**Meet the editor**

Dr. Farzad Ebrahimi was born in Qazvin, Iran, in 1979. He graduated in Mechanical Engineering from the University of Tehran, Iran, in 2002. He received his MSc and PhD degrees in Mechanical Engineering with a specialization in applied design from the University of Tehran, Iran, in 2009. Since 2002, he has been working at the Smart Materials and Structures Research Center,

Faculty of Mechanical Engineering, University of Tehran, where he is a researcher of smart functionally graded materials and structures. He joined the Department of Mechanical Engineering at the Imam Khomeini International University as an associate professor in 2010. He is involved in several international journals as an Editor and a reviewer. He serves on the Editorial board of the SAGE Publication: *Journal of Advances in Mechanical Engineering*. He is the author of books *Smart Functionally Graded Plates* and *Progress in Analysis of Functionally Graded Structures* (Nova Science Publishers, NY). He also served as the editor of the book *Nanocomposites—New Trends and Developments* (InTech—Open Access Publisher). His research interests focus on the areas of smart materials and structures, nanostructures, vibration nanocomposites, composite materials, and structures, and he has published several researches in these fields. His research in these areas has been presented at international conferences and appeared in academic journals such as *Composite Structures*, *Composites Part B*, *Journal of Mechanical Science and Technology*, *Smart Materials and Structures*, *European Journal of Mechanics*, and *Archive of Applied Mechanics*. He also has a strong collaboration with the Iranian industries on gas and oil projects, and he serves as ad hoc referee in several top academic journals as well.

Contents

**Preface VII**

**Applications 1** Shotaro Ishino

Piotr Kiełczyński

**on the Edge 43**

Chapter 1 **Novel Waveguide Technologies and Its Future System**

Chapter 2 **Properties and Applications of Love Surface Waves in**

Chapter 4 **Measurement of Sea Wave Spatial Spectra from High-Resolution Optical Aerospace Imagery 71** Valery G. Bondur and Alexander B. Murynin

Chapter 3 **Dyakonov Surface Waves: Anisotropy-Enabling Confinement**

Chapter 5 **Electromagnetic Polarization: A New Approach on the Linear**

Chapter 6 **Modal Phenomena of Surface and Bulk Polaritons in Magnetic-**

Vladimir R. Tuz, Illia V. Fedorin and Volodymyr I. Fesenko

Chapter 7 **Video Measurements and Analysis of Surface Gravity Waves in**

Charles R. Bostater Jr, Bingyu Yang and Tyler Rotkiske

Jobson de Araújo Nascimento, Regina Maria De Lima Neta, José Moraes Gurgel Neto, Adi Neves Rocha and Alexsandro Aleixo

Carlos J. Zapata-Rodríguez, Slobodan Vuković, Juan J. Miret, Mahin

**Seismology and Biosensors 17**

Naserpour and Milivoj R. Belić

**Component Method 89**

**Semiconductor Superlattices 99**

Pereira Da Silva

**Shallow Water 127**

## Contents

#### **Preface XI**


Preface

dia having different dielectric constants.

of open problems are in this area.

ments in *surface waves* aspects.

process of creating this book.

ed by fundamental and applied research studies.

A surface wave is a mechanical wave that propagates along the interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occur within liquids at the interface between two fluids with different densities. Elastic surface waves can travel along the surface of solids, such as Ray‐ leigh or Love waves. Electromagnetic waves can also propagate as "surface waves" in which they can be guided along a refractive index gradient or along an interface between two me‐

Surface waves have drawn a significant attention and interest in the recent years in a broad range of commercial applications, while their commercial developments have been support‐

This book is a result of contributions of experts from international scientific community working in different aspects of *surface waves* and reports on the state-of-the-art research and

The book contains up-to-date publications of leading experts, and the edition is intended to furnish valuable recent information to the professionals involved in *surface wave* aspects and technologies. The text is addressed not only to researchers but also to professional engi‐ neers, students, and other experts in various disciplines, both academic and industrial, seek‐ ing to gain a better understanding of what has been done in the field recently and what kind

I hope that readers will find the book useful and inspiring by examining the recent develop‐

Finally, I would like to thank all the authors for their excellent contributions in different areas covered in this book, and I would also like to thank the InTech team, especially Ms. Renata Sliva, Publishing Process Manager, for her support and patience during the whole

> **Dr. Farzad Ebrahimi** Associate Professor

> > Qazvin, Iran

Department of Mechanical Engineering Imam Khomeini International University

development findings on this topic through original and innovative research studies.

## Preface

A surface wave is a mechanical wave that propagates along the interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occur within liquids at the interface between two fluids with different densities. Elastic surface waves can travel along the surface of solids, such as Ray‐ leigh or Love waves. Electromagnetic waves can also propagate as "surface waves" in which they can be guided along a refractive index gradient or along an interface between two me‐ dia having different dielectric constants.

Surface waves have drawn a significant attention and interest in the recent years in a broad range of commercial applications, while their commercial developments have been support‐ ed by fundamental and applied research studies.

This book is a result of contributions of experts from international scientific community working in different aspects of *surface waves* and reports on the state-of-the-art research and development findings on this topic through original and innovative research studies.

The book contains up-to-date publications of leading experts, and the edition is intended to furnish valuable recent information to the professionals involved in *surface wave* aspects and technologies. The text is addressed not only to researchers but also to professional engi‐ neers, students, and other experts in various disciplines, both academic and industrial, seek‐ ing to gain a better understanding of what has been done in the field recently and what kind of open problems are in this area.

I hope that readers will find the book useful and inspiring by examining the recent develop‐ ments in *surface waves* aspects.

Finally, I would like to thank all the authors for their excellent contributions in different areas covered in this book, and I would also like to thank the InTech team, especially Ms. Renata Sliva, Publishing Process Manager, for her support and patience during the whole process of creating this book.

> **Dr. Farzad Ebrahimi** Associate Professor Department of Mechanical Engineering Imam Khomeini International University Qazvin, Iran

**Chapter 1**

**Provisional chapter**

**Novel Waveguide Technologies and Its Future System**

Radio waves are widely used in the fields of communication and sensing, and technologies for sending wireless power are currently being put to practical use. The barriers that have so far limited these technologies are about to disappear completely. In the present study, we examine waveguides, which are a key component of the next-generation wireless technologies. A waveguide is a metal pipe through which radio waves transfer. Although a waveguide is a very heavy component, due to technological innovations, waveguides will undergo drastic modifications in the near future. This chapter introduces trends in innovative waveguide technologies and the latest wireless systems,

including communication and power transfer system, that use waveguides.

communication, wireless power transfer, wireless systems

**Keywords:** microwaves, radio waves, wave propagation, electromagnetic theory, surface transmission, evanescent wave, components, waveguide, antenna, wireless

**Novel Waveguide Technologies and Its Future System** 

DOI: 10.5772/intechopen.72039

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

Electromagnetic waves are waves formed by changing electric and magnetic fields in space. Electromagnetic waves refer to waves with a wavelength of 100 μm or more (3 THz or less). They are described as microwaves or millimeter waves, depending on the wavelength. The existence of electromagnetic waves was predicted by J. C. Maxwell in 1864. J. C. Maxwell proved that the speed at which electromagnetic waves propagate is equal to the speed of light and revealed the fundamental principle that light is propagated in the form of electromagnetic waves [1]. In 1888, H. R. Herth confirmed the presence of electromagnetic waves. This experimentally demonstrated the existence of the electromagnetic waves that was theoretically explained by Maxwell and was shown by the air propagation that Maxwell had not revealed [2]. In 1895, G. Marconi

**Applications**

**Applications**

Shotaro Ishino

**Abstract**

**1. Introduction**

Shotaro Ishino

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72039

**Provisional chapter**

#### **Novel Waveguide Technologies and Its Future System Applications Applications**

**Novel Waveguide Technologies and Its Future System** 

DOI: 10.5772/intechopen.72039

#### Shotaro Ishino Additional information is available at the end of the chapter

Shotaro Ishino

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72039

#### **Abstract**

Radio waves are widely used in the fields of communication and sensing, and technologies for sending wireless power are currently being put to practical use. The barriers that have so far limited these technologies are about to disappear completely. In the present study, we examine waveguides, which are a key component of the next-generation wireless technologies. A waveguide is a metal pipe through which radio waves transfer. Although a waveguide is a very heavy component, due to technological innovations, waveguides will undergo drastic modifications in the near future. This chapter introduces trends in innovative waveguide technologies and the latest wireless systems, including communication and power transfer system, that use waveguides.

**Keywords:** microwaves, radio waves, wave propagation, electromagnetic theory, surface transmission, evanescent wave, components, waveguide, antenna, wireless communication, wireless power transfer, wireless systems

#### **1. Introduction**

Electromagnetic waves are waves formed by changing electric and magnetic fields in space. Electromagnetic waves refer to waves with a wavelength of 100 μm or more (3 THz or less). They are described as microwaves or millimeter waves, depending on the wavelength. The existence of electromagnetic waves was predicted by J. C. Maxwell in 1864. J. C. Maxwell proved that the speed at which electromagnetic waves propagate is equal to the speed of light and revealed the fundamental principle that light is propagated in the form of electromagnetic waves [1]. In 1888, H. R. Herth confirmed the presence of electromagnetic waves. This experimentally demonstrated the existence of the electromagnetic waves that was theoretically explained by Maxwell and was shown by the air propagation that Maxwell had not revealed [2]. In 1895, G. Marconi

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

succeeded in wireless telegraphy [3–5]. In Japan, radio broadcasting began in 1925, and television broadcasting began in in 1953. Moreover, to date, electromagnetic waves are used for various purposes ranging from communication and sensing to microwave ovens. Electromagnetic waves are colloquially described as "fluttering in space," and it can be said that life is established by these waves. In recent years, attention has been paid to a technology for wireless power transfer. This technology converts electric power that was previously sent by wire into electromagnetic waves to transmit electricity in space [6–9]. In around the 1900s, N. Tesla tried wireless transmission at a frequency of 150 kHz but failed in his attempts. However, in the 1960s, W. Brown succeeded with his experiments by using microwaves at 2.45 GHz [10]. Research on wireless power transfer is being actively conducted for the range of several-microwatts, used for energy harvesting [11–14] and RFID [15, 16], to the several-kilowatts, used for applications in space in solar power satellites [17–19]. Ultimately, a perfect wireless smart society (**Figure 1**) may be realized in which all wires are unnecessary. As G. Marconi said, "It is dangerous to put limits on wireless." The possibilities of wireless are, indeed, infinite.

However, electromagnetic waves have several drawbacks. As electromagnetic waves propagate, the propagation loss increases because they spread out in space when radiated. This is indicated by the Friis formula [20, 21] and is a physically fixed loss. When the transmission power is *Pt*, the received power is *Pr*, the wavelength is *λ*, and the transmission distance is *d*, then the transmission equation is as follows.

$$Pr = \left(\frac{\lambda}{4\pi d}\right)^2 Pt\tag{1}$$

The received power is inversely proportional to the square of the distance and it attenuates. Moreover, if shields are present between transmission and reception of power or if the line of sight is bad, then the attenuation increases further or is completely cut off. Therefore, efficient and reliable transmission is an issue. In the future wireless society, a transmission path that assists transmission lines will play an important role. In this study, we examine waveguides, which are a key component of the next-generation wireless technologies. A waveguide is a metal pipe through which radio waves transfer. Despite being a very heavy component, due to technological innovations, waveguides will undergo drastic modifications in the near future. This chapter introduces trends in innovative waveguide technologies and the latest wireless systems, including communication and power transfer system, that use waveguides.

Novel Waveguide Technologies and Its Future System Applications

http://dx.doi.org/10.5772/intechopen.72039

3

A waveguide is a transmission line that transmits electromagnetic waves in a hollow tube (**Figure 2**). Initially, J. J. Thomson and L. Rayleigh et al. came up with the first proposal for such a system [22–30]. Since a waveguide is installed within an enclosed tube, the problem of blocking transmission is solved, thus contributing to improved reliability. There is no fear of power spreading in space; thus, there is no transmission loss. Compared to other forms of transmission, better transmission efficiency is offered by a waveguide. For utilizing the features of waveguides, they are widely used as components for high-power transmission, such as for feeding to an antenna for broadcasting and application between a magnetron and a chamber in a microwave oven. Microwave heating applications are not limited to domestic microwave ovens but extend to industrial applications such as food processing [31–33] and smelting of iron ores [34, 35]. A waveguide is designed to be approximately *λ*/2 with respect to the wavelength of the electromagnetic wave to be used if the waveguide is circular in diameter. Moreover, in the case of rectangular waveguide, a waveguide is long side dimension.

The electromagnetic field is a wave that exhibits sinusoidal variation in time. By solving Maxwell's equations and the Helmholtz equations, the solution of the electromagnetic field

*Ez* = 0, *Hz* = 0; Transverse electric and magnetic (TEM) (2)

*Ez* = 0, *Hz* ≠ 0; Transverse electric (TE) (3)

propagating in the +z direction can be classified into the following three types [36].

**2. What is a waveguide**

**Figure 2.** Examples of waveguides.

**Figure 1.** Our dream: wireless smart society [19].

The received power is inversely proportional to the square of the distance and it attenuates. Moreover, if shields are present between transmission and reception of power or if the line of sight is bad, then the attenuation increases further or is completely cut off. Therefore, efficient and reliable transmission is an issue. In the future wireless society, a transmission path that assists transmission lines will play an important role. In this study, we examine waveguides, which are a key component of the next-generation wireless technologies. A waveguide is a metal pipe through which radio waves transfer. Despite being a very heavy component, due to technological innovations, waveguides will undergo drastic modifications in the near future. This chapter introduces trends in innovative waveguide technologies and the latest wireless systems, including communication and power transfer system, that use waveguides.

#### **2. What is a waveguide**

succeeded in wireless telegraphy [3–5]. In Japan, radio broadcasting began in 1925, and television broadcasting began in in 1953. Moreover, to date, electromagnetic waves are used for various purposes ranging from communication and sensing to microwave ovens. Electromagnetic waves are colloquially described as "fluttering in space," and it can be said that life is established by these waves. In recent years, attention has been paid to a technology for wireless power transfer. This technology converts electric power that was previously sent by wire into electromagnetic waves to transmit electricity in space [6–9]. In around the 1900s, N. Tesla tried wireless transmission at a frequency of 150 kHz but failed in his attempts. However, in the 1960s, W. Brown succeeded with his experiments by using microwaves at 2.45 GHz [10]. Research on wireless power transfer is being actively conducted for the range of several-microwatts, used for energy harvesting [11–14] and RFID [15, 16], to the several-kilowatts, used for applications in space in solar power satellites [17–19]. Ultimately, a perfect wireless smart society (**Figure 1**) may be realized in which all wires are unnecessary. As G. Marconi said, "It is dangerous to put

However, electromagnetic waves have several drawbacks. As electromagnetic waves propagate, the propagation loss increases because they spread out in space when radiated. This is indicated by the Friis formula [20, 21] and is a physically fixed loss. When the transmission power is *Pt*, the received power is *Pr*, the wavelength is *λ*, and the transmission distance is *d*,

> \_\_\_\_ *λ* <sup>4</sup>*d*) 2

*Pt* (1)

limits on wireless." The possibilities of wireless are, indeed, infinite.

then the transmission equation is as follows.

2 Surface Waves - New Trends and Developments

*Pr* = (

**Figure 1.** Our dream: wireless smart society [19].

A waveguide is a transmission line that transmits electromagnetic waves in a hollow tube (**Figure 2**). Initially, J. J. Thomson and L. Rayleigh et al. came up with the first proposal for such a system [22–30]. Since a waveguide is installed within an enclosed tube, the problem of blocking transmission is solved, thus contributing to improved reliability. There is no fear of power spreading in space; thus, there is no transmission loss. Compared to other forms of transmission, better transmission efficiency is offered by a waveguide. For utilizing the features of waveguides, they are widely used as components for high-power transmission, such as for feeding to an antenna for broadcasting and application between a magnetron and a chamber in a microwave oven. Microwave heating applications are not limited to domestic microwave ovens but extend to industrial applications such as food processing [31–33] and smelting of iron ores [34, 35]. A waveguide is designed to be approximately *λ*/2 with respect to the wavelength of the electromagnetic wave to be used if the waveguide is circular in diameter. Moreover, in the case of rectangular waveguide, a waveguide is long side dimension.

The electromagnetic field is a wave that exhibits sinusoidal variation in time. By solving Maxwell's equations and the Helmholtz equations, the solution of the electromagnetic field propagating in the +z direction can be classified into the following three types [36].

$$E\_z = 0, \quad H\_z = 0; \text{ Transverse electric and magnetic (TEM)}\tag{2}$$

$$E\_z = 0, \quad H\_z \neq 0; \text{ Transverse electric (TE)}\tag{3}$$

**Figure 2.** Examples of waveguides.

**Figure 3.** Propagation modes.

$$E\_{\pm} \neq 0, \quad H\_{\pm} = 0; \text{ Transverse magnetic (TM)}\tag{4}$$

3D printers based on thermal melting lamination are also available for sale. Moreover, for business use, machines that employ the inkjet method, optical shaping, and powder sintering molding are used in the development department of the manufacturing industry. Because 3D prototypes

Novel Waveguide Technologies and Its Future System Applications

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5

Various reports have been produced on prototypes of waveguides and peripheral components made by resin molding 3D printers [39–43]. Because electromagnetic waves cannot be confined in plastic tubes, it is necessary to make additional conductive membranes on the surface of the pipe. Thus, although a film having high conductivity can be formed by the plating method, the film thickness is approximately 1 μm and so the microwave penetrates into the inside of the film before finally being transmitted. Thus, adequate shielding properties cannot be obtained. There are also examples that employ a conductive paint to achieve a film thickness of approximately 10 μm; however, the conductivity is poor and the loss due to the conductor becomes large. Moreover, these are mainly microwave components. For millimeter wave components, fine pro-

cessing is required, which is difficult to realize with the current processing precision.

microwave components.

**3.2. Hose-type waveguide**

**Figure 4.** (a) Metallic rectangular waveguide and (b) polished surface.

Moreover, evaluation of several resin materials of the acrylonitrile butadiene styrene such as the resin used in the optical fabrication method revealed that the value of the imaginary part of the dielectric constant, which is a factor of the loss of electromagnetic waves, is relatively large. In the future, it is desirable to develop low-loss materials and molding methods for

Moreover, 3D printers capable of directly molding metallic materials are also being used. Metal powder can be sintered by selective laser sintering or selective laser melting. As a result, processing of the conductive film and losses due to resin are eliminated. However, unevenness is formed on the surface, and there can be a problem with the surface becoming very rough. As surface roughness decreases, the conductivity of the surface decreases conduction loss increases. Currently, aluminum alloys are mainly used in 3D printing as materials, but a practical use of copper-based materials is progressing. If the conductivity of the material improves, this loss can be expected to decrease. We fabricated a 10-GHz rectangular waveguide and evaluated its characteristics (**Figure 4**). As a result, there was a transmission loss of approximately 1.5 times than that of the usual waveguide. There was also a leak from the flange portion. The connection was improved by polishing unevenness, but additional work is still required.

Weight reduction of the waveguide is done by using resin, but the structure of the waveguide has remained as a non-hollow, solid pipe. Therefore, we are developing a flexible waveguide that is like a water hose. Besides improving convenience by making the pipe flexible like a

can be made without a mold, they can be made using simple prototyping.

An electromagnetic field can be expressed as a combination of three types of waves. The TEM wave has no electromagnetic field component in the propagation direction. The wave is an entirely transverse electromagnetic wave. A plane wave propagating in space, a flat plate line, and an electromagnetic wave transmitted inside the coaxial line are all TEM waves. A plane wave propagating in an electromagnetic field can be expressed as a combination of three types of waves. Space, a flat plate line, and an electromagnetic wave transmitted inside a coaxial line are TEM waves. The states of the electric and magnetic fields in the x–y plane perpendicular to the propagation direction of the TEM wave are the same as those of the electrostatic field and the static magnetic field. Because there is no electrostatic field in the tube surrounded by the conductor wall of the same potential, the TEM wave does not propagate to the waveguide. In order to propagate the TEM wave, it is necessary to use a transmission path comprising two or more insulated conductors.

TE and TM waves are generated in the waveguide. The TE wave is also known as the H wave. The z component of the electric field *E* is an electromagnetic wave with *Ez* = 0. The electric field is a transverse wave. The magnetic field is a longitudinal wave. In a rectangular waveguide, electromagnetic waves are transmitted with the TE wave as a basic mode. The TM wave is also known as the E wave. The z component of the magnetic field *H* is an electromagnetic wave with *Hz* = 0. The electric field is the longitudinal and transverse waves, and the magnetic field is the transverse wave. The spherical wave propagating in space is a TM wave (**Figure 3**).

A cut-off frequency exists in the waveguide, and a frequency lower than the cut-off frequency is in the attenuation mode (evanescent mode) and cannot be transmitted. That is, it functions as a high-pass filter. Conversely, in the TEM wave transmission, the frequency is arbitrary and there is no cut-off frequency.

#### **3. Novel waveguide technologies**

#### **3.1. 3D printing waveguides**

3D printers were invented in the 1980s [37, 38], and their applications are spreading rapidly. Originally known as rapid prototyping machine, a 3D printer is a molding machine that specializes in rapid shaping. In recent years, the price of 3D printers has reduced, and home-use 3D printers based on thermal melting lamination are also available for sale. Moreover, for business use, machines that employ the inkjet method, optical shaping, and powder sintering molding are used in the development department of the manufacturing industry. Because 3D prototypes can be made without a mold, they can be made using simple prototyping.

Various reports have been produced on prototypes of waveguides and peripheral components made by resin molding 3D printers [39–43]. Because electromagnetic waves cannot be confined in plastic tubes, it is necessary to make additional conductive membranes on the surface of the pipe. Thus, although a film having high conductivity can be formed by the plating method, the film thickness is approximately 1 μm and so the microwave penetrates into the inside of the film before finally being transmitted. Thus, adequate shielding properties cannot be obtained. There are also examples that employ a conductive paint to achieve a film thickness of approximately 10 μm; however, the conductivity is poor and the loss due to the conductor becomes large. Moreover, these are mainly microwave components. For millimeter wave components, fine processing is required, which is difficult to realize with the current processing precision.

Moreover, evaluation of several resin materials of the acrylonitrile butadiene styrene such as the resin used in the optical fabrication method revealed that the value of the imaginary part of the dielectric constant, which is a factor of the loss of electromagnetic waves, is relatively large. In the future, it is desirable to develop low-loss materials and molding methods for microwave components.

Moreover, 3D printers capable of directly molding metallic materials are also being used. Metal powder can be sintered by selective laser sintering or selective laser melting. As a result, processing of the conductive film and losses due to resin are eliminated. However, unevenness is formed on the surface, and there can be a problem with the surface becoming very rough. As surface roughness decreases, the conductivity of the surface decreases conduction loss increases. Currently, aluminum alloys are mainly used in 3D printing as materials, but a practical use of copper-based materials is progressing. If the conductivity of the material improves, this loss can be expected to decrease. We fabricated a 10-GHz rectangular waveguide and evaluated its characteristics (**Figure 4**). As a result, there was a transmission loss of approximately 1.5 times than that of the usual waveguide. There was also a leak from the flange portion. The connection was improved by polishing unevenness, but additional work is still required.

#### **3.2. Hose-type waveguide**

**Figure 3.** Propagation modes.

4 Surface Waves - New Trends and Developments

there is no cut-off frequency.

**3.1. 3D printing waveguides**

**3. Novel waveguide technologies**

*Ez* ≠ 0, *Hz* = 0; Transverse magnetic (TM) (4)

An electromagnetic field can be expressed as a combination of three types of waves. The TEM wave has no electromagnetic field component in the propagation direction. The wave is an entirely transverse electromagnetic wave. A plane wave propagating in space, a flat plate line, and an electromagnetic wave transmitted inside the coaxial line are all TEM waves. A plane wave propagating in an electromagnetic field can be expressed as a combination of three types of waves. Space, a flat plate line, and an electromagnetic wave transmitted inside a coaxial line are TEM waves. The states of the electric and magnetic fields in the x–y plane perpendicular to the propagation direction of the TEM wave are the same as those of the electrostatic field and the static magnetic field. Because there is no electrostatic field in the tube surrounded by the conductor wall of the same potential, the TEM wave does not propagate to the waveguide. In order to propagate the TEM wave, it is necessary to use a transmission path comprising two or more insulated conductors.

TE and TM waves are generated in the waveguide. The TE wave is also known as the H wave. The z component of the electric field *E* is an electromagnetic wave with *Ez* = 0. The electric field is a transverse wave. The magnetic field is a longitudinal wave. In a rectangular waveguide, electromagnetic waves are transmitted with the TE wave as a basic mode. The TM wave is also known as the E wave. The z component of the magnetic field *H* is an electromagnetic wave with *Hz* = 0. The electric field is the longitudinal and transverse waves, and the magnetic field is the transverse wave. The spherical wave propagating in space is a TM wave (**Figure 3**). A cut-off frequency exists in the waveguide, and a frequency lower than the cut-off frequency is in the attenuation mode (evanescent mode) and cannot be transmitted. That is, it functions as a high-pass filter. Conversely, in the TEM wave transmission, the frequency is arbitrary and

3D printers were invented in the 1980s [37, 38], and their applications are spreading rapidly. Originally known as rapid prototyping machine, a 3D printer is a molding machine that specializes in rapid shaping. In recent years, the price of 3D printers has reduced, and home-use Weight reduction of the waveguide is done by using resin, but the structure of the waveguide has remained as a non-hollow, solid pipe. Therefore, we are developing a flexible waveguide that is like a water hose. Besides improving convenience by making the pipe flexible like a

**Figure 4.** (a) Metallic rectangular waveguide and (b) polished surface.

hose, the image changes and the application range may expand. We introduce an example in which a waveguide is made by winding a copper foil in a hollow resin hose [44].

Our waveguide is a hollow, soft-resin hose with a conductive coating on the outside for electromagnetic wave transfer. Conventional metal waveguides undergo passage, return, and conductive losses, which should be reduced as far as possible. Resin waveguides generate additional dielectric and radiation losses. The dielectric losses are due to absorption by the resin, and radiation losses occurs by leakage due to the insufficient shielding of the thin-film conductor. Dielectric and radiation losses are the dominant loss components in resin waveguides (**Figure 5**).

In this study, we use a soft elastomer material with excellent properties for forming flexible waveguides. In the 10-GHz band, the relative permittivity and dielectric loss tangent of the resin are *ε'r* = 2.28 and tan*δ* = 0.00072, respectively, ensuring very low losses as in a Teflon.

Conversely, the conventional metal-film-deposition techniques of plating, sputtering, and vapor deposition are limited to conductive films with submicron thickness. The required thickness at 10 GHz, estimated from the skin-depth relationship, is at least 10 μm. Therefore, the film in our prototype was formed by winding an 18-μm-thick copper foil around the aforementioned resin hose. We investigated several types of foil winding and found that the lowest radiation loss occurs in the H-center configuration of the waveguide.

**3.3. Sheet-type waveguide**

**Figure 6.** Hose-type waveguides.

considered [55].

**3.4. DC waveguide**

**Figure 7.** Inter-vehicle communication system.

Research on two-dimensional communication by using electromagnetic waves that propagate in a thin sheet is progressing [52, 53]. It is assumed that the communication distance is up to several meters. Moreover, by placing a type of antenna known as a coupler at an arbitrary point in the sheet form, close proximity communication inside and outside the seat can be made possible (**Figure 8**). It uses evanescent waves that ooze out of the sheet. The evanescent wave is an electromagnetic wave propagating only near the surface of the sheet. In this way, the sheet-shaped waveguide does not require wiring for each sensor terminal and does not radiate electromagnetic waves to space. In addition to contribute to improving communication security, a relatively large power can be transmitted without exposing people or objects that are not close to the seat to a strong electromagnetic field. Applications for wireless power transmission are also under consideration. Moreover, in the case of Japan, standard specifications for wireless power transfer in the seat are also in place [54]. For future applications, a power supply for a car while in motion and wearable sensor devices are being

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7

The conventional waveguide is a high-pass filter and cannot transmit bands below the cutoff frequency. However, if the structure can be revised for direct current (DC) propagation, then

The prototype (**Figure 6**) weighs 67 g/m and costs \$1.3 per meter, enabling a lightweight and inexpensive waveguide. The waveguide has a low loss and low emission, with a transmission characteristic of −0.39 dB/m in the 10-GHz band.

In future application to automated vehicles, it is necessary to install various sensors, e.g., high-quality inter-vehicle cameras [45–50] that requires transmission speeds on the order of several Gbps [51] with high security. Conventional wire harnesses cannot tolerate external noise in transfers on the order of several Gbps. Because the influence of noise increases with transmission speed, we believe that it is necessary to review the transmission line design. As shown in **Figure 7**, the proposed waveguide is laid from the front to the back to transmit a camera image. The camera image was transmitted inside the waveguide by using high-speed communication between sensors.

**Figure 5.** Loss factors in the resin waveguide.

**Figure 6.** Hose-type waveguides.

#### **3.3. Sheet-type waveguide**

Research on two-dimensional communication by using electromagnetic waves that propagate in a thin sheet is progressing [52, 53]. It is assumed that the communication distance is up to several meters. Moreover, by placing a type of antenna known as a coupler at an arbitrary point in the sheet form, close proximity communication inside and outside the seat can be made possible (**Figure 8**). It uses evanescent waves that ooze out of the sheet. The evanescent wave is an electromagnetic wave propagating only near the surface of the sheet. In this way, the sheet-shaped waveguide does not require wiring for each sensor terminal and does not radiate electromagnetic waves to space. In addition to contribute to improving communication security, a relatively large power can be transmitted without exposing people or objects that are not close to the seat to a strong electromagnetic field. Applications for wireless power transmission are also under consideration. Moreover, in the case of Japan, standard specifications for wireless power transfer in the seat are also in place [54]. For future applications, a power supply for a car while in motion and wearable sensor devices are being considered [55].

#### **3.4. DC waveguide**

Dielectric Loss

characteristic of −0.39 dB/m in the 10-GHz band.

Return Loss

Conductor Conductive Loss

hose, the image changes and the application range may expand. We introduce an example in

Our waveguide is a hollow, soft-resin hose with a conductive coating on the outside for electromagnetic wave transfer. Conventional metal waveguides undergo passage, return, and conductive losses, which should be reduced as far as possible. Resin waveguides generate additional dielectric and radiation losses. The dielectric losses are due to absorption by the resin, and radiation losses occurs by leakage due to the insufficient shielding of the thin-film conductor. Dielectric and radiation losses are the dominant loss components in resin wave-

In this study, we use a soft elastomer material with excellent properties for forming flexible waveguides. In the 10-GHz band, the relative permittivity and dielectric loss tangent of the resin are *ε'r* = 2.28 and tan*δ* = 0.00072, respectively, ensuring very low losses as in a Teflon.

Conversely, the conventional metal-film-deposition techniques of plating, sputtering, and vapor deposition are limited to conductive films with submicron thickness. The required thickness at 10 GHz, estimated from the skin-depth relationship, is at least 10 μm. Therefore, the film in our prototype was formed by winding an 18-μm-thick copper foil around the aforementioned resin hose. We investigated several types of foil winding and found that the

The prototype (**Figure 6**) weighs 67 g/m and costs \$1.3 per meter, enabling a lightweight and inexpensive waveguide. The waveguide has a low loss and low emission, with a transmission

In future application to automated vehicles, it is necessary to install various sensors, e.g., high-quality inter-vehicle cameras [45–50] that requires transmission speeds on the order of several Gbps [51] with high security. Conventional wire harnesses cannot tolerate external noise in transfers on the order of several Gbps. Because the influence of noise increases with transmission speed, we believe that it is necessary to review the transmission line design. As shown in **Figure 7**, the proposed waveguide is laid from the front to the back to transmit a camera image. The camera image was transmitted inside the waveguide by using high-speed

lowest radiation loss occurs in the H-center configuration of the waveguide.

which a waveguide is made by winding a copper foil in a hollow resin hose [44].

Pass Loss

Radiation Loss

Resin

**Figure 5.** Loss factors in the resin waveguide.

communication between sensors.

guides (**Figure 5**).

6 Surface Waves - New Trends and Developments

The conventional waveguide is a high-pass filter and cannot transmit bands below the cutoff frequency. However, if the structure can be revised for direct current (DC) propagation, then

**Figure 7.** Inter-vehicle communication system.

**Figure 8.** Sheet-type waveguide [52].

high-power transmission becomes possible with a sufficient pipe thickness. We also consider that if a stop band sufficiently far from the pass band can be transmitted, then we can achieve low-frequency communication and sharing in addition to broadband communication. We propose a waveguide with a divided structure that operates not only in a conventional (**Figure 9a**) but also in DC (**Figure 9b**) and parallel line (**Figure 9c**) modes. Subsequently, we investigated whether the waveguide realizes DC in DC mode and can transmit the stop band in a parallel line mode. The conventional mode is a TE10 mode, and the parallel line mode is a TEM mode. It is known as DC waveguide [56].

[57–62]. SIWs are planar structures fabricated using metallic via-hole arrays connecting the top and bottom ground planes of a dielectric substrate (**Figure 11**). A non-planar conventional waveguide can be modeled into a substrate integrated circuit. They are compact, lightweight, cost-effective, and easy to fabricate. Microfabrication of SIW of several microns is also possible,

10 100 1000 10000

S11-Meas S21-Meas S11-Sim S21-Sim

Novel Waveguide Technologies and Its Future System Applications

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9

Frequency [MHz]

An example in which a resonance structure is provided in a tube like a conventional waveguide to form a band-pass filter has also been reported [63–66]. We also report an example of

When the microwave is input to the SIW filter, reflection occurs due to a mismatch of the characteristic impedance at the input portion. The microwave input to the filter is distributed at a distribution ratio in the microstrip layer and the waveguide layer. Microwaves entering the microstrip layer ideally do not reflect and propagate. When the microwave entering the waveguide layer is at the frequency of the cutoff region of the waveguide, it propagates while attenuating. Thus, the attenuated portion becomes a reflected wave. Subsequently, at the output of the multilayer substrate filter, microwaves output from the microstrip layer and the waveguide layer are synthesized. Therefore, due to the phase difference of the microwaves output from each layer, propagation waves are canceled at a certain frequency, resulting in

and usage in the terahertz-band order is also expected to be promising.


*Ŷ*

*Ŷ*

S parameter [dB]

**Figure 10.** Measured results of DC waveguide.

fabricating band-stop filters with stacked waveguide structures by SIW [67].

reflection. This is the principle that enables an SIW to function as a band-stop filter.

**Figure 11.** Configuration of an SIW structure synthesized using metallic via-hole arrays [57].

Although the result (**Figure 10**) differs from simulation results, transmission in the stop band was, at least to some extent, experimentally confirmed. In DC mode, the resistance was approximately 0 Ω, confirming that high-efficiency DC transmission is also possible.

In future work, we will assess the performance of the waveguide in industrial applications. Such plural transmission modes are desired for high-power transmission and broadband communications in automatic driving.

#### **3.5. Substrate integrated waveguide (SIW)**

A conventional waveguide is a non-planar three-dimensional circuit, and it is a challenge to fabricate such a waveguide in bulk. SIWs act as an alternative option to conventional waveguides

**Figure 9.** Propagation modes of the split waveguide.

Novel Waveguide Technologies and Its Future System Applications http://dx.doi.org/10.5772/intechopen.72039 9

**Figure 10.** Measured results of DC waveguide.

[57–62]. SIWs are planar structures fabricated using metallic via-hole arrays connecting the top and bottom ground planes of a dielectric substrate (**Figure 11**). A non-planar conventional waveguide can be modeled into a substrate integrated circuit. They are compact, lightweight, cost-effective, and easy to fabricate. Microfabrication of SIW of several microns is also possible, and usage in the terahertz-band order is also expected to be promising.

An example in which a resonance structure is provided in a tube like a conventional waveguide to form a band-pass filter has also been reported [63–66]. We also report an example of fabricating band-stop filters with stacked waveguide structures by SIW [67].

When the microwave is input to the SIW filter, reflection occurs due to a mismatch of the characteristic impedance at the input portion. The microwave input to the filter is distributed at a distribution ratio in the microstrip layer and the waveguide layer. Microwaves entering the microstrip layer ideally do not reflect and propagate. When the microwave entering the waveguide layer is at the frequency of the cutoff region of the waveguide, it propagates while attenuating. Thus, the attenuated portion becomes a reflected wave. Subsequently, at the output of the multilayer substrate filter, microwaves output from the microstrip layer and the waveguide layer are synthesized. Therefore, due to the phase difference of the microwaves output from each layer, propagation waves are canceled at a certain frequency, resulting in reflection. This is the principle that enables an SIW to function as a band-stop filter.

**Figure 11.** Configuration of an SIW structure synthesized using metallic via-hole arrays [57].

䠇 䠉

**Figure 9.** Propagation modes of the split waveguide.

a TEM mode. It is known as DC waveguide [56].

communications in automatic driving.

**Figure 8.** Sheet-type waveguide [52].

8 Surface Waves - New Trends and Developments

**3.5. Substrate integrated waveguide (SIW)**

(aa)) ᑟἼ⟶䝰䞊䝗 (TE10) (b) ┤ὶ䝰䞊䝗 (c)) ᖹ⾜⥺㊰䝰䞊䝗䠄TEM䠅

A conventional waveguide is a non-planar three-dimensional circuit, and it is a challenge to fabricate such a waveguide in bulk. SIWs act as an alternative option to conventional waveguides

high-power transmission becomes possible with a sufficient pipe thickness. We also consider that if a stop band sufficiently far from the pass band can be transmitted, then we can achieve low-frequency communication and sharing in addition to broadband communication. We propose a waveguide with a divided structure that operates not only in a conventional (**Figure 9a**) but also in DC (**Figure 9b**) and parallel line (**Figure 9c**) modes. Subsequently, we investigated whether the waveguide realizes DC in DC mode and can transmit the stop band in a parallel line mode. The conventional mode is a TE10 mode, and the parallel line mode is

Although the result (**Figure 10**) differs from simulation results, transmission in the stop band was, at least to some extent, experimentally confirmed. In DC mode, the resistance was

In future work, we will assess the performance of the waveguide in industrial applications. Such plural transmission modes are desired for high-power transmission and broadband

approximately 0 Ω, confirming that high-efficiency DC transmission is also possible.

**References**

pp. 28-29

Press; 1922. p. 27

2002;**50**(7):1784-1789

Magazine. 2013;**14**(2):55-62

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[4] Baker WJ. A History of the Marconi Company 1874-1965. London: Methuen; 1970.

[5] Record of the Development of Wireless Telegraphy, The Year Book of Wireless Telegraphy and Telephony. London, Pub. for the Marconi Press Agency Ltd., by the St. Catherine

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[9] Popovic Z. Cut the cord: Low-power far-field wireless powering. IEEE Microwave

[10] Brown WC. The history of power transmission by radio waves. IEEE Transactions on

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[14] Furuta T, Ito M, Nambo N, Ito K, Noguchi K, Ida J. The 500MHz band low power rectenna for DTV in the Tokyo area. In: IEEE Wireless Power Transfer Conf.; 2016

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Royal Society of London. 1865;**155**:459-512

**Figure 12.** Multilayer SIW filter [67].

In this way, a SIW can easily realize a complicated circuit like a laminated structure. Structures and characteristics that were impossible with a stereo waveguide are obtained and can be expected to be used in various applications. The use of the band-stop filter as the harmonic circuit of the F-class amplifier [68–70] and rectifier [71, 72] are being studied. It can be expected that the efficiency of the microwave circuit can be improved, thus contributing to a low-fuelconsumption society (**Figure 12**).

#### **4. Conclusion**

This chapter introduced the novel waveguide technology. The conventional waveguide is characterized as being a large mass of metal, but the proposed waveguide is light, thin, cheap, can change its shape. Thus, waveguides are drastically renewed by the proposed novel technology. Thus, it is time for classic circuits to make a big leap forward.

In addition, along with the technical improvement to the machining technology, a waveguide circuit with a new function can also be realized. We will continue to fuse semiconductor and micro electro mechanical systems (MEMS) processes to develop fine and precise circuit technologies. In addition, we also introduced some application examples. From the microwave band to the terahertz band, the waveguide will be widely used more than ever. In order to realize a sustainable wireless society, the proposed waveguide will prove to be a key component.

#### **Acknowledgements**

I am grateful to Dr. Farzad Ebrahimi (University of Tehran, Iran) of the book editor who gave me the opportunity to write this chapter. Moreover, I respect the great achievements of my predecessors whose studies I have cited.

#### **Author details**

Shotaro Ishino

Address all correspondence to: shotaroh.ishino.qx@furuno.co.jp

Furuno Electric, Nishinomiya, Hyōgo Prefecture, Japan

#### **References**

In this way, a SIW can easily realize a complicated circuit like a laminated structure. Structures and characteristics that were impossible with a stereo waveguide are obtained and can be expected to be used in various applications. The use of the band-stop filter as the harmonic circuit of the F-class amplifier [68–70] and rectifier [71, 72] are being studied. It can be expected that the efficiency of the microwave circuit can be improved, thus contributing to a low-fuel-

This chapter introduced the novel waveguide technology. The conventional waveguide is characterized as being a large mass of metal, but the proposed waveguide is light, thin, cheap, can change its shape. Thus, waveguides are drastically renewed by the proposed novel tech-

In addition, along with the technical improvement to the machining technology, a waveguide circuit with a new function can also be realized. We will continue to fuse semiconductor and micro electro mechanical systems (MEMS) processes to develop fine and precise circuit technologies. In addition, we also introduced some application examples. From the microwave band to the terahertz band, the waveguide will be widely used more than ever. In order to realize a sustainable wireless society, the proposed waveguide will prove to be a key component.

I am grateful to Dr. Farzad Ebrahimi (University of Tehran, Iran) of the book editor who gave me the opportunity to write this chapter. Moreover, I respect the great achievements of my

nology. Thus, it is time for classic circuits to make a big leap forward.

Address all correspondence to: shotaroh.ishino.qx@furuno.co.jp

Furuno Electric, Nishinomiya, Hyōgo Prefecture, Japan

consumption society (**Figure 12**).

**Figure 12.** Multilayer SIW filter [67].

10 Surface Waves - New Trends and Developments

**4. Conclusion**

**Acknowledgements**

**Author details**

Shotaro Ishino

predecessors whose studies I have cited.


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**Chapter 2**

**Provisional chapter**

**Properties and Applications of Love Surface Waves in**

**Properties and Applications of Love Surface Waves in** 

Shear horizontal (SH) surface waves of the Love type are elastic surface waves propagating in layered waveguides, in which surface layer is "slower" than the substrate. Love surface waves are of primary importance in geophysics and seismology, since most structural damages in the wake of earthquakes are attributed to the devastating SH motion inherent to the Love surface waves. On the other hand, Love surface waves found benign applications in biosensors used in biology, medicine, and chemistry. In this chapter, we briefly sketch a mathematical model for Love surface waves and present examples of the resulting dispersion curves for phase and group velocities, attenuation as well as the amplitude distribution as a function of the depth. We illustrate damages due to Love surface waves generated by earthquakes on real-life examples. In the following of this chapter, we present a number of representative examples for Love wave biosensors, which have been already used to DNA characterization, bacteria and virus detection, measurements of toxic substances, etc. We hope that the reader, after studying this chapter, will have a clear idea that deadly earthquakes and a beneficiary biosensor technology share the same physical phenomenon, which is the basis of a fascinating interdisciplinary research. **Keywords:** Love waves, biosensors, earthquakes, surface acoustic waves, wireless

It is interesting to note that many outstanding physicists (Kelvin, Michelson, and Jolly) expressed in the second half of the nineteenth century an opinion that classical physics (how we name it nowadays) is in principle completed and nothing interesting or significant rests to be discovered. Needless to say, forecasting development of future events was always and

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

DOI: 10.5772/intechopen.75479

**Seismology and Biosensors**

**Seismology and Biosensors**

http://dx.doi.org/10.5772/intechopen.75479

sensors, dispersion curves

**1. Introduction**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Piotr Kiełczyński

Piotr Kiełczyński

**Abstract**

#### **Properties and Applications of Love Surface Waves in Seismology and Biosensors Properties and Applications of Love Surface Waves in Seismology and Biosensors**

DOI: 10.5772/intechopen.75479

#### Piotr Kiełczyński Piotr Kiełczyński

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.75479

#### **Abstract**

Shear horizontal (SH) surface waves of the Love type are elastic surface waves propagating in layered waveguides, in which surface layer is "slower" than the substrate. Love surface waves are of primary importance in geophysics and seismology, since most structural damages in the wake of earthquakes are attributed to the devastating SH motion inherent to the Love surface waves. On the other hand, Love surface waves found benign applications in biosensors used in biology, medicine, and chemistry. In this chapter, we briefly sketch a mathematical model for Love surface waves and present examples of the resulting dispersion curves for phase and group velocities, attenuation as well as the amplitude distribution as a function of the depth. We illustrate damages due to Love surface waves generated by earthquakes on real-life examples. In the following of this chapter, we present a number of representative examples for Love wave biosensors, which have been already used to DNA characterization, bacteria and virus detection, measurements of toxic substances, etc. We hope that the reader, after studying this chapter, will have a clear idea that deadly earthquakes and a beneficiary biosensor technology share the same physical phenomenon, which is the basis of a fascinating interdisciplinary research.

**Keywords:** Love waves, biosensors, earthquakes, surface acoustic waves, wireless sensors, dispersion curves

#### **1. Introduction**

It is interesting to note that many outstanding physicists (Kelvin, Michelson, and Jolly) expressed in the second half of the nineteenth century an opinion that classical physics (how we name it nowadays) is in principle completed and nothing interesting or significant rests to be discovered. Needless to say, forecasting development of future events was always and

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

still is a very risky business, especially in physical sciences and engineering. Indeed, in these disciplines of human endeavors, one must take into account not only an inherently volatile human factor but also the impact of potential discoveries of unknown yet laws of nature, which often open new unanticipated possibilities and horizons. We may try to justify such an obvious complacency, attributed to the abovementioned scientists, by the historical spirit of the Belle Époque (1870–1914), that believed in harmony, good taste, optimism, unlimited progress and generally in positivistic philosophical ideas.

It is noteworthy that the existence of Rayleigh and Love surface waves was first predicted mathematically prior to their experimental confirmation. This shows how beneficial can be the mutual interaction between the theory and experiment. Indeed, the theory indicates directions of future experimental research and the experiment confirms or renders the theory obsolete. It is worth noticing that the existence of a new type of electromagnetic surface waves was predicted mathematically quite recently, i.e., in 1988, and soon confirmed experimentally.

Properties and Applications of Love Surface Waves in Seismology and Biosensors

http://dx.doi.org/10.5772/intechopen.75479

19

It is interesting to note that Love surface waves have direct counterparts in electromagnetism (optical planar waveguides) and quantum mechanics (particle motion in a quantum well). By contrast, a similar statement is not true for Rayleigh surface waves, which therefore remain a

Surface waves of the Love type have a number of unique features. Firstly, they have only one SH component of vibrations. As a result, Love surface waves are insensitive to the loading with liquids of zero or negligible viscosities. Thus, Love surface waves can propagate long distances without a significant attenuation. Indeed, Love waves propagating many times around the Earth's circumference have been observed experimentally. On the other hand, it was discovered much later (1981) that Love waves are very well suited for measurements of viscoelastic properties of liquids. Secondly, the mathematical description of Love surface waves is much simpler than that for Rayleigh surface waves. A relative simplicity of the mathematical model enables for direct physical insight in the process of Love wave propagation,

The idea to employ Love surface waves for measurements of viscoelastic properties of liquids was presented for the first time in 1981 by Kiełczyński and Płowiec in their Polish patent [2]. In 1987, the theory of the new method was presented by Kiełczyński and Pajewski on the international arena at the European Mechanics Colloquium 226 in Nottingham, UK [3]. In 1988, they presented this new method with equations and experimental results at the IEEE 1988 Ultrasonic Symposium in Chicago [4]. In 1989, Kiełczyński and Płowiec published a detailed theory and experimental results in the prestigious Journal of the Acoustical Society of America [5]. It is noteworthy that subsequent publications on Love wave sensors for liquid characterization appeared in USA not earlier than in 1992 [6], but nowadays, we witness

We hope that the reader, after studying this chapter, will agree that the nature has many different faces and that the same physical phenomenon can be sometimes deadly (earthquakes) and in different circumstances, can be beneficiary (biosensor technology). As a consequence, SH surface waves of the Love type are an interesting example of an interdisciplinary research. This chapter is organized as follows. Section 2 presents main characteristics and properties of Love surface waves, including basic mathematical model and examples of dispersion curves and amplitude distributions. More advanced mathematical treatment of the Love surface waves can be found, for example, in [8]. Section 3 shows the importance of Love surface waves in geophysics and seismology. Section 4 describes applications of Love surface waves in biosensors used in biology, medicine, chemistry, etc. Section 5 contains discussion of the chronological development of SH ultrasonic sensors starting from bulk wave sensors and then first surface

unique phenomenon within the frame of the classical theory of elasticity.

attenuation, etc.

about 100 publications per year on that subject [7].

Anyway, not waiting for the revolution heralded by quantum mechanics (1900) or general theory of relativity (1917), classical physics was already shaken by the emergence of the theory of chaos (Poincaré 1882 and Hadamard 1898), which later on in the twentieth century will effectively eliminate deterministic description from many physical problems, such as weather forecasting, etc. Another new significant achievement of the classical physics (although not revolutionary) was the discovery of surface waves. At first, elastic surface waves were discovered in solids (Rayleigh 1885 and Love 1911) and then in electromagnetism (Zenneck 1907 and Sommerfeld 1909).

In fact, the existence of surface waves in solids was predicted mathematically by the celebrated British scientist Lord Rayleigh in 1885, who showed that elastic surface waves can propagate along a free surface of a semi-infinite body. By contrast to bulk waves, the amplitude of surface waves is confined to a narrow area adjacent to the guiding surface. Since surface waves are a type of guided waves, they can propagate often longer distances than their bulk counterparts and in addition, they are inherently sensitive to material properties in the vicinity of the guiding surface. It will be shown in the following of this chapter that these two properties of surface waves are of crucial importance in geophysics and sensor technology.

First, seismographs were constructed by British engineers in 1880, working in Japan for Meiji government. Consequently, the first long distance seismogram was registered in 1889 by German astronomer Ernst von Rebeur-Paschwitz in Potsdam (Germany), who was able to detect seismic signals generated by an earthquake occurred in Japan, some 9000 km away from Potsdam (Berlin). It was obvious soon that long distance seismograms display two different phases. First (preliminary tremor), a relatively weak signal arriving with the velocity of bulk waves (P and S) and second (main shock) with a much higher amplitude arriving with the velocity close to that of Rayleigh surface waves. However, this Rayleigh wave hypothesis was not satisfactory, since large part of the main shock energy was associated with the shear horizontal (SH) component of vibrations, absent by definition in Rayleigh surface waves composed of shear vertical (SV) and longitudinal (L) displacements. This dilemma was resolved in 1911 by the British physicist and mathematician Augustus Edward Hough Love by a brilliant stroke of thought [1]. Firstly, Love postulated that the SH component in the main shock is due to the arrival of a new type of surface waves (named later after his name) with only one SH component of vibrations. Secondly, Love assumed that SH surface waves are guided by an extra surface layer existing on the Earth's surface, with properties different than those in the Earth's interior. Using contemporary language, we can say that he made a direct hit.

It is noteworthy that the existence of Rayleigh and Love surface waves was first predicted mathematically prior to their experimental confirmation. This shows how beneficial can be the mutual interaction between the theory and experiment. Indeed, the theory indicates directions of future experimental research and the experiment confirms or renders the theory obsolete. It is worth noticing that the existence of a new type of electromagnetic surface waves was predicted mathematically quite recently, i.e., in 1988, and soon confirmed experimentally.

still is a very risky business, especially in physical sciences and engineering. Indeed, in these disciplines of human endeavors, one must take into account not only an inherently volatile human factor but also the impact of potential discoveries of unknown yet laws of nature, which often open new unanticipated possibilities and horizons. We may try to justify such an obvious complacency, attributed to the abovementioned scientists, by the historical spirit of the Belle Époque (1870–1914), that believed in harmony, good taste, optimism, unlimited

Anyway, not waiting for the revolution heralded by quantum mechanics (1900) or general theory of relativity (1917), classical physics was already shaken by the emergence of the theory of chaos (Poincaré 1882 and Hadamard 1898), which later on in the twentieth century will effectively eliminate deterministic description from many physical problems, such as weather forecasting, etc. Another new significant achievement of the classical physics (although not revolutionary) was the discovery of surface waves. At first, elastic surface waves were discovered in solids (Rayleigh 1885 and Love 1911) and then in electromagnetism (Zenneck 1907 and

In fact, the existence of surface waves in solids was predicted mathematically by the celebrated British scientist Lord Rayleigh in 1885, who showed that elastic surface waves can propagate along a free surface of a semi-infinite body. By contrast to bulk waves, the amplitude of surface waves is confined to a narrow area adjacent to the guiding surface. Since surface waves are a type of guided waves, they can propagate often longer distances than their bulk counterparts and in addition, they are inherently sensitive to material properties in the vicinity of the guiding surface. It will be shown in the following of this chapter that these two properties of surface waves are of crucial importance in geophysics and sensor

First, seismographs were constructed by British engineers in 1880, working in Japan for Meiji government. Consequently, the first long distance seismogram was registered in 1889 by German astronomer Ernst von Rebeur-Paschwitz in Potsdam (Germany), who was able to detect seismic signals generated by an earthquake occurred in Japan, some 9000 km away from Potsdam (Berlin). It was obvious soon that long distance seismograms display two different phases. First (preliminary tremor), a relatively weak signal arriving with the velocity of bulk waves (P and S) and second (main shock) with a much higher amplitude arriving with the velocity close to that of Rayleigh surface waves. However, this Rayleigh wave hypothesis was not satisfactory, since large part of the main shock energy was associated with the shear horizontal (SH) component of vibrations, absent by definition in Rayleigh surface waves composed of shear vertical (SV) and longitudinal (L) displacements. This dilemma was resolved in 1911 by the British physicist and mathematician Augustus Edward Hough Love by a brilliant stroke of thought [1]. Firstly, Love postulated that the SH component in the main shock is due to the arrival of a new type of surface waves (named later after his name) with only one SH component of vibrations. Secondly, Love assumed that SH surface waves are guided by an extra surface layer existing on the Earth's surface, with properties different than those in the Earth's interior. Using contemporary language, we can

progress and generally in positivistic philosophical ideas.

18 Surface Waves - New Trends and Developments

Sommerfeld 1909).

technology.

say that he made a direct hit.

It is interesting to note that Love surface waves have direct counterparts in electromagnetism (optical planar waveguides) and quantum mechanics (particle motion in a quantum well). By contrast, a similar statement is not true for Rayleigh surface waves, which therefore remain a unique phenomenon within the frame of the classical theory of elasticity.

Surface waves of the Love type have a number of unique features. Firstly, they have only one SH component of vibrations. As a result, Love surface waves are insensitive to the loading with liquids of zero or negligible viscosities. Thus, Love surface waves can propagate long distances without a significant attenuation. Indeed, Love waves propagating many times around the Earth's circumference have been observed experimentally. On the other hand, it was discovered much later (1981) that Love waves are very well suited for measurements of viscoelastic properties of liquids. Secondly, the mathematical description of Love surface waves is much simpler than that for Rayleigh surface waves. A relative simplicity of the mathematical model enables for direct physical insight in the process of Love wave propagation, attenuation, etc.

The idea to employ Love surface waves for measurements of viscoelastic properties of liquids was presented for the first time in 1981 by Kiełczyński and Płowiec in their Polish patent [2]. In 1987, the theory of the new method was presented by Kiełczyński and Pajewski on the international arena at the European Mechanics Colloquium 226 in Nottingham, UK [3]. In 1988, they presented this new method with equations and experimental results at the IEEE 1988 Ultrasonic Symposium in Chicago [4]. In 1989, Kiełczyński and Płowiec published a detailed theory and experimental results in the prestigious Journal of the Acoustical Society of America [5]. It is noteworthy that subsequent publications on Love wave sensors for liquid characterization appeared in USA not earlier than in 1992 [6], but nowadays, we witness about 100 publications per year on that subject [7].

We hope that the reader, after studying this chapter, will agree that the nature has many different faces and that the same physical phenomenon can be sometimes deadly (earthquakes) and in different circumstances, can be beneficiary (biosensor technology). As a consequence, SH surface waves of the Love type are an interesting example of an interdisciplinary research.

This chapter is organized as follows. Section 2 presents main characteristics and properties of Love surface waves, including basic mathematical model and examples of dispersion curves and amplitude distributions. More advanced mathematical treatment of the Love surface waves can be found, for example, in [8]. Section 3 shows the importance of Love surface waves in geophysics and seismology. Section 4 describes applications of Love surface waves in biosensors used in biology, medicine, chemistry, etc. Section 5 contains discussion of the chronological development of SH ultrasonic sensors starting from bulk wave sensors and then first surface wave sensors. We show also that the results of research conducted in Seismology and geophysics can be transferred to biosensor technology and vice versa. Conclusions and propositions for future research in biosensor technology employing Love surface waves are given in Section 6.

surface waves cannot exist. The extra surface layer must also be "slower" than the substrate,

*v*<sup>1</sup> < *v*<sup>2</sup> (1)

respectively. In fact, the condition expressed by Eq. (1) allows for entrapment of partial waves in the surface layer due to the total reflection phenomenon occurring at the layer-substrate

where the function *f*(*<sup>x</sup>*2) describes the amplitude of the Love wave as a function of the depth

Substitution of Eq. (2) into Newton's equation of motion leads to the Helmholtz differential equation for the transverse amplitude *f*(*x*2). Solutions of the resulting Helmholtz differential

> <sup>2</sup> = *k*<sup>1</sup> <sup>2</sup> − *k*<sup>2</sup>

> > *k*1 <sup>2</sup> = *<sup>ω</sup>*<sup>2</sup> \_\_\_ *v*1 2

> > *k*2 <sup>2</sup> = *<sup>ω</sup>*<sup>2</sup> \_\_\_ *v*2 2

and A is an arbitrary constant. In isotropic solids, Love surface waves have two stress compo-

−*q*<sup>2</sup> <sup>2</sup> = *k*<sup>2</sup>

\_\_\_

is the wavenumber of the Love wave, *<sup>ω</sup>* <sup>=</sup> <sup>2</sup>*f* is its angular frequency, *vp*

, *for* 0 ≤ *x*<sup>2</sup> < *h* (*surface layer*) *<sup>A</sup> <sup>e</sup>* <sup>−</sup>*q*2(*x*<sup>2</sup>

interface (*x*<sup>2</sup> <sup>=</sup> *<sup>h</sup>*). By contrast, if the condition given by Eq. (1) is not satisfied (*v*<sup>1</sup> <sup>&</sup>gt; *<sup>v</sup>*<sup>2</sup>

are phase velocities of bulk shear waves in the surface layer and substrate,

of a time-harmonic Love surface wave propagating in the direc-

Properties and Applications of Love Surface Waves in Seismology and Biosensors

, *t*) = *f*(*x*2) *exp*[*j*(*k x*<sup>1</sup> − *t*)] (2)

−1. Since the surface waveguide is assumed to be

, *for <sup>h</sup>* <sup>≤</sup> *<sup>x</sup>*<sup>2</sup> (*substrate*) (3)

<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> (4)

. From Eq. (3), it follows that stress *τ*<sup>23</sup>

), then

21

is the

(5)

and, on average, no net power is

http://dx.doi.org/10.5772/intechopen.75479

i.e., the following condition must hold [15]:

transmitted along the surface waveguide.

**2.1. Dispersion equation of the Love surface wave**

Love waves are evanescent in the direction of propagation *x*<sup>1</sup>

, *x*2

where *v*<sup>1</sup>

tion *x*<sup>1</sup>

(*x*2

where

axis), *<sup>k</sup>* <sup>=</sup> *<sup>ω</sup>*/*vp*

and *v*<sup>2</sup>

Mechanical displacement *u*<sup>3</sup>

has the following form:

*u*3(*x*<sup>1</sup>

phase velocity of the Love wave and *j* = √

equation have the following form [16]:

*<sup>q</sup>*<sup>1</sup>

can be expressed by the following formula:

*f*(*x*2) =

lossless, the wavenumber *k* in Eq. (2) is a real quantity.

⎧ ⎪ ⎨ ⎪ ⎩

nents, *τ*<sup>23</sup> *and <sup>τ</sup>*13, associated with the SH displacement *<sup>u</sup>*<sup>3</sup>

*A*

*cos*(*q*<sup>1</sup> *<sup>x</sup>*2) \_\_\_\_\_\_\_\_ *cos*(*q*<sup>1</sup> *h*)

−*h*)

In addition to biosensors, Love surface waves are used in chemosensors, in non-destructive testing (NDT) of materials, and in sensors of various physical quantities such as:


Recently, Love surface waves were also employed in the construction of the magnetic field sensor system with outstanding characteristics (sensitivity, dynamic range, etc.) [14].

#### **2. Properties of Love surface waves**

Shear horizontal (SH) surface waves of the Love type are elastic waves propagating in a surface waveguide, which is composed of a surface layer rigidly bonded to an elastic substrate, see **Figure 1**. The existence of an elastic surface layer is a necessary condition for propagation of Love surface waves, since it can be easily shown that on an elastic half-space alone, SH

**Figure 1.** Basic structure of a free Love wave waveguide, not loaded with a viscoelastic liquid. An elastic surface layer of thickness "h" and a shear velocity *v*<sup>1</sup> is rigidly bonded to the underlying semi-infinitive substrate with a shear velocity *v*<sup>2</sup> .

surface waves cannot exist. The extra surface layer must also be "slower" than the substrate, i.e., the following condition must hold [15]:

$$v\_1 \le v\_2 \tag{1}$$

where *v*<sup>1</sup> and *v*<sup>2</sup> are phase velocities of bulk shear waves in the surface layer and substrate, respectively. In fact, the condition expressed by Eq. (1) allows for entrapment of partial waves in the surface layer due to the total reflection phenomenon occurring at the layer-substrate interface (*x*<sup>2</sup> <sup>=</sup> *<sup>h</sup>*). By contrast, if the condition given by Eq. (1) is not satisfied (*v*<sup>1</sup> <sup>&</sup>gt; *<sup>v</sup>*<sup>2</sup> ), then Love waves are evanescent in the direction of propagation *x*<sup>1</sup> and, on average, no net power is transmitted along the surface waveguide.

#### **2.1. Dispersion equation of the Love surface wave**

Mechanical displacement *u*<sup>3</sup> of a time-harmonic Love surface wave propagating in the direction *x*<sup>1</sup> has the following form:

$$u\_{\sf s}(\mathbf{x}\_1, \mathbf{x}\_2, t) = f(\mathbf{x}\_2) \exp\left[j(\mathbf{k} \,\mathbf{x}\_1 - \alpha t)\right] \tag{2}$$

where the function *f*(*<sup>x</sup>*2) describes the amplitude of the Love wave as a function of the depth (*x*2 axis), *<sup>k</sup>* <sup>=</sup> *<sup>ω</sup>*/*vp* is the wavenumber of the Love wave, *<sup>ω</sup>* <sup>=</sup> <sup>2</sup>*f* is its angular frequency, *vp* is the phase velocity of the Love wave and *j* = √ \_\_\_ −1. Since the surface waveguide is assumed to be lossless, the wavenumber *k* in Eq. (2) is a real quantity.

Substitution of Eq. (2) into Newton's equation of motion leads to the Helmholtz differential equation for the transverse amplitude *f*(*x*2). Solutions of the resulting Helmholtz differential equation have the following form [16]:

$$f\_{\{\mathbf{x}\_2\}} = \begin{cases} \text{equation have the following form [16]:}\\ f(\mathbf{x}\_2) = \begin{cases} A \frac{\cos\left(q\_i \mathbf{x}\_2\right)}{\cos\left(q\_i \mathbf{h}\right)} & \text{for } 0 \le \mathbf{x}\_2 < h \text{ (surface layer)}\\ A \, e^{-q\_i \left(\mathbf{x}\_i - \mathbf{h}\right)} & \text{for } h \le \mathbf{x}\_2 \text{ (substrate)} \end{cases} \tag{3}$$

where

wave sensors. We show also that the results of research conducted in Seismology and geophysics can be transferred to biosensor technology and vice versa. Conclusions and propositions for future research in biosensor technology employing Love surface waves are given in Section 6. In addition to biosensors, Love surface waves are used in chemosensors, in non-destructive

**2.** spatial distribution of elastic parameters in solid functionally graded materials (FGM) [10];

Recently, Love surface waves were also employed in the construction of the magnetic field

Shear horizontal (SH) surface waves of the Love type are elastic waves propagating in a surface waveguide, which is composed of a surface layer rigidly bonded to an elastic substrate, see **Figure 1**. The existence of an elastic surface layer is a necessary condition for propagation of Love surface waves, since it can be easily shown that on an elastic half-space alone, SH

**Figure 1.** Basic structure of a free Love wave waveguide, not loaded with a viscoelastic liquid. An elastic surface layer

is rigidly bonded to the underlying semi-infinitive substrate with a shear

sensor system with outstanding characteristics (sensitivity, dynamic range, etc.) [14].

testing (NDT) of materials, and in sensors of various physical quantities such as:

**1.** humidity of air [9];

**3.** elastic parameters of nanolayers [11];

**2. Properties of Love surface waves**

of thickness "h" and a shear velocity *v*<sup>1</sup>

velocity *v*<sup>2</sup> .

