**1. Introduction**

12 Will-be-set-by-IN-TECH

102 Advanced Holography – Metrology and Imaging

Schreiber, H.; Bruning, J.H.; Greivenkamp, J.E. (2007). Phase Shifting Interferometry,

Takeda, M.; Ina, H.; Kobayashi, S. (1982). Fourier-transform method of fringe-pattern analysis

Toge, H.; Fujiwara, H.; Sato, K. (2008). One-shot digital holography for recording color 3-d

Wyant, J.C. (2003). Dynamic Interferometry, *Optics & Photonics News* Vol.14, No.4, 36-41. Yamaguchi, I. & Zhang, T. (1997). Phase-shifting digital holography, *Optics Letters*, Vol.22,

images, *Proceedings of SPIE*, Vol.69120, 69120U-1 - 69120U-7.

ISBN:978-0-471-48404-2, Hoboken, pp.547-666.

*America*, Vol.72, No.1, 156-160.

No.16, 1268-1270.

*in* Maralaca, D., ed., *Optical Shop Testing, Third Edition*, John Wiley & Sons,

for computer-based topography and interferometry, *Journal of Optical Society of*

Optical interferometry was always considered as one of the most flexible and sensitive techniques for measuring mechanical vibrations (Hariharan, 1990; Osterberg, 1932). The number of applications of this method tremendously increased after the discovery of lasers (in 1960) and the development of low-loss optical fibres (in 1966). A laser as a source of coherent radiation assures high sensitivity of an interferometer while an optical fibre guarantees compactness, light weight, immunity to electromagnetic influence, and capability of the measuring system to operate in hazardous conditions (high temperature, radiation, etc.). As known, the phase of a light field cannot be directly detected. It is an interferometer, which provides the detection of the phase difference by combining the signal wave (which bears information about the measurand) with the coherent reference wave and measuring changes of the resultant light flux by a photodetector. The sensitivity of an interferometer to phase difference is limited by shot noise of the photoelectrons and can be extremely high: the theoretical minimum detectable displacement (corresponding to the phase difference) is 1.110-16 m·Hz-1/2 for 10 mW of detected laser power at the wavelength of 500 nm (Forward, 1978; Wagner & Spicer, 1987). However, two main problems should be overcome to achieve such high sensitivity in non-laboratory conditions. The first is the necessity of precise adjustment of the object and the reference wavefronts interfering at the photodetector. This requirement makes difficult, in particular, the use of multimode optical fibres in interferometric sensors. The second is the need to keep constant the average phase shift between the interfering wavefronts. This is more fundamental than the first one because the induced phase shift (which is assumed to be proportional to influence of the measurand on the optical paths) is nonlinearly transferred into the light-flux change at the detector via the familiar cosine interference function.

A simple and elegant solution of both these problems was achieved when a conventional beam-splitter, which serves for combining the reference and the object waves, was replaced by a dynamic hologram continuously recorded in a photorefractive crystal (PRC). Since the configuration in this case involves only two waves that interfere inside the crystal, it was called two-wave mixing, TWM (Huignard & Marrakchi, 1981). Considering that the photorefractive dynamic hologram adapts not only the signal wavefront to the reference one but it is also self-adapted to slow temporal variations of the phase difference, this type of optical system is referred to as adaptive interferometry (Stepanov, 1991). The idea of using

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 105

environment keeping the sensitivity at the high level approaching the classical homodyne detection limit. More specifically, adaptive interferometers are used in so-called laser ultrasound systems for non-destructive remote inspection of internal defects inside materials and constructions (Dewhurst & Shan, 1999). In these systems an acoustic wave (a sound pulse) generated by very short and powerful laser pulse propagates through a sample under the test and produces very week vibrations of the backside surface with the tiny amplitude of the order of 1 nm or even smaller. These vibrations (if they are detected by an adaptive interferometer) allow quality estimations of the tested object along the path of

Another area of applications of phase demodulators based on photorefractive dynamic holograms is stabilization of fibre-optical interferometric sensors eliminating all unwanted influences of the environment (e.g., temperature changes) (Di Girolamo, et al., 2007a; Fomitchov, et al., 2002; Kamshilin, et al., 1995; Kamshilin, et al., 1998; Qiao, et al., 2006; Romashko, et al., 2005b). Adaptive interferometers find also application for real-time analysis of deformations and vibrations of technical constructions (such as space reflectors (Pauliat, et al., 2006) or even artworks (Thizy, et al., 2007)), for molecular recognition (Peng, et al., 2007), for acousto-optical imaging of biological tissues (Delaye, et al., 2005), for refractive index measurement (Lichtenberg, et al., 2005), in optical coherence-domain

Among those applications there is a class of practical problems where several simultaneous measurements are required (Kulchin, 2001; Peiponen, et al., 2009). Thereby a development

