of control signals and the processing of the electron density signal in real time

Several factors affect the propagation of electromagnetic waves in the plasmas and they affect the phase shift suffered by these waves. In most cases, the cold plasma approximation is valid. However, for electron temperatures in the order of tens of keV, relativistic effects become important, and a correction factor must be included in the equation that relates the phase

The cutoff plasma frequency, mechanical vibrations and refraction are limiting parameters for the probing wavelengths. For high plasma densities, these limiting factors force

2*π*[(*i* − *j*)Δ*t* − *τn*] *tn*+<sup>1</sup> − *tn*

. (41)

*ϕ<sup>i</sup>* = 2*πn* +

Esteban et al. (2011); Esteban, Sánchez, López, Nieto-Taladriz & Sánchez (2010).

different numbers of samples per period is illustrated

shifts with the line integrated electron density.

Number of samples per period

10−3

correspondent at sample time i is:

**8. Conclusions**

Period relative error

10−2

10−1

j is the number of zero crossings, Δ*t* is the sample period and |*Sj*| and |*Sj*+1| are the absolute values of leading and trailing samples of a zero crossing. The total time elapsed at the j zero crossing is given by:

$$t\_{\rm ll} = j\Delta t + \tau\_{\rm ll}.\tag{40}$$

14 Will-be-set-by-IN-TECH

i sample

Fig. 10. Ilustration of the parameters involved in the interpolation algorithm. In this example

the signals in its In-Phase & Quadrature components (IQ scheme). As the phase shifts in IR-interferometer operating as fusion diagnostic are of several periods the use of this technique requires of an additional fringe counting mechanism. Pulse width modulation based schemes are also used for phase measurements in interferometry Choi et al. (2005); Mlynek et al. (2010) Another method based on Analog-to-Digital conversion, ADC, zero crossing detection and linear interpolation Zubarev & Khilchenko (2003) was used in the TJ-II interferometer with good results Sanchez et al. (2004) this has been the deciding factor to implement this phase measuring algorithm in the real time measuring FPGA-based system.

During several TJ-II campaigns an interpolation algorithm has been used to compute the phase differences of the interferometric signals Sanchez et al. (2004). This algorithm counts both the zero crossings of the signals n, and the periods of time between zero crossings, Figure 10. Counting the number of samples in a period and adding it to the fraction of sample correspondent to each zero crossing, the phase at each sampling time can be obtained easily. The sample time fractions are computed interpolating linearly two consecutive samples around a zero cross, each value is given by Sanchez et al. (2004); Zubarev & Khilchenko (2003):

*<sup>τ</sup><sup>n</sup>* <sup>=</sup> <sup>|</sup>*Sj*<sup>|</sup>


j is the number of zero crossings, Δ*t* is the sample period and |*Sj*| and |*Sj*+1| are the absolute values of leading and trailing samples of a zero crossing. The total time elapsed at the j zero

Δ*t*, (39)

*tn* = *j*Δ*t* + *τn*. (40)

Below the IQ scheme and the interpolation algorithm are explained in detail.

just the negative edge zero crossings are considered.

**7.1 Interpolation algorithm**

crossing is given by:

n zero n+1 zero

Fig. 11. Interpolation error according to the number of samples per period of the signals, as it can be seen in the figure with less than four samples per period the error increases abruptly.

Using the same procedure, *tn*+<sup>1</sup> can be calculated. The calculation of the phase value correspondent at sample time i is:

$$\varphi\_{i} = 2\pi n + \frac{2\pi [(i-j)\Delta t - \tau\_{n}]}{t\_{n+1} - t\_{n}}.\tag{41}$$

Where *tn*+<sup>1</sup> − *tn* is the current period. in Figure 11 the error introduced by the algorithm for different numbers of samples per period is illustrated

Precisions of 1/400 of CO2 fringe have been achieved with this algorithm . This precision is equivalent to an error in the line integrated density of <sup>±</sup>1017m−3. The algorithm was implemented in an off-line processing system and has been routinely working since autumn of 2003. Currently the FPGA-based processing is installed in the TJ-II IR-interferometer, allowing the readout of control signals and the processing of the electron density signal in real time Esteban et al. (2011); Esteban, Sánchez, López, Nieto-Taladriz & Sánchez (2010).

### **8. Conclusions**

Several factors affect the propagation of electromagnetic waves in the plasmas and they affect the phase shift suffered by these waves. In most cases, the cold plasma approximation is valid. However, for electron temperatures in the order of tens of keV, relativistic effects become important, and a correction factor must be included in the equation that relates the phase shifts with the line integrated electron density.

The cutoff plasma frequency, mechanical vibrations and refraction are limiting parameters for the probing wavelengths. For high plasma densities, these limiting factors force

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Sanchez, M., Sanchez, J., Estrada, T., Sanchez, E., Acedo, P. & Lamela, H. (2004). High

Sánchez, M. & Sánchez, J. (2005). Thermo-optical effect in zinc selenide windows for two-color interferometer for fusion plasma diagnostics, *Rev. Sci. Instrum.* 76: 046104–046104. Zeeland, M. A. V., Boivin, R. L., Carlstrom, T. N., Deterly, T. & Finkenthal, D. K. (2006).

Self, S. A. (1983). Focusing of spherical gaussian beams, *Applied Optics* 22(5): 658–661.

Kogelnik, H. & Li, T. (1966). Laser beams and resonators, *Appl. Opt.* 5(10): 1550–1567.

Hutchinson, I. H. (1990). *Principles of plasma diagnostics*, Cambridge University Press.

Sánchez, J. & Tribaldos, V. (2000). Profile structures of tj-ii stellarator plasmas, *Phys.*

for the frascati tokamak upgrade fusion test device, *Applied Optics* 45(36): 9105–9114.

interferometer using in-vessel optics for extended edge resolution, *Review of Scientific*

far infrared laser interferometer for the lhd, *Fusion Engineering and Design* 34-35: 393

URL: *http://www.sciencedirect.com/science/article/B6V3C-3SN5TJ5-2H/2/2f9f024cce400*

Eheberg, J. & Team, A. U. (2010). Design of a digital multiradian phase detector and its application in fusion plasma interferometry, *Review of Scientific Instruments*

a two colour interferometer for plasma density diagnostics. a reduction algorithm.,

resolution co[sub 2] interferometry on the tj-ii stellarator by using an adc-based phase

Fiber optic two-color vibration compensated interferometer for plasma density

to use short wavelengths, that are specially sensitive to mechanical vibrations. These mechanical vibrations are canceled using another interferometer with a shorter wavelength, and subtracting both results.

The optical setup of both, the two-color one-channel interferometer at the TJ-II and the two color interferometer planned to be installed at the W7-X, have been described. They both operate in the middle infrared range to measure high density plasmas, and use heterodyne frequency displacement to cope with the phase sensitivity problem. In the particular case of the W7-X interferometer, the possibility of using a single detector is opened, which represents a challenge in the design of a frequency diplexor. The basic principles of gaussian beam propagation have been summarized, and illustrated for the TJ-II one-channel heterodyne interferometer. Also, an estimation of the signal-to-noise ratio at the output of both detectors has been calculated for the TJ-II interferometer. In addition several phase measuring algorithms for these type of interferometers have been listed, and the linear interpolation based algorithm used at the the TJ-II interferometer and tested at the W7-X one has been described.

## **9. References**

(2004). *Measurement Science and Technology* 15: 2341–2348.

Alejaldre, C. (1989). Tj-ii project: A flexible heliac stellarator, *Fusion Technology* 17.


16 Will-be-set-by-IN-TECH

to use short wavelengths, that are specially sensitive to mechanical vibrations. These mechanical vibrations are canceled using another interferometer with a shorter wavelength,

The optical setup of both, the two-color one-channel interferometer at the TJ-II and the two color interferometer planned to be installed at the W7-X, have been described. They both operate in the middle infrared range to measure high density plasmas, and use heterodyne frequency displacement to cope with the phase sensitivity problem. In the particular case of the W7-X interferometer, the possibility of using a single detector is opened, which represents a challenge in the design of a frequency diplexor. The basic principles of gaussian beam propagation have been summarized, and illustrated for the TJ-II one-channel heterodyne interferometer. Also, an estimation of the signal-to-noise ratio at the output of both detectors has been calculated for the TJ-II interferometer. In addition several phase measuring algorithms for these type of interferometers have been listed, and the linear interpolation based algorithm used at the the TJ-II interferometer and tested at the W7-X one has been

and subtracting both results.

described.

**9. References**

*Technology* .

*81427fb48803f2d8b58*

(2004). *Measurement Science and Technology* 15: 2341–2348.

URL: *http://link.aip.org/link/?RSI/60/2825/1*

*Review of Scientific Instruments* 77(10): 10E911. URL: *http://link.aip.org/link/?RSI/77/10E911/1*

URL: *http://link.aip.org/link/?RSI/76/093105/1*

Alejaldre, C. (1989). Tj-ii project: A flexible heliac stellarator, *Fusion Technology* 17.

Braithwaite, G., Gottardi, N., Magyar, G., O'Rourke, J., Ryan, J. & Véron, D. (1989). Jet polari-interferometer, *Review of Scientific Instruments* 60(9): 2825–2834.

Brower, D. L., Deng, B. H. & Ding, W. X. (2006). Divertor interferometer diagnostic for iter,

Choi, H., Park, K. & La, J. (2005). Novel phase measurement technique of the heterodyne laser

Esteban, L., Sánchez, M., López, J. A., Nieto-Taladriz, O., Kornejew, P. & Hirsch, M.

Esteban, L., Sánchez, M., Sánchez, J., Kornejew, P., Hirsch, M., López, J. A., Fernández,

Gil, C., Elbeze, D. & Portafaix, C. (2001). Development of the ir interfero-polarimeter for long pulse operation at tore supra, *Fusion Engineering and Design* 56-57: 969 – 973.

Hartfuss, H. J., Geist, T. & Hirsch, M. (1997). Heterodyne methods in millimetre wave plasma

ir-interferometers, *IEEE Transactions on Nuclear Science* 58: 1562 – 1569. Esteban, L., Sánchez, M., López, J. A., Nieto-Taladriz, O. & Sánchez, J. (2010). Continuous

(2011). Development of efficient fpga-based multi-channel phase meters for

plasma density measurement in tj-ii infrared interferometer-advanced signal

A. & Nieto-Taladriz, O. (2010). Continous phase measurement in the w7-x ir-interferometer by means of an fpga and high speed adcs, *Fusion Science and*

URL: *http://www.sciencedirect.com/science/article/B6V3C-44JDTMG-5G/2/6ec2412225de2*

diagnostics with applications to ece, interferometry and reflectometry, *Plasma Physics*

Alejaldre, C. (2005). Tj-ii the flexible heliac, *Europhysics News* pp. 232–233.

interferometer, *Review of Scientific Instruments* 76(9): 093105.

processing based on fpgas, *Fusion Engineering and Design* .

*and Controlled Fusion* 39(11): 1693.

URL: *http://stacks.iop.org/0741-3335/39/i=11/a=001*


Hutchinson, I. H. (1990). *Principles of plasma diagnostics*, Cambridge University Press.


URL: *http://link.aip.org/link/?RSI/68/694/1*


URL: *http://www.sciencedirect.com/science/article/B6V3C-3SN5TJ5-2H/2/2f9f024cce400 6745e3f824b0609ad87*


URL: *http://link.aip.org/link/?RSI/81/033507/1*


**13** 

*México* 

**Simultaneous Phase Shifting Shearing** 

Noel-Ivan Toto-Arellano1, David-Ignacio Serrano-García1, Amalia Martínez-García1 and Gustavo Rodríguez-Zurita2

**and Dynamic Phase Objects** 

*1Centro de Investigaciones en Óptica A.C. León, Gto., 2Benemérita Universidad Autónoma de Puebla,* 

**Interferometry for Measurement of Static** 

Through the use of non-destructive optical techniques to measure the surface with high accuracy, two shearing interferometers were implemented to analyze the slope of phase objects using simultaneous phase-shifting shearing interferometry [Toto et al., 2010]. These optical configurations allow obtaining *n*-shearograms simultaneously to retrieve the optical

Several optical systems have been developed to retrieve the optical phase data in a single capture employing polarization, for example, through the use of micro-polarizer arrays [Novak et al.,2005], a point diffraction interferometer[Neal, 2003 ], a two-beam grating interferometer [Rodriguez, 2008] and a liquid-crystal spatial modulator [Mercer and Creath, 1996], among others. These systems have been employed in several fields of application, such as optical metrology[Wyant, 2004; Cheng and Wyant, 1984], holography[Yamaguchi and Zhang, 1997; Nomura, 2006], optical tomography[Meneses, 2006], ESPI [Chen, 2010], etc. In this chapter, we present theoretical analyses and experimental results obtained by implementing several shearing simultaneous phaseshifting interferometers capable of retrieving the directional derivatives in the *x*direction, or adapting the systems to obtain radial derivatives [Serrano et al., 2011], allowing direct slope measurements; in each case, the optical phase data are processed by means of the capture of *n*-interferograms in a single shot. To achieve this, we propose an alternative to the interferometers developed for phase shifting that use micro polarizer arrays, replacing them with a 4-*f* system with high frequency Ronchi gratings or Phase

In order to obtain the partial derivative of the phase object, a Cyclic Shear Interferometer (CSI) of lateral shear with crossed circular polarization has been implemented; the CSI is coupled to a 4-*f* system with high frequency Ronchi gratings, a pair of crossed gratings

),( placed as the system's pupil with a spatial period *d.* The high frequency Bi-Ronchi gratings generate replicas of the patterns required to process the derivative of the phase. The

phase data map with higher accuracy [Rodriguez et al., 2009; Toto et al 2009].

gratings coupled at the end of the interferometer.

**1. Introduction** 

*G* 

measurements, *Review of Scientific Instruments* 77(10): 10F325. URL: *http://link.aip.org/link/?RSI/77/10F325/1*

Zubarev, P. V. & Khilchenko, A. D. (2003). Precision phase discriminator for the heterodyne interferometric plasma-density measurement technique, *Instruments and Experimental Techniques* 46: 171–176. 10.1023/A:1023661531511. URL: *http://dx.doi.org/10.1023/A:1023661531511*

## **Simultaneous Phase Shifting Shearing Interferometry for Measurement of Static and Dynamic Phase Objects**

Noel-Ivan Toto-Arellano1, David-Ignacio Serrano-García1, Amalia Martínez-García1 and Gustavo Rodríguez-Zurita2 *1Centro de Investigaciones en Óptica A.C. León, Gto., 2Benemérita Universidad Autónoma de Puebla, México* 

## **1. Introduction**

18 Will-be-set-by-IN-TECH

264 Interferometry – Research and Applications in Science and Technology

Zubarev, P. V. & Khilchenko, A. D. (2003). Precision phase discriminator for the heterodyne

interferometric plasma-density measurement technique, *Instruments and Experimental*

measurements, *Review of Scientific Instruments* 77(10): 10F325.

URL: *http://link.aip.org/link/?RSI/77/10F325/1*

*Techniques* 46: 171–176. 10.1023/A:1023661531511. URL: *http://dx.doi.org/10.1023/A:1023661531511*

> Through the use of non-destructive optical techniques to measure the surface with high accuracy, two shearing interferometers were implemented to analyze the slope of phase objects using simultaneous phase-shifting shearing interferometry [Toto et al., 2010]. These optical configurations allow obtaining *n*-shearograms simultaneously to retrieve the optical phase data map with higher accuracy [Rodriguez et al., 2009; Toto et al 2009].

> Several optical systems have been developed to retrieve the optical phase data in a single capture employing polarization, for example, through the use of micro-polarizer arrays [Novak et al.,2005], a point diffraction interferometer[Neal, 2003 ], a two-beam grating interferometer [Rodriguez, 2008] and a liquid-crystal spatial modulator [Mercer and Creath, 1996], among others. These systems have been employed in several fields of application, such as optical metrology[Wyant, 2004; Cheng and Wyant, 1984], holography[Yamaguchi and Zhang, 1997; Nomura, 2006], optical tomography[Meneses, 2006], ESPI [Chen, 2010], etc. In this chapter, we present theoretical analyses and experimental results obtained by implementing several shearing simultaneous phaseshifting interferometers capable of retrieving the directional derivatives in the *x*direction, or adapting the systems to obtain radial derivatives [Serrano et al., 2011], allowing direct slope measurements; in each case, the optical phase data are processed by means of the capture of *n*-interferograms in a single shot. To achieve this, we propose an alternative to the interferometers developed for phase shifting that use micro polarizer arrays, replacing them with a 4-*f* system with high frequency Ronchi gratings or Phase gratings coupled at the end of the interferometer.

> In order to obtain the partial derivative of the phase object, a Cyclic Shear Interferometer (CSI) of lateral shear with crossed circular polarization has been implemented; the CSI is coupled to a 4-*f* system with high frequency Ronchi gratings, a pair of crossed gratings *G* ),( placed as the system's pupil with a spatial period *d.* The high frequency Bi-Ronchi gratings generate replicas of the patterns required to process the derivative of the phase. The

Simultaneous Phase Shifting Shearing Interferometry

on each diffraction order [Rodriguez, 2009].

*x*: linear shear. *x*0: beam separation. *F fd* <sup>0</sup>

**2.2 Mach Zehnder interferometer** 

for Measurement of Static and Dynamic Phase Objects 267

superposition of the two beam replicas with mutual shear *x* would appear around each diffraction order. Since the interferograms retain their polarization characteristics, polarizing filters are placed in each generated replica in order to generate the phase shifts required to obtain the phase. Fig. 1(b) shows the two configurations of the polarizing arrangement proposed to generate the phase shifts in a single shot, used to process the optical phase. Some replicas of the interference patterns have different intensities; however, there are at least four patterns with similar intensities in each configuration. As will be shown in the following sections, the replicas of the pattern generated by the CSI are centered

Fig. 1. Simultaneous phase shifting cyclic path interferometer. (a) CSI with variable shear coupled to 4*-f* system with high frequency Bi-Ronchi grating. Q, Q': Quarter wave retarders.

In several cases, such as in ophthalmic lens analysis, radial symmetry is encountered, and it is strongly necessary to obtain the radial derivative of the incident wave front in order to analyze the radial variations of the objects. For that reason, we propose the use of another optical system to obtain the radial derivative and associate it with its corresponding radial slope. Fig. 2 shows the experimental setup for the measurement of the radial slope of transparent objects. The interferometric system uses a *He*-*Ne* laser operating at = 632.8 nm. Polarized light at 45° enters the interferometer by using a quarter-wave retarder plate Q0 and a linear polarizer P0. The Mach-Zehnder radial-shear interferometer (MZRI) presented comprises two telescope systems, (S1, S2) [Lago and de la Fuente R. ,2008] one on each arm with vertical and horizontal linear polarized light, respectively. Their variants are widely used as part of different applications, so they can be considered to be representative cases of potential adaptations for simultaneous phase shifting interferometry. A 4-*f* system is coupled at the end of the interferometer; this system consists of two similar achromatic

/ is the order of separation. (b) Arrangement

of polarizing filters. *P*i: polarizing filters, ψi: transmission angle of polarization.

processing of the interferograms is performed using conventional methods of phase extraction with the algorithm of four shifts. For the case when the object presents radial symmetry, it is more convenient to use a radial-shear interferometer. Applications of radialshear interferometers have been previously reported in optical testing [ Malacara, 1974; Steel 1965], aspherical surface measuring [Hariharan, 1984], corneal topographic inspection [Kowalik et al., 2002], adaptive optics [Shirai et al., 2001], wavefront sensing [Geary, 2000; Hutchin 1985] and beam characterization [Hernandez, 2000; Liu, 2007], among others, for these cases, we propose the use of a Mach-Zehnder Radial-Shear Interferometer (MZRI), consisting of one telescope system on each arm.

The advantage of both systems is its ability to obtain the derivative of the object wavefront in real time, which can be associated with the partial or radial slope of the phase object, as will be seen in the following sections; additionally, the phase gratings used in these systems can be replaced by high frequency Bi-Ronchi gratings. Experimental results are presented for phase objects.

## **2. Lateral and radial shearing interferometers**

Obtaining the derivative of the wavefront brings numerous advantages due to the high sensitivity against abrupt phase changes. In some cases, if the derivatives in the *x* and *y* directions are known, we can calculate the gradient information of the object, but this implies the design of optical systems that generate shearing in both directions, or we can obtain shearing in each direction separately. If the object under study presents radial symmetry, it is more convenient to use a radial-shear interferometer. In this section, we present in detail two interferometric systems used to generate directional derivatives and the slope associated, using simultaneous phase shifting interferometry.

## **2.1 Cyclic path interferometer**

Figure 1(a) shows the experimental set-up to generate the derivative of the incident wave front in the x-direction. It consists of a cyclic path interferometer where the illumination comes from a polarized Verdi laser underneath, operating at λ= 532 nm. The system is illuminated with polarized light at 45° with respect to the *x*-axis (using a quarter-wave retarding plate, Q, and a linear polarizer, P).

The PBS generates two beams with crossed linear polarization states, which become circular after going through a wave retardation plate of λ/4 (Q'). Thus, the output of the CSI consists of two sheared wavefronts, each one with mutually orthogonal circular polarizations(Left and Right). The interference pattern generated has circular crossed polarizations, and the shear can be varied by moving mirror *M*'. At the output of the CSI, a 4-*f* system has been implemented, with two crossed high frequency Ronchi gratings placed on the Fourier plane [Toto et al., 2010](The Ronchi gratings can be substituted by Phase grids, taking into account the proper considerations at the moment of selecting the interference replicas to calculate the phase data map). For this case, shear *x* = *x*0 has a smaller value than size **a** of the beam's cross section, which is smaller than order separation *F*0 ( Two neighboring diffraction orders have a distance of <sup>0</sup> / *dfF* on the image plane for a grating); thus, two beams with shear *x* enter the 4-*f* system close to the optical axes. At the image plane, the

processing of the interferograms is performed using conventional methods of phase extraction with the algorithm of four shifts. For the case when the object presents radial symmetry, it is more convenient to use a radial-shear interferometer. Applications of radialshear interferometers have been previously reported in optical testing [ Malacara, 1974; Steel 1965], aspherical surface measuring [Hariharan, 1984], corneal topographic inspection [Kowalik et al., 2002], adaptive optics [Shirai et al., 2001], wavefront sensing [Geary, 2000; Hutchin 1985] and beam characterization [Hernandez, 2000; Liu, 2007], among others, for these cases, we propose the use of a Mach-Zehnder Radial-Shear Interferometer (MZRI),

The advantage of both systems is its ability to obtain the derivative of the object wavefront in real time, which can be associated with the partial or radial slope of the phase object, as will be seen in the following sections; additionally, the phase gratings used in these systems can be replaced by high frequency Bi-Ronchi gratings. Experimental results are presented

Obtaining the derivative of the wavefront brings numerous advantages due to the high sensitivity against abrupt phase changes. In some cases, if the derivatives in the *x* and *y* directions are known, we can calculate the gradient information of the object, but this implies the design of optical systems that generate shearing in both directions, or we can obtain shearing in each direction separately. If the object under study presents radial symmetry, it is more convenient to use a radial-shear interferometer. In this section, we present in detail two interferometric systems used to generate directional derivatives and

Figure 1(a) shows the experimental set-up to generate the derivative of the incident wave front in the x-direction. It consists of a cyclic path interferometer where the illumination comes from a polarized Verdi laser underneath, operating at λ= 532 nm. The system is illuminated with polarized light at 45° with respect to the *x*-axis (using a quarter-wave

The PBS generates two beams with crossed linear polarization states, which become circular after going through a wave retardation plate of λ/4 (Q'). Thus, the output of the CSI consists of two sheared wavefronts, each one with mutually orthogonal circular polarizations(Left and Right). The interference pattern generated has circular crossed polarizations, and the shear can be varied by moving mirror *M*'. At the output of the CSI, a 4-*f* system has been implemented, with two crossed high frequency Ronchi gratings placed on the Fourier plane [Toto et al., 2010](The Ronchi gratings can be substituted by Phase grids, taking into account the proper considerations at the moment of selecting the interference replicas to calculate the phase data map). For this case, shear *x* = *x*0 has a smaller value than size **a** of the beam's cross section, which is smaller than order separation *F*0 ( Two neighboring

beams with shear *x* enter the 4-*f* system close to the optical axes. At the image plane, the

/ *dfF* on the image plane for a grating); thus, two

consisting of one telescope system on each arm.

**2. Lateral and radial shearing interferometers** 

the slope associated, using simultaneous phase shifting interferometry.

for phase objects.

**2.1 Cyclic path interferometer** 

retarding plate, Q, and a linear polarizer, P).

diffraction orders have a distance of <sup>0</sup>

superposition of the two beam replicas with mutual shear *x* would appear around each diffraction order. Since the interferograms retain their polarization characteristics, polarizing filters are placed in each generated replica in order to generate the phase shifts required to obtain the phase. Fig. 1(b) shows the two configurations of the polarizing arrangement proposed to generate the phase shifts in a single shot, used to process the optical phase. Some replicas of the interference patterns have different intensities; however, there are at least four patterns with similar intensities in each configuration. As will be shown in the following sections, the replicas of the pattern generated by the CSI are centered on each diffraction order [Rodriguez, 2009].

Fig. 1. Simultaneous phase shifting cyclic path interferometer. (a) CSI with variable shear coupled to 4*-f* system with high frequency Bi-Ronchi grating. Q, Q': Quarter wave retarders. *x*: linear shear. *x*0: beam separation. *F fd* <sup>0</sup> / is the order of separation. (b) Arrangement of polarizing filters. *P*i: polarizing filters, ψi: transmission angle of polarization.

## **2.2 Mach Zehnder interferometer**

In several cases, such as in ophthalmic lens analysis, radial symmetry is encountered, and it is strongly necessary to obtain the radial derivative of the incident wave front in order to analyze the radial variations of the objects. For that reason, we propose the use of another optical system to obtain the radial derivative and associate it with its corresponding radial slope. Fig. 2 shows the experimental setup for the measurement of the radial slope of transparent objects. The interferometric system uses a *He*-*Ne* laser operating at = 632.8 nm. Polarized light at 45° enters the interferometer by using a quarter-wave retarder plate Q0 and a linear polarizer P0. The Mach-Zehnder radial-shear interferometer (MZRI) presented comprises two telescope systems, (S1, S2) [Lago and de la Fuente R. ,2008] one on each arm with vertical and horizontal linear polarized light, respectively. Their variants are widely used as part of different applications, so they can be considered to be representative cases of potential adaptations for simultaneous phase shifting interferometry. A 4-*f* system is coupled at the end of the interferometer; this system consists of two similar achromatic

Simultaneous Phase Shifting Shearing Interferometry

, the new polarization states are

*yxxOPyxO* ),(),(' ,

> 

<sup>2</sup> ( , ) '( , ) ''( , )

*i i*

*i i*

*yx <sup>x</sup>* 

given by 577.460 989.92 42.136

3

**4. Polarizing phase-shifting radial shear interferometry** 

1

When the two fields interfere, the resulting irradiance can be written as

*Ixy O xy O xy*

an angle

where *P*

by

1

 4/ *x* .

with *c*<sup>1</sup>

2

*O r r circ r r i r r* ( ) exp

new polarization states are:

for Measurement of Static and Dynamic Phase Objects 269

Eq. (1) represents cross circular polarization states of same beams with lateral shear *x*. When each field is observed through a linear polarizing filter whose transmission axis is at

is the matrix for a linear polarizer with its axis of transmission at angle

( , ) 2 2 cos 2 , ,

*Ixy x xy xy*

When *i* = 1…4, the relative phase can be calculated as [Sharma, 1984; Barrientos, 1999]

 

where *<sup>i</sup> I* are the intensity measurements captured in a single shot, with the values of

 

1 tan 1),(

Assuming each beam section at the end of the MZRI as *O r circ r i r* ( ) exp

1 1 *O r Or Or r* () () ( ) *i i*

representing cross circular polarization states of the same beam radially sheared. When each field is observed through a linear polarizing filter whose transmission axis is at angle

*cx*

 

> 

<sup>1</sup> <sup>31</sup> tan),( *II*

( , ) 2 1 cos 2 ,

*Ixy x xy*

 2 2

cossin sin

 

cos cossin *<sup>P</sup>* . (3)

*x*

 

*II*

42

 

> 

*II II*

42 1 31

, the field entering the 4-*f* system is given by

, (5)

<sup>4</sup> . Hence, the slope is:

*yxw* , (6)

, (7)

 

*yxOPyxO* ),(),('' , (2)

, given

. (4)

and

, the

lenses of focal length *f* 20 cm and a phase grid G(µ,ν) placed as the system's pupil with spatial period *d*=110 *ln/mm*. In the phase grid used, /*fu* and v/*f* are the frequency coordinates (*u,v*) scaled to wavelength and focal length *f*. The output of the MZRI consists of two versions of the same wavefront, but slightly enlarged in relation to the copy, and each one with mutually orthogonal linear polarizations (vertical and horizontal) too. A wave retarder plate of (Q1) is used to obtain cross circular polarization for each radial-shear wavefront (left and right, *<sup>L</sup> J* and *RJ* ) with equal amplitudes[Toto et al., 2009].

A linear polarized filter array is placed on the image plane, where each one is centered on its respective replicas of the interference pattern, as it is shown in Fig. 2 of the polarized linear array. The cross sections of the two sheared beams are 1*a mm* 7.0 and 2 *a mm* 8.6 , and the relative magnification of the pupils is *Ma* = 1.23.

Fig. 2. Simultaneous phase shifting Mach-Zehnder radial-shear interferometer. *P*i: Polarizers; *Q*i: QWP operating at 632.8 *nm*; PBS: Polarizing Beam splitter; *M*i: Mirrors; G(µ,ν): Phase grid. Translation of coordinates around the order position: *xq = x − qF0* and *yr = y − rF0.*

In the present section, we analyze the general case of the interference patterns obtained by the shearing interferometers presented, as well as the modulation by polarization generated in each replica in both cases (lateral or radial shear) by the Jones's Matrix representation.

## **3. Polarizing phase-shifting lateral shear interferometry**

At the output of the CSI, two images are generated, each with circular polarization; the two fields generated for the system are described by

$$O(\mathbf{x} + \Delta \mathbf{x}, \mathbf{y}) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix} e^{i\boldsymbol{\phi}(\mathbf{x} + \Delta \mathbf{x}, \mathbf{y})} \quad O(\mathbf{x}, \mathbf{y}) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix} e^{i\boldsymbol{\phi}(\mathbf{x}, \mathbf{y})} \tag{1}$$

lenses of focal length *f* 20 cm and a phase grid G(µ,ν) placed as the system's pupil with

frequency coordinates (*u,v*) scaled to wavelength and focal length *f*. The output of the MZRI consists of two versions of the same wavefront, but slightly enlarged in relation to the copy, and each one with mutually orthogonal linear polarizations (vertical and horizontal) too. A wave retarder plate of (Q1) is used to obtain cross circular polarization for each

A linear polarized filter array is placed on the image plane, where each one is centered on its respective replicas of the interference pattern, as it is shown in Fig. 2 of the polarized linear array. The cross sections of the two sheared beams are 1*a mm* 7.0 and 2 *a mm* 8.6 , and

Fig. 2. Simultaneous phase shifting Mach-Zehnder radial-shear interferometer. *P*i: Polarizers; *Q*i: QWP operating at 632.8 *nm*; PBS: Polarizing Beam splitter; *M*i: Mirrors; G(µ,ν): Phase grid. Translation of coordinates around the order position: *xq = x − qF0* and *yr = y − rF0.*

In the present section, we analyze the general case of the interference patterns obtained by the shearing interferometers presented, as well as the modulation by polarization generated in each replica in both cases (lateral or radial shear) by the Jones's Matrix

At the output of the CSI, two images are generated, each with circular polarization; the two

 ),( <sup>1</sup> 2 <sup>1</sup> ),( *yxi <sup>e</sup> <sup>i</sup> yxO*

(1)

 

),( 1

**3. Polarizing phase-shifting lateral shear interferometry** 

2 <sup>1</sup> ),( *yxxi <sup>e</sup> <sup>i</sup> yxxO*

 

fields generated for the system are described by

and *RJ*

 /*fu* and  v/

) with equal amplitudes[Toto et al., 2009].

*f* are the

spatial period *d*=110 *ln/mm*. In the phase grid used,

radial-shear wavefront (left and right, *<sup>L</sup> J*

representation.

the relative magnification of the pupils is *Ma* = 1.23.

Eq. (1) represents cross circular polarization states of same beams with lateral shear *x*. When each field is observed through a linear polarizing filter whose transmission axis is at an angle , the new polarization states are

$$O^\prime(\mathbf{x}, \mathbf{y}) = P\_{\psi} \cdot O(\mathbf{x} + \Delta \mathbf{x}, \mathbf{y}) \quad , \quad O^\prime(\mathbf{x}, \mathbf{y}) = P\_{\psi} \cdot O(\mathbf{x}, \mathbf{y}) \quad , \tag{2}$$

where *P* is the matrix for a linear polarizer with its axis of transmission at angle , given by

$$P\_{\psi} = \begin{pmatrix} \cos^2 \psi & \sin \psi \cos \psi \\ \sin \psi \cos \psi & \sin^2 \psi \end{pmatrix} \tag{3}$$

When the two fields interfere, the resulting irradiance can be written as

$$\begin{aligned} I(\mathbf{x}, y) &= \left\| O^\circ(\mathbf{x}, y) + O^\circ(\mathbf{x}, y) \right\|^2 \\ I(\mathbf{x}, y)\_i &= 2 + 2 \cos \left\{ 2\boldsymbol{\nu}\_i - \left[ \boldsymbol{\phi}(\mathbf{x} + \Delta \mathbf{x}, y) - \boldsymbol{\phi}(\mathbf{x}, y) \right] \right\}. \end{aligned} \tag{4}$$

$$I(\mathbf{x}, y)\_i = 2 \cdot \left\{ 1 + \cos \left[ 2\boldsymbol{\nu}\_i - \Delta \mathbf{x} \frac{\partial}{\partial \mathbf{x}} \boldsymbol{\phi}(\mathbf{x}, y) \right] \right\}$$

When *i* = 1…4, the relative phase can be calculated as [Sharma, 1984; Barrientos, 1999]

$$\frac{\partial}{\partial \mathbf{x}} \phi(\mathbf{x}, \mathbf{y}) = \tan^{-1} \left( \frac{I\_1 - I\_3}{I\_2 - I\_4} \right),\tag{5}$$

where *<sup>i</sup> I* are the intensity measurements captured in a single shot, with the values of given by 577.460 989.92 42.136 1 2 3 <sup>4</sup> . Hence, the slope is:

$$\frac{\partial \mathbf{w}(\mathbf{x}, \mathbf{y})}{\partial \mathbf{x}} = \frac{1}{c\_1} \cdot \tan^{-1} \left( \frac{I\_1 - I\_3}{I\_2 - I\_4} \right),\tag{6}$$

with *c*<sup>1</sup> 4/ *x* .

## **4. Polarizing phase-shifting radial shear interferometry**

Assuming each beam section at the end of the MZRI as *O r circ r i r* ( ) exp and *O r r circ r r i r r* ( ) exp , the field entering the 4-*f* system is given by

$$O\_1(r) = \begin{pmatrix} 1 \\ i \end{pmatrix} \cdot O(r) + \begin{pmatrix} 1 \\ -i \end{pmatrix} \cdot O(r + \Delta r) \tag{7}$$

representing cross circular polarization states of the same beam radially sheared. When each field is observed through a linear polarizing filter whose transmission axis is at angle , the new polarization states are:

Simultaneous Phase Shifting Shearing Interferometry

*"* and "

(,) () ()

*Gxy Gx Gy*

**5.1.1 Interference pattern replication** 

operating linear polarizers placed at an angle

*yxI* ),(' on the image plane of the system will be:

, as well as

corresponding spectrum is:

along directions "

 *ddd* 

( ,) ( ) ()

 

were *N*, *L* are the numbers of components of the grating,

2 2

 

", and *aw*

*N L*

*n N l L*

*xn yl*

pattern with comparable intensities, the fringe modulation it is close to unity.

 *aw*

for Measurement of Static and Dynamic Phase Objects 271

*n d l d G G G rect rect*

both directions. The gratings studied have equal periods in both directions; then,

sinc sinc

*n l C x Cy d d*

Fig. 3. Simulated diffraction pattern generated by high frequency Bi-Ronchi grating. (a) Diffraction spectra. (B) *x*-*y* components of diffraction spectra of each grating component.

The 4-*f* system with high frequency Bi-Ronchi grating only multiplexes the interference pattern generated at the output of CSI; the shifts in the interference pattern can be generated

*N L w w <sup>w</sup> n N l L*

For the experimental case where gratings (100 ln/*mm*) have the same period and 01.0 *mmd* , the diffraction patterns generated using Eq. (15) are shown in Fig. 3. Fig. 3(a) shows the simulated diffraction pattern generated by the Bi-Ronchi grating. In Fig. 3(b), it can be seen that the amplitude spectra for each axis show more than three orders of diffraction; however, only orders 0 and ±1 allow one to obtain replicas of the interference

*a a n l a x x y y <sup>d</sup> d d dd*

 

*N L*

 

*n N w w l L*

*a a*

 *d* , *d*

*aaa www* . Due to the properties of the Fourier transform, the

are the respective periods

(15)

 

are the widths in clear strips along each one of

 

Considering Eq. (1), the interference pattern

(14)

$$
\hat{\rho}O\_2(r) = \cdot P\_{\psi} \left[ \begin{pmatrix} 1 \\ i \end{pmatrix} \cdot O(r) + \begin{pmatrix} 1 \\ -i \end{pmatrix} \cdot O(r + \Delta r) \right] \tag{8}
$$

As it was shown in section 3, we can show that the interference pattern can be represented as:

$$I(r)\_l = 2 \cdot \left[ 1 + \cos\left[2\psi\_l - \Delta r \frac{\partial}{\partial r} \phi(r) \right] \right] \tag{9}$$

and considering also that *i* = 1…4, the relative phase data map can be calculated as

$$\frac{\partial}{\partial r}\phi(r) = \tan^{-1}\left(\frac{I\_1 - I\_3}{I\_2 - I\_4}\right),\tag{10}$$

where *<sup>i</sup> I* are the intensity measurements captured in a single shot. In this particular case, the radial slope can be obtained as:

$$\frac{\partial \mathcal{W}(r)}{\partial r} = \frac{1}{c\_1} \cdot \tan^{-1} \left( \frac{I\_1 - I\_3}{I\_2 - I\_4} \right) \tag{11}$$

## **5. Grating interferometry: pattern replication and modulation of polarization**

## **5.1 Interferometer with Bi-Ronchi gratings**

In the one-dimensional case, with denoting the object space, the amplitude transmission of a Ronchi grating [Ronchi ,1964.] can be written as the periodic function

$$G(\mu) = \text{rect}\left[\frac{\mu}{a\_w}\right] \* \sum\_{n=-N}^{N} \delta\left[\frac{\mu - n \cdot d}{a\_w}\right],\tag{12}$$

with *d = aw* + *bw* being the grating period, *a*w and *bw* being the widths of the clear and dark bands respectively, denoting the Dirac delta function, and the convolution. /*fu* is the frequency coordinate scaled to the relevant wavelength and the focal length *f*. The Fourier transform of Eq. (12) with *N* is given by

$$\begin{split} \tilde{G}\left(\mathbf{x}\right) &= \frac{a\_w}{d} \text{sinc}\left(a\_w \mathbf{x}\right) \sum\_{n=-\infty}^{\infty} \delta\left(\mathbf{x} - \frac{n}{d}\right) \\ &= \frac{a\_w}{d} \sum\_{n=-\infty}^{\infty} \text{sinc}\left(a\_w \frac{n}{d}\right) \delta\left(\mathbf{x} - \frac{n}{d}\right) = \sum\_{n=-\infty}^{\infty} \mathbf{C}\_n \cdot \delta\left(\mathbf{x} - \frac{n}{d}\right) \end{split} \tag{13}$$

where *Cn* results as the *n*-Fourier complex coefficient of *G* . In general, when Bi-Ronchi gratings[Cordero, 1998] are placed on the Fourier plane, they can be considered to be two cross amplitude Ronchi gratings,

 )( 1

*i*

42

 

> 

*II II*

42 1 31

)(

As it was shown in section 3, we can show that the interference pattern can be represented

 

> 

> >

*N*

*n d G rect*

with *d = aw* + *bw* being the grating period, *a*w and *bw* being the widths of the clear and dark

*<sup>w</sup> w n n n*

gratings[Cordero, 1998] are placed on the Fourier plane, they can be considered to be two

*<sup>a</sup> nn n a x Cx d dd d*

*fu* is the frequency coordinate scaled to the relevant wavelength

*n*

 

*w w n N*

   

denoting the Dirac delta function, and the convolution.

*a a*

<sup>1</sup> <sup>31</sup> tan)( *II II <sup>r</sup> <sup>r</sup>*

where *<sup>i</sup> I* are the intensity measurements captured in a single shot. In this particular case,

 

**5. Grating interferometry: pattern replication and modulation of polarization** 

1 tan 1)(

*cr*

length *f*. The Fourier transform of Eq. (12) with *N* is given by

sinc

*<sup>a</sup> <sup>n</sup> Gx ax x d d*

sinc

where *Cn* results as the *n*-Fourier complex coefficient of *G*

*<sup>w</sup> <sup>w</sup>*

a Ronchi grating [Ronchi ,1964.] can be written as the periodic function

*<sup>r</sup> <sup>r</sup> rI <sup>r</sup> <sup>i</sup>* 2cos12)( *<sup>i</sup>*

)( <sup>2</sup> *rrO*

*rO i*

 

*PrO*

the radial slope can be obtained as:

In the one-dimensional case, with

bands respectively,

 /

**5.1 Interferometer with Bi-Ronchi gratings** 

 

cross amplitude Ronchi gratings,

as:

 

1

 

and considering also that *i* = 1…4, the relative phase data map can be calculated as

 

 

 

, (10)

*rw* (11)

denoting the object space, the amplitude transmission of

, (12)

 

. In general, when Bi-Ronchi

and the focal

(13)

. (8)

(9)

 

$$\mathbf{G}(\mu, \nu) = \mathbf{G}(\mu) \cdot \mathbf{G}(\nu) = \sum\_{n=-N}^{N} \operatorname{rect} \left[ \frac{\mu - n \cdot d\_{\mu}}{a\_{w\mu}} \right] \cdot \sum\_{l=-L}^{L} \operatorname{rect} \left[ \frac{\nu - l \cdot d\_{\nu}}{a\_{w\mu}} \right] \tag{14}$$

were *N*, *L* are the numbers of components of the grating, *d* , *d* are the respective periods along directions "*"* and "", and *aw aw* are the widths in clear strips along each one of both directions. The gratings studied have equal periods in both directions; then, *ddd* , as well as *aaa www* . Due to the properties of the Fourier transform, the corresponding spectrum is:

$$\begin{split} \tilde{\mathbf{G}}(\mathbf{x},\mathbf{y}) &= \tilde{\mathbf{G}}(\mathbf{x}) \cdot \tilde{\mathbf{G}}(\mathbf{y}) \\ &= \frac{a\_w^2}{d^2} \sum\_{n=-N}^{N} \text{sinc}\left(\frac{a\_w}{d} \cdot \mathbf{x}\right) \cdot \boldsymbol{\mathcal{S}}\left(\mathbf{x} - \frac{n}{d}\right) \sum\_{l=-L}^{L} \text{sinc}\left(\frac{a\_w}{d} \cdot \mathbf{y}\right) \cdot \boldsymbol{\mathcal{S}}\left(\mathbf{y} - \frac{l}{d}\right) \\ &= \sum\_{n=-N}^{N} \mathbf{C}\_{\text{x}n} \cdot \boldsymbol{\mathcal{S}}\left(\mathbf{x} - \frac{n}{d}\right) \sum\_{l=-L}^{L} \mathbf{C}\_{yl} \cdot \boldsymbol{\mathcal{S}}\left(\mathbf{y} - \frac{l}{d}\right) \end{split} \tag{15}$$

For the experimental case where gratings (100 ln/*mm*) have the same period and 01.0 *mmd* , the diffraction patterns generated using Eq. (15) are shown in Fig. 3. Fig. 3(a) shows the simulated diffraction pattern generated by the Bi-Ronchi grating. In Fig. 3(b), it can be seen that the amplitude spectra for each axis show more than three orders of diffraction; however, only orders 0 and ±1 allow one to obtain replicas of the interference pattern with comparable intensities, the fringe modulation it is close to unity.

Fig. 3. Simulated diffraction pattern generated by high frequency Bi-Ronchi grating. (a) Diffraction spectra. (B) *x*-*y* components of diffraction spectra of each grating component.

## **5.1.1 Interference pattern replication**

The 4-*f* system with high frequency Bi-Ronchi grating only multiplexes the interference pattern generated at the output of CSI; the shifts in the interference pattern can be generated operating linear polarizers placed at an angle Considering Eq. (1), the interference pattern *yxI* ),(' on the image plane of the system will be:

Simultaneous Phase Shifting Shearing Interferometry

vectors. Taking the rulings of one grating along the "

*q r*

*q gr g L R*

*L <sup>R</sup> JJ* ' **<sup>J</sup>**

*q r*

**5.2 Interferometer with phase grids** 

*wxy Ox Oy i xy* ( , ) ( ) ( ) exp ,

second grating along the "

phase grid can be written as

can be written as

*o i*

*q r*

where the order separation is <sup>0</sup>

(,) (,) (,)

*t xy t xy G xy*

2

 

*L L <sup>L</sup> JJ* ' **<sup>J</sup>**

, *<sup>R</sup>*

by

for Measurement of Static and Dynamic Phase Objects 273

Previous works have shown that when the interferometer is coupled to a 4-*f* system using phase grids, it is possible to obtain replicas of the interference pattern, and each one of these replicas can be modulated by polarization [Rodriguez 2008; Toto, 2008; Serrano, 2011]. A sinusoidal phase grid generated by the multiplication of two sinusoidal phase gratings, whose respective grating vectors are crossed, generates diffraction orders modulated by Bessel functions *Jq* and *Jr* [Rodriguez, 2008;Toto, 2008], and the image of the interference pattern generated by the interferometer is found centered around each of them. If we assume, for simplicity, that each beam section is of the form

), <sup>2</sup> (), <sup>2</sup> (),( *<sup>y</sup> <sup>x</sup> xwJy <sup>x</sup> xwJyxti <sup>L</sup> <sup>R</sup>*

The displacement *x* between the wavefronts can be adjusted by translating the mirror *M* of the CSI. To draw theoretical conclusions, we assume that the phase grid is made up of two phase gratings (with 2πAg the grating's phase amplitude) and with orthogonal gratings

2 0 <sup>0</sup> , 22 ,

 

(2 ) (2 ) ( , ) ( , ) 2 2

,

 2 2

cossin sin

*x x J A J A J w x qF y rF J w x qF y rF*

 

*q gr g*

kind and integer order *q,r* [Goodman, 1988]. On the image plane of the CSI, the amplitude

with ( ) denoting convolution. On the image plane of the system, a series of replicated beams can be observed. Because of their polarization, an interference pattern can be detected when a linear polarizing filter is placed before the detector. Using Jones calculation with

the pattern irradiance turns out to be proportional to the squared modulus of Eq. (21) in the general case. Considering the shear <sup>0</sup> *xx* of a value smaller than the side *a* of the beam section, which in turn is smaller than the order separation *F*0 (Fig. 1), two beams with shear <sup>0</sup> *x* enter the 4-*f* system close to the optical axes. At the image plane, the superposition of the

*G xy J A J A x qF y rF* 

, the transmittance in front of the entrance lens is given

" direction, the resulting Fourier transform of the centered

/ *dfF* , and *<sup>q</sup> J* , *<sup>r</sup> J* denote the Bessel function of the first

00 00

 

*<sup>L</sup>* cos cossin **<sup>J</sup>** , (22)

(21)

" direction, and the rulings of the

 (19)

(20)

$$\begin{aligned} I'(\mathbf{x}, \mathbf{y}) &= \left| \vec{P}\_{\mathbf{y}} \left[ \vec{J}\_L O(\mathbf{x} + \Delta \mathbf{x}, \mathbf{y}) + \vec{J}\_R O(\mathbf{x} + \mathbf{y}) \right] \* \widetilde{G}(\mathbf{x}, \mathbf{y}) \right|^2 \\ &= I(\mathbf{x}, \mathbf{y}) \* \left| \widetilde{G}(\mathbf{x}, \mathbf{y}) \right|^2 \end{aligned} \tag{16}$$

where each sheared wavefront has a circular polarization, *LJ* (left) and *RJ* (right) respectively, and *Ixy* (,) , the interference pattern generated by the CSI, having the form

$$I = 1 + \cos\left[2 \cdot \nu - \Delta x \frac{\partial}{\partial x} \phi(x, y)\right];\tag{17}$$

so, the interference pattern on the image plane, centered on each diffraction order, will be:

$$\begin{split} I'(\mathbf{x}, \mathbf{y}) &= \sum\_{n=-N}^{N} \sum\_{l=-L}^{L} C\_{yl} \,^2 \,^C C\_{\mathbf{x}n} \,^2 \cdot I \left( \mathbf{x} \cdot \frac{n}{d\_x}, \mathbf{y} \cdot \frac{l}{d\_y} \right) \\ &= \sum\_{n=-N}^{N} \sum\_{l=-L}^{L} C\_{yl} \,^2 \,^C C\_{\mathbf{x}n} \,^2 \cdot \left\{ \mathbf{l} + \cos \left[ 2 \cdot \mathbf{y} - \Delta \mathbf{x} \frac{\partial}{\partial \mathbf{x}} \phi \left( \mathbf{x} - \frac{n}{d\_x}, \mathbf{y} - \frac{n}{d\_y} \right) \right] \right\} \end{split} \tag{18}$$

Eq. (18) shows that the results are replicas of the pattern generated by the CSI displaced to the *n*th and *l*th orders of diffraction; each replica of the main pattern maintains an intensity modulated by Fourier Coefficients *Cn* and *Cl* corresponding to each direction. It should be noted that the contrast of the fringes depends on the degree of circular polarization of the interfering beams [Rodriguez, 2008]. In this case, the Ronchi gratings only multiplex the pattern, so the amplitudes of the orders only increase or decrease the intensity of the interference patterns, as it is shown in Fig. 4. Figure 4 shows the experimental results obtained with a Bi-Ronchi grating. The interference patterns used are shown enclosed in a circle, independent phase shifting, can be generated operating linear polarizers. Fig. 4(a) shows the diffraction orders' overlap, indicated by a subscript. Fig. 4(b) shows the diffraction orders generated by the grating; it can be seen that they have at least 9 diffraction orders with comparable intensities. This result is due to the modulation generated by coefficients *Cn* and *Cl* (see Eq. 18). Fig. 4(c) shows the replicas of the interference patterns centered on each diffraction order.

Fig. 4. Replicas of the interference patterns. (a) Fourier coefficients for each replica of the interference pattern. (b) Experimental diffraction orders generated by high frequency Bi-Ronchi gratings (100 ln/mm). (c) Replicas of the interference patterns centered on each diffraction order (Circle).

## **5.2 Interferometer with phase grids**

272 Interferometry – Research and Applications in Science and Technology

 2

),( <sup>~</sup> ),(' ),(),(

*yxGyxOJyxxOJPyxI <sup>L</sup> <sup>R</sup>*

respectively, and *Ixy* (,) , the interference pattern generated by the CSI, having the form

*I x* 1 cos 2 *x*,*y*

so, the interference pattern on the image plane, centered on each diffraction order, will be:

 

Fig. 4. Replicas of the interference patterns. (a) Fourier coefficients for each replica of the interference pattern. (b) Experimental diffraction orders generated by high frequency Bi-Ronchi gratings (100 ln/mm). (c) Replicas of the interference patterns centered on each

*<sup>n</sup> <sup>x</sup> <sup>x</sup> CC <sup>x</sup>*

Eq. (18) shows that the results are replicas of the pattern generated by the CSI displaced to the *n*th and *l*th orders of diffraction; each replica of the main pattern maintains an intensity modulated by Fourier Coefficients *Cn* and *Cl* corresponding to each direction. It should be noted that the contrast of the fringes depends on the degree of circular polarization of the interfering beams [Rodriguez, 2008]. In this case, the Ronchi gratings only multiplex the pattern, so the amplitudes of the orders only increase or decrease the intensity of the interference patterns, as it is shown in Fig. 4. Figure 4 shows the experimental results obtained with a Bi-Ronchi grating. The interference patterns used are shown enclosed in a circle, independent phase shifting, can be generated operating linear polarizers. Fig. 4(a) shows the diffraction orders' overlap, indicated by a subscript. Fig. 4(b) shows the diffraction orders generated by the grating; it can be seen that they have at least 9 diffraction orders with comparable intensities. This result is due to the modulation generated by coefficients *Cn* and *Cl* (see Eq. 18). Fig. 4(c) shows the replicas of the interference patterns

*xn*

22

 

*x*

*<sup>y</sup> <sup>d</sup>*


2cos1 ,

 

> 

*d l*

*yx*

 

),( <sup>~</sup> ),(

 

*xn*

22

*L*

*Ll*

*N*

*Nn*

centered on each diffraction order.

diffraction order (Circle).

),(' ,

*<sup>n</sup> yxI xICC*

*yl*

*L*

*Ll*

*N*

*Nn*

*yl*

*yxGyxI*

where each sheared wavefront has a circular polarization, *LJ*

2

(left) and *RJ*

; (17)

 

 

 

*yx*

*d <sup>n</sup> <sup>y</sup> <sup>d</sup>* (right)

(18)

, (16)

Previous works have shown that when the interferometer is coupled to a 4-*f* system using phase grids, it is possible to obtain replicas of the interference pattern, and each one of these replicas can be modulated by polarization [Rodriguez 2008; Toto, 2008; Serrano, 2011]. A sinusoidal phase grid generated by the multiplication of two sinusoidal phase gratings, whose respective grating vectors are crossed, generates diffraction orders modulated by Bessel functions *Jq* and *Jr* [Rodriguez, 2008;Toto, 2008], and the image of the interference pattern generated by the interferometer is found centered around each of them. If we assume, for simplicity, that each beam section is of the form *wxy Ox Oy i xy* ( , ) ( ) ( ) exp , , the transmittance in front of the entrance lens is given by

$$\vec{J}\_l(\mathbf{x}, \mathbf{y}) = \vec{J}\_L \cdot \mathbf{w}(\mathbf{x} - \frac{\Delta \mathbf{x}}{2}, \mathbf{y}) + \vec{J}\_R \mathbf{w}(\mathbf{x} + \frac{\Delta \mathbf{x}}{2}, \mathbf{y}) \tag{19}$$

The displacement *x* between the wavefronts can be adjusted by translating the mirror *M* of the CSI. To draw theoretical conclusions, we assume that the phase grid is made up of two phase gratings (with 2πAg the grating's phase amplitude) and with orthogonal gratings vectors. Taking the rulings of one grating along the " " direction, and the rulings of the second grating along the " " direction, the resulting Fourier transform of the centered phase grid can be written as

$$\tilde{\mathbf{G}}\_2(\mathbf{x}, \mathbf{y}) = \sum\_{q=-\infty}^{q=\infty} \sum\_{r=-\infty}^{r=\infty} I\_q \left( 2\pi A\_{\frac{q}{\mathcal{S}}} \right) I\_r \left( 2\pi A\_{\frac{q}{\mathcal{S}}} \right) \delta \left( \mathbf{x} - q \mathbf{F}\_0, \mathbf{y} - r \mathbf{F}\_0 \right) \tag{20}$$

where the order separation is <sup>0</sup> / *dfF* , and *<sup>q</sup> J* , *<sup>r</sup> J* denote the Bessel function of the first kind and integer order *q,r* [Goodman, 1988]. On the image plane of the CSI, the amplitude can be written as

$$\begin{aligned} \vec{t}\_o(\mathbf{x}, \boldsymbol{y}) &= \vec{t}\_i(\mathbf{x}, \boldsymbol{y}) \ast \tilde{\mathbf{G}}\_2(\mathbf{x}, \boldsymbol{y}) \\ &= \sum\_{q=-\alpha}^{\alpha} \sum\_{r=-\alpha}^{\alpha} I\_q(2\pi A\_g) I\_r(2\pi A\_g) \left\{ \vec{J}\_L \cdot w(\mathbf{x} - q\mathbf{F}\_0 - \frac{\Delta \mathbf{x}}{2}, \boldsymbol{y} - r\mathbf{F}\_0) + \vec{J}\_R \cdot w(\mathbf{x} - q\mathbf{F}\_0 + \frac{\Delta \mathbf{x}}{2}, \boldsymbol{y} - r\mathbf{F}\_0) \right\} \end{aligned} \tag{21}$$

with ( ) denoting convolution. On the image plane of the system, a series of replicated beams can be observed. Because of their polarization, an interference pattern can be detected when a linear polarizing filter is placed before the detector. Using Jones calculation with

$$\mathbf{J}\_L \mathbf{\dot{J}} = \mathbf{J}\_\psi^L \mathbf{\dot{J}}\_L \quad \mathbf{\dot{J}}\_R \mathbf{\dot{=}} = \mathbf{J}\_\psi^L \mathbf{\dot{J}}\_R \quad \mathbf{J}\_\psi^L = \begin{pmatrix} \cos^2 \psi & \sin \psi \cos \psi \\ \sin \psi \cos \psi & \sin^2 \psi \end{pmatrix} \tag{22}$$

the pattern irradiance turns out to be proportional to the squared modulus of Eq. (21) in the general case. Considering the shear <sup>0</sup> *xx* of a value smaller than the side *a* of the beam section, which in turn is smaller than the order separation *F*0 (Fig. 1), two beams with shear <sup>0</sup> *x* enter the 4-*f* system close to the optical axes. At the image plane, the superposition of the

Simultaneous Phase Shifting Shearing Interferometry

consider the special case where

where 

relationship

in the normal direction (

with *c*<sup>1</sup>

dependency.

 4/ *x* .

**6.1 Static distributions** 

2/ because

for Measurement of Static and Dynamic Phase Objects 275

),( ),( *yxi eyxO* 

),(1),( ),( ),( *yxiyxO eyxO yxi*

> ),(),( <sup>0</sup> *yx x*

with *ic*0 being a constant. It can be noted that the derivative of the phase object is proportional to the derivative of the phase, while additionally maintaining a constant phase

single collimated beam, the relationship between the phase difference and the change in

*<sup>x</sup>*

*x yxw*

<sup>1</sup> *yxO xc c*

*yx*

*x du*

where *u* and *w* specify the displacement components. The fringe pattern has contributions from the deformation / *xu* and the slope / *xw* . However, when the object is illuminated

*<sup>x</sup>*

*yx*

 , 4 ),( 

 

),( ),( 0

For the case of the radial slope, Eq. (29) can be used only with the consideration of the radial

For the case of measurements of the slope in the x-direction, experimental results are presented in Figures 6 and 7 for a microscope slide and an acetate film, respectively. The

, (26)

*ei* . It is well known that when an object is illuminated by a

*x w*

 

=0°), the fringe pattern represents the *x*-partial slope, / *xw* . This is:

, (28)

*cyxO*

 

and taking the partial derivative of Eq. (26) with respect to *x* , we obtain

*x*

displacement [Rastogi, 1996; Ng , 1995; Bhaduri, 2006] is obtained from:

*x*

*x yxw*

 

 )cos1(sin , <sup>2</sup> 

2/ *<sup>i</sup>*

*x*

giving us the slope of a phase object as

transparent objects known as phase objects. Accepting the approach

*yx* ),( is a real function; this object is known as a phase object. For this analysis, we

*yx* 1),( for both 1),( <sup>2</sup>

(24)

*yx* , resulting in the kind of

(25)

(27)

(29)

sheared beams would appear around each diffraction order, each superposition isolated from the others. The sheared beams around a given order *qr* will have counter rotating circular polarization. A diffraction order *qr* passes through a filter designed to block out all of the remaining orders. Under these conditions, by detecting the irradiance with a linear polarizing filter with the transmission axis at an angle , only the contribution of an isolated term of order *qr* can be considered, and its irradiance would be proportional to

$$\begin{split} I(\mathbf{x}, \boldsymbol{\mathcal{y}}) &= \left( J\_q \boldsymbol{J}\_r \right)^2 + \left( J\_q \boldsymbol{J}\_r \right)^2 + 2 \boldsymbol{J}\_q \boldsymbol{J}\_q \boldsymbol{J}\_r \boldsymbol{J}\_r \cdot \cos \left[ \xi(\boldsymbol{\nu}) - \Delta \phi(\mathbf{x}\_q, \mathbf{y}\_r) \right] \\ &= 2 \boldsymbol{J}\_q^2 \cdot \boldsymbol{J}\_r^2 \left( \mathbf{l} + \cos \left[ \xi(\boldsymbol{\nu}) - \Delta \phi(\mathbf{x}\_q, \mathbf{y}\_r) \right] \right) \end{split} \tag{23}$$

where a translation of coordinates was used around the order position ( <sup>0</sup> *qFxxq* and <sup>0</sup> *rFyyr* ), and ),2/(),2/(),( <sup>0</sup> <sup>0</sup> *yxxyxxyx* . The pattern would present a phase shift () 2 and fringe modulation of unity. This configuration uses the CSI as a beam divider and to produce the shear.

Figure 5 shows the diffraction pattern generated by a phase grid. Fig. 5(a) shows the diffraction orders' overlap, indicated by a subscript. Fig. 5(b) shows the diffraction orders generated by the grating; it can be seen that there are at least 9 diffraction orders with comparable intensities. This result is due to the modulation generated by coefficients *Jn,l*. Fig. 5(c) shows the replicas of the interference patterns centered on each diffraction order.

Fig. 5. Replicas of the interference patterns. (a) Fourier coefficients for each replica of the interference pattern. (b) Experimental diffraction orders generated by high frequency Bi-Ronchi gratings (100 ln/mm). (c) Replicas of the interference patterns centered on each diffraction order (Circle).

## **6. Slope measurements of a phase object**

In general, if an object without absorption (transparent) is placed on one arm of the interferometer, it is expressed as:

sheared beams would appear around each diffraction order, each superposition isolated from the others. The sheared beams around a given order *qr* will have counter rotating circular polarization. A diffraction order *qr* passes through a filter designed to block out all of the remaining orders. Under these conditions, by detecting the irradiance with a linear polarizing filter with the transmission axis at an angle , only the contribution of an isolated term of order *qr* can be considered, and its irradiance would be proportional to

(23)

*yxxyxxyx* . The pattern would present a

and fringe modulation of unity. This configuration uses the CSI as a

*rq rq rrqq rq*

),( 2 ),()(cos

where a translation of coordinates was used around the order position ( <sup>0</sup> *qFxxq* and

Figure 5 shows the diffraction pattern generated by a phase grid. Fig. 5(a) shows the diffraction orders' overlap, indicated by a subscript. Fig. 5(b) shows the diffraction orders generated by the grating; it can be seen that there are at least 9 diffraction orders with comparable intensities. This result is due to the modulation generated by coefficients *Jn,l*. Fig. 5(c) shows the replicas of the interference patterns centered on each diffraction order.

Fig. 5. Replicas of the interference patterns. (a) Fourier coefficients for each replica of the interference pattern. (b) Experimental diffraction orders generated by high frequency Bi-Ronchi gratings (100 ln/mm). (c) Replicas of the interference patterns centered on each

In general, if an object without absorption (transparent) is placed on one arm of the

*JJJJJJJJyxI yx*

),()(cos12

*JJ yx*

*rq rq*

2 2

<sup>0</sup> *rFyyr* ), and ),2/(),2/(),( <sup>0</sup> <sup>0</sup>

22

 

beam divider and to produce the shear.

phase shift

() 2 

diffraction order (Circle).

interferometer, it is expressed as:

**6. Slope measurements of a phase object** 

$$O(\mathbf{x}, \mathbf{y}) = e^{\iota \cdot \phi(\mathbf{x}, \mathbf{y})} \tag{24}$$

where *yx* ),( is a real function; this object is known as a phase object. For this analysis, we consider the special case where *yx* 1),( for both 1),( <sup>2</sup> *yx* , resulting in the kind of transparent objects known as phase objects. Accepting the approach

$$\begin{aligned} O(\mathbf{x}, \mathbf{y}) &= e^{l \cdot \phi(\mathbf{x}, \mathbf{y})} \\ O(\mathbf{x}, \mathbf{y}) &= \mathbf{l} + i \cdot \phi(\mathbf{x}, \mathbf{y}) \end{aligned} \tag{25}$$

and taking the partial derivative of Eq. (26) with respect to *x* , we obtain

$$\frac{\partial}{\partial \mathbf{x}} \mathcal{O}(\mathbf{x}, y) = \mathbf{c}\_0 \cdot \frac{\partial}{\partial \mathbf{x}} \phi(\mathbf{x}, y) \tag{26}$$

with *ic*0 being a constant. It can be noted that the derivative of the phase object is proportional to the derivative of the phase, while additionally maintaining a constant phase relationship 2/ because 2/ *<sup>i</sup> ei* . It is well known that when an object is illuminated by a single collimated beam, the relationship between the phase difference and the change in displacement [Rastogi, 1996; Ng , 1995; Bhaduri, 2006] is obtained from:

$$\frac{\partial \phi(\mathbf{x}, y)}{\partial \mathbf{x}} = \frac{2\pi}{\mathcal{A}} \cdot \left[ \sin \theta \frac{\partial d\mathbf{u}}{\partial \mathbf{x}} + (\mathbf{l} + \cos \theta) \frac{\partial w}{\partial \mathbf{x}} \right] \Delta \mathbf{x} \tag{27}$$

where *u* and *w* specify the displacement components. The fringe pattern has contributions from the deformation / *xu* and the slope / *xw* . However, when the object is illuminated in the normal direction (=0°), the fringe pattern represents the *x*-partial slope, / *xw* . This is:

$$\frac{\partial \phi(\mathbf{x}, \mathbf{y})}{\partial \mathbf{x}} = \frac{4\pi}{\mathcal{A}} \cdot \left[\frac{\partial w(\mathbf{x}, \mathbf{y})}{\partial \mathbf{x}}\right] \Delta \mathbf{x} \tag{28}$$

giving us the slope of a phase object as

$$\frac{\partial w(\mathbf{x}, \mathbf{y})}{\partial \mathbf{x}} = \frac{c\_1}{c\_0} \cdot \frac{\partial}{\partial \mathbf{x}} O(\mathbf{x}, \mathbf{y}) \tag{29}$$

with *c*<sup>1</sup> 4/ *x* .

For the case of the radial slope, Eq. (29) can be used only with the consideration of the radial dependency.

### **6.1 Static distributions**

For the case of measurements of the slope in the x-direction, experimental results are presented in Figures 6 and 7 for a microscope slide and an acetate film, respectively. The

Simultaneous Phase Shifting Shearing Interferometry

for Measurement of Static and Dynamic Phase Objects 277

The experimental results for an acetate sample (average thickness of about 100 microns), on which a deformation in the *z* direction was caused, are shown in Fig. 8, where the patterns obtained in simultaneous capture and the slope associated with the derivative of the phase for configuration **I** of the polarizer array are presented. In both cases, it can be seen that the

Fig. 8. Acetate. The relative shifts corresponding to polarizing filter angles in degrees were

Experimental results are presented using the optical system described in section 2.2 (Fig. 2). In order to extract the radial slope of a phase object, it is only analyzed using four patterns obtained in a single shot, and applying the classical method of phase extraction. Fig. 9 shows experimental results corresponding to several samples under study and their associated radial slopes. The upper row in Fig. 9 presents the radial slope of an ophthalmic lens presenting a cross section of 5 mm, the center row shows the deformed interference patterns of one drop of immersion oil placed on a microscope slide, and the lower row presents the deformation caused by lubricating oil over a microscope slide. The fringe pattern shows stress associated with the stabilization on the slide (perpendicular lines in the concentric

<sup>4</sup> (a) Interference patterns obtained in a

pattern). This variation can be observed due to the radial slope variation.

577.460 989.92 42.136

3

1 2

single shot (b) Retrieved Slope.

fringe pattern changes to generate a deformation around a point on the acetate.

polarizer arrays used for these two proposed cases are presented in section 2.1 (See Fig. 1). In Fig. 6 (Case: configuration **I**), the upper row shows a microscope slide blocking half of the beam, with an inclination of 5 ° to axis *z*, resulting in a modification of the slope of the fringes. The lower row shows the results obtained with a water drop flowing into a microscope slide (since it modifies the surface of slide, the phase introduced by the drop is equivalent to obtaining a deformation in the z direction, out of plane). Fig. 7 shows the case of Configuration II of the polarizer array, where the microscope slide has been placed with a minimum inclination (above); as a result, the unwrapped phase presents the slope obtained. The second case shows water deposited on a microscope slide presenting variations in the surface. It can be seen that the patterns used with polarizer arrangement **I** have an apparent decrease of the contrast of the fringes; this is because in both cases, the samples had residual dust particles, and the water was not entirely transparent, so there was minimal absorption of the medium.

Fig. 6. Microscope slide (Configuration **I** of polarizer's array). The relative shifts corresponding to polarizing filter angles in degrees were

12 3 4 0 46.577 92.989 136.42 (a) Interference patterns obtained in a single shot (b) Retrieved Slope.

Fig. 7. Microscope slide (Configuration **II** of polarizer's array). The relative shifts corresponring to polarizing filter angles in degrees were

12 3 4 0 46.577 92.989 136.42 (a) Interference patterns obtained in a single shot (b) Retrieved Slope.

polarizer arrays used for these two proposed cases are presented in section 2.1 (See Fig. 1). In Fig. 6 (Case: configuration **I**), the upper row shows a microscope slide blocking half of the beam, with an inclination of 5 ° to axis *z*, resulting in a modification of the slope of the fringes. The lower row shows the results obtained with a water drop flowing into a microscope slide (since it modifies the surface of slide, the phase introduced by the drop is equivalent to obtaining a deformation in the z direction, out of plane). Fig. 7 shows the case of Configuration II of the polarizer array, where the microscope slide has been placed with a minimum inclination (above); as a result, the unwrapped phase presents the slope obtained. The second case shows water deposited on a microscope slide presenting variations in the surface. It can be seen that the patterns used with polarizer arrangement **I** have an apparent decrease of the contrast of the fringes; this is because in both cases, the samples had residual dust particles, and the water was not entirely transparent, so there was minimal absorption

Fig. 6. Microscope slide (Configuration **I** of polarizer's array). The relative shifts

Fig. 7. Microscope slide (Configuration **II** of polarizer's array). The relative shifts

(a) Interference patterns obtained in a

0 46.577 92.989 136.42 (a) Interference patterns obtained in a

 

corresponding to polarizing filter angles in degrees were

corresponring to polarizing filter angles in degrees were

12 3 4

single shot (b) Retrieved Slope.

 0 46.577 92.989 136.42 

12 3 4

single shot (b) Retrieved Slope.

 

of the medium.

The experimental results for an acetate sample (average thickness of about 100 microns), on which a deformation in the *z* direction was caused, are shown in Fig. 8, where the patterns obtained in simultaneous capture and the slope associated with the derivative of the phase for configuration **I** of the polarizer array are presented. In both cases, it can be seen that the fringe pattern changes to generate a deformation around a point on the acetate.

Fig. 8. Acetate. The relative shifts corresponding to polarizing filter angles in degrees were 577.460 989.92 42.136 1 2 3 <sup>4</sup> (a) Interference patterns obtained in a single shot (b) Retrieved Slope.

Experimental results are presented using the optical system described in section 2.2 (Fig. 2). In order to extract the radial slope of a phase object, it is only analyzed using four patterns obtained in a single shot, and applying the classical method of phase extraction. Fig. 9 shows experimental results corresponding to several samples under study and their associated radial slopes. The upper row in Fig. 9 presents the radial slope of an ophthalmic lens presenting a cross section of 5 mm, the center row shows the deformed interference patterns of one drop of immersion oil placed on a microscope slide, and the lower row presents the deformation caused by lubricating oil over a microscope slide. The fringe pattern shows stress associated with the stabilization on the slide (perpendicular lines in the concentric pattern). This variation can be observed due to the radial slope variation.

Simultaneous Phase Shifting Shearing Interferometry

Representative Frames.

for Measurement of Static and Dynamic Phase Objects 279

Fig. 10. Dynamic phase object. Transparent bubble moving on a microscope slide.

Fig. 11. Dynamic distributions of the radial slope are presented in 4-D. Representative Frames. Upper Row: Interference patters. Lower row: Evolution of the radial slope.

Fig. 9. Radial slope measurements. (a) Interference pattern captured in a single shot and its respective (b) radial slope retrieved.

## **6.2 Dynamic distributions**

A dynamic phase object is shown in Fig. 10 where a transparent bubble is crossing the field of the CCD camera. The corresponding interferograms for successive captures are presented. These results show the curvature of the semi-sphere of the phase object producing a characteristic distortion of a spherical wavefront in the primary interference pattern (circle). In the animation, it is possible to see the transition that takes place when the bubble bursts and the interference patterns recover their initial profile (dot circle). The slope was calculated obtaining the phase unwrapping, according to Eq. (27). Before calculating the unwrapped phase, each interferogram was normalized and filtered using a low-pass filter.

For the case of radial symmetry, a dynamic event (4-D) is presented. Evaporation of lubricating oil placed on a microscope slide by using a tin soldiering iron is presented in Figure 11. It shows the change in the interference pattern (upper row) and its associated radial slope (lower row); the radial slope transition that occurs when the oil is evaporated can be seen. It is important to show that the results are obtained using an optical table without pneumatic suspension, and that the polarized array used is formed by conventional polarized film placed at different angles, having the advantage of not using a micropolarized array.

Fig. 9. Radial slope measurements. (a) Interference pattern captured in a single shot and its

A dynamic phase object is shown in Fig. 10 where a transparent bubble is crossing the field of the CCD camera. The corresponding interferograms for successive captures are presented. These results show the curvature of the semi-sphere of the phase object producing a characteristic distortion of a spherical wavefront in the primary interference pattern (circle). In the animation, it is possible to see the transition that takes place when the bubble bursts and the interference patterns recover their initial profile (dot circle). The slope was calculated obtaining the phase unwrapping, according to Eq. (27). Before calculating the unwrapped phase, each interferogram was normalized and filtered using a low-pass filter. For the case of radial symmetry, a dynamic event (4-D) is presented. Evaporation of lubricating oil placed on a microscope slide by using a tin soldiering iron is presented in Figure 11. It shows the change in the interference pattern (upper row) and its associated radial slope (lower row); the radial slope transition that occurs when the oil is evaporated can be seen. It is important to show that the results are obtained using an optical table without pneumatic suspension, and that the polarized array used is formed by conventional polarized film placed at different angles, having the advantage of not using a micro-

respective (b) radial slope retrieved.

**6.2 Dynamic distributions** 

polarized array.

Fig. 10. Dynamic phase object. Transparent bubble moving on a microscope slide. Representative Frames.

Fig. 11. Dynamic distributions of the radial slope are presented in 4-D. Representative Frames. Upper Row: Interference patters. Lower row: Evolution of the radial slope.

Simultaneous Phase Shifting Shearing Interferometry

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## **7. Conclusion**

With the presented interferometric systems, the slope of phase objects can be obtained in a single shot in real time by using the diffraction properties of high frequency Ronchi gratings. It is suitable to replace the Ronchi gratings used in other systems by phase gratings when it is necessary to implement systems capable of obtaining *N*-interferograms. It should be noted that the elements used are inexpensive and easily accessible; as a result, the system presented can be easily implemented for various applications in single shot polarizing phase shifting interferometry.

For the case of the radial derivative, an adaptation of the MZRI with a phase grid to simultaneously capture several shearograms for slope measurements through the use of phase-shifting techniques has been demonstrated. We present different experimental results obtained by a radial shear interferometer capable of studying events.

The advantage of using these systems is that the lateral or radial slope of the phase object is obtained in real time, allowing us to study transparent samples with this symmetry or dynamic events of transparent samples, which requires obtaining real-time directional derivatives. As future work, we intend to use these systems as wave front sensors, for temperature measurements, or in optical tomography applications.

## **8. Acknowledgments**

Authors thank M. A. Ruiz for his contribution in proofreading the manuscript. Partial support from "Consejo Nacional de Ciencia y Tecnología (CONACYT)" and "Centro de Investigaciones en Óptica A.C. (CIO)" through projects 290597 (CONACYT-CIO) and 124145 (CONACYT-BUAP) is also acknowledged. Author **N. I. Toto-Arellano** expresses sincere appreciation to **Lichita, Miguelon and Damian-El,** for the support provided, and to CONACYT for grant 102137/43055. Author **D. I. Serrano-García** (Grant:227470/31458) is very grateful to CONACyT for the graduate scholarship granted, and expresses sincere appreciation to **Geliztle**.

## **9. References**


With the presented interferometric systems, the slope of phase objects can be obtained in a single shot in real time by using the diffraction properties of high frequency Ronchi gratings. It is suitable to replace the Ronchi gratings used in other systems by phase gratings when it is necessary to implement systems capable of obtaining *N*-interferograms. It should be noted that the elements used are inexpensive and easily accessible; as a result, the system presented can be easily implemented for various applications in single shot polarizing phase

For the case of the radial derivative, an adaptation of the MZRI with a phase grid to simultaneously capture several shearograms for slope measurements through the use of phase-shifting techniques has been demonstrated. We present different experimental results

The advantage of using these systems is that the lateral or radial slope of the phase object is obtained in real time, allowing us to study transparent samples with this symmetry or dynamic events of transparent samples, which requires obtaining real-time directional derivatives. As future work, we intend to use these systems as wave front sensors, for

Authors thank M. A. Ruiz for his contribution in proofreading the manuscript. Partial support from "Consejo Nacional de Ciencia y Tecnología (CONACYT)" and "Centro de Investigaciones en Óptica A.C. (CIO)" through projects 290597 (CONACYT-CIO) and 124145 (CONACYT-BUAP) is also acknowledged. Author **N. I. Toto-Arellano** expresses sincere appreciation to **Lichita, Miguelon and Damian-El,** for the support provided, and to CONACYT for grant 102137/43055. Author **D. I. Serrano-García** (Grant:227470/31458) is very grateful to CONACyT for the graduate scholarship granted, and expresses sincere

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**14** 

*Poland* 

**Laser Interferometric Determination** 

Michał Arabski1, Sławomir Wąsik1, Zuzanna Drulis-Kawa2, Hubert Grześkiewicz1, Jerzy Gubernator2 and Wiesław Kaca1

Cellulose membranes have been used in the medical field as wound dressing and artificial skin material for the treatment of skin wounds such as burns, ulcers, and grafts and as an adjuvant in dermal abrasions (Cheng et al., 2009; Fontana et al., 1990). Using liposomes as drug-delivery strategies in connection with a cellulose-based wound dressing might have important impact on increasing the efficacy of wound healing. The concentration of liposomes used in this therapy is crucial. The amount of liposomes might permit increasing the efficacy of their diffusion through the cellulose membrane. In the present study we propose a laser interferometric method as a novel technique to investigate liposome diffusion through membranes and cellulose-based wound dressing to optimize drug

Laser interferometric methods are widely used for studying diffusion through membranes or in polymeric media (Dworecki et al., 2003; Dworecki et al., 2006; Wąsik et al., 2010). Substance transport through any membrane or release from gel structure are connected with the formation of concentration boundary layers (CBLs). The analysis of thickness and hydrodynamic properties of the CBLs play an important role in determining parameters of substance transport. These parameters might be analyzed by laser interferometry, for example, determination of antibiotics transfer through cellulose biomembrane in the presence of native and O-deacylated (lack of ester-bound fatty acids) forms of lipopolysaccharide (Arabski et al., 2007). Lipopolysaccharides (LPSs), a constituent of the outer membrane of gram-negative bacterial cell walls, are one of the important factors of pathogenicity of bacteria. LPS characterized amphiphilic biopolymeric compounds combining, in a single molecule, hydrophilic (O-specific chains, core oligosaccharide, etc.) and hydrophobic (lipid A) entities. The interferometric technique combined with digital image analysis of the antibiotic CBLs showed that the amount of colistin transported through the cellulose membrane was not influenced by the presence of native LPS, in

**1. Introduction** 

delivery.

**1.1 Analysis of diffusion by laser interferometry** 

**of Liposomes Diffusion Through** 

**Artificial Membranes** 

*1Jan Kochanowski University, 2University of Wrocław* 


## **Laser Interferometric Determination of Liposomes Diffusion Through Artificial Membranes**

Michał Arabski1, Sławomir Wąsik1, Zuzanna Drulis-Kawa2, Hubert Grześkiewicz1, Jerzy Gubernator2 and Wiesław Kaca1 *1Jan Kochanowski University, 2University of Wrocław Poland* 

## **1. Introduction**

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Montes-Perez A., (2010). Slope measurement of a phase object using a polarizing phase-shifting high-frequency Ronchi grating interferometer, *Appl. Opt.*, Vol. 49, Cellulose membranes have been used in the medical field as wound dressing and artificial skin material for the treatment of skin wounds such as burns, ulcers, and grafts and as an adjuvant in dermal abrasions (Cheng et al., 2009; Fontana et al., 1990). Using liposomes as drug-delivery strategies in connection with a cellulose-based wound dressing might have important impact on increasing the efficacy of wound healing. The concentration of liposomes used in this therapy is crucial. The amount of liposomes might permit increasing the efficacy of their diffusion through the cellulose membrane. In the present study we propose a laser interferometric method as a novel technique to investigate liposome diffusion through membranes and cellulose-based wound dressing to optimize drug delivery.

## **1.1 Analysis of diffusion by laser interferometry**

Laser interferometric methods are widely used for studying diffusion through membranes or in polymeric media (Dworecki et al., 2003; Dworecki et al., 2006; Wąsik et al., 2010). Substance transport through any membrane or release from gel structure are connected with the formation of concentration boundary layers (CBLs). The analysis of thickness and hydrodynamic properties of the CBLs play an important role in determining parameters of substance transport. These parameters might be analyzed by laser interferometry, for example, determination of antibiotics transfer through cellulose biomembrane in the presence of native and O-deacylated (lack of ester-bound fatty acids) forms of lipopolysaccharide (Arabski et al., 2007). Lipopolysaccharides (LPSs), a constituent of the outer membrane of gram-negative bacterial cell walls, are one of the important factors of pathogenicity of bacteria. LPS characterized amphiphilic biopolymeric compounds combining, in a single molecule, hydrophilic (O-specific chains, core oligosaccharide, etc.) and hydrophobic (lipid A) entities. The interferometric technique combined with digital image analysis of the antibiotic CBLs showed that the amount of colistin transported through the cellulose membrane was not influenced by the presence of native LPS, in

Laser Interferometric Determination of Liposomes Diffusion Through Artificial Membranes 285

toxicity, enhanced activity against intracellular pathogens, target selectivity and enhanced activity against extracellular pathogens, in particular to overcome bacterial drug resistance. The variety of liposomal formulations allows the design of effective antibiotic forms and subsequent therapeutic success (Abeylath & Turos, 2008; Schiffelers et al., 2001d; Sharma & Sharma 1997; Yimei et al., 2008). There is much evidence of the benefits of liposomes as antibiotic delivery systems. The advantage of liposomal carriers is the possibility of a gradual and sustained release of antibiotics during drug circulation in the body. This allows maintaining the proper drug concentration for a relatively long term. In comparison, administration of the free antibiotic exhibits a quick and short effect and requires several doses per day (Hamidi et al., 2006). Drug encapsulation in liposomal vesicles improves the pharmacokinetics and also protects drug against the hydrolytic activity of enzymes and chemical and immunological deactivation (Allen, 1998; Omri & Ravaoarinoro 1996a,b; Schiffelers et al., 2001). Conventional liposomes applied by intravenous administration are recognized as foreign antigens by the immunological system and are opsonised. This activates nonspecific defence mechanisms and the liposomes are taken up by the mononuclear phagocyte system (MPS), which leads to lower blood circulation time and fast blood clearance. Liposomes accumulate in the liver, spleen, lung, and kidneys (Bakker-Woudenberg, 1994, 2002; Schiffelers et al., 2001). This phenomenon (phagocytosis of liposomes) is desirable for intracellular pathogen eradication, but unfavourable for other kinds of infection (Lasic, 1998; Voinea & Simionescu, 2002). The MPS uptake rate depends on several liposomal properties, such as size, charge, and fluidity. The blood clearance of small vesicles (~100 nm) rises to several hours, in comparison with several minutes for MLV formulations. Rigid and uncharged vesicles circulate longer than fluid and charged ones (Beaulac et al., 1997; Scherphof et al., 1997). The plasma circulation time of antibiotics can be improved by encapsulation in polyethylene glycol-coated ("pegylated") (STEALTH) liposomes. The hydrophilic layer composed of PEG protects the vesicles from the MPS and allows a long liposome circulation in the blood system. Sterically stabilised STEALTH liposomes exhibit sustained release of drug and are able to accumulate selectively at sites of infection (Bakker-Woudenberg et al., 1993; Ceh et al., 1997). Liposomes are currently in common use as universal drug carriers in the cosmetic and pharmaceutical industries. In healthcare there are antitumor anthracyclines such doxorubicin and antifungal amphotericin B liposomal formulations available (Allen & Martin, 2004; Bakker-Woudenberg et al., 1994). Intensive research is focused on antibiotics entrapped in liposomes to enhance their antibacterial activity and pharmacokinetic properties. Lipid vesicles as drug carriers significantly influence drug distribution and reduce toxic side effects during antibiotic therapy (Allen 1998; Bakker-Woudenberg 1993, 1994, 2002; Drulis et al., 2006ab, 2009;

The present study was designed to examine the physical properties of PC:Chol:DOTAP (3:4:3) liposomes, especially the diffusive transport of the liposome through different membranes, as a model of a liposomal drug system for local applications. Interferomentric description of diffusive transport of the liposome through different membranes would be

The laser interferometry method for the investigations of substance transport was presented previously (Arabski et al., 2007). A sketch of the measuring system is presented in Fig. 1.

useful, as a model of a liposomal drug system for local applications.

Gubernator et al., 2007; Sapra & Allen 2003).

**2. Laser interferometric method** 

contrast to presence of O-deacylated form. This effect might be associated with the phenomenon of micelles formed by native LPSs (Arabski et al., 2007).

The modification of laser interferometric technique by immobilising the tested molecules in agarose gel and measuring the amount of released substances allowed investigating interactions of partially insoluble mixtures as lipopolisaccharide with chitosan (Arabski et al., 2009a). The results of studies with laser interferometry indicated that chitosan's binding to LPS and this interaction is weaker than that of colistin. It might be associated with presence of fatty acid residues as colistin components. Chitosan reduced the speed of colistin diffusion from the gel measured by laser interferometry. These results were confirmed with electron microscopy and precipitation assay (Arabski et al., 2009a)

Additionally saponins enhance the interaction of colistin with the S and Re types of *P. mirabilis* LPSs. Analysis by laser interferometry method shows that less colistin diffused from the S1959 and R45 LPSs aggregates pre-incubated with saponins. This effect might be associated with disaggregation of LPSs micelles by saponin and facilitation of colistin binding to the lipid A part of the LPSs. These results were confirmed in a whole bacterial cell experiment (Arabski et al., 2009b).

Because this method was successfully used in our previously studies we decided to apply interferometry to examine the physical properties of PC:Chol:DOTAP (3:4:3) liposomes.

## **1.2 Liposomes as antibiotic carriers**

Liposomes are spherical vesicles consisting of one or more phospholipid bilayers surrounding a water space. The diameter of the liposome varies from 0.02 to 10 μm. Vesicle formulations are usually based on natural and synthetic phospholipids and cholesterol. The structure may also possess lipoproteins (Urlich, 2002). The base property of these compounds is to form bilayer structure in water environment. The physicochemical properties of liposomes can be modified by changing: the types of lipids, the composition and proportions of lipids in the liposomal formulation, the size of the liposome, the charge of the liposomal surface, pH and temperature sensitivity and the fluidity of the liposomal membrane.

Regarding the variety of liposomal formulations, the vesicles are universal carriers for both hydrophilic and hydrophobic compounds. Hydrophilic elements are dissolved in the water space inside the vesicles. The most useful for this are LUVs (*Large Unilamellar Vesicles*) because the volume of encapsulated water is relatively high (Gregoriadis, 1995; Sharma & Sharma, 1997). Hydrophobic compounds are located in the lipid bilayer, and MLVs (*Multilamellar Vesicles*) or SUVs (*Small Unilamellar Vesicles*) may be applied. Charged drugs can be associated to the lipid surface (Gregoriadis, 1995; Sharma & Sharma, 1997). The size of the liposomal vesicles significantly influences drug distribution. Large (>1 μm) MLV formulations are usually not used as drug carriers, but SUVs of ~100 nm exhibited high efficacy in the eradication of bacterial pathogens (Krieger et al., 1999; Drulis & Dorotkiewicz, 2010). Encapsulation of the drugs in lipid vesicles is a good solution for designing the required pharmacokinetic and pharmacodynamic properties (Allen, 1998; Bakker-Woudenberg 1993, 1994, 2002; Swenson et al., 1988). There are many advantages of liposomes as antibiotic carriers: improved pharmacokinetics and biodistribution, decreased

contrast to presence of O-deacylated form. This effect might be associated with the

The modification of laser interferometric technique by immobilising the tested molecules in agarose gel and measuring the amount of released substances allowed investigating interactions of partially insoluble mixtures as lipopolisaccharide with chitosan (Arabski et al., 2009a). The results of studies with laser interferometry indicated that chitosan's binding to LPS and this interaction is weaker than that of colistin. It might be associated with presence of fatty acid residues as colistin components. Chitosan reduced the speed of colistin diffusion from the gel measured by laser interferometry. These results were

Additionally saponins enhance the interaction of colistin with the S and Re types of *P. mirabilis* LPSs. Analysis by laser interferometry method shows that less colistin diffused from the S1959 and R45 LPSs aggregates pre-incubated with saponins. This effect might be associated with disaggregation of LPSs micelles by saponin and facilitation of colistin binding to the lipid A part of the LPSs. These results were confirmed in a whole bacterial

Because this method was successfully used in our previously studies we decided to apply interferometry to examine the physical properties of PC:Chol:DOTAP (3:4:3)

Liposomes are spherical vesicles consisting of one or more phospholipid bilayers surrounding a water space. The diameter of the liposome varies from 0.02 to 10 μm. Vesicle formulations are usually based on natural and synthetic phospholipids and cholesterol. The structure may also possess lipoproteins (Urlich, 2002). The base property of these compounds is to form bilayer structure in water environment. The physicochemical properties of liposomes can be modified by changing: the types of lipids, the composition and proportions of lipids in the liposomal formulation, the size of the liposome, the charge of the liposomal surface, pH and temperature sensitivity and the fluidity of the liposomal

Regarding the variety of liposomal formulations, the vesicles are universal carriers for both hydrophilic and hydrophobic compounds. Hydrophilic elements are dissolved in the water space inside the vesicles. The most useful for this are LUVs (*Large Unilamellar Vesicles*) because the volume of encapsulated water is relatively high (Gregoriadis, 1995; Sharma & Sharma, 1997). Hydrophobic compounds are located in the lipid bilayer, and MLVs (*Multilamellar Vesicles*) or SUVs (*Small Unilamellar Vesicles*) may be applied. Charged drugs can be associated to the lipid surface (Gregoriadis, 1995; Sharma & Sharma, 1997). The size of the liposomal vesicles significantly influences drug distribution. Large (>1 μm) MLV formulations are usually not used as drug carriers, but SUVs of ~100 nm exhibited high efficacy in the eradication of bacterial pathogens (Krieger et al., 1999; Drulis & Dorotkiewicz, 2010). Encapsulation of the drugs in lipid vesicles is a good solution for designing the required pharmacokinetic and pharmacodynamic properties (Allen, 1998; Bakker-Woudenberg 1993, 1994, 2002; Swenson et al., 1988). There are many advantages of liposomes as antibiotic carriers: improved pharmacokinetics and biodistribution, decreased

confirmed with electron microscopy and precipitation assay (Arabski et al., 2009a)

phenomenon of micelles formed by native LPSs (Arabski et al., 2007).

cell experiment (Arabski et al., 2009b).