**4.** porosity of the medium [12]; and **5.** dielectric constant of liquids [13].

20 Surface Waves - New Trends and Developments

$$\begin{aligned} q\_1^2 &= \, ^k\_1 - k^2\\ -q\_2^2 &= \, ^k\_2 - k^2 \end{aligned} \tag{4}$$

$$\begin{aligned} k\_1^2 &= \frac{\omega^2}{\upsilon\_1^2} \\ k\_2^2 &= \frac{\omega^2}{\upsilon\_2^2} \end{aligned} \tag{5}$$

and A is an arbitrary constant. In isotropic solids, Love surface waves have two stress components, *τ*<sup>23</sup> *and <sup>τ</sup>*13, associated with the SH displacement *<sup>u</sup>*<sup>3</sup> . From Eq. (3), it follows that stress *τ*<sup>23</sup> can be expressed by the following formula:

$$\pi\_{z3}(\mathbf{x\_2}) = \mu\_{1,2} \frac{\partial f(\mathbf{x\_2})}{\partial \mathbf{x\_2}} = \begin{cases} -\mu\_1 q\_1 A \frac{\sin(q\_1 \mathbf{x\_2})}{\cos(q\_1 h)}, & \text{for } 0 \le \mathbf{x\_2} < h \text{ (surface layer)}\\ -\mu\_2 q\_2 A e^{-q\_2 (\mathbf{x\_2} - \mathbf{b})}, & \text{for } h \le \mathbf{x\_2} \text{ (substrate)} \end{cases} \tag{6}$$

where *μ*<sup>1</sup> *and <sup>μ</sup>*<sup>2</sup> are shear moduli of elasticity in the surface layer and substrate, respectively.

The mechanical displacement *u*<sup>3</sup> and the associated stress *τ*23 must satisfy the appropriate boundary conditions, i.e., the continuity of *u*<sup>3</sup> and *τ*23 at interfaces *<sup>x</sup>*<sup>2</sup> <sup>=</sup> 0 (free guiding surface) and *x*<sup>2</sup> <sup>=</sup> *<sup>h</sup>* (the interface between the surface layer and the substrate). Substituting Eqs. (3) and (6) into the boundary conditions at *x*<sup>2</sup> <sup>=</sup> 0 and *x*<sup>2</sup> <sup>=</sup> *<sup>h</sup>*, one obtains the following dispersion relation [16], for Love surface waves propagating in a planar waveguide shown in **Figure 1**:

$$F[\omega, k(\omega)] = \mu\_1 \, q\_1 \tan(q\_1 h) - \mu\_2 q\_2 = 0 \tag{7}$$

**2.3. Phase and group velocity of the Love surface wave**

<sup>∂</sup>*<sup>ω</sup> <sup>F</sup>*[*ω*, *<sup>k</sup>*(*ω*)] <sup>+</sup> \_\_\_<sup>∂</sup>

envelope of the Love surface wave propagates, is defined as \_\_\_ *<sup>d</sup>*

*v*2 <sup>2</sup> =

respect to the angular frequency *ω* equals:

*vg* <sup>=</sup> \_\_\_ *<sup>d</sup>*

\_\_\_<sup>∂</sup>

*vg <sup>v</sup>* \_\_\_\_*<sup>p</sup>*

Eqs. (7) and (11) show that phase *vp*

Since group velocity *vg*

The total derivative of the implicit function *F*[*ω*, *k*(*ω*)] in the dispersion relation [Eq. (7)] with

**Figure 2.** Amplitude of the fundamental (n = 0) Love wave mode, as a function of the normalized depth *x*<sup>2</sup> /*h*, in a copper-

steel waveguide, for different wave frequencies f = 3, 5, and 7 MHz, and surface layer thickness h = 100 μm.

*dk* <sup>=</sup> <sup>−</sup>

velocities of the Love surface wave are connected via the following algebraic equation:

*sin*(2 *<sup>q</sup>*<sup>1</sup> *<sup>h</sup>*) \_\_\_\_\_\_\_\_

The phase velocity resulting from the solution of Eq. (8) and the group velocity determined by Eq. (11) of the fundamental mode of Love surface waves, as a function of the normalized frequency *fh*, are given in **Figure 3**. From **Figure 3**, it is evident that for low frequencies, the

<sup>2</sup> *<sup>q</sup>*<sup>1</sup> *<sup>h</sup>* <sup>+</sup> <sup>1</sup>] <sup>+</sup> *<sup>μ</sup>*<sup>2</sup> cos2 (*q*<sup>1</sup> *<sup>h</sup>*) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

<sup>2</sup> *<sup>q</sup>*<sup>1</sup> *<sup>h</sup>* <sup>+</sup> <sup>1</sup>] <sup>+</sup> *<sup>μ</sup>*<sup>2</sup> cos2

(*q*<sup>1</sup> *h*)

As a consequence, using Eqs. (7) and (10), one can show [8, 15–19] that group *vg*

*μ*<sup>1</sup> *q*<sup>2</sup> *h*[

and group *vg*

(*<sup>f</sup>* <sup>→</sup> 0) and high (*<sup>f</sup>* <sup>→</sup> <sup>∞</sup>) frequency limits are the same and equal, respectively, *v*<sup>2</sup>

*μ*<sup>1</sup> *q*<sup>2</sup> *h* ( *v* \_\_2 *v*1) 2 [ *sin*(2 *<sup>q</sup>*<sup>1</sup> *<sup>h</sup>*) \_\_\_\_\_\_\_\_

\_\_\_∂ <sup>∂</sup>*<sup>k</sup> <sup>F</sup>*[*ω*, *<sup>k</sup>*(*ω*)] \_\_\_\_\_\_\_\_\_\_

\_\_\_∂

<sup>∂</sup>*<sup>k</sup> <sup>F</sup>*[*ω*, *<sup>k</sup>*(*ω*)] *dk*(*ω*) \_\_\_\_\_

of the Love surface wave, which describes the speed at which pulse

Properties and Applications of Love Surface Waves in Seismology and Biosensors

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23

*<sup>d</sup>* <sup>=</sup> <sup>0</sup> (9)

*dk* from Eq. 9, it is clear that:

and phase *vp*

 and *v*<sup>1</sup> . (11)

<sup>∂</sup>*<sup>ω</sup> <sup>F</sup>*[*ω*, *<sup>k</sup>*(*ω*)] (10)

velocities of the Love surface wave in the low

Using Eq. (4), one can rewrite Eq. (7) in a more explicit form as:

$$F[\omega, k(\omega)] = \mu\_1 \left( \sqrt{\frac{1}{\nu\_1^2} - \frac{1}{\nu\_p^2}} \right) \tan \left[ 2\pi \left( \sqrt{\frac{1}{\nu\_1^2} - \frac{1}{\nu\_p^2}} \right) \left( \text{f} \mathfrak{h} \right) \right] - \mu\_2 \left( \sqrt{\frac{1}{\nu\_p^2} - \frac{1}{\nu\_2^2}} \right) \tag{8}$$

Eq. (8) shows that the unknown phase velocity *vp* of the Love surface wave is de facto an explicit function of the normalized product frequency-thickness (*fh*), with *v*<sup>1</sup> and *<sup>v</sup>*<sup>2</sup> , being parameters. This property does not, however, hold for lossy Love wave waveguides where the elastic moduli *μ*<sup>1</sup> , *μ*2 as well as the velocities *v*<sup>1</sup> and *<sup>v</sup>*<sup>2</sup> are implicit functions of the frequency *f* and are obviously independent of the surface layer thickness *h*.

The dispersion relation Eq. (8) is a transcendental algebraic equation for the unknown phase velocity *vp* and therefore can be solved only numerically using, for example, the Newton-Raphson iterative method [17].

#### **2.2. Modal structure of the Love surface wave**

The dispersion relation [Eq. (8)] reveals that phase velocity *vp* of the Love surface wave is a function of frequency. Hence, Love surface waves are dispersive. Moreover, since the function tangent in Eq. (8) is periodic, i.e., *tan*(*q*<sup>1</sup> *<sup>h</sup>*) <sup>=</sup> *tan*(*q*<sup>1</sup> *<sup>h</sup>* <sup>+</sup> *<sup>n</sup>*), where *<sup>n</sup>* <sup>=</sup> 0, 1, 2, …,etc., Love surface waves display a multimode structure.

The amplitude *f*(*<sup>x</sup>*2) of the fundamental (*<sup>n</sup>* <sup>=</sup> 0) mode of the Love surface wave, as a function of the distance *x*<sup>2</sup> from the guiding surface *x*<sup>2</sup> <sup>=</sup> 0, is shown in **Figure 2**. It is clear that for sufficiently high frequencies, the energy of the Love wave is concentrated mostly in the surface layer in the vicinity of the guiding surface *x*<sup>2</sup> <sup>=</sup> <sup>0</sup>. By differentiation of Eq. (3), it is easy to show that the maximum of the amplitude *f*(*<sup>x</sup>*2) occurs exactly at the free surface *<sup>x</sup>*<sup>2</sup> <sup>=</sup> 0. By contrast, the associated stress *τ*23 vanishes at *<sup>x</sup>*<sup>2</sup> <sup>=</sup> 0, i.e., at the free surface of the waveguide.

Properties and Applications of Love Surface Waves in Seismology and Biosensors http://dx.doi.org/10.5772/intechopen.75479 23

**Figure 2.** Amplitude of the fundamental (n = 0) Love wave mode, as a function of the normalized depth *x*<sup>2</sup> /*h*, in a coppersteel waveguide, for different wave frequencies f = 3, 5, and 7 MHz, and surface layer thickness h = 100 μm.

#### **2.3. Phase and group velocity of the Love surface wave**

*τ*23(*x*2) = *μ*1,2

The mechanical displacement *u*<sup>3</sup>

22 Surface Waves - New Trends and Developments

*<sup>F</sup>*[*ω*, *<sup>k</sup>*(*ω*)] <sup>=</sup> *<sup>μ</sup>*1(<sup>√</sup>

Raphson iterative method [17].

the elastic moduli *μ*<sup>1</sup>

velocity *vp*

of the distance *x*<sup>2</sup>

boundary conditions, i.e., the continuity of *u*<sup>3</sup>

where *μ*<sup>1</sup> *and <sup>μ</sup>*<sup>2</sup>

**Figure 1**:

<sup>∂</sup>*f*(*x*2) \_\_\_\_\_ ∂ *x*<sup>2</sup> = ⎧ ⎪ ⎨ ⎪ ⎩

Using Eq. (4), one can rewrite Eq. (7) in a more explicit form as:

and are obviously independent of the surface layer thickness *h*.

The dispersion relation [Eq. (8)] reveals that phase velocity *vp*

Eq. (8) shows that the unknown phase velocity *vp*

, *μ*2

**2.2. Modal structure of the Love surface wave**

waves display a multimode structure.

\_\_\_\_\_ \_\_1 *v*1 <sup>2</sup> <sup>−</sup> \_\_1 *vp* <sup>2</sup> ) *tan*[

−*μ*<sup>1</sup> *q*<sup>1</sup> *A*

−*μ*<sup>2</sup> *q*<sup>2</sup> *A e* <sup>−</sup>*q*2(*x*<sup>2</sup>

*sin*(*q*<sup>1</sup> *<sup>x</sup>*2) \_\_\_\_\_\_\_ *cos*(*q*<sup>1</sup> *h*)

−*h*)

and *x*<sup>2</sup> <sup>=</sup> *<sup>h</sup>* (the interface between the surface layer and the substrate). Substituting Eqs. (3) and (6) into the boundary conditions at *x*<sup>2</sup> <sup>=</sup> 0 and *x*<sup>2</sup> <sup>=</sup> *<sup>h</sup>*, one obtains the following dispersion relation [16], for Love surface waves propagating in a planar waveguide shown in

*F*[*ω*, *k*(*ω*)] = *μ*<sup>1</sup> *q*<sup>1</sup> *tan*(*q*<sup>1</sup> *h*) − *μ*<sup>2</sup> *q*<sup>2</sup> = 0 (7)

<sup>2</sup>*π*(<sup>√</sup>

parameters. This property does not, however, hold for lossy Love wave waveguides where

The dispersion relation Eq. (8) is a transcendental algebraic equation for the unknown phase

function of frequency. Hence, Love surface waves are dispersive. Moreover, since the function tangent in Eq. (8) is periodic, i.e., *tan*(*q*<sup>1</sup> *<sup>h</sup>*) <sup>=</sup> *tan*(*q*<sup>1</sup> *<sup>h</sup>* <sup>+</sup> *<sup>n</sup>*), where *<sup>n</sup>* <sup>=</sup> 0, 1, 2, …,etc., Love surface

The amplitude *f*(*<sup>x</sup>*2) of the fundamental (*<sup>n</sup>* <sup>=</sup> 0) mode of the Love surface wave, as a function

ficiently high frequencies, the energy of the Love wave is concentrated mostly in the surface layer in the vicinity of the guiding surface *x*<sup>2</sup> <sup>=</sup> <sup>0</sup>. By differentiation of Eq. (3), it is easy to show that the maximum of the amplitude *f*(*<sup>x</sup>*2) occurs exactly at the free surface *<sup>x</sup>*<sup>2</sup> <sup>=</sup> 0. By contrast, the

associated stress *τ*23 vanishes at *<sup>x</sup>*<sup>2</sup> <sup>=</sup> 0, i.e., at the free surface of the waveguide.

from the guiding surface *x*<sup>2</sup> <sup>=</sup> 0, is shown in **Figure 2**. It is clear that for suf-

and therefore can be solved only numerically using, for example, the Newton-

explicit function of the normalized product frequency-thickness (*fh*), with *v*<sup>1</sup> and *<sup>v</sup>*<sup>2</sup>

as well as the velocities *v*<sup>1</sup> and *<sup>v</sup>*<sup>2</sup>

\_\_\_\_\_ \_\_1 *v*1 <sup>2</sup> <sup>−</sup> \_\_1 *vp* 2 ) (*fh*)

are shear moduli of elasticity in the surface layer and substrate, respectively.

, *for* 0 ≤ *x*<sup>2</sup> < *h* (*surface layer*)

, *for h* ≤ *x*<sup>2</sup> (*substrate*)

and the associated stress *τ*23 must satisfy the appropriate

] <sup>−</sup> *<sup>μ</sup>*2(<sup>√</sup>

\_\_\_\_\_ \_\_1 *vp* <sup>2</sup> <sup>−</sup> \_\_1 *v*2 2

of the Love surface wave is de facto an

are implicit functions of the frequency *f*

of the Love surface wave is a

) <sup>=</sup> <sup>0</sup> (8)

, being

and *τ*23 at interfaces *<sup>x</sup>*<sup>2</sup> <sup>=</sup> 0 (free guiding surface)

(6)

The total derivative of the implicit function *F*[*ω*, *k*(*ω*)] in the dispersion relation [Eq. (7)] with respect to the angular frequency *ω* equals:

$$\frac{\partial}{\partial \omega} F[\omega, k(\omega)] + \frac{\partial}{\partial k} F[\omega, k(\omega)] \frac{dk(\omega)}{d\omega} = 0 \tag{9}$$

Since group velocity *vg* of the Love surface wave, which describes the speed at which pulse envelope of the Love surface wave propagates, is defined as \_\_\_ *<sup>d</sup> dk* from Eq. 9, it is clear that:

$$\boldsymbol{\upsilon}\_{s} = \frac{d\boldsymbol{\alpha}}{d\boldsymbol{k}} = -\frac{\frac{\partial}{\partial\boldsymbol{k}}\,\mathrm{F}[\boldsymbol{\omega},\boldsymbol{k}(\boldsymbol{\omega})]}{\frac{\partial}{\partial\boldsymbol{\omega}}\,\mathrm{F}[\boldsymbol{\omega},\boldsymbol{k}(\boldsymbol{\omega})]}\tag{10}$$

As a consequence, using Eqs. (7) and (10), one can show [8, 15–19] that group *vg* and phase *vp* velocities of the Love surface wave are connected via the following algebraic equation:

$$\begin{aligned} \text{Illumination on the above parameters, have the same assumption } \text{arg}\max\_{\mathbf{q}} \text{arg}\max\_{\mathbf{q}} \text{arg}\max\_{\mathbf{q}} \\\\ \frac{\boldsymbol{v}\_{g} \cdot \boldsymbol{v}\_{g}}{\boldsymbol{v}\_{2}^{2}} = \frac{\mu\_{1} q\_{2} h \left[ \frac{\sin(2 \, q\_{1} h)}{2 \, q\_{1} h} + 1 \right] + \mu\_{2} \cos^{2}(q\_{1} h)}{\mu\_{1} q\_{2} h \left( \frac{\boldsymbol{v}\_{2}}{\boldsymbol{v}\_{1}} \right)^{2} \left[ \frac{\sin(2 \, q\_{1} h)}{2 \, q\_{1} h} + 1 \right] + \mu\_{2} \cos^{2}(q\_{1} h)} \end{aligned} \tag{11}$$

Eqs. (7) and (11) show that phase *vp* and group *vg* velocities of the Love surface wave in the low (*<sup>f</sup>* <sup>→</sup> 0) and high (*<sup>f</sup>* <sup>→</sup> <sup>∞</sup>) frequency limits are the same and equal, respectively, *v*<sup>2</sup> and *v*<sup>1</sup> .

The phase velocity resulting from the solution of Eq. (8) and the group velocity determined by Eq. (11) of the fundamental mode of Love surface waves, as a function of the normalized frequency *fh*, are given in **Figure 3**. From **Figure 3**, it is evident that for low frequencies, the

Propagation of Love surface waves on the Earth's surface is made possible by layered structure of the Earth. The outermost layer of the Earth, the crust, is made of solid rocks composed of lighter elements. Thickness of the crust varies from 5 to 10 km under oceans (oceanic crust) to 30–70 km under continents (continental crust). The crust sits on mantle, which in turn covers the outer and inner core. The destructive power of earthquakes is mainly due to waves

Properties and Applications of Love Surface Waves in Seismology and Biosensors

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25

As predicted by Love, the velocity of SH bulk waves increases with depth [24], i.e., as a func-

The frequency of Love waves generated by earthquakes is rather low comparing to that used

Love and Rayleigh surface waves travel along great circle paths around the globe. Surface waves from strong earthquakes may travel several times around the Earth without a significant attenuation. They are termed global Rayleigh wave impulses [25]. An example of surface

Seismic waves, generated both by natural earthquakes and by man-made sources, have delivered an enormous amount of information about the Earth's interior (subsurface properties of Earth's crust). In classical seismology, Earth is modeled as a sequence of uniform horizontal layers (or spherical shells) having different elastic properties and one determines these prop-

Love surface waves have been successfully employed in a tomographic reconstruction of the physical properties of Earth's upper mantle [28] as well as in diamond, gold, and copper

**Figure 4.** Illustration of a seismogram of Rayleigh surface waves triggered by an earthquake. Note that, Rayleigh wave

traveling in this thin crustal layer [23].

tion of distance from the free surface of the Earth.

in sensor technology and ranges typically from 10 mHz to 10 Hz.

**3.1. Investigation of the Earth's interior with Love surface waves**

waves traveling multiply around the Earth [26] is given in **Figure 4**.

erties from travel times and dispersion of seismic waves [27].

exploration in Australia, South America, and South Africa [29].

packet traveled 8 times around the Earth's circumference.

**Figure 3.** Phase *vp* and group *vg* velocities of the fundamental mode of the Love surface wave propagating in a copperstainless steel waveguide, as a function of the normalized frequency-thickness product *fh* [MHz-mm], *v*<sup>1</sup> = 2223.5 *m*/*s*, and *v*<sup>2</sup> = 3017 *m*/*s*.

phase and group velocities of Love surface waves approach asymptotically that of bulk shear waves *v*<sup>2</sup> in the substrate. On the other hand, at high frequency limit, the phase and group velocities of the Love wave tend to the velocity *v*<sup>1</sup> , namely to the velocity of bulk shear waves in the surface layer.

#### **2.4. Influence of a viscoelastic liquid loading Love wave waveguides**

It is noteworthy that in waveguides loaded with a lossy, viscoelastic liquid, the wavenumber k of the Love surface wave is a complex quantity, i.e., *<sup>k</sup>* <sup>=</sup> *<sup>ω</sup>*/*vp* <sup>+</sup> *<sup>j</sup>*, where *α* is the coefficient of attenuation of the Love wave. Three most popular viscoelastic liquids are described by Kelvin-Voigt, Newton and Maxwell models, respectively [20]. The dispersion relation of Love surface waves propagating in waveguides loaded with a viscous liquid can be found in [21]. In lossy waveguides, the group velocity of Love waves cannot be rigorously defined [22]. As a result, the formula 11 is valid only approximately in lossy Love wave waveguides. As a matter of fact, in a waveguide loaded with a viscoelastic liquid, the amplitude of the Love wave is non-zero in a thin layer of the liquid adjacent to the surface layer of the waveguide. The penetration of the Love wave energy into the adjacent liquid is of crucial importance in understanding the operation of Love wave biosensors. Indeed, if Love wave energy was not penetrating in the measured liquid, the parameters of the Love wave might not be affected by the liquid and the operation of the whole sensor would be essentially impossible.

#### **3. Love surface waves in seismology**

Since Love surface waves were originally discovered in seismology, we give here a brief description of their applications in seismic and geophysical research.

Propagation of Love surface waves on the Earth's surface is made possible by layered structure of the Earth. The outermost layer of the Earth, the crust, is made of solid rocks composed of lighter elements. Thickness of the crust varies from 5 to 10 km under oceans (oceanic crust) to 30–70 km under continents (continental crust). The crust sits on mantle, which in turn covers the outer and inner core. The destructive power of earthquakes is mainly due to waves traveling in this thin crustal layer [23].

As predicted by Love, the velocity of SH bulk waves increases with depth [24], i.e., as a function of distance from the free surface of the Earth.

The frequency of Love waves generated by earthquakes is rather low comparing to that used in sensor technology and ranges typically from 10 mHz to 10 Hz.

#### **3.1. Investigation of the Earth's interior with Love surface waves**

phase and group velocities of Love surface waves approach asymptotically that of bulk shear

stainless steel waveguide, as a function of the normalized frequency-thickness product *fh* [MHz-mm], *v*<sup>1</sup> = 2223.5 *m*/*s*, and

It is noteworthy that in waveguides loaded with a lossy, viscoelastic liquid, the wavenumber k of the Love surface wave is a complex quantity, i.e., *<sup>k</sup>* <sup>=</sup> *<sup>ω</sup>*/*vp* <sup>+</sup> *<sup>j</sup>*, where *α* is the coefficient of attenuation of the Love wave. Three most popular viscoelastic liquids are described by Kelvin-Voigt, Newton and Maxwell models, respectively [20]. The dispersion relation of Love surface waves propagating in waveguides loaded with a viscous liquid can be found in [21]. In lossy waveguides, the group velocity of Love waves cannot be rigorously defined [22]. As a result, the formula 11 is valid only approximately in lossy Love wave waveguides. As a matter of fact, in a waveguide loaded with a viscoelastic liquid, the amplitude of the Love wave is non-zero in a thin layer of the liquid adjacent to the surface layer of the waveguide. The penetration of the Love wave energy into the adjacent liquid is of crucial importance in understanding the operation of Love wave biosensors. Indeed, if Love wave energy was not penetrating in the measured liquid, the parameters of the Love wave might not be affected by

the liquid and the operation of the whole sensor would be essentially impossible.

description of their applications in seismic and geophysical research.

Since Love surface waves were originally discovered in seismology, we give here a brief

in the substrate. On the other hand, at high frequency limit, the phase and group

velocities of the fundamental mode of the Love surface wave propagating in a copper-

, namely to the velocity of bulk shear waves

waves *v*<sup>2</sup>

**Figure 3.** Phase *vp*

*v*<sup>2</sup> = 3017 *m*/*s*.

in the surface layer.

velocities of the Love wave tend to the velocity *v*<sup>1</sup>

and group *vg*

24 Surface Waves - New Trends and Developments

**3. Love surface waves in seismology**

**2.4. Influence of a viscoelastic liquid loading Love wave waveguides**

Love and Rayleigh surface waves travel along great circle paths around the globe. Surface waves from strong earthquakes may travel several times around the Earth without a significant attenuation. They are termed global Rayleigh wave impulses [25]. An example of surface waves traveling multiply around the Earth [26] is given in **Figure 4**.

Seismic waves, generated both by natural earthquakes and by man-made sources, have delivered an enormous amount of information about the Earth's interior (subsurface properties of Earth's crust). In classical seismology, Earth is modeled as a sequence of uniform horizontal layers (or spherical shells) having different elastic properties and one determines these properties from travel times and dispersion of seismic waves [27].

Love surface waves have been successfully employed in a tomographic reconstruction of the physical properties of Earth's upper mantle [28] as well as in diamond, gold, and copper exploration in Australia, South America, and South Africa [29].

**Figure 4.** Illustration of a seismogram of Rayleigh surface waves triggered by an earthquake. Note that, Rayleigh wave packet traveled 8 times around the Earth's circumference.

Surface waves generated by earthquakes or man-made explosions were used in quantitative recovery of Earth's parameters as a function of depth. These seismic inverse problems helped to discover many fine details of the Earth's interior [30–32].

It is noteworthy that many theoretical methods were initially originated in seismology and geophysics before their transfer to the surface wave sensor technology (see **Table 1** in Section 5.5).

#### **3.2. Structural damages due to Love surface waves generated by earthquakes**

An example of structural damages made by surface waves of the Love type is shown in **Figure 5**. It is apparent that railway tracks were deformed by strong shear horizontal SH forces parallel to the Earth's surface. Love surface waves together with Rayleigh surface waves are the most devastating waves occurring during earthquakes.

#### **3.3. Application of metamaterials to minimize devastating effects of Love surface waves in the aftermath of earthquakes**

It is interesting to note that recently developed earthquake engineered metamaterials open a new way to counterattack seismic waves [33, 34]. The metamaterials actively control the seismic waves by providing an additional shield around the protected building rather than reconstructing the building structure. Compared with common engineering solutions, the advantage of the metamaterial method is that it can not only attenuate seismic waves before they reach critical targets, but also protect a distributed area rather than an individual building. The periodic arrangement of metamaterial structure creates frequency band gaps, which effectively prevent surface waves propagation on the Earth's surface via a Bragg scattering mechanism.

**Figure 5.** Twisted railroad tracks, an example of structural damages due to SH displacement of Love surface waves in

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27

A biosensor can be described as a device which can generate a signal (usually electrical) that is proportional to the concentration of a particular biomaterial or chemicals in the presence of a number of interfering species [35]. This can be accomplished using biological recognition elements such as enzymes, antibodies, receptors, tissues, and microorganisms as sensitive materials because of their selective functionality for target analytes along with an appropriate transducer.

High sensitivity of Love surface wave sensors can be explained by spatial concentration of the energy of Love waves. Indeed, it was shown in Section 2 that the energy of Love surface waves is localized mostly in the vicinity of the free guiding surface (**Figure 1**), looking in both sides from it. Moreover, the amplitude of Love surface waves reaches maximum at the free guiding surface *x*<sup>2</sup> <sup>=</sup> 0 (**Figure 2**). Therefore, we can expect that propagation of Love surface waves will be to a lesser or higher extent perturbed by a material (such as liquid) being in contact with the guiding surface. This feature of Love waves was exploited in the construction of various biosensors used for detection and quantification of many important parameters of biological

**4.1. Confinement of the energy of Love surface waves near the free surface of the** 

**4.2. Correlation between concentration of the measured analyte and parameters of** 

Surface waves of the Love type are especially suited to measure parameters of viscoelastic liquids, polymers, gels, etc., providing that they can form a good mechanical contact

**4. Biosensors employing Love surface waves**

**waveguide**

the aftermath of an earthquake.

and chemical substances [21, 36–40].

**the Love surface wave**


**Table 1.** Chronology of developments in Love wave biosensors and Love wave seismology.

Properties and Applications of Love Surface Waves in Seismology and Biosensors http://dx.doi.org/10.5772/intechopen.75479 27

**Figure 5.** Twisted railroad tracks, an example of structural damages due to SH displacement of Love surface waves in the aftermath of an earthquake.

periodic arrangement of metamaterial structure creates frequency band gaps, which effectively prevent surface waves propagation on the Earth's surface via a Bragg scattering mechanism.

#### **4. Biosensors employing Love surface waves**

Surface waves generated by earthquakes or man-made explosions were used in quantitative recovery of Earth's parameters as a function of depth. These seismic inverse problems helped

It is noteworthy that many theoretical methods were initially originated in seismology and geophysics before their transfer to the surface wave sensor technology (see **Table 1** in Section 5.5).

An example of structural damages made by surface waves of the Love type is shown in **Figure 5**. It is apparent that railway tracks were deformed by strong shear horizontal SH forces parallel to the Earth's surface. Love surface waves together with Rayleigh surface waves are the most

It is interesting to note that recently developed earthquake engineered metamaterials open a new way to counterattack seismic waves [33, 34]. The metamaterials actively control the seismic waves by providing an additional shield around the protected building rather than reconstructing the building structure. Compared with common engineering solutions, the advantage of the metamaterial method is that it can not only attenuate seismic waves before they reach critical targets, but also protect a distributed area rather than an individual building. The

**3.2. Structural damages due to Love surface waves generated by earthquakes**

**3.3. Application of metamaterials to minimize devastating effects of Love surface** 

**Developments Seismology Biosensors** Basic theory Love [1] Kiełczyński [3] Multilayered waveguides (transfer matrix method) Haskell [72] Kiełczyński [8] Viscoelastic waveguides (theoretical analysis) Sezawa [73] Kiełczyński [74] Inverse problems Dorman [76] Kiełczyński [77]

Nonlinear waves Kalyanasundarm [78] — Phased arrays Frosch [79] — Tomography Nakanishi [80] — Higher-order modes Haskell [81] — Solitary waves Bataille [82] — Energy harvesting Qu [83] —

Waveguides with nanomaterials — Penza [84] Piezoelectric waveguides — Kovacs [6] Resonators — Kovacs [67] Delay lines — Tournois [19]

**Table 1.** Chronology of developments in Love wave biosensors and Love wave seismology.

to discover many fine details of the Earth's interior [30–32].

devastating waves occurring during earthquakes.

**waves in the aftermath of earthquakes**

26 Surface Waves - New Trends and Developments

A biosensor can be described as a device which can generate a signal (usually electrical) that is proportional to the concentration of a particular biomaterial or chemicals in the presence of a number of interfering species [35]. This can be accomplished using biological recognition elements such as enzymes, antibodies, receptors, tissues, and microorganisms as sensitive materials because of their selective functionality for target analytes along with an appropriate transducer.

#### **4.1. Confinement of the energy of Love surface waves near the free surface of the waveguide**

High sensitivity of Love surface wave sensors can be explained by spatial concentration of the energy of Love waves. Indeed, it was shown in Section 2 that the energy of Love surface waves is localized mostly in the vicinity of the free guiding surface (**Figure 1**), looking in both sides from it. Moreover, the amplitude of Love surface waves reaches maximum at the free guiding surface *x*<sup>2</sup> <sup>=</sup> 0 (**Figure 2**). Therefore, we can expect that propagation of Love surface waves will be to a lesser or higher extent perturbed by a material (such as liquid) being in contact with the guiding surface. This feature of Love waves was exploited in the construction of various biosensors used for detection and quantification of many important parameters of biological and chemical substances [21, 36–40].

#### **4.2. Correlation between concentration of the measured analyte and parameters of the Love surface wave**

Surface waves of the Love type are especially suited to measure parameters of viscoelastic liquids, polymers, gels, etc., providing that they can form a good mechanical contact (absorption and adhesion) with free surface of the waveguide. Since Love surface waves are, in principle, mechanical waves, they can measure the following mechanical parameters of an adjacent medium: density, modulus of elasticity, and viscosity. In waveguides composed of piezoelectric elements (substrate and/or surface layer), dielectric constant of the adjacent medium will also affect the propagation of Love surface waves. In practice, we are interested in detection and quantification other more specific properties of biological and chemical materials, such as concentration and presence of proteins, antibodies, toxins, bacteria, viruses, size and shape of DNA, etc. Therefore, the next step in the development of Love wave sensors is to correlate (experimentally or analytically) the abovementioned specific properties of the measured analytes with changes in density, viscosity, and elastic moduli of the surface (sensing) layer. Finally, we have to measure changes in phase velocity and attenuation of Love surface waves, which are due to changes in density, viscosity, and elastic moduli of this surface layer. It should be noticed that part of Love wave energy enters into the measured liquid to some distance (penetration depth) from the guiding surface. Such an energy redistribution changes certainly the phase velocity and attenuation of the Love surface wave. In practice, we often adopt a more empirical approach, i.e., we measure directly changes in phase velocity and attenuation of Love waves, as a function of the aforementioned specific properties of the measured material, such as the concentration of proteins and so on, without referring to changes in density, viscosity or elastic modulus of the measured material. However, the former step is indispensable during modeling, design, and optimization of Love surface wave sensors.

#### **4.3. Parameters of the Love surface wave measured**

As with other types of wave motion, we can measure in principle two parameters of Love surface waves, i.e., their phase and amplitude. Polarization of SH surface waves of the Love type is constant and therefore does not provide any additional information about the medium of propagation. Phase Φ(*x*<sup>1</sup> , *t*) measurements in radians are directly related to the phase velocity *vp* of Love surface waves via the following equation:

$$\Phi(\mathbf{x}\_{\mathbf{r}'}t) = k\mathbf{x}\_{\mathbf{r}} - \alpha t = \omega \left(\frac{\mathbf{x}\_{\mathbf{r}}}{\overline{v}\_{\mathbf{r}}} - t\right) \tag{12}$$

**4.4. Sensors working in a resonator and delay line configurations**

sors is from 50 MHz to 500 MHz [7].

sensor. The guiding layer (SiO2

**4.5. Sensors controlled remotely by wireless devices**

ronmental monitoring, food quality control, and defense.

section of this sensor structure + loading liquid.

Phase and amplitude characteristics of Love surface waves can be measured in a closed loop configuration by placing Love wave delay line in a feedback circuit of an electrical oscillator (resonator). Another possibility is to use network analyzer, which provides phase shift and insertion loss of the Love wave sensor working in an open loop configuration, due to the load of the sensor with a measured material. The typical frequency range used by Love wave sen-

The structure and cross section of a typical Love wave biosensor is shown in **Figure 6a** and **b**. A relatively thick (0.5–1.0 mm) substrate provides mechanical support for the whole sensor. Often the substrate material is piezoelectric (AT-cut quartz material [41]). In this case, a pair of interdigital transducers (IDTs) can be deposited on the substrate to form a delay line of the

vides entrapment for surface wave energy. The sensing layer, made of gold (Au) or a polymer, usually very thin (~50-100 nm), serves as an immobilization area for the measured biological material. This thin-sensing layer interacts directly with the measured material (liquid) and serves often as a selector of the specific target substance, such as antigen, to be measured.

An interesting solution for Love wave sensors was proposed in [42], where the Love wave sensor works in a wireless configuration without an external power supply. This design has many unique advantages, i.e., the sensor can be permanently implanted in a patient body to monitor continuously the selected property of a biological liquid. Readings of the sensor can be made on demand, totally noninvasively by a reading device connected to a broader computer system of patient monitoring. Another implementation of a remotely controlled wireless Love wave sensor was presented in [43]. The proposed sensor can measure simultaneously two different analytes using Love surface waves with a frequency of 440 MHz.

Wireless bioelectronics sensors may be used in a variety of fields including: healthcare, envi-

**Figure 6.** a) Layered structure of a typical Love wave sensor not yet connected to the external driving circuit and b) cross-

, ZnO, PMMA, etc.), deposited directly on the substrate, pro-

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Similarly, amplitude A measurements are correlated with the coefficient of attenuation *α* (in Np/m) of Love surface waves as follows:

$$\alpha = \frac{1}{\left(\mathbf{x}\_2 - \mathbf{x}\_1\right)} \ln\left(\frac{A\_1}{A\_2}\right) \left[\mathrm{Np}/m\right] \tag{13}$$

where *A*<sup>1</sup> and *A*<sup>2</sup> are two amplitudes of the wave measured at points *x*<sup>1</sup> and *<sup>x</sup>*<sup>2</sup> , respectively (*x*<sup>2</sup> <sup>&</sup>gt; *x*1 ). In order to obtain the coefficient of attenuation in *dB*/*m*, the coefficient *α* given by Eq. (13) must be multiplied by 20 *log*(*e*) ≈ 8.686.

#### **4.4. Sensors working in a resonator and delay line configurations**

(absorption and adhesion) with free surface of the waveguide. Since Love surface waves are, in principle, mechanical waves, they can measure the following mechanical parameters of an adjacent medium: density, modulus of elasticity, and viscosity. In waveguides composed of piezoelectric elements (substrate and/or surface layer), dielectric constant of the adjacent medium will also affect the propagation of Love surface waves. In practice, we are interested in detection and quantification other more specific properties of biological and chemical materials, such as concentration and presence of proteins, antibodies, toxins, bacteria, viruses, size and shape of DNA, etc. Therefore, the next step in the development of Love wave sensors is to correlate (experimentally or analytically) the abovementioned specific properties of the measured analytes with changes in density, viscosity, and elastic moduli of the surface (sensing) layer. Finally, we have to measure changes in phase velocity and attenuation of Love surface waves, which are due to changes in density, viscosity, and elastic moduli of this surface layer. It should be noticed that part of Love wave energy enters into the measured liquid to some distance (penetration depth) from the guiding surface. Such an energy redistribution changes certainly the phase velocity and attenuation of the Love surface wave. In practice, we often adopt a more empirical approach, i.e., we measure directly changes in phase velocity and attenuation of Love waves, as a function of the aforementioned specific properties of the measured material, such as the concentration of proteins and so on, without referring to changes in density, viscosity or elastic modulus of the measured material. However, the former step is indispensable during modeling, design, and optimization of Love surface

As with other types of wave motion, we can measure in principle two parameters of Love surface waves, i.e., their phase and amplitude. Polarization of SH surface waves of the Love type is constant and therefore does not provide any additional information about the medium

, *t*) = *k x*<sup>1</sup> − *t* = *ω*(

Similarly, amplitude A measurements are correlated with the coefficient of attenuation *α* (in

\_\_\_ *A*1

(*x*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*1) *ln*(

are two amplitudes of the wave measured at points *x*<sup>1</sup> and *<sup>x</sup>*<sup>2</sup>

). In order to obtain the coefficient of attenuation in *dB*/*m*, the coefficient *α* given by Eq. (13)

, *t*) measurements in radians are directly related to the phase velocity

) (12)

, respectively (*x*<sup>2</sup> <sup>&</sup>gt;

*<sup>A</sup>*2) [*Np*/*m*] (13)

*x* \_\_1 *vp* − *t*

wave sensors.

*vp*

where *A*<sup>1</sup>

*x*1

and *A*<sup>2</sup>

of propagation. Phase Φ(*x*<sup>1</sup>

Φ(*x*<sup>1</sup>

28 Surface Waves - New Trends and Developments

Np/m) of Love surface waves as follows:

must be multiplied by 20 *log*(*e*) ≈ 8.686.

*<sup>α</sup>* <sup>=</sup> \_\_\_\_\_ <sup>1</sup>

**4.3. Parameters of the Love surface wave measured**

of Love surface waves via the following equation:

Phase and amplitude characteristics of Love surface waves can be measured in a closed loop configuration by placing Love wave delay line in a feedback circuit of an electrical oscillator (resonator). Another possibility is to use network analyzer, which provides phase shift and insertion loss of the Love wave sensor working in an open loop configuration, due to the load of the sensor with a measured material. The typical frequency range used by Love wave sensors is from 50 MHz to 500 MHz [7].

The structure and cross section of a typical Love wave biosensor is shown in **Figure 6a** and **b**. A relatively thick (0.5–1.0 mm) substrate provides mechanical support for the whole sensor. Often the substrate material is piezoelectric (AT-cut quartz material [41]). In this case, a pair of interdigital transducers (IDTs) can be deposited on the substrate to form a delay line of the sensor. The guiding layer (SiO2 , ZnO, PMMA, etc.), deposited directly on the substrate, provides entrapment for surface wave energy. The sensing layer, made of gold (Au) or a polymer, usually very thin (~50-100 nm), serves as an immobilization area for the measured biological material. This thin-sensing layer interacts directly with the measured material (liquid) and serves often as a selector of the specific target substance, such as antigen, to be measured.

#### **4.5. Sensors controlled remotely by wireless devices**

An interesting solution for Love wave sensors was proposed in [42], where the Love wave sensor works in a wireless configuration without an external power supply. This design has many unique advantages, i.e., the sensor can be permanently implanted in a patient body to monitor continuously the selected property of a biological liquid. Readings of the sensor can be made on demand, totally noninvasively by a reading device connected to a broader computer system of patient monitoring. Another implementation of a remotely controlled wireless Love wave sensor was presented in [43]. The proposed sensor can measure simultaneously two different analytes using Love surface waves with a frequency of 440 MHz.

Wireless bioelectronics sensors may be used in a variety of fields including: healthcare, environmental monitoring, food quality control, and defense.

**Figure 6.** a) Layered structure of a typical Love wave sensor not yet connected to the external driving circuit and b) crosssection of this sensor structure + loading liquid.

#### **4.6. Examples of laboratory and industrial grade Love wave sensors**

To apply the measured analyte to the Love wave sensor, the sensor is often equipped with a flow cell, which separates interdigital transducers from sensing area of the waveguide [44]. A laboratory grade Love wave sensor equipped with a flow cell is shown in **Figure 7**.

• detection of nanoparticles in liquid media [53];

• okadaic acid detection [54]; • study of protein layers [55];

• antibody binding detection [56];

• real-time detection of hepatitis B [59];

• immunosensors for detection of pesticide residues and metabolites in fruit juices [61];

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This rather impressive list of achievements in R&D activities on biosensor technology suggests that biosensors employing Love surface waves have a huge potential. However, in order to compete with other types of biosensors, such as optical sensors based on the surface plasmon resonance [64], the biosensors employing Love surface waves should possess the follow-

At present, none of the above targets have been fully achieved. Love wave biosensors are, in general, still in the laboratory research phase, where most developments are focused on the proof of concept and construction of a working prototype. Only one European company offers today commercially available Love wave sensors [7]. Nevertheless, as it was shown in this section, Love wave biosensors can be used to measurements of a surprisingly large number of biological substances (analytes) with a quite remarkable accuracy and sensitivity. Therefore, in our opinion, Love wave biosensors will reach soon an industrial grade level with

numerous real-life applications in biology, medicine (clinical practice), and chemistry.

**4.8. Desired characteristics (features) of industrial grade Love wave sensors**

• high sensitivity to the measured property (measurand); • high selectivity to the measured property (measurand);

• zero temperature coefficient (high-thermal stability);

• toxicity of heavy metals [57]; • size and shape of DNA [58];

• liquid chromatography [60];

• detection of cocaine [62]; and

ing characteristics:

• low limit of detection;

• cost-effectiveness.

• high repeatability and stability; • possibility of multiple reuse; and

• detection of carbaryl pesticide [63].

A prototype of an commercial ready Love wave sensor was presented in 2015 in Ref. [45]. A 250 MHz delay line Love wave immunosensor was designed on the ST quartz substrate with a thin gold layer of thickness ~90 nm used as a guiding and sensing area, for antibodies or antigens can be easily immobilized on a gold surface. The changes of Love wave velocity and attenuation were due to antibodies-antigens interactions. A disposable test cassette with embedded Love wave immunosensor is connected to a handheld electronic reader, which in turn is connected wirelessly via bluetooth to a smartphone or a computer. This device is a strong candidate for clinical and personnel healthcare applications.

#### **4.7. Examples of analytes measured by Love wave biosensors**

Love wave biosensors have been used in measurement and detection of a large number of substances (analytes) [44]. As representative examples, we can mention the following:


**Figure 7.** An example of a laboratory grade Love wave sensor with a flow cell [42].


**4.6. Examples of laboratory and industrial grade Love wave sensors**

30 Surface Waves - New Trends and Developments

strong candidate for clinical and personnel healthcare applications.

**4.7. Examples of analytes measured by Love wave biosensors**

• simultaneous detection of Legionella and *E. coli* bacteria [48];

• investigation of lipid specificity of human antimicrobial peptides [51];

• concentration of bovine serum albumin [46];

• virus and bacteria detection in liquids [49];

from hantavirus cardiopulmonary syndrome [52];

**Figure 7.** An example of a laboratory grade Love wave sensor with a flow cell [42].

To apply the measured analyte to the Love wave sensor, the sensor is often equipped with a flow cell, which separates interdigital transducers from sensing area of the waveguide [44].

A prototype of an commercial ready Love wave sensor was presented in 2015 in Ref. [45]. A 250 MHz delay line Love wave immunosensor was designed on the ST quartz substrate with a thin gold layer of thickness ~90 nm used as a guiding and sensing area, for antibodies or antigens can be easily immobilized on a gold surface. The changes of Love wave velocity and attenuation were due to antibodies-antigens interactions. A disposable test cassette with embedded Love wave immunosensor is connected to a handheld electronic reader, which in turn is connected wirelessly via bluetooth to a smartphone or a computer. This device is a

Love wave biosensors have been used in measurement and detection of a large number of

substances (analytes) [44]. As representative examples, we can mention the following:

• real-time detection of antigen-antibody interactions in liquids (immunosensor) [47];

• detection of pathogenic spores *Bacillus anthracis* below inhalation infectious levels [50];

• Sin Nombre Virus detection at levels lower than those typical for human patients suffering

A laboratory grade Love wave sensor equipped with a flow cell is shown in **Figure 7**.


#### **4.8. Desired characteristics (features) of industrial grade Love wave sensors**

This rather impressive list of achievements in R&D activities on biosensor technology suggests that biosensors employing Love surface waves have a huge potential. However, in order to compete with other types of biosensors, such as optical sensors based on the surface plasmon resonance [64], the biosensors employing Love surface waves should possess the following characteristics:


At present, none of the above targets have been fully achieved. Love wave biosensors are, in general, still in the laboratory research phase, where most developments are focused on the proof of concept and construction of a working prototype. Only one European company offers today commercially available Love wave sensors [7]. Nevertheless, as it was shown in this section, Love wave biosensors can be used to measurements of a surprisingly large number of biological substances (analytes) with a quite remarkable accuracy and sensitivity. Therefore, in our opinion, Love wave biosensors will reach soon an industrial grade level with numerous real-life applications in biology, medicine (clinical practice), and chemistry.

## **5. Discussion**

#### **5.1. Older sensors using bulk SH waves**

It is interesting to note that first acoustic sensors for measurements of viscoelastic properties of liquids used to this end bulk (not surface) SH waves propagating in a solid buffer, loaded on one side with a measured viscoelastic liquid. This idea appeared in 1950 in works of such prominent ultrasonic scientists Mason and McSkimmin [65]. However, the main drawback of the bulk wave sensors was their inherent low sensitivity. For example, to perform measurements with a water-loaded sensor, one had to observe about 50 consecutive reflections in the solid buffer.

viscosity sensors were of inherently low sensitivity, difficult in practical realization and difficult in theoretical analysis (leaky SH SAW waves). In fact, the energy of SH plate waves is uniformly distributed across the whole thickness of the plate. Therefore, SH plate waves are not so sensitive to viscous loading as Love surface waves, whose energy is highly concentrated in the surface layer of the waveguide. On the other hand, leaky SH SAW waves are not pure SH waves and contains in principle all three components of vibrations, not only the SH one. In particular, the component perpendicular to free surface of the waveguide will continuously radiate energy into the adjacent liquid. This will cause an additional attenuation for leaky SH SAW waves, which will be indistinguishable from that due to the viscous

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**5.3. Mathematical apparatus and numerical methods used in analysis of Love** 

**5.4. Milestones in developments of Love wave seismology and Love wave** 

Examination of **Table 1** reveals that a number of R&D activities already well established in Seismology were not yet initiated in biosensor technology. As examples, one can mention the applications of nonlinear Love waves, higher-order Love wave modes or solitary waves. This suggests that in future research, it may be advantageous to employ higher-order modes, nonlinear Love waves, metamaterials, etc., to increase biosensors sensitivity [75] or lower their limit of detection. Other technologies not yet used in biosensor technology are phased array

R&D activities in seismology and biosensor technology using Love surface waves focus inevitably on different problems and challenges. The main reason for these differences is the nature and scale of Love surface waves used in seismology and biosensor technology, i.e., in seismology, they are a natural phenomenon and in biosensors, they are controlled within man-made devices. It is instructive to compare the chronology of developments made in seismology and in biosensor technology (see **Table 1**). In fact, the theory of Love waves published in 1911 [1] was developed for the simplest surface wave waveguide, namely for that composed of linear, isotropic, and lossless materials (surface layer on a substrate). Since loading viscoelastic liquids are always lossy, the corresponding theory of Love wave sensors had to use perturbative [3] or numerical methods [37]. The theory of Love waves in multilayered waveguides, developed in Seismology [72], uses a conventional transfer-matrix method based on the elementary matrix algebra. By contrast, the theory developed for biosensors extends the transfer-matrix method to a more advanced formalism of matrix differential equations with eigenvectors and eigenvalues and operator functions [8]. First theories of Love waves propagating in viscoelastic waveguides, were developed in Seismology [73], long before the advent of modern fast digital computers. By contrast, the corresponding theory developed for biosensors [74] in 2016 heavily relates on

loading measured.

numerical methods.

**biosensors**

**surface waves**

#### **5.2. Emergence of new sensors using SH surface waves of the Love type**

The breakthrough came with a proposition to employ to this end SH surface waves of the Love and Bleustein-Gulyaev types. This idea was first articulated by Kiełczyński and Płowiec in 1981 in their Polish patent no 130040 [2]. In 1987, the theory of the new method was presented by Kiełczyński and Pajewski on the international arena at the European Mechanics Colloquium 226 in Nottingham, UK [3]. In 1988, this new method, with equations and experimental results, was presented by Kiełczyński and Pajewski at IEEE 1988 Ultrasonic Symposium in Chicago [4]. In 1989, Kiełczyński and Płowiec published detailed theory and experimental results in the prestigious Journal of the Acoustical Society of America [5]. Their theory [3–5] was based on the Auld's perturbative technique [66] and gave satisfactory results for liquids of viscosities up to ~10 Pas. The main advantage of the Love surface wave sensors is their very high sensitivity, namely the sensitivity of a few orders of magnitude (102 to 104 ) higher than that of their bulk SH waves counterparts [3–5]. As a result, measurements of the viscosity of water (~1 mPas) and other biological substances (based largely on water) was no longer a challenge, what was the case with bulk SH wave sensors. In other words, due to the employment of SH surface waves, the way for development of the corresponding biosensors was widely open.

It should be noticed that next publications on the Love wave sensors for liquid characterization appeared in the open literature not earlier than in 1992 [6]. In fact, in papers published in 1992, Kovacs and Venema [67], and, in 1993, Gizeli et al. [68] confirmed our earlier discovery [3–5] that Love surface waves are much more sensitive to viscous loading than other types of SH waves. In another paper published in 1992, Gizeli et al. [69] developed theoretical analysis for Love wave sensors, using the same Auld's perturbative technique [66] as that employed by us in papers [3–5].

It is interesting to note that two other types of SH waves, i.e., leaky SH SAW waves and plate SH waves, were also tried to measure viscosity of liquids. Leaky SH SAW waves were proposed in 1987 [70] by Moriizumi et al. and SH plate waves in 1988 by Martin et al. [71]. However, these two types of SH waves were quickly abandoned, since the corresponding viscosity sensors were of inherently low sensitivity, difficult in practical realization and difficult in theoretical analysis (leaky SH SAW waves). In fact, the energy of SH plate waves is uniformly distributed across the whole thickness of the plate. Therefore, SH plate waves are not so sensitive to viscous loading as Love surface waves, whose energy is highly concentrated in the surface layer of the waveguide. On the other hand, leaky SH SAW waves are not pure SH waves and contains in principle all three components of vibrations, not only the SH one. In particular, the component perpendicular to free surface of the waveguide will continuously radiate energy into the adjacent liquid. This will cause an additional attenuation for leaky SH SAW waves, which will be indistinguishable from that due to the viscous loading measured.

#### **5.3. Mathematical apparatus and numerical methods used in analysis of Love surface waves**

**5. Discussion**

solid buffer.

was widely open.

by us in papers [3–5].

**5.1. Older sensors using bulk SH waves**

32 Surface Waves - New Trends and Developments

It is interesting to note that first acoustic sensors for measurements of viscoelastic properties of liquids used to this end bulk (not surface) SH waves propagating in a solid buffer, loaded on one side with a measured viscoelastic liquid. This idea appeared in 1950 in works of such prominent ultrasonic scientists Mason and McSkimmin [65]. However, the main drawback of the bulk wave sensors was their inherent low sensitivity. For example, to perform measurements with a water-loaded sensor, one had to observe about 50 consecutive reflections in the

The breakthrough came with a proposition to employ to this end SH surface waves of the Love and Bleustein-Gulyaev types. This idea was first articulated by Kiełczyński and Płowiec in 1981 in their Polish patent no 130040 [2]. In 1987, the theory of the new method was presented by Kiełczyński and Pajewski on the international arena at the European Mechanics Colloquium 226 in Nottingham, UK [3]. In 1988, this new method, with equations and experimental results, was presented by Kiełczyński and Pajewski at IEEE 1988 Ultrasonic Symposium in Chicago [4]. In 1989, Kiełczyński and Płowiec published detailed theory and experimental results in the prestigious Journal of the Acoustical Society of America [5]. Their theory [3–5] was based on the Auld's perturbative technique [66] and gave satisfactory results for liquids of viscosities up to ~10 Pas. The main advantage of the Love surface wave sensors

is their very high sensitivity, namely the sensitivity of a few orders of magnitude (102

higher than that of their bulk SH waves counterparts [3–5]. As a result, measurements of the viscosity of water (~1 mPas) and other biological substances (based largely on water) was no longer a challenge, what was the case with bulk SH wave sensors. In other words, due to the employment of SH surface waves, the way for development of the corresponding biosensors

It should be noticed that next publications on the Love wave sensors for liquid characterization appeared in the open literature not earlier than in 1992 [6]. In fact, in papers published in 1992, Kovacs and Venema [67], and, in 1993, Gizeli et al. [68] confirmed our earlier discovery [3–5] that Love surface waves are much more sensitive to viscous loading than other types of SH waves. In another paper published in 1992, Gizeli et al. [69] developed theoretical analysis for Love wave sensors, using the same Auld's perturbative technique [66] as that employed

It is interesting to note that two other types of SH waves, i.e., leaky SH SAW waves and plate SH waves, were also tried to measure viscosity of liquids. Leaky SH SAW waves were proposed in 1987 [70] by Moriizumi et al. and SH plate waves in 1988 by Martin et al. [71]. However, these two types of SH waves were quickly abandoned, since the corresponding

 to 104 )

**5.2. Emergence of new sensors using SH surface waves of the Love type**

R&D activities in seismology and biosensor technology using Love surface waves focus inevitably on different problems and challenges. The main reason for these differences is the nature and scale of Love surface waves used in seismology and biosensor technology, i.e., in seismology, they are a natural phenomenon and in biosensors, they are controlled within man-made devices. It is instructive to compare the chronology of developments made in seismology and in biosensor technology (see **Table 1**). In fact, the theory of Love waves published in 1911 [1] was developed for the simplest surface wave waveguide, namely for that composed of linear, isotropic, and lossless materials (surface layer on a substrate). Since loading viscoelastic liquids are always lossy, the corresponding theory of Love wave sensors had to use perturbative [3] or numerical methods [37]. The theory of Love waves in multilayered waveguides, developed in Seismology [72], uses a conventional transfer-matrix method based on the elementary matrix algebra. By contrast, the theory developed for biosensors extends the transfer-matrix method to a more advanced formalism of matrix differential equations with eigenvectors and eigenvalues and operator functions [8]. First theories of Love waves propagating in viscoelastic waveguides, were developed in Seismology [73], long before the advent of modern fast digital computers. By contrast, the corresponding theory developed for biosensors [74] in 2016 heavily relates on numerical methods.

#### **5.4. Milestones in developments of Love wave seismology and Love wave biosensors**

Examination of **Table 1** reveals that a number of R&D activities already well established in Seismology were not yet initiated in biosensor technology. As examples, one can mention the applications of nonlinear Love waves, higher-order Love wave modes or solitary waves. This suggests that in future research, it may be advantageous to employ higher-order modes, nonlinear Love waves, metamaterials, etc., to increase biosensors sensitivity [75] or lower their limit of detection. Other technologies not yet used in biosensor technology are phased array and tomography. Indeed, applied to biosensors they may allow for a 2D characterization of the analyte distribution, electronic beam steering, focusing, etc. These indications for future research in biosensor technology show clearly advantages of multidisciplinary R&D activities, in this case seismology and biosensor technology. Indeed, it is much easier to adapt an existing technology already developed in other fields to a new domain than to invent a new technology from scratch without any prior feedback.

perspective gives an invaluable insight in the process of developments made in this fascinat-

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35

In this chapter, we attempted to present a variety of aspects that can be attributed to SH surface waves of the Love type. As a matter of fact, Dr. Jekyll and Mr. Hyde Love surface waves possess simultaneously two diametrically different faces, i.e., first benign (biosensors) and second deadly (earthquakes). The good news is that developments made in one of these domains can be easily transferred to the second one and vice versa. In fact, Love surface waves were first discovered in seismology (1911). They finally enabled for precise interpretation of seismograms registered in the aftermath of earthquakes. Beneficiary applications (biosensors) of Love surface waves were announced exactly 70 years later (1981) in a Polish

Since earthquake is a natural phenomenon, we have little or no influence on its occurrence and dynamics. By contrast, the construction and the operation of biosensors can be optimized by mathematical modeling and experimental studies. At present, the mathematical modeling of Love wave biosensors is an active domain of research. On the other hand, progress in electronics and computer technology will lead to development of new compact and reliable

Despite their centennial heritage, Love surface waves are subject of an intensive research activity. For example, one can mention the application of inverse problem techniques to recover material parameters of surface layers from measurements of velocity and attenuation of Love surface waves. Inverse problem techniques have been successfully employed in seismology and geophysics [25] and recently also pioneered by the authors [74, 77, 85] and others [86, 87]

Other open problems in the theory and technique of Love surface waves are non-linear Love waves, extremely slow Love waves [88], Love waves in layered nanostructures [89], energy

Finally, coming back to the idea expressed at the beginning of the introduction in this chapter, we want to assure the reader that there exist still many significant unresolved problems in the theory and technique of the Love surface waves, which deserve to be addressed in future

Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw,

harvesting with Love waves, and metamaterial-based seismic shielding, [33, 34], etc.

R&D activities. We hope that this chapter may be helpful in this endeavor.

Address all correspondence to: pkielczy@ippt.pan.pl

instrumentation working in conjunction with Love wave biosensors.

ing interdisciplinary domain of research.

patent.

in the biosensor technology.

**Author details**

Piotr Kiełczyński

Poland

#### **5.5. Novelty of the present chapter**

Despite the fact that the first theory of Love surface waves was published as early as in 1911 [1], surprisingly, a large number of problems concerning the theory of Love surface waves have not yet been solved.

This chapter contains theoretical foundations and calculation results regarding the propagation of the Love wave in various media. A new interpretation of the Love wave dispersion equation was given. This equation is presented in the form of an implicit function of two variables, i.e., (*ω*, *k*). This allowed to evaluate the analytical dependencies on group velocity of Love wave propagating in a wide class of layered waveguides, e.g., in graded waveguides. This problem will be the subject of future author's works.

The obtained results can be employed in the design and optimization of not only biosensors but also chemosensors and sensors of physical quantities that use Love waves. In addition, the obtained results can be used in seismology and geophysics for the interpretation of seismograms and determining the distribution of elastic parameters of the Earth's crust.

This chapter contains also a novel comparison of milestones in developments made in Love wave seismology and Love wave biosensors (see Section 5.4). Since Love wave biosensors appeared exactly 70 years [2] after emergence of Love surface waves in seismology [1], it is not surprising that many discoveries and developments were made first in seismology and then transferred to biosensors (see **Table 1**). This cross-pollination between the two seemingly distant branches of science is very beneficial and can significantly accelerate developments made in either of them.

#### **6. Conclusions**

In this limited space chapter, it was impossible to address or even mention all interesting problems relevant to the properties and applications of Love surface waves in seismology and biosensor technology. Instead, we tried to present only main properties of the Love surface waves, such as their dispersive nature, phase and group velocities, amplitude distribution, etc., as well as their most iconic applications in seismology and biosensor technology. We think that presentation of the Love surface waves R&D activities in a broader historical perspective gives an invaluable insight in the process of developments made in this fascinating interdisciplinary domain of research.

In this chapter, we attempted to present a variety of aspects that can be attributed to SH surface waves of the Love type. As a matter of fact, Dr. Jekyll and Mr. Hyde Love surface waves possess simultaneously two diametrically different faces, i.e., first benign (biosensors) and second deadly (earthquakes). The good news is that developments made in one of these domains can be easily transferred to the second one and vice versa. In fact, Love surface waves were first discovered in seismology (1911). They finally enabled for precise interpretation of seismograms registered in the aftermath of earthquakes. Beneficiary applications (biosensors) of Love surface waves were announced exactly 70 years later (1981) in a Polish patent.

Since earthquake is a natural phenomenon, we have little or no influence on its occurrence and dynamics. By contrast, the construction and the operation of biosensors can be optimized by mathematical modeling and experimental studies. At present, the mathematical modeling of Love wave biosensors is an active domain of research. On the other hand, progress in electronics and computer technology will lead to development of new compact and reliable instrumentation working in conjunction with Love wave biosensors.

Despite their centennial heritage, Love surface waves are subject of an intensive research activity. For example, one can mention the application of inverse problem techniques to recover material parameters of surface layers from measurements of velocity and attenuation of Love surface waves. Inverse problem techniques have been successfully employed in seismology and geophysics [25] and recently also pioneered by the authors [74, 77, 85] and others [86, 87] in the biosensor technology.

Other open problems in the theory and technique of Love surface waves are non-linear Love waves, extremely slow Love waves [88], Love waves in layered nanostructures [89], energy harvesting with Love waves, and metamaterial-based seismic shielding, [33, 34], etc.

Finally, coming back to the idea expressed at the beginning of the introduction in this chapter, we want to assure the reader that there exist still many significant unresolved problems in the theory and technique of the Love surface waves, which deserve to be addressed in future R&D activities. We hope that this chapter may be helpful in this endeavor.

#### **Author details**

and tomography. Indeed, applied to biosensors they may allow for a 2D characterization of the analyte distribution, electronic beam steering, focusing, etc. These indications for future research in biosensor technology show clearly advantages of multidisciplinary R&D activities, in this case seismology and biosensor technology. Indeed, it is much easier to adapt an existing technology already developed in other fields to a new domain than to invent a new

Despite the fact that the first theory of Love surface waves was published as early as in 1911 [1], surprisingly, a large number of problems concerning the theory of Love surface waves

This chapter contains theoretical foundations and calculation results regarding the propagation of the Love wave in various media. A new interpretation of the Love wave dispersion equation was given. This equation is presented in the form of an implicit function of two variables, i.e., (*ω*, *k*). This allowed to evaluate the analytical dependencies on group velocity of Love wave propagating in a wide class of layered waveguides, e.g., in graded waveguides.

The obtained results can be employed in the design and optimization of not only biosensors but also chemosensors and sensors of physical quantities that use Love waves. In addition, the obtained results can be used in seismology and geophysics for the interpretation of seismograms and determining the distribution of elastic parameters of the Earth's

This chapter contains also a novel comparison of milestones in developments made in Love wave seismology and Love wave biosensors (see Section 5.4). Since Love wave biosensors appeared exactly 70 years [2] after emergence of Love surface waves in seismology [1], it is not surprising that many discoveries and developments were made first in seismology and then transferred to biosensors (see **Table 1**). This cross-pollination between the two seemingly distant branches of science is very beneficial and can significantly accelerate developments

In this limited space chapter, it was impossible to address or even mention all interesting problems relevant to the properties and applications of Love surface waves in seismology and biosensor technology. Instead, we tried to present only main properties of the Love surface waves, such as their dispersive nature, phase and group velocities, amplitude distribution, etc., as well as their most iconic applications in seismology and biosensor technology. We think that presentation of the Love surface waves R&D activities in a broader historical

technology from scratch without any prior feedback.

This problem will be the subject of future author's works.

**5.5. Novelty of the present chapter**

34 Surface Waves - New Trends and Developments

have not yet been solved.

made in either of them.

**6. Conclusions**

crust.