Taking into account that a key element of an adaptive interferometer is a dynamic hologram (DH) recorded in PRC, two possible strategies for development of multichannel systems can be considered. The first is based on using of several crystals, one for each channel. The advantage of this extensive approach is a possibility of providing complete independency of the system channels performance. However it leads to system complexity (due to the large number of reference beams, optical elements, etc.), disproportional increase of system cost, increase of energy consumption, reduced reliability and so on. An alternative strategy for development of a multi-channel adaptive interferometer consists in multiplexing of DHs in single PRC. Note that multiple holograms recording in same volume of photosensitive material is widely used for data storage in holography memory systems (Peiponen, et al., 2009; Steckman, et al., 2001), for correlation processing and pattern recognition (Feng, et al., 2000; Wen & Yang, 1997), in multichannel optical communication systems (An, et al., 2001; Petrov, et al., 2001), in multichannel measurement systems (Andersen, et al., 2009;

It is worth noting that static (or permanent) holograms can be multiplexed in a photosensitive material one by one. On contrary, a distinctive feature of both recording and reading out dynamic holograms is that the hologram disappears if light waves are switched off. Therefore a multiplexing of dynamic holograms is possible only in case of their simultaneous recording. This could leads to additional cross-talk between channels due to pair wise interaction of two arbitrary signal waves in the PRC. Moreover, an overlap of light fields from neighboring channels in a crystal volume could affect the parameters of the holograms and, as a consequence, the sensitivity of particular holographic channels. All these circumstances must be taken into account in the development of multichannel

reflectometry systems (Peng, et al., 2003), and many others.

of multichannel adaptive system is a topical problem.

Kujawinska & Robinson, 1988) and many others.

adaptive interferometric system.

the sound pulse.

photorefractive crystals in an adaptive interferometer with multimode optical fibres was first proposed in 1980 by Hall et al. (Hall, et al., 1980). It was pointed out in this early paper that the physical mechanism of the dynamic grating formation in photorefractive crystals strongly affects phase-to-intensity transformation. The highest sensitivity to small phase excursions is achieved in the linear mode of phase detection when spatial variations of the refractive index inside the crystal are either non-shifted or shifted by a half of the grating period with respect to the interference fringes (Hall, et al., 1980). Such spatial shift occurs when the hologram is recorded in a photorefractive crystal under strong DC-field, in the so called drift regime (Young, et al., 1974). The main disadvantage of this approach is screening of the external electric field which leads to serious suppression of coupling of interfering beams in the illuminated part of the crystal. Another technical problem is overheating of the crystal under strong DC electric field. It is especially serious for fast photorefractive crystals because of their high photoconductivity. Strong overheating may even lead to the breakdown of the crystal. To prevent this DC-field should be applied to the crystal just during a short time (typically tens of milliseconds) followed by the relaxation period (typically tens of seconds). Thus the measurement can be done only in the pulse regime, which could be unacceptable for certain applications. Moreover, realization of this regime requires use of special synchronizing electronics that also complicates the measuring system as a whole. Alternatively, holographic recording in the diffusion regime (without external electric field) leads to the less sensitive, quadratic regime of phase demodulation (Kamshilin, et al., 1986). However, as it was shown later by Kamshilin and Grachev, the linear phase-to-intensity transformation can be achieved even in the diffusion regime of the hologram recording if the interfering waves have different polarization states (Kamshilin & Grachev, 2002). The last approach is based on vectorial-wave mixing (VWM) in photorefractive crystals for which the theory was developed by Sturman et al. (Sturman, et al., 1999). It is worth noting that an adaptive interferometer based on the technique proposed in (Kamshilin & Grachev, 2002) operates in so called enhanced diffusion mode when a strong AC-electric field is applied to a crystal for enhancing wave coupling. Further development of VWM-based approach has allowed one to find solutions which it made it possible:


These solutions have allowed to significantly simplify the scheme of adaptive interferometer (by removing high-voltage suppliers, synchronizing feedback loops, etc.), improve performance stability, reduce optical loses, noise and energy-consumption. As a result, adaptive interferometers become more promising tools for different practical applications.