**1.2 Liposomes as antibiotic carriers** 

liposomes.

membrane.

toxicity, enhanced activity against intracellular pathogens, target selectivity and enhanced activity against extracellular pathogens, in particular to overcome bacterial drug resistance.

The variety of liposomal formulations allows the design of effective antibiotic forms and subsequent therapeutic success (Abeylath & Turos, 2008; Schiffelers et al., 2001d; Sharma & Sharma 1997; Yimei et al., 2008). There is much evidence of the benefits of liposomes as antibiotic delivery systems. The advantage of liposomal carriers is the possibility of a gradual and sustained release of antibiotics during drug circulation in the body. This allows maintaining the proper drug concentration for a relatively long term. In comparison, administration of the free antibiotic exhibits a quick and short effect and requires several doses per day (Hamidi et al., 2006). Drug encapsulation in liposomal vesicles improves the pharmacokinetics and also protects drug against the hydrolytic activity of enzymes and chemical and immunological deactivation (Allen, 1998; Omri & Ravaoarinoro 1996a,b; Schiffelers et al., 2001). Conventional liposomes applied by intravenous administration are recognized as foreign antigens by the immunological system and are opsonised. This activates nonspecific defence mechanisms and the liposomes are taken up by the mononuclear phagocyte system (MPS), which leads to lower blood circulation time and fast blood clearance. Liposomes accumulate in the liver, spleen, lung, and kidneys (Bakker-Woudenberg, 1994, 2002; Schiffelers et al., 2001). This phenomenon (phagocytosis of liposomes) is desirable for intracellular pathogen eradication, but unfavourable for other kinds of infection (Lasic, 1998; Voinea & Simionescu, 2002). The MPS uptake rate depends on several liposomal properties, such as size, charge, and fluidity. The blood clearance of small vesicles (~100 nm) rises to several hours, in comparison with several minutes for MLV formulations. Rigid and uncharged vesicles circulate longer than fluid and charged ones (Beaulac et al., 1997; Scherphof et al., 1997). The plasma circulation time of antibiotics can be improved by encapsulation in polyethylene glycol-coated ("pegylated") (STEALTH) liposomes. The hydrophilic layer composed of PEG protects the vesicles from the MPS and allows a long liposome circulation in the blood system. Sterically stabilised STEALTH liposomes exhibit sustained release of drug and are able to accumulate selectively at sites of infection (Bakker-Woudenberg et al., 1993; Ceh et al., 1997). Liposomes are currently in common use as universal drug carriers in the cosmetic and pharmaceutical industries. In healthcare there are antitumor anthracyclines such doxorubicin and antifungal amphotericin B liposomal formulations available (Allen & Martin, 2004; Bakker-Woudenberg et al., 1994). Intensive research is focused on antibiotics entrapped in liposomes to enhance their antibacterial activity and pharmacokinetic properties. Lipid vesicles as drug carriers significantly influence drug distribution and reduce toxic side effects during antibiotic therapy (Allen 1998; Bakker-Woudenberg 1993, 1994, 2002; Drulis et al., 2006ab, 2009; Gubernator et al., 2007; Sapra & Allen 2003).

The present study was designed to examine the physical properties of PC:Chol:DOTAP (3:4:3) liposomes, especially the diffusive transport of the liposome through different membranes, as a model of a liposomal drug system for local applications. Interferomentric description of diffusive transport of the liposome through different membranes would be useful, as a model of a liposomal drug system for local applications.

## **2. Laser interferometric method**

The laser interferometry method for the investigations of substance transport was presented previously (Arabski et al., 2007). A sketch of the measuring system is presented in Fig. 1.

Laser Interferometric Determination of Liposomes Diffusion Through Artificial Membranes 287

(internal dimensions: 70 mm high × 10 mm wide, optical path length: 7 mm) separated by the horizontally located membrane. The cuvettes are made of optical glass of high uniformity. In our experiments the upper cuvette held pure water while the lower was filled

The laser light (Fig.1A,B) is spatially filtered and is transformed by the beam expander into a parallel beam ca. 80 mm wide and then split into two beams. The first beam goes through the investigated membrane system parallel to the membrane surface, while the second goes directly through the compensation plate to the light detection system. As a consequence of the superimposition of these beams, respective interference images are generated (Fig.1C). The images depend on the refraction coefficient of the solute, which in turn depends on the substance concentration. When the solute is uniform the interference fringes are straight, and they bend when a concentration gradient appears. In this system, water and dissolved liposomes diffusing through the membrane will lead to the formation of concentration boundary layers (CBLs). The applied computer program to analyse these images allows, among other things, to find the concentration profiles and the CBL thicknesses. The concentration profile *C*(*x*,*t*) is determined by the deviation of the fringes from their straight line run *d*(*x*,*t*). Since the concentration *C* and the refraction coefficient are assumed to be

> 0 ( ,) ( ,) *dxt Cxt C a*

where *C*0 is the initial liposomes concentration, *a* the proportionality constant between the concentration and the refraction index, the wavelength of the laser light, *h* the distance between the fringes in the field where they are straight lines, and *f* the thickness of the

> *m <sup>C</sup> <sup>k</sup> C*

> > ( )

In our investigations we took arbitrary k=0.08. The flux *Js* of the solute which flows through

where *N*(*t*) is the amount of liposomes which diffuses after time *t* through a membrane of area *S* from one compartment of the membrane system to the other. *N*(*t*) at any time *t* was

> 1 0 *N t S C x t dx* () ( ,)

and *N*(*t*) in the given time interval *t* was calculated by integrating the concentration

*S*

concentration C is a certain part (*k*) of concentration on the membrane surface Cm:

*hf* 

(1)

(2)

*N t <sup>J</sup> St* (3)

(4)

is the distance from the membrane to the point at which the

was defined according the

with aqueous liposome solution (0.5, 1 or 2 mg/ml).

linear (Robinson & Reid, 1993), we have:

following criterion.

the membrane is given by:

profile according to:

solution layer in the measurement cuvette. The CBL thickness

calculated by integrating the concentration profile according to:

Fig. 1. Experimental set-up (A) of the interferometric investigations of the transport processes in a membrane system and its photo (B). Examples of interferograms (C) obtained for PC:Chol:DOTAP 3:4:3 liposomes at 0.5, 1, and 2 mg/ml initial concentrations transported through BioFill membrane after 10 min. C1 denotes the initial solution concentration in the upper cuvette (water C1=0), C2 is the initial solution concentration in the lower cuvette (aqueous liposome solution at 0.5, 1 or 2 mg/ml). B→A- direction of liposomes diffusion (A).

It consist of a Mach-Zehnder interferometer with a He-Ne laser, a membrane system, a TV-CCD camera and a computer with a system for the acquisition and processing of interference images. The membrane system under study consists of two glass cuvettes

Fig. 1. Experimental set-up (A) of the interferometric investigations of the transport

for PC:Chol:DOTAP 3:4:3 liposomes at 0.5, 1, and 2 mg/ml initial concentrations transported through BioFill membrane after 10 min. C1 denotes the initial solution

lower cuvette (aqueous liposome solution at 0.5, 1 or 2 mg/ml). B→A- direction of

liposomes diffusion (A).

processes in a membrane system and its photo (B). Examples of interferograms (C) obtained

concentration in the upper cuvette (water C1=0), C2 is the initial solution concentration in the

It consist of a Mach-Zehnder interferometer with a He-Ne laser, a membrane system, a TV-CCD camera and a computer with a system for the acquisition and processing of interference images. The membrane system under study consists of two glass cuvettes (internal dimensions: 70 mm high × 10 mm wide, optical path length: 7 mm) separated by the horizontally located membrane. The cuvettes are made of optical glass of high uniformity. In our experiments the upper cuvette held pure water while the lower was filled with aqueous liposome solution (0.5, 1 or 2 mg/ml).

The laser light (Fig.1A,B) is spatially filtered and is transformed by the beam expander into a parallel beam ca. 80 mm wide and then split into two beams. The first beam goes through the investigated membrane system parallel to the membrane surface, while the second goes directly through the compensation plate to the light detection system. As a consequence of the superimposition of these beams, respective interference images are generated (Fig.1C). The images depend on the refraction coefficient of the solute, which in turn depends on the substance concentration. When the solute is uniform the interference fringes are straight, and they bend when a concentration gradient appears. In this system, water and dissolved liposomes diffusing through the membrane will lead to the formation of concentration boundary layers (CBLs). The applied computer program to analyse these images allows, among other things, to find the concentration profiles and the CBL thicknesses. The concentration profile *C*(*x*,*t*) is determined by the deviation of the fringes from their straight line run *d*(*x*,*t*). Since the concentration *C* and the refraction coefficient are assumed to be linear (Robinson & Reid, 1993), we have:

$$\mathbf{C}(\mathbf{x},t) = \mathbf{C}\_0 + a \frac{\lambda d(\mathbf{x},t)}{hf} \tag{1}$$

where *C*0 is the initial liposomes concentration, *a* the proportionality constant between the concentration and the refraction index, the wavelength of the laser light, *h* the distance between the fringes in the field where they are straight lines, and *f* the thickness of the solution layer in the measurement cuvette. The CBL thickness was defined according the following criterion. is the distance from the membrane to the point at which the concentration C is a certain part (*k*) of concentration on the membrane surface Cm:

$$k = \frac{\mathbf{C}\_{\delta}}{\mathbf{C}\_{m}} \tag{2}$$

In our investigations we took arbitrary k=0.08. The flux *Js* of the solute which flows through the membrane is given by:

$$J\_S = \frac{N(t)}{St} \tag{3}$$

where *N*(*t*) is the amount of liposomes which diffuses after time *t* through a membrane of area *S* from one compartment of the membrane system to the other. *N*(*t*) at any time *t* was calculated by integrating the concentration profile according to:

$$N(t) = S \bigcup\_{0}^{\delta} \mathbb{C}\_{1}(\mathbf{x}, t) d\mathbf{x} \tag{4}$$

and *N*(*t*) in the given time interval *t* was calculated by integrating the concentration profile according to:

Laser Interferometric Determination of Liposomes Diffusion Through Artificial Membranes 289

() () ( ) 4 ( ( )) *t t D t*

The interferograms were recorded from 120 to 2400 s with a time interval of *t* = 120 s and the profiles for a given initial liposome concentration (0.5, 1 or 2 mg/ml) were reconstructed. Such profiles were used to calculate the time dependencies *N*(*t*), *N*(*t*), *Js*(*t*),

*KALPROST* application descript by Mr. Grześkiewicz using *VISUAL C++ 2005 EXPRESS* 

A cationic liposomal formulation (PC:Chol:DOTAP 3:4:3) was prepared using a thin lipid film method. Appropriate amounts of lipids dissolved in chloroform (10 mg/ml) were mixed in a 100-ml round-bottom flask. Then the chloroform was evaporated under vacuum on a rotary evaporator and the resulting thin lipid film was hydrated by agitation with 1 ml of PBS at room temperature. The MLV liposomes were then extruded 10 times through a 100-nm-pore filter using a 10-ml extruder. The size of the liposomes was determined with a ZetaSizer Nano-ZS (Malvern Instruments, UK) and was usually in the range of 105-110 nm. The stability of liposomal formulation was previously determined by carboxyfluorescein release profile. The liposome leakage reached approximately 6, 8, 14, and 24% after 1, 3, 12, and 24 h, respectively. In the experiments of vesicles stability Sephadex columns were used and no lipid exchange was observed (Drulis-Kawa, 2006b). 1,2-dioleoyloxy-3 trimethylammonium-propane (DOTAP) and phosphatidylcholine (PC) were purchased from Northern Lipids Inc. (Vancouver, BC, Canada). Cholesterol (Chol) was obtained from E. Merck (Darmstadt, Germany). HPLC solvents were supplied by J. T. Baker (Deventer, the

The liposome diffusion efficiency was examined using commercial membranes such as Biofill and Nephrophane. The bacterial cellulose membrane (BioFill) is a transduced film with a gram-meter of 9 gm-2 to 20 gm-2, pH between 6.0 and 7.0, obtained through biosynthesis from bacteria of the genus *Acetobacter* (Klemm et al. , 2001). Its thickness varies between 20 m and 52 m in dehydrated and in hydrated conditions, respectively. Bacterial cellulose has a microfibrous structure and it is a hypoallergic, nontoxic, nonirritant, biodegradable, non-pyrogenic, highly hydrophilic, and biocompatible material (Pitanguy et al., 1988; Kucharzewski et al., 2003). A cellulose acetate membrane Nephrophane is a hydrophilic membrane and its thickness varies from 25 m in the dehydrated state to 200 m under hydrated conditions (Kargol, 2001). Nucleopores, a polymeric nuclear track membranes, with different pore diameters (0.092-0.1 m, 0.2 m, 0.9 m, and 1.27-1.35 m) were purchased from Joint Institute for Nuclear Research in

The results of liposome diffusion at three concentrations (0.5, 1, and 2 mg/ml) through nucleopore membranes of different pore diameter, bacterial cellulose (BioFill), and cellulose

acetate (Nephrophane) membranes are shown in Table 1.

*EDITION.* All experiments were performed at a temperature of 37ºC.

D(t), D(t) and

Netherlands).

Dubna, Russia.

**3. Results** 

2 2 22 11 1 2

*t erfc k*

 

*<sup>m</sup>*. All parameters were measured on the basic of interferograms by

(12)

$$N(\Delta t) = S \int\_0^\delta \mathbb{C}\_1(\mathbf{x}, t + \Delta t) d\mathbf{x} - S \int\_0^\delta \mathbb{C}\_1(\mathbf{x}, t) d\mathbf{x} \tag{5}$$

On the basis of (3) and (5) we obtain:

$$\int\_{S}^{\delta} \mathbf{C}\_{1}(\mathbf{x}, t + \Delta t) d\mathbf{x} - \int\_{0}^{\delta} \mathbf{C}\_{1}(\mathbf{x}, t) d\mathbf{x}$$

where *C*1(*x*,*t+Δt*), δ' and *C*1(*x*,*t*), δ denote the concentration profiles and the thickness of CBL for times *t+Δt* and *t*, respectively.

The membrane permeability coefficient was determined on the basis of Fick's first law. In the case of diffusive transport of binary solutions, this law can be written as:

$$J\_S = RTo\_m \left[ \mathbf{C\_2(x=0,t)} - \mathbf{C\_1(x=0,t)} \right] \tag{7}$$

where *Js* is the solute flux and *C*1(*x*=0,*t*) and *C*2(*x*=0,*t*) the concentrations of the solutions on the membrane surfaces at time *t*. On the basis of Eqs. (6) and (7) we obtain the formula for the membrane permeability coefficient:

$$\begin{aligned} \stackrel{\circ}{\rho} \mathbf{C}\_1(\mathbf{x}, t + \Delta t)d\mathbf{x} - \stackrel{\circ}{\rho} \mathbf{C}\_1(\mathbf{x}, t)d\mathbf{x} \\ \rho \mathbf{o}\_m = \frac{\stackrel{\circ}{0}}{RT\left[\mathbf{C}\_2(\mathbf{x} = 0, t) - \mathbf{C}\_1(\mathbf{x} = 0, t)\right] \Delta t} \end{aligned} \tag{8}$$

We present method of diffusion coefficient calculations based time evolution of CBLs measured by laser interferometry. The diffusion coefficient D is determined by the CBLs thickness criterion mentioned above and the theoretical concentration profile (Dworecki et al., 2000):

$$\mathcal{C}\_1(\mathbf{x}, t) = \mathcal{C}\_2 \frac{1 - \sigma}{2} \text{erfc}(\frac{\mathbf{x}}{2\sqrt{Dt}}) \tag{9}$$

where denotes the membrane selectivity coefficient, C2 is the initial solution concentration in the lower cuvette, *erfc* is complementary error function defined as:

$$
\text{erfc}(\mathbf{x}) = \frac{2}{\sqrt{\prod\_x}} \stackrel{\Rightarrow}{\int} e^{-\eta/2} d\eta \tag{10}
$$

On the basis of the Eqs. (2) and (9) we obtain the formula for the diffusion coefficient D(t) after time t for the membrane sytems in whose one compartment held pure solvent.

$$D(t) = \frac{\delta^2(t)}{4t(erfc^{-1}(k))^2} \tag{11}$$

where *erfc*-1 is the inverse function to *erfc*. The formula of diffusion coefficient D(*t*) in the time interval *t* is given:

$$D(\Delta t) = \frac{\delta\_2^2(t\_2) - \delta\_1^2(t\_1)}{4\Delta t (erfc^{-1}(k))^2} \tag{12}$$

The interferograms were recorded from 120 to 2400 s with a time interval of *t* = 120 s and the profiles for a given initial liposome concentration (0.5, 1 or 2 mg/ml) were reconstructed. Such profiles were used to calculate the time dependencies *N*(*t*), *N*(*t*), *Js*(*t*), D(t), D(t) and *<sup>m</sup>*. All parameters were measured on the basic of interferograms by *KALPROST* application descript by Mr. Grześkiewicz using *VISUAL C++ 2005 EXPRESS EDITION.* All experiments were performed at a temperature of 37ºC.

A cationic liposomal formulation (PC:Chol:DOTAP 3:4:3) was prepared using a thin lipid film method. Appropriate amounts of lipids dissolved in chloroform (10 mg/ml) were mixed in a 100-ml round-bottom flask. Then the chloroform was evaporated under vacuum on a rotary evaporator and the resulting thin lipid film was hydrated by agitation with 1 ml of PBS at room temperature. The MLV liposomes were then extruded 10 times through a 100-nm-pore filter using a 10-ml extruder. The size of the liposomes was determined with a ZetaSizer Nano-ZS (Malvern Instruments, UK) and was usually in the range of 105-110 nm. The stability of liposomal formulation was previously determined by carboxyfluorescein release profile. The liposome leakage reached approximately 6, 8, 14, and 24% after 1, 3, 12, and 24 h, respectively. In the experiments of vesicles stability Sephadex columns were used and no lipid exchange was observed (Drulis-Kawa, 2006b). 1,2-dioleoyloxy-3 trimethylammonium-propane (DOTAP) and phosphatidylcholine (PC) were purchased from Northern Lipids Inc. (Vancouver, BC, Canada). Cholesterol (Chol) was obtained from E. Merck (Darmstadt, Germany). HPLC solvents were supplied by J. T. Baker (Deventer, the Netherlands).

The liposome diffusion efficiency was examined using commercial membranes such as Biofill and Nephrophane. The bacterial cellulose membrane (BioFill) is a transduced film with a gram-meter of 9 gm-2 to 20 gm-2, pH between 6.0 and 7.0, obtained through biosynthesis from bacteria of the genus *Acetobacter* (Klemm et al. , 2001). Its thickness varies between 20 m and 52 m in dehydrated and in hydrated conditions, respectively. Bacterial cellulose has a microfibrous structure and it is a hypoallergic, nontoxic, nonirritant, biodegradable, non-pyrogenic, highly hydrophilic, and biocompatible material (Pitanguy et al., 1988; Kucharzewski et al., 2003). A cellulose acetate membrane Nephrophane is a hydrophilic membrane and its thickness varies from 25 m in the dehydrated state to 200 m under hydrated conditions (Kargol, 2001). Nucleopores, a polymeric nuclear track membranes, with different pore diameters (0.092-0.1 m, 0.2 m, 0.9 m, and 1.27-1.35 m) were purchased from Joint Institute for Nuclear Research in Dubna, Russia.

## **3. Results**

288 Interferometry – Research and Applications in Science and Technology

1 1 0 0

The membrane permeability coefficient was determined on the basis of Fick's first law. In

*J RT C x t C x t S m*

where *Js* is the solute flux and *C*1(*x*=0,*t*) and *C*2(*x*=0,*t*) the concentrations of the solutions on the membrane surfaces at time *t*. On the basis of Eqs. (6) and (7) we obtain the formula for

> 1 1 0 0 2 1

 

( 0, ) ( 0, ) *<sup>m</sup>*

We present method of diffusion coefficient calculations based time evolution of CBLs measured by laser interferometry. The diffusion coefficient D is determined by the CBLs thickness criterion mentioned above and the theoretical concentration profile (Dworecki et

> <sup>1</sup> ( ,) ( ) <sup>2</sup> <sup>2</sup> *<sup>x</sup> C x t C erfc Dt*

where denotes the membrane selectivity coefficient, C2 is the initial solution concentration

<sup>2</sup> <sup>2</sup> ( ) *x erfcx e d*

On the basis of the Eqs. (2) and (9) we obtain the formula for the diffusion coefficient D(t)

( ) ( ) 4 ( ( )) *<sup>t</sup> D t*

where *erfc*-1 is the inverse function to *erfc*. The formula of diffusion coefficient D(*t*) in the

2

*t erfc k* 

after time t for the membrane sytems in whose one compartment held pure solvent.

 

1 2

*RT C x t C x t t*

( , ) ( ,)

 

*C x t t dx C x t dx*

( , ) ( ,)

 

*C x t t dx C x t dx*

1 1 0 0 *N t S C x t t dx S C x t dx* ( ) ( , ) ( ,)

 

and *C*1(*x*,*t*), δ denote the concentration profiles and the thickness of CBL

(5)

2 1 ( 0, ) ( 0, ) (7)

(9)

(10)

(11)

(6)

(8)

'

'

*<sup>J</sup> <sup>t</sup>*

the case of diffusive transport of binary solutions, this law can be written as:

'

1 2

in the lower cuvette, *erfc* is complementary error function defined as:

*S*

On the basis of (3) and (5) we obtain:

for times *t+Δt* and *t*, respectively.

the membrane permeability coefficient:

where *C*1(*x*,*t+Δt*), δ'

al., 2000):

time interval *t* is given:

The results of liposome diffusion at three concentrations (0.5, 1, and 2 mg/ml) through nucleopore membranes of different pore diameter, bacterial cellulose (BioFill), and cellulose acetate (Nephrophane) membranes are shown in Table 1.

Laser Interferometric Determination of Liposomes Diffusion Through Artificial Membranes 291

Fig. 2. The amount *N(t)* of PC:Chol:DOTAP 3:4:3 liposomes transported through nucleopore membrane of pore diameter 0.9 m (A), BioFill membrane (B) and Nephrophane membrane

(C) in 40 min at 37ºC measured by the laser interferometric system.


Table 1. Diffusion of liposomes through different membranes.

Filtration of the liposome vesicles was noted for nucleopore membrane of pore diameters 1.27-1.35 m. In the case of 0.2 m pores, diffusion was blocked after 2, 8, and 12 min. for the 0.5, 1, and 2 mg/ml concentrations of lipids, respectively, although the vesicle sizes were lower than the pore diameters. No diffusion through 0.092-0.1 µm pores was observed although fluid formulation of liposome was tested and the plastic deformation of vesicles thus the passing across smaller pores would be possible. Mathematical analysis of the timedependent concentration profiles indicates that the amount of liposomes (0.069 mg) transported through the nucleopore membrane of 0.9 m pore diameter at the initial concentration of 2 mg/ml was the same as for 1 mg/ml after 40 min in distance of 0.995 mm from membrane (Fig. 2A).

In the first phase of diffusion we observed throw of liposomes through nucleopore membrane of 0.9 m pore diameter and then (after ~10 min.) diffusion was relatively constant (Fig. 3).

The same kinetics of liposomes transport across nucleopore 0.9 m after analysis of flux *Js* through a membrane area was observed (Fig. 4).

0.092-0.1 µm no diffusion observed no diffusion observed no diffusion observed

1.27-1.35 µm filtration filtration filtration

Filtration of the liposome vesicles was noted for nucleopore membrane of pore diameters 1.27-1.35 m. In the case of 0.2 m pores, diffusion was blocked after 2, 8, and 12 min. for the 0.5, 1, and 2 mg/ml concentrations of lipids, respectively, although the vesicle sizes were lower than the pore diameters. No diffusion through 0.092-0.1 µm pores was observed although fluid formulation of liposome was tested and the plastic deformation of vesicles thus the passing across smaller pores would be possible. Mathematical analysis of the timedependent concentration profiles indicates that the amount of liposomes (0.069 mg) transported through the nucleopore membrane of 0.9 m pore diameter at the initial concentration of 2 mg/ml was the same as for 1 mg/ml after 40 min in distance of 0.995 mm

In the first phase of diffusion we observed throw of liposomes through nucleopore membrane of 0.9 m pore diameter and then (after ~10 min.) diffusion was relatively

The same kinetics of liposomes transport across nucleopore 0.9 m after analysis of flux *Js*

0.031 mg of liposomes pass membrane, blocked after 2 min of diffusion

diffusion of 0.069 mg of liposomes after 40 min. (Fig. 2A●)

diffusion of 0.187 mg of liposomes after 40 min. (Fig. 2B●)

diffusion of 0.144 mg of liposomes after 40 min. (Fig. 2C●)

Table 1. Diffusion of liposomes through different membranes.

Liposome concentration

2 mg/ml 1 mg/ml 0.5 mg/ml

0.030 mg of liposomes pass membrane, blocked after 8 min of diffusion

diffusion of 0.069 mg of liposomes after 40 min. (Fig. 2A●)

diffusion of 0.082 mg of liposomes after 40 min. (Fig. 2B●)

diffusion of 0.088 mg of liposomes after 40 min. (Fig. 2C●)

0.035 mg of liposomes pass membrane, blocked after 12 min of diffusion

diffusion of 0.036 mg of liposomes after 40 min. Fig. 2A○)

diffusion of 0.034 mg of liposomes after 40 min. (Fig. 2B○)

diffusion of 0.028 mg of liposomes after 40 min. (Fig. 2C○)

Membrane

Nucleopore

Nucleopore 0.2 µm

Nucleopore 0.9 µm

Nucleopore

Biofill

Nephrophane

from membrane (Fig. 2A).

through a membrane area was observed (Fig. 4).

constant (Fig. 3).

Fig. 2. The amount *N(t)* of PC:Chol:DOTAP 3:4:3 liposomes transported through nucleopore membrane of pore diameter 0.9 m (A), BioFill membrane (B) and Nephrophane membrane (C) in 40 min at 37ºC measured by the laser interferometric system.

Laser Interferometric Determination of Liposomes Diffusion Through Artificial Membranes 293

The next step of our mathematical analysis was calculations of the average membrane permeability coefficient for aqueous solution of liposomes. The greater amounts of liposomes transported through BioFill, Nephrophane in contrast to nucleopore membrane was associated with the different values of this coefficients for above membranes (1.21×10-10, 1.06×10-10, and 4.31×10-11 mol N-1s-1, respectively). We observed different kinetics of liposomes diffusion through both cellulose membranes. The amount of transported substances was positively correlated with the initial concentration of liposomes in dose-

Additionally, analysis of the membrane permeability coefficient for different liposomes concentrations shown the lower value of this coefficient for 2 mg/ml than for 1 and 0.5

The same conclusions are based on analysis of diffusion coefficient D values for liposomes solutions measured both after time t and in the given time interval *t*. Analysis of diffusion coefficient for liposomes solution in the time interval *t* (Fig. 6) was more precise than after time t (Fig. 5). Therefore, we could noted that throw of liposomes from initial concentration

Fig. 5. The diffusion coefficient in water solution of liposomes measured in time *t* by the

dependent manner and increase was regular for 40 min.

2 mg/ml took place in the first ~10 minutes of diffusion (Fig. 6)

mg/ml.

laser interferometric system.

Fig. 3. The amount *N(t)* of PC:Chol:DOTAP 3:4:3 liposomes transported through the nucleopore membrane of 0.9 m pore diameter measured by the laser interferometric system.

Fig. 4. The flux *Js* of the PC:Chol:DOTAP 3:4:3 liposomes which cross the nucleopore membrane of 0.9 m pore diameter measured by the laser interferometric system.

More efficient diffusion of liposomes was noticed in regards to BioFill (Fig. 2B) and Nephrophane (Fig. 2C) membranes than to 0.9 m nucleopore (Fig. 2A). For the highest liposome concentration, 0.187 and 0.144 mg of lipids were transported across BioFill and Nephrophane, respectively.

*t)* of PC:Chol:DOTAP 3:4:3 liposomes transported through the

nucleopore membrane of 0.9 m pore diameter measured by the laser interferometric

Fig. 4. The flux *Js* of the PC:Chol:DOTAP 3:4:3 liposomes which cross the nucleopore membrane of 0.9 m pore diameter measured by the laser interferometric system.

More efficient diffusion of liposomes was noticed in regards to BioFill (Fig. 2B) and Nephrophane (Fig. 2C) membranes than to 0.9 m nucleopore (Fig. 2A). For the highest liposome concentration, 0.187 and 0.144 mg of lipids were transported across BioFill and

Fig. 3. The amount *N(*

Nephrophane, respectively.

system.

The next step of our mathematical analysis was calculations of the average membrane permeability coefficient for aqueous solution of liposomes. The greater amounts of liposomes transported through BioFill, Nephrophane in contrast to nucleopore membrane was associated with the different values of this coefficients for above membranes (1.21×10-10, 1.06×10-10, and 4.31×10-11 mol N-1s-1, respectively). We observed different kinetics of liposomes diffusion through both cellulose membranes. The amount of transported substances was positively correlated with the initial concentration of liposomes in dosedependent manner and increase was regular for 40 min.

Additionally, analysis of the membrane permeability coefficient for different liposomes concentrations shown the lower value of this coefficient for 2 mg/ml than for 1 and 0.5 mg/ml.

The same conclusions are based on analysis of diffusion coefficient D values for liposomes solutions measured both after time t and in the given time interval *t*. Analysis of diffusion coefficient for liposomes solution in the time interval *t* (Fig. 6) was more precise than after time t (Fig. 5). Therefore, we could noted that throw of liposomes from initial concentration 2 mg/ml took place in the first ~10 minutes of diffusion (Fig. 6)

Fig. 5. The diffusion coefficient in water solution of liposomes measured in time *t* by the laser interferometric system.

Laser Interferometric Determination of Liposomes Diffusion Through Artificial Membranes 295

In the first phase of diffusion across nucleopore membrane of 0.9 m pore diameter throw of liposomes was observed and then diffusion started to be constant (Fig. 7A and Fig. 2A). This effect might be associated with different polarizations of the liposome concentrations on both sides of the nucleopore membrane. The difference of substance concentrations influenced on physical properties of diffusion process, and amount of transported particles. The lower concentration (1 mg/ml) might promote constant transport of liposomes through the membrane. At the greater difference in liposomes concentration (2 mg/ml) on the two membrane sides, reciprocal inhibition of liposome diffusion was observed. The bacterial cellulose membrane transfered greater amounts of liposomal vesicles in a gradual manner and no inhibition of diffusion was detected (Fig. 7B and Fig. 2BC). In conclusion, in the first step of diffusion process cellulose membranes were saturated by liposomes and thus releasing of substance to aqueous solution in upper side of cuvette was observed. Both

In conclusion, the determination of dependence between the concentration of the liposome vesicles and membrane diffusion efficiency by laser interferometry method might be important from a clinical point of view. The effect of liposome concentration polarization on diffusion process through a wound dressing and artificial skin materials might be useful in the optimization of therapy using liposome-encapsulated drugs. The wound dressing based on cellulose membrane might promote more effective transport of liposomes as antibiotics carriers in comparison to standard nucleopore membranes. Additionally, laser interferometry is a very precise and quick method for determining of liposomes diffusion.

This work was supported by a grant from the Jan Kochanowski University, Kielce, Poland

Abeylath, S.C. & Turos, E. (2008). Drug delivery approaches to overcome bacterial resistance

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commercial membranes exhibited efficient diffusion and no filtration effect.

**5. Acknowledgment** 

**6. References** 

(062/R/11 for M. Arabski).

1744-7593

0012-6667

1532-8708

2009), pp.481-487, ISSN 0144-8617

2007), pp.268-75, ISSN 0376-7388

Fig. 6. The diffusion coefficient in water solution of liposomes measured in the given time interval *t* by the laser interferometric system.

## **4. Conclusions**

Liposome diffusion through the membrane separating solutions of lipid concentration leaded to build-up of the concentration boundary layers (CBLs) on both membrane sides measured by laser interferometry. The CBL dependents on the distance from the membrane, the physical properties of the membrane, and the type of solutes. These factors leaded to concentration polarization on both sides of the membrane (Grzegorczyn & Ślęzak, 2007).

In this study the measurements of the concentration profiles and liposome fluxes were done using laser interferometry. The analysis of liposome diffusion was performed for 40 min and in the given time interval *t*. Analysis of diffusion of liposomes through nucleopore and cellulose membranes showed different kinetics of diffusion process which determined amount of transported substance (Fig.7).

Fig. 7. Schematic presentation of liposomes diffusion through nucleopore (A) and cellulose (B) membranes proposed by the authors. Grey arrow denotes direction of diffusion. Grey curves denote layout of substance concentration.

In the first phase of diffusion across nucleopore membrane of 0.9 m pore diameter throw of liposomes was observed and then diffusion started to be constant (Fig. 7A and Fig. 2A). This effect might be associated with different polarizations of the liposome concentrations on both sides of the nucleopore membrane. The difference of substance concentrations influenced on physical properties of diffusion process, and amount of transported particles. The lower concentration (1 mg/ml) might promote constant transport of liposomes through the membrane. At the greater difference in liposomes concentration (2 mg/ml) on the two membrane sides, reciprocal inhibition of liposome diffusion was observed. The bacterial cellulose membrane transfered greater amounts of liposomal vesicles in a gradual manner and no inhibition of diffusion was detected (Fig. 7B and Fig. 2BC). In conclusion, in the first step of diffusion process cellulose membranes were saturated by liposomes and thus releasing of substance to aqueous solution in upper side of cuvette was observed. Both commercial membranes exhibited efficient diffusion and no filtration effect.

In conclusion, the determination of dependence between the concentration of the liposome vesicles and membrane diffusion efficiency by laser interferometry method might be important from a clinical point of view. The effect of liposome concentration polarization on diffusion process through a wound dressing and artificial skin materials might be useful in the optimization of therapy using liposome-encapsulated drugs. The wound dressing based on cellulose membrane might promote more effective transport of liposomes as antibiotics carriers in comparison to standard nucleopore membranes. Additionally, laser interferometry is a very precise and quick method for determining of liposomes diffusion.

## **5. Acknowledgment**

This work was supported by a grant from the Jan Kochanowski University, Kielce, Poland (062/R/11 for M. Arabski).

## **6. References**

294 Interferometry – Research and Applications in Science and Technology

Fig. 6. The diffusion coefficient in water solution of liposomes measured in the given time

Liposome diffusion through the membrane separating solutions of lipid concentration leaded to build-up of the concentration boundary layers (CBLs) on both membrane sides measured by laser interferometry. The CBL dependents on the distance from the membrane, the physical properties of the membrane, and the type of solutes. These factors leaded to concentration polarization on both sides of the membrane (Grzegorczyn & Ślęzak, 2007).

In this study the measurements of the concentration profiles and liposome fluxes were done using laser interferometry. The analysis of liposome diffusion was performed for 40 min and in the given time interval *t*. Analysis of diffusion of liposomes through nucleopore and cellulose membranes showed different kinetics of diffusion process which determined

Fig. 7. Schematic presentation of liposomes diffusion through nucleopore (A) and cellulose (B) membranes proposed by the authors. Grey arrow denotes direction of diffusion. Grey

interval *t* by the laser interferometric system.

amount of transported substance (Fig.7).

curves denote layout of substance concentration.

**4. Conclusions**


Laser Interferometric Determination of Liposomes Diffusion Through Artificial Membranes 297

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**15** 

*México* 

**Experimental π Phase-Shifts Observed in the** 

Through the use of a great variety of interferometric techniques, simultaneous phase shifting interferometry can be implemented to study dynamic events. Most of these techniques use expensive components; therefore, it is convenient to develop simultaneous

Recent studies show that a grating can be used as a beam splitter to attain several interference patterns around each diffraction order. Since each pattern has to show a different phase-shift, a suitable shifting technique must be employed, and phase gratings are

As it is very well known, the Fourier coefficients of only-phase gratings are integer order Bessel functions of the first kind. The values of these real-valued functions oscillate around zero, so they can adopt negative values, thereby introducing phase shifts of π at certain diffraction orders. Because of this, an almost trivial fact seems to have been overlooked in

In this chapter such phase shifts are presented in the description of interference patterns obtained with grating interferometers. These patterns are obtained by means of a two beam interferometer using a Mach Zehnder arrangement. The beams interfere at the object plane of a 4-*f* system with a sinusoidal grating/grid on the Fourier plane. It is shown the corresponding experimental observations of the fringe modulation, as well as the corresponding phase measurements, are all in agreement with the proposed description. A simultaneous phase shifting interferometer is finally proposed taking into account these

attractive to perform the former task due to their higher diffraction efficiencies.

phase shifting techniques that employ more accessible elements.

properties after proper incorporation of polarization modulation.

the literature regarding its practical implications.

**1. Introduction** 

**Fourier Spectra of Phase Gratings and** 

**Applications in Simultaneous PSI** 

and María-Graciela Hernández-Orduña3 *1Centro de Investigaciones en Óptica A.C. León, Gto., 2Benemérita Universidad Autónoma de Puebla, 3Instituto Tecnológico Superior de Misanta, Ver.* 

Noel-Ivan Toto-Arellano1, Gustavo Rodríguez-Zurita2, Amalia Martínez-García1, David-Ignacio Serrano-García1


## **Experimental π Phase-Shifts Observed in the Fourier Spectra of Phase Gratings and Applications in Simultaneous PSI**

Noel-Ivan Toto-Arellano1, Gustavo Rodríguez-Zurita2, Amalia Martínez-García1, David-Ignacio Serrano-García1 and María-Graciela Hernández-Orduña3 *1Centro de Investigaciones en Óptica A.C. León, Gto., 2Benemérita Universidad Autónoma de Puebla, 3Instituto Tecnológico Superior de Misanta, Ver. México* 

## **1. Introduction**

298 Interferometry – Research and Applications in Science and Technology

Sharma, A. & Sharma, U.S. (1997). Liposomes in drug delivery: progress and limitations. *Int* 

Swenson, C.E., Popescu, M.C. & Ginsberg, R.S. (1988). Preparation and use of liposomes in

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31, ISSN 1549-7828

,pp.341-346, ISSN 0376-7388

Through the use of a great variety of interferometric techniques, simultaneous phase shifting interferometry can be implemented to study dynamic events. Most of these techniques use expensive components; therefore, it is convenient to develop simultaneous phase shifting techniques that employ more accessible elements.

Recent studies show that a grating can be used as a beam splitter to attain several interference patterns around each diffraction order. Since each pattern has to show a different phase-shift, a suitable shifting technique must be employed, and phase gratings are attractive to perform the former task due to their higher diffraction efficiencies.

As it is very well known, the Fourier coefficients of only-phase gratings are integer order Bessel functions of the first kind. The values of these real-valued functions oscillate around zero, so they can adopt negative values, thereby introducing phase shifts of π at certain diffraction orders. Because of this, an almost trivial fact seems to have been overlooked in the literature regarding its practical implications.

In this chapter such phase shifts are presented in the description of interference patterns obtained with grating interferometers. These patterns are obtained by means of a two beam interferometer using a Mach Zehnder arrangement. The beams interfere at the object plane of a 4-*f* system with a sinusoidal grating/grid on the Fourier plane. It is shown the corresponding experimental observations of the fringe modulation, as well as the corresponding phase measurements, are all in agreement with the proposed description. A simultaneous phase shifting interferometer is finally proposed taking into account these properties after proper incorporation of polarization modulation.

Experimental π Phase-Shifts Observed

grating *G*

with

with 

the " **3.1 Phase grids** 

rulings of one grating along the "

configuration.

 

**3. Sinusoidal phase gratings** 

4-*f* system. Its complex amplitude can be expressed as

in the Fourier Spectra of Phase Gratings and Applications in Simultaneous PSI 301

Fig. 1. Experimental setup for the Two Beam Phase Grating Interferometer using a phase

),( of period *d* placed in a 4-*f* system at the end of a Mach Zehnder

For simplicity, we will consider a sinusoidal phase grating centered on the Fourier plane of a

 <sup>0</sup> <sup>0</sup> 2sin2 <sup>2</sup> , <sup>2</sup> *qFi q*

*yqFxAJyxG*

A sinusoidal phase grid can be generated through the multiplication of two sinusoidal phase gratings whose respective grating vectors are forming a 90-degree angle. Taking the

*gq* , <sup>2</sup> , <sup>~</sup> <sup>0</sup>

*q*

" direction, the resulting centered phase grid can be written as

Bessel functions. Such spectrum can be detected at the image plane of the 4-*f* system.

*Ag* 2 being the grating's phase amplitude and *<sup>q</sup> J* the Bessel function of the first kind

 *yx* ),( as the two-dimensional Dirac delta function. Thus, the Fourier spectrum of a sinusoidal phase grating comprises point-like diffraction orders of amplitude weighted by

(2)

" direction and the rulings of the second grating along

(1)

*gq*

*FAi eG eAJ <sup>g</sup>*

of integer order *q*. The Fourier transform of Eq. 1 is then given by

## **2. Basic considerations**

Phase gratings have been employed as an optical element with more efficiency than absorption gratings to perform a variety of tasks. Among them are beam splitting for interferometry, intensity measuring (Azzam, 1982) and optical shop testing (Ronchi, 1964; Cornejo, 1992). Their performance strongly depends upon their Fourier spectra, and for the case of sinusoidal phase grating has been extensively discussed (Barrekette and Freitag, 1963) It is well known that their Fourier coefficients are Bessel functions of the first kind of integer order *q*, *Jq* (Goodman, 1988) ; such functions are real valued, and their values oscillate around zero, so they can eventually introduce -phase shifts into a given grating Fourier spectrum. These shifts were of little relevance, if any, to applications where power spectrum is the main concern, as is often the case in spectroscopy (Kneubühl, 1969) , but recently, it has been pointed out that a grating interferometer with two windows on the object plane performs a common path interferometer (Arrizón and Sánchez, 2004). Several advantages have been shown about this interferometers, i.e. their mechanical stability (Meneses et al., 2006). Moreover, in conjunction with a suitable modulation of polarization, single-shot phase-shifting interferometric systems can be implemented with phase gratings (Rodriguez et al., 2008). Twobeam phase-grating interferometers (TBPGI) are based on the interference between neighboring diffraction orders. Thus, the fringe modulation of each interference pattern can be affected when -phase shifts are present (Rodriguez, 2008; Thomas, 1976). Furthermore, these phase shifts have to be taken into account for the overall performance of the system, and their practical advantages can even influence its design. This chapter is first aimed at the phase shifts between diffraction orders that have been observed in the Fourier spectra of a phase grating. An example of phase sinusoidal grating is calculated with a standard FFT routine that is useful for the interpretation of later experimental observations.

Experimental observations in agreement with the previous discussions, are shown by using commercial phase gratings, and finally, as an application of the above, a system able to obtain n-interferograms captured in one shot.

## **2.1 Experimental setup**

The optical system proposed is shown in Fig. 1. Basically, it consists of a combination of a quarter-wave plate *Q* and a linear polarizing filter *P*0 that generates linearly polarized light oriented at a 45 angle entering the Mach Zehnder configuration from a YVO3 laser operating at 532 *nm* (see Fig. 1). This configuration generates two symmetrically displaced beams by moving mirrors *M* and *M'*, enabling one to change the spacing <sup>0</sup> *x* between beam centers. Two retardation plates (Q*L* and Q*R*) with mutually orthogonal fast axes are placed in front of the two beams (A, B) to generate left and right nearly-circular polarized light (Novak et al., 2005; Rodrigurez 2008). A phase-grating *fvfuG* )/,/( is placed on the frequency plane (*u*,*v*) of the 4-*f* Fourier optical system that is coupled to the Mach Zehnder configuration, with *f* being the focal length of each transforming lens. Then, / *fu* and / *fv* are the frequency coordinates scaled to the wavelength and the focal length. On plane ),( , the period of *G* is denoted by *d* , and its spatial frequency by /1 *d* . Two neighboring diffraction orders have a distance of <sup>0</sup> / *dfF* at the image plane for a grating; then, *Fu* <sup>0</sup> and *F*0 can be used as a frequency. In the following sections, phase shifts at the image plane of this system due to the grating are discussed.

Fig. 1. Experimental setup for the Two Beam Phase Grating Interferometer using a phase grating *G* ),( of period *d* placed in a 4-*f* system at the end of a Mach Zehnder configuration.

## **3. Sinusoidal phase gratings**

300 Interferometry – Research and Applications in Science and Technology

Phase gratings have been employed as an optical element with more efficiency than absorption gratings to perform a variety of tasks. Among them are beam splitting for interferometry, intensity measuring (Azzam, 1982) and optical shop testing (Ronchi, 1964; Cornejo, 1992). Their performance strongly depends upon their Fourier spectra, and for the case of sinusoidal phase grating has been extensively discussed (Barrekette and Freitag, 1963) It is well known that their Fourier coefficients are Bessel functions of the first kind of integer order *q*, *Jq* (Goodman, 1988) ; such functions are real valued, and their values oscillate around zero, so they can eventually introduce -phase shifts into a given grating Fourier spectrum. These shifts were of little relevance, if any, to applications where power spectrum is the main concern, as is often the case in spectroscopy (Kneubühl, 1969) , but recently, it has been pointed out that a grating interferometer with two windows on the object plane performs a common path interferometer (Arrizón and Sánchez, 2004). Several advantages have been shown about this interferometers, i.e. their mechanical stability (Meneses et al., 2006). Moreover, in conjunction with a suitable modulation of polarization, single-shot phase-shifting interferometric systems can be implemented with phase gratings (Rodriguez et al., 2008). Twobeam phase-grating interferometers (TBPGI) are based on the interference between neighboring diffraction orders. Thus, the fringe modulation of each interference pattern can be affected when -phase shifts are present (Rodriguez, 2008; Thomas, 1976). Furthermore, these phase shifts have to be taken into account for the overall performance of the system, and their practical advantages can even influence its design. This chapter is first aimed at the phase shifts between diffraction orders that have been observed in the Fourier spectra of a phase grating. An example of phase sinusoidal grating is calculated with a standard FFT routine that

Experimental observations in agreement with the previous discussions, are shown by using commercial phase gratings, and finally, as an application of the above, a system able to

The optical system proposed is shown in Fig. 1. Basically, it consists of a combination of a quarter-wave plate *Q* and a linear polarizing filter *P*0 that generates linearly polarized light oriented at a 45 angle entering the Mach Zehnder configuration from a YVO3 laser operating at 532 *nm* (see Fig. 1). This configuration generates two symmetrically displaced beams by moving mirrors *M* and *M'*, enabling one to change the spacing <sup>0</sup> *x* between beam centers. Two retardation plates (Q*L* and Q*R*) with mutually orthogonal fast axes are placed in front of the two beams (A, B) to generate left and right nearly-circular polarized light

frequency plane (*u*,*v*) of the 4-*f* Fourier optical system that is coupled to the Mach Zehnder

*fv* are the frequency coordinates scaled to the wavelength and the focal length. On

and *F*0 can be used as a frequency. In the following sections, phase shifts

configuration, with *f* being the focal length of each transforming lens. Then, /

),( , the period of *G* is denoted by *d* , and its spatial frequency by

 / *dfF* at the image plane for a grating;

*fvfuG* )/,/( is placed on the

*fu* and

/1 *d* . Two

is useful for the interpretation of later experimental observations.

(Novak et al., 2005; Rodrigurez 2008). A phase-grating

neighboring diffraction orders have a distance of <sup>0</sup>

at the image plane of this system due to the grating are discussed.

obtain n-interferograms captured in one shot.

**2.1 Experimental setup** 

 / 

> 

plane

then, *Fu* <sup>0</sup>

**2. Basic considerations** 

For simplicity, we will consider a sinusoidal phase grating centered on the Fourier plane of a 4-*f* system. Its complex amplitude can be expressed as

$$G(\mu, \nu) = e^{i2\pi \cdot A\_{\frac{\mu}{\beta}} \sin \left[2\pi F\_0 \mu\right]} = \sum\_{q = -\infty}^{\infty} J\_q \left(2\pi A\_{\frac{\mu}{\beta}}\right) e^{i2\pi \cdot qF\_0 \mu} \tag{1}$$

with *Ag* 2 being the grating's phase amplitude and *<sup>q</sup> J* the Bessel function of the first kind of integer order *q*. The Fourier transform of Eq. 1 is then given by

$$\mathfrak{G}(\mathbf{x}, y) = \sum\_{q = -\infty}^{\infty} I\_q \left( 2\pi \mathbf{A}\_{\mathcal{g}} \right) \delta(\mathbf{x} - qF\_0, y) \tag{2}$$

with *yx* ),( as the two-dimensional Dirac delta function. Thus, the Fourier spectrum of a sinusoidal phase grating comprises point-like diffraction orders of amplitude weighted by Bessel functions. Such spectrum can be detected at the image plane of the 4-*f* system.

## **3.1 Phase grids**

A sinusoidal phase grid can be generated through the multiplication of two sinusoidal phase gratings whose respective grating vectors are forming a 90-degree angle. Taking the rulings of one grating along the " " direction and the rulings of the second grating along the "" direction, the resulting centered phase grid can be written as

Experimental π Phase-Shifts Observed

*q*

is expected; otherwise, it would be positive.

simplifies to

Fig. 1.

0 *J*<sup>4</sup> is shown.

in the Fourier Spectra of Phase Gratings and Applications in Simultaneous PSI 303

the beams' positions with the diffraction order's positions ( *F x* 0 0 ), the previous equation

<sup>1</sup> ´( , ) (2 ) (2 ) , <sup>2</sup>

Thus, an interference pattern between fields associated to each window must appear within each replicated window. The fringe modulation *mq* of each pattern would be of the form

> 2 2 <sup>1</sup>

Further phase-shifting techniques must be used in order to introduce proper additional shifts into each pattern. One possible technique to be used is through an appropriate displacement of the grating (Meneses 2006), and an alternative approach is the use of the polarization-induced phase shifting (Rodriguez 2008a) method used and the one depicted in

According to Eq. 8, *mq* and the contrast of each interference pattern depend on the signs of

relationship is shown on Table 1 for some cases of *q*. For *q* = 0*,-*1,-2,-3,-4 , a negative contrast

Interference fringe modulation is positive for one half of the diffraction orders if the grating's Fourier coefficients are all positive for *q* > 0, whereas the other half would show alternating fringe modulation due to the odd parity of *q*12*J* . These results can also be depicted as in Fig. 2 for a hypothetical case, where only the signs of the amplitudes of the diffraction orders from a given grating are separately shown displaced from the origin due to the respective displacement of windows A and B (first two plots from above). The case of

 *qq qq <sup>q</sup> JJ JJ*

*O x y J A J Ae wx x q y* 

 

*q gq g*

**4.1** *π***–shifts of the Fourier spectra of sinusoidal phase gratings** 

*qJ* and *q*<sup>1</sup> *J* ; thus, the signs of *qJ* are also relevant. For the case of

Table 1. Bessel coeficients. Sign relations of superimposed orders. Case

0 <sup>1</sup> ( ,) <sup>2</sup> 1 0

2 1

 

*i xx q y*

(7)

*m* (8)

(,) 0 *x y* .

*yx* 0),( , this

$$G\_2\left(\mu,\nu\right) = e^{i2\pi A\_\xi \sin\left[2\pi \cdot X\_0 \mu\right]} e^{i2\pi A\_\xi \sin\left[2\pi \cdot Y\_0 \nu\right]} = \sum\_{q=-\infty}^{\infty} J\_q\left(2\pi A\_\xi\right) e^{i2\pi \cdot q \cdot \mathbb{F}\_0 \mu} \sum\_{r=-\infty}^{\infty} J\_r\left(2\pi A\_\xi\right) e^{i2\pi \cdot r \cdot \mathbb{F}\_0 \nu} \tag{3}$$

where the frequencies along each axis direction are taken as *FYX* <sup>000</sup> . The Fourier transform of the phase grid becomes

$$\mathfrak{G}\_2(\mathbf{x}, y) = \sum\_{q = -\infty}^{q = \infty} \sum\_{r = -\infty}^{r = \infty} \mathfrak{J}\_q \left( 2\pi \mathbf{A}\_{\mathcal{g}} \right) \mathfrak{J}\_r \left( 2\pi \mathbf{A}\_{\mathcal{g}} \right) \delta(\mathbf{x} - q \mathbf{F}\_0, y - r \mathbf{F}\_0) \tag{4}$$

which also consists of point-like diffraction orders distributed on the image plane on the nodes of a lattice with a period given by value *F*<sup>0</sup> .

## **4. Two-windows phase-grating interferometry: interference-pattern contrasts and modulation**

Phase grating interferometry is based on a phase grid placed as the pupil of a 4-*f* Fourier optical system. The use of two windows at the object plane, in conjunction with phase grating interferometry, allows interference between the optical fields associated to each window with higher diffraction efficiency (Arrizón, 2004 and Meneses, 2006); such system performs as a common path interferometer (Fig. 1). A convenient pair of windows for a grating interferometer implies an amplitude transmittance given by

$$O(\mathbf{x}, y) = w(\mathbf{x} + \frac{\mathbf{x}\_0}{2}, y) + w'(\mathbf{x} - \frac{\mathbf{x}\_0}{2}, y) \tag{5}$$

where x0 is the separation from center to center between two beams. For simplicity, considering a rectangular aperture *w x* ,*y* , written as *w x*(,) / / *y rect x a rect y b* and *w x* ´( , ) ( , )exp , *y w x y i x y* , with *x*,*y* as the object phase function. As shown in Fig. 2, *a* and *b* represent the side lengths of each beam (A and B). Placing a grating of spatial period / <sup>0</sup> *d fF* at the Fourier plane, with a corresponding transmittance given by *G*, . The image formed by the system consists basically of replications of each window at distances *F*<sup>0</sup> . This image is defined by *O x* ´ , *y* ; that is, the convolution of *O x* ,*y* with the Fourier transform of the phase grating, *Gxy* , represented by

$$\begin{split} O^\circ(\mathbf{x},\mathbf{y}) &= O(\mathbf{x},\mathbf{y}) \ast \hat{\mathbf{G}}(\mathbf{x},\mathbf{y}) \\ &= w\Big(\mathbf{x} + \frac{\mathbf{x}\_0}{2}, \mathbf{y}\Big) \ast \sum\_{q=-\infty}^{\infty} I\_q(2\pi A\_{\mathcal{S}}) \delta\left(\mathbf{x} - qF\_0, \mathbf{y}\right) \\ &+ \left[ w\left(\mathbf{x} - \frac{\mathbf{x}\_0}{2}, \mathbf{y}\right) \varepsilon^{i\phi\left(\mathbf{x} - \frac{\mathbf{x}\_0}{2}, \mathbf{y}\right)} \right] \ast \sum\_{q=-\infty}^{\infty} I\_q(2\pi A\_{\mathcal{S}}) \delta\left(\mathbf{x} - qF\_0, \mathbf{y}\right). \end{split} \tag{6}$$

where (\*) denotes the convolution. By adding the terms of Eq. 6, taking *q* and *q*-1 (both located within the same replicated window <sup>0</sup> <sup>0</sup> , <sup>2</sup> *x w x qF y* ), and for the case of matching

 0 0 <sup>0</sup> <sup>0</sup> 2 sin 2 2 sin 2 <sup>2</sup> <sup>2</sup> <sup>2</sup> , 2 <sup>2</sup> *g g iA X iA Y i qF i rF*

where the frequencies along each axis direction are taken as *FYX* <sup>000</sup> . The Fourier

which also consists of point-like diffraction orders distributed on the image plane on the

**4. Two-windows phase-grating interferometry: interference-pattern contrasts** 

Phase grating interferometry is based on a phase grid placed as the pupil of a 4-*f* Fourier optical system. The use of two windows at the object plane, in conjunction with phase grating interferometry, allows interference between the optical fields associated to each window with higher diffraction efficiency (Arrizón, 2004 and Meneses, 2006); such system performs as a common path interferometer (Fig. 1). A convenient pair of windows for a

0 0 ( , ) ( , ) ´( , ) 2 2

where x0 is the separation from center to center between two beams. For simplicity, considering a rectangular aperture *w x* ,*y* , written as *w x*(,) / / *y rect x a rect y b* and

2, *a* and *b* represent the side lengths of each beam (A and B). Placing a grating of spatial

 . The image formed by the system consists basically of replications of each window at distances *F*<sup>0</sup> . This image is defined by *O x* ´ , *y* ; that is, the convolution of *O x* ,*y* with

, (2 ) , <sup>2</sup>

*q g*

where (\*) denotes the convolution. By adding the terms of Eq. 6, taking *q* and *q*-1 (both

 

*grgq yxG rFyqFxAJAJ* <sup>2</sup> <sup>00</sup> , <sup>22</sup> , <sup>~</sup> 

*G e e J Ae J Ae*

 

grating interferometer implies an amplitude transmittance given by

( ,) <sup>0</sup> <sup>2</sup>

*<sup>x</sup> ix y*

*<sup>x</sup> w x y J A x qF y*

*q*

*q*

*q*

*r*

*r*

transform of the phase grid becomes

 

**and modulation** 

*w x* ´( , ) ( , )exp , *y w x y i x*

period / <sup>0</sup> *d fF* 

*G*, *y* , with

´( , ) ( , ) ,

located within the same replicated window <sup>0</sup>

*O xy Oxy G xy*

the Fourier transform of the phase grating, *Gxy* , represented by

0

*x*

 

nodes of a lattice with a period given by value *F*<sup>0</sup> .

*qg rg*

, (4)

 

 

*q r*

*x x Oxy wx y w x y* (5)

at the Fourier plane, with a corresponding transmittance given by

0

<sup>0</sup>

*q g*

 

, (2 ) , . <sup>2</sup>

<sup>0</sup> , <sup>2</sup> *x w x qF y* 

*q*

*wx ye J A x qF y*

0

), and for the case of matching

(6)

*x*,*y* as the object phase function. As shown in Fig.

(3)

the beams' positions with the diffraction order's positions ( *F x* 0 0 ), the previous equation simplifies to

$$O'(\mathbf{x}, y) = \sum\_{q=-\infty}^{\infty} \left[ J\_q(2\pi A\_{\mathcal{g}}) + J\_{q-1}(2\pi A\_{\mathcal{g}}) e^{i\phi(\mathbf{x} - \mathbf{x}\_0 \left[ q - \frac{1}{2} \right], y)} \right] w \left( \mathbf{x} - \mathbf{x}\_0 \left[ q - \frac{1}{2} \right], y \right) \tag{7}$$

Thus, an interference pattern between fields associated to each window must appear within each replicated window. The fringe modulation *mq* of each pattern would be of the form

$$m\_q = \frac{2J\_q J\_{q-1}}{J\_q^2 + J\_{q-1}^2} \tag{8}$$

Further phase-shifting techniques must be used in order to introduce proper additional shifts into each pattern. One possible technique to be used is through an appropriate displacement of the grating (Meneses 2006), and an alternative approach is the use of the polarization-induced phase shifting (Rodriguez 2008a) method used and the one depicted in Fig. 1.

## **4.1** *π***–shifts of the Fourier spectra of sinusoidal phase gratings**

According to Eq. 8, *mq* and the contrast of each interference pattern depend on the signs of *qJ* and *q*<sup>1</sup> *J* ; thus, the signs of *qJ* are also relevant. For the case of *yx* 0),( , this relationship is shown on Table 1 for some cases of *q*. For *q* = 0*,-*1,-2,-3,-4 , a negative contrast is expected; otherwise, it would be positive.


Table 1. Bessel coeficients. Sign relations of superimposed orders. Case (,) 0 *x y* .

Interference fringe modulation is positive for one half of the diffraction orders if the grating's Fourier coefficients are all positive for *q* > 0, whereas the other half would show alternating fringe modulation due to the odd parity of *q*12*J* . These results can also be depicted as in Fig. 2 for a hypothetical case, where only the signs of the amplitudes of the diffraction orders from a given grating are separately shown displaced from the origin due to the respective displacement of windows A and B (first two plots from above). The case of 0 *J*<sup>4</sup> is shown.

Experimental π Phase-Shifts Observed

phase grating (right).

region within the 9th lobule of *J*0.

A rectangular phase grid *G*

**4.3** 

written as:

in the Fourier Spectra of Phase Gratings and Applications in Simultaneous PSI 305

Fig. 3. Bessel domain for the ninth lobule of *J*0 (left). Corresponding Fourier spectrum of the

its Fourier components in phase, while the other half shows phase shifts of in an alternating fashion. In this region, *J*0 is positive and in phase with the Fourier components of the first half above mentioned. For the second case, *J*0 is negative. Taking a value of *A*g such that the Bessel functions with *q* = 1…6 take only positive values, the Fourier spectrum of the resulting grating turns out to be similar to the first case, with *J*0 being negative as the only different phase shift. Within the 3th lobule, it is possible to pick up negative values for *q* = 1, 2, 3, with the resulting Fourier spectrum showing π-shifts accordingly. In Fig.4 , a case of positive values for *q* = 1,2,3 and negative for *q* = 4,5 is shown. This situation corresponds to a

frequency. Fig. 5(a) depicts the signs of the diffraction orders of a grid made up from two crossed gratings having spectra as those of the one in Fig. 2. Positive signs are denoted by hollow circles, whereas negative signs are marked with crosses. The dashed lines form regions enclosing diffraction orders of index pairs 0,*n* or *m*,0. Then, order 0,0 is found at the intersection of these regions. Two possible window configurations can be considered for a TBPGI; these configurations are shown in Fig. 5(b), and they are denoted by **W**1 and **W**2. The respective displacements of the diffraction patterns are indicated with displaced dashed lines too. For the case of phase grids with windows in configuration **W2**, the image can be

, (2 ) (2 ) , 2 2

*w x y J AJ A x qF y rF*

*w x y J AJ A x qF y rF*

*q gr g*

 

 

´ , (2 ) (2 ) , . 2 2

*q gr g*

can be generated with two phase gratings of equal spatial

0 0

(9)

0 0

**shifts of the Fourier spectra of sinusoidal phase grids** 

0 0

*x x*

0 0

*x x*

2

´( , ) ( , ) ,

*O xy Oxy G xy*

 

*q*

*q*

,

Fig. 2. Amplitude signs of diffraction orders (hypothetical grating) on the image plane of a TBPGI resulting from a window displacement of 2/ <sup>0</sup> *x* . Upper left: window configuration.

For simplicity, the replicated windows are not plotted. The third row on the graph exhibits the superposition of the two previous spectra. Fringe modulation changes must appear in both halves of the image plane. In this case, the expected signs of each fringe modulation would be +,-,-,-,+,+,+,- from left to right. The Fig. 2 then shows that odd diffraction orders have odd-order Bessel function parity, and conversely for even diffraction orders and even parity. Figure 3 shows a case of positive *Jq* values for *q* = 0,1,4,5, and negative for *q* = 2, 3, 6 as an example (left). This situation corresponds to the value of *Ag* 5.262 , which belongs to a range in the Bessel domain within the 9th lobule of *J*0. A phase grating of the type represented by Eq. 1 (with the same amplitude 2*Ag* as before) has a Fourier spectrum whose components show shifts accordingly (see the positive orders on the right of Fig. 3). -shifts such as these can be detected as changes in the signs of fringe modulations in the experimental interferometers discussed later.

## **4.2 Some examples of calculated Fourier spectra: sinusoidal phase gratings.**

In order to show some cases of phase-shifts in grating spectra, several fast Fourier transforms have been calculated for gratings of different phase amplitudes *A*g. We use the lobules of *J*0 as a reference to indicate the Bessel region within which the amplitude *A*g of the corresponding phase grating is to be found. Only seven Bessel functions are shown in Fig. 4. In the first case, within the first lobule of *J*0, the associated spectrum results with one half of

Fig. 2. Amplitude signs of diffraction orders (hypothetical grating) on the image plane of a TBPGI resulting from a window displacement of 2/ <sup>0</sup> *x* . Upper left: window configuration.

For simplicity, the replicated windows are not plotted. The third row on the graph exhibits the superposition of the two previous spectra. Fringe modulation changes must appear in both halves of the image plane. In this case, the expected signs of each fringe modulation would be +,-,-,-,+,+,+,- from left to right. The Fig. 2 then shows that odd diffraction orders have odd-order Bessel function parity, and conversely for even diffraction orders and even parity. Figure 3 shows a case of positive *Jq* values for *q* = 0,1,4,5, and negative for *q* = 2, 3, 6 as

range in the Bessel domain within the 9th lobule of *J*0. A phase grating of the type

whose components show shifts accordingly (see the positive orders on the right of Fig. 3). -shifts such as these can be detected as changes in the signs of fringe modulations in the

In order to show some cases of phase-shifts in grating spectra, several fast Fourier transforms have been calculated for gratings of different phase amplitudes *A*g. We use the lobules of *J*0 as a reference to indicate the Bessel region within which the amplitude *A*g of the corresponding phase grating is to be found. Only seven Bessel functions are shown in Fig. 4. In the first case, within the first lobule of *J*0, the associated spectrum results with one half of

**4.2 Some examples of calculated Fourier spectra: sinusoidal phase gratings.** 

*Ag* 5.262 , which belongs to a

*Ag* as before) has a Fourier spectrum

an example (left). This situation corresponds to the value of

represented by Eq. 1 (with the same amplitude 2

experimental interferometers discussed later.

Fig. 3. Bessel domain for the ninth lobule of *J*0 (left). Corresponding Fourier spectrum of the phase grating (right).

its Fourier components in phase, while the other half shows phase shifts of in an alternating fashion. In this region, *J*0 is positive and in phase with the Fourier components of the first half above mentioned. For the second case, *J*0 is negative. Taking a value of *A*g such that the Bessel functions with *q* = 1…6 take only positive values, the Fourier spectrum of the resulting grating turns out to be similar to the first case, with *J*0 being negative as the only different phase shift. Within the 3th lobule, it is possible to pick up negative values for *q* = 1, 2, 3, with the resulting Fourier spectrum showing π-shifts accordingly. In Fig.4 , a case of positive values for *q* = 1,2,3 and negative for *q* = 4,5 is shown. This situation corresponds to a region within the 9th lobule of *J*0.

### **4.3 shifts of the Fourier spectra of sinusoidal phase grids**

A rectangular phase grid *G*, can be generated with two phase gratings of equal spatial frequency. Fig. 5(a) depicts the signs of the diffraction orders of a grid made up from two crossed gratings having spectra as those of the one in Fig. 2. Positive signs are denoted by hollow circles, whereas negative signs are marked with crosses. The dashed lines form regions enclosing diffraction orders of index pairs 0,*n* or *m*,0. Then, order 0,0 is found at the intersection of these regions. Two possible window configurations can be considered for a TBPGI; these configurations are shown in Fig. 5(b), and they are denoted by **W**1 and **W**2. The respective displacements of the diffraction patterns are indicated with displaced dashed lines too. For the case of phase grids with windows in configuration **W2**, the image can be written as:

$$\begin{split} O^\*(\mathbf{x}, \mathbf{y}) &= O(\mathbf{x}, \mathbf{y}) \* \tilde{\mathbf{G}}\_2(\mathbf{x}, \mathbf{y}) \\ &= \sum\_{q=-\alpha}^{\alpha} w \Big( \mathbf{x} + \frac{\mathbf{x}\_0}{2}, \mathbf{y} + \frac{\mathbf{x}\_0}{2} \Big) \* I\_q(2\pi A\_{\mathcal{S}}) I\_r(2\pi A\_{\mathcal{S}}) \delta \Big( \mathbf{x} - qF\_0, \mathbf{y} - rF\_0 \Big) \\ &+ \sum\_{q=-\alpha}^{\alpha} w \Big( \mathbf{x} - \frac{\mathbf{x}\_0}{2}, \mathbf{y} - \frac{\mathbf{x}\_0}{2} \Big) \* I\_q(2\pi A\_{\mathcal{S}}) I\_r(2\pi A\_{\mathcal{S}}) \delta \Big( \mathbf{x} - qF\_0, \mathbf{y} - rF\_0 \Big). \end{split} \tag{9}$$

Experimental π Phase-Shifts Observed

 

´( , )

in

1 <sup>0</sup> <sup>0</sup> *rxqx* is

 <sup>2</sup> <sup>1</sup> , <sup>2</sup>

  *O xy*

*q r*

where

 

 

and cross patterns is to be found in the text.

in the Fourier Spectra of Phase Gratings and Applications in Simultaneous PSI 307

1 1

As it is for gratings, fringe modulation of the interference pattern within a window centered

<sup>2</sup>

*rqrq rrqq*

*JJJJ*

2 <sup>11</sup> 2

Fig. 5. (a) -phase distribution of diffraction orders of grids. Dashed lines enclose diffraction orders of indexes 0,*n* or *m*,0. b) TBPGI order superposition: Configuration **W**1: interference

displacements of diffraction patterns are highlighted by dashed lines. An explanation of dot

pattern signs for windows displaced along the horizontal axis. Configuration **W**2: interference patterns signs for displaced windows along a line at 45°. Respective

*qr JJJJ*

11

 

 

*x*

*<sup>x</sup> xwyxO* . This result simplifies to

 

(2 ) (2 ) (2 ) (2 )

*J AJ A J AJ Ae*

*q gr g q gr g*

1 1 , . 2 2

*xw*

0 0

*wx x q y x r*

<sup>2</sup> , <sup>2</sup> ´ <sup>2</sup> , <sup>2</sup> ),( <sup>00</sup> *<sup>x</sup>*<sup>00</sup> *<sup>y</sup>*

*x y*

 

(11)

0 0 1 1 , 2 2

*i xx q yx r*

*m* (12)

Fig. 4. Fourier spectra of phase gratings: left column, Bessel region (first seven Bessel functions shown); right column, grating spectra.

Again, with <sup>00</sup> *xF*

$$\begin{split} O'(\mathbf{x}, \mathbf{y}) &= \sum\_{q = -\alpha}^{\infty} \sum\_{r = -\alpha}^{\alpha} I\_q(2\pi A\_{\mathcal{g}}) I\_r(2\pi A\_{\mathcal{g}}) \, w \Big( \mathbf{x} + \frac{\mathbf{x}\_0}{2} [1 - 2q]\_r \mathbf{y} + \frac{\mathbf{x}\_0}{2} [1 - 2r] \Big) + \\ &\sum\_{q = -\alpha}^{\infty} \sum\_{r = -\alpha}^{\infty} I\_{q-1}(2\pi A\_{\mathcal{g}}) I\_{r-1}(2\pi A\_{\mathcal{g}}) \, w \Big( \mathbf{x} - \frac{\mathbf{x}\_0}{2} [1 - 2q]\_r \mathbf{y} - \frac{\mathbf{x}\_0}{2} [1 - 2r] \Big) \end{split} \tag{10}$$

Fig. 4. Fourier spectra of phase gratings: left column, Bessel region (first seven Bessel

´( , ) (2 ) (2 ) 1 2 , 1 2 2 2

 

*x x O xy J A J A w x q y r*

 

1 1

*q gr g*

*q gr g*

0 0

 

0 0

(2 ) (2 ) ´ 1 2 , 1 2 , 2 2

*x x J AJ A wx q y <sup>r</sup>*

(10)

functions shown); right column, grating spectra.

 

*q r*

*q r*

Again, with <sup>00</sup> *xF*

where <sup>2</sup> , <sup>2</sup> ´ <sup>2</sup> , <sup>2</sup> ),( <sup>00</sup> *<sup>x</sup>*<sup>00</sup> *<sup>y</sup> x xw x y <sup>x</sup> xwyxO* . This result simplifies to

$$\begin{split} O^r(\mathbf{x}, y) &= \\ \sum\_{q = -\alpha}^{\infty} \sum\_{r = -\alpha}^{\infty} \Bigg[ I\_q(2\pi A\_{\mathcal{S}}) I\_r(2\pi A\_{\mathcal{S}}) + I\_{q-1}(2\pi A\_{\mathcal{S}}) I\_{r-1}(2\pi A\_{\mathcal{S}}) e^{i\boldsymbol{\theta} \cdot \left[ \left( \mathbf{x} - \mathbf{x}\_0 \left[ q - \frac{1}{2} \right] \right) \left( \mathbf{y} - \mathbf{x}\_0 \left[ r - \frac{1}{2} \right] \right) \right]} \\ & \times w \Big( \mathbf{x} - \mathbf{x}\_0 \left[ q - \frac{1}{2} \right], y - \mathbf{x}\_0 \left[ r - \frac{1}{2} \right] \Big). \end{split} \tag{11}$$

As it is for gratings, fringe modulation of the interference pattern within a window centered in <sup>2</sup> <sup>1</sup> , <sup>2</sup> 1 <sup>0</sup> <sup>0</sup> *rxqx* is

Fig. 5. (a) -phase distribution of diffraction orders of grids. Dashed lines enclose diffraction orders of indexes 0,*n* or *m*,0. b) TBPGI order superposition: Configuration **W**1: interference pattern signs for windows displaced along the horizontal axis. Configuration **W**2: interference patterns signs for displaced windows along a line at 45°. Respective displacements of diffraction patterns are highlighted by dashed lines. An explanation of dot and cross patterns is to be found in the text.

Experimental π Phase-Shifts Observed

resulting in similar conclusions.

  2 1*J* , <sup>2</sup> 2*J* , <sup>2</sup> 3*J* and

( , /2) 2 

where

in the Fourier Spectra of Phase Gratings and Applications in Simultaneous PSI 309

 

been simplified to *x*,*y*. The same fringe modulation *mq* results as in Eq. 8. Therefore, the discussion about fringe modulation given in previous sections is retained when introducing the modulation of polarization. Such polarization modulation can also be made for grids,

**5. Experimental testing of the phase-shifts in phase gratings and phase grids**  Figure 6 shows the superposition of Fourier amplitude spectra under window configuration *W1* for a phase grating with 110 lines/mm. The contrast of the corresponding experimental interference patterns can be interpreted as if its first four Fourier coefficients had phase relations as the ones sketched in Fig. 2. They were obtained with the system of Fig.1 before placing the retardation plates QR, QL and the polarizing filters on the image plane. The patterns show the relative phases of the diffraction orders discussed in previous sections. The phase-shifted steps of the experimental fringe patterns can be calculated by applying the algorithm proposed by Kreis, 1986. The resulting mean values are shown in Table 2. It

Table 2. Phase shifts measured from experimental patterns in Fig.6 according to the method

Due to that, the interference patterns are obtained from the interference between the replicas of each beam, centered around each diffraction order. Fig. 7(a) presents the replicas of window A with right circular polarization, and the replicas of window B with left circular polarization; each order is superposed depending on separation <sup>0</sup> *x* . Fig. 7(b) presents the interference pattern generated by the interference of windows AB, due to the cross circular polarization, a linear polarizer is placed to modulate the phase shift on each replicated

are the intensity measurements with the values of

on the coordinates of the centered point has

 , respectively. Note that

, so a good choice for the retarders is quarter-wave

2 4*J*

 

( ) 0, ( ) /2, ( ) , ( ) 3 /2

 

can be seen from the table that they depart by small amounts from or 0.

such that 12 3 4

 

Fig. 6. Experimental results with a phase-grating.

from Kreis, 1986.

interference pattern.

 and *A*( , /2) 1 

retarders, as it is well known. Dependence of

Each corresponding fringe modulation depends on the relative phases between *qJ* , *<sup>q</sup>*<sup>1</sup> *J* , *rJ* and *r*<sup>1</sup> *J* . These relationships are discussed with the same example of Fig. 5(a). Fig. 5(b) shows the relative phases (and thus, expected signs of *mqr*) of each diffraction order arising from configuration **W**1 or **W**2. If positive signs or negative ones coincide in a diffraction order, fringe modulation will be positive, and this is plotted with one symbol (cross or hollow dot). Only when a cross with a hollow dot appears, modulation is expected to be negative. "Vertical" bands whose sign is equal to that of *mqr* are thus expected in configuration **W**1. Regions with a sign that is equal to that of *mqr* can be seen in configuration **W**2.

## **4.4 Phase shifting interferometry with modulation of polarization**

Turning the attention to gratings, in order to introduce additional phase shifts in the interference pattern centered at ,)2/1( *yqx* <sup>0</sup> for 4-step phase-shifting (Schwieder et al., 1983), each of the windows is illuminated with different polarizations using retarding plates QR and QL (Fig. 1). This arrangement introduces Jones polarization vectors *JR* and *LJ* into the interference terms of Eq. 6 through appropriate modification of Eq. 5. After placing a linear polarizing filter with the transmission axis at an angle , its irradiance results as being proportional to

$$\left\|\vec{f}\_{T}\right\|^{2} = A(\xi, \alpha') \left\langle \left(I\_{q}^{\ 2} + I\_{q-1}{}^{2}\right) + 2J\_{q}J\_{q-1}\cos\left[\xi(\nu', \alpha') - \phi(\mathbf{x}, y)\right] \right\rangle \tag{13}$$

Where is the linear polarizer angle, ' the retardation of each plate, and ( , ') denotes the phase shifting term induced by modulation of polarization given by

$$\xi(\boldsymbol{\nu},\alpha^{\circ}) = \operatorname{ArcTan}\left[\frac{\sin[2\boldsymbol{\nu}]\cdot\sin[\alpha^{\circ}] + \sin^{2}[\boldsymbol{\nu}]\cdot\sin[2\alpha^{\circ}]}{\cos^{2}[\boldsymbol{\nu}] + \sin^{2}[\boldsymbol{\nu}]\cdot\cos[2\alpha^{\circ}] + \sin[2\boldsymbol{\nu}]\cdot\cos[\alpha^{\circ}]}\right],\tag{14}$$

and *A* )',( is defined as

$$A(\boldsymbol{\psi}, \alpha^{\prime}) = \mathbb{I} + \sin[2\boldsymbol{\psi}\,\boldsymbol{\zeta}\cos[\alpha^{\prime}]\,\mathrm{d}\,\mathrm{s}^{\prime}]\,\mathrm{d}\,\mathrm{s}^{\prime}\,\tag{15}$$

For phase-shifting interferometry with four patterns, four irradiances can be used, each one taken at a different angle. Denoting each pattern as

$$\left\|\vec{j}\_{i}\right\|^{2} = A(\xi, \alpha^{\prime}) \left\{ \left(I\_{q}^{\ 2} + I\_{q-1}{}^{2}\right) + 2I\_{q}I\_{q-1}\cos[\xi(\nu\_{i}, \alpha^{\prime}) - \phi(\mathbf{x}, y)]\right\} \tag{16}$$

with *i* = 1…4, the relative phase can be calculated as (J. Schwieder 1983)

$$\tan \phi = \frac{\left\| \mathbf{J}\_1 \right\|^2 - \left\| \mathbf{J}\_3 \right\|^2}{\left\| \mathbf{J}\_2 \right\|^2 - \left\| \mathbf{J}\_4 \right\|^2} \quad \text{{} \text{ } \tag{17}$$

Each corresponding fringe modulation depends on the relative phases between *qJ* , *<sup>q</sup>*<sup>1</sup> *J* , *rJ* and *r*<sup>1</sup> *J* . These relationships are discussed with the same example of Fig. 5(a). Fig. 5(b) shows the relative phases (and thus, expected signs of *mqr*) of each diffraction order arising from configuration **W**1 or **W**2. If positive signs or negative ones coincide in a diffraction order, fringe modulation will be positive, and this is plotted with one symbol (cross or hollow dot). Only when a cross with a hollow dot appears, modulation is expected to be negative. "Vertical" bands whose sign is equal to that of *mqr* are thus expected in configuration **W**1. Regions with a sign that is equal to that of *mqr* can be seen in configuration

Turning the attention to gratings, in order to introduce additional phase shifts in the interference pattern centered at ,)2/1( *yqx* <sup>0</sup> for 4-step phase-shifting (Schwieder et al., 1983), each of the windows is illuminated with different polarizations using retarding plates

the interference terms of Eq. 6 through appropriate modification of Eq. 5. After placing a linear polarizing filter with the transmission axis at an angle , its irradiance results as

> <sup>2</sup> 2 2 1 1 ( , ') 2 cos[ ( , ') ( , )] *T q q q <sup>q</sup> J A J J JJ*

*x y*

sin[2 ] sin[ '] sin [ ] sin[2 '] ( , ') cos [ ] sin [ ] cos[2 '] sin[2 ] cos[ ']

 

]'cos[]2sin[1)',(

For phase-shifting interferometry with four patterns, four irradiances can be used, each one

 <sup>2</sup> 2 2 1 1 ( , ') 2 cos[ ( , ') ( , )] *i q q q q i J A J J JJ*

> 2 2

*JJ JJ* 

 

<sup>1</sup> tan

2

denotes the phase shifting term induced by modulation of polarization given by

2 2

angle. Denoting each pattern as

with *i* = 1…4, the relative phase can be calculated as (J. Schwieder 1983)

*A*   

2

 

 

 

*<sup>x</sup> <sup>y</sup>* (16)

2 4

2 3

the retardation of each plate, and

 

 

, (17)

and *LJ*

into

(13)

( , ') 

, (14)

. (15)

**4.4 Phase shifting interferometry with modulation of polarization** 

 

is the linear polarizer angle, '

*ArcTan*

 

QR and QL (Fig. 1). This arrangement introduces Jones polarization vectors *JR*

**W**2.

being proportional to

Where

and *A* 

taken at a different

 

)',( is defined as

where 2 1*J* , <sup>2</sup> 2*J* , <sup>2</sup> 3*J* and 2 4*J* are the intensity measurements with the values of such that 12 3 4 ( ) 0, ( ) /2, ( ) , ( ) 3 /2 , respectively. Note that ( , /2) 2 and *A*( , /2) 1 , so a good choice for the retarders is quarter-wave retarders, as it is well known. Dependence of on the coordinates of the centered point has been simplified to *x*,*y*. The same fringe modulation *mq* results as in Eq. 8. Therefore, the discussion about fringe modulation given in previous sections is retained when introducing the modulation of polarization. Such polarization modulation can also be made for grids, resulting in similar conclusions.

## **5. Experimental testing of the phase-shifts in phase gratings and phase grids**

Figure 6 shows the superposition of Fourier amplitude spectra under window configuration *W1* for a phase grating with 110 lines/mm. The contrast of the corresponding experimental interference patterns can be interpreted as if its first four Fourier coefficients had phase relations as the ones sketched in Fig. 2. They were obtained with the system of Fig.1 before placing the retardation plates QR, QL and the polarizing filters on the image plane. The patterns show the relative phases of the diffraction orders discussed in previous sections. The phase-shifted steps of the experimental fringe patterns can be calculated by applying the algorithm proposed by Kreis, 1986. The resulting mean values are shown in Table 2. It can be seen from the table that they depart by small amounts from or 0.

Fig. 6. Experimental results with a phase-grating.


Table 2. Phase shifts measured from experimental patterns in Fig.6 according to the method from Kreis, 1986.

Due to that, the interference patterns are obtained from the interference between the replicas of each beam, centered around each diffraction order. Fig. 7(a) presents the replicas of window A with right circular polarization, and the replicas of window B with left circular polarization; each order is superposed depending on separation <sup>0</sup> *x* . Fig. 7(b) presents the interference pattern generated by the interference of windows AB, due to the cross circular polarization, a linear polarizer is placed to modulate the phase shift on each replicated interference pattern.

Experimental π Phase-Shifts Observed

needed values of

 can be 1 

Fig. 9. Polarizing filters array for 90° phase stepping.

**7. Simultaneous phase shifting interferometry** 

compared with the similar square of Fig. 8(b) (dotted lines).

our system is not a shearing interferometer of any type.

in the Fourier Spectra of Phase Gratings and Applications in Simultaneous PSI 311

Incorporating modulation of polarization, a TBPGI can be used for dynamic interferometry measurements. This system is able to obtain n-interferograms phase-shifted with a single shot. Phase evolving in time can be calculated and displayed on the basis of phase-shifting techniques with four or more interferograms. The system performs as previous proposals to attain four interferograms with a simultaneos capture (Barrientos et al., 1999; Novak et al., 2005). The system uses a grid as a beam splitter in a way that resembles the well-known double-frequency shearing interferometer as proposed by Wyant 1986, but our proposal differs from it not only due to its modulation of polarization, the use of a single frequency and the use of two windows, but also in the phase steps that our system introduces. Besides,

Figure 1 shows the arrangement of a simultaneous phase-shifting grid interferometer including modulation of polarization with retarders for the beams and linear polarizers on the image plane. The system generates several diffraction orders of similar irradiances, but not of equal fringe modulations, as expected (Eq. 12). In order to use Eq. 17 properly, each interferogram image was scaled to the same values of grey levels (from 0 to 255) and normalized. In the previous section, it was shown that a simplification for the polarizing

. For configuration **W2** (Fig. 8), due to the -shifts, only two linear polarizing filters have to be placed (instead of four filters, as when without the -shifts). The transmission axes of filter pairs P1, P3 and P2, P4 can be the same for each as long as they cover two patterns that are 180° phase apart (Fig. 9). Considering the polarizers angles , according with Sec. 4.3, the

sketched in Fig. 9. The square enclosing the 16 window replicas in the same figure is to be

In order to extract phase distributions that evolve in time using phase-shifting interferometry, simultaneous capture of several interferograms with a prescribed shift has to be done. The well-known four-step phase shifting algorithm is presented. Due to the

46.577 (G Rodriguez 2008a). They are

of 0, and

filter array can be attained when using the phase shifts obtain values of

0 and 2

**6. Phase-grid interference patterns with modulation of polarization** 

Fig. 7. Replicated windows generated by the Phase grating. (a) Replicas of A and B. (b) Interference pattern obtained through order diffraction superposition. (window configuration **W**1).

Fig. 8. Replicated windows generated by the Phase grating. (a) Replicas of A and B. (b) Interference pattern obtained through order diffraction superposition. (window configuration **W**2).

For the case of the diffraction orders belonging to a phase-grid constructed with two crossed gratings of equal frequency, the corresponding interference patterns are shown in Fig.8 for window configuration **W**2. Each grating gives patterns as in Fig. 6 when placed alone in the system of Fig. 1, with neither plate retarders nor polarizers. In each case, contrasts are in agreement with the conclusions derived from Fig. 5(b). The relative phase values of the 16 patterns within the square drawn in the patterns of Fig. 8(b) employing the method from Kreis, 1986 can be seen in Table 3.


Table 3. Phase shifts measured by means of the Kreis method for **W**2 configuration presented in Fig. 8.

Fig. 7. Replicated windows generated by the Phase grating. (a) Replicas of A and B. (b)

Fig. 8. Replicated windows generated by the Phase grating. (a) Replicas of A and B. (b)

For the case of the diffraction orders belonging to a phase-grid constructed with two crossed gratings of equal frequency, the corresponding interference patterns are shown in Fig.8 for window configuration **W**2. Each grating gives patterns as in Fig. 6 when placed alone in the system of Fig. 1, with neither plate retarders nor polarizers. In each case, contrasts are in agreement with the conclusions derived from Fig. 5(b). The relative phase values of the 16 patterns within the square drawn in the patterns of Fig. 8(b) employing the method from

Table 3. Phase shifts measured by means of the Kreis method for **W**2 configuration presented

Interference pattern obtained through order diffraction superposition. (window

Interference pattern obtained through order diffraction superposition. (window

configuration **W**1).

configuration **W**2).

in Fig. 8.

Kreis, 1986 can be seen in Table 3.

## **6. Phase-grid interference patterns with modulation of polarization**

Incorporating modulation of polarization, a TBPGI can be used for dynamic interferometry measurements. This system is able to obtain n-interferograms phase-shifted with a single shot. Phase evolving in time can be calculated and displayed on the basis of phase-shifting techniques with four or more interferograms. The system performs as previous proposals to attain four interferograms with a simultaneos capture (Barrientos et al., 1999; Novak et al., 2005). The system uses a grid as a beam splitter in a way that resembles the well-known double-frequency shearing interferometer as proposed by Wyant 1986, but our proposal differs from it not only due to its modulation of polarization, the use of a single frequency and the use of two windows, but also in the phase steps that our system introduces. Besides, our system is not a shearing interferometer of any type.

Figure 1 shows the arrangement of a simultaneous phase-shifting grid interferometer including modulation of polarization with retarders for the beams and linear polarizers on the image plane. The system generates several diffraction orders of similar irradiances, but not of equal fringe modulations, as expected (Eq. 12). In order to use Eq. 17 properly, each interferogram image was scaled to the same values of grey levels (from 0 to 255) and normalized. In the previous section, it was shown that a simplification for the polarizing filter array can be attained when using the phase shifts obtain values of of 0, and . For configuration **W2** (Fig. 8), due to the -shifts, only two linear polarizing filters have to be placed (instead of four filters, as when without the -shifts). The transmission axes of filter pairs P1, P3 and P2, P4 can be the same for each as long as they cover two patterns that are 180° phase apart (Fig. 9). Considering the polarizers angles , according with Sec. 4.3, the needed values of can be 1 0 and 2 46.577 (G Rodriguez 2008a). They are sketched in Fig. 9. The square enclosing the 16 window replicas in the same figure is to be compared with the similar square of Fig. 8(b) (dotted lines).