Piotr Kiełczyński

Address all correspondence to: pkielczy@ippt.pan.pl

Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland

#### **References**

[1] Love AEH. Some Problems of Geodynamics. UK: Cambridge University Press; 1911. pp. 89-104 and 149-152

[14] Kittmann A, Durdaut P, Zabel S, Reermann J, Schmalz J, Spetzler B, Meyners D, Sun NX, McCord J, Gerken M, Schmidt G, Höft M, Knöchel R, Faupel F, Quandt E. Wide band low noise love wave magnetic field sensor system. Scientific Reports. 2018;**8**:278. DOI:

Properties and Applications of Love Surface Waves in Seismology and Biosensors

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37

[15] Auld BA. Acoustics Fields and Waves in Solids, Vol. II. Florida: Krieger Publishing

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**Chapter 3**

Provisional chapter

**Dyakonov Surface Waves: Anisotropy-Enabling**

DOI: 10.5772/intechopen.74126

The title "Dyakonov surface waves: anisotropy enabling confinement on the edge" plainly sets the scope for this chapter. The focus here is on the formation of bounded waves at the interface of two distinct media, at least one of them exhibiting optical anisotropy, which are coined as Dyakonov surface waves (DSWs) in recognition to the physicist who reported their existence for the first time. First, the general aspects of the topic are discussed. It also treats the characterization of bounded waves in isotropicuniaxial multilayered structures, allowing not only the derivation of the dispersion equation of DSWs but also that of surface plasmons polaritons (SPPs), for instance. Furthermore, the interaction of such surfaces waves, with the possibility of including guided waves in a given planar layer and external sources mimicking experimental setups, can be accounted for by using the transfer matrix formalism introduced here. Finally, special attention is devoted to hyperbolic media with indefinite anisotropy-

enabling hybridized scenarios integrating the prototypical DSWs and SPPs.

The planar interface of two dissimilar materials plays a relevant role in many optical phenomena. In recent years, particularly, evanescent waves have been used in newly developing technologies such as near-field spectroscopy. The electromagnetic surface wave, which is intimately tied to the interface, travels in a direction parallel to the interface but, on either side of the interface, its amplitude is imperceptible after a certain distance from the interface. The notion of an electromagnetic surface wave made a significant appearance in 1907 when

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Keywords: surface waves, anisotropy, transfer matrix formulation

Dyakonov Surface Waves: Anisotropy-Enabling

**Confinement on the Edge**

Confinement on the Edge

http://dx.doi.org/10.5772/intechopen.74126

Abstract

1. Introduction

Carlos J. Zapata-Rodríguez, Slobodan Vuković, Juan J. Miret, Mahin Naserpour and Milivoj R. Belić

Carlos J. Zapata-Rodríguez, Slobodan Vuković, Juan J. Miret, Mahin Naserpour and Milivoj R. Belić

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter


#### **Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge** Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

DOI: 10.5772/intechopen.74126

Carlos J. Zapata-Rodríguez, Slobodan Vuković, Juan J. Miret, Mahin Naserpour and Milivoj R. Belić Carlos J. Zapata-Rodríguez, Slobodan Vuković, Juan J. Miret, Mahin Naserpour and Milivoj R. Belić

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74126

#### Abstract

[87] Ueda K, Kondoh J. Estimation of liquid properties by inverse problems analysis based on shear horizontal surface acoustic wave sensor responses. Japanese Journal of Applied

[88] Chen X, Jiu Hui W, Kuan L, Cao P. Slow light and stored light of Love wave within a thin

[89] Zhang S, Bin G, Zhang H, Feng X-Q, Pan R, Alamusi NH. Propagation of Love waves with surface effects in an electrically-shorted piezoelectric nanofilm on a half-space elas-

Physics. 2017;**56**:07 JD08

42 Surface Waves - New Trends and Developments

film. Optik. 2017;**132**:329-336

tic substrate. Ultrasonics. 2016;**66**:65-71

The title "Dyakonov surface waves: anisotropy enabling confinement on the edge" plainly sets the scope for this chapter. The focus here is on the formation of bounded waves at the interface of two distinct media, at least one of them exhibiting optical anisotropy, which are coined as Dyakonov surface waves (DSWs) in recognition to the physicist who reported their existence for the first time. First, the general aspects of the topic are discussed. It also treats the characterization of bounded waves in isotropicuniaxial multilayered structures, allowing not only the derivation of the dispersion equation of DSWs but also that of surface plasmons polaritons (SPPs), for instance. Furthermore, the interaction of such surfaces waves, with the possibility of including guided waves in a given planar layer and external sources mimicking experimental setups, can be accounted for by using the transfer matrix formalism introduced here. Finally, special attention is devoted to hyperbolic media with indefinite anisotropyenabling hybridized scenarios integrating the prototypical DSWs and SPPs.

Keywords: surface waves, anisotropy, transfer matrix formulation

#### 1. Introduction

The planar interface of two dissimilar materials plays a relevant role in many optical phenomena. In recent years, particularly, evanescent waves have been used in newly developing technologies such as near-field spectroscopy. The electromagnetic surface wave, which is intimately tied to the interface, travels in a direction parallel to the interface but, on either side of the interface, its amplitude is imperceptible after a certain distance from the interface. The notion of an electromagnetic surface wave made a significant appearance in 1907 when

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Zen-neck [1] authored a theoretical paper exploring the possibility of a wave guided by the interface of the atmosphere and either Earth or a large body of water. His focus was on radio waves, a region of the electromagnetic spectrum far from the optical regime in which we are particularly interested, but the principles involved are the same, owing to the scale invariance of the Maxwell postulates.

2.1. Wave propagation in isotropic media

the speed of the waves in the medium.

spatial frequency k ¼ kx; ky; kz

strated here, <sup>b</sup>e<sup>2</sup> is related to TM<sup>x</sup>

f g ^e1; ^e2; k form an orthogonal trihedron.

In this section, we consider the propagation of electromagnetic waves in linear, homogeneous, and isotropic dielectrics. Under these conditions, the relative permittivity e relating E and D is a scalar constant. Considering that the medium is free of electric charges and currents, and taking into account the medium equation B ¼ μ0H, Maxwell's equations can be written as

> ∂E ∂t

> > ∂H

, (1a)

http://dx.doi.org/10.5772/intechopen.74126

<sup>∂</sup><sup>t</sup> , (1b)

<sup>t</sup> u ¼ 0,

45

I þ k 2 <sup>0</sup>EI.

∇ � E ¼ 0, (1c)

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

∇ � H ¼ 0: (1d)

E rð Þ¼ ; t E<sup>0</sup> exp ð Þ ik � r exp ð Þ �iωt , (2a)

H rð Þ¼ ; t H<sup>0</sup> exp ð Þ ik � r exp ð Þ �iωt , (2b)

� �, (3a)

� �: (3b)


∇�H ¼ ee<sup>0</sup>

∇�E ¼ �μ<sup>0</sup>

Now, each of the scalar components of <sup>E</sup> and <sup>H</sup> satisfies the wave equation <sup>∇</sup><sup>2</sup><sup>u</sup> � <sup>c</sup>�<sup>2</sup>∂<sup>2</sup>

where u represents any one of the six scalar components of the electromagnetic field and c is

When the electromagnetic wave is plane and monochromatic, all components of the electric and magnetic fields are harmonic functions in time and space at the time frequency ω and

where E0ð Þr and H0ð Þr are the complex amplitudes of the electric and magnetic fields. If we substitute the vector wave fields of Eqs. (2) in Maxwell's equations (1), with the help of the transformations ∇ ! ik and ∂<sup>t</sup> ! �iω, we might attain a simplified expression of k � ð Þ k � E<sup>0</sup> ,

Here I is the 3 � 3 identity matrix, and k is the modulus of the wavevector k and k<sup>0</sup> ¼ ω=c0. In order to obtain the dispersion equation, we look for nontrivial solutions of the electric field E<sup>0</sup> by imposing that detð Þ¼ <sup>M</sup> 0. Its solution leads to <sup>k</sup><sup>2</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>=c2. The electric field amplitude <sup>E</sup><sup>0</sup>

<sup>b</sup>e<sup>1</sup> <sup>¼</sup> <sup>0</sup>; kz; �ky

We point out that <sup>b</sup>e<sup>1</sup> is associated with TE<sup>x</sup> modes and that <sup>b</sup>e1:be<sup>2</sup> <sup>¼</sup> 0. Although not demon-

to the following orthogonality relations-hips: k � ^e<sup>1</sup> ¼ k � ^e<sup>2</sup> ¼ 0. As a result, the vectors

<sup>z</sup> ; �kykx; �kzkx

enabling to obtain the following wave equation: <sup>M</sup> � <sup>E</sup><sup>0</sup> <sup>¼</sup> 0, where <sup>M</sup> � <sup>k</sup> <sup>⊗</sup> <sup>k</sup> � <sup>k</sup><sup>2</sup>

can be written as a linear combination of the following vectors

<sup>b</sup>e<sup>2</sup> <sup>¼</sup> <sup>k</sup> 2 <sup>y</sup> þ k 2

� �, respectively. Particularly they may be set as

Yet, nearly a century later, a unique type of wave, the surface plasmon polariton (SPP) wave, that dominates the nanotechnology scene, at least at optical frequencies, resulted in wonderful developments with the creation of extremely sensitive bio/chemical sensors, and improvements in this mature technology continue to this day [2]. Even in this highly developed application, the two partnering materials which meet at the interface may be simple: one is a typical metal, a plasmonic material at optical frequencies, and the other is a homogeneous, isotropic, dielectric material. While the interface of a plasmonic material and a polarizable material supports SPPs, a variety of other types of surface waves can be supported by the interface of two polarizable materials. Since polarizable materials such as dielectric materials are less dissipative, in general, than plasmonic materials such as metals, the advantage of these materials for long-range propagation of surface waves is apparent.

The interface of two homogeneous dielectric materials of which at least one is anisotropic may support surface-wave propagation of another type, even though the real parts of all components of the permittivity dyadics of both materials are positive. Interest in surface waves guided by the interface of two dielectric materials began to take after Dyakonov in 1988 [3] explored the propagation of a surface wave guided by the interface of a uniaxial dielectric material and an isotropic dielectric material. The Dyakonov surface wave (DSWs) is the focus of this chapter.

In this chapter, we perform a thorough analysis of DSWs taking place in semi-infinite anisotropic media. Basic concepts related to the propagation of electromagnetic waves in homogeneous media will be introduced, including isotropic and anisotropic materials. Birefringent metal-dielectric (MD) lattices will be also considered as a contribution of meta-materials in the development of DSWs [4]. Special emphasis will be put when the effective-medium approaches induce unsatisfactory results, which occur in most experimental configurations. Practical cases will be analyzed including dissipative effects due to Ohmic losses of the metal.

#### 2. Wave propagation in bulk media and interfaces

In this section we introduce the basic concepts related to the propagation of electromagnetic waves in homogeneous media, including isotropic and anisotropic materials. We describe in detail complex multilayered structures. For that purpose, we introduce a transfer matrix formulation that applies to isotropic and uniaxial media simultaneously. Finally, we discuss the conditions that give rise to surface waves at the interface of two isotropic media; the case of dealing with anisotropic media is considered in Section 5. Moreover, we obtain the dispersion equation for SPPs, which appears at the interface between a dielectric and a metal.

#### 2.1. Wave propagation in isotropic media

Zen-neck [1] authored a theoretical paper exploring the possibility of a wave guided by the interface of the atmosphere and either Earth or a large body of water. His focus was on radio waves, a region of the electromagnetic spectrum far from the optical regime in which we are particularly interested, but the principles involved are the same, owing to the scale invariance

Yet, nearly a century later, a unique type of wave, the surface plasmon polariton (SPP) wave, that dominates the nanotechnology scene, at least at optical frequencies, resulted in wonderful developments with the creation of extremely sensitive bio/chemical sensors, and improvements in this mature technology continue to this day [2]. Even in this highly developed application, the two partnering materials which meet at the interface may be simple: one is a typical metal, a plasmonic material at optical frequencies, and the other is a homogeneous, isotropic, dielectric material. While the interface of a plasmonic material and a polarizable material supports SPPs, a variety of other types of surface waves can be supported by the interface of two polarizable materials. Since polarizable materials such as dielectric materials are less dissipative, in general, than plasmonic materials such as metals, the advantage of these

The interface of two homogeneous dielectric materials of which at least one is anisotropic may support surface-wave propagation of another type, even though the real parts of all components of the permittivity dyadics of both materials are positive. Interest in surface waves guided by the interface of two dielectric materials began to take after Dyakonov in 1988 [3] explored the propagation of a surface wave guided by the interface of a uniaxial dielectric material and an isotropic dielectric material. The Dyakonov surface wave (DSWs) is the focus

In this chapter, we perform a thorough analysis of DSWs taking place in semi-infinite anisotropic media. Basic concepts related to the propagation of electromagnetic waves in homogeneous media will be introduced, including isotropic and anisotropic materials. Birefringent metal-dielectric (MD) lattices will be also considered as a contribution of meta-materials in the development of DSWs [4]. Special emphasis will be put when the effective-medium approaches induce unsatisfactory results, which occur in most experimental configurations. Practical cases will be analyzed including dissipative effects due to Ohmic losses of the metal.

In this section we introduce the basic concepts related to the propagation of electromagnetic waves in homogeneous media, including isotropic and anisotropic materials. We describe in detail complex multilayered structures. For that purpose, we introduce a transfer matrix formulation that applies to isotropic and uniaxial media simultaneously. Finally, we discuss the conditions that give rise to surface waves at the interface of two isotropic media; the case of dealing with anisotropic media is considered in Section 5. Moreover, we obtain the dispersion

equation for SPPs, which appears at the interface between a dielectric and a metal.

materials for long-range propagation of surface waves is apparent.

2. Wave propagation in bulk media and interfaces

of the Maxwell postulates.

44 Surface Waves - New Trends and Developments

of this chapter.

In this section, we consider the propagation of electromagnetic waves in linear, homogeneous, and isotropic dielectrics. Under these conditions, the relative permittivity e relating E and D is a scalar constant. Considering that the medium is free of electric charges and currents, and taking into account the medium equation B ¼ μ0H, Maxwell's equations can be written as

$$
\nabla \times \mathbf{H} = \epsilon \epsilon\_0 \frac{\partial \mathbf{E}}{\partial t} \tag{1a}
$$

$$\nabla \times \mathbf{E} = -\mu\_0 \frac{\partial \mathbf{H}}{\partial t} \tag{1b}$$

$$
\nabla \cdot \mathbf{E} = \mathbf{0},
\tag{1c}
$$

$$
\nabla \cdot \mathbf{H} = 0.\tag{1d}
$$

Now, each of the scalar components of <sup>E</sup> and <sup>H</sup> satisfies the wave equation <sup>∇</sup><sup>2</sup><sup>u</sup> � <sup>c</sup>�<sup>2</sup>∂<sup>2</sup> <sup>t</sup> u ¼ 0, where u represents any one of the six scalar components of the electromagnetic field and c is the speed of the waves in the medium.

When the electromagnetic wave is plane and monochromatic, all components of the electric and magnetic fields are harmonic functions in time and space at the time frequency ω and spatial frequency k ¼ kx; ky; kz � �, respectively. Particularly they may be set as

$$\mathbf{E}(\mathbf{r}, \mathbf{t}) = \mathbf{E}\_0 \exp\left(i\mathbf{k} \cdot \mathbf{r}\right) \exp\left(-i\omega t\right),\tag{2a}$$

$$\mathbf{H}(\mathbf{r}, \mathbf{t}) = \mathbf{H}\_0 \exp\left(i\mathbf{k} \cdot \mathbf{r}\right) \exp\left(-i\omega t\right),\tag{2b}$$

where E0ð Þr and H0ð Þr are the complex amplitudes of the electric and magnetic fields. If we substitute the vector wave fields of Eqs. (2) in Maxwell's equations (1), with the help of the transformations ∇ ! ik and ∂<sup>t</sup> ! �iω, we might attain a simplified expression of k � ð Þ k � E<sup>0</sup> , enabling to obtain the following wave equation: <sup>M</sup> � <sup>E</sup><sup>0</sup> <sup>¼</sup> 0, where <sup>M</sup> � <sup>k</sup> <sup>⊗</sup> <sup>k</sup> � <sup>k</sup><sup>2</sup> I þ k 2 <sup>0</sup>EI. Here I is the 3 � 3 identity matrix, and k is the modulus of the wavevector k and k<sup>0</sup> ¼ ω=c0. In order to obtain the dispersion equation, we look for nontrivial solutions of the electric field E<sup>0</sup> by imposing that detð Þ¼ <sup>M</sup> 0. Its solution leads to <sup>k</sup><sup>2</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>=c2. The electric field amplitude <sup>E</sup><sup>0</sup> can be written as a linear combination of the following vectors

$$
\hat{\mathcal{e}}\_1 = \{0, k\_z, -k\_y\}, \tag{3a}
$$

$$
\widehat{e}\_2 = \left(k\_y^2 + k\_z^2, -k\_y k\_x, -k\_z k\_x\right). \tag{3b}
$$

We point out that <sup>b</sup>e<sup>1</sup> is associated with TE<sup>x</sup> modes and that <sup>b</sup>e1:be<sup>2</sup> <sup>¼</sup> 0. Although not demonstrated here, <sup>b</sup>e<sup>2</sup> is related to TM<sup>x</sup> -polarized plane waves. Note also that Eqs. (1c) and (1d) lead to the following orthogonality relations-hips: k � ^e<sup>1</sup> ¼ k � ^e<sup>2</sup> ¼ 0. As a result, the vectors f g ^e1; ^e2; k form an orthogonal trihedron.

#### 2.2. Wave propagation in uniaxial media

Uniaxial crystals are media with certain symmetries that make them have two equal principal refractive indices: nx ¼ ny � no (ordinary index) and nz � ne (extraordinary index). The crystal is to be a positive uniaxial if ne > no and negative uniaxial if ne < no. The z axis of a uniaxial crystal is called the optic axis. We call it the ordinary polarization direction if the wave has an eigenindex of refraction no and extraordinary, if the wave has an eigenindex of refraction ne. In Table 1 we show the values of no and ne for some natural birefringent materials (uniaxial crystals).

Let us now consider the propagation of wave planes in uniaxial media. To obtain the eigenvalues associated with plane-wave propagation, we proceed in a similar way as in Section 2.1 for isotropic media, but taking into account that now the relative permittivity e ¼ Exðx ⊗ x þ y ⊗ yÞ þ Ezð Þ z ⊗ z is a tensor. Therefore, we solve detð Þ¼ M 0, where the matrix M � k ⊗ k � k 2 I þ k 2 <sup>0</sup>e. Then, after solving the abovementioned determinant, we obtain two solutions. The first of them is k<sup>2</sup> <sup>x</sup> þ k 2 <sup>y</sup> þ k 2 <sup>z</sup> <sup>¼</sup> <sup>E</sup>xk<sup>2</sup> <sup>0</sup> which corresponds to the dispersion equation of the ordinary plane waves. The electric field for this kind of plane waves is proportional to the vector <sup>b</sup>eo ¼ �ky; kx; <sup>0</sup> � �. As a consequence, ordinary plane waves are TEz -polarized waves. The second solution gives us the dispersion equation of the extraordinary plane waves:

$$\frac{k\_x^2 + k\_y^2}{\epsilon\_z} + \frac{k\_z^2}{\epsilon\_x} = k\_0^2. \tag{4}$$

2.3.1. Electromagnetic fields in a uniaxial elementary layer

polarized (ATEi) and TM-polarized (ATMi) waves.

Figure 1. Schematic arrangement of the multilayered media. The amplitudes Ai and A<sup>0</sup>

ordinary waves, we can write

taken from Eq. (4).

and for extraordinary waves (here kxi � kei)

Let us first consider a multilayered media of uniaxial materials. For simplicity, let us take into account only relative permittivities of the form e<sup>i</sup> ¼ Exiðx ⊗ x þ y ⊗ yÞ þ Ezið Þ z ⊗ z for a given medium "i." As we demonstrated in Section 2.2, the dispersion equation for extraordinary waves propagating in bulk uniaxial media is given by Eq. (4). Due to the boundary conditions at the interfaces between different media, the components of the wave vector ky and kz are conserved, but not its projection upon the x-axis. More specifically, if we rename kxi � koi for

propagating along the positive (negative) x-axis. These amplitudes characterize a given state of polarization: For uniaxial media we may deal with ordinary (for example, Aoi) and extraordinary (Aei) waves. For isotropic materials we have TE-

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>r</sup> � �, (5)

<sup>i</sup> (Bi and B<sup>0</sup>

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

http://dx.doi.org/10.5772/intechopen.74126

<sup>i</sup>) correspond to waves

47

<sup>E</sup>xi <sup>s</sup> � �, (6)

tot <sup>¼</sup> <sup>E</sup>ð Þ<sup>i</sup> ð Þ<sup>x</sup> exp ikyy <sup>þ</sup> ikzz � <sup>i</sup>ω<sup>t</sup> � �: (7)

<sup>þ</sup>Aei^aei exp ikeið Þ <sup>x</sup> � xi ½ �þ Bei^bei exp �ikeið Þ <sup>x</sup> � xi ½ �, (8)

Exik 2 <sup>0</sup> � k 2 <sup>y</sup> þ k 2 z

Ezik 2 <sup>0</sup> � <sup>k</sup><sup>2</sup> <sup>y</sup> þ Ezik 2 z

The part of the electric field that varies along with the spatial coordinate x can be written as

<sup>E</sup>ð Þ<sup>i</sup> ð Þ¼ <sup>x</sup> Aoi^aoi exp ikoið Þ <sup>x</sup> � xi ½ �þ Boi^boi exp �ikoið Þ <sup>x</sup> � xi ½ �

where the amplitude Aoi (and Aei) corresponds to propagating ordinary (and extraordinary) waves, and Boi (and Bei) is related with counter-propagating ordinary (and extraordinary)

koi ¼

kei ¼

The total electric field of the elementary layer "i" is set as

Eð Þ<sup>i</sup>

In this case, the electric field is proportional to the vector ^ee <sup>¼</sup> kxkz; kykz; <sup>k</sup><sup>2</sup> <sup>z</sup> � <sup>E</sup>xk<sup>2</sup> 0 � �.

#### 2.3. Matrix formulation for multilayered media

In this section we look at the case of multilayered media composed of different nonmagnetic materials which are separated by planar parallel interfaces, displaced on x ¼ xi as shown in Figure 1. In particular, we deal with either uniaxial or isotropic materials, including metals. We make a detailed description of the electromagnetic fields inside a given medium "i", which lies in xi�<sup>1</sup> < x < xi. Our objective is the analysis of the appropriate conditions for the propagation of surface waves at the abovementioned interfaces.


Table 1. Birefringence Δn of some natural materials [5].

Figure 1. Schematic arrangement of the multilayered media. The amplitudes Ai and A<sup>0</sup> <sup>i</sup> (Bi and B<sup>0</sup> <sup>i</sup>) correspond to waves propagating along the positive (negative) x-axis. These amplitudes characterize a given state of polarization: For uniaxial media we may deal with ordinary (for example, Aoi) and extraordinary (Aei) waves. For isotropic materials we have TEpolarized (ATEi) and TM-polarized (ATMi) waves.

#### 2.3.1. Electromagnetic fields in a uniaxial elementary layer

Let us first consider a multilayered media of uniaxial materials. For simplicity, let us take into account only relative permittivities of the form e<sup>i</sup> ¼ Exiðx ⊗ x þ y ⊗ yÞ þ Ezið Þ z ⊗ z for a given medium "i." As we demonstrated in Section 2.2, the dispersion equation for extraordinary waves propagating in bulk uniaxial media is given by Eq. (4). Due to the boundary conditions at the interfaces between different media, the components of the wave vector ky and kz are conserved, but not its projection upon the x-axis. More specifically, if we rename kxi � koi for ordinary waves, we can write

$$k\_{oi} = \sqrt{\epsilon\_{xi} k\_0^2 - \left(k\_y^2 + k\_z^2\right)},\tag{5}$$

and for extraordinary waves (here kxi � kei)

$$k\_{\epsilon i} = \sqrt{\epsilon\_{zi}k\_0^2 - \left(k\_y^2 + \frac{\epsilon\_{zi}k\_z^2}{\epsilon\_{xi}}\right)},\tag{6}$$

taken from Eq. (4).

2.2. Wave propagation in uniaxial media

46 Surface Waves - New Trends and Developments

crystals).

M � k ⊗ k � k

2 I þ k 2

solutions. The first of them is k<sup>2</sup>

<sup>x</sup> þ k 2 <sup>y</sup> þ k 2 <sup>z</sup> <sup>¼</sup> <sup>E</sup>xk<sup>2</sup>

the vector <sup>b</sup>eo ¼ �ky; kx; <sup>0</sup> � �. As a consequence, ordinary plane waves are TEz

k 2 <sup>x</sup> þ k 2 y Ez

In this case, the electric field is proportional to the vector ^ee <sup>¼</sup> kxkz; kykz; <sup>k</sup><sup>2</sup>

2.3. Matrix formulation for multilayered media

of surface waves at the abovementioned interfaces.

Table 1. Birefringence Δn of some natural materials [5].

Uniaxial crystals are media with certain symmetries that make them have two equal principal refractive indices: nx ¼ ny � no (ordinary index) and nz � ne (extraordinary index). The crystal is to be a positive uniaxial if ne > no and negative uniaxial if ne < no. The z axis of a uniaxial crystal is called the optic axis. We call it the ordinary polarization direction if the wave has an eigenindex of refraction no and extraordinary, if the wave has an eigenindex of refraction ne. In Table 1 we show the values of no and ne for some natural birefringent materials (uniaxial

Let us now consider the propagation of wave planes in uniaxial media. To obtain the eigenvalues associated with plane-wave propagation, we proceed in a similar way as in Section 2.1 for isotropic media, but taking into account that now the relative permittivity e ¼ Exðx ⊗ x þ y ⊗ yÞ þ Ezð Þ z ⊗ z is a tensor. Therefore, we solve detð Þ¼ M 0, where the matrix

of the ordinary plane waves. The electric field for this kind of plane waves is proportional to

þ k 2 z Ex ¼ k 2

In this section we look at the case of multilayered media composed of different nonmagnetic materials which are separated by planar parallel interfaces, displaced on x ¼ xi as shown in Figure 1. In particular, we deal with either uniaxial or isotropic materials, including metals. We make a detailed description of the electromagnetic fields inside a given medium "i", which lies in xi�<sup>1</sup> < x < xi. Our objective is the analysis of the appropriate conditions for the propagation

Birefringent material no ne Δn ¼ ne � no Crystal quartz 1.547 1.556 0.009 MgF 1.3786 1.3904 0.0118 YVO4 1.9929 2.2154 0.2225 Rutile (TiO2) 2.65 2.95 0.3 E7 liquid crystal 1.520 1.725 0.205 Calomel (Hg2Cl2) 1.96 2.62 0.68

The second solution gives us the dispersion equation of the extraordinary plane waves:

<sup>0</sup>e. Then, after solving the abovementioned determinant, we obtain two

<sup>0</sup> which corresponds to the dispersion equation

<sup>0</sup>: (4)

� �.

<sup>z</sup> � <sup>E</sup>xk<sup>2</sup> 0


The total electric field of the elementary layer "i" is set as

$$\mathbf{E}\_{tot}^{(i)} = \mathbf{E}^{(i)}(\mathbf{x}) \exp\left(ik\_y y + ik\_z z - i\omega t\right). \tag{7}$$

The part of the electric field that varies along with the spatial coordinate x can be written as

$$\begin{split} \mathbf{E}^{(i)}(\mathbf{x}) &= \ \ A\_{\textit{vi}} \hat{a}\_{\textit{vi}} \exp\left[i\mathbf{k}\_{\textit{vi}}(\mathbf{x}-\mathbf{x}\_{i})\right] + B\_{\textit{vi}} \hat{b}\_{\textit{ai}} \exp\left[-i\mathbf{k}\_{\textit{vi}}(\mathbf{x}-\mathbf{x}\_{i})\right] \\ &+ A\_{\textit{ei}} \hat{a}\_{\textit{ei}} \exp\left[i\mathbf{k}\_{\textit{vi}}(\mathbf{x}-\mathbf{x}\_{i})\right] + B\_{\textit{ei}} \hat{b}\_{\textit{ei}} \exp\left[-i\mathbf{k}\_{\textit{vi}}(\mathbf{x}-\mathbf{x}\_{i})\right]. \end{split} \tag{8}$$

where the amplitude Aoi (and Aei) corresponds to propagating ordinary (and extraordinary) waves, and Boi (and Bei) is related with counter-propagating ordinary (and extraordinary) waves. Note that all these amplitudes have zero dephase at x ¼ xi. Finally, the vectors ^aoi and ^aei are rewritten as:

$$
\hat{a}\_{oi} = \begin{pmatrix} -k\_y, k\_{oi}, 0 \end{pmatrix} \tag{9a}
$$

^doi ¼� �koikz; kykz; <sup>k</sup><sup>2</sup>

^dei ¼ �Exik<sup>2</sup>

Once we have a complete description of the wave fields in every elementary layer of our metamaterial, we have to impose some boundary conditions at the interfaces x ¼ xi. The

tot and the magnetic field <sup>H</sup>ð Þ<sup>i</sup>

koi �koi kykz kykz

2 <sup>z</sup> � Exik 2 <sup>0</sup> k 2 <sup>z</sup> � Exik 2 0

A0 oi B0 oi A0 ei B0 ei

<sup>0</sup>kei �Exik<sup>2</sup>

<sup>0</sup> 0 0

be continuous. Particularly these boundary conditions may be expressed in a matrix form as

Div<sup>i</sup> ¼ D<sup>i</sup>þ<sup>1</sup>v<sup>0</sup>

0 0 k

Aoi Boi Aei Bei

v0

3 7 7 7 5 , v<sup>0</sup> <sup>i</sup> ¼

The matrix formulation can also be used to relate the amplitude vector v<sup>i</sup> with zero-phase shift

established in Eq. (12a), (12b). For that purpose, we introduce the propagation matrix Pi, which takes into account the amplitude dephasing between the boundaries of each layer. Explicitly

> e�ikoiwi 000 0 eikoiwi 0 0 0 0 e�ikeiwi 0 000 eikeiwi

<sup>0</sup> �k 2 <sup>z</sup> <sup>þ</sup> <sup>E</sup>xik<sup>2</sup>

�kykz �kykz <sup>E</sup>xik<sup>2</sup>

We point out that the vector fields ^coi and ^doi can be given in units of k<sup>2</sup>

0.

2.3.2. Boundary conditions for anisotropic layered media

We defined the following matrix, explicitly given as

�k 2 <sup>z</sup> <sup>þ</sup> <sup>E</sup>xik<sup>2</sup>

On the other hand, we introduced the amplitude column vectors

P<sup>i</sup> ¼

v<sup>i</sup> ¼

D<sup>i</sup> ¼

at x ¼ xi with the amplitude vector v<sup>0</sup>

we may write

being

be expressed in units of k<sup>3</sup>

components of the electric field Eð Þ<sup>i</sup>

<sup>z</sup> � <sup>E</sup>xik<sup>2</sup> 0

� �, (14c)

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

<sup>0</sup> ky; kei; <sup>0</sup> � �: (14d)

http://dx.doi.org/10.5772/intechopen.74126

<sup>0</sup>, whereas ^cei and ^dei can

49

tot lying on the planes x ¼ xi must

<sup>i</sup>þ<sup>1</sup>: (15)

: (17)

, (19)

: (16)

<sup>0</sup>kei

<sup>i</sup> exhibiting zero dephase at x ¼ xi�1, previously

<sup>i</sup> ¼ P<sup>i</sup> � vi, (18)

$$
\hat{a}\_{\rm ef} = \left( k\_{\rm ef} k\_z, k\_y k\_z, k\_z^2 - \epsilon\_{\rm xi} k\_0^2 \right). \tag{9b}
$$

In the case of counter-propagating waves, we take into consideration that kxi ¼ �koi for ordinary waves and kxi ¼ �kei for extraordinary waves. This fact leads us to introduce the vector fields

$$
\hat{b}\_{o\dot{i}} = \begin{pmatrix} -k\_y, -k\_{o\dot{i}}, 0 \end{pmatrix} . \tag{10a}
$$

$$
\hat{\boldsymbol{b}}\_{\rm ei} = \begin{pmatrix} -k\_{\rm ci}k\_z, k\_yk\_z, k\_z^2 - \epsilon\_{\rm xi}k\_0^2 \end{pmatrix}. \tag{10b}
$$

The vector fields ^aoi and ^boi can be given in units of k0, whereas ^aei and ^bei can be expressed in units of k<sup>2</sup> 0.

For convenience, the field function Eð Þ<sup>i</sup> can be set in terms of the wave amplitudes A<sup>0</sup> oi, A<sup>0</sup> ei and B0 oi and B<sup>0</sup> ei with zero dephase at x ¼ xi�1, namely:

$$\begin{split} \mathbf{E}^{(i)}(\mathbf{x}) &= A'\_{\,\dot{\alpha}} \hat{a}\_{\,\dot{\alpha}} \exp\left[i\mathbf{k}\_{\dot{\alpha}}(\mathbf{x} - \mathbf{x}\_{i-1})\right] + B'\_{\,\dot{\alpha}} \hat{b}\_{\,\dot{\alpha}+1} \exp\left[-i\mathbf{k}\_{\dot{\alpha}}(\mathbf{x} - \mathbf{x}\_{i-1})\right] \\ &+ A'\_{\,\dot{\alpha}} \hat{a}\_{\,\dot{\alpha}} \exp\left[i\mathbf{k}\_{\dot{\alpha}}(\mathbf{x} - \mathbf{x}\_{i-1})\right] + B'\_{\,\dot{\alpha}} \hat{b}\_{\,\dot{\alpha}} \exp\left[-i\mathbf{k}\_{\dot{\alpha}}(\mathbf{x} - \mathbf{x}\_{i-1})\right]. \end{split} \tag{11}$$

The complete set of amplitudes is Aqi, A<sup>0</sup> qi and Bqi and B<sup>0</sup> qi where q ¼ f g o;e satisfies the following relationships:

$$A'\_{qi} = A\_{qi} \exp\left[-ik\_{qi}(\mathbf{x}\_i - \mathbf{x}\_{i-1})\right],\tag{12a}$$

$$B'\_{q\dot{\imath}} = B\_{q\dot{\imath}} \exp\left[\mathrm{i}k\_{q\dot{\imath}}(\mathbf{x}\_{i} - \mathbf{x}\_{i-1})\right].\tag{12b}$$

For completeness we calculate the magnetic field in a given elementary layer "i." By using the Maxwell's equation Hð Þ<sup>i</sup> tot ¼ iωμ<sup>0</sup> �<sup>1</sup> <sup>∇</sup>�Eð Þ<sup>i</sup> tot, and considering that the magnetic field may be written as Hð Þ<sup>i</sup> tot <sup>¼</sup> <sup>H</sup>ð Þ<sup>i</sup> ð Þ<sup>x</sup> exp ikyy <sup>þ</sup> ikzz � <sup>i</sup>ω<sup>t</sup> , we finally obtain the following expression for the variation of the field along the x direction, namely:

$$\begin{split} \omega \mu\_{0} \mathbf{H}^{(i)}(\mathbf{x}) &= A\_{\text{el}} \hat{\mathbf{c}}\_{\text{el}} \exp\left[i \mathbf{k}\_{\text{el}}(\mathbf{x} - \mathbf{x}\_{i})\right] + B\_{\text{el}} \hat{d}\_{\text{el}} \exp\left[-i \mathbf{k}\_{\text{el}}(\mathbf{x} - \mathbf{x}\_{i})\right] \\ &+ A\_{\text{el}} \hat{\mathbf{c}}\_{\text{el}} \exp\left[i \mathbf{k}\_{\text{el}}(\mathbf{x} - \mathbf{x}\_{i})\right] + B\_{\text{el}} \hat{d}\_{\text{el}} \exp\left[-i \mathbf{k}\_{\text{el}}(\mathbf{x} - \mathbf{x}\_{i})\right]. \end{split} \tag{13}$$

In the previous equations, the new vector fields are set as

$$\hat{\mathfrak{c}}\_{oi} = -\left(k\_{oi}k\_z, k\_yk\_z, k\_z^2 - \epsilon\_{xi}k\_0^2\right),\tag{14a}$$

$$
\hat{\epsilon}\_{\rm ef} = \epsilon\_{\rm xi} k\_0^2 \langle -k\_{\rm y}, k\_{\rm ei}, 0 \rangle,\tag{14b}
$$

$$\hat{d}\_{oi} = -\left(-k\_{oi}k\_z, k\_yk\_z, k\_z^2 - \epsilon\_{xi}k\_0^2\right),\tag{14c}$$

$$
\hat{d}\_{\text{ei}} = -\epsilon\_{\text{xi}} k\_0^2 \{ k\_{\text{y}}, k\_{\text{ei}}, \mathbf{0} \}. \tag{14d}
$$

We point out that the vector fields ^coi and ^doi can be given in units of k<sup>2</sup> <sup>0</sup>, whereas ^cei and ^dei can be expressed in units of k<sup>3</sup> 0.

#### 2.3.2. Boundary conditions for anisotropic layered media

Once we have a complete description of the wave fields in every elementary layer of our metamaterial, we have to impose some boundary conditions at the interfaces x ¼ xi. The components of the electric field Eð Þ<sup>i</sup> tot and the magnetic field <sup>H</sup>ð Þ<sup>i</sup> tot lying on the planes x ¼ xi must be continuous. Particularly these boundary conditions may be expressed in a matrix form as

$$\mathbf{D}\_i \mathbf{v}\_i = \mathbf{D}\_{i+1} \mathbf{v}'\_{i+1}.\tag{15}$$

We defined the following matrix, explicitly given as

$$\mathbf{D}\_{i} = \begin{bmatrix} k\_{oi} & -k\_{oi} & k\_{y}k\_{z} & k\_{y}k\_{z} \\ 0 & 0 & k\_{z}^{2} - \epsilon\_{xi}k\_{0}^{2} & k\_{z}^{2} - \epsilon\_{xi}k\_{0}^{2} \\ -k\_{y}k\_{z} & -k\_{y}k\_{z} & \epsilon\_{xi}k\_{0}^{2}k\_{ci} & -\epsilon\_{xi}k\_{0}^{2}k\_{ci} \\ -k\_{z}^{2} + \epsilon\_{xi}k\_{0}^{2} & -k\_{z}^{2} + \epsilon\_{xi}k\_{0}^{2} & 0 & 0 \end{bmatrix}. \tag{16}$$

On the other hand, we introduced the amplitude column vectors

$$\mathbf{v}\_{i} = \begin{bmatrix} A\_{oi} \\ B\_{oi} \\ A\_{ci} \\ B\_{ei} \end{bmatrix}, \mathbf{v}'\_{i} = \begin{bmatrix} A'\_{oi} \\ B'\_{oi} \\ A'\_{ci} \\ B'\_{ei} \end{bmatrix}. \tag{17}$$

The matrix formulation can also be used to relate the amplitude vector v<sup>i</sup> with zero-phase shift at x ¼ xi with the amplitude vector v<sup>0</sup> <sup>i</sup> exhibiting zero dephase at x ¼ xi�1, previously established in Eq. (12a), (12b). For that purpose, we introduce the propagation matrix Pi, which takes into account the amplitude dephasing between the boundaries of each layer. Explicitly we may write

$$\mathbf{v}'\_i = \mathbf{P}\_i \cdot \mathbf{v}\_{i\prime} \tag{18}$$

being

waves. Note that all these amplitudes have zero dephase at x ¼ xi. Finally, the vectors ^aoi and

In the case of counter-propagating waves, we take into consideration that kxi ¼ �koi for ordinary waves and kxi ¼ �kei for extraordinary waves. This fact leads us to introduce the vector

The vector fields ^aoi and ^boi can be given in units of k0, whereas ^aei and ^bei can be expressed in

<sup>z</sup> � <sup>E</sup>xik<sup>2</sup> 0

> <sup>z</sup> � Exik 2 0

qi and Bqi and B<sup>0</sup>

tot <sup>¼</sup> <sup>H</sup>ð Þ<sup>i</sup> ð Þ<sup>x</sup> exp ikyy <sup>þ</sup> ikzz � <sup>i</sup>ω<sup>t</sup> , we finally obtain the following expression for

<sup>þ</sup> Aei^cei exp ikeið Þ <sup>x</sup> � xi ½ �þ Bei^dei exp �ikeið Þ <sup>x</sup> � xi ½ �:

2 <sup>z</sup> � <sup>E</sup>xik<sup>2</sup> 0

qi ¼ Aqi exp �ikqið Þ xi � xi�<sup>1</sup>

qi ¼ Bqi exp ikqið Þ xi � xi�<sup>1</sup>

For completeness we calculate the magnetic field in a given elementary layer "i." By using the

ωμ0Hð Þ<sup>i</sup> ð Þ¼ <sup>x</sup> Aoi^coi exp ikoið Þ <sup>x</sup> � xi ½ �þ Boi^doi exp �ikoið Þ <sup>x</sup> � xi ½ �

^aei <sup>¼</sup> keikz; kykz; <sup>k</sup><sup>2</sup>

^bei ¼ �keikz; kykz; <sup>k</sup><sup>2</sup>

For convenience, the field function Eð Þ<sup>i</sup> can be set in terms of the wave amplitudes A<sup>0</sup>

oi^aoi exp ½ikoið Þ x � xi�<sup>1</sup> � þ B<sup>0</sup>

ei^aei exp ½ikeið Þ x � xi�<sup>1</sup> � þ B<sup>0</sup>

<sup>∇</sup>�Eð Þ<sup>i</sup>

^coi ¼ � koikz; kykz; k

2

^cei ¼ Exik

ei with zero dephase at x ¼ xi�1, namely:

A0

tot ¼ iωμ<sup>0</sup> �<sup>1</sup>

the variation of the field along the x direction, namely:

In the previous equations, the new vector fields are set as

B0

<sup>E</sup>ð Þ<sup>i</sup> ð Þ¼ <sup>x</sup> <sup>A</sup><sup>0</sup>

The complete set of amplitudes is Aqi, A<sup>0</sup>

þ A<sup>0</sup>

^aoi ¼ �ky; koi; <sup>0</sup> , (9a)

: (9b)

^boi ¼ �ky; �koi; <sup>0</sup> , (10a)

: (10b)

oi^boiþ<sup>1</sup> exp ½ � �ikoið Þ <sup>x</sup> � xi�<sup>1</sup>

, (12a)

: (12b)

tot, and considering that the magnetic field may be

, (14a)

<sup>0</sup> �ky; kei; <sup>0</sup> , (14b)

ei^bei exp ½ � �ikeið Þ <sup>x</sup> � xi�<sup>1</sup> :

oi, A<sup>0</sup>

qi where q ¼ f g o;e satisfies the

ei and

(11)

(13)

^aei are rewritten as:

48 Surface Waves - New Trends and Developments

fields

units of k<sup>2</sup>

oi and B<sup>0</sup>

B0

0.

following relationships:

Maxwell's equation Hð Þ<sup>i</sup>

written as Hð Þ<sup>i</sup>

$$\mathbf{P}\_i = \begin{bmatrix} e^{-ik\_i w\_i} & 0 & 0 & 0 \\ 0 & e^{ik\_i w\_i} & 0 & 0 \\ 0 & 0 & e^{-ik\_i w\_i} & 0 \\ 0 & 0 & 0 & e^{ik\_i w\_i} \end{bmatrix} \tag{19}$$

where wi ¼ xi � xi�<sup>1</sup> denotes the width of the elementary layer "i:" We note that equivalent matrix formulations for anisotropic multilayered media can be found elsewhere [6, 7].

#### 2.3.3. Electromagnetic fields in layered isotropic media

At this point, once we have described the electromagnetic fields in uniaxial media, let us study a multilayered media composed of isotropic materials. Considering an isotropic medium of relative permittivity E<sup>i</sup> , once again, the projection of the wave vector along the positive x direction is set as

$$k\_{\rm TEi} = k\_{\rm TMi} = \sqrt{\epsilon\_i k\_0^2 - \left(k\_y^2 + k\_z^2\right)}.\tag{20}$$

In the previous equation we introduced the vector fields

Note that ^cTMi ¼ Eik

tudes of TM<sup>x</sup> waves.

2

2.3.4. Application of the boundary conditions

In the previous matrix equation, we introduced the element

v<sup>i</sup> ¼

we introduce the propagation matrix for the TE<sup>x</sup> and TM<sup>x</sup> modes

P<sup>i</sup> ¼

D<sup>i</sup> ¼

Finally, the amplitude vectors now are represented as

at x ¼ xi with the amplitude vector v<sup>0</sup>

tions in the following matrix form:

^aTEi <sup>¼</sup> ^bTEi <sup>¼</sup> <sup>0</sup>; kz; �ky

2 <sup>y</sup> þ k 2

<sup>0</sup>^aTEi. We conclude that the amplitudes ATEi and B<sup>0</sup>

At a given interface x ¼ xi, the electromagnetic fields should accomplish the continuity boundary conditions. Note that we can write the four equations derived from the boundary condi-

Div<sup>i</sup> ¼ D<sup>i</sup>þ<sup>1</sup>v<sup>0</sup>

kykTEi �kykTEi <sup>E</sup>ik<sup>2</sup>

kzkTEi �kzkTEi �Eik<sup>2</sup>

3 7 7 7 5, v0 <sup>i</sup> ¼

The matrix formulation can also be used to relate the amplitude vector v<sup>i</sup> with zero-phase shift

e�ikTEiwi 000 0 eikTEiwi 0 0 0 0 e�ikTMiwi 0 000 eikTMiwi

ATEi BTEi ATMi BTMi

kz kz �kykTMi kykTMi �ky �ky �kzkTMi kzkTMi

<sup>0</sup>kz <sup>E</sup>ik<sup>2</sup>

<sup>0</sup>ky �Eik

A0 TEi B0 TEi A0 TMi B0 TMi

0kz

2 0ky

<sup>i</sup> exhibiting zero dephase at x ¼ xi�1. For that purpose,

2 <sup>y</sup> � <sup>k</sup><sup>2</sup>

2

<sup>0</sup> 0; kz; �ky

^aTMi ¼ �^cTEi ¼� �k

^bTMi ¼ �^dTEi <sup>¼</sup> <sup>k</sup>

TE-polarized waves along the x-axis, that is, TE<sup>x</sup> waves, and ATMi and B<sup>0</sup>

^cTMi <sup>¼</sup> ^dTMi <sup>¼</sup> <sup>E</sup>ik

� �, (24a)

http://dx.doi.org/10.5772/intechopen.74126

<sup>z</sup> ; kykTEi; kzkTEi � �, (24b)

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

<sup>z</sup> ; kykTEi; kzkTEi � �, (24c)

� �: (24d)

<sup>i</sup>þ<sup>1</sup>: (25)

TEi are associated with

: (26)

: (27)

5, (28)

TMi are field ampli-

51

Formally, kTMi applies to TM-polarized waves and kTEi corresponds to TE-polarized waves. The total electric field of the elementary layer "i" can be set, again, as given in Eq. (7). The part of the electric field that varies along with the spatial coordinate x can be written using the amplitude ATEi (and ATMi), which corresponds to propagating TE (and TM) waves and BTEi (and BTMi) that is related with counter-propagating TE (and TM) waves. This finally reads as

$$\begin{split} \mathbf{E}^{(i)}(\mathbf{x}) &= A\_{\text{TEi}} \hat{a}\_{\text{TEi}} \exp\left[i\mathbf{k}\_{\text{TEi}}(\mathbf{x} - \mathbf{x}\_{i})\right] + B\_{\text{TEi}} \hat{b}\_{\text{TEi}} \exp\left[-i\mathbf{k}\_{\text{TEi}}(\mathbf{x} - \mathbf{x}\_{i})\right] \\ &+ A\_{\text{TMi}} \hat{a}\_{\text{TMi}} \exp\left[i\mathbf{k}\_{\text{TM}}(\mathbf{x} - \mathbf{x}\_{i})\right] + B\_{\text{TMi}} \hat{b}\_{\text{TMi}} \exp\left[-i\mathbf{k}\_{\text{TM}}(\mathbf{x} - \mathbf{x}\_{i})\right]. \end{split} \tag{21}$$

Note that all these amplitudes have zero dephase at x ¼ xi. Finally, the vectors ^aTEi and ^aTMi are given in Eq. (3a), (3b) by ^e<sup>1</sup> and ^e2, respectively, which we rewrite as

$$
\hat{a}\_{TEi} = \bar{b}\_{TEi} = \begin{pmatrix} 0, k\_z, -k\_y \end{pmatrix} \tag{22a}
$$

$$
\hat{a}\_{\rm TM} = \left(k\_y^2 + k\_z^2, -k\_y k\_{\rm TM}, -k\_z k\_{\rm TM}\right),
\tag{22b}
$$

$$
\hat{\boldsymbol{b}}\_{TMi} = \left(\boldsymbol{k}\_y^2 + \boldsymbol{k}\_z^2, \boldsymbol{k}\_y \boldsymbol{k}\_{TMi}, \boldsymbol{k}\_z \boldsymbol{k}\_{TMi}\right). \tag{22c}
$$

In Eq. (22a) and (22c) we have included the field vectors ^bTEi and ^bTMi which are associated with counter-propagating waves. Again, we point out that the vector fields ^aTEi and ^bTEi can be given in units of k0, and the vectors ^aTMi and ^bTMi can be expressed in units of k<sup>2</sup> <sup>0</sup>. Similarly as performed in the previous section, the field function Eð Þ<sup>i</sup> can also be set in terms of the wave amplitudes A<sup>0</sup> TEi, A<sup>0</sup> TEi, B<sup>0</sup> TMi, and B<sup>0</sup> TMi with zero dephase at x ¼ xi�1. As in the previous section, the amplitudes Aqi, A<sup>0</sup> qi, Bqi, and B<sup>0</sup> qi, where q ¼ f g TE; TM , are related by A0 qi ¼ Aqi exp �ikqiwi � � and B<sup>0</sup> qi ¼ Bqi exp ikqiwi � �. Following the same procedure seen earlier, we calculate the magnetic field in every elementary layer "i." We finally obtain the following expression for the variation of the field along the x direction, namely

$$\begin{split} \omega \mu\_{0} \mathbf{H}^{(i)}(\mathbf{x}) &= A\_{T\text{Ei}} \, \hat{c}\_{T\text{Ei}} \exp\left[i\mathbf{k}\_{T\text{Ei}}(\mathbf{x} - \mathbf{x}\_{i})\right] + B\_{T\text{Ei}} \, \hat{d}\_{T\text{Ei}} \exp\left[-i\mathbf{k}\_{T\text{Ei}}(\mathbf{x} - \mathbf{x}\_{i})\right] \\ &+ A\_{T\text{Mi}} \, \hat{c}\_{T\text{Mi}} \exp\left[i\mathbf{k}\_{T\text{Mi}}(\mathbf{x} - \mathbf{x}\_{i})\right] + B\_{T\text{Mi}} \, \hat{d}\_{T\text{Mi}} \exp\left[-i\mathbf{k}\_{T\text{Mi}}(\mathbf{x} - \mathbf{x}\_{i})\right]. \end{split} \tag{23}$$

In the previous equation we introduced the vector fields

where wi ¼ xi � xi�<sup>1</sup> denotes the width of the elementary layer "i:" We note that equivalent

At this point, once we have described the electromagnetic fields in uniaxial media, let us study a multilayered media composed of isotropic materials. Considering an isotropic medium of

> Eik<sup>2</sup> <sup>0</sup> � <sup>k</sup><sup>2</sup>

Formally, kTMi applies to TM-polarized waves and kTEi corresponds to TE-polarized waves. The total electric field of the elementary layer "i" can be set, again, as given in Eq. (7). The part of the electric field that varies along with the spatial coordinate x can be written using the amplitude ATEi (and ATMi), which corresponds to propagating TE (and TM) waves and BTEi (and BTMi) that is related with counter-propagating TE (and TM) waves. This finally reads as

<sup>E</sup>ð Þ<sup>i</sup> ð Þ¼ <sup>x</sup> ATEi ^aTEi exp ikTEið Þ <sup>x</sup> � xi ½ �þ BTEi ^bTEi exp �ikTEið Þ <sup>x</sup> � xi ½ �

Note that all these amplitudes have zero dephase at x ¼ xi. Finally, the vectors ^aTEi and ^aTMi are

^aTEi <sup>¼</sup> ^bTEi <sup>¼</sup> <sup>0</sup>; kz; �ky

In Eq. (22a) and (22c) we have included the field vectors ^bTEi and ^bTMi which are associated with counter-propagating waves. Again, we point out that the vector fields ^aTEi and ^bTEi can be

performed in the previous section, the field function Eð Þ<sup>i</sup> can also be set in terms of the wave

we calculate the magnetic field in every elementary layer "i." We finally obtain the following

<sup>þ</sup> ATMi ^cTMi exp ikTMið Þ <sup>x</sup> � xi <sup>½</sup> � þ BTMi ^dTMi exp �ikTMið Þ <sup>x</sup> � xi ½ �:

ωμ0Hð Þ<sup>i</sup> ð Þ¼ <sup>x</sup> ATEi ^cTEi exp ikTEið Þ <sup>x</sup> � xi ½ �þ BTEi ^dTEi exp �ikTEið Þ <sup>x</sup> � xi ½ �

<sup>þ</sup> ATMi ^aTMi exp ikTMið Þ <sup>x</sup> � xi <sup>½</sup> � þ BTMi ^bTMi exp �ikTMið Þ <sup>x</sup> � xi ½ �:

<sup>z</sup> ; �kykTMi; �kzkTMi � �

<sup>z</sup> ; kykTMi; kzkTMi � �

, once again, the projection of the wave vector along the positive x

<sup>y</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> z

: (20)

� �, (22a)

TMi with zero dephase at x ¼ xi�1. As in the previous

� �. Following the same procedure seen earlier,

qi, where q ¼ f g TE; TM , are related by

, (22b)

: (22c)

<sup>0</sup>. Similarly as

(23)

(21)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r � �

matrix formulations for anisotropic multilayered media can be found elsewhere [6, 7].

kTEi ¼ kTMi ¼

given in Eq. (3a), (3b) by ^e<sup>1</sup> and ^e2, respectively, which we rewrite as

^aTMi ¼ k

TMi, and B<sup>0</sup>

expression for the variation of the field along the x direction, namely

^bTMi <sup>¼</sup> <sup>k</sup>

2 <sup>y</sup> þ k 2

> 2 <sup>y</sup> þ k 2

given in units of k0, and the vectors ^aTMi and ^bTMi can be expressed in units of k<sup>2</sup>

qi, Bqi, and B<sup>0</sup>

qi ¼ Bqi exp ikqiwi

2.3.3. Electromagnetic fields in layered isotropic media

relative permittivity E<sup>i</sup>

50 Surface Waves - New Trends and Developments

direction is set as

amplitudes A<sup>0</sup>

A0

TEi, A<sup>0</sup>

section, the amplitudes Aqi, A<sup>0</sup>

qi ¼ Aqi exp �ikqiwi

TEi, B<sup>0</sup>

� � and B<sup>0</sup>

$$
\hat{a}\_{TEi} = \bar{b}\_{TEi} = \begin{pmatrix} 0, k\_z, -k\_y \end{pmatrix}, \tag{24a}
$$

$$
\hat{a}\_{T\text{Mi}} = -\hat{c}\_{T\text{EI}} = -\left(-k\_y^2 - k\_z^2, k\_y k\_{T\text{EI}}, k\_z k\_{T\text{EI}}\right) \tag{24b}
$$

$$
\hat{\boldsymbol{b}}\_{\rm TMi} = -\hat{\boldsymbol{d}}\_{\rm TEI} = \left(\boldsymbol{k}\_y^2 + \boldsymbol{k}\_z^2, \boldsymbol{k}\_y \boldsymbol{k}\_{\rm TEI}, \boldsymbol{k}\_z \boldsymbol{k}\_{\rm TEI}\right), \tag{24c}
$$

$$
\hat{\boldsymbol{c}}\_{\rm TMi} = \hat{\boldsymbol{d}}\_{\rm TMi} = \epsilon\_i k\_0^2 \{ \mathbf{0}, k\_z, -k\_y \}. \tag{24d}
$$

Note that ^cTMi ¼ Eik 2 <sup>0</sup>^aTEi. We conclude that the amplitudes ATEi and B<sup>0</sup> TEi are associated with TE-polarized waves along the x-axis, that is, TE<sup>x</sup> waves, and ATMi and B<sup>0</sup> TMi are field amplitudes of TM<sup>x</sup> waves.

#### 2.3.4. Application of the boundary conditions

At a given interface x ¼ xi, the electromagnetic fields should accomplish the continuity boundary conditions. Note that we can write the four equations derived from the boundary conditions in the following matrix form:

$$\mathbf{D}\_i \mathbf{v}\_i = \mathbf{D}\_{i+1} \mathbf{v}\_{i+1}'.\tag{25}$$

In the previous matrix equation, we introduced the element

$$\mathbf{D}\_{i} = \begin{bmatrix} k\_{z} & k\_{z} & -k\_{y}k\_{\rm TMi} & k\_{y}k\_{\rm TMi} \\ -k\_{y} & -k\_{y} & -k\_{z}k\_{\rm TMi} & k\_{z}k\_{\rm TMi} \\ k\_{y}k\_{\rm TFi} & -k\_{y}k\_{\rm TFi} & \epsilon\_{i}k\_{0}^{2}k\_{z} & \epsilon\_{i}k\_{0}^{2}k\_{z} \\ k\_{z}k\_{\rm TFi} & -k\_{z}k\_{\rm TFi} & -\epsilon\_{i}k\_{0}^{2}k\_{y} & -\epsilon\_{i}k\_{0}^{2}k\_{y} \end{bmatrix} . \tag{26}$$

Finally, the amplitude vectors now are represented as

$$\mathbf{v}\_{i} = \begin{bmatrix} A\_{T\to i} \\ B\_{T\to i} \\ A\_{T\text{M}i} \\ B\_{T\text{M}i} \end{bmatrix}, \mathbf{v}\_{i}' = \begin{bmatrix} A'\_{T\to i} \\ B'\_{T\to i} \\ A'\_{T\text{M}i} \\ B'\_{T\text{M}i} \end{bmatrix}. \tag{27}$$

The matrix formulation can also be used to relate the amplitude vector v<sup>i</sup> with zero-phase shift at x ¼ xi with the amplitude vector v<sup>0</sup> <sup>i</sup> exhibiting zero dephase at x ¼ xi�1. For that purpose, we introduce the propagation matrix for the TE<sup>x</sup> and TM<sup>x</sup> modes

$$\mathbf{P}\_{i} = \begin{bmatrix} e^{-ik\_{\rm TE}w\_{i}} & 0 & 0 & 0 \\ 0 & e^{ik\_{\rm TE}w\_{i}} & 0 & 0 \\ 0 & 0 & e^{-ik\_{\rm T\&\rm H\&}w\_{i}} & 0 \\ 0 & 0 & 0 & e^{ik\_{\rm T\&\rm H\&}w\_{i}} \end{bmatrix} \tag{28}$$

which takes into account the amplitude phase shift due the finite width wi ¼ xi � xi�<sup>1</sup> of each layer. Explicitly we will write v<sup>0</sup> <sup>i</sup> ¼ P<sup>i</sup> � vi, which is formally the same as Eq. (18), previously derived for uniaxial media.

To accomplish Eq. (30a), the following equation must be satisfied, kTM2=kTM<sup>1</sup> ¼ �E2=E1. This is

vanishing real part and a positive imaginary part, this requires that E2=E<sup>1</sup> < 0, that is, one of the relative permittivities must be negative. Once we have obtained the relationship between the relative permittivities and the purely imaginary wavenumbers, we may obtain a new

Eq. (30b). We may rewrite the dispersion equation, making use of the definition of kTMi given in

r

dispersion relation for TM-bounded modes at frequencies lower than the surface plasmon

<sup>ω</sup>SPP <sup>¼</sup> <sup>ω</sup><sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

At ω ! 0 the SPP wavenumber tends to zero; however, when ω ! ωSPP we find that kSPP ! ∞. In addition, radiative modes may arise at higher frequencies, typically ω ≥ ωp; in such cases, the

wave field is not confined near the interface and it will lose its energy by radiation.

Figure 2. SPP dispersion relation at the interface between a lossless Drude metal (E2ð Þ¼ <sup>ω</sup> <sup>1</sup> � <sup>ω</sup><sup>2</sup>

in blue (E<sup>1</sup> ¼ 1) and GaAs in red (E<sup>1</sup> ¼ 11:55). Note that kp ¼ ωp=c.

1 þ E<sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi E1E<sup>2</sup> E<sup>1</sup> þ E<sup>2</sup>

is the SPP wavenumber. In Figure 2 we represent Eq. (31) giving the

kSPP ¼ k<sup>0</sup>


Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

TM2, which have been derived from

http://dx.doi.org/10.5772/intechopen.74126

53

, (31)

p : (32)

<sup>p</sup>=ω2) and a dielectric: air

the dispersion equation of the TM<sup>x</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 <sup>y</sup> þ k 2 z

q

Eq. (20), resulting:

where kSPP ¼

frequency:

ligature involving the field amplitudes, BTM<sup>1</sup> ¼ ð Þ E2=E<sup>1</sup> A<sup>0</sup>

#### 3. Surface modes in isotropic media

The main purpose of this chapter is the analysis of Dyakonov surface waves, which originally was formulated for an isotropic medium and a uniaxial crystal. However, this analysis is developed in Section 5. Here we introduce the most well-known surface waves arisen at the interface between isotropic media of different dielectric constants. In addition, these surface waves will play a relevant role when dealing with metal-dielectric multilayered structures.

The so-called surface plasmon polaritons are waves that propagate along the surface of a conductor, usually a metal [2, 8, 9]. These are essentially light waves that are trapped on the surface, evanescently confined in the perpendicular direction and caused by their interaction with the free electrons of the conductor, the latter oscillating in resonance with the electromagnetic field. To describe these wave fields, we use the matrix formalism applied in the vicinity of a single interface between two isotropic media with different dielectric permittivities.

Let us consider the propagation of bound waves on the interface between two semi-infinite media, which are denoted as medium 1 and medium 2 with dielectric permittivities E<sup>1</sup> i E2, respectively. This interface is located at x<sup>1</sup> ¼ 0. For medium 1, the electric and magnetic fields varying along the x-axis are given by Eqs. (22a), (22b), (22c) and (24a), (24b), (24c), (24d) respectively.

As we are only interested in bound states; the elements of the remaining field vectors read as

$$\mathbf{v}\_1 = \begin{bmatrix} 0\\B\_{T\mathbb{E}1} \\ 0\\ 0\\ B\_{TM1} \end{bmatrix} \text{and } \mathbf{v}\_2' = \begin{bmatrix} A\_{T\mathbb{E}2}'\\ 0\\ A\_{TM2}'\\ 0 \end{bmatrix}. \tag{29}$$

The values of the amplitudes ATE1, ATM1, B<sup>0</sup> TE2, and B<sup>0</sup> TM<sup>2</sup> are identically zero in case of lack of interaction with external sources, as we assume here. Since we are dealing with bound states, the wavenumbers kTE1, kTM1, kTE2, , and kTM<sup>2</sup> are purely imaginary. The application of the boundary conditions, D1v<sup>1</sup> ¼ D2v<sup>0</sup> 2, gives us the following two equations:

$$0 = \frac{\epsilon\_2 k\_{\rm TM1} + \epsilon\_1 k\_{\rm TM2}}{2\epsilon\_1 k\_{\rm TM1}} A'\_{\rm TM2} \tag{30a}$$

$$B\_{\rm TM1} = \frac{\epsilon\_2 k\_{\rm TM1} - \epsilon\_1 k\_{\rm TM2}}{2\epsilon\_1 k\_{\rm TM1}} A'\_{\rm TM2}. \tag{30b}$$

To accomplish Eq. (30a), the following equation must be satisfied, kTM2=kTM<sup>1</sup> ¼ �E2=E1. This is the dispersion equation of the TM<sup>x</sup> -polarized surface modes. As kTM<sup>1</sup> and kTM<sup>2</sup> have a vanishing real part and a positive imaginary part, this requires that E2=E<sup>1</sup> < 0, that is, one of the relative permittivities must be negative. Once we have obtained the relationship between the relative permittivities and the purely imaginary wavenumbers, we may obtain a new ligature involving the field amplitudes, BTM<sup>1</sup> ¼ ð Þ E2=E<sup>1</sup> A<sup>0</sup> TM2, which have been derived from Eq. (30b). We may rewrite the dispersion equation, making use of the definition of kTMi given in Eq. (20), resulting:

which takes into account the amplitude phase shift due the finite width wi ¼ xi � xi�<sup>1</sup> of each

The main purpose of this chapter is the analysis of Dyakonov surface waves, which originally was formulated for an isotropic medium and a uniaxial crystal. However, this analysis is developed in Section 5. Here we introduce the most well-known surface waves arisen at the interface between isotropic media of different dielectric constants. In addition, these surface waves will play a relevant role when dealing with metal-dielectric multilayered structures.

The so-called surface plasmon polaritons are waves that propagate along the surface of a conductor, usually a metal [2, 8, 9]. These are essentially light waves that are trapped on the surface, evanescently confined in the perpendicular direction and caused by their interaction with the free electrons of the conductor, the latter oscillating in resonance with the electromagnetic field. To describe these wave fields, we use the matrix formalism applied in the vicinity of

Let us consider the propagation of bound waves on the interface between two semi-infinite media, which are denoted as medium 1 and medium 2 with dielectric permittivities E<sup>1</sup> i E2, respectively. This interface is located at x<sup>1</sup> ¼ 0. For medium 1, the electric and magnetic fields varying along the x-axis are given by Eqs. (22a), (22b), (22c) and (24a), (24b), (24c), (24d)

As we are only interested in bound states; the elements of the remaining field vectors read as

and v<sup>0</sup>

TE2, and B<sup>0</sup>

interaction with external sources, as we assume here. Since we are dealing with bound states, the wavenumbers kTE1, kTM1, kTE2, , and kTM<sup>2</sup> are purely imaginary. The application of the

> <sup>0</sup> <sup>¼</sup> <sup>E</sup>2kTM<sup>1</sup> <sup>þ</sup> <sup>E</sup>1kTM<sup>2</sup> 2E1kTM<sup>1</sup>

BTM<sup>1</sup> <sup>¼</sup> <sup>E</sup>2kTM<sup>1</sup> � <sup>E</sup>1kTM<sup>2</sup> 2E1kTM<sup>1</sup>

<sup>2</sup> ¼

A0 TE2 0

: (29)

TM<sup>2</sup> are identically zero in case of lack of

TM2, (30a)

TM2: (30b)

2, gives us the following two equations:

A0

A0

A0 TM2 0

0 BTE<sup>1</sup> 0 BTM<sup>1</sup>

v<sup>1</sup> ¼

The values of the amplitudes ATE1, ATM1, B<sup>0</sup>

boundary conditions, D1v<sup>1</sup> ¼ D2v<sup>0</sup>

a single interface between two isotropic media with different dielectric permittivities.