It is worth noting that nowadays the adaptive interferometers are intensively developed and find new and new application areas. First of all is non-destructive testing of materials and elements of technical constructions. In this area adaptive interferometry becomes one of the most promising techniques due to its possibility to operate reliably in an industrial

photorefractive crystals in an adaptive interferometer with multimode optical fibres was first proposed in 1980 by Hall et al. (Hall, et al., 1980). It was pointed out in this early paper that the physical mechanism of the dynamic grating formation in photorefractive crystals strongly affects phase-to-intensity transformation. The highest sensitivity to small phase excursions is achieved in the linear mode of phase detection when spatial variations of the refractive index inside the crystal are either non-shifted or shifted by a half of the grating period with respect to the interference fringes (Hall, et al., 1980). Such spatial shift occurs when the hologram is recorded in a photorefractive crystal under strong DC-field, in the so called drift regime (Young, et al., 1974). The main disadvantage of this approach is screening of the external electric field which leads to serious suppression of coupling of interfering beams in the illuminated part of the crystal. Another technical problem is overheating of the crystal under strong DC electric field. It is especially serious for fast photorefractive crystals because of their high photoconductivity. Strong overheating may even lead to the breakdown of the crystal. To prevent this DC-field should be applied to the crystal just during a short time (typically tens of milliseconds) followed by the relaxation period (typically tens of seconds). Thus the measurement can be done only in the pulse regime, which could be unacceptable for certain applications. Moreover, realization of this regime requires use of special synchronizing electronics that also complicates the measuring system as a whole. Alternatively, holographic recording in the diffusion regime (without external electric field) leads to the less sensitive, quadratic regime of phase demodulation (Kamshilin, et al., 1986). However, as it was shown later by Kamshilin and Grachev, the linear phase-to-intensity transformation can be achieved even in the diffusion regime of the hologram recording if the interfering waves have different polarization states (Kamshilin & Grachev, 2002). The last approach is based on vectorial-wave mixing (VWM) in photorefractive crystals for which the theory was developed by Sturman et al. (Sturman, et al., 1999). It is worth noting that an adaptive interferometer based on the technique proposed in (Kamshilin & Grachev, 2002) operates in so called enhanced diffusion mode when a strong AC-electric field is applied to a crystal for enhancing wave coupling. Further development of VWM-based approach has allowed one to find solutions which it made it




Girolamo, et al., 2007a; Romashko, et al., 2005a);

VWM (Di Girolamo, et al., 2010);

possible:

environment keeping the sensitivity at the high level approaching the classical homodyne detection limit. More specifically, adaptive interferometers are used in so-called laser ultrasound systems for non-destructive remote inspection of internal defects inside materials and constructions (Dewhurst & Shan, 1999). In these systems an acoustic wave (a sound pulse) generated by very short and powerful laser pulse propagates through a sample under the test and produces very week vibrations of the backside surface with the tiny amplitude of the order of 1 nm or even smaller. These vibrations (if they are detected by an adaptive interferometer) allow quality estimations of the tested object along the path of the sound pulse.

Another area of applications of phase demodulators based on photorefractive dynamic holograms is stabilization of fibre-optical interferometric sensors eliminating all unwanted influences of the environment (e.g., temperature changes) (Di Girolamo, et al., 2007a; Fomitchov, et al., 2002; Kamshilin, et al., 1995; Kamshilin, et al., 1998; Qiao, et al., 2006; Romashko, et al., 2005b). Adaptive interferometers find also application for real-time analysis of deformations and vibrations of technical constructions (such as space reflectors (Pauliat, et al., 2006) or even artworks (Thizy, et al., 2007)), for molecular recognition (Peng, et al., 2007), for acousto-optical imaging of biological tissues (Delaye, et al., 2005), for refractive index measurement (Lichtenberg, et al., 2005), in optical coherence-domain reflectometry systems (Peng, et al., 2003), and many others.

Among those applications there is a class of practical problems where several simultaneous measurements are required (Kulchin, 2001; Peiponen, et al., 2009). Thereby a development of multichannel adaptive system is a topical problem.

Taking into account that a key element of an adaptive interferometer is a dynamic hologram (DH) recorded in PRC, two possible strategies for development of multichannel systems can be considered. The first is based on using of several crystals, one for each channel. The advantage of this extensive approach is a possibility of providing complete independency of the system channels performance. However it leads to system complexity (due to the large number of reference beams, optical elements, etc.), disproportional increase of system cost, increase of energy consumption, reduced reliability and so on. An alternative strategy for development of a multi-channel adaptive interferometer consists in multiplexing of DHs in single PRC. Note that multiple holograms recording in same volume of photosensitive material is widely used for data storage in holography memory systems (Peiponen, et al., 2009; Steckman, et al., 2001), for correlation processing and pattern recognition (Feng, et al., 2000; Wen & Yang, 1997), in multichannel optical communication systems (An, et al., 2001; Petrov, et al., 2001), in multichannel measurement systems (Andersen, et al., 2009; Kujawinska & Robinson, 1988) and many others.