Fig. 9. Polarizing filters array for 90° phase stepping.

## **7. Simultaneous phase shifting interferometry**

In order to extract phase distributions that evolve in time using phase-shifting interferometry, simultaneous capture of several interferograms with a prescribed shift has to be done. The well-known four-step phase shifting algorithm is presented. Due to the

Experimental π Phase-Shifts Observed

**7.2 Case of nine interferograms** 

simplifying the filter array (See Fig. 13).

Fig. 13. Interference patterns detected with a polarizing filter at

symmetrical interferograms.

patterns used are enclosed in the dotted rectangle. (b) Polarizer filter array for N= 8, nine

25 . (a)The interference

second.

in the Fourier Spectra of Phase Gratings and Applications in Simultaneous PSI 313

Fig. 12. Moving distributions. Oil drops moving under gravity on a slide. One capture per

To demonstrate the use of several interferograms, we choose the symmetrical n=*N*+1 phase

employed when using these techniques. The phase shifts of p due to the grid spectra allow the use of a number of polarizing filters that is less than the number of interferograms,

/2 *N* is

steps algorithms, for data processing ( case *N*=8). A constant phase shift value of

capability of TBPGI to capture more than four interferograms in a single shot by placing a phase grid, we have the advantage of analyzing the case of more than four interferograms; the case of *Nn* 1 interferograms has been chosen. This method reduces errors in phase calculations when noisy interferograms are involved (Malacara and Servin, 2005). It the next sections of the chapter, we will show experimental results for *n* = 9 interferograms.

## **7.1 Four interferogram case**

A phase dot was placed in the path of beam B, while beam A was used as a reference; the results obtained are shown in Fig. 10. Fig. 10(a) shows the four patterns with 2/ phase shifts obtained in a single shot, and Fig. 10(b) presents the phase profile of the object, in false color coding. However, more than four interferograms could be used, whether for *N*-steps phaseshifting interferometry (Rodriguez, 2008b) or for averaging images with the same shift.

Fig. 10. Static test objects: Phase dot. (a) Four 90° phase-shifted interferograms. (b) Unwrapped phase.

Figure 11 shows the case of moving distributions, corresponding to water flowing on a microscope slide; it shows the phase profile resulting of phase evolution *tyx* ),,( . Fig. 12, by analyzing flow motion by gravity, shows the temporal evolution of oil flow, clearly showing the phase changes induced by the motion of oil; the evolution of the drops as they go by in front of the camera can be shown. Images are presented in 4-D (Wyant 2003).

Fig. 11. Moving distributions. Representative Frames. Evolution of the phase, one capture per second.

Fig. 12. Moving distributions. Oil drops moving under gravity on a slide. One capture per second.

## **7.2 Case of nine interferograms**

312 Interferometry – Research and Applications in Science and Technology

capability of TBPGI to capture more than four interferograms in a single shot by placing a phase grid, we have the advantage of analyzing the case of more than four interferograms; the case of *Nn* 1 interferograms has been chosen. This method reduces errors in phase calculations when noisy interferograms are involved (Malacara and Servin, 2005). It the next

A phase dot was placed in the path of beam B, while beam A was used as a reference; the

obtained in a single shot, and Fig. 10(b) presents the phase profile of the object, in false color coding. However, more than four interferograms could be used, whether for *N*-steps phaseshifting interferometry (Rodriguez, 2008b) or for averaging images with the same shift.

*tyx* ),,( . Fig. 12,

2/ phase shifts

sections of the chapter, we will show experimental results for *n* = 9 interferograms.

Fig. 10. Static test objects: Phase dot. (a) Four 90° phase-shifted interferograms. (b)

microscope slide; it shows the phase profile resulting of phase evolution

Figure 11 shows the case of moving distributions, corresponding to water flowing on a

by analyzing flow motion by gravity, shows the temporal evolution of oil flow, clearly showing the phase changes induced by the motion of oil; the evolution of the drops as they go by in front of the camera can be shown. Images are presented in 4-D (Wyant 2003).

Fig. 11. Moving distributions. Representative Frames. Evolution of the phase, one capture

results obtained are shown in Fig. 10. Fig. 10(a) shows the four patterns with

**7.1 Four interferogram case** 

Unwrapped phase.

per second.

To demonstrate the use of several interferograms, we choose the symmetrical n=*N*+1 phase steps algorithms, for data processing ( case *N*=8). A constant phase shift value of /2 *N* is employed when using these techniques. The phase shifts of p due to the grid spectra allow the use of a number of polarizing filters that is less than the number of interferograms, simplifying the filter array (See Fig. 13).

Fig. 13. Interference patterns detected with a polarizing filter at 25 . (a)The interference patterns used are enclosed in the dotted rectangle. (b) Polarizer filter array for N= 8, nine symmetrical interferograms.

Experimental π Phase-Shifts Observed

29, pp. 685-689.

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38, pp. 2069-2074.

*Optical Testing.*

16, 7806-7817.

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*Opt.*, Vol. 44, pp. 6861-6868.

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in the Fourier Spectra of Phase Gratings and Applications in Simultaneous PSI 315

Investigaciones en Óptica A.C. (CIO)" through projects 290597 (CONACYT-CIO) and 124145 (CONACYT-BUAP) is also acknowledged. Author *NITA* expresses sincere appreciation to *Luisa*, M*iguel* and *Damian-El* for the support provided, and to CONACYT for grant 102137/43055. Author DISG (Grant:227470/31458) is very grateful to CONACyT

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Kneubühl F.,(1969). Diffraction Grating Spectroscopy, *Appl. Opt.*, Vol. 8, pp. 505-519.

Fig. 14. Tilted wave-front. (a) Nine 45° phase-shifted interferograms (b) Unwrapped phase data map.

Considering the retarders at disposal, according to Rodriguez 2008b, it can be shown that for symmetrical nine, 0 <sup>1</sup> , 989.92 <sup>2</sup> , 975.22 <sup>3</sup> , 577.46 <sup>4</sup> and <sup>5</sup> 903.157 . In this case, each resulting step corresponds to a 45° phase-shift. The corresponding results and calculated phases are shown in Fig. 14, the experimental results shows the interference pattern generated by a tilted wavefront. That is said before, this systems are capable of obtaining *n* = *N*+1 interferograms in a single capture (this case, *n* 12 ).

## **8. Final remark**

Theoretical and experimental evidence ofshifts in the Fourier spectra of phase gratings and phase grids were presented; these shifts are not discussed in the literature as far as we know. When the power spectrum is focused, it tends to hide the effect presented, but it can be of considerable relevance when gratings or grids are used for interferometric applications. For example: in phase shift interferometry, these results are convenient since only one grating displacement is necessary to capture four interferograms, and for the case of polarization phase shifting interferometry, the characteristics of the phase gratings used simplify the placement of polarization filters and allow us to obtain *n*-patterns in one shot.

As a final remark the system presented does not use a physical double window; it generates two beams with variable separation according to the characteristics of the grid used. The combination of circular polarization states and the use of phase-grid enable the generation of *n*-interference patterns with independent phase shifts. This characteristic optimizes the interferometric system used, and allows the analysis of static and dynamic phase distributions.

## **9. Acknowledgments**

Authors thank M. A. Ruiz for his contribution in proofreading the manuscript. Partial support from "Consejo Nacional de Ciencia y Tecnología (CONACYT)" and "Centro de Investigaciones en Óptica A.C. (CIO)" through projects 290597 (CONACYT-CIO) and 124145 (CONACYT-BUAP) is also acknowledged. Author *NITA* expresses sincere appreciation to *Luisa*, M*iguel* and *Damian-El* for the support provided, and to CONACYT for grant 102137/43055. Author DISG (Grant:227470/31458) is very grateful to CONACyT for the graduate scholarship granted and expresses sincere appreciation to *Geliztle*.

## **10. References**

314 Interferometry – Research and Applications in Science and Technology

(a) (b)

data map.

for symmetrical nine, 0

**8. Final remark** 

distributions.

**9. Acknowledgments** 

In this case, each resulting step

<sup>1</sup> , 989.92 

Fig. 14. Tilted wave-front. (a) Nine 45° phase-shifted interferograms (b) Unwrapped phase

Considering the retarders at disposal, according to Rodriguez 2008b, it can be shown that

results and calculated phases are shown in Fig. 14, the experimental results shows the interference pattern generated by a tilted wavefront. That is said before, this systems are

Theoretical and experimental evidence ofshifts in the Fourier spectra of phase gratings and phase grids were presented; these shifts are not discussed in the literature as far as we know. When the power spectrum is focused, it tends to hide the effect presented, but it can be of considerable relevance when gratings or grids are used for interferometric applications. For example: in phase shift interferometry, these results are convenient since only one grating displacement is necessary to capture four interferograms, and for the case of polarization phase shifting interferometry, the characteristics of the phase gratings used simplify the placement of polarization filters and allow us to obtain *n*-patterns in one shot. As a final remark the system presented does not use a physical double window; it generates two beams with variable separation according to the characteristics of the grid used. The combination of circular polarization states and the use of phase-grid enable the generation of *n*-interference patterns with independent phase shifts. This characteristic optimizes the interferometric system used, and allows the analysis of static and dynamic phase

Authors thank M. A. Ruiz for his contribution in proofreading the manuscript. Partial support from "Consejo Nacional de Ciencia y Tecnología (CONACYT)" and "Centro de

 <sup>3</sup> , 577.46 

<sup>4</sup> and

corresponds to a 45° phase-shift. The corresponding

<sup>5</sup> 903.157 .

 <sup>2</sup> , 975.22 

capable of obtaining *n* = *N*+1 interferograms in a single capture (this case, *n* 12 ).


**16** 

*3NTT corporation* 

**ONU**

*Japan* 

**Length Measurement for Optical Transmission** 

*2Department of Information and Systems Engineering, Fukuoka Institute of Technology* 

More than half a million kilometres of aerial/underground optical telecommunication cable have been installed in Japan. A guaranteed telecommunication service is replacing the besteffort service as real-time streaming delivery increases. Fiber-to-the-home (FTTH) systems are mainly designed by passive optical network (PON) topology (ITU-T Recommendation G.983.1, 1998; Hornung et al., 1990; Sankawa et al., 2006). They are not allowed for even an instantaneous interruption day and night, because numerous customers suffer damage from

Telecommunication cables are required to be relocated in road construction and work on the water supply. Each optical fiber leading from an optical line terminal (OLT) in a telephone office to a customer's optical network unit (ONU) must be cut and reconnected as shown in Fig. 1. Customers expect real-time transmission for high-quality communications to

Several studies have reported on protection of PON system (Xu & Ho, 2001; Tanaka & Horiuchi, 2008). Electrical transmission apparatus can maintain communication without interruption, even when optical cables are temporarily cut. The system is complicated and any transmission delay during O/E conversion is fatal to real-time communication. Although it is desirable to directly switch the transmission medium itself, it had been thought that some data

**1. Introduction** 

**O L T**

ONU: Optical Network Unit

Fig. 1. Optical fiber cable replacement.

it as 1 x 109 bits of data will be lost every second.

continue uninterrupted, especially for video transmission services.

New cable OLT : Optical Line Terminal

Takeshi Tsujimura1, Koichi Yoshida2 and Kuniaki Tanaka3

*1Department of Mechanical Engineering, Saga University* 

**Line Using Interferometry** 

Wyant J. C., (1973). Double Frequency Grating Lateral Shear Interferometer, *Appl. Opt.*, Vol. 12, pp. 2057-2060.

Wyant J. C., (2003). Dynamic Interferometry, *Optics & Photonics News*, Vol. 14, pp. 36-41.

## **Length Measurement for Optical Transmission Line Using Interferometry**

Takeshi Tsujimura1, Koichi Yoshida2 and Kuniaki Tanaka3

*1Department of Mechanical Engineering, Saga University 2Department of Information and Systems Engineering, Fukuoka Institute of Technology 3NTT corporation Japan* 

## **1. Introduction**

316 Interferometry – Research and Applications in Science and Technology

Wyant J. C., (1973). Double Frequency Grating Lateral Shear Interferometer, *Appl. Opt.*, Vol.

Wyant J. C., (2003). Dynamic Interferometry, *Optics & Photonics News*, Vol. 14, pp. 36-41.

12, pp. 2057-2060.

More than half a million kilometres of aerial/underground optical telecommunication cable have been installed in Japan. A guaranteed telecommunication service is replacing the besteffort service as real-time streaming delivery increases. Fiber-to-the-home (FTTH) systems are mainly designed by passive optical network (PON) topology (ITU-T Recommendation G.983.1, 1998; Hornung et al., 1990; Sankawa et al., 2006). They are not allowed for even an instantaneous interruption day and night, because numerous customers suffer damage from it as 1 x 109 bits of data will be lost every second.

Telecommunication cables are required to be relocated in road construction and work on the water supply. Each optical fiber leading from an optical line terminal (OLT) in a telephone office to a customer's optical network unit (ONU) must be cut and reconnected as shown in Fig. 1. Customers expect real-time transmission for high-quality communications to continue uninterrupted, especially for video transmission services.

Fig. 1. Optical fiber cable replacement.

Several studies have reported on protection of PON system (Xu & Ho, 2001; Tanaka & Horiuchi, 2008). Electrical transmission apparatus can maintain communication without interruption, even when optical cables are temporarily cut. The system is complicated and any transmission delay during O/E conversion is fatal to real-time communication. Although it is desirable to directly switch the transmission medium itself, it had been thought that some data

Length Measurement for Optical Transmission Line Using Interferometry 319

switching method is necessary to maintain communications completely during optical cable

107 103 105 10 interruption time (ns)

0 2 4 6 8

We propose an optical line switching technique that uses duplex transmission lines as shown in Fig. 4. Normal transmission system consists of an OLT, ONUs, and an optical fiber labeled 'regular line' in Fig. 4. Signals are bidirectionally transmitted along the line through a wavelength independent optical coupler (WIC) and an optical splitter. Our proposed

Number of ONUs

replacement and thus provide a highly reliable network.

0

0

R

**3. Optical fiber line switching 3.1 Optical line switching technique** 

e

storatio

n tim

e (s)

10

20

Fig. 2. Relationship between interruption time and restoration time.

Fig. 3. Relationship between number of ONUs and restoration time.

R

e

storatio

n tim

e (s)

10

20

bits would inevitably be lost during the replacement of optical fibers. An optical fiber cable transfer splicing system has been developed to minimize cable replacing time (Watanabe et al., 1992; Tanaka et al., 2002). It disconnects an optical fiber and exchanges it for another in a flash. It takes 30 ms to switch a transmission line, and more than 2 seconds to restore communications with, for example, a gigabit Ethernet passive optical network (GE-PON) (Azuma et al., 2008). Neither the PON protection technique nor the splicing system avoids communication failures because they cannot reduce the disconnecting time absolutely to zero.

The authors have developed an interruption-free replacement method for in-service telecommunication lines, which can be applied to the current PON system equipped with conventional OLTs and ONUs (Tsujimura et al., 2010; Tanaka et al., 2009; Yoshida & Tsujimura, 2010). Two essential techniques were newly proposed; a measurement method and an adjustment system for the transmission line length. The latter continuously lengthens/shortens the transmission line over very long distances without losing transmitted data based on free space optics (FSO) system (Tsujimura et al., 2009b; Yoshida et al., 2009). The former distinguishes the difference between the duplicated line lengths using interferometry (Tsujimura et al., 2009a). Interferometry is the technique of diagnosing the properties of two or more lasers or waves by studying the pattern of interference created by their superposition. It is an important investigative technique in the fields of astronomy, fiber optics, optical metrology and so on. Studies on optical interferometry are reported to improve tiny optical devices (Saunders & Hardcastle, 1994; Cao & Cartledge, 2002; Torregrosa et al., 2007). We have applied the optical interferometry technique to measure length of several kilometers of transmission lines with a 10 mm resolution in this study.

This paper reveals the problems in replacing optical cables first. It proposes a switching method of optical fiber transmission lines next. We design a shunt system for in-service optical fibers. Then an optical line length measurement method is studied to distinguish the difference of two lines by evaluating interfered optical pulses. An optical line switching system is designed, and a line length adjustment system is prototyped. The proposed system is applied to a 15 km GE-PON optical fiber network while adding a 10 m extension to show the efficiency of this approach when replacing in-service optical cables. Finally, we discuss some proposal for improvement of line length measurement.

## **2. Optical cable replacement**

First of all, fundamental investigations were conducted on the effects of transmission interruption on the telecommunication service.

When an optical fiber line was intensionally disconnected in a short time, transmission broke off and was restored by PON transmission system. Figure 2 and 3 show the experimental results of transmission characteristics when GE-PON optical communication line is switched using high-speed line transfer. Figure 2 expresses the relationship between interruption time and optical network unit (ONU) restoration time for 4 ONUs. Even if an optical fiber line was reconnected only in 100 ns, the transmission system took about 6 seconds to restore the telecommunication service. Up to 8 ONUs can be connected to an OLT. It took more than 10 seconds if eight ONUs were connected to an OLT as shown in Fig. 3, which indicates that the restoration time increases in relation to the number of ONUs.

These results suggest that communication service is inevitably interrupted no matter how quick the conventional method reconnects the transmission line. That is why a new

bits would inevitably be lost during the replacement of optical fibers. An optical fiber cable transfer splicing system has been developed to minimize cable replacing time (Watanabe et al., 1992; Tanaka et al., 2002). It disconnects an optical fiber and exchanges it for another in a flash. It takes 30 ms to switch a transmission line, and more than 2 seconds to restore communications with, for example, a gigabit Ethernet passive optical network (GE-PON) (Azuma et al., 2008). Neither the PON protection technique nor the splicing system avoids communication failures because they cannot reduce the disconnecting time absolutely to zero. The authors have developed an interruption-free replacement method for in-service telecommunication lines, which can be applied to the current PON system equipped with conventional OLTs and ONUs (Tsujimura et al., 2010; Tanaka et al., 2009; Yoshida & Tsujimura, 2010). Two essential techniques were newly proposed; a measurement method and an adjustment system for the transmission line length. The latter continuously lengthens/shortens the transmission line over very long distances without losing transmitted data based on free space optics (FSO) system (Tsujimura et al., 2009b; Yoshida et al., 2009). The former distinguishes the difference between the duplicated line lengths using interferometry (Tsujimura et al., 2009a). Interferometry is the technique of diagnosing the properties of two or more lasers or waves by studying the pattern of interference created by their superposition. It is an important investigative technique in the fields of astronomy, fiber optics, optical metrology and so on. Studies on optical interferometry are reported to improve tiny optical devices (Saunders & Hardcastle, 1994; Cao & Cartledge, 2002; Torregrosa et al., 2007). We have applied the optical interferometry technique to measure length of several kilometers of transmission lines with a 10 mm resolution in this study.

This paper reveals the problems in replacing optical cables first. It proposes a switching method of optical fiber transmission lines next. We design a shunt system for in-service optical fibers. Then an optical line length measurement method is studied to distinguish the difference of two lines by evaluating interfered optical pulses. An optical line switching system is designed, and a line length adjustment system is prototyped. The proposed system is applied to a 15 km GE-PON optical fiber network while adding a 10 m extension to show the efficiency of this approach when replacing in-service optical cables. Finally, we

First of all, fundamental investigations were conducted on the effects of transmission

When an optical fiber line was intensionally disconnected in a short time, transmission broke off and was restored by PON transmission system. Figure 2 and 3 show the experimental results of transmission characteristics when GE-PON optical communication line is switched using high-speed line transfer. Figure 2 expresses the relationship between interruption time and optical network unit (ONU) restoration time for 4 ONUs. Even if an optical fiber line was reconnected only in 100 ns, the transmission system took about 6 seconds to restore the telecommunication service. Up to 8 ONUs can be connected to an OLT. It took more than 10 seconds if eight ONUs were connected to an OLT as shown in Fig. 3, which indicates that the restoration time increases in relation to the number of ONUs.

These results suggest that communication service is inevitably interrupted no matter how quick the conventional method reconnects the transmission line. That is why a new

discuss some proposal for improvement of line length measurement.

**2. Optical cable replacement** 

interruption on the telecommunication service.

switching method is necessary to maintain communications completely during optical cable replacement and thus provide a highly reliable network.

Fig. 2. Relationship between interruption time and restoration time.

Fig. 3. Relationship between number of ONUs and restoration time.

## **3. Optical fiber line switching**

## **3.1 Optical line switching technique**

We propose an optical line switching technique that uses duplex transmission lines as shown in Fig. 4. Normal transmission system consists of an OLT, ONUs, and an optical fiber labeled 'regular line' in Fig. 4. Signals are bidirectionally transmitted along the line through a wavelength independent optical coupler (WIC) and an optical splitter. Our proposed

Length Measurement for Optical Transmission Line Using Interferometry 321

**Step 3.** Once the lengths of the two lines coincide, the transmission signals are also

**Step 4.** The regular line is cut. Transmission keeps connected because the signals travels

**Step 6.** The test light measures the lengths of the new line and the detour line. The detour

**Step 7.** The LWPF is then removed and the transmission is duplicated. The detour line is

**Step 8.** Replacement of the optical fiber line is completed without any interruption.

through the detour line. A long-wavelength pass filter (LWPF) is temporarily

line is adjusted to the new line while communications are maintained. The LWPF keeps preventing the optical pulses for telecommunication from traveling through

**WIC**

Fig. 5. Part I

OLT ONU

**Detour line** OLT ONU

OLT ONU

OLT ONU

**length adjuster**

**Regular line**

**Regular line** ONU **Optical splitter**

1650nmLD **Optical line**

2x8

2x8

2x8

2x8

ONU

ONU ONU

1650nmLD

ONU ONU

ONU ONU

1650nmLD

launched into the detour line.

**Step 5.** The LWPF passes the test light alone through the new line.

installed in the new line.

the new line.

finally cut off.

Step 1: Set detour line.

Step 2: Adjust detour line to regular line.

Step 3: Multiplex optical signals.

Step 0: Signals travel through regular line.

technique requires a detour optical fiber to temporarily transmit signals. It also uses a test light whose wavelength is different from ones for signal transmission. The test light is created by an optical frequency-chirped pulse light source, and reaches an oscilloscope through the optical splitter, the regular/detour line, and the WIC. An optical line length measuring method detects the optical path difference between the detour and the regular line with these arrangements. An optical line length adjusting method controls the detour line length in virtue of FSO device.

Because either of two lines are managed to keep connection and to transmit signals at any time, the entire transmission system are secured against interruption. One of optical cables can be freely cut or reconnected, while the other cable maintains connected and transmits signals.

In order to avoid suspending transmission, the communication conditions have to be kept in the physical layer, for example, carrier power or signal phase. Thus, the following two conditions must be met while exchanging an in-service line for a spare line,


Fig. 4. Duplex optical fiber transmission system.

The conditions are satisfied by equalizing the transmission times of both signals transmitted through the duplicated lines. We accomplish this by tuning the length of the detour line.

The optical line replacement procedure, illustrated in Fig. 5 where the transmission lines are simplified, is as follows:


technique requires a detour optical fiber to temporarily transmit signals. It also uses a test light whose wavelength is different from ones for signal transmission. The test light is created by an optical frequency-chirped pulse light source, and reaches an oscilloscope through the optical splitter, the regular/detour line, and the WIC. An optical line length measuring method detects the optical path difference between the detour and the regular line with these arrangements. An optical line length adjusting method controls the detour

Because either of two lines are managed to keep connection and to transmit signals at any time, the entire transmission system are secured against interruption. One of optical cables can be freely cut or reconnected, while the other cable maintains connected and transmits signals. In order to avoid suspending transmission, the communication conditions have to be kept in the physical layer, for example, carrier power or signal phase. Thus, the following two

> OLT : Optical Line Terminal ONU: Optical Network Unit O/E : Optic/Electric converter

Optical frequency-chirped pulse light source

2 x 8

ONU

conditions must be met while exchanging an in-service line for a spare line,

Regular line

Optical line length detector

The conditions are satisfied by equalizing the transmission times of both signals transmitted through the duplicated lines. We accomplish this by tuning the length of the detour line.

The optical line replacement procedure, illustrated in Fig. 5 where the transmission lines are

**Step 0.** Signals are ordinarily transmitted through an optical fiber (regular line) between an

**Step 1.** A detour line is established between the WIC and the 2x8 optical splitter. A

transmission system in the telephone office commonly equips with a WIC next to an OLT. A 1650 nm LD is connected to one of the branches of the 2x8 optical splitter. **Step 2.** The detour line length is measured with a 1650 nm test light using an optical line

length measuring method, and is adjusted to the same length as the regular line

Detour line

WIC ONU

Optical splitter New line

line length in virtue of FSO device.

a. No transmission signals must be lost,

Oscilloscope

Fig. 4. Duplex optical fiber transmission system.

OLT

simplified, is as follows:

b. Transmission time in a protocol must be maintained.

O/E Optical line length adjuster

OLT and ONUs through a 2x8 optical splitter.

using an optical line length adjusting method.


Length Measurement for Optical Transmission Line Using Interferometry 323

The proposed method temporarily provides a duplicate transmission line as shown in Fig. 6 to replace optical fiber lines. A detour line is prepared in advance through which to divert

*Telephone office Outdoors Customers*

New line

Detour line

**Optical line length adjuster**

> Test light source

1650nm DFB-LD

2x8 SP

ONU

ONU

・・ ONU

signals while the existing line (regular line) is replaced with a new one.

FSO

Path #0

*L0 2L0 3L0* FS-0

This system transfers signals between the two lines. Signals are duplicated at the moment of changeover to maintain continuous communications. The signals travel separately through the two lines to a receiver. A difference in the line lengths leads to a difference in the signals' arrival times. A communication fault occurs if, as a result of their proximity, the waveforms of the two arriving signals are too blurred for the signals to be identified as discrete. Thus it

We investigated the tolerance of the multiplexed signal synchronicity in advance. The transmission quality is observed by changing the difference between the duplicated line lengths. The results show that the transmission linkage is maintained if the difference is within 80 mm as with GE-PON. A multiplexed signal cannot be perceived as a single bit when the duplicated line lengths have a larger gap for 1 Gbit/s transmission. Because these characteristics depend on the periodic length of a transmission bit, the requirement is assumed to be more severe when the method is applied to higher-speed communication services.

Experiments determined that the tolerance of the difference in line length is 80 mm with

Accordingly, the proposed system controls the adjustment procedure so that the difference

We next constructed a prototype of the optical line shunt system to apply to a GE-PON

in length between the detour and regular lines is adjusted within 80 mm.

optical fiber line replacement according to the procedure described above.

WDM

SW-1 SW-0

Regular line

**Optical line shunt system**

LWPF

Oscilloscope

FS-1

Path #1

SW-2

Fig. 6. Shunt system of optical fiber transmission line.

is important to adjust the lengths of both lines precisely.

regard to the GE-PON transmission system.

1310/1490nm Amplifier

1650nm Amplifier

WIC

OLT

**Optical line length detector**

Part II

Fig. 5. Optical line replacement procedure.

## **3.2 Optical line shunt system**

The authors have designed an optical line shunt system, and a switching procedure for three wavelengths, namely 1310 and 1490 nm for GE-PON transmission, and 1650 nm for measurement. The optical line shunt system is installed in a telephone office. It is composed of an optical line length detector and an optical line length adjuster. Figure 6 shows an individual optical fiber line in a GE-PON transmission system. Optical pulse signals at two wavelengths are bidirectionally transmitted through a regular line between customers' ONUs and an OLT in a telephone office via a wavelength independent optical coupler (WIC) and a 2x8 optical splitter (2x8 SP), respectively.

A test light at a wavelength different from those of the transmission signals is sent from one of the optical splitter's ports to the duplicated lines. We use a 1650 distributed feedback laser diode (DFB-LD) whose pulse width is 200 ns as a test light source. An oscilloscope is connected to the optical coupler to detect the test light through a long-wavelength pass filter (LWPF). The optical line length adjuster is an FSO application (Yoshida et al., 2011; Willebrand et al., 1999). Some optical switches (SW) and optical fiber selectors (FS) control the flow of the optical signals managed by a PC. The optical pulses are compensated by 1650 and 1310/1490 nm amplifiers (Fukada et al., 2008).

Part II

The authors have designed an optical line shunt system, and a switching procedure for three wavelengths, namely 1310 and 1490 nm for GE-PON transmission, and 1650 nm for measurement. The optical line shunt system is installed in a telephone office. It is composed of an optical line length detector and an optical line length adjuster. Figure 6 shows an individual optical fiber line in a GE-PON transmission system. Optical pulse signals at two wavelengths are bidirectionally transmitted through a regular line between customers' ONUs and an OLT in a telephone office via a wavelength independent optical coupler

A test light at a wavelength different from those of the transmission signals is sent from one of the optical splitter's ports to the duplicated lines. We use a 1650 distributed feedback laser diode (DFB-LD) whose pulse width is 200 ns as a test light source. An oscilloscope is connected to the optical coupler to detect the test light through a long-wavelength pass filter (LWPF). The optical line length adjuster is an FSO application (Yoshida et al., 2011; Willebrand et al., 1999). Some optical switches (SW) and optical fiber selectors (FS) control the flow of the optical signals managed by a PC. The optical pulses are compensated by 1650

OLT ONU

OLT ONU

**New line LWPF**

OLT ONU

OLT ONU

OLT ONU

ONU ONU

ONU ONU

1650nmLD

ONU ONU

1650nmLD

2x8

2x8

**LWPF** 2x8

2x8

2x8

1650nmLD

ONU ONU

1650nmLD

ONU ONU

1650nmLD

Step 6: Adjust detour line to new line.

Step 7: Cut off detour line.

Step 4: Cut regular line.

Step 5: Connect new line.

Step 8: Complete replacement.

**3.2 Optical line shunt system** 

Fig. 5. Optical line replacement procedure.

(WIC) and a 2x8 optical splitter (2x8 SP), respectively.

and 1310/1490 nm amplifiers (Fukada et al., 2008).

The proposed method temporarily provides a duplicate transmission line as shown in Fig. 6 to replace optical fiber lines. A detour line is prepared in advance through which to divert signals while the existing line (regular line) is replaced with a new one.

Fig. 6. Shunt system of optical fiber transmission line.

This system transfers signals between the two lines. Signals are duplicated at the moment of changeover to maintain continuous communications. The signals travel separately through the two lines to a receiver. A difference in the line lengths leads to a difference in the signals' arrival times. A communication fault occurs if, as a result of their proximity, the waveforms of the two arriving signals are too blurred for the signals to be identified as discrete. Thus it is important to adjust the lengths of both lines precisely.

We investigated the tolerance of the multiplexed signal synchronicity in advance. The transmission quality is observed by changing the difference between the duplicated line lengths. The results show that the transmission linkage is maintained if the difference is within 80 mm as with GE-PON. A multiplexed signal cannot be perceived as a single bit when the duplicated line lengths have a larger gap for 1 Gbit/s transmission. Because these characteristics depend on the periodic length of a transmission bit, the requirement is assumed to be more severe when the method is applied to higher-speed communication services.

Experiments determined that the tolerance of the difference in line length is 80 mm with regard to the GE-PON transmission system.

Accordingly, the proposed system controls the adjustment procedure so that the difference in length between the detour and regular lines is adjusted within 80 mm.

We next constructed a prototype of the optical line shunt system to apply to a GE-PON optical fiber line replacement according to the procedure described above.

Length Measurement for Optical Transmission Line Using Interferometry 325

where *k, W, n* are the number of journeys between the retroreflectors, the retroreflector

where *Wmax* and *Wmin* are the maximum and minimum retroreflector intervals. The retroreflector stroke of our prototype, *Wmax- Wmin*, was 0.3 m, the refractive index, *n*, of the optical fiber was 1.46, the number of journeys, *k* was 10, and the optical line span, *L0*, tuned by the adjuster was 2 m. Our prototype adjusted the detour line length with a resolution of

The limit of the adjustable range is a practical problem when this system is applied to several kilometers of access network. Therefore, we designed an optical line length accumulator. The optical line length adjuster contains two optical paths, #0 and #1, as shown in Fig. 6. An optical switch (SW) and an optical fiber selector (FS) are installed in each path. Optical switches control the optical pulse flow. Each optical fiber selector is equipped with various lengths of optical fiber, for example L0, 2L0 and 3L0. The path length can be

The optical line length adjuster can extend the detour line as much as required using the operation shown in Fig. 9. First, the FSO system lengthens path #0 by L0 by gradually increasing the retroreflector interval. After the optical fiber selector, FS-1, has selected an optical fiber of length L0, the active line is switched from path #0 to path #1 at time t0. The FSO system then returns to the origin, and the optical fiber selector, FS-0, selects an optical fiber of length L0 instead to keep the length of path #0 at L0. The FSO system increases the retroreflector interval again at time 2t0 to repeat the same operation. In this way the adjuster accumulates spans extended by the FSO system. The scanning time of our prototype, t0, was 10 seconds, because the retroreflector moved along the motorized sliding stage at 30 mm/s. The optical line length adjuster enables us to lengthen/shorten the detour line while

On line Off line

t0 2t0 3t0 4t0 Time

5t0

*L0* **=** *k* **(***Wmax- Wmin***)/** *n*, (2)

interval, and the refractive index of optical fiber, respectively.

The FSO lengthens the optical line length up to *L0*.

discretely changed by choosing any one of them.

continuing to transmit optical signals.

L0

0 0

Fig. 9. Accumulation of optics line length.

2L0

Accumulated optical line length

Path #0 Path #1

0.1 mm.

An optical line length adjuster, shown in Fig. 7, was installed along the detour line. The adjuster was equipped with two retroreflectors, which directly faced each other as illustrated in Fig. 8. Optical pulses were transmitted through an optical fiber, divided into three wavelengths by wavelength division multiplexing (WDM) couplers, and discharged separately into the air from collimators. They traveled 10 times between the retroreflectors, and were introduced into the opposite optical fiber. The number of reflections was determined based on the retroreflector arrangement.

Fig. 7. Photos of optics line length adjuster.

Fig. 8. Free-space optics line length adjuster.

The detour line between the retroreflectors consisted of an FSO system. The detour line length could be easily adjusted by controlling the retroreflector interval with a resolution of 0.14 mm with a motorized sliding stage. Optical pulses travel **n**-times faster in the air than in an optical fiber, where **n** is the refractive index of the optical fiber. Thus the optical line length adjuster lengthens/shortens the corresponding optical fiber length, *L*

$$L \equiv k \ W / n\_r \tag{1}$$

where *k, W, n* are the number of journeys between the retroreflectors, the retroreflector interval, and the refractive index of optical fiber, respectively.

The FSO lengthens the optical line length up to *L0*.

324 Interferometry – Research and Applications in Science and Technology

An optical line length adjuster, shown in Fig. 7, was installed along the detour line. The adjuster was equipped with two retroreflectors, which directly faced each other as illustrated in Fig. 8. Optical pulses were transmitted through an optical fiber, divided into three wavelengths by wavelength division multiplexing (WDM) couplers, and discharged separately into the air from collimators. They traveled 10 times between the retroreflectors, and were introduced into the opposite optical fiber. The number of reflections was

> **WDM coupler**

> **WDM coupler**

The detour line between the retroreflectors consisted of an FSO system. The detour line length could be easily adjusted by controlling the retroreflector interval with a resolution of 0.14 mm with a motorized sliding stage. Optical pulses travel **n**-times faster in the air than in an optical fiber, where **n** is the refractive index of the optical fiber. Thus the optical line

length adjuster lengthens/shortens the corresponding optical fiber length, *L*

Collimators

**Collimator**

**To optical splitter**

*L* **=** *k W* **/** *n***,** (1)

determined based on the retroreflector arrangement.

Retroreflectors

**Retroreflector**

**To optical coupler**

Fig. 7. Photos of optics line length adjuster.

Fig. 8. Free-space optics line length adjuster.

$$\mathbf{L}\_0 \equiv \mathbf{k} \text{ (}\mathbf{W}\_{\text{max}}\text{- }\mathbf{W}\_{\text{min}}\text{)}\prime\prime\prime\,,\tag{2}$$

where *Wmax* and *Wmin* are the maximum and minimum retroreflector intervals. The retroreflector stroke of our prototype, *Wmax- Wmin*, was 0.3 m, the refractive index, *n*, of the optical fiber was 1.46, the number of journeys, *k* was 10, and the optical line span, *L0*, tuned by the adjuster was 2 m. Our prototype adjusted the detour line length with a resolution of 0.1 mm.

The limit of the adjustable range is a practical problem when this system is applied to several kilometers of access network. Therefore, we designed an optical line length accumulator. The optical line length adjuster contains two optical paths, #0 and #1, as shown in Fig. 6. An optical switch (SW) and an optical fiber selector (FS) are installed in each path. Optical switches control the optical pulse flow. Each optical fiber selector is equipped with various lengths of optical fiber, for example L0, 2L0 and 3L0. The path length can be discretely changed by choosing any one of them.

The optical line length adjuster can extend the detour line as much as required using the operation shown in Fig. 9. First, the FSO system lengthens path #0 by L0 by gradually increasing the retroreflector interval. After the optical fiber selector, FS-1, has selected an optical fiber of length L0, the active line is switched from path #0 to path #1 at time t0. The FSO system then returns to the origin, and the optical fiber selector, FS-0, selects an optical fiber of length L0 instead to keep the length of path #0 at L0. The FSO system increases the retroreflector interval again at time 2t0 to repeat the same operation. In this way the adjuster accumulates spans extended by the FSO system. The scanning time of our prototype, t0, was 10 seconds, because the retroreflector moved along the motorized sliding stage at 30 mm/s.

The optical line length adjuster enables us to lengthen/shorten the detour line while continuing to transmit optical signals.

Fig. 9. Accumulation of optics line length.

Length Measurement for Optical Transmission Line Using Interferometry 327

(a) Optical path difference 100m.

(b) Optical path difference 20m.

(c) Optical path difference 1m.

Fig. 10. Delay of duplicated received pulse.

## **4. Optical line length measurement**

The proposed system uses laser pulses at a wavelength of 1650 mm to measure the optical path length. They are introduced from an optical splitter, duplicated, and transmitted toward the OLT through the active and detour lines.

They are distributed by an optical coupler just in front of the OLT, and observed with an oscilloscope. The conventional measurement method evaluates the arrival time interval between the duplicated signals, and converts it to the difference between the lengths of the regular line and the detour line at a resolution of 1 m.

The difference in line length, *L* is described as

$$\mathbf{AL} = \mathbf{c} \cdot \mathbf{At} \text{ / n,} \tag{3}$$

where *c* is the speed of light, *t* is the difference between the signal arrival times for the regular and detour lines, and *n* is the refractive index of optical fiber.

Figure 10 shows the received pulses observed with an oscilloscope. When the detour line was 100 m shorter than the regular line, pulses traveling through the detour reached the oscilloscope about 500 ns earlier than through the regular line as shown in Fig 10 (a). The proposed system lengthened the detour line using the optical path length adjuster. Consequently, the former pulse approached the latter. Figure 10 (b) is an example when the gap was shorten to 20 m, where the regular line pulse arrived after a delay of 100 ns. This method failed if the difference between the line lengths was less than 1 m, because the two pulses combined as shown in Fig. 10 (c).

That is why the authors have developed an advanced technique for measuring a difference of less than 1 m between optical line lengths. Interferometry enables us to obtain more detailed measurements even when the optical pulses combine. A chirped light source generates interference in the waveform of a unified pulse.

A 1650 nm laser pulse, discharged from the test light source, is devided by the 2x8 optical splitter. Two pulses travels through the detour and regular line separately toward the oscilloscope.

Both pulses along the regular line, *E (L1, t)*, and the detour line, *E (L2, t)* are expressed as

$$E\left(L\_1, \mathbf{t}\right) = A\_1 \exp\left\{-i\left(\mathbf{k}\cdot\mathbf{n}\cdot\mathbf{L}\_1\cdot\mathbf{a}\mathbf{p}\cdot\mathbf{t} + \boldsymbol{\Phi}\right)\right\},\tag{4}$$

$$E\left(L\_2, t\right) = A\_2 \exp\left\{-i\left(\mathbf{k}\cdot\mathbf{n}\cdot\mathbf{L}\_2\cdot\mathbf{a}\mathbf{p}\cdot\mathbf{t} + \boldsymbol{\Phi}\_0\right)\right\},\tag{5}$$

where *Aj, k, n, Lj, j, t,* and *<sup>0</sup>* denote amplitude, wavenumber in a vacuum, refractive index of optical fiber, line length, frequency, time, and initial phase, respectively. Subscript *j* represents the regular line, 1, or the detour line, 2.

The intensity of a waveform with interference, *I*, is calculated by taking the square sum as

$$\mathbf{I} = \begin{vmatrix} \mathbf{E} \ (\mathbf{L}\_{I\prime} \ \mathbf{t}) + \mathbf{E} \ (\mathbf{L}\_{2\prime} \ \mathbf{t}) \ \ \end{vmatrix}^2$$

$$\mathbf{I} = A\_1 \mathbf{i}^2 + A\_2 \mathbf{i}^2 + 2 \ A\_1 \mathbf{A}\_2 \cos \ \begin{pmatrix} \mathbf{k} \ \cdot \mathbf{n} \ \cdot \mathbf{d} \ \mathbf{L} - \mathbf{A} \mathbf{p} \cdot \mathbf{t} \end{pmatrix}, \tag{6}$$

where *L* and represent the differences between line lengths and frequencies, respectively.

The proposed system uses laser pulses at a wavelength of 1650 mm to measure the optical path length. They are introduced from an optical splitter, duplicated, and transmitted

They are distributed by an optical coupler just in front of the OLT, and observed with an oscilloscope. The conventional measurement method evaluates the arrival time interval between the duplicated signals, and converts it to the difference between the lengths of the

Figure 10 shows the received pulses observed with an oscilloscope. When the detour line was 100 m shorter than the regular line, pulses traveling through the detour reached the oscilloscope about 500 ns earlier than through the regular line as shown in Fig 10 (a). The proposed system lengthened the detour line using the optical path length adjuster. Consequently, the former pulse approached the latter. Figure 10 (b) is an example when the gap was shorten to 20 m, where the regular line pulse arrived after a delay of 100 ns. This method failed if the difference between the line lengths was less than 1 m, because the two

That is why the authors have developed an advanced technique for measuring a difference of less than 1 m between optical line lengths. Interferometry enables us to obtain more detailed measurements even when the optical pulses combine. A chirped light source

A 1650 nm laser pulse, discharged from the test light source, is devided by the 2x8 optical splitter. Two pulses travels through the detour and regular line separately toward the oscilloscope. Both pulses along the regular line, *E (L1, t)*, and the detour line, *E (L2, t)* are expressed as

of optical fiber, line length, frequency, time, and initial phase, respectively. Subscript *j*

The intensity of a waveform with interference, *I*, is calculated by taking the square sum as

*I = | E (L1, t) + E (L2, t) | 2* 

*2 + 2 A1 A2 cos (k ·n ·*

*1 ·t +*

*2 ·t +*

*L –* 

represent the differences between line lengths and frequencies,

*<sup>0</sup>* denote amplitude, wavenumber in a vacuum, refractive index

*E (L1, t) = A1 exp [ -i (k ·n ·L1 –*

*E (L2, t) = A2 exp [ -i (k ·n ·L2 –*

*t* / *n*, (3)

*0 ) ]* , (4)

*0 ) ]* , (5)

 *·t)* , (6)

*t* is the difference between the signal arrival times for the

**4. Optical line length measurement** 

The difference in line length,

where *c* is the speed of light,

pulses combined as shown in Fig. 10 (c).

*j, t,* and

where *Aj, k, n, Lj,* 

*L* and

respectively.

where

toward the OLT through the active and detour lines.

regular line and the detour line at a resolution of 1 m.

generates interference in the waveform of a unified pulse.

 *= A1*

*2+ A2*

represents the regular line, 1, or the detour line, 2.

regular and detour lines, and *n* is the refractive index of optical fiber.

*L* is described as

*L* = *c ·* 

(c) Optical path difference 1m.

Fig. 10. Delay of duplicated received pulse.

Length Measurement for Optical Transmission Line Using Interferometry 329



0 200 400 600 800 1000 1200

Frequency (MHz) (a) Optical path difference 1 m.

0 200 400 600 800 1000 1200

(b) Optical path difference 0.5 m.

Frequency (MHz)

Time (ns) (d) Optical path difference 0 m.

(c) Optical path difference 0.1 m.

Time (ns)

1.0 0.8 0.6 0.4 0.2 0

> 1.0 0.8 0.6 0.4 0.2 0

Voltage (V)

Fig. 11. Time-domain unified optical pulses.

0

0

0.1

0.2

Voltage (mV)

0.3

0.1

0.2

Voltage (mV)

0.3

Voltage (V)

The waveform with interference depends on the delay between the pulses' arrival times. Experiments were carried out to obtain time-domain waveforms as shown in Fig. 11. When the optical path difference was 1 m, the waveform contained high-frequency waves as shown in Fig. 11 (a). The less the gap became, the lower-frequency the interfered waveform was composed of. Figures 11 (b) and (c) express that phenomenon. When the lengths of two lines coincided, a quite low-frequency waveform was observed as shown in Fig. 11 (d).

A Fourier-transform spectrum reveals the characteristics. When the optical path difference was 1 m, the waveform with interference was composed of the power spectrum shown in Fig. 12 (a). The peak power represented that the major frequency component was around 700 MHz. Figure 12 (b) and (c) indicate that the peak powers for gaps of 0.5 and 0.1 m were around 400 and 70 MHz, respectively. It became difficult to determine the peak for smaller gaps, because the frequency peak became so low that it was hidden by the near directcurrent part of the frequency component. When the lengths of duplicated lines coincided, the power spectrum was obtained as Fig. 12 (d).

An evaluation of the frequency characteristics in the interfered waveforms showed that the peak power frequencies are proportional to the difference between the line lengths from -1 to 1 m as shown in Fig. 13, where the horizontal and vertical axes express the optical path difference and the frequency for the maximum power of Fourier-transformed spectrum, respectively. This result helps us to determine the optimal position for adjustment. The optimal position where the line lengths coincide can be estimated by extrapolating the data.

The waveform with interference depends on the delay between the pulses' arrival times. Experiments were carried out to obtain time-domain waveforms as shown in Fig. 11. When the optical path difference was 1 m, the waveform contained high-frequency waves as shown in Fig. 11 (a). The less the gap became, the lower-frequency the interfered waveform was composed of. Figures 11 (b) and (c) express that phenomenon. When the lengths of two lines coincided, a quite low-frequency waveform was observed as shown in Fig. 11 (d).

A Fourier-transform spectrum reveals the characteristics. When the optical path difference was 1 m, the waveform with interference was composed of the power spectrum shown in Fig. 12 (a). The peak power represented that the major frequency component was around 700 MHz. Figure 12 (b) and (c) indicate that the peak powers for gaps of 0.5 and 0.1 m were around 400 and 70 MHz, respectively. It became difficult to determine the peak for smaller gaps, because the frequency peak became so low that it was hidden by the near directcurrent part of the frequency component. When the lengths of duplicated lines coincided,

An evaluation of the frequency characteristics in the interfered waveforms showed that the peak power frequencies are proportional to the difference between the line lengths from -1 to 1 m as shown in Fig. 13, where the horizontal and vertical axes express the optical path difference and the frequency for the maximum power of Fourier-transformed spectrum, respectively. This result helps us to determine the optimal position for adjustment. The optimal position where the line lengths coincide can be estimated by extrapolating the data.



(b) Optical path difference 0.5 m.

Time (ns)

(a) Optical path difference 1 m.

Time (ns)

the power spectrum was obtained as Fig. 12 (d).

1.0 0.8 0.6 0.4 0.2

Voltage (V)

0

1.0 0.8 0.6 0.4 0.2

Voltage (V)

0

Fig. 11. Time-domain unified optical pulses.

Length Measurement for Optical Transmission Line Using Interferometry 331

We finally applied the prototype of the optical line shunt system to a 15 km GE-PON optical transmission line replacement. Photographs of our experimental setup are shown in Fig. 14. Figure 14 (a), (b), and (c) display a whole system for experimental optical transmission line, the side view, and the perspective view of the optical line length adjuster, respectively.

A 10 m optical fiber extension was added to the transmission line, while optical signals were switched between the duplicated lines during transmission. We have evaluated the frame loss that occurred during optical line replacement by measuring with a SmartBit network performance analyzer (Spilent Communications, 2011). Optical line length difference was purposely set at 0, 50, 80 and 120 mm in order to verify the effects of measurement accuracy. Same optical line replacement operation, illustrated in Fig. 5, was conducted for each case. No frame loss was observed at any stage of the replacement procedure if the difference between the duplicated line lengths was less than 80 mm. If the difference exceeded 80 mm,

We also evaluated communication quality of the transmission signals through duplicated lines. A 1488.6 nm optical pulse was actually transmitted along both detour and regular lines. Figure 15 shows the eye diagrams of the received signals with regard to optical path difference, L, 1, 22, 44, 66, 88, and 110 mm. If two line lengths were close, communication quality was satisfactory as the eye opened wide in the diagram. When the difference became larger, the transmitted signals turned out erroneous because binary bit patterns were breaking. As a result, we confirmed that the optical signals were completely switched between the regular, detour, and new lines on condition that the line length was adjusted with sufficient accuracy. The experimental results proved that our proposed system successfully relocated

We used a 1650 nm DFB-LD as a test light source. Because it was chirped nonlinearly, interference depended on the pulse width. Experiments were conducted to evaluate the effects on the power spectrum of the interfered waveform with regard to several pulse

The results indicated that the test light pulse with narrower width provided higher sensitivity in estimating optical line length coincidence as shown in Fig. 16, where the horizontal and vertical axes express the optical path difference and the frequency for the maximum power of Fourier-transformed spectrum, respectively. Note that in the practical usage of the narrow width pulse with an oscilloscope, measurable range is restricted by the upper limit of the Fourier-transformed frequency. For example, a test light with 20 ns pulse width has a sensitivity of 6 MHz/mm while 1 GHz oscilloscope is necessary to identify an

As discussed in Section 4, it is difficult for the above proposed method to directly identify the point where the lengths of the detour and regular lines coincide because the frequency

signal multiplexing caused frame loss at step 3 in Fig. 5.

an in-service broadband network without any service interruption.

**5.2 Deterministic line length detection using frequency shift method** 

**5. Improvement of line length measurement** 

widths 20, 50, 100, 200, and 500 ns.

optical path difference of 150 mm.

**5.1 Sensitivity improvement due to narrow pulse width** 

Fig. 12. Frequency-domain unified optical pulses.

Fig. 13. Estimation of line length coincidence.

Our investigation has established a technique for distinguishing the difference between line lengths to an accuracy of better than 10 mm by analyzing interfering waveforms created by chirped laser pulses. We have realized a complete length measurement for optical transmission lines from 100 m to 10 mm.

0 200 400 600 800 1000 1200

0 200 400 600 800 1000 1200

0


Our investigation has established a technique for distinguishing the difference between line lengths to an accuracy of better than 10 mm by analyzing interfering waveforms created by chirped laser pulses. We have realized a complete length measurement for optical

Optical path difference (m)

P

e

a

k fre

q

u

e

n

c

H

z)

y (M

200

400

600

800

Frequency (MHz) (d) Optical path difference 0 m.

Frequency (MHz) (c) Optical path difference 0.1 m.

0

0

0.1

0.2

Voltage (mV)

Fig. 12. Frequency-domain unified optical pulses.

Fig. 13. Estimation of line length coincidence.

transmission lines from 100 m to 10 mm.

0.3

0.1

0.2

Voltage (mV)

0.3

We finally applied the prototype of the optical line shunt system to a 15 km GE-PON optical transmission line replacement. Photographs of our experimental setup are shown in Fig. 14. Figure 14 (a), (b), and (c) display a whole system for experimental optical transmission line, the side view, and the perspective view of the optical line length adjuster, respectively.

A 10 m optical fiber extension was added to the transmission line, while optical signals were switched between the duplicated lines during transmission. We have evaluated the frame loss that occurred during optical line replacement by measuring with a SmartBit network performance analyzer (Spilent Communications, 2011). Optical line length difference was purposely set at 0, 50, 80 and 120 mm in order to verify the effects of measurement accuracy. Same optical line replacement operation, illustrated in Fig. 5, was conducted for each case. No frame loss was observed at any stage of the replacement procedure if the difference between the duplicated line lengths was less than 80 mm. If the difference exceeded 80 mm, signal multiplexing caused frame loss at step 3 in Fig. 5.

We also evaluated communication quality of the transmission signals through duplicated lines. A 1488.6 nm optical pulse was actually transmitted along both detour and regular lines. Figure 15 shows the eye diagrams of the received signals with regard to optical path difference, L, 1, 22, 44, 66, 88, and 110 mm. If two line lengths were close, communication quality was satisfactory as the eye opened wide in the diagram. When the difference became larger, the transmitted signals turned out erroneous because binary bit patterns were breaking.

As a result, we confirmed that the optical signals were completely switched between the regular, detour, and new lines on condition that the line length was adjusted with sufficient accuracy. The experimental results proved that our proposed system successfully relocated an in-service broadband network without any service interruption.

## **5. Improvement of line length measurement**

## **5.1 Sensitivity improvement due to narrow pulse width**

We used a 1650 nm DFB-LD as a test light source. Because it was chirped nonlinearly, interference depended on the pulse width. Experiments were conducted to evaluate the effects on the power spectrum of the interfered waveform with regard to several pulse widths 20, 50, 100, 200, and 500 ns.

The results indicated that the test light pulse with narrower width provided higher sensitivity in estimating optical line length coincidence as shown in Fig. 16, where the horizontal and vertical axes express the optical path difference and the frequency for the maximum power of Fourier-transformed spectrum, respectively. Note that in the practical usage of the narrow width pulse with an oscilloscope, measurable range is restricted by the upper limit of the Fourier-transformed frequency. For example, a test light with 20 ns pulse width has a sensitivity of 6 MHz/mm while 1 GHz oscilloscope is necessary to identify an optical path difference of 150 mm.

## **5.2 Deterministic line length detection using frequency shift method**

As discussed in Section 4, it is difficult for the above proposed method to directly identify the point where the lengths of the detour and regular lines coincide because the frequency

Length Measurement for Optical Transmission Line Using Interferometry 333

30 mV

Voltage

Voltage

Voltage

Voltage

Voltage

Voltage

30 mV

30 mV

30 mV

30 mV

30 mV

Fig. 15. Eyediagram of duplicated transmission signal.

10 ns

10 ns

10 ns

10 ns

10 ns

10 ns

Time

L=22 mm

L1 mm

L=44 mm

L=66 mm

L=88 mm

L=110 mm

peak is hidden by the near direct-current part of the frequency component in the power spectrum.

(a) Whole system for experimental optical transmission line.

(b) Side view of optical line length adjuster.

(c) Perspective view of optical line length adjuster.

Fig. 14. Photographs of optical line replacement experiments.

peak is hidden by the near direct-current part of the frequency component in the power

Regular

Detour line

line

(a) Whole system for experimental optical transmission line.

(b) Side view of optical line length adjuster.

Collimators

Retroreflectors

(c) Perspective view of optical line length adjuster.

Collimators

Sliding stage Retroreflectors

ONUs

Optical line length adjuster

spectrum.

OLTs

Oscilloscope

Sliding stage

Fig. 14. Photographs of optical line replacement experiments.

Fig. 15. Eyediagram of duplicated transmission signal.

Length Measurement for Optical Transmission Line Using Interferometry 335

white circles in Fig. 20, where the horizontal and vertical axes express the optical path difference and the frequency for the maximum power of Fourier-transformed spectrum,

Black circles in the figure represent the remainders of subtracting 80 MHz from the peak frequency. They also show the linearity as good as we obtained without the frequency filter. If adopting this method, we can determine the optimal position where the line lengths coincide in high accuracy only by finding the specified value of the frequency shifter instead

Fig. 17. Advanced duplex optical fiber transmission system with frequency shifter.

O/E

Optical line length adjuster

Detour (15 km)

Regular line (15 km)

WIC

Optical splitter

Oscilloscope

80 MHz frequency shifter

light source

respectively.

of extrapolation.

Fig. 16. Estimation of line length coincidence.

We are developing an advanced method to resolve the problem. Our new proposal is to deliberately shift the interference frequency. Figure 17 shows an advanced duplex optical fiber transmission system with frequency shifter. It equips an 80 MHz frequency shifter along the detour line. Other devices are the same as those described in Section 3.2.

We have conducted the same experiments with this advanced transmission system as before. We evaluated interference of the duplicated signals by changing the optical path difference with the optical line length adjuster.

Experimental results are shown in Figs. 18 and 19. Time-domain waveform of unified optical pulses indicates in Fig. 18 that signals along the detour and regular lines have interfered. A Fourier-transformation provided the frequency-domain characteristics of the interference as shown in Fig. 19. It reveals that each interfered waveform has the peak of spectrum power. In case the detour line length agrees with the regular line, the frequency of the peak power is 80 MHz which is as much as we have assigned with the frequency shifter.

An evaluation of the frequency of the peak power in terms of the optical path difference confirmed that the frequencies were proportional to the optical path difference as shown by

150 100 50 0 50 100 150

Optical path difference(mm)

along the detour line. Other devices are the same as those described in Section 3.2.

We are developing an advanced method to resolve the problem. Our new proposal is to deliberately shift the interference frequency. Figure 17 shows an advanced duplex optical fiber transmission system with frequency shifter. It equips an 80 MHz frequency shifter

We have conducted the same experiments with this advanced transmission system as before. We evaluated interference of the duplicated signals by changing the optical path

Experimental results are shown in Figs. 18 and 19. Time-domain waveform of unified optical pulses indicates in Fig. 18 that signals along the detour and regular lines have interfered. A Fourier-transformation provided the frequency-domain characteristics of the interference as shown in Fig. 19. It reveals that each interfered waveform has the peak of spectrum power. In case the detour line length agrees with the regular line, the frequency of the peak power is 80 MHz which is as much as we have assigned with the frequency

An evaluation of the frequency of the peak power in terms of the optical path difference confirmed that the frequencies were proportional to the optical path difference as shown by

0

Fig. 16. Estimation of line length coincidence.

difference with the optical line length adjuster.

500

Frequency(MHz)

shifter.

1000

:500 ns :200 ns :100 ns :50 ns :20 ns

Pulse width

white circles in Fig. 20, where the horizontal and vertical axes express the optical path difference and the frequency for the maximum power of Fourier-transformed spectrum, respectively.

Black circles in the figure represent the remainders of subtracting 80 MHz from the peak frequency. They also show the linearity as good as we obtained without the frequency filter.

If adopting this method, we can determine the optimal position where the line lengths coincide in high accuracy only by finding the specified value of the frequency shifter instead of extrapolation.

Fig. 17. Advanced duplex optical fiber transmission system with frequency shifter.

Length Measurement for Optical Transmission Line Using Interferometry 337

0 100 200 300 400 500 Frequency (MHz)

(a) Optical path difference 0.31 m.

173.7 MHz

0 100 200 300 400 500 Frequency(MHz)

(b) Optical path difference 0.14 m.

0 100 200 300 400 500 Frequency(MHz)

(c) Optical path difference 0 m.

285.06 MHz

=205.06 MHz

=93.7 MHz

0

0

0

10

20

Voltage(mV)

Fig. 19. Frequency-domain unified optical pulses.

30

40

50

10

20

Voltage(mV)

30

40

80 MHz

80 MHz

50

10

20

Voltage (mV)

30

40

80 MHz

50

Fig. 18. Time-domain unified optical pulses.

0 100 200 400300

0 100 200 400300

Time (ns)

(b) Optical path difference 0.14 m.


Time (ns)

(a) Optical path difference 0.31 m.

0

100

0

0

20

40

Voltage(mV)

Fig. 18. Time-domain unified optical pulses.

60

80

100


20

40

60

Voltage (mV)

80


20

40

60

Voltage (mV)

80

100

Fig. 19. Frequency-domain unified optical pulses.

Length Measurement for Optical Transmission Line Using Interferometry 339

Azuma, Y., Tanaka, K., Yoshida, K., Katayama, K., Tsujimura, T., & Shimizu, M. (2008).

Cao, S. C. & Cartledge, J.C. (2002). Measurement-Based Method for Characterizing the

Hornung, S., Wood, R., & Keeble, P. (1990). Single-mode optical networks to the home,

ITU-T Recommendation G.983.1 (1998). Broadband optical access systems based on Passive

Sankawa, I., Yamamoto, F., Okumura, Y., & Ogura, Y. (2006). Cost and quantity analysis of

Tanaka, K., & Horiuchi, Y. (2008). 1:N OLT Redundant Protection Architecture in Ethernet

Tanaka, K., Tsujimura, T., Yoshida, K., Katayama, K., & Azuma, Y. (2009). Frame-loss-free

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Tsujimura, T., Tanaka, K., Yoshida, K., Katayama, K., & Azuma, Y. (2009a). Infallible Layer-

Tsujimura, T., Yoshida, K., Tanaka, K., & Azuma, Y. (2010). Transmission length

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*IEEE Photonics Technology Letters*, Vol.14, No.11, (2002), pp.1578- 1580 Fukada, Y., Suzuki, K., Nakamura, H., Yoshimoto, N., & Tsubokawa, M. (2008). First

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*Optical Communications 2008*, paper WE.2.F.4, 2008

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demonstration of fast automatic-gain-control (AGC) PDFA for amplifying burstmode PON upstream signal, *Proceedings of European Conference on Networks and* 

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April 1990

Optical Networks (PON), 1998

*2009*, paper PDPD6, 2009

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Spilent Communications. (2011). *SmartBits,* Available from:

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Fig. 20. Deterministic identification of line length coincidence.

## **6. Conclusion**

This paper proposed the high-resolution optical line length measurement technique, which was applied to the switching method for optical transmission lines transferring live optical signals. The method exchanged optical fibers instead of using electric apparatus to control transmission speed.

We have disclosed the requirements for replacement of in-service optical transmission cables after investigating problems of the existing methods.

We next proposed the optical line switching technique and the optical cable replacement procedure that used duplex transmission lines. It was determined that the tolerance of the difference in duplex line length is 80 mm as for the GE-PON transmission system.

The optical line shunt system was designed, and its prototype was actually constructed. An optical line length adjuster, designed based on an FSO system, continuously lengthened the optical line up to 100 m with a resolution of 0.1 mm.

An optical line length measurement technique was investigated in detail by evaluating the duplicated test light pulses. As a result, it successfully identified the difference in length between the duplicated lines from 100 m to 10 mm. An interferometry measurement distinguished the difference between line lengths to an accuracy of better than 10 mm by analyzing interfering waveforms created by chirped laser pulses. This system was applied to a 15 km GE-PON network and succeeded in replacing the communication lines without inducing any frame loss.

We also discussed the improvement of line length measurement. It was suggested that the test light with narrow pulse width could give a higher sensitivity. If using the frequency shifter within the detour transmission line, we could identify the optimal position where the line lengths coincide in high accuracy without extrapolation.

## **7. References**

338 Interferometry – Research and Applications in Science and Technology

: measured value : remainder

0 0.1 0.2 0.3 0.4

This paper proposed the high-resolution optical line length measurement technique, which was applied to the switching method for optical transmission lines transferring live optical signals. The method exchanged optical fibers instead of using electric apparatus to control

We have disclosed the requirements for replacement of in-service optical transmission

We next proposed the optical line switching technique and the optical cable replacement procedure that used duplex transmission lines. It was determined that the tolerance of the

The optical line shunt system was designed, and its prototype was actually constructed. An optical line length adjuster, designed based on an FSO system, continuously lengthened the

An optical line length measurement technique was investigated in detail by evaluating the duplicated test light pulses. As a result, it successfully identified the difference in length between the duplicated lines from 100 m to 10 mm. An interferometry measurement distinguished the difference between line lengths to an accuracy of better than 10 mm by analyzing interfering waveforms created by chirped laser pulses. This system was applied to a 15 km GE-PON network and succeeded in replacing the communication lines without

We also discussed the improvement of line length measurement. It was suggested that the test light with narrow pulse width could give a higher sensitivity. If using the frequency shifter within the detour transmission line, we could identify the optimal position where the

difference in duplex line length is 80 mm as for the GE-PON transmission system.

0

Fig. 20. Deterministic identification of line length coincidence.

cables after investigating problems of the existing methods.

optical line up to 100 m with a resolution of 0.1 mm.

line lengths coincide in high accuracy without extrapolation.

200

Peak

**6. Conclusion** 

transmission speed.

inducing any frame loss.

frequency (M

Hz)

400

Optical path difference (m)

0.5


**1. Introduction**

standing wave scattering, respectively.

Atomic interferometry is very useful in fundamental studies of coherence, decoherence and phase shifts and for practical precision measurements, with the example of gravimeters, gyroscopes, and gradiometers (Cronin et al., 2009). Bose-Einstein Condensate (BEC) based atomic interferometry provides high contrasts, long integration times and the possible use of small devices (Baumgärtner et al., 2010; Sapiro et al., 2009a; Torii et al., 2000). In such atomic interferometry, coherent momentum manipulation is very effective for splitting and recombining the condensate (Deng et al., 1999; Kozuma, Deng et al., 1999; Ovchinnikov et al., 1999), for realizing the interference. In some precision measurements, the accumulated phase is positively correlated to the atomic velocity, so that the larger the atomic momentum is, the

**High Order Momentum States by** 

*School of Electronics Engineering and Computer Science,* 

**Light Wave Scattering** 

Xiaoji Zhou and Xuguang Yue

*Peking University, Beijing* 

*China* 

**17**

Based on our previous experimental and theoretical work about the interaction between laser pulses and a BEC, in this chapter we mainly present methods of getting high order of momentum states by resonant superradiance (Zhou et al., 2009) and by multi-pulse Kapitza-Dirac scattering (Xiong et al., 2011), corresponding to travelling wave scattering and

In the superradiant Rayleigh scattering (Bar-Gill et al., 2007; Guo et al., 2008; Inouye et al., 1999; Kozuma, Suzuki et al., 1999; Moore & Meystre, 1999; Pu et al., 2003; Sadler et al., 2007; Schneble et al., 2003; Slama et al., 2007; Zobay & Nikolopoulos, 2006), the spatial and time evolutions of superradiant scattering are studied for a weak pump beam with different frequency components traveling along the long axis of an elongated Bose-Einstein condensate. Through the analysis of the mode competition between the different resonant channels and the local depletion of the spatial distribution, we can get a large number of high-order forward

On the other hand, the atomic diffraction from a standing wave light grating is a primary method of atomic momentum manipulation. Different to usual separation of the Kapitza-Dirac regime, the Bragg regime and the channeling regime (Keller et al., 1999), the scattering process is described by the projection of atomic states between the momentum presentation and Bloch states form by the scattering standing waves. According to this

modes by resonant frequency components of the pump beam (Zhou et al., 2009).

more precise the measurements can be if the measuring time is unchanged.


## **High Order Momentum States by Light Wave Scattering**

Xiaoji Zhou and Xuguang Yue

*School of Electronics Engineering and Computer Science, Peking University, Beijing China* 

## **1. Introduction**

340 Interferometry – Research and Applications in Science and Technology

Tsujimura, T., Yoshida, K., Tanaka, K., Katayama, K., & Azuma, Y. (2009b). Interruption-free

Watanabe, I., Saito, H., Kobayashi, H., & Takashima, S. (1992). Subscriber optical fiber cable

Xu, D. J., Yen, W., & Ho, E. (2001). Proposal of a New Protection Mechanism for ATM PON

Yoshida, K., & Tsujimura, T. (2010). Seamless Transmission Between Single-mode Optical

Yoshida, K., Tanaka, K., & Tsujimura, T. (2011). Robotic Waveguide by Free Space Optics, In: *Advances in Mechatronics,* Martínez-Alfaro, H., InTech, ISBN 978-953-307-373-6 Yoshida, K., Tanaka, K., Katayama, K., Tsujimura, T., & Azuma, Y. (2009). Collimator Focus

*Proceedings of ICROS-SICE International Joint Conference,* pp. 1338-1341, 2009

*Application,* Vol. 8, No. 16, (2009), pp. 125-131

2160-2165, 2001

*Technology*, Vol. 10, No. 6, (June, 1992), pp. 720-727 Willebrand, H. et al. (1999), *Free-Space Optics*, Sams Publishing

*System Integration*, Vol. 3, No. 2, (2010), pp.94-100

Shunt System for Fiber Optics Transmission Line, *SICE Transactions on Industrial* 

transfer splicing using mechanically transferable connectors, *IEEE Journal of Lightwave* 

Interface, *Proceedings of IEEE International Conference on Communications*, Vol. 7, pp.

Fibers Using Free Space Optics System, *SICE Journal of Control Measurement and* 

Adjustment for Free Space Optics System Using Single-mode Optical Fiber,

Atomic interferometry is very useful in fundamental studies of coherence, decoherence and phase shifts and for practical precision measurements, with the example of gravimeters, gyroscopes, and gradiometers (Cronin et al., 2009). Bose-Einstein Condensate (BEC) based atomic interferometry provides high contrasts, long integration times and the possible use of small devices (Baumgärtner et al., 2010; Sapiro et al., 2009a; Torii et al., 2000). In such atomic interferometry, coherent momentum manipulation is very effective for splitting and recombining the condensate (Deng et al., 1999; Kozuma, Deng et al., 1999; Ovchinnikov et al., 1999), for realizing the interference. In some precision measurements, the accumulated phase is positively correlated to the atomic velocity, so that the larger the atomic momentum is, the more precise the measurements can be if the measuring time is unchanged.

Based on our previous experimental and theoretical work about the interaction between laser pulses and a BEC, in this chapter we mainly present methods of getting high order of momentum states by resonant superradiance (Zhou et al., 2009) and by multi-pulse Kapitza-Dirac scattering (Xiong et al., 2011), corresponding to travelling wave scattering and standing wave scattering, respectively.

In the superradiant Rayleigh scattering (Bar-Gill et al., 2007; Guo et al., 2008; Inouye et al., 1999; Kozuma, Suzuki et al., 1999; Moore & Meystre, 1999; Pu et al., 2003; Sadler et al., 2007; Schneble et al., 2003; Slama et al., 2007; Zobay & Nikolopoulos, 2006), the spatial and time evolutions of superradiant scattering are studied for a weak pump beam with different frequency components traveling along the long axis of an elongated Bose-Einstein condensate. Through the analysis of the mode competition between the different resonant channels and the local depletion of the spatial distribution, we can get a large number of high-order forward modes by resonant frequency components of the pump beam (Zhou et al., 2009).

On the other hand, the atomic diffraction from a standing wave light grating is a primary method of atomic momentum manipulation. Different to usual separation of the Kapitza-Dirac regime, the Bragg regime and the channeling regime (Keller et al., 1999), the scattering process is described by the projection of atomic states between the momentum presentation and Bloch states form by the scattering standing waves. According to this

Fig. 1. (Color online) Our experimental scheme. A cigar-shape BEC is illuminated by a far off-resonant laser pulse along its long axis **zˆ**. Collective Rayleigh scattering induces superradiance. Two end-fire modes, which are also along **zˆ** axis, form in superradiance

High Order Momentum States by Light Wave Scattering 343

order scattering mode is suppressed at the resonant condition Δ*ω* = 8*ω<sup>r</sup>* and the forward second order mode is enhanced, resulting from the combination of mode competition effects and spatial distribution of the modes. The case of the three-frequency pump beams for a large number of the forward third order scattering modes, and the higher modes for more resonant frequencies are studied, which supplies a new method to get a large number of atoms in higher

We consider the pump laser, with amplitude E*l*(*t*), polarization **ey**, wave vector **k***l*, frequencies *ω<sup>l</sup>* and *ω<sup>l</sup>* − Δ*ωn*, propagating along the long axis **zˆ** of an elongated BEC, **E***<sup>l</sup>* = E*l*(*t*)**ey**[(1 +

happens, end-fire modes spread along the same axis. The E<sup>+</sup> mode has the same direction as the incident light and mainly interacts with the right part of the condensate, and the E− mode overlaps with the left part of the condensate. The atoms are recoiled to some discrete momentum states with momentum 2*mh*¯ **k**, where *m* is an integer and the wave vector of end-fire mode light *k* is approximated as *kl* for energy conservation. The total electric field **E**(**r**, *t*) = **E**(+) + **E**(−) is given by (Bar-Gill et al., 2007; Li, Zhou et al., 2008; Yang et al., 2008;

)E*l*(*t*)*e*−i(*ωlt*−*klz*)

where *<sup>ω</sup>* <sup>=</sup> *ck*, **<sup>E</sup>**(−) <sup>=</sup> **<sup>E</sup>**(+)∗, and <sup>E</sup><sup>+</sup> is ignored because it has the same wave vector as the pump beam but is very small in comparison to E*l*. Δ*ω<sup>n</sup>* satisfies the condition Δ*ω<sup>n</sup>* � *ω<sup>l</sup>* (Bar-Gill et al., 2007) and the initial phases of the different frequency components

Since the BEC is tightly constrained in its short axis (**xˆ**, **yˆ**) in the present superradiance setup and the Fresnel number of the optical field is around 1, one dimensional approximation is usually used (Bar-Gill et al., 2007; Hilliard et al., 2008; Inouye et al., 1999; Li, Zhou et al.,

/2 <sup>+</sup> E−(*z*, *<sup>t</sup>*)*e*−i(*ωt*+*kz*)

]**ey** (1)

)*ei*(*klz*−*ωlt*) + *c*.*c*.]/2, as shown in Fig. 1. When supperradiant Rayleigh scattering

process and the 1st-order recoiled atoms obtain a momentum of 2¯*h***k**.

order forward modes. Finally, some discussion and conclusion are given.

*ei*Δ*ωnt*

**2.1 Model for a multiple-frequency end-pumped beam**

Σ*nei*Δ*ωnt*

Zobay & Nikolopoulos, 2006)

are assumed to be zero.

**<sup>E</sup>**(+)(**r**, *<sup>t</sup>*) = [(<sup>1</sup> <sup>+</sup> <sup>∑</sup>*<sup>n</sup>*

method, we are able to design and realize several specific high order momentum states for the atomic interferometry (Xiong et al., 2011).

## **2. High-order momentum modes by resonant superradiant scattering**

A typical superradiance experiment consists in a far off-resonant laser pulse traveling along the short axis of a cigar-shaped BEC sample (Inouye et al., 1999), the scattered lights, called end-fire modes, propagate along the long axis of the condensate, and the recoiled atoms are refereed to as side modes. A series of experiments (Bar-Gill et al., 2007; Kozuma, Suzuki et al., 1999; Sadler et al., 2007; Schneble et al., 2003; Slama et al., 2007) have sparked related interests in phase-coherent amplification of matter waves (Kozuma, Suzuki et al., 1999; Schneble et al., 2003), quantum information (Bar-Gill et al., 2007), collective scattering instability (Slama et al., 2007), and coherent imaging (Sadler et al., 2007). Several theoretical descriptions of these cooperative scattering in BEC with single-frequency pump have also been presented (Guo et al., 2008; Moore & Meystre, 1999; Pu et al., 2003; Zobay & Nikolopoulos, 2006).

For the long and weak pump beam, we can observe the forward peaks correspond to Bragg diffraction of atoms (Inouye et al., 1999), where the high order scattering is limited by detuning barriers for the end-fire mode radiation (Zobay & Nikolopoulos, 2007). On the other hand, a X-shaped recoiling pattern is demonstrated in a short and strong pulse as Kapitza-Dirac diffraction of atoms (Schneble et al., 2003), where an atom in the condensate absorbs a photon from the pump laser, then emits a photon into an end-fire mode, and recoils forwardly. Meanwhile another atom absorbs a photon from the end-fire modes, emits into the pump beam and finally recoils backwardly. In this case, there is an energy mismatch of four times the one-photon recoil kinetic energy ¯*hωr* in backward scattering, which then remains very weak unless a short pumping pulse with a broad spectrum is used. Hence, two phase-locked incident lasers with the frequency difference Δ*ω* compensating for the energy mismatch has been used (Bar-Gill et al., 2007; Cola et al., 2009; Yang et al., 2008), which is named resonant superradiance, where a large number of backward recoiling atoms can be produced.

Followed that, it is attractive to extent this idea to achieve a high momentum transfer by overcoming the detuning barriers, by a weak and long pump beams with the resonant frequency. It requires to analysis the competition between the different transition channels and the spatial distribution of different modes. Because the above traditional superradiant-scattering configuration involves many atomic side modes coupled together, to simplify it, we chose another configuration where a pump beam travels along the long axis of the BEC. This scheme is widely studied in photon echo (Piovella et al., 2003), decoherence (Fallani et al., 2005), spatial distribution effects (Li, Zhou et al., 2008) and self-organized formation of dynamic gratings (Hilliard et al., 2008). Since the pulse length is far longer than the initial spontaneous process (Zobay & Nikolopoulos, 2006), we choose the semi-classical theory which can well describe the experimental results (Bar-Gill et al., 2007; Yang et al., 2008; Zobay & Nikolopoulos, 2006).

In this section, we first introduce the semi-classical theory for the superradiance scattering with a several-frequency pump in the weak coupling. Then the spatial and time evolutions of scattered modes are analyzed for two-frequency pump beam, we find the backward first 2 Will-be-set-by-IN-TECH

method, we are able to design and realize several specific high order momentum states for

A typical superradiance experiment consists in a far off-resonant laser pulse traveling along the short axis of a cigar-shaped BEC sample (Inouye et al., 1999), the scattered lights, called end-fire modes, propagate along the long axis of the condensate, and the recoiled atoms are refereed to as side modes. A series of experiments (Bar-Gill et al., 2007; Kozuma, Suzuki et al., 1999; Sadler et al., 2007; Schneble et al., 2003; Slama et al., 2007) have sparked related interests in phase-coherent amplification of matter waves (Kozuma, Suzuki et al., 1999; Schneble et al., 2003), quantum information (Bar-Gill et al., 2007), collective scattering instability (Slama et al., 2007), and coherent imaging (Sadler et al., 2007). Several theoretical descriptions of these cooperative scattering in BEC with single-frequency pump have also been presented (Guo et al., 2008; Moore & Meystre, 1999;

For the long and weak pump beam, we can observe the forward peaks correspond to Bragg diffraction of atoms (Inouye et al., 1999), where the high order scattering is limited by detuning barriers for the end-fire mode radiation (Zobay & Nikolopoulos, 2007). On the other hand, a X-shaped recoiling pattern is demonstrated in a short and strong pulse as Kapitza-Dirac diffraction of atoms (Schneble et al., 2003), where an atom in the condensate absorbs a photon from the pump laser, then emits a photon into an end-fire mode, and recoils forwardly. Meanwhile another atom absorbs a photon from the end-fire modes, emits into the pump beam and finally recoils backwardly. In this case, there is an energy mismatch of four times the one-photon recoil kinetic energy ¯*hωr* in backward scattering, which then remains very weak unless a short pumping pulse with a broad spectrum is used. Hence, two phase-locked incident lasers with the frequency difference Δ*ω* compensating for the energy mismatch has been used (Bar-Gill et al., 2007; Cola et al., 2009; Yang et al., 2008), which is named resonant superradiance, where a large number of backward recoiling atoms can be

Followed that, it is attractive to extent this idea to achieve a high momentum transfer by overcoming the detuning barriers, by a weak and long pump beams with the resonant frequency. It requires to analysis the competition between the different transition channels and the spatial distribution of different modes. Because the above traditional superradiant-scattering configuration involves many atomic side modes coupled together, to simplify it, we chose another configuration where a pump beam travels along the long axis of the BEC. This scheme is widely studied in photon echo (Piovella et al., 2003), decoherence (Fallani et al., 2005), spatial distribution effects (Li, Zhou et al., 2008) and self-organized formation of dynamic gratings (Hilliard et al., 2008). Since the pulse length is far longer than the initial spontaneous process (Zobay & Nikolopoulos, 2006), we choose the semi-classical theory which can well describe the experimental results (Bar-Gill et al., 2007;

In this section, we first introduce the semi-classical theory for the superradiance scattering with a several-frequency pump in the weak coupling. Then the spatial and time evolutions of scattered modes are analyzed for two-frequency pump beam, we find the backward first

**2. High-order momentum modes by resonant superradiant scattering**

the atomic interferometry (Xiong et al., 2011).

Pu et al., 2003; Zobay & Nikolopoulos, 2006).

Yang et al., 2008; Zobay & Nikolopoulos, 2006).

produced.

Fig. 1. (Color online) Our experimental scheme. A cigar-shape BEC is illuminated by a far off-resonant laser pulse along its long axis **zˆ**. Collective Rayleigh scattering induces superradiance. Two end-fire modes, which are also along **zˆ** axis, form in superradiance process and the 1st-order recoiled atoms obtain a momentum of 2¯*h***k**.

order scattering mode is suppressed at the resonant condition Δ*ω* = 8*ω<sup>r</sup>* and the forward second order mode is enhanced, resulting from the combination of mode competition effects and spatial distribution of the modes. The case of the three-frequency pump beams for a large number of the forward third order scattering modes, and the higher modes for more resonant frequencies are studied, which supplies a new method to get a large number of atoms in higher order forward modes. Finally, some discussion and conclusion are given.

### **2.1 Model for a multiple-frequency end-pumped beam**

We consider the pump laser, with amplitude E*l*(*t*), polarization **ey**, wave vector **k***l*, frequencies *ω<sup>l</sup>* and *ω<sup>l</sup>* − Δ*ωn*, propagating along the long axis **zˆ** of an elongated BEC, **E***<sup>l</sup>* = E*l*(*t*)**ey**[(1 + Σ*nei*Δ*ωnt* )*ei*(*klz*−*ωlt*) + *c*.*c*.]/2, as shown in Fig. 1. When supperradiant Rayleigh scattering happens, end-fire modes spread along the same axis. The E<sup>+</sup> mode has the same direction as the incident light and mainly interacts with the right part of the condensate, and the E− mode overlaps with the left part of the condensate. The atoms are recoiled to some discrete momentum states with momentum 2*mh*¯ **k**, where *m* is an integer and the wave vector of end-fire mode light *k* is approximated as *kl* for energy conservation. The total electric field **E**(**r**, *t*) = **E**(+) + **E**(−) is given by (Bar-Gill et al., 2007; Li, Zhou et al., 2008; Yang et al., 2008; Zobay & Nikolopoulos, 2006)

$$\mathbf{E}^{(+)} (\mathbf{r}, t) = [(1 + \sum\_{\mathbf{n}} e^{i\Lambda \omega\_{\mathbf{n}} t}) \mathcal{E}\_l(t) e^{-\mathbf{i} \left(\omega\_l t - k\_l z\right)}/2 + \mathcal{E}\_-(z, t) e^{-\mathbf{i} \left(\omega t + kz\right)}] \mathbf{e}\_{\mathbf{y}} \tag{1}$$

where *<sup>ω</sup>* <sup>=</sup> *ck*, **<sup>E</sup>**(−) <sup>=</sup> **<sup>E</sup>**(+)∗, and <sup>E</sup><sup>+</sup> is ignored because it has the same wave vector as the pump beam but is very small in comparison to E*l*. Δ*ω<sup>n</sup>* satisfies the condition Δ*ω<sup>n</sup>* � *ω<sup>l</sup>* (Bar-Gill et al., 2007) and the initial phases of the different frequency components are assumed to be zero.

Since the BEC is tightly constrained in its short axis (**xˆ**, **yˆ**) in the present superradiance setup and the Fresnel number of the optical field is around 1, one dimensional approximation is usually used (Bar-Gill et al., 2007; Hilliard et al., 2008; Inouye et al., 1999; Li, Zhou et al., 2008). We expand the wavefunction of the condensate *ψ*(**r**, *t*) in momentum space, *ψ*(**r**, *t*) = <sup>∑</sup>*<sup>m</sup> <sup>φ</sup>m*(*z*, *<sup>t</sup>*) <sup>×</sup>*e*−*i*(*ωmt*−2*mkz*) , where *φm*(*z*, *t*) = *ψm*(*z*, *t*)/ <sup>√</sup>*A*, *<sup>ω</sup><sup>m</sup>* <sup>=</sup> 2¯*hm*2*k*2/*<sup>M</sup>* with *<sup>M</sup>* the atomic mass, *m* = 0 corresponds to the residual condensates, *m* �= 0 denotes the side modes, and *A* is the average cross area of the condensate perpendicular to **ˆz**. Using the Maxwell-Schrödinger equations, we obtain dynamics equations for *φm*(*z*, *t*),

$$\mathrm{i}\frac{\partial\phi\_{m}}{\partial t} = -\frac{\hbar}{2M}\frac{\partial^{2}\phi\_{m}}{\partial z^{2}} - \frac{2\mathrm{im}\hbar\mathrm{k}}{M}\frac{\partial\phi\_{m}}{\partial z} + \overline{\mathrm{g}}\left[\mathcal{E}\_{-}^{\*}\phi\_{m-1}e^{-4\mathrm{i}(1-2m)\omega\cdot t} + \mathcal{E}\_{-}\phi\_{m+1}e^{-4\mathrm{i}(1+2m)\omega\cdot t}\right],\tag{2}$$

where *ω<sup>r</sup>* = *hk*¯ <sup>2</sup> *<sup>l</sup>* /2*M* is the recoil frequency, the coupling between modes is given by

$$\bar{\mathbf{g}}(t) = \mathbf{g} \left( 1 + \sum\_{n} e^{i\Delta\omega\_{n}t} \right) \tag{3}$$

Fig. 2. (Color online) Light-field components of a two-frequency pump laser. The broad arrows are the pump laser and narrow ones are the end-fire mode (scattering optical field). In a spontaneous process, atoms in the condensate absorb photons from the pump laser with frequencies *ω<sup>l</sup>* and *ω<sup>l</sup>* − 8*ωr*, are scattered to side mode *m* = 1 and emit end-fire mode photons with frequency *ω<sup>l</sup>* − 4*ω<sup>r</sup>* (dashed arrow) and *ω<sup>l</sup>* − 12*ω<sup>r</sup>* (dotted arrow), respectively. Meanwhile, atoms in the condensate can also absorb end-fire mode photons with frequency

High Order Momentum States by Light Wave Scattering 345

*ω<sup>l</sup>* − 4*ωr*, be scattered back to side mode *m* = −1 and emit photons with frequency *ω<sup>l</sup>* − 8*ωr*(solid arrow), resonating to one of the pump laser components. The side mode *m* = 1 can absorb pump laser photons with frequency *ω<sup>l</sup>* and be scattered to mode *m* = 2, emitting photons with frequency *ω<sup>l</sup>* − 12*ω<sup>r</sup>* resonating to the existing end-fire mode.

by the pump laser.

mode *m* = 2 is enhanced.

due to the energy barrier. However, if we use the two components pump laser with frequency difference 8*ωr*, i.e. resonant frequency difference, the energy mismatch can be compensated

Although the resonant condition for the backward mode is satisfied, it should be noticed that two scattering channels exist almost simultaneously. One is atoms scattered from side mode *m* = 0 to *m* = −1 and the other is from *m* = 1 to *m* = 2, resulting in mode competition. The transition from mode *m* = 1 to *m* = 2 requires absorbtion of photons from pump laser, while the backward transition takes photons from the end-fire mode. Because the intensity of the pump laser is far greater than that of the end-fire mode, the transition from *m* = 1 to *m* = 2 has a bigger probability than the transition from *m* = 0 to *m* = −1. Thus the population of the backward mode *m* = −1 is suppressed even at the resonant condition, while the forward

However, the existence of competition between these two channels may not lead to the suppression of the backward mode. If these two channels happen in different spacial parts

with the coupling factor *g* = 3*πc*3*R*/(2*ω*<sup>2</sup> *<sup>l</sup> AL*), *R* is the Rayleigh scattering rate of the pump components, and *L* is the BEC length.

The first term on the right-hand-side of Eq.(2) describes the dispersion of *φm*, and the second term gives rise to their translation. The terms in square brackets describe the atom exchange between *φ<sup>m</sup>* and *φm*+<sup>1</sup> or *φm*−<sup>1</sup> through the pump laser and end-fire mode fields. An atom in mode *m* may absorb a laser photon and emit it into end-fire mode E−, and the accompanying recoil drives the atom into *m* + 1 mode, hence atoms with mode *m* + 1 can emerge in forward scattering. On the other hand, in the backward scattering, atoms with mode *m* absorb one E− mode photon, deposit it into the laser mode and go into mode *m* − 1. The envelope function of end-fire mode E− is given by

E− = −i *ωrg*¯ 2*cε*<sup>0</sup> +∞ *z* d*z*� ∑ *m φm*(*z*� , *t*)*φ*∗ *m*+1(*z*� , *t*)*e* i4(2*m*+1)*ωrt* , (4)

indicating that the end-fire mode field E− is due to the transition between *m* and *m* + 1 mode and the magnitude of E− depends on the spatial overlap between the two modes. In addition, there is a frequency difference of 8*ωr* between adjacent modes.