<sup>i</sup> ¼ P<sup>i</sup> � vi, which is formally the same as Eq. (18), previously

layer. Explicitly we will write v<sup>0</sup>

52 Surface Waves - New Trends and Developments

3. Surface modes in isotropic media

derived for uniaxial media.

respectively.

$$k\_{\rm SPP} = k\_0 \sqrt{\frac{\epsilon\_1 \epsilon\_2}{\epsilon\_1 + \epsilon\_2}}\tag{31}$$

where kSPP ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 <sup>y</sup> þ k 2 z q is the SPP wavenumber. In Figure 2 we represent Eq. (31) giving the dispersion relation for TM-bounded modes at frequencies lower than the surface plasmon frequency:

$$
\omega\_{SPP} = \frac{\omega\_p}{\sqrt{1 + \epsilon\_1}}.\tag{32}
$$

At ω ! 0 the SPP wavenumber tends to zero; however, when ω ! ωSPP we find that kSPP ! ∞. In addition, radiative modes may arise at higher frequencies, typically ω ≥ ωp; in such cases, the wave field is not confined near the interface and it will lose its energy by radiation.

Figure 2. SPP dispersion relation at the interface between a lossless Drude metal (E2ð Þ¼ <sup>ω</sup> <sup>1</sup> � <sup>ω</sup><sup>2</sup> <sup>p</sup>=ω2) and a dielectric: air in blue (E<sup>1</sup> ¼ 1) and GaAs in red (E<sup>1</sup> ¼ 11:55). Note that kp ¼ ωp=c.

#### 4. Multilayered plasmonic lattices

Wave propagation in periodic media can be treated as the motion of electrons in crystalline solids. In fact, formulation of the Kronig-Penney model used in the elementary band theory of solids is mathematically identical to that of the electromagnetic radiation in periodic layered media. Thus, some of the physical concepts used in material physics such as Bloch waves, Brillouin zones, and forbidden bands can also be used here. A periodic layered medium is equivalent to a one-dimensional lattice that is invariant under lattice translation.

Here we will treat the propagation of electromagnetic radiation in a simple periodic layered medium that consists of alternating layers of transparent nonmagnetic materials with different electric permittivities. The layers are set in a way that the x-axis points along the perpendicular direction of the layers. The permittivity profile is given by E1, for x<sup>0</sup> < x < x1, and E2, for x<sup>1</sup> < x < x2. In addition, the relative permittivity satisfies the condition of periodicity, Eð Þ¼ x Eð Þ x þ Λ , where w<sup>1</sup> ¼ x<sup>1</sup> � x<sup>0</sup> ð Þ w<sup>2</sup> ¼ x<sup>2</sup> � x<sup>1</sup> is the thickness of the layers of permittivity E<sup>1</sup> ð Þ E<sup>2</sup> and Λ ¼ w<sup>1</sup> þ w<sup>2</sup> represents the period of the structure.

According to the Floquet theorem, solutions of the wave equation for a periodic medium may be set in the form <sup>E</sup>ð Þ <sup>i</sup>þ<sup>2</sup> ð Þ¼ <sup>x</sup> <sup>þ</sup> <sup>Λ</sup> <sup>E</sup>ð Þ<sup>i</sup> ð Þ� <sup>x</sup> exp ð Þ iK<sup>Λ</sup> . The constant <sup>K</sup> is known as the Bloch wavenumber. The problem is thus that of determining <sup>K</sup> and <sup>E</sup>ð Þ<sup>i</sup> ð Þ<sup>x</sup> . Finally, the equation

$$\cos\left(K\_{\emptyset}\Lambda\right) = \frac{1}{2}\operatorname{tr}\_{\emptyset}.\tag{33}$$

metal width, some deviations are evident. Reaching a given value of wm, a large band gap centered at kt ¼ 0 surges. Particularly for TM modes, at KTM ¼ 0, the two curves collapse. This fact may be understood as that the symmetric and antisymmetric surface modes in the metallic layer approach for SPP in a single metal-dielectric interface, giving kt ! kSPP [see Eq. (31)]. For illustration, the wavenumber of the SPP propagating on the interface of our Drude metal/GaAs materials yields kSPP ¼ 0:746 � 2π=Λ, assuming that Λ ¼ 325 nm (associated with the period of

Figure 3. Exact dispersion curves derived from Eq. (33) for a lossless Drude metal/GaAs composite medium with <sup>λ</sup> <sup>¼</sup> <sup>1</sup>:55<sup>μ</sup> m, <sup>ω</sup><sup>p</sup> <sup>¼</sup> <sup>13</sup>:94 fs�<sup>1</sup> and <sup>E</sup><sup>1</sup> <sup>¼</sup> <sup>11</sup>:55, for (a) TM modes and (b) TE modes. GaAs layer is always 12 times thicker

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For near-infrared and visible wavelengths, nanolayered metal-dielectric compounds enable a simplified description of the medium by using the long-wavelength approximation, which involves a homogenization of the structured metamaterial [10, 11]. The effective medium approach (EMA), as Rytov exposed in his seminal paper [12], involves representing MD multilayered metamaterial as an uniaxial plasmonic crystal, whose optical axis is normal to the layers (in our case, the x-axis is the optical axis), a procedure that requires the metallic elements to have a size of a few nanometers. This is caused by the fact that transparency of noble metals is restrained to a propagation distance not surpassing the metal skin depth. In Ref. to this point, recent development of nanofabrication technology makes it possible to create such subwavelength structures. Under this condition, the plasmonic lattice behaves as a uniaxial crystal

characterized by a relative permittivity tensor e ¼ E∥ð Þþ x ⊗ x E⊥ð Þ y ⊗ y þ z ⊗ z , where

<sup>E</sup><sup>∥</sup> <sup>¼</sup> <sup>E</sup>1E<sup>2</sup>

ð Þ 1 � f E<sup>2</sup> þ fE<sup>1</sup>

gives the permittivity along the optical axis, and E<sup>⊥</sup> ¼ ð Þ 1 � f E<sup>1</sup> þ fE<sup>2</sup> corresponds to the permittivity in the transversal direction. In the previous equations, f ¼ w2=ð Þ w<sup>1</sup> þ w<sup>2</sup> , denotes the

, (36)

a multilayer with metal thickness wm ¼ 25 nm).

4.1. Effective medium approach

than the metallic layer.

represents the dispersion relation for TE and TM modes, written in a compact way. Representing KTE (KTM), the Bloch wavenumber K associated with the mode TE (TM), and writing

$$\text{tr}\_{\text{TE}} = 2 \cos \left( w\_1 k\_{\text{TE1}} \right) \cos \left( w\_2 k\_{\text{TE2}} \right) - \frac{\left( k\_{\text{TE1}}^2 + k\_{\text{TE2}}^2 \right)}{k\_{\text{TE1}} k\_{\text{TE2}}} \sin \left( w\_1 k\_{\text{TE1}} \right) \sin \left( w\_2 k\_{\text{TE2}} \right), \tag{34a}$$

$$\text{tr}\_{\text{TM}} = 2\cos\left(w\_1 k\_{\text{TM1}}\right)\cos\left(w\_2 k\_{\text{TM2}}\right) - \frac{\left(\epsilon\_2^2 k\_{\text{TM1}}^2 + \epsilon\_1^2 k\_{\text{TM2}}^2\right)}{\epsilon\_1 \epsilon\_2 k\_{\text{TM1}} k\_{\text{TM2}}} \sin\left(w\_1 k\_{\text{TM1}}\right)\sin\left(w\_2 k\_{\text{TM2}}\right). \tag{34b}$$

we find the dispersion equation of a binary periodic medium for each polarization.

Neglecting losses in the materials, regimes where ∣trq∣ < 2 correspond to real Kq and thus to propagating Bloch waves, when ∣trq∣ > 2; however, Kq ¼ mπ=Λ þ iKqi, where m is an integer and Kqi is the imaginary part of Kq, which gives an evanescent behavior to the Bloch wave. These are the so-called forbidden bands of the periodic medium. The band edges are set for ∣trq∣ ¼ 2.

In Figure 3, the transverse wavenumber reads as

$$k\_t = \sqrt{k\_y^2 + k\_z^2}.\tag{35}$$

For ultra-thin metallic layers, for instance, wm ¼ 3 nm, the curves resembles ellipses and circumferences for TM and TE modes, respectively. In the particular case of TM modes, a secondary curve surges, in relation with the excitation of SPPs. For increasing values of the Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge http://dx.doi.org/10.5772/intechopen.74126 55

Figure 3. Exact dispersion curves derived from Eq. (33) for a lossless Drude metal/GaAs composite medium with <sup>λ</sup> <sup>¼</sup> <sup>1</sup>:55<sup>μ</sup> m, <sup>ω</sup><sup>p</sup> <sup>¼</sup> <sup>13</sup>:94 fs�<sup>1</sup> and <sup>E</sup><sup>1</sup> <sup>¼</sup> <sup>11</sup>:55, for (a) TM modes and (b) TE modes. GaAs layer is always 12 times thicker than the metallic layer.

metal width, some deviations are evident. Reaching a given value of wm, a large band gap centered at kt ¼ 0 surges. Particularly for TM modes, at KTM ¼ 0, the two curves collapse. This fact may be understood as that the symmetric and antisymmetric surface modes in the metallic layer approach for SPP in a single metal-dielectric interface, giving kt ! kSPP [see Eq. (31)]. For illustration, the wavenumber of the SPP propagating on the interface of our Drude metal/GaAs materials yields kSPP ¼ 0:746 � 2π=Λ, assuming that Λ ¼ 325 nm (associated with the period of a multilayer with metal thickness wm ¼ 25 nm).

#### 4.1. Effective medium approach

4. Multilayered plasmonic lattices

54 Surface Waves - New Trends and Developments

Wave propagation in periodic media can be treated as the motion of electrons in crystalline solids. In fact, formulation of the Kronig-Penney model used in the elementary band theory of solids is mathematically identical to that of the electromagnetic radiation in periodic layered media. Thus, some of the physical concepts used in material physics such as Bloch waves, Brillouin zones, and forbidden bands can also be used here. A periodic layered medium is

Here we will treat the propagation of electromagnetic radiation in a simple periodic layered medium that consists of alternating layers of transparent nonmagnetic materials with different electric permittivities. The layers are set in a way that the x-axis points along the perpendicular direction of the layers. The permittivity profile is given by E1, for x<sup>0</sup> < x < x1, and E2, for x<sup>1</sup> < x < x2. In addition, the relative permittivity satisfies the condition of periodicity, Eð Þ¼ x Eð Þ x þ Λ , where w<sup>1</sup> ¼ x<sup>1</sup> � x<sup>0</sup> ð Þ w<sup>2</sup> ¼ x<sup>2</sup> � x<sup>1</sup> is the thickness of the layers of permittivity

According to the Floquet theorem, solutions of the wave equation for a periodic medium may be set in the form <sup>E</sup>ð Þ <sup>i</sup>þ<sup>2</sup> ð Þ¼ <sup>x</sup> <sup>þ</sup> <sup>Λ</sup> <sup>E</sup>ð Þ<sup>i</sup> ð Þ� <sup>x</sup> exp ð Þ iK<sup>Λ</sup> . The constant <sup>K</sup> is known as the Bloch wavenumber. The problem is thus that of determining <sup>K</sup> and <sup>E</sup>ð Þ<sup>i</sup> ð Þ<sup>x</sup> . Finally, the equation

cos Kq<sup>Λ</sup> � � <sup>¼</sup> <sup>1</sup>

represents the dispersion relation for TE and TM modes, written in a compact way. Representing

k 2 TE1 þ k 2 TE2 � � kTE1kTE2

Neglecting losses in the materials, regimes where ∣trq∣ < 2 correspond to real Kq and thus to propagating Bloch waves, when ∣trq∣ > 2; however, Kq ¼ mπ=Λ þ iKqi, where m is an integer and Kqi is the imaginary part of Kq, which gives an evanescent behavior to the Bloch wave. These are the so-called forbidden bands of the periodic medium. The band edges are set for ∣trq∣ ¼ 2.

� � E1E2kTM1kTM<sup>2</sup>

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 <sup>y</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> z

KTE (KTM), the Bloch wavenumber K associated with the mode TE (TM), and writing

E2 2k 2 TM<sup>1</sup> <sup>þ</sup> <sup>E</sup><sup>2</sup> 1k 2 TM2

we find the dispersion equation of a binary periodic medium for each polarization.

kt ¼

q

For ultra-thin metallic layers, for instance, wm ¼ 3 nm, the curves resembles ellipses and circumferences for TM and TE modes, respectively. In the particular case of TM modes, a secondary curve surges, in relation with the excitation of SPPs. For increasing values of the

2

trq: (33)

sin ð Þ w1kTE1 sin ð Þ w2kTE2 , (34a)

sin ð Þ w1kTM<sup>1</sup> sin ð Þ w2kTM<sup>2</sup> : (34b)

: (35)

equivalent to a one-dimensional lattice that is invariant under lattice translation.

E<sup>1</sup> ð Þ E<sup>2</sup> and Λ ¼ w<sup>1</sup> þ w<sup>2</sup> represents the period of the structure.

trTE ¼ 2 cos ð Þ w1kTE1 cos ð Þ� w2kTE2

trTM ¼ 2 cos ð Þ w1kTM1 cos ð Þ� w2kTM<sup>2</sup>

In Figure 3, the transverse wavenumber reads as

For near-infrared and visible wavelengths, nanolayered metal-dielectric compounds enable a simplified description of the medium by using the long-wavelength approximation, which involves a homogenization of the structured metamaterial [10, 11]. The effective medium approach (EMA), as Rytov exposed in his seminal paper [12], involves representing MD multilayered metamaterial as an uniaxial plasmonic crystal, whose optical axis is normal to the layers (in our case, the x-axis is the optical axis), a procedure that requires the metallic elements to have a size of a few nanometers. This is caused by the fact that transparency of noble metals is restrained to a propagation distance not surpassing the metal skin depth. In Ref. to this point, recent development of nanofabrication technology makes it possible to create such subwavelength structures. Under this condition, the plasmonic lattice behaves as a uniaxial crystal characterized by a relative permittivity tensor e ¼ E∥ð Þþ x ⊗ x E⊥ð Þ y ⊗ y þ z ⊗ z , where

$$
\epsilon\_{\parallel} = \frac{\epsilon\_1 \epsilon\_2}{(1 - f)\epsilon\_2 + f\epsilon\_1} \tag{36}
$$

gives the permittivity along the optical axis, and E<sup>⊥</sup> ¼ ð Þ 1 � f E<sup>1</sup> þ fE<sup>2</sup> corresponds to the permittivity in the transversal direction. In the previous equations, f ¼ w2=ð Þ w<sup>1</sup> þ w<sup>2</sup> , denotes the filling factor of medium "2" providing the metal rate in a unit cell. The dispersion equations given by the EMA are k 2 <sup>x</sup> þ k 2 <sup>t</sup> ¼ E⊥k 2 <sup>0</sup>, corresponding to TE (o-) waves, where kx represents the Bloch wavenumber KTE, and for TM (e-) waves, we have

$$\frac{k\_\pi^2}{\epsilon\_\perp} + \frac{k\_t^2}{\epsilon\_\parallel} = k\_{0\prime}^2\tag{37}$$

In Figure 4(a), we represent the permittivities E<sup>∥</sup> and E<sup>⊥</sup> of our plasmonic crystals for a wide range of frequencies. Note that the metal-filling factor takes control on the dissipative effects in the metamaterial; accordingly low values of f are of great convenience. We set f ¼ 1=4 in our numerical simulations. For low frequencies, Ω ≪ 1, the following approximations can be used: <sup>E</sup><sup>⊥</sup> <sup>≈</sup> <sup>f</sup>E<sup>2</sup> <sup>&</sup>lt; 0 and 0 <sup>&</sup>lt; <sup>E</sup><sup>∥</sup> <sup>≈</sup> <sup>E</sup>1=ð Þ <sup>1</sup> � <sup>f</sup> . Therefore, propagating TE<sup>x</sup> modes (Ex <sup>¼</sup> 0) cannot exist in the bulk crystal since it behaves like a metal in these circumstances. On the other hand, TM<sup>x</sup> waves propagate following the spatial dispersion curve of Eq. (37). This is a characteristic of Type II hyperbolic media. As mentioned earlier, Eq. (37) denotes a hyperboloid of one sheet

Figure 4. (a) Variation of relative permittivities E<sup>∥</sup> (blue solid line) and E<sup>⊥</sup> (magenta solid line) as a function of normalized frequency Ω, for the plasmonic crystal including a lossless Drude metal and fused silica (E<sup>1</sup> ¼ 2:25) as dielectric material. Ω<sup>1</sup> and Ω<sup>2</sup> yield 0.359 and 0.759, respectively. (b) and (c) plot Eq. (37) in the kxky plane for extraordinary waves (TM<sup>x</sup> modes) for a plasmonic effective crystal, including a dielectric of permittivity E<sup>1</sup> ¼ 2:25, in the range Ω < 1. Solid line corresponds to kz ¼ 0 and shaded regions are associated with harmonic waves with kz > 0 (nonevanescent fields).

<sup>Ω</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

for which E<sup>⊥</sup> ¼ 0. For slightly higher frequencies, both E<sup>∥</sup> and E<sup>⊥</sup> are positive and Eq. (37)

simulates a uniaxial medium with positive birefringence. Raising the frequency even more, E<sup>∥</sup>

<sup>Ω</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

leading to the so-called canalization regime. In general, Ω<sup>1</sup> < Ω<sup>2</sup> provided that f < 1=2. Beyond Ω2, Eq. (37) turns to a hyperboloidal shape. In the range Ω<sup>2</sup> < Ω < 1; however, the dispersion curve has two sheets (Type I hyperbolic medium). Figure 4(c) illustrates this case. Note that the upper limit of this hyperbolic band is determined by the condition E<sup>∥</sup> ¼ 0 or in an

<sup>1</sup> <sup>þ</sup> <sup>E</sup>1ð Þ <sup>1</sup> � <sup>f</sup> <sup>=</sup><sup>f</sup> <sup>p</sup> , (38)

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E⊥

<sup>1</sup> <sup>þ</sup> <sup>E</sup>1<sup>f</sup> <sup>=</sup>ð Þ <sup>1</sup> � <sup>f</sup> <sup>p</sup> , (39)

p , the periodic multilayer

(see Figure 4(b) for Ω ¼ 0:20).

diverges at

Furthermore, the hyperbolic dispersion exists up to a frequency

becomes an ellipsoid of revolution. Since its minor semi-axis is Ω ffiffiffiffiffi

equivalent way, E<sup>2</sup> ¼ 0, occurring at the plasma frequency.

being now kx, the Bloch wavenumber KTM.

The validity of the EMA is related on the assumption that the period Λ is much shorter than the wavelength, that is, Λ ≪ λ0. Apparently, Eq. (33) is in good agreement with the EMA in the vicinity of kx <sup>¼</sup> 0 for TM<sup>x</sup> waves only. In contrast, propagation along the <sup>x</sup>-axis, where ky ¼ kz ¼ 0, results in large discrepancies. Even small-filling factors of the metallic composite lead to enormous birefringences. Such metamaterials enlarge the birefringence of the effectiveuniaxial crystal at least in one order of magnitude in comparison with values shown in Table 1. However, the size of birefringence displayed by extraordinary waves is reduced if w<sup>2</sup> increases. On the other hand, the isotropy of the isofrequency curve is practically conserved for ordinary waves.

#### 4.2. Hyperbolic media

As we have seen in Section 3.3, nanolayered metal-dielectric compounds behave like plasmonic crystals enabling a simplified description of the medium by using the longwavelength approximation [10–12]. Under certain conditions, the permittivity of the medium set in the form of a second-rank tensor includes elements of opposite signs, leading to a metamaterial of extreme anisotropy [13, 14]. This class of nanostructured media with hyperbolic dispersion is promising metamaterials with a plethora of practical applications from biosensing to fluorescence engineering [15].

Type I hyperbolic media refers to a special kind of uniaxially anisotropic media, that can be described by a permittivity tensor where element E<sup>∥</sup> is negative and E<sup>⊥</sup> is positive. In this case, Eq. (37) leads to a two-sheet hyperboloid. Type II hyperbolic media lead to positive E<sup>∥</sup> and negative E⊥, and Eq. (37) gives us a one-sheet hyperboloid [16]. The fulfillment of hyperbolic dispersion allows wave propagation over a wide spatial spectrum that would be evanescent in an ordinary isotropic dielectric. At the optical range, hyperbolic media can be manufactured with metal-dielectric multilayers or metallic nanowires. Multilayered hyperbolic metamaterials at an optical range take advantage of the wide frequency band in which metals exhibit negative permittivity and support plasmonic modes.

The system under analysis is a periodic binary medium, where we take as medium 1 a transparent dielectric medium that is ideally nondispersive. In our three numerical simulations we take a lossless Drude metal where its permittivity is <sup>E</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>Ω</sup>�<sup>2</sup> and dielectric media with permittivities: (a) E<sup>1</sup> ¼ 1, (b) E<sup>1</sup> ¼ 2:25 and (c) E<sup>1</sup> ¼ 11:55. Note that frequencies can be expressed in units of the plasma frequency, Ω ¼ ω=ωp.

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge http://dx.doi.org/10.5772/intechopen.74126 57

Figure 4. (a) Variation of relative permittivities E<sup>∥</sup> (blue solid line) and E<sup>⊥</sup> (magenta solid line) as a function of normalized frequency Ω, for the plasmonic crystal including a lossless Drude metal and fused silica (E<sup>1</sup> ¼ 2:25) as dielectric material. Ω<sup>1</sup> and Ω<sup>2</sup> yield 0.359 and 0.759, respectively. (b) and (c) plot Eq. (37) in the kxky plane for extraordinary waves (TM<sup>x</sup> modes) for a plasmonic effective crystal, including a dielectric of permittivity E<sup>1</sup> ¼ 2:25, in the range Ω < 1. Solid line corresponds to kz ¼ 0 and shaded regions are associated with harmonic waves with kz > 0 (nonevanescent fields).

In Figure 4(a), we represent the permittivities E<sup>∥</sup> and E<sup>⊥</sup> of our plasmonic crystals for a wide range of frequencies. Note that the metal-filling factor takes control on the dissipative effects in the metamaterial; accordingly low values of f are of great convenience. We set f ¼ 1=4 in our numerical simulations. For low frequencies, Ω ≪ 1, the following approximations can be used: <sup>E</sup><sup>⊥</sup> <sup>≈</sup> <sup>f</sup>E<sup>2</sup> <sup>&</sup>lt; 0 and 0 <sup>&</sup>lt; <sup>E</sup><sup>∥</sup> <sup>≈</sup> <sup>E</sup>1=ð Þ <sup>1</sup> � <sup>f</sup> . Therefore, propagating TE<sup>x</sup> modes (Ex <sup>¼</sup> 0) cannot exist in the bulk crystal since it behaves like a metal in these circumstances. On the other hand, TM<sup>x</sup> waves propagate following the spatial dispersion curve of Eq. (37). This is a characteristic of Type II hyperbolic media. As mentioned earlier, Eq. (37) denotes a hyperboloid of one sheet (see Figure 4(b) for Ω ¼ 0:20).

Furthermore, the hyperbolic dispersion exists up to a frequency

filling factor of medium "2" providing the metal rate in a unit cell. The dispersion equations

The validity of the EMA is related on the assumption that the period Λ is much shorter than the wavelength, that is, Λ ≪ λ0. Apparently, Eq. (33) is in good agreement with the EMA in the vicinity of kx <sup>¼</sup> 0 for TM<sup>x</sup> waves only. In contrast, propagation along the <sup>x</sup>-axis, where ky ¼ kz ¼ 0, results in large discrepancies. Even small-filling factors of the metallic composite lead to enormous birefringences. Such metamaterials enlarge the birefringence of the effectiveuniaxial crystal at least in one order of magnitude in comparison with values shown in Table 1. However, the size of birefringence displayed by extraordinary waves is reduced if w<sup>2</sup> increases. On the other hand, the isotropy of the isofrequency curve is practically conserved

As we have seen in Section 3.3, nanolayered metal-dielectric compounds behave like plasmonic crystals enabling a simplified description of the medium by using the longwavelength approximation [10–12]. Under certain conditions, the permittivity of the medium set in the form of a second-rank tensor includes elements of opposite signs, leading to a metamaterial of extreme anisotropy [13, 14]. This class of nanostructured media with hyperbolic dispersion is promising metamaterials with a plethora of practical applications from

Type I hyperbolic media refers to a special kind of uniaxially anisotropic media, that can be described by a permittivity tensor where element E<sup>∥</sup> is negative and E<sup>⊥</sup> is positive. In this case, Eq. (37) leads to a two-sheet hyperboloid. Type II hyperbolic media lead to positive E<sup>∥</sup> and negative E⊥, and Eq. (37) gives us a one-sheet hyperboloid [16]. The fulfillment of hyperbolic dispersion allows wave propagation over a wide spatial spectrum that would be evanescent in an ordinary isotropic dielectric. At the optical range, hyperbolic media can be manufactured with metal-dielectric multilayers or metallic nanowires. Multilayered hyperbolic metamaterials at an optical range take advantage of the wide frequency band in which metals exhibit negative

The system under analysis is a periodic binary medium, where we take as medium 1 a transparent dielectric medium that is ideally nondispersive. In our three numerical simulations we take a lossless Drude metal where its permittivity is <sup>E</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>Ω</sup>�<sup>2</sup> and dielectric media with permittivities: (a) E<sup>1</sup> ¼ 1, (b) E<sup>1</sup> ¼ 2:25 and (c) E<sup>1</sup> ¼ 11:55. Note that frequencies can be

k 2 x E⊥ þ k2 t E∥ <sup>¼</sup> <sup>k</sup><sup>2</sup>

<sup>0</sup>, corresponding to TE (o-) waves, where kx represents the

<sup>0</sup>, (37)

given by the EMA are k

56 Surface Waves - New Trends and Developments

for ordinary waves.

4.2. Hyperbolic media

2 <sup>x</sup> þ k 2 <sup>t</sup> ¼ E⊥k 2

being now kx, the Bloch wavenumber KTM.

biosensing to fluorescence engineering [15].

permittivity and support plasmonic modes.

expressed in units of the plasma frequency, Ω ¼ ω=ωp.

Bloch wavenumber KTE, and for TM (e-) waves, we have

$$
\Omega\_1 = \frac{1}{\sqrt{1 + \epsilon\_1 (1 - f)/f}},\tag{38}
$$

for which E<sup>⊥</sup> ¼ 0. For slightly higher frequencies, both E<sup>∥</sup> and E<sup>⊥</sup> are positive and Eq. (37) becomes an ellipsoid of revolution. Since its minor semi-axis is Ω ffiffiffiffiffi E⊥ p , the periodic multilayer simulates a uniaxial medium with positive birefringence. Raising the frequency even more, E<sup>∥</sup> diverges at

$$
\Omega\_2 = \frac{1}{\sqrt{1 + \epsilon\_1 f/(1 - f)}},
\tag{39}
$$

leading to the so-called canalization regime. In general, Ω<sup>1</sup> < Ω<sup>2</sup> provided that f < 1=2. Beyond Ω2, Eq. (37) turns to a hyperboloidal shape. In the range Ω<sup>2</sup> < Ω < 1; however, the dispersion curve has two sheets (Type I hyperbolic medium). Figure 4(c) illustrates this case. Note that the upper limit of this hyperbolic band is determined by the condition E<sup>∥</sup> ¼ 0 or in an equivalent way, E<sup>2</sup> ¼ 0, occurring at the plasma frequency.

#### 5. Dyakonov surface waves

Dyakonov surface waves (DSWs) are another kind of surface waves, supported at the interface between an optically isotropic medium and a uniaxial-birefringent material. In the original work by Dyakonov (English version was reported in 1988 [3]), the optical axis of the uniaxial medium was assumed in-plane with respect to the interface. This is the case we deal with here. Since we treat the plasmonic lattice as a uniaxial crystal, we may establish analytically the

follow Dyakonov [3] by considering hybrid-polarized surface modes. In the isotropic medium we consider TE<sup>x</sup> (Ex <sup>¼</sup> 0) and TM<sup>x</sup> (Hx <sup>¼</sup> 0) waves whose wave vectors have the same real components ky and kz in the plane x ¼ 0. Therefore the electric field in both media may be set as

Moreover, these fields are evanescent in the isotropic medium and in the superlattice. In the

where the ordinary and extraordinary waves in the effective uniaxial medium decay exponentially with rates given by κ<sup>o</sup> ¼ �iko<sup>1</sup> and κ<sup>e</sup> ¼ �ike1, respectively. Taking the formulation given in Section 2.3.1, the amplitudes Ao<sup>1</sup> and Ae<sup>1</sup> are identically zero. In the isotropic medium

where the evanescent decay for TE and TM modes is κ ¼ �ikTE<sup>2</sup> ¼ �ikTM2. Now the ampli-

Once we have the amplitudes in both sides of the interface, we apply the boundary conditions

A0 TE2 0 A0 TM2 0

of hybrid polarization modes. Using the elements Mij of the matrix Mh, and defining M<sup>i</sup> and M<sup>a</sup> as

� ikzð Þ <sup>κ</sup><sup>o</sup> <sup>þ</sup> <sup>κ</sup> 2 k<sup>2</sup> <sup>z</sup> � <sup>k</sup><sup>2</sup> <sup>0</sup>E<sup>⊥</sup> � �

� ð Þ <sup>κ</sup><sup>e</sup> <sup>þ</sup> <sup>κ</sup> ky 2κ<sup>e</sup> k 2 <sup>z</sup> � k 2 <sup>0</sup>E<sup>⊥</sup>

ikzð Þ κ<sup>o</sup> � κ

ð Þ κ � κ<sup>e</sup> ky

2 k<sup>2</sup> <sup>z</sup> � k 2 <sup>0</sup>E<sup>⊥</sup> � �

2κ<sup>e</sup> k 2 <sup>z</sup> � k 2 <sup>0</sup>E<sup>⊥</sup>

anisotropic medium (x < 0) the evanescent electric amplitude can be written as

TE<sup>2</sup> ^aTE<sup>2</sup> exp ð Þþ ikTE2x A<sup>0</sup>

0 Bo<sup>1</sup> 0 Be<sup>1</sup>

� � in <sup>x</sup> <sup>¼</sup> 0. For that purpose, we

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59

TM<sup>2</sup> ^aTM<sup>2</sup> exp ð Þ ikTM2x , (42)

, (43)

o

o

, (44a)

: (44b)

<sup>E</sup>tot <sup>¼</sup> <sup>E</sup>ð Þ<sup>x</sup> exp ikyy <sup>þ</sup> ikzz � <sup>i</sup>ω<sup>t</sup> � �: (40)

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

<sup>E</sup>ð Þ¼ <sup>x</sup> Bo<sup>1</sup> ^bo<sup>1</sup> exp ð Þþ �iko1<sup>x</sup> Be<sup>1</sup> ^be<sup>1</sup> exp ð Þ �ike1<sup>x</sup> , (41)

2, where D<sup>1</sup> is provided by Eq. (16), D<sup>2</sup> from Eq. (26), v<sup>1</sup> from Eq. (17),

<sup>1</sup> � D<sup>2</sup> establishes a relationship between the amplitudes

<sup>0</sup>kyð Þ Eκ<sup>o</sup> þ κE<sup>⊥</sup>

<sup>0</sup>kyð Þ Eκ<sup>o</sup> � κE<sup>⊥</sup>

� �

<sup>z</sup> � k 2 <sup>0</sup>E<sup>⊥</sup> � �

� �

k2

� � � ikz κκeE<sup>⊥</sup> <sup>þ</sup> <sup>E</sup>κ<sup>2</sup>

k2

� � � ikz κκeE<sup>⊥</sup> � <sup>E</sup>κ<sup>2</sup>

2κ<sup>o</sup> k 2 <sup>z</sup> � k 2 <sup>0</sup>E<sup>⊥</sup> � �

2κeE<sup>⊥</sup> k<sup>2</sup>

2κ<sup>o</sup> k 2 <sup>z</sup> � k 2 <sup>0</sup>E<sup>⊥</sup> � �

2κeE<sup>⊥</sup> k 2 <sup>z</sup> � <sup>k</sup><sup>2</sup> <sup>0</sup>E<sup>⊥</sup> � �

diffraction equation that gives the 2D wave vector k<sup>D</sup> ¼ 0; ky; kz

(x > 0) the amplitude of the electric field is

tudes B<sup>0</sup>

and v<sup>0</sup>

TE<sup>2</sup> and B<sup>0</sup>

at x ¼ 0, D<sup>1</sup> � v<sup>1</sup> ¼ D<sup>2</sup> � v<sup>0</sup>

Eð Þ¼ x A<sup>0</sup>

TM<sup>2</sup> are zero.

where the transmission matrix <sup>M</sup><sup>h</sup> <sup>¼</sup> <sup>D</sup>�<sup>1</sup>

<sup>M</sup><sup>i</sup> <sup>¼</sup> <sup>M</sup><sup>11</sup> <sup>M</sup><sup>13</sup>

<sup>M</sup><sup>a</sup> <sup>¼</sup> <sup>M</sup><sup>21</sup> <sup>M</sup><sup>23</sup>

<sup>M</sup><sup>31</sup> <sup>M</sup><sup>33</sup> � � <sup>¼</sup>

<sup>M</sup><sup>41</sup> <sup>M</sup><sup>43</sup> � � <sup>¼</sup>

<sup>2</sup> from Eq. (27). This equation reduces to:

The importance of DSWs for integrated optical applications, such as sensing and nanowaveguiding, was appreciated in a series of papers [17, 18]. Indeed Dyakonov-like surface waves also emerge in the case that a biaxial crystal [19] or a structurally chiral material [20] takes the place of the uniaxial medium. The case of metal-dielectric (MD) multilayers as structurally anisotropic media is especially convenient since small-filling fractions of the metallic inclusions enable metamaterials with an enormous birefringence, thus enhancing density of DSWs and relaxing their prominent directivity [21–23].

#### 5.1. Dispersion equation of DSWs

The system under study is the plotted in Figure 5, where we have two semi-infinite media, one of them is isotropic and the second one is an MD lattice. In our case, the indices "1" and "2" make reference to the plasmonic lattice and the isotropic medium, respectively. We have previously reported a comprehensive analysis of this case in [4]. As we have seen earlier, the plasmonic lattice can be taken as an effective uniaxial crystal. In this case, the permittivity along its optical axis, Ez<sup>1</sup> ¼ E∥, is given by Eq. (36); also, the permittivity in the transverse direction Ex<sup>1</sup> ¼ Ey<sup>1</sup> ¼ E<sup>⊥</sup> may be appropriately averaged. From hereon, the permittivity E<sup>2</sup> of the isotropic medium in x > 0 will be denoted by E. Note that our analysis serves for natural birefringent materials characterized by permittivities E<sup>∥</sup> and E⊥.

Figure 5. Schematic setup under study, consisting in a semi-infinite dielectric-metal superlattice ð Þ x < 0 and an isotropic substrate ð Þ x > 0 .

Since we treat the plasmonic lattice as a uniaxial crystal, we may establish analytically the diffraction equation that gives the 2D wave vector k<sup>D</sup> ¼ 0; ky; kz � � in <sup>x</sup> <sup>¼</sup> 0. For that purpose, we follow Dyakonov [3] by considering hybrid-polarized surface modes. In the isotropic medium we consider TE<sup>x</sup> (Ex <sup>¼</sup> 0) and TM<sup>x</sup> (Hx <sup>¼</sup> 0) waves whose wave vectors have the same real components ky and kz in the plane x ¼ 0. Therefore the electric field in both media may be set as

5. Dyakonov surface waves

58 Surface Waves - New Trends and Developments

5.1. Dispersion equation of DSWs

substrate ð Þ x > 0 .

Dyakonov surface waves (DSWs) are another kind of surface waves, supported at the interface between an optically isotropic medium and a uniaxial-birefringent material. In the original work by Dyakonov (English version was reported in 1988 [3]), the optical axis of the uniaxial medium was assumed in-plane with respect to the interface. This is the case we deal with here. The importance of DSWs for integrated optical applications, such as sensing and nanowaveguiding, was appreciated in a series of papers [17, 18]. Indeed Dyakonov-like surface waves also emerge in the case that a biaxial crystal [19] or a structurally chiral material [20] takes the place of the uniaxial medium. The case of metal-dielectric (MD) multilayers as structurally anisotropic media is especially convenient since small-filling fractions of the metallic inclusions enable metamaterials with an enormous birefringence, thus enhancing

The system under study is the plotted in Figure 5, where we have two semi-infinite media, one of them is isotropic and the second one is an MD lattice. In our case, the indices "1" and "2" make reference to the plasmonic lattice and the isotropic medium, respectively. We have previously reported a comprehensive analysis of this case in [4]. As we have seen earlier, the plasmonic lattice can be taken as an effective uniaxial crystal. In this case, the permittivity along its optical axis, Ez<sup>1</sup> ¼ E∥, is given by Eq. (36); also, the permittivity in the transverse direction Ex<sup>1</sup> ¼ Ey<sup>1</sup> ¼ E<sup>⊥</sup> may be appropriately averaged. From hereon, the permittivity E<sup>2</sup> of the isotropic medium in x > 0 will be denoted by E. Note that our analysis serves for natural

Figure 5. Schematic setup under study, consisting in a semi-infinite dielectric-metal superlattice ð Þ x < 0 and an isotropic

density of DSWs and relaxing their prominent directivity [21–23].

birefringent materials characterized by permittivities E<sup>∥</sup> and E⊥.

$$\mathbf{E}\_{\text{tot}} = \mathbf{E}(\mathbf{x}) \exp\left(i\mathbf{k}\_y y + i\mathbf{k}\_z z - i\omega t\right). \tag{40}$$

Moreover, these fields are evanescent in the isotropic medium and in the superlattice. In the anisotropic medium (x < 0) the evanescent electric amplitude can be written as

$$\mathbf{E}(\mathbf{x}) = B\_{o1} \ddot{b}\_{o1} \exp\left(-i\mathbf{k}\_{o1}\mathbf{x}\right) + B\_{e1} \ddot{b}\_{e1} \exp\left(-i\mathbf{k}\_{e1}\mathbf{x}\right),\tag{41}$$

where the ordinary and extraordinary waves in the effective uniaxial medium decay exponentially with rates given by κ<sup>o</sup> ¼ �iko<sup>1</sup> and κ<sup>e</sup> ¼ �ike1, respectively. Taking the formulation given in Section 2.3.1, the amplitudes Ao<sup>1</sup> and Ae<sup>1</sup> are identically zero. In the isotropic medium (x > 0) the amplitude of the electric field is

$$\mathbf{E(x)} = A'\_{T\mathbf{E2}} \lor\_{T\mathbf{E2}} \exp\left(i\mathbf{k}\_{T\mathbf{E2}}\mathbf{x}\right) + A'\_{T\mathbf{M2}} \lor\_{T\mathbf{M2}} \exp\left(i\mathbf{k}\_{T\mathbf{M2}}\mathbf{x}\right),\tag{42}$$

where the evanescent decay for TE and TM modes is κ ¼ �ikTE<sup>2</sup> ¼ �ikTM2. Now the amplitudes B<sup>0</sup> TE<sup>2</sup> and B<sup>0</sup> TM<sup>2</sup> are zero.

Once we have the amplitudes in both sides of the interface, we apply the boundary conditions at x ¼ 0, D<sup>1</sup> � v<sup>1</sup> ¼ D<sup>2</sup> � v<sup>0</sup> 2, where D<sup>1</sup> is provided by Eq. (16), D<sup>2</sup> from Eq. (26), v<sup>1</sup> from Eq. (17), and v<sup>0</sup> <sup>2</sup> from Eq. (27). This equation reduces to:

$$
\begin{bmatrix} 0\\ B\_{o1} \\ 0 \\ B\_{e1} \end{bmatrix} = \mathbf{M}\_{\mathbb{H}} \begin{bmatrix} A'\_{T\mathbb{E}2} \\ 0 \\ A'\_{T\mathbb{M}2} \\ 0 \end{bmatrix}' \tag{43}
$$

where the transmission matrix <sup>M</sup><sup>h</sup> <sup>¼</sup> <sup>D</sup>�<sup>1</sup> <sup>1</sup> � D<sup>2</sup> establishes a relationship between the amplitudes of hybrid polarization modes. Using the elements Mij of the matrix Mh, and defining M<sup>i</sup> and M<sup>a</sup> as

$$\mathbf{M}\_{i} = \begin{bmatrix} M\_{11} & M\_{13} \\ M\_{31} & M\_{33} \end{bmatrix} = \begin{bmatrix} -\frac{i\mathbf{k}\_{z}(\kappa\_{o} + \kappa)}{2\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} & \frac{k\_{0}^{2}k\_{y}(\kappa\epsilon\_{o} + \kappa\epsilon\_{\perp})}{2\kappa\_{o}\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} \\ -\frac{(\kappa\_{e} + \kappa)k\_{y}}{2\kappa\_{e}\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} & -\frac{i\mathbf{k}\_{z}(\kappa\kappa\_{e}\epsilon\_{\perp} + \epsilon\kappa\_{o}^{2})}{2\kappa\_{e}\epsilon\_{\perp}\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} \end{bmatrix},\tag{44a}$$

$$\mathbf{M}\_{4} = \begin{bmatrix} M\_{21} & M\_{23} \\ M\_{41} & M\_{43} \end{bmatrix} = \begin{bmatrix} \frac{i\mathbf{k}\_{\varepsilon}(\kappa\_{o} - \kappa)}{2\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} & \frac{k\_{0}^{2}k\_{y}(\epsilon\kappa\_{o} - \kappa\epsilon\_{\perp})}{2\kappa\_{o}\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} \\ \frac{(\kappa - \kappa\_{e})k\_{y}}{2\kappa\_{\varepsilon}\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} & -\frac{i\mathbf{k}\_{z}(\kappa\kappa\_{e}\epsilon\_{\perp} - \kappa\kappa\_{o}^{2})}{2\kappa\_{\varepsilon}\epsilon\_{\perp}\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} \end{bmatrix}. \tag{44b}$$

Eq. (43) can be rewritten as a set of two independent matrix equations, namely

$$
\begin{bmatrix} 0 \\ 0 \end{bmatrix} = \mathbf{M}\_i \cdot \begin{bmatrix} A'\_{\text{TE2}} \\ A'\_{\text{TM2}} \end{bmatrix} \prime \tag{45a}
$$

$$
\begin{bmatrix} B\_{o1} \\ B\_{\epsilon 1} \end{bmatrix} = \mathbf{M}\_{\mathfrak{a}} \cdot \begin{bmatrix} A'\_{T\mathbf{E}2} \\ A'\_{T\mathbf{M}2} \end{bmatrix}. \tag{45b}
$$

the substrate x > 0. Looking at the other side of the dispersion curve, θmax is established by ke ¼ 0, shown as a black solid line, for which the extraordinary wave will not decay spatially at

Figure 6. Dispersion Eq. (47) for DSWs (dotted-dashed line) propagating on the interface of a semi-infinite dielectricmetal lattice with metal filling factor f = 0.1. The solid elliptical line and the black dashed line are associated with homogeneous extraordinary waves ð Þ κ<sup>e</sup> ¼ 0 and homogeneous ordinary waves ð Þ κ<sup>o</sup> ¼ 0 , respectively. The isofrequency curve ð Þ <sup>κ</sup> <sup>¼</sup> <sup>0</sup> of isotropic N-BAK1 are represented in by the red solid line, which applies for TE<sup>x</sup> and TM<sup>x</sup> waves.

Consequently, the solution for Eq. (47) can be traced near the curves κ ¼ 0 and κ<sup>e</sup> ¼ 0; thus,

As we discussed in Section 4.1, the EMA is limited to metallic slabs' width wm ≪ λ0. However, this condition must be taken into account with care, since the skin depth of noble metals is extremely short, δ ≈ c=ωp. For instance, we estimate δ ¼ 24 nm in the case of silver. If the metal thickness is comparable to its skin depth, the EMA will substantially deviate from exact calculations. Note that experimental studies from multilayer optics rarely incorporate metallic

We emphasize that moderate changes in the birefringence of the plasmonic crystal will substantially affect the existence of DSWs. More specifically, an enlargement of E<sup>⊥</sup> driven by increasing wm, provided f is fixed (see Section 4.1), will lead to a significant modification of the DSW dispersion curves. Ultimately, this phenomenon is clearly attributed to nonlocal effects in the effective-medium response of nanolayered metamaterials [24], which is associ-

We conclude that, in order to excite DSWs, one may counterbalance the decrease of birefringence in the plasmonic lattice by means of a dielectric substrate of higher index of refraction. To illustrate this matter, the dispersion Eq. (47) for DSWs is represented in Figure 7, in addition to using the values of E<sup>∥</sup> and E<sup>⊥</sup> from nonlocal estimators [4]. When wm grows but f is kept fixed, the dispersion curve of the Dyakonov surface waves tends to approach the optic axis.

ated with a strong variation of the fields on the scale of a single layer.

of both curves.

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

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61

DSWs are always found close to the crosspoint P<sup>0</sup> ky0; kz<sup>0</sup>

Adapted with permission from [4] of copyright ©2013 IEEE photonics society.

x ! �∞.

5.2.1. Nonlocal effects

slabs with a thickness below 10 nm.

Note that M<sup>i</sup> governs the amplitudes A<sup>0</sup> TE<sup>2</sup> and A<sup>0</sup> TM<sup>2</sup> of the isotropic medium, and M<sup>a</sup> (also Mh) may be used to determine the amplitudes Bo<sup>1</sup> and Be<sup>1</sup> of the anisotropic medium.

Dyakonov equation is obtained by means of letting the determinant of M<sup>i</sup> equal to zero, giving

$$(k\_0^2 k\_y^2 \epsilon\_\perp(\kappa + \kappa\_\epsilon)(\epsilon \kappa\_o + \epsilon\_\perp \kappa) = \kappa\_o k\_z^2 (\kappa + \kappa\_o)(\epsilon \kappa\_o^2 + \epsilon\_\perp \kappa \kappa\_\epsilon),\tag{46}$$

which provides a spectral map of allowed values ky; kz � �. After fairly tedious algebraic transformations we can reduce Eq. (46) to a more convenient form [3].

$$(\kappa + \kappa\_{\epsilon})(\kappa + \kappa\_{o})(\epsilon \kappa\_{o} + \epsilon\_{\bot}\kappa\_{\epsilon}) = (\epsilon\_{\parallel} - \epsilon)(\epsilon - \epsilon\_{\bot})k\_{0}^{2}\kappa\_{o}.\tag{47}$$

Assuming that E∥, E<sup>⊥</sup> and all decay rates are positive, the additional restriction E<sup>⊥</sup> < E < E<sup>∥</sup> can be deduced for the existence of surface waves. As a consequence, positive birefringence is mandatory to ensure a stationary solution of Maxwell's equations. Therefore, layered superlattices supporting Dyakonov-like surface waves cannot be formed by all dielectric materials [21].

#### 5.2. DSWs in nano-engineered materials

To illustrate the difference between using conventional birefringent materials and plasmonic crystals, we solve Eq. (47) for liquid crystal E7 with E<sup>∥</sup> ¼ 2:98 i E<sup>⊥</sup> ¼ 2:31 at a wavelength of λ<sup>0</sup> ¼ 1:55μ m and N-BAK1 substrate of dielectric constant E ¼ 2:42. In this case, DSWs propagate in a narrow angular region Δθ ¼ θmax � θmin, where θ stands for the angle between the inplane vector ky; kz � � and the optical axis. More specifically, the angular range yields Δθ <sup>¼</sup> <sup>0</sup>:92<sup>∘</sup> around the mean angle <sup>θ</sup> <sup>¼</sup> <sup>26</sup>:6<sup>∘</sup> . It appears that the resulting angular range Δθ is short. But this range would become much smaller when using other optical crystals like quartz, exhibiting a common birefringence. In order to gain in angular extent Δθ, we consider a GaAs-Ag crystal (E<sup>1</sup> ¼ 12:5 and E<sup>2</sup> ¼ �103:3, where we neglect losses) that leads to values of E<sup>∥</sup> ¼ 14:08, derived from Eq. (36), and E<sup>⊥</sup> ¼ 0:92 with a metal-filling factor f ¼ 0:10. Form birefringence now yields Δn ¼ 2:79. Moreover, solutions to Eq. (47) can be found in the region of angles comprised between <sup>θ</sup>min <sup>¼</sup> <sup>39</sup>:0<sup>∘</sup> and <sup>θ</sup>max <sup>¼</sup> <sup>71</sup>:3<sup>∘</sup> . It is important to point out that the total angular range of existence of DSWs, Δθ <sup>¼</sup> <sup>32</sup>:3<sup>∘</sup> , grows by more than an order of magnitude. Figure 6 shows the dispersion curve for DSWs; one can observe that θmin is attained under the condition <sup>κ</sup> <sup>¼</sup> 0 (red solid line), where TE<sup>x</sup> and TM<sup>x</sup> waves are uniform in

Figure 6. Dispersion Eq. (47) for DSWs (dotted-dashed line) propagating on the interface of a semi-infinite dielectricmetal lattice with metal filling factor f = 0.1. The solid elliptical line and the black dashed line are associated with homogeneous extraordinary waves ð Þ κ<sup>e</sup> ¼ 0 and homogeneous ordinary waves ð Þ κ<sup>o</sup> ¼ 0 , respectively. The isofrequency curve ð Þ <sup>κ</sup> <sup>¼</sup> <sup>0</sup> of isotropic N-BAK1 are represented in by the red solid line, which applies for TE<sup>x</sup> and TM<sup>x</sup> waves. Adapted with permission from [4] of copyright ©2013 IEEE photonics society.

the substrate x > 0. Looking at the other side of the dispersion curve, θmax is established by ke ¼ 0, shown as a black solid line, for which the extraordinary wave will not decay spatially at x ! �∞.

Consequently, the solution for Eq. (47) can be traced near the curves κ ¼ 0 and κ<sup>e</sup> ¼ 0; thus, DSWs are always found close to the crosspoint P<sup>0</sup> ky0; kz<sup>0</sup> of both curves.

#### 5.2.1. Nonlocal effects

Eq. (43) can be rewritten as a set of two independent matrix equations, namely

0 0 � �

Bo<sup>1</sup> Be<sup>1</sup> � �

<sup>y</sup>E⊥ð Þ <sup>κ</sup> <sup>þ</sup> <sup>κ</sup><sup>e</sup> ð Þ¼ <sup>E</sup>κ<sup>o</sup> <sup>þ</sup> <sup>E</sup>⊥<sup>κ</sup> <sup>κ</sup>ok<sup>2</sup>

Note that M<sup>i</sup> governs the amplitudes A<sup>0</sup>

60 Surface Waves - New Trends and Developments

k 2 0k 2

5.2. DSWs in nano-engineered materials

rials [21].

plane vector ky; kz

around the mean angle <sup>θ</sup> <sup>¼</sup> <sup>26</sup>:6<sup>∘</sup>

which provides a spectral map of allowed values ky; kz

formations we can reduce Eq. (46) to a more convenient form [3].

of angles comprised between <sup>θ</sup>min <sup>¼</sup> <sup>39</sup>:0<sup>∘</sup> and <sup>θ</sup>max <sup>¼</sup> <sup>71</sup>:3<sup>∘</sup>

<sup>¼</sup> <sup>M</sup><sup>i</sup> � <sup>A</sup><sup>0</sup>

<sup>¼</sup> <sup>M</sup><sup>a</sup> � <sup>A</sup><sup>0</sup>

TE<sup>2</sup> and A<sup>0</sup>

Dyakonov equation is obtained by means of letting the determinant of M<sup>i</sup> equal to zero, giving

ð Þ <sup>κ</sup> <sup>þ</sup> <sup>κ</sup><sup>e</sup> ð Þ <sup>κ</sup> <sup>þ</sup> <sup>κ</sup><sup>o</sup> ð Þ¼ <sup>E</sup>κ<sup>o</sup> <sup>þ</sup> <sup>E</sup>⊥κ<sup>e</sup> <sup>E</sup><sup>∥</sup> � <sup>E</sup> � �ð Þ <sup>E</sup> � <sup>E</sup><sup>⊥</sup> <sup>k</sup>

Assuming that E∥, E<sup>⊥</sup> and all decay rates are positive, the additional restriction E<sup>⊥</sup> < E < E<sup>∥</sup> can be deduced for the existence of surface waves. As a consequence, positive birefringence is mandatory to ensure a stationary solution of Maxwell's equations. Therefore, layered superlattices supporting Dyakonov-like surface waves cannot be formed by all dielectric mate-

To illustrate the difference between using conventional birefringent materials and plasmonic crystals, we solve Eq. (47) for liquid crystal E7 with E<sup>∥</sup> ¼ 2:98 i E<sup>⊥</sup> ¼ 2:31 at a wavelength of λ<sup>0</sup> ¼ 1:55μ m and N-BAK1 substrate of dielectric constant E ¼ 2:42. In this case, DSWs propagate in a narrow angular region Δθ ¼ θmax � θmin, where θ stands for the angle between the in-

this range would become much smaller when using other optical crystals like quartz, exhibiting a common birefringence. In order to gain in angular extent Δθ, we consider a GaAs-Ag crystal (E<sup>1</sup> ¼ 12:5 and E<sup>2</sup> ¼ �103:3, where we neglect losses) that leads to values of E<sup>∥</sup> ¼ 14:08, derived from Eq. (36), and E<sup>⊥</sup> ¼ 0:92 with a metal-filling factor f ¼ 0:10. Form birefringence now yields Δn ¼ 2:79. Moreover, solutions to Eq. (47) can be found in the region

the total angular range of existence of DSWs, Δθ <sup>¼</sup> <sup>32</sup>:3<sup>∘</sup> , grows by more than an order of magnitude. Figure 6 shows the dispersion curve for DSWs; one can observe that θmin is attained under the condition <sup>κ</sup> <sup>¼</sup> 0 (red solid line), where TE<sup>x</sup> and TM<sup>x</sup> waves are uniform in

� � and the optical axis. More specifically, the angular range yields Δθ <sup>¼</sup> <sup>0</sup>:92<sup>∘</sup>

. It appears that the resulting angular range Δθ is short. But

Mh) may be used to determine the amplitudes Bo<sup>1</sup> and Be<sup>1</sup> of the anisotropic medium.

TE2 A0 TM2

> TE2 A0 TM2

<sup>z</sup> ð Þ <sup>κ</sup> <sup>þ</sup> <sup>κ</sup><sup>o</sup> <sup>E</sup>κ<sup>2</sup>

" #

, (45a)

: (45b)

� �, (46)

. It is important to point out that

<sup>0</sup>κo: (47)

TM<sup>2</sup> of the isotropic medium, and M<sup>a</sup> (also

� �. After fairly tedious algebraic trans-

<sup>o</sup> þ E⊥κκ<sup>e</sup>

2

" #

As we discussed in Section 4.1, the EMA is limited to metallic slabs' width wm ≪ λ0. However, this condition must be taken into account with care, since the skin depth of noble metals is extremely short, δ ≈ c=ωp. For instance, we estimate δ ¼ 24 nm in the case of silver. If the metal thickness is comparable to its skin depth, the EMA will substantially deviate from exact calculations. Note that experimental studies from multilayer optics rarely incorporate metallic slabs with a thickness below 10 nm.

We emphasize that moderate changes in the birefringence of the plasmonic crystal will substantially affect the existence of DSWs. More specifically, an enlargement of E<sup>⊥</sup> driven by increasing wm, provided f is fixed (see Section 4.1), will lead to a significant modification of the DSW dispersion curves. Ultimately, this phenomenon is clearly attributed to nonlocal effects in the effective-medium response of nanolayered metamaterials [24], which is associated with a strong variation of the fields on the scale of a single layer.

We conclude that, in order to excite DSWs, one may counterbalance the decrease of birefringence in the plasmonic lattice by means of a dielectric substrate of higher index of refraction. To illustrate this matter, the dispersion Eq. (47) for DSWs is represented in Figure 7, in addition to using the values of E<sup>∥</sup> and E<sup>⊥</sup> from nonlocal estimators [4]. When wm grows but f is kept fixed, the dispersion curve of the Dyakonov surface waves tends to approach the optic axis.

Figure 7. Solutions to Dyakonov equation, drawn in dotted-dashed lines, for a MD lattice with the same filling factor f ¼ 0:10 but different wm, using estimates from the nonlocal birefringence approach described in [4]. Note that the red solid line designates the isofrequency curve of isotropic substrate (a) N-BAK1 and (b) P-SF68. Adapted with permission from [4] of copyright ©2013 IEEE photonics society.

For an N-BAK1 substrate, as shown in Figure 7(a), <sup>θ</sup>max <sup>¼</sup> <sup>68</sup>:2<sup>∘</sup> and 58:7<sup>∘</sup> for wm <sup>¼</sup> 3 nm and 6 nm, respectively. Also, <sup>θ</sup>min <sup>¼</sup> <sup>37</sup>:6<sup>∘</sup> and 32:1<sup>∘</sup> for these two cases. As a consequence the angular range Δθ shrinks when wm increases. In the limit E<sup>⊥</sup> ! E, which occurs for wm ¼ 10:3 nm using a N-BAK1 substrate, DSWs are not supported at the interface of the MD lattice and the isotropic dielectric. A substrate with greater relative permittivity E would be necessary. For example, if we use a substrate with greater relative permittivity E as P-SF68 [see Figure 7(b)], DSWs exist for wm <sup>¼</sup> 12 nm with an angular range Δθ <sup>¼</sup> <sup>12</sup>:5<sup>∘</sup> .

#### 5.2.2. Dissipative effects

Up to now, we have avoided another important aspect of plasmonic devices namely dissipation in metallic elements. In this regard, effective permittivities are fundamentally complex, and consequently the Dyakonov Eq. (47) is expected to give complex values of ky; kz . This procedure has been discussed by Sorni et al. [25] recently. In order to tackle this problem, we evaluate numerically the value of the Bloch wavenumber kz for a given real value ky. The spatial frequency kz becomes complex since Im½ �¼ E<sup>m</sup> 8:1. As a consequence, the surface wave cannot propagate indefinitely, undergoing an energy attenuation given by <sup>l</sup> <sup>¼</sup> ð Þ 2Im½ � kz �<sup>1</sup> . Furthermore, we naturally assume that the real part of the parameters κ, κo, and κ<sup>e</sup> are all positive. These positive values correlate with a decay at ∣x∣ ! ∞ and thus with a confinement of the wave near x ¼ 0.

We observe that the dispersion curve for dissipative DSWs is flatter and larger than the curve

Figure 8. (a) Isofrequency curve that corresponds to hybrid surface waves existing at the boundary between a semiinfinite P-SF68 substrate and a lossy MD superlattice of f ¼ 0:10 and wm ¼ 12 nm. (b) Ratio of Im(kz) over re(kz) representing dissipation effects in the propagation of DSWs. (c) Three contour plots of the magnetic field ∣Hx∣ computed using the finite-element method. The superlattice is set on the left, for which only one period is represented. Capital letters a, B, and C, designate the transverse spatial frequencies ky ¼ 0:8k0, 1:2k0, and 1:6 k0, respectively. Adapted with permis-

waves. In these two figures, capital letters A, B, and C designate the transverse spatial frequencies ky ¼ 0:8k0, 1:2k0, and 1:6k0, respectively. Figure 8(c) shows the magnetic field ∣Hx∣ for the three different cases denoted by capital letters A, B, and C. Note that in the case of paraxial surface waves, for which ky reaches a minimum value (case A), one achieves Imð Þ kz ≪ Reð Þ kz as depicted in Figure 8(b). This is induced by an enormous shift undergone by the field maximum in the direction to the isotropic medium, as shown in Figure 8(c), where dissipation effects are barely disadvantageous on surface-wave propagation. Moreover, this would be consistent with a condition Reð Þ κ ≪ Reð Þ κ<sup>e</sup> . On the other hand, for nonparaxial waves, having the largest values of ky, the fields show slow energy decay inside the plasmonic superlattice. In case C, the magnetic field ∣Hx∣ is localized around the metallic layer and takes significant values far from the boundary of the substrate. As a consequence, losses in the metal translate

. Figure 8(b) shows Im(kz)/Re(kz) in the range of existence of the surface

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

http://dx.doi.org/10.5772/intechopen.74126

63

, giving an angular

obtained by neglecting losses. Specifically <sup>θ</sup>max <sup>¼</sup> <sup>49</sup>:9<sup>∘</sup> and <sup>θ</sup>min <sup>¼</sup> <sup>23</sup>:7<sup>∘</sup>

range Δθ <sup>¼</sup> <sup>26</sup>:2<sup>∘</sup>

into a significant rise in the values of Im(kz).

sion from [4] of copyright ©2013 IEEE photonics society.

Figure 8(a) depicts the dispersion curve corresponding to dissipative DSWs, for the case of a plasmonic MD lattice with f ¼ 0:10 and wm ¼ 12 nm. We used a commercial software (COMSOL Multiphysics) based on the finite-element method (FEM) in order to perform our numerical simulations. We cannot observe surface waves by setting an N-BAK1 substrate with n ¼ 1:56, suggesting that this is a retardation effect. More specifically, Figure 8(a) shows the isofrequency curves when n ¼ 1:95, corresponding to P-SF68.

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge http://dx.doi.org/10.5772/intechopen.74126 63

For an N-BAK1 substrate, as shown in Figure 7(a), <sup>θ</sup>max <sup>¼</sup> <sup>68</sup>:2<sup>∘</sup> and 58:7<sup>∘</sup> for wm <sup>¼</sup> 3 nm and 6 nm, respectively. Also, <sup>θ</sup>min <sup>¼</sup> <sup>37</sup>:6<sup>∘</sup> and 32:1<sup>∘</sup> for these two cases. As a consequence the angular range Δθ shrinks when wm increases. In the limit E<sup>⊥</sup> ! E, which occurs for wm ¼ 10:3 nm using a N-BAK1 substrate, DSWs are not supported at the interface of the MD lattice and the isotropic dielectric. A substrate with greater relative permittivity E would be necessary. For example, if we use a substrate with greater relative permittivity E as P-SF68 [see Figure 7(b)],

Figure 7. Solutions to Dyakonov equation, drawn in dotted-dashed lines, for a MD lattice with the same filling factor f ¼ 0:10 but different wm, using estimates from the nonlocal birefringence approach described in [4]. Note that the red solid line designates the isofrequency curve of isotropic substrate (a) N-BAK1 and (b) P-SF68. Adapted with permission

Up to now, we have avoided another important aspect of plasmonic devices namely dissipation in metallic elements. In this regard, effective permittivities are fundamentally complex, and consequently the Dyakonov Eq. (47) is expected to give complex values of ky; kz

procedure has been discussed by Sorni et al. [25] recently. In order to tackle this problem, we evaluate numerically the value of the Bloch wavenumber kz for a given real value ky. The spatial frequency kz becomes complex since Im½ �¼ E<sup>m</sup> 8:1. As a consequence, the surface wave cannot propagate indefinitely, undergoing an energy attenuation given by <sup>l</sup> <sup>¼</sup> ð Þ 2Im½ � kz �<sup>1</sup>

Furthermore, we naturally assume that the real part of the parameters κ, κo, and κ<sup>e</sup> are all positive. These positive values correlate with a decay at ∣x∣ ! ∞ and thus with a confinement

Figure 8(a) depicts the dispersion curve corresponding to dissipative DSWs, for the case of a plasmonic MD lattice with f ¼ 0:10 and wm ¼ 12 nm. We used a commercial software (COMSOL Multiphysics) based on the finite-element method (FEM) in order to perform our numerical simulations. We cannot observe surface waves by setting an N-BAK1 substrate with n ¼ 1:56, suggesting that this is a retardation effect. More specifically, Figure 8(a) shows the

.

. This

.

DSWs exist for wm <sup>¼</sup> 12 nm with an angular range Δθ <sup>¼</sup> <sup>12</sup>:5<sup>∘</sup>

isofrequency curves when n ¼ 1:95, corresponding to P-SF68.

5.2.2. Dissipative effects

from [4] of copyright ©2013 IEEE photonics society.

62 Surface Waves - New Trends and Developments

of the wave near x ¼ 0.

Figure 8. (a) Isofrequency curve that corresponds to hybrid surface waves existing at the boundary between a semiinfinite P-SF68 substrate and a lossy MD superlattice of f ¼ 0:10 and wm ¼ 12 nm. (b) Ratio of Im(kz) over re(kz) representing dissipation effects in the propagation of DSWs. (c) Three contour plots of the magnetic field ∣Hx∣ computed using the finite-element method. The superlattice is set on the left, for which only one period is represented. Capital letters a, B, and C, designate the transverse spatial frequencies ky ¼ 0:8k0, 1:2k0, and 1:6 k0, respectively. Adapted with permission from [4] of copyright ©2013 IEEE photonics society.