It is worth noting that static (or permanent) holograms can be multiplexed in a photosensitive material one by one. On contrary, a distinctive feature of both recording and reading out dynamic holograms is that the hologram disappears if light waves are switched off. Therefore a multiplexing of dynamic holograms is possible only in case of their simultaneous recording. This could leads to additional cross-talk between channels due to pair wise interaction of two arbitrary signal waves in the PRC. Moreover, an overlap of light fields from neighboring channels in a crystal volume could affect the parameters of the holograms and, as a consequence, the sensitivity of particular holographic channels. All these circumstances must be taken into account in the development of multichannel adaptive interferometric system.

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 107

The first theoretical model of the PR effect was proposed by Kukhtarev etal. in 1979 (Kukhtarev, et al., 1979). Later in 1999, Sturman et.al. developed more general and more rigorous theoretical model of two-wave mixing in PR crystal which takes into account a vectorial nature of light waves, anisotropic properties of photorefractive materials and anisotropy of light diffraction at a dynamic holographic grating (Sturman, et al., 1999). After decades of studying the PR effect a number of fundamental monographs devoted to wave-mixing in PR materials were written (Petrov, et al., 1983; Petrov, et al., 1991; Solymar, et al., 1996; Sturman & Fridkin, 1992). These books can be recommended to the reader interested in detailed study of PR effect. In this Chapter we give just the main points of the theoretical model of vectorial wave mixing in PR crystal of cubic symmetry (Sturman, et al.,

Consider two coherent waves with vectorial amplitudes **A1** and **A2** entering a PR crystal of cubic symmetry under an external electric field **E0** through its (*xy*)-face (Fig.1). In paraxial approximation valid for small angles between wave vectors **k1** and **k2** and axis *z* the wave

The waves' interference will result in the appearance of space charge field **EK** which alters the crystal refractive index and forms a dynamic holographic grating with wave vector **K** = **k1** – **k2** in plane (*xy*). Diffraction of mixed waves at the dynamic grating provides their coupling which results in change of wave amplitudes at the crystal output. This process is

1 2

*K*

<sup>ˆ</sup> <sup>ˆ</sup> , <sup>2</sup>

**G A VA**

*i iE*

*g i iE*

where *g* is the parameter which takes into account relative direction of wave propagation in the crystal (*g* = +1 in transmission geometry where waves propagate in same direction, and

absorption coefficient; *EK* is the amplitude of space-charge field **EK** . Using the conventional one-trap–one-band model for the charge transfer (Petrov, et al., 1991) and assuming

<sup>ˆ</sup> <sup>ˆ</sup> , <sup>2</sup>

**G A VA**

\* 2 1 (1)

is the light

*K*

1999) which can help better understanding of further material.

amplitudes can be considered as 2D-vectors with *x*- and *y*-components.

Fig. 1. Geometry of vectorial wave mixing in a photorefractive crystal

*z*

*z*

 

g = –1 in reflection geometry where waves propagate in opposite directions);

 

described by the system of coupled-wave equations:

All approaches for multiple hologram recording can be grouped into three classes: spatial, angular and spectral multiplexing techniques. In the first approach, holograms are recorded in different parts of the same crystal. This technique allows one to provide maximal independence of channel performance; the cross-talk is practically precluded. However, the number of multiplexed holograms (and therefore number of channels) is limited by the crystal size, and in most cases does not exceed a few tens (Kulchin, et al., 2000b). Moreover, an equivalent number of individual reference beams is required for recording of each hologram in this case, which also leads to complexity of the measurement system and brings its efficiency down.

Angular multiplexing of holograms allows significant increase in the number of channels formed in a single crystal. In this case the same volume of crystal can be used for recording several holograms, signal light beams can partially or even completely overlap in a crystal and a common reference beam can be used for all holograms. However cross-talk between channels becomes more probable in such a scheme. This defines one of the important problems which should be solved. It is necessary to find conditions which preclude a crosstalk between angular multiplexed holograms or the cross-talk level will be below the inherent noise of the system.

The spectral multiplexing approach is based on using different wavelengths for dynamic hologram recording. This approach naturally fits with the WDM-technique of fiber-optical sensors (especially FBG) multiplexing. As a result effective multichannel measurement systems can be created by combined these two principles. However a realization of the spectral multiplexing approach requires keeping a number of peculiarities. In particular, the spectral working range should match the spectral sensitivity of a photosensitive material (e.g. PRC). Moreover, it is necessary to take into account the possible appearance of crosstalk due to overlap of multiplexed channel spectra. The last requirement together with the first one constrains the number of channels which could be realized in a multichannel system based on spectral multiplexing.

Below in the following Sections 4-6 we consider more detailed practical realizations of multichannel adaptive interferometers and measurement systems based on the above listed principles of dynamic hologram multiplexing in a photorefractive crystal. First we give briefly the basics of wave mixing in photorefractive crystal (Section 2) and introduce a relative detection limit as a parameter which characterizes adaptive interferometer sensitivity (Section 3).