### **2.2 Mode competition for a two-frequency pump beam**

In the case of a single-frequency pump in the weak coupling regime, the evolution of the side modes and the end-fire mode indicates that the scattering is a localized process. For this end-pumping configuration, the scattering first starts on the leading edge of the BEC and then moves towards the tailing edge. To investigate the effect of the two-frequency pump beam, the different frequency components of the end-fire mode which indicate the energy change during the scattering are depicted in Fig. 2. The momentum of side mode *m* = *n* is 2*nh*¯**k**, and its kinetic energy is 4*n*2*h*¯ <sup>2</sup>**k**2/2*<sup>M</sup>* = <sup>4</sup>*n*2*h*¯ *<sup>ω</sup>r*. For the pump component with frequency *<sup>ω</sup>l*, atoms from the condensate are pumped to the side mode *m* = 1 and emit end-fire mode photons with frequency *ω<sup>l</sup>* − 4*ω<sup>r</sup>* spontaneously. However, in the backward scattering process, an atom in the condensate absorbs the end-fire mode (*ω<sup>l</sup>* − 4*ωr*) and emits a photon with frequency *ω<sup>l</sup>* back into the pump laser. Since energy is not conserved in backward-scattering, the backward side mode is not populated in weak-pulse regime. Side mode *m* = 2 is also not populated 4 Will-be-set-by-IN-TECH

2008). We expand the wavefunction of the condensate *ψ*(**r**, *t*) in momentum space, *ψ*(**r**, *t*) =

atomic mass, *m* = 0 corresponds to the residual condensates, *m* �= 0 denotes the side modes, and *A* is the average cross area of the condensate perpendicular to **ˆz**. Using the

*<sup>l</sup>* /2*M* is the recoil frequency, the coupling between modes is given by

*ei*Δ*ωnt* 

<sup>1</sup> <sup>+</sup> <sup>∑</sup>*<sup>n</sup>*

The first term on the right-hand-side of Eq.(2) describes the dispersion of *φm*, and the second term gives rise to their translation. The terms in square brackets describe the atom exchange between *φ<sup>m</sup>* and *φm*+<sup>1</sup> or *φm*−<sup>1</sup> through the pump laser and end-fire mode fields. An atom in mode *m* may absorb a laser photon and emit it into end-fire mode E−, and the accompanying recoil drives the atom into *m* + 1 mode, hence atoms with mode *m* + 1 can emerge in forward scattering. On the other hand, in the backward scattering, atoms with mode *m* absorb one E− mode photon, deposit it into the laser mode and go into mode *m* − 1. The envelope function

*φm*(*z*�

indicating that the end-fire mode field E− is due to the transition between *m* and *m* + 1 mode and the magnitude of E− depends on the spatial overlap between the two modes. In addition,

In the case of a single-frequency pump in the weak coupling regime, the evolution of the side modes and the end-fire mode indicates that the scattering is a localized process. For this end-pumping configuration, the scattering first starts on the leading edge of the BEC and then moves towards the tailing edge. To investigate the effect of the two-frequency pump beam, the different frequency components of the end-fire mode which indicate the energy change during the scattering are depicted in Fig. 2. The momentum of side mode *m* = *n* is 2*nh*¯**k**, and its kinetic energy is 4*n*2*h*¯ <sup>2</sup>**k**2/2*<sup>M</sup>* = <sup>4</sup>*n*2*h*¯ *<sup>ω</sup>r*. For the pump component with frequency *<sup>ω</sup>l*, atoms from the condensate are pumped to the side mode *m* = 1 and emit end-fire mode photons with frequency *ω<sup>l</sup>* − 4*ω<sup>r</sup>* spontaneously. However, in the backward scattering process, an atom in the condensate absorbs the end-fire mode (*ω<sup>l</sup>* − 4*ωr*) and emits a photon with frequency *ω<sup>l</sup>* back into the pump laser. Since energy is not conserved in backward-scattering, the backward side mode is not populated in weak-pulse regime. Side mode *m* = 2 is also not populated

, *t*)*φ*∗

*m*+1(*z*�

, *t*)*e*

i4(2*m*+1)*ωrt*

, (4)

<sup>√</sup>*A*, *<sup>ω</sup><sup>m</sup>* <sup>=</sup> 2¯*hm*2*k*2/*<sup>M</sup>* with *<sup>M</sup>* the

, (3)

 , (2)

<sup>−</sup>*φm*−1*e*−4i(1−2*m*)*ωrt* <sup>+</sup> E−*φm*+1*e*−4i(1+2*m*)*ωrt*

*<sup>l</sup> AL*), *R* is the Rayleigh scattering rate of the pump

, where *φm*(*z*, *t*) = *ψm*(*z*, *t*)/

Maxwell-Schrödinger equations, we obtain dynamics equations for *φm*(*z*, *t*),

+ *g*¯ E∗

*g*¯(*t*) = *g*

3*πc*3*R*/(2*ω*<sup>2</sup>

*∂φm ∂z*

<sup>∑</sup>*<sup>m</sup> <sup>φ</sup>m*(*z*, *<sup>t</sup>*) <sup>×</sup>*e*−*i*(*ωmt*−2*mkz*)

2*M*

with the coupling factor *g* =

of end-fire mode E− is given by

E− = −i

*ωrg*¯ 2*cε*<sup>0</sup>

there is a frequency difference of 8*ωr* between adjacent modes.

**2.2 Mode competition for a two-frequency pump beam**

 +∞ *z*

d*z*� ∑ *m*

components, and *L* is the BEC length.

*∂*2*φ<sup>m</sup>*

*<sup>∂</sup>z*<sup>2</sup> <sup>−</sup> 2i*mhk*¯ *M*

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup> *<sup>h</sup>*¯

where *ω<sup>r</sup>* = *hk*¯ <sup>2</sup>

i *∂φm*

Fig. 2. (Color online) Light-field components of a two-frequency pump laser. The broad arrows are the pump laser and narrow ones are the end-fire mode (scattering optical field). In a spontaneous process, atoms in the condensate absorb photons from the pump laser with frequencies *ω<sup>l</sup>* and *ω<sup>l</sup>* − 8*ωr*, are scattered to side mode *m* = 1 and emit end-fire mode photons with frequency *ω<sup>l</sup>* − 4*ω<sup>r</sup>* (dashed arrow) and *ω<sup>l</sup>* − 12*ω<sup>r</sup>* (dotted arrow), respectively. Meanwhile, atoms in the condensate can also absorb end-fire mode photons with frequency *ω<sup>l</sup>* − 4*ωr*, be scattered back to side mode *m* = −1 and emit photons with frequency *ω<sup>l</sup>* − 8*ωr*(solid arrow), resonating to one of the pump laser components. The side mode *m* = 1 can absorb pump laser photons with frequency *ω<sup>l</sup>* and be scattered to mode *m* = 2, emitting photons with frequency *ω<sup>l</sup>* − 12*ω<sup>r</sup>* resonating to the existing end-fire mode.

due to the energy barrier. However, if we use the two components pump laser with frequency difference 8*ωr*, i.e. resonant frequency difference, the energy mismatch can be compensated by the pump laser.

Although the resonant condition for the backward mode is satisfied, it should be noticed that two scattering channels exist almost simultaneously. One is atoms scattered from side mode *m* = 0 to *m* = −1 and the other is from *m* = 1 to *m* = 2, resulting in mode competition. The transition from mode *m* = 1 to *m* = 2 requires absorbtion of photons from pump laser, while the backward transition takes photons from the end-fire mode. Because the intensity of the pump laser is far greater than that of the end-fire mode, the transition from *m* = 1 to *m* = 2 has a bigger probability than the transition from *m* = 0 to *m* = −1. Thus the population of the backward mode *m* = −1 is suppressed even at the resonant condition, while the forward mode *m* = 2 is enhanced.

However, the existence of competition between these two channels may not lead to the suppression of the backward mode. If these two channels happen in different spacial parts

−50 −25 <sup>0</sup> <sup>25</sup> <sup>50</sup> <sup>0</sup>

2

(a)

1

Fig. 4. (Color online) Normalized side mode populations versus time: (a) for a

single-frequency pump beam; (b) for a two-frequency resonant pump beam. In both cases the coupling constant is kept *<sup>g</sup>* <sup>=</sup> 1.55 <sup>×</sup> 106*s*−1. The side mode are: m=-1 (solid); m=1

Fig. 5. (Color online) The light-field components of a three-frequency pump laser. The broad

mode *m* = 1 and *m* = 2. The apparent regrowth of side mode *m* = 1 on the leading-edge shown in Fig.3(d) indicates that there are Rabi oscillations between side modes *m* = 1 and

The time evolution of several side modes populations normalized by the total atom number are depicted by Fig.4. Fig.4 (a) shows that using a single-frequency pump laser cannot produce backward mode *m* = −1 or forward higher mode *m* = 2 in the weak-pulse regime. Using a resonant two-frequency pump beam with the same intensity, modes *m* = −1 and *m* = 2 increased, as shown in Fig.4 (b), however, the forward mode is greatly enhanced while the

The second forward side mode *m* = 2 is greatly enhanced with a resonant two-frequency pump beam, however, the populations of higher forward modes such as *m* = 3 are very small as the channel from the second forward mode to the third forward mode is not resonant with the exiting optical field. To get a large number mode for *m* = 3, Fig.5 depicts the scheme

**2.3 The third order forward modes enhanced with a three-frequency pump beam**

arrows are the pump laser and narrow ones are the end-fire mode.

*m* = 2 in the depleted area of the condensate.

backward mode remains very small.

−50 −25 <sup>0</sup> <sup>25</sup> <sup>50</sup> <sup>0</sup>

2

(b)

1

z (μm)

0.5 1 1.5 2 2.5 3 3.5 x 109

**|**ψ**2| , |**ε

**|** 

High Order Momentum States by Light Wave Scattering 347

**−**

z (μm)

0.5 1 1.5 2 2.5 3 3.5 x 109

(dotted); m=2 (dashed).

**|**ψ**2| , |**ε

**|**

**−**

Fig. 3. (Color online) Spatial distribution of the side modes <sup>|</sup>*ψ*2<sup>|</sup> and the end-fire mode <sup>|</sup>*ε*−| in the weak coupling regime (*<sup>g</sup>* <sup>=</sup> 1.25 <sup>×</sup> 106*s*−1) with the two-frequency pump for different pulse durations: 150*μs* (a); 200*μs* (b); 250*μs* (c); 300*μs* (d). Condensate mode *m* = 0 is the solid line-1, backward first-order side mode *m* = −1 is the solid line-2, forward first-order side mode *m* = 1 is the dash-dotted line, forward second-order side mode *m* = 2 is the dashed line, and end-fire mode is the dotted line.

of the condensate, then both of side mode *m* = −1 and *m* = 2 will be enhanced. The suppression of backward mode *m* = −1 and the enhancement of mode *m* = 2 need that these two scattering channels happen in the same area. Therefore, the spatial distribution effect should be considered.

We analyze the spatial effect when second-order forward side mode and backward side mode are populated at the resonant condition Δ*ω* = 8*ωr*. The evolution of spatial distribution of side modes and end-fire mode is shown in Fig.3. Superradiance first starts on the leading edge of the BEC, as shown in Fig.3(a). Although the backward first-order side mode *m* = −1 is populated through the overlap between end-fire mode E− and side mode *m* = 0, it is very small and emerges at the leading-edge of the BEC. Since the overlap between end-fire mode and side mode *m* = 1 is in the same area, the population of side mode *m* = 2 is obvious on this edge, as shown in Fig.3(b). Side mode *m* = 2 grows more rapidly than side mode *m* = −1, which means more atoms are scattered from side mode *m* = 1 to *m* = 2 than that from *m* = 0 to *m* = −1.Then the first peaks of side modes *m* = 1 and *m* = 2 move to the center of the BEC, as shown in Fig.3(c). Though the movement of the peaks is similar to that in the case of a single-frequency pump laser, one major difference is that the regrowth of side mode *m* = 0 is very little, hence nearly all the atoms on this edge are forwardly scattered. Due to the nearly-complete depletion of the condensate, atoms are mainly transferred between side 6 Will-be-set-by-IN-TECH

0.5 1 1.5 2 2.5 3 3.5 x 109

<sup>3</sup> x 109

0.5 1 1.5 2 2.5

**|**ψ**2| , |**ε

Fig. 3. (Color online) Spatial distribution of the side modes <sup>|</sup>*ψ*2<sup>|</sup> and the end-fire mode <sup>|</sup>*ε*−| in the weak coupling regime (*<sup>g</sup>* <sup>=</sup> 1.25 <sup>×</sup> 106*s*−1) with the two-frequency pump for different pulse durations: 150*μs* (a); 200*μs* (b); 250*μs* (c); 300*μs* (d). Condensate mode *m* = 0 is the solid line-1, backward first-order side mode *m* = −1 is the solid line-2, forward first-order side mode *m* = 1 is the dash-dotted line, forward second-order side mode *m* = 2 is the

of the condensate, then both of side mode *m* = −1 and *m* = 2 will be enhanced. The suppression of backward mode *m* = −1 and the enhancement of mode *m* = 2 need that these two scattering channels happen in the same area. Therefore, the spatial distribution

We analyze the spatial effect when second-order forward side mode and backward side mode are populated at the resonant condition Δ*ω* = 8*ωr*. The evolution of spatial distribution of side modes and end-fire mode is shown in Fig.3. Superradiance first starts on the leading edge of the BEC, as shown in Fig.3(a). Although the backward first-order side mode *m* = −1 is populated through the overlap between end-fire mode E− and side mode *m* = 0, it is very small and emerges at the leading-edge of the BEC. Since the overlap between end-fire mode and side mode *m* = 1 is in the same area, the population of side mode *m* = 2 is obvious on this edge, as shown in Fig.3(b). Side mode *m* = 2 grows more rapidly than side mode *m* = −1, which means more atoms are scattered from side mode *m* = 1 to *m* = 2 than that from *m* = 0 to *m* = −1.Then the first peaks of side modes *m* = 1 and *m* = 2 move to the center of the BEC, as shown in Fig.3(c). Though the movement of the peaks is similar to that in the case of a single-frequency pump laser, one major difference is that the regrowth of side mode *m* = 0 is very little, hence nearly all the atoms on this edge are forwardly scattered. Due to the nearly-complete depletion of the condensate, atoms are mainly transferred between side

**|** 

**−**

**|**ψ**2| , |**ε

**|** 

**−**

−50 −25 <sup>0</sup> <sup>25</sup> <sup>50</sup> <sup>0</sup>

2

(b)

1

1

z (μm)

(d)

−50 −25 <sup>0</sup> <sup>25</sup> <sup>50</sup> <sup>0</sup>

2

z (μm)

−50 −25 <sup>0</sup> <sup>25</sup> <sup>50</sup> <sup>0</sup>

2

(a)

1

1

z (μm)

(c)

−50 −25 <sup>0</sup> <sup>25</sup> <sup>50</sup> <sup>0</sup>

2

z (μm)

dashed line, and end-fire mode is the dotted line.

0.5 1 1.5 2 2.5 3 3.5 x 109

0.5 1 1.5 2 2.5 3 3.5 x 109

effect should be considered.

**|**ψ**2| , |**ε

**|**

**−**

**|**ψ**2| , |**ε

**|**

**−**

Fig. 4. (Color online) Normalized side mode populations versus time: (a) for a single-frequency pump beam; (b) for a two-frequency resonant pump beam. In both cases the coupling constant is kept *<sup>g</sup>* <sup>=</sup> 1.55 <sup>×</sup> 106*s*−1. The side mode are: m=-1 (solid); m=1 (dotted); m=2 (dashed).

Fig. 5. (Color online) The light-field components of a three-frequency pump laser. The broad arrows are the pump laser and narrow ones are the end-fire mode.

mode *m* = 1 and *m* = 2. The apparent regrowth of side mode *m* = 1 on the leading-edge shown in Fig.3(d) indicates that there are Rabi oscillations between side modes *m* = 1 and *m* = 2 in the depleted area of the condensate.

The time evolution of several side modes populations normalized by the total atom number are depicted by Fig.4. Fig.4 (a) shows that using a single-frequency pump laser cannot produce backward mode *m* = −1 or forward higher mode *m* = 2 in the weak-pulse regime. Using a resonant two-frequency pump beam with the same intensity, modes *m* = −1 and *m* = 2 increased, as shown in Fig.4 (b), however, the forward mode is greatly enhanced while the backward mode remains very small.

### **2.3 The third order forward modes enhanced with a three-frequency pump beam**

The second forward side mode *m* = 2 is greatly enhanced with a resonant two-frequency pump beam, however, the populations of higher forward modes such as *m* = 3 are very small as the channel from the second forward mode to the third forward mode is not resonant with the exiting optical field. To get a large number mode for *m* = 3, Fig.5 depicts the scheme

mode *m* = *n* cannot emit resonant end-fire mode, so mode *m* = *n* will be enhanced. To show it, Fig.6(b) is the simulated result of the time evolution of normalized side mode populations for a five-frequency pump beam. We could see that side mode *m* = 5 would be strongly

High Order Momentum States by Light Wave Scattering 349

In the experiment, to get several resonant frequencies, the laser beam from an external cavity diode laser can be split into several parts, and their frequencies are shifted individually by acoustic-optical modulators (AOMs) which are driven by phase-locked radio frequency signals, as demonstrated in the case of two resonant frequency (Bar-Gill et al., 2007; Yang et al., 2008). Therefore, the frequency difference between the beams can be controlled precisely. Furthermore, to avoid the reflection from the glass tube and formation of Bragg scattering in the experiment, the pump beam can actually deviate a few degrees from the long axis, as shown in the experiments (Fallani et al., 2005; Hilliard et al., 2008; Li, Zhou et al., 2008).

Different to the works in the configuration where the pump beam travels along the short axis of the condensate with the resonant frequency (Yang et al., 2008), where a large number of backward scattering is obvious in a two-frequency pump beam, the backward scattering is suppressed and the forward second-order mode is obviously enhanced in our case. This is due to mode competition between the forward second-order mode and the backward mode

We have not considered the initial quantum process because its time scale is very small, shorter than 1*μs*. In this quantum process there is also mode competition to form the end-fire modes along the long axis and suppress the emission on the other direction. This is different concept from what has been discussed above, in which case mode competition exists in the

For the pump beam with several resonant frequencies, not only can we obtain the high order momentum transfer which is important in the momentum manipulation for atom interferometry, but also the above analysis is useful to understand the interplay between the matter wave and light in the matter wave amplification (Kozuma, Suzuki et al., 1999; Schneble et al., 2003), atomic cooperative scattering in the optical lattice (Xu et al., 2009), and

In this section, we apply a method for flexible manipulation of the atomic momentum states with the standing wave pulses. The atomic diffraction from standing wave pulses is demonstrated in the experiments and systematically analyzed by the band structure theory of one-dimension optical lattice. With this method, we are able to design and realize several specific momentum states, which may be applied in atomic interferometry. In principle, this

We consider a non-interacting condensate being diffracted by a sequence of square shaped standing wave pulses with the successive durations *τi*(*i* = 1, 2, ...,*s* + 1), separated by the

enhanced.

**2.5 Discussion**

**3.1 Theoretical model**

and local depletion of the superradiant process.

different channels satisfying the energy match and spatial condition.

by the pump with a noisy laser (Robb & Firth, 2007; Zhou, 2009).

**3. High order momentum states by scattering of standing wave pulses**

method could be used for designing a wide range of possible target states.

Fig. 6. (Color online) Normalized side mode populations versus time with the coupling constant *<sup>g</sup>* <sup>=</sup> 1.55 <sup>×</sup> 106*s*−1: (a) for a three-frequency pump laser: *<sup>m</sup>* <sup>=</sup> <sup>−</sup>1 (dash-dotted), *m* = 1 (dotted), *m* = 2 (dashed), *m* = 3 (solid) ; (b) for a five-frequency pump laser: *m* = 1 (dotted), *m* = 3 (dash-dotted), *m* = 4 (dashed), *m* = 5 (solid) .

of the three-frequency pump beam with the frequencies of the pump laser *ωl*, *ω<sup>l</sup>* − 8*ω<sup>r</sup>* and *ω<sup>l</sup>* − 16*ωr*. The frequency components *ωl*, *ω<sup>l</sup>* − 8*ω<sup>r</sup>* and *ω<sup>l</sup>* − 16*ω<sup>r</sup>* both have the resonant frequency difference. Hence, there could be two channels to form the backward side mode *m* = −1 but the enhancement of the backward scattering is small because of the formation of higher forward side modes. There are also two channels to form side mode *m* = 2. One thing different from the two-frequency pump beam is that there is also a channel to form side mode *m* = 3 for the reason that atoms in side mode *m* = 2 absorb pump laser photons with frequency *ωl*, are then scattered to mode *m* = 3 and eventually emit end-fire mode photons with frequency *ω<sup>l</sup>* − 20*ω<sup>r</sup>* which is resonant to an existing end-fire mode. This means that more atoms in side mode *m* = 2 will be pumped to side mode *m* = 3 and less will be transferred back to side mode *m* = 1, a competition between side mode *m* = 3 and *m* = 1 is set up. As a result, side mode *m* = 3 will be enhanced and *m* = 1 will be reduced relatively.

Fig.6(a) is the simulated result of the time evolution of normalized side mode populations for a three-frequency pump beam. We could see that side mode *m* = 3 would be strongly enhanced at long time while side mode *m* = 1 reduced.

### **2.4 Momentum transfer in the high order forward modes**

From the above discussion we know that using multi-resonant frequencies is a promising way to get a large number of higher forward modes. When a pump laser has frequency components *ωl*, *ω<sup>l</sup>* − 8*ωr*, ··· , *ω<sup>l</sup>* − (*n* − 1) × 8*ωr*, satisfying (*n* − 1) × 8*ω<sup>r</sup>* � *ωl*, with the kinetic energy of mode *m* = *n* equal to 4*n*2*h*¯ *ωr*, then after the condensate atoms spontaneously scattered to mode *m* = 1, the end-fire mode will have frequency components *ω<sup>l</sup>* − 4*ωr*, *ω<sup>l</sup>* − 12*ωr*, ··· , *ω<sup>l</sup>* − (2*n* − 1) × 4*ωr*. For resonance concern, mode *m* = 1 will absorb photons from the pump components *ωl*, *ω<sup>l</sup>* − 8*ωr*, ··· , *ω<sup>l</sup>* − (*n* − 2) × 8*ω<sup>r</sup>* and emits end-fire mode photons with frequency *ω<sup>l</sup>* − 12*ωr*, ··· , *ω<sup>l</sup>* − (2*n* − 1) × 4*ω<sup>r</sup>* which are resonant with existing end-fire mode, so mode *m* = 2 is produced. Like mode *m* = 1, modes *m* = 2, *m* = 3, ··· , *m* = *n* − 1 can absorb pump photons and emit photons resonant to the existing end-fire mode. For example, mode *m* = *n* − 1 will absorb photons with frequency *ω<sup>l</sup>* and emits photons with frequency *ω<sup>l</sup>* − (2*n* − 1) × 4*ωr*. Therefore atoms could finally be transferred to mode *m* = *n*. Note that

## **2.5 Discussion**

8 Will-be-set-by-IN-TECH

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

**Normalized Mode Population**

Fig. 6. (Color online) Normalized side mode populations versus time with the coupling constant *<sup>g</sup>* <sup>=</sup> 1.55 <sup>×</sup> 106*s*−1: (a) for a three-frequency pump laser: *<sup>m</sup>* <sup>=</sup> <sup>−</sup>1 (dash-dotted), *m* = 1 (dotted), *m* = 2 (dashed), *m* = 3 (solid) ; (b) for a five-frequency pump laser: *m* = 1

of the three-frequency pump beam with the frequencies of the pump laser *ωl*, *ω<sup>l</sup>* − 8*ω<sup>r</sup>* and *ω<sup>l</sup>* − 16*ωr*. The frequency components *ωl*, *ω<sup>l</sup>* − 8*ω<sup>r</sup>* and *ω<sup>l</sup>* − 16*ω<sup>r</sup>* both have the resonant frequency difference. Hence, there could be two channels to form the backward side mode *m* = −1 but the enhancement of the backward scattering is small because of the formation of higher forward side modes. There are also two channels to form side mode *m* = 2. One thing different from the two-frequency pump beam is that there is also a channel to form side mode *m* = 3 for the reason that atoms in side mode *m* = 2 absorb pump laser photons with frequency *ωl*, are then scattered to mode *m* = 3 and eventually emit end-fire mode photons with frequency *ω<sup>l</sup>* − 20*ω<sup>r</sup>* which is resonant to an existing end-fire mode. This means that more atoms in side mode *m* = 2 will be pumped to side mode *m* = 3 and less will be transferred back to side mode *m* = 1, a competition between side mode *m* = 3 and *m* = 1 is set up. As a result, side mode *m* = 3 will be enhanced and *m* = 1 will be reduced relatively. Fig.6(a) is the simulated result of the time evolution of normalized side mode populations for a three-frequency pump beam. We could see that side mode *m* = 3 would be strongly

From the above discussion we know that using multi-resonant frequencies is a promising way to get a large number of higher forward modes. When a pump laser has frequency components *ωl*, *ω<sup>l</sup>* − 8*ωr*, ··· , *ω<sup>l</sup>* − (*n* − 1) × 8*ωr*, satisfying (*n* − 1) × 8*ω<sup>r</sup>* � *ωl*, with the kinetic energy of mode *m* = *n* equal to 4*n*2*h*¯ *ωr*, then after the condensate atoms spontaneously scattered to mode *m* = 1, the end-fire mode will have frequency components *ω<sup>l</sup>* − 4*ωr*, *ω<sup>l</sup>* − 12*ωr*, ··· , *ω<sup>l</sup>* − (2*n* − 1) × 4*ωr*. For resonance concern, mode *m* = 1 will absorb photons from the pump components *ωl*, *ω<sup>l</sup>* − 8*ωr*, ··· , *ω<sup>l</sup>* − (*n* − 2) × 8*ω<sup>r</sup>* and emits end-fire mode photons with frequency *ω<sup>l</sup>* − 12*ωr*, ··· , *ω<sup>l</sup>* − (2*n* − 1) × 4*ω<sup>r</sup>* which are resonant with existing end-fire mode, so mode *m* = 2 is produced. Like mode *m* = 1, modes *m* = 2, *m* = 3, ··· , *m* = *n* − 1 can absorb pump photons and emit photons resonant to the existing end-fire mode. For example, mode *m* = *n* − 1 will absorb photons with frequency *ω<sup>l</sup>* and emits photons with frequency *ω<sup>l</sup>* − (2*n* − 1) × 4*ωr*. Therefore atoms could finally be transferred to mode *m* = *n*. Note that

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup> <sup>400</sup> <sup>450</sup> <sup>0</sup>

**(b)**

**t (**μ**s)**

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup> <sup>0</sup>

**(a)**

**t (**μ**s)**

(dotted), *m* = 3 (dash-dotted), *m* = 4 (dashed), *m* = 5 (solid) .

enhanced at long time while side mode *m* = 1 reduced.

**2.4 Momentum transfer in the high order forward modes**

0.05

0.1

0.15

**Normalized Mode Population**

0.2

0.25

In the experiment, to get several resonant frequencies, the laser beam from an external cavity diode laser can be split into several parts, and their frequencies are shifted individually by acoustic-optical modulators (AOMs) which are driven by phase-locked radio frequency signals, as demonstrated in the case of two resonant frequency (Bar-Gill et al., 2007; Yang et al., 2008). Therefore, the frequency difference between the beams can be controlled precisely. Furthermore, to avoid the reflection from the glass tube and formation of Bragg scattering in the experiment, the pump beam can actually deviate a few degrees from the long axis, as shown in the experiments (Fallani et al., 2005; Hilliard et al., 2008; Li, Zhou et al., 2008).

Different to the works in the configuration where the pump beam travels along the short axis of the condensate with the resonant frequency (Yang et al., 2008), where a large number of backward scattering is obvious in a two-frequency pump beam, the backward scattering is suppressed and the forward second-order mode is obviously enhanced in our case. This is due to mode competition between the forward second-order mode and the backward mode and local depletion of the superradiant process.

We have not considered the initial quantum process because its time scale is very small, shorter than 1*μs*. In this quantum process there is also mode competition to form the end-fire modes along the long axis and suppress the emission on the other direction. This is different concept from what has been discussed above, in which case mode competition exists in the different channels satisfying the energy match and spatial condition.

For the pump beam with several resonant frequencies, not only can we obtain the high order momentum transfer which is important in the momentum manipulation for atom interferometry, but also the above analysis is useful to understand the interplay between the matter wave and light in the matter wave amplification (Kozuma, Suzuki et al., 1999; Schneble et al., 2003), atomic cooperative scattering in the optical lattice (Xu et al., 2009), and by the pump with a noisy laser (Robb & Firth, 2007; Zhou, 2009).

## **3. High order momentum states by scattering of standing wave pulses**

In this section, we apply a method for flexible manipulation of the atomic momentum states with the standing wave pulses. The atomic diffraction from standing wave pulses is demonstrated in the experiments and systematically analyzed by the band structure theory of one-dimension optical lattice. With this method, we are able to design and realize several specific momentum states, which may be applied in atomic interferometry. In principle, this method could be used for designing a wide range of possible target states.

## **3.1 Theoretical model**

We consider a non-interacting condensate being diffracted by a sequence of square shaped standing wave pulses with the successive durations *τi*(*i* = 1, 2, ...,*s* + 1), separated by the

After the first interval *τ<sup>f</sup>* <sup>1</sup> and the second pulse *τ*2, the population of the |*pm*� state can be

High Order Momentum States by Light Wave Scattering 351

As shown in Eq. (10),the population is affected by the two pulses, the first one corresponding to *b*(*m*0, *m*1, *τ*1), and the second one corresponding to *b*(*m*1, *m*, *τ*2). During the interval *τ<sup>f</sup>* 1, the phase of the <sup>|</sup>*pm*<sup>1</sup> � state evolves along the time as *<sup>e</sup>*−*iE*(*m*1)*t*/¯*h*, where *<sup>E</sup>*(*m*1) = (2*m*1*hk*¯ *<sup>L</sup>*)2/2*<sup>M</sup>* <sup>=</sup>

<sup>1</sup>*ER* is the kinetic energy, and *ER* = (*hk*¯ *<sup>L</sup>*)2/2*<sup>M</sup>* is the single photon recoil energy. The

*<sup>b</sup>*(*mi*−1, *mi*, *<sup>τ</sup>i*)

From the analysis above, the momentum distribution after a sequence of pulses' scattering

populations of the momentum states do not change during the intervals, the phase-evolution rates of the momentum states with different kinetic energies are not identical. The phase deviations between the states oscillate from 0 to 2*π* with the interval, and the heterogeneously accumulated phases change the distribution of the condensate in the quasi-momentum space.

We performed the experiments of a condensate in a magnetic trap (MT) (see Fig. 7(a)) being scattered by a sequence of standing wave pulses (see Fig. 7(b)). As shown in Fig. 7(c), after pre-cooling, a cigar shaped 87Rb condensate of 2 <sup>×</sup> 105 atoms in 52*S*1/2 <sup>|</sup>*<sup>F</sup>* <sup>=</sup> 2, *MF* <sup>=</sup> <sup>2</sup>� state was achieved by the radio frequency (RF) cooling in the magnetic trap, of which the axial frequency is 20 Hz and the radial frequency is 220 Hz (Yang et al., 2008; Zhou et al., 2010). A pair of counter-propagating laser beams, of which the durations were controlled by an acousto-optical modulator, and the amplitudes were adjusted by the injection current of a tapered-amplifier, were applied to the condensate along the axial direction. The linear polarized incident light at the wavelength *λ<sup>L</sup>* = 852 nm was focused with a waist of 110 *μ*m to cover the condensate. The trap depth, which was calibrated by Kapitza-Dirac scattering experimentally, reached 120*ER*, corresponding to the light power of 320 mW. The incident light and the magnetic trap were simultaneously shut after the BEC-light interaction. After 30 *ms* free falling and ballistic expansion, the atomic gas was pictured by absorption imaging. Since the minimum gap between different momentum states is 2¯*hk*, which is much larger than the momentum width of a single momentum state, the components with different momenta will be separated in the TOF (time of flight) signal (see Fig. 7(d)), and the atomic number of each momentum state is possible to be read separately as *Nm* (the momentum

In the same way, the population of the |*pm*� state after (*s* + 1) pulses can be achieved as:

*s*+1 ∏ *i*=1 <sup>−</sup>*iE*(*m*1)*τ<sup>f</sup>* 1/¯*hb*(*m*1, *<sup>m</sup>*, *<sup>τ</sup>*2)

*s* ∏ *i*=1 *e* <sup>−</sup>*iE*(*mi* ) *τf i* /¯*h* 

 

*<sup>τ</sup><sup>f</sup>* 1/¯*<sup>h</sup>* and contributes to the momentum distribution.

*<sup>i</sup>*=<sup>1</sup> *e*

<sup>−</sup>*iE*(*mi*)*τ<sup>f</sup> i* /¯*h*

2

2

. (10)

, (11)

. Although the

*<sup>i</sup>*=<sup>1</sup> *<sup>b</sup>*(*mi*−1, *mi*, *<sup>τ</sup>i*), but also

*b*(*m*0, *m*1, *τ*1)*e*

achieved as

4*m*<sup>2</sup>

*<sup>P</sup>*(2) *<sup>m</sup>* <sup>=</sup>

interval *<sup>τ</sup><sup>f</sup>* <sup>1</sup> produces a phase shift *<sup>e</sup>*−*iE*(*m*)

with *m*<sup>0</sup> = 0, and *ms*+<sup>1</sup> = *m*.

*<sup>P</sup>*(*s*+1) *<sup>m</sup>* <sup>=</sup>

 

<sup>∑</sup> *<sup>m</sup>*1,*m*2,···,*ms*

is influenced by not only the lattice pulses with the term ∏*s*+<sup>1</sup>

the intervals among the pulses as reflected in the term ∏*<sup>s</sup>*

**3.2 Experiments of standing wave pulse sequences**

 ∑ *m*<sup>1</sup>

intervals *τf i*(*i* = 1, 2, ...,*s*). The standing wave consists of a pair of laser beams far-detuned enough to suppress the spontaneous emission.

The periodic potential (Denschlag et al., 2002; Morsch & Oberthaler, 2006) introduced by the ac Stark shift can be described as *V*(*x*) = *U*<sup>0</sup> cos2(*kLx*), with the trap depth *U*<sup>0</sup> and the laser's wave vector *kL* = 2*π*/*λ<sup>L</sup>* (*λ<sup>L</sup>* is the wavelength of the laser). The lattice leads to a band structure of the energy spectra, of which the eigenvalues of the energy *En*,*q* and eigenvectors |*n*, *q*� (Bloch states) are labeled by the quasi-momentum *q* and the band index *n*, and they satisfy the equation:

$$
\hat{H} \left| n, q \right> = E\_{\text{n.q.}} \left| n, q \right> , \tag{5}
$$

where the Hamiltonian *H*ˆ = *p*ˆ2/2*M* + *U*0cos<sup>2</sup>(*kLx*), with the atomic momentum *p*ˆ and the atomic mass *M*. The Bloch states form a quasi-momentum space. In the lattice, the spatial periodicity of the wave function results in separated peaks in momentum space, corresponding to the reciprocal lattice vector 2*kL*.

When a condensate with an initial momentum *pm*<sup>0</sup> = *h*¯(*q* + 2*m*0*kL*) (¯*h* is the Plank constant, −*kL* ≤ *q* ≤ *kL* , *m*<sup>0</sup> = ..., −1, 0, 1, ...) is abruptly loaded into a lattice, the wave packet can be described as a superposition of the Bloch states:

$$|\Psi(t=0)\rangle = \sum\_{n=0}^{\infty} |n,q\rangle \langle n,q|p\_{m\_0}\rangle\_{\prime} \tag{6}$$

where �*n*, *<sup>q</sup>*|*pm*<sup>0</sup> � <sup>=</sup> *cn*,*q*(*m*0). The *<sup>n</sup>th* Bloch state evolves independently as *<sup>e</sup>*−*iEn*,*qt*/¯*h*, and the total wave function evolves as

$$\left|\Psi(t)\right\rangle = \sum\_{n=0}^{\infty} c\_{n,q}(m\_0) e^{-iE\_{n,q}t/\hbar} \left|n\right.\tag{7}$$

While the incident light is switched off after the duration *τ*1, the wave function is projected back to the momentum space from the quasi-momentum space. The coefficient *b*(*m*0, *m*, *τ*1) of each |*pm*� state (*m* = ..., −1, 0, 1, ...) can be acquainted as:

$$b(m\_0, m, \tau\_1) = \sum\_{n=0}^{\infty} c\_{n,q}(m\_0) c\_{n,q}(m) e^{-iE\_{n,q}\tau\_1/\hbar}.\tag{8}$$

For a zero initial momentum of the condensate, the subscript *q* can be omitted for simplification and *m*<sup>0</sup> = 0. For one pulse scattering, the population of the |*pm*� state is *<sup>P</sup>*(1) *<sup>m</sup>* <sup>=</sup> <sup>|</sup>*b*(0, *<sup>m</sup>*, *<sup>τ</sup>*1)<sup>|</sup> 2. It can be seen that the probabilities of the momentum states after one scattering pulse depend on the lattice depth and the pulse duration. The lattice depth determines the band structure and is reflected in the terms *cn*,*q*. The pulse duration influences the phase evolution of each Bloch state as *e*−*iEn*,*qτ*1/¯*h*.

The multi-pulse process, which consists of a number of single pulses and intervals can be solved as follows. The wave function of the condensate after the first pulse *τ*<sup>1</sup> can be derived from Eq. (8) as

$$|\Psi(\tau\_1, t)\rangle = \sum\_m b(m\_0, m\_\prime \tau\_1) e^{-iE^{(m)}t/\hbar} |2m\hbar k\_L\rangle. \tag{9}$$

10 Will-be-set-by-IN-TECH

intervals *τf i*(*i* = 1, 2, ...,*s*). The standing wave consists of a pair of laser beams far-detuned

The periodic potential (Denschlag et al., 2002; Morsch & Oberthaler, 2006) introduced by the ac Stark shift can be described as *V*(*x*) = *U*<sup>0</sup> cos2(*kLx*), with the trap depth *U*<sup>0</sup> and the laser's wave vector *kL* = 2*π*/*λ<sup>L</sup>* (*λ<sup>L</sup>* is the wavelength of the laser). The lattice leads to a band structure of the energy spectra, of which the eigenvalues of the energy *En*,*q* and eigenvectors |*n*, *q*� (Bloch states) are labeled by the quasi-momentum *q* and the band index *n*, and they

where the Hamiltonian *H*ˆ = *p*ˆ2/2*M* + *U*0cos<sup>2</sup>(*kLx*), with the atomic momentum *p*ˆ and the atomic mass *M*. The Bloch states form a quasi-momentum space. In the lattice, the spatial periodicity of the wave function results in separated peaks in momentum space,

When a condensate with an initial momentum *pm*<sup>0</sup> = *h*¯(*q* + 2*m*0*kL*) (¯*h* is the Plank constant, −*kL* ≤ *q* ≤ *kL* , *m*<sup>0</sup> = ..., −1, 0, 1, ...) is abruptly loaded into a lattice, the wave packet can be

> ∞ ∑ *n*=0

where �*n*, *<sup>q</sup>*|*pm*<sup>0</sup> � <sup>=</sup> *cn*,*q*(*m*0). The *<sup>n</sup>th* Bloch state evolves independently as *<sup>e</sup>*−*iEn*,*qt*/¯*h*, and the

*cn*,*q*(*m*0)*e*

While the incident light is switched off after the duration *τ*1, the wave function is projected back to the momentum space from the quasi-momentum space. The coefficient *b*(*m*0, *m*, *τ*1) of

For a zero initial momentum of the condensate, the subscript *q* can be omitted for simplification and *m*<sup>0</sup> = 0. For one pulse scattering, the population of the |*pm*� state is

one scattering pulse depend on the lattice depth and the pulse duration. The lattice depth determines the band structure and is reflected in the terms *cn*,*q*. The pulse duration influences

The multi-pulse process, which consists of a number of single pulses and intervals can be solved as follows. The wave function of the condensate after the first pulse *τ*<sup>1</sup> can be derived

*b*(*m*0, *m*, *τ*1)*e*

2. It can be seen that the probabilities of the momentum states after

<sup>−</sup>*iE*(*m*)


∞ ∑ *n*=0

> ∞ ∑ *n*=0


each |*pm*� state (*m* = ..., −1, 0, 1, ...) can be acquainted as:

the phase evolution of each Bloch state as *e*−*iEn*,*qτ*1/¯*h*.

*b*(*m*0, *m*, *τ*1) =


*m*

*<sup>H</sup>*<sup>ˆ</sup> <sup>|</sup>*n*, *<sup>q</sup>*� <sup>=</sup> *En*,*<sup>q</sup>* <sup>|</sup>*n*, *<sup>q</sup>*�, (5)


<sup>−</sup>*iEn*,*qt*/¯*h*|*n*, *<sup>q</sup>*�. (7)

*<sup>t</sup>*/¯*h*|2*mhk*¯ *<sup>L</sup>*�. (9)

*cn*,*q*(*m*0)*cn*,*q*(*m*)*e*−*iEn*,*qτ*1/¯*h*. (8)

enough to suppress the spontaneous emission.

corresponding to the reciprocal lattice vector 2*kL*.

described as a superposition of the Bloch states:

total wave function evolves as

*<sup>P</sup>*(1) *<sup>m</sup>* <sup>=</sup> <sup>|</sup>*b*(0, *<sup>m</sup>*, *<sup>τ</sup>*1)<sup>|</sup>

from Eq. (8) as

satisfy the equation:

After the first interval *τ<sup>f</sup>* <sup>1</sup> and the second pulse *τ*2, the population of the |*pm*� state can be achieved as

$$P\_m^{(2)} = \left| \sum\_{m\_1} b(m\_0, m\_1, \tau\_1) e^{-iE^{(m\_1)}\tau\_{f1}/\hbar} b(m\_1, m, \tau\_2) \right|^2. \tag{10}$$

As shown in Eq. (10),the population is affected by the two pulses, the first one corresponding to *b*(*m*0, *m*1, *τ*1), and the second one corresponding to *b*(*m*1, *m*, *τ*2). During the interval *τ<sup>f</sup>* 1, the phase of the <sup>|</sup>*pm*<sup>1</sup> � state evolves along the time as *<sup>e</sup>*−*iE*(*m*1)*t*/¯*h*, where *<sup>E</sup>*(*m*1) = (2*m*1*hk*¯ *<sup>L</sup>*)2/2*<sup>M</sup>* <sup>=</sup> 4*m*<sup>2</sup> <sup>1</sup>*ER* is the kinetic energy, and *ER* = (*hk*¯ *<sup>L</sup>*)2/2*<sup>M</sup>* is the single photon recoil energy. The interval *<sup>τ</sup><sup>f</sup>* <sup>1</sup> produces a phase shift *<sup>e</sup>*−*iE*(*m*) *<sup>τ</sup><sup>f</sup>* 1/¯*<sup>h</sup>* and contributes to the momentum distribution.

In the same way, the population of the |*pm*� state after (*s* + 1) pulses can be achieved as:

$$P\_{m}^{(s+1)} = \left| \sum\_{m\_1, m\_2, \dots, m\_s} \prod\_{i=1}^{s+1} b(m\_{i-1}, m\_i, \tau\_i) \prod\_{i=1}^s e^{-iE^{(m\_i)}\tau\_{f\_i}/\hbar} \right|^2,\tag{11}$$

with *m*<sup>0</sup> = 0, and *ms*+<sup>1</sup> = *m*.

From the analysis above, the momentum distribution after a sequence of pulses' scattering is influenced by not only the lattice pulses with the term ∏*s*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *<sup>b</sup>*(*mi*−1, *mi*, *<sup>τ</sup>i*), but also the intervals among the pulses as reflected in the term ∏*<sup>s</sup> <sup>i</sup>*=<sup>1</sup> *e* <sup>−</sup>*iE*(*mi*)*τ<sup>f</sup> i* /¯*h* . Although the populations of the momentum states do not change during the intervals, the phase-evolution rates of the momentum states with different kinetic energies are not identical. The phase deviations between the states oscillate from 0 to 2*π* with the interval, and the heterogeneously accumulated phases change the distribution of the condensate in the quasi-momentum space.

### **3.2 Experiments of standing wave pulse sequences**

We performed the experiments of a condensate in a magnetic trap (MT) (see Fig. 7(a)) being scattered by a sequence of standing wave pulses (see Fig. 7(b)). As shown in Fig. 7(c), after pre-cooling, a cigar shaped 87Rb condensate of 2 <sup>×</sup> 105 atoms in 52*S*1/2 <sup>|</sup>*<sup>F</sup>* <sup>=</sup> 2, *MF* <sup>=</sup> <sup>2</sup>� state was achieved by the radio frequency (RF) cooling in the magnetic trap, of which the axial frequency is 20 Hz and the radial frequency is 220 Hz (Yang et al., 2008; Zhou et al., 2010). A pair of counter-propagating laser beams, of which the durations were controlled by an acousto-optical modulator, and the amplitudes were adjusted by the injection current of a tapered-amplifier, were applied to the condensate along the axial direction. The linear polarized incident light at the wavelength *λ<sup>L</sup>* = 852 nm was focused with a waist of 110 *μ*m to cover the condensate. The trap depth, which was calibrated by Kapitza-Dirac scattering experimentally, reached 120*ER*, corresponding to the light power of 320 mW. The incident light and the magnetic trap were simultaneously shut after the BEC-light interaction. After 30 *ms* free falling and ballistic expansion, the atomic gas was pictured by absorption imaging. Since the minimum gap between different momentum states is 2¯*hk*, which is much larger than the momentum width of a single momentum state, the components with different momenta will be separated in the TOF (time of flight) signal (see Fig. 7(d)), and the atomic number of each momentum state is possible to be read separately as *Nm* (the momentum

*Nhk*

*relative population P*

momenta 0¯*hk*, ±2¯*hk*, ±4¯*hk* and ±6¯*hk* respectively.

*pulse width s* W P( )

Fig. 8. (Color online) Single pulse scattering of condensate: The black dots represent the experiment results. The blue solid line is the theoretical analysis with Raman-Nath

approximation. The red dashed line is the numerical simulation with band structure theory. Figure (a), (b),(c) and (d) correspond to the relative populations of the condensates with the

First we demonstrate a one-pulse scattering experiment. A condensate is exposed to a standing wave pulse with depth 100*ER* and duration varying from 0 to 30*μs*. The relative populations of the condensates with the momenta 0¯*hk*, ±2¯*hk*, ±4¯*hk* and ±6¯*hk*, corresponding to Fig. 8(a), (b), (c) and (d) respectively, are measured and theoretically analyzed. In addition, the theoretical analysis with the Raman-Nath approximation, is also shown in the figure for comparison. It can be seen that within 3*μs* the theoretical analysis with the Raman-Nath approximation (blue solid line) is close to the experimental results (black dots), and so is the theoretical analysis with band structure theory (red dashed line). When the pulse duration exceeds 3*μs*, the analysis with the Raman-Nath approximation gradually goes far away from the experimental results, while the numerical simulation with band structure theory still agrees with the experimental results along the entire time scale. As shown in Fig. 8, the probability of each momentum state oscillates with the pulse duration as described by the band structure theory. It is clear that, in the single pulse scattering process, the band structure theory works well not only for the short pulse but also for the longer pulse, because the atomic motion has been taken into account. So the atomic diffraction by a single standing wave pulse

can be predicted in a wider range of pulse duration with the band structure theory.

Then we increase the number of pulses in the experiments to explore the extra factors influencing the momentum distributions. In every sequence, all the pulses are the same and all the intervals are identical to make the experiments more convenient to carry out. For further

<sup>4</sup> ( ) *<sup>k</sup> c P*<sup>r</sup> <sup>=</sup> <sup>6</sup> ( ) *<sup>k</sup> d P*<sup>r</sup> <sup>=</sup>

<sup>0</sup> ( ) *<sup>k</sup> a P* <sup>=</sup> <sup>2</sup> ( ) *<sup>k</sup> b P*<sup>r</sup> <sup>=</sup>

High Order Momentum States by Light Wave Scattering 353

Fig. 7. (Color online) (a) A pair of counter-propagating light beams are applied to a condensate in magnetic trap. (b) The scattering process consists of a sequence of standing wave pulses, which contained *s* intervals with widths *τf i* (i=1,2,...*s*) and *s* + 1 light pulses with durations *τ<sup>i</sup>* (i=1,2,...*s* + 1). The incident light's wavelength is 852 *nm* and its maximum intensity can reach 120*ER*. (c) The procedure for the experiments is shown. The condensate is exposed to a sequence of standing wave pulses and then released from the magnetic trap. The absorption images of the condensate can be observed after the free falling. (d) A TOF signal obtained in our experiment.

order *m* = ..., −2, −1, 0, 1, 2). The relative population of the momentum state |2*mhk*¯ � can be evaluated as *Nm*/ ∑ *Nm*.

The lattice in our experiments is quite deep, so we concentrate on the short-pulse diffractions to avoid the de-coherence and heating effects of long pulses relevant for Bragg scattering. However, for more flexible momentum manipulation, our pulses are not so short as the Raman-Nath pulses (Huckans et al., 2009) used in previous works.

A brief introduction to the Raman-Nath regime is given in the following for comparison. In the scattering process, the evolution during the free evolution intervals is analyzed as in previous section, while the effect of lattice with adequately short duration *τ* can be analytically solved by the *Schrodinger* ¨ equation *ih*¯ *<sup>∂</sup>* <sup>|</sup>Ψ(*t*)� /*∂<sup>t</sup>* <sup>=</sup> *<sup>H</sup>*<sup>ˆ</sup> <sup>|</sup>Ψ(*t*)�, after omitting the atomic kinetic energy term *p*ˆ2/2*M* in the Hamiltonian. This approximation can be made while the displacement of the scattered atoms during the interaction time is much smaller than the spatial period of the standing wave. Equivalently, the standing wave duration *τ* and the single photon recoil frequency *ω<sup>r</sup>* = *hk*¯ <sup>2</sup> *<sup>L</sup>*/2*M* have to fulfill *τ* � 1/*ωr*. The pulse is able to split a stationary condensate into components with symmetrical momenta *pn* = 2*nhk*¯ *<sup>L</sup>*(*n* = 0, ±1, ±2, ...), with corresponding populations *Pn* = *J*<sup>2</sup> *<sup>n</sup>*(*U*0*τ*/2¯*h*), where *Jn*(*z*) are Bessel functions of the first kind.

12 Will-be-set-by-IN-TECH

*s*

*Lattice sequence*

*MT off, BEC released*

8=*k* 6=*k* 4=*k* 2=*k* 0=*k* 2=*k* 4=*k* 6=*k* 8=*k*

order *m* = ..., −2, −1, 0, 1, 2). The relative population of the momentum state |2*mhk*¯ � can be

The lattice in our experiments is quite deep, so we concentrate on the short-pulse diffractions to avoid the de-coherence and heating effects of long pulses relevant for Bragg scattering. However, for more flexible momentum manipulation, our pulses are not so short as the

A brief introduction to the Raman-Nath regime is given in the following for comparison. In the scattering process, the evolution during the free evolution intervals is analyzed as in previous section, while the effect of lattice with adequately short duration *τ* can be analytically solved by the *Schrodinger* ¨ equation *ih*¯ *<sup>∂</sup>* <sup>|</sup>Ψ(*t*)� /*∂<sup>t</sup>* <sup>=</sup> *<sup>H</sup>*<sup>ˆ</sup> <sup>|</sup>Ψ(*t*)�, after omitting the atomic kinetic energy term *p*ˆ2/2*M* in the Hamiltonian. This approximation can be made while the displacement of the scattered atoms during the interaction time is much smaller than the spatial period of the standing wave. Equivalently, the standing wave duration *τ* and the single photon recoil

condensate into components with symmetrical momenta *pn* = 2*nhk*¯ *<sup>L</sup>*(*n* = 0, ±1, ±2, ...), with

*<sup>L</sup>*/2*M* have to fulfill *τ* � 1/*ωr*. The pulse is able to split a stationary

*<sup>n</sup>*(*U*0*τ*/2¯*h*), where *Jn*(*z*) are Bessel functions of the first

Raman-Nath pulses (Huckans et al., 2009) used in previous works.

Fig. 7. (Color online) (a) A pair of counter-propagating light beams are applied to a condensate in magnetic trap. (b) The scattering process consists of a sequence of standing wave pulses, which contained *s* intervals with widths *τf i* (i=1,2,...*s*) and *s* + 1 light pulses with durations *τ<sup>i</sup>* (i=1,2,...*s* + 1). The incident light's wavelength is 852 *nm* and its maximum intensity can reach 120*ER*. (c) The procedure for the experiments is shown. The condensate is exposed to a sequence of standing wave pulses and then released from the magnetic trap. The absorption images of the condensate can be observed after the free falling. (d) A TOF

W *fs* W *s* 1 W

> *absorption imaging*

1 W

( ) *c RF Cooling in MT*

*BEC achieved*

( ) *a*

( ) *b*

( ) *d*

signal obtained in our experiment.

evaluated as *Nm*/ ∑ *Nm*.

frequency *ω<sup>r</sup>* = *hk*¯ <sup>2</sup>

kind.

corresponding populations *Pn* = *J*<sup>2</sup>

 2 W*f* 1

W

Fig. 8. (Color online) Single pulse scattering of condensate: The black dots represent the experiment results. The blue solid line is the theoretical analysis with Raman-Nath approximation. The red dashed line is the numerical simulation with band structure theory. Figure (a), (b),(c) and (d) correspond to the relative populations of the condensates with the momenta 0¯*hk*, ±2¯*hk*, ±4¯*hk* and ±6¯*hk* respectively.

First we demonstrate a one-pulse scattering experiment. A condensate is exposed to a standing wave pulse with depth 100*ER* and duration varying from 0 to 30*μs*. The relative populations of the condensates with the momenta 0¯*hk*, ±2¯*hk*, ±4¯*hk* and ±6¯*hk*, corresponding to Fig. 8(a), (b), (c) and (d) respectively, are measured and theoretically analyzed. In addition, the theoretical analysis with the Raman-Nath approximation, is also shown in the figure for comparison. It can be seen that within 3*μs* the theoretical analysis with the Raman-Nath approximation (blue solid line) is close to the experimental results (black dots), and so is the theoretical analysis with band structure theory (red dashed line). When the pulse duration exceeds 3*μs*, the analysis with the Raman-Nath approximation gradually goes far away from the experimental results, while the numerical simulation with band structure theory still agrees with the experimental results along the entire time scale. As shown in Fig. 8, the probability of each momentum state oscillates with the pulse duration as described by the band structure theory. It is clear that, in the single pulse scattering process, the band structure theory works well not only for the short pulse but also for the longer pulse, because the atomic motion has been taken into account. So the atomic diffraction by a single standing wave pulse can be predicted in a wider range of pulse duration with the band structure theory.

Then we increase the number of pulses in the experiments to explore the extra factors influencing the momentum distributions. In every sequence, all the pulses are the same and all the intervals are identical to make the experiments more convenient to carry out. For further

pulse to make the probability *P*0¯*hk* the minimum. While the phase shift is *π* with the interval *πh*¯ /4*ER* (about 40*μs*), the second pulse produces an effect opposite of the first one and diffract the non-stationary components of the condensate back to the stationary one and make the

High Order Momentum States by Light Wave Scattering 355

It also can be acquainted from Fig. 9 that the numerical simulation with band structure theory is much closer to the experimental results than the analytical solution with Raman-Nath approximation. It is conjectured that the phase evolution during the scattering process makes the difference. The phase evolution in the scattering process is neglected in the Raman-Nath approximation, but not in the numerical simulation with band structure theory. Although the duration of the scattering process is short, the phase shifts in the scattering process still increase. The phase shift in the scattering process needs to be taken into account and influences the final momentum distribution. As a result, the longer the scattering pulse is, the larger the difference is. Although the maximum of the probability *P*0¯*hk* corresponds to the interval *πh*¯ /4*ER*, the two-pulse experiments in Fig. 9 can clearly show that the longer pulse leads to the larger difference. In Fig. 9 (a), the pulse duration is 6*μs*, the probability *P*0¯*hk* reaches the maximum with the interval 34*μs*. In Fig. 9 (b), the pulse duration is 2*μs*, the

Nevertheless, there is still some obvious deviation between the simulation and the experimental results. It is observed that the momentum width has been expanded after the former pulse (see Fig. 9(c)), because of the *s*-wave scattering between the different momentum states. Consequently, this dispersion process is approximated to an initial momentum width of ∼ 0.1¯*hkL* on average to optimize the numerical simulation. Unlike the analysis without momentum width, phase evolution is different for different initial momenta and results in a phase dispersion. The quasi modes obtained at the end of the diffraction process result from the linear superposition of final states obtained after time evolutions of the different momenta populated the initial BEC. It can be seen from Fig. 9 that the approximation is effective.

As discussed in (Li, Deng et al., 2008), the maximum of the probability *P*0¯*hk* will never reach 1 thanks to the imperfect optical lattice. In our case, the momentum expansion is an explanation of the similar situation as shown in Fig. 9. Since the momentum width is considered, the phase shift is populated around *π* with a width, instead of a definite *π*, with the interval *πh*¯ /4*ER*. In other words, there is no interval that accumulates a phase shift exactly equal to *π*, so with any interval, the second standing wave pulse is not able to diffract all the condensates back to

The experiments and the numerical simulations above have shown the possibility and feasibility of the manipulation of a condensate's momentum states. We manage to design several two-pulse sequences to achieve high contrast momentum states such as |±2¯*hk*�, |±4¯*hk*� and |±6¯*hk*�, which may be useful in atomic interferometry (Beattie et al., 2009; Rohwedder, 2001). For each state, we apply two totally different two-pulse sequences to show the flexibility of the method. The general method to achieve the target states is to find

is the probability of |2*mhk*¯ *<sup>L</sup>*� in the goal state, and *Pm* is that generated by the sequence. A

<sup>∑</sup> *<sup>m</sup>*=−<sup>∞</sup>

(*P<sup>g</sup> <sup>m</sup>* − *Pm*)

2

, where *P<sup>g</sup>*

*m*

probability *P*0¯*hk* the maximum.

the stationary part.

probability *P*0¯*hk* gets to the top with the interval 38*μs*.

**3.3 Manipulate the momentum states as design**

out the condition of the minimum of the square deviation <sup>Δ</sup><sup>2</sup> <sup>=</sup> <sup>+</sup><sup>∞</sup>

comparison between the band structure theory and the analysis in the Raman-Nath regime, every single pulse is made short enough for the Raman-Nath approximation.

Fig. 9. (Color online) Two-pulse scattering of the condensate: The relative populations of the stationary condensate *P*0¯*hk* versus the varied intervals *τ<sup>f</sup>* is shown. The parameters of the experiments are described above each figure. The black dots are the experimental results. The blue dashed line is the analysis with Raman-Nath approximation. The magenta dotted line is the numerical simulation with band structure theory. The red solid line is a numerical simulation taking into account the momentum dispersion. (c) A demonstration of momentum expansion introduced by *s*-wave scattering. The TOF signal (i) corresponds to the dashed curve and pictures the momentum distribution before the former lattice pulse. The signal (ii) corresponds to the solid curve and shows the momentum distribution after the former pulse. The momentum width along Z direction (the lattice direction) is evaluated based on the parts in the dashed boxes which include the momenta from −*hk*¯ to ¯*hk*.

Two experiments of two-pulse scattering are demonstrated in Fig. 9, in which the relative populations of the stationary condensate *P*0¯*hk* versus the varied intervals *τ* is shown. The parameters of the scattering pulses used in different sequences are chosen to be of the same products of the lattice depth and the pulse duration, so that each pulse affects the condensate equivalently. As shown in the figure, the intervals actually affect the final momentum distribution, and the theoretical analysis with the band structure theory and Raman-Nath approximation both picture well the evolution of the atomic distributions versus the interval between the two pulses. The results of two-pulse scattering can be explained as the fact that since the phase shift accumulated during the interval varies harmonically from 0 to 2*π*, the probability of the stationary condensate oscillates between the minimum and the maximum. When the phase shift is 2*π* with the interval *πh*¯ /2*ER* (around 80*μs*), the wave function is little affected by the interval and the two pulses diffract the condensate as one combined 14 Will-be-set-by-IN-TECH

comparison between the band structure theory and the analysis in the Raman-Nath regime,

*f*

P

( ) *<sup>f</sup> interval ti s me* W P

0 0.2 0.4 0.6 0.8 1.0

( )*i* ( ) *ii*

40 60 80

*position along Z direction m* ()

Fig. 9. (Color online) Two-pulse scattering of the condensate: The relative populations of the stationary condensate *P*0¯*hk* versus the varied intervals *τ<sup>f</sup>* is shown. The parameters of the experiments are described above each figure. The black dots are the experimental results. The blue dashed line is the analysis with Raman-Nath approximation. The magenta dotted line is the numerical simulation with band structure theory. The red solid line is a numerical

momentum expansion introduced by *s*-wave scattering. The TOF signal (i) corresponds to the dashed curve and pictures the momentum distribution before the former lattice pulse. The signal (ii) corresponds to the solid curve and shows the momentum distribution after the former pulse. The momentum width along Z direction (the lattice direction) is evaluated based on the parts in the dashed boxes which include the momenta from −*hk*¯ to ¯*hk*.

Two experiments of two-pulse scattering are demonstrated in Fig. 9, in which the relative populations of the stationary condensate *P*0¯*hk* versus the varied intervals *τ* is shown. The parameters of the scattering pulses used in different sequences are chosen to be of the same products of the lattice depth and the pulse duration, so that each pulse affects the condensate equivalently. As shown in the figure, the intervals actually affect the final momentum distribution, and the theoretical analysis with the band structure theory and Raman-Nath approximation both picture well the evolution of the atomic distributions versus the interval between the two pulses. The results of two-pulse scattering can be explained as the fact that since the phase shift accumulated during the interval varies harmonically from 0 to 2*π*, the probability of the stationary condensate oscillates between the minimum and the maximum. When the phase shift is 2*π* with the interval *πh*¯ /2*ER* (around 80*μs*), the wave function is little affected by the interval and the two pulses diffract the condensate as one combined

simulation taking into account the momentum dispersion. (c) A demonstration of

2=*k* 2=*k*

P

0 20 40 60 80

60 20*ER ER*

*s <sup>f</sup>* W

every single pulse is made short enough for the Raman-Nath approximation.

*s* 2

( ) *a* ( ) *b*

6P

( ) *c*

0 20

*0hk*

*relative population P*

0 0.2 0.4 0.6 0.8 1.0

*integrated*

 *optical density*

*along Y direction*

W

pulse to make the probability *P*0¯*hk* the minimum. While the phase shift is *π* with the interval *πh*¯ /4*ER* (about 40*μs*), the second pulse produces an effect opposite of the first one and diffract the non-stationary components of the condensate back to the stationary one and make the probability *P*0¯*hk* the maximum.

It also can be acquainted from Fig. 9 that the numerical simulation with band structure theory is much closer to the experimental results than the analytical solution with Raman-Nath approximation. It is conjectured that the phase evolution during the scattering process makes the difference. The phase evolution in the scattering process is neglected in the Raman-Nath approximation, but not in the numerical simulation with band structure theory. Although the duration of the scattering process is short, the phase shifts in the scattering process still increase. The phase shift in the scattering process needs to be taken into account and influences the final momentum distribution. As a result, the longer the scattering pulse is, the larger the difference is. Although the maximum of the probability *P*0¯*hk* corresponds to the interval *πh*¯ /4*ER*, the two-pulse experiments in Fig. 9 can clearly show that the longer pulse leads to the larger difference. In Fig. 9 (a), the pulse duration is 6*μs*, the probability *P*0¯*hk* reaches the maximum with the interval 34*μs*. In Fig. 9 (b), the pulse duration is 2*μs*, the probability *P*0¯*hk* gets to the top with the interval 38*μs*.

Nevertheless, there is still some obvious deviation between the simulation and the experimental results. It is observed that the momentum width has been expanded after the former pulse (see Fig. 9(c)), because of the *s*-wave scattering between the different momentum states. Consequently, this dispersion process is approximated to an initial momentum width of ∼ 0.1¯*hkL* on average to optimize the numerical simulation. Unlike the analysis without momentum width, phase evolution is different for different initial momenta and results in a phase dispersion. The quasi modes obtained at the end of the diffraction process result from the linear superposition of final states obtained after time evolutions of the different momenta populated the initial BEC. It can be seen from Fig. 9 that the approximation is effective.

As discussed in (Li, Deng et al., 2008), the maximum of the probability *P*0¯*hk* will never reach 1 thanks to the imperfect optical lattice. In our case, the momentum expansion is an explanation of the similar situation as shown in Fig. 9. Since the momentum width is considered, the phase shift is populated around *π* with a width, instead of a definite *π*, with the interval *πh*¯ /4*ER*. In other words, there is no interval that accumulates a phase shift exactly equal to *π*, so with any interval, the second standing wave pulse is not able to diffract all the condensates back to the stationary part.

## **3.3 Manipulate the momentum states as design**

The experiments and the numerical simulations above have shown the possibility and feasibility of the manipulation of a condensate's momentum states. We manage to design several two-pulse sequences to achieve high contrast momentum states such as |±2¯*hk*�, |±4¯*hk*� and |±6¯*hk*�, which may be useful in atomic interferometry (Beattie et al., 2009; Rohwedder, 2001). For each state, we apply two totally different two-pulse sequences to show the flexibility of the method. The general method to achieve the target states is to find out the condition of the minimum of the square deviation <sup>Δ</sup><sup>2</sup> <sup>=</sup> <sup>+</sup><sup>∞</sup> <sup>∑</sup> *<sup>m</sup>*=−<sup>∞</sup> (*P<sup>g</sup> <sup>m</sup>* − *Pm*) 2 , where *P<sup>g</sup> m* is the probability of |2*mhk*¯ *<sup>L</sup>*� in the goal state, and *Pm* is that generated by the sequence. A

of generating higher order momentum states. In the Raman-Nath regime, the intensity of the standing wave is not limited so that higher order momentum states can be generated symmetrically (Gadway et al., 2009; Sapiro et al., 2009b). However the pulse duration has to be short enough to neglect the atomic motion, so the momentum states can not be predicted in this regime if the pulse duration is slightly longer. In our work, the scattering can be well explained and numerically analyzed within a much wider range of pulse intensity and

High Order Momentum States by Light Wave Scattering 357

Some valuable states, such as |±2¯*hk*�, |±4¯*hk*� and |±6¯*hk*� states with high contrast, can not be realized by the single pulse scattering, while they can be realized by a sequence of standing wave pulses. A sequence of lattice pulses is a more effective and flexible tool for momentum manipulation. It can generate many useful momentum states, in addition to the

Although the numerical simulation is corrected to take into account the momentum dispersion, some deviations between the experiments and the simulation still exist. The inaccuracy of the lattice-depth calibration, which is 5% at least, is one of the reasons. The phase shift introduced by the magnetic trap is another one, while the influence is estimated to be within 0.03%, which is below the experimental uncertainty. The heating and momenta

We have shown in this chapter two methods of getting high order momentum states by resonant superradiant scattering and by a sequences of pulsed standing waves, corresponding to traveling wave scattering and standing wave scattering, respectively. Different to Bragg scattering, we can get a large number of high-order forward modes by resonant frequency components of the pump beam by the former method through the mode competition between the different resonant channels and the local depletion of the spatial distribution. While with the latter method a symmetric momentum distribution is generated, with one pair of momentum states designed to be mostly populated. Both methods are beneficial to a lager momentum transfer in atom manipulation for the atom interferometry and atomic optics.

Bar-Gill, N., Rowen, E. E. & Davidson, N. (2007). Spectroscopy of strong-pulse superradiance

Baumgärtner, F., Sewell, R. J., Eriksson, S., Llorente-Garcia, I., Dingjan, J., Cotter, J. P. & Hinds,

Beattie, S., Barrett, B., Chan, I., Mok, C., Yavin, I. & Kumarakrishnan, A. (2009). Atom-interferometric studies of light scattering, *Phys. Rev. A* 80: 013618. Cola, M. M., Volpe, L. & Piovella, N. (2009). Accelerated superradiance and collective atomic

Cronin, A. D., Schmiedmayer, J. & Pritchard, D. E. (2009). Optics and interferometry with

recoil lasing with a two-frequency pump, *Phys. Rev. A* 79: 013613.

E. A. (2010). Measuring energy differences by BEC interferometry on a chip, *Phys.*

duration. So it is natural that more momentum states can be generated.

exchange during the s-wave scattering may also lead to some differences.

in a Bose-Einstein condensate, *Phys. Rev. A* 76: 043603.

atoms and molecules, *Rev. Mod. Phys.* 81: 1051.

ones demonstrated in our work.

**4. Conclusion**

**5. References**

*Rev. Lett.* 105: 243003.

second method, as the target is to obtain the highest population of some certain momentum state, consists in scanning the set of initial conditions and choose the one corresponding to the maximum value of the desired population. We apply the two methods above separately and obtain the same pulse sequences. As shown in Fig. 10, the experimental results (the black

Fig. 10. Experimental realization of designed momentum states. The expected momentum state is ±2¯*hkL*((a) and (b)), ±4¯*hkL*((c) and (d)) and ±6¯*hkL*((e) and (f)). The pulse sequences are shown above each figure. The black round dots are experimental results. The blue diamond dots are the expectations based on the design. The red square dots are the modified design with momentum width, which agree with the experiments better.

round dots) agree well with the expectations of the designs (the blue diamond dots), whether the pulses are in the Raman-Nath regime (see Fig. 10(b)) or not (see others in Fig. 10). When the momentum dispersion is being considered, the expected momentum distributions (the red square dots) get closer to the experiments, where the figures only display the relative populations of the target states and omit the others for the figures being more clear.

It can be seen from Fig. 10 that the momentum width correction can improve the precision of the prediction with our method. The average relative deviation between the experimental results and the expected values without including the momentum width is 25.03%, while the deviation is decreased to 13.15% with the correction.

An asymmetry of the momenta can be observed in Fig. 10, and it may ascribe to the following factors. Besides the measurement error, there is an imperfection of the standing wave, brought forth by the unbalanced intensity of the laser beams. External field (such as the magnetic trap) fluctuations during the scattering process may also affect the momentum distribution.

### **3.4 Discussion**

The band structure theory is a global method to deal with the standing wave scattering a condensate, while Bragg and Raman-Nath scattering are two special situations which can be analytically solved with their respective approximations. In the Bragg regime, the potential height introduced by the standing wave is restrained below 4*ER* and that leads to the difficulty of generating higher order momentum states. In the Raman-Nath regime, the intensity of the standing wave is not limited so that higher order momentum states can be generated symmetrically (Gadway et al., 2009; Sapiro et al., 2009b). However the pulse duration has to be short enough to neglect the atomic motion, so the momentum states can not be predicted in this regime if the pulse duration is slightly longer. In our work, the scattering can be well explained and numerically analyzed within a much wider range of pulse intensity and duration. So it is natural that more momentum states can be generated.

Some valuable states, such as |±2¯*hk*�, |±4¯*hk*� and |±6¯*hk*� states with high contrast, can not be realized by the single pulse scattering, while they can be realized by a sequence of standing wave pulses. A sequence of lattice pulses is a more effective and flexible tool for momentum manipulation. It can generate many useful momentum states, in addition to the ones demonstrated in our work.

Although the numerical simulation is corrected to take into account the momentum dispersion, some deviations between the experiments and the simulation still exist. The inaccuracy of the lattice-depth calibration, which is 5% at least, is one of the reasons. The phase shift introduced by the magnetic trap is another one, while the influence is estimated to be within 0.03%, which is below the experimental uncertainty. The heating and momenta exchange during the s-wave scattering may also lead to some differences.

## **4. Conclusion**

16 Will-be-set-by-IN-TECH

second method, as the target is to obtain the highest population of some certain momentum state, consists in scanning the set of initial conditions and choose the one corresponding to the maximum value of the desired population. We apply the two methods above separately and obtain the same pulse sequences. As shown in Fig. 10, the experimental results (the black

90*ER*

( ) *a* ( ) *b*

*fi* 1.8P*s*

0 *k*

 =

*relative population P*

**3.4 Discussion**

W

( ) *fi total interval time s* ¦

Fig. 10. Experimental realization of designed momentum states. The expected momentum state is ±2¯*hkL*((a) and (b)), ±4¯*hkL*((c) and (d)) and ±6¯*hkL*((e) and (f)). The pulse sequences are shown above each figure. The black round dots are experimental results. The blue diamond dots are the expectations based on the design. The red square dots are the modified

round dots) agree well with the expectations of the designs (the blue diamond dots), whether the pulses are in the Raman-Nath regime (see Fig. 10(b)) or not (see others in Fig. 10). When the momentum dispersion is being considered, the expected momentum distributions (the red square dots) get closer to the experiments, where the figures only display the relative

It can be seen from Fig. 10 that the momentum width correction can improve the precision of the prediction with our method. The average relative deviation between the experimental results and the expected values without including the momentum width is 25.03%, while the

An asymmetry of the momenta can be observed in Fig. 10, and it may ascribe to the following factors. Besides the measurement error, there is an imperfection of the standing wave, brought forth by the unbalanced intensity of the laser beams. External field (such as the magnetic trap) fluctuations during the scattering process may also affect the momentum distribution.

The band structure theory is a global method to deal with the standing wave scattering a condensate, while Bragg and Raman-Nath scattering are two special situations which can be analytically solved with their respective approximations. In the Bragg regime, the potential height introduced by the standing wave is restrained below 4*ER* and that leads to the difficulty

populations of the target states and omit the others for the figures being more clear.

design with momentum width, which agree with the experiments better.

deviation is decreased to 13.15% with the correction.

W P

1.35P

*s fi* W

90*ER*

We have shown in this chapter two methods of getting high order momentum states by resonant superradiant scattering and by a sequences of pulsed standing waves, corresponding to traveling wave scattering and standing wave scattering, respectively. Different to Bragg scattering, we can get a large number of high-order forward modes by resonant frequency components of the pump beam by the former method through the mode competition between the different resonant channels and the local depletion of the spatial distribution. While with the latter method a symmetric momentum distribution is generated, with one pair of momentum states designed to be mostly populated. Both methods are beneficial to a lager momentum transfer in atom manipulation for the atom interferometry and atomic optics.

## **5. References**


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**18** 

*Cyprus* 

**Thickness Measurement of Photoresist** 

Fotis Kossivas, Charalabos Doumanidis and Andreas Kyprianou

Photoresists are polymers sensitive to light, usually in the ultraviolet (UV) range of the electromagnetic radiation spectrum. They are classified into positive and negative. In positive types, the exposed to light region of the photoresist becomes solvable to a developer, while the unexposed one remains unsolvable. In negative types the opposite happens. Photoresists are widely used in micro and nano fabrication such as photo and ebeam lithography. In these techniques the substrate, usually a silicon (Si), wafer, is spin coated with a thin film of photoresist and then is selectively exposed to the proper radiation. The substrate is then developed with a suitable chemical solution which etches the exposed (positive) or unexposed (negative) regions, resulting in transferring the desired pattern on it. For these, knowledge of the photoresist exact thickness is crucial for the calculation of the

Traditionally in photolithography practice, the thickness of photoresist films is deduced indirectly by the volume of the resin dispensed during spin coating, divided by the wafer surface. However, this method is subject to errors due to uncertainty of the solvent evaporation and thickness distribution nonuniformities, e.g. near the wafer edges due to wetting effects etc. Recently, the measurement of liquid crystal (LC) films thickness was achieved using a relative simple processing of common interferometer images (optical profilometer), considering the phase shift of light as propagated through LCs. The technique was tested and verified with atomic force microscopy (AFM) in micron and sub micron film

This work proposes an extension of this technique to thin films made from photoresists. Thin films of various thicknesses have been fabricated, measured with the proposed method and verified with AFM technique. The results further confirmed the validity of the proposed technique, as the difference of the two methods was within the statistical error bounds. Further testing has been also performed in standard photolithography processes to investigate if the proposed technique affects the performance of photoresists. It was found that the influence in photoresist performance is negligible as they are sensitive only to the i-line of the spectrum from a mercury vapour lamp used for photolithography. The interferometer used here was

thickness, and was found to be in good agreement to the AFM measurements [1].

operated with a low intensity green (550 nm wavelength) laser source.

**1. Introduction** 

necessary radiation energy dose.

**Thin Films Using Interferometry** 

*University of Cyprus, Mechanical and Manufacturing* 

*Engineering Department, Nicosia,* 


## **Thickness Measurement of Photoresist Thin Films Using Interferometry**

Fotis Kossivas, Charalabos Doumanidis and Andreas Kyprianou *University of Cyprus, Mechanical and Manufacturing Engineering Department, Nicosia, Cyprus* 

## **1. Introduction**

20 Will-be-set-by-IN-TECH

360 Interferometry – Research and Applications in Science and Technology

Zobay, O. & Nikolopoulos, G. (2007). Sequential superradiant scattering from atomic

Zobay, O. & Nikolopoulos, G. M. (2006). Spatial effects in superradiant Rayleigh scattering

Bose-Einstein condensates, *Laser Physics* 17: 180.

from Bose-Einstein condensates, *Phys. Rev. A* 73: 013620.

Photoresists are polymers sensitive to light, usually in the ultraviolet (UV) range of the electromagnetic radiation spectrum. They are classified into positive and negative. In positive types, the exposed to light region of the photoresist becomes solvable to a developer, while the unexposed one remains unsolvable. In negative types the opposite happens. Photoresists are widely used in micro and nano fabrication such as photo and ebeam lithography. In these techniques the substrate, usually a silicon (Si), wafer, is spin coated with a thin film of photoresist and then is selectively exposed to the proper radiation. The substrate is then developed with a suitable chemical solution which etches the exposed (positive) or unexposed (negative) regions, resulting in transferring the desired pattern on it. For these, knowledge of the photoresist exact thickness is crucial for the calculation of the necessary radiation energy dose.

Traditionally in photolithography practice, the thickness of photoresist films is deduced indirectly by the volume of the resin dispensed during spin coating, divided by the wafer surface. However, this method is subject to errors due to uncertainty of the solvent evaporation and thickness distribution nonuniformities, e.g. near the wafer edges due to wetting effects etc. Recently, the measurement of liquid crystal (LC) films thickness was achieved using a relative simple processing of common interferometer images (optical profilometer), considering the phase shift of light as propagated through LCs. The technique was tested and verified with atomic force microscopy (AFM) in micron and sub micron film thickness, and was found to be in good agreement to the AFM measurements [1].

This work proposes an extension of this technique to thin films made from photoresists. Thin films of various thicknesses have been fabricated, measured with the proposed method and verified with AFM technique. The results further confirmed the validity of the proposed technique, as the difference of the two methods was within the statistical error bounds. Further testing has been also performed in standard photolithography processes to investigate if the proposed technique affects the performance of photoresists. It was found that the influence in photoresist performance is negligible as they are sensitive only to the i-line of the spectrum from a mercury vapour lamp used for photolithography. The interferometer used here was operated with a low intensity green (550 nm wavelength) laser source.

Thickness Measurement of Photoresist Thin Films Using Interferometry 363

Generally, in optical profilometry (or interference profilometry) the measurement starts with the operator focusing to the point where the fringe pattern occurs. Then this pattern is subjected to appropriate processing by the software of the instrument in order to extract the 3D topographical image of the sample. In the case of a transparent film with thicknesses bigger than about one micron, two fringe patterns are generated (figure 1), from exactly the same area of the sample. The first is taken by focusing on the upper surface of the transparent film and the second by focusing on the substrate. Furthermore, for thicknesses less than a micron there is only one fringe pattern which may be attributed to the substrate,

as its reflectivity is significantly higher than that of the upper boundary of the film.

Fig. 1. A) First fringe pattern generated from the upper surface of the transparent film. B) fringe pattern generated from the substrate. The right image is distorted in the region of film because of the phase shift that light undergoes while it travels through the film.

changes in the film in order to correct the initially measured image follows.

**2.3 Understanding the initial image** 

In the case where the first fringe pattern is used, the topology of the upper boundary of the film is taken. However in the case where the second pattern is used, an image which corresponds to the substrate is obtained. Here the region that is coated by the transparent film appears to be at a lower height than the non-coated substrate. Such an image is shown in figure 8 in section 6. Definitely, this does not reflect the reality. This happens because to the instrument the existence of the transparent film is unaccounted for, and hence the phase difference registered by the instrument is taken to be due to height differences and not due to the delay of light propagation in the film. A method that considers the optical path

Here, the principle of operation of the instrument and the optical path of its light source are considered for the case of measuring the boundary of a transparent film. Figure 2 shows a

**2.2 Initial imaging of the borders of a transparent film** 

## **2. Optical profilometry theory and thickness measurement methodology**

This section starts with an overview of how light interference is implemented into a singlewavelength optical profilometer and it is used for the extraction of three dimensional (3D) topography images of substrates. Then it analytically describes the methodology of using the phenomenon of phase shift that appears in light propagated through transparent films, together with optical profilometry to extract the thickness of the film. Finally an overview of fringe order theory is given, which is useful in the determination of the maximum possible height the instrument is capable to measure.

## **2.1 Existing knowledge**

In general, optical profilometry exploits the interference between a reference light beam and a measuring beam reflected from the sample surface in order to extract a 3D topographical image. If the travelled distances of the two beams differ by an even number of halfwavelengths then a bright fringe (constructive interference) is created, whilst a difference equal to an odd number of half-wavelengths creates a dark one (destructive interference). In intermediate cases, gray lines are created. The height of a particular point on the sample surface is determined by the type of interference, which in the resulting image of the surface is captured by the colour of the corresponding pixel. In this mode, the number of fringes defines the maximum possible measured height. It is generally recommended to use less than three fringes (which correspond to a height difference equal to three times the source wavelength), in order to avoid problems with the coherence length of the laser source. However, with some care slightly larger numbers of fringes can also be used.

Measurements of thicknesses and refractive indices of transparent thick polymer films based on light interference phenomena are well known and developed. They are based on the fact that fringes from the upper and lower boundaries of the film are well separated, and hence the film thickness or the refractive index can be calculated from the distance between the envelopes of the two fringe patterns. However, as the film thickness decreases, the envelopes of the two fringes come closer to each other, and for thicknesses less than about 1µm they overlap. This complicates significantly the calculation of the distance between fringes, and thus the film thickness. Recently, the helical conjugate field function (HCF) has been utilized in coherence correlation interferometry to circumvent this limitation [8, 9,10].

In many areas of engineering and physics, the above technique has been incorporated in methodologies of measuring physical properties and their response to the presence of external stimuli. In the field of mechanical engineering, this constituted the underlying principle in holographic interferometry and Moire inteferometry. The former concerns measuring the properties of transparent specimens [3], whereas the latter is used to measure small deformations of non-transparent solids due to external forces and thermal expansions, as for example is described in [4, 5]. Recently, a methodology which uses two sample probes for simultaneously measuring the refractive index and thickness of transparent specimens has been developed and demonstrated in [2]. Interferometry has also been used to measure parameters of transparent films made from nematic liquid crystals, such as birefringence, dielectric anisotropy, refractive index change in response to applied DC voltage [6] and thickness [1]. In this article, the methodology of [1] is extended on other transparent materials that are mainly used as photoresists in micro-fabrication.

## **2.2 Initial imaging of the borders of a transparent film**

362 Interferometry – Research and Applications in Science and Technology

This section starts with an overview of how light interference is implemented into a singlewavelength optical profilometer and it is used for the extraction of three dimensional (3D) topography images of substrates. Then it analytically describes the methodology of using the phenomenon of phase shift that appears in light propagated through transparent films, together with optical profilometry to extract the thickness of the film. Finally an overview of fringe order theory is given, which is useful in the determination of the maximum possible

In general, optical profilometry exploits the interference between a reference light beam and a measuring beam reflected from the sample surface in order to extract a 3D topographical image. If the travelled distances of the two beams differ by an even number of halfwavelengths then a bright fringe (constructive interference) is created, whilst a difference equal to an odd number of half-wavelengths creates a dark one (destructive interference). In intermediate cases, gray lines are created. The height of a particular point on the sample surface is determined by the type of interference, which in the resulting image of the surface is captured by the colour of the corresponding pixel. In this mode, the number of fringes defines the maximum possible measured height. It is generally recommended to use less than three fringes (which correspond to a height difference equal to three times the source wavelength), in order to avoid problems with the coherence length of the laser source.

Measurements of thicknesses and refractive indices of transparent thick polymer films based on light interference phenomena are well known and developed. They are based on the fact that fringes from the upper and lower boundaries of the film are well separated, and hence the film thickness or the refractive index can be calculated from the distance between the envelopes of the two fringe patterns. However, as the film thickness decreases, the envelopes of the two fringes come closer to each other, and for thicknesses less than about 1µm they overlap. This complicates significantly the calculation of the distance between fringes, and thus the film thickness. Recently, the helical conjugate field function (HCF) has been utilized in coherence correlation interferometry to circumvent this limitation [8, 9,10]. In many areas of engineering and physics, the above technique has been incorporated in methodologies of measuring physical properties and their response to the presence of external stimuli. In the field of mechanical engineering, this constituted the underlying principle in holographic interferometry and Moire inteferometry. The former concerns measuring the properties of transparent specimens [3], whereas the latter is used to measure small deformations of non-transparent solids due to external forces and thermal expansions, as for example is described in [4, 5]. Recently, a methodology which uses two sample probes for simultaneously measuring the refractive index and thickness of transparent specimens has been developed and demonstrated in [2]. Interferometry has also been used to measure parameters of transparent films made from nematic liquid crystals, such as birefringence, dielectric anisotropy, refractive index change in response to applied DC voltage [6] and thickness [1]. In this article, the methodology of [1] is extended on other transparent

However, with some care slightly larger numbers of fringes can also be used.

materials that are mainly used as photoresists in micro-fabrication.

**2. Optical profilometry theory and thickness measurement methodology** 

height the instrument is capable to measure.

**2.1 Existing knowledge** 

Generally, in optical profilometry (or interference profilometry) the measurement starts with the operator focusing to the point where the fringe pattern occurs. Then this pattern is subjected to appropriate processing by the software of the instrument in order to extract the 3D topographical image of the sample. In the case of a transparent film with thicknesses bigger than about one micron, two fringe patterns are generated (figure 1), from exactly the same area of the sample. The first is taken by focusing on the upper surface of the transparent film and the second by focusing on the substrate. Furthermore, for thicknesses less than a micron there is only one fringe pattern which may be attributed to the substrate, as its reflectivity is significantly higher than that of the upper boundary of the film.

Fig. 1. A) First fringe pattern generated from the upper surface of the transparent film. B) fringe pattern generated from the substrate. The right image is distorted in the region of film because of the phase shift that light undergoes while it travels through the film.

In the case where the first fringe pattern is used, the topology of the upper boundary of the film is taken. However in the case where the second pattern is used, an image which corresponds to the substrate is obtained. Here the region that is coated by the transparent film appears to be at a lower height than the non-coated substrate. Such an image is shown in figure 8 in section 6. Definitely, this does not reflect the reality. This happens because to the instrument the existence of the transparent film is unaccounted for, and hence the phase difference registered by the instrument is taken to be due to height differences and not due to the delay of light propagation in the film. A method that considers the optical path changes in the film in order to correct the initially measured image follows.

## **2.3 Understanding the initial image**

Here, the principle of operation of the instrument and the optical path of its light source are considered for the case of measuring the boundary of a transparent film. Figure 2 shows a

Thickness Measurement of Photoresist Thin Films Using Interferometry 365

where Zfilm corrected is the actual height, z, value of the transparent film. This correction is implemented at the pixels corresponding to the film region. The correction step is depicted

The benefit of this technique is that it does not need any instrumentation modification, but only a relatively simple image processing. Furthermore, only the lower fringe pattern is required. This overcomes the common interferometer issue where for very thin films only one fringe pattern appears, which is taken to be that of the lower boundary (substrate). The intensity of the fringe pattern due to the upper boundary (transparent film) is much weaker.

Fig. 2. Schematic representation of the principle of operation of the optical profilometry for

both non coated substrate and coated with thin transparent film substrate.

in figure 3 (B).

schematic presentation of an optical profilometer, operated both on a flat substrate and a transparent film. In both cases, monochromatic light from the light source impinges on the beam splitter where the beam splits into two halves. One half of the beam travels to the objective lens and then to the second beam splitter. At this point, the light beam splits into two identical beams. The one beam travels upwards to the reference mirror and reflected back towards the second beam splitter (optical path A′B′C′ in figure 2). The other travels downwards and reflected from the sample surface also back to the second beam splitter (optical path A′D′C′). The two beams interfere at point C′ and the interference pattern is captured by the camera. Further processing of the fringes pattern image from the instrument software, defines the type of interference in each pixel according to its intensity, and therefore its height. Now for the typical case of the non coated sample, the optical path difference of the two beams is given by A′B′C′ - A′D′C′. Since the distance A′B′C′ and the type of interferences are known, the distance A′C′D′ and therefore the height of point D′ can be extracted as follows:

$$\mathbf{Z}\_D \text{'} = 0.5 \text{(A'B'C' - A'D'C')} \tag{1}$$

In the second case, where the substrate is coated by a transparent film, of refractive index *n*, the optical path difference of the two beams becomes ABC-(AD +*n*DEF +FC). In this situation the problem with the instrument is that it takes the delay of light travelling in the transparent film as a larger distance travelled in the air. However, the delay of light is due to the presence of a transparent film with a refractive index higher than that of air. As a result, in the final topographical image the region of the substrate that is coated with the thin transparent film, erroneously appears to be at a lower position than the non coated substrate. The height of point E in the initial image is calculated as:

$$Z\_{\mathbb{E}} = 0.5(\text{ABC} \cdot (\text{AD} + \eta \text{DEF} + \text{FC})) \tag{2}$$

Subtracting the two heights and taking that A′B′C′ = ABC and for the case of a non tilted substrate A′D′C′ = AEC, then:

$$\mathbf{Z}\_E \mathbf{\cdot} \mathbf{Z}\_{\mathrm{II}} = \begin{pmatrix} n \ -1 \end{pmatrix} \mathbf{DE} \tag{3}$$

## **2.4 Correcting the initial image**

In subsection 2.3, the optical paths of instrument light during the measurement of both substrate and transparent film were explained and computed. In what follows, these optical paths are considered in a methodology that corrects the images presented in section 2.2 and thereby extracts the thickness of transparent films. This methodology can be used for instruments that use single wavelength optical (interference) profilometry.