We observe that the dispersion curve for dissipative DSWs is flatter and larger than the curve obtained by neglecting losses. Specifically <sup>θ</sup>max <sup>¼</sup> <sup>49</sup>:9<sup>∘</sup> and <sup>θ</sup>min <sup>¼</sup> <sup>23</sup>:7<sup>∘</sup> , giving an angular range Δθ <sup>¼</sup> <sup>26</sup>:2<sup>∘</sup> . Figure 8(b) shows Im(kz)/Re(kz) in the range of existence of the surface waves. In these two figures, capital letters A, B, and C designate the transverse spatial frequencies ky ¼ 0:8k0, 1:2k0, and 1:6k0, respectively. Figure 8(c) shows the magnetic field ∣Hx∣ for the three different cases denoted by capital letters A, B, and C. Note that in the case of paraxial surface waves, for which ky reaches a minimum value (case A), one achieves Imð Þ kz ≪ Reð Þ kz as depicted in Figure 8(b). This is induced by an enormous shift undergone by the field maximum in the direction to the isotropic medium, as shown in Figure 8(c), where dissipation effects are barely disadvantageous on surface-wave propagation. Moreover, this would be consistent with a condition Reð Þ κ ≪ Reð Þ κ<sup>e</sup> . On the other hand, for nonparaxial waves, having the largest values of ky, the fields show slow energy decay inside the plasmonic superlattice. In case C, the magnetic field ∣Hx∣ is localized around the metallic layer and takes significant values far from the boundary of the substrate. As a consequence, losses in the metal translate into a significant rise in the values of Im(kz).

#### 5.3. New families of DSWs in lossy media

In this section we carry out a thorough analysis of DSWs that takes place in lossy uniaxial metamaterials. Special emphasis is put when the effective-medium approach induces satisfactory results. The introduction of losses leads to a transformation of the isofrequency curves, which deviates from spheres and ellipsoids, as commonly considered by ordinary and extraordinary waves, respectively. As a consequence, one can find two different families of surface waves as reported by Sorni et al. [25]. One family of surface waves is directly related with the well-known solutions derived by Dyakonov [3]. Importantly, the existence of a new family of surface waves is revealed, closely connected to the presence of losses in the uniaxial effective crystal. We point out that the solutions to Dyakonov equation presented earlier are partial ones insofar as the z-component of the wavevector is kept real valued. Nevertheless, the whole set of solutions includes all possible wavevectors that feature a complex-valued kx.

#### 6. Dyakonov surface waves in hyperbolic media

In this section we perform a thorough analysis of DSWs taking place in semi-infinite MD lattices exhibiting hyperbolic dispersion. Part of this section was previously reported by Zapata-Rodríguez et al. [26]; we point out that recently further studies on DSW in hyperbolic metamaterials have been reported by other authors [27]. Our approach puts emphasis on the EMA. Under these conditions, different regimes can be found including DSWs with nonhyperbolic dispersion. The system under analysis is again as depicted in Figure 5. For simplicity, we assume that dielectric materials are nondispersive; indeed, we set E ¼ 1 and E<sup>d</sup> ¼ 2:25 in our numerical simulations. Furthermore, Drude metals are included, and frequencies will be expressed in units of its plasma frequency, Ω ¼ ω=ωp. Again, Dyakonov Eq. (47) provides the spectral map of wave vectors k<sup>D</sup> ¼ 0; ky; kz � �. Note that in this section, spatial frequencies will be expressed in units of kp.

In the special case of the surface wave propagation perpendicular to the optical axis (kz ¼ 0), Eq. (47) reveals the following solution: Eκ<sup>o</sup> þ E⊥κ ¼ 0. In the case: E<sup>⊥</sup> < 0 and E < ∣E⊥∣, this equation has the well-known solution

$$k\_y = \Omega \sqrt{\frac{\epsilon \epsilon\_\perp}{\epsilon + \epsilon\_\perp}}\tag{48}$$

other cases are treated elsewhere [26]. In the effective-uniaxial medium, it is easy to realize that κ < κ<sup>o</sup> and also κ<sup>e</sup> < κo. Under these circumstances, all brackets in Dyakonov Eq. (47) are positive provided Eκ<sup>o</sup> þ E⊥κ<sup>e</sup> > 0. This happens within the spectral band Ω<sup>0</sup> < Ω < Ω1, where

<sup>Ω</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

In Figure 9(a) and (b), we illustrate the dispersion equation of DSWs for two different frequencies within the spectral range 0 < Ω < Ω0. In these cases, the dispersion curve approaches a hyperbola. We find a bandgap around kz ¼ 0 in Figure 9(a), unlike what occurs in (b). Note that hybrid solutions near kz ¼ 0 are additionally constrained to the condition ky ≥ Ω ffiffiffiffi

also Eq. (48)], which is a necessary condition for κ<sup>e</sup> to exhibit real and positive values. We consider the quasi-static regime (Ω ! 0) where ∣kD∣ ¼ kD ≫ Ω to determine the asymptotes of the hyperbolic-like DSW dispersion curve. Under this approximation, κ ¼ kD, κ<sup>o</sup> ¼ kD and

being ky ¼ kD cos θ and kz ¼ kD sin θ. These asymptotes establish a canalization regime leading to a collective directional propagation of DSW beams [4, 28]. At this point it is necessary to remind that the asymptotes of the e-waves dispersion curve, in the kykz plane, have slopes

In the high-frequency band Ω<sup>2</sup> < Ω < 1 we find that E<sup>∥</sup> < 0 < E⊥, as occurs in Figure 9(c). Note the relevant proximity of DSW dispersion curve to κ<sup>e</sup> ¼ 0, the same way we also find in Figure 9(a) and (b). Conversely it crosses the e-wave hyperbolic curve at two different points,

Figure 9. Solutions to Eq. (47), drawn in solid line, providing the spatial dispersion of DSWs which can exist in the arrangement of Figure 5, at different frequencies: (a) Ω ¼ 0:20, (b) Ω ¼ 0:28, and (c) Ω ¼ 0:85. Here, the isotropic medium is air and the multi-layered metamaterial has a filling factor f ¼ 0:25 and. Also, we include equations κ ¼ 0 (dotted line)

and κ<sup>e</sup> ¼ 0 (dashed line). Adapted with permission from [26] of copyright ©2013 Optical Society of America.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos <sup>2</sup>θ þ E∥=E<sup>⊥</sup>

� � sin <sup>2</sup>θ

Note that Ω<sup>0</sup> ¼ 0:292 in our numerical simulation.

Θ ¼

satisfying the condition θ<sup>D</sup> < θe, as illustrated in Figure 9(b).

q

κ<sup>e</sup> ¼ ΘkD, where

<sup>1</sup> <sup>þ</sup> <sup>E</sup>=<sup>f</sup> <sup>þ</sup> <sup>E</sup>dð Þ <sup>1</sup> � <sup>f</sup> <sup>=</sup><sup>f</sup> <sup>p</sup> : (49)

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

http://dx.doi.org/10.5772/intechopen.74126

E∥ p [see 65

, (50)

which resembles the dispersion equation of conventional SPPs [see Eq. (31)]. Here we have purely TM<sup>x</sup> polarized waves, as expected. Note that no solutions to Eq. (47) can be found, in the form of surface waves, considering wave propagation parallel to the optical axis (ky ¼ 0) for hyperbolic metamaterials: E⊥E<sup>∥</sup> < 0. That means that a threshold value of ky can be found for the existence of surface waves.

Next we describe a specific configuration governing DSWs, subject to a low value of the refractive index <sup>n</sup> <sup>¼</sup> ffiffi <sup>E</sup> <sup>p</sup> , namely <sup>E</sup> <sup>&</sup>lt; <sup>E</sup><sup>∥</sup> (<sup>E</sup> <sup>&</sup>lt; <sup>E</sup>⊥) occurring at low and moderate frequencies; other cases are treated elsewhere [26]. In the effective-uniaxial medium, it is easy to realize that κ < κ<sup>o</sup> and also κ<sup>e</sup> < κo. Under these circumstances, all brackets in Dyakonov Eq. (47) are positive provided Eκ<sup>o</sup> þ E⊥κ<sup>e</sup> > 0. This happens within the spectral band Ω<sup>0</sup> < Ω < Ω1, where

$$\Omega\_0 = \frac{1}{\sqrt{1 + \epsilon/f + \epsilon\_d(1 - f)/f}}.\tag{49}$$

Note that Ω<sup>0</sup> ¼ 0:292 in our numerical simulation.

5.3. New families of DSWs in lossy media

64 Surface Waves - New Trends and Developments

In this section we carry out a thorough analysis of DSWs that takes place in lossy uniaxial metamaterials. Special emphasis is put when the effective-medium approach induces satisfactory results. The introduction of losses leads to a transformation of the isofrequency curves, which deviates from spheres and ellipsoids, as commonly considered by ordinary and extraordinary waves, respectively. As a consequence, one can find two different families of surface waves as reported by Sorni et al. [25]. One family of surface waves is directly related with the well-known solutions derived by Dyakonov [3]. Importantly, the existence of a new family of surface waves is revealed, closely connected to the presence of losses in the uniaxial effective crystal. We point out that the solutions to Dyakonov equation presented earlier are partial ones insofar as the z-component of the wavevector is kept real valued. Nevertheless, the whole set

In this section we perform a thorough analysis of DSWs taking place in semi-infinite MD lattices exhibiting hyperbolic dispersion. Part of this section was previously reported by Zapata-Rodríguez et al. [26]; we point out that recently further studies on DSW in hyperbolic metamaterials have been reported by other authors [27]. Our approach puts emphasis on the EMA. Under these conditions, different regimes can be found including DSWs with nonhyperbolic dispersion. The system under analysis is again as depicted in Figure 5. For simplicity, we assume that dielectric materials are nondispersive; indeed, we set E ¼ 1 and E<sup>d</sup> ¼ 2:25 in our numerical simulations. Furthermore, Drude metals are included, and frequencies will be expressed in units of its plasma frequency, Ω ¼ ω=ωp. Again, Dyakonov Eq. (47)

In the special case of the surface wave propagation perpendicular to the optical axis (kz ¼ 0), Eq. (47) reveals the following solution: Eκ<sup>o</sup> þ E⊥κ ¼ 0. In the case: E<sup>⊥</sup> < 0 and E < ∣E⊥∣, this

r

which resembles the dispersion equation of conventional SPPs [see Eq. (31)]. Here we have purely TM<sup>x</sup> polarized waves, as expected. Note that no solutions to Eq. (47) can be found, in the form of surface waves, considering wave propagation parallel to the optical axis (ky ¼ 0) for hyperbolic metamaterials: E⊥E<sup>∥</sup> < 0. That means that a threshold value of ky can be found

Next we describe a specific configuration governing DSWs, subject to a low value of the

<sup>E</sup> <sup>p</sup> , namely <sup>E</sup> <sup>&</sup>lt; <sup>E</sup><sup>∥</sup> (<sup>E</sup> <sup>&</sup>lt; <sup>E</sup>⊥) occurring at low and moderate frequencies;

ffiffiffiffiffiffiffiffiffiffiffiffi EE<sup>⊥</sup> E þ E<sup>⊥</sup>

ky ¼ Ω

� �. Note that in this section, spatial

, (48)

of solutions includes all possible wavevectors that feature a complex-valued kx.

6. Dyakonov surface waves in hyperbolic media

provides the spectral map of wave vectors k<sup>D</sup> ¼ 0; ky; kz

frequencies will be expressed in units of kp.

equation has the well-known solution

for the existence of surface waves.

refractive index <sup>n</sup> <sup>¼</sup> ffiffi

In Figure 9(a) and (b), we illustrate the dispersion equation of DSWs for two different frequencies within the spectral range 0 < Ω < Ω0. In these cases, the dispersion curve approaches a hyperbola. We find a bandgap around kz ¼ 0 in Figure 9(a), unlike what occurs in (b). Note that hybrid solutions near kz ¼ 0 are additionally constrained to the condition ky ≥ Ω ffiffiffiffi E∥ p [see also Eq. (48)], which is a necessary condition for κ<sup>e</sup> to exhibit real and positive values. We consider the quasi-static regime (Ω ! 0) where ∣kD∣ ¼ kD ≫ Ω to determine the asymptotes of the hyperbolic-like DSW dispersion curve. Under this approximation, κ ¼ kD, κ<sup>o</sup> ¼ kD and κ<sup>e</sup> ¼ ΘkD, where

$$\Theta = \sqrt{\cos^2 \theta + \left(\epsilon\_{\parallel}/\epsilon\_{\perp}\right) \sin^2 \theta},\tag{50}$$

being ky ¼ kD cos θ and kz ¼ kD sin θ. These asymptotes establish a canalization regime leading to a collective directional propagation of DSW beams [4, 28]. At this point it is necessary to remind that the asymptotes of the e-waves dispersion curve, in the kykz plane, have slopes satisfying the condition θ<sup>D</sup> < θe, as illustrated in Figure 9(b).

In the high-frequency band Ω<sup>2</sup> < Ω < 1 we find that E<sup>∥</sup> < 0 < E⊥, as occurs in Figure 9(c). Note the relevant proximity of DSW dispersion curve to κ<sup>e</sup> ¼ 0, the same way we also find in Figure 9(a) and (b). Conversely it crosses the e-wave hyperbolic curve at two different points,

Figure 9. Solutions to Eq. (47), drawn in solid line, providing the spatial dispersion of DSWs which can exist in the arrangement of Figure 5, at different frequencies: (a) Ω ¼ 0:20, (b) Ω ¼ 0:28, and (c) Ω ¼ 0:85. Here, the isotropic medium is air and the multi-layered metamaterial has a filling factor f ¼ 0:25 and. Also, we include equations κ ¼ 0 (dotted line) and κ<sup>e</sup> ¼ 0 (dashed line). Adapted with permission from [26] of copyright ©2013 Optical Society of America.

where solutions to the Dyakonov equation begin and end, respectively. It is clear that the angular range of DSWs now turns to be significantly low.

showed that hybrid-polarized surface waves may propagate obliquely at the boundary between a plasmonic bilayer superlattice and an isotropic loss-free material. We revealed that realistic widths of the slabs might lead to solutions which deviate significantly from the results derived directly from the EMA and Dyakonov analysis. Finally, we showed that excitation of DSWs at the boundary of an isotropic dielectric and a hyperbolic metamaterial enables a distinct regime of propagation. It is important to note that the properties of the resulting bound states change drastically with the index of refraction of the surrounding medium, suggesting potential applications in chemical and biological sensing and nanoimaging.

This chapter was supported by the Qatar National Research Fund (Grant No. NPRP 8-028-1- 001) and the Spanish Ministry of Economy and Competitiveness (Grants No. TEC2014-53727-

1 Departament of Optics and Optometry and Vision Science, University of Valencia, Burjassot,

2 Center of Microelectronic Technologies, Institute of Chemistry, Technology and Metallurgy

4 Department of Optics, Pharmacology and Anatomy, University of Alicante, Alicante, Spain

[1] Zenneck J. Über die Fortpflanzung ebener elektromagnetischer wellen längs einer ebenen Lieterfläche und ihre Beziehung zur drahtlosen Telegraphie. Annalen der Physik (Leip-

[3] D'yakonov MI. New type of electromagnetic wave propagating at an interface. Sov. Phys.

[4] Zapata-Rodríguez CJ, Miret JJ, Sorni JA, Vuković SM. Propagation of dyakonon wavepackets at the boundary of metallodielectric lattices. IEEE Journal of Selected Topics in

[2] Maier SA. Plasmonics: Fundamentals and Applications. New York: Springer; 2007

2,3, Juan J. Miret4

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

http://dx.doi.org/10.5772/intechopen.74126

67

, Mahin Naserpour1,5 and

\*, Slobodan Vuković

5 Department of Physics, College of Sciences, Shiraz University, Shiraz, Iran

C2-1-R and TEC2017-86102-C2-1R).

Author details

Milivoj R. Belić

References

zig). 1907;23:846-866

JETP. 1988;67:714-716

Quantum Electronics. 2013;19:4601408

Spain

Carlos J. Zapata-Rodríguez<sup>1</sup>

3

\*Address all correspondence to: carlos.zapata@uv.es

(IHTM), University of Belgrade, Belgrade, Serbia 3 Texas A&M University at Qatar, Doha, Qatar

#### 6.1. DSWs in band-gap hyperbolic media

In previous sections we demonstrated that the presence of metallic nano-elements leads to nonlocal effects and dissipation effects which reshape the propagation dynamics of the surface signal. Here, we briefly discuss the extraordinary favorable conditions which may appear in band-gap metal–insulator-layered media for the existence of DSWs. As reported thoroughly by Miret et al. [29], engineering secondary bands by tuning the plasmonic-crystal geometry may lead to a controlled optical anisotropy, which is markedly dissimilar to the prescribed hyperbolic regime that is derived by the EMA, however, assisting the presence of DSWs on the interface between such hyperbolic metamaterial and an insulator.

In particular, a surface wave propagating on an Ag-Ge grating was considered, where the environment medium that is set above the metallic grating is formed by SiO2. If the metalfilling factor was f ¼ 0:25, the effective permittivities of the anisotropic metamaterial would be estimated as E<sup>⊥</sup> ¼ �11:48 þ i2:05 and E<sup>k</sup> ¼ 25:96 þ i0:14 at a wavelength of λ<sup>0</sup> ¼ 1550 nm. Disregarding losses, the DSW dispersion curve describes an incomplete hyperbolic curve, finding an endpoint under the condition κ<sup>e</sup> ¼ 0, where the extraordinary wave breaks its confinement in the vicinities of the isotropic-uniaxial interface [26].

Considering now a realistic nanostructure consisting of Ag layers of w<sup>2</sup> ¼ 40 nm interspersed between Ge layers of w<sup>1</sup> ¼ 120 nm, thus maintaining a metal-filling factor of f ¼ 0:25 as analyzed earlier, a first TM band with hyperbolic-like characteristics dominates at high inplane frequencies kt. Additionally, a second band emerges for TM Bloch modes, which exhibits a moderate anisotropy, demonstrating near-elliptical dispersion curves (with positive effective permittivities) and positive birefringence. Furthermore, TE modal dispersion is roughly isotropic. Therefore, satisfactory conditions are found near the second TM band for the existence of Dyakonov-like surface waves. Finally, in order to numerically obtain the dispersion curves and wave fields associated with DSWs, one may follow the same computational procedure followed by Zapata-Rodríguez et al. and [4, 26].

#### 7. Summary

In this chapter we provide several methods to analytically calculate and numerically simulate modal propagation of DSWs governed by material anisotropy. We focused on the spatial properties of DSWs at optical and telecom wavelengths, particularly using uniaxial metamaterials formed of dielectric and metallic nanolayers. We developed an electromagnetic matrix procedure enabling different aspects reviewed in this chapter, specially adapted to complex multilayered configurations. The EMA results are particularly appropriate for the characterization of the form birefringence of a multilayered nanostructure, though limitations driven by the layers width have been discussed. Through a rigorous full-wave analysis, we showed that hybrid-polarized surface waves may propagate obliquely at the boundary between a plasmonic bilayer superlattice and an isotropic loss-free material. We revealed that realistic widths of the slabs might lead to solutions which deviate significantly from the results derived directly from the EMA and Dyakonov analysis. Finally, we showed that excitation of DSWs at the boundary of an isotropic dielectric and a hyperbolic metamaterial enables a distinct regime of propagation. It is important to note that the properties of the resulting bound states change drastically with the index of refraction of the surrounding medium, suggesting potential applications in chemical and biological sensing and nanoimaging.

This chapter was supported by the Qatar National Research Fund (Grant No. NPRP 8-028-1- 001) and the Spanish Ministry of Economy and Competitiveness (Grants No. TEC2014-53727- C2-1-R and TEC2017-86102-C2-1R).

## Author details

where solutions to the Dyakonov equation begin and end, respectively. It is clear that the

In previous sections we demonstrated that the presence of metallic nano-elements leads to nonlocal effects and dissipation effects which reshape the propagation dynamics of the surface signal. Here, we briefly discuss the extraordinary favorable conditions which may appear in band-gap metal–insulator-layered media for the existence of DSWs. As reported thoroughly by Miret et al. [29], engineering secondary bands by tuning the plasmonic-crystal geometry may lead to a controlled optical anisotropy, which is markedly dissimilar to the prescribed hyperbolic regime that is derived by the EMA, however, assisting the presence of DSWs on the

In particular, a surface wave propagating on an Ag-Ge grating was considered, where the environment medium that is set above the metallic grating is formed by SiO2. If the metalfilling factor was f ¼ 0:25, the effective permittivities of the anisotropic metamaterial would be estimated as E<sup>⊥</sup> ¼ �11:48 þ i2:05 and E<sup>k</sup> ¼ 25:96 þ i0:14 at a wavelength of λ<sup>0</sup> ¼ 1550 nm. Disregarding losses, the DSW dispersion curve describes an incomplete hyperbolic curve, finding an endpoint under the condition κ<sup>e</sup> ¼ 0, where the extraordinary wave breaks its

Considering now a realistic nanostructure consisting of Ag layers of w<sup>2</sup> ¼ 40 nm interspersed between Ge layers of w<sup>1</sup> ¼ 120 nm, thus maintaining a metal-filling factor of f ¼ 0:25 as analyzed earlier, a first TM band with hyperbolic-like characteristics dominates at high inplane frequencies kt. Additionally, a second band emerges for TM Bloch modes, which exhibits a moderate anisotropy, demonstrating near-elliptical dispersion curves (with positive effective permittivities) and positive birefringence. Furthermore, TE modal dispersion is roughly isotropic. Therefore, satisfactory conditions are found near the second TM band for the existence of Dyakonov-like surface waves. Finally, in order to numerically obtain the dispersion curves and wave fields associated with DSWs, one may follow the same computational procedure

In this chapter we provide several methods to analytically calculate and numerically simulate modal propagation of DSWs governed by material anisotropy. We focused on the spatial properties of DSWs at optical and telecom wavelengths, particularly using uniaxial metamaterials formed of dielectric and metallic nanolayers. We developed an electromagnetic matrix procedure enabling different aspects reviewed in this chapter, specially adapted to complex multilayered configurations. The EMA results are particularly appropriate for the characterization of the form birefringence of a multilayered nanostructure, though limitations driven by the layers width have been discussed. Through a rigorous full-wave analysis, we

angular range of DSWs now turns to be significantly low.

interface between such hyperbolic metamaterial and an insulator.

confinement in the vicinities of the isotropic-uniaxial interface [26].

followed by Zapata-Rodríguez et al. and [4, 26].

7. Summary

6.1. DSWs in band-gap hyperbolic media

66 Surface Waves - New Trends and Developments

Carlos J. Zapata-Rodríguez<sup>1</sup> \*, Slobodan Vuković 2,3, Juan J. Miret4 , Mahin Naserpour1,5 and Milivoj R. Belić 3

\*Address all correspondence to: carlos.zapata@uv.es

1 Departament of Optics and Optometry and Vision Science, University of Valencia, Burjassot, Spain

2 Center of Microelectronic Technologies, Institute of Chemistry, Technology and Metallurgy (IHTM), University of Belgrade, Belgrade, Serbia


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[5] Takayama O, Crasovan LC, Johansen SK, Mihalache D, Artigas D, Torner L. Dyakonov surface waves:A review. Electromagnetics. 2008;28:126-145

[21] Vuković SM, Miret JJ, Zapata-Rodríguez CJ, Jaksić Z. Oblique surface waves at an interface of metal-dielectric superlattice and isotropic dielectric. Physica Scripta. 2012;T149:

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

http://dx.doi.org/10.5772/intechopen.74126

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[22] Ghasempour Ardakani A, Naserpour M, Zapata Rodríguez CJ. Dyakonov-like surface waves in the THz regime. Photonics and Nanostructures - Fundamentals and Applica-

[23] Miret JJ, Zapata-Rodrïguez CJ, Jaksić Z, Vuković SM, Belić MR. Substantial enlargement of angular existence range for Dyakonov-like surface waves at semi-infinite metal-dielectric

[24] Elser J, Podolskiy VA, Salakhutdinov I, Avrutsky I. Nonlocal effects in effective-medium response of nanolayered metamaterials. Applied Physics Letters. 2007;90:191109

[25] Sorni JA, Naserpour M, Zapata-Rodríguez CJ, Miret JJ. Dyakonov surface waves in lossy

[26] Zapata-Rodríguez CJ, Miret JJ, Vuković S, Belić MR. Engineered surface waves in hyper-

[27] Xiang Y, Guo J, Dai X, Wen S, Tang D. Engineered surface Bloch waves in graphene-based

[28] Jacob Z, Narimanov EE. Optical hyperspace for plasmons: Dyakonov states in metama-

[29] Miret JJ, Sorní JA, Naserpour M, Ghasempour Ardakani A, Zapata-Rodríguez CJ. Nonlocal dispersion anomalies of Dyakonov-like surface waves at hyperbolic media interfaces. Photonics and Nanostructures - Fundamentals and Applications. 2016;18:16-22

superlattice. Journal of Nanophotonics. 2012;6. DOI: 063525

metamaterials. Optics Communication. 2015;355:251-255

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[21] Vuković SM, Miret JJ, Zapata-Rodríguez CJ, Jaksić Z. Oblique surface waves at an interface of metal-dielectric superlattice and isotropic dielectric. Physica Scripta. 2012;T149: 014041

[5] Takayama O, Crasovan LC, Johansen SK, Mihalache D, Artigas D, Torner L. Dyakonov

[6] Berreman DW. Optics in stratified and anisotropic media: 44 matrix formulation. Journal

[7] Hodgkinson IJ, Kassam S, Wu QH. Eigenequations and compact algorithms for bulk and layered anisotropic optical media: Reflection and refraction at a crystal-crystal interface.

[8] Barnes WL, Dereux A, Ebbesen TW. Surface plasmon subwavelength optics. Nature. 2003;

[9] Zayats AV, Smolyaninov II, Maradudin AA. Nano-optics of surface plasmon polaritons.

[10] Yariv A, Yeh P. Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers. Journal of the Optical Society of America. 1977;

[11] Vuković SM, Shadrivov IV, Kivshar YS. Surface Bloch waves in metamaterial and metal-

[12] Rytov SM. Electromagnetic properties of a finely stratified medium. Soviet Physics - JETP.

[13] Smith DR, Schurig D, Rosenbluth M, Schultz S. Limitations on subdiffraction imaging with a negative refractive index slab. Applied Physics Letters. 2003;82:1506-1508

[14] Smolyaninov II, Hwang E, Narimanov E. Hyperbolic metamaterial interfaces: Hawking radiation from Rindler horizons and spacetime signature transitions. Physical Review B.

[15] Guo Y, Newman W, Cortes CL, Jacob Z. Applications of hyperbolic metamaterial sub-

[16] Cortes CL, Newman W, Molesky S, Jacob Z. Quantum nanophotonics using hyperbolic

[17] Takayama O, Crasovan L, Artigas D, Torner L. Observation of Dyakonov surface waves.

[18] Takayama O, Artigas D, Torner L. Lossless directional guiding of light in dielectric nanosheets using Dyakonov surface waves. Nature Nanotechnology. 2014;9:419-424 [19] Walker DB, Glytsis EN, Gaylord TK. Surface mode at isotropic uniaxial and isotropicbiaxial interfaces. Journal of the Optical Society of America. A. 1998;15:248-260

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Physics Reports. 2005;408:131-314


**Chapter 4**

Provisional chapter

**Measurement of Sea Wave Spatial Spectra from High-**

DOI: 10.5772/intechopen.71834

The chapter is devoted to the development of methods for remote measurement of spatial spectra of waves arising on marine and ocean surface. It is shown that in most natural conditions of optical image formation, a nonlinear modulation of the brightness field occurs by slopes of water surface elements. Methods for reconstructing the spectra of surface waves from optical image spectra with allowance for such modulation are proposed. The methods are based on the numerical simulation of water surface taking into account wave formation conditions and conditions of light entering the sea surface from the upper and lower hemispheres. Using the results of numerical simulation, special operators are built to retrieve wave spectra from the spectra of aerospace images. These retrieving operators are presented in the form of analytical expressions, depending on the sets of parameters, which are determined by the conditions for the formation of images. The results of experimental studies of the sea wave spectra in various water areas using satellite optical images of high spatial resolution are presented. In the experimental studies, the spatial spectral characteristics of sea waves estimated from remote sensing data were compared with the corresponding characteristics measured by contact assets under

Keywords: wave spectra, surface waves, remote sensing, image processing

Registration of spatial spectra of surface waves is actual in solving many fundamental and applied problems of modern oceanology [1–3]. Obtaining information about such spectra is important for studying various physical processes occurring near the ocean-atmosphere interface, detecting water pollution, and monitoring anthropogenic impacts on the marine areas [1, 4–12].

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

Measurement of Sea Wave Spatial Spectra from

High-Resolution Optical Aerospace Imagery

**Resolution Optical Aerospace Imagery**

Valery G. Bondur and Alexander B. Murynin

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Valery G. Bondur and Alexander B. Murynin

http://dx.doi.org/10.5772/intechopen.71834

Abstract

controlled conditions.

1. Introduction

Provisional chapter

#### **Measurement of Sea Wave Spatial Spectra from High-Resolution Optical Aerospace Imagery** Measurement of Sea Wave Spatial Spectra from

DOI: 10.5772/intechopen.71834

Valery G. Bondur and Alexander B. Murynin

High-Resolution Optical Aerospace Imagery

Additional information is available at the end of the chapter Valery G. Bondur and Alexander B. Murynin

http://dx.doi.org/10.5772/intechopen.71834 Additional information is available at the end of the chapter

#### Abstract

The chapter is devoted to the development of methods for remote measurement of spatial spectra of waves arising on marine and ocean surface. It is shown that in most natural conditions of optical image formation, a nonlinear modulation of the brightness field occurs by slopes of water surface elements. Methods for reconstructing the spectra of surface waves from optical image spectra with allowance for such modulation are proposed. The methods are based on the numerical simulation of water surface taking into account wave formation conditions and conditions of light entering the sea surface from the upper and lower hemispheres. Using the results of numerical simulation, special operators are built to retrieve wave spectra from the spectra of aerospace images. These retrieving operators are presented in the form of analytical expressions, depending on the sets of parameters, which are determined by the conditions for the formation of images. The results of experimental studies of the sea wave spectra in various water areas using satellite optical images of high spatial resolution are presented. In the experimental studies, the spatial spectral characteristics of sea waves estimated from remote sensing data were compared with the corresponding characteristics measured by contact assets under controlled conditions.

Keywords: wave spectra, surface waves, remote sensing, image processing

#### 1. Introduction

Registration of spatial spectra of surface waves is actual in solving many fundamental and applied problems of modern oceanology [1–3]. Obtaining information about such spectra is important for studying various physical processes occurring near the ocean-atmosphere interface, detecting water pollution, and monitoring anthropogenic impacts on the marine areas [1, 4–12].

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

To obtain two-dimensional spectra of surface waves in large water areas, including hard-to-reach ones, the use of remote sensing methods based on the processing of various images obtained from air carriers and also space images of high spatial resolution is promising [4, 13–15].

An adequate estimation of such spectra formed by image processing obtained in the process of aerospace monitoring of marine areas of the seas and oceans requires the use of reconstructing operators that are functions that allow the spatial spectra of brightness fields recorded in optical aerospace images to be transformed into sea wave spectra [4, 16–18, 24]. These operators are built using numerical simulation methods based on the various conditions for the formation of aerospace images and the characteristics of remote sensing equipment [4, 16–21]. Initially, these methods were used to construct retrieving operators, which allow us to obtain wavelet spectra in the equilibrium interval [16–18]. At the present time, the development of modified retrieving operators, which are a superposition of the high-frequency and lowfrequency components, is being developed and is suitable for use in the low-frequency region, including near the spectral maximum [17, 18].

This chapter describes a method for retrieving sea wave spectra from the spectra of aerospace optical images over a wide range of spatial frequencies, including the equilibrium interval, the spectral maximum, and the low-frequency region. To calibrate and verify the adequacy of the developed method, the contact data obtained in synchronous measurements with the help of an array of string wave recorders are used. The results of experimental studies carried out using the developed method are presented.

#### 2. An approach to retrieve marine surface spectra

Rough sea surface is a random field of elevations (wave applications).

$$z = \zeta \left( x, y, t \right) \tag{1}$$

Ψð Þ¼ k S½ � ξ ð Þ k (3)

http://dx.doi.org/10.5772/intechopen.71834

<sup>ξ</sup><sup>x</sup> <sup>x</sup>; <sup>y</sup> <sup>¼</sup> <sup>∂</sup><sup>ξ</sup> <sup>x</sup>; <sup>y</sup> <sup>=</sup>∂x, <sup>ξ</sup><sup>y</sup> <sup>x</sup>; <sup>y</sup> <sup>¼</sup> <sup>∂</sup><sup>ξ</sup> <sup>x</sup>; <sup>y</sup> <sup>=</sup>∂<sup>y</sup> (4)

Measurement of Sea Wave Spatial Spectra from High-Resolution Optical Aerospace Imagery

ξϕ <sup>x</sup>; <sup>y</sup> <sup>¼</sup> cosϕξ<sup>x</sup> <sup>x</sup>; <sup>y</sup> <sup>þ</sup> sinϕξ<sup>y</sup> <sup>x</sup>; <sup>y</sup> (5)

<sup>Ψ</sup>ð Þ <sup>k</sup> (6)

Φð Þ¼ k RSð Þ k (8)

,

73

where S is the spectral density operator, which is proportional to the square of the modulus of

Since optical images of the sea surface are formed as a result of reflection and refraction of light according to the laws of geometrical optics, for their analysis, the structure of the sea surface along with the field of elevations (x, y) is conveniently characterized by fields of slopes (or

The gradient of the sea surface in an arbitrary direction ϕ, with allowance for (3), can be

Taking into account the properties of the Fourier transform, one can associate the spectrum of

The brightness field, recorded by the remote sensing equipment at a fixed time, can be expanded in a power series along the surface slopes and is represented in the form [1, 11, 12]:

where N is the nonlinear component of the signal, containing terms proportional to (ξx(x,y))<sup>2</sup>

The contribution to the detected signal of the nonlinear component N (x, y, ξx, ξy) is determined by a number of parameters: lighting conditions, wave state, and recording equipment

In order to change from the spectrum of the optical image S(k), obtained under known conditions, to the slope spectrum of the sea surface Φ(k) in the direction determined by these

As a rule, analytical estimates of the contribution of the nonlinear component of N to the spatial spectrum of the luminosity field are difficult; therefore, the numerical simulation

The method of constructing the retrieving operator developed and described in this chapter is an extension of the method proposed in [1, 5, 6, 10–14]. To construct a retrieving operator corresponding to certain image acquisition conditions, direct numerical modeling of optical

method is used to solve the problem of constructing the recovery operator [1, 11–13].

and ξy(x,y) are fields of slopes (gradients of the elevation field) of the sea surface.

conditions, the definition of the retrieving operator R is introduced as:

L x; <sup>y</sup> <sup>¼</sup> C0 <sup>þ</sup> Cxξ<sup>x</sup> <sup>x</sup>; <sup>y</sup> <sup>þ</sup> Cyξ<sup>y</sup> <sup>x</sup>; <sup>y</sup> <sup>þ</sup> N x; <sup>y</sup>; <sup>ξ</sup><sup>x</sup> <sup>x</sup>; <sup>y</sup> ; <sup>ξ</sup><sup>y</sup> <sup>x</sup>; <sup>y</sup> (7)

, and so on. C0, Cx, and Cy are coefficients of the linear part of the expansion and ξx(x,y)

Φð Þ¼ k cosϕ kx þ sinϕ ky

the Fourier transform of the field ξ(x, y); k = (kx, ky) is the wave vector.

such a field of slopes with the spectrum of the field of elevations:

gradients) along the axes [2, 4, 17].

expressed as follows:

(ξy(x,y))2

characteristics [4, 17].

where ζ (x, y, t) is random function of sea surface elevations (elevation field); (x, y, z) is a rectangular Cartesian coordinate system in which the (x, y) plane coincides with the level of a calm (undisturbed) water surface; t is time.

Fixing the time instant t = t<sup>0</sup> in (1), we obtain a two-dimensional random function of spatial coordinates:

$$z = \zeta(\mathbf{x}, y, t)|\_{t=t0} = \xi(\mathbf{x}, y) \tag{2}$$

Aerospace images, which are recorded by remote methods, are used to study the characteristics of the sea surface elevation field at a fixed time z = ξ (x, y). Two-dimensional signal fields that are represented in aerospace images are associated with the sea surface elevation field and can be used to estimate significant characteristics of this surface.

Since the sea surface elevation field (x, y) is a Gaussian quasistationary field, it is described quite adequately by the spectral density [2]:

Measurement of Sea Wave Spatial Spectra from High-Resolution Optical Aerospace Imagery http://dx.doi.org/10.5772/intechopen.71834 73

$$\Psi(\mathbf{k}) = \mathbf{S}[\xi](\mathbf{k}) \tag{3}$$

where S is the spectral density operator, which is proportional to the square of the modulus of the Fourier transform of the field ξ(x, y); k = (kx, ky) is the wave vector.

To obtain two-dimensional spectra of surface waves in large water areas, including hard-to-reach ones, the use of remote sensing methods based on the processing of various images obtained

An adequate estimation of such spectra formed by image processing obtained in the process of aerospace monitoring of marine areas of the seas and oceans requires the use of reconstructing operators that are functions that allow the spatial spectra of brightness fields recorded in optical aerospace images to be transformed into sea wave spectra [4, 16–18, 24]. These operators are built using numerical simulation methods based on the various conditions for the formation of aerospace images and the characteristics of remote sensing equipment [4, 16–21]. Initially, these methods were used to construct retrieving operators, which allow us to obtain wavelet spectra in the equilibrium interval [16–18]. At the present time, the development of modified retrieving operators, which are a superposition of the high-frequency and lowfrequency components, is being developed and is suitable for use in the low-frequency region,

This chapter describes a method for retrieving sea wave spectra from the spectra of aerospace optical images over a wide range of spatial frequencies, including the equilibrium interval, the spectral maximum, and the low-frequency region. To calibrate and verify the adequacy of the developed method, the contact data obtained in synchronous measurements with the help of an array of string wave recorders are used. The results of experimental studies carried out

where ζ (x, y, t) is random function of sea surface elevations (elevation field); (x, y, z) is a rectangular Cartesian coordinate system in which the (x, y) plane coincides with the level of a

Fixing the time instant t = t<sup>0</sup> in (1), we obtain a two-dimensional random function of spatial

Aerospace images, which are recorded by remote methods, are used to study the characteristics of the sea surface elevation field at a fixed time z = ξ (x, y). Two-dimensional signal fields that are represented in aerospace images are associated with the sea surface elevation field and

Since the sea surface elevation field (x, y) is a Gaussian quasistationary field, it is described

z ¼ ζ ð Þ x; y; t (1)

<sup>z</sup> <sup>¼</sup> <sup>ζ</sup> ð Þj <sup>x</sup>; <sup>y</sup>; <sup>t</sup> <sup>t</sup>¼t<sup>0</sup> <sup>¼</sup> <sup>ξ</sup>ð Þ <sup>x</sup>; <sup>y</sup> (2)

from air carriers and also space images of high spatial resolution is promising [4, 13–15].

including near the spectral maximum [17, 18].

72 Surface Waves - New Trends and Developments

using the developed method are presented.

calm (undisturbed) water surface; t is time.

quite adequately by the spectral density [2]:

coordinates:

2. An approach to retrieve marine surface spectra

can be used to estimate significant characteristics of this surface.

Rough sea surface is a random field of elevations (wave applications).

Since optical images of the sea surface are formed as a result of reflection and refraction of light according to the laws of geometrical optics, for their analysis, the structure of the sea surface along with the field of elevations (x, y) is conveniently characterized by fields of slopes (or gradients) along the axes [2, 4, 17].

$$
\mathfrak{E}\_{\mathbf{x}}(\mathbf{x}, \mathbf{y}) = \mathfrak{d}\xi(\mathbf{x}, \mathbf{y}) / \mathfrak{d}\mathfrak{x}, \mathfrak{E}\_{\mathbf{y}}(\mathbf{x}, \mathbf{y}) = \mathfrak{d}\xi(\mathbf{x}, \mathbf{y}) / \mathfrak{d}\mathfrak{y} \tag{4}
$$

The gradient of the sea surface in an arbitrary direction ϕ, with allowance for (3), can be expressed as follows:

$$\mathfrak{E}\_{\phi}(\mathbf{x}, \mathbf{y}) = \cos \phi \mathfrak{E}\_{\mathbf{x}}(\mathbf{x}, \mathbf{y}) + \sin \phi \mathfrak{E}\_{\mathbf{y}}(\mathbf{x}, \mathbf{y}) \tag{5}$$

Taking into account the properties of the Fourier transform, one can associate the spectrum of such a field of slopes with the spectrum of the field of elevations:

$$\boldsymbol{\Phi}(\mathbf{k}) = \left(\cos\phi \,\mathbf{k}\_{\mathbf{x}} + \sin\phi \,\mathbf{k}\_{\mathbf{y}}\right) \,\Psi(\mathbf{k})\tag{6}$$

The brightness field, recorded by the remote sensing equipment at a fixed time, can be expanded in a power series along the surface slopes and is represented in the form [1, 11, 12]:

$$\mathbf{L}(\mathbf{x}, \mathbf{y}) = \mathbf{C}\_0 + \mathbf{C}\_\mathbf{x} \xi\_\mathbf{x}(\mathbf{x}, \mathbf{y}) + \mathbf{C}\_\mathbf{y} \xi\_\mathbf{y}(\mathbf{x}, \mathbf{y}) + \mathbf{N}(\mathbf{x}, \mathbf{y}, \xi\_\mathbf{x}(\mathbf{x}, \mathbf{y}), \xi\_\mathbf{y}(\mathbf{x}, \mathbf{y})) \tag{7}$$

where N is the nonlinear component of the signal, containing terms proportional to (ξx(x,y))<sup>2</sup> , (ξy(x,y))2 , and so on. C0, Cx, and Cy are coefficients of the linear part of the expansion and ξx(x,y) and ξy(x,y) are fields of slopes (gradients of the elevation field) of the sea surface.

The contribution to the detected signal of the nonlinear component N (x, y, ξx, ξy) is determined by a number of parameters: lighting conditions, wave state, and recording equipment characteristics [4, 17].

In order to change from the spectrum of the optical image S(k), obtained under known conditions, to the slope spectrum of the sea surface Φ(k) in the direction determined by these conditions, the definition of the retrieving operator R is introduced as:

$$\boldsymbol{\Phi}(\mathbf{k}) = \mathbf{R}\mathbf{S}(\mathbf{k})\tag{8}$$

As a rule, analytical estimates of the contribution of the nonlinear component of N to the spatial spectrum of the luminosity field are difficult; therefore, the numerical simulation method is used to solve the problem of constructing the recovery operator [1, 11–13].

The method of constructing the retrieving operator developed and described in this chapter is an extension of the method proposed in [1, 5, 6, 10–14]. To construct a retrieving operator corresponding to certain image acquisition conditions, direct numerical modeling of optical images is performed under a given set of conditions [8–11], after which an approximation of the spatial frequency filter (transfer function) [11–14] is constructed, which allows obtaining a spatial spectrum of sea surface slopes from the aerospace image spectrum.

Parametrization of the spatial-frequency filter using a set of parameters that depends on the experimental conditions was proposed in [21]. The adequacy of the parametrization was experimentally verified in the equilibrium interval using contact data obtained by wave recorders and also by stereoscopic photography from the oceanographic platform [22].

However, the operator obtained in [21] does not allow for reconstructing the wave spectra in the region of low spatial frequencies and near the spectral maximum. To eliminate this shortcoming, an improvement was made to the method of constructing the retrieving operator and its approbation in various conditions [20, 19].

The modified retrieving operator Rmod is represented as the product of two operators (transfer functions):

$$\mathbf{R}\_{\text{mod}}(\mathbf{k}) = \mathbf{R}\_{\text{low}}(\mathbf{k}) \mathbf{R}\_{\text{high}}(\mathbf{k}) \tag{9}$$

and temporal characteristics of sea waves based on the corresponding hydrodynamic models, it is possible to calibrate remote sensing assets for measuring sea surface characteristics.

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As sources of information for comparing the results of remote and direct contact measure-

• Measurements in which temporal sequences of elevations of the sea surface are formed

• Stereo-photogrammetric measurements performed from close distances and allowing us to directly measure the two-dimensional realizations of the sea surface elevation field with

For direct determination of characteristics of sea waves, contact methods usually employ arrays of sensors (wave recorders) measuring the parameters of sea surface (gradient, elevation, and acceleration) at one or several points spaced from each other by some distance. To estimate spatiotemporal spectra from the arrays of wave recorders, indirect iterative calculation methods are widely used, in which some hypotheses about the statistical properties of

The stereo-photogrammetric processing of synchronously registered optical images of the sea surface from two different points of view (stereopairs) makes it possible to directly measure the realizations of two-dimensional fields of sea surface elevations, and calculate spatial spec-

The stereo-photogrammetric method allows measuring the shape of rough sea surface. The drawbacks of this method include the greater complexity of processing primary data and high

To compare the frequency and frequency-directed wave spectra recorded by wave recorders with two-dimensional and one-dimensional spatial spectra of sea surface elevations recorded by remote data, it is necessary to develop a special approach that takes into account the

Taking into account the features considered, the present chapter proposes an approach to

• Conducting experiments under controlled conditions, including satellite imagery of sea surface test areas and synchronous measurements using contact equipment and/or stereo-

• Retrieving sea surface spectra from space images under fixed conditions for obtaining

• Calculation of frequency spectra of waves from contact data obtained by wave recorders. • Formation of wave spectra from the satellite data of stereo-photogrammetric image

comparing satellite and contact measurements, which consists of the following:

photogrammetric imagery of the sea surface from a low altitude.

ments of sea surface characteristics, we will use the following types of measurements:

• Measurements using wave buoys drifting in the space survey area.

using stationary contact sensors.

tra of elevations on the base of these realizations.

features of gravitational and gravitational-capillary waves.

demands on the computing resources used.

high spatial resolution.

wave are postulated.

these images.

processing.

where Rhigh is a recovery operator (transfer function) in the high-frequency region; Rlow is a recovery operator (transfer function) in the low-frequency range.

An approximation of the retrieving operator in the form

$$\mathbf{R(k)} = a\_0 \left( \exp \left( a\_4 k^{a\_5} \right) \right) \left( \cos \left( \wp - \wp\_c \right) \right)^{a\_3} k^{a\_1 + a\_2 \cos \left( \wp - \wp\_c \right)} \tag{10}$$

where the parameter vector a = (a0, a1, a2, a3, a4, a5) is formed on the basis of experimental data obtained in complex experiments, including remote and in situ measurements of wave spectra.

#### 3. Calibration and verification of the adequacy of remote methods

When verifying the adequacy of remote methods of measuring wave spectra, sea truth data obtained in specially conducted experiments under controlled conditions using reference techniques and means ensuring the measurement of the sea surface spectra with sufficient accuracy should be used. As such methods and means in this chapter, measurements of sea surface elevations made using a grid of string wave recorders, stereo photography from low altitudes, and measurements using wave buoys will be used.

In an experimental verification of the reliability of methods for recording wave spectra from the spectra of satellite images, we will make a quantitative comparison of the results of remote measurements with data obtained by direct measurements by contact methods.

Having obtained under the same conditions the results of remote measurements of instantaneous two-dimensional fields of slopes and elevations, as well as data of direct (contact) measurements of local time series of heights, and also using the method of comparing spatial and temporal characteristics of sea waves based on the corresponding hydrodynamic models, it is possible to calibrate remote sensing assets for measuring sea surface characteristics.

As sources of information for comparing the results of remote and direct contact measurements of sea surface characteristics, we will use the following types of measurements:


images is performed under a given set of conditions [8–11], after which an approximation of the spatial frequency filter (transfer function) [11–14] is constructed, which allows obtaining a

Parametrization of the spatial-frequency filter using a set of parameters that depends on the experimental conditions was proposed in [21]. The adequacy of the parametrization was experimentally verified in the equilibrium interval using contact data obtained by wave

However, the operator obtained in [21] does not allow for reconstructing the wave spectra in the region of low spatial frequencies and near the spectral maximum. To eliminate this shortcoming, an improvement was made to the method of constructing the retrieving operator and

The modified retrieving operator Rmod is represented as the product of two operators (transfer

where Rhigh is a recovery operator (transfer function) in the high-frequency region; Rlow is a

where the parameter vector a = (a0, a1, a2, a3, a4, a5) is formed on the basis of experimental data obtained in complex experiments, including remote and in situ measurements of wave spectra.

When verifying the adequacy of remote methods of measuring wave spectra, sea truth data obtained in specially conducted experiments under controlled conditions using reference techniques and means ensuring the measurement of the sea surface spectra with sufficient accuracy should be used. As such methods and means in this chapter, measurements of sea surface elevations made using a grid of string wave recorders, stereo photography from low

In an experimental verification of the reliability of methods for recording wave spectra from the spectra of satellite images, we will make a quantitative comparison of the results of remote

Having obtained under the same conditions the results of remote measurements of instantaneous two-dimensional fields of slopes and elevations, as well as data of direct (contact) measurements of local time series of heights, and also using the method of comparing spatial

measurements with data obtained by direct measurements by contact methods.

<sup>a</sup><sup>3</sup>

<sup>a</sup><sup>5</sup> ð Þ cos <sup>φ</sup> � <sup>φ</sup><sup>c</sup>

3. Calibration and verification of the adequacy of remote methods

Rmodð Þ¼ k Rlowð Þ k Rhighð Þ k (9)

<sup>a</sup>1þa<sup>2</sup> cos ð Þ <sup>φ</sup>�φ<sup>c</sup> (10)

k

recorders and also by stereoscopic photography from the oceanographic platform [22].

spatial spectrum of sea surface slopes from the aerospace image spectrum.

its approbation in various conditions [20, 19].

74 Surface Waves - New Trends and Developments

recovery operator (transfer function) in the low-frequency range.

An approximation of the retrieving operator in the form

R kð Þ¼ a<sup>0</sup> exp a4k

altitudes, and measurements using wave buoys will be used.

functions):

• Stereo-photogrammetric measurements performed from close distances and allowing us to directly measure the two-dimensional realizations of the sea surface elevation field with high spatial resolution.

For direct determination of characteristics of sea waves, contact methods usually employ arrays of sensors (wave recorders) measuring the parameters of sea surface (gradient, elevation, and acceleration) at one or several points spaced from each other by some distance. To estimate spatiotemporal spectra from the arrays of wave recorders, indirect iterative calculation methods are widely used, in which some hypotheses about the statistical properties of wave are postulated.

The stereo-photogrammetric processing of synchronously registered optical images of the sea surface from two different points of view (stereopairs) makes it possible to directly measure the realizations of two-dimensional fields of sea surface elevations, and calculate spatial spectra of elevations on the base of these realizations.

The stereo-photogrammetric method allows measuring the shape of rough sea surface. The drawbacks of this method include the greater complexity of processing primary data and high demands on the computing resources used.

To compare the frequency and frequency-directed wave spectra recorded by wave recorders with two-dimensional and one-dimensional spatial spectra of sea surface elevations recorded by remote data, it is necessary to develop a special approach that takes into account the features of gravitational and gravitational-capillary waves.

Taking into account the features considered, the present chapter proposes an approach to comparing satellite and contact measurements, which consists of the following:


• Comparison of wave spectra obtained remotely with contact and/or stereo-photogrammetric measurements.

Consider the relationship between the spatial spectra of the sea surface, reconstructed from optical images, as well as frequency and frequency-directed wave spectra, measured by contact sensors, and characterize the fluctuations in sea level over time at a fixed point.

The frequency spectrum of sea surface elevations Sξ(ω) characterizes the distribution of wave oscillations at a given point along the cyclic frequencies ω, ω = 2π/τ, where τ is the period of the wave oscillation.

The spatial wave spectrum characterizes the energy distribution at a fixed time instant with respect to wave numbers k (or spatial frequencies ν = 1/Λ), k = 2π/Λ, where Λ is the length of the surface wave.

Spatial and frequency spectra of waves are functions of different arguments. They have different physical meanings and require the use of different methods of measurement. Therefore, to compare the spectra measured by different methods, a dispersion relation is used that describes the relationship between the time and spatial frequencies of waves, depending on the physical mechanisms that form the surface waves of the range in question. Within the framework of the linear hydrodynamic model, the components of the wave spectrum can be considered as elementary plane waves, for which the dispersion relation of the theory of potential waves of small amplitude that relates the cyclic frequency of the wave (ω) to the wave number k is valid. In deep water (for kh ≫ 1, where h is the depth), the dispersion relation taking into account the contribution of gravitational and capillary forces to wave formation has the form

$$
\omega(k) = \sqrt{\text{gk} + (\text{T}/\rho)k^3} \tag{11}
$$

4. The results of experimental studies

types of experiments were used [19, 22]:

drifting wave buoys.

top of the pentagon, were used.

ment zone.

• Resolution—no worse than 3 mm • Number of measuring channels—6

To solve the problems of validation of the developed methods, the results obtained in three

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• The first type of experiments consisted of carrying out space surveys and synchronous sea truth measurements of wave spectra near the stationary oceanographic platform.

• The second type of experiments consisted of carrying out a space survey of the investigated water area and simultaneous measurements of wave spectra with the help of

• The third type of experiments consisted of carrying out studies in the short-wave region of wave spectra (Λ = 0.04–1.0 m) with the help of string wave recorders, object photography

Experimental work of the first type was carried out in the area of the oceanographic platform

For the measurement of space-time wave spectra, contact data obtained using a wave measuring unit based on an array of string wave recorders, which were a set of six resistive wave recorders measuring the elevations of the sea surface at points located in the center and at the

The distance from the central string to each of the external strings was 25 cm. The main

During the first type of experiments, a special space imagery was conducted in the vicinity of the oceanographic platform using high-spatial resolution (0.5 m) optic-electronic equipment installed on board the GEOEYE spacecraft. A brief overview of the complex experiments conducted on September 24, 2015 at 11:52 (LT) and on September 12, 2011 at 12:06 is presented below. The presented experiments were characterized by different conditions of wave formation. During the first complex experiment, developing wind waves were observed at a nearsurface wind speed of about 4 m/s. During the second complex experiment, the velocity of the near-surface wind varied from 0 to 2 m/s; however, swell waves were present in the experi-

Figures 1 and 2 show the results of a joint analysis of sea wave spectra obtained from satellite and contact data under various wave formation conditions. Figure 1 shows the results of an experiment conducted on September 24, 2015 under the conditions of developing wind waves, and Figure 2 shows the results experiment, conducted on September 12, 2012 with weak wind

(Katsiveli settlement, Crimea), installed at a distance of about 600 m from the shore.

and stereo survey from the deck of the oceanographic platform.

Let us briefly summarize some experimental results of these types.

technical characteristics of this complex were as follows:

• Frequency of interrogation of channels—10, 20, 50, and 100 Hz • The maximum height of the measured waves was up to 4 m.

For gravitational waves (for Λ > 10�<sup>1</sup> m), the relation (11) is simplified and takes the form

$$
\omega(k) = \sqrt{gk} \tag{12}
$$

The dispersion relation makes it possible to establish a relationship between the spatial spectrum of χ(k) characterizing the total energy of waves with a wave number k propagating in all possible directions and a frequency spectrum S(ω) characterizing the total energy of waves with a cyclic frequency ω propagating in all possible directions

$$
\chi(k) = \mathcal{S}\left(\omega(k)\right)d\omega(k)\,dk.\tag{13}
$$

On the other hand, a one-dimensional spatial spectrum can be obtained from the twodimensional spectrum of sea surface elevations Ψ(k) by integrating over the azimuth

$$\chi(k) = \int\_{-\pi/2}^{\pi/2} \Psi(k\cos\varphi, k\sin\varphi) k d\varphi \tag{14}$$

Thus, the spectrum χð Þk can be used as a base function and recalculate the spectra obtained by various measurement methods in this spectrum by means of appropriate transformations.

## 4. The results of experimental studies

• Comparison of wave spectra obtained remotely with contact and/or stereo-photogrammetric

Consider the relationship between the spatial spectra of the sea surface, reconstructed from optical images, as well as frequency and frequency-directed wave spectra, measured by con-

The frequency spectrum of sea surface elevations Sξ(ω) characterizes the distribution of wave oscillations at a given point along the cyclic frequencies ω, ω = 2π/τ, where τ is the period of

The spatial wave spectrum characterizes the energy distribution at a fixed time instant with respect to wave numbers k (or spatial frequencies ν = 1/Λ), k = 2π/Λ, where Λ is the length of

Spatial and frequency spectra of waves are functions of different arguments. They have different physical meanings and require the use of different methods of measurement. Therefore, to compare the spectra measured by different methods, a dispersion relation is used that describes the relationship between the time and spatial frequencies of waves, depending on the physical mechanisms that form the surface waves of the range in question. Within the framework of the linear hydrodynamic model, the components of the wave spectrum can be considered as elementary plane waves, for which the dispersion relation of the theory of potential waves of small amplitude that relates the cyclic frequency of the wave (ω) to the wave number k is valid. In deep water (for kh ≫ 1, where h is the depth), the dispersion relation taking into account the contribution of gravitational and capillary forces to wave formation has the form

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gk <sup>þ</sup> ð Þ <sup>Τ</sup>=<sup>r</sup> <sup>k</sup><sup>3</sup>

> > gk p (12)

χð Þ¼ k S ð Þ ωð Þk dωð Þk dk: (13)

Ψð Þ kcosφ; ksinφ kdφ (14)

(11)

ωð Þ¼ k

with a cyclic frequency ω propagating in all possible directions

χð Þ¼ k

q

For gravitational waves (for Λ > 10�<sup>1</sup> m), the relation (11) is simplified and takes the form

<sup>ω</sup>ð Þ¼ <sup>k</sup> ffiffiffiffiffi

The dispersion relation makes it possible to establish a relationship between the spatial spectrum of χ(k) characterizing the total energy of waves with a wave number k propagating in all possible directions and a frequency spectrum S(ω) characterizing the total energy of waves

On the other hand, a one-dimensional spatial spectrum can be obtained from the two-

Thus, the spectrum χð Þk can be used as a base function and recalculate the spectra obtained by various measurement methods in this spectrum by means of appropriate transformations.

dimensional spectrum of sea surface elevations Ψ(k) by integrating over the azimuth

ð<sup>π</sup>=<sup>2</sup> �π=2

tact sensors, and characterize the fluctuations in sea level over time at a fixed point.

measurements.

76 Surface Waves - New Trends and Developments

the wave oscillation.

the surface wave.

To solve the problems of validation of the developed methods, the results obtained in three types of experiments were used [19, 22]:


Let us briefly summarize some experimental results of these types.

Experimental work of the first type was carried out in the area of the oceanographic platform (Katsiveli settlement, Crimea), installed at a distance of about 600 m from the shore.

For the measurement of space-time wave spectra, contact data obtained using a wave measuring unit based on an array of string wave recorders, which were a set of six resistive wave recorders measuring the elevations of the sea surface at points located in the center and at the top of the pentagon, were used.

The distance from the central string to each of the external strings was 25 cm. The main technical characteristics of this complex were as follows:


During the first type of experiments, a special space imagery was conducted in the vicinity of the oceanographic platform using high-spatial resolution (0.5 m) optic-electronic equipment installed on board the GEOEYE spacecraft. A brief overview of the complex experiments conducted on September 24, 2015 at 11:52 (LT) and on September 12, 2011 at 12:06 is presented below. The presented experiments were characterized by different conditions of wave formation. During the first complex experiment, developing wind waves were observed at a nearsurface wind speed of about 4 m/s. During the second complex experiment, the velocity of the near-surface wind varied from 0 to 2 m/s; however, swell waves were present in the experiment zone.

Figures 1 and 2 show the results of a joint analysis of sea wave spectra obtained from satellite and contact data under various wave formation conditions. Figure 1 shows the results of an experiment conducted on September 24, 2015 under the conditions of developing wind waves, and Figure 2 shows the results experiment, conducted on September 12, 2012 with weak wind

Figure 1. Comparison of the spectra of developing wind waves measured from remote and contact data of the complex experiment. September 24, 2015: (a) and (b) two-dimensional slope spectrum recovered in two intervals of spatial frequencies; (c) comparison of the one-dimensional frequency spectrum of the elevations obtained from the twodimensional spectrum of slopes by the retrieving operator R, with the spectrum obtained from the data of the string wave recorder. The dashed lines denote the Toba approximation [23].

waves in the presence of swell waves [19, 20]. Figures 1(a) and 2(a) show the reconstructed spatial spectra of sea surface slopes in a wide range of spatial frequencies, and Figures 1(b) and 2(b) show the slope spectra in the region of the spectral maximum and low spatial frequencies are enlarged.

Figures 1(c) and 2(c) show a comparison of the one-dimensional frequency spectrum of the elevations obtained from the two-dimensional spectrum of slopes with the frequency spectrum of the elevations obtained from the data of the array of string wave recorders. The conjugated spectra conditionally show the boundary ω<sup>0</sup> between the frequency domains corresponding to the regions of action of the high-frequency and low-frequency reconstructing operators included in the formula (9) [20].

The comparison of the graphs of one-dimensional frequency spectra of waves, shown in Figures 1 and 2, allows us to visually evaluate the good correspondence between the spectra measured by contact data and spectra reconstructed from remote data.

For the quantitative analysis of the correspondence between the results of remote and contact measurements, a measure of discrepancy was used, calculated from formula

$$\Delta = \sqrt{\left(1/N\right)\sum\_{n=1}^{N} \left(1 - \Psi\_{cn}(\omega\_n)/\Psi\_{contact}(\omega\_n)\right)^2} \tag{15}$$

2012, this measure of divergence amounted to Δ ≈ 0.08. The obtained results indicate the adequacy of the determination of wave spectra from satellite images of high spatial resolution. During the experiments of the second type, a space survey was conducted in the vicinity of Oahu Island (Hawaii, USA) from the Ikonos and QuickBird satellites, which provided space images of high spatial resolution (0.65–1.0 m) and synchronous measurements of the frequency-angular

Figure 2. Comparison of the spectra of weak wind waves in the presence of swell waves measured from remote and contact experimental data. September 12, 2012: (a) and (b) two-dimensional slope spectrum recovered in two intervals of spatial frequencies; (c) comparison of the one-dimensional frequency spectrum of the elevations obtained from the twodimensional spectrum of slopes by the retrieving operator R, with the spectrum obtained from the data of the string wave

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The experiments were carried out to assess the anthropogenic impacts on the water area of Mamala bay, using satellite and contact data [19]. In the course of complex experiments, space surveys and synchronous sea truth measurements were carried out, including with the help of drifting wave buoys, which ensure the registration of frequency-angle spectra of surface waves. The following circumstance was taken into account when processing the data of the complex experiment. IKONOS and QuickBird satellites were photographed in these experiments far from the sunlight, and the resolution of the survey cameras of these satellites allowed us to fix surface waves of the submeter range (0.6–1.0 m). Under such conditions, the nonlinear components of

wave spectra with the help of two wave buoys, drifting in the space survey area.

recorder.

For the complex experiment carried out on September 24, 2010, the measure of discrepancy estimated by formula (15) was Δ ≈ 0.07, and for the experiment conducted on September 12, Measurement of Sea Wave Spatial Spectra from High-Resolution Optical Aerospace Imagery http://dx.doi.org/10.5772/intechopen.71834 79

Figure 2. Comparison of the spectra of weak wind waves in the presence of swell waves measured from remote and contact experimental data. September 12, 2012: (a) and (b) two-dimensional slope spectrum recovered in two intervals of spatial frequencies; (c) comparison of the one-dimensional frequency spectrum of the elevations obtained from the twodimensional spectrum of slopes by the retrieving operator R, with the spectrum obtained from the data of the string wave recorder.

waves in the presence of swell waves [19, 20]. Figures 1(a) and 2(a) show the reconstructed spatial spectra of sea surface slopes in a wide range of spatial frequencies, and Figures 1(b) and 2(b) show the slope spectra in the region of the spectral maximum and low spatial frequencies

Figure 1. Comparison of the spectra of developing wind waves measured from remote and contact data of the complex experiment. September 24, 2015: (a) and (b) two-dimensional slope spectrum recovered in two intervals of spatial frequencies; (c) comparison of the one-dimensional frequency spectrum of the elevations obtained from the twodimensional spectrum of slopes by the retrieving operator R, with the spectrum obtained from the data of the string wave

Figures 1(c) and 2(c) show a comparison of the one-dimensional frequency spectrum of the elevations obtained from the two-dimensional spectrum of slopes with the frequency spectrum of the elevations obtained from the data of the array of string wave recorders. The conjugated spectra conditionally show the boundary ω<sup>0</sup> between the frequency domains corresponding to the regions of action of the high-frequency and low-frequency reconstructing operators

The comparison of the graphs of one-dimensional frequency spectra of waves, shown in Figures 1 and 2, allows us to visually evaluate the good correspondence between the spectra

For the quantitative analysis of the correspondence between the results of remote and contact

For the complex experiment carried out on September 24, 2010, the measure of discrepancy estimated by formula (15) was Δ ≈ 0.07, and for the experiment conducted on September 12,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ <sup>1</sup> � <sup>Ψ</sup>спð Þ <sup>ω</sup><sup>n</sup> <sup>=</sup>Ψcontactð Þ <sup>ω</sup><sup>n</sup> <sup>2</sup>

vuut (15)

measured by contact data and spectra reconstructed from remote data.