The goal of this methodology is the correction of the z height values of pixels that correspond to the substrate area that is coated by a transparent film. Figure 3 depicts schematically the correction step. Figure 3 (A) represents the situation after the measurement but before the correction where the mean height of the film erroneously appears lower than the average height of non coated substrate, Zsubstrate average. The correction equation 4 is obtained from the application of equation 3 to the general case of figure 3:

$$\mathbf{Z}\_{\text{film corrected}} = (n \cdot 1) \cdot \mathbf{1} \cdot (\mathbf{Z}\_{\text{substrate average}} - \mathbf{Z}\_{\text{film}}) \tag{4}$$

schematic presentation of an optical profilometer, operated both on a flat substrate and a transparent film. In both cases, monochromatic light from the light source impinges on the beam splitter where the beam splits into two halves. One half of the beam travels to the objective lens and then to the second beam splitter. At this point, the light beam splits into two identical beams. The one beam travels upwards to the reference mirror and reflected back towards the second beam splitter (optical path A′B′C′ in figure 2). The other travels downwards and reflected from the sample surface also back to the second beam splitter (optical path A′D′C′). The two beams interfere at point C′ and the interference pattern is captured by the camera. Further processing of the fringes pattern image from the instrument software, defines the type of interference in each pixel according to its intensity, and therefore its height. Now for the typical case of the non coated sample, the optical path difference of the two beams is given by A′B′C′ - A′D′C′. Since the distance A′B′C′ and the type of interferences are known, the distance A′C′D′ and therefore the height of point D′ can

 ZD′ = 0.5(A′B′C′ - A′D′C′) (1) In the second case, where the substrate is coated by a transparent film, of refractive index *n*, the optical path difference of the two beams becomes ABC-(AD +*n*DEF +FC). In this situation the problem with the instrument is that it takes the delay of light travelling in the transparent film as a larger distance travelled in the air. However, the delay of light is due to the presence of a transparent film with a refractive index higher than that of air. As a result, in the final topographical image the region of the substrate that is coated with the thin transparent film, erroneously appears to be at a lower position than the non coated

 ZE = 0.5(ABC - (AD + *n*DEF + FC)) (2) Subtracting the two heights and taking that A′B′C′ = ABC and for the case of a non tilted

ZE - ZD′ = (*n* - 1)DE (3)

In subsection 2.3, the optical paths of instrument light during the measurement of both substrate and transparent film were explained and computed. In what follows, these optical paths are considered in a methodology that corrects the images presented in section 2.2 and thereby extracts the thickness of transparent films. This methodology can be used for

The goal of this methodology is the correction of the z height values of pixels that correspond to the substrate area that is coated by a transparent film. Figure 3 depicts schematically the correction step. Figure 3 (A) represents the situation after the measurement but before the correction where the mean height of the film erroneously appears lower than the average height of non coated substrate, Zsubstrate average. The correction equation 4 is obtained from the application of equation 3 to the general case of figure 3:

Zfilm corrected =(*n* - 1) -1(Zsubstrate average – Zfilm) (4)

substrate. The height of point E in the initial image is calculated as:

instruments that use single wavelength optical (interference) profilometry.

be extracted as follows:

substrate A′D′C′ = AEC, then:

**2.4 Correcting the initial image** 

where Zfilm corrected is the actual height, z, value of the transparent film. This correction is implemented at the pixels corresponding to the film region. The correction step is depicted in figure 3 (B).

The benefit of this technique is that it does not need any instrumentation modification, but only a relatively simple image processing. Furthermore, only the lower fringe pattern is required. This overcomes the common interferometer issue where for very thin films only one fringe pattern appears, which is taken to be that of the lower boundary (substrate). The intensity of the fringe pattern due to the upper boundary (transparent film) is much weaker.

Fig. 2. Schematic representation of the principle of operation of the optical profilometry for both non coated substrate and coated with thin transparent film substrate.

Thickness Measurement of Photoresist Thin Films Using Interferometry 367

The optical profilometer used in thickness measurements was an Ambios Xi-100 with monochromatic light source. The light source was a green laser with wavelength equal to *550nm*. The atomic force microscope (AFM) used for the verification of film thickness measurements was a Quesant Qscope 350, operated in contact mode, whose the cantilever stiffness was *50N/m* and *10nm* tip diameter. The influence of the thickness measurement to photolithography performance was tested using a Suss MicroTech MA6 mask aligner. The spin coating machine (has been also supplied from Suss MicroTech). The negative AZ® nLOF 2070 supplied by MicroChemicals was used as photoresist. The value for the photoresist refractive index was taken from the supplier and was *n=1,64*. The photoresists were spin coated at 4000 rpms for 30sec. For the formation of thinner films the photoresist was thinned using AZ® EBR solvent. AZ® 826 MIF was finally used as a developer. The exact photolithography process started with soft bake at 100ºC in a hot plate for 5min,

This section, based on the transparent film boundary and geometry, clarifies the rationale for selecting the correct operation mode and identifies the limitations associated with height measurements. This is very important step for correct measurement of thicknesses in general. Consider the maximum height difference that an optical profilometer may accurately measure in the *z* range. In the case where only one interference pattern is used for the extraction of the sample topography, the maximum *z* range that may be measured is

1. Surfaces with high aspect ratio: Consider surfaces where the height of features increases (or decreases) abruptly, i.e. regions that are separated with only very few microns along any horizontal direction and their height difference is higher than half the wavelength. In such a case, the instrument confuses the fringe order and determines erroneous values for heights. The example in figure 5, represents a graded by strips silicon wafer where the edges of the strips are very sharp. The strips were fabricated on the surface by etching. The height of the graded steps is 1.2 microns and the sample was measured with an optical monochromatic profilometer of a light source generating light of 550 nm wavelength (figure 6). Therefore, by neglecting the sample tilt, the height difference at the edges of the strips is equal to about 4.18 times λ/2. The problem here is that the instrument can not determine if the optical path difference is 0.18 or 4.18 times λ/2, i.e. it is not possible to determine the fringe order with this mode. The instrument is programmed to consider smooth surfaces where all fringes are of the same order and therefore the step is taken as about 0.18 times λ/2. That is why the gradation in topographical images appears with wrong heights. As a result this mode is not suitable

2. Smooth surfaces: In this case, the features on the surface are separated by relative small inclinations. Therefore, the fringe order is always the same and the height of all features can be accurately measured. Here the *z* range may extend from few nanometers to several tens of micrometers without losing any precision. In figure 7, an example of measuring height differences of several microns on a smooth spherical sample is shown. The apex and base of a spherical surface are separated by a relatively large distance; however, the order of all fringes remains the same and this is reflected on the

exposure 15sec, post-exposure bake 115ºC for 2sec and develop for 45sec.

**3. Experimental preparation** 

**4. Height difference limit** 

different for the various cases described below:

for surfaces with high aspect ratio.

correct value of their intermediary distance.

Fig. 3. 2D schematic presentation of (A) the initial image and of (B) the corrected image.

## **2.5 Fringe order**

Let's consider figure 2, where for example the distances between the beam splitter and the reference mirror are taken as equal to the distance between the beam splitter and a sample point, while the profilometer is in focus range. In such a case, the optical path difference is equal to zero and constructive interference occurs, and therefore it is the zeroth order bright fringe. The zero order bright fringe appears with maximum intensity, compared to other fringes. Now consider an increase in the optical path equal to the wavelength, *λ,* due to sample moving a corresponding distance away from the beam splitter. Now, the first order bright fringe appears with lower brightness than the zeroth order fringe. The same will happen with the second, third and higher order fringes until moving out of the coherence length of the microscope where the intensity will drop to zero. A schematic curve that shows this variation in light intensity, *I*, with respect to the distance between the sample and lens, *dx*, is shown in figure 4. The optimum condition occurs when the instrument is focussed as near as possible to the zeroth order bright fringe. However as it is discussed in section 5, the confusion in fringe order for non smooth samples may lead to wrong measurements.

Fig. 4. Schematic representative of light intensity (I) with respect the distance between the sample and lens.

## **3. Experimental preparation**

366 Interferometry – Research and Applications in Science and Technology

Fig. 3. 2D schematic presentation of (A) the initial image and of (B) the corrected image.

Let's consider figure 2, where for example the distances between the beam splitter and the reference mirror are taken as equal to the distance between the beam splitter and a sample point, while the profilometer is in focus range. In such a case, the optical path difference is equal to zero and constructive interference occurs, and therefore it is the zeroth order bright fringe. The zero order bright fringe appears with maximum intensity, compared to other fringes. Now consider an increase in the optical path equal to the wavelength, *λ,* due to sample moving a corresponding distance away from the beam splitter. Now, the first order bright fringe appears with lower brightness than the zeroth order fringe. The same will happen with the second, third and higher order fringes until moving out of the coherence length of the microscope where the intensity will drop to zero. A schematic curve that shows this variation in light intensity, *I*, with respect to the distance between the sample and lens, *dx*, is shown in figure 4. The optimum condition occurs when the instrument is focussed as near as possible to the zeroth order bright fringe. However as it is discussed in section 5, the confusion in fringe order for non smooth samples may lead to wrong

Fig. 4. Schematic representative of light intensity (I) with respect the distance between the

**2.5 Fringe order** 

measurements.

sample and lens.

The optical profilometer used in thickness measurements was an Ambios Xi-100 with monochromatic light source. The light source was a green laser with wavelength equal to *550nm*. The atomic force microscope (AFM) used for the verification of film thickness measurements was a Quesant Qscope 350, operated in contact mode, whose the cantilever stiffness was *50N/m* and *10nm* tip diameter. The influence of the thickness measurement to photolithography performance was tested using a Suss MicroTech MA6 mask aligner. The spin coating machine (has been also supplied from Suss MicroTech). The negative AZ® nLOF 2070 supplied by MicroChemicals was used as photoresist. The value for the photoresist refractive index was taken from the supplier and was *n=1,64*. The photoresists were spin coated at 4000 rpms for 30sec. For the formation of thinner films the photoresist was thinned using AZ® EBR solvent. AZ® 826 MIF was finally used as a developer. The exact photolithography process started with soft bake at 100ºC in a hot plate for 5min, exposure 15sec, post-exposure bake 115ºC for 2sec and develop for 45sec.

## **4. Height difference limit**

This section, based on the transparent film boundary and geometry, clarifies the rationale for selecting the correct operation mode and identifies the limitations associated with height measurements. This is very important step for correct measurement of thicknesses in general. Consider the maximum height difference that an optical profilometer may accurately measure in the *z* range. In the case where only one interference pattern is used for the extraction of the sample topography, the maximum *z* range that may be measured is different for the various cases described below:


Thickness Measurement of Photoresist Thin Films Using Interferometry 369

Fig. 7. The fringe pattern and the extracted topography image and profile of a smooth

Finally, it is worth mentioning that further to the mode where only one fringe pattern is used, there is another mode usually called scanning mode. In this mode, a piezoelectric transducer accurately moves the beam splitter while the camera monitors the change in each pixel colour with respect to the travelled distance of the beam splitter. In this way, the instrument identifies the position of the zeroth order fringe (maximum intensity) for each pixel. The height of each pixel is defined by the position of the zeroth order fringe. The scanning mode *z* range is much higher and may be used up to several hundreds of microns. The general conclusion of this section is that the technique for transparent thin film thickness measurement presented is applicable only in the smooth mode. Therefore, the technique can be applied only to films with thickness below λ/2, or to films of which the

This section implements the technique described in section 2 by measuring the thickness of a photoresist thin film. It follows a detailed description of the steps required to implement

sphere with optical profilometry (diameter D= 10mm).

boundaries are smooth enough to avoid jumps on fringe order.

**5. Method implementation to photoresist films** 

this technique.

Fig. 5. 3D topography image of the sample with graded strips taken by atomic force microscopy. The steps are formed by sharp edges of 1.2μm height

Fig. 6. Optical profilometer fringe pattern and topography images of the sample of figure 5. It is clear that the height dimensions are wrong because of the confusion of fringe order.

Fig. 5. 3D topography image of the sample with graded strips taken by atomic force

Fig. 6. Optical profilometer fringe pattern and topography images of the sample of figure 5. It is clear that the height dimensions are wrong because of the confusion of fringe order.

microscopy. The steps are formed by sharp edges of 1.2μm height

Fig. 7. The fringe pattern and the extracted topography image and profile of a smooth sphere with optical profilometry (diameter D= 10mm).

Finally, it is worth mentioning that further to the mode where only one fringe pattern is used, there is another mode usually called scanning mode. In this mode, a piezoelectric transducer accurately moves the beam splitter while the camera monitors the change in each pixel colour with respect to the travelled distance of the beam splitter. In this way, the instrument identifies the position of the zeroth order fringe (maximum intensity) for each pixel. The height of each pixel is defined by the position of the zeroth order fringe. The scanning mode *z* range is much higher and may be used up to several hundreds of microns.

The general conclusion of this section is that the technique for transparent thin film thickness measurement presented is applicable only in the smooth mode. Therefore, the technique can be applied only to films with thickness below λ/2, or to films of which the boundaries are smooth enough to avoid jumps on fringe order.

## **5. Method implementation to photoresist films**

This section implements the technique described in section 2 by measuring the thickness of a photoresist thin film. It follows a detailed description of the steps required to implement this technique.

Thickness Measurement of Photoresist Thin Films Using Interferometry 371

In the third step the boundaries of the transparent film are identified in the profilometer image using a code which scans across image lines that transverse the film boundaries and identifies where the tilt between three pixels of the image is higher than a threshold value. This value varies according to each image, and usually several values have to be tested to extract the optimal one. The image of figure 8 after tilt removal and film boundary definition

Fig. 10. The image of figure 8 with no tilt and film boundaries identified, with the curve

Finally, the height of all pixels that belong to the transparent film region are subjected to the correction of equation (4) and demonstrated in figure 3. The result of this height correction code after implementation on the image in figure 8 is shown in figure 11. Furthermore the film thickness is calculated by subtracting the average height of an indicative film region from that of the substrate. Indicative examples of these regions are presented in figure 11

The image processing is not always necessary. In the general case where the exact topography of the film boundaries is not needed and only a thickness value is sought, the image processing may be avoided. In such a case the height difference of the substrate and film may be measured in the uncorrected optical profilometer image and then entered into equation (4) for the thickness measurement determination. This speeds up the process significantly as the image processing usually requires several hours while the alternative method needs only a few seconds. For thickness measurement both methods are equally

accurate and therefore in the following section the image processing is avoided.

are presented in figure 10.

consisting of small cycles.

with red and blue colours respectively.

It starts by imaging a region at the boundaries of the transparent film of which the thickness has to be measured. In this step it is important to choose a region which: (a) is smooth enough to overcome the height limitation mentioned in section 5 and (b) includes both the substrate and the flat (thickness height equilibrium) area of the film. In this step, an image such as the one given in figure 8 is obtained. As it can be seen, the film appears to be lower than the substrate, as discussed in section 2.2. In order to circumvent this, the implementation of the height correction technique described in section 2.3 is applied.

Fig. 8. Initial image taken by the optical profilometer at the boundaries of a photoresist thin film. Both the film and the substrate are also represented.

The second step includes the extraction of the profilometer image to a suitable format (.txt) for Matlab® processing. This removes the tilt that always appears in such images, due to the relative small (but significant for nano-scale measurements) natural tilt of the instrument base, or sometimes the sample apex with respect to the base surface. Tilt removal is performed with a code that fits image lines that are transverse to film boundaries with straight lines, but only to the region of the substrate. This fitting is demonstrated for a single profile in figure 9. The line extracted from this fitting is then subtracted from the corresponding image profile.

Fig. 9. Schematic representative of image fitting for tilt removal.

It starts by imaging a region at the boundaries of the transparent film of which the thickness has to be measured. In this step it is important to choose a region which: (a) is smooth enough to overcome the height limitation mentioned in section 5 and (b) includes both the substrate and the flat (thickness height equilibrium) area of the film. In this step, an image such as the one given in figure 8 is obtained. As it can be seen, the film appears to be lower than the substrate, as discussed in section 2.2. In order to circumvent this, the

implementation of the height correction technique described in section 2.3 is applied.

Fig. 8. Initial image taken by the optical profilometer at the boundaries of a photoresist thin

The second step includes the extraction of the profilometer image to a suitable format (.txt) for Matlab® processing. This removes the tilt that always appears in such images, due to the relative small (but significant for nano-scale measurements) natural tilt of the instrument base, or sometimes the sample apex with respect to the base surface. Tilt removal is performed with a code that fits image lines that are transverse to film boundaries with straight lines, but only to the region of the substrate. This fitting is demonstrated for a single profile in figure 9. The line extracted from this fitting is then subtracted from the

film. Both the film and the substrate are also represented.

Fig. 9. Schematic representative of image fitting for tilt removal.

corresponding image profile.

In the third step the boundaries of the transparent film are identified in the profilometer image using a code which scans across image lines that transverse the film boundaries and identifies where the tilt between three pixels of the image is higher than a threshold value. This value varies according to each image, and usually several values have to be tested to extract the optimal one. The image of figure 8 after tilt removal and film boundary definition are presented in figure 10.

Fig. 10. The image of figure 8 with no tilt and film boundaries identified, with the curve consisting of small cycles.

Finally, the height of all pixels that belong to the transparent film region are subjected to the correction of equation (4) and demonstrated in figure 3. The result of this height correction code after implementation on the image in figure 8 is shown in figure 11. Furthermore the film thickness is calculated by subtracting the average height of an indicative film region from that of the substrate. Indicative examples of these regions are presented in figure 11 with red and blue colours respectively.

The image processing is not always necessary. In the general case where the exact topography of the film boundaries is not needed and only a thickness value is sought, the image processing may be avoided. In such a case the height difference of the substrate and film may be measured in the uncorrected optical profilometer image and then entered into equation (4) for the thickness measurement determination. This speeds up the process significantly as the image processing usually requires several hours while the alternative method needs only a few seconds. For thickness measurement both methods are equally accurate and therefore in the following section the image processing is avoided.

Thickness Measurement of Photoresist Thin Films Using Interferometry 373

As an alternative method to measure the thickness of the photoresist film, AFM was used to scan the region in the boundary of the film with the substrate. A typical set of AFM images taken with this method is presented in figure 12. In this figures, it is obvious that due to the relatively small scanning area of the AFM image it is not possible to measure an area that completely contains both uncoated substrate and a film region with smooth and flat surface. It should be considered in this case that the film forms a hill at its borders as a result of the spin coating process. The width of this hill is typically more than 100μm, while the AFM maximum scanning area is 80μmx80μm. This limitation is overcome by scanning three adjacent areas of the sample using the motorized x-y translation stage of the AFM instrument. The three images were then subjected to the same tilt removal and tiled together using an image code developed in Matlab®. The image resulted from tiling of the three images of figure 12 is shown in figure 13. The thickness was calculated by subtraction of the substrate average height from the film average height using the same image. It was important to exclude the upward projected (hill) region of the film from the average film

Fig. 13. Image resulted from the union of images in figure 12. It represents the borders of the

Five measurements performed separately on two films using both AFM and the proposed technique based on optical profilometry. The one film has been made from the spin coating of a negative AZ® nLOF 2070 photoresist and the other from a solution of AZ® nLOF 2070 with AZ® EBR (volume concentration 1:1). As shown in table 1, the results were in good agreement and inside the statistical error. Further to the good agreement of the two techniques in the photoresist film thickness measurement, it is important to note further agreement of the profile and geometry of films boundaries such as the hill shape and its

height calculation.

same photoresist film as in figure 11.

dimensions.

Fig. 11. Corrected image depicted by processing the image of figure 8. The regions used for thickness measurement are also highlighted.

## **6. Validation using atomic force microscopy**

The technique for measuring transparent thin films such as photoresists, already described in section 6, has been validated here using atomic force microscopy (AFM). The atomic force microscope belongs to the scanning probe microscope (SPM) family of instruments. It mainly consists of a cantilever with a sharp tip (radius ~ 10nm), a scanner element and a laser with photo-diodes. The scanner consists of three piezoelectric elements and moves the tip in the x-y-z directions with sub nanometer precision. While the cantilever scans the sample surface, the laser light is reflected its light from the cantilever into a split photodiode and thus the cantilever deflections at every sample point can be monitored. The whole procedure is controlled and monitored by dedicated software in a computer which reconstructs the three dimensional topographical image of the sample surface.

Fig. 12. Successive atomic force microscopy images of the boundaries of the photoresist film at the same region as in figure 11.

Fig. 11. Corrected image depicted by processing the image of figure 8. The regions used for

The technique for measuring transparent thin films such as photoresists, already described in section 6, has been validated here using atomic force microscopy (AFM). The atomic force microscope belongs to the scanning probe microscope (SPM) family of instruments. It mainly consists of a cantilever with a sharp tip (radius ~ 10nm), a scanner element and a laser with photo-diodes. The scanner consists of three piezoelectric elements and moves the tip in the x-y-z directions with sub nanometer precision. While the cantilever scans the sample surface, the laser light is reflected its light from the cantilever into a split photodiode and thus the cantilever deflections at every sample point can be monitored. The whole procedure is controlled and monitored by dedicated software in a computer which

Fig. 12. Successive atomic force microscopy images of the boundaries of the photoresist film

reconstructs the three dimensional topographical image of the sample surface.

thickness measurement are also highlighted.

at the same region as in figure 11.

**6. Validation using atomic force microscopy** 

As an alternative method to measure the thickness of the photoresist film, AFM was used to scan the region in the boundary of the film with the substrate. A typical set of AFM images taken with this method is presented in figure 12. In this figures, it is obvious that due to the relatively small scanning area of the AFM image it is not possible to measure an area that completely contains both uncoated substrate and a film region with smooth and flat surface. It should be considered in this case that the film forms a hill at its borders as a result of the spin coating process. The width of this hill is typically more than 100μm, while the AFM maximum scanning area is 80μmx80μm. This limitation is overcome by scanning three adjacent areas of the sample using the motorized x-y translation stage of the AFM instrument. The three images were then subjected to the same tilt removal and tiled together using an image code developed in Matlab®. The image resulted from tiling of the three images of figure 12 is shown in figure 13. The thickness was calculated by subtraction of the substrate average height from the film average height using the same image. It was important to exclude the upward projected (hill) region of the film from the average film height calculation.

Fig. 13. Image resulted from the union of images in figure 12. It represents the borders of the same photoresist film as in figure 11.

Five measurements performed separately on two films using both AFM and the proposed technique based on optical profilometry. The one film has been made from the spin coating of a negative AZ® nLOF 2070 photoresist and the other from a solution of AZ® nLOF 2070 with AZ® EBR (volume concentration 1:1). As shown in table 1, the results were in good agreement and inside the statistical error. Further to the good agreement of the two techniques in the photoresist film thickness measurement, it is important to note further agreement of the profile and geometry of films boundaries such as the hill shape and its dimensions.

Thickness Measurement of Photoresist Thin Films Using Interferometry 375

to be transferred, figure 14. Finally, the sample has been exposed to i-line (wavelength equal to 365nm) using a mask with a pattern consisted of black lines with 20*μm* width separated by 40*μm* spacing. The sample was then developed and imaged with optical microscopy,

Fig. 14. Left image represents the light spot from the instrument and the sample area that was exposed to it. Right image shows an optical microscopy image of the transferred to the

As depicted from figure 14, the influence of optical profilometry was negligible. This is reasonable as in this case single wavelength optical profilometry was used with wavelength

This work has further developed and tested on photoresist thin films a technique of measuring the thickness of LCs films [1]. It was shown that the thcikness of thin films of transparent films, like photoresists, can be measured accurately and the limitation of the the technique was also pointed out and analysed. This provides a cheap and quick nondestructive methdology of measuring photoresist thickness that could easily be adopted in the MEMS fabrication industry. In particular, the optical profilometry-based non-contact method introduced here for measurement of photoresist and other thin films is well suited not only to single and batch wafer fabrication, but also to continuous roll-to-roll (R2R) manufacturing on flexible substrates. Such R2R manufacturing, based on new deposition processes other than spin coating (e.g. doctor blading, transfer and jet printing etc). is becoming prevalent for flexible electronics and organic/hybrid photovoltaics production. The uncorrected (without image processing) version of the profilometric thickness measurement technique is fast enough for in-process monitoring of R2R production and quality control of the thin films. Such a real-time sensing tool therefore shows promise and could be valuable for feedback control of the photoresist deposition process, with

sample pattern after the exposure to instrument light.

considerable improvements in throughput and cost.

equals to 550nm while the photoresist used is sensitive to 365nm.

figure 14.

**8. Conclusions** 


Table 1. Average values of film thickness measured on the same film (n=5) with both AFM and optical profilometry.

The advantages and disadvantages of optical profilometry in comparison with AFM in film thickness measurement and in general are the following:

Pros:


Cons:


## **7. Effect to photolithography performance**

In the previous sections, the applicability and validity of a technique for measuring transparent films thickness has been proven based on optical profilometry on unexposed photoresists. As discussed in the introduction, photoresists are photo-sensitive polymers usually used in micro and nano patterning methods. Therefore it is possible that the exposure of photoresist films to the optical profilometer laser source for several minutes that measurements take, may influence the photolithography process. This section studies the influence of the thickness measurement technique in photolithography performance.

To investigate this, a plate substrate made from silicon wafer was spin coated with negative AZ® nLOF 2070 resist. The sample was subsequently exposed to the light of the optical profilometer. The exposure duration was 60sec in each point while the source scanned the sample at the successive points in lines to cover the area where the mask pattern was going to be transferred, figure 14. Finally, the sample has been exposed to i-line (wavelength equal to 365nm) using a mask with a pattern consisted of black lines with 20*μm* width separated by 40*μm* spacing. The sample was then developed and imaged with optical microscopy, figure 14.

Fig. 14. Left image represents the light spot from the instrument and the sample area that was exposed to it. Right image shows an optical microscopy image of the transferred to the sample pattern after the exposure to instrument light.

As depicted from figure 14, the influence of optical profilometry was negligible. This is reasonable as in this case single wavelength optical profilometry was used with wavelength equals to 550nm while the photoresist used is sensitive to 365nm.

## **8. Conclusions**

374 Interferometry – Research and Applications in Science and Technology

Thickness obtained with optical Profilometer (nm)

531±87 551±39

AZ® nLOF 2070 2059±97 1966±90

Table 1. Average values of film thickness measured on the same film (n=5) with both AFM

The advantages and disadvantages of optical profilometry in comparison with AFM in film

 Optical profilometry has a much higher imaging area than AFM. With the appropriate lens of small magnification it can image areas with several millimetre width. It is a non destructive method as further shown in section 8. AFM is also non destructive for this application. However in other soft or wet transparent films like liquid crystals and elastomer polymers, the interaction and adhesion forces of AFM cantilever with the

film may cause changes to the sample surface or even make imaging impossible.

 The imaging of films with thickness of several hundreds of nanometers, according to section 5, is not possible by optical profilometry if the film edges are not sufficiently

 Optical profilometry in general requires surfaces with good reflectivity. Also if the sample surface consists of more than one material with different reflectivity this may

 The resolution of two techniques in film thickness measurement is practically the same. However in optical profilometry the usage of lenses with small magnification to

In the previous sections, the applicability and validity of a technique for measuring transparent films thickness has been proven based on optical profilometry on unexposed photoresists. As discussed in the introduction, photoresists are photo-sensitive polymers usually used in micro and nano patterning methods. Therefore it is possible that the exposure of photoresist films to the optical profilometer laser source for several minutes that measurements take, may influence the photolithography process. This section studies the

To investigate this, a plate substrate made from silicon wafer was spin coated with negative AZ® nLOF 2070 resist. The sample was subsequently exposed to the light of the optical profilometer. The exposure duration was 60sec in each point while the source scanned the sample at the successive points in lines to cover the area where the mask pattern was going

influence of the thickness measurement technique in photolithography performance.

Solution of AZ® nLOF 2070 and AZ® EBR (1:1)

and optical profilometry.

Pros:

Cons:

smooth.

result in poor imaging.

thickness measurement and in general are the following:

Optical profilometry is easier, simpler and faster.

**7. Effect to photolithography performance** 

increase the image area significantly decreases resolution.

Thickness obtained with AFM (nm)

> This work has further developed and tested on photoresist thin films a technique of measuring the thickness of LCs films [1]. It was shown that the thcikness of thin films of transparent films, like photoresists, can be measured accurately and the limitation of the the technique was also pointed out and analysed. This provides a cheap and quick nondestructive methdology of measuring photoresist thickness that could easily be adopted in the MEMS fabrication industry. In particular, the optical profilometry-based non-contact method introduced here for measurement of photoresist and other thin films is well suited not only to single and batch wafer fabrication, but also to continuous roll-to-roll (R2R) manufacturing on flexible substrates. Such R2R manufacturing, based on new deposition processes other than spin coating (e.g. doctor blading, transfer and jet printing etc). is becoming prevalent for flexible electronics and organic/hybrid photovoltaics production. The uncorrected (without image processing) version of the profilometric thickness measurement technique is fast enough for in-process monitoring of R2R production and quality control of the thin films. Such a real-time sensing tool therefore shows promise and could be valuable for feedback control of the photoresist deposition process, with considerable improvements in throughput and cost.

**0**

**19**

Akira Kimachi

*Japan*

**Real-Time Heterodyne Interferometry with**

*Department of Engineering Informatics, Osaka Electro-Communication University*

Optical heterodyne interfereometry (HI) (R. Dändliker, 1973; N.A. Massie, 1979) is an interferometric technique that employs two interfering beams of slightly different frequencies. Interference of these beams yields a time-varying sinusoidal intensity signal whose frequency equals the difference between the frequencies of the two beams. The signal and frequency are called beat signal and beat frequency, respectively. One of the most significant properties of HI is that the phase difference between the two beams, which is usually the most important quantity in interferometry, equals the phase of the beat signal. In conventional HI, the phase of the beat signal is measured by a single photodetector and a demodulation circuit or instrument. It is said that accuracy of phase measurement in HI reaches 2*π*/1000 rad. Because of the nature of point-wise measurement, however, this method requires two-dimensional (2-D) scanning of the measured point to realize 2-D measurement, resulting in a trade-off

The objective of this article is to introduce a real-time 2-D HI technique. The key idea is the use of the time-domain correlation image sensor (CIS) (S. Ando, 2003) as a 2-D array of phase-demodulating pixels for incident heterodyne beams (A. Kimachi, 2007; 2010a). The CIS produces temporal correlations between the intensity signal of incident light and global reference signals over a frame period and outputs them as images at an ordinary frame rate. This property enables the CIS to simultaneously demodulate the amplitude and phase of the wavefront of incident heterodyne beams at each pixel in every frame using sinusoidal reference signals of the beat frequency. Consequently, as the most significant advantage, this real-time 2-D HI can ensure both maximal temporal and spatial resolution of the image sensor and high accuracy of phase measurement. Note that other than real-time 2-D HI, the CIS has opened a very wide range of novel applications such as image feature extraction (A. Kimachi, 1998), three-dimensional measurement (A. Kimachi, 2001; T. Kurihara, 2003; S. Ando, 2008a; A. Kimachi, 2009), optical flow measurement (S. Ando, 2008b), spectral matching imaging (A. Kimachi, 2002; 2004; 2010b), magneto-optic leakage flux imaging (S. Ando, 2007), and so

The remainder of this article is organized as follows. Section 2 reviews previous studies on real-time 2-D HI and phase-shifting interferometry (PSI). Section 3 describes the principle of the CIS and the real-time 2-D HI. Section 4 shows experimental results of the real-time 2-D HI for a polarizing Michelson interferometer and speckle pattern interferometers for in-plane and out-of-plane deformation measurement. Section 5 briefly describes ongoing research on a

**1. Introduction**

forth.

between temporal and spatial resolution.

**Correlation Image Sensor**

## **9. References**


## **Real-Time Heterodyne Interferometry with Correlation Image Sensor**

Akira Kimachi *Department of Engineering Informatics, Osaka Electro-Communication University Japan*

## **1. Introduction**

376 Interferometry – Research and Applications in Science and Technology

[1] Kossivas, F. & Kyprianou, A. (2010). Measurement of liquid crystal film thickness using

[2] Park, J. S. ; Park, S. K. ; Kim, H. Y & Lee, B. H. (2011). Simultaneous measurements of

[3] Xia, T. H. ; Guo, X. R. ; Fan B. Z.; Cheng, M. H. & Yang C. B. (2011). Non-invasive

[4] Czarnek, R. ; Skrzat, A. & Lin, Y. S. (2011). Application of Moire interferometry to

[5] Chen, B. & Basaran, C (2011). Measuring Joule heating and strain induced by elctrical current with Moire interferometry, *Journal of Applied Physics* Vol. 109 074908 [6] Inam, M. ; Singh, G. ; Srivastava, V. ; Prakash, J ; Joshi, T. ; Biradar, B. I. & Mehta, S. D.

[8] H. M. Shabana (2004). Determination of film thickness and refractive index by

[9] M. Conroy (2009). Advances in thick and thin film analysis using interferometry, *Wear*

[10] D. I. Mansfield (2006). Thin film extraction from scanning white light interferometry,

[7] Ambios Technology Corporation (2009). Xi-100 plus operator's manual, V1.00

interferometer, *Polymer Testing* Vol. 23 pp. 695-702

*Proceedings of the 21st Annual ASPE Meeting*

refractive index and thickness by spectral-domain low coherence interferometry having dual sample probes, *IEEE Photonics Technology Letters,* Vol. 23, No. 15, pp.

mechanical measurement for transparent objects by digital holographic interferometry based on iterative least squares phase unwrapping, *Experimental* 

reconstruction of residual stress in cut railroad car wheels, *Measurement* Vol. 44 pp.

(2001) Two dimensional cell parameter measurement of nematic liquid crystal using optical interferometry and Fourier transform fringe analysis technique, *Optics* 

interferometry*, Meausurement Science and Technology*, 105707

**9. References** 

1076-1078

*Mechanics*

569-579

*Communications,* Optics-16416

Vol. 266 pp. 502-506

Optical heterodyne interfereometry (HI) (R. Dändliker, 1973; N.A. Massie, 1979) is an interferometric technique that employs two interfering beams of slightly different frequencies. Interference of these beams yields a time-varying sinusoidal intensity signal whose frequency equals the difference between the frequencies of the two beams. The signal and frequency are called beat signal and beat frequency, respectively. One of the most significant properties of HI is that the phase difference between the two beams, which is usually the most important quantity in interferometry, equals the phase of the beat signal. In conventional HI, the phase of the beat signal is measured by a single photodetector and a demodulation circuit or instrument. It is said that accuracy of phase measurement in HI reaches 2*π*/1000 rad. Because of the nature of point-wise measurement, however, this method requires two-dimensional (2-D) scanning of the measured point to realize 2-D measurement, resulting in a trade-off between temporal and spatial resolution.

The objective of this article is to introduce a real-time 2-D HI technique. The key idea is the use of the time-domain correlation image sensor (CIS) (S. Ando, 2003) as a 2-D array of phase-demodulating pixels for incident heterodyne beams (A. Kimachi, 2007; 2010a). The CIS produces temporal correlations between the intensity signal of incident light and global reference signals over a frame period and outputs them as images at an ordinary frame rate. This property enables the CIS to simultaneously demodulate the amplitude and phase of the wavefront of incident heterodyne beams at each pixel in every frame using sinusoidal reference signals of the beat frequency. Consequently, as the most significant advantage, this real-time 2-D HI can ensure both maximal temporal and spatial resolution of the image sensor and high accuracy of phase measurement. Note that other than real-time 2-D HI, the CIS has opened a very wide range of novel applications such as image feature extraction (A. Kimachi, 1998), three-dimensional measurement (A. Kimachi, 2001; T. Kurihara, 2003; S. Ando, 2008a; A. Kimachi, 2009), optical flow measurement (S. Ando, 2008b), spectral matching imaging (A. Kimachi, 2002; 2004; 2010b), magneto-optic leakage flux imaging (S. Ando, 2007), and so forth.

The remainder of this article is organized as follows. Section 2 reviews previous studies on real-time 2-D HI and phase-shifting interferometry (PSI). Section 3 describes the principle of the CIS and the real-time 2-D HI. Section 4 shows experimental results of the real-time 2-D HI for a polarizing Michelson interferometer and speckle pattern interferometers for in-plane and out-of-plane deformation measurement. Section 5 briefly describes ongoing research on a

higher-order harmonic components in the beat signal. Moreover, the two shutter-CCD pairs

Real-Time Heterodyne Interferometry with Correlation Image Sensor 379

Other studies focus on integrating real-time phase-demodulating pixels for HI into 2-D arrays. It is a straightforward idea to employ high-speed cameras as 2-D phase-demodulating arrays for sampling continuous beat signals at discrete time intervals (P. Haible, 2000; M.V. Aguanno, 2007). The higher the frame rate, the higher the accuracy of phase demodulation because of a larger number of samples, while the spatial resolution of the image sensor is maintained. The use of a high-speed camera, however, must be accompanied by a high-power laser source; otherwise it degrades the accuracy of phase demodulation because of shorter exposure time, lower light intensity, and hence lower signal-to-noise ratio (SNR) in the interference images. Moreover, high-speed cameras require high-speed frame grabber interface with large memory,

Another group of studies employ 2-D arrays of phase-demodulating sensors developed on their own. A type of phase-demodulating image sensor called lock-in CCD, originally proposed by Spirig *et al.* (T. Spirig, 1995), outputs a set of discrete samples of the incident beat signal from each pixel, instead of a sequence of fringe images. The multiple samples are transferred to separate bins within a pixel as charge packets and accumulated over a frame period. The phase of the beat signal is demodulated from the set of samples at each pixel on a computer by the same algorithm as used in PSI. Dändliker *et al.* (R.D. Dändliker, 1998) used this sensor for distance interferometry. Earlier than these reports Povel *et al.* (H. Povel, 1990) devised a similar sensor for imaging polarimetry, though it was not named lock-in CCD.

It follows that in lock-in CCDs, the accuracy of phase demodulation, which increases for a larger number of samples or a higher sampling rate, is in a trade-off with spatial resolution because storing a large number of samples requires more space within the pixel on the sensor chip. Note also that the nature of discrete temporal sampling in lock-in CCDs as well as in high-speed cameras, represented by a rectangular temporal window function, again brings about the problem of sensitivity to higher-order harmonic components in the incident beat signal. This degradation becomes severer for a smaller number of samples or a lower sampling

Another type of 2-D phase-demodulating pixel array was developed by Bourquin *et al.* for optical coherence tomography (S. Bourquin, 2001a). The pixel consists of a photodetector followed by a feedback circuit with a low-pass filter (LPF) and a rectifier (S. Bourquin, 2001b). For an incident beat signal of a frequency above the LPF cutoff frequency, the pixel can demodulate the signal amplitude as a result of open-loop operation of the feedback circuit. This sensor, however, cannot perform phase demodulation because it does not have the circuit

Now the real-time 2-D HI presented in this article is compared to the techniques discussed so far. The CIS used in the real-time 2-D HI is exactly an ideal type of 2-D phase-demodulating pixel array. As a significant advantage over other techniques employing 2-D phase-demodulating pixel arrays and high-speed cameras, the CIS demodulates the phase of a continuous beat signal by continuous operation on analog circuitry under an ordinary frame rate, instead of discrete sampling and off-line computation. This property makes the real-time 2-D HI immune to the problems of rectangular windowing, light SNR, and frame

make the system rather bulky and demand cumbersome optical alignment.

**2.2 Integration of 2-D phase-demodulating pixel arrays**

which gives rise to a problem in terms of commercial availability.

rate.

for it.

networked sensing system for health monitoring of large-scale structures based on multi-zero beams and the real-time 2-D HI. Finally, Section 6 concludes this article.

## **2. Review of real-time 2-D heterodyne and phase-shifting interferometry**

A great number of studies have been devoted to the problem with HI mentioned in Section 1. Realization of real-time 2-D HI can be divided into two categories: 1) development of real-time PSI methods using image sensors, and 2) integration of phase-demodulating sensors into 2-D arrays. The following subsections briefly review studies belonging to each category.

## **2.1 Real-time PSI using image sensors**

PSI, a widely-used interferometry technique realized with a commercial image sensor, is easier to deal with than the development of 2-D arrays of phase-demodulating sensors. PSI captures a sequence of fringe images formed by two interfering beams of the same frequency, while the phase difference between the beams is shifted stepwise with a piezoelectric actuator or a variable retarder in each frame (K.J. Gåsvik, 2002; J.H. Bruning, 1974). The original phase difference is estimated by fitting the intensity of the image sequence to a sine wave individually at each pixel (C.J. Morgan, 1982). PSI is used not only in ordinary interferometry for specularly reflective objects or transparent objects, but also in speckle pattern interferometry (SPI) (K. Creath, 1985; J. Kato, 1993; M. Takeda, 1994), optical coherence tomography (M. Akiba, 2003), or holography (I. Yamaguchi, 1997) for objects with rough surfaces or made of turbid media.

PSI requires three images at the minimum, and frequently employs four images with a phase shift increment by 90◦. Higher accuracy is achieved for phase demodulation with a larger number of images; however, it results in a longer time for image capture. In PSI, temporal resolution and measurement accuracy are thus in a trade-off.

It seems that this problem with PSI could be solved by capturing fringe images with different phase shifts at the same time. North-Morris *et al.* (M. North-Morris, 2005) developed a phase mask consisting of an array of 2 × 2-pixel retarder matrices with *π*/2 steps of retardance for phase-shifting SPI. Four-step PSI is thus realized from a single-frame image by attaching this mask in front of an image sensor. A similar phase mask was proposed for phase-shifting holography by Awatsuji *et al.* (Y. Awatsuji, 2004). Kiire *et al.* (T. Kiire, 2008) proposed a method for simultaneously producing four separate images of speckle interference for different phase shifts on an image sensor using a polarizing prism, a diffraction grating, and two lasers of different wavelengths. The price for these single-frame PSI methods is, of course, loss of spatial resolution down to half of the image sensor.

Akiba *et al.* (M. Akiba, 2003) proposed a system of optical coherence tomography using two pairs of a liquid-crystal shutter and a CCD (charge coupled device) camera. The beat signal of low coherence interference exerted by a translating mirror is sampled in synchrony by the two shutters and integrated over a frame period of the CCDs at each pixel. This system captures a pair of phase-shifted fringe images at a video frame rate, though it may be categorized into 2-D discrete sampling of a continous beat signal as discussed in the next subsection rather than phase shifting. The phase of the beat signal can be obtained from the pair of integrated images and a dc component image captured beforehand, though phase demodulation was not performed in the paper. The accuracy of this method, however, can be degraded because of temporal sampling with a rectangular window, which has a certain degree of sensitivity to 2 Will-be-set-by-IN-TECH

networked sensing system for health monitoring of large-scale structures based on multi-zero

A great number of studies have been devoted to the problem with HI mentioned in Section 1. Realization of real-time 2-D HI can be divided into two categories: 1) development of real-time PSI methods using image sensors, and 2) integration of phase-demodulating sensors into 2-D

PSI, a widely-used interferometry technique realized with a commercial image sensor, is easier to deal with than the development of 2-D arrays of phase-demodulating sensors. PSI captures a sequence of fringe images formed by two interfering beams of the same frequency, while the phase difference between the beams is shifted stepwise with a piezoelectric actuator or a variable retarder in each frame (K.J. Gåsvik, 2002; J.H. Bruning, 1974). The original phase difference is estimated by fitting the intensity of the image sequence to a sine wave individually at each pixel (C.J. Morgan, 1982). PSI is used not only in ordinary interferometry for specularly reflective objects or transparent objects, but also in speckle pattern interferometry (SPI) (K. Creath, 1985; J. Kato, 1993; M. Takeda, 1994), optical coherence tomography (M. Akiba, 2003), or holography (I. Yamaguchi, 1997) for objects with rough

PSI requires three images at the minimum, and frequently employs four images with a phase shift increment by 90◦. Higher accuracy is achieved for phase demodulation with a larger number of images; however, it results in a longer time for image capture. In PSI, temporal

It seems that this problem with PSI could be solved by capturing fringe images with different phase shifts at the same time. North-Morris *et al.* (M. North-Morris, 2005) developed a phase mask consisting of an array of 2 × 2-pixel retarder matrices with *π*/2 steps of retardance for phase-shifting SPI. Four-step PSI is thus realized from a single-frame image by attaching this mask in front of an image sensor. A similar phase mask was proposed for phase-shifting holography by Awatsuji *et al.* (Y. Awatsuji, 2004). Kiire *et al.* (T. Kiire, 2008) proposed a method for simultaneously producing four separate images of speckle interference for different phase shifts on an image sensor using a polarizing prism, a diffraction grating, and two lasers of different wavelengths. The price for these single-frame PSI methods is, of course, loss of

Akiba *et al.* (M. Akiba, 2003) proposed a system of optical coherence tomography using two pairs of a liquid-crystal shutter and a CCD (charge coupled device) camera. The beat signal of low coherence interference exerted by a translating mirror is sampled in synchrony by the two shutters and integrated over a frame period of the CCDs at each pixel. This system captures a pair of phase-shifted fringe images at a video frame rate, though it may be categorized into 2-D discrete sampling of a continous beat signal as discussed in the next subsection rather than phase shifting. The phase of the beat signal can be obtained from the pair of integrated images and a dc component image captured beforehand, though phase demodulation was not performed in the paper. The accuracy of this method, however, can be degraded because of temporal sampling with a rectangular window, which has a certain degree of sensitivity to

beams and the real-time 2-D HI. Finally, Section 6 concludes this article.

**2.1 Real-time PSI using image sensors**

surfaces or made of turbid media.

resolution and measurement accuracy are thus in a trade-off.

spatial resolution down to half of the image sensor.

**2. Review of real-time 2-D heterodyne and phase-shifting interferometry**

arrays. The following subsections briefly review studies belonging to each category.

higher-order harmonic components in the beat signal. Moreover, the two shutter-CCD pairs make the system rather bulky and demand cumbersome optical alignment.

## **2.2 Integration of 2-D phase-demodulating pixel arrays**

Other studies focus on integrating real-time phase-demodulating pixels for HI into 2-D arrays. It is a straightforward idea to employ high-speed cameras as 2-D phase-demodulating arrays for sampling continuous beat signals at discrete time intervals (P. Haible, 2000; M.V. Aguanno, 2007). The higher the frame rate, the higher the accuracy of phase demodulation because of a larger number of samples, while the spatial resolution of the image sensor is maintained. The use of a high-speed camera, however, must be accompanied by a high-power laser source; otherwise it degrades the accuracy of phase demodulation because of shorter exposure time, lower light intensity, and hence lower signal-to-noise ratio (SNR) in the interference images. Moreover, high-speed cameras require high-speed frame grabber interface with large memory, which gives rise to a problem in terms of commercial availability.

Another group of studies employ 2-D arrays of phase-demodulating sensors developed on their own. A type of phase-demodulating image sensor called lock-in CCD, originally proposed by Spirig *et al.* (T. Spirig, 1995), outputs a set of discrete samples of the incident beat signal from each pixel, instead of a sequence of fringe images. The multiple samples are transferred to separate bins within a pixel as charge packets and accumulated over a frame period. The phase of the beat signal is demodulated from the set of samples at each pixel on a computer by the same algorithm as used in PSI. Dändliker *et al.* (R.D. Dändliker, 1998) used this sensor for distance interferometry. Earlier than these reports Povel *et al.* (H. Povel, 1990) devised a similar sensor for imaging polarimetry, though it was not named lock-in CCD.

It follows that in lock-in CCDs, the accuracy of phase demodulation, which increases for a larger number of samples or a higher sampling rate, is in a trade-off with spatial resolution because storing a large number of samples requires more space within the pixel on the sensor chip. Note also that the nature of discrete temporal sampling in lock-in CCDs as well as in high-speed cameras, represented by a rectangular temporal window function, again brings about the problem of sensitivity to higher-order harmonic components in the incident beat signal. This degradation becomes severer for a smaller number of samples or a lower sampling rate.

Another type of 2-D phase-demodulating pixel array was developed by Bourquin *et al.* for optical coherence tomography (S. Bourquin, 2001a). The pixel consists of a photodetector followed by a feedback circuit with a low-pass filter (LPF) and a rectifier (S. Bourquin, 2001b). For an incident beat signal of a frequency above the LPF cutoff frequency, the pixel can demodulate the signal amplitude as a result of open-loop operation of the feedback circuit. This sensor, however, cannot perform phase demodulation because it does not have the circuit for it.

Now the real-time 2-D HI presented in this article is compared to the techniques discussed so far. The CIS used in the real-time 2-D HI is exactly an ideal type of 2-D phase-demodulating pixel array. As a significant advantage over other techniques employing 2-D phase-demodulating pixel arrays and high-speed cameras, the CIS demodulates the phase of a continuous beat signal by continuous operation on analog circuitry under an ordinary frame rate, instead of discrete sampling and off-line computation. This property makes the real-time 2-D HI immune to the problems of rectangular windowing, light SNR, and frame

correlation image sensor

ysel

SW1

Q1 *<sup>V</sup>*<sup>1</sup>

the accumulated charges *Qk*(*i*, *j*) as

⎡ ⎣

*Q*1(*i*, *j*) *Q*2(*i*, *j*) *Q*3(*i*, *j*)

⎤ ⎦ = ⎡ ⎣

= ⎡ ⎣

�*fij*(*t*)� =

where �·� denotes an integral over *T*. The term �*fij*(*t*)�, obtained by

PD

*V*2

*CCC*

lens

light intensity

*f ij*(*t*) *g*1 (*t*) *g*2 (*t*) *g*3 (*t*)

Real-Time Heterodyne Interferometry with Correlation Image Sensor 381

(a)

reference signals

*Q*1

SW2 SW3

ysel ysel

*V*3

Q2 Q3

out1 out2 out3

*<sup>V</sup>*SS = 0V

*I*

(b) Fig. 1. Schematic of the CIS. (a) Overall architecture. (b) Pixel circuit for passive readout.

over *T* while redefining *ρgk*(*t*) as *gk*(*t*) and replacing *I*(*t*) with *fij*(*t*) yields an expression for

⎤ ⎦

> ⎤ ⎦ +

⎡ ⎢ ⎣

⎤ ⎥

*Qk*(*i*, *j*), (3)

<sup>⎦</sup> , (2)

�*I*1(*t*)� �*I*2(*t*)� �*I*3(*t*)�

Δ*Q*1(*i*, *j*) Δ*Q*2(*i*, *j*) Δ*Q*3(*i*, *j*)

> 3 ∑ *k*=1

*f ij*(*t*)

(*i*,*j*) *Q*<sup>2</sup>

*g*1 (*t*) *g*2 (*t*) *g*3 (*t*)

(*i*,*j*) *Q*<sup>3</sup>

*<sup>V</sup>*DD = 5V

output images

(*i*,*j*)

*Q*1 (*i*,*j*)

*Q*2 (*i*,*j*)

*Q*3 (*i*,*j*)


Table 1. Comparison of 2-D HI and PSI techniques. The boxes with an asterisk in each row indicate a trade-off relation.

grabber interface. Moreover, spatial resolution can be maximally utilized as far as the pixel size permits, without any loss by sharing among multiple phase-shifted images, as is the case with the real-time PSI methods. The real-time 2-D HI and PSI techniques discussed above are summarized in Table 1.

## **3. Principle**

This section first describes the principle of the CIS, and then explains how the CIS demodulates amplitude and phase images of heterodyne beams in real time.

## **3.1 Correlation image sensor**

Figure 1(a) depicts the architecture of the CIS, which consists of a pixel array and three external inputs of reference signals supplied to all of the pixels. The pixel circuit for passive readout is illustrated in Figure 1(b) (S. Ando, 2003). The photodiode (PD) converts the incident light into a photocurrent *I*(*t*) in proportion to its intensity *fij*(*t*) at pixel (*i*, *j*). *I*(*t*) is divided into three currents *Ik*(*t*) (*k* = 1, 2, 3, ∑<sup>3</sup> *<sup>k</sup>*=<sup>1</sup> *Ik*(*t*) = *I*(*t*)) that flow through three NMOS transistors Q*k*, and accumulated in three capacitors connected to the drains of Q*<sup>k</sup>* over a frame period *T*. The balance among *Ik*(*t*) depends on the gate voltages *Vk*(*t*) as

$$I\_k(t) = \rho g\_k(t) I(t) + \frac{1}{3} I(t) \quad (k = 1, 2, 3), \tag{1}$$

where *gk*(*t*) = *Vk*(*t*) <sup>−</sup> *<sup>V</sup>* denotes the differential component of *Vk*(*t*) from the average *<sup>V</sup>* <sup>=</sup> <sup>1</sup> <sup>3</sup> <sup>∑</sup><sup>3</sup> *<sup>k</sup>*=<sup>1</sup> *Vk*(*t*), and *ρ* is a constant. The signals *gk*(*t*), called reference signals, are supplied to the gates of Q*<sup>k</sup>* in each pixel under a constraint ∑<sup>3</sup> *<sup>k</sup>*=<sup>1</sup> *gk*(*t*) = 0. In Eq. (1), integrating *Ik*(*t*) 4 Will-be-set-by-IN-TECH

Phase shifting Two pairs of image sensor Good Good Fair Alignment,

2-D heterodyne High-speed camera Good Good Fair Light S/N, (discrete) camera I/F

Table 1. Comparison of 2-D HI and PSI techniques. The boxes with an asterisk in each row

grabber interface. Moreover, spatial resolution can be maximally utilized as far as the pixel size permits, without any loss by sharing among multiple phase-shifted images, as is the case with the real-time PSI methods. The real-time 2-D HI and PSI techniques discussed above are

This section first describes the principle of the CIS, and then explains how the CIS

Figure 1(a) depicts the architecture of the CIS, which consists of a pixel array and three external inputs of reference signals supplied to all of the pixels. The pixel circuit for passive readout is illustrated in Figure 1(b) (S. Ando, 2003). The photodiode (PD) converts the incident light into a photocurrent *I*(*t*) in proportion to its intensity *fij*(*t*) at pixel (*i*, *j*). *I*(*t*) is divided into

Q*k*, and accumulated in three capacitors connected to the drains of Q*<sup>k</sup>* over a frame period *T*.

where *gk*(*t*) = *Vk*(*t*) <sup>−</sup> *<sup>V</sup>* denotes the differential component of *Vk*(*t*) from the average *<sup>V</sup>* <sup>=</sup> <sup>1</sup>

*<sup>k</sup>*=<sup>1</sup> *Vk*(*t*), and *ρ* is a constant. The signals *gk*(*t*), called reference signals, are supplied to

*<sup>k</sup>*=<sup>1</sup> *Ik*(*t*) = *I*(*t*)) that flow through three NMOS transistors

<sup>3</sup> *<sup>I</sup>*(*t*) (*<sup>k</sup>* <sup>=</sup> 1, 2, 3), (1)

*<sup>k</sup>*=<sup>1</sup> *gk*(*t*) = 0. In Eq. (1), integrating *Ik*(*t*)

and electronic shutter bulkiness

Temporal Spatial issue

Type Method/tool Resolution Accuracy System

2-D heterodyne Photodetector, 2-D scan \* \* Good Phase shifting Image sensor, phase shifter \* Good \*

Phase shifting Image sensor with phase mask Good Poor Fair Phase shifting Phase-shifted images on one Good Poor Fair

2-D heterodyne Lock-in CCD Good \* \*

2-D heterodyne Correlation image sensor Good Good Good

demodulates amplitude and phase images of heterodyne beams in real time.

The balance among *Ik*(*t*) depends on the gate voltages *Vk*(*t*) as

the gates of Q*<sup>k</sup>* in each pixel under a constraint ∑<sup>3</sup>

*Ik*(*t*) = *<sup>ρ</sup>gk*(*t*)*I*(*t*) + <sup>1</sup>

image sensor

(discrete)

indicate a trade-off relation.

summarized in Table 1.

**3.1 Correlation image sensor**

three currents *Ik*(*t*) (*k* = 1, 2, 3, ∑<sup>3</sup>

**3. Principle**

<sup>3</sup> <sup>∑</sup><sup>3</sup>

Fig. 1. Schematic of the CIS. (a) Overall architecture. (b) Pixel circuit for passive readout.

over *T* while redefining *ρgk*(*t*) as *gk*(*t*) and replacing *I*(*t*) with *fij*(*t*) yields an expression for the accumulated charges *Qk*(*i*, *j*) as

$$
\begin{bmatrix} Q\_1(i,j) \\ Q\_2(i,j) \\ Q\_3(i,j) \end{bmatrix} = \begin{bmatrix} \langle I\_1(t) \rangle \\ \langle I\_2(t) \rangle \\ \langle I\_3(t) \rangle \end{bmatrix}
$$

$$
= \begin{bmatrix} \Delta Q\_1(i,j) \\ \Delta Q\_2(i,j) \\ \Delta Q\_3(i,j) \end{bmatrix} + \begin{bmatrix} \frac{1}{3} \langle f\_{lj}(t) \rangle \\ \frac{1}{3} \langle f\_{lj}(t) \rangle \\ \frac{1}{3} \langle f\_{lj}(t) \rangle \end{bmatrix}.\tag{2}
$$

where �·� denotes an integral over *T*. The term �*fij*(*t*)�, obtained by

$$
\langle f\_{\vec{l}\vec{j}}(t) \rangle = \sum\_{k=1}^{3} Q\_k(\vec{i}, \vec{j}) , \tag{3}
$$

angular frequency *ω* expressed by

⎡ ⎣

*θij* = tan−<sup>1</sup>

√2

where (*i*, *j*) is dropped for notational symplicity.

and out-of-plane deformation measurement.

**4.1 Real-time heterodyne Michelson interferometry**

*Aij* <sup>=</sup> <sup>2</sup>

**4. Experimental results**

Δ*Q*1(*i*, *j*) Δ*Q*2(*i*, *j*) Δ*Q*3(*i*, *j*) ⎡ ⎣

yields the single-frame temporal correlation images Δ*Qk*(*i*, *j*) as

⎤

<sup>⎦</sup> <sup>=</sup> *TAij* 2

phase *θij* are obtained from Δ*Qk*(*i*, *j*) by least-squares estimation as

<sup>3</sup>*<sup>T</sup>* [(Δ*Q*<sup>1</sup> <sup>−</sup> <sup>Δ</sup>*Q*2)

<sup>√</sup>3(Δ*Q*<sup>2</sup> <sup>−</sup> <sup>Δ</sup>*Q*3) 2Δ*Q*<sup>1</sup> − Δ*Q*<sup>2</sup> − Δ*Q*<sup>3</sup>

*g*1(*t*) *g*2(*t*) *g*3(*t*)

⎤ ⎦ = ⎡ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎣

cos *θij* cos �

cos �

where noise contributions *ξk*(*i*, *j*) are taken into account. The interference amplitude *Aij* and

The theory on real-time 2-D HI using the CIS described in Section 3 are verified on two types of interferometer—a Michelson interferometer and speckle pattern interferometers for in-plane

To study the feasibility of the proposed real-time 2-D HI, a heterodyne polarizing Michelson interferometer was constructed with the CIS as depicted in Figure 3(a) (A. Kimachi, 2007). A pair of orthogonally-polarized heterodyne beams coaxially enter the interferometer through a beam expander L1. The beams are split by the polarizing beam splitter PBS1 and reflected on the mirrors M1 and M2 while passing through the quarter wave plates QWP1 and QWP2 twice, which rotate the polarization of each reflected beam by 90◦. The reflected beams are merged by PBS1 and interfere on the CIS through an analyzer POL1 and an imaging lens L2. A reference beat signal is generated by taking out a small portion of the heterodyne beams with a beam splitter BS and detecting the interference intensity with a photodiode PD. This signal is converted into three-phase sinusoidal reference signals for the CIS. The heterodyne beams are emitted from a commercial two-frequency He-Ne laser source shown in Figure 3(b) (wavelength *λ* = 632.8 nm), which consists of two acousto-optic modulators driven at slightly different frequencies to yield a beat frequency of Δ*ν* = 25 kHz. The mirror M1 is longitudinally moved with a piezoelectric actuator under open-loop control to change the

interference phase *θij* between the two beams linearly to the displacement *δz*.

*<sup>θ</sup>ij* <sup>−</sup> <sup>2</sup> 3*π* �

*<sup>θ</sup>ij* <sup>−</sup> <sup>4</sup> 3*π* �

cos *ωt* cos �

Real-Time Heterodyne Interferometry with Correlation Image Sensor 383

cos �

Noting that the average intensity *Bij* and the background illumination *nij*(*t*) are both uncorrelated with the sinusoudal reference signals *gk*(*t*), i.e. �*gk*(*t*)*Bij*� = �*gk*(*t*)*nij*(*t*)� = 0, substitution of Eqs. (5) and (6) into Eq. (4) under the condition <sup>Δ</sup>*<sup>ν</sup>* <sup>=</sup> *mT*−<sup>1</sup> or <sup>Δ</sup>*<sup>ν</sup>* � *<sup>T</sup>*−<sup>1</sup>

*ωt* + <sup>2</sup> 3*π* �

⎤ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎦ + ⎡ ⎣

<sup>2</sup> + (Δ*Q*<sup>2</sup> <sup>−</sup> <sup>Δ</sup>*Q*3)<sup>2</sup> + (Δ*Q*<sup>3</sup> <sup>−</sup> <sup>Δ</sup>*Q*1)

*ξ*1(*i*, *j*) *ξ*2(*i*, *j*) *ξ*3(*i*, *j*)

⎤

, (8)

2]

. (6)

⎦ , (7)

1/2, (9)

*ωt* + <sup>4</sup> 3*π* �

Fig. 2. Photographs of CIS cameras with CMOS chips of (a) 64 × 64-pixel and (b) 200 × 200-pixel sensors.

is proportional to the average light intensity, as in conventional image sensors. The most significant difference from them is the term

$$\begin{split} \Delta Q\_k(i,j) &= \langle f\_{\vec{l}\vec{j}}(t)g\_k(t) \rangle \\ &= Q\_k(i,j) - \frac{1}{3} \langle f\_{\vec{l}\vec{j}}(t) \rangle \quad (k=1,2,3), \end{split} \tag{4}$$

which represents the temporal correlation between the intensity of the incident light *fij*(*t*) and each of the reference signals *gk*(*t*). The three temporal correlations Δ*Qk*(*i*, *j*) have two degrees of freedom under a constraint ∑<sup>3</sup> *<sup>k</sup>*=<sup>1</sup> <sup>Δ</sup>*Qk*(*i*, *<sup>j</sup>*) = 0 caused by <sup>∑</sup><sup>3</sup> *<sup>k</sup>*=<sup>1</sup> *gk*(*t*) = 0. The charges *Qk*(*i*, *j*) are read out either passively through PMOS switches as shown in Figure 1(b) (S. Ando, 2003), or actively with voltage source followers (S. Han, 2010). Figure 2 shows photographs of CIS cameras with 64 × 64-pixel (S. Ando, 2003) and 200 × 200-pixel (S. Ando, 2007) CMOS chips of passive type.

### **3.2 Real-time amplitude and phase demodulation of heterodyne beams**

Consider two heterodyne coherent beams with complex wavefronts *uij*, *vij* and a beat frequency Δ*ν* to interfere on the focal plane of the CIS. The intensity *fij*(*t*) of the interfering beams is given by

$$\begin{split} f\_{\vec{ij}}(t) &= |u\_{\vec{ij}} + v\_{\vec{ij}} \exp[i2\pi\Delta\nu t]|^2 + n\_{\vec{ij}}(t) \\ &= A\_{\vec{ij}} \cos(\omega t + \theta\_{\vec{ij}}) + B\_{\vec{ij}} + n\_{\vec{ij}}(t), \\ A\_{\vec{ij}} &= 2|u\_{\vec{ij}}||v\_{\vec{ij}}|, \quad \theta\_{\vec{ij}} = \angle v\_{\vec{ij}} - \angle u\_{\vec{ij}}, \quad B\_{\vec{ij}} = |u\_{\vec{ij}}|^2 + |v\_{\vec{ij}}|^2, \quad \omega = 2\pi\Delta v, \end{split} \tag{5}$$

where i denotes the imaginary unit (i<sup>2</sup> <sup>=</sup> <sup>−</sup>1) and background illumination *nij*(*t*) is taken into account. The beat frequency Δ*ν* is chosen to be a multiple of the frame rate of the CIS, Δ*ν* = *mT*−<sup>1</sup> (*m*: integer), or much higher than the frame rate, <sup>Δ</sup>*<sup>ν</sup>* � *<sup>T</sup>*−1. The interference amplitude *Aij* and phase *θij* are simultaneously demodulated from single-frame temporal correlation images Δ*Qk*(*i*, *j*), by supplying the CIS with three-phase reference signals *gk*(*t*) of the same angular frequency *ω* expressed by

6 Will-be-set-by-IN-TECH

(a) (b)

is proportional to the average light intensity, as in conventional image sensors. The most

3

which represents the temporal correlation between the intensity of the incident light *fij*(*t*) and each of the reference signals *gk*(*t*). The three temporal correlations Δ*Qk*(*i*, *j*) have two degrees

*Qk*(*i*, *j*) are read out either passively through PMOS switches as shown in Figure 1(b) (S. Ando, 2003), or actively with voltage source followers (S. Han, 2010). Figure 2 shows photographs of CIS cameras with 64 × 64-pixel (S. Ando, 2003) and 200 × 200-pixel (S. Ando, 2007) CMOS

Consider two heterodyne coherent beams with complex wavefronts *uij*, *vij* and a beat frequency Δ*ν* to interfere on the focal plane of the CIS. The intensity *fij*(*t*) of the interfering

<sup>2</sup> + *nij*(*t*)

where i denotes the imaginary unit (i<sup>2</sup> <sup>=</sup> <sup>−</sup>1) and background illumination *nij*(*t*) is taken into account. The beat frequency Δ*ν* is chosen to be a multiple of the frame rate of the CIS, Δ*ν* = *mT*−<sup>1</sup> (*m*: integer), or much higher than the frame rate, <sup>Δ</sup>*<sup>ν</sup>* � *<sup>T</sup>*−1. The interference amplitude *Aij* and phase *θij* are simultaneously demodulated from single-frame temporal correlation images Δ*Qk*(*i*, *j*), by supplying the CIS with three-phase reference signals *gk*(*t*) of the same

= *Aij* cos(*ωt* + *θij*) + *Bij* + *nij*(*t*), (5)

<sup>2</sup> <sup>+</sup> <sup>|</sup>*vij*<sup>|</sup>

*<sup>k</sup>*=<sup>1</sup> <sup>Δ</sup>*Qk*(*i*, *<sup>j</sup>*) = 0 caused by <sup>∑</sup><sup>3</sup>

�*fij*(*t*)� (*k* = 1, 2, 3), (4)

*<sup>k</sup>*=<sup>1</sup> *gk*(*t*) = 0. The charges

2, *ω* = 2*π*Δ*ν*,

Fig. 2. Photographs of CIS cameras with CMOS chips of (a) 64 × 64-pixel and (b)

<sup>=</sup> *Qk*(*i*, *<sup>j</sup>*) <sup>−</sup> <sup>1</sup>

Δ*Qk*(*i*, *j*) = �*fij*(*t*)*gk*(*t*)�

**3.2 Real-time amplitude and phase demodulation of heterodyne beams**

*Aij* = 2|*uij*||*vij*|, *θij* = ∠*vij* − ∠*uij*, *Bij* = |*uij*|

*fij*(*t*) = |*uij* + *vij* exp[i2*π*Δ*νt*]|

200 × 200-pixel sensors.

significant difference from them is the term

of freedom under a constraint ∑<sup>3</sup>

chips of passive type.

beams is given by

$$
\begin{bmatrix}
\mathbf{g}\_1(t) \\
\mathbf{g}\_2(t) \\
\mathbf{g}\_3(t)
\end{bmatrix} = \begin{bmatrix}
\cos\omega t \\
\cos\left(\omega t + \frac{2}{3}\pi\right) \\
\cos\left(\omega t + \frac{4}{3}\pi\right)
\end{bmatrix} . \tag{6}
$$

Noting that the average intensity *Bij* and the background illumination *nij*(*t*) are both uncorrelated with the sinusoudal reference signals *gk*(*t*), i.e. �*gk*(*t*)*Bij*� = �*gk*(*t*)*nij*(*t*)� = 0, substitution of Eqs. (5) and (6) into Eq. (4) under the condition <sup>Δ</sup>*<sup>ν</sup>* <sup>=</sup> *mT*−<sup>1</sup> or <sup>Δ</sup>*<sup>ν</sup>* � *<sup>T</sup>*−<sup>1</sup> yields the single-frame temporal correlation images Δ*Qk*(*i*, *j*) as

$$
\begin{bmatrix}
\Delta Q\_1(i,j) \\
\Delta Q\_2(i,j) \\
\Delta Q\_3(i,j)
\end{bmatrix} = \frac{TA\_{ij}}{2} \begin{bmatrix}
\cos\theta\_{ij} \\
\cos\left(\theta\_{ij} - \frac{2}{3}\pi\right) \\
\cos\left(\theta\_{ij} - \frac{4}{3}\pi\right)
\end{bmatrix} + \begin{bmatrix}
\mathbb{G}\_1(i,j) \\
\mathbb{G}\_2(i,j) \\
\mathbb{G}\_3(i,j)
\end{bmatrix},
\tag{7}
$$

where noise contributions *ξk*(*i*, *j*) are taken into account. The interference amplitude *Aij* and phase *θij* are obtained from Δ*Qk*(*i*, *j*) by least-squares estimation as

$$\theta\_{\rm ij} = \tan^{-1} \frac{\sqrt{3}(\Delta Q\_2 - \Delta Q\_3)}{2\Delta Q\_1 - \Delta Q\_2 - \Delta Q\_3},\tag{8}$$

$$A\_{\rm ij} = \frac{2\sqrt{2}}{3T} [\left(\Delta Q\_1 - \Delta Q\_2\right)^2 + \left(\Delta Q\_2 - \Delta Q\_3\right)^2 + \left(\Delta Q\_3 - \Delta Q\_1\right)^2]^{1/2},\tag{9}$$

where (*i*, *j*) is dropped for notational symplicity.

## **4. Experimental results**

The theory on real-time 2-D HI using the CIS described in Section 3 are verified on two types of interferometer—a Michelson interferometer and speckle pattern interferometers for in-plane and out-of-plane deformation measurement.

### **4.1 Real-time heterodyne Michelson interferometry**

To study the feasibility of the proposed real-time 2-D HI, a heterodyne polarizing Michelson interferometer was constructed with the CIS as depicted in Figure 3(a) (A. Kimachi, 2007). A pair of orthogonally-polarized heterodyne beams coaxially enter the interferometer through a beam expander L1. The beams are split by the polarizing beam splitter PBS1 and reflected on the mirrors M1 and M2 while passing through the quarter wave plates QWP1 and QWP2 twice, which rotate the polarization of each reflected beam by 90◦. The reflected beams are merged by PBS1 and interfere on the CIS through an analyzer POL1 and an imaging lens L2. A reference beat signal is generated by taking out a small portion of the heterodyne beams with a beam splitter BS and detecting the interference intensity with a photodiode PD. This signal is converted into three-phase sinusoidal reference signals for the CIS. The heterodyne beams are emitted from a commercial two-frequency He-Ne laser source shown in Figure 3(b) (wavelength *λ* = 632.8 nm), which consists of two acousto-optic modulators driven at slightly different frequencies to yield a beat frequency of Δ*ν* = 25 kHz. The mirror M1 is longitudinally moved with a piezoelectric actuator under open-loop control to change the interference phase *θij* between the two beams linearly to the displacement *δz*.

(a) (b) (c)

Michelson interfereometer (A. Kimachi, 2007). (a) Average intensity �*fij*(*t*)�. (b)

length, which is close to the characteristic of the piezoelectric actuator (61 nm/V).

as convection that brings heated silicone oil up toward the surface.

**4.2 Real-time heterodyne speckle pattern interferometry**

phase.

Fig. 4. An example of output images from the 64 × 64-pixel CIS in the real-time heterodyne

Real-Time Heterodyne Interferometry with Correlation Image Sensor 385

Figure 5 shows the results when the mirror M1 was moved by the deformation of the piezoelectric actuator for changes in voltage (A. Kimachi, 2007). Figure 5(a) shows a sequence of demodulated phase images for voltages from 30.0 V to 36.5 V in 0.5-V steps. The difference of each phase image from that for 30.0 V is shown in Figure 5(b). The phase difference for the pixel at the image center is unwrapped and plotted in Figure 5(c) up to 50.0 V, along with a linear fit. The plot in Figure 5(c) exhibits a good linear response to the applied voltage, which is ideally linear to the displacement of M1, despite the open-loop control of the piezoelectric actuator. The slope of the linear fit in Figure 5(c) amounts to 65 nm/V in terms of optical path

The real-time heterodyne Michelson interfereometer in Figure 3 was applied to monitoring dynamic change in the interference phase of the heterodyne beams, which was caused by silicone oil inserted in front of the mirror M1 as depicted in Figures 6(a) and (b). Figure 6(c) shows a real-time sequence of phase difference images in every 0.2 s, i.e. every six frames, with respect to the initial state, while the silicone oil was heated by the heating cable (A. Kimachi, 2007). It is observed that gradation is buidling up near the surface of the silicone oil as time passes, suggesting changes in refractive index due to increase in temperature as well

The real-time 2-D HI was also applied to SPI. Figure 7(a) depicts a real-time heterodyne SPI system for in-plane deformation measurement (A. Kimachi, 2010a). The heterodyne beams are generated by a He-Ne laser and an electro-optic modulator (EOM), which shifts the phase between a pair of orthogonally-polarized coaxial beams from 0 to 2*π* linearly to time at a frequency of Δ*ν* = 100 Hz by a sawtooth-shaped driving voltage signal. The heterodyne beams are split by the polarizing beam splitter PBS and projected onto the object from symmetrically opposite directions with an inclination angle *α* = 45◦. The interference speckle pattern generated on the object is imaged through an analyzer POL onto the CIS, which is supplied with three-phase reference signals of the same frequency Δ*ν* as that of the EOM driving signal. The CIS demodulates the phase of the interference speckle pattern from single-frame output images, yielding a random pattern. The difference of the demodulated phase from that of the object at an initial state changes linearly to the in-plane deformation *δx*

Demodulated amplitude *Aij*. (c) Demodulated phase *θij*. (d) Gray-scale representation of

0

/2 (d)

/2

Fig. 3. Schematic of the real-time heterodyne polarizing Michelson interferometer with the CIS (A. Kimachi, 2007). (a) Interferometer optics. BS: beam splitter; PBS: polarizing beam splitter; QWP: quarter-wave plate; POL: polarizer/analyzer; PD: photodiode. (b) Two-frequency He-Ne laser source. AOM: acousto-optic modulator; HWP: half-wave plate.

The 64 × 64-pixel CIS in Figure 2(a) was used in the interferometer in Figure 3(a), operating at a frame rate of *<sup>T</sup>*−<sup>1</sup> <sup>=</sup> 30 fps (� <sup>Δ</sup>*ν*). Figure 4 shows an example of average intensity, demodulated amplitude, and demodulated phase images computed from the raw output images *Qk*(*i*, *j*) with Eqs. (3), (4), (8) and (9) (A. Kimachi, 2007). The demodulated phase *θij* in Figure 4(c) is coded in gray scale as depicted in Figure 4(d). The abrupt edge from black to white in Figure 4(c) corresponds to a 2*π* jump.

8 Will-be-set-by-IN-TECH

QWP2 (135°)

+ν<sup>2</sup>

M2

PBS1 L1

0 +2π/3

+4π/3

PBS2

splitter; QWP: quarter-wave plate; POL: polarizer/analyzer; PD: photodiode. (b)

(a)

(b) Fig. 3. Schematic of the real-time heterodyne polarizing Michelson interferometer with the CIS (A. Kimachi, 2007). (a) Interferometer optics. BS: beam splitter; PBS: polarizing beam

Two-frequency He-Ne laser source. AOM: acousto-optic modulator; HWP: half-wave plate.

The 64 × 64-pixel CIS in Figure 2(a) was used in the interferometer in Figure 3(a), operating at a frame rate of *<sup>T</sup>*−<sup>1</sup> <sup>=</sup> 30 fps (� <sup>Δ</sup>*ν*). Figure 4 shows an example of average intensity, demodulated amplitude, and demodulated phase images computed from the raw output images *Qk*(*i*, *j*) with Eqs. (3), (4), (8) and (9) (A. Kimachi, 2007). The demodulated phase *θij* in Figure 4(c) is coded in gray scale as depicted in Figure 4(d). The abrupt edge from black to

(45°) (45°)

BS

two-freq. He-Ne laser

+ν<sup>2</sup>

+ν<sup>1</sup>

POL2

PD

(90°) (0°)

> phase shifter

He-Ne HWP

frequency shifter module

white in Figure 4(c) corresponds to a 2*π* jump.

L2

CIS

AOM2

+ν<sup>2</sup>

+ν<sup>1</sup>

AOM1

−ν<sup>2</sup> Δν | POL1

Δν = |ν<sup>1</sup>

QWP1

(45°)

A/D

PBS3

+ν<sup>1</sup> <sup>δ</sup>*<sup>z</sup>*

PC

M1

Fig. 4. An example of output images from the 64 × 64-pixel CIS in the real-time heterodyne Michelson interfereometer (A. Kimachi, 2007). (a) Average intensity �*fij*(*t*)�. (b) Demodulated amplitude *Aij*. (c) Demodulated phase *θij*. (d) Gray-scale representation of phase.

Figure 5 shows the results when the mirror M1 was moved by the deformation of the piezoelectric actuator for changes in voltage (A. Kimachi, 2007). Figure 5(a) shows a sequence of demodulated phase images for voltages from 30.0 V to 36.5 V in 0.5-V steps. The difference of each phase image from that for 30.0 V is shown in Figure 5(b). The phase difference for the pixel at the image center is unwrapped and plotted in Figure 5(c) up to 50.0 V, along with a linear fit. The plot in Figure 5(c) exhibits a good linear response to the applied voltage, which is ideally linear to the displacement of M1, despite the open-loop control of the piezoelectric actuator. The slope of the linear fit in Figure 5(c) amounts to 65 nm/V in terms of optical path length, which is close to the characteristic of the piezoelectric actuator (61 nm/V).

The real-time heterodyne Michelson interfereometer in Figure 3 was applied to monitoring dynamic change in the interference phase of the heterodyne beams, which was caused by silicone oil inserted in front of the mirror M1 as depicted in Figures 6(a) and (b). Figure 6(c) shows a real-time sequence of phase difference images in every 0.2 s, i.e. every six frames, with respect to the initial state, while the silicone oil was heated by the heating cable (A. Kimachi, 2007). It is observed that gradation is buidling up near the surface of the silicone oil as time passes, suggesting changes in refractive index due to increase in temperature as well as convection that brings heated silicone oil up toward the surface.

## **4.2 Real-time heterodyne speckle pattern interferometry**

The real-time 2-D HI was also applied to SPI. Figure 7(a) depicts a real-time heterodyne SPI system for in-plane deformation measurement (A. Kimachi, 2010a). The heterodyne beams are generated by a He-Ne laser and an electro-optic modulator (EOM), which shifts the phase between a pair of orthogonally-polarized coaxial beams from 0 to 2*π* linearly to time at a frequency of Δ*ν* = 100 Hz by a sawtooth-shaped driving voltage signal. The heterodyne beams are split by the polarizing beam splitter PBS and projected onto the object from symmetrically opposite directions with an inclination angle *α* = 45◦. The interference speckle pattern generated on the object is imaged through an analyzer POL onto the CIS, which is supplied with three-phase reference signals of the same frequency Δ*ν* as that of the EOM driving signal. The CIS demodulates the phase of the interference speckle pattern from single-frame output images, yielding a random pattern. The difference of the demodulated phase from that of the object at an initial state changes linearly to the in-plane deformation *δx*

silicone oil

glass cell

object was heated.

range.

heater

(a)

PBS1

Real-Time Heterodyne Interferometry with Correlation Image Sensor 387

0.0 s 0.2 s 0.4 s 0.6 s 0.8 s 1.0 s 1.2 s

1.4 s 1.6 s 1.8 s 2.0 s 2.2 s 2.4 s 2.6 s (c)

flat object with a rough surface while it was moved in an in-plane direction in 40-nm steps by a nano-scale linear stage (A. Kimachi, 2010a). Figure 8(a) shows a sequence of phase difference images with respect to the initial position of the object, smoothed with a 5 × 5-pixel Gaussian filter. The value of phase difference is coded again as in Figure 4(d). The images exhibit gradual changes in phase difference in accordance with the in-plane displacement. The average and standard deviation of unwrapped phase difference over the central 50 × 50-pixel region are plotted in Figure 8(b) along with a theoretical response, up to the displacement of 2 *μ*m. The plot of the average phase difference agrees well with the theoretical response. The standard deviation was 0.13 rad, or 9 nm in terms of displacement, in average over the plotted

A similar experiment was carried out as well for the real-time heterodyne SPI system for out-of-plane deformation measurement depicted in Figure 7(b), which is based on a Michelson interferometer with the reflection mirrors replaced by the object and a reference plate with a rough surface (A. Kimachi, 2010a). The same 200 × 200-pixel CIS camera, heterodyne laser source, and flat object were used as in the system in Figure 7(a). Figure 9(a) shows a sequence of phase difference images with respect to the initial position of the object, while it underwent out-of-plane displacement along the optical axis in 40-nm steps. The images exhibit gradual

Fig. 6. Results on the real-time heterodyne Michelson interferometer for an object under dynamic behavior (A. Kimachi, 2007). (a) Illustration of the object. (b) Optical arrangement of the object. (c) Real-time sequence of phase difference images with respect to 0.0 s while the

sample (silicone oil)

M1

QWP1(45 )

(b)

imaging area

Fig. 5. Results of the real-time heterodyne Michelson interferometer for changes in the piezoelectric actuator's voltage (A. Kimachi, 2007). (a) Demodulated phase images. (b) Images of phase difference with respect to 30.0 V. (c) Plot of the unwrapped phase difference at a pixel at the image center.

at each pixel, as long as the deformation does not exceed the average speckle size to escape speckle decorrelation.

The 200 × 200-pixel CIS camera in Figure 2(b) was used in the SPI system in Figure 7(a), operating at a frame rate of *T*−<sup>1</sup> = 8.33 fps (= Δ*ν*/12). Figure 8 shows the results for a 10 Will-be-set-by-IN-TECH

30.0 V 30.5 V 31.0 V 31.5 V 32.0 V 32.5 V 33.0 V

33.5 V 34.0 V 34.5 V 35.0 V 35.5 V 36.0 V 36.5 V (a)

33.5 V 34.0 V 34.5 V 35.0 V 35.5 V 36.0 V 36.5 V (b)

Phase

at a pixel at the image center.

speckle decorrelation.

 [rad]

> 0.0

8.0

 16.0 Estimate Measurement

24.0

Voltage [V] 30 34 38 42 46 50

(c)

at each pixel, as long as the deformation does not exceed the average speckle size to escape

The 200 × 200-pixel CIS camera in Figure 2(b) was used in the SPI system in Figure 7(a), operating at a frame rate of *T*−<sup>1</sup> = 8.33 fps (= Δ*ν*/12). Figure 8 shows the results for a

Fig. 5. Results of the real-time heterodyne Michelson interferometer for changes in the piezoelectric actuator's voltage (A. Kimachi, 2007). (a) Demodulated phase images. (b) Images of phase difference with respect to 30.0 V. (c) Plot of the unwrapped phase difference

30.5 V 31.0 V 31.5 V 32.0 V 32.5 V 33.0 V

Fig. 6. Results on the real-time heterodyne Michelson interferometer for an object under dynamic behavior (A. Kimachi, 2007). (a) Illustration of the object. (b) Optical arrangement of the object. (c) Real-time sequence of phase difference images with respect to 0.0 s while the object was heated.

flat object with a rough surface while it was moved in an in-plane direction in 40-nm steps by a nano-scale linear stage (A. Kimachi, 2010a). Figure 8(a) shows a sequence of phase difference images with respect to the initial position of the object, smoothed with a 5 × 5-pixel Gaussian filter. The value of phase difference is coded again as in Figure 4(d). The images exhibit gradual changes in phase difference in accordance with the in-plane displacement. The average and standard deviation of unwrapped phase difference over the central 50 × 50-pixel region are plotted in Figure 8(b) along with a theoretical response, up to the displacement of 2 *μ*m. The plot of the average phase difference agrees well with the theoretical response. The standard deviation was 0.13 rad, or 9 nm in terms of displacement, in average over the plotted range.

A similar experiment was carried out as well for the real-time heterodyne SPI system for out-of-plane deformation measurement depicted in Figure 7(b), which is based on a Michelson interferometer with the reflection mirrors replaced by the object and a reference plate with a rough surface (A. Kimachi, 2010a). The same 200 × 200-pixel CIS camera, heterodyne laser source, and flat object were used as in the system in Figure 7(a). Figure 9(a) shows a sequence of phase difference images with respect to the initial position of the object, while it underwent out-of-plane displacement along the optical axis in 40-nm steps. The images exhibit gradual

(a)

system for in-plane deformation in Figure 7(a).

direction.

central 50 × 50-pixel region, along with a theoretical response.

Standard deviation [2

0

0.25

0.5

Displacement [μm]

(b)

012

Mean phase [2

Fig. 8. Results of the real-time heterodyne SPI system for in-plane displacement of a flat object with a rough surface (A. Kimachi, 2010a). (a) Sequence of phase difference images with respect to the initial position of the object, for displacement increasing in 40-nm steps. (b) Average and standard deviation of the images of unwrapped phase difference within a

theoretical plots is not as good as that for the in-plane displacement in Figure 8(b), exhibiting a drift of the experimental plot from the theoretical one. These observations can be attributed to the nature of the Michelson interferometer the SPI system in Figure 7(b) is based on, which involves two separate reflective objects and is thus more sensitive to disturbance than the SPI

The real-time heterodyne SPI system for in-plane deformation in Figure 7(a) was employed for monitoring an object under dynamic deformation. The object is a piezoelectric actuator placed horizontally and attached to a fixed frame, as depicted in Figure 10(a). Figure 10(b) shows a real-time sequence of phase difference images in every 0.12-s frame with respect to the initial state, featuring the junction between the object and fixed frame. The profile of the images along the horizontal white line is plotted in Figure 10(c). The left-hand side of the plot, corresponding to the actuator, exhibits a continuous increase in the slope as well as a gradual upward shift. The increasing slope indicates that the actuator is expanding toward the left, whereas the upward shift implies that the actuator itself also slides to the left. The right-hand side of the plot, corresponding to the fixed frame, exhibits a gradual downward shift of much smaller steps than that for the actuator. This suggests the existence of a strong force by the actuator that slightly pushes the fixed frame in the opposite direction to its expanding

4

0

 8

mean phase

measured std. deviation

theoretical

π rad]

Real-Time Heterodyne Interferometry with Correlation Image Sensor 389

π rad]

Fig. 7. Schematics of the real-time heterodyne SPI systems with the CIS for (a) in-plane, and (b) out-of-plane deformation measurement (A. Kimachi, 2010a).

changes in phase difference, as observed in Figure 8(a), though they look a little noisier. The average and standard deviation of unwrapped phase difference over the central 50 × 50-pixel region are plotted in Figure 9(b), along with a theoretical response. The standard deviation was 0.33 rad, or 17 nm in terms of displacement, in average over the plotted range, which is larger than the plot in Figure 8(b). Moreover, the agreement between the experimental and 12 Will-be-set-by-IN-TECH

+240°

ν0

δ*x*

+240°

+120°

*gk* (*t*)

+120°

POL

*gk* (*t*)

He-Ne

He-Ne

Δν

EOM

PBS

EOM

Δν

CIS

αα

+Δν ν<sup>0</sup>

(a)

PBS

ν0

(b) Fig. 7. Schematics of the real-time heterodyne SPI systems with the CIS for (a) in-plane, and

changes in phase difference, as observed in Figure 8(a), though they look a little noisier. The average and standard deviation of unwrapped phase difference over the central 50 × 50-pixel region are plotted in Figure 9(b), along with a theoretical response. The standard deviation was 0.33 rad, or 17 nm in terms of displacement, in average over the plotted range, which is larger than the plot in Figure 8(b). Moreover, the agreement between the experimental and

δ*z*

(b) out-of-plane deformation measurement (A. Kimachi, 2010a).

POL

CIS

*Qk* (*i*,*j*)

output images PC

object

PC

reference

*Qk* (*i*,*j*)

output images

ν0 +Δν

object

Fig. 8. Results of the real-time heterodyne SPI system for in-plane displacement of a flat object with a rough surface (A. Kimachi, 2010a). (a) Sequence of phase difference images with respect to the initial position of the object, for displacement increasing in 40-nm steps. (b) Average and standard deviation of the images of unwrapped phase difference within a central 50 × 50-pixel region, along with a theoretical response.

theoretical plots is not as good as that for the in-plane displacement in Figure 8(b), exhibiting a drift of the experimental plot from the theoretical one. These observations can be attributed to the nature of the Michelson interferometer the SPI system in Figure 7(b) is based on, which involves two separate reflective objects and is thus more sensitive to disturbance than the SPI system for in-plane deformation in Figure 7(a).

The real-time heterodyne SPI system for in-plane deformation in Figure 7(a) was employed for monitoring an object under dynamic deformation. The object is a piezoelectric actuator placed horizontally and attached to a fixed frame, as depicted in Figure 10(a). Figure 10(b) shows a real-time sequence of phase difference images in every 0.12-s frame with respect to the initial state, featuring the junction between the object and fixed frame. The profile of the images along the horizontal white line is plotted in Figure 10(c). The left-hand side of the plot, corresponding to the actuator, exhibits a continuous increase in the slope as well as a gradual upward shift. The increasing slope indicates that the actuator is expanding toward the left, whereas the upward shift implies that the actuator itself also slides to the left. The right-hand side of the plot, corresponding to the fixed frame, exhibits a gradual downward shift of much smaller steps than that for the actuator. This suggests the existence of a strong force by the actuator that slightly pushes the fixed frame in the opposite direction to its expanding direction.

piezoelectric actuator

(b)

imaged area

Real-Time Heterodyne Interferometry with Correlation Image Sensor 391

(a)

Unwrapped phase [2


Fig. 10. Results of real-time heterodyne SPI for an object under dynamic in-plane deformation (A. Kimachi, 2010a). (a) Illustration of the object. (b) Real-time sequence of phase difference images with respect to 0.0 s while the object was horizontally expanded. (c) Profiles of unwrapped phase difference along the horizontal white lines in the images.

entire deformation of a large-scale structure can be obtained by integrating the relative six-axis deformation between each transmitter-receiver pair of sensing units over the network.

The key idea for measuring the relative deformation between a pair of sensing units lies in the use of multi-zero beams (S. Sato, 2008a;b;c), which have a wavefront with multiple

 0

 0.5

 1

1.5

π rad]

fixed frame

X axis [pix] 0 100 200

(c)

piezoelectric actuator

1.80 s

fixed frame

1.80 s 1.68 s 1.56 s 1.44 s 1.32 s 1.20 s 1.08 s 0.96 s 0.84 s 0.72 s 0.60 s 0.48 s 0.36 s 0.24 s 0.12 s 0.12 s

Fig. 9. Results on the real-time heterodyne SPI system for out-of-plane displacement of a flat object with a rough surface (A. Kimachi, 2010a). (a) Sequence of phase difference images with respect to the initial position of the object, for displacement increasing in 40-nm steps. (b) Average and standard deviation of the images of unwrapped phase difference within a central 50 × 50-pixel region, along with a theoretical response.

## **5. Application to a health monitoring system for large-scale structures**

The real-time 2-D HI described in the previous sections is particularly effective in applications that require not only high accuracy but also real-time 2-D phase measurement. As one of such applications, a research project on health monitoring of large-scale structures has been promoted by Ando *et al.* (S. Sato, 2008a;b;c; S. Ando, 2009). This project aims at developing a networked sensing system that can measure relative deformation of large-scale structures such as bridges, tunnels, industrial plants and tall buildings in real time and at high accuracy over three axes of displacement and three axes of rotation. This kind of sensing system is drawing increasing attention because large-scale structures can cause catastrophic disasters when they are broken down from aging, fatigue, or defects. Health monitoring of large-scale structures, however, is not easy because they are located outdoors in tough enviroment and their size makes it very laborious to inspect the whole entity for defects.

As a solution to this problem, a networked sensing system that employs the real-time 2-D HI has been proposed (S. Sato, 2008a;b;c). The sensor network is comprised of a train of sensing units placed over a large-scale structure. The sensing unit consists of a pair of a CIS chip and a heterodyne laser source, and transmits coaxial heterodyne laser beams to one neighboring unit while receiving coaxial heterodyne laser beams from another neighboring unit. The 14 Will-be-set-by-IN-TECH

Mean phase [2

Fig. 9. Results on the real-time heterodyne SPI system for out-of-plane displacement of a flat object with a rough surface (A. Kimachi, 2010a). (a) Sequence of phase difference images with respect to the initial position of the object, for displacement increasing in 40-nm steps. (b) Average and standard deviation of the images of unwrapped phase difference within a

The real-time 2-D HI described in the previous sections is particularly effective in applications that require not only high accuracy but also real-time 2-D phase measurement. As one of such applications, a research project on health monitoring of large-scale structures has been promoted by Ando *et al.* (S. Sato, 2008a;b;c; S. Ando, 2009). This project aims at developing a networked sensing system that can measure relative deformation of large-scale structures such as bridges, tunnels, industrial plants and tall buildings in real time and at high accuracy over three axes of displacement and three axes of rotation. This kind of sensing system is drawing increasing attention because large-scale structures can cause catastrophic disasters when they are broken down from aging, fatigue, or defects. Health monitoring of large-scale structures, however, is not easy because they are located outdoors in tough enviroment and

As a solution to this problem, a networked sensing system that employs the real-time 2-D HI has been proposed (S. Sato, 2008a;b;c). The sensor network is comprised of a train of sensing units placed over a large-scale structure. The sensing unit consists of a pair of a CIS chip and a heterodyne laser source, and transmits coaxial heterodyne laser beams to one neighboring unit while receiving coaxial heterodyne laser beams from another neighboring unit. The

**5. Application to a health monitoring system for large-scale structures**

their size makes it very laborious to inspect the whole entity for defects.

4

0

 8

mean phase

measured std. deviation

theoretical

π rad]

Standard deviation [2

0

Displacement [μm] 012

(b)

0.25

0.5

π rad]

(a)

central 50 × 50-pixel region, along with a theoretical response.

Fig. 10. Results of real-time heterodyne SPI for an object under dynamic in-plane deformation (A. Kimachi, 2010a). (a) Illustration of the object. (b) Real-time sequence of phase difference images with respect to 0.0 s while the object was horizontally expanded. (c) Profiles of unwrapped phase difference along the horizontal white lines in the images.

entire deformation of a large-scale structure can be obtained by integrating the relative six-axis deformation between each transmitter-receiver pair of sensing units over the network.

The key idea for measuring the relative deformation between a pair of sensing units lies in the use of multi-zero beams (S. Sato, 2008a;b;c), which have a wavefront with multiple

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Lenses," *Appl. Opt.*, vol. 13, pp. 2693-2703, 1974.

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Isolated Surfaces," *Appl. Opt.*, vol. 33, pp. 7829-7837, 1994.

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U.K., 2008.

2007.

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zeros distributed over the beam area. The relative deformation can be known from a change in the amplitude and phase pattern of a multi-zero wavefront, which is easily detected on the CIS chip by the real-time 2-D HI at high accuracy. The multi-zero beams, which are generated by multiplexing Laguerre-Gaussian (LG) beams of different orders, have been found from extensive studies to possess many characteristics suitable for measurement and sensing (S. Sato, 2008a;b;c).

Based on these ideas, several experimental systems have been so far developed and tested in experiments on real-time tracking of multi-zeros (S. Ando, 2009). This project is currently at the stage of the development of a prototype system and the experimental evaluation on actual large-scale structures.

## **6. Conclusion**

The real-time 2-D HI technique presented in this article is based on the use of the CIS as a two-dimensional array of pixels, each of which simultaneously demodulate the amplitude and phase of incident heterodyne beams at an ordinary frame rate. As the most significant advantage over other real-time 2-D HI or PSI techniques, the CIS-based technique ensures both maximal temporal and spatial resolution and high accuracy of phase measurement. The experimental results obtained on a polarizing Michelson interferometer and speckle pattern interferometers for in-plane and out-of-plane deformation measurement confirm the feasibility and usefulness of the real-time 2-D HI. The real-time 2-D HI is expected to find many application areas such as the health monitoring system for large-scale structures based on multi-zero beams.

## **7. References**


16 Will-be-set-by-IN-TECH

zeros distributed over the beam area. The relative deformation can be known from a change in the amplitude and phase pattern of a multi-zero wavefront, which is easily detected on the CIS chip by the real-time 2-D HI at high accuracy. The multi-zero beams, which are generated by multiplexing Laguerre-Gaussian (LG) beams of different orders, have been found from extensive studies to possess many characteristics suitable for measurement and

Based on these ideas, several experimental systems have been so far developed and tested in experiments on real-time tracking of multi-zeros (S. Ando, 2009). This project is currently at the stage of the development of a prototype system and the experimental evaluation on actual

The real-time 2-D HI technique presented in this article is based on the use of the CIS as a two-dimensional array of pixels, each of which simultaneously demodulate the amplitude and phase of incident heterodyne beams at an ordinary frame rate. As the most significant advantage over other real-time 2-D HI or PSI techniques, the CIS-based technique ensures both maximal temporal and spatial resolution and high accuracy of phase measurement. The experimental results obtained on a polarizing Michelson interferometer and speckle pattern interferometers for in-plane and out-of-plane deformation measurement confirm the feasibility and usefulness of the real-time 2-D HI. The real-time 2-D HI is expected to find many application areas such as the health monitoring system for large-scale structures based

R. Dändliker, B. Ineichen and F.M. Mottier, "High-Resolution Hologram Interferometry by Electronic Phase Measurement," *Opt. Commun.*, vol. 9, pp. 412-416, 1973. N.A. Massie, R.D. Nelson and S. Holly, "High-Performance Real-Time Heterodyne

S. Ando and A. Kimachi, "Correlation Image Sensor: Two-Dimensional Matched Detection

A. Kimachi, "Real-Time Heterodyne Imaging Interferometry: Focal-Plane Amplitude and

A. Kimachi, "Real-Time Heterodyne Speckle Pattern Interferometry Using the Correlation

A. Kimachi, R. Imaizumi and S. Ando, "Intelligent Image Sensor with a Vibratory Mirror

A. Kimachi, T. Kurihara, M. Takamoto and S. Ando, "A Novel Range Finding System Using Correlation Image Sensor," *Trans. IEE Japan*, vol. 121-E, pp. 367-375, 2001. T. Kurihara, N. Ono and S. Ando, "Surface Orientation Imager Using Three-Phase

of Amplitude-Modulated Light," *IEEE Trans. Electron Devices*, vol. 50, pp. 2059-2066,

Phase Demodulation Using a Three-Phase Correlation Image Sensor," *Appl. Opt.*, vol.

Mimicking Involuntary Eye Movement," in *Tech. Digest of 16th Sensor Symp.*, pp.

Amplitude-Modulated Illumination and Correlation Image Sensor," in *Proc. SPIE*,

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Image Sensor," *Appl. Opt.*, vol. 49, pp. 6808-6815, 2010.

sensing (S. Sato, 2008a;b;c).

large-scale structures.

on multi-zero beams.

2003.

46, pp. 87-94, 2007.

171-176, Kawasaki, Japan, 1998.

vol. 5013, pp. 95-102, 2003.

**7. References**

**6. Conclusion**


**20** 

*Russia* 

**Application of Interferometry** 

Veronica Makarova and Valery Kulichikhin

**to Analysis of Polymer-Polymer** 

**and Polymer-Solvent Interactions** 

*A.V. Topchiev Institute of Petrochemical Synthesis, Russian Academy of Sciences,* 

For prediction of polymer systems behavior on processing and exploiting stages knowledge of phase diagrams is very useful and in some cases – necessary. They characterize equilibrium phase states and for particular systems – phase structure. Among other methods to estimate compatibility of two substances including polymeric one, and consequently to obtain reasonable information about phase state and structure of the final heterophase system, the optical interferometry method in a version of optical wedge (Borovskii et al, 1973; Malkin et al., 1983; Malkin&Chalykh, 1979; Avdeev, 1990; Chalykh & Gerasimov, 2004; Chalykh et al., 1998; Makarova, 2007) engages specific space. Each of components forms the intrinsic interference pattern with a pitch reciprocally proportional to its refraction index. Two components form two interference patterns separated by interface. As a result of interdiffusion, interference fringes close to the interface are bending. This process indicates on formation of solutions of various concentrations with their own refraction indices. Evolution of fringes shape in time gives the information about intensity of interpenetration of components allowing us to calculate the interdiffusion coefficients. Plotting concentration profiles at different temperatures and accepting that concentrations at the interface are equilibrium ones,

In addition, kinetics of interference patterns evolution reflects constants of mass-transfer, i.e. kinetic stability of systems. This means that in one experiment we can obtain information on interaction kinetics and, finally, on composition of equilibrium phases. Such combination of kinetic and thermodynamic information is exclusively important for correct organization of

Simultaneously with high informativity, this method has some additional advantages compared with other methods for investigation of interdiffusion processes: simplicity, compactness of equipment, high enough preciseness of measurements, small amount of testing material. The main condition of this method consists in requirement that at least one of interacting components has to be transparent and difference of refraction indices should

Method of optical interferometry in application to interaction of two polymeric components is based on multipath interference from two surfaces of the film (or films) under

it is possible to construct phase diagrams of binary systems.

blending stage and processing of polymer solutions, blends and alloys.

**1. Introduction** 

be not less than 10-3.


## **Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions**

Veronica Makarova and Valery Kulichikhin *A.V. Topchiev Institute of Petrochemical Synthesis, Russian Academy of Sciences, Russia* 

## **1. Introduction**

18 Will-be-set-by-IN-TECH

394 Interferometry – Research and Applications in Science and Technology

T. Spirig, P. Seitz, O. Vietze and F. Heitger, "The Lock-in CCD—Two-Dimensional

R. Dändliker, Y. Salvadé and E. Zimmermann, "Distance-Measurement by Multiple-Wavelength Interferometry," *J. Opt.*, vol. 29, pp. 105-114, 1998. H. Povel, H. Aebersold and J.O. Stenflo, "Charge-Coupled Device Image Sensor as a

S. Bourquin, P. Seitz, and R.P. Salathé, "Optical Coherence Topography Based on a Two-Dimensional Smart Detector Array," *Opt. Lett.*, vol. 26, pp. 512-514, 2001. S. Bourquin, P. Seitz, and R.P. Salathé, "Two-Dimensional Smart Detector Array for Interferometric Applications," *Electron. Lett.*, vol. 37, pp. 975-976, 2001. S. Han, T. Sawada, T. Iwahori, S. Kawahito and S. Ando, "Three-Phase Time-Correlation

S. Sato, I. Fujimoto, T. Kurihara and S. Ando, "Remote Six-Axis Deformation Sensing with

S. Sato, I. Fujimoto, T. Kurihara and S. Ando, "Optical Vortex and Correlation Image Sensor for

S. Sato, I. Fujimoto, T. Kurihara and S. Ando, "Remote Full-Axis Deformation Sensing with

S. Ando, S. Sato and T. Kurihara, "Real-Time Tracking Experiment of Higher-Order

*Int. Conf. Networked Sensing Systems*, pp. 1-4, Pittsburgh, PA, 2009.

Optical Vortex Beams," in *Proc. SPIE*, vol. 6877, 2008.

*Sensing Systems*, pp. 39-42, Kanazawa, Japan, 2008.

*Sensing Technology*, pp. 452-456, Tainan, Taiwan, 2008.

1995.

29, pp. 1186-1190, 1990.

Synchronous Detection of Light," *IEEE J. Quantum Electronics*, vol. 31, pp. 1705-1708,

Demodulator in a 2-D Polarimeter with a Piezoelastic Modulator," *Appl. Opt.*, vol.

Image Sensor Using Pinned Photodiode Active Pixels," in *Proc. SPIE*, vol. 7536, 2010.

Networked Deformation Sensing of Infrastructures," in *Proc. 5th Int. Conf. Networked*

Optical Vortex Beam for Health Monitoring of Infrastructures," in *Proc. 3rd Int. Conf.*

Laguerre-Gaussian Beam for Remote Six-Axis Deformation Sensing," in *Proc. Sixth*

For prediction of polymer systems behavior on processing and exploiting stages knowledge of phase diagrams is very useful and in some cases – necessary. They characterize equilibrium phase states and for particular systems – phase structure. Among other methods to estimate compatibility of two substances including polymeric one, and consequently to obtain reasonable information about phase state and structure of the final heterophase system, the optical interferometry method in a version of optical wedge (Borovskii et al, 1973; Malkin et al., 1983; Malkin&Chalykh, 1979; Avdeev, 1990; Chalykh & Gerasimov, 2004; Chalykh et al., 1998; Makarova, 2007) engages specific space. Each of components forms the intrinsic interference pattern with a pitch reciprocally proportional to its refraction index. Two components form two interference patterns separated by interface. As a result of interdiffusion, interference fringes close to the interface are bending. This process indicates on formation of solutions of various concentrations with their own refraction indices. Evolution of fringes shape in time gives the information about intensity of interpenetration of components allowing us to calculate the interdiffusion coefficients. Plotting concentration profiles at different temperatures and accepting that concentrations at the interface are equilibrium ones, it is possible to construct phase diagrams of binary systems.

In addition, kinetics of interference patterns evolution reflects constants of mass-transfer, i.e. kinetic stability of systems. This means that in one experiment we can obtain information on interaction kinetics and, finally, on composition of equilibrium phases. Such combination of kinetic and thermodynamic information is exclusively important for correct organization of blending stage and processing of polymer solutions, blends and alloys.

Simultaneously with high informativity, this method has some additional advantages compared with other methods for investigation of interdiffusion processes: simplicity, compactness of equipment, high enough preciseness of measurements, small amount of testing material. The main condition of this method consists in requirement that at least one of interacting components has to be transparent and difference of refraction indices should be not less than 10-3.

Method of optical interferometry in application to interaction of two polymeric components is based on multipath interference from two surfaces of the film (or films) under

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 397

Binary polymer systems are thermodynamically reversible and submit to phase rule as lowmolecular weight formulations. The following kinds of the phase equilibrium are typical for these systems (the rank with increase of phase ordering) (Chalykh&Gerasimov, 2004;

 amorphous (liquid-liquid) equilibrium realizing in the domain of bounded solubility of components where the system decomposes on two phases, and each of them is the

 liquid-crystalline (mesomorphic) equilibrium characterizing with transition of isotropic solution to liquid crystal phase via biphasic zone where disordered and ordered phases

crystalline equilibrium in that the saturated solution of one component into another

In many cases the different kinds of the phase equilibrium can superimpose leading to complex phase equilibrium. Let consider in short each individual kind of the phase

At mixing of two partially compatible liquids in the definite temperature-concentration domain a system decomposes on two liquid coexisting phases – saturated solutions of one component to another. Change of temperature results in change of components solubility and causes variation of coexisting phase compositions. At definite temperature, known as critical, boundaries between phases disappear and a system passes from heterophase to homophase. Return to previous conditions causes phase decomposition again. Such transitions can be performing unrestricted times without any shift of critical temperature.

Phase equilibria in binary systems are expressed by phase diagrams. Usually they are plotted in coordinates "temperature-composition" at constant pressure. Schematic images of basic kinds of phase diagrams of binary polymer systems characterizing with amorphous equilibrium are presented in Fig. 3. The equilibrium values of coexisting phases compositions form unique curve – binodal. Convex upwards binodal (Fig. 3a) corresponds to the case of increasing compatibility with increase of temperature. Its cupola reflects the limiting temperature higher that a system becomes single-phase at any concentration of components. This temperature is known as the Upper Critical Solubility Temperature (UCST). The phase composition corresponding to UCST is also named as

If compatibility of components increases with decrease of temperature, the binodal has a shape of convex downwards curve (Fig. 3b) and critical temperature is named as Low Critical Solubility Temperature (LCST). Some systems can have both critical temperatures and their amorphous phase equilibriums are expressed by circle-like or closed (UCST>LCST) diagram (Fig. 3c), non-closed diagram (UCST<LCST) (Fig. 3d) or the diagram

Chalykh et al., 1998; Papkov, 1981; Tager, 1978; Lipatov, 1984; Klenin, 1995):

coexists with crystalline or crystal solvate phase of this component.

saturated solution of one component into another;

equilibrium and typical cases of their superposition.

of sand-glass shape (UCMT~LCMT) (Fig. 3f).

**3. Amorphous equilibrium in polymer systems** 

**2. Results and discussion** 

are in equilibrium;

critical.

investigation (Fig.1). Films of testing polymers are placing in side-by-side (edge-to-edge) manner between two glass plates with a small wedge angle (~2o) (Fig.2). Preliminary, semitransparent metallic layer (Ag, Au, Ni-Cr alloy) is applied on inner surfaces of glass plates for reinforcing their reflection capability. At passing bunch of monochromatic beams through films of alternating thickness, the interference pattern appears as a result of pathlength difference.

In the beginning of experiment two systems of interference fringes (bands) with different pitches corresponding to refraction indices of each component are separated by interface. As a result of mutual dissolution, concentration gradients appear which lead to refraction index gradients and, consequently, to fringes bending. Evolution of fringes shape in time gives information about redistribution of components concentration in the diffusion zone as well as about the direction and the rate of the diffusion stream.

The optical interferometer used in this research was designed on a base of UV microscope (Avdeev, 1990; Makarova, 2007). The interferometer consists of light source, optical system, diffusion cell, oven and microscope equipped with video-camera. The diffusion cell consists of frame with optical channel, heating chamber, and the cell as it. Surface of glass plates with a thickness of 3-5 mm and plain dimensions 20x25 mm is treated with the accuracy rating 14. The necessary wedge angle and gap thickness was done by means of metallic (foil) ribbons with a thickness from 80 to 110 .

Fig. 1. Interference at multipass light refelection.

Fig. 2. Schematic image of the experimental procedure.

## **2. Results and discussion**

396 Interferometry – Research and Applications in Science and Technology

investigation (Fig.1). Films of testing polymers are placing in side-by-side (edge-to-edge) manner between two glass plates with a small wedge angle (~2o) (Fig.2). Preliminary, semitransparent metallic layer (Ag, Au, Ni-Cr alloy) is applied on inner surfaces of glass plates for reinforcing their reflection capability. At passing bunch of monochromatic beams through films of alternating thickness, the interference pattern appears as a result of path-

In the beginning of experiment two systems of interference fringes (bands) with different pitches corresponding to refraction indices of each component are separated by interface. As a result of mutual dissolution, concentration gradients appear which lead to refraction index gradients and, consequently, to fringes bending. Evolution of fringes shape in time gives information about redistribution of components concentration in the diffusion zone as well

The optical interferometer used in this research was designed on a base of UV microscope (Avdeev, 1990; Makarova, 2007). The interferometer consists of light source, optical system, diffusion cell, oven and microscope equipped with video-camera. The diffusion cell consists of frame with optical channel, heating chamber, and the cell as it. Surface of glass plates with a thickness of 3-5 mm and plain dimensions 20x25 mm is treated with the accuracy rating 14. The necessary wedge angle and gap thickness was done by means of metallic (foil)

as about the direction and the rate of the diffusion stream.

ribbons with a thickness from 80 to 110 .

Fig. 1. Interference at multipass light refelection.

Fig. 2. Schematic image of the experimental procedure.

length difference.

Binary polymer systems are thermodynamically reversible and submit to phase rule as lowmolecular weight formulations. The following kinds of the phase equilibrium are typical for these systems (the rank with increase of phase ordering) (Chalykh&Gerasimov, 2004; Chalykh et al., 1998; Papkov, 1981; Tager, 1978; Lipatov, 1984; Klenin, 1995):


In many cases the different kinds of the phase equilibrium can superimpose leading to complex phase equilibrium. Let consider in short each individual kind of the phase equilibrium and typical cases of their superposition.

## **3. Amorphous equilibrium in polymer systems**

At mixing of two partially compatible liquids in the definite temperature-concentration domain a system decomposes on two liquid coexisting phases – saturated solutions of one component to another. Change of temperature results in change of components solubility and causes variation of coexisting phase compositions. At definite temperature, known as critical, boundaries between phases disappear and a system passes from heterophase to homophase. Return to previous conditions causes phase decomposition again. Such transitions can be performing unrestricted times without any shift of critical temperature.

Phase equilibria in binary systems are expressed by phase diagrams. Usually they are plotted in coordinates "temperature-composition" at constant pressure. Schematic images of basic kinds of phase diagrams of binary polymer systems characterizing with amorphous equilibrium are presented in Fig. 3. The equilibrium values of coexisting phases compositions form unique curve – binodal. Convex upwards binodal (Fig. 3a) corresponds to the case of increasing compatibility with increase of temperature. Its cupola reflects the limiting temperature higher that a system becomes single-phase at any concentration of components. This temperature is known as the Upper Critical Solubility Temperature (UCST). The phase composition corresponding to UCST is also named as critical.

If compatibility of components increases with decrease of temperature, the binodal has a shape of convex downwards curve (Fig. 3b) and critical temperature is named as Low Critical Solubility Temperature (LCST). Some systems can have both critical temperatures and their amorphous phase equilibriums are expressed by circle-like or closed (UCST>LCST) diagram (Fig. 3c), non-closed diagram (UCST<LCST) (Fig. 3d) or the diagram of sand-glass shape (UCMT~LCMT) (Fig. 3f).

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 399

rheological data that allows us to obtain more informative picture of mass-transfer in

Amorphous phase equilibrium will be considered on examples of polycarbonate (PC) – polymethylmethacrylate (PMMA) pair. PC and PMMA are transparent polymers (the coefficient of light transmission is ~90%) with similar exploiting properties. That is why fields of their application in many cases coincides (light-technical products, engineering plastics, materials for medicine, etc.). However, mechanical characteristics of PC, partially impact strength, are much better compared with PMMA. Keeping in mind that PC is more expensive than PMMA, development of blending materials based on these polymers can be interesting and important, but blends should be transparent and this means that the main

Literature data concerning compatibility of these components are rather contradictive (Chiou et al, 1987; Kim&Paul, 1992; Butzbach&Wendorff, 1991; Landry & Henrichs, 1989; Debier, 1994; Kyu, 1993; Nishimoto, 1991; Rabeony, 1992; Woo&So, 1996): from conclusion about their immiscibility to partial or even total compatibility. Basic methods used in cited papers were light transmission, light scattering, DSC, and specimens for investigation were prepared from mutual for both polymers solutions. The nature of solvent, thermal prehistory and long relaxation times could influence significantly on formed heterophase structure (Woo&So, 1996). Such uncertainty was the driving force for this investigation using interferometric method allowing us to observe direct penetration of melted

104; Tg=150oС, Тm=267oС) and series of PMMA (Мw=1.5.

104; Тg=114оС; Тflow=180oС) were used as objects under investigation. Experiments were carried out in temperature diapason from 150oC (glass point of PC) to 280oC (decomposition point of PMMA). Films with a thickness of 100-120 were prepared by hot-pressing at 150oС (PMMA) and 240oС (PMMA) with subsequent cooling with a rate of 5-10 oC/min.

method of squeezing flow allowing us to work with micro-amount of materials. Interval of shear rates and stresses realized in this method at 240oC corresponds to Newtonian flow. Blends for rheological measurements were obtained via mutual solvent – chloroform that

Interferograms obtained present two interference patterns with a different pitches corresponding to the refraction indices of each component. In vicinity of interface "bands of equal thickness" bend that indicates on formation of solutions of various compositions PC in PMMA, on the one hand (Fig. 4, left) and PMMA in PC, on the other hand (right). Based on connection of the refraction index with composition, the profiles of concentration distribution in diffusion zone were plotted for different temperatures and observation times. Components concentrations reached on both sides of interface were accepted for solubility

Various curvature of bands near the interface for PMMA of different molecular weight reflects strong dependence of compatibility on Mw PMMA. For interaction of PC with

curvature is more significant but calculation shows that maximum solubility does not exceed

dissolution of components is not higher than 0.5%. In the case of PMMA with Mw=7.4.

104) the bands bending is practically absent and this means that the mutual

104; 7.4.

104) were measured by

104;

104 the

task is to find temperature-composition regions of their compatibility.

components (Avdeev et al., 2001; Eastmond&Kotomin, 1994).

Viscosities of the neat and blended samples PC-PMMA (Mw=1.5.

was removed in vacuum at 60oC during 8 h.

vicinity of interfaces.

Samples of PC (Mw=2.5.

one into another.

PMMA (Mw=30.

30.

Fig. 3. Basic types of the phase diagrams of polymer system with amorphous equilibrium according to (Chalykh et al., 1998). See explanations in the text.

Under binodal a system decomposes on two phases. Analysis of kinetics of new phases formation indicates on unusual behavior of microheterophase system in vicinity of binodal and necessity to draw additional line – spinodal (Fig. 3a, curve 2). This line separates unstable (under binodal) and meta-stable (between binodal and spinodal) compositions. The critical point is the same as for binodal, as for spinodal. In meta-stable domain formation of novel phases proceeds very quickly at small fluctuations of composition.

In some cases also useful to show on phase diagrams boundaries of relaxation states of binary system (glass and brittleness points, flow temperature, limits of thermal stability of components, etc) (Papkov, 1971; Chalykh et al, 1990). Such diagrams are named "generalized phase diagrams" or "diagrams of phase and relaxation states".

Equilibrium curves shown in Fig. 3 are idealized. Actually, the total binodal curve cannot be registered ever because of temperature limitations. For example, in whole temperature diapason either the single-phase domain (complete compatibility of components) can exist or its coexistence with biphasic domain but without achievement of critical temperatures (partial compatibility of components). The most typical are phase diagrams with UCST because for majority systems increase of temperature leads to increase of kinetic mobility of molecules and macromolecules improving their mutual solubility. Diagrams with LCST meet for aqueous solutions of various polysaccharides

In the case of amorphous equilibrium at T>UCST (or T<LCST) the continuous change of composition at transition from one component to another is realized in diffusion zone. At temperatures of phase transitions in diffusion zone the border between two solutions appears. We will consider further some examples of amorphous phase equilibrium registered by interference method. As a rule, optical measurements were added by

Fig. 3. Basic types of the phase diagrams of polymer system with amorphous equilibrium

Under binodal a system decomposes on two phases. Analysis of kinetics of new phases formation indicates on unusual behavior of microheterophase system in vicinity of binodal and necessity to draw additional line – spinodal (Fig. 3a, curve 2). This line separates unstable (under binodal) and meta-stable (between binodal and spinodal) compositions. The critical point is the same as for binodal, as for spinodal. In meta-stable domain formation of

In some cases also useful to show on phase diagrams boundaries of relaxation states of binary system (glass and brittleness points, flow temperature, limits of thermal stability of components, etc) (Papkov, 1971; Chalykh et al, 1990). Such diagrams are named

Equilibrium curves shown in Fig. 3 are idealized. Actually, the total binodal curve cannot be registered ever because of temperature limitations. For example, in whole temperature diapason either the single-phase domain (complete compatibility of components) can exist or its coexistence with biphasic domain but without achievement of critical temperatures (partial compatibility of components). The most typical are phase diagrams with UCST because for majority systems increase of temperature leads to increase of kinetic mobility of molecules and macromolecules improving their mutual solubility. Diagrams with LCST

In the case of amorphous equilibrium at T>UCST (or T<LCST) the continuous change of composition at transition from one component to another is realized in diffusion zone. At temperatures of phase transitions in diffusion zone the border between two solutions appears. We will consider further some examples of amorphous phase equilibrium registered by interference method. As a rule, optical measurements were added by

according to (Chalykh et al., 1998). See explanations in the text.

meet for aqueous solutions of various polysaccharides

novel phases proceeds very quickly at small fluctuations of composition.

"generalized phase diagrams" or "diagrams of phase and relaxation states".

rheological data that allows us to obtain more informative picture of mass-transfer in vicinity of interfaces.

Amorphous phase equilibrium will be considered on examples of polycarbonate (PC) – polymethylmethacrylate (PMMA) pair. PC and PMMA are transparent polymers (the coefficient of light transmission is ~90%) with similar exploiting properties. That is why fields of their application in many cases coincides (light-technical products, engineering plastics, materials for medicine, etc.). However, mechanical characteristics of PC, partially impact strength, are much better compared with PMMA. Keeping in mind that PC is more expensive than PMMA, development of blending materials based on these polymers can be interesting and important, but blends should be transparent and this means that the main task is to find temperature-composition regions of their compatibility.

Literature data concerning compatibility of these components are rather contradictive (Chiou et al, 1987; Kim&Paul, 1992; Butzbach&Wendorff, 1991; Landry & Henrichs, 1989; Debier, 1994; Kyu, 1993; Nishimoto, 1991; Rabeony, 1992; Woo&So, 1996): from conclusion about their immiscibility to partial or even total compatibility. Basic methods used in cited papers were light transmission, light scattering, DSC, and specimens for investigation were prepared from mutual for both polymers solutions. The nature of solvent, thermal prehistory and long relaxation times could influence significantly on formed heterophase structure (Woo&So, 1996). Such uncertainty was the driving force for this investigation using interferometric method allowing us to observe direct penetration of melted components (Avdeev et al., 2001; Eastmond&Kotomin, 1994).

Samples of PC (Mw=2.5. 104; Tg=150oС, Тm=267oС) and series of PMMA (Мw=1.5. 104; 7.4. 104; 30. 104; Тg=114оС; Тflow=180oС) were used as objects under investigation. Experiments were carried out in temperature diapason from 150oC (glass point of PC) to 280oC (decomposition point of PMMA). Films with a thickness of 100-120 were prepared by hot-pressing at 150oС (PMMA) and 240oС (PMMA) with subsequent cooling with a rate of 5-10 oC/min.

Viscosities of the neat and blended samples PC-PMMA (Mw=1.5. 104) were measured by method of squeezing flow allowing us to work with micro-amount of materials. Interval of shear rates and stresses realized in this method at 240oC corresponds to Newtonian flow. Blends for rheological measurements were obtained via mutual solvent – chloroform that was removed in vacuum at 60oC during 8 h.

Interferograms obtained present two interference patterns with a different pitches corresponding to the refraction indices of each component. In vicinity of interface "bands of equal thickness" bend that indicates on formation of solutions of various compositions PC in PMMA, on the one hand (Fig. 4, left) and PMMA in PC, on the other hand (right). Based on connection of the refraction index with composition, the profiles of concentration distribution in diffusion zone were plotted for different temperatures and observation times. Components concentrations reached on both sides of interface were accepted for solubility one into another.

Various curvature of bands near the interface for PMMA of different molecular weight reflects strong dependence of compatibility on Mw PMMA. For interaction of PC with PMMA (Mw=30. 104) the bands bending is practically absent and this means that the mutual dissolution of components is not higher than 0.5%. In the case of PMMA with Mw=7.4. 104 the curvature is more significant but calculation shows that maximum solubility does not exceed

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 401

spinodal) systems. *а* and *b* – calculated values of UCST with accepting of concentration

 PMMA

20 40 60 80

from 200 (a) to 260oC (b) and at decrease of temperature from 260 to 220 (c) and 160oC (d).

For the system with lowest Mw of PMMA the spinodal was determined as well. For this aim the blend was heated to 270oC for significant compatibility of components (Fig. 6a, b), and then it was cooled step-by-step keeping at each temperature several hours (Fig. 6c, d).

10-4=30 (1), 7.4 (2) and 1.5 (3 – binodal, 3\* -

104) system obtained at increase of temperature

Fig. 5. Phase diagrams of PC-PMMA (Мw.

Tg PC

Tdecomp PMMA

Fig. 6. Interferograms of PC-PMMA (Мw=1.5.

, and ignoring it, respectively.

C b a

dependence of

16 0

20 0

24 0

28 0

T, <sup>0</sup>

Fig. 4. Interferograms and concentration profiles at interaction of PC with PMMA (Мw=30. 104 (а); 7.4. 104 (b); 1.5. 104 (c)). Temperature 240оС.

3%. The highest compatibility was observed for the system PC-PMMA with Mw=1.5. 104. At permissible temperature of 280oC maximum solubility for PMMA in PC reaches 40% and for PC in PMMA – 20%. This result corresponds to well-known, but subjective opinion about increase of mutual solubility for polymers, if one component has Мw<104 [3].

Using data on solubility of components at different temperatures the phase diagrams were plotted (Fig. 5). The temperature diapason was limited by glass point of PC and decomposition point of PMMA. One can see that there exists domains of mutual solubility of components and with growth of temperature the compatibility increases. Consequently, these diagrams belong to amorphous equilibrium with UCMT though the critical point was not achieved experimentally for any system because of decomposition of PMMA. Since the solubility of PMMA in PC is higher than vice versa, the binodal is asymmetric and shifted in direction of higher content of PMMA.

Fig. 4. Interferograms and concentration profiles at interaction of PC with PMMA

increase of mutual solubility for polymers, if one component has Мw<104 [3].

104 (c)). Temperature 240оС.

3%. The highest compatibility was observed for the system PC-PMMA with Mw=1.5.

permissible temperature of 280oC maximum solubility for PMMA in PC reaches 40% and for PC in PMMA – 20%. This result corresponds to well-known, but subjective opinion about

Using data on solubility of components at different temperatures the phase diagrams were plotted (Fig. 5). The temperature diapason was limited by glass point of PC and decomposition point of PMMA. One can see that there exists domains of mutual solubility of components and with growth of temperature the compatibility increases. Consequently, these diagrams belong to amorphous equilibrium with UCMT though the critical point was not achieved experimentally for any system because of decomposition of PMMA. Since the solubility of PMMA in PC is higher than vice versa, the binodal is asymmetric and shifted in

104. At

(Мw=30.

104 (а); 7.4.

104 (b); 1.5.

direction of higher content of PMMA.

Fig. 5. Phase diagrams of PC-PMMA (Мw. 10-4=30 (1), 7.4 (2) and 1.5 (3 – binodal, 3\* spinodal) systems. *а* and *b* – calculated values of UCST with accepting of concentration dependence of , and ignoring it, respectively.

For the system with lowest Mw of PMMA the spinodal was determined as well. For this aim the blend was heated to 270oC for significant compatibility of components (Fig. 6a, b), and then it was cooled step-by-step keeping at each temperature several hours (Fig. 6c, d).

Fig. 6. Interferograms of PC-PMMA (Мw=1.5. 104) system obtained at increase of temperature from 200 (a) to 260oC (b) and at decrease of temperature from 260 to 220 (c) and 160oC (d).

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 403

calculated value of UCMT allowed us to plot the complete phase diagram of this system.

Fig. 7. Concentration dependence of the interaction parameter for the system PC-PMMA

104). Т=212 (а), 226 (b), 240 (c), 254 (d), 268(e) и 282oС (f). The dotted line

20 40 60 80

PMMA

f e d c

b

a

220 240 260 280

Т, <sup>0</sup> С

constant in Eq. (1) for the same system.

*cr*=0,61,

*cr*, determine the critical temperature equal to 296oC. The

*cr*=0,017 and by

These equations allowed us to calculate critical values:

extrapolation of

(Мw=1.5.

corresponds to the *χ* values on the binodal.

0.002

Fig. 8. Temperature dependence of the

0.003

0.004

0.020

0.025

0.030

 *at =cr* to 

At cooling the compatibility decreases and in the solutions domain close to the interface the concentration of one component becomes higher than thermodynamically equilibrium (super-saturation) that causes appearance of the disperse phase enriched by this component. Due to high dispersity degree this two-phase region scatters light intensively and it looks more dark than single-phase regions. With time on boundaries between homophase and heterophase domains the concentrations corresponding to spinodal branches are realized (Fig 5, curve 3\*) and bending of interference bands decreases (Fig. 6 d). The concentration width of the meta-stable region located between binodal and spinodal monotonously decreases at moving from low to higher temperatures: from 8% at 190oC to 2% at 270oC from the side of the higher content of PC of and from 5 to 2% at the same temperatures from the side of PMMA.

Since the compatibility of PC and PMMA of higher molecular weights is very low, the further calculations and experiments on kinetics of dissolution and measuring the viscous properties were carried out for the system PC-PMMA with Mw=1.5. 104.

For the quantitative interpretation of experimental data on interdiffusion the theory of Flory-Huggins-Scott (FHS) (Scott, 1949; Tompa, 1956) was applied. Partially, from classical expression for the Gibbs free energy the interaction parameter was calculated. Calculation of interaction parameter was performed assuming its linear dependence on concentration (Koningsweld& Kleintjens, 1977; Palatnik&Landau, 1961):

$$
\mathcal{X} = \mathcal{X}\_0 + \alpha \mathfrak{w}\_1,\tag{1}
$$

where *χo* – the interaction parameter at *1*=0, - the constant, reflecting influence of blend composition on intermolecular interaction.

Based on concentrations corresponding to binodal branches at definite temperatures, the values of *0* and were calculated using FHS equation and the expression (1). The family of linear dependences *(1)* for different temperatures is shown in Fig. 7. The calculated values of the interaction parameter lie in limits of 0.017-0.030, that is inherent for good compatibility of components. At moving from PMMA to PC, the value decreases that confirms higher solubility of PMMA in PC than vice versa.

At relatively low temperatures the concentration dependence of the interaction parameter is strong enough (the value in Eq. (1) at 212oC is equal to 0.004), but it decreases with increase of temperature reaching the constant level 0.002 (Fig. 8). Using this fact as well as zero-meaning of the second and the third derivatives in critical point, the expression for and at UCMT was obtained (the dependence *(χ)* was also taken into account):

$$\varphi\_{2\text{-}up} = \frac{\sqrt{\frac{\chi\_1}{\chi\_2} - 6a\chi\_1}}{1 + \sqrt{\frac{\chi\_1}{\chi\_2} - 6a\chi\_1}}\tag{2}$$

$$\mathcal{X}\_{12} = \frac{1}{2} \left( \frac{1}{\sqrt{\mathbf{x}\_2}} + \frac{1}{\sqrt{\mathbf{x}\_1 - 6a\mathbf{x}\_1}} \right)^2 + a \left( 5\varphi\_{2\text{op}} - 1 \right) \tag{3}$$

At cooling the compatibility decreases and in the solutions domain close to the interface the concentration of one component becomes higher than thermodynamically equilibrium (super-saturation) that causes appearance of the disperse phase enriched by this component. Due to high dispersity degree this two-phase region scatters light intensively and it looks more dark than single-phase regions. With time on boundaries between homophase and heterophase domains the concentrations corresponding to spinodal branches are realized (Fig 5, curve 3\*) and bending of interference bands decreases (Fig. 6 d). The concentration width of the meta-stable region located between binodal and spinodal monotonously decreases at moving from low to higher temperatures: from 8% at 190oC to 2% at 270oC from the side of the higher content of PC of and from 5 to 2% at the same temperatures from the

Since the compatibility of PC and PMMA of higher molecular weights is very low, the further calculations and experiments on kinetics of dissolution and measuring the viscous

For the quantitative interpretation of experimental data on interdiffusion the theory of Flory-Huggins-Scott (FHS) (Scott, 1949; Tompa, 1956) was applied. Partially, from classical

of interaction parameter was performed assuming its linear dependence on concentration

Based on concentrations corresponding to binodal branches at definite temperatures, the

At relatively low temperatures the concentration dependence of the interaction parameter

is strong enough (the value in Eq. (1) at 212oC is equal to 0.004), but it decreases with increase of temperature reaching the constant level 0.002 (Fig. 8). Using this fact as well as zero-meaning of the second and the third derivatives in critical point, the expression for

1

*х*

1

 

*х*

2

1

*х*

*х х*

12 2 21 1

2

1 6

*х*

1

6

2

11 1 5 1 2 6 *кр xx x*

were calculated using FHS equation and the expression (1). The family of

*1)* for different temperatures is shown in Fig. 7. The calculated values

lie in limits of 0.017-0.030, that is inherent for good

*(χ)* was also taken into account):

 

> *1*=0,

104.

0 1 , (1)


was calculated. Calculation

value decreases that

(2)

(3)

properties were carried out for the system PC-PMMA with Mw=1.5.

expression for the Gibbs free energy the interaction parameter

confirms higher solubility of PMMA in PC than vice versa.

at UCMT was obtained (the dependence

compatibility of components. At moving from PMMA to PC, the

2

*кр*

(Koningsweld& Kleintjens, 1977; Palatnik&Landau, 1961):

where *χo* – the interaction parameter at

composition on intermolecular interaction.

*(*

of the interaction parameter

side of PMMA.

values of

and  *0* and 

linear dependences

These equations allowed us to calculate critical values: *cr*=0,61, *cr*=0,017 and by extrapolation of  *at =cr* to *cr*, determine the critical temperature equal to 296oC. The calculated value of UCMT allowed us to plot the complete phase diagram of this system.

Fig. 7. Concentration dependence of the interaction parameter for the system PC-PMMA (Мw=1.5. 104). Т=212 (а), 226 (b), 240 (c), 254 (d), 268(e) и 282oС (f). The dotted line corresponds to the *χ* values on the binodal.

Fig. 8. Temperature dependence of the constant in Eq. (1) for the same system.

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 405

mixed solutions of PC and PMMA in a common solvent. After removing a solvent the thermodynamically unstable system at heating suffers very slow noticible decomposition at

One of the practical application of obtained data on interdiffusion coefficients and mutual solubilities of partially compatible components consists in estimation of time needed for formation of the interphase layer of the definite thickness that should influence on properties of the heterophase system including rheological and mechanical ones. A volume of the interphase layer was accepted as the tenth part of disperse particle volume. Calculation was done using the model of dissolution of a particle of spherical shape (Crank,

> <sup>0</sup> 2 *<sup>v</sup> a ra erfc r D*

the interphase layer, *a* – a radius of a particle, *r* – a distance from particle center to the border of the layer, *erfc* – empirical probability function (taken from the reference book). The PMMA was chosen as the disperse phase and PC as the disperse medium. The concentration of PC onto interface was accepted as in *e* times less than the initial concentration. Results of the calculation for 240oC are presented in Table 1. One can see that time of spontaneous formation of such a layer depends strongly on the particle dimensions and can differ from a few minutes to several years. This difference underlines again the determining role of

– concentrations of the component on the inner and the outer boundaries of

, (4)

Time of formation of the diffusion layer of the definite thickness

104).

T\*>Tg only, because of low Dv values. The T\* value could be accepted as LCST.

Diameter of a particle,

0,3 10 5 min 3 100 8,3 h 15 500 8,7 days 30 1000 34,7 days 60 2000 4,6 months 150 5000 2,4 year Table 1. Time of the diffusion layer formation at mixing of PC with PMMA (Мw=1.5.

1 10 s 30 h 10 20 min 4 months 100 30 h 30 years

The time of total dissolution of PMMA particle in PC was estimated as well. For comparison, calculations were performed for two temperatures: 160 and 240oC. Results shown in Table 2, illustrate significant delay of interpenetration process at approaching to the glass point. Simultaneously, these data allow us to make choice for the mixing regime to obtain either homogeneous solution of PMMA in PC or three-phase system consisting

Table 2. Time of complete dissolution of the PMMA particle in PC.

Diameter of a particle, Т=240С Т=160С

1956):

where and 

dispersity at dissolution of polymers.

Thickness of the diffusion layer,

The *cr* value (Fig. 5, point *a*) calculated with taking into account the concentration dependence of the interaction parameter differs almost on 10% on *cr* value estimated if this dependence is ignored (Fig. 5, point *b*) (Koningsweld& Kleintjens, 1977; Palatnik&Landau, 1961). This means that the critical concentration depends not only on dimensions of macromolecules, but on their nature as well.

For description of kinetics of mixing the interdiffusion coefficients Dv, characterizing the rate of interpenetration of components, were calculated. The calculation was done by graphic method from the curves of concentration distribution by means of Matano-Boltzman method (Borovskii, 1973) at three temperatures and concentration domains of compatibility (Fig. 10). Interdiffusion coefficients are rather high for polymer pairs and change from 1,5. 10-12 (220oC) to 4. 10-11 m2/c (260oC). Dv values in the region of predominant PMMA content are a little lower than for the region of higher PC content.

Fig. 9. Interdiffusion coefficients vs. composition for PC-PMMA (Мw=1.5. 104) system at 260 (1), 240 (2) and 220oC (3). The domain restricted with dotted line corresponds to the biphasic region of the phase diagram.

The concentration dependence of the interdiffusion coefficient far from binodal is rather weak. With increase of the one component content in another Dv values a little increase but at approaching to binodal decrease again. Deceleration of diffusion process in vicinity of saturation state for both solutions is natural, since Dv is proportional to derivative of chemical potential on concentration that becomes equal to zero on the spinodal.

With increase of temperature interdiffusion coefficients become higher for all compositions (Fig. 9). Activation energy of diffusion is high enough for polymer systems and equal to 170 kJ/mole. Extrapolation of the temperature dependence of Dv to 160oC gives a value of 8.1. 10- 15 m2/c. Probably, the high difference in diffusion mobility of components at high (4. 10-11 m2/c at 260oC) and low temperatures close to the glass point could be the explanation of false LCST's [14-19]. This situation can be realized in attempts to create the homophase

dependence is ignored (Fig. 5, point *b*) (Koningsweld& Kleintjens, 1977; Palatnik&Landau, 1961). This means that the critical concentration depends not only on dimensions of

For description of kinetics of mixing the interdiffusion coefficients Dv, characterizing the rate of interpenetration of components, were calculated. The calculation was done by graphic method from the curves of concentration distribution by means of Matano-Boltzman method (Borovskii, 1973) at three temperatures and concentration domains of compatibility (Fig. 10). Interdiffusion coefficients are rather high for polymer pairs and

dependence of the interaction parameter differs almost on 10% on

PMMA content are a little lower than for the region of higher PC content.

Fig. 9. Interdiffusion coefficients vs. composition for PC-PMMA (Мw=1.5.

chemical potential on concentration that becomes equal to zero on the spinodal.

(1), 240 (2) and 220oC (3). The domain restricted with dotted line corresponds to the biphasic

20 80

PMMA

The concentration dependence of the interdiffusion coefficient far from binodal is rather weak. With increase of the one component content in another Dv values a little increase but at approaching to binodal decrease again. Deceleration of diffusion process in vicinity of saturation state for both solutions is natural, since Dv is proportional to derivative of

With increase of temperature interdiffusion coefficients become higher for all compositions (Fig. 9). Activation energy of diffusion is high enough for polymer systems and equal to 170 kJ/mole. Extrapolation of the temperature dependence of Dv to 160oC gives a value of 8.1.

m2/c at 260oC) and low temperatures close to the glass point could be the explanation of false LCST's [14-19]. This situation can be realized in attempts to create the homophase

15 m2/c. Probably, the high difference in diffusion mobility of components at high (4.

macromolecules, but on their nature as well.

10-12 (220oC) to 4.

*cr* value (Fig. 5, point *a*) calculated with taking into account the concentration

10-11 m2/c (260oC). Dv values in the region of predominant

 1 2 3 *cr* value estimated if this

104) system at 260

10-

10-11

The 

change from 1,5.

region of the phase diagram.




lgDv [m2 /s] mixed solutions of PC and PMMA in a common solvent. After removing a solvent the thermodynamically unstable system at heating suffers very slow noticible decomposition at T\*>Tg only, because of low Dv values. The T\* value could be accepted as LCST.

One of the practical application of obtained data on interdiffusion coefficients and mutual solubilities of partially compatible components consists in estimation of time needed for formation of the interphase layer of the definite thickness that should influence on properties of the heterophase system including rheological and mechanical ones. A volume of the interphase layer was accepted as the tenth part of disperse particle volume. Calculation was done using the model of dissolution of a particle of spherical shape (Crank, 1956):

$$\frac{\partial \mathcal{P}}{\partial \rho\_0} = \frac{a}{r} \operatorname{erfc} \frac{r - a}{2\sqrt{D\_v \pi}} \,. \tag{4}$$

where and – concentrations of the component on the inner and the outer boundaries of the interphase layer, *a* – a radius of a particle, *r* – a distance from particle center to the border of the layer, *erfc* – empirical probability function (taken from the reference book). The PMMA was chosen as the disperse phase and PC as the disperse medium. The concentration of PC onto interface was accepted as in *e* times less than the initial concentration. Results of the calculation for 240oC are presented in Table 1. One can see that time of spontaneous formation of such a layer depends strongly on the particle dimensions and can differ from a few minutes to several years. This difference underlines again the determining role of dispersity at dissolution of polymers.


Table 1. Time of the diffusion layer formation at mixing of PC with PMMA (Мw=1.5. 104).


Table 2. Time of complete dissolution of the PMMA particle in PC.

The time of total dissolution of PMMA particle in PC was estimated as well. For comparison, calculations were performed for two temperatures: 160 and 240oC. Results shown in Table 2, illustrate significant delay of interpenetration process at approaching to the glass point. Simultaneously, these data allow us to make choice for the mixing regime to obtain either homogeneous solution of PMMA in PC or three-phase system consisting

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 407

22 2 1 lg

*2* – viscosities of disperse phase and medium, respectively,

fraction of dispersion medium that for compatible polymers coincides with relative content of the component forming the medium. But if at least partial mutual dissolution proceeds,

the meanings of concentrations on binodal and "the lever rule". The calculated curve 3 (Fig.

So, the couple PC – low molecular weight PMMA is a good example of polymer systems with amorphous phase equilibrium. Method of interferometry is very useful and fruitful for such kind of systems due to informativity and combination of joint approaches: kinetics of mass-transfer and thermodynamics of phase states. Increase of the mutual solubility with temperature indicates on UCST that could not be measured experimentally but was

Comparison of the phase diagram with concentration dependence of viscosity allowed us to observe local minimum of viscosity near the binodal (sooner in meta-stable region between binodal and spinodal where microemulsion appears as the first step of the phase decomposition) and to apply the empiric equation for calculation of the blends viscosity by means of accurate determination of the composition from binodal branches. Joint consideration of diffusion and rheological properties seems to be very fruitful for judging about structure, kinetic and thermodynamic behaviors of complex, multiphase polymer systems. This approach will be used further for analysis of systems with LC and crystalline

Calculations of the interphase layers thickness depending on dimensions of drops and time, based on the knowledge of interdiffusion coefficients can be very useful for correct organization of technological process of mixing and, consequently, development of new

Crystalline equilibrium is inherent for polymer systems if at least one component is capable to crystallize. At cooling from the region of the total compatibility (isotropic solution) at definite temperature the system decomposes on two equilibrium phases: deposition of the

The curve describing sum of transition (melting or crystallization) points corresponding to various compositions is named "liquidus". Higher of the liquidus line the single-phase region (liquid solution) exists, and lower the liquidus line – biphasic system (crystalline phase distributed in solution). Basic kinds of phase diagrams with crystalline equilibrium

Cases a) and b) are typical for components with a weal interaction. If one component of such a system is amorphous and the other – crystalline, the melting point of a system as whole decreases monotonously as a result of dilution of the crystalline component by amorphous one (depression of melting points) as is seen in the graph a). In systems containing two components capable to form crystalline phases two liquidus line cross in the eutectic point

   

lg 1 lg , (5)

*<sup>2</sup>* value was found from the phase diagram using

*<sup>2</sup>* - the volume

 

2 2

materials with three-phase structure and new properties.

this coincidence is absent. It is why the

where *<sup>1</sup>* and 

equilibria.

are shown in Fig. 11.

11) coincides in limits of equipment error with the experimental one.

calculated using evolution of the interaction parameter with concentration.

**4. Crystalline and complex phase equilibria in polymer systems** 

crystalline component and its saturated solution into the other component.

of PMMA, PC and solution of PMMA in PC (as the interphase layer of the definite thickness).

Data on kinetics (interdiffusion coefficients) and thermodynamics (phase equiblibria) of polymers under investigation have been compared with rheological results obtained by squeeze flow method (Kotomin&Kulichikhin, 1996) - the concentration dependence of the Newtonian viscosity, presented in Fig. 10. The matter is that in both processes (diffusion and flow) a segmental motion takes place. In the case of diffusion it is stipulated by difference of chemical potentials, and in the case of rheology – applying the external force. That is why segments participating in the diffusion and the rheology can be different, but mechanisms are similar (Malkin&Chalykh, 1979), and joint consideration of data of both methods are undoubtedly valuable.

The concentration dependence of viscosity of PC-PMMA (Мw=1.5. 104) system at 240oC at majority compositions excluding specific domain is located higher of the logarithmic additivity line that indicates on active interaction of components (Kuleznev, 1980). At small content of PMMA the characteristic minimum was observed. At comparing with the phase diagram it is evident that its position is nearst binodal. The viscosity decrease can be explained by appearance of microemulsion with high fraction of interfacial layer with lower density compared with neat polymer melts (Kuleznev, 1980; Kuleznev&Kandyrin, 2000; Kuleznev, 1987). Presumably, this minimum is connected with meta-stable state of the system where a great number of heterophase fluctuations exists.

Fig. 10. Viscosity vs. composition curves for the system PC - PMMA (Мw=1.5. 104): 1 – experimental curve; 2 – line of logarithmic additivity; 3 – the curve, calculated using formulae (5). Vertical lines designate domains of solutions (a), meta-stable state (b) and biphasic (c).

The concentration dependence of viscosity in biphasic region was analyzed based on knowledge of the phase diagram. For this aim the empirical formulae was used (Kuleznev&Kandyrin, 2000):

of PMMA, PC and solution of PMMA in PC (as the interphase layer of the definite

Data on kinetics (interdiffusion coefficients) and thermodynamics (phase equiblibria) of polymers under investigation have been compared with rheological results obtained by squeeze flow method (Kotomin&Kulichikhin, 1996) - the concentration dependence of the Newtonian viscosity, presented in Fig. 10. The matter is that in both processes (diffusion and flow) a segmental motion takes place. In the case of diffusion it is stipulated by difference of chemical potentials, and in the case of rheology – applying the external force. That is why segments participating in the diffusion and the rheology can be different, but mechanisms are similar (Malkin&Chalykh, 1979), and joint consideration of data of both methods are

majority compositions excluding specific domain is located higher of the logarithmic additivity line that indicates on active interaction of components (Kuleznev, 1980). At small content of PMMA the characteristic minimum was observed. At comparing with the phase diagram it is evident that its position is nearst binodal. The viscosity decrease can be explained by appearance of microemulsion with high fraction of interfacial layer with lower density compared with neat polymer melts (Kuleznev, 1980; Kuleznev&Kandyrin, 2000; Kuleznev, 1987). Presumably, this minimum is connected with meta-stable state of the

104) system at 240oC at

104): 1 –

The concentration dependence of viscosity of PC-PMMA (Мw=1.5.

system where a great number of heterophase fluctuations exists.

Fig. 10. Viscosity vs. composition curves for the system PC - PMMA (Мw=1.5.

experimental curve; 2 – line of logarithmic additivity; 3 – the curve, calculated using formulae (5). Vertical lines designate domains of solutions (a), meta-stable state (b) and

The concentration dependence of viscosity in biphasic region was analyzed based on knowledge of the phase diagram. For this aim the empirical formulae was used

20 40 60 80

PMMA

a c b c a

 1 2 3

thickness).

undoubtedly valuable.

biphasic (c).

(Kuleznev&Kandyrin, 2000):

0

2

4

6

l g [Pa\*s]

$$\lg \eta = \nu\_2^2 \lg \eta\_2 + \left(1 - \nu\_2^2\right) \lg \eta\_{1'} \tag{5}$$

where *<sup>1</sup>* and *2* – viscosities of disperse phase and medium, respectively, *<sup>2</sup>* - the volume fraction of dispersion medium that for compatible polymers coincides with relative content of the component forming the medium. But if at least partial mutual dissolution proceeds, this coincidence is absent. It is why the *<sup>2</sup>* value was found from the phase diagram using the meanings of concentrations on binodal and "the lever rule". The calculated curve 3 (Fig. 11) coincides in limits of equipment error with the experimental one.

So, the couple PC – low molecular weight PMMA is a good example of polymer systems with amorphous phase equilibrium. Method of interferometry is very useful and fruitful for such kind of systems due to informativity and combination of joint approaches: kinetics of mass-transfer and thermodynamics of phase states. Increase of the mutual solubility with temperature indicates on UCST that could not be measured experimentally but was calculated using evolution of the interaction parameter with concentration.

Comparison of the phase diagram with concentration dependence of viscosity allowed us to observe local minimum of viscosity near the binodal (sooner in meta-stable region between binodal and spinodal where microemulsion appears as the first step of the phase decomposition) and to apply the empiric equation for calculation of the blends viscosity by means of accurate determination of the composition from binodal branches. Joint consideration of diffusion and rheological properties seems to be very fruitful for judging about structure, kinetic and thermodynamic behaviors of complex, multiphase polymer systems. This approach will be used further for analysis of systems with LC and crystalline equilibria.

Calculations of the interphase layers thickness depending on dimensions of drops and time, based on the knowledge of interdiffusion coefficients can be very useful for correct organization of technological process of mixing and, consequently, development of new materials with three-phase structure and new properties.

## **4. Crystalline and complex phase equilibria in polymer systems**

Crystalline equilibrium is inherent for polymer systems if at least one component is capable to crystallize. At cooling from the region of the total compatibility (isotropic solution) at definite temperature the system decomposes on two equilibrium phases: deposition of the crystalline component and its saturated solution into the other component.

The curve describing sum of transition (melting or crystallization) points corresponding to various compositions is named "liquidus". Higher of the liquidus line the single-phase region (liquid solution) exists, and lower the liquidus line – biphasic system (crystalline phase distributed in solution). Basic kinds of phase diagrams with crystalline equilibrium are shown in Fig. 11.

Cases a) and b) are typical for components with a weal interaction. If one component of such a system is amorphous and the other – crystalline, the melting point of a system as whole decreases monotonously as a result of dilution of the crystalline component by amorphous one (depression of melting points) as is seen in the graph a). In systems containing two components capable to form crystalline phases two liquidus line cross in the eutectic point

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 409

The maximum number of phase diagrams for polymer systems with crystalline equilibrium is devoted to polymer-solvent systems (Papkov, 1971; Tager, 1978; Papkov, 1981). One of the classical examples is the phase diagram of poly-p-phenyleneterephthalamide (PPTA) - H2SO4 schematically shown in Fig. 12. (Iovleva&Papkov, 1982; Papkov et al., 1986; Kulichikhin, 1989). But in the particular case of stiff-chain polymers solutions the LC phase appears at definite concentration of polymer separated from isotropic solutions by narrow biphasic "corridor". The situation with LC equilibrium will be considered later, but we should indicate here on the right part of the diagram where formation of the CS phase with congruent melting is realized. In this case the knowledge of the phase diagram was used for practical aim of choosing the most appropriate concentration of dopes and the temperature regime of spinning the super-strong fibers of Kevlar (Terlon) type (Papkov&Kulichikhin,

Fig. 12. Schematic image of the phase diagram for PPTA-H2SO4 system.

high-temperature region where chemical interaction becomes significant.

**4.1 Interaction between LC- and isotropic polyesters melts** 

Coming back to polymer-polymer interaction let us consider now several binary systems based on polyesters. These polymers are of interest from the viewpoint of mixing just to reach the new complex of properties hoping at least on partial physical compatibility. Simultaneously, we should keep in mind a possibility of chemical interaction of polyesters via mechanisms of acidolysis or alkoholysis which involve ester-ester exchange. So, in this case interdiffusion can be stipulated as chemical similarity as chemical interaction leading to formation of copolyesters. Let us accept that 280-290oC is the border temperature that separates the low-temperature region where physical interaction (diffusion) prevails, and

At low-temperature region two couples of polyesters have been studied. In the first one two components were in interaction: polydecamethyleneterephthaloil-bis-4-hydroxybenzoate LC polyester and polybutyleneterephthalate (PBT) (Kulichikhin et al., 1994). PBT is semi-

1977).

Fig. 11. Basic types of phase diagrams with crystalline equilibrium [5,6,9].

located in position lower than melting points of neat components (type b). Under the left liquidus curve up to *xe* point the equilibrium of the crystalline phase A and its saturated solution exists, and after *xe* – the equilibrium of the crystalline phase B with its saturated solution. In the temperature region lower of the eutectic point the biphasic domain consists of crystals A and B only.

Phase equilibria of types c) and d) are inherent for systems with polar components and their interaction leads to formation of crystalline additive compounds – crystal-solvates (CS). At congruent melting of CS (without its decomposition on components) the maximum melting point TmCS exists (type c). But if the energy of the additive compound formation is low, (incongruent melting) its decomposition takes place until reaching TmCS (type d). Under liquidus curve in region of compositions less than *xCS* there exist in equilibrium phases of CS of *xCS* composition and saturated solution (S) B in A (T>TmA) or crystals (C) A (T<TmA). Passing through *xCS* in the temperature region lower the eutectic point, the CS phase coexists with saturated solution of A in B.

Fig. 11. Basic types of phase diagrams with crystalline equilibrium [5,6,9].

of crystals A and B only.

with saturated solution of A in B.

located in position lower than melting points of neat components (type b). Under the left liquidus curve up to *xe* point the equilibrium of the crystalline phase A and its saturated solution exists, and after *xe* – the equilibrium of the crystalline phase B with its saturated solution. In the temperature region lower of the eutectic point the biphasic domain consists

Phase equilibria of types c) and d) are inherent for systems with polar components and their interaction leads to formation of crystalline additive compounds – crystal-solvates (CS). At congruent melting of CS (without its decomposition on components) the maximum melting point TmCS exists (type c). But if the energy of the additive compound formation is low, (incongruent melting) its decomposition takes place until reaching TmCS (type d). Under liquidus curve in region of compositions less than *xCS* there exist in equilibrium phases of CS of *xCS* composition and saturated solution (S) B in A (T>TmA) or crystals (C) A (T<TmA). Passing through *xCS* in the temperature region lower the eutectic point, the CS phase coexists The maximum number of phase diagrams for polymer systems with crystalline equilibrium is devoted to polymer-solvent systems (Papkov, 1971; Tager, 1978; Papkov, 1981). One of the classical examples is the phase diagram of poly-p-phenyleneterephthalamide (PPTA) - H2SO4 schematically shown in Fig. 12. (Iovleva&Papkov, 1982; Papkov et al., 1986; Kulichikhin, 1989). But in the particular case of stiff-chain polymers solutions the LC phase appears at definite concentration of polymer separated from isotropic solutions by narrow biphasic "corridor". The situation with LC equilibrium will be considered later, but we should indicate here on the right part of the diagram where formation of the CS phase with congruent melting is realized. In this case the knowledge of the phase diagram was used for practical aim of choosing the most appropriate concentration of dopes and the temperature regime of spinning the super-strong fibers of Kevlar (Terlon) type (Papkov&Kulichikhin, 1977).

Fig. 12. Schematic image of the phase diagram for PPTA-H2SO4 system.

Coming back to polymer-polymer interaction let us consider now several binary systems based on polyesters. These polymers are of interest from the viewpoint of mixing just to reach the new complex of properties hoping at least on partial physical compatibility. Simultaneously, we should keep in mind a possibility of chemical interaction of polyesters via mechanisms of acidolysis or alkoholysis which involve ester-ester exchange. So, in this case interdiffusion can be stipulated as chemical similarity as chemical interaction leading to formation of copolyesters. Let us accept that 280-290oC is the border temperature that separates the low-temperature region where physical interaction (diffusion) prevails, and high-temperature region where chemical interaction becomes significant.

## **4.1 Interaction between LC- and isotropic polyesters melts**

At low-temperature region two couples of polyesters have been studied. In the first one two components were in interaction: polydecamethyleneterephthaloil-bis-4-hydroxybenzoate LC polyester and polybutyleneterephthalate (PBT) (Kulichikhin et al., 1994). PBT is semi-

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 411

Fig. 14. Evolution of the interference patterns in the diffusion zone at cooling the system LC

that indicates on continuation of diffusion interaction in this time interval (Tereshin at al., 2000). So, in both cases the method of observation on one half of the interference pattern related with transparent polymer melt was developed. This approach could be very useful

So, in previous cases the experiments on optical interferometry were carried out up to temperature of 290oC. Presumably, this is lower than it is possible to expect active chemical interaction between two polyesters, especially if one of them is LCP. The matter is that due to orientation of LC domains onto interface and formation more dense boundary layer, both processes – diffusion and chemical interaction can be slowed. For isotropic polyesters the

polyester – PBT. Temperature: 310 (a), 270 (b, LC polyester crystallizes), 220 (c, PBT

for analysis of diffusion interaction between two polymers, of one of them is turbid.

crystallizes), 210 (d, the eutectic zone is lighted up), and 20oC (e).

situation with physical and chemical interactions should be more clear.

crystalline polymer. This pair has some specific features from the viewpoint of application of method of the optical interferometry. In the case of LC polyester-PBT couple the melt of the first polymer has LC ordering up to clearing (isotropization) point. This means that LC melt is not transparent that is why the evolution of interference bands during interaction of two above indicated melts was controlled from one side – transparent PBT melt only.

At room temperature both polymers are non-transparent. At 230oC PBT melts and intrinsic for it interference bands appear while LC polyester even in melt state remains dark. Bending of PBT interference bands in vicinity of the interface indicates on dissolution of LC polyester in PBT, and in concentration profile gently sloping section is seen. At T~290oC LC polyester becomes isotropic and the interference picture indicates on complete compatibility of components in these conditions. The inereference data allowed us to construct the phase diagram (Fig. 13).

Fig. 13. Phase diagram for the system LC polyester – PBT. I – isotropic solutions, II – LC state of LC polyester solutions in PBT, III crystalline state of PBT saturated solutions, IV – crystalline state of LC polyester.

There are two liquidus lines with eutectics at volume fraction of PBT=0.7. The presence of eutectics confirms by preservation of transparent zone for this composition up to 170oC, i.e. much lower of crystallization points of neat components and solutions of other compositions (Fig. 14). In heating cycle the eutectic zone is clearing the first and then zones of other compositions. This situation is very similar to the classical case of crystalline equilibrium expressed by pattern b) in Fig. 11.

The second system was a pair of polyehyleneterephthalate (PET) and LC copolyester of PET with p-hydroxybenzoic acid (PET/HBA) one half of the interference picture is dark, but sloping the interference bands of transparent PET near interface indicates on partial dissolution of PET/HBA anisotropic melt in PET at 290oC forming interphase layer with a thickness increasing in time. Unfortunately, the clearing point of PET/HBA copolyester is higher than 350oC and higher than decomposition point of this polymer. That is why we cannot say definitely about degree of compatibility, but an increase of bending is observed

crystalline polymer. This pair has some specific features from the viewpoint of application of method of the optical interferometry. In the case of LC polyester-PBT couple the melt of the first polymer has LC ordering up to clearing (isotropization) point. This means that LC melt is not transparent that is why the evolution of interference bands during interaction of two above indicated melts was controlled from one side – transparent PBT melt only.

At room temperature both polymers are non-transparent. At 230oC PBT melts and intrinsic for it interference bands appear while LC polyester even in melt state remains dark. Bending of PBT interference bands in vicinity of the interface indicates on dissolution of LC polyester in PBT, and in concentration profile gently sloping section is seen. At T~290oC LC polyester becomes isotropic and the interference picture indicates on complete compatibility of components in these conditions. The inereference data allowed us to construct the phase

T, °C

Fig. 13. Phase diagram for the system LC polyester – PBT. I – isotropic solutions, II – LC state of LC polyester solutions in PBT, III crystalline state of PBT saturated solutions, IV –

There are two liquidus lines with eutectics at volume fraction of PBT=0.7. The presence of eutectics confirms by preservation of transparent zone for this composition up to 170oC, i.e. much lower of crystallization points of neat components and solutions of other compositions (Fig. 14). In heating cycle the eutectic zone is clearing the first and then zones of other compositions. This situation is very similar to the classical case of crystalline

φPES

The second system was a pair of polyehyleneterephthalate (PET) and LC copolyester of PET with p-hydroxybenzoic acid (PET/HBA) one half of the interference picture is dark, but sloping the interference bands of transparent PET near interface indicates on partial dissolution of PET/HBA anisotropic melt in PET at 290oC forming interphase layer with a thickness increasing in time. Unfortunately, the clearing point of PET/HBA copolyester is higher than 350oC and higher than decomposition point of this polymer. That is why we cannot say definitely about degree of compatibility, but an increase of bending is observed

diagram (Fig. 13).

crystalline state of LC polyester.

equilibrium expressed by pattern b) in Fig. 11.

Fig. 14. Evolution of the interference patterns in the diffusion zone at cooling the system LC polyester – PBT. Temperature: 310 (a), 270 (b, LC polyester crystallizes), 220 (c, PBT crystallizes), 210 (d, the eutectic zone is lighted up), and 20oC (e).

that indicates on continuation of diffusion interaction in this time interval (Tereshin at al., 2000). So, in both cases the method of observation on one half of the interference pattern related with transparent polymer melt was developed. This approach could be very useful for analysis of diffusion interaction between two polymers, of one of them is turbid.

So, in previous cases the experiments on optical interferometry were carried out up to temperature of 290oC. Presumably, this is lower than it is possible to expect active chemical interaction between two polyesters, especially if one of them is LCP. The matter is that due to orientation of LC domains onto interface and formation more dense boundary layer, both processes – diffusion and chemical interaction can be slowed. For isotropic polyesters the situation with physical and chemical interactions should be more clear.

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 413

Analogous conclusions concerning the incompatibility of the physical blend and the decisive role of trans-esterification in the compatibility of PET and PEN were reported in (Ihm et al., 1996). If the degree of interchain exchange is greater than 50%, blends do not crystallize and a single glass transition point is observed only. The rate of trans-esterification proves to be temperature- and time-dependent; however, it is unaffected by the blend composition.

Thus, the above seemingly simple and evident approach to modifying the properties of PET, which involves its blending with a certain amount of PEN, is complicated by the chemical interaction of these two polyesters. However, kinetics of physical and chemical processes and its determining factors, allowing us to estimate all stages of interaction are still unknown completely. The compatibility of molten PET and PEN calls for further investigation, and in this Chapter we focus our attention just on this problem (Makarova,

**PEN PET** 

**200** 

Fig. 15. Interferograms of PET - PEN system at 290oC in 5 (a) and 10 min (b) after contact. The pattern (b) corresponds to the combined specimen cooled to ambient temperature from

At contacting PET and PEN after the melting of PET (~260oC) the weak bending of the interference bands occur that is caused by penetration to PET of macromolecules of still crystalline PEN (Tm=266oC). A solubility of PEN in PET in such conditions is not more 5- 10%. After PEN melting a lot of sloped bands appear from both sides of the interface, and

2005).

**а**

**c** 

**b** 

290oC.

## **4.2 Pair of PET with polyethylenenaphtalate (PEN)**

In the case of PET-PEN pair a special attention was devoted to superposition of two processes: interdiffusion and typical for polyesters interchain chemical reaction of the esterester exchange. These components are miscible completely, finally forming copolyester, and the main problem consisted in comparison of diffusion and chemical reaction rates. In our opinion, the first stage of interaction of molten polyesters is interdiffusion, and the second one – chemical trans-esterification. Transition from one stage to another is reflected on kinetics of concentration profile suffering significant change that time.

PET is a large-capacity thermoplastic used for the production of fibers, films, engineering plastics, and packaging articles. PET articles demonstrate excellent mechanical properties, with their manufacture technique being relatively simple. Nevertheless, some properties of PET do not suit consumers. Specifically, this refers to the phase state of PET, which belongs to the family of semi-crystalline polymers and possesses a high crystallinity degree (40– 45%). Crystallites resulting from the rapid cooling of thin-walled articles are too small in size to affect the optical properties of PET articles. However, in the case of bulky thickwalled products, crystalline structures forming in the course of cooling lead to the loss of transparency. As a consequence, the scope of potential applications of PET appears to be limited. Moreover, a relatively low heat resistance (the glass transition temperature of PET is ~80°C) provides no way of using PET articles at high temperatures.

It is advisable to use as a partner for PET poly(ethylenenaphthalate) (PEN) both for physical (interdiffusion at relatively low melt temperatures) and chemical interaction (at high temperatures). The most universally adopted mechanism of trans-esterification relies on the decisive role of hydroxyl end terminal groups that participate in alcoholysis and acidolysis reactions and enter to neighboring chains of polyesters (Fakirov, 1999). The block copolymers thus produced improve the compatibility of polyesters with different chemical structures. Subsequently, neighboring polyester chains undergo scission and recombination reactions (direct ester–ester exchange), making copolymers random in structure (Stewart et al., 1993).

Over many years, it has been agreed that this reaction is responsible for compatibility of blend components (Litmanovich et al., 2002; Ihm et al., 1996). However, Guo&Brittain, 1998, revealed that trans-esterification is not prerequisite to compatibility. According to solid-state NMR spectroscopy, a mixture of melts became compatible during the first 1.5 min of injection molding and only after that could signs of the trans-esterification reaction be detected. The authors of the cited work proposed that, for the chemical interaction to occur, chains should be situated close to each other (at least within the action of dispersion forces). It is natural that for the incompatible blends of molten polyesters, the existence of interfaces limits coming together of potentially reactive macromolecules.

Thus, polymers making up the blend should form single-phase regions so that the effective chemical interaction can take place. Okamato&Kotaka, 1997, have demonstrated that the rapid cooling (quenching) of a PET/PEN blend being extruded at 280°C leads to the heterophase melt morphology (probably, because of spinodal phase decomposition). Further melting and annealing initiate the occurrence of trans-esterification at the boundaries of domains and their gradual disappearance as a result of compatibility. The use of a random PET–PEN copolymer as a third component accelerates compatibility.

In the case of PET-PEN pair a special attention was devoted to superposition of two processes: interdiffusion and typical for polyesters interchain chemical reaction of the esterester exchange. These components are miscible completely, finally forming copolyester, and the main problem consisted in comparison of diffusion and chemical reaction rates. In our opinion, the first stage of interaction of molten polyesters is interdiffusion, and the second one – chemical trans-esterification. Transition from one stage to another is reflected on

PET is a large-capacity thermoplastic used for the production of fibers, films, engineering plastics, and packaging articles. PET articles demonstrate excellent mechanical properties, with their manufacture technique being relatively simple. Nevertheless, some properties of PET do not suit consumers. Specifically, this refers to the phase state of PET, which belongs to the family of semi-crystalline polymers and possesses a high crystallinity degree (40– 45%). Crystallites resulting from the rapid cooling of thin-walled articles are too small in size to affect the optical properties of PET articles. However, in the case of bulky thickwalled products, crystalline structures forming in the course of cooling lead to the loss of transparency. As a consequence, the scope of potential applications of PET appears to be limited. Moreover, a relatively low heat resistance (the glass transition temperature of PET is

It is advisable to use as a partner for PET poly(ethylenenaphthalate) (PEN) both for physical (interdiffusion at relatively low melt temperatures) and chemical interaction (at high temperatures). The most universally adopted mechanism of trans-esterification relies on the decisive role of hydroxyl end terminal groups that participate in alcoholysis and acidolysis reactions and enter to neighboring chains of polyesters (Fakirov, 1999). The block copolymers thus produced improve the compatibility of polyesters with different chemical structures. Subsequently, neighboring polyester chains undergo scission and recombination reactions (direct ester–ester exchange), making copolymers random in structure (Stewart et

Over many years, it has been agreed that this reaction is responsible for compatibility of blend components (Litmanovich et al., 2002; Ihm et al., 1996). However, Guo&Brittain, 1998, revealed that trans-esterification is not prerequisite to compatibility. According to solid-state NMR spectroscopy, a mixture of melts became compatible during the first 1.5 min of injection molding and only after that could signs of the trans-esterification reaction be detected. The authors of the cited work proposed that, for the chemical interaction to occur, chains should be situated close to each other (at least within the action of dispersion forces). It is natural that for the incompatible blends of molten polyesters, the existence of interfaces

Thus, polymers making up the blend should form single-phase regions so that the effective chemical interaction can take place. Okamato&Kotaka, 1997, have demonstrated that the rapid cooling (quenching) of a PET/PEN blend being extruded at 280°C leads to the heterophase melt morphology (probably, because of spinodal phase decomposition). Further melting and annealing initiate the occurrence of trans-esterification at the boundaries of domains and their gradual disappearance as a result of compatibility. The use

of a random PET–PEN copolymer as a third component accelerates compatibility.

**4.2 Pair of PET with polyethylenenaphtalate (PEN)** 

kinetics of concentration profile suffering significant change that time.

~80°C) provides no way of using PET articles at high temperatures.

limits coming together of potentially reactive macromolecules.

al., 1993).

Analogous conclusions concerning the incompatibility of the physical blend and the decisive role of trans-esterification in the compatibility of PET and PEN were reported in (Ihm et al., 1996). If the degree of interchain exchange is greater than 50%, blends do not crystallize and a single glass transition point is observed only. The rate of trans-esterification proves to be temperature- and time-dependent; however, it is unaffected by the blend composition.

Thus, the above seemingly simple and evident approach to modifying the properties of PET, which involves its blending with a certain amount of PEN, is complicated by the chemical interaction of these two polyesters. However, kinetics of physical and chemical processes and its determining factors, allowing us to estimate all stages of interaction are still unknown completely. The compatibility of molten PET and PEN calls for further investigation, and in this Chapter we focus our attention just on this problem (Makarova, 2005).

Fig. 15. Interferograms of PET - PEN system at 290oC in 5 (a) and 10 min (b) after contact. The pattern (b) corresponds to the combined specimen cooled to ambient temperature from 290oC.

At contacting PET and PEN after the melting of PET (~260oC) the weak bending of the interference bands occur that is caused by penetration to PET of macromolecules of still crystalline PEN (Tm=266oC). A solubility of PEN in PET in such conditions is not more 5- 10%. After PEN melting a lot of sloped bands appear from both sides of the interface, and

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 415

**PET PEN copolyester** 

**200**

**200** 

Fig. 17. Interferograms of PET – PEN - copolyester PET/HBA=80/20 at 268 (a) and 272oC in

Combination of PEN with mechanical blend PET/PEN=90/10 and copolyester containing 10% of HNA at 280oC (Fig. 18) has shown an absence of interfaces from both sides. However at duration of observation 20 min the bending of interference bands preserves from a side of the blend. Disappearance of such bending from a side of the copolyester indicates on fast

**blend PEN copolymer**

Fig. 18. Interferogram of the combined system: mechanical blend PET/PEN=90/10 – PEN –

10 (b) and 30 min (c).

redistribution of concentrations in this diffusion zone.

**а**

**b**

**c**

copolymer PET/HNA=90/10 at 280oC in 20 min after contact.

number of them increases with time (Fig. 15a). In 10 min (this time depends on temperature) the interface disappears completely (Fig. 15b). After cooling the cell to ambient temperature the region in vicinity of the former interface remains light (Fig. 15c). Presumably, either amorphous solution of components is formed in this region (eutectic point) or amorphous transparent copolyester.

Presence of the definite time lag (or "time barrier") as well continuous increasing a number of interference bands in isothermal conditions is not typical for the case of the physical compatibility, since at full compatibility the interface is absent already at first contacting of components. So, such behavior can be caused by simultaneous passage of interdiffusion and chemical interaction, promoting compatibility of components in time.

Fig. 16. Movement of iso-concentration fronts for the system PET – PEN at 300oC.

Plotted on the base of interferograms, concentration profiles in the transient zone are asymmetric from the PEN side that witnesses about prevail penetration of PET into PEN. Movement of the iso-concentration plains *x()* calculated from concentration profiles, follows to diffusion laws (*x~t0.5*) on the initial sections only (Fig. 16) which length does not exceed 10 min. This time is comparable with disappearance of the interface. The further nonlinearity can be connected with chemical interaction. In the figure this stage is noted as "chemical diffusion" because traditional diffusion rules do not work for chemically different elementary units.

For analysis of the difference in behavior of copolymers and complementary blends the interdiffusion was investigated on specimens consisting of three components: PET, PEN and copolyester or blend of the same composition. Copolyesters were synthesized via chemical reaction of PET with 2,6-hydroxynaphthoic acid (HNA) (Makarova et al, 2005). For the combined specimen "PET-PEN-copolyester" the interferograms are shown in Fig. 17. After PEN melting the interface is presented only from the PET side, while from copolyester side the continuous concentration profile is observed. Probably, the above discussed "time lag" is stipulated by necessity of copolyesters formation accelerating the further compatibility.

number of them increases with time (Fig. 15a). In 10 min (this time depends on temperature) the interface disappears completely (Fig. 15b). After cooling the cell to ambient temperature the region in vicinity of the former interface remains light (Fig. 15c). Presumably, either amorphous solution of components is formed in this region (eutectic point) or amorphous

Presence of the definite time lag (or "time barrier") as well continuous increasing a number of interference bands in isothermal conditions is not typical for the case of the physical compatibility, since at full compatibility the interface is absent already at first contacting of components. So, such behavior can be caused by simultaneous passage of interdiffusion and

> **PET** 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1

*)* calculated from concentration profiles,

0 2 4 6 8 10 12

Plotted on the base of interferograms, concentration profiles in the transient zone are asymmetric from the PEN side that witnesses about prevail penetration of PET into PEN.

follows to diffusion laws (*x~t0.5*) on the initial sections only (Fig. 16) which length does not exceed 10 min. This time is comparable with disappearance of the interface. The further nonlinearity can be connected with chemical interaction. In the figure this stage is noted as "chemical diffusion" because traditional diffusion rules do not work for chemically different

For analysis of the difference in behavior of copolymers and complementary blends the interdiffusion was investigated on specimens consisting of three components: PET, PEN and copolyester or blend of the same composition. Copolyesters were synthesized via chemical reaction of PET with 2,6-hydroxynaphthoic acid (HNA) (Makarova et al, 2005). For the combined specimen "PET-PEN-copolyester" the interferograms are shown in Fig. 17. After PEN melting the interface is presented only from the PET side, while from copolyester side the continuous concentration profile is observed. Probably, the above discussed "time lag" is stipulated by necessity of copolyesters formation accelerating the further compatibility.

**t 1/2, min1/2**

Fig. 16. Movement of iso-concentration fronts for the system PET – PEN at 300oC.

"chemical" diffusion

chemical interaction, promoting compatibility of components in time.

transparent copolyester.


Movement of the iso-concentration plains *x(*

elementary units.




"physical" diffusion

0,0

0,2

0,4

**x, mm**

Fig. 17. Interferograms of PET – PEN - copolyester PET/HBA=80/20 at 268 (a) and 272oC in 10 (b) and 30 min (c).

Combination of PEN with mechanical blend PET/PEN=90/10 and copolyester containing 10% of HNA at 280oC (Fig. 18) has shown an absence of interfaces from both sides. However at duration of observation 20 min the bending of interference bands preserves from a side of the blend. Disappearance of such bending from a side of the copolyester indicates on fast redistribution of concentrations in this diffusion zone.

Fig. 18. Interferogram of the combined system: mechanical blend PET/PEN=90/10 – PEN – copolymer PET/HNA=90/10 at 280oC in 20 min after contact.

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 417

PET – PEN 2,0

Table 3. Interdiffusion coefficients for different interacting components at 280oC.

d, Т=2800С 1 0,1 s 10 10 s 100 20 min 1000 30 h

Table 4. Time of complete dissolution of PEN drop of various size in PET.

macromolecules of different sorts. At their closing on a distance permitted for covalent exchange, the chemical process starts leading to formation of copolyesters in vicinity of interface. Further, copolyesters promote as diffusion interaction, as propagation of chemical

We can estimate the dissolution time of a particle of PEN or PET in an unrestricted volume of another polymer. Calculations were done in the same way as in the case of PC-PMMA pair (Avdeev, 2001; Makarova, 2007). The corresponding data are presented in Table 4.

These data show that in any industrial mixing method only droplets with a size of <10 m can be dissolved completely. For larger droplets the interphase layer of the definite thickness will be formed consisting of mutual solutions of homopolymers and copolymers.

Liquid-crystalline state is stable, thermodynamically equilibrium and engages a gap between amorphous and crystalline states (Papkov et al., 1974; Papkov&Kulichikhin, 1977; Collyer, 1996; Kulichikhin, 2000]. In LC state macromolecules form uni-dimensional or twodimensional order that distinguishes it from full disorder in amorphous state and threedimensional order in crystalline state. Such state is named often as anisotropic since presence of the order in macromolecular arrangement leads to appearance of anisotropy of many physical properties, partially optical, rheological (majority of these systems are

LC phase in polymers can be realized either in melt state, if 3D crystalline order disappears not completely, but step-by-step destruction at melting (thermotropic LC polymers) or in

At cooling the system will contain 3 or 4 phases, influencing its exploring properties.

PET – blend PET/PEN=80/20

PEN – blend PET/PEN=90/10

PEN – blend PET/PEN=80/20

PET – copolyester PET/HNA=80/20

PEN – copolyester PET/HBA=90/10

PEN – copolyester PET/HBA=80/20

reaction front in depths of both reacting components.

**5. LC phase equilibrium in polymer systems** 

capable to flow) and many others.

Systems DV·1012, м2/с

0,2 3,0

3,5 7,0 0,5 4,5

In some cases, at slow cooling as PET itself as the combined specimen PET – PEN the formation of tree-like morphology was observed similar to fragments of dendrite spherolites. "Branches" of PET were more monolithic and dense compared with graceful branches formed in PET containing 5% of PEN (Fig. 19). This fact can be explained by decrease of crystallinity degree of PET at adding minor amount of PEN and formation of amorphous copolyester. In this circumstances PET crystallites can grow and form spherolites.

Fig. 19. Micrograph of the diffusion zone at contact of PET with mechanical blend PET/5% of PEN.

Using concentration profiles in the diffusion zone PET-PEN, interdiffusion coefficients were calculated at different temperatures. The Dv values in this temperature range lie in diapason 1.4. 10-125.4. 10-11 m2/s. The apparent activation energy of the diffusion process is closed to 410 kJ/mole. It is likely that so high value of the temperature coefficient can be explained by additional input of trans-esterification into the process of mutual penetration of components. Since dependence of Dv on concentration is rather weak, further calculations for various compositions at one temperature were carried out assuming independence of Dv on composition. The algorithm of calculations was the same as for PC-PMMA system using the probability diagram (Crank, 1956; Avdeev, 1990).

Calculated values of Dv initial polyesters and their combination with mechanical blends and copolyesters of complementary compositions are presented in Table 3. These data reflect more fast interpenetration of homo- and copolyesters compared with blends. The corresponding Dv values differ in 2-10 times.

It is rather strange that at contact of homopolyesters with mechanical blends the diffusion penetration is less intensive even than for homopolymers. Since blends were obtained at temperatures higher 280oC during 10 min and more, it is reasonable to expect the definite fraction of copolyesters formation. Nevertheless their interaction with homopolymers is not too intensive as for pure copolyesters. Probably, this can be explained by chaotic morphology of blends that leads to formation of heterogeneous interfaces and to interaction with homopolymer melts different species. So, presence in the mixture of copolyesters not ever accelerates the interpenetration. They should be just at interface as in the case of interaction of homo- and copolyesters.

The hypothetic mechanism of the general interaction in polyesters should be the following. First of all, the interdiffusion between components proceeds resulting in interpenetration of

In some cases, at slow cooling as PET itself as the combined specimen PET – PEN the formation of tree-like morphology was observed similar to fragments of dendrite spherolites. "Branches" of PET were more monolithic and dense compared with graceful branches formed in PET containing 5% of PEN (Fig. 19). This fact can be explained by decrease of crystallinity degree of PET at adding minor amount of PEN and formation of amorphous copolyester. In this circumstances PET crystallites can grow and form

**PET blend** 

Fig. 19. Micrograph of the diffusion zone at contact of PET with mechanical blend PET/5%

Using concentration profiles in the diffusion zone PET-PEN, interdiffusion coefficients were calculated at different temperatures. The Dv values in this temperature range lie in diapason

410 kJ/mole. It is likely that so high value of the temperature coefficient can be explained by additional input of trans-esterification into the process of mutual penetration of components. Since dependence of Dv on concentration is rather weak, further calculations for various compositions at one temperature were carried out assuming independence of Dv on composition. The algorithm of calculations was the same as for PC-PMMA system using

Calculated values of Dv initial polyesters and their combination with mechanical blends and copolyesters of complementary compositions are presented in Table 3. These data reflect more fast interpenetration of homo- and copolyesters compared with blends. The

It is rather strange that at contact of homopolyesters with mechanical blends the diffusion penetration is less intensive even than for homopolymers. Since blends were obtained at temperatures higher 280oC during 10 min and more, it is reasonable to expect the definite fraction of copolyesters formation. Nevertheless their interaction with homopolymers is not too intensive as for pure copolyesters. Probably, this can be explained by chaotic morphology of blends that leads to formation of heterogeneous interfaces and to interaction with homopolymer melts different species. So, presence in the mixture of copolyesters not ever accelerates the interpenetration. They should be just at interface as in the case of

The hypothetic mechanism of the general interaction in polyesters should be the following. First of all, the interdiffusion between components proceeds resulting in interpenetration of

10-11 m2/s. The apparent activation energy of the diffusion process is closed to

**200** 

spherolites.

of PEN.

10-125.4.

the probability diagram (Crank, 1956; Avdeev, 1990).

corresponding Dv values differ in 2-10 times.

interaction of homo- and copolyesters.