Δ ¼

measurements, a measure of discrepancy was used, calculated from formula

ð Þ <sup>1</sup>=<sup>N</sup> <sup>X</sup> N

n¼1

are enlarged.

included in the formula (9) [20].

recorder. The dashed lines denote the Toba approximation [23].

78 Surface Waves - New Trends and Developments

2012, this measure of divergence amounted to Δ ≈ 0.08. The obtained results indicate the adequacy of the determination of wave spectra from satellite images of high spatial resolution.

During the experiments of the second type, a space survey was conducted in the vicinity of Oahu Island (Hawaii, USA) from the Ikonos and QuickBird satellites, which provided space images of high spatial resolution (0.65–1.0 m) and synchronous measurements of the frequency-angular wave spectra with the help of two wave buoys, drifting in the space survey area.

The experiments were carried out to assess the anthropogenic impacts on the water area of Mamala bay, using satellite and contact data [19]. In the course of complex experiments, space surveys and synchronous sea truth measurements were carried out, including with the help of drifting wave buoys, which ensure the registration of frequency-angle spectra of surface waves.

The following circumstance was taken into account when processing the data of the complex experiment. IKONOS and QuickBird satellites were photographed in these experiments far from the sunlight, and the resolution of the survey cameras of these satellites allowed us to fix surface waves of the submeter range (0.6–1.0 m). Under such conditions, the nonlinear components of the brightness field of the sea surface make an insignificant contribution to the spectra of satellite images. In this connection, in the analysis of complex experiment data, it is permissible to use the linear model, according to which the two-dimensional spatial spectra of satellite images S (kx, ky) are linearly related to the spatial spectra of sea wave inclinations Ψφ (νx, νy).

the time of shooting (see Figure 3a). The data obtained by measurements with a wave buoy (see Figure 3c) were selected in such a way that the moment of the survey was as close as possible to the end of the half-hour accumulation of data obtained by the buoy. For the selected image fragments, two-dimensional spatial spectra S (kx, ky) were formed, examples of which are shown in Figure 3b, and according to measurements by wave buoys, the frequency-angle

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Two-dimensional spectra of satellite images S (kx, ky) (see Figure 3b) also yielded one-dimensional spatial spectra S (k) (see Figure 3d), and from the frequency-angle spectra of the elevations Ψ (ω, θ) obtained by wave buoys, the frequency spectra of Ψ (ω) (see Figure 3e) were

From the one-dimensional spatial spectra obtained from two-dimensional spectra of satellite images using the dispersion relation for gravitational waves, the frequency spectra S (ω) were

An important stage in the comparison of wave spectra obtained from space images and data from wave buoys is the detection of additional spectral harmonics due to anthropogenic effects on the surface waves caused by buried wastewater outfalls in the region of the experiments [7–12]. Such spectral harmonics arise under the action of high-frequency internal waves generated by turbulent jets of nonsalty water discharged into the saline marine environment through the diffusers of deep outfalls. Similar effects are caused by various physical mechanisms, analyzed in [14, 15].

The conducted experiments' results and comparison of additional spectral harmonics detected on the base of satellite image spectra and wave buoy data have confirmed the efficiency of remote methods for detection of manifestations of anthropogenic impacts on the water envi-

Experimental work of the third type was performed completely on the oceanographic platform

To verify the adequacy of the methods for retrieving sea wave spectra in the 0.1–1 m shortwave range, the results of complex field experiments of the third type were analyzed. These experiments were carried out at an oceanographic platform, including conventional and stereo survey, as well as contact measurements of wave spectra using a wave recorder array. Stereo surveying of the sea surface was carried out from the working platform, which was located at a height of 16 m above sea level with a 10.2 m basis using stereocameras with a focal length of 99 mm. From the measured elevation field samples, the spectra Ψϕ (kϕ) were calculated using the interpolation and discrete Fourier transform procedures. Simultaneously, with the stereoscopic shooting on the platform, contact measurements of the frequency spectra of waves were performed using a wave spectrograph providing registration in the frequency range of 0.1– 15 Hz with a tunable filter bandwidth of 0.1 Hz. In the wave spectrograph, a method for sequential analysis of the frequency spectrum was realized by automatically tuning the trans-

Images of the sea surface obtained during stereo photography were also used to retrieve the wave spectra by a nonlinear multiposition method [22]. At the same time, fragments of two

spectra of the elevations Ψε (ω, θ) are constructed.

ronment taking into account wave spectra changes.

mission frequency of a narrowband filter in a given interval [22].

constructed in the chosen directions.

determined from the formula (12).

in Katsiveli settlement (Crimea).

The results of comparison of the wave spectra obtained on the basis of remote data and drifting buoy data are shown in Figure 3.

During the processing, fragments of satellite images with dimensions of 1024 1024 or 2048 2048 pixels were selected in such a way that the fragment overlapped the wave buoy path at

Figure 3. The results of a joint analysis of sea wave spectra obtained from space images and from drifting wave buoys: (a) the initial space image and its fragment used for processing; (b) two-dimensional spatial spectrum of the selected fragment and its enlarged central part (right); (c) installation of a wave buoy; (d) a one-dimensional spatial spectrum obtained from the two-dimensional spectrum of the satellite image; (e) the frequency spectrum obtained by a wave buoy; (f) comparison of frequency spectra obtained by different methods.

the time of shooting (see Figure 3a). The data obtained by measurements with a wave buoy (see Figure 3c) were selected in such a way that the moment of the survey was as close as possible to the end of the half-hour accumulation of data obtained by the buoy. For the selected image fragments, two-dimensional spatial spectra S (kx, ky) were formed, examples of which are shown in Figure 3b, and according to measurements by wave buoys, the frequency-angle spectra of the elevations Ψε (ω, θ) are constructed.

the brightness field of the sea surface make an insignificant contribution to the spectra of satellite images. In this connection, in the analysis of complex experiment data, it is permissible to use the linear model, according to which the two-dimensional spatial spectra of satellite images S (kx, ky)

The results of comparison of the wave spectra obtained on the basis of remote data and drifting

During the processing, fragments of satellite images with dimensions of 1024 1024 or 2048 2048 pixels were selected in such a way that the fragment overlapped the wave buoy path at

Figure 3. The results of a joint analysis of sea wave spectra obtained from space images and from drifting wave buoys: (a) the initial space image and its fragment used for processing; (b) two-dimensional spatial spectrum of the selected fragment and its enlarged central part (right); (c) installation of a wave buoy; (d) a one-dimensional spatial spectrum obtained from the two-dimensional spectrum of the satellite image; (e) the frequency spectrum obtained by a wave buoy;

(f) comparison of frequency spectra obtained by different methods.

are linearly related to the spatial spectra of sea wave inclinations Ψφ (νx, νy).

buoy data are shown in Figure 3.

80 Surface Waves - New Trends and Developments

Two-dimensional spectra of satellite images S (kx, ky) (see Figure 3b) also yielded one-dimensional spatial spectra S (k) (see Figure 3d), and from the frequency-angle spectra of the elevations Ψ (ω, θ) obtained by wave buoys, the frequency spectra of Ψ (ω) (see Figure 3e) were constructed in the chosen directions.

From the one-dimensional spatial spectra obtained from two-dimensional spectra of satellite images using the dispersion relation for gravitational waves, the frequency spectra S (ω) were determined from the formula (12).

An important stage in the comparison of wave spectra obtained from space images and data from wave buoys is the detection of additional spectral harmonics due to anthropogenic effects on the surface waves caused by buried wastewater outfalls in the region of the experiments [7–12]. Such spectral harmonics arise under the action of high-frequency internal waves generated by turbulent jets of nonsalty water discharged into the saline marine environment through the diffusers of deep outfalls. Similar effects are caused by various physical mechanisms, analyzed in [14, 15].

The conducted experiments' results and comparison of additional spectral harmonics detected on the base of satellite image spectra and wave buoy data have confirmed the efficiency of remote methods for detection of manifestations of anthropogenic impacts on the water environment taking into account wave spectra changes.

Experimental work of the third type was performed completely on the oceanographic platform in Katsiveli settlement (Crimea).

To verify the adequacy of the methods for retrieving sea wave spectra in the 0.1–1 m shortwave range, the results of complex field experiments of the third type were analyzed. These experiments were carried out at an oceanographic platform, including conventional and stereo survey, as well as contact measurements of wave spectra using a wave recorder array. Stereo surveying of the sea surface was carried out from the working platform, which was located at a height of 16 m above sea level with a 10.2 m basis using stereocameras with a focal length of 99 mm. From the measured elevation field samples, the spectra Ψϕ (kϕ) were calculated using the interpolation and discrete Fourier transform procedures. Simultaneously, with the stereoscopic shooting on the platform, contact measurements of the frequency spectra of waves were performed using a wave spectrograph providing registration in the frequency range of 0.1– 15 Hz with a tunable filter bandwidth of 0.1 Hz. In the wave spectrograph, a method for sequential analysis of the frequency spectrum was realized by automatically tuning the transmission frequency of a narrowband filter in a given interval [22].

Images of the sea surface obtained during stereo photography were also used to retrieve the wave spectra by a nonlinear multiposition method [22]. At the same time, fragments of two stereopair images were analyzed on which a surface area processed by a stereo-photogrammetric method is presented. Experimental conditions: the zenith angle of the Sun was 30 (the images were recorded with a cloudless sky), and the wind speed was 5 m/s. During the experiments, the wind blew from the shore, so the acceleration of wind waves did not exceed the distance from the platform to the shore, which was ~600 m.

As the main characteristics for comparing the wave spectra obtained by different methods, the rms elevation σξ and the power exponent px in two wavelength intervals Λ were chosen: from 0.1 to 1 m and from 0.04 to 0.4 m. The wave spectra, measured by different methods, were recalculated in spatial spectra of χ (k).

In the presence of solar glitters in the image, which cause a significant nonlinearity of the transfer function, which connects the slopes and the brightness of the elements of the sea surface, systematic errors arise in retrieving the spectra of waves. Such errors were eliminated by means of nonlinear retrieval filters adapted to the wave characteristics using an iterative procedure for retrieving the wave spectra [20, 25].

Quantitative estimates of these errors were given in [17, 21], where it was shown that even with the use of images containing solar patches, two iterations of retrieving obtained from different positions are sufficient to reduce the relative error in measuring the integral spectral energy to ≈3% and the error determines the exponent of the power-law approximation of the wave spectrum to ≈0.03.

To correct the spectra of elevations obtained by the stereo method, the noise of digitization resulting from errors in measuring the elevations of the surface during stereo-photogrammetric image processing was taken into account. It was assumed that the noise is additive, so their spectrum was subtracted from the spectra measured during processing [22].

The form of short-wave spatial spectra of sea surface elevations χ (k), obtained from data of different measurement methods, is shown in Figure 4. The spectra are constructed by averaging over six realizations. The analysis of Figure 4 indicates a good correspondence of the wave spectra obtained by different methods in the short-wave range of wavelengths (Λ = 0.04– 1.0 m). The character of the obtained spatial spectra of the elevations testifies to the possibility of their power-law approximation by the Toba formula [23], recalculated taking into account the dispersion relation (12).

Analysis of the experimental data has shown that there is a coincidence of statistical estimates of the characteristics of sea waves measured by different methods within the mean square scatter of samples of these parameters. This indicates that the developed remote methods

Figure 4. Spatial spectra of sea surface elevations, obtained by different methods: (1) retrieving of images by the nonlinear multiposition method; (2) contact measurements by a string wave; (3) stereo-photogrammetric measurements; red lines

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The most important for practice is the determination of the exponent p in power approximation. An analysis of the data obtained in the experiments shows that the parameters of the spectra of waves retrieved from the image spectra correspond best to the Toba approxima-

The obtained results of the studies by remote methods made it possible to reveal the following

p = 2.22 0.08 for the wavelength interval Λ = 1.0–5.0 m; p = 2.23 0.09 for the shortwavelength wavelength interval Λ = 0.1–1.0 m; p = 2.1 0.08 for the short-wavelength

sufficiently accurately determine the characteristics of sea waves.

values of the parameters p for the experimental conditions:

wavelength interval Λ = 0.04–0.4 m.

tion [23].

denote Toba approximation.

To generalize the data obtained, such characteristics of the spectra as the exponents of power approximation and the dispersion of wave energy in different wavelength ranges were analyzed. For the wavelength range Λ = 1.0–5.0 m, wave spectra were used, which were determined from the space image obtained from the GEOEYE satellite by the nonlinear recovery approach. For shortwave waves (Λ = 0.04–1.0 m), experimental data obtained in six experiments of the third type by three methods in two bands were used: Λ = 0.04–0.4 m and Λ = 0.1–1.0 m.

For comparison with known literature data, the parameters of wave spectra were calculated from the well-known approximations: Phillips, Pierson-Moskovitz, Toba, Leikin, and Rosenberg [19]. The values of the variances and exponents of power approximations of the spatial spectra of sea surface elevations, obtained experimentally by various methods, as well as obtained from known approximations, were analyzed in [19].

Measurement of Sea Wave Spatial Spectra from High-Resolution Optical Aerospace Imagery http://dx.doi.org/10.5772/intechopen.71834 83

stereopair images were analyzed on which a surface area processed by a stereo-photogrammetric method is presented. Experimental conditions: the zenith angle of the Sun was 30 (the images were recorded with a cloudless sky), and the wind speed was 5 m/s. During the experiments, the wind blew from the shore, so the acceleration of wind waves did not exceed the distance from

As the main characteristics for comparing the wave spectra obtained by different methods, the rms elevation σξ and the power exponent px in two wavelength intervals Λ were chosen: from 0.1 to 1 m and from 0.04 to 0.4 m. The wave spectra, measured by different methods, were

In the presence of solar glitters in the image, which cause a significant nonlinearity of the transfer function, which connects the slopes and the brightness of the elements of the sea surface, systematic errors arise in retrieving the spectra of waves. Such errors were eliminated by means of nonlinear retrieval filters adapted to the wave characteristics using an iterative

Quantitative estimates of these errors were given in [17, 21], where it was shown that even with the use of images containing solar patches, two iterations of retrieving obtained from different positions are sufficient to reduce the relative error in measuring the integral spectral energy to ≈3% and the error determines the exponent of the power-law approximation of the

To correct the spectra of elevations obtained by the stereo method, the noise of digitization resulting from errors in measuring the elevations of the surface during stereo-photogrammetric image processing was taken into account. It was assumed that the noise is additive, so

The form of short-wave spatial spectra of sea surface elevations χ (k), obtained from data of different measurement methods, is shown in Figure 4. The spectra are constructed by averaging over six realizations. The analysis of Figure 4 indicates a good correspondence of the wave spectra obtained by different methods in the short-wave range of wavelengths (Λ = 0.04– 1.0 m). The character of the obtained spatial spectra of the elevations testifies to the possibility of their power-law approximation by the Toba formula [23], recalculated taking into account

To generalize the data obtained, such characteristics of the spectra as the exponents of power approximation and the dispersion of wave energy in different wavelength ranges were analyzed. For the wavelength range Λ = 1.0–5.0 m, wave spectra were used, which were determined from the space image obtained from the GEOEYE satellite by the nonlinear recovery approach. For shortwave waves (Λ = 0.04–1.0 m), experimental data obtained in six experiments of the third

For comparison with known literature data, the parameters of wave spectra were calculated from the well-known approximations: Phillips, Pierson-Moskovitz, Toba, Leikin, and Rosenberg [19]. The values of the variances and exponents of power approximations of the spatial spectra of sea surface elevations, obtained experimentally by various methods, as well as

their spectrum was subtracted from the spectra measured during processing [22].

type by three methods in two bands were used: Λ = 0.04–0.4 m and Λ = 0.1–1.0 m.

obtained from known approximations, were analyzed in [19].

the platform to the shore, which was ~600 m.

procedure for retrieving the wave spectra [20, 25].

recalculated in spatial spectra of χ (k).

82 Surface Waves - New Trends and Developments

wave spectrum to ≈0.03.

the dispersion relation (12).

Figure 4. Spatial spectra of sea surface elevations, obtained by different methods: (1) retrieving of images by the nonlinear multiposition method; (2) contact measurements by a string wave; (3) stereo-photogrammetric measurements; red lines denote Toba approximation.

Analysis of the experimental data has shown that there is a coincidence of statistical estimates of the characteristics of sea waves measured by different methods within the mean square scatter of samples of these parameters. This indicates that the developed remote methods sufficiently accurately determine the characteristics of sea waves.

The most important for practice is the determination of the exponent p in power approximation. An analysis of the data obtained in the experiments shows that the parameters of the spectra of waves retrieved from the image spectra correspond best to the Toba approximation [23].

The obtained results of the studies by remote methods made it possible to reveal the following values of the parameters p for the experimental conditions:

p = 2.22 0.08 for the wavelength interval Λ = 1.0–5.0 m; p = 2.23 0.09 for the shortwavelength wavelength interval Λ = 0.1–1.0 m; p = 2.1 0.08 for the short-wavelength wavelength interval Λ = 0.04–0.4 m.

To determine the effect of sea surface disturbances associated with the nonstationary nature of the wave, correlation analysis was used to obtain the estimates. Correlation coefficients between sea surface characteristics in the short-wave range were studied in [19].

The correlation of the estimates carried out by various methods is quite high, and averages were 0.8–0.9. Those cases in which there is a decrease in the correlation coefficient are explained by the peculiarities of the measurement methods. For example, a slight decrease in the correlation coefficient of estimates of the exponent px obtained by the contact method, with estimates obtained from the images, is due to the time diversity of the registration of the spectral density at different frequencies ω. During the tuning of the filter transmission frequency, a change in the spectral density of the wave at different frequencies ω can occur, which leads to a deviation in the estimate of the exponent of the elevation spectrum with respect to estimates obtained for a fixed time in image processing [22].

Good correlation between independently measured wave spectra allows us to conclude that the developed remote measurement methods reliably reflect real processes in the oceanatmosphere boundary layer.

Retrieving operators constructed by the numerical method and having passed the calibration procedure using contact data were used for experimental studies in various water areas [16, 18–20].

One of the important experiments was to study the development of wind waves at various distances from the shore. An experiment of this type was conducted in the water area of Mamala Bay near the Oahu Island (Hawaii, USA) [19].

Figure 5 shows four fragments of the space image obtained by the QuickBird satellite with 0.65 m spatial resolution, and the spectra of slopes retrieved on their base.

Figure 6 presents one-dimensional spatial elevation spectra obtained by averaging over the direction of two-dimensional spatial spectra of elevations in the 70 angular sector, recovered

Figure 6. One-dimensional spatial spectra of sea wave elevation, retrieved from fragments of the satellite image, obtained at different distances from the shore at different wind speeds in the water area of Mamala Bay near Oahu (Hawaii, USA).

Measurement of Sea Wave Spatial Spectra from High-Resolution Optical Aerospace Imagery

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85

The analysis of Figure 6 shows that the course of one-dimensional spectra retrieved from the two-dimensional spatial spectra of fragments of the satellite image depends on wind speed and the distance of the corresponding sections from the shore. The shape and position of the spectral maxima depends not only on the speed of the near-surface wind but also on the acceleration from the leeward shore, which in this experiment was between 10 and 50 km.

A method is developed to retrieve the spectra of gradients and elevations of sea waves from the spectra of aerospace optical images over a wide frequency range, based on the formation of a retrieving operator that takes into account the nonlinear character of the modulation of the

The conducted studies demonstrated the effectiveness and adequacy of the application of the method of recovering the spectra of gradients and elevations of sea waves from satellite optical

using the developed retrieving operator.

brightness field by sea surface slopes.

images of high spatial resolution.

5. Conclusion

An analysis of the two-dimensional retrieved slope spectra shows that, as the distance from the coast and the wind regime change, characteristic additional wave systems develop. At the same time, the energy of sea waves increases with the distance from the coast and the increase in wind speed.

Figure 5. Fragments of the satellite image obtained at various distances from the shore in the water area of Mamala Bay off the Oahu Island and retrieved two-dimensional spatial spectra of sea wave slopes.

Measurement of Sea Wave Spatial Spectra from High-Resolution Optical Aerospace Imagery http://dx.doi.org/10.5772/intechopen.71834 85

Figure 6. One-dimensional spatial spectra of sea wave elevation, retrieved from fragments of the satellite image, obtained at different distances from the shore at different wind speeds in the water area of Mamala Bay near Oahu (Hawaii, USA).

Figure 6 presents one-dimensional spatial elevation spectra obtained by averaging over the direction of two-dimensional spatial spectra of elevations in the 70 angular sector, recovered using the developed retrieving operator.

The analysis of Figure 6 shows that the course of one-dimensional spectra retrieved from the two-dimensional spatial spectra of fragments of the satellite image depends on wind speed and the distance of the corresponding sections from the shore. The shape and position of the spectral maxima depends not only on the speed of the near-surface wind but also on the acceleration from the leeward shore, which in this experiment was between 10 and 50 km.

#### 5. Conclusion

To determine the effect of sea surface disturbances associated with the nonstationary nature of the wave, correlation analysis was used to obtain the estimates. Correlation coefficients

The correlation of the estimates carried out by various methods is quite high, and averages were 0.8–0.9. Those cases in which there is a decrease in the correlation coefficient are explained by the peculiarities of the measurement methods. For example, a slight decrease in the correlation coefficient of estimates of the exponent px obtained by the contact method, with estimates obtained from the images, is due to the time diversity of the registration of the spectral density at different frequencies ω. During the tuning of the filter transmission frequency, a change in the spectral density of the wave at different frequencies ω can occur, which leads to a deviation in the estimate of the exponent of the elevation spectrum with respect to

Good correlation between independently measured wave spectra allows us to conclude that the developed remote measurement methods reliably reflect real processes in the ocean-

Retrieving operators constructed by the numerical method and having passed the calibration procedure using contact data were used for experimental studies in various water areas [16, 18–20]. One of the important experiments was to study the development of wind waves at various distances from the shore. An experiment of this type was conducted in the water area of

Figure 5 shows four fragments of the space image obtained by the QuickBird satellite with

An analysis of the two-dimensional retrieved slope spectra shows that, as the distance from the coast and the wind regime change, characteristic additional wave systems develop. At the same time, the energy of sea waves increases with the distance from the coast and the increase

Figure 5. Fragments of the satellite image obtained at various distances from the shore in the water area of Mamala Bay

off the Oahu Island and retrieved two-dimensional spatial spectra of sea wave slopes.

0.65 m spatial resolution, and the spectra of slopes retrieved on their base.

between sea surface characteristics in the short-wave range were studied in [19].

estimates obtained for a fixed time in image processing [22].

Mamala Bay near the Oahu Island (Hawaii, USA) [19].

atmosphere boundary layer.

84 Surface Waves - New Trends and Developments

in wind speed.

A method is developed to retrieve the spectra of gradients and elevations of sea waves from the spectra of aerospace optical images over a wide frequency range, based on the formation of a retrieving operator that takes into account the nonlinear character of the modulation of the brightness field by sea surface slopes.

The conducted studies demonstrated the effectiveness and adequacy of the application of the method of recovering the spectra of gradients and elevations of sea waves from satellite optical images of high spatial resolution.

The optimal parameters of such a retrieving operator are determined by an iterative method when comparing the spectra of aerospace images with spectra with sea wave characteristics measured with high accuracy by string waveforms under controlled conditions.

Author details

References

Press; 1977. 336 p

Gidrometeoizdat; 1985. 289 p

Kosmosa. 2001;6:46-67

45(6):779-790

46(4):482-491

2005;32:L12610. DOI: 10.1029/2005GL022390

Valery G. Bondur\* and Alexander B. Murynin\*

\*Address all correspondence to: vgbondur@aerocosmos.info and amurynin@bk.ru

[1] Monin AS, Krasitsky VP. Ocean Surface Phenomena. Leningrad: Gidrometeoizdat; 1985. 375 p [2] Phillips OM. Dynamics of the Upper Ocean Layer. Cambridge: Cambridge University

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[3] Davidan IN, Lopatukhin LI, Rozhkov VA. Wind Waves in the World Ocean. Leningrad:

[4] Bondur VG. Aerospace methods in modern oceanology. In: New Ideas in Oceanology,

[5] Bondur VG, Grebenyuk YV. Remote indication of anthropogenic influence on marine environment caused by depth wastewater plume: Modeling, experiments. Issledovaniya Zemli iz

[6] Keeler R, Bondur V, Gibson C. Optical satellite imagery detection of internal wave effects from a submerged turbulent outfall in the stratified ocean. Geophysical Research Letters.

[7] Bondur VG, Zhurbas VM, Grebenyuk YV. Mathematical modeling of turbulent jets of deepwater sewage discharge into coastal basins. Оceanology. 2006;46(6):757-771

[8] Bondur VG, Sabinin KD, Grebenyuk YV. Anomalous variation of the Ocean's Inertial

[9] Bondur VG, Grebenyuk YV, Sabinin KD. Variability of internal tides in the coastal water

[10] Bondur VG, Grebenyuk YV, Ezhova EV, Kazakov VI, Sergeev DA, Soustova IA, Troitskaya YI. Surface manifestations of internal waves investigated by a subsurface buoyant jet: 1. The mechanism of internal-wave generation. Izvestiya, Atmospheric and Oceanic Physics. 2009;

[11] Bondur VG, Grebenyuk YV, Ezhova EV, Kazakov VI, Sergeev DA, Soustova IA, Troitskaya YI. Surface manifestations of internal waves investigated by a subsurface buoyant jet: 3. Surface manifestations of internal waves. Izvestiya, Atmospheric and Oceanic Physics. 2010;

Oscillations at the Hawaii Shelf. Doklady Earth Sciences. 2013;450:526-530

area of Oahu Island (Hawaii). Oceanology. 2008;48(5):611-621

Physics, Chemistry, and Biology. Vol. 1. Moscow: Nauka; 2004. pp. 55-117

AEROCOSMOS Research Institute for Aerospace Monitoring, Moscow, Russia

During the research, the spatial spectral characteristics of sea waves estimated from remote sensing data were compared with the corresponding characteristics measured by contact means under controlled conditions. The satellite data used for the comparison were the arrays of string wave recorders mounted on a stationary oceanographic platform, the data of a stereo survey performed with a high spatial resolution from a low altitude above the sea surface, as well as data of drifting wave buoys. Comparison of the spectra of waves and their statistical characteristics demonstrated the consistency of the results obtained in the processing of satellite images of high spatial resolution and the results of processing data obtained by various sea truth assets.

Experiments carried out in different water areas demonstrated the possibility of using a retrieving operator with optimal values of parameters found under certain conditions for obtaining satellite optical images of the sea surface for a wide range of wave formation conditions.

As a result of the application of numerical optimization procedures, the values of the parameters of nonlinear retrieving filters that are effective in both developed and developing waves, as well as in the presence of swell waves, are chosen. In this case, the measurement of the divergence of the wave spectra obtained from satellite images of high spatial resolution and subsatellite data at optimal values of these parameters is 0.08–0.12. This testifies to the adequacy of the proposed method for recording wave spectra from the spectra of satellite images over a wide frequency range.

Thus, the adequacy of remote measurement of both the time-averaged spectra of the sea surface and the variations of these spectra caused by wave nonstationarity using high spatial resolution images and nonlinear recovering operators has been experimentally confirmed. The conducted researches made it possible to develop methods and technologies for comprehensive ground-space monitoring of marine areas to obtain such important oceanographic characteristics as surface wave spectra.

The developed method can be used to study surface wave conditions, including in the space monitoring of natural and anthropogenic impacts on the marine waters.

### Acknowledgements

The work was carried out with the financial support of the state represented by the Ministry of Education and Science of the Russian Federation in the framework of the Federal Program "Research and development in priority areas of development of the scientific and technological complex of Russia for 2014-2020" (unique identifier of the project RFMEFI57716 X0234).

#### Author details

The optimal parameters of such a retrieving operator are determined by an iterative method when comparing the spectra of aerospace images with spectra with sea wave characteristics

During the research, the spatial spectral characteristics of sea waves estimated from remote sensing data were compared with the corresponding characteristics measured by contact means under controlled conditions. The satellite data used for the comparison were the arrays of string wave recorders mounted on a stationary oceanographic platform, the data of a stereo survey performed with a high spatial resolution from a low altitude above the sea surface, as well as data of drifting wave buoys. Comparison of the spectra of waves and their statistical characteristics demonstrated the consistency of the results obtained in the processing of satellite images of high spatial resolution and the results of processing data obtained by various sea

Experiments carried out in different water areas demonstrated the possibility of using a retrieving operator with optimal values of parameters found under certain conditions for obtaining satellite

As a result of the application of numerical optimization procedures, the values of the parameters of nonlinear retrieving filters that are effective in both developed and developing waves, as well as in the presence of swell waves, are chosen. In this case, the measurement of the divergence of the wave spectra obtained from satellite images of high spatial resolution and subsatellite data at optimal values of these parameters is 0.08–0.12. This testifies to the adequacy of the proposed method for recording wave spectra from the spectra of satellite images

Thus, the adequacy of remote measurement of both the time-averaged spectra of the sea surface and the variations of these spectra caused by wave nonstationarity using high spatial resolution images and nonlinear recovering operators has been experimentally confirmed. The conducted researches made it possible to develop methods and technologies for comprehensive ground-space monitoring of marine areas to obtain such important oceanographic char-

The developed method can be used to study surface wave conditions, including in the space

The work was carried out with the financial support of the state represented by the Ministry of Education and Science of the Russian Federation in the framework of the Federal Program "Research and development in priority areas of development of the scientific and technological complex of Russia for 2014-2020" (unique identifier of the project RFMEFI57716

monitoring of natural and anthropogenic impacts on the marine waters.

measured with high accuracy by string waveforms under controlled conditions.

optical images of the sea surface for a wide range of wave formation conditions.

truth assets.

over a wide frequency range.

86 Surface Waves - New Trends and Developments

acteristics as surface wave spectra.

Acknowledgements

X0234).

Valery G. Bondur\* and Alexander B. Murynin\*

\*Address all correspondence to: vgbondur@aerocosmos.info and amurynin@bk.ru

AEROCOSMOS Research Institute for Aerospace Monitoring, Moscow, Russia

#### References


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**Chapter 5**

Provisional chapter

**Electromagnetic Polarization: A New Approach on the**

DOI: 10.5772/intechopen.71836

Electromagnetic Polarization: A New Approach on the

In this chapter it is elucidated how the analytical solution was obtained to obtain the electromagnetic polarization profile of a completely polarized wave in the region of distant fields. The analytical solution was obtained from the interpretation of the physical phenomenon associated with the method of the linear component, which was adapted for the use of discrete and miniaturized elements, reducing the physical space for the measurement circuit and the expenses associated to the delay circuit. From the analytical solution it is possible to observe that with only one phase measurement in the delay circuit it is possible to obtain the polarization profile of the wave, with the axial

Nowadays, in many industries, industries, offices, companies, supermarket chains among others, there is a very large diversity of radio frequency (RF) equipment. In these sectors, several RF equipment can coexist, operating simultaneously in a small space, even having different emission/reception characteristics. Because there are several devices, some of them may have their performances harmed or even compromised due to electromagnetic interference (EM) caused by other equipment that may be emitting an undesirable EM wave to the apparatus that is undergoing the interference process. In addition, it is also observed that, in the technological market of the present day, the size of the antennas existing in these

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

**Linear Component Method**

Linear Component Method

José Moraes Gurgel Neto, Adi Neves Rocha and

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Keywords: polarization, phased shifter, microfite, dipole

José Moraes Gurgel Neto, Adi Neves Rocha

Jobson de Araújo Nascimento, Regina Maria De Lima Neta,

Jobson de Araújo Nascimento, Regina Maria De Lima Neta,

Alexsandro Aleixo Pereira Da Silva

and Alexsandro Aleixo Pereira Da Silva

http://dx.doi.org/10.5772/intechopen.71836

ratio and phase, respectively.

Abstract

1. Introduction


#### **Electromagnetic Polarization: A New Approach on the Linear Component Method** Electromagnetic Polarization: A New Approach on the Linear Component Method

DOI: 10.5772/intechopen.71836

Jobson de Araújo Nascimento, Regina Maria De Lima Neta, José Moraes Gurgel Neto, Adi Neves Rocha and Alexsandro Aleixo Pereira Da Silva Jobson de Araújo Nascimento, Regina Maria De Lima Neta, José Moraes Gurgel Neto, Adi Neves Rocha

Additional information is available at the end of the chapter and Alexsandro Aleixo Pereira Da Silva

http://dx.doi.org/10.5772/intechopen.71836 Additional information is available at the end of the chapter

#### Abstract

[12] Bondur VG, Grebenyuk YV, Sabinin KD. The spectral characteristics and kinematics of shortperiod internal waves on the Hawaiian shelf. Izvestiya, Atmospheric and Oceanic

[13] Bondur VG. Modern approaches to processing large hyperspectral and multispectral aerospace data flows. Izvestiya, Atmospheric and Oceanic Physics. 2014;50(9):840-852 [14] Bondur VG. Satellite monitoring and mathematical modelling of deep runoff turbulent jets in coastal water areas. In: Waste Water—Evaluation and Management. Croatia: InTech;

[15] Bondur VG, Zubkov EV. Showing up the small-scale ocean upper layer optical inhomogeneities by the multispectral space images with the high surface resolution. Part 1. The canals and channels drainage effects at the coastal zone. Issledovaniya Zemli iz Kosmosa.

[16] Bondur VG, Murynin AB. Reconstruction of the spectra of the surface waves from the spectra of their images with an account of the nonlinear modulation of the brightness

[17] Murynin AB. Restoration of the spatial spectrum of the sea surface from the optical images in a nonlinear model of brightness field. Issledovaniya Zemli iz Kosmosa. 1990;6:60-70 [18] Bondur VG, Murynin AB. Methods for retrieval of sea wave spectra from aerospace image spectra. Izvestiya, Atmospheric and Oceanic Physics. 2016;52(9):877-887

[19] Bondur VG, Dulov VA, Murynin AB, Yurovsky YY. Studying marine wave spectra in a wide range of wavelengths using satellite and in situ data. Izvestiya, Atmospheric and

[20] Bondur VG, Dulov VA, Murynin AB, Ignatiev VY. Retrieving sea-wave spectra using satellite-imagery spectra in a wide range of frequencies. Izvestiya, Atmospheric and

[21] Murynin AB. Parametrization of filters retrieving spatial spectra of sea surface slopes

[22] Baranovskii VD, Bondur VG, Kulakov VV, et al. Calibration of remote measurements of 2- D spatial wave spectra from optical images. Issledovaniya Zemli iz Kosmosa. 1992;2:59-67

[23] Toba J. Local balance in the air-sea boundary process. Journal of the Oceanographi-

[24] Yurovskaya MV, Dulov VA, Chapron B, Kudryavtsev VN. Directional short wind wave spectra derived from the sea surface photography. Journal of Geophysical Research.

[25] Bondur VG, Murynin AB, Ignatiev VY. Parameters optimization in the problem of seawave spectra recovery by airspace images. Journal of Machine Learning and Data Anal-

from optical images. Issledovaniya Zemli iz Kosmosa. 1991;5:31-38

field. Optika Atmosfery i Okeana. 1991;4(4):387-393

Oceanic Physics. 2016;52(9):888-903

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2013;118(9):4380-4394. DOI: 10.1002/jgrc.20296

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88 Surface Waves - New Trends and Developments

2011. pp. 155-180

2005;(4):54-61

In this chapter it is elucidated how the analytical solution was obtained to obtain the electromagnetic polarization profile of a completely polarized wave in the region of distant fields. The analytical solution was obtained from the interpretation of the physical phenomenon associated with the method of the linear component, which was adapted for the use of discrete and miniaturized elements, reducing the physical space for the measurement circuit and the expenses associated to the delay circuit. From the analytical solution it is possible to observe that with only one phase measurement in the delay circuit it is possible to obtain the polarization profile of the wave, with the axial ratio and phase, respectively.

Keywords: polarization, phased shifter, microfite, dipole

#### 1. Introduction

Nowadays, in many industries, industries, offices, companies, supermarket chains among others, there is a very large diversity of radio frequency (RF) equipment. In these sectors, several RF equipment can coexist, operating simultaneously in a small space, even having different emission/reception characteristics. Because there are several devices, some of them may have their performances harmed or even compromised due to electromagnetic interference (EM) caused by other equipment that may be emitting an undesirable EM wave to the apparatus that is undergoing the interference process. In addition, it is also observed that, in the technological market of the present day, the size of the antennas existing in these

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

equipments becomes less and less. The miniaturization of antennas in many cases makes it difficult to locate the emitter system, which is possibly causing EM interference, since the antenna is often built into the equipment. Therefore, knowledge of the direction of propagation of the EM wave emitted by these antennas or EM field emitting elements will facilitate the discovery of the EM interference source (s) and the determination of possible engineering actions that help to attenuate the same. The study of techniques that aim to reduce the damage caused by EM interference between electronic devices has increased in recent years. Through a study of the polarization of electromagnetic waves, one can identify the behavior of the irradiated electric field. The knowledge of the amplitude and phase of the electric field [1] allows to identify the polarization pattern of the wave The information of the phase of the electric field in the region of far fields (FFR) is of great value for the discovery and suggestions of overcoming problems caused By EM interference. An experimental technique was developed to confirm the linear component method, proposed by Kraus [2], for the reconstruction of the electromagnetic polarization in the FFR. The method was adapted using miniaturized discrete elements for the lagging circuit and an experimental and analytical confirmation was obtained in order to obtain the phase of the electric field radiated to electromagnetic waves that are completely polarized in the FFR using discrete components.

#### 2. Methodology

The model for obtaining the polarization profile of an electromagnetic (EM) wave in the RCD was initially proposed by [2]. An adaptation in the phase-shift circuit to obtain the phase of polarization is proposed in an analytical way below.

The mathematical approach to gain access to the polarization phase of the EM wave in the FFR (Far Field Region) is made using the phasor analysis of the signal that is received by each dipole, and the resulting signal at the receiver of Figure 1(b). The discrepancy introduced by the discrete components does not identify the phase of the transmitted signal, and it allows to deduce this phase after a phasor sum of the two components. What would be simpler is the construction of a system to identify the phase of the emitted electric field.

In the delay circuit Figure 1(b) it was proposed a modification in the physical structure of the phase shift, replacing the split line with planar transmission lines at the end of the connection with the dipoles. And in a discrete way the delay was supposed from the interpretation of the associated physical phenomenon and the supposed theoretical formulation. The proposed delay circuit can be observed in Figure (2).

The particular cases were tested in order to validate the method. But for the general case, where the polarization phase of the signal emitted by the transmitting antenna is not known, the following reasoning is given. From the consideration that the fields captured by the dipoles are of the form

$$
\stackrel{\rightharpoonup}{E\_v} = E\_v. \mathcal{e}^{\mathcal{O}^\circ} \widehat{\mathcal{v}} \tag{1}
$$

Where ψ is the phase difference between the two signals picked up by the dipoles, i.e., the phase of the polarization of the transmitted signal, and θ is the delay intentionally inserted in order to find the phase of the emitted signal, and <sup>b</sup>v and <sup>b</sup><sup>h</sup> are the unit vectors in the vertical and horizontal directions, respectively. The receiver is the one who performs a phasor sum of the

Figure 1. Set up for measuring: (a) the axial ratio; (b) the phase difference ψ, using the method of linear components [1].

Electromagnetic Polarization: A New Approach on the Linear Component Method

http://dx.doi.org/10.5772/intechopen.71836

91

(3)

ER ! ¼Ev ! þ Eh !

signals at high frequencies, has its resulting field given by

Figure 2. Circuit validation of adjustments made in the method of the linear component.

In phasor form one has to

$$
\overrightarrow{E\_h} = E\_h.e^{j(\psi + \theta)}\widehat{h} \tag{2}
$$

Figure 1. Set up for measuring: (a) the axial ratio; (b) the phase difference ψ, using the method of linear components [1].

Figure 2. Circuit validation of adjustments made in the method of the linear component.

Where ψ is the phase difference between the two signals picked up by the dipoles, i.e., the phase of the polarization of the transmitted signal, and θ is the delay intentionally inserted in order to find the phase of the emitted signal, and <sup>b</sup>v and <sup>b</sup><sup>h</sup> are the unit vectors in the vertical and horizontal directions, respectively. The receiver is the one who performs a phasor sum of the signals at high frequencies, has its resulting field given by

$$
\overrightarrow{E\_R} = \overrightarrow{E\_v} + \overrightarrow{E\_h} \tag{3}
$$

In phasor form one has to

equipments becomes less and less. The miniaturization of antennas in many cases makes it difficult to locate the emitter system, which is possibly causing EM interference, since the antenna is often built into the equipment. Therefore, knowledge of the direction of propagation of the EM wave emitted by these antennas or EM field emitting elements will facilitate the discovery of the EM interference source (s) and the determination of possible engineering actions that help to attenuate the same. The study of techniques that aim to reduce the damage caused by EM interference between electronic devices has increased in recent years. Through a study of the polarization of electromagnetic waves, one can identify the behavior of the irradiated electric field. The knowledge of the amplitude and phase of the electric field [1] allows to identify the polarization pattern of the wave The information of the phase of the electric field in the region of far fields (FFR) is of great value for the discovery and suggestions of overcoming problems caused By EM interference. An experimental technique was developed to confirm the linear component method, proposed by Kraus [2], for the reconstruction of the electromagnetic polarization in the FFR. The method was adapted using miniaturized discrete elements for the lagging circuit and an experimental and analytical confirmation was obtained in order to obtain the phase of the electric field radiated to electromagnetic waves

The model for obtaining the polarization profile of an electromagnetic (EM) wave in the RCD was initially proposed by [2]. An adaptation in the phase-shift circuit to obtain the phase of

The mathematical approach to gain access to the polarization phase of the EM wave in the FFR (Far Field Region) is made using the phasor analysis of the signal that is received by each dipole, and the resulting signal at the receiver of Figure 1(b). The discrepancy introduced by the discrete components does not identify the phase of the transmitted signal, and it allows to deduce this phase after a phasor sum of the two components. What would be simpler is the

In the delay circuit Figure 1(b) it was proposed a modification in the physical structure of the phase shift, replacing the split line with planar transmission lines at the end of the connection with the dipoles. And in a discrete way the delay was supposed from the interpretation of the associated physical phenomenon and the supposed theoretical formulation. The proposed

The particular cases were tested in order to validate the method. But for the general case, where the polarization phase of the signal emitted by the transmitting antenna is not known, the following reasoning is given. From the consideration that the fields captured by the dipoles are

<sup>b</sup><sup>v</sup> (1)

<sup>j</sup>ð Þ <sup>ψ</sup>þ<sup>θ</sup> <sup>b</sup><sup>h</sup> (2)

Ev ! ¼ Ev:e j0�

Eh ! ¼ Eh:e

that are completely polarized in the FFR using discrete components.

construction of a system to identify the phase of the emitted electric field.

polarization is proposed in an analytical way below.

delay circuit can be observed in Figure (2).

2. Methodology

90 Surface Waves - New Trends and Developments

of the form

$$E\_R = E\_v e^{j0^\circ} + E\_h e^{j(\psi + \theta)} \tag{4}$$

cos ð Þ¼� ψ

q

D ¼ A:

D ¼

Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ψ ¼ acos

Denoting x ¼ senð Þ ψ , one can write Eq. (16) as

known constants. In solving Eq. (23) have

Figure 3. This is achieved as the phase of the polarization.

In (14) one has to

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � sen<sup>2</sup>ð Þ ψ

Electromagnetic Polarization: A New Approach on the Linear Component Method

B A

cos ψ þ atan

B A

B A

> B A

� atan

� B:senð Þ ψ : (16)

http://dx.doi.org/10.5772/intechopen.71836

� � (17)

¼ OP: cos ð Þ α (18)

¼ D (19)

� � � � (20)

� � � � (21)

<sup>1</sup> � <sup>x</sup><sup>2</sup> <sup>p</sup> � <sup>B</sup>:x: (23)

� � (22)

(15)

93

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � sen<sup>2</sup>ð Þ ψ

The geometric development of the analytic model is elucidated in the following expressions. From the geometric analysis to Eq. (16) illustrated in the Figure 3 is possible to verify that:

α ¼ ψ þ atan

OP<sup>0</sup>

<sup>A</sup><sup>2</sup> <sup>þ</sup> <sup>B</sup><sup>2</sup> <sup>p</sup> <sup>¼</sup> cos <sup>ψ</sup> <sup>þ</sup> atan

Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>A</sup><sup>2</sup> <sup>þ</sup> <sup>B</sup><sup>2</sup> <sup>p</sup>

Then it found a closed expression to find the value of phase difference ψ. The discovery phase can be analytically or by using the parameters measured experimentally as done at work.

In developing Eq. (23), one arrives at the expression that relates the unknown angle ψ, with

<sup>D</sup> <sup>¼</sup> <sup>A</sup>: ffiffiffiffiffiffiffiffiffiffiffiffiffi

!

OP<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>A</sup><sup>2</sup> <sup>þ</sup> <sup>B</sup><sup>2</sup> <sup>p</sup>

In developing this expression one has to

$$E\_{\mathbb{R}} = E\_v \cos \left( 0^\circ \right) + E\_h \cos \left( \theta + \psi \right) + jE\_h [\sin(\theta + \psi)]$$

$$E\_{\mathbb{R}} = E\_v + E\_h [\cos \left( \theta \right) \cos \left( \psi \right) - \sin(\theta) \sin(\psi)] + jE\_h [\sin(\theta) \cos \left( \psi \right) + \sin(\psi) \cos(\theta)] \tag{5}$$

Once ER, Eh, Ev e ψ are known it can be adopted that

$$\begin{cases} A = E\_h. \cos\left(\theta\right) \\ B = E\_h. \text{sen}(\theta) \\ C = E\_R \end{cases} \tag{6}$$

Is that z ¼ a þ bi;

$$a = E\_v + A.\cos\left(\psi\right) - B\text{sen}(\psi)\tag{7}$$

$$b = A\cos(\psi) + B\text{sen}(\psi)\tag{8}$$

$$\mathcal{C} = E\_v + A.\cos\left(\psi\right) - B\text{sen}(\psi) + j[A\text{sen}(\psi) + B\cos(\psi)].\tag{9}$$

The value of C in this equation is the electric field module ER and this module is given by

$$|z| = \sqrt{a^2 + b^2} \tag{10}$$

Thus it is to develop the expression (9) in (10) that:

$$\begin{aligned} |\mathbf{C}| &= \sqrt{[\mathbf{E}\_v + A.\cos\left(\psi\right) - B\epsilon n(\psi)]^2 + \left[A\epsilon n(\psi) + B\cos\left(\psi\right)\right]^2} \\ \mathbf{C}^2 &= E\_v^2 + A^2 + B^2 + 2.E\_v[A.\cos\left(\psi\right) - B.\epsilon n(\psi)] \end{aligned} \tag{11}$$

In the case where the angle ψ it is unknown to have:

$$\begin{aligned} \text{C}^2 - E\_v^2 - A^2 - B^2 &= 2. E\_v [A.\cos\left(\psi\right) - B.sen(\psi)]\\ \frac{\text{C}^2 - E\_v^2 - A^2 - B^2}{2.E\_v} &= A.\cos\left(\psi\right) - B.sen(\psi) \end{aligned} \tag{12}$$

Doing

$$D = \frac{\left(\mathcal{C}^2 - E\_v^2 - A^2 - B^2\right)}{2E\_v} \tag{13}$$

it has been

$$D = A.\cos\left(\psi\right) - B.sen(\psi)\tag{14}$$

Using the expression

Electromagnetic Polarization: A New Approach on the Linear Component Method http://dx.doi.org/10.5772/intechopen.71836 93

$$\cos\left(\psi\right) = \pm\sqrt{1 - \text{sen}^2(\psi)}\tag{15}$$

In (14) one has to

ER ¼ Eve

8 >><

>>:

In developing this expression one has to

92 Surface Waves - New Trends and Developments

Is that z ¼ a þ bi;

Doing

it has been

Using the expression

ER <sup>¼</sup> Ev cos 0� ��

Once ER, Eh, Ev e ψ are known it can be adopted that

Thus it is to develop the expression (9) in (10) that:

q

<sup>C</sup><sup>2</sup> <sup>¼</sup> <sup>E</sup><sup>2</sup>

In the case where the angle ψ it is unknown to have:

<sup>C</sup><sup>2</sup> � <sup>E</sup><sup>2</sup>

<sup>C</sup><sup>2</sup> � <sup>E</sup><sup>2</sup>

<sup>v</sup> � <sup>A</sup><sup>2</sup> � <sup>B</sup><sup>2</sup> 2:Ev

<sup>D</sup> <sup>¼</sup> <sup>C</sup><sup>2</sup> � <sup>E</sup><sup>2</sup>

∣C∣ ¼

j0� þ Ehe

þ Eh cos ð Þþ θ þ ψ jEh½ � senð Þ θ þ ψ

a ¼ Ev þ A: cos ð Þ� ψ Bsenð Þ ψ (7)

C ¼ Ev þ A: cos ð Þ� ψ Bsenð Þþ ψ j Asen ½ � ð Þþ ψ Bcosð Þ ψ : (9)

b ¼ Acosð Þþ ψ Bsenð Þ ψ (8)

ER <sup>¼</sup> Ev <sup>þ</sup> Eh<sup>½</sup> cos ð Þ <sup>θ</sup> : cos ð Þ� <sup>ψ</sup> senð Þ <sup>θ</sup> :senð Þ <sup>ψ</sup> � þ jEh½ � senð Þ <sup>θ</sup> : cos ð Þþ <sup>ψ</sup> senð Þ <sup>ψ</sup> : cos ð Þ <sup>θ</sup> (5)

A ¼ Eh: cos ð Þ θ B ¼ Eh:senð Þ θ

C ¼ ER

The value of C in this equation is the electric field module ER and this module is given by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> <sup>p</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ � Ev <sup>þ</sup> <sup>A</sup>: cos ð Þ� <sup>ψ</sup> Bsenð Þ <sup>ψ</sup> <sup>2</sup> <sup>þ</sup> ½ � Asenð Þþ <sup>ψ</sup> Bcosð Þ <sup>ψ</sup> <sup>2</sup>

<sup>v</sup> <sup>þ</sup> <sup>A</sup><sup>2</sup> <sup>þ</sup> <sup>B</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>:Ev½ � <sup>A</sup>: cos ð Þ� <sup>ψ</sup> <sup>B</sup>:senð Þ <sup>ψ</sup>

<sup>v</sup> � <sup>A</sup><sup>2</sup> � <sup>B</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>:Ev½ � <sup>A</sup>: cos ð Þ� <sup>ψ</sup> <sup>B</sup>:senð Þ <sup>ψ</sup>

<sup>v</sup> � <sup>A</sup><sup>2</sup> � <sup>B</sup><sup>2</sup> � � 2Ev

¼ A: cos ð Þ� ψ B:senð Þ ψ

D ¼ A: cos ð Þ� ψ B:senð Þ ψ (14)

j j z ¼

<sup>j</sup>ð Þ <sup>ψ</sup>þ<sup>θ</sup> (4)

(6)

(10)

(11)

(12)

(13)

$$D = A \sqrt{1 - \operatorname{sen}^2(\psi)} - B \operatorname{sen}(\psi). \tag{16}$$

The geometric development of the analytic model is elucidated in the following expressions. From the geometric analysis to Eq. (16) illustrated in the Figure 3 is possible to verify that:

$$
\alpha = \psi + \operatorname{atan}\left(\frac{B}{A}\right) \tag{17}
$$

$$
\overline{OP}' = \overline{OP}.\cos\left(a\right)\tag{18}
$$

$$
\overrightarrow{OP} = D \tag{19}
$$

$$D = \sqrt{A^2 + B^2} \cos\left(\psi + \operatorname{atan}\left(\frac{B}{A}\right)\right) \tag{20}$$

$$\frac{D}{\sqrt{A^2 + B^2}} = \cos\left(\psi + \operatorname{atan}\left(\frac{B}{A}\right)\right) \tag{21}$$

$$\psi = a \cos \left(\frac{D}{\sqrt{A^2 + B^2}}\right) - a \tan \left(\frac{B}{A}\right) \tag{22}$$

Then it found a closed expression to find the value of phase difference ψ. The discovery phase can be analytically or by using the parameters measured experimentally as done at work.

Denoting x ¼ senð Þ ψ , one can write Eq. (16) as

$$D = A\sqrt{1 - \mathbf{x}^2} - B\mathbf{x}.\tag{23}$$

In developing Eq. (23), one arrives at the expression that relates the unknown angle ψ, with known constants. In solving Eq. (23) have

Figure 3. This is achieved as the phase of the polarization.

$$\text{sen}(\psi) = \frac{-D \pm \left[ (DB)^2 - \left( A^2 + B^2 \right) . \left( D^2 - A^2 \right) \right]^{1/2}}{\left( A^2 + B^2 \right)} \tag{24}$$

An analysis was performed to validate Eq. (11) for elementary cases. The results can be seen in Table 1 and all the mathematical development that follows.

$$\psi = \arcsin\left(\frac{-D \pm \left[ (DB)^2 - \left(A^2 + B^2\right) . \left(D^2 - A^2\right) \right]^{1/2}}{\left(A^2 + B^2\right)}\right) \tag{25}$$

• For the left circular case, we have:

Table 1. Polarization patterns tested.

Substituting this data into Eq. (C.4.3) gives:

Eh <sup>¼</sup> 1 horizontal dipole amplitude � � Ev <sup>¼</sup> 1 vertical dipole amplitude � �

Polarization profile Eh Ev θ j j ER ψ

Linear to 135� 100 100 45� 0 135�

Linear to 45� 100 100 45� ffiffiffi

Left circular 100 100 45� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Right circular 100 100 45� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

phase difference � �

ffiffiffi 2 p <sup>2</sup> <sup>þ</sup> <sup>j</sup>:

Eh <sup>¼</sup> 1 horizontal dipole amplitude � � Ev <sup>¼</sup> 1 vertical dipole amplitude � �

phase difference � �

generic phase of the line � �

<sup>þ</sup> <sup>45</sup> � � � ��

j j¼ ER

<sup>þ</sup> <sup>45</sup> � � � ��

j j ER ¼

ffiffiffi 2 p

Therefore what will be done is a comparison of the values of θ, with the possible results of ER,

<sup>2</sup> <sup>þ</sup> <sup>j</sup>: �

p q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup> <sup>þ</sup> ffiffiffi 2

ER ¼ 1 þ

ER ¼ 1 �

generic phase of the line � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup> � ffiffiffi 2 p p

<sup>þ</sup> <sup>j</sup>:1, <sup>0</sup>: sen <sup>90</sup>�

<sup>þ</sup> <sup>j</sup>:1, <sup>0</sup>: sen <sup>270</sup>�

ffiffiffi 2 p 2 !

<sup>þ</sup> <sup>45</sup> � � � ��

ffiffiffi 2 p 2 ! <sup>þ</sup> <sup>45</sup> � � � ��

2

Electromagnetic Polarization: A New Approach on the Linear Component Method

<sup>2</sup> � ffiffiffi 2 p p 90�

http://dx.doi.org/10.5772/intechopen.71836

<sup>2</sup> <sup>þ</sup> ffiffiffi 2 p p 270�

p 45�

95

(28)

(29)

<sup>ψ</sup> <sup>¼</sup> <sup>90</sup>�

8 >>>>><

>>>>>:

• For the Circular case on the right, we have that:

8 >>>>><

>>>>>:

ER <sup>¼</sup> <sup>1</sup>, <sup>0</sup> <sup>þ</sup> <sup>1</sup>, <sup>0</sup>: cos 270�

Substituting this data into Eq. (C.4.3) gives:

and we have:

<sup>θ</sup> <sup>¼</sup> <sup>45</sup>�

<sup>ψ</sup> <sup>¼</sup> <sup>270</sup>�

<sup>θ</sup> <sup>¼</sup> <sup>45</sup>�

ER <sup>¼</sup> <sup>1</sup>, <sup>0</sup> <sup>þ</sup> <sup>1</sup>, <sup>0</sup>: cos 90�

To validate the cases where there is ambiguity in the polarization profile of the EM wave, which are those in which the axial ratio is unitary, Eq. (9) and tested a random phase in the delay circuit, in order to prove that it is enough to only insert a delay line to validate the adapted linear component method [3].

The elementary cases were tested in Eq. (9), the whole mathematical development for these cases is next

• For the Linear case at 45�, it is possible to write:

$$\begin{cases} E\_h = 1 \text{(horizontal dipole amplitude)}\\ E\_v = 1 \text{(vertical dipole amplitude)}\\ \psi = 45^\circ \text{(phase difference)}\\ \theta = 45^\circ \text{ (generic phase of the line)} \end{cases}$$

Substituting this data into Eq. (9) we have that:

$$E\_R = 1,0+1,0.\left[\cos\left(45^\circ + 45^\circ\right)\right] + j1,0.\left[\text{sen}\left(45^\circ + 45^\circ\right)\right]$$

$$\begin{aligned} E\_R &= 1+j\\ |E\_R| &= \sqrt{2} \end{aligned} \tag{26}$$

• For the linear case at 135�, we have that:

$$\begin{cases} E\_h = 1 \text{(horizontal dipole amplitude)}\\ E\_v = 1 \text{(vertical dipole amplitude)}\\ \psi = 135^\circ \text{(phase difference)}\\ \theta = 45^\circ \text{ (generic phase of the line)} \end{cases}$$

Substituting this data into Eq. (C.4.3) gives:

$$E\_R = 1,0+1,0.\left[\cos\left(135^\circ + 45^\circ\right)\right] + j1,0.\left[\sin\left(135^\circ + 45^\circ\right)\right]$$

$$\begin{aligned} E\_R &= 0\\ |E\_R| &= 0 \end{aligned} \tag{27}$$


Table 1. Polarization patterns tested.

senð Þ¼ ψ

94 Surface Waves - New Trends and Developments

ψ ¼ arcsen

• For the Linear case at 45�, it is possible to write:

8 >>>>><

>>>>>:

8 >>>>><

>>>>>:

ER <sup>¼</sup> <sup>1</sup>, <sup>0</sup> <sup>þ</sup> <sup>1</sup>, <sup>0</sup>: cos 135�

Substituting this data into Eq. (9) we have that:

• For the linear case at 135�, we have that:

Substituting this data into Eq. (C.4.3) gives:

<sup>ψ</sup> <sup>¼</sup> <sup>45</sup>�

<sup>θ</sup> <sup>¼</sup> <sup>45</sup>�

<sup>ψ</sup> <sup>¼</sup> <sup>135</sup>�

<sup>θ</sup> <sup>¼</sup> <sup>45</sup>�

ER <sup>¼</sup> <sup>1</sup>, <sup>0</sup> <sup>þ</sup> <sup>1</sup>, <sup>0</sup>: cos 45�

adapted linear component method [3].

cases is next

Table 1 and all the mathematical development that follows.

0 B@

�<sup>D</sup> � ð Þ DB <sup>2</sup> � <sup>A</sup><sup>2</sup> <sup>þ</sup> <sup>B</sup><sup>2</sup> � �: <sup>D</sup><sup>2</sup> � <sup>A</sup><sup>2</sup> � � h i<sup>1</sup>=<sup>2</sup>

�<sup>D</sup> � ð Þ DB <sup>2</sup> � <sup>A</sup><sup>2</sup> <sup>þ</sup> <sup>B</sup><sup>2</sup> � �: <sup>D</sup><sup>2</sup> � <sup>A</sup><sup>2</sup> � � h i<sup>1</sup>=<sup>2</sup> <sup>A</sup><sup>2</sup> <sup>þ</sup> <sup>B</sup><sup>2</sup> � �

An analysis was performed to validate Eq. (11) for elementary cases. The results can be seen in

To validate the cases where there is ambiguity in the polarization profile of the EM wave, which are those in which the axial ratio is unitary, Eq. (9) and tested a random phase in the delay circuit, in order to prove that it is enough to only insert a delay line to validate the

The elementary cases were tested in Eq. (9), the whole mathematical development for these

Eh ¼ 1 horizontal dipole amplitude � �

Ev ¼ 1 vertical dipole amplitude � �

> phase difference � �

> > ER ¼ 1 þ j j j ER <sup>¼</sup> ffiffiffi 2 p

Eh ¼ 1 horizontal dipole amplitude � �

> phase difference � �

generic phase of the line � �

Ev ¼ 1 vertical dipole amplitude � �

<sup>þ</sup> <sup>45</sup> � � � ��

ER ¼ 0 j j ER ¼ 0

<sup>þ</sup> <sup>45</sup> � � � ��

generic phase of the line � �

<sup>þ</sup> <sup>j</sup>:1, <sup>0</sup>: sen <sup>45</sup>�

<sup>þ</sup> <sup>j</sup>:1, <sup>0</sup>: sen <sup>135</sup>�

<sup>þ</sup> <sup>45</sup> � � � ��

<sup>þ</sup> <sup>45</sup> � � � ��

<sup>A</sup><sup>2</sup> <sup>þ</sup> <sup>B</sup><sup>2</sup> � � (24)

1

CA (25)

(26)

(27)

• For the left circular case, we have:

$$\begin{cases} E\_{\mathbb{H}} = 1 \text{(horizontal dipole amplitude)}\\ E\_{v} = 1 \text{(vertical dipole amplitude)}\\ \psi = 90^{\circ} \text{(phase difference)}\\ \theta = 45^{\circ} \text{ (generic phase of the line)} \end{cases}$$

Substituting this data into Eq. (C.4.3) gives:

$$E\_R = 1,0+1,0.\left[\cos\left(90^\circ + 45^\circ\right)\right] + j1,0.\left[\sin\left(90^\circ + 45^\circ\right)\right]$$

$$E\_R = 1 - \frac{\sqrt{2}}{2} + j,\left(\frac{\sqrt{2}}{2}\right) \tag{28}$$

$$|E\_R| = \sqrt{2 - \sqrt{2}}$$

• For the Circular case on the right, we have that:

$$\begin{cases} E\_h = 1 \text{(horizontal dipole amplitude)}\\ E\_v = 1 \text{(vertical dipole amplitude)}\\ \psi = 270^\circ \text{(phase difference)}\\ \theta = 45^\circ \text{ (generic phase of the line)} \end{cases}$$

Substituting this data into Eq. (C.4.3) gives:

$$E\_R = 1,0+1,0.\left[\cos\left(270^\circ + 45^\circ\right)\right] + j.1,0.\left[\text{sen}\left(270^\circ + 45^\circ\right)\right]$$

$$E\_R = 1 + \frac{\sqrt{2}}{2} + j.\left(-\frac{\sqrt{2}}{2}\right)$$

$$|E\_R| = \sqrt{2 + \sqrt{2}}\tag{29}$$

Therefore what will be done is a comparison of the values of θ, with the possible results of ER, and we have:


When applying in (9) we have:

$$
\sqrt{2}^2 = 1 + \left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 + \cos(2\psi) \left[ \left(\frac{\sqrt{2}}{2}\right)^2 - \left(\frac{\sqrt{2}}{2}\right)^2 \right] + 2.1.
\left[\frac{\sqrt{2}}{2}, \cos\left(\psi\right) - \frac{\sqrt{2}}{2}\sin(\psi)\right]
$$

$$
2 = 2 + 2.\frac{\sqrt{2}}{2}[\cos(\psi) - \sin(\psi)]
$$

$$
\text{sen}(\frac{\pi}{2} - \psi) = \text{sen}(\psi),
$$

$$
\frac{\pi}{2} - \psi = \psi + 2.k.\pi
$$

$$
\psi = \frac{\pi}{4} + 2.k.\pi.\tag{30}
$$

<sup>ψ</sup> <sup>¼</sup> <sup>3</sup>:<sup>π</sup>

• Which validates the phase difference of 135�.

• Circular left

• Eh ¼ 1; • Ev ¼ 1;

• <sup>ψ</sup> <sup>¼</sup> <sup>45</sup>�

• j j ER ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup> � ffiffiffi 2 p � � q <sup>2</sup>

• Circular right:

;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup> <sup>þ</sup> ffiffiffi 2 <sup>p</sup> <sup>p</sup> ;

When replacing in (9), we have that:

• Eh ¼ 1; • Ev ¼ 1;

• <sup>ψ</sup> <sup>¼</sup> <sup>45</sup>�

• j j¼ ER

• ψ =?

• ψ =?

;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup> � ffiffiffi 2 <sup>p</sup> <sup>p</sup> ;

When applying in (9), we have that:

ffiffiffi 2 p 2 !<sup>2</sup>

Which gives the phase difference of 90�.

þ

ffiffiffi 2 p 2 !<sup>2</sup>

<sup>2</sup> � ffiffiffi 2 <sup>p</sup> <sup>¼</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>:

þ cos 2ð Þ ψ

sen <sup>ψ</sup> � <sup>π</sup> 4 � � <sup>¼</sup> sen

<sup>ψ</sup> <sup>¼</sup> <sup>π</sup>

ffiffiffi 2 p 2 !<sup>2</sup>

> π 4 � � <sup>þ</sup> <sup>2</sup>:k:π,

4

ffiffiffi 2 p

�

<sup>2</sup> ½ � cos ð Þ� <sup>ψ</sup> senð Þ <sup>ψ</sup> ,

!<sup>2</sup> 2

ffiffiffi 2 p 2

3 5 þ 2:1:

ffiffiffi 2 p

<sup>2</sup> <sup>þ</sup> <sup>2</sup>:k:π: (32)

<sup>2</sup> : cos ð Þ� <sup>ψ</sup>

ffiffiffi 2 p

" #

<sup>2</sup> senð Þ <sup>ψ</sup>

¼ 1 þ

<sup>4</sup> <sup>þ</sup> <sup>2</sup>:k:π: (31)

http://dx.doi.org/10.5772/intechopen.71836

97

Electromagnetic Polarization: A New Approach on the Linear Component Method

Which certifies the phase difference of 45�.


When replacing in (9) we have:

$$0 = 1 + \left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 + \cos(2\psi)\left[\left(\frac{\sqrt{2}}{2}\right)^2 - \left(\frac{\sqrt{2}}{2}\right)^2\right] + 2.1. \left[\frac{\sqrt{2}}{2}. \cos(\psi) - \frac{\sqrt{2}}{2}\sin(\psi)\right],$$

$$-2 = 2.\frac{\sqrt{2}}{2}[\cos(\psi) - \sin(\psi)],$$

$$\text{sen}(\psi - \frac{\pi}{4}) = \text{sen}\left(\frac{\pi}{2}\right) + 2.k.\pi,$$

Electromagnetic Polarization: A New Approach on the Linear Component Method http://dx.doi.org/10.5772/intechopen.71836 97

$$
\psi = \frac{3.\pi}{4} + 2.k.\pi.\tag{31}
$$


• Linear at 45�:

;

2 <sup>p</sup> ;

96 Surface Waves - New Trends and Developments

When applying in (9) we have:

ffiffiffi 2 p 2 !<sup>2</sup>

þ

Which certifies the phase difference of 45�.

ffiffiffi 2 p 2 !<sup>2</sup>

þ cos 2ð Þ ψ

2 ¼ 2 þ 2:

sen π <sup>2</sup> � <sup>ψ</sup> � �

π

ffiffiffi 2 p 2 !<sup>2</sup>

ffiffiffi 2 p

<sup>2</sup> � <sup>ψ</sup> <sup>¼</sup> <sup>ψ</sup> <sup>þ</sup> <sup>2</sup>:k:<sup>π</sup>

<sup>ψ</sup> <sup>¼</sup> <sup>π</sup>

ffiffiffi 2 p 2 !<sup>2</sup>

ffiffiffi 2 p

4

�

<sup>2</sup> ½ � cos ð Þ� <sup>ψ</sup> senð Þ <sup>ψ</sup> ,

π 2 � �

!<sup>2</sup> 2

¼ sen

ffiffiffi 2 p 2

3

þ 2:k:π,

5 þ 2:1:

ffiffiffi 2 p

<sup>2</sup> : cos ð Þ� <sup>ψ</sup>

ffiffiffi 2 p

" #

<sup>2</sup> senð Þ <sup>ψ</sup>

,

4

�

<sup>2</sup> ½ � cos ð Þ� <sup>ψ</sup> senð Þ <sup>ψ</sup>

¼ senð Þ ψ ,

!<sup>2</sup> 2

ffiffiffi 2 p 2

3

5 þ 2:1:

ffiffiffi 2 p

<sup>4</sup> <sup>þ</sup> <sup>2</sup>:k:π: (30)

<sup>2</sup> : cos ð Þ� <sup>ψ</sup>

ffiffiffi 2 p

" #

<sup>2</sup> senð Þ <sup>ψ</sup>

• Eh ¼ 1; • Ev ¼ 1;

• <sup>ψ</sup> <sup>¼</sup> <sup>45</sup>�

• ψ =?

ffiffiffi 2 p <sup>2</sup>

• j j ER <sup>¼</sup> ffiffiffi

¼ 1 þ

• Linear at 135�:

;

When replacing in (9) we have:

þ

ffiffiffi 2 p 2 !<sup>2</sup>

þ cos 2ð Þ ψ

�2 ¼ 2:

sen <sup>ψ</sup> � <sup>π</sup> 4 � �

ffiffiffi 2 p 2 !<sup>2</sup>

• Eh ¼ 1; • Ev ¼ 1;

• <sup>ψ</sup> <sup>¼</sup> <sup>45</sup>�

• j j ER ¼ 0;

• ψ=?

0 ¼ 1 þ


When applying in (9), we have that:

$$
\left(\sqrt{2-\sqrt{2}}\right)^2 = 1 + \left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 + \cos(2\psi) \left[\left(\frac{\sqrt{2}}{2}\right)^2 - \left(\frac{\sqrt{2}}{2}\right)^2\right] + 2.1.
\left[\frac{\sqrt{2}}{2}\cos(\psi) - \frac{\sqrt{2}}{2}\sin(\psi)\right]
$$

$$
2 - \sqrt{2} = 2 + 2.\frac{\sqrt{2}}{2} [\cos(\psi) - \sin(\psi)],
$$

$$
\operatorname{sen}(\psi - \frac{\pi}{4}) = \operatorname{sen}(\frac{\pi}{4}) + 2.k.\pi,
$$

$$
\psi = \frac{\pi}{2} + 2.k.\pi.\tag{32}
$$

Which gives the phase difference of 90�.


When replacing in (9), we have that:

$$
\left(\sqrt{2+\sqrt{2}}\right)^2 = 1 + \left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 + \cos\left(2\psi\right)\left[\left(\frac{\sqrt{2}}{2}\right)^2 - \left(\frac{\sqrt{2}}{2}\right)^2\right] + 2.1.
\left[\frac{\sqrt{2}}{2}\cos\left(\psi\right) - \frac{\sqrt{2}}{2}\sin(\psi)\right]
$$

$$
2 + \sqrt{2} = 2 + \ 2.\frac{\sqrt{2}}{2}[\cos\left(\psi\right) - \sin(\psi)],
$$

$$
\operatorname{sen}(\psi - \frac{\pi}{4}) = \operatorname{sen}\left(\frac{5.\pi}{4}\right) + 2.k.\pi,
$$

$$
\psi = \frac{3.\pi}{2} + 2.k.\pi.\tag{33}
$$

**Chapter 6**

Provisional chapter

**Modal Phenomena of Surface and Bulk Polaritons in**

DOI: 10.5772/intechopen.71837

We discuss peculiarities of bulk and surface polaritons propagating in a composite magnetic-semiconductor superlattice influenced by an external static magnetic field. Three particular configurations of magnetization, namely, the Voigt, polar, and Faraday geometries, are considered. In the long-wavelength limit, involving the effective medium theory, the proposed superlattice is described as an anisotropic uniform medium defined by the tensors of effective permittivity and effective permeability. The study is carried out in the frequency band where the characteristic resonant frequencies of underlying constitutive magnetic and semiconductor materials of the superlattice are different but closely spaced. The effects of mode crossing and anti-crossing in dispersion characteristics of both bulk and surface polaritons are revealed and explained with an assistance of the concept of

Surface polaritons are a special type of electromagnetic waves propagating along the interface of two partnering materials whose material functions (e.g., permittivities) have opposite signs that are typical for a metal-dielectric boundary [1]. These waves are strongly localized at the interface and penetrate into the surrounding space over a distance of wavelength order in a medium, and their amplitudes fall exponentially away from the surface. Observed strong confinement of electromagnetic field in small volumes beyond the diffraction limit leads to enormous increasing matter-field interaction, and it makes attractive using surface waves in the wide fields from the microwave and photonic devices to solar cells [2, 3]. Furthermore, surface electromagnetic waves are highly promising from physical point of view because from

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

Keywords: electromagnetic theory, polaritons, magneto-optical materials,

Modal Phenomena of Surface and Bulk Polaritons in

**Magnetic-Semiconductor Superlattices**

Magnetic-Semiconductor Superlattices

Vladimir R. Tuz, Illia V. Fedorin and

Vladimir R. Tuz, Illia V. Fedorin and

http://dx.doi.org/10.5772/intechopen.71837

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Morse critical points from the catastrophe theory.

superlattices, metamaterials

Volodymyr I. Fesenko

Volodymyr I. Fesenko

Abstract

1. Introduction

Which certifies the phase difference of 270�.

#### Author details

Jobson de Araújo Nascimento<sup>1</sup> \*, Regina Maria De Lima Neta2 , José Moraes Gurgel Neto<sup>3</sup> , Adi Neves Rocha<sup>3</sup> and Alexsandro Aleixo Pereira Da Silva4

\*Address all correspondence to: job.nascimento@gmail.com

1 Universidade Estadual de Ciências da Saúde de Alagoas (UNCISAL), Trapiche da Barra, Maceió-Alagoas, Brazil


#### References


#### **Modal Phenomena of Surface and Bulk Polaritons in Magnetic-Semiconductor Superlattices** Modal Phenomena of Surface and Bulk Polaritons in Magnetic-Semiconductor Superlattices

DOI: 10.5772/intechopen.71837

Vladimir R. Tuz, Illia V. Fedorin and Volodymyr I. Fesenko Vladimir R. Tuz, Illia V. Fedorin and Volodymyr I. Fesenko

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.71837

#### Abstract

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup> <sup>þ</sup> ffiffiffi 2 p � � q <sup>2</sup>

Author details

Jobson de Araújo Nascimento<sup>1</sup>

Maceió-Alagoas, Brazil

References

¼ 1 þ

98 Surface Waves - New Trends and Developments

ffiffiffi 2 p 2 !<sup>2</sup>

Which certifies the phase difference of 270�.

2002, Kharkiv, Ukraine. pp. 188-196

ponent modified; 2013. pp. 1-6

þ

ffiffiffi 2 p 2 !<sup>2</sup>

<sup>2</sup> <sup>þ</sup> ffiffiffi 2 <sup>p</sup> <sup>¼</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>:

Adi Neves Rocha<sup>3</sup> and Alexsandro Aleixo Pereira Da Silva4

\*Address all correspondence to: job.nascimento@gmail.com

2 IFPE-Rua Edson Barbosa de Araújo, Afogados da Ingazeira, PE, Brazil

4 Universidade Federal de Pernambuco (UFPE), Recife, PE, Brazil

[2] Kraus JD. Antennas, 1 ed. McGraw–Hill book Company; 1950

3 Centro de Estudos Superiores de Maceió Cesmac, Maceió-Alagoas, Brazil

sen <sup>ψ</sup> � <sup>π</sup>

þ cos 2ð Þ ψ

4 � �

<sup>ψ</sup> <sup>¼</sup> <sup>3</sup>:<sup>π</sup>

ffiffiffi 2 p 2 !<sup>2</sup>

> 5:π 4 � �

4

ffiffiffi 2 p

¼ sen

\*, Regina Maria De Lima Neta2

1 Universidade Estadual de Ciências da Saúde de Alagoas (UNCISAL), Trapiche da Barra,

[1] Peter K, Yuri M. Measurement of Polarization and Applications. LFNM 2002, 3–5 June,

[3] Nascimento JA, Assis FM, Serres A. Determination of the electromagnetic wave polarization from unknown sources by the method of linear component modified. In: Internacional Microwave and Optoelectronics Conference, 2013, Rio de Janeiro. Determination of the electromagnetic wave polarization from unknown sources by the method of linear com-

�

<sup>2</sup> ½ � cos ð Þ� <sup>ψ</sup> senð Þ <sup>ψ</sup> ,

!<sup>2</sup> 2

ffiffiffi 2 p 2

þ 2:k:π,

3 5 þ 2:1:

ffiffiffi 2 p

<sup>2</sup> <sup>þ</sup> <sup>2</sup>:k:π: (33)

, José Moraes Gurgel Neto<sup>3</sup>

<sup>2</sup> : cos ð Þ� <sup>ψ</sup>

ffiffiffi 2 p

" #

<sup>2</sup> senð Þ <sup>ψ</sup>

,

We discuss peculiarities of bulk and surface polaritons propagating in a composite magnetic-semiconductor superlattice influenced by an external static magnetic field. Three particular configurations of magnetization, namely, the Voigt, polar, and Faraday geometries, are considered. In the long-wavelength limit, involving the effective medium theory, the proposed superlattice is described as an anisotropic uniform medium defined by the tensors of effective permittivity and effective permeability. The study is carried out in the frequency band where the characteristic resonant frequencies of underlying constitutive magnetic and semiconductor materials of the superlattice are different but closely spaced. The effects of mode crossing and anti-crossing in dispersion characteristics of both bulk and surface polaritons are revealed and explained with an assistance of the concept of Morse critical points from the catastrophe theory.

Keywords: electromagnetic theory, polaritons, magneto-optical materials, superlattices, metamaterials

#### 1. Introduction

Surface polaritons are a special type of electromagnetic waves propagating along the interface of two partnering materials whose material functions (e.g., permittivities) have opposite signs that are typical for a metal-dielectric boundary [1]. These waves are strongly localized at the interface and penetrate into the surrounding space over a distance of wavelength order in a medium, and their amplitudes fall exponentially away from the surface. Observed strong confinement of electromagnetic field in small volumes beyond the diffraction limit leads to enormous increasing matter-field interaction, and it makes attractive using surface waves in the wide fields from the microwave and photonic devices to solar cells [2, 3]. Furthermore, surface electromagnetic waves are highly promising from physical point of view because from

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

the character of their propagation, one can derive information about both interface quality and electromagnetic properties of partnering materials (such as permittivity and permeability). High sensitivity to the electromagnetic properties of media enables utilization of surface waves in the sensing applications, particularly in both chemical and biological systems [4]. Thus, studying characteristics of surface waves is essential in the physics of surfaces and optics; in the latter case, the research has led to the emergence of a new science—plasmonics.

microwave range, whereas characteristic frequencies of permittivity of semiconductors com-

Modal Phenomena of Surface and Bulk Polaritons in Magnetic-Semiconductor Superlattices

http://dx.doi.org/10.5772/intechopen.71837

101

At the same time, it is evident that combining together magnetic and semiconductor materials into a single gyroelectromagnetic superlattice in which both permeability and permittivity simultaneously are tensor quantities allows additional possibilities in the control of polaritons using the magnetic field that are unattainable in convenient either gyromagnetic or gyroelectric media. Fortunately, it is possible to design heterostructures in which both characteristic resonant frequencies of semiconductor and magnetic materials are different but, nevertheless, closely spaced in the same frequency band. As a relevant example, the magnetic-semiconductor heterostructures proposed in [17–20] can be mentioned that are able to exhibit a gyroelectromagnetic effect from gigahertz up to tens of terahertz [21]. Thus investigation of electromagnetic properties of such structures in view of their promising

This chapter is devoted to the discussion of dispersion peculiarities of both bulk and surface polaritons propagating in a finely-stratified magnetic-semiconductor superlattice influenced by an external static magnetic field. It is organized as follows. In Section 2, we formulate the problem to be solved and derive effective medium expressions suitable for calculation of the properties of modes under the long-wavelength approximation. Section 3 describes the problem solution in a general case assuming an arbitrary orientation of the external magnetic field with respect to the direction of wave propagation and interface of the structure. The discussion about manifestation of mode crossing and anti-crossing effects is presented in Section 4 involving a concept of the Morse critical points from the catastrophe theory. In Section 5, we reveal dispersion peculiarities of bulk and surface polaritons in the given superlattice for three particular cases of the vector orientation of the external magnetic field with respect to the superlattice's interface and wave vector, namely we study the configurations where the external magnetic field is influenced in the Voigt, polar, and Faraday geometries. Finally, Section 6

Thereby, in this chapter, we study dispersion features of surface and bulk polaritons propagating in a semi-infinite stack of identical composite double-layered slabs arranged along the y-axis that forms a superlattice (Figure 1). Each composite slab within the superlattice includes magnetic (with constitutive parameters <sup>ε</sup>m, <sup>μ</sup>bm) and semiconductor (with constitutive parameters <sup>b</sup>εs, <sup>μ</sup>s) layers with thicknesses dm and ds, respectively. The stack possesses a periodic structure (with period <sup>L</sup> <sup>¼</sup> dm <sup>þ</sup> ds) that fills half-space <sup>y</sup> <sup>&</sup>lt; 0 and adjoins a vacuum <sup>ε</sup><sup>0</sup> <sup>¼</sup> <sup>μ</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> � ) occupying half-space y > 0. Therefore, the superlattice's interface lies in the x � z plane, and along this plane, the system is considered to be infinite. The structure under investigation is

!

angle θ with the y-axis. It is supposed that the strength of this field is high enough to form a homogeneous saturated state of magnetic as well as semiconductor subsystems. Finally, the

that lies in the y � z plane and makes an

application as a part of plasmonic devices is a significant task.

monly are in the infrared range.

summarizes the chapter.

2. Outline of problem

influenced by an external static magnetic field M

Today plasmonics is a rapidly developing field characterized by enormous variety of possible practical applications. In many of them, an ability to control and guide surface waves is a crucial characteristic. Thereby in last decades, many efforts have been made to realize active tunable components for plasmonic integrated circuits such as switchers, active couplers, modulators, etc. In this regard, searching effective ways to control characteristics of plasmon-polariton propagation by utilizing external driving agents is a very important task. In particular, the nonlinear, thermo-optical, and electro-optical effects are proposed to be used in the tunable plasmonic devices for the control of plasmon-polariton propagation [5–8]. In such devices, the tuning mechanism is conditioned by changing the permittivity of the dielectric medium due to applying external electric field or temperature control. At the same time, utilization of an external magnetic field as a driving agent to gain a control over polariton dispersion features is very promising, since it allows changing both permeability of magnetic materials (e.g., ferrites) and permittivity of conducting materials (e.g., metals or semiconductors). It is worth mentioning that the uniqueness of this controlling mechanism lies in the fact that the polariton properties depend not only on the magnitude of the magnetic field but also on its direction. An applied magnetic field also produces additional branches in spectra of magnetic plasmon-polariton resulting in the multiband propagation accompanied by nonreciprocal effects [9–16]. Thus, a combination of plasmonic and magnetic functionalities opens a prospect toward active devices with an additional degrees of freedom in the control of plasmon-polariton properties, and such systems have already found a number of practical applications in integrated photonic devices for telecommunications (see, for instance [3, 8] and references therein).

In this framework, using superlattices (which typically consist of alternating layers of two partnering materials) that are capable to provide a combined plasmon and magnetic functionality instead of traditional plasmonic systems (in which the presence of a metal-dielectric interface is implied) has great prospects. Particularly, it conditioned by the fact that the superlattices demonstrate many exotic electronic and optical properties uncommon to the homogeneous (bulk) samples due to the presence of additional periodic potential, which period is greater than the original lattice constant [14]. The application of magnetic field to a superlattice leads to the so-called magneto-plasmon-polariton excitations. Properties of the magnetic polaritons in the superlattices of different kinds being under the action of an external static magnetic field have been studied by many authors during several last decades [10–16]. The problem is usually solved within two distinct considerations of gyroelectric media (e.g., semiconductors) with magneto-plasmons [10, 14] and gyromagnetic media (e.g., ferrites) with magnons [11–13, 16], which involve the medium characterization with either permittivity or permeability tensor having asymmetric off-diagonal parts. This distinction is governed by the fact that the resonant frequencies of permeability of magnetic materials usually lie in the microwave range, whereas characteristic frequencies of permittivity of semiconductors commonly are in the infrared range.

At the same time, it is evident that combining together magnetic and semiconductor materials into a single gyroelectromagnetic superlattice in which both permeability and permittivity simultaneously are tensor quantities allows additional possibilities in the control of polaritons using the magnetic field that are unattainable in convenient either gyromagnetic or gyroelectric media. Fortunately, it is possible to design heterostructures in which both characteristic resonant frequencies of semiconductor and magnetic materials are different but, nevertheless, closely spaced in the same frequency band. As a relevant example, the magnetic-semiconductor heterostructures proposed in [17–20] can be mentioned that are able to exhibit a gyroelectromagnetic effect from gigahertz up to tens of terahertz [21]. Thus investigation of electromagnetic properties of such structures in view of their promising application as a part of plasmonic devices is a significant task.

This chapter is devoted to the discussion of dispersion peculiarities of both bulk and surface polaritons propagating in a finely-stratified magnetic-semiconductor superlattice influenced by an external static magnetic field. It is organized as follows. In Section 2, we formulate the problem to be solved and derive effective medium expressions suitable for calculation of the properties of modes under the long-wavelength approximation. Section 3 describes the problem solution in a general case assuming an arbitrary orientation of the external magnetic field with respect to the direction of wave propagation and interface of the structure. The discussion about manifestation of mode crossing and anti-crossing effects is presented in Section 4 involving a concept of the Morse critical points from the catastrophe theory. In Section 5, we reveal dispersion peculiarities of bulk and surface polaritons in the given superlattice for three particular cases of the vector orientation of the external magnetic field with respect to the superlattice's interface and wave vector, namely we study the configurations where the external magnetic field is influenced in the Voigt, polar, and Faraday geometries. Finally, Section 6 summarizes the chapter.

#### 2. Outline of problem

the character of their propagation, one can derive information about both interface quality and electromagnetic properties of partnering materials (such as permittivity and permeability). High sensitivity to the electromagnetic properties of media enables utilization of surface waves in the sensing applications, particularly in both chemical and biological systems [4]. Thus, studying characteristics of surface waves is essential in the physics of surfaces and optics; in

Today plasmonics is a rapidly developing field characterized by enormous variety of possible practical applications. In many of them, an ability to control and guide surface waves is a crucial characteristic. Thereby in last decades, many efforts have been made to realize active tunable components for plasmonic integrated circuits such as switchers, active couplers, modulators, etc. In this regard, searching effective ways to control characteristics of plasmon-polariton propagation by utilizing external driving agents is a very important task. In particular, the nonlinear, thermo-optical, and electro-optical effects are proposed to be used in the tunable plasmonic devices for the control of plasmon-polariton propagation [5–8]. In such devices, the tuning mechanism is conditioned by changing the permittivity of the dielectric medium due to applying external electric field or temperature control. At the same time, utilization of an external magnetic field as a driving agent to gain a control over polariton dispersion features is very promising, since it allows changing both permeability of magnetic materials (e.g., ferrites) and permittivity of conducting materials (e.g., metals or semiconductors). It is worth mentioning that the uniqueness of this controlling mechanism lies in the fact that the polariton properties depend not only on the magnitude of the magnetic field but also on its direction. An applied magnetic field also produces additional branches in spectra of magnetic plasmon-polariton resulting in the multiband propagation accompanied by nonreciprocal effects [9–16]. Thus, a combination of plasmonic and magnetic functionalities opens a prospect toward active devices with an additional degrees of freedom in the control of plasmon-polariton properties, and such systems have already found a number of practical applications in integrated photonic devices for telecommunications (see, for instance [3, 8]

In this framework, using superlattices (which typically consist of alternating layers of two partnering materials) that are capable to provide a combined plasmon and magnetic functionality instead of traditional plasmonic systems (in which the presence of a metal-dielectric interface is implied) has great prospects. Particularly, it conditioned by the fact that the superlattices demonstrate many exotic electronic and optical properties uncommon to the homogeneous (bulk) samples due to the presence of additional periodic potential, which period is greater than the original lattice constant [14]. The application of magnetic field to a superlattice leads to the so-called magneto-plasmon-polariton excitations. Properties of the magnetic polaritons in the superlattices of different kinds being under the action of an external static magnetic field have been studied by many authors during several last decades [10–16]. The problem is usually solved within two distinct considerations of gyroelectric media (e.g., semiconductors) with magneto-plasmons [10, 14] and gyromagnetic media (e.g., ferrites) with magnons [11–13, 16], which involve the medium characterization with either permittivity or permeability tensor having asymmetric off-diagonal parts. This distinction is governed by the fact that the resonant frequencies of permeability of magnetic materials usually lie in the

the latter case, the research has led to the emergence of a new science—plasmonics.

and references therein).

100 Surface Waves - New Trends and Developments

Thereby, in this chapter, we study dispersion features of surface and bulk polaritons propagating in a semi-infinite stack of identical composite double-layered slabs arranged along the y-axis that forms a superlattice (Figure 1). Each composite slab within the superlattice includes magnetic (with constitutive parameters <sup>ε</sup>m, <sup>μ</sup>bm) and semiconductor (with constitutive parameters <sup>b</sup>εs, <sup>μ</sup>s) layers with thicknesses dm and ds, respectively. The stack possesses a periodic structure (with period <sup>L</sup> <sup>¼</sup> dm <sup>þ</sup> ds) that fills half-space <sup>y</sup> <sup>&</sup>lt; 0 and adjoins a vacuum <sup>ε</sup><sup>0</sup> <sup>¼</sup> <sup>μ</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> � ) occupying half-space y > 0. Therefore, the superlattice's interface lies in the x � z plane, and along this plane, the system is considered to be infinite. The structure under investigation is influenced by an external static magnetic field M ! that lies in the y � z plane and makes an angle θ with the y-axis. It is supposed that the strength of this field is high enough to form a homogeneous saturated state of magnetic as well as semiconductor subsystems. Finally, the

permeability and permittivity μ and ε; the superscript j is introduced to distinguish between

In the given structure geometry, the interfaces between adjacent layers within the superlattice

<sup>x</sup> and Qð Þ<sup>j</sup> z :

Modal Phenomena of Surface and Bulk Polaritons in Magnetic-Semiconductor Superlattices

xz � <sup>g</sup> ð Þj xyg ð Þj yz g ð Þj yy !Pð Þ<sup>j</sup>

zz � <sup>g</sup> ð Þj zy g ð Þj yz g ð Þj yy !Pð Þ<sup>j</sup>

!

<sup>y</sup> , it follows that

D E <sup>¼</sup> <sup>Q</sup>ð Þ<sup>j</sup>

D E <sup>þ</sup> <sup>α</sup>xzh i Pz ,

D E <sup>þ</sup> <sup>β</sup>yzh i Pz ,

D E <sup>þ</sup> <sup>α</sup>zzh i Pz :

yy � �δj, <sup>δ</sup><sup>j</sup> <sup>¼</sup> dj=<sup>L</sup> is filling factor, and νν<sup>0</sup> iterates over <sup>x</sup> and <sup>z</sup>.

<sup>j</sup> g ð Þj νν<sup>0</sup> � g ð Þj <sup>ν</sup><sup>y</sup> g ð Þj <sup>y</sup>ν0=g ð Þj yy � �δj, <sup>β</sup>yy <sup>¼</sup> <sup>P</sup>

ð Þj and Q !

D E !

<sup>x</sup> , Pð Þ<sup>j</sup>

<sup>y</sup> can be expressed

, (3)

ð Þj inside the layers

can be deter-

(6)

<sup>j</sup> 1=g ð Þj yy � �δj,

<sup>z</sup> , (2)

http://dx.doi.org/10.5772/intechopen.71837

z

z :

and P D E!

ð Þj dj: (4)

<sup>y</sup> , (5)

<sup>z</sup> , and

103

magnetic (m ! j) and semiconductor ( s ! j) layers; and νν<sup>0</sup> iterates over x, y, and z.

lie in the <sup>x</sup> � <sup>z</sup> plane, and they are normal to the <sup>y</sup>-axis; thus, the field components <sup>P</sup>ð Þ<sup>j</sup>

<sup>y</sup> are continuous at the interfaces. Therefore, the normal component <sup>P</sup>ð Þ<sup>j</sup>

ð Þj yx g ð Þj yy Pð Þ<sup>j</sup> <sup>x</sup> þ 1 g ð Þj yy Qð Þ<sup>j</sup> <sup>y</sup> � <sup>g</sup> ð Þj yz g ð Þj yy Pð Þ<sup>j</sup>

> <sup>x</sup> <sup>þ</sup> <sup>g</sup> ð Þj xy g ð Þj yy Qð Þ<sup>j</sup> <sup>y</sup> <sup>þ</sup> <sup>g</sup>ð Þ<sup>j</sup>

<sup>x</sup> <sup>þ</sup> <sup>g</sup> ð Þj zy g ð Þj yy Qð Þ<sup>j</sup> <sup>y</sup> <sup>þ</sup> <sup>g</sup>ð Þ<sup>j</sup>

ð Þ<sup>j</sup> dj, QD E !

<sup>x</sup> , Pð Þ<sup>j</sup>

and with using Eqs. (2) and (3), we can obtain the relations between the averaged field compo-

<sup>x</sup> , Ph i<sup>z</sup> <sup>¼</sup> <sup>P</sup>ð Þ<sup>j</sup>

Qx h i¼ αxxh iþ Px γxy Qy

� � <sup>¼</sup> <sup>β</sup>yxh iþ Px <sup>β</sup>yy Qy

Qz h i ¼ αzzh i Px þ γzy Qy

¼ 1 L X j Q !

<sup>z</sup> , and Qð Þ<sup>j</sup>

<sup>z</sup> , Qy

from (1) in terms of the continuous components of the field as follow:

Pð Þ<sup>j</sup> <sup>y</sup> ¼ � <sup>g</sup>

and substituted into equations for components Qð Þ<sup>j</sup>

ð Þj xx � <sup>g</sup> ð Þj xy g ð Þj yx g ð Þj yy !Pð Þ<sup>j</sup>

ð Þj zx � <sup>g</sup> ð Þj zy g ð Þj yx g ð Þj yy !Pð Þ<sup>j</sup>

P D E! ¼ 1 L X j P !

In view of the continuity of components Pð Þ<sup>j</sup>

Relations (2) and (3) are then used for the field averaging [24].

With taking into account the long-wavelength limit, the fields P

h i Px <sup>¼</sup> <sup>P</sup>ð Þ<sup>j</sup>

Py

<sup>j</sup> g ð Þj <sup>ν</sup><sup>y</sup> =g ð Þj

Here, we used the following designations ανν<sup>0</sup> <sup>¼</sup> <sup>P</sup>

are considered to be constant, and the averaged (Maxwell) fields Q

Qð Þ<sup>j</sup> <sup>x</sup> ¼ g

Qð Þ<sup>j</sup> <sup>z</sup> ¼ g

mined by the equalities:

nents in the next form:

<sup>β</sup>yy <sup>¼</sup> <sup>P</sup>

<sup>j</sup> g ð Þj <sup>y</sup>ν0=g ð Þj yy � �δj, γν<sup>y</sup> <sup>¼</sup> <sup>P</sup>

Qð Þ<sup>j</sup>

Figure 1. Schematic representation of a magnetic-semiconductor superlattice influenced by an external static magnetic field M ! and a visual representation of the tangential electric field distribution of the surface polariton propagating over the interface between the given structure and free space.

wave vector k ! of the macroscopic electric field lies in the x � z plane and makes an angle w with the x-axis.

In the general case, when no restrictions are imposed on characteristic dimensions (dm, ds, and L) of the superlattice compared to the wavelength of propagating modes, the transfer matrix formalism [22] is usually involved in order to reveal the dispersion features of polaritons. It implies a numerical solution of a canonical boundary value problem formulated for each layer within the period of superlattice and then performing a subsequent multiplication of the obtained transfer matrices to form a semi-infinite extent. On the other hand, when all characteristic dimensions of the superlattice satisfy the long-wavelength limit, i.e., they are all much smaller than the wavelength in the corresponding layer and period of structure (dm ≪ λ, ds ≪ λ, L ≪ λ), homogenization procedures from the effective medium theory can be involved in order to derive dispersion characteristics in an explicit form [24–28] that is suitable for identifying the main features of interest. Therefore, further only the modes under the long-wavelength approximation are studied in this chapter, i.e., the structure is considered to be a finely-stratified one.

In order to obtain expressions for the tensors of effective permeability and permittivity of the superlattice in a general form, constitutive equations, B ! ¼ μH ! and D ! ¼ εE ! , for magnetic (0 < z < dm) and semiconductor (dm < z < L) layers are represented as follow [26]:

$$Q\_{\nu}^{(j)} = \sum\_{\nu'} g\_{\nu\nu'}^{(j)} P\_{\nu'}^{(j)} \tag{1}$$

where Q ! takes values of the magnetic and electric flux densities B ! and D ! , respectively; P ! is varied between the magnetic and electric field strengths H ! and E ! ; g is substituted for permeability and permittivity μ and ε; the superscript j is introduced to distinguish between magnetic (m ! j) and semiconductor ( s ! j) layers; and νν<sup>0</sup> iterates over x, y, and z.

In the given structure geometry, the interfaces between adjacent layers within the superlattice lie in the <sup>x</sup> � <sup>z</sup> plane, and they are normal to the <sup>y</sup>-axis; thus, the field components <sup>P</sup>ð Þ<sup>j</sup> <sup>x</sup> , Pð Þ<sup>j</sup> <sup>z</sup> , and Qð Þ<sup>j</sup> <sup>y</sup> are continuous at the interfaces. Therefore, the normal component <sup>P</sup>ð Þ<sup>j</sup> <sup>y</sup> can be expressed from (1) in terms of the continuous components of the field as follow:

$$P\_y^{(j)} = -\frac{\mathcal{g}\_{yx}^{(j)}}{\mathcal{g}\_{yy}^{(j)}} P\_x^{(j)} + \frac{1}{\mathcal{g}\_{yy}^{(j)}} \mathcal{Q}\_y^{(j)} - \frac{\mathcal{g}\_{yz}^{(j)}}{\mathcal{g}\_{yy}^{(j)}} P\_z^{(j)},\tag{2}$$

and substituted into equations for components Qð Þ<sup>j</sup> <sup>x</sup> and Qð Þ<sup>j</sup> z :

$$\begin{split} \mathbf{Q}\_{x}^{(j)} &= \left( \mathbf{g}\_{xx}^{(j)} - \frac{\mathbf{g}\_{xy}^{(j)} \mathbf{g}\_{yx}^{(j)}}{\mathbf{g}\_{yy}^{(j)}} \right) \mathbf{P}\_{x}^{(j)} + \frac{\mathbf{g}\_{xy}^{(j)}}{\mathbf{g}\_{yy}^{(j)}} \mathbf{Q}\_{y}^{(j)} + \left( \mathbf{g}\_{xz}^{(j)} - \frac{\mathbf{g}\_{xy}^{(j)} \mathbf{g}\_{yz}^{(j)}}{\mathbf{g}\_{yy}^{(j)}} \right) \mathbf{P}\_{z}^{(j)} \\ \mathbf{Q}\_{z}^{(j)} &= \left( \mathbf{g}\_{xx}^{(j)} - \frac{\mathbf{g}\_{xy}^{(j)} \mathbf{g}\_{yx}^{(j)}}{\mathbf{g}\_{yy}^{(j)}} \right) \mathbf{P}\_{x}^{(j)} + \frac{\mathbf{g}\_{xy}^{(j)}}{\mathbf{g}\_{yy}^{(j)}} \mathbf{Q}\_{y}^{(j)} + \left( \mathbf{g}\_{zz}^{(j)} - \frac{\mathbf{g}\_{xy}^{(j)} \mathbf{g}\_{yz}^{(j)}}{\mathbf{g}\_{yy}^{(j)}} \right) \mathbf{P}\_{z}^{(j)}. \end{split} \tag{3}$$

Relations (2) and (3) are then used for the field averaging [24].

wave vector k

field M !

with the x-axis.

where Q ! !

102 Surface Waves - New Trends and Developments

the interface between the given structure and free space.

of the macroscopic electric field lies in the x � z plane and makes an angle w

In the general case, when no restrictions are imposed on characteristic dimensions (dm, ds, and L) of the superlattice compared to the wavelength of propagating modes, the transfer matrix formalism [22] is usually involved in order to reveal the dispersion features of polaritons. It implies a numerical solution of a canonical boundary value problem formulated for each layer within the period of superlattice and then performing a subsequent multiplication of the obtained transfer matrices to form a semi-infinite extent. On the other hand, when all characteristic dimensions of the superlattice satisfy the long-wavelength limit, i.e., they are all much smaller than the wavelength in the corresponding layer and period of structure (dm ≪ λ, ds ≪ λ, L ≪ λ), homogenization procedures from the effective medium theory can be involved in order to derive dispersion characteristics in an explicit form [24–28] that is suitable for identifying the main features of interest. Therefore, further only the modes under the long-wavelength approximation are studied in this chapter, i.e., the structure is considered to be a finely-stratified one.

Figure 1. Schematic representation of a magnetic-semiconductor superlattice influenced by an external static magnetic

and a visual representation of the tangential electric field distribution of the surface polariton propagating over

In order to obtain expressions for the tensors of effective permeability and permittivity of the

(0 < z < dm) and semiconductor (dm < z < L) layers are represented as follow [26]:

Qð Þ<sup>j</sup> <sup>ν</sup> <sup>¼</sup> <sup>X</sup> ν0 g ð Þj νν0Pð Þ<sup>j</sup>

takes values of the magnetic and electric flux densities B

! ¼ μH ! and D ! ¼ εE !

!

and E !

!

<sup>ν</sup><sup>0</sup> , (1)

and D ! , for magnetic

, respectively; P

; g is substituted for

!

superlattice in a general form, constitutive equations, B

is varied between the magnetic and electric field strengths H

With taking into account the long-wavelength limit, the fields P ! ð Þj and Q ! ð Þj inside the layers are considered to be constant, and the averaged (Maxwell) fields Q D E ! and P D E! can be determined by the equalities:

$$
\left\langle \vec{P} \right\rangle = \frac{1}{L} \sum\_{j} \vec{P} \text{ (j)} d\_{\rangle} \quad \left\langle \vec{Q} \right\rangle = \frac{1}{L} \sum\_{j} \vec{Q} \text{ (j)} d\_{\rangle}.\tag{4}
$$

In view of the continuity of components Pð Þ<sup>j</sup> <sup>x</sup> , Pð Þ<sup>j</sup> <sup>z</sup> , and Qð Þ<sup>j</sup> <sup>y</sup> , it follows that

$$
\langle P\_x \rangle = P\_x^{(j)}, \quad \langle P\_z \rangle = P\_z^{(j)}, \quad \left\langle Q\_y \right\rangle = Q\_y^{(j)}.\tag{5}
$$

and with using Eqs. (2) and (3), we can obtain the relations between the averaged field components in the next form:

$$
\begin{aligned}
\langle Q\_x \rangle &= \alpha\_{xx} \langle P\_x \rangle + \mathcal{V}\_{xy} \left\langle Q\_y \right\rangle + \alpha\_{xz} \langle P\_z \rangle, \\
\langle P\_y \rangle &= \beta\_{yx} \langle P\_x \rangle + \beta\_{yy} \left\langle Q\_y \right\rangle + \beta\_{yz} \langle P\_z \rangle, \\
\langle Q\_z \rangle &= \alpha\_{zz} \langle P\_x \rangle + \mathcal{V}\_{zy} \left\langle Q\_y \right\rangle + \alpha\_{zz} \langle P\_z \rangle.
\end{aligned} \tag{6}
$$

Here, we used the following designations ανν<sup>0</sup> <sup>¼</sup> <sup>P</sup> <sup>j</sup> g ð Þj νν<sup>0</sup> � g ð Þj <sup>ν</sup><sup>y</sup> g ð Þj <sup>y</sup>ν0=g ð Þj yy � �δj, <sup>β</sup>yy <sup>¼</sup> <sup>P</sup> <sup>j</sup> 1=g ð Þj yy � �δj, <sup>β</sup>yy <sup>¼</sup> <sup>P</sup> <sup>j</sup> g ð Þj <sup>y</sup>ν0=g ð Þj yy � �δj, γν<sup>y</sup> <sup>¼</sup> <sup>P</sup> <sup>j</sup> g ð Þj <sup>ν</sup><sup>y</sup> =g ð Þj yy � �δj, <sup>δ</sup><sup>j</sup> <sup>¼</sup> dj=<sup>L</sup> is filling factor, and νν<sup>0</sup> iterates over <sup>x</sup> and <sup>z</sup>. Expressing Qy D E from the second equation in system (6) and substituting it into the rest two equations, the constitutive equations for the flux densities of the effective medium Q D E ! <sup>¼</sup> <sup>b</sup>geff <sup>P</sup> D E! can be derived, where <sup>b</sup>geff is a tensor quantity:

$$
\begin{pmatrix} \hat{\mathcal{a}}\_{xx} & \hat{\mathcal{V}}\_{xy} & \hat{\mathcal{a}}\_{xz} \\ \end{pmatrix} = \begin{pmatrix} \begin{matrix} \hat{\mathcal{a}}\_{xx} & \hat{\mathcal{V}}\_{xy} & \hat{\mathcal{a}}\_{xz} \\ \end{matrix} \\ \end{pmatrix} = \begin{pmatrix} \begin{matrix} \tilde{\mathcal{g}}\_{xx} & \tilde{\mathcal{g}}\_{xy} & \tilde{\mathcal{g}}\_{xz} \\ \tilde{\mathcal{g}}\_{yx} & \tilde{\mathcal{g}}\_{yy} & \tilde{\mathcal{g}}\_{yz} \\ \tilde{\mathcal{g}}\_{zx} & \tilde{\mathcal{g}}\_{zy} & \tilde{\mathcal{g}}\_{zz} \end{matrix} \\ \end{pmatrix}, \tag{7}
$$

When M !

β~

g ð Þ m yy g ð Þs

<sup>γ</sup>~zy <sup>¼</sup> <sup>β</sup><sup>~</sup>

g ð Þ m xy g ð Þs yy δ<sup>m</sup>

�

yz <sup>¼</sup> 0, <sup>α</sup>~xx <sup>¼</sup> <sup>g</sup>ð Þ <sup>m</sup>

yy τ, where τ ¼ g

In the second case, when M

yz <sup>¼</sup> 0, <sup>α</sup>~xx <sup>¼</sup> <sup>g</sup>ð Þ <sup>m</sup>

þg ð Þs xy g ð Þ m

xx <sup>δ</sup><sup>m</sup> <sup>þ</sup> <sup>g</sup>ð Þ<sup>s</sup>

ð Þ m yy δ<sup>s</sup> þ g

!

xx <sup>δ</sup><sup>m</sup> <sup>þ</sup> <sup>g</sup>ð Þ<sup>s</sup>

yy <sup>δ</sup>sÞτ, and <sup>β</sup><sup>~</sup>

spaced within the same frequency band.

components are met: g~xx ¼ g~zz 6¼ g~yy and g~xz ¼ �g~zx ¼6 0.

anisotropic uniform medium when an external static magnetic field M

∥y, then ζ ¼ 0, ξ ¼ 1, and tensor (8) is reduced to the form

0 B@

and for the components of tensor (7), we have following expressions: <sup>γ</sup>~xy <sup>¼</sup> <sup>γ</sup>~zy <sup>¼</sup> <sup>β</sup><sup>~</sup>

zz <sup>δ</sup><sup>m</sup> <sup>þ</sup> <sup>g</sup>ð Þ<sup>s</sup>

:

0 B@

ð Þ m xy � g ð Þs xy � �<sup>2</sup>

<sup>b</sup>gð Þ<sup>j</sup> <sup>¼</sup>

xx δ<sup>s</sup> þ g

yy ¼ g ð Þ m yy g ð Þs yy τ:

g<sup>1</sup> 0 ig<sup>2</sup> 0 g<sup>3</sup> 0 �ig<sup>2</sup> 0 g<sup>1</sup>

∥z, then ζ ¼ 1, ξ ¼ 0, and tensor (8) has the form

g<sup>1</sup> ig<sup>2</sup> 0 �ig<sup>2</sup> g<sup>1</sup> 0 0 0 g<sup>3</sup>

In this configuration, the components of tensor (7) can be written as follows: α~xz ¼ α~zx ¼

For further reference, the dispersion curves of the tensor components of relative effective permeability <sup>μ</sup>beff and relative effective permittivity <sup>b</sup>εeff of the homogenized medium (with filling factors δ<sup>m</sup> ¼ δ<sup>s</sup> ¼ 0:5) are presented in Figure 2. Figure 2(a) and (b) represents constitutive parameters for the polar configuration, whereas Figure 2(c) and (d) represents those for the Voigt and Faraday configurations of magnetization. For these calculations, we used typical constitutive parameters for magnetic and semiconductor materials. In particular, here we follow the results of paper [30], where a magnetic-semiconductor composite is made in the form of a barium-cobalt/doped-silicon superlattice for operating in the microwave part of spectrum. A distinct peculiarity of such a superlattice is that the characteristic resonant frequencies of the underlying constitutive magnetic and semiconductor materials are closely

From Figure 2, one can conclude that in both the Voigt and Faraday geometries, the next relations between the components of effective tensor (7) hold g~xx ¼6 g~yy ¼6 g~zz and g~xy ¼ �g~yx ¼6 0, so it means that the obtained homogenized medium is a biaxial bigyrotropic crystal [1]. In the polar geometry, it is a uniaxial bigyrotropic crystal, and the following relations between tensor

To sum up, with an assistance of the homogenization procedures from the effective medium theory, the superlattice under study is approximately represented as a uniaxial or biaxial

1

Modal Phenomena of Surface and Bulk Polaritons in Magnetic-Semiconductor Superlattices

zz <sup>δ</sup>s, <sup>α</sup>~xz ¼ �α~zx <sup>¼</sup> <sup>g</sup>ð Þ <sup>m</sup>

1

<sup>δ</sup>mδsτ, <sup>α</sup>~zz <sup>¼</sup> <sup>g</sup>ð Þ <sup>m</sup>

CA, (9)

http://dx.doi.org/10.5772/intechopen.71837

zx <sup>δ</sup><sup>m</sup> <sup>þ</sup> <sup>g</sup>ð Þ<sup>s</sup>

CA: (10)

zz <sup>δ</sup><sup>m</sup> <sup>þ</sup> <sup>g</sup>ð Þ<sup>s</sup>

!

is directed along or

yx ¼

105

yy ¼

yx ¼

zx <sup>δ</sup>s, and <sup>β</sup><sup>~</sup>

zz <sup>δ</sup>s, <sup>γ</sup>~xy ¼ �β<sup>~</sup>

<sup>b</sup>gð Þ<sup>j</sup> <sup>¼</sup>

xx <sup>δ</sup>s, <sup>α</sup>~zz <sup>¼</sup> <sup>g</sup>ð Þ <sup>m</sup>

ð Þs yy δ<sup>m</sup> � ��<sup>1</sup>

with components <sup>α</sup>~νν<sup>0</sup> <sup>¼</sup> ανν<sup>0</sup> � <sup>β</sup><sup>y</sup>ν0γνy=βyy, <sup>β</sup><sup>~</sup> yy <sup>¼</sup> <sup>1</sup>=β<sup>~</sup> yy, <sup>β</sup><sup>~</sup> <sup>y</sup>ν<sup>0</sup> ¼ �β<sup>y</sup>ν0=βyy, and γ~ν<sup>y</sup> ¼ �γνy=βyy: The expressions for tensor components of the underlying constitutive parameters of magnetic

(μb<sup>m</sup> ! <sup>b</sup>gð Þ <sup>m</sup> ) and semiconductor (bε<sup>s</sup> ! <sup>b</sup>gð Þ<sup>s</sup> ) layers depend on the orientation of the external magnetic field M ! in the y � z plane which is defined by the angle θ in the form:

$$
\widehat{\mathbf{g}}^{(j)} = \begin{pmatrix} \mathbf{g}\_1 & i\zeta \mathbf{g}\_2 & i\xi \mathbf{g}\_2 \\ -i\zeta \mathbf{g}\_2 & \zeta^2 \mathbf{g}\_1 + \xi^2 \mathbf{g}\_3 & \zeta \xi (\mathbf{g}\_1 - \mathbf{g}\_3) \\ -i\xi \mathbf{g}\_2 & \zeta \xi (\mathbf{g}\_1 - \mathbf{g}\_3) & \xi^2 \mathbf{g}\_1 + \zeta^2 \mathbf{g}\_3 \end{pmatrix} \tag{8}
$$

where ζ ¼ sin θ and ξ ¼ cos θ.

For magnetic layers [29], the components of tensor <sup>b</sup>gð Þ <sup>m</sup> are <sup>g</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>χ</sup><sup>0</sup> <sup>þ</sup> <sup>i</sup><sup>χ</sup> 00 , <sup>g</sup><sup>2</sup> <sup>¼</sup> <sup>Ω</sup><sup>0</sup> <sup>þ</sup> <sup>i</sup>Ω<sup>00</sup> , <sup>g</sup><sup>3</sup> <sup>¼</sup> 1 and <sup>χ</sup><sup>0</sup> <sup>¼</sup> <sup>ω</sup>0ω<sup>m</sup> <sup>ω</sup><sup>2</sup> <sup>0</sup> � <sup>ω</sup><sup>2</sup> <sup>1</sup> � <sup>b</sup><sup>2</sup> � � � � <sup>D</sup>�<sup>1</sup> , χ 00 <sup>¼</sup> ωωmb <sup>ω</sup><sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � � � � <sup>D</sup>�<sup>1</sup> , <sup>Ω</sup><sup>0</sup> <sup>¼</sup> ωω<sup>m</sup> <sup>ω</sup><sup>2</sup> <sup>0</sup>� � <sup>ω</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � ��D�<sup>1</sup> , Ω<sup>00</sup> <sup>¼</sup> <sup>2</sup>ω<sup>2</sup>ω0ωmbD�<sup>1</sup> , and <sup>D</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> <sup>0</sup> � <sup>ω</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � � � � <sup>2</sup> <sup>þ</sup> <sup>4</sup>ω<sup>2</sup> 0ωb<sup>2</sup> , where ω<sup>0</sup> is the Larmor frequency and b is a dimensionless damping constant.

For semiconductor layers [23], the components of tensor <sup>b</sup>gð Þ<sup>s</sup> are <sup>g</sup><sup>1</sup> <sup>¼</sup> <sup>ε</sup><sup>l</sup> <sup>1</sup> � <sup>ω</sup><sup>2</sup> <sup>p</sup>ð Þ ω þ iν ½ω ωðð þ h iνÞ <sup>2</sup> � <sup>ω</sup><sup>2</sup> <sup>c</sup> Þ��<sup>1</sup> i , g<sup>2</sup> <sup>¼</sup> <sup>ε</sup>lω<sup>2</sup> <sup>p</sup>ω<sup>c</sup> ω ωð Þ þ iν <sup>2</sup> � <sup>ω</sup><sup>2</sup> c h i � � �<sup>1</sup> , and <sup>g</sup><sup>3</sup> <sup>¼</sup> <sup>ε</sup><sup>l</sup> <sup>1</sup> � <sup>ω</sup><sup>2</sup> <sup>p</sup>ð Þ ω þ iν h i�<sup>1</sup> � �, where <sup>ε</sup><sup>l</sup>

is the part of permittivity attributed to the lattice, ω<sup>p</sup> is the plasma frequency, ω<sup>c</sup> is the cyclotron frequency, and ν is the electron collision frequency in plasma.

Permittivity ε<sup>m</sup> of the magnetic layers as well as permeability μ<sup>s</sup> of the semiconductor layers are scalar quantities.

Hereinafter, we consider two specific orientations of the external magnetic field vector M ! with respect to the superlattice's interface (see Figure 1), namely (i) the polar configuration in which θ ¼ 0 and the vector M ! is parallel to the surface normal (M ! ∥y) and (ii) θ ¼ π=2 and the vector M ! is parallel to the surface plane (M ! ∥zÞ, which is inherent in both the Voigt and Faraday configurations.

When M ! ∥y, then ζ ¼ 0, ξ ¼ 1, and tensor (8) is reduced to the form

Expressing Qy

magnetic field M

!

where ζ ¼ sin θ and ξ ¼ cos θ.

, Ω<sup>00</sup>

, g<sup>2</sup> <sup>¼</sup> <sup>ε</sup>lω<sup>2</sup>

!

is parallel to the surface plane (M

<sup>g</sup><sup>3</sup> <sup>¼</sup> 1 and <sup>χ</sup><sup>0</sup> <sup>¼</sup> <sup>ω</sup>0ω<sup>m</sup> <sup>ω</sup><sup>2</sup>

<sup>ω</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � ��D�<sup>1</sup>

are scalar quantities.

θ ¼ 0 and the vector M

configurations.

iνÞ <sup>2</sup> � <sup>ω</sup><sup>2</sup> <sup>c</sup> Þ��<sup>1</sup> i

M !

D E

104 Surface Waves - New Trends and Developments

can be derived, where <sup>b</sup>geff is a tensor quantity:

with components <sup>α</sup>~νν<sup>0</sup> <sup>¼</sup> ανν<sup>0</sup> � <sup>β</sup><sup>y</sup>ν0γνy=βyy, <sup>β</sup><sup>~</sup>

<sup>b</sup>geff <sup>¼</sup>

<sup>b</sup>gð Þ<sup>j</sup> <sup>¼</sup>

0

BBB@

<sup>0</sup> � <sup>ω</sup><sup>2</sup> <sup>1</sup> � <sup>b</sup><sup>2</sup> � � � � <sup>D</sup>�<sup>1</sup>

Larmor frequency and b is a dimensionless damping constant.

<sup>p</sup>ω<sup>c</sup> ω ωð Þ þ iν

frequency, and ν is the electron collision frequency in plasma.

<sup>¼</sup> <sup>2</sup>ω<sup>2</sup>ω0ωmbD�<sup>1</sup>

β~ yx <sup>β</sup><sup>~</sup>

0

BBB@

from the second equation in system (6) and substituting it into the rest two

g~xx g~xy g~xz g~yx g~yy g~yz g~zx g~zy g~zz 1

1

<sup>0</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � � � � <sup>D</sup>�<sup>1</sup>

<sup>þ</sup> <sup>4</sup>ω<sup>2</sup> 0ωb<sup>2</sup>

h

<sup>p</sup>ð Þ ω þ iν h i�<sup>1</sup> � �

∥y) and (ii) θ ¼ π=2 and the vector

D E !

CA, (7)

CCCA, (8)

00

, <sup>g</sup><sup>2</sup> <sup>¼</sup> <sup>Ω</sup><sup>0</sup> <sup>þ</sup> <sup>i</sup>Ω<sup>00</sup>

, <sup>Ω</sup><sup>0</sup> <sup>¼</sup> ωω<sup>m</sup> <sup>ω</sup><sup>2</sup>

, where ω<sup>0</sup> is the

<sup>p</sup>ð Þ ω þ iν ½ω ωðð þ

, where ε<sup>l</sup>

! with

,

<sup>0</sup>� �

<sup>y</sup>ν<sup>0</sup> ¼ �β<sup>y</sup>ν0=βyy, and γ~ν<sup>y</sup> ¼ �γνy=βyy:

<sup>¼</sup> <sup>b</sup>geff <sup>P</sup> D E!

equations, the constitutive equations for the flux densities of the effective medium Q

α~xx γ~xy α~xz

α~zx γ~zy α~zz

�iζg<sup>2</sup> <sup>ζ</sup><sup>2</sup>

For magnetic layers [29], the components of tensor <sup>b</sup>gð Þ <sup>m</sup> are <sup>g</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>χ</sup><sup>0</sup> <sup>þ</sup> <sup>i</sup><sup>χ</sup>

For semiconductor layers [23], the components of tensor <sup>b</sup>gð Þ<sup>s</sup> are <sup>g</sup><sup>1</sup> <sup>¼</sup> <sup>ε</sup><sup>l</sup> <sup>1</sup> � <sup>ω</sup><sup>2</sup>

h i � � �<sup>1</sup>

<sup>2</sup> � <sup>ω</sup><sup>2</sup> c

is the part of permittivity attributed to the lattice, ω<sup>p</sup> is the plasma frequency, ω<sup>c</sup> is the cyclotron

Permittivity ε<sup>m</sup> of the magnetic layers as well as permeability μ<sup>s</sup> of the semiconductor layers

respect to the superlattice's interface (see Figure 1), namely (i) the polar configuration in which

Hereinafter, we consider two specific orientations of the external magnetic field vector M

is parallel to the surface normal (M

!

�iξg<sup>2</sup> ζξ g<sup>1</sup> � g<sup>3</sup>

, χ 00

, and <sup>D</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>

yy <sup>β</sup><sup>~</sup> yz 1

CCCA ¼

yy <sup>¼</sup> <sup>1</sup>=β<sup>~</sup>

The expressions for tensor components of the underlying constitutive parameters of magnetic (μb<sup>m</sup> ! <sup>b</sup>gð Þ <sup>m</sup> ) and semiconductor (bε<sup>s</sup> ! <sup>b</sup>gð Þ<sup>s</sup> ) layers depend on the orientation of the external

in the y � z plane which is defined by the angle θ in the form:

g<sup>1</sup> iζg<sup>2</sup> iξg<sup>2</sup>

� � ξ<sup>2</sup>

<sup>¼</sup> ωωmb <sup>ω</sup><sup>2</sup>

<sup>0</sup> � <sup>ω</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � � � � <sup>2</sup>

, and <sup>g</sup><sup>3</sup> <sup>¼</sup> <sup>ε</sup><sup>l</sup> <sup>1</sup> � <sup>ω</sup><sup>2</sup>

!

∥zÞ, which is inherent in both the Voigt and Faraday

<sup>g</sup><sup>1</sup> <sup>þ</sup> <sup>ξ</sup><sup>2</sup>

0 B@

yy, <sup>β</sup><sup>~</sup>

g<sup>3</sup> ζξ g<sup>1</sup> � g<sup>3</sup> � �

> <sup>g</sup><sup>1</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup> g3

$$
\hat{\mathbf{g}}^{(j)} = \begin{pmatrix}
\mathbf{g}\_1 & \mathbf{0} & \mathbf{i}\mathbf{g}\_2 \\
\mathbf{0} & \mathbf{g}\_3 & \mathbf{0} \\
\end{pmatrix},
\tag{9}
$$

and for the components of tensor (7), we have following expressions: <sup>γ</sup>~xy <sup>¼</sup> <sup>γ</sup>~zy <sup>¼</sup> <sup>β</sup><sup>~</sup> yx ¼ β~ yz <sup>¼</sup> 0, <sup>α</sup>~xx <sup>¼</sup> <sup>g</sup>ð Þ <sup>m</sup> xx <sup>δ</sup><sup>m</sup> <sup>þ</sup> <sup>g</sup>ð Þ<sup>s</sup> xx <sup>δ</sup>s, <sup>α</sup>~zz <sup>¼</sup> <sup>g</sup>ð Þ <sup>m</sup> zz <sup>δ</sup><sup>m</sup> <sup>þ</sup> <sup>g</sup>ð Þ<sup>s</sup> zz <sup>δ</sup>s, <sup>α</sup>~xz ¼ �α~zx <sup>¼</sup> <sup>g</sup>ð Þ <sup>m</sup> zx <sup>δ</sup><sup>m</sup> <sup>þ</sup> <sup>g</sup>ð Þ<sup>s</sup> zx <sup>δ</sup>s, and <sup>β</sup><sup>~</sup> yy ¼ g ð Þ m yy g ð Þs yy τ, where τ ¼ g ð Þ m yy δ<sup>s</sup> þ g ð Þs yy δ<sup>m</sup> � ��<sup>1</sup> :

In the second case, when M ! ∥z, then ζ ¼ 1, ξ ¼ 0, and tensor (8) has the form

$$
\hat{\mathbf{g}}^{(j)} = \begin{pmatrix} \mathbf{g}\_1 & \mathbf{i}\mathbf{g}\_2 & \mathbf{0} \\ -\mathbf{i}\mathbf{g}\_2 & \mathbf{g}\_1 & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{g}\_3 \end{pmatrix}. \tag{10}
$$

In this configuration, the components of tensor (7) can be written as follows: α~xz ¼ α~zx ¼ <sup>γ</sup>~zy <sup>¼</sup> <sup>β</sup><sup>~</sup> yz <sup>¼</sup> 0, <sup>α</sup>~xx <sup>¼</sup> <sup>g</sup>ð Þ <sup>m</sup> xx <sup>δ</sup><sup>m</sup> <sup>þ</sup> <sup>g</sup>ð Þ<sup>s</sup> xx δ<sup>s</sup> þ g ð Þ m xy � g ð Þs xy � �<sup>2</sup> <sup>δ</sup>mδsτ, <sup>α</sup>~zz <sup>¼</sup> <sup>g</sup>ð Þ <sup>m</sup> zz <sup>δ</sup><sup>m</sup> <sup>þ</sup> <sup>g</sup>ð Þ<sup>s</sup> zz <sup>δ</sup>s, <sup>γ</sup>~xy ¼ �β<sup>~</sup> yx ¼ g ð Þ m xy g ð Þs yy δ<sup>m</sup> � þg ð Þs xy g ð Þ m yy <sup>δ</sup>sÞτ, and <sup>β</sup><sup>~</sup> yy ¼ g ð Þ m yy g ð Þs yy τ:

For further reference, the dispersion curves of the tensor components of relative effective permeability <sup>μ</sup>beff and relative effective permittivity <sup>b</sup>εeff of the homogenized medium (with filling factors δ<sup>m</sup> ¼ δ<sup>s</sup> ¼ 0:5) are presented in Figure 2. Figure 2(a) and (b) represents constitutive parameters for the polar configuration, whereas Figure 2(c) and (d) represents those for the Voigt and Faraday configurations of magnetization. For these calculations, we used typical constitutive parameters for magnetic and semiconductor materials. In particular, here we follow the results of paper [30], where a magnetic-semiconductor composite is made in the form of a barium-cobalt/doped-silicon superlattice for operating in the microwave part of spectrum. A distinct peculiarity of such a superlattice is that the characteristic resonant frequencies of the underlying constitutive magnetic and semiconductor materials are closely spaced within the same frequency band.

From Figure 2, one can conclude that in both the Voigt and Faraday geometries, the next relations between the components of effective tensor (7) hold g~xx ¼6 g~yy ¼6 g~zz and g~xy ¼ �g~yx ¼6 0, so it means that the obtained homogenized medium is a biaxial bigyrotropic crystal [1]. In the polar geometry, it is a uniaxial bigyrotropic crystal, and the following relations between tensor components are met: g~xx ¼ g~zz 6¼ g~yy and g~xz ¼ �g~zx ¼6 0.

To sum up, with an assistance of the homogenization procedures from the effective medium theory, the superlattice under study is approximately represented as a uniaxial or biaxial anisotropic uniform medium when an external static magnetic field M ! is directed along or

In a general form [26], the electric and magnetic field vectors E

where a time factor exp ð Þ �iω<sup>t</sup> is also supposed and omitted, and sign " � " is related to the fields in the upper medium (<sup>y</sup> <sup>&</sup>gt; <sup>0</sup>, j <sup>¼</sup> 0), while sign " <sup>þ</sup> " is related to the fields in the composite medium (y < 0, j ¼ 1), respectively, which provide required wave attenuation

> ! ¼ ik<sup>0</sup> B !

P !ð Þ<sup>j</sup> ¼ p !ð Þ<sup>j</sup>

From a pair of the curl Maxwell's equations ∇� E

<sup>b</sup>εð Þ<sup>j</sup> made in the appropriate order.

sponding constitutive parameters (bςð Þ<sup>0</sup>

tions for the rest two components of P

Anmð Þ¼ <sup>κ</sup> <sup>k</sup><sup>2</sup>

2 0ς ð Þ1

Bnmð Þ¼� κ knkm þ ς

way, we arrive at the following equation for the macroscopic field:

∇ � ∇ � P

For the upper medium (j ¼ 0), direct substitution of expression (11) with P

in Cartesian coordinates) into Eq. (12) gives us the relation with respect to κ0:

!ð Þ<sup>1</sup> :

AxzPð Þ<sup>1</sup>

BzxPð Þ<sup>1</sup>

<sup>x</sup> <sup>þ</sup> BxzPð Þ<sup>1</sup>

<sup>x</sup> <sup>þ</sup> AzxPð Þ<sup>1</sup>

<sup>y</sup> <sup>þ</sup> knkmκ<sup>2</sup> <sup>þ</sup> <sup>i</sup><sup>κ</sup> kn<sup>ς</sup>

yy , and subscripts m and n iterate over indexes x and z.

<sup>y</sup> þ k 2

For the composite medium (j ¼ 1), substitution of (11) with P

where Anm and Bnm are functions of κ derived in the form [10]:

ð Þ1 nm � �ϰ<sup>2</sup>

<sup>m</sup> � k 2 0ς ð Þ1 nn � <sup>κ</sup><sup>2</sup> � �ϰ<sup>2</sup>

κ2 <sup>0</sup> ¼ k

!ð Þ<sup>j</sup> � k 2 <sup>0</sup>bςð Þ<sup>j</sup> <sup>P</sup> !ð Þ<sup>j</sup>

where <sup>k</sup><sup>0</sup> <sup>¼</sup> <sup>ω</sup>=<sup>c</sup> is the free-space wavenumber and <sup>b</sup>ςð Þ<sup>j</sup> is introduced as the product of <sup>μ</sup>bð Þ<sup>j</sup> and

νν<sup>0</sup> <sup>¼</sup> 1 for <sup>ν</sup> <sup>¼</sup> <sup>ν</sup><sup>0</sup> and <sup>b</sup>ςð Þ<sup>0</sup>

<sup>2</sup> � <sup>k</sup> 2

subscripts ν and ν<sup>0</sup> are substituted to iterate over indexes of the tensor components x, y, and z

represented as

along the y-axis.

where k

with ϰ<sup>2</sup>

<sup>y</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>k</sup>

<sup>2</sup> <sup>¼</sup> <sup>k</sup> 2 <sup>x</sup> þ k 2 z .

quent elimination of Pð Þ<sup>1</sup>

!

exp ½ � i kð Þ xx þ kzz exp ð Þ ∓κy , (11)

Modal Phenomena of Surface and Bulk Polaritons in Magnetic-Semiconductor Superlattices

and ∇� H !

νν<sup>0</sup> ¼ 0 for ν 6¼ ν<sup>0</sup>

!ð Þ<sup>1</sup>

ny þ ς ð Þ1 yn � �k<sup>2</sup>

ny � �<sup>k</sup>

ð Þ1

ð Þ1 ym þ kmς <sup>0</sup> � k 4 0ς ð Þ1 ny ς ð Þ1 yn ,

2 <sup>0</sup> � k 4 0ς ð Þ1 ny ς ð Þ1 ym,

<sup>y</sup> yield us the following system of two homogeneous algebraic equa-

<sup>z</sup> ¼ 0,

<sup>z</sup> ¼ 0,

<sup>n</sup>κ<sup>2</sup> <sup>þ</sup> iknκ ςð Þ<sup>1</sup>

<sup>0</sup>, (13)

and H !

http://dx.doi.org/10.5772/intechopen.71837

¼ �ik<sup>0</sup> D !