1.4.


Table 3. Interdiffusion coefficients for different interacting components at 280oC.

macromolecules of different sorts. At their closing on a distance permitted for covalent exchange, the chemical process starts leading to formation of copolyesters in vicinity of interface. Further, copolyesters promote as diffusion interaction, as propagation of chemical reaction front in depths of both reacting components.

We can estimate the dissolution time of a particle of PEN or PET in an unrestricted volume of another polymer. Calculations were done in the same way as in the case of PC-PMMA pair (Avdeev, 2001; Makarova, 2007). The corresponding data are presented in Table 4.


Table 4. Time of complete dissolution of PEN drop of various size in PET.

These data show that in any industrial mixing method only droplets with a size of <10 m can be dissolved completely. For larger droplets the interphase layer of the definite thickness will be formed consisting of mutual solutions of homopolymers and copolymers. At cooling the system will contain 3 or 4 phases, influencing its exploring properties.

## **5. LC phase equilibrium in polymer systems**

Liquid-crystalline state is stable, thermodynamically equilibrium and engages a gap between amorphous and crystalline states (Papkov et al., 1974; Papkov&Kulichikhin, 1977; Collyer, 1996; Kulichikhin, 2000]. In LC state macromolecules form uni-dimensional or twodimensional order that distinguishes it from full disorder in amorphous state and threedimensional order in crystalline state. Such state is named often as anisotropic since presence of the order in macromolecular arrangement leads to appearance of anisotropy of many physical properties, partially optical, rheological (majority of these systems are capable to flow) and many others.

LC phase in polymers can be realized either in melt state, if 3D crystalline order disappears not completely, but step-by-step destruction at melting (thermotropic LC polymers) or in

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 419

TLC-I. In addition appearance of crystalline phase is possible in this region and equilibrium

Roughly speaking, the phase diagram for polymer systems with LC equilibrium is combination of LC-liquidus line and binodal. With increase of macromolecular flexibility the critical concentration growths and the width of biphasic domain extends. In the limiting case this phase diagram transforms to state diagram typical for amorphous equilibrium

For several polymer systems the superposition is possible of the different types of phase equilibria in various combinations: amorphous and crystalline, amorphous and LC, LC and crystalline. The principle of the mutual independence of at least two versions of phase equilibria is fulfilled (Papkov&Kulichikhin, 1977; Papkov, 1981). In a system non-equlibrium in relation to one kind of phase equilibrium, the other kind can be realized. The sequence of phase states exchange depends on an ordering degree of forming phase. The higher ordering level, the slower a process of the new phase nuclei formation proceeds. The fastest case – formation of amorphous phases since for this case just fluctuating collision of molecules is enough without their any ordering. LC phase is more ordered that is why its nuclei form slower. Much more inhibition is observed for creation of a crystalline phase.

Do not discuss in details different partial versions of superposition of different kinds of phase equilibria there is sense to show one of the most complex to present time phase diagrams (Fig. 21). It combines LC and crystalline (CS) equilibria, as well as two binodals

Fig. 21. Superposition of different kinds of phase equilibria for a system polymer-solvent.

semi-stiff-chain polymers. Hydroxypropylcellulose (HPC) is one of such polymers.

In principle, an existence in one polymer a capability to form both thermotropic and lyotropic phases meets very rarely because of high values of TLC-I, lying higher decomposition points of polymers or difficulty to obtain very concentrated solutions for

(LC+C) – LC should be considered.

(biphasic part disappears completely).

with UCST and LCST (Iovleva et al., 1989).

solutions of stiff-chain polymers (lyotropic LC state). In the last case LC equilibrium takes place at full compatibility of components. Formation of LC solutions is stipulated predominantly by geometric asymmetry of rigid macromolecules which can engage any position in solution volume up to definite concentration only. After critical concentration adding new macromolecule is possible at condition of partial (in the beginning) or full ordering of other macromolecules. Using a term "partial ordering" means that the transition from isotropic to LC state along concentration axis passes via biphasic region where isotropic and LC state coexists.

The general view of the state diagram for solution of stiff-chain polymer with rod-like macromolecules, based on theoretical paper of Flory (Flory, 1956) and Papkov (Papkov et al., 1974; Papkov, 1992) is shown in Fig. 20. The diagram contains a combination of narrow and wide biphasic regions separating isotropic and anisotropic solutions. According to Flory, the critical concentration of LC phase appearance is related with molecular stiffness by equation:

$$
\rho\_2^\* \approx \frac{8}{\pi} \left( 1 - \frac{2}{\pi} \right) \tag{6}
$$

where *x* is asymmetry degree of macromolecules or, in common case, the ratio of the Kuhn segment to diameter *<sup>A</sup> x <sup>d</sup>* .

Fig. 20. Theoretical phase diagram for solutions of stiff-chain polymers according to (Flory, 1956) and (Papkov, 1992)

For rod-like macromolecules the location of the narrow biphasic region practically does not depend on temperature. But decrease of macromolecular stiffness leads to increase of the critical concentration \* 2 and to appearance of significant slope of the equilibrium line T\*( \* <sup>2</sup> ). Papkov, 1992, hypothetically extended this line in direction of 100% polymer (dotted line in Fig. 20). According to his opinion, the width of the biphasic region has to be narrower at closing to 100% polymer phase, and in a limiting case crosses the temperature axis inherent for polymer. The crossing point is the temperature of thermotropic transition, i.e.

solutions of stiff-chain polymers (lyotropic LC state). In the last case LC equilibrium takes place at full compatibility of components. Formation of LC solutions is stipulated predominantly by geometric asymmetry of rigid macromolecules which can engage any position in solution volume up to definite concentration only. After critical concentration adding new macromolecule is possible at condition of partial (in the beginning) or full ordering of other macromolecules. Using a term "partial ordering" means that the transition from isotropic to LC state along concentration axis passes via biphasic region where

The general view of the state diagram for solution of stiff-chain polymer with rod-like macromolecules, based on theoretical paper of Flory (Flory, 1956) and Papkov (Papkov et al., 1974; Papkov, 1992) is shown in Fig. 20. The diagram contains a combination of narrow and wide biphasic regions separating isotropic and anisotropic solutions. According to Flory, the critical concentration of LC phase appearance is related with molecular stiffness

> 8 2 <sup>1</sup> *x x*

(6)

where *x* is asymmetry degree of macromolecules or, in common case, the ratio of the Kuhn

Fig. 20. Theoretical phase diagram for solutions of stiff-chain polymers according to (Flory,

For rod-like macromolecules the location of the narrow biphasic region practically does not depend on temperature. But decrease of macromolecular stiffness leads to increase of the

<sup>2</sup> ). Papkov, 1992, hypothetically extended this line in direction of 100% polymer (dotted line in Fig. 20). According to his opinion, the width of the biphasic region has to be narrower at closing to 100% polymer phase, and in a limiting case crosses the temperature axis inherent for polymer. The crossing point is the temperature of thermotropic transition, i.e.

2 and to appearance of significant slope of the equilibrium line

\* 2

isotropic and LC state coexists.

segment to diameter *<sup>A</sup>*

1956) and (Papkov, 1992)

critical concentration \*

T\*( \* 

*x <sup>d</sup>* .

by equation:

TLC-I. In addition appearance of crystalline phase is possible in this region and equilibrium (LC+C) – LC should be considered.

Roughly speaking, the phase diagram for polymer systems with LC equilibrium is combination of LC-liquidus line and binodal. With increase of macromolecular flexibility the critical concentration growths and the width of biphasic domain extends. In the limiting case this phase diagram transforms to state diagram typical for amorphous equilibrium (biphasic part disappears completely).

For several polymer systems the superposition is possible of the different types of phase equilibria in various combinations: amorphous and crystalline, amorphous and LC, LC and crystalline. The principle of the mutual independence of at least two versions of phase equilibria is fulfilled (Papkov&Kulichikhin, 1977; Papkov, 1981). In a system non-equlibrium in relation to one kind of phase equilibrium, the other kind can be realized. The sequence of phase states exchange depends on an ordering degree of forming phase. The higher ordering level, the slower a process of the new phase nuclei formation proceeds. The fastest case – formation of amorphous phases since for this case just fluctuating collision of molecules is enough without their any ordering. LC phase is more ordered that is why its nuclei form slower. Much more inhibition is observed for creation of a crystalline phase.

Do not discuss in details different partial versions of superposition of different kinds of phase equilibria there is sense to show one of the most complex to present time phase diagrams (Fig. 21). It combines LC and crystalline (CS) equilibria, as well as two binodals with UCST and LCST (Iovleva et al., 1989).

Fig. 21. Superposition of different kinds of phase equilibria for a system polymer-solvent.

In principle, an existence in one polymer a capability to form both thermotropic and lyotropic phases meets very rarely because of high values of TLC-I, lying higher decomposition points of polymers or difficulty to obtain very concentrated solutions for semi-stiff-chain polymers. Hydroxypropylcellulose (HPC) is one of such polymers.

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 421

In parallel, the viscosity of solutions of different concentrations was measured. This information can be very useful for understanding not only compositions of various phases,

For all systems under consideration, an increase in the concentration of HPC is accompanied by formation of the LC phase; in some cases, this process is complicated by amorphous separation of the system. In addition, for a number of systems, the CS phase can form in the high-concentration range. It is reasonable to examine various systems separately and then to generalize the data on phase equilibrium and rheological properties of HPC-based solutions. Glycols are among the solvents used in this study. Let us first all analyze the phase equilibrium in low-molecular-weight liquids and oligomers containing terminal hydroxyl

The interferograms measured at 75oC (Fig. 22) exhibit two phase boundaries (dotted lines): the first boundary separates isotropic and two-phase solutions, while the second boundary separates fully LC solutions from the crystalline (or the CS) phase. The presence of the LC phase is confirmed by the polarization microscopy data. Thus, under crossed polarizers the region of the LC phase of 60% solution exhibits a characteristic rainbow luminescence typical for cholesteric liquid crystals. Cellulose derivatives containing the asymmetric carbon atom (the chiral center) form just the cholesteric mesophase. As temperature is increased, the fully LC solution transforms into the two-phase region, in which LC and isotropic phases coexist. Given this, the solution gradually darkens and, after full

**HPC** 

Fig. 22. Interferogram of the transition zone for the HPC-PG system at 75oC. The phase

corridor separating the two-phase region and the region of fully LC solutions.

The resulting phase diagram is the superposition of LC and crystalline equilibria (Fig. 23). The results of polarization microscopy measurements made it possible to refine position of the LC liquidus and to supplement the diagram with the second boundary of the two-phase

but their properties as well.

**5.1.1 HPC-PG system** 

isotropization, the field of vision becomes dark.

boundaries are marked with dotted line.

groups.

## **5.1 LC and complex phase equilibria in hydroxypropylcellulose solutions**

So, HPC is among few polymers capable of forming both thermotropic and lyotropic LC phases. This is why HPC, as none other natural polymer, is accessible for analysis of phase equilibrium both in melt and solutions in a wide concentration range and in the region of easily achievable temperatures. This situation is primarily explained by the chemical structure of HPC, which contains primary and secondary hydroxypropyl and residual hydroxyl group moieties.

Therefore, a system of H- bonds inherent for cellulose is destabilized and the level of intermolecular interaction is reduced. As a result, the polymer shows good solubility in solvents of various nature. The above features of the chemical structure are responsible for a rather high skeletal rigidity of HPC macromolecules (the Kuhn segment length is ~13 nm in dimethylaceteamide (Kulichikhin&Golova, 1985), which ensures transition to the LC melt at rather low temperatures (~120oC) with an isotropization (clearing) temperature of ~205oC (Makarova, 2007; Makarova et al., 2007).

HPC of the Klucel EF brand (Hercules-Aqualon, Unites States) with Mw= 8. 104 was used. Solvents were: two oligomeric PEG samples (Spectrum, Unites States) with Mw=400 (PEG 400) and 1500 (PEG 1500), distilled water, propylene glycol (PG) (Labtex, Russia), triethyl citrate (TEC) (Morflex, United States), and DMSO (Gaylord Chemical Corp., United States).


Table 5. Physical-chemical characteristics of solvents used.

Table 5 lists some characteristics of the used solvents (Woolley, 1974; Kolotyrkin, 1974; Allan, 1990; Geller et al., 1996; Uusi-Penttila, 1997; Kulvinder, 2000; Gallyamov, 2009; Mali, 2007) which may be useful for analysis of phase equilibria in HPC--solvent systems. Solutions of HPC with a HPC concentration of 30, 35, 40, 45, 50, 55, 60, 70, 75, and 80 wt % were prepared in various solvents in accordance with (Fischer et al., 1995).

The phase equilibrium in solutions was studied by the microinterference method at 18- 210oC which renders it possible to analyze concentration profiles in the diffusion zone. On the basis of interferograms and relationships between refractive index and composition in the transition zone, the profiles of component concentration distributions were constructed and the lines of phase equilibria were derived from the boundary concentrations of components at various temperatures. To confirm the equilibrium state and reversibility of the boundary concentrations, measurements were performed at both ways: increasing and decreasing temperatures. The LC phase in solutions was identified by polarization microscopy.

In parallel, the viscosity of solutions of different concentrations was measured. This information can be very useful for understanding not only compositions of various phases, but their properties as well.

For all systems under consideration, an increase in the concentration of HPC is accompanied by formation of the LC phase; in some cases, this process is complicated by amorphous separation of the system. In addition, for a number of systems, the CS phase can form in the high-concentration range. It is reasonable to examine various systems separately and then to generalize the data on phase equilibrium and rheological properties of HPC-based solutions.

Glycols are among the solvents used in this study. Let us first all analyze the phase equilibrium in low-molecular-weight liquids and oligomers containing terminal hydroxyl groups.

## **5.1.1 HPC-PG system**

420 Interferometry – Research and Applications in Science and Technology

So, HPC is among few polymers capable of forming both thermotropic and lyotropic LC phases. This is why HPC, as none other natural polymer, is accessible for analysis of phase equilibrium both in melt and solutions in a wide concentration range and in the region of easily achievable temperatures. This situation is primarily explained by the chemical structure of HPC, which contains primary and secondary hydroxypropyl and residual

Therefore, a system of H- bonds inherent for cellulose is destabilized and the level of intermolecular interaction is reduced. As a result, the polymer shows good solubility in solvents of various nature. The above features of the chemical structure are responsible for a rather high skeletal rigidity of HPC macromolecules (the Kuhn segment length is ~13 nm in dimethylaceteamide (Kulichikhin&Golova, 1985), which ensures transition to the LC melt at rather low temperatures (~120oC) with an isotropization (clearing) temperature of ~205oC

Solvents were: two oligomeric PEG samples (Spectrum, Unites States) with Mw=400 (PEG 400) and 1500 (PEG 1500), distilled water, propylene glycol (PG) (Labtex, Russia), triethyl citrate (TEC) (Morflex, United States), and DMSO (Gaylord Chemical Corp., United States).

Table 5 lists some characteristics of the used solvents (Woolley, 1974; Kolotyrkin, 1974; Allan, 1990; Geller et al., 1996; Uusi-Penttila, 1997; Kulvinder, 2000; Gallyamov, 2009; Mali, 2007) which may be useful for analysis of phase equilibria in HPC--solvent systems. Solutions of HPC with a HPC concentration of 30, 35, 40, 45, 50, 55, 60, 70, 75, and 80 wt %

The phase equilibrium in solutions was studied by the microinterference method at 18- 210oC which renders it possible to analyze concentration profiles in the diffusion zone. On the basis of interferograms and relationships between refractive index and composition in the transition zone, the profiles of component concentration distributions were constructed and the lines of phase equilibria were derived from the boundary concentrations of components at various temperatures. To confirm the equilibrium state and reversibility of the boundary concentrations, measurements were performed at both ways: increasing and decreasing temperatures. The LC phase in solutions was identified by polarization

104 was used.

**5.1 LC and complex phase equilibria in hydroxypropylcellulose solutions** 

HPC of the Klucel EF brand (Hercules-Aqualon, Unites States) with Mw= 8.

Table 5. Physical-chemical characteristics of solvents used.

were prepared in various solvents in accordance with (Fischer et al., 1995).

hydroxyl group moieties.

microscopy.

(Makarova, 2007; Makarova et al., 2007).

The interferograms measured at 75oC (Fig. 22) exhibit two phase boundaries (dotted lines): the first boundary separates isotropic and two-phase solutions, while the second boundary separates fully LC solutions from the crystalline (or the CS) phase. The presence of the LC phase is confirmed by the polarization microscopy data. Thus, under crossed polarizers the region of the LC phase of 60% solution exhibits a characteristic rainbow luminescence typical for cholesteric liquid crystals. Cellulose derivatives containing the asymmetric carbon atom (the chiral center) form just the cholesteric mesophase. As temperature is increased, the fully LC solution transforms into the two-phase region, in which LC and isotropic phases coexist. Given this, the solution gradually darkens and, after full isotropization, the field of vision becomes dark.

Fig. 22. Interferogram of the transition zone for the HPC-PG system at 75oC. The phase boundaries are marked with dotted line.

The resulting phase diagram is the superposition of LC and crystalline equilibria (Fig. 23). The results of polarization microscopy measurements made it possible to refine position of the LC liquidus and to supplement the diagram with the second boundary of the two-phase corridor separating the two-phase region and the region of fully LC solutions.

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 423

The phase equilibrium changes substantially with an increase in the molecular weight of PEG. The main feature of HPC solutions in PEG 1500 is a superposition of LC and amorphous phase equilibria (Fig. 25) (Makarova, 2007; Kulichikhin et al., 2010; Tolstykh et al., 2010). Separation on amorphous phases is observed in the region of small and moderate concentrations with a UCMT of ~185oC, while the liquid-crystalline equilibrium occurs at higher concentrations of HPC. Within the experimental error, the position of the liquidus for

Nevertheless, PEG 1500 is a less strong solvent for HPC than PEG 400. This is evidenced by the absence of solubility in the concentration and temperature region below the binodal. Such behavior can be explained by the intramolecular association of PEG molecules, as

Fig. 24. Phase diagram of HPC-PEG 400 pair.

a given system coincides with that for the HPC-PEG 400 system.

Fig. 25. Phase diagram of HPC-PEG 1500 system.

**5.1.3 HPC-PEG 1500 system** 

Fig. 23. Phase diagram of HPC-PG system, determined by interferometry (1) and polarizing microscopy (2). I – isotropic, LC – liquid-crystalline, C – crystalline state.

As opposed to extremely rigid-chain polymers with rod-like macromolecules, for which the I-LC transition is almost independent of temperature and concentration (the line of phase equilibrium is almost parallel to the ordinate axis), in the case of HPC solutions in PG, the position of boundary lines strongly depends on temperature and concentration of HPC. This finding is indicative of the semi-rigid nature of HPC in this solvent,

At high concentrations of the polymer, the LC phase occurs in equilibrium with the crystalline phase of HPC (C) or the CS phase. Note that in the neat HPC the degree of crystallinity does not exceed 20% (Shimamura, 1981). On the whole, the pattern of the phase diagram corresponds to the upper part of the hypothetical diagram shown in Fig. 20.

## **5.1.2 HPC-PEG 400 system**

The phase diagram for HPC-PEG 400 solutions was described in (Makarova, 2007; Makarova et al., 2007). Throughout the studied temperature range, this system is characterized by LC and crystalline transitions (Fig. 24). On the left of the boundary line, the solutions are isotropic, while on the right to the boundary line, they are mesomorphic. As in the case of HPC solutions in PG, the position of the boundary line is markedly temperaturedependent.

Given almost full qualitative analogy, there are marked quantitative differences between HPC-PEG 400 and HPC-PG systems. This primarily concerns smaller critical concentration corresponding to formation of the LC phase: ~25% for HPC solutions in PEG 400 compared to ~40% solutions of HPC in PG.

Fig. 24. Phase diagram of HPC-PEG 400 pair.

## **5.1.3 HPC-PEG 1500 system**

422 Interferometry – Research and Applications in Science and Technology

Fig. 23. Phase diagram of HPC-PG system, determined by interferometry (1) and polarizing

As opposed to extremely rigid-chain polymers with rod-like macromolecules, for which the I-LC transition is almost independent of temperature and concentration (the line of phase equilibrium is almost parallel to the ordinate axis), in the case of HPC solutions in PG, the position of boundary lines strongly depends on temperature and concentration of HPC. This

At high concentrations of the polymer, the LC phase occurs in equilibrium with the crystalline phase of HPC (C) or the CS phase. Note that in the neat HPC the degree of crystallinity does not exceed 20% (Shimamura, 1981). On the whole, the pattern of the phase

The phase diagram for HPC-PEG 400 solutions was described in (Makarova, 2007; Makarova et al., 2007). Throughout the studied temperature range, this system is characterized by LC and crystalline transitions (Fig. 24). On the left of the boundary line, the solutions are isotropic, while on the right to the boundary line, they are mesomorphic. As in the case of HPC solutions in PG, the position of the boundary line is markedly temperature-

Given almost full qualitative analogy, there are marked quantitative differences between HPC-PEG 400 and HPC-PG systems. This primarily concerns smaller critical concentration corresponding to formation of the LC phase: ~25% for HPC solutions in PEG 400 compared

diagram corresponds to the upper part of the hypothetical diagram shown in Fig. 20.

microscopy (2). I – isotropic, LC – liquid-crystalline, C – crystalline state.

finding is indicative of the semi-rigid nature of HPC in this solvent,

**5.1.2 HPC-PEG 400 system** 

to ~40% solutions of HPC in PG.

dependent.

The phase equilibrium changes substantially with an increase in the molecular weight of PEG. The main feature of HPC solutions in PEG 1500 is a superposition of LC and amorphous phase equilibria (Fig. 25) (Makarova, 2007; Kulichikhin et al., 2010; Tolstykh et al., 2010). Separation on amorphous phases is observed in the region of small and moderate concentrations with a UCMT of ~185oC, while the liquid-crystalline equilibrium occurs at higher concentrations of HPC. Within the experimental error, the position of the liquidus for a given system coincides with that for the HPC-PEG 400 system.

Fig. 25. Phase diagram of HPC-PEG 1500 system.

Nevertheless, PEG 1500 is a less strong solvent for HPC than PEG 400. This is evidenced by the absence of solubility in the concentration and temperature region below the binodal. Such behavior can be explained by the intramolecular association of PEG molecules, as

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 425

One of them (I) is related to the amorphous separation of the components, while the other boundary separates isotropic and mesophase regions. Above 70oC, phase boundary I disappears, whereas boundary II is preserved, thus indicating full dissolution of HPC in

On the basis of the interferograms with the use of the boundary concentrations of components estimated at various temperatures, the phase diagram was constructed (Fig. 27). As the HPC-PEG 1500 system, the HPC-TEC pair is characterized by the superposition of amorphous and LC equilibria. This situation is reflected on the diagram as the binodal with the HCST and the LC liquidus, respectively. Separation on amorphous phases is observed in the region of small and moderate concentrations, while the liquid-crystalline

Fig. 27. Phase diagram of the HPC-TEC system constructed from data of microinterference (1)

An analysis of the phase equilibrium in the HPC-TEC system shows that this system behaves as HPC solutions in PEG 1500. It appears that TEC molecules are characterized by self-association via formation of H-bonds between hydroxyl and ester groups. Furthermore, the steric factor may play an important role in this case. These structure features of TEC molecules affect the dissolving capacity of this solvent, which depends on solution concentration, as demonstrated by amorphous separation in the dilute-solution region and

Water is the next solvent containing hydroxyl groups, and solubility in water is a indubitable advantage of HPC. However, water possesses the lowest dipole moment and the highest dielectric constant; therefore, a certain specifics of phase equilibrium in aqueous

The phase behavior of HPC-water solutions has more complex than those described above. Several versions of diagrams for this system were obtained by DSC, X-ray diffraction, NMR,

formation of the LC phase at increased concentrations of HPC.

TEC at elevated temperatures with the concomitant formation of LC solutions.

equilibrium is seen at high concentrations of HPC.

and polarizing microscopy (2)

solutions of HPC should be expected.

**5.1.5 HPC-water system** 

confirmed by the IR data (Makarova et al., 2007). As molecular weight is increased, this effect becomes more pronounced and the probability of interaction between terminal hydroxyls of PEG and the functional groups of HPC tends to decrease.

Thus, solvents containing end hydroxyl groups dissolve well HPC and give rise to isotropic and anisotropic solutions. Owing to the presence of oxygen atoms in oligomeric PEG 400 chains and their interaction with ether or residual hydroxyl groups of HPC, the rigidity of a HPC macromolecule is somewhat higher; as a result, the transition to the LC state occurs at a lower concentration than that in the case of the low-molecular-weight PG. However, an increase in the molecular weight of PEG to 1500 causes a significant reduction in the dissolving capacity apparently due to cyclization of sufficiently long oligomeric molecules which worsens the efficiency of interaction between functional groups of the solvent and the polymer. It is not inconceivable that cyclization likewise depends on the concentration of solution since at high concentrations of HPC the LC equilibrium predominates and gives rise to LC solutions.

The hydroxyl group is also contained in a molecule of TEC, citric ester. Moreover, PEG and TEC have the highest dipole moments and the lowest dielectric permittivities among the used solvents (Table 5). Therefore, in what follows, it is advisable to examine the phase equilibrium in the HPC-TEC system.

## **5.1.4 HPC-TEC system**

Fig. 26 shows the interferograms of the transition zone obtained for this system. At temperatures below 70oC, two phase boundaries appear in the diffusion zone.

Fig. 26. Interferograms of transition zone for the HPC-TEC system registered at 55 (a) and 77°C (b). Phase boundaries are marked with dotted lines.

One of them (I) is related to the amorphous separation of the components, while the other boundary separates isotropic and mesophase regions. Above 70oC, phase boundary I disappears, whereas boundary II is preserved, thus indicating full dissolution of HPC in TEC at elevated temperatures with the concomitant formation of LC solutions.

On the basis of the interferograms with the use of the boundary concentrations of components estimated at various temperatures, the phase diagram was constructed (Fig. 27). As the HPC-PEG 1500 system, the HPC-TEC pair is characterized by the superposition of amorphous and LC equilibria. This situation is reflected on the diagram as the binodal with the HCST and the LC liquidus, respectively. Separation on amorphous phases is observed in the region of small and moderate concentrations, while the liquid-crystalline equilibrium is seen at high concentrations of HPC.

Fig. 27. Phase diagram of the HPC-TEC system constructed from data of microinterference (1) and polarizing microscopy (2)

An analysis of the phase equilibrium in the HPC-TEC system shows that this system behaves as HPC solutions in PEG 1500. It appears that TEC molecules are characterized by self-association via formation of H-bonds between hydroxyl and ester groups. Furthermore, the steric factor may play an important role in this case. These structure features of TEC molecules affect the dissolving capacity of this solvent, which depends on solution concentration, as demonstrated by amorphous separation in the dilute-solution region and formation of the LC phase at increased concentrations of HPC.

Water is the next solvent containing hydroxyl groups, and solubility in water is a indubitable advantage of HPC. However, water possesses the lowest dipole moment and the highest dielectric constant; therefore, a certain specifics of phase equilibrium in aqueous solutions of HPC should be expected.

## **5.1.5 HPC-water system**

424 Interferometry – Research and Applications in Science and Technology

confirmed by the IR data (Makarova et al., 2007). As molecular weight is increased, this effect becomes more pronounced and the probability of interaction between terminal

Thus, solvents containing end hydroxyl groups dissolve well HPC and give rise to isotropic and anisotropic solutions. Owing to the presence of oxygen atoms in oligomeric PEG 400 chains and their interaction with ether or residual hydroxyl groups of HPC, the rigidity of a HPC macromolecule is somewhat higher; as a result, the transition to the LC state occurs at a lower concentration than that in the case of the low-molecular-weight PG. However, an increase in the molecular weight of PEG to 1500 causes a significant reduction in the dissolving capacity apparently due to cyclization of sufficiently long oligomeric molecules which worsens the efficiency of interaction between functional groups of the solvent and the polymer. It is not inconceivable that cyclization likewise depends on the concentration of solution since at high concentrations of HPC the LC equilibrium predominates and gives

The hydroxyl group is also contained in a molecule of TEC, citric ester. Moreover, PEG and TEC have the highest dipole moments and the lowest dielectric permittivities among the used solvents (Table 5). Therefore, in what follows, it is advisable to examine the phase

Fig. 26 shows the interferograms of the transition zone obtained for this system. At

Fig. 26. Interferograms of transition zone for the HPC-TEC system registered at 55 (a) and

77°C (b). Phase boundaries are marked with dotted lines.

temperatures below 70oC, two phase boundaries appear in the diffusion zone.

hydroxyls of PEG and the functional groups of HPC tends to decrease.

rise to LC solutions.

**5.1.4 HPC-TEC system** 

equilibrium in the HPC-TEC system.

**TEC**

The phase behavior of HPC-water solutions has more complex than those described above. Several versions of diagrams for this system were obtained by DSC, X-ray diffraction, NMR,

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 427

Fig. 29. Interferograms of transition zones for the HPC-water system at 29 (a) and 43°C (b). Figures denote regions corresponding to different phase states (see explanations in text).

LCST. As a rule, such phase diagrams are observed for polymer-solvent components with strong intermolecular interaction and corresponds to the upper part of the theoretical phase diagram shown in Fig. 21. In this sense, glycols and esters are more active solvents in

The next solvent from the selected series is aprotonic dipolar DMSO, which stands somewhat apart from other solvents since it contains no hydroxyl groups. The DMSO molecule has unshared electron pairs on oxygen and sulfur atoms, and this fact explains high dissolving capacity of this donor solvent. In terms of dipole moment and dielectric

relation to formation of H-bonds with the functional groups of HPC than water.

Fig. 30. The corrected phase diagram of the HPC-H2O system .

constant, DMSO is rather close to PG.

nefelometry, and rheology. A set of some versions of diagrams is shown in Fig. 28. At temperatures below 40oC, with an increase in the concentration of HPC, solutions transform from the isotropic region (I) to the two-phase region containing isotropic and LC phases (II) and then to fully anisotropic region (III). At high concentrations of HPC, CS (IV) forms.

Fig. 28. A set of phase diagrams for the HPC-water system according to (Fischer et al., 1995; Guido, 1995; Larez et al., 1995; Vshivkov et al., 2007)

Although the diagrams follow the general pattern, the positions of phase equilibrium lines differ appreciably; moreover, it is unclear which is the character of phase separation at temperatures above 40-45oC. It is known that elevated temperatures solutions experience reversible gel-formation; however, the mechanism controlling formation of the physical network remains vague up to now. Two mechanisms of this process are discussed in the literature: phase decomposition of the liquid-liquid type and formation of the HPC. 6H2O crystal solvate. Unfortunately, structural methods do not assist in understanding this mechanism. This circumstance forced us to refine the lines of phase equilibrium for this system and to gain insight into the reasons of gelation at elevated temperatures.

At temperatures below 37oC, the interferograms show all four above-mentioned regions in the transition zone (Fig. 29). As temperature is increased, the first two regions disappear and the phase boundary responsible for amorphous separation appears. The phase diagram constructed from concentration profiles in the diffusion region (Fig. 30) is the superposition of three types of phase equilibria, namely, amorphous, LC, and crystalline.

From the refined phase diagram, the mechanism of gelation of aqueous solutions at elevated temperatures becomes understandable, namely, amorphous separation and, as a result, "incomplete separation into phases" that leads to formation of the structural network. The fact that in the case of TEC and PEG 1500, solubility increases with temperature (the binodal with the HCST) and, in the case of water, decreases (the binodal with the LCST) suggest that temperature dependences of the interaction parameter are different. However, it should be mentioned that the Flory-Huggins theory cannot predict the existence of systems with the

nefelometry, and rheology. A set of some versions of diagrams is shown in Fig. 28. At temperatures below 40oC, with an increase in the concentration of HPC, solutions transform from the isotropic region (I) to the two-phase region containing isotropic and LC phases (II) and then to fully anisotropic region (III). At high concentrations of HPC, CS (IV) forms.

Fig. 28. A set of phase diagrams for the HPC-water system according to (Fischer et al., 1995;

Although the diagrams follow the general pattern, the positions of phase equilibrium lines differ appreciably; moreover, it is unclear which is the character of phase separation at temperatures above 40-45oC. It is known that elevated temperatures solutions experience reversible gel-formation; however, the mechanism controlling formation of the physical network remains vague up to now. Two mechanisms of this process are discussed in the literature: phase decomposition of the liquid-liquid type and formation of the HPC.

crystal solvate. Unfortunately, structural methods do not assist in understanding this mechanism. This circumstance forced us to refine the lines of phase equilibrium for this

At temperatures below 37oC, the interferograms show all four above-mentioned regions in the transition zone (Fig. 29). As temperature is increased, the first two regions disappear and the phase boundary responsible for amorphous separation appears. The phase diagram constructed from concentration profiles in the diffusion region (Fig. 30) is the superposition

From the refined phase diagram, the mechanism of gelation of aqueous solutions at elevated temperatures becomes understandable, namely, amorphous separation and, as a result, "incomplete separation into phases" that leads to formation of the structural network. The fact that in the case of TEC and PEG 1500, solubility increases with temperature (the binodal with the HCST) and, in the case of water, decreases (the binodal with the LCST) suggest that temperature dependences of the interaction parameter are different. However, it should be mentioned that the Flory-Huggins theory cannot predict the existence of systems with the

system and to gain insight into the reasons of gelation at elevated temperatures.

of three types of phase equilibria, namely, amorphous, LC, and crystalline.

6H2O

Guido, 1995; Larez et al., 1995; Vshivkov et al., 2007)

Fig. 29. Interferograms of transition zones for the HPC-water system at 29 (a) and 43°C (b). Figures denote regions corresponding to different phase states (see explanations in text).

LCST. As a rule, such phase diagrams are observed for polymer-solvent components with strong intermolecular interaction and corresponds to the upper part of the theoretical phase diagram shown in Fig. 21. In this sense, glycols and esters are more active solvents in relation to formation of H-bonds with the functional groups of HPC than water.

Fig. 30. The corrected phase diagram of the HPC-H2O system .

The next solvent from the selected series is aprotonic dipolar DMSO, which stands somewhat apart from other solvents since it contains no hydroxyl groups. The DMSO molecule has unshared electron pairs on oxygen and sulfur atoms, and this fact explains high dissolving capacity of this donor solvent. In terms of dipole moment and dielectric constant, DMSO is rather close to PG.

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 429

Table 6. Critical concentrations of phase transitions for considered systems at 25oC.

of HPC.

rather close to the

somewhat smaller than the

situation is observed (disregarding water).

fractions for PEG, DMSO, PG, and TEC, respectively.

It is possible that this fact explains why HPC is not dissolved in water at temperatures above 36.5oC (the LCST) but it cannot explain why HPC shows excellent solubility at reduced temperatures. The latter phenomenon may be attributed to redistribution of H-bonds formed by water molecules to those formed between water molecules and functional groups

As regards other solvents, their solubility parameters range from 8.2 to 12.8; that is, they are

somewhat higher (12.6 and 12.8). It is pertinent to note that for the first pair of solvents, the superposition of amorphous (with the HCST) and LC equilibria is observed, while for the second pair, only the LC equilibrium is implemented. PEG 400 stands somewhat apart from the rest solvents since in this case no amorphous separation is detected. These differences may be due to different polarizabilities of molecules: for TEC and PEG, high dipole moments coexist with low dielectric constants, while for PG and DMSO, the reverse

If we examine the critical concentrations corresponding to formation of the LC phase (C\*) and the achievement of its 100% content in the system (C\*\*), then the highest rigidity of HPC macromolecules is realized in PEG, as evidenced by the lowest value of C\* (0.25 vol. fractions at 25oC). In the case of water, PG, and DMSO, these parameters are in the range 0.38-0.39. At a temperature of 140oC, the values of C\* are 0.64, 0.70, 0.73, and 0.85 vol.

Table 7. Critical concentrations of phase transitions for considered systems at 140oC.

of HPC. Note that for TEC and PEG 1500, solubility parameters are

of HPC (8.2 and 8.4), while for PG and DMSO, they are

## **5.1.6 HPC-DMSO system**

The typical interferogram of the region of interdiffusion of HPC and DMSO is shown in Fig. 31. Throughout the studied temperature range, the phase boundary (marked by the dotted line) is seen which separates the regions of isotropic and mesomorphic states.

Fig. 31. Interferogram of the transition zone for the HPC-DMSO system measured at 55°C. The phase boundary is denoted with a dotted line.

The phase diagram of the HPC-DMSO system constructed from the data of interference rheology and polarizing microscopy is characterized by the LC equilibrium (Fig. 32). Throughout the temperature range under study, the two-phase corridor manifests itself which separates the regions of isotropic and LC solutions.

Fig. 32. Phase diagram for the HPC-DMSO system constructed from the data of microinterference (1), microscopy (2) and rheology (3).

Tables 6 and 7 list some parameters of the phase diagrams obtained at 25 and 140oC, respectively. Let compare these data with the characteristics of the tested solvents (Table 5). Strong intermolecular interactions are responsible for the highest solubility parameter of water, as calculated in accordance Hildebrand and Small [62], = 21.2 (cal/cm3)0.5. This value is much higher than the of HPC (approximately10.72 (cal/cm3)0.5 (Allan, 1990).

The typical interferogram of the region of interdiffusion of HPC and DMSO is shown in Fig. 31. Throughout the studied temperature range, the phase boundary (marked by the dotted

Fig. 31. Interferogram of the transition zone for the HPC-DMSO system measured at 55°C.

The phase diagram of the HPC-DMSO system constructed from the data of interference rheology and polarizing microscopy is characterized by the LC equilibrium (Fig. 32). Throughout the temperature range under study, the two-phase corridor manifests itself

Fig. 32. Phase diagram for the HPC-DMSO system constructed from the data of

Tables 6 and 7 list some parameters of the phase diagrams obtained at 25 and 140oC, respectively. Let compare these data with the characteristics of the tested solvents (Table 5). Strong intermolecular interactions are responsible for the highest solubility parameter

water, as calculated in accordance Hildebrand and Small [62], = 21.2 (cal/cm3)0.5. This

of HPC (approximately10.72 (cal/cm3)0.5 (Allan, 1990).

of

The phase boundary is denoted with a dotted line.

which separates the regions of isotropic and LC solutions.

microinterference (1), microscopy (2) and rheology (3).

value is much higher than the

line) is seen which separates the regions of isotropic and mesomorphic states.

**5.1.6 HPC-DMSO system** 

**DMSO**


Table 6. Critical concentrations of phase transitions for considered systems at 25oC.

It is possible that this fact explains why HPC is not dissolved in water at temperatures above 36.5oC (the LCST) but it cannot explain why HPC shows excellent solubility at reduced temperatures. The latter phenomenon may be attributed to redistribution of H-bonds formed by water molecules to those formed between water molecules and functional groups of HPC.

As regards other solvents, their solubility parameters range from 8.2 to 12.8; that is, they are rather close to the of HPC. Note that for TEC and PEG 1500, solubility parameters are somewhat smaller than the of HPC (8.2 and 8.4), while for PG and DMSO, they are somewhat higher (12.6 and 12.8). It is pertinent to note that for the first pair of solvents, the superposition of amorphous (with the HCST) and LC equilibria is observed, while for the second pair, only the LC equilibrium is implemented. PEG 400 stands somewhat apart from the rest solvents since in this case no amorphous separation is detected. These differences may be due to different polarizabilities of molecules: for TEC and PEG, high dipole moments coexist with low dielectric constants, while for PG and DMSO, the reverse situation is observed (disregarding water).

If we examine the critical concentrations corresponding to formation of the LC phase (C\*) and the achievement of its 100% content in the system (C\*\*), then the highest rigidity of HPC macromolecules is realized in PEG, as evidenced by the lowest value of C\* (0.25 vol. fractions at 25oC). In the case of water, PG, and DMSO, these parameters are in the range 0.38-0.39. At a temperature of 140oC, the values of C\* are 0.64, 0.70, 0.73, and 0.85 vol. fractions for PEG, DMSO, PG, and TEC, respectively.


Table 7. Critical concentrations of phase transitions for considered systems at 140oC.

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 431

biphasic region

Fig. 33. Concentration dependences of viscosity for solutions of HPC in PG at 23 (a), TEC at 50 (b), water at 23 (c), DMSO at 23oC (d). Dotted lines show phase boundaries according to

on the curve corresponds to the critical concentration at which the system becomes fully LC. After further increase in concentration, viscosity increases again due to a reduction in the free volume. Thus, an increase in viscosity with concentration is typical for single-phase (isotropic and LC) solutions, whereas in the two-phase corridor, viscosity drops with

It is pertinent to note the specifics inherent in the HPC-TEC system in the concentration range neighboring amorphous phase separation. In this case, the abnormal decrease in viscosity with a decrease in the concentration of HPC is apparently related to formation of nuclei (species) of the amorphous phase highly concentrated with respect to the polymer. The above dependences of viscosity on concentration vividly reflect the specifics inherent in solutions of rigid-chain polymers, which form the LC phase in a certain temperatureconcentration range. Exactly for these systems viscosity is as a parameter suitable to refine the position of boundary lines of phase equilibrium. Very likely this is the key point in a given analysis. The region of amorphous equilibrium is practically unfeasible for the correct study by means of rheological methods since the system experiences phase separation below

The different kinds of phase equilibria (amorphous, crystalline, liquid-crystalline) are considered. Each of them has definite futures that were analyzed on examples of various

phase diagrams constructed.

the binodal.

**6. Conclusion** 

increase of concentration (the fraction of the LC phase).

For all systems, the width of the two-phase corridor, which was estimated from the ratio of critical concentrations C\*\*/C\*, is much higher at 25oC (1.24-1.48) than that at 140oC (1.02- 1.16). This narrowing of the corridor with temperature, first, is indicative of the semi-rigid nature of HPC (the rigidity of chains is strongly temperature-dependent) and second, is consistent with Papkov's predictions (Fig. 20) that lines C\*(CHPC) and C\*\*(CHPC) come closer and in the limit merge on approach to the 100% polymer.

For such solvents as PG, PEG 400, PEG 1500, and water, the crystalline phase (CS or the crystalline HPC) is detected at a concentration of Cc. It appears that the rigidity of HPC macromolecules implemented in various solvents may affect the parameters of amorphous separation of systems. At least, this is true for HPC-PEG 1500 and HPC-TEC systems. In PEG 1500, chain rigidity, as estimated from C\*, is higher and its HCST (185oC) is situated at a concentration of 0.18 vol. fractions. For HPC solutions in TEC, C\* is lower and the HCST is 71oC at a concentration of 0.09 vol. fractions.

The above analysis does not pretend to be full and does not reveal all features of phase equilibrium in HPC-solvent systems. Nevertheless, it demonstrates the peculiar phase behavior of these systems in relation to the structure and properties of solvents.

## **5.1.7 Rheological properties of HPC solutions**

The rheological properties of LC systems are characterized by some distinctive features (Papkov et al., 1974; Kulichikhin et al., 1982; Kulichikhin, 1989). As opposed to isotropic polymers, LC polymers manifest the abnormal viscosity throughout the studied shear rate range. This effect is associated with the polydomain structure of LC systems and the presence of a system of disclinations at rest. With an increase in the shear rate as a result of orientation processes, the system gradually transforms into the monodomain structure. Furthermore, in the two-phase region, interphase boundaries between drops of isotropic phase and LC disperse medium, or LC drops and isotropic matrix make a substantial contribution to the rheological response.

The main result at analysis of rheological properties of solutions under consideration consists in conclusion that viscosity is a parameter sensitive to LC transitions. The flow curves reflect the existence of the polydomain structure in the initial LC system and its transformation into the monodomain structure with an increase in the intensity of shear deformation. Presence of polydomain structure and such defects as disclinations is stipulated the viscoplastic behavior with a yield stress. However, the most vivid dependence that makes it possible to estimate the critical concentrations of transitions is the concentration dependence of viscosity. In this sense rheological data are good addition to results of interferometric measurements to prove the location of boundary lines between liquid phases.

Fig. 33 displays dependences of this type for several systems under study in the region of the Newtonian (isotropic solutions) or quasi-Newtonian (anisotropic solutions) behavior. As is seen, these dependences contain extreme points. Despite of different scales of maxima and minima, extremes are observed for all systems. As is seen from comparison with phase diagrams, the maximum of viscosity corresponds to the critical concentration, at which the anisotropic phase appears in solutions. When the system contains isotropic and LC phases, an increase in the fraction of the anisotropic phase causes a drop in viscosity. The minimum

Fig. 33. Concentration dependences of viscosity for solutions of HPC in PG at 23 (a), TEC at 50 (b), water at 23 (c), DMSO at 23oC (d). Dotted lines show phase boundaries according to phase diagrams constructed.

on the curve corresponds to the critical concentration at which the system becomes fully LC. After further increase in concentration, viscosity increases again due to a reduction in the free volume. Thus, an increase in viscosity with concentration is typical for single-phase (isotropic and LC) solutions, whereas in the two-phase corridor, viscosity drops with increase of concentration (the fraction of the LC phase).

It is pertinent to note the specifics inherent in the HPC-TEC system in the concentration range neighboring amorphous phase separation. In this case, the abnormal decrease in viscosity with a decrease in the concentration of HPC is apparently related to formation of nuclei (species) of the amorphous phase highly concentrated with respect to the polymer.

The above dependences of viscosity on concentration vividly reflect the specifics inherent in solutions of rigid-chain polymers, which form the LC phase in a certain temperatureconcentration range. Exactly for these systems viscosity is as a parameter suitable to refine the position of boundary lines of phase equilibrium. Very likely this is the key point in a given analysis. The region of amorphous equilibrium is practically unfeasible for the correct study by means of rheological methods since the system experiences phase separation below the binodal.

## **6. Conclusion**

430 Interferometry – Research and Applications in Science and Technology

For all systems, the width of the two-phase corridor, which was estimated from the ratio of critical concentrations C\*\*/C\*, is much higher at 25oC (1.24-1.48) than that at 140oC (1.02- 1.16). This narrowing of the corridor with temperature, first, is indicative of the semi-rigid nature of HPC (the rigidity of chains is strongly temperature-dependent) and second, is consistent with Papkov's predictions (Fig. 20) that lines C\*(CHPC) and C\*\*(CHPC) come closer

For such solvents as PG, PEG 400, PEG 1500, and water, the crystalline phase (CS or the crystalline HPC) is detected at a concentration of Cc. It appears that the rigidity of HPC macromolecules implemented in various solvents may affect the parameters of amorphous separation of systems. At least, this is true for HPC-PEG 1500 and HPC-TEC systems. In PEG 1500, chain rigidity, as estimated from C\*, is higher and its HCST (185oC) is situated at a concentration of 0.18 vol. fractions. For HPC solutions in TEC, C\* is lower and the HCST is

The above analysis does not pretend to be full and does not reveal all features of phase equilibrium in HPC-solvent systems. Nevertheless, it demonstrates the peculiar phase

The rheological properties of LC systems are characterized by some distinctive features (Papkov et al., 1974; Kulichikhin et al., 1982; Kulichikhin, 1989). As opposed to isotropic polymers, LC polymers manifest the abnormal viscosity throughout the studied shear rate range. This effect is associated with the polydomain structure of LC systems and the presence of a system of disclinations at rest. With an increase in the shear rate as a result of orientation processes, the system gradually transforms into the monodomain structure. Furthermore, in the two-phase region, interphase boundaries between drops of isotropic phase and LC disperse medium, or LC drops and isotropic matrix make a substantial

The main result at analysis of rheological properties of solutions under consideration consists in conclusion that viscosity is a parameter sensitive to LC transitions. The flow curves reflect the existence of the polydomain structure in the initial LC system and its transformation into the monodomain structure with an increase in the intensity of shear deformation. Presence of polydomain structure and such defects as disclinations is stipulated the viscoplastic behavior with a yield stress. However, the most vivid dependence that makes it possible to estimate the critical concentrations of transitions is the concentration dependence of viscosity. In this sense rheological data are good addition to results of interferometric measurements to prove the location of boundary lines between

Fig. 33 displays dependences of this type for several systems under study in the region of the Newtonian (isotropic solutions) or quasi-Newtonian (anisotropic solutions) behavior. As is seen, these dependences contain extreme points. Despite of different scales of maxima and minima, extremes are observed for all systems. As is seen from comparison with phase diagrams, the maximum of viscosity corresponds to the critical concentration, at which the anisotropic phase appears in solutions. When the system contains isotropic and LC phases, an increase in the fraction of the anisotropic phase causes a drop in viscosity. The minimum

behavior of these systems in relation to the structure and properties of solvents.

and in the limit merge on approach to the 100% polymer.

71oC at a concentration of 0.09 vol. fractions.

**5.1.7 Rheological properties of HPC solutions** 

contribution to the rheological response.

liquid phases.

The different kinds of phase equilibria (amorphous, crystalline, liquid-crystalline) are considered. Each of them has definite futures that were analyzed on examples of various

Application of Interferometry to Analysis of Polymer-Polymer and Polymer-Solvent Interactions 433

informativity of interferometry for such systems. The main portion of equilibria lines in solutions were detected by interferometric method. Of course, for LC systems method of polarizing microscopy has to be obligatory supporting method for confirmation of the LC phase existence. In addition, viscometry seemed to be very useful again not only for description of features of these systems at flow, but also for additional detection of LC

As opposed to diagrams obtained for other systems, for which two types of phase equilibrium are located in different concentration regions, in the case of aqueous systems, all transitions, including formation of crystal solvates, take place below the convex part of the binodal, that is, at the same concentrations but at different temperatures. This result indicates that the activity of water with respect to formation of hetero H-bonds tends to

Authors appreciate the partial financial support from Russian Foundation for Basic Research

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Avdeev, N.N. (1990). Phase equilibria and diffusion in oligomer-polymer systems. PhD Thesis.

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Borovskii, I.; Gurov, K.; Marchukova, I. & Ugaste, Yu. (1973). Processes of interdiffusion in

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transitions as well as for formation of CS phase.

increase with a decrease in temperature.

**7. Acknowledgement** 

**8. References** 

polymer-polymer and polymer-solvent systems. All above mentioned results relate to thermodynamic properties for binary systems containing polymers, and all of them were obtained by interferometry method. In addition to thermodynamic data, this method gives information about kinetics of components interactions that allows us to measure simultaneously interdiffusion coefficients. This dualism of optical interferometry method in application to interaction in polymer containing systems is its great advantage. That why its application in scientific practice is very useful and fruitful.

The couple PC – low molecular weight PMMA is a good example of polymer system with amorphous phase equilibrium. Method of interferometry is very useful and fruitful for such kind of systems due to informativity and combination of joint approaches: kinetics of masstransfer and thermodynamics of phase states. Increase of the mutual solubility with temperature indicates on UCST that could not be measured experimentally but was calculated using evolution of the interaction parameter with concentration.

Comparison of the phase diagram with concentration dependence of viscosity allowed us to observe local minimum of viscosity near the binodal (sooner in meta-stable region between binodal and spinodal where microemulsion appears as the first step of the phase decomposition) and to apply the empiric equation for calculation of the blends viscosity by means of accurate determination of the composition from binodal branches. Joint consideration of diffusion and rheological properties seems to be important for judging about structure, kinetic and thermodynamic behaviors of complex, multiphase polymer systems. This approach will be used further for analysis of systems with LC and crystalline equilibria.

Calculations of the interphase layers thickness depending on dimensions of drops and time, based on knowledge of interdiffusion coefficients can be very useful for correct organization of technological process of mixing and, consequently, development of new materials with three-phase structure and new properties.

Several examples of crystalline equilibrium in couples polyester-polyester are considered a well. The neat polymers are semi-crystalline and phase equilibrium for their pairs is describing by liquidus lines, sometimes with eutectic points. But this situation takes place at T<280-290oC, when chemical interaction can be neglected. The original feature of these data consists in using as one component of pair thermotropic LC polyesters and copolyesters. Up to clearing point of LC component the interdiffusion process is registered using one part of the interference patterns. Such approach was demonstrated first time.

At using LC polymer melt as one component, both interdiffusion process and ester-ester exchange is delayed because of high molecular orientation in LC polymer along interface. This dense layer prevents fast penetration of coil-like macromolecules of isotropic polyester melt. But in the case of PET-PEN pair both melts are isotropic and transparent, that is why interaction process can be visualized easy. The hypothetic mechanism of joint action of physical and chemical interaction is proposed. The first stage is diffusion, but after penetration of macromolecules of one polymer into another the trans-esterification starts and transition between these two processes can be estimated from kinetics of movement of iso-concentration planes.

Method of optical interferometry is not so often used for describing phase equilibria in solutions of stiff-chain polymers forming in solutions LC phase. In this part was shown its application for several solutions the same polymer in different solvents. We were wonder to informativity of interferometry for such systems. The main portion of equilibria lines in solutions were detected by interferometric method. Of course, for LC systems method of polarizing microscopy has to be obligatory supporting method for confirmation of the LC phase existence. In addition, viscometry seemed to be very useful again not only for description of features of these systems at flow, but also for additional detection of LC transitions as well as for formation of CS phase.

As opposed to diagrams obtained for other systems, for which two types of phase equilibrium are located in different concentration regions, in the case of aqueous systems, all transitions, including formation of crystal solvates, take place below the convex part of the binodal, that is, at the same concentrations but at different temperatures. This result indicates that the activity of water with respect to formation of hetero H-bonds tends to increase with a decrease in temperature.

## **7. Acknowledgement**

Authors appreciate the partial financial support from Russian Foundation for Basic Research (grants 08-03-12035, 09-03-00285a, 10-03-00994a, 11-03-12042-ofi-m).

## **8. References**

432 Interferometry – Research and Applications in Science and Technology

polymer-polymer and polymer-solvent systems. All above mentioned results relate to thermodynamic properties for binary systems containing polymers, and all of them were obtained by interferometry method. In addition to thermodynamic data, this method gives information about kinetics of components interactions that allows us to measure simultaneously interdiffusion coefficients. This dualism of optical interferometry method in application to interaction in polymer containing systems is its great advantage. That why its

The couple PC – low molecular weight PMMA is a good example of polymer system with amorphous phase equilibrium. Method of interferometry is very useful and fruitful for such kind of systems due to informativity and combination of joint approaches: kinetics of masstransfer and thermodynamics of phase states. Increase of the mutual solubility with temperature indicates on UCST that could not be measured experimentally but was

Comparison of the phase diagram with concentration dependence of viscosity allowed us to observe local minimum of viscosity near the binodal (sooner in meta-stable region between binodal and spinodal where microemulsion appears as the first step of the phase decomposition) and to apply the empiric equation for calculation of the blends viscosity by means of accurate determination of the composition from binodal branches. Joint consideration of diffusion and rheological properties seems to be important for judging about structure, kinetic and thermodynamic behaviors of complex, multiphase polymer systems. This approach will be used further for analysis of systems with LC and crystalline equilibria. Calculations of the interphase layers thickness depending on dimensions of drops and time, based on knowledge of interdiffusion coefficients can be very useful for correct organization of technological process of mixing and, consequently, development of new materials with

Several examples of crystalline equilibrium in couples polyester-polyester are considered a well. The neat polymers are semi-crystalline and phase equilibrium for their pairs is describing by liquidus lines, sometimes with eutectic points. But this situation takes place at T<280-290oC, when chemical interaction can be neglected. The original feature of these data consists in using as one component of pair thermotropic LC polyesters and copolyesters. Up to clearing point of LC component the interdiffusion process is registered using one part of

At using LC polymer melt as one component, both interdiffusion process and ester-ester exchange is delayed because of high molecular orientation in LC polymer along interface. This dense layer prevents fast penetration of coil-like macromolecules of isotropic polyester melt. But in the case of PET-PEN pair both melts are isotropic and transparent, that is why interaction process can be visualized easy. The hypothetic mechanism of joint action of physical and chemical interaction is proposed. The first stage is diffusion, but after penetration of macromolecules of one polymer into another the trans-esterification starts and transition between these two processes can be estimated from kinetics of movement of

Method of optical interferometry is not so often used for describing phase equilibria in solutions of stiff-chain polymers forming in solutions LC phase. In this part was shown its application for several solutions the same polymer in different solvents. We were wonder to

calculated using evolution of the interaction parameter with concentration.

the interference patterns. Such approach was demonstrated first time.

application in scientific practice is very useful and fruitful.

three-phase structure and new properties.

iso-concentration planes.


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Wireless Sensor Networks (WSN) consist of small, low-cost, resource-constrained embedded computers equipped with low-power radios and various sensors. When deployed, they form an ad-hoc wireless network and sense the environment cooperatively. Application areas of WSNs include environmental monitoring, healthcare, structural monitoring and the military among others. Many WSN applications rely on the location of sensor nodes in order to make consistent spatio-temporal observations. Sensor node localization, therefore, has been an

**Interferometry in Wireless Sensor Networks** 

Sandor Szilvasi1, Peter Volgyesi1, Janos Sallai1,

*1Institute for Software Integrated Systems, Vanderbilt University* 

Akos Ledeczi1 and Miklos Maroti2

*2Bolyai Institute, University of Szeged* 

*1USA 2Hungary* 

**21**

Consequently, a wide variety of localization methods has been proposed for WSNs. Among the most popular are acoustic and radio signal-based approaches. The limited range of acoustic methods make them impractical for anything but small scale deployments. The best known radio-based method is the Global Positioning System (GPS). For outdoor applications that require absolute localization accuracy in the order of meters, GPS is generally a good solution. However, its price and power requirement restrict its applicability to high-end WSNs. In indoor and urban environments, heavily affected by multipath effects, Ultra Wide band (UWB) methods offer high accuracy by measuring the Time of Flight (TOF) or Time Difference of Arrival (TDOA) of a short pulse-waveform to deduce ranges Tuchler et al. (2005), Fontana et al. (2003). The UWB-based techniques typically achieve sub-meter ranging error where the accuracy depends mainly on the available bandwidth, which in turn is constrained by emission regulations. UWB has not become popular in WSNs primarily due

Techniques based on received radio signal strength (RSS) gained significant popularity in WSNs as they require no specialized hardware; they use the radio chip readily available on WSN nodes. RSS-based ranging measurements are also simple as they typically consist of averaging the value of the radio chip RSS signal. The operating range is limited to the communication range of the radio with a precision of a few meters Taubenheim et al. (2005).

This research was partially supported by ARO MURI grant W911NF-06-1-0076 and the

TÁMOP-4.2.2/08/1/2008-0008 program of the Hungarian National Development Agency.

active research area for the past decade.

to its high cost.

**1. Introduction**


## **Interferometry in Wireless Sensor Networks**

Sandor Szilvasi1, Peter Volgyesi1, Janos Sallai1, Akos Ledeczi1 and Miklos Maroti2 *1Institute for Software Integrated Systems, Vanderbilt University 2Bolyai Institute, University of Szeged 1USA 2Hungary* 

## **1. Introduction**

436 Interferometry – Research and Applications in Science and Technology

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Effects of Interphase Interactions on Rheological Properties of PET-LC Polyester

Wireless Sensor Networks (WSN) consist of small, low-cost, resource-constrained embedded computers equipped with low-power radios and various sensors. When deployed, they form an ad-hoc wireless network and sense the environment cooperatively. Application areas of WSNs include environmental monitoring, healthcare, structural monitoring and the military among others. Many WSN applications rely on the location of sensor nodes in order to make consistent spatio-temporal observations. Sensor node localization, therefore, has been an active research area for the past decade.

Consequently, a wide variety of localization methods has been proposed for WSNs. Among the most popular are acoustic and radio signal-based approaches. The limited range of acoustic methods make them impractical for anything but small scale deployments. The best known radio-based method is the Global Positioning System (GPS). For outdoor applications that require absolute localization accuracy in the order of meters, GPS is generally a good solution. However, its price and power requirement restrict its applicability to high-end WSNs. In indoor and urban environments, heavily affected by multipath effects, Ultra Wide band (UWB) methods offer high accuracy by measuring the Time of Flight (TOF) or Time Difference of Arrival (TDOA) of a short pulse-waveform to deduce ranges Tuchler et al. (2005), Fontana et al. (2003). The UWB-based techniques typically achieve sub-meter ranging error where the accuracy depends mainly on the available bandwidth, which in turn is constrained by emission regulations. UWB has not become popular in WSNs primarily due to its high cost.

Techniques based on received radio signal strength (RSS) gained significant popularity in WSNs as they require no specialized hardware; they use the radio chip readily available on WSN nodes. RSS-based ranging measurements are also simple as they typically consist of averaging the value of the radio chip RSS signal. The operating range is limited to the communication range of the radio with a precision of a few meters Taubenheim et al. (2005).

This research was partially supported by ARO MURI grant W911NF-06-1-0076 and the TÁMOP-4.2.2/08/1/2008-0008 program of the Hungarian National Development Agency.

mathematical foundation of the localization method, which is then referenced throughout the rest of the chapter. It also presents the first platform using this approach, called the Radio Interferometric Positioning System (RIPS), along with a discussion of the initial results. Section 3 introduces a method for tracking mobile nodes in WSNs utilizing the phenomena of Doppler effects and radio interferometry. The technique shown in Section 4 exploits the Doppler effect in a different way to determine the location of static nodes in a WSN. It relies on a few distinct nodes with known locations (anchors) to first calculate the bearing to the rest of the nodes and to deduce location information based on the bearing estimates. The technique presented in Section 5 also calculates bearings from anchor nodes, but instead of using Doppler shifts, it uses a special arrangement of RIPS which allows for an enhanced,

Interferometry in Wireless Sensor Networks 439

The basic idea behind interferometric localization is to use the phase information of a radio signal to measure distance. In practice, however, this is hard to achieve, because it would require (a) nanosecond-precision timing, (b) precise control over the phase of the transmitted signal, and (c) precise phase detection on the receiver. COTS radio transceivers commonly used in sensor nodes do not offer these features. The novelty of the method presented in this chapter is to use a transmitter pair and a receiver pair to eliminate the unknown initial phase of the respective local oscillators (LO), as well well as to allow for a low-speed ADC to process

The interferometric approach thus utilizes a pair of wireless sensor nodes to generate interfering radio signals by transmitting pure sinusoids at slightly different frequencies. The envelope of this composite signal has a beat frequency equal to the frequency difference of the sinusoids, as illustrated in Figure 2. It is the phase of the envelope signal that is of most interest, as it carries location related information. When measured at two receivers at the same time, the instantaneous phase difference (also referred to as the *phase offset*) of the respective interference signal envelopes is directly related to the carrier wavelength and the relative distance between the transmitters and receivers. Working with the envelope signal instead of the "raw" carrier signal has several advantages. On one hand, the envelope frequency depends on only the frequency difference of the transmitted sinusoids. This means that the received envelope frequency is the same at all receivers, and it is independent of the frequency and initial phase of the receivers' local oscillators. On the other hand, the envelope frequency can be precisely controlled (in short term), thus it is adjustable to meet the available time synchronization accuracy, limited sampling rate and processing capability of the wireless

The rest of the section discusses the theory of radio interferometric localization in detail.

These sinusoids we refer to as carrier signals, though they are not modulated.

Let us consider, hypothetically, that the transmitted signal's phase is controllable, and the

distributed method of node localization.

**2. Radio interferometric positioning**

the incoming signal on the receiver.

**2.1 Idealistic phase-based localization**

received phase can be detected accurately.

sensor nodes.

However, the measurement is highly environment dependent. Achieving reasonable accuracy requires extensive calibration prior to ranging. Uncalibrated RSS ranging can exhibit 10-20% errors.

A novel ranging method based on radio interferometry (RI) for static WSN node localization was introduced in Maróti et al. (2005). This RI ranging method measures the phase of the radio signals to obtain information on the relative distance between the sensor nodes. It achieves sub-meter localization accuracy and long range simultaneously. As this RI approach also relies solely on the radio chip readily available on popular, low-cost commercial off-the-shelf (COTS) sensor nodes, it became an alternative to RSS-based methods and, thus, garnered considerable attention in the WSN community. Since its original debut, improvements in several directions have been proposed. Dil & Havinga (2011) drops the requirement that a low frequency interference signal need to be synthesized through tuning and copes with the problem taking a stochastic approach. Contrary, Amundson et al. (2010) assumes a priori knowledge of certain node locations and transforms the range estimation into a bearing estimation problem. RI phase measurement based techniques generally outperform other currently available methods that use no specialized hardware, such as RSS-based methods. However, they also require more sophisticated ranging measurements and computationally more intensive fusion of the ranging data as shown in Figure 1. Furthermore, as multipath propagation heavily impacts the phase measurement accuracy, their performance degrades significantly in indoor environments.


Table 1. Comparison of received signal strength (RSS) and signal phase based localization

While the above techniques measure either the signal strength or phase, radio-interferomentric methods exploiting the Doppler effect deduce information from the signal frequency. The measured frequency change of the radio signal produced by a moving transmitter is a function of its velocity relative to the receiver. In a WSN with a few static nodes with known locations, this relative velocity can be used to track mobile transmitters Kusý et al. (2007). Inverting the roles, the Doppler-shift technique is used for localization Lédeczi et al. (2008): the technique replaces the static receivers with rotating transmitters around a known point with a known angular speed. The receiver nodes, in turn, can calculate their bearings from these anchors. RI is applied in both cases to allow precise measurement of the frequency change on resource constrained WSN nodes. Unfortunately, these methods still suffer from the adverse effect of multipath propagation. Furthermore, the rotation of the receivers calls for special hardware, usually unavailable on WSN nodes.

This chapter describes the baseline approach of radio interferometry based localization and details the many developments that occurred since its introduction. Section 2 lays down the 2 Will-be-set-by-IN-TECH

However, the measurement is highly environment dependent. Achieving reasonable accuracy requires extensive calibration prior to ranging. Uncalibrated RSS ranging can exhibit 10-20%

A novel ranging method based on radio interferometry (RI) for static WSN node localization was introduced in Maróti et al. (2005). This RI ranging method measures the phase of the radio signals to obtain information on the relative distance between the sensor nodes. It achieves sub-meter localization accuracy and long range simultaneously. As this RI approach also relies solely on the radio chip readily available on popular, low-cost commercial off-the-shelf (COTS) sensor nodes, it became an alternative to RSS-based methods and, thus, garnered considerable attention in the WSN community. Since its original debut, improvements in several directions have been proposed. Dil & Havinga (2011) drops the requirement that a low frequency interference signal need to be synthesized through tuning and copes with the problem taking a stochastic approach. Contrary, Amundson et al. (2010) assumes a priori knowledge of certain node locations and transforms the range estimation into a bearing estimation problem. RI phase measurement based techniques generally outperform other currently available methods that use no specialized hardware, such as RSS-based methods. However, they also require more sophisticated ranging measurements and computationally more intensive fusion of the ranging data as shown in Figure 1. Furthermore, as multipath propagation heavily impacts the phase measurement accuracy, their performance degrades

**Signal strength based Signal phase based**

High (up to hundreds of meters)

errors.

significantly in indoor environments.

Accuracy Low (beyond a few

Ranging measurement complexity Very low High Localization complexity Low, trilateration High Additional hardware required No No Bandwidth requirement Low Medium

meters)

Table 1. Comparison of received signal strength (RSS) and signal phase based localization

While the above techniques measure either the signal strength or phase, radio-interferomentric methods exploiting the Doppler effect deduce information from the signal frequency. The measured frequency change of the radio signal produced by a moving transmitter is a function of its velocity relative to the receiver. In a WSN with a few static nodes with known locations, this relative velocity can be used to track mobile transmitters Kusý et al. (2007). Inverting the roles, the Doppler-shift technique is used for localization Lédeczi et al. (2008): the technique replaces the static receivers with rotating transmitters around a known point with a known angular speed. The receiver nodes, in turn, can calculate their bearings from these anchors. RI is applied in both cases to allow precise measurement of the frequency change on resource constrained WSN nodes. Unfortunately, these methods still suffer from the adverse effect of multipath propagation. Furthermore, the rotation of the receivers calls for special hardware, usually unavailable on WSN nodes.

This chapter describes the baseline approach of radio interferometry based localization and details the many developments that occurred since its introduction. Section 2 lays down the mathematical foundation of the localization method, which is then referenced throughout the rest of the chapter. It also presents the first platform using this approach, called the Radio Interferometric Positioning System (RIPS), along with a discussion of the initial results. Section 3 introduces a method for tracking mobile nodes in WSNs utilizing the phenomena of Doppler effects and radio interferometry. The technique shown in Section 4 exploits the Doppler effect in a different way to determine the location of static nodes in a WSN. It relies on a few distinct nodes with known locations (anchors) to first calculate the bearing to the rest of the nodes and to deduce location information based on the bearing estimates. The technique presented in Section 5 also calculates bearings from anchor nodes, but instead of using Doppler shifts, it uses a special arrangement of RIPS which allows for an enhanced, distributed method of node localization.

## **2. Radio interferometric positioning**

The basic idea behind interferometric localization is to use the phase information of a radio signal to measure distance. In practice, however, this is hard to achieve, because it would require (a) nanosecond-precision timing, (b) precise control over the phase of the transmitted signal, and (c) precise phase detection on the receiver. COTS radio transceivers commonly used in sensor nodes do not offer these features. The novelty of the method presented in this chapter is to use a transmitter pair and a receiver pair to eliminate the unknown initial phase of the respective local oscillators (LO), as well well as to allow for a low-speed ADC to process the incoming signal on the receiver.

The interferometric approach thus utilizes a pair of wireless sensor nodes to generate interfering radio signals by transmitting pure sinusoids at slightly different frequencies. The envelope of this composite signal has a beat frequency equal to the frequency difference of the sinusoids, as illustrated in Figure 2. It is the phase of the envelope signal that is of most interest, as it carries location related information. When measured at two receivers at the same time, the instantaneous phase difference (also referred to as the *phase offset*) of the respective interference signal envelopes is directly related to the carrier wavelength and the relative distance between the transmitters and receivers. Working with the envelope signal instead of the "raw" carrier signal has several advantages. On one hand, the envelope frequency depends on only the frequency difference of the transmitted sinusoids. This means that the received envelope frequency is the same at all receivers, and it is independent of the frequency and initial phase of the receivers' local oscillators. On the other hand, the envelope frequency can be precisely controlled (in short term), thus it is adjustable to meet the available time synchronization accuracy, limited sampling rate and processing capability of the wireless sensor nodes.

The rest of the section discusses the theory of radio interferometric localization in detail.

## **2.1 Idealistic phase-based localization**

Let us consider, hypothetically, that the transmitted signal's phase is controllable, and the received phase can be detected accurately.

These sinusoids we refer to as carrier signals, though they are not modulated.

Here, the measured phase at the receiver as a function of the distance from the receiver is

$$
\varphi = 2\pi \left(\frac{d}{\lambda}\right) \mod (2\pi),
\tag{1}
$$

precise tuning of the frequency difference *f*<sup>1</sup> − *f*<sup>2</sup> is required, which is achieved through a *calibration* process. Once the calibration is done, the received composite signal at a receiver

Interferometry in Wireless Sensor Networks 441

Interestingly, the envelope of *s*(*t*) preserves important phase information such as the *ϕ*<sup>1</sup> − *ϕ*<sup>2</sup>

Fig. 2. The radio interferometric signal generated by two independent transmitters received

phase offset the two carriers. To show this, let the carrier signal be down-mixed to the intermediate frequency *f*IF. Denoting the mean of the two transmitted frequencies (*f*<sup>1</sup> and *f*2) with *δ* = (*f*<sup>1</sup> − *f*2)/2 and assuming *f*<sup>2</sup> < *f*1, *δ* � *f*<sup>2</sup> and that the down-mixing introduces

 + *a*<sup>2</sup>

2 +

4*π*(*f*IF + *δ*)*t* + 2*ϕ*<sup>1</sup>

+ *a*1*a*<sup>2</sup> cos

 cos

<sup>2</sup> <sup>+</sup>

+ *a*<sup>2</sup> cos

cos(2*x*)

 + *a*2 2 <sup>2</sup> cos

<sup>2</sup> cos<sup>2</sup>

2*π*(*f*IF − *δ*)*t* + *ϕ*<sup>2</sup>

2*π*(*f*IF − *δ*)*t* + *ϕ*<sup>2</sup>

2*π*(*f*IF − *δ*)*t* + *ϕ*<sup>2</sup>

cos(*x* − *y*)

4*πδt* + *ϕ*<sup>1</sup> − *ϕ*<sup>2</sup>

<sup>2</sup> (6)

<sup>2</sup> , (7)

4*π*(*f*IF − *δ*)*t* + 2*ϕ*<sup>2</sup>

. (4)

. (5)

, (8)

zero phase shift, the intermediate frequency signal can be written as

The envelope of this signal is accessed by first calculating its power:

2*π*(*f*IF + *δ*)*t* + *ϕ*<sup>1</sup>

2*π*(*f*IF + *δ*)*t* + *ϕ*<sup>1</sup>

2*π*(*f*IF + *δ*)*t* + *ϕ*<sup>1</sup>

cos2(*x*) = <sup>1</sup>

cos(*x*) cos(*y*) = cos(*<sup>x</sup>* <sup>+</sup> *<sup>y</sup>*)

*s*IF(*t*) = *a*<sup>1</sup> cos

<sup>1</sup> cos<sup>2</sup>

<sup>2</sup>) + *<sup>a</sup>*<sup>2</sup> 1 <sup>2</sup> cos

4*π f*IF*t* + *ϕ*<sup>1</sup> + *ϕ*<sup>2</sup>

+ 2*a*1*a*<sup>2</sup> cos

Applying the following trigonometric identities

the square of *s*IF(*t*) can be rewritten as

+ *a*1*a*<sup>2</sup> cos

*s* 2 IF(*t*) = <sup>1</sup> 2 (*a*<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup>

*s* 2 IF(*t*) = *<sup>a</sup>*<sup>2</sup>

*s*(*t*) = *a*<sup>1</sup> cos(2*π f*1*t* + *ϕ*1) + *a*<sup>2</sup> cos(2*π f*2*t* + *ϕ*2). (3)

node, see Figure 2, takes the following form:

by a third node

where *ϕ<sup>i</sup>* is the measured phase, *d* is the receiver's distance from the transmitter and *λ<sup>i</sup>* = *c*/ *f* is the wavelength of the RF signal. A representative received phase map is is shown in Figure 1.

Fig. 1. Phase map of an unmodulated carrier signals assuming controllable transmitter phase and accurate phase detection at the receiver.

From here, we can express the distance as follows.

$$d\_{\dot{l}} = \frac{\varrho\_{\dot{l}}}{2\pi} \lambda\_{\dot{l}} + k \cdot \lambda\_{\dot{l}} \qquad k \in \mathbb{Z} \tag{2}$$

This means that the receiver is capable of measuring its distance from the transmitter with a mod (*λi*) ambiguity. By repeating the measurement at different wavelengths, it is possible to determine the unambiguous distance from the transmitter within the effective radio range of the transmitter, thereby restricting the possible receiver positions to a single circle around the receiver. Then, this is repeated for several other transmitters to gain multiple circles intersecting at a single point, at the position of the receiver.

## **2.2 Interferometric localization**

In WSNs, however, the nodes have independent LOs and neither their initial phase, nor their relative phase to each other can be assumed to be known. Maróti et al. (2005) suggests the use of transmitter and receiver pairs along with radio interferometry to deduct location information from the phase of propagating radio waves without assumptions on the LO initial phases.

### **2.2.1 Phase measurement**

A ranging measurement starts with two transmitters emitting high-frequency pure sinusoid radio signals with slightly different frequencies *f*<sup>1</sup> and *f*2. Due to variations of the LO crystals, 4 Will-be-set-by-IN-TECH

where *ϕ<sup>i</sup>* is the measured phase, *d* is the receiver's distance from the transmitter and *λ<sup>i</sup>* = *c*/ *f* is the wavelength of the RF signal. A representative received phase map is is shown in

Fig. 1. Phase map of an unmodulated carrier signals assuming controllable transmitter phase

This means that the receiver is capable of measuring its distance from the transmitter with a mod (*λi*) ambiguity. By repeating the measurement at different wavelengths, it is possible to determine the unambiguous distance from the transmitter within the effective radio range of the transmitter, thereby restricting the possible receiver positions to a single circle around the receiver. Then, this is repeated for several other transmitters to gain multiple circles

In WSNs, however, the nodes have independent LOs and neither their initial phase, nor their relative phase to each other can be assumed to be known. Maróti et al. (2005) suggests the use of transmitter and receiver pairs along with radio interferometry to deduct location information from the phase of propagating radio waves without assumptions on the LO initial

A ranging measurement starts with two transmitters emitting high-frequency pure sinusoid radio signals with slightly different frequencies *f*<sup>1</sup> and *f*2. Due to variations of the LO crystals,

and accurate phase detection at the receiver.

**2.2 Interferometric localization**

**2.2.1 Phase measurement**

phases.

From here, we can express the distance as follows.

intersecting at a single point, at the position of the receiver.

*di* <sup>=</sup> *<sup>ϕ</sup><sup>i</sup>* 2*π* mod (2*π*), (1)

*λ<sup>i</sup>* + *k* · *λ<sup>i</sup> k* ∈ **Z** (2)

Here, the measured phase at the receiver as a function of the distance from the receiver is

 *d λ* 

*ϕ* = 2*π*

Figure 1.

precise tuning of the frequency difference *f*<sup>1</sup> − *f*<sup>2</sup> is required, which is achieved through a *calibration* process. Once the calibration is done, the received composite signal at a receiver node, see Figure 2, takes the following form:

$$s(t) = a\_1 \cos(2\pi f\_1 t + \varphi\_1) + a\_2 \cos(2\pi f\_2 t + \varphi\_2). \tag{3}$$

Interestingly, the envelope of *s*(*t*) preserves important phase information such as the *ϕ*<sup>1</sup> − *ϕ*<sup>2</sup>

Fig. 2. The radio interferometric signal generated by two independent transmitters received by a third node

phase offset the two carriers. To show this, let the carrier signal be down-mixed to the intermediate frequency *f*IF. Denoting the mean of the two transmitted frequencies (*f*<sup>1</sup> and *f*2) with *δ* = (*f*<sup>1</sup> − *f*2)/2 and assuming *f*<sup>2</sup> < *f*1, *δ* � *f*<sup>2</sup> and that the down-mixing introduces zero phase shift, the intermediate frequency signal can be written as

$$s\_{\rm IF}(t) = a\_1 \cos \left(2\pi (f\_{\rm IF} + \delta)t + \varphi\_1 \right) + a\_2 \cos \left(2\pi (f\_{\rm IF} - \delta)t + \varphi\_2 \right). \tag{4}$$

The envelope of this signal is accessed by first calculating its power:

$$\begin{split} s\_{\text{IF}}^2(t) &= a\_1^2 \cos^2 \left( 2\pi (f\_{\text{IF}} + \delta)t + \varphi\_1 \right) + a\_2^2 \cos^2 \left( 2\pi (f\_{\text{IF}} - \delta)t + \varphi\_2 \right) \\ &+ 2a\_1 a\_2 \cos \left( 2\pi (f\_{\text{IF}} + \delta)t + \varphi\_1 \right) \cos \left( 2\pi (f\_{\text{IF}} - \delta)t + \varphi\_2 \right). \end{split} \tag{5}$$

Applying the following trigonometric identities

$$\cos^2(\mathbf{x}) = \frac{1}{2} + \frac{\cos(2\mathbf{x})}{2} \tag{6}$$

$$
\cos(\mathbf{x})\cos(y) = \frac{\cos(\mathbf{x}+y)}{2} + \frac{\cos(\mathbf{x}-y)}{2},
\tag{7}
$$

the square of *s*IF(*t*) can be rewritten as

$$\begin{split} s\_{\rm IF}^2(t) &= \frac{1}{2} (a\_1^2 + a\_2^2) + \frac{a\_1^2}{2} \cos\left(4\pi (f\_{\rm IF} + \delta)t + 2\varphi\_1\right) + \frac{a\_2^2}{2} \cos\left(4\pi (f\_{\rm IF} - \delta)t + 2\varphi\_2\right) \\ &+ a\_1 a\_2 \cos\left(4\pi f\_{\rm IF} t + \varphi\_1 + \varphi\_2\right) + a\_1 a\_2 \cos\left(4\pi \delta t + \varphi\_1 - \varphi\_2\right), \end{split} \tag{8}$$

after sufficient amount of time, that is when *t* is greater than *tA* + *dAY*/*c* and *tB* + *dBY*/*c*. Using

Interferometry in Wireless Sensor Networks 443

+ 2*π fA*(*tA* + *dAD*/*c*) − 2*π fB*(*tB* + *dBD*/*c*) = 2*π fA*/*c* · (*dAD* − *dAC*) + 2*π fB*/*c* · (*dBC* − *dBD*).

> *dBC* − *dBD c*/ *fB*

Due to the limited range of wireless sensor nodes and their high carrier frequency relative to the envelope frequency, the above formula of the measured relative phase offset can be simplified. Consider Figure 3 again, where nodes *A* and *B* transmit pure sine waves at two close frequencies *fA* > *fB*, and two other nodes *C* and *D* measure the envelope of the signal.

Assuming that *fA* − *fB* < 2 kHz and *dAC*, *dAD*, *dBC*, *dBD* ≤ 1 km, it follows that *c*/*δ* ≥ 300 km and thus the second term can be neglected. Now, if for any four nodes *A*, *B*, *C* and *D* we

*dABCD*

Thus, if adequately precise time synchronization is available, the receiver can effectively

Similarly to Figure 1, the phase ambiguity corresponds to equi-phase-offset curves (or surfaces in three dimension) in Figure 4. Note, that while Figure 1 plots the phase of the signal transmitted by a single transmitter, Figure 4 depicts the phase offset of the interference signal transmitted by a transmitter pair relative to that of a reference receiver. That is, the equi-phase-offset curves with respect to a reference receiver are multiple hyperbolas in Figure 4. If the phase offset measurement is repeated at *i* different carrier wavelengths *λ<sup>i</sup>* for the same set of nodes *A*, *B*, *C* and *D*, we get the following set of equations by reorganizing

Now consider the two receivers *C* and *D*. The relative phase offset of *rC*(*t*) and *rD*(*t*) is

+

Using the notation *δ* = (*fA* − *fB*)/2, the relative phase offset of *rC*(*t*) and *rD*(*t*) is

*<sup>c</sup>*/ *<sup>f</sup>* <sup>+</sup> <sup>2</sup>*<sup>π</sup>*

*ϑ<sup>Y</sup>* = −2*π fA*(*tA* + *dAY*/*c*) + 2*π fB*(*tB* + *dBY*/*c*). (11)

*ϑ<sup>C</sup>* − *ϑ<sup>D</sup>* = −2*π fA*(*tA* + *dAC*/*c*) + 2*π fB*(*tB* + *dBC*/*c*) (12)

*dAD* − *dAC* − *dBC* + *dBD*

*dABCD* = *dAD* − *dBD* + *dBC* − *dAC*, (14)

*<sup>c</sup>*/ *<sup>f</sup>* (mod 2*π*). (15)

*<sup>λ</sup>* <sup>+</sup> *<sup>k</sup><sup>λ</sup>* (*<sup>k</sup>* <sup>∈</sup> <sup>I</sup>) (16)

(mod 2*π*). (13)

*<sup>c</sup>*/*<sup>δ</sup>* (mod 2*π*).

Equation 9, the absolute phase offset of the envelope signal *rY*(*t*) is

 *dAD* − *dAC c*/ *fA*

*dAD* − *dAC* + *dBC* − *dBD*

then for any frequency *f* the relative phase offset simplifies to

*ϑABCD* = 2*π*

measure *ϑABCD* and calculate *dABCD* with a *λ* = *c*/ *f* wave length ambiguity.

*dABCD* <sup>=</sup> *<sup>ϑ</sup>ABCD*

2*π*

From this the relative phase offset of *rC*(*t*) and *rD*(*t*) is

2*π*

*ϑ<sup>C</sup>* − *ϑ<sup>D</sup>* = 2*π*

Equation 15:

define a quantity called the *q-range* as

where the components are either at zero frequency (DC), 2(*f*IF ± *δ*), 2 *f*IF or 2*δ*. Removing the DC, and the 2 *f*IF ± *δ* and 2 *f*IF double-IF frequency components by appropriate filtering we obtain the envelope signal

$$r(t) = a\_1 a\_2 \cos\left(2\pi(2\delta)t + \varphi\_1 - \varphi\_2\right),\tag{9}$$

where the frequency of *r*(*t*) is 2*δ* = *f*<sup>1</sup> − *f*<sup>2</sup> and the *ϕ*<sup>1</sup> − *ϕ*<sup>2</sup> phase corresponds to the phase offset of the two carrier signals. That is, by measuring the phase of the the envelope signal, *r*(*t*), the phase difference of the two carriers can be obtained *independently of the receiver LO initial phase*.

## **2.2.2 Q-range estimation**

The carrier phase offset in the form of Equation 9 is unsuitable to deduct location information directly. Interstingly, however, when it is measured at two different nodes in a time synchronized manner, it allows to gain information on the relative distances between the transmitters and receivers.

To show this, consider Figure 3, where transmitters *A* and *B* generate the interfering carrier signals and receivers *C* and *D* extract the envelope signals *rC*(*t*) and *rD*(*t*), respectively.

Fig. 3. Radio interferometric ranging performed by a pair of transmitters and a pair of receivers.

Let *Y* be a receiver node, and let *A* and *B* be the two transmitters. Let *tA* and *tB* be the time when node *A* and *B* start to transmit, respectively. Furthermore, let *aAY* and *aBY* the amplitude of the attenuated signal as received at node *Y* from node *A* and *B*, respectively. The received composite signal at node *Y* is

$$\begin{split} s\_Y(t) &= a\_{AY} \cos \left( 2\pi f\_A (t - t\_A - d\_{AY}/c) \right) \\ &+ a\_{BY} \cos \left( 2\pi f\_B (t - t\_B - d\_{BY}/c) \right) \\ &= a\_{AY} \cos \left( 2\pi f\_A t - 2\pi f\_A (t\_A + d\_{AY}/c) \right) \\ &+ a\_{BY} \cos \left( 2\pi f\_B t - 2\pi f\_B (t\_B + d\_{BY}/c) \right) \end{split} \tag{10}$$

6 Will-be-set-by-IN-TECH

where the components are either at zero frequency (DC), 2(*f*IF ± *δ*), 2 *f*IF or 2*δ*. Removing the DC, and the 2 *f*IF ± *δ* and 2 *f*IF double-IF frequency components by appropriate filtering we

where the frequency of *r*(*t*) is 2*δ* = *f*<sup>1</sup> − *f*<sup>2</sup> and the *ϕ*<sup>1</sup> − *ϕ*<sup>2</sup> phase corresponds to the phase offset of the two carrier signals. That is, by measuring the phase of the the envelope signal, *r*(*t*), the phase difference of the two carriers can be obtained *independently of the receiver LO*

The carrier phase offset in the form of Equation 9 is unsuitable to deduct location information directly. Interstingly, however, when it is measured at two different nodes in a time synchronized manner, it allows to gain information on the relative distances between the

To show this, consider Figure 3, where transmitters *A* and *B* generate the interfering carrier signals and receivers *C* and *D* extract the envelope signals *rC*(*t*) and *rD*(*t*), respectively.

Fig. 3. Radio interferometric ranging performed by a pair of transmitters and a pair of

*sY*(*t*) = *aAY* cos

+ *aBY* cos

+ *aBY* cos

= *aAY* cos

Let *Y* be a receiver node, and let *A* and *B* be the two transmitters. Let *tA* and *tB* be the time when node *A* and *B* start to transmit, respectively. Furthermore, let *aAY* and *aBY* the amplitude of the attenuated signal as received at node *Y* from node *A* and *B*, respectively. The received

2*π fA*(*t* − *tA* − *dAY*/*c*)

2*π fB*(*t* − *tB* − *dBY*/*c*)

2*π fAt* − 2*π fA*(*tA* + *dAY*/*c*)

2*π fBt* − 2*π fB*(*tB* + *dBY*/*c*)

2*π*(2*δ*)*t* + *ϕ*<sup>1</sup> − *ϕ*<sup>2</sup>

, (9)

(10)

*r*(*t*) = *a*1*a*<sup>2</sup> cos

obtain the envelope signal

**2.2.2 Q-range estimation**

transmitters and receivers.

*initial phase*.

receivers.

composite signal at node *Y* is

after sufficient amount of time, that is when *t* is greater than *tA* + *dAY*/*c* and *tB* + *dBY*/*c*. Using Equation 9, the absolute phase offset of the envelope signal *rY*(*t*) is

$$\mathfrak{d}\_{Y} = -2\pi f\_{A}(t\_{A} + d\_{AY}/c) + 2\pi f\_{B}(t\_{B} + d\_{BY}/c). \tag{11}$$

Now consider the two receivers *C* and *D*. The relative phase offset of *rC*(*t*) and *rD*(*t*) is

$$
\begin{split}
\boldsymbol{\theta\_{C}} - \boldsymbol{\theta\_{D}} &= -2\pi f\_{A} (\boldsymbol{t}\_{A} + \boldsymbol{d}\_{A\complement}/\boldsymbol{c}) + 2\pi f\_{B} (\boldsymbol{t}\_{B} + \boldsymbol{d}\_{B\complement}/\boldsymbol{c}) \\ &+ 2\pi f\_{A} (\boldsymbol{t}\_{A} + \boldsymbol{d}\_{AD}/\boldsymbol{c}) - 2\pi f\_{B} (\boldsymbol{t}\_{B} + \boldsymbol{d}\_{BD}/\boldsymbol{c}) \\ &= 2\pi f\_{A}/\boldsymbol{c} \cdot (\boldsymbol{d}\_{AD} - \boldsymbol{d}\_{AC}) + 2\pi f\_{B}/\boldsymbol{c} \cdot (\boldsymbol{d}\_{BC} - \boldsymbol{d}\_{BD}).
\end{split}
\tag{12}
$$

From this the relative phase offset of *rC*(*t*) and *rD*(*t*) is

$$2\pi \left( \frac{d\_{AD} - d\_{AC}}{c/f\_A} + \frac{d\_{BC} - d\_{BD}}{c/f\_B} \right) \pmod{2\pi}.\tag{13}$$

Due to the limited range of wireless sensor nodes and their high carrier frequency relative to the envelope frequency, the above formula of the measured relative phase offset can be simplified. Consider Figure 3 again, where nodes *A* and *B* transmit pure sine waves at two close frequencies *fA* > *fB*, and two other nodes *C* and *D* measure the envelope of the signal. Using the notation *δ* = (*fA* − *fB*)/2, the relative phase offset of *rC*(*t*) and *rD*(*t*) is

$$
\theta\_{\text{C}} - \theta\_{\text{D}} = 2\pi \frac{d\_{AD} - d\_{AC} + d\_{BC} - d\_{BD}}{c/f} + 2\pi \frac{d\_{AD} - d\_{AC} - d\_{BC} + d\_{BD}}{c/\delta} \pmod{2\pi}.
$$

Assuming that *fA* − *fB* < 2 kHz and *dAC*, *dAD*, *dBC*, *dBD* ≤ 1 km, it follows that *c*/*δ* ≥ 300 km and thus the second term can be neglected. Now, if for any four nodes *A*, *B*, *C* and *D* we define a quantity called the *q-range* as

$$d\_{ABCD} = d\_{AD} - d\_{BD} + d\_{BC} - d\_{AC\prime} \tag{14}$$

then for any frequency *f* the relative phase offset simplifies to

$$
\theta\_{ABCD} = 2\pi \frac{d\_{ABCD}}{c/f} \pmod{2\pi}.\tag{15}
$$

Thus, if adequately precise time synchronization is available, the receiver can effectively measure *ϑABCD* and calculate *dABCD* with a *λ* = *c*/ *f* wave length ambiguity.

Similarly to Figure 1, the phase ambiguity corresponds to equi-phase-offset curves (or surfaces in three dimension) in Figure 4. Note, that while Figure 1 plots the phase of the signal transmitted by a single transmitter, Figure 4 depicts the phase offset of the interference signal transmitted by a transmitter pair relative to that of a reference receiver. That is, the equi-phase-offset curves with respect to a reference receiver are multiple hyperbolas in Figure 4. If the phase offset measurement is repeated at *i* different carrier wavelengths *λ<sup>i</sup>* for the same set of nodes *A*, *B*, *C* and *D*, we get the following set of equations by reorganizing Equation 15:

$$d\_{ABCD} = \frac{\theta\_{ABCD}}{2\pi} \lambda + k\lambda \text{ (}k \in \mathbb{I}\text{)}\tag{16}$$

**2.3 Platform description and results**

signal is limited to 9 kHz by the ADC found on the MICA2.

no a priori knowledge on any of the node locations.

**3. Doppler shift-based tracking**

offers an alternative.

velocity.

The first interferometry based localization system, the Radio Interferometric Localization System, was presented in Maróti et al. (2005). The RIPS uses the popular MICA2 wireless sensor nodes equipped with CC1000 radio chip Texas Instruments (2007a). The fine grained LO tuning capability of the CC1000 allows to precisely set the interference frequency to 300-700 Hz in the 433 MHz band. On the downside, the sampling rate of the interference

Interferometry in Wireless Sensor Networks 445

Nonetheless, this relatively low sampling rate proves to be sufficient for accurate phase offset measurements when accompanied with precise, microsecond order time synchronization among the independent sensor nodes. In RIPS, two transmitter nodes generate the interference signal and the receiver nodes measure its envelope phase in a time-synchronized manner, at particular time instant. This is repeated on 11 different radio channels in the 400 MHz to 430 MHz range. The measured phase values are sent to a PC using WSN communication infrastructure. (Notice that the CC1000 radio chip is used for both localization and communication.) For a given pair of transmitters, the PC calculates the phase offsets of all possible pairs of receivers. As the number of participating recivers, and thus the phase measurements, is limited only by the communication range, the PC calculates the q-ranges for the same transmitter pair and all possible combination of the receivers with valid phase measurements. The whole process is then repeated with different combination of transmitters to acquire a large number of q-ranges. The q-ranges are then used as input for a genetic algorithm (GA) to optimize the heavily non-linear localization problem expressed by Equation 18. The output of the GA is the relative coordinates of the sensor nodes, assuming

The 16-node experiment on a 18×18 m outdoors area showed that RIPS was able to localize the nodes with an average error less than 5 cm. For that, 240 different transmitter combinations

Tracking of mobile sensor nodes in a WSN has been another active area of research in the past years. It is typically more challenging that localizing static nodes as the latency needs to be low enough to keep up with the mobility. On the other hand, estimating the current location of a node is aided by the already obtained estimate of the previous location. Similarly to static localization, a number of different methods have been proposed in the literature. When the most natural solution, GPS is not available due to cost or power constraints, interferometry

The original idea of the Doppler shift-based tracking presented in Kusý et al. (2007) assumes a set of stationary infrastructure nodes, *anchor nodes*, and another set of *mobile nodes* that need to be tracked – both with known locations initially. The tracking system aims for power-efficiency by remaining inactive as long as the tracked nodes stay still. Tracked nodes are responsible for maintaining their location, detecting their movement and invoking the tracking service only when movement is detected. Once a tracked node reports movement, the tracking system becomes active and assists the mobile node to determine its location and

were used and the overall localization process took approximately 80 minutes.

Fig. 4. Map of phase difference with respect to a reference receiver at (0,-4), when the interference signal is created by two independent transmitters located at (-2,0) and (2,0), respectively.

To compute the q-range estimate *dABCD*, we have to solve not only for *dABCD*, but also for integers *ki*. Clearly, no closed form solution exists, given the integer ambiguity in the equations. However, direct search techniques are possible, since in practice the q-range is constrained by the effective radio communications range, thereby restricting the search space. In practice, the measured q-range is always between −2*r* and 2*r*, *r* being the radio's maximum transmission distance. Kusý et al. (2006) describe a search heuristic that finds the q-range by minimizing the error of the q-range estimate with respect to the available phase offset measurements:

$$\mathcal{d}\_{ABCD} = \underset{d\_{ABCD} \in [-2r, 2r]}{\text{arg min}} \sum\_{i=1}^{N} \left| \frac{\varrho\_i}{2\pi} \lambda\_i + k\_i \lambda\_i - \hat{d}\_{ABCD} \right| \tag{17}$$

### **2.2.3 Position estimation**

Once the necessary amount of q-ranges is obtained from the radio-interferometric phase measurements, the position of the individual nodes is determined by finding the optimum of a set of constrained non-linear equations. For this optimization problem Maróti et al. (2005) suggests to use least-squares (LS) error fitting of the *j* q-range measurements:

$$\hat{\mathbf{y}}(\hat{\mathbf{x}},\hat{\mathbf{y}}) = \underset{\mathbf{x},\mathbf{\hat{y}}}{\text{arg min}} \sum\_{j=1}^{M} (d\_{ABCD,j} - \hat{d}\_{ABCD,j})^2. \tag{18}$$

Maróti et al. (2005) describe a genetic search algorithm to localize the network. Dil & Havinga (2011) observed that LS approach is highly suspectible to outliers, and propose to use the Least Absolute Deviation (LAD) instead:

$$\hat{\mathbf{x}}(\hat{\mathbf{x}},\hat{\mathbf{y}}) = \operatorname\*{arg\,min}\_{\hat{\mathbf{x}},\hat{\mathbf{y}}} \sum\_{j=1}^{M} |d\_{ABCD,j} - \hat{d}\_{ABCD,j}|.\tag{19}$$

## **2.3 Platform description and results**

8 Will-be-set-by-IN-TECH

Fig. 4. Map of phase difference with respect to a reference receiver at (0,-4), when the interference signal is created by two independent transmitters located at (-2,0) and (2,0),

<sup>ˆ</sup>*dABCD* = arg min

<sup>ˆ</sup>*dABCD*∈[−2*r*,2*r*]

suggests to use least-squares (LS) error fitting of the *j* q-range measurements:

*x*ˆ,*y*ˆ

*x*ˆ,*y*ˆ

(*x*ˆ, *y*ˆ) = arg min

(*x*ˆ, *y*ˆ) = arg min

To compute the q-range estimate *dABCD*, we have to solve not only for *dABCD*, but also for integers *ki*. Clearly, no closed form solution exists, given the integer ambiguity in the equations. However, direct search techniques are possible, since in practice the q-range is constrained by the effective radio communications range, thereby restricting the search space. In practice, the measured q-range is always between −2*r* and 2*r*, *r* being the radio's maximum transmission distance. Kusý et al. (2006) describe a search heuristic that finds the q-range by minimizing the error of the q-range estimate with respect to the available phase offset

> *N* ∑ *i*=1 *ϕi* 2*π*

Once the necessary amount of q-ranges is obtained from the radio-interferometric phase measurements, the position of the individual nodes is determined by finding the optimum of a set of constrained non-linear equations. For this optimization problem Maróti et al. (2005)

> *M* ∑ *j*=1

Maróti et al. (2005) describe a genetic search algorithm to localize the network. Dil & Havinga (2011) observed that LS approach is highly suspectible to outliers, and propose to use the Least

> *M* ∑ *j*=1

*<sup>λ</sup><sup>i</sup>* <sup>+</sup> *kiλ<sup>i</sup>* <sup>−</sup> <sup>ˆ</sup>*dABCD*

 

(*dABCD*,*<sup>j</sup>* <sup>−</sup> <sup>ˆ</sup>*dABCD*,*j*)2. (18)

<sup>|</sup>*dABCD*,*<sup>j</sup>* <sup>−</sup> <sup>ˆ</sup>*dABCD*,*j*|. (19)

(17)

respectively.

measurements:

**2.2.3 Position estimation**

Absolute Deviation (LAD) instead:

The first interferometry based localization system, the Radio Interferometric Localization System, was presented in Maróti et al. (2005). The RIPS uses the popular MICA2 wireless sensor nodes equipped with CC1000 radio chip Texas Instruments (2007a). The fine grained LO tuning capability of the CC1000 allows to precisely set the interference frequency to 300-700 Hz in the 433 MHz band. On the downside, the sampling rate of the interference signal is limited to 9 kHz by the ADC found on the MICA2.

Nonetheless, this relatively low sampling rate proves to be sufficient for accurate phase offset measurements when accompanied with precise, microsecond order time synchronization among the independent sensor nodes. In RIPS, two transmitter nodes generate the interference signal and the receiver nodes measure its envelope phase in a time-synchronized manner, at particular time instant. This is repeated on 11 different radio channels in the 400 MHz to 430 MHz range. The measured phase values are sent to a PC using WSN communication infrastructure. (Notice that the CC1000 radio chip is used for both localization and communication.) For a given pair of transmitters, the PC calculates the phase offsets of all possible pairs of receivers. As the number of participating recivers, and thus the phase measurements, is limited only by the communication range, the PC calculates the q-ranges for the same transmitter pair and all possible combination of the receivers with valid phase measurements. The whole process is then repeated with different combination of transmitters to acquire a large number of q-ranges. The q-ranges are then used as input for a genetic algorithm (GA) to optimize the heavily non-linear localization problem expressed by Equation 18. The output of the GA is the relative coordinates of the sensor nodes, assuming no a priori knowledge on any of the node locations.

The 16-node experiment on a 18×18 m outdoors area showed that RIPS was able to localize the nodes with an average error less than 5 cm. For that, 240 different transmitter combinations were used and the overall localization process took approximately 80 minutes.

## **3. Doppler shift-based tracking**

Tracking of mobile sensor nodes in a WSN has been another active area of research in the past years. It is typically more challenging that localizing static nodes as the latency needs to be low enough to keep up with the mobility. On the other hand, estimating the current location of a node is aided by the already obtained estimate of the previous location. Similarly to static localization, a number of different methods have been proposed in the literature. When the most natural solution, GPS is not available due to cost or power constraints, interferometry offers an alternative.

The original idea of the Doppler shift-based tracking presented in Kusý et al. (2007) assumes a set of stationary infrastructure nodes, *anchor nodes*, and another set of *mobile nodes* that need to be tracked – both with known locations initially. The tracking system aims for power-efficiency by remaining inactive as long as the tracked nodes stay still. Tracked nodes are responsible for maintaining their location, detecting their movement and invoking the tracking service only when movement is detected. Once a tracked node reports movement, the tracking system becomes active and assists the mobile node to determine its location and velocity.

the movement of a transmitting mobile node. The radio interferometric Doppler shift measurement begins with an anchor node and the mobile node to be localized transmitting pure sinusoid waveforms at slightly different frequencies. Consider Figure 6, where *A* denotes an anchor node and *M* the mobile node, both acting as transmitters, and the rest as receiver anchor nodes. Let *M* and *A* transmit an unmodulated carrier with frequencies *fM*, and *FA* respectively, such that *fM* > *fA*. The two carriers interfere with each other and create a signal with an envelope frequency of *fM* − *fA*. This interference signal is measured by a number of anchor nodes *Ai*. As *M* moves among the anchor nodes *Ai*, they observe *fM* to be Doppler

Interferometry in Wireless Sensor Networks 447

Transmitter *A* is stationary and its singal is not affected by the doppler shift, thus the measured

*fi* <sup>=</sup> *fM* <sup>−</sup> *fA* <sup>+</sup> <sup>Δ</sup>*<sup>f</sup> <sup>i</sup>*

This allows for the calculation of the Doppler shift measured at node *Ai* and, consequently,

Fig. 6. Doppler shift-based tracking of a mobile node (M) using several stationary nodes

Let �*<sup>v</sup>* denote the velocity of the tracked mobile node *<sup>M</sup>* and �*ui* <sup>=</sup> −−→*AiM*/�*AiM*� the unit length vector pointing from anchor node *Ai* to *M*. The relative speed of *Ai* and *M* can then be defined

where *vi* is a scalar value with positive sign if �*v* points away from *Ai* and negative sign

The Doppler equation states that if *f* is the frequency of the transmitted radio signal, *c* is the speed of light, and *v* � *c* is the speed of the source with respect to the observer (with negative sign of *v* if the source was going towards the observer), then the observed frequency

*<sup>M</sup>* depends on the relative speed of *M* to *Ai*.

*<sup>M</sup>*. (20)

*vi* = �*ui* ·�*v*, (21)

Δ*f* = *f* � − *f* = −*v f* /*c*. (22)

shifted by Δ*f <sup>i</sup>*

(A-E).

otherwise.

as the following dot product

is *f* � = (1 − *v*/*c*)*f* . Therefore,

*<sup>M</sup>*, where the value of <sup>Δ</sup>*<sup>f</sup> <sup>i</sup>*

envelope frequency at receiver node *Ai* becomes

the relative speed of the tracked node.

On the fundamental level, the provided tracking service is based on the phenomenon of Doppler effect or Doppler shift. The movement of a radio signal transmitter relative to a stationary receiver introduces a change in the observed signal frequency as illustrated in Figure 5.

Fig. 5. Illustration of the Doppler frequency shift in the signal introduced by the relative velocity of the moving transmitter to the stationary receiver.

The velocity of the mobile transmitter is directly related to the change in the received signal frequency, thus, by precisely measuring the frequency deviation, the velocity can be estimated. In order to improve measurement accuracy by canceling the receiver side LO uncertainty, an interference signal similar to the one introduced in Section 2 is utilized. When the tracking service is invoked, the mobile node and *one* anchor node transmit their unmodulated carrier at close frequencies. The rest of the anchor nodes measure the frequency change in the interference signal envelope and report the estimates to a central unit. The central unit fuses frequency change information to calculate the location and velocity of the mobile node simultaneously. The derived location and velocity estimates are then fed to a Kalman filter to keep the state information of the nodes up-to-date.

The remainder of the section starts by detailing the interferometry based Doppler shift measurement. That is followed by the description of the location and velocity estimation formulated as a non-linear optimization problem, and the Kalman filter used for tracking. Finally, a prototype implementation on the CC1000 radio chip based MICA2 platform is shown.

## **3.1 Interferometric tracking**

Radio interferometric tracking based on doppler shifts is naturally divided into the *measurement* of doppler shifts and *tracking* phases, that is, the fusion of the doppler shift measurements. Throughout the discussion, nodes are classified either as *anchor nodes* with known positions, or as *mobile nodes* that need to be localized. The interferometry based Doppler shift measurement utilizes an interference signal similar to the one described in Section 2. However, while interferometric ranging estimates the phase of the interference signal, interferometric tracking exploits the frequency shift of the same signal induced by 10 Will-be-set-by-IN-TECH

On the fundamental level, the provided tracking service is based on the phenomenon of Doppler effect or Doppler shift. The movement of a radio signal transmitter relative to a stationary receiver introduces a change in the observed signal frequency as illustrated in

Fig. 5. Illustration of the Doppler frequency shift in the signal introduced by the relative

The velocity of the mobile transmitter is directly related to the change in the received signal frequency, thus, by precisely measuring the frequency deviation, the velocity can be estimated. In order to improve measurement accuracy by canceling the receiver side LO uncertainty, an interference signal similar to the one introduced in Section 2 is utilized. When the tracking service is invoked, the mobile node and *one* anchor node transmit their unmodulated carrier at close frequencies. The rest of the anchor nodes measure the frequency change in the interference signal envelope and report the estimates to a central unit. The central unit fuses frequency change information to calculate the location and velocity of the mobile node simultaneously. The derived location and velocity estimates are then fed to a Kalman filter to

The remainder of the section starts by detailing the interferometry based Doppler shift measurement. That is followed by the description of the location and velocity estimation formulated as a non-linear optimization problem, and the Kalman filter used for tracking. Finally, a prototype implementation on the CC1000 radio chip based MICA2 platform is

Radio interferometric tracking based on doppler shifts is naturally divided into the *measurement* of doppler shifts and *tracking* phases, that is, the fusion of the doppler shift measurements. Throughout the discussion, nodes are classified either as *anchor nodes* with known positions, or as *mobile nodes* that need to be localized. The interferometry based Doppler shift measurement utilizes an interference signal similar to the one described in Section 2. However, while interferometric ranging estimates the phase of the interference signal, interferometric tracking exploits the frequency shift of the same signal induced by

velocity of the moving transmitter to the stationary receiver.

keep the state information of the nodes up-to-date.

Figure 5.

shown.

**3.1 Interferometric tracking**

the movement of a transmitting mobile node. The radio interferometric Doppler shift measurement begins with an anchor node and the mobile node to be localized transmitting pure sinusoid waveforms at slightly different frequencies. Consider Figure 6, where *A* denotes an anchor node and *M* the mobile node, both acting as transmitters, and the rest as receiver anchor nodes. Let *M* and *A* transmit an unmodulated carrier with frequencies *fM*, and *FA* respectively, such that *fM* > *fA*. The two carriers interfere with each other and create a signal with an envelope frequency of *fM* − *fA*. This interference signal is measured by a number of anchor nodes *Ai*. As *M* moves among the anchor nodes *Ai*, they observe *fM* to be Doppler shifted by Δ*f <sup>i</sup> <sup>M</sup>*, where the value of <sup>Δ</sup>*<sup>f</sup> <sup>i</sup> <sup>M</sup>* depends on the relative speed of *M* to *Ai*.

Transmitter *A* is stationary and its singal is not affected by the doppler shift, thus the measured envelope frequency at receiver node *Ai* becomes

$$f\_i = f\_M - f\_A + \Delta f\_M^i. \tag{20}$$

This allows for the calculation of the Doppler shift measured at node *Ai* and, consequently, the relative speed of the tracked node.

Fig. 6. Doppler shift-based tracking of a mobile node (M) using several stationary nodes (A-E).

Let �*<sup>v</sup>* denote the velocity of the tracked mobile node *<sup>M</sup>* and �*ui* <sup>=</sup> −−→*AiM*/�*AiM*� the unit length vector pointing from anchor node *Ai* to *M*. The relative speed of *Ai* and *M* can then be defined as the following dot product

$$
\vec{v}\_{l} = \vec{u}\_{l} \cdot \vec{v}\_{\prime} \tag{21}
$$

where *vi* is a scalar value with positive sign if �*v* points away from *Ai* and negative sign otherwise.

The Doppler equation states that if *f* is the frequency of the transmitted radio signal, *c* is the speed of light, and *v* � *c* is the speed of the source with respect to the observer (with negative sign of *v* if the source was going towards the observer), then the observed frequency is *f* � = (1 − *v*/*c*)*f* . Therefore,

$$
\Delta f = f' - f = -vf/c.\tag{22}
$$

Using *f* = *fM* − *fA* and *λ<sup>M</sup>* = *c*/ *fM*, the node *Ai* observes the interference signal with frequency

$$f\_{\mathbf{i}} = \widehat{f} - v\_{\mathbf{i}}/\lambda\_{\mathbf{M}}\tag{23}$$

Fig. 7. Vector decomposition of the mobile node velocity *v* = *vt* +*vr* as observed by the

Interferometry in Wireless Sensor Networks 449

As due to measurement errors, there may exist no **x** such that *H*(**x**) = **c**, the parameters are estimated by finding **<sup>x</sup>** ∈ R<sup>5</sup> such that �*H*(**x**) <sup>−</sup> **<sup>c</sup>**� is minimized. Note that components of the objective function *H* are non-linear functions, requiring the use of non-linear optimization

The Doppler shift-based tracking uses radio interferometry to enable frequency shift measurements on resource constrained WSN platforms, such as the CC1000 equipped MICA2. The latter platform is capable of synthesizing a 300-700 Hz interference frequency in the 433 MHz band on the transmitter side, and to accurately measure frequency shifts in the envelope of the interference signal despite its limited, 9 kHz, sampling rate. Therefore,

Kusý et al. (2007) suggests to find the initial locations at deployment time with an accurate and possibly computationally more expensive algorithm, such Maróti et al. (2005), and switch to Doppler shift-based tracking afterwards. Once the initial positions are available, the tracking system starts in listening mode and operates on a per-request basis. A mobile target node can invoke the tracking service of the anchor nodes by sending a request message and starting to transmit an unmodulated carrier signal. In response, one of the anchor nodes start transmitting at a close frequency, while the rest measure frequency changes in the interference signal caused by the movement of the mobile node. The relative speed is calculated from the frequency shift by Equation 23 at every anchor node, which is then fed into an extended Kalman filter (EKF) running on a PC to keep an up-to-date model of the node location.

Measurements with eight anchor nodes and a single mobile node in a 50×30 m field indicated that the Doppler shift-based tracking of a mobile node with a speed between 1 and 3 m/s can be performed with an average location error of 1.5 m, speed error of 0.14 m/s and heading error of 7.2◦. The update interval of the EKF is the sum of the 0.3 s coordination phase, 0.4 s measurement time and 1.0 s to route the measurement results to the PC, resulting in 1.7 s

*vx*(*xi* − *x*) + *vy*(*yi* − *y*)

(*xi* <sup>−</sup> *<sup>x</sup>*)<sup>2</sup> + (*yi* <sup>−</sup> *<sup>y</sup>*)<sup>2</sup> (29)

the parameter vector **x** and the known quantities *λM*, *xi*, *yi*:

*Hi*(**x**) = *f*

<sup>−</sup> <sup>1</sup> *λ<sup>M</sup>*

identical sensor nodes can be used both as anchor nodes and as mobile nodes.

receiver anchor node *A*.

**3.2 Platform description and results**

methods.

latency.

which can be measured at node *Ai*. Consequently, if the *f* difference of the two transmitted frequencies is known, the relative speed of the tracked node *M* and the anchor node *Ai* can be calculated.

Note that estimating the frequency difference *f* with sufficient accuracy becomes a problem when using low-cost radio transceivers due to LO inaccuracies. The use of radio interferometry instead of direct carrier signals partially alleviates this issue by removing the receiver side LO uncertainty. However, as the transmitter side LO may still drift, *f* is treated as an unknown parameter in the tracking phase.

In the tracking phase the Doppler shifts measured by multiple infrastructure nodes are utilized to calculate the location and the velocity of a tracked node. The tracking problem is modeled as a non-linear optimization problem where the location coordinates (*x*, *y*) and the velocity vector *v* = (*vx*, *vy*) of the tracked node need to be determined. Due to the uncertainty in the transmitted signal frequencies *fM* and *fA*, the interference frequency *f* = *fM* − *fA* is also treated as a parameter to be estimated. Consequently, the full parameter vector **x** of the optimization problem is

$$\mathbf{x} = (\mathbf{x}\_{\prime} y\_{\prime} v\_{\mathbf{x}\_{\prime}} v\_{y\_{\prime}} \widehat{f})^{\top}. \tag{24}$$

Assuming that *n* anchor nodes measure the Doppler shifted radio signal, there are *n* frequency observations *fi*. Therefore, the observation vector **c** is defined as

$$\mathbf{c} = \begin{pmatrix} f\_1, f\_2, \dots, f\_n \end{pmatrix}^\top. \tag{25}$$

The relation of the parameter vector **x** and the observation vector **c** can be formalized as a function *<sup>H</sup>* : <sup>R</sup><sup>5</sup> → R*n*, such that

$$\mathbf{c} = H(\mathbf{x}).\tag{26}$$

The function *<sup>H</sup>* is a vector-vector function consisting of *<sup>n</sup>* functions *Hi* : <sup>R</sup><sup>5</sup> → R, each of them calculating the Doppler shifted interference frequency *fi* measured at an infrastructure node *Ai*. From Equation 23, *Hi*(**x**) is defined as

$$H\_{\rm i}(\mathbf{x}) = \dot{f} - v\_{\rm i}/\lambda\_{\rm M}.\tag{27}$$

The relative speed *vi* of the mobile node *M* to an anchor node *Ai* can be calculated from the location of the two nodes and the velocity *v* of the tracked node *M*, as shown in Figure 7. Using the distributive property of the dot product and the identities *v* = *vx* +*vy* and |*ui*| = 1 the relative velocity becomes

$$
\upsilon\_i = \vec{u\_i} \cdot \vec{v} = \vec{u\_i} \cdot \vec{v}\_x + \vec{u\_i} \cdot \vec{v}\_y = |\vec{v}\_x| \cos \alpha + |\vec{v}\_y| \sin \alpha,\tag{28}
$$

where sin *α* and cos *α* can be calculated from coordinates (*x*, *y*) and (*xi*, *yi*) of nodes *M* and *Ai*, respectively. This allows for the calculation of the expected measurements *fi* = *Hi*(**x**) from 12 Will-be-set-by-IN-TECH

frequencies is known, the relative speed of the tracked node *M* and the anchor node *Ai* can be

when using low-cost radio transceivers due to LO inaccuracies. The use of radio interferometry instead of direct carrier signals partially alleviates this issue by removing the

In the tracking phase the Doppler shifts measured by multiple infrastructure nodes are utilized to calculate the location and the velocity of a tracked node. The tracking problem is modeled as a non-linear optimization problem where the location coordinates (*x*, *y*) and the velocity vector *v* = (*vx*, *vy*) of the tracked node need to be determined. Due to the uncertainty

also treated as a parameter to be estimated. Consequently, the full parameter vector **x** of the

Assuming that *n* anchor nodes measure the Doppler shifted radio signal, there are *n* frequency

**c** = (*f*1, *f*2,..., *fn*)

The relation of the parameter vector **x** and the observation vector **c** can be formalized as a

The function *<sup>H</sup>* is a vector-vector function consisting of *<sup>n</sup>* functions *Hi* : <sup>R</sup><sup>5</sup> → R, each of them calculating the Doppler shifted interference frequency *fi* measured at an infrastructure

The relative speed *vi* of the mobile node *M* to an anchor node *Ai* can be calculated from the location of the two nodes and the velocity *v* of the tracked node *M*, as shown in Figure 7. Using the distributive property of the dot product and the identities *v* = *vx* +*vy* and |*ui*| = 1

where sin *α* and cos *α* can be calculated from coordinates (*x*, *y*) and (*xi*, *yi*) of nodes *M* and *Ai*, respectively. This allows for the calculation of the expected measurements *fi* = *Hi*(**x**) from

*vi* = *u<sup>i</sup>* ·*v* = *ui* ·*vx* +*ui* ·*vy* = |*vx*| cos *α* + |*vy*| sin *α*, (28)

*Hi*(**x**) = *f*

)

**x** = (*x*, *y*, *vx*, *vy*, *f*

*fi* = *f*

receiver side LO uncertainty. However, as the transmitter side LO may still drift, *f*

in the transmitted signal frequencies *fM* and *fA*, the interference frequency *f*

observations *fi*. Therefore, the observation vector **c** is defined as

which can be measured at node *Ai*. Consequently, if the *f*

Note that estimating the frequency difference *f*

as an unknown parameter in the tracking phase.

= *fM* − *fA* and *λ<sup>M</sup>* = *c*/ *fM*, the node *Ai* observes the interference signal with

− *vi*/*λ<sup>M</sup>* (23)

with sufficient accuracy becomes a problem

difference of the two transmitted

�. (24)

�. (25)

**c** = *H*(**x**). (26)

− *vi*/*λM*. (27)

is treated

= *fM* − *fA* is

Using *f*

frequency

calculated.

optimization problem is

function *<sup>H</sup>* : <sup>R</sup><sup>5</sup> → R*n*, such that

the relative velocity becomes

node *Ai*. From Equation 23, *Hi*(**x**) is defined as

Fig. 7. Vector decomposition of the mobile node velocity *v* = *vt* +*vr* as observed by the receiver anchor node *A*.

the parameter vector **x** and the known quantities *λM*, *xi*, *yi*:

$$H\_{\bar{l}}(\mathbf{x}) = \hat{f} - \frac{1}{\lambda\_M} \frac{v\_{\mathbf{x}}(\mathbf{x}\_{\bar{l}} - \mathbf{x}) + v\_{\mathbf{y}}(y\_{\bar{l}} - y)}{\sqrt{(\mathbf{x}\_{\bar{l}} - \mathbf{x})^2 + (y\_{\bar{l}} - y)^2}} \tag{29}$$

As due to measurement errors, there may exist no **x** such that *H*(**x**) = **c**, the parameters are estimated by finding **<sup>x</sup>** ∈ R<sup>5</sup> such that �*H*(**x**) <sup>−</sup> **<sup>c</sup>**� is minimized. Note that components of the objective function *H* are non-linear functions, requiring the use of non-linear optimization methods.

## **3.2 Platform description and results**

The Doppler shift-based tracking uses radio interferometry to enable frequency shift measurements on resource constrained WSN platforms, such as the CC1000 equipped MICA2. The latter platform is capable of synthesizing a 300-700 Hz interference frequency in the 433 MHz band on the transmitter side, and to accurately measure frequency shifts in the envelope of the interference signal despite its limited, 9 kHz, sampling rate. Therefore, identical sensor nodes can be used both as anchor nodes and as mobile nodes.

Kusý et al. (2007) suggests to find the initial locations at deployment time with an accurate and possibly computationally more expensive algorithm, such Maróti et al. (2005), and switch to Doppler shift-based tracking afterwards. Once the initial positions are available, the tracking system starts in listening mode and operates on a per-request basis. A mobile target node can invoke the tracking service of the anchor nodes by sending a request message and starting to transmit an unmodulated carrier signal. In response, one of the anchor nodes start transmitting at a close frequency, while the rest measure frequency changes in the interference signal caused by the movement of the mobile node. The relative speed is calculated from the frequency shift by Equation 23 at every anchor node, which is then fed into an extended Kalman filter (EKF) running on a PC to keep an up-to-date model of the node location.

Measurements with eight anchor nodes and a single mobile node in a 50×30 m field indicated that the Doppler shift-based tracking of a mobile node with a speed between 1 and 3 m/s can be performed with an average location error of 1.5 m, speed error of 0.14 m/s and heading error of 7.2◦. The update interval of the EKF is the sum of the 0.3 s coordination phase, 0.4 s measurement time and 1.0 s to route the measurement results to the PC, resulting in 1.7 s latency.

Fig. 8. Doppler shift generated by the rotating-antenna anchor node.

Therefore, the receiver node *R* observes the projected velocity component *vp*(*t*):

*vp*(*t*) = *ur* ·*<sup>v</sup>*(*t*) = <sup>−</sup>*ωrd* sin(*ω<sup>t</sup>* <sup>+</sup> *<sup>ϕ</sup>*)

and *R* can then be defined as the following dot product

Consequently, the observed frequency change at *R* is

<sup>Δ</sup>*<sup>f</sup>* <sup>=</sup> *vp*(*t*)

expression of Δ*f* simplifies to

<sup>=</sup> <sup>−</sup>*ωr f c*

*<sup>c</sup> <sup>f</sup>* <sup>=</sup> *<sup>f</sup> c*

> lim *r*/*d*→0

The argument of the expression in Equation 34 then becomes

where

The measured frequency change is related to the movement of the transmitter through the Doppler effect, Δ*f* = *v f* /*c*, where *v* is the tangential component of the transmitter velocity relative to the receiver, *f* is the carrier frequency and *c* is the speed of light. To discuss the case where the transmitter antenna is rotating, consider Figure 8. The antenna of the anchor node, *Arot*, is rotating counterclockwise around the origin with a radius *r* and a constant angular velocity *<sup>ω</sup>*. Let *<sup>v</sup>*(*t*) denote the velocity of the rotating antenna *Arot* and *ur* <sup>=</sup> −→*RArot*/�*RArot*� the unit length vector pointing from the receiver node *R* to *Arot*. The relative speed of *Arot*

Interferometry in Wireless Sensor Networks 451

*vp*(*t*) = *ur* ·*v*(*t*), (30)

*<sup>d</sup>*<sup>2</sup> <sup>+</sup> *<sup>r</sup>*<sup>2</sup> <sup>−</sup> <sup>2</sup>*dr* cos(*ω<sup>t</sup>* <sup>+</sup> *<sup>ϕ</sup>*) (32)

sin(*ωt* + *ϕ*). (34)

. (35)

(33)

*v*(*t*) = (−*ωr* sin(*ωt* + *ϕ*), *ωr* cos(*ωt* + *ϕ*)). (31)

−*ωrd* sin(*ωt* + *ϕ*) *<sup>d</sup>*<sup>2</sup> <sup>+</sup> *<sup>r</sup>*<sup>2</sup> <sup>−</sup> <sup>2</sup>*dr* cos(*ω<sup>t</sup>* <sup>+</sup> *<sup>ϕ</sup>* <sup>−</sup> *<sup>α</sup>*)

> <sup>−</sup>*c*Δ*<sup>f</sup> <sup>ω</sup>r f*

sin(*ωt* + *ϕ* − *α*) <sup>1</sup> + (*r*/*d*)<sup>2</sup> <sup>−</sup> <sup>2</sup>(*r*/*d*) cos(*ω<sup>t</sup>* <sup>+</sup> *<sup>ϕ</sup>* <sup>−</sup> *<sup>α</sup>*).

Assuming that the rotational radius is significantly smaller than the node distance, *r* � *d*, the

Thus, instead of estimating the *ϕ* initial phase of the rotating anchor, a second receiver, a *reference* node can be used to eliminate the (*ωt* + *ϕ*) term. Let *R*<sup>2</sup> denote the reference node as

<sup>Δ</sup>*<sup>f</sup>* <sup>=</sup> <sup>−</sup>*ωr f c*

*<sup>ω</sup><sup>t</sup>* <sup>+</sup> *<sup>ϕ</sup>* <sup>−</sup> *<sup>α</sup>* <sup>=</sup> sin−<sup>1</sup>

## **4. Doppler shift-based localization**

The methods presented in Section 2 and Section 3 for localization and tracking both have the adventage that they use WSN nodes equipped with the same hardware throughout the entire system. The distinction between any two nodes is their actual role (e.g. *transmitter/receiver, mobile/anchor*) and the a priori knowledge of their location. Also, both methods call for computationally expensive non-linear optimization at the sensor fusion phase. However, by specializing certain nodes in the system the "workload" of the fusion phase can be significantly decreased.

Lédeczi et al. (2008) and Chang et al. (2008) proposed the idea to use physically *rotating anchor* nodes in WSNs for node self-localization. Their underlying approach is similar to the one used for mobile node tracking, described in Section 3, in many aspects. Anchor nodes represent key elements of the localization service and measurements rely on the Doppler effect, which in turn utilize radio interferometry. However, the Doppler effect is used in a different way. Instead of having the tracked node and an anchor node transmitting the interfering carrier signals it is two anchor nodes that transmit. One of the anchor nodes is equipped with no specialized hardware, while the other one has a "spinning" antenna. This spinning either refers to the physical rotation of a single antenna or its imitation using an antenna array. In both cases, the target nodes observe a Doppler shift in the received signal due to the actual or virtual rotational movement of the transmitter antenna. The measured Doppler shift is used by the target nodes to determine their bearing from the anchor nodes. The bearing estimates from multiple anchor node pairs are then used to determine the actual location of the target node.

The remainder of this section details the theory behind the Doppler-shift based localization methods using rotating-antenna and antenna-array anchor nodes, their first WSN implementation, and experimental results.

## **4.1 Bearing estimation with rotating-antenna anchor nodes**

The Doppler shift-based radio interferometric localization with rotating-antenna anchor nodes is divided into two distinct phases, *bearing estimation* and *localization*. In the bearing estimation phase, the nodes to be localized, the *target nodes*, measure the Doppler shift in the carrier signal introduced by the rotational movement of the anchor node antenna and deduct bearing information. The localization phase takes the bearing estimates and the anchor node locations to determine the target node location via triangulation.

The bearing estimation starts with two anchor nodes, a rotating and a fixed one, transmitting unmodulated sinusoid carriers at close frequencies, where the fixed node can be an arbitrarily selected transmitter node with known location. Receiver nodes extract the envelope of the interference signal, see Equation 9, and measure its frequency.

Note that radio interferometry is utilized to make frequency estimation feasible on resource constrained sensor nodes as any frequency change in the envelope signal equals to that of the original carrier. Thus, assuming the additional transmitter node has ideal accuracy, its presence and the details of radio interferometry will be temporarily ignored in the following discussion.

Fig. 8. Doppler shift generated by the rotating-antenna anchor node.

The measured frequency change is related to the movement of the transmitter through the Doppler effect, Δ*f* = *v f* /*c*, where *v* is the tangential component of the transmitter velocity relative to the receiver, *f* is the carrier frequency and *c* is the speed of light. To discuss the case where the transmitter antenna is rotating, consider Figure 8. The antenna of the anchor node, *Arot*, is rotating counterclockwise around the origin with a radius *r* and a constant angular velocity *<sup>ω</sup>*. Let *<sup>v</sup>*(*t*) denote the velocity of the rotating antenna *Arot* and *ur* <sup>=</sup> −→*RArot*/�*RArot*� the unit length vector pointing from the receiver node *R* to *Arot*. The relative speed of *Arot* and *R* can then be defined as the following dot product

$$
v\_p(t) = \vec{u}\_r \cdot \vec{v}(t),\tag{30}$$

where

14 Will-be-set-by-IN-TECH

The methods presented in Section 2 and Section 3 for localization and tracking both have the adventage that they use WSN nodes equipped with the same hardware throughout the entire system. The distinction between any two nodes is their actual role (e.g. *transmitter/receiver, mobile/anchor*) and the a priori knowledge of their location. Also, both methods call for computationally expensive non-linear optimization at the sensor fusion phase. However, by specializing certain nodes in the system the "workload" of the fusion phase can be

Lédeczi et al. (2008) and Chang et al. (2008) proposed the idea to use physically *rotating anchor* nodes in WSNs for node self-localization. Their underlying approach is similar to the one used for mobile node tracking, described in Section 3, in many aspects. Anchor nodes represent key elements of the localization service and measurements rely on the Doppler effect, which in turn utilize radio interferometry. However, the Doppler effect is used in a different way. Instead of having the tracked node and an anchor node transmitting the interfering carrier signals it is two anchor nodes that transmit. One of the anchor nodes is equipped with no specialized hardware, while the other one has a "spinning" antenna. This spinning either refers to the physical rotation of a single antenna or its imitation using an antenna array. In both cases, the target nodes observe a Doppler shift in the received signal due to the actual or virtual rotational movement of the transmitter antenna. The measured Doppler shift is used by the target nodes to determine their bearing from the anchor nodes. The bearing estimates from multiple anchor node pairs are then used to determine the actual location of the target

The remainder of this section details the theory behind the Doppler-shift based localization methods using rotating-antenna and antenna-array anchor nodes, their first WSN

The Doppler shift-based radio interferometric localization with rotating-antenna anchor nodes is divided into two distinct phases, *bearing estimation* and *localization*. In the bearing estimation phase, the nodes to be localized, the *target nodes*, measure the Doppler shift in the carrier signal introduced by the rotational movement of the anchor node antenna and deduct bearing information. The localization phase takes the bearing estimates and the anchor node

The bearing estimation starts with two anchor nodes, a rotating and a fixed one, transmitting unmodulated sinusoid carriers at close frequencies, where the fixed node can be an arbitrarily selected transmitter node with known location. Receiver nodes extract the envelope of the

Note that radio interferometry is utilized to make frequency estimation feasible on resource constrained sensor nodes as any frequency change in the envelope signal equals to that of the original carrier. Thus, assuming the additional transmitter node has ideal accuracy, its presence and the details of radio interferometry will be temporarily ignored in the following

**4. Doppler shift-based localization**

implementation, and experimental results.

**4.1 Bearing estimation with rotating-antenna anchor nodes**

locations to determine the target node location via triangulation.

interference signal, see Equation 9, and measure its frequency.

significantly decreased.

node.

discussion.

$$\vec{v}(t) = \left(-\omega r \sin(\omega t + \varphi), \omega r \cos(\omega t + \varphi)\right). \tag{31}$$

Therefore, the receiver node *R* observes the projected velocity component *vp*(*t*):

$$v\_p(t) = \vec{u}\_r \cdot \vec{v}(t) = \frac{-\omega r d \sin(\omega t + \varphi)}{\sqrt{d^2 + r^2 - 2dr \cos(\omega t + \varphi)}}\tag{32}$$

Consequently, the observed frequency change at *R* is

$$\begin{split} \Delta f &= \frac{v\_p(t)}{c} f = \frac{f}{c} \frac{-\omega r d \sin(\omega t + \varphi)}{\sqrt{d^2 + r^2 - 2dr \cos(\omega t + \varphi - \alpha)}} \\ &= \frac{-\omega r f}{c} \frac{\sin(\omega t + \varphi - \alpha)}{\sqrt{1 + (r/d)^2 - 2(r/d)\cos(\omega t + \varphi - \alpha)}}. \end{split} \tag{33}$$

Assuming that the rotational radius is significantly smaller than the node distance, *r* � *d*, the expression of Δ*f* simplifies to

$$\lim\_{\omega \ne d \to 0} \Delta f = \frac{-\omega rf}{c} \sin(\omega t + \varphi). \tag{34}$$

The argument of the expression in Equation 34 then becomes

$$
\omega t + \varphi - \mathfrak{a} = \sin^{-1}\left(\frac{-c\Delta f}{\omega rf}\right). \tag{35}
$$

Thus, instead of estimating the *ϕ* initial phase of the rotating anchor, a second receiver, a *reference* node can be used to eliminate the (*ωt* + *ϕ*) term. Let *R*<sup>2</sup> denote the reference node as in Figure 9. The *α* angle between the two nodes is obtained by subtracting the two arguments:

Fig. 9. Estimation of the angle between two receivers and a rotating anchor node.

$$\alpha = \sin^{-1}\left(\frac{c\Delta f\_{R\_1}}{\omega rf}\right) - \sin^{-1}\left(\frac{c\Delta f\_{R\_2}}{\omega rf}\right) \tag{36}$$

Fig. 10. Quasi doppler bearing estimation using only four discrete (N=4) steps instead of

where *ω* = 1/(*N*Δ*t*). Thus, by measuring the N phase jumps throughout a full turn of the antenna, the sin(*ωt* + *ϕ*) term can be estimated. The phase of this signal is then used to calculate the bearing to the anchor nodes in a similar fashion as described in Section 4.1.

Interferometry in Wireless Sensor Networks 453

The localization phase takes multiple angle estimates, *αi*, between the target node, rotating-antenna or antenna-array node *i* and the reference node, as depicted in Figure 11, and deduces the location of node *Q*. In the optimal case, the bearing estimates intersect at a single point. Due to measurement errors, however, bearing estimates *αi*(*i* > 2) to target node *Q* may have multiple intersections. In this case, an optimization algorithm can be used to

The Doppler shift-based SpinLoc localization system presented in Chang et al. (2008) uses MICA2 type CC1000 equipped sensor nodes both for anchor nodes and target nodes, where the rotating-antenna transmitter anchors are physically rotated on a 50 cm radius arm, at 133 revolutions per minute (RPM) by a servo motor. The measurements use the 900 MHz band to

The measurement round starts with time synchronization initiated by the base station. The rotating anchor node and a static node starts transmitting the unmodulated carriers, which is received and processed by a reference node and the target nodes. The latter two calculate the signal frequencies and send it to the base station. A PC attached to the base station then calculates the orientation of the rotating anchors, the relative bearings and the position of the

generate the 600 Hz interference signal and measure its frequency change.

continuous rotation.

**4.3 Localizaton based on bearing estimates**

estimate the most probable target node location.

**4.4 Platform description and results**

target nodes.

### **4.2 Bearing estimation with antenna-array anchor nodes**

Similarly to the rotating-antenna case, bearing estimation starts with two anchor nodes transmitting unmodulated sinusoids at close frequencies to construct the interference signal. One of these anchor nodes is equipped with multiple circularly arranged antennas, while the other one can be any node of the WSN with a priori known location. The antenna array mimics the physical rotation of a single antenna by switching between the antennas in sequence enabling only one at a time. The receivers process the envelope of the interference signal to obtain bearing information through the quasi Doppler shift caused by the imitated movement of the antenna. Note, that in case of ideal transmitters, any change in the carrier frequency introduces the same amount of frequency shift in the detected envelope signal. Therefore, the presence of the interferometry is temporarily disregarded in the following discussion, despite its heavy use in real WSN measurements. Consider an N element uniform circular array (UCA) as depicted in Figure 10. The virtual rotation of the antenna over the N locations results in a constant frequency signal with sudden phase jumps. As a corner case, when *N* → ∞ the magnitude of phase changes converge to the Doppler shift frequency, essentially giving the same results as the physically rotating antenna discussed in Section 4.1. Thus, by measuring the N discrete phase changes over a full turn of the virtual antenna, the frequency of the mimicked rotating antenna can be calculated at N different locations along the UCA. Assume that the radius of an N-element UCA is negligible compared to receiver to transmitter distance and let the angular speed of the virtually rotating antenna be *ω*. The phase change between time instants *t* and Δ*t* then becomes:

$$
\Delta\phi(t, t + \Delta t) = \frac{\int\_{t}^{t + \Delta t} \sin(\omega t)dt}{\Delta t} = \sin\left(\omega t + \frac{\Delta t}{2}\right) \frac{2\sin(\frac{\Delta t}{2})}{\omega \Delta t},\tag{37}
$$

16 Will-be-set-by-IN-TECH

in Figure 9. The *α* angle between the two nodes is obtained by subtracting the two arguments:

Fig. 9. Estimation of the angle between two receivers and a rotating anchor node.

 *c*Δ*fR*<sup>1</sup> *ωr f*

Similarly to the rotating-antenna case, bearing estimation starts with two anchor nodes transmitting unmodulated sinusoids at close frequencies to construct the interference signal. One of these anchor nodes is equipped with multiple circularly arranged antennas, while the other one can be any node of the WSN with a priori known location. The antenna array mimics the physical rotation of a single antenna by switching between the antennas in sequence enabling only one at a time. The receivers process the envelope of the interference signal to obtain bearing information through the quasi Doppler shift caused by the imitated movement of the antenna. Note, that in case of ideal transmitters, any change in the carrier frequency introduces the same amount of frequency shift in the detected envelope signal. Therefore, the presence of the interferometry is temporarily disregarded in the following discussion, despite its heavy use in real WSN measurements. Consider an N element uniform circular array (UCA) as depicted in Figure 10. The virtual rotation of the antenna over the N locations results in a constant frequency signal with sudden phase jumps. As a corner case, when *N* → ∞ the magnitude of phase changes converge to the Doppler shift frequency, essentially giving the same results as the physically rotating antenna discussed in Section 4.1. Thus, by measuring the N discrete phase changes over a full turn of the virtual antenna, the frequency of the mimicked rotating antenna can be calculated at N different locations along the UCA. Assume that the radius of an N-element UCA is negligible compared to receiver to transmitter distance and let the angular speed of the virtually rotating antenna be *ω*. The phase change

<sup>−</sup> sin−<sup>1</sup>

 *c*Δ*fR*<sup>2</sup> *ωr f*

(36)

*α* = sin−<sup>1</sup>

**4.2 Bearing estimation with antenna-array anchor nodes**

between time instants *t* and Δ*t* then becomes:

Δ*φ*(*t*, *t* + Δ*t*) =

*<sup>t</sup>*+Δ*<sup>t</sup>*

*<sup>t</sup> sin*(*ωt*)*dt*

<sup>Δ</sup>*<sup>t</sup>* <sup>=</sup> *sin*

 *ωt* + Δ*t* 2

2*sin*( <sup>Δ</sup>*<sup>t</sup>* 2 )

*<sup>ω</sup>*Δ*<sup>t</sup>* , (37)

Fig. 10. Quasi doppler bearing estimation using only four discrete (N=4) steps instead of continuous rotation.

where *ω* = 1/(*N*Δ*t*). Thus, by measuring the N phase jumps throughout a full turn of the antenna, the sin(*ωt* + *ϕ*) term can be estimated. The phase of this signal is then used to calculate the bearing to the anchor nodes in a similar fashion as described in Section 4.1.

## **4.3 Localizaton based on bearing estimates**

The localization phase takes multiple angle estimates, *αi*, between the target node, rotating-antenna or antenna-array node *i* and the reference node, as depicted in Figure 11, and deduces the location of node *Q*. In the optimal case, the bearing estimates intersect at a single point. Due to measurement errors, however, bearing estimates *αi*(*i* > 2) to target node *Q* may have multiple intersections. In this case, an optimization algorithm can be used to estimate the most probable target node location.

## **4.4 Platform description and results**

The Doppler shift-based SpinLoc localization system presented in Chang et al. (2008) uses MICA2 type CC1000 equipped sensor nodes both for anchor nodes and target nodes, where the rotating-antenna transmitter anchors are physically rotated on a 50 cm radius arm, at 133 revolutions per minute (RPM) by a servo motor. The measurements use the 900 MHz band to generate the 600 Hz interference signal and measure its frequency change.

The measurement round starts with time synchronization initiated by the base station. The rotating anchor node and a static node starts transmitting the unmodulated carriers, which is received and processed by a reference node and the target nodes. The latter two calculate the signal frequencies and send it to the base station. A PC attached to the base station then calculates the orientation of the rotating anchors, the relative bearings and the position of the target nodes.

estimated phase difference between the receiver of the composite anchor node and the target node constraints the location of the latter to a hyperbola. The bearing of the target node to the

Interferometry in Wireless Sensor Networks 455

The rest of the section details the theoretical background of the bearing estimation approach,

The bearing estimation phase starts with a regular interferometric measurement where the location of three closely placed nodes is known a priori. These three nodes, *M*, *A*<sup>1</sup> and *A*2, form a composite anchor from which a receiver node, *R*, with unknown location estimates its bearing, see Figure 12 (a). Two nodes of the anchor, *A*<sup>1</sup> and *M*, transmit unmodulated sinusoid signals at close frequencies, generating an interference signal with a low-frequency envelope. The receiver nodes, *A*<sup>2</sup> and *R* measure the phase of the envelope signal according to Equation 9. The measured *ϑMA*1*A*2*<sup>R</sup>* = *ϑ<sup>R</sup>* − *ϑA*<sup>2</sup> phase difference is related to the linear combination of the distances between the transmitters according to Equation 15. Thus,

*dMR* − *dA*1*<sup>R</sup>* + *dA*1*A*<sup>2</sup> − *dMA*<sup>2</sup>

where *λ* = *c*/ *f* is the wavelength of the carrier signal and *d* is the distance between the nodes denoted in the index. Assuming that the pairwise distance between the three nodes in the anchor is the same, *dA*1*A*<sup>2</sup> − *dMA*<sup>2</sup> = 0, the expression of the q-range gets simpler, and

*dMR* − *dA*1*<sup>R</sup>*

The distance difference *dMR* − *dA*1*<sup>R</sup>* is of high importance, thus it is shortened as *dA*1*MR*, and referred to as *t-range*. The *dA*1*MR* t-range takes its maximum when *R* is colinear with the two transmitters *A*<sup>1</sup> and *M*. According to Equation 39 the measured distance difference is 2*π*-ambiguous if the distance between the two transmitters is larger than *λ*/2. Thus, with the assumption that the antennas of the anchor nodes are less than half wavelength apart, the

*dA*1*MR* <sup>=</sup> *<sup>ϑ</sup>MA*1*A*2*<sup>R</sup>*

According to Equation 40, the distances between the receiver node and the two transmitter nodes have a constant difference. Thus, *dA*1*MR* in Equation 40 defines the arm of a hyperbola

The possible location coordinates (*x*, *y*) of *R* can be expressed with the general equation of the

Assuming *dA*1*R*, *dMR* � *dA*1*M*, the bearing of the *R* can estimated with the asymptote of the hyperbola. The bearing *α* of *R* relative to the origin can then be written using the two

*x*2 *<sup>a</sup>*<sup>2</sup> <sup>−</sup> *<sup>y</sup>*<sup>2</sup>

*<sup>c</sup>*/ *<sup>f</sup>* (mod 2*π*), (38)

*<sup>c</sup>*/ *<sup>f</sup>* (mod 2*π*). (39)

<sup>2</sup>*<sup>π</sup>* · *<sup>λ</sup>*. (40)

*<sup>b</sup>*<sup>2</sup> <sup>=</sup> 1. (41)

the prototype system used for experimental measurements and the reported results.

anchor node is estimated by the asymptote of the hyperbola.

**5.1 Bearing estimation using 3-node anchors**

*ϑMA*1*A*2*<sup>R</sup>* = 2*π*

modulo 2*π* is removed and t-range is simplified to

*ϑMA*1*A*2*<sup>R</sup>* = 2*π*

that intersects with the position of node *R* as shown in Figure 12 (b).

Equation 38 becomes

origin centered hyperbola:

Fig. 11. Localization based on bearing estimates towards rotating transmitter nodes.

SpinLoc experimental measurements in an 8 × 10 m indoor garage area showed an average bearing estimation accuracy of 3◦ leading to a localization accuracy of approximately 40 cm. The data collection time is reported by Chang et al. (2008) to be less than 1.5 s, while the per target node localization to be approximately 0.5 s.

The quasi-Doppler localization system proposed in Sallai et al. (2009) generates the interference signal with two anchor nodes. One of these anchor nodes switch the continuous, unmodulated carrier between the antennas of the array. A third anchor node and a target node act as receiver and measure the phase jumps caused by the virtual re-location of the transmitter antenna and reconstruct the quasi Doppler-shift of the signal. The Doppler-shift estimates are then sent to the a PC to estimate the bearings and calculate the target node position. Sallai et al. (2009) present experiments that verify the feasibility of the approach, but no thorough evaluation of the expected accuracy is available.

## **5. Radio interferometric bearing estimation**

The first techniques for bearing estimation in WSNs relied both on radio interferometry and a secondary phenomenon, the Doppler-effect. While these techniques provided bearing estimates of adequate precision for node localization and significantly simplified the sensor fusion algorithm, they called for specialized hardware on a subset of the WSN nodes. To mitigate this unwelcome requirement, Amundson et al. (2010) presents another approach that uses the same uniform, simple WSN nodes as the original RIPS discussed in Section 2, yet it is capable of accurately estimating the bearing from anchor nodes. Note, however, that even though specialized hardware is not required, the method assumes a specific arrangement of certain nodes.

The baseline approach of the method presented in Amundson et al. (2010) is to group together three of the four nodes participating in a typical radio interferometric measurement described in 2. The three nodes, two transmitters and a receiver, placed together in a known and fixed geometry, form a composite anchor node. During the interferometric measurement, the 18 Will-be-set-by-IN-TECH

Fig. 11. Localization based on bearing estimates towards rotating transmitter nodes.

target node localization to be approximately 0.5 s.

**5. Radio interferometric bearing estimation**

certain nodes.

no thorough evaluation of the expected accuracy is available.

SpinLoc experimental measurements in an 8 × 10 m indoor garage area showed an average bearing estimation accuracy of 3◦ leading to a localization accuracy of approximately 40 cm. The data collection time is reported by Chang et al. (2008) to be less than 1.5 s, while the per

The quasi-Doppler localization system proposed in Sallai et al. (2009) generates the interference signal with two anchor nodes. One of these anchor nodes switch the continuous, unmodulated carrier between the antennas of the array. A third anchor node and a target node act as receiver and measure the phase jumps caused by the virtual re-location of the transmitter antenna and reconstruct the quasi Doppler-shift of the signal. The Doppler-shift estimates are then sent to the a PC to estimate the bearings and calculate the target node position. Sallai et al. (2009) present experiments that verify the feasibility of the approach, but

The first techniques for bearing estimation in WSNs relied both on radio interferometry and a secondary phenomenon, the Doppler-effect. While these techniques provided bearing estimates of adequate precision for node localization and significantly simplified the sensor fusion algorithm, they called for specialized hardware on a subset of the WSN nodes. To mitigate this unwelcome requirement, Amundson et al. (2010) presents another approach that uses the same uniform, simple WSN nodes as the original RIPS discussed in Section 2, yet it is capable of accurately estimating the bearing from anchor nodes. Note, however, that even though specialized hardware is not required, the method assumes a specific arrangement of

The baseline approach of the method presented in Amundson et al. (2010) is to group together three of the four nodes participating in a typical radio interferometric measurement described in 2. The three nodes, two transmitters and a receiver, placed together in a known and fixed geometry, form a composite anchor node. During the interferometric measurement, the estimated phase difference between the receiver of the composite anchor node and the target node constraints the location of the latter to a hyperbola. The bearing of the target node to the anchor node is estimated by the asymptote of the hyperbola.

The rest of the section details the theoretical background of the bearing estimation approach, the prototype system used for experimental measurements and the reported results.

## **5.1 Bearing estimation using 3-node anchors**

The bearing estimation phase starts with a regular interferometric measurement where the location of three closely placed nodes is known a priori. These three nodes, *M*, *A*<sup>1</sup> and *A*2, form a composite anchor from which a receiver node, *R*, with unknown location estimates its bearing, see Figure 12 (a). Two nodes of the anchor, *A*<sup>1</sup> and *M*, transmit unmodulated sinusoid signals at close frequencies, generating an interference signal with a low-frequency envelope. The receiver nodes, *A*<sup>2</sup> and *R* measure the phase of the envelope signal according to Equation 9. The measured *ϑMA*1*A*2*<sup>R</sup>* = *ϑ<sup>R</sup>* − *ϑA*<sup>2</sup> phase difference is related to the linear combination of the distances between the transmitters according to Equation 15. Thus,

$$\text{tg}\_{MA\_1A\_2R} = 2\pi \frac{d\_{MR} - d\_{A\_1R} + d\_{A\_1A\_2} - d\_{MA\_2}}{c/f} \pmod{2\pi} \,\text{,}\tag{38}$$

where *λ* = *c*/ *f* is the wavelength of the carrier signal and *d* is the distance between the nodes denoted in the index. Assuming that the pairwise distance between the three nodes in the anchor is the same, *dA*1*A*<sup>2</sup> − *dMA*<sup>2</sup> = 0, the expression of the q-range gets simpler, and Equation 38 becomes

$$
\theta\_{MA\_1A\_2R} = 2\pi \frac{d\_{MR} - d\_{A\_1R}}{\mathfrak{c}/f} \pmod{2\pi}.\tag{39}
$$

The distance difference *dMR* − *dA*1*<sup>R</sup>* is of high importance, thus it is shortened as *dA*1*MR*, and referred to as *t-range*. The *dA*1*MR* t-range takes its maximum when *R* is colinear with the two transmitters *A*<sup>1</sup> and *M*. According to Equation 39 the measured distance difference is 2*π*-ambiguous if the distance between the two transmitters is larger than *λ*/2. Thus, with the assumption that the antennas of the anchor nodes are less than half wavelength apart, the modulo 2*π* is removed and t-range is simplified to

$$d\_{A\_1MR} = \frac{\theta\_{MA\_1A\_2R}}{2\pi} \cdot \lambda. \tag{40}$$

According to Equation 40, the distances between the receiver node and the two transmitter nodes have a constant difference. Thus, *dA*1*MR* in Equation 40 defines the arm of a hyperbola that intersects with the position of node *R* as shown in Figure 12 (b).

The possible location coordinates (*x*, *y*) of *R* can be expressed with the general equation of the origin centered hyperbola:

$$
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.\tag{41}
$$

Assuming *dA*1*R*, *dMR* � *dA*1*M*, the bearing of the *R* can estimated with the asymptote of the hyperbola. The bearing *α* of *R* relative to the origin can then be written using the two

ambiguity. Node *A*<sup>1</sup> and *A*<sup>2</sup> switch transmitter-receiver roles at the anchor, defining a new transverse axis and ±*α* values relative to it. When the ±*α* values from the two measurements are compared to the same reference orientation, two of them coincide, yielding the final bearing estimate. Once the bearing estimates are available from a target node to a set of anchor

Interferometry in Wireless Sensor Networks 457

The first experimental implementation of the radio interferometry based bearing estimation was named TripLoc presented in Amundson et al. (2010). The nodes in TripLoc are equipped with fine tunable CC1000 radio chips operating in the 433 MHz band and processing it with a

The basic TripLoc measurement is a series of radio interferometric phase measurements with four participants, three of which form an anchor node with a priori known location. The measurement starts with two nodes of the anchor transmitting unmodulated carriers at close frequencies. The third node of the anchor and another node with unknown location then synchronize their clocks and measure the phase of the envelope signal. The phase measured by the anchor-receiver is transmitted to the receiver to be localized, which in turn calculates the phase offset. The same measurement is then repeated with the anchor-receiver switching role with one of the transmitters. Using the two phase offsets the receiver node is able to calculate its t-range, see Equation 40, and estimate its bearing to the anchor node according to Equation 46. After repeating the bearing estimation relative to another anchor node, TripLoc uses triangulation to determine the non-anchor receiver node location. As the non-anchor node always acts as a "passive receiver", multiple receivers can participate in the same

TripLoc experiments using a 16-node setup on a 20×20 m low-multipath environment, with four anchor nodes at the corners, showed that receiver nodes are capable of bearing estimation with an average error of approximately 3◦. After triangulation, the average overall position

The localization algorithms presented in Sections 2, 4 and 5 are all based on radio interferometry, where two independent transmitter nodes generate an interference signal by transmitting unmodulated carriers at close frequencies. The radio interferometric localization discussed in Section 2 and the radio interferometric bearing estimation in Section 5 measure the phase of this interference signal, while the Doppler-shift based bearing the frequency of that. However, regardless of the way this interference signals is processed at further stages, all of these methods assume that the interference signal has a low frequency. The fine grained control over the transmit frequencies available on the 433 MHz/950 MHz ISM band CC1000 radio chip based platforms is not present on the increasingly popular 2.4 GHz radio chips, which renders this assumption invalid. The recently presented radio interferometric localization method introduced in Dil & Havinga (2011), therefore, drops this assumption and places most of its processing steps on stochastic bases. The rest of the section presents the theory behind the stochastic radio interferometric positioning by highlighting the key

nodes, simple triangulation can be used for localization.

measurement and calculate their own location simultaneously.

**6. Stochastic radio interferometric positioning**

error was 78 cm, while the whole localization took less then 1 second.

**5.2 Platform description and results**

microcontroller at a 9 kHz rate.

(a) Relative bearing from the 3-node anchor (b) The 3-node anchor exaggerated to show distances

### Fig. 12. Bearing estimation using 3-node anchors

parameters of the hyperbola, *a* and *b*:

$$
\alpha = \tan^{-1} \left( \frac{b}{a} \right). \tag{42}
$$

To find the value of *a* and *b* consider the point where the hyperbola arm intersects the line connecting the two focus points. As the distance differences between the foci and *all* points of the hyperbola are the same constant, this point is no exception. Therefore, denoting the distance between the origin and the foci as *c* and using the t-range definition:

$$d\_{A\_1R} - d\_{MR} = (c+a) - (c-a) = 2a,\tag{43}$$

where *<sup>c</sup>* is also known as the linear eccentricity of the hyperbola, for which *<sup>c</sup>* <sup>=</sup> <sup>√</sup>*a*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*<sup>2</sup> and *<sup>c</sup>* <sup>=</sup> *dA*1*M*. Thus, *<sup>b</sup>* <sup>=</sup> <sup>√</sup>*c*<sup>2</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup> and in terms of distances, the value of *<sup>a</sup>* and *<sup>b</sup>* and the bearing estimate becomes

$$a = (d\_{A\_1R} - d\_{MR})/2\tag{44}$$

$$b = \sqrt{(d\_{A\_1M}/2)^2 - ((d\_{A\_1R} - d\_{MR})/2)^2} \tag{45}$$

$$\alpha = \tan^{-1}\left(\frac{\sqrt{(d\_{A\_1M}/2)^2 - ((d\_{A\_1R} - d\_{MR})/2)^2}}{(d\_{A\_1R} - d\_{MR})/2}\right). \tag{46}$$

The bearing estimate *α* is, however, ambiguous. First, the hyperbola has two arms. The sign of *dA*1*<sup>R</sup>* − *dMR*, and thus the sign of the phase difference *ϑ<sup>R</sup>* − *ϑA*<sup>2</sup> selects the arm of the hyperbola. In case *ϑ<sup>R</sup>* < *ϑA*<sup>2</sup> , *R* lies on the left arm, otherwise it lies on the right arm. Second, the ±*α* values, relative to the transverse axis of the hyperbola, are both solutions of Equation 46. A second measurement with swapped anchor node roles is used to resolve this 20 Will-be-set-by-IN-TECH

*α* = tan−<sup>1</sup>

distance between the origin and the foci as *c* and using the t-range definition:

To find the value of *a* and *b* consider the point where the hyperbola arm intersects the line connecting the two focus points. As the distance differences between the foci and *all* points of the hyperbola are the same constant, this point is no exception. Therefore, denoting the

where *<sup>c</sup>* is also known as the linear eccentricity of the hyperbola, for which *<sup>c</sup>* <sup>=</sup> <sup>√</sup>*a*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*<sup>2</sup> and *<sup>c</sup>* <sup>=</sup> *dA*1*M*. Thus, *<sup>b</sup>* <sup>=</sup> <sup>√</sup>*c*<sup>2</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup> and in terms of distances, the value of *<sup>a</sup>* and *<sup>b</sup>* and the bearing

The bearing estimate *α* is, however, ambiguous. First, the hyperbola has two arms. The sign of *dA*1*<sup>R</sup>* − *dMR*, and thus the sign of the phase difference *ϑ<sup>R</sup>* − *ϑA*<sup>2</sup> selects the arm of the hyperbola. In case *ϑ<sup>R</sup>* < *ϑA*<sup>2</sup> , *R* lies on the left arm, otherwise it lies on the right arm. Second, the ±*α* values, relative to the transverse axis of the hyperbola, are both solutions of Equation 46. A second measurement with swapped anchor node roles is used to resolve this

� *b a* �

(b) The 3-node anchor exaggerated to show distances

*dA*1*<sup>R</sup>* − *dMR* = (*c* + *a*) − (*c* − *a*) = 2*a*, (43)

(*dA*1*M*/2)<sup>2</sup> − ((*dA*1*<sup>R</sup>* − *dMR*)/2)<sup>2</sup> (45)

⎞

⎠ . (46)

*a* = (*dA*1*<sup>R</sup>* − *dMR*)/2 (44)

(*dA*1*M*/2)<sup>2</sup> − ((*dA*1*<sup>R</sup>* − *dMR*)/2)<sup>2</sup> (*dA*1*<sup>R</sup>* − *dMR*)/2

. (42)

(a) Relative bearing from the 3-node

parameters of the hyperbola, *a* and *b*:

Fig. 12. Bearing estimation using 3-node anchors

*b* = �

*α* = tan−<sup>1</sup>

⎛ ⎝ �

anchor

estimate becomes

ambiguity. Node *A*<sup>1</sup> and *A*<sup>2</sup> switch transmitter-receiver roles at the anchor, defining a new transverse axis and ±*α* values relative to it. When the ±*α* values from the two measurements are compared to the same reference orientation, two of them coincide, yielding the final bearing estimate. Once the bearing estimates are available from a target node to a set of anchor nodes, simple triangulation can be used for localization.

## **5.2 Platform description and results**

The first experimental implementation of the radio interferometry based bearing estimation was named TripLoc presented in Amundson et al. (2010). The nodes in TripLoc are equipped with fine tunable CC1000 radio chips operating in the 433 MHz band and processing it with a microcontroller at a 9 kHz rate.

The basic TripLoc measurement is a series of radio interferometric phase measurements with four participants, three of which form an anchor node with a priori known location. The measurement starts with two nodes of the anchor transmitting unmodulated carriers at close frequencies. The third node of the anchor and another node with unknown location then synchronize their clocks and measure the phase of the envelope signal. The phase measured by the anchor-receiver is transmitted to the receiver to be localized, which in turn calculates the phase offset. The same measurement is then repeated with the anchor-receiver switching role with one of the transmitters. Using the two phase offsets the receiver node is able to calculate its t-range, see Equation 40, and estimate its bearing to the anchor node according to Equation 46. After repeating the bearing estimation relative to another anchor node, TripLoc uses triangulation to determine the non-anchor receiver node location. As the non-anchor node always acts as a "passive receiver", multiple receivers can participate in the same measurement and calculate their own location simultaneously.

TripLoc experiments using a 16-node setup on a 20×20 m low-multipath environment, with four anchor nodes at the corners, showed that receiver nodes are capable of bearing estimation with an average error of approximately 3◦. After triangulation, the average overall position error was 78 cm, while the whole localization took less then 1 second.

## **6. Stochastic radio interferometric positioning**

The localization algorithms presented in Sections 2, 4 and 5 are all based on radio interferometry, where two independent transmitter nodes generate an interference signal by transmitting unmodulated carriers at close frequencies. The radio interferometric localization discussed in Section 2 and the radio interferometric bearing estimation in Section 5 measure the phase of this interference signal, while the Doppler-shift based bearing the frequency of that. However, regardless of the way this interference signals is processed at further stages, all of these methods assume that the interference signal has a low frequency. The fine grained control over the transmit frequencies available on the 433 MHz/950 MHz ISM band CC1000 radio chip based platforms is not present on the increasingly popular 2.4 GHz radio chips, which renders this assumption invalid. The recently presented radio interferometric localization method introduced in Dil & Havinga (2011), therefore, drops this assumption and places most of its processing steps on stochastic bases. The rest of the section presents the theory behind the stochastic radio interferometric positioning by highlighting the key differences with the original "deterministic" approach, shows a 2.4 GHz band CC2430 radio chip based implementation and the corresponding localization experimentation results.

## **6.1 Stochastic interferometric localization**

Similarly to the original radio interferometric positioning method, the stochastic approach starts with two nodes transmitting unmodulated carriers at close frequencies. As fine-grained control over the transmit frequencies is not assumed, the calibration phase is omitted as shown in Figure 13. As a result, the frequency value of the two transmitted signals becomes unknown and the difference of the two has a significantly higher variance than in case of the original method after calibration. The high variance of the interference frequency, in turn, manifests in numerous measurements with interference frequency too low or too high for precise phase estimation. This ultimately generates a significantly higher number of inaccurate phase measurements, hence, the phase estimate is treated by the upcoming stochastic processing stages as a random variable. Despite the fact that a significant number of the phase measurements are likely to be corrupted, the phase measurement process is identical to that of the one presented in Section 2. That is, with carrier frequencies *fA* and *fB* of transmitters *A* and *B*, where *fA* > *fB* is assumed:

$$
\Delta \varphi\_{\rm l} = 2\pi \left( \frac{d\_{AD} - d\_{A\mathcal{C}}}{\lambda\_A} - \frac{d\_{BD} - d\_{B\mathcal{C}}}{\lambda\_B} \right) \mod (2\pi) \tag{47}
$$

$$
\approx 2\pi \left( \frac{d\_{ABCD}}{\lambda\_i} \right) \mod (2\pi),
\tag{48}
$$

Fig. 13. Comparison of the original (RIPS) and the stochastic interferometric localization

minimized directly. Rather, it is smoothed first by taking its envelope and then, the minimum

Interferometry in Wireless Sensor Networks 459

The key motivation behind the Stochastic Radio Interferometric Positioning System (SRIPS), detailed in Dil & Havinga (2011), was to make inteferometric localization available on WSN platforms using radio chips operating in the 2.4 GHz band. Popular 2.4 GHz radio chips, such as the CC2430 Texas Instruments (2007b), offer no mechanism to fine tune the transmit frequency. However, they support much higher sampling rates than the CC1000 radio. As a result, the synthesized interference signal envelope frequency has a larger variance, with high

The SRIPS phase measurement starts with two nodes transmitting unmodulated carriers at nominally same frequencies – without making attempts to fine tuning. The receiver nodes measure the envelope phase and transmit it to a PC. The same phase measurement is carried out over a set of frequencies and is repeated for different transmitter pair combinations. Due to the inaccurately set interference signals, several of the phase measurements are invalid. To cope with the large number of inaccurate measurements, SRIPS skips the q-range calculation

The algorithms presented in this chapter all rely on radio interferometry in order to localize or track nodes in WSNs. In each case, the interference signal is generated the same way,

confidence in the 200 Hz to 14 kHz range, but it is sampled at a rate of 62.5 kHz.

step and optimizes Equation 51 on the PC using the phase offsets directly.

SRIPS experiments on a 20×20 m field resulted in an average error of 50 cm.

algorithms

**7. Summary**

of the envelope is searched.

**6.2 Platform description and results**

where Δ*ϕ<sup>i</sup>* is, again, the relative phase offset of the envelope frequencies at the two receivers, *dXY* is the distance of the corresponding nodes *X* and *Y*, and *λ<sup>i</sup>* is defined as *λ<sup>i</sup>* = 2*c*/(*fA* + *fB*).

Again, Equation 48 provides the q-range value with a 2*π* ambiguity, which is removed in the method in Section 2 by repeating the same measurement over a set of different carrier frequencies to obtain the q-range. The stochastic method takes a different approach here, as shown in Figure 13. Rather than calculating the q-range error distribution first, then estimating the q-range values and then the node positions, it calculates only the q-range error distributions and uses them to estimate the positions directly:

$$\mathbf{f}(\pounds, \mathfrak{H}) = \underset{\pounds, \mathfrak{H}}{\text{arg min}} \sum\_{j=1}^{M} \sum\_{i=1}^{N} \left(\Delta \varphi\_{i,j} - \Delta \phi\_{i,j}\right)^2 \tag{49}$$

$$=\underset{\mathfrak{k},\mathfrak{\emptyset}}{\arg\min} \sum\_{j=1}^{M} \sum\_{i=1}^{N} \left( d\_{i,j} - \mathfrak{q}\_{j} \right)^{2} \tag{50}$$

$$\mathbf{y} = \underset{\mathbf{f}, \mathbf{f}}{\text{arg min}} \sum\_{j=1}^{M} \text{error}(\mathbf{\hat{q}}\_{j}) \tag{51}$$

That is, the stochastic method minimizes Equation 51, where Δ*ϕ*ˆ*i*,*<sup>j</sup>* is the estimated phase offset calculated by the location estimate, *di*,*<sup>j</sup>* is defined by Equation 17, *q*ˆ*<sup>j</sup>* is the q-range estimate and error(*q*ˆ*j*) is the q-range error distribution. As the latter one oscillates rapidly, it is not 22 Will-be-set-by-IN-TECH

differences with the original "deterministic" approach, shows a 2.4 GHz band CC2430 radio chip based implementation and the corresponding localization experimentation results.

Similarly to the original radio interferometric positioning method, the stochastic approach starts with two nodes transmitting unmodulated carriers at close frequencies. As fine-grained control over the transmit frequencies is not assumed, the calibration phase is omitted as shown in Figure 13. As a result, the frequency value of the two transmitted signals becomes unknown and the difference of the two has a significantly higher variance than in case of the original method after calibration. The high variance of the interference frequency, in turn, manifests in numerous measurements with interference frequency too low or too high for precise phase estimation. This ultimately generates a significantly higher number of inaccurate phase measurements, hence, the phase estimate is treated by the upcoming stochastic processing stages as a random variable. Despite the fact that a significant number of the phase measurements are likely to be corrupted, the phase measurement process is identical to that of the one presented in Section 2. That is, with carrier frequencies *fA* and

> <sup>−</sup> *dBD* <sup>−</sup> *dBC λ<sup>B</sup>*

where Δ*ϕ<sup>i</sup>* is, again, the relative phase offset of the envelope frequencies at the two receivers, *dXY* is the distance of the corresponding nodes *X* and *Y*, and *λ<sup>i</sup>* is defined as *λ<sup>i</sup>* = 2*c*/(*fA* +

Again, Equation 48 provides the q-range value with a 2*π* ambiguity, which is removed in the method in Section 2 by repeating the same measurement over a set of different carrier frequencies to obtain the q-range. The stochastic method takes a different approach here, as shown in Figure 13. Rather than calculating the q-range error distribution first, then estimating the q-range values and then the node positions, it calculates only the q-range error

> *M* ∑ *j*=1

> *M* ∑ *j*=1

> *M* ∑ *j*=1

That is, the stochastic method minimizes Equation 51, where Δ*ϕ*ˆ*i*,*<sup>j</sup>* is the estimated phase offset calculated by the location estimate, *di*,*<sup>j</sup>* is defined by Equation 17, *q*ˆ*<sup>j</sup>* is the q-range estimate and error(*q*ˆ*j*) is the q-range error distribution. As the latter one oscillates rapidly, it is not

*N* ∑ *i*=1 

*N* ∑ *i*=1 *di*,*<sup>j</sup>* − qˆ*<sup>j</sup>*

error(qˆ*<sup>j</sup>*

Δ*ϕi*,*<sup>j</sup>* − Δ*ϕ*ˆ*i*,*<sup>j</sup>*

2

2

) (51)

mod (2*π*) (47)

(49)

(50)

mod (2*π*), (48)

**6.1 Stochastic interferometric localization**

*fB* of transmitters *A* and *B*, where *fA* > *fB* is assumed:

≈ 2*π*

distributions and uses them to estimate the positions directly:

(*x*ˆ, *y*ˆ) = arg min

*x*ˆ,*y*ˆ

= arg min *x*ˆ,*y*ˆ

= arg min *x*ˆ,*y*ˆ

 *dAD* <sup>−</sup> *dAC λ<sup>A</sup>*

 *dABCD λi*

Δ*ϕ<sup>i</sup>* = 2*π*

*fB*).

Fig. 13. Comparison of the original (RIPS) and the stochastic interferometric localization algorithms

minimized directly. Rather, it is smoothed first by taking its envelope and then, the minimum of the envelope is searched.

## **6.2 Platform description and results**

The key motivation behind the Stochastic Radio Interferometric Positioning System (SRIPS), detailed in Dil & Havinga (2011), was to make inteferometric localization available on WSN platforms using radio chips operating in the 2.4 GHz band. Popular 2.4 GHz radio chips, such as the CC2430 Texas Instruments (2007b), offer no mechanism to fine tune the transmit frequency. However, they support much higher sampling rates than the CC1000 radio. As a result, the synthesized interference signal envelope frequency has a larger variance, with high confidence in the 200 Hz to 14 kHz range, but it is sampled at a rate of 62.5 kHz.

The SRIPS phase measurement starts with two nodes transmitting unmodulated carriers at nominally same frequencies – without making attempts to fine tuning. The receiver nodes measure the envelope phase and transmit it to a PC. The same phase measurement is carried out over a set of frequencies and is repeated for different transmitter pair combinations. Due to the inaccurately set interference signals, several of the phase measurements are invalid. To cope with the large number of inaccurate measurements, SRIPS skips the q-range calculation step and optimizes Equation 51 on the PC using the phase offsets directly.

SRIPS experiments on a 20×20 m field resulted in an average error of 50 cm.

## **7. Summary**

The algorithms presented in this chapter all rely on radio interferometry in order to localize or track nodes in WSNs. In each case, the interference signal is generated the same way,

RIPS bearing, Doppler tracking and Doppler bearing leverage the prior knowledge of the anchor node locations and allow for distributed calculation of the locations, e.g. the nodes can

Interferometry in Wireless Sensor Networks 461

The operating range of the presented radio interferometric platforms is generally in the order of tens of meters up to a few hundred meters. In multipath free environments, the original RIPS proves to be the most accurate, which is, again, mainly due to the favorable properties of the CC1000. This is followed by the Doppler bearing, SRIPS and RIPS bearing, each of which achieves sub-meter accuracy. The average error of Doppler tracking is difficult to compare as it assumes the initial positions to be known and because the error is calculated over several

**RIPS** Radio Interferometric Positioning System

Amundson, I., Sallai, J., Koutsoukos, X. & Lédeczi, A. (2010). Radio Interferometric Angle

Chang, H.-l., Tian, J.-b., Lai, T.-T., Chu, H.-H. & Huang, P. (2008). *Spinning beacons for precise*

Dil, B. J. & Havinga, P. J. M. (2011). Stochastic Radio Interferometric Positioning in the 2.4

Fontana, R. J., Richley, E. & Barney, J. (2003). Commercialization of an ultra wideband

URL: *http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=1267866* Kusý, B., Balogh, G., Völgyesi, P., Sallai, J., Nádas, A., Maróti, M., Meertens, L. & Lédeczi,

Kusý, B., Lédeczi, A. & Koutsoukos, X. (2007). *Tracking mobile nodes using RF Doppler shifts*,

of Arrival Estimation, *7th European Conference on Wireless Sensor Networks*, Springer,

GHz Range, *Proceedings of the 9th ACM conference on Embedded network sensor systems*

precision asset location system, *IEEE Conference on Ultra Wideband Systems and*

A. (2006). Node-density independent localization, *Information Processing in Sensor*

**SRIPS** Stochastic Radio Interferometric Positioning System

**Anchor node** Node with a priori known location **COTS** Commercial Off-The Shelf **IF** Intermediate Frequency

**RSS** Received Signal Strength

**WSN** Wireless Sensor Network

*indoor localization*, ACM Press, New York, New York, USA. URL: *http://portal.acm.org/citation.cfm?id=1460412.1460426*

*Technologies 2003*, number November, IEEE, IEEE, pp. 369–373.

URL: *http://portal.acm.org/citation.cfm?id=1322263.1322267*

**LO** Local Oscillator **PC** Personal Computer

URL: *http://www.isis.vanderbilt.edu/node/4139*

*- SenSys'11*, ACM Press, Seattle, WA, USA.

*Networks (IPSN 06) SPOTS Track*.

ACM Press, New York, New York, USA.

estimate their own positions.

iterations.

**8. Nomenclature**

Table 3. List of acronyms

Coimbra, Portugal.

**9. References**

by two independent transmitters, but the different algorithms exploit features of this signal in different ways. Table 2 compares the algorithms with an emphasis on the experimental implementation and key results. All the prototype systems expect for SRIPS rely on the CC1000 radio chipset, which is designed operate in the 433 MHz and 900 MHz bands. They are characterized by a precisely tuned interference frequency which allows for processing at the limited 9 kHz sampling rate. Clearly, it is the unique fine-tuning capability of the CC1000 LO that made this radio chip - along with the MICA2 platform - the most favorable choice. SRIPS, on the other hand, aimed to make interferometric localization available on platforms that lack a fine-tunable LO. Driven by the reduced control over the interference frequency in the CC2430, SRIPS decided to drop tuning altogether. The available higher 62.5 kHz sampling rate made the processing of the highly varying frequency interference signal possible. This came at the cost of significantly increased number of inaccurate phase measurements, which SRIPS deals with in later processing stages.


Table 2. Comparison of RI localization systems

An important aspect in several applications is whether the localization or tracking service requires additional hardware. RIPS, SRIPS, RIPS bearing and Doppler tracking are especially favorable as they rely on no extra hardware. RIPS bearing, however, uses a special arrangement of certain nodes, which might be unfavorable in certain scenarios. Doppler bearing approaches utilize either a physically rotating antenna, or a relatively complex antenna array—both of which are impractical in low-cost WSNs.

RIPS and SRIPS are unique in the sense that they are able to calculate the relative node locations without anchor nodes, that is, without prior knowledge of locations of a subset of nodes. In this mode, they generally require orders of magnitude more time to do the self-localization. The presence of anchor nodes may potentially speed up the localization process both for RIPS and SRIPS, while it is essential for the other systems. In general, RIPS bearing, Doppler tracking and Doppler bearing leverage the prior knowledge of the anchor node locations and allow for distributed calculation of the locations, e.g. the nodes can estimate their own positions.

The operating range of the presented radio interferometric platforms is generally in the order of tens of meters up to a few hundred meters. In multipath free environments, the original RIPS proves to be the most accurate, which is, again, mainly due to the favorable properties of the CC1000. This is followed by the Doppler bearing, SRIPS and RIPS bearing, each of which achieves sub-meter accuracy. The average error of Doppler tracking is difficult to compare as it assumes the initial positions to be known and because the error is calculated over several iterations.

## **8. Nomenclature**

24 Will-be-set-by-IN-TECH

by two independent transmitters, but the different algorithms exploit features of this signal in different ways. Table 2 compares the algorithms with an emphasis on the experimental implementation and key results. All the prototype systems expect for SRIPS rely on the CC1000 radio chipset, which is designed operate in the 433 MHz and 900 MHz bands. They are characterized by a precisely tuned interference frequency which allows for processing at the limited 9 kHz sampling rate. Clearly, it is the unique fine-tuning capability of the CC1000 LO that made this radio chip - along with the MICA2 platform - the most favorable choice. SRIPS, on the other hand, aimed to make interferometric localization available on platforms that lack a fine-tunable LO. Driven by the reduced control over the interference frequency in the CC2430, SRIPS decided to drop tuning altogether. The available higher 62.5 kHz sampling rate made the processing of the highly varying frequency interference signal possible. This came at the cost of significantly increased number of inaccurate phase measurements, which

bearing

433 MHz 2400 MHz 433 MHz 433 MHz 900 MHz

300-700 Hz 200-12000 Hz 300-700 Hz 300-700 Hz 300-700 Hz

9 kHz 62.5 kHz 9 kHz 9 kHz 9 kHz

No No No No Yes

No No Yes Yes Yes

Few seconds

< 5 cm 50 cm 80 cm 1.5 m 40 cm

An important aspect in several applications is whether the localization or tracking service requires additional hardware. RIPS, SRIPS, RIPS bearing and Doppler tracking are especially favorable as they rely on no extra hardware. RIPS bearing, however, uses a special arrangement of certain nodes, which might be unfavorable in certain scenarios. Doppler bearing approaches utilize either a physically rotating antenna, or a relatively complex

RIPS and SRIPS are unique in the sense that they are able to calculate the relative node locations without anchor nodes, that is, without prior knowledge of locations of a subset of nodes. In this mode, they generally require orders of magnitude more time to do the self-localization. The presence of anchor nodes may potentially speed up the localization process both for RIPS and SRIPS, while it is essential for the other systems. In general,

Radio chip CC1000 CC2430 CC1000 CC1000 CC1000

Doppler tracking

Few seconds Doppler bearing

Few seconds

SRIPS deals with in later processing stages.

Operating band

Sampling frequency

Special hardware required

Anchor nodes required

time

Average error

Localization

Several minutes

Table 2. Comparison of RI localization systems

Interference frequency

RIPS SRIPS RIPS

Several minutes

antenna array—both of which are impractical in low-cost WSNs.

**Anchor node** Node with a priori known location

	- **IF** Intermediate Frequency
	- **LO** Local Oscillator
	- **PC** Personal Computer

Table 3. List of acronyms

## **9. References**

Amundson, I., Sallai, J., Koutsoukos, X. & Lédeczi, A. (2010). Radio Interferometric Angle of Arrival Estimation, *7th European Conference on Wireless Sensor Networks*, Springer, Coimbra, Portugal.

URL: *http://www.isis.vanderbilt.edu/node/4139*


URL: *http://portal.acm.org/citation.cfm?id=1602165.1602195*


URL: *http://www.ti.com/lit/ds/symlink/cc1000.pdf*


URL: *http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=1570023*

26 Will-be-set-by-IN-TECH

462 Interferometry – Research and Applications in Science and Technology

Lédeczi, A., Völgyesi, P., Sallai, J. & Thibodeaux, R. (2008). *A novel RF ranging method*, IEEE.

Maróti, M., Völgyesi, P., Dóra, S., Kusý, B., Nádas, A., Lédeczi, A., Balogh, G. & Molnár, K. (2005). *Radio interferometric geolocation*, ACM Press, New York, New York, USA.

Sallai, J., Völgyesi, P. & Lédeczi, A. (2009). Radio interferometric Quasi Doppler bearing

Taubenheim, D., Kyperountas, S. & Correal, N. (2005). Distributed radiolocation hardware core for ieee 802.15.4, *Technical report*, Motorola Labs, Plantation, Florida. Texas Instruments (2007a). Single-chip very low power RF transceiver, *Technical report*, Texas

Texas Instruments (2007b). A true system-on-chip solution for 2.4 GHz IEEE 802.15.4 /

Tuchler, M., Schwarz, V. & Huber, A. (2005). Location accuracy of an UWB localization system

URL: *http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=1570023*

in a multi-path environment, *2005 IEEE International Conference on UltraWideband*

estimation, *IPSN '09 Proceedings of the 2009 International Conference on Information Processing in Sensor Networks*, IEEE Computer Society, Washington, DC, USA,

URL: *http://ieeexplore.ieee.org/xpl/freeabs\_all.jsp?arnumber=4623303*

URL: *http://portal.acm.org/citation.cfm?id=1098918.1098920*

URL: *http://portal.acm.org/citation.cfm?id=1602165.1602195*

URL: *http://www.ti.com/lit/ds/symlink/cc1000.pdf*

zigbee(tm), *Technical report*, Texas Instruments. URL: *http://www.ti.com/lit/ds/symlink/cc2430.pdf*

pp. 325–336.

Instruments.

(September): 414–419.

## *Edited by Ivan Padron*

This book provides the most recent studies on interferometry and its applications in science and technology. It is an outline of theoretical and experimental aspects of interferometry and their applications. The book is divided in two sections. The first one is an overview of different interferometry techniques and their general applications, while the second section is devoted to more specific interferometry applications comprising from interferometry for magnetic fusion plasmas to interferometry in wireless networks. The book is an excellent reference of current interferometry applications in science and technology. It offers the opportunity to increase our knowledge about interferometry and encourage researchers in development of new applications.

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Interferometry - Research and Applications in Science and Technology

Interferometry

Research and Applications in

Science and Technology