¼ 0, (12)

!ð Þ<sup>0</sup>

and <sup>b</sup>ςð Þ<sup>1</sup> into (12) and subse-

used here are

107

, in a standard

and corre-

(14)

(15)

; here and further

Figure 2. Dispersion curves of the tensor components of (a, c) relative effective permeability <sup>μ</sup>beff and (b, d) relative effective permittivity <sup>b</sup>εeff of the homogenized medium. Panels (a) and (b) correspond to the polar geometry and panels (c) and (d) correspond to the Voigt and Faraday geometries. For the magnetic constitutive layers, under saturation magnetization of 2930 G, parameters are f <sup>0</sup> ¼ ω0=2π ¼ 3:9 GHz, f <sup>m</sup> ¼ ωm=2π ¼ 8:2 GHz, b ¼ 0, and ε<sup>m</sup> ¼ 5:5; for the semiconductor constitutive layers, parameters are f <sup>p</sup> ¼ ωp=2π ¼ 5:5 GHz, f <sup>c</sup> ¼ ωc=2π ¼ 4:5 GHz, ν ¼ 0, ε<sup>l</sup> ¼ 1:0, and μ<sup>s</sup> ¼ 1:0. Filling factors are δ<sup>m</sup> ¼ δ<sup>s</sup> ¼ 0:5.

orthogonal to the structure periodicity, respectively. In the latter case, the first optical axis of the biaxial medium is directed along the structure periodicity, whereas the second one coincides with the direction of the external static magnetic field M ! .

#### 3. General solution for bulk and surface polaritons

In order to obtain a general solution for both bulk and surface polaritons, we follow the approach developed in Ref. [10] where dispersion characteristics of polaritons in a uniaxial anisotropic dielectric medium have been studied. Here, we extend this approach to the case of a gyroelectromagnetic medium whose permittivity and permeability simultaneously are tensor quantities.

In a general form [26], the electric and magnetic field vectors E ! and H ! used here are represented as

$$\overrightarrow{\boldsymbol{P}}^{(j)} = \overrightarrow{\boldsymbol{p}}^{(j)} \exp\left[i(k\_x \mathbf{x} + k\_z \mathbf{z})\right] \exp\left(\mp \kappa \mathbf{y}\right),\tag{11}$$

where a time factor exp ð Þ �iω<sup>t</sup> is also supposed and omitted, and sign " � " is related to the fields in the upper medium (<sup>y</sup> <sup>&</sup>gt; <sup>0</sup>, j <sup>¼</sup> 0), while sign " <sup>þ</sup> " is related to the fields in the composite medium (y < 0, j ¼ 1), respectively, which provide required wave attenuation along the y-axis.

From a pair of the curl Maxwell's equations ∇� E ! ¼ ik<sup>0</sup> B ! and ∇� H ! ¼ �ik<sup>0</sup> D ! , in a standard way, we arrive at the following equation for the macroscopic field:

$$\nabla \times \nabla \times \overrightarrow{\boldsymbol{P}}^{(j)} - k\_0^2 \hat{\boldsymbol{\zeta}}^{(j)} \overrightarrow{\boldsymbol{P}}^{(j)} = \mathbf{0},\tag{12}$$

where <sup>k</sup><sup>0</sup> <sup>¼</sup> <sup>ω</sup>=<sup>c</sup> is the free-space wavenumber and <sup>b</sup>ςð Þ<sup>j</sup> is introduced as the product of <sup>μ</sup>bð Þ<sup>j</sup> and <sup>b</sup>εð Þ<sup>j</sup> made in the appropriate order.

For the upper medium (j ¼ 0), direct substitution of expression (11) with P !ð Þ<sup>0</sup> and corresponding constitutive parameters (bςð Þ<sup>0</sup> νν<sup>0</sup> <sup>¼</sup> 1 for <sup>ν</sup> <sup>¼</sup> <sup>ν</sup><sup>0</sup> and <sup>b</sup>ςð Þ<sup>0</sup> νν<sup>0</sup> ¼ 0 for ν 6¼ ν<sup>0</sup> ; here and further subscripts ν and ν<sup>0</sup> are substituted to iterate over indexes of the tensor components x, y, and z in Cartesian coordinates) into Eq. (12) gives us the relation with respect to κ0:

$$
\kappa\_0^2 = k^2 - k\_0^2 \tag{13}
$$

where k <sup>2</sup> <sup>¼</sup> <sup>k</sup> 2 <sup>x</sup> þ k 2 z .

orthogonal to the structure periodicity, respectively. In the latter case, the first optical axis of the biaxial medium is directed along the structure periodicity, whereas the second one coin-

Figure 2. Dispersion curves of the tensor components of (a, c) relative effective permeability <sup>μ</sup>beff and (b, d) relative effective permittivity <sup>b</sup>εeff of the homogenized medium. Panels (a) and (b) correspond to the polar geometry and panels (c) and (d) correspond to the Voigt and Faraday geometries. For the magnetic constitutive layers, under saturation magnetization of 2930 G, parameters are f <sup>0</sup> ¼ ω0=2π ¼ 3:9 GHz, f <sup>m</sup> ¼ ωm=2π ¼ 8:2 GHz, b ¼ 0, and ε<sup>m</sup> ¼ 5:5; for the semiconductor constitutive layers, parameters are f <sup>p</sup> ¼ ωp=2π ¼ 5:5 GHz, f <sup>c</sup> ¼ ωc=2π ¼ 4:5 GHz, ν ¼ 0, ε<sup>l</sup> ¼ 1:0, and

In order to obtain a general solution for both bulk and surface polaritons, we follow the approach developed in Ref. [10] where dispersion characteristics of polaritons in a uniaxial anisotropic dielectric medium have been studied. Here, we extend this approach to the case of a gyroelectromagnetic medium whose permittivity and permeability simultaneously are ten-

! .

cides with the direction of the external static magnetic field M

3. General solution for bulk and surface polaritons

sor quantities.

μ<sup>s</sup> ¼ 1:0. Filling factors are δ<sup>m</sup> ¼ δ<sup>s</sup> ¼ 0:5.

106 Surface Waves - New Trends and Developments

For the composite medium (j ¼ 1), substitution of (11) with P !ð Þ<sup>1</sup> and <sup>b</sup>ςð Þ<sup>1</sup> into (12) and subsequent elimination of Pð Þ<sup>1</sup> <sup>y</sup> yield us the following system of two homogeneous algebraic equations for the rest two components of P !ð Þ<sup>1</sup> :

$$\begin{aligned} A\_{xz}P\_x^{(1)} + B\_{xz}P\_z^{(1)} &= 0, \\ B\_{zx}P\_x^{(1)} + A\_{zx}P\_z^{(1)} &= 0, \end{aligned} \tag{14}$$

where Anm and Bnm are functions of κ derived in the form [10]:

$$\begin{aligned} A\_{nm}(\kappa) &= \left(k\_m^2 - k\_0^2 \boldsymbol{\zeta}\_m^{(1)} - \kappa^2\right) \varkappa\_y^2 + k\_n^2 \kappa^2 + ik\_n \kappa \left(\boldsymbol{\zeta}\_{ny}^{(1)} + \boldsymbol{\zeta}\_{yn}^{(1)}\right) k\_0^2 - k\_0^4 \boldsymbol{\zeta}\_{ny}^{(1)} \boldsymbol{\zeta}\_{yn}^{(1)}, \\ B\_{nm}(\kappa) &= -\left(k\_n k\_m + \boldsymbol{\zeta}\_{nm}^{(1)}\right) \varkappa\_y^2 + k\_n k\_m \kappa^2 + i\kappa \left(k\_n \boldsymbol{\zeta}\_{yn}^{(1)} + k\_m \boldsymbol{\zeta}\_{ny}^{(1)}\right) k\_0^2 - k\_0^4 \boldsymbol{\zeta}\_{ny}^{(1)} \boldsymbol{\zeta}\_{ym}^{(1)}. \end{aligned} \tag{15}$$

with ϰ<sup>2</sup> <sup>y</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>k</sup> 2 0ς ð Þ1 yy , and subscripts m and n iterate over indexes x and z. In order to find a nontrivial solution of system (15), we set its determinant of coefficients to zero. After disclosure of the determinant, we obtain an equation of the fourth degree with respect to κ:

$$
\varepsilon\_{yy}^{(1)}\kappa^4 + a\kappa^3 + b\kappa^2 + c\kappa + d = 0,\tag{16}
$$

Taking into consideration that two appropriate roots κ<sup>1</sup> and κ<sup>2</sup> of (16) are selected, the compo-

can be rewritten as the linear superposition of two terms:

KwAzxð Þ κ<sup>w</sup> exp ð Þ κwy ,

Modal Phenomena of Surface and Bulk Polaritons in Magnetic-Semiconductor Superlattices

KwCð Þ κ<sup>w</sup> exp ð Þ κwy ,

(19)

109

(22)

KwBzxð Þ κ<sup>w</sup> exp ð Þ κwy ,

!

flux densities; one can immediately obtain the relations between

KwAzxð Þ κ<sup>w</sup>

KwBzxð Þ κ<sup>w</sup> ,

X <sup>w</sup>¼<sup>1</sup>, <sup>2</sup>

KwBzxð Þ κ<sup>w</sup> :

<sup>x</sup> <sup>þ</sup> kzPð Þ<sup>0</sup> z

¼ 0 and ∇� D

<sup>¼</sup> <sup>∇</sup> � <sup>b</sup>gð Þ<sup>j</sup> <sup>P</sup>ð Þ<sup>j</sup> � � <sup>¼</sup> <sup>0</sup>, (20)

!

� �: (21)

, i.e., in our notations these compo-

!

http://dx.doi.org/10.5772/intechopen.71837

¼ 0 in the form

is substituted for the

nents of the field P

magnetic B

of E ! and H !

!

and electric D

with (21) gives us the next set of equations:

Pð Þ<sup>0</sup> <sup>x</sup> <sup>¼</sup> <sup>X</sup> <sup>w</sup>¼<sup>1</sup>, <sup>2</sup>

Pð Þ<sup>0</sup>

kzPð Þ<sup>0</sup>

<sup>z</sup> ¼ � <sup>X</sup>

<sup>x</sup> � kxPð Þ<sup>0</sup>

ig0=κ<sup>0</sup> � � kxPð Þ<sup>0</sup>

!

the field components in the upper (y > 0) medium as follows:

Qð Þ<sup>0</sup>

as well as the normal components of D

<sup>þ</sup> <sup>g</sup>~yx <sup>X</sup> <sup>w</sup>¼<sup>1</sup>, <sup>2</sup>

<sup>w</sup>¼<sup>1</sup>, <sup>2</sup>

<sup>z</sup> ¼ kz

condition gives us the required dispersion equation for surface polaritons.

!ð Þ<sup>1</sup>

Pð Þ<sup>1</sup> <sup>x</sup> <sup>¼</sup> <sup>X</sup> <sup>w</sup>¼<sup>1</sup>, <sup>2</sup>

Pð Þ<sup>1</sup> <sup>y</sup> <sup>¼</sup> <sup>X</sup> <sup>w</sup>¼<sup>1</sup>, <sup>2</sup>

Pð Þ<sup>1</sup> <sup>z</sup> <sup>¼</sup> <sup>X</sup> <sup>w</sup>¼<sup>1</sup>, <sup>2</sup>

where y < 0 and the factor exp ½ � i kð Þ xx þ kzz � ωt is omitted.

Involving a pair of the divergent Maxwell's equations ∇� B

∇ � Q !ð Þ<sup>j</sup>

where g is substituted for permeability μ and permittivity ε and Q

<sup>y</sup> ¼ ig0=κ<sup>0</sup>

<sup>x</sup> <sup>þ</sup> kzPð Þ<sup>0</sup> z � � <sup>¼</sup> <sup>g</sup>~yx <sup>X</sup>

KwAzxð Þ κ<sup>w</sup> ,

KwBzxð Þ κ<sup>w</sup> ,

X <sup>w</sup>¼<sup>1</sup>, <sup>2</sup>

� � kxPð Þ<sup>0</sup>

The boundary conditions at the interface require the continuity of the tangential components

nents are Px, Pz, and Qy, respectively. Thus, application of the boundary conditions together

KwCð Þ� <sup>κ</sup><sup>w</sup> <sup>g</sup>~yz <sup>X</sup>

! and B !

<sup>w</sup>¼<sup>1</sup>, <sup>2</sup>

<sup>w</sup>¼<sup>1</sup>, <sup>2</sup>

KwAzxð Þþ κ<sup>w</sup> kx

The system of Eq. (22) has a nontrivial solution only if its determinant vanishes. Applying this

whose coefficients a, b, c, and d are

a ¼ ikx � ς ð1Þ yz þ ς ð1Þ zy � þ ikz � ς ð1Þ yx þ ς ð1Þ xy � , b ¼ k 2 <sup>0</sup> ς ð1Þ yy � ς ð1Þ zz þ ς ð1Þ xx � � ς ð1Þ zy ς ð1Þ yz � ς ð1Þ xy ς ð1Þ yx h i � k 2 ς ð1Þ yy þ k 2 zς ð1Þ zz þ k 2 xς ð1Þ xx þ kxkz � ς ð1Þ zx þ ς ð1Þ xz h i� , <sup>c</sup> ¼ �ik<sup>2</sup> <sup>0</sup> kz ς ð1Þ xy ς ð1Þ zx þ ς ð1Þ yx ς ð1Þ xz � ς ð1Þ xx � ς ð1Þ yz þ ς ð1Þ zy <sup>n</sup> h i� þkx ς ð1Þ yz ς ð1Þ zx þ ς ð1Þ zy ς ð1Þ xz � ς ð1Þ zz � ς ð1Þ yx þ ς ð1Þ xy <sup>h</sup> �io �ik<sup>2</sup> kz � ς ð1Þ yz þ ς ð1Þ zy � þ kx � ς ð1Þ yx þ ς ð1Þ xy h i� , <sup>d</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> <sup>k</sup><sup>2</sup> zς ð1Þ zz þ k 2 xς ð1Þ xx þ kxkz � ς ð1Þ zx þ ς ð1Þ xz h i� <sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>0</sup>k<sup>2</sup> <sup>ς</sup> ð1Þ zx ς ð1Þ xz � ς ð1Þ zz ς ð1Þ xx h i þk 2 <sup>0</sup> k 2 x � ς ð1Þ xy ς ð1Þ yx � ς ð1Þ yy ς ð1Þ xx � þ k 2 z � ς ð1Þ zy ς ð1Þ yz � ς ð1Þ yy ς ð1Þ zz <sup>n</sup> � þkxkz ς ð1Þ xy ς ð1Þ yz þ ς ð1Þ zy ς ð1Þ yx � ς ð1Þ yy � ς ð1Þ zx þ ς ð1Þ xz <sup>h</sup> �io þk<sup>4</sup> <sup>0</sup> ς ð1Þ xx ς ð1Þ yy ς ð1Þ zz � ς ð1Þ yy ς ð1Þ zx ς ð1Þ xz þ ς ð1Þ xy � ς ð1Þ yz ς ð1Þ zx � ς ð1Þ zz ς ð1Þ yx <sup>h</sup> � þς ð1Þ zy � ς ð1Þ xz ς ð1Þ yx � ς ð1Þ yz ς ð1Þ xx �i: (17)

The dispersion relation for bulk polaritons is then obtained straightforwardly from (16) by putting κ ¼ 0 inside it.

In order to find the dispersion law of surface polaritons from four roots of (16), two physically correct ones must be selected. In general, two such roots are required to satisfy the electromagnetic boundary conditions at the surface of the composite medium. We define these roots as κ<sup>1</sup> and κ<sup>2</sup> and then following [10] introduce the quantities Kw (w ¼ 1, 2) in the form:

$$\begin{aligned} P\_x^{(1)}(\kappa\_w) &= K\_w A\_{zx}(\kappa\_w)\_\prime \\ P\_y^{(1)}(\kappa\_w) &= K\_w C(\kappa\_w)\_\prime \\ P\_z^{(1)}(\kappa\_w) &= -K\_w B\_{zx}(\kappa\_w)\_\prime \end{aligned} \tag{18}$$

where <sup>C</sup>ð Þ¼� <sup>κ</sup><sup>w</sup> <sup>1</sup>=ϰ<sup>2</sup> y � � ikxκ<sup>w</sup> � <sup>k</sup> 2 0ς ð Þ1 yx � �Azxð Þþ <sup>κ</sup><sup>w</sup> ikzκ<sup>w</sup> � <sup>k</sup> 2 0ς ð Þ1 yz � �Bzxð Þ <sup>κ</sup><sup>w</sup> h i.

In (18), unknown quantities Kw need to be determined from the boundary conditions.

Taking into consideration that two appropriate roots κ<sup>1</sup> and κ<sup>2</sup> of (16) are selected, the components of the field P !ð Þ<sup>1</sup> can be rewritten as the linear superposition of two terms:

$$\begin{aligned} P\_x^{(1)} &= \sum\_{w=1,2} K\_w A\_{zx}(\kappa\_w) \exp\left(\kappa\_w y\right), \\ P\_y^{(1)} &= \sum\_{w=1,2} K\_w \mathbb{C}(\kappa\_w) \exp\left(\kappa\_w y\right), \\ P\_z^{(1)} &= \sum\_{w=1,2} K\_w B\_{zx}(\kappa\_w) \exp\left(\kappa\_w y\right), \end{aligned} \tag{19}$$

where y < 0 and the factor exp ½ � i kð Þ xx þ kzz � ωt is omitted.

In order to find a nontrivial solution of system (15), we set its determinant of coefficients to zero. After disclosure of the determinant, we obtain an equation of the fourth degree with

> � ς ð1Þ zx þ ς ð1Þ xz

> > ,

<sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>0</sup>k<sup>2</sup> <sup>ς</sup> ð1Þ zx ς ð1Þ xz � ς ð1Þ zz ς ð1Þ xx

yy <sup>κ</sup><sup>4</sup> <sup>þ</sup> <sup>a</sup>κ<sup>3</sup> <sup>þ</sup> <sup>b</sup>κ<sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>κ</sup> <sup>þ</sup> <sup>d</sup> <sup>¼</sup> <sup>0</sup>, (16)

,

h i

(17)

(18)

ςð Þ<sup>1</sup>

h i

h i�

h �io

� ς ð1Þ zx þ ς ð1Þ xz

h �io

n �

h �

The dispersion relation for bulk polaritons is then obtained straightforwardly from (16) by

In order to find the dispersion law of surface polaritons from four roots of (16), two physically correct ones must be selected. In general, two such roots are required to satisfy the electromagnetic boundary conditions at the surface of the composite medium. We define these roots as κ<sup>1</sup>

<sup>x</sup> ð Þ¼ κ<sup>w</sup> KwAzxð Þ κ<sup>w</sup> ,

<sup>y</sup> ð Þ¼ κ<sup>w</sup> KwCð Þ κ<sup>w</sup> ,

In (18), unknown quantities Kw need to be determined from the boundary conditions.

<sup>z</sup> ð Þ¼� κ<sup>w</sup> KwBzxð Þ κ<sup>w</sup> ,

Azxð Þþ κ<sup>w</sup> ikzκ<sup>w</sup> � k

h i

2 0ς ð Þ1 yz

Bzxð Þ κ<sup>w</sup>

.

� �

and κ<sup>2</sup> and then following [10] introduce the quantities Kw (w ¼ 1, 2) in the form:

Pð Þ<sup>1</sup>

Pð Þ<sup>1</sup>

Pð Þ<sup>1</sup>

2 0ς ð Þ1 yx

ikxκ<sup>w</sup> � k

� �

h i�

h i�

n h i�

respect to κ:

whose coefficients a, b, c, and d are

108 Surface Waves - New Trends and Developments

a ¼ ikx � ς ð1Þ yz þ ς ð1Þ zy � þ ikz � ς ð1Þ yx þ ς ð1Þ xy � ,

b ¼ k 2 <sup>0</sup> ς ð1Þ yy � ς ð1Þ zz þ ς ð1Þ xx � � ς ð1Þ zy ς ð1Þ yz � ς ð1Þ xy ς ð1Þ yx

> � k 2 ς ð1Þ yy þ k 2 zς ð1Þ zz þ k 2 xς ð1Þ xx þ kxkz

þkx ς ð1Þ yz ς ð1Þ zx þ ς ð1Þ zy ς ð1Þ xz � ς ð1Þ zz � ς ð1Þ yx þ ς ð1Þ xy

�ik<sup>2</sup> kz � ς ð1Þ yz þ ς ð1Þ zy � þ kx � ς ð1Þ yx þ ς ð1Þ xy

<sup>0</sup> kz ς ð1Þ xy ς ð1Þ zx þ ς ð1Þ yx ς ð1Þ xz � ς ð1Þ xx � ς ð1Þ yz þ ς ð1Þ zy

<sup>c</sup> ¼ �ik<sup>2</sup>

<sup>d</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> <sup>k</sup><sup>2</sup> zς ð1Þ zz þ k 2 xς ð1Þ xx þ kxkz

> þk 2 <sup>0</sup> k 2 x � ς ð1Þ xy ς ð1Þ yx � ς ð1Þ yy ς ð1Þ xx � þ k 2 z � ς ð1Þ zy ς ð1Þ yz � ς ð1Þ yy ς ð1Þ zz

þk<sup>4</sup> <sup>0</sup> ς ð1Þ xx ς ð1Þ yy ς ð1Þ zz � ς ð1Þ yy ς ð1Þ zx ς ð1Þ xz þ ς ð1Þ xy � ς ð1Þ yz ς ð1Þ zx � ς ð1Þ zz ς ð1Þ yx

þς ð1Þ zy � ς ð1Þ xz ς ð1Þ yx � ς ð1Þ yz ς ð1Þ xx �i :

putting κ ¼ 0 inside it.

where <sup>C</sup>ð Þ¼� <sup>κ</sup><sup>w</sup> <sup>1</sup>=ϰ<sup>2</sup>

y � �

þkxkz ς

ð1Þ xy ς ð1Þ yz þ ς ð1Þ zy ς ð1Þ yx � ς ð1Þ yy � ς ð1Þ zx þ ς ð1Þ xz

Involving a pair of the divergent Maxwell's equations ∇� B ! ¼ 0 and ∇� D ! ¼ 0 in the form

$$\nabla \cdot \overrightarrow{\mathbf{Q}}^{(j)} = \nabla \cdot \left(\widehat{\mathbf{g}}^{(j)} P^{(j)}\right) = \mathbf{0},\tag{20}$$

where g is substituted for permeability μ and permittivity ε and Q ! is substituted for the magnetic B ! and electric D ! flux densities; one can immediately obtain the relations between the field components in the upper (y > 0) medium as follows:

$$Q\_y^{(0)} = \left(\mathrm{ig}\_0/\kappa\_0\right) \left(k\_\mathrm{x} P\_\mathrm{x}^{(0)} + k\_\mathrm{z} P\_\mathrm{z}^{(0)}\right). \tag{21}$$

The boundary conditions at the interface require the continuity of the tangential components of E ! and H ! as well as the normal components of D ! and B ! , i.e., in our notations these components are Px, Pz, and Qy, respectively. Thus, application of the boundary conditions together with (21) gives us the next set of equations:

$$\begin{aligned} \left(\mathrm{ig}\_{0}/\kappa\_{0}\right) \left(k\_{x}P\_{x}^{(0)} + k\_{z}P\_{z}^{(0)}\right) &= \bar{\mathcal{g}}\_{yx} \sum\_{w=1,2} K\_{w} A\_{zx}(\kappa\_{w}) \\ &+ \bar{\mathcal{g}}\_{yx} \sum\_{w=1,2} K\_{w} \mathsf{C}(\kappa\_{w}) - \bar{\mathcal{g}}\_{yz} \sum\_{w=1,2} K\_{w} B\_{zx}(\kappa\_{w}), \\ P\_{x}^{(0)} &= \sum\_{w=1,2} K\_{w} A\_{zx}(\kappa\_{w}), \\ P\_{z}^{(0)} &= -\sum\_{w=1,2} K\_{w} B\_{zx}(\kappa\_{w}), \\ k\_{z} P\_{x}^{(0)} - k\_{x} P\_{z}^{(0)} &= k\_{z} \sum\_{w=1,2} K\_{w} A\_{zx}(\kappa\_{w}) + k\_{x} \sum\_{w=1,2} K\_{w} B\_{zx}(\kappa\_{w}). \end{aligned} \tag{22}$$

The system of Eq. (22) has a nontrivial solution only if its determinant vanishes. Applying this condition gives us the required dispersion equation for surface polaritons.

Finally, the amplitudes K<sup>1</sup> and K<sup>2</sup> can be found by solving set of linear homogeneous Eq. (22). They are

$$\begin{aligned} K\_1 &= [k\_\text{x} A\_{\text{zx}}(\kappa\_2) + k\_z B\_{\text{zx}}(\kappa\_2)](\kappa\_0 + \kappa\_2), \\ K\_2 &= -[k\_\text{x} B\_{\text{zx}}(\kappa\_1) + k\_z A\_{\text{zx}}(\kappa\_1)](\kappa\_0 + \kappa\_1). \end{aligned} \tag{23}$$

D<sup>0</sup> <sup>k</sup>ð Þ k; k<sup>0</sup> k <sup>m</sup>;k <sup>m</sup> ð Þ<sup>0</sup>

h<sup>11</sup> ¼ D<sup>00</sup>

h<sup>21</sup> ¼ D<sup>00</sup>

corresponding Morse critical point can be defined as follows [33]:

and H is the Hessian determinant with elements:

where k

<sup>m</sup>; km 0

shown in Figure 3.

ities at k ¼ 0 (Figure 3d).

¼ D<sup>0</sup> k0 ð Þ k; k<sup>0</sup> k <sup>m</sup>;k <sup>m</sup> ð Þ<sup>0</sup>

H ¼ ½h11h<sup>22</sup> � h12h21�j <sup>k</sup>m;<sup>k</sup>

¼ 0,

6¼ <sup>0</sup>, (25)

http://dx.doi.org/10.5772/intechopen.71837

111

ð Þ <sup>k</sup>; <sup>k</sup><sup>0</sup> : (26)

<sup>m</sup> ð Þ<sup>0</sup>

Modal Phenomena of Surface and Bulk Polaritons in Magnetic-Semiconductor Superlattices

kk<sup>0</sup> ð Þ k; k<sup>0</sup> ,

k0k<sup>0</sup>

Codirectional forward : h12=h<sup>11</sup> < 0, h22=h<sup>11</sup> > 0; (27)

Codirectional backward : h12=h<sup>11</sup> > 0, h22=h<sup>11</sup> > 0; (28)

Contradirectional : h12=h<sup>11</sup> > 0, h22=h<sup>11</sup> < 0: (29)

 are coordinates in the <sup>k</sup> � <sup>k</sup><sup>0</sup> plane of a particular <sup>m</sup>-th Morse critical point, the subscripts k and k<sup>0</sup> near the letter D define corresponding partial derivatives ∂=∂k and ∂=∂k0,

kkð Þ k; k<sup>0</sup> , h<sup>12</sup> ¼ D<sup>00</sup>

<sup>k</sup>0<sup>k</sup>ð Þ k; k<sup>0</sup> , h<sup>22</sup> ¼ D<sup>00</sup>

The type of each extreme state defined by set of Eq. (25) can be uniquely identified from the sign of the Hessian determinant [35]. For instance, when H < 0, the corresponding Morse critical point represents a saddle point, which occurs in the region of a modal coupling (the anti-crossing effect), whereas in the case of degeneracy, when H ¼ 0, it is a non-isolated critical point (the crossing effect). In the case when H > 0, the Morse critical point defines either a local minimum or maximum (this case is not considered here). In what follows we distinguish found critical points by circles for both crossing and anti-crossing effects, as

In general, when conditions (25) are met, the type of interacting modes in the vicinity of the

The strength of modes interaction within the found extreme states in the region of their coupling can be identified considering the classification introduced in paper [37], which concerns on the mode behaviors appearing in axial waveguides. In particular, (i) a weak interaction takes place when frequency band gap between dispersion curves of interacting modes is high enough (Figure 3a), (ii) an intermediate interaction of modes leads to formation of very flattened parts in the dispersion curves (Figure 3b), (iii) a strong interaction appears when the repulsion between modes is strong enough resulting in formation of dispersion curve having anomalous dispersion line (Figure 3c), and (iv) an accidental degeneracy arises when two dispersion branches are merged within the critical point which leads to nonzero group veloc-

The strong interaction with forming negative-slope region in dispersion curve of one of interacting modes (as it is depicted in Figure 3c and d) leads to some unusual effects [37]. First, zero group velocity (vg ¼ ∂k0=∂k) can appear at a nonzero value of k. The flattened region of the dispersion curve around that extreme point (Figure 3c) is very useful for applications in nonlinear optics, where a small group velocity is suitable for enhancing nonlinear effects, while the phase-matching criterion can still be satisfied because the wave vector is nonzero. Second,

Here, the problem is considered to be formally solved, and the dispersion relations are derived in a general form for both bulk and surface polaritons.

#### 4. Theory of Morse critical points: mode coupling phenomena

Further, for brevity, obtained dispersion equations for the bulk and surface polaritons are denoted in the form:

$$
\mathfrak{D}(k, k\_0) = 0.\tag{24}
$$

Numerical solution of Eq. (24) gives a set of dispersion curves k0ð Þk of polaritons, which can contain both regular and singular (critical) points. The regular points draw dispersion curves of a classical form possessing either normal or anomalous dispersion line, at which a small variation in k<sup>0</sup> results in a smooth changing in the form of the curves. Besides, some situations are possible when a slight variation in k<sup>0</sup> leads to a very sharp (catastrophic) changing in the form of dispersion curves. Such singularities (extreme states) can be accompanied by mutual coupling phenomena of modes, which are further of our interest.

From the mathematical point of view the found extreme states in dispersion curves exist in the region where the differential D<sup>0</sup> ð Þ k; k<sup>0</sup> of dispersion Eq. (24) vanishes (see, e.g., Figure 3). These extreme states can be carefully identified and studied involving the approach based on the theory of the Morse critical points from the catastrophe theory [31–36]. This treatment has been originally applied to study open waveguides and resonators [31, 32], and later it has been extended to more complex waveguide structures [33–35]. From viewpoint of this theory, the presence of the Morse critical points is generally defined by a set of nonlinear differential equations written in the form [33]:

Figure 3. Sketch of the band diagrams presenting different kinds of interaction between two neighboring modes: (a) weak interaction, (b) intermediate interaction, (c) strong interaction, and (d) accidental degeneracy. Areas where the critical points exist are pointed by circles.

Modal Phenomena of Surface and Bulk Polaritons in Magnetic-Semiconductor Superlattices http://dx.doi.org/10.5772/intechopen.71837 111

$$\begin{aligned} \left. \mathfrak{D}'\_k(k, k\_0) \right|\_{\left(k'', k''\_0\right)} &= \mathfrak{D}'\_{k\_0}(k, k\_0) \vert\_{\left(k'', k''\_0\right)} = 0, \\ \mathbb{H} &= \left[ h\_{11} h\_{22} - h\_{12} h\_{21} \right] \vert\_{\left(k'', k''\_0\right)} \neq 0, \end{aligned} \tag{25}$$

where k <sup>m</sup>; km 0 are coordinates in the <sup>k</sup> � <sup>k</sup><sup>0</sup> plane of a particular <sup>m</sup>-th Morse critical point, the subscripts k and k<sup>0</sup> near the letter D define corresponding partial derivatives ∂=∂k and ∂=∂k0, and H is the Hessian determinant with elements:

Finally, the amplitudes K<sup>1</sup> and K<sup>2</sup> can be found by solving set of linear homogeneous Eq. (22).

K<sup>1</sup> ¼ ½ � kxAzxð Þþ κ<sup>2</sup> kzBzxð Þ κ<sup>2</sup> ð Þ κ<sup>0</sup> þ κ<sup>2</sup> ,

Here, the problem is considered to be formally solved, and the dispersion relations are derived

Further, for brevity, obtained dispersion equations for the bulk and surface polaritons are

Numerical solution of Eq. (24) gives a set of dispersion curves k0ð Þk of polaritons, which can contain both regular and singular (critical) points. The regular points draw dispersion curves of a classical form possessing either normal or anomalous dispersion line, at which a small variation in k<sup>0</sup> results in a smooth changing in the form of the curves. Besides, some situations are possible when a slight variation in k<sup>0</sup> leads to a very sharp (catastrophic) changing in the form of dispersion curves. Such singularities (extreme states) can be accompanied by mutual

From the mathematical point of view the found extreme states in dispersion curves exist in the

extreme states can be carefully identified and studied involving the approach based on the theory of the Morse critical points from the catastrophe theory [31–36]. This treatment has been originally applied to study open waveguides and resonators [31, 32], and later it has been extended to more complex waveguide structures [33–35]. From viewpoint of this theory, the presence of the Morse critical points is generally defined by a set of nonlinear differential

Figure 3. Sketch of the band diagrams presenting different kinds of interaction between two neighboring modes: (a) weak interaction, (b) intermediate interaction, (c) strong interaction, and (d) accidental degeneracy. Areas where the

4. Theory of Morse critical points: mode coupling phenomena

coupling phenomena of modes, which are further of our interest.

in a general form for both bulk and surface polaritons.

<sup>K</sup><sup>2</sup> ¼ �½ � kxBzxð Þþ <sup>κ</sup><sup>1</sup> kzAzxð Þ <sup>κ</sup><sup>1</sup> ð Þ <sup>κ</sup><sup>0</sup> <sup>þ</sup> <sup>κ</sup><sup>1</sup> : (23)

Dð Þ¼ k; k<sup>0</sup> 0: (24)

ð Þ k; k<sup>0</sup> of dispersion Eq. (24) vanishes (see, e.g., Figure 3). These

They are

110 Surface Waves - New Trends and Developments

denoted in the form:

region where the differential D<sup>0</sup>

equations written in the form [33]:

critical points exist are pointed by circles.

$$\begin{aligned} h\_{11} &= \mathfrak{D}''\_{kk}(k, k\_0), \quad h\_{12} = \mathfrak{D}''\_{kk\_0}(k, k\_0), \\ h\_{21} &= \mathfrak{D}''\_{k\_0k}(k, k\_0), \quad h\_{22} = \mathfrak{D}''\_{k\_0k\_0}(k, k\_0). \end{aligned} \tag{26}$$

The type of each extreme state defined by set of Eq. (25) can be uniquely identified from the sign of the Hessian determinant [35]. For instance, when H < 0, the corresponding Morse critical point represents a saddle point, which occurs in the region of a modal coupling (the anti-crossing effect), whereas in the case of degeneracy, when H ¼ 0, it is a non-isolated critical point (the crossing effect). In the case when H > 0, the Morse critical point defines either a local minimum or maximum (this case is not considered here). In what follows we distinguish found critical points by circles for both crossing and anti-crossing effects, as shown in Figure 3.

In general, when conditions (25) are met, the type of interacting modes in the vicinity of the corresponding Morse critical point can be defined as follows [33]:

$$\text{Coefficiental forward}: \qquad h\_{12}/h\_{11} < 0, \quad h\_{22}/h\_{11} > 0; \tag{27}$$

$$\text{Coefficientual backwards}: \quad h\_{12}/h\_{11} > 0, \quad h\_{22}/h\_{11} > 0; \tag{28}$$

$$\text{Contrad directional}: \qquad h\_{12}/h\_{11} > 0, \quad h\_{22}/h\_{11} < 0. \tag{29}$$

The strength of modes interaction within the found extreme states in the region of their coupling can be identified considering the classification introduced in paper [37], which concerns on the mode behaviors appearing in axial waveguides. In particular, (i) a weak interaction takes place when frequency band gap between dispersion curves of interacting modes is high enough (Figure 3a), (ii) an intermediate interaction of modes leads to formation of very flattened parts in the dispersion curves (Figure 3b), (iii) a strong interaction appears when the repulsion between modes is strong enough resulting in formation of dispersion curve having anomalous dispersion line (Figure 3c), and (iv) an accidental degeneracy arises when two dispersion branches are merged within the critical point which leads to nonzero group velocities at k ¼ 0 (Figure 3d).

The strong interaction with forming negative-slope region in dispersion curve of one of interacting modes (as it is depicted in Figure 3c and d) leads to some unusual effects [37]. First, zero group velocity (vg ¼ ∂k0=∂k) can appear at a nonzero value of k. The flattened region of the dispersion curve around that extreme point (Figure 3c) is very useful for applications in nonlinear optics, where a small group velocity is suitable for enhancing nonlinear effects, while the phase-matching criterion can still be satisfied because the wave vector is nonzero. Second, when interacting modes are exactly degenerated, then their group velocities appear to be nonzero and roughly constant values as k ! 0 (Figure 3d). From viewpoint of practical applications, this situation is useful, because regular modes in the vicinity of k ¼ 0 have extremely small value of their group velocity and thus become unusable. Since both dispersion and losses are inversely proportional to vg, they diverge when k ! 0.

for some particular configurations (e.g., for the Voigt geometry), and generally, the field has all six components. Such surface waves are classified as hybrid EH modes and HE modes, and these modes appear as some superposition of longitudinal and transverse waves. By analogy with [38], we classify hybrid modes depending on the magnitude ratio between the longitudinal electric and magnetic field components (Px and Pz components for the polar and Faraday geometries, respectively). For instance, for the polar configuration, it is supposed that the wave has the EH type if Ex > Hx and the HE type if Hx > Ex. Contrariwise, in the Faraday geometry, we stipulate that the wave has the EH type if Ez > Hz and the HE type if Hz > Ez. In the Voigt geometry, the waves appear as transverse electric (TE) and transverse magnetic (TM) modes,

Modal Phenomena of Surface and Bulk Polaritons in Magnetic-Semiconductor Superlattices

The polariton dispersion relations for these specific cases of magnetization can be obtained using general results derived in Section 3 with application of appropriate boundary and initial conditions.

solution of Eq. (16) for both bulk and surface waves splits apart into two independent equations for distinct polarizations [10], namely, TE modes with field components Hx; Hy; Ez

polaritons are uniquely determined by solutions of two separated equations related to the TE

0εzzμvμyyμ�<sup>1</sup>

0μzzεvεyyε�<sup>1</sup>

Under such magnetization, the dispersion equation for the surface polaritons at the interface

We should note that in two particular cases of the gyroelectric and gyromagnetic superlattices, dispersion relation (32) coincides with Eq. (33) of Ref. [10] and Eq. (21) of Ref. [15], respectively,

It follows from Figure 4b that in the Voigt geometry, the magnetic field vector in the TM mode

the absence of its interaction with the magnetic subsystem [39–41]. Thus, hereinafter dispersion features only of the TE mode are of interest for which the dispersion equation for surface

<sup>κ</sup>1μ<sup>v</sup> <sup>þ</sup> <sup>κ</sup>2μ<sup>0</sup> <sup>þ</sup> ikxμ0μxyμ�<sup>1</sup>

<sup>κ</sup>1gv <sup>þ</sup> <sup>κ</sup>2g<sup>0</sup> <sup>þ</sup> ikxg0g~xyg~�<sup>1</sup>

where substitutions μ ! g and ε ! g are related to the TE and TM modes, respectively.

has components 0f g ; 0; Hz , and it is parallel to the external magnetic field M

!

http://dx.doi.org/10.5772/intechopen.71837

. Therefore, regions of existence of bulk

xx ¼ 0, (30)

xx ¼ 0, (31)

yy ¼ 0, (32)

!

yy ¼ 0: (33)

, which results in

xy=μyy are the Voigt relative permittivity and perme-

∥z, M!

and

⊥ k ! ), 113

When an external static magnetic field is influenced in the Voigt geometry (M

k 2 <sup>x</sup> � <sup>k</sup><sup>2</sup>

k 2 <sup>x</sup> � <sup>k</sup><sup>2</sup>

xy=εyy and <sup>μ</sup><sup>v</sup> <sup>¼</sup> <sup>μ</sup>xx <sup>þ</sup> <sup>μ</sup><sup>2</sup>

between vacuum and the given structure has the form [26]:

where each of them has three field components.

TM modes with field components Ex; Ey; Hz

and the TM modes as follows [26, 38]:

which verifies the obtained solution.

polaritons (32) can be rewritten as

where <sup>ε</sup><sup>v</sup> <sup>¼</sup> <sup>ε</sup>xx <sup>þ</sup> <sup>ε</sup><sup>2</sup>

ability, respectively.

At the same time, the intermediate interaction with forming extremely flattened part in the mode dispersion curve (see Figure 3b) leads to strong divergence in density of states, and it can be utilized in designs of low-threshold lasers [37].

### 5. Dispersion features of bulk and surface polaritons for three particular orientations of magnetization

Since further our goal is to elucidate the dispersion laws of the bulk and surface polaritons (which are in fact eigenwaves), we are interested in real solutions of Eq. (24). In order to find the real solutions, the absence of losses in constitutive parameters of the underlying layers is supposed.

We consider three particular orientations of the external magnetic field M ! with respect to the superlattice's interface (the x � z plane) and to the wave vector k ! , namely, (i) the polar geometry in which the external magnetic field is applied perpendicular to both the direction of wave propagation (M ! ⊥ k ! ) and structure's interface (M ! ∥y) as shown in Figure 4a; (ii) the Voigt geometry in which the external magnetic field is applied parallel to the structure's interface, and it is perpendicular to the direction of the wave propagation, so M ! ∥z and M ! ⊥ k ! as presented in Figure 4b; (iii) the Faraday geometry in which the external magnetic field is applied parallel to both the direction of wave propagation and structure's interface, i.e., M ! ∥z and M ! ∥ k ! as presented in Figure 4c.

With respect to the problem of polaritons, in any kind of an unbounded gyrotropic medium, there are two distinct eigenwaves (the bulk waves), whereas the surface waves split apart only

Figure 4. Three particular orientations of the external magnetic field vector M ! with respect to the superlattice's interface and wave vector; (a) polar geometry, M ! ∥y, M! ⊥ k ! ; (b) Voigt geometry, M ! ∥z, M! ⊥ k ! ; and (c) Faraday geometry M ! ∥z, M ! ∥ k ! .

for some particular configurations (e.g., for the Voigt geometry), and generally, the field has all six components. Such surface waves are classified as hybrid EH modes and HE modes, and these modes appear as some superposition of longitudinal and transverse waves. By analogy with [38], we classify hybrid modes depending on the magnitude ratio between the longitudinal electric and magnetic field components (Px and Pz components for the polar and Faraday geometries, respectively). For instance, for the polar configuration, it is supposed that the wave has the EH type if Ex > Hx and the HE type if Hx > Ex. Contrariwise, in the Faraday geometry, we stipulate that the wave has the EH type if Ez > Hz and the HE type if Hz > Ez. In the Voigt geometry, the waves appear as transverse electric (TE) and transverse magnetic (TM) modes, where each of them has three field components.

when interacting modes are exactly degenerated, then their group velocities appear to be nonzero and roughly constant values as k ! 0 (Figure 3d). From viewpoint of practical applications, this situation is useful, because regular modes in the vicinity of k ¼ 0 have extremely small value of their group velocity and thus become unusable. Since both dispersion and losses

At the same time, the intermediate interaction with forming extremely flattened part in the mode dispersion curve (see Figure 3b) leads to strong divergence in density of states, and it

5. Dispersion features of bulk and surface polaritons for three particular

Since further our goal is to elucidate the dispersion laws of the bulk and surface polaritons (which are in fact eigenwaves), we are interested in real solutions of Eq. (24). In order to find the real solutions, the absence of losses in constitutive parameters of the underlying layers is

in which the external magnetic field is applied perpendicular to both the direction of wave

geometry in which the external magnetic field is applied parallel to the structure's interface,

presented in Figure 4b; (iii) the Faraday geometry in which the external magnetic field is applied

With respect to the problem of polaritons, in any kind of an unbounded gyrotropic medium, there are two distinct eigenwaves (the bulk waves), whereas the surface waves split apart only

; (b) Voigt geometry, M

parallel to both the direction of wave propagation and structure's interface, i.e., M

!

!

∥y) as shown in Figure 4a; (ii) the Voigt

!

!

!

! ∥z, M! ⊥ k ! with respect to the

, namely, (i) the polar geometry

∥z and M ! ⊥ k ! as

with respect to the superlattice's interface

; and (c) Faraday geometry M

!

∥z and

! ∥z,

We consider three particular orientations of the external magnetic field M

) and structure's interface (M

and it is perpendicular to the direction of the wave propagation, so M

superlattice's interface (the x � z plane) and to the wave vector k

Figure 4. Three particular orientations of the external magnetic field vector M

! ∥y, M! ⊥ k !

are inversely proportional to vg, they diverge when k ! 0.

can be utilized in designs of low-threshold lasers [37].

orientations of magnetization

112 Surface Waves - New Trends and Developments

supposed.

propagation (M

M ! ∥ k !

M ! ∥ k ! . ! ⊥ k !

as presented in Figure 4c.

and wave vector; (a) polar geometry, M

The polariton dispersion relations for these specific cases of magnetization can be obtained using general results derived in Section 3 with application of appropriate boundary and initial conditions.

When an external static magnetic field is influenced in the Voigt geometry (M ! ∥z, M! ⊥ k ! ), solution of Eq. (16) for both bulk and surface waves splits apart into two independent equations for distinct polarizations [10], namely, TE modes with field components Hx; Hy; Ez and TM modes with field components Ex; Ey; Hz . Therefore, regions of existence of bulk polaritons are uniquely determined by solutions of two separated equations related to the TE and the TM modes as follows [26, 38]:

$$k\_x^2 - k\_0^2 \varepsilon\_{zz} \mu\_v \mu\_{yy} \mu\_{xx}^{-1} = 0,\tag{30}$$

$$k\_x^2 - k\_0^2 \mu\_{zz} \varepsilon\_v \varepsilon\_{yy} \varepsilon\_{xx}^{-1} = 0,\tag{31}$$

where <sup>ε</sup><sup>v</sup> <sup>¼</sup> <sup>ε</sup>xx <sup>þ</sup> <sup>ε</sup><sup>2</sup> xy=εyy and <sup>μ</sup><sup>v</sup> <sup>¼</sup> <sup>μ</sup>xx <sup>þ</sup> <sup>μ</sup><sup>2</sup> xy=μyy are the Voigt relative permittivity and permeability, respectively.

Under such magnetization, the dispersion equation for the surface polaritons at the interface between vacuum and the given structure has the form [26]:

$$
\kappa\_1 \mathbf{g}\_v + \kappa\_2 \mathbf{g}\_0 + \mathrm{ik}\_x \mathbf{g}\_0 \tilde{\mathbf{g}}\_{xy} \tilde{\mathbf{g}}\_{yy}^{-1} = \mathbf{0},\tag{32}
$$

where substitutions μ ! g and ε ! g are related to the TE and TM modes, respectively.

We should note that in two particular cases of the gyroelectric and gyromagnetic superlattices, dispersion relation (32) coincides with Eq. (33) of Ref. [10] and Eq. (21) of Ref. [15], respectively, which verifies the obtained solution.

It follows from Figure 4b that in the Voigt geometry, the magnetic field vector in the TM mode has components 0f g ; 0; Hz , and it is parallel to the external magnetic field M ! , which results in the absence of its interaction with the magnetic subsystem [39–41]. Thus, hereinafter dispersion features only of the TE mode are of interest for which the dispersion equation for surface polaritons (32) can be rewritten as

$$
\kappa \kappa\_1 \mu\_v + \kappa\_2 \mu\_0 + i k\_x \mu\_0 \mu\_{xy} \mu\_{yy}^{-1} = 0. \tag{33}
$$

Importantly, since the dispersion equation consists of a term which is linearly depended on kx (the last term in (33)), the spectral characteristics of the surface polaritons in the structure under study possess the nonreciprocal nature, i.e., k0ð Þ kx 6¼ k0ð Þ �kx .

In the polar geometry (M

where <sup>ε</sup><sup>v</sup> <sup>¼</sup> <sup>ε</sup>xx <sup>þ</sup> <sup>ε</sup><sup>2</sup>

E !

the form:

where ϰ<sup>2</sup>

and vector H

obtained solution.

κ2 0g~xxg�<sup>1</sup> <sup>0</sup> κ<sup>2</sup>

<sup>þ</sup> <sup>g</sup>~xzς�<sup>1</sup>

<sup>ν</sup> ¼ k 2 <sup>x</sup> � <sup>k</sup><sup>2</sup>

!

!

waves with field components Ex; Hy; Ez

ability and permittivity, respectively.

tinua is outlined by two dispersion relations:

∥y, M!

k 2 <sup>x</sup> � k 2

<sup>1</sup> <sup>þ</sup> <sup>κ</sup>1κ<sup>2</sup> <sup>þ</sup> <sup>κ</sup><sup>2</sup>

<sup>z</sup> � <sup>ϰ</sup><sup>2</sup>

<sup>þ</sup> <sup>κ</sup>0ð Þ <sup>κ</sup><sup>1</sup> <sup>þ</sup> <sup>κ</sup><sup>2</sup> <sup>κ</sup>1κ<sup>2</sup> <sup>þ</sup> <sup>ϰ</sup><sup>2</sup>

xz ϰ<sup>4</sup>

xz=εxx and <sup>μ</sup><sup>v</sup> <sup>¼</sup> <sup>μ</sup>xx <sup>þ</sup> <sup>μ</sup><sup>2</sup>

decomposed into two particular solutions with respect to the vector H

<sup>2</sup> � <sup>ϰ</sup><sup>2</sup> z

<sup>z</sup> ð Þþ <sup>κ</sup><sup>1</sup> <sup>þ</sup> <sup>κ</sup><sup>2</sup> <sup>κ</sup><sup>2</sup>

� � <sup>þ</sup> <sup>ϰ</sup><sup>2</sup>

<sup>y</sup>g~xxgv <sup>g</sup>0ςyy � ��<sup>1</sup> h i

<sup>y</sup>ςxzg~xz <sup>ς</sup>yyg~xx � ��<sup>1</sup> h i

1κ2 2

εvv<sup>0</sup> ! g~vv<sup>0</sup> , μ<sup>0</sup> ! g0, μ<sup>v</sup> ! gv correspond to the problem resolving with respect to the vector E

Complete sets of dispersion curves calculated from the solution of Eqs. (34) and (35) that outline the passbands of both ordinary (blue curves) and extraordinary (red curves) bulk polaritons as functions of the filling factor δ<sup>m</sup> are presented in Figure 6a and b. Moreover, in order to discuss the observed crossing and anti-crossing effects more clearly, dispersion curves of both ordinary and extraordinary bulk polaritons are plotted in the k<sup>0</sup> � kx plane at the

From Figure 6a one can conclude that there are two isolated areas of existence of the ordinary bulk polaritons. The upper passband starts at the frequency where εyy ¼ 0, while the bottom passband is bounded above by the asymptotic line where εyy ! ∞. Remarkably, the anti-

, respectively; ε<sup>v</sup> and μ<sup>v</sup> are the same shown in Eqs. (34) and (35). For two particular cases of the gyroelectric and gyromagnetic superlattices, dispersion relation (36) coincides with Eq. (23) of Ref. [10] and Eq. (21) of Ref. [13], respectively, which verifies the

� � <sup>þ</sup> <sup>ϰ</sup><sup>2</sup>

particular values of filling factor δ<sup>m</sup> as shown in Figure 6c–f.

⊥ k !

> k2 <sup>x</sup> � k 2

whose optical axis is directed along the y-axis. In this case, bulk polaritons split onto two

Hereinafter, we distinguish these two kinds of waves as ordinary and extraordinary bulk

In order to elucidate the dispersion features of hybrid surface polaritons, the initial problem is

(HE modes) [10, 42]. In this way, the dispersion equation for surface polaritons is derived in

<sup>y</sup> ϰ<sup>2</sup>

� �ςxzg~zz <sup>ς</sup>yyg~xz � ��<sup>1</sup> h i <sup>¼</sup> <sup>0</sup>,

<sup>z</sup> þ κ1κ<sup>2</sup>

<sup>0</sup>ςνν and two distinct substitutions εvv<sup>0</sup> ! g~vv<sup>0</sup> , ε<sup>0</sup> ! g0, ε<sup>v</sup> ! gv and

polaritons, respectively (note: such a definition is common in the plasma physics [43]).

� � and Hx; Ey; Hz

), composite medium under study is a uniaxial crystal

Modal Phenomena of Surface and Bulk Polaritons in Magnetic-Semiconductor Superlattices

� �, respectively [40], and their con-

http://dx.doi.org/10.5772/intechopen.71837

<sup>0</sup>μyyεyy ¼ 0, (34)

xz=μxx are introduced as the effective bulk perme-

!

(EH modes) and vector

(36)

115

!

<sup>0</sup>μvε<sup>v</sup> <sup>1</sup> � <sup>μ</sup>xzεxz<sup>=</sup> <sup>μ</sup>xxεxx � � � � �<sup>1</sup> <sup>¼</sup> <sup>0</sup>, (35)

A complete set of dispersion curves obtained from solution of Eq. (30) that outlines the passbands of the TE bulk polaritons as a function of the filling factor δ<sup>m</sup> is presented in Figure 5a. One can see that behaviors of the TE bulk polaritons are quite trivial in the overwhelming majority of structure's configurations, namely, there are two isolated passbands separated by a forbidden band. The upper passband is bounded laterally by the light line, and its lower limit is restricted by the line at which μ<sup>v</sup> ¼ 0. The bottom passband starts on the line where εzz ¼ 0 and then approaches the asymptotic frequency (kx ! ∞) at which μxx ¼ 0 (see Eq. (25)).

In this study, we are mainly interested in those curves of the set which have greatly sloping branches and exhibit the closest approaching each other (i.e., they manifest the anti-crossing effect) or have a crossing point, since such dispersion behaviors correspond to the existence of the Morse critical points. Hereinafter, the areas of interest in which these extreme states exist are denoted in figures by orange circles.

It follows from Figure 5a that in the Voigt geometry, both the anti-crossing (H < 0) and crossing ðH ¼ 0) effects with forming the bottom branch characterized by the anomalous dispersion (i.e., the strong mode interaction occurs) can be achieved in the composite structure with a predominant impact of the semiconductor subsystem (further we stipulate that the composite system has a predominant impact of either magnetic subsystem or semiconductor subsystem if δ<sup>m</sup> ≫ δ<sup>s</sup> or δ<sup>s</sup> ≫ δm, respectively). Moreover, since condition (29) is met near the critical point, the bulk waves appear to be contradirectional.

The crossing effect is found to be at kx ! 0 (accidentally degenerate modes) for the particular configuration of the structure, when δ<sup>m</sup> ¼ 0:132 (Figure 5b). Remarkably, such an extreme state corresponds to a particular frequency where εzz and μ<sup>v</sup> simultaneously acquire zero [26].

Figure 5. (a) A set of dispersion curves of the TE bulk polaritons for different filling factors δ<sup>m</sup> for the structure being in the Voigt geometry and (b) manifestation of crossing effect in dispersion curves of the TE bulk polaritons at the particular value of filling factor δ<sup>m</sup> ¼ 0:132. Parameters of magnetic constitutive layers are shown in Figure 2. For the semiconductor constitutive layers, parameters are f <sup>p</sup> ¼ ωp=2π ¼ 10:5 GHz, f <sup>c</sup> ¼ ωc=2π ¼ 9:5 GHz, ν ¼ 0, ε<sup>l</sup> ¼ 1:0, and μ<sup>s</sup> ¼ 1:0.

In the polar geometry (M ! ∥y, M! ⊥ k ! ), composite medium under study is a uniaxial crystal whose optical axis is directed along the y-axis. In this case, bulk polaritons split onto two waves with field components Ex; Hy; Ez � � and Hx; Ey; Hz � �, respectively [40], and their continua is outlined by two dispersion relations:

Importantly, since the dispersion equation consists of a term which is linearly depended on kx (the last term in (33)), the spectral characteristics of the surface polaritons in the structure

A complete set of dispersion curves obtained from solution of Eq. (30) that outlines the passbands of the TE bulk polaritons as a function of the filling factor δ<sup>m</sup> is presented in Figure 5a. One can see that behaviors of the TE bulk polaritons are quite trivial in the overwhelming majority of structure's configurations, namely, there are two isolated passbands separated by a forbidden band. The upper passband is bounded laterally by the light line, and its lower limit is restricted by the line at which μ<sup>v</sup> ¼ 0. The bottom passband starts on the line where εzz ¼ 0 and then approaches the asymptotic frequency (kx ! ∞) at which μxx ¼ 0 (see

In this study, we are mainly interested in those curves of the set which have greatly sloping branches and exhibit the closest approaching each other (i.e., they manifest the anti-crossing effect) or have a crossing point, since such dispersion behaviors correspond to the existence of the Morse critical points. Hereinafter, the areas of interest in which these extreme states exist

It follows from Figure 5a that in the Voigt geometry, both the anti-crossing (H < 0) and crossing ðH ¼ 0) effects with forming the bottom branch characterized by the anomalous dispersion (i.e., the strong mode interaction occurs) can be achieved in the composite structure with a predominant impact of the semiconductor subsystem (further we stipulate that the composite system has a predominant impact of either magnetic subsystem or semiconductor subsystem if δ<sup>m</sup> ≫ δ<sup>s</sup> or δ<sup>s</sup> ≫ δm, respectively). Moreover, since condition (29) is met near the

The crossing effect is found to be at kx ! 0 (accidentally degenerate modes) for the particular configuration of the structure, when δ<sup>m</sup> ¼ 0:132 (Figure 5b). Remarkably, such an extreme state corresponds to a particular frequency where εzz and μ<sup>v</sup> simultaneously acquire zero [26].

Figure 5. (a) A set of dispersion curves of the TE bulk polaritons for different filling factors δ<sup>m</sup> for the structure being in the Voigt geometry and (b) manifestation of crossing effect in dispersion curves of the TE bulk polaritons at the particular value of filling factor δ<sup>m</sup> ¼ 0:132. Parameters of magnetic constitutive layers are shown in Figure 2. For the semiconductor constitutive layers, parameters are f <sup>p</sup> ¼ ωp=2π ¼ 10:5 GHz, f <sup>c</sup> ¼ ωc=2π ¼ 9:5 GHz, ν ¼ 0, ε<sup>l</sup> ¼ 1:0, and μ<sup>s</sup> ¼ 1:0.

under study possess the nonreciprocal nature, i.e., k0ð Þ kx 6¼ k0ð Þ �kx .

Eq. (25)).

are denoted in figures by orange circles.

114 Surface Waves - New Trends and Developments

critical point, the bulk waves appear to be contradirectional.

$$k\_{\pi}^{2} - k\_{0}^{2} \mu\_{yy} \varepsilon\_{yy} = 0,\tag{34}$$

$$k\_x^2 - k\_0^2 \mu\_v \varepsilon\_v \left[1 - \mu\_{xz} \varepsilon\_{xz} / \left(\mu\_{xx} \varepsilon\_{xx}\right)\right]^{-1} = 0,\tag{35}$$

where <sup>ε</sup><sup>v</sup> <sup>¼</sup> <sup>ε</sup>xx <sup>þ</sup> <sup>ε</sup><sup>2</sup> xz=εxx and <sup>μ</sup><sup>v</sup> <sup>¼</sup> <sup>μ</sup>xx <sup>þ</sup> <sup>μ</sup><sup>2</sup> xz=μxx are introduced as the effective bulk permeability and permittivity, respectively.

Hereinafter, we distinguish these two kinds of waves as ordinary and extraordinary bulk polaritons, respectively (note: such a definition is common in the plasma physics [43]).

In order to elucidate the dispersion features of hybrid surface polaritons, the initial problem is decomposed into two particular solutions with respect to the vector H ! (EH modes) and vector E ! (HE modes) [10, 42]. In this way, the dispersion equation for surface polaritons is derived in the form:

$$\begin{split} &\kappa\_{0}^{2}\tilde{g}\_{xx}\mathbf{g}\_{0}^{-1}\left[\left(\kappa\_{1}^{2}+\kappa\_{1}\kappa\_{2}+\kappa\_{2}^{2}-\varkappa\_{z}^{2}\right)+\varkappa\_{y}^{2}\boldsymbol{\varsigma}\_{xz}\tilde{g}\_{xx}\left(\boldsymbol{\varsigma}\_{yy}\tilde{g}\_{xx}\right)^{-1}\right] \\ &+\kappa\_{0}(\kappa\_{1}+\kappa\_{2})\left[\kappa\_{1}\kappa\_{2}+\varkappa\_{y}^{2}\tilde{g}\_{xx}\mathbf{g}\_{v}\left(\boldsymbol{g}\_{0}\boldsymbol{\varsigma}\_{yy}\right)^{-1}\right] \\ &+\tilde{g}\_{xz}\boldsymbol{\varsigma}\_{zz}^{-1}\left[\left(\varkappa\_{z}^{4}-\varkappa\_{z}^{2}(\kappa\_{1}+\kappa\_{2})+\varkappa\_{1}^{2}\kappa\_{2}^{2}\right)+\varkappa\_{y}^{2}\left(\varkappa\_{z}^{2}+\kappa\_{1}\kappa\_{2}\right)\boldsymbol{\varsigma}\_{xz}\tilde{g}\_{zz}\left(\boldsymbol{\varsigma}\_{yy}\tilde{g}\_{xx}\right)^{-1}\right] = 0,\end{split} \tag{36}$$

where ϰ<sup>2</sup> <sup>ν</sup> ¼ k 2 <sup>x</sup> � k 2 <sup>0</sup>ςνν and two distinct substitutions εvv<sup>0</sup> ! g~vv<sup>0</sup> , ε<sup>0</sup> ! g0, ε<sup>v</sup> ! gv and εvv<sup>0</sup> ! g~vv<sup>0</sup> , μ<sup>0</sup> ! g0, μ<sup>v</sup> ! gv correspond to the problem resolving with respect to the vector E ! and vector H ! , respectively; ε<sup>v</sup> and μ<sup>v</sup> are the same shown in Eqs. (34) and (35).

For two particular cases of the gyroelectric and gyromagnetic superlattices, dispersion relation (36) coincides with Eq. (23) of Ref. [10] and Eq. (21) of Ref. [13], respectively, which verifies the obtained solution.

Complete sets of dispersion curves calculated from the solution of Eqs. (34) and (35) that outline the passbands of both ordinary (blue curves) and extraordinary (red curves) bulk polaritons as functions of the filling factor δ<sup>m</sup> are presented in Figure 6a and b. Moreover, in order to discuss the observed crossing and anti-crossing effects more clearly, dispersion curves of both ordinary and extraordinary bulk polaritons are plotted in the k<sup>0</sup> � kx plane at the particular values of filling factor δ<sup>m</sup> as shown in Figure 6c–f.

From Figure 6a one can conclude that there are two isolated areas of existence of the ordinary bulk polaritons. The upper passband starts at the frequency where εyy ¼ 0, while the bottom passband is bounded above by the asymptotic line where εyy ! ∞. Remarkably, the anti-

dispersion curves is peculiar (see, for instance, Figure 3b). Besides, condition (27) is met, so

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As shown in Figure 6b, there are two separated areas of existence of extraordinary bulk polaritons for each particular filling factor δm. Moreover, there are two possible combinations of conditions for their passbands, which depend strictly on the value of filling factor δ<sup>m</sup> [42]. Therefore, the particular critical filling factor is denoted here as δ<sup>c</sup> at which ε<sup>v</sup> ¼ μ<sup>v</sup> ¼ 0 at kx ¼ 0, and it is considered as a separation point between these two combinations. Thus, the overall range of values δ<sup>m</sup> can be separated onto two subranges 0 ≤ δ<sup>m</sup> < δ<sup>c</sup> and δ<sup>c</sup> < δ<sup>m</sup> ≤ 1,

The value of the critical filling factor δ<sup>c</sup> depends on the constitutive parameters of magnetic and semiconductor layers, and for the structure under study, it is δ<sup>с</sup> ¼ 0:267. Such critical state in the effective parameters of the superlattice leads to the absence of the forbidden band between the upper and bottom passbands as shown in Figure 6c. Remarkably, these dispersion branches appear to be coupled exactly at the frequency where ε<sup>v</sup> ¼ μ<sup>v</sup> ¼ 0, i.e., the crossing effect (H ¼ 0) is found to be at kx ! 0. Besides, such coupled waves are contradirectional, and

At the same time, for all present values of filling factor δ<sup>m</sup> from the first subrange 0 ≤ δ<sup>m</sup> < δc, the lower limits of both the upper and bottom passbands are restricted by the lines at which ε<sup>v</sup> ¼ 0, as well as the bottom passband is bounded above by the asymptotic line where ε<sup>v</sup> ! ∞. The upper passband is outlined by the set of following conditions [40]: ε<sup>v</sup> > 0, μ<sup>v</sup> > 0, εxzμxz=εxxμxx < 1. The bottom passband exists when the set of following conditions holds

For all other values of filling factor δ<sup>m</sup> (i.e., δ<sup>c</sup> < δ<sup>m</sup> ≤ 1), the conditions for the upper passband are the same (i.e., ε<sup>v</sup> > 0, μ<sup>v</sup> > 0, εxzμxz=εxxμxx < 1), whereas its lower limit is at the line where μ<sup>v</sup> ¼ 0. The bottom passband exists when either the set of conditions ε<sup>v</sup> < 0, μ<sup>v</sup> > 0 and εxzμxz=εxxμxx > 1 or ε<sup>v</sup> > 0, μ<sup>v</sup> > 0 and εxzμxz=εxxμxx < 1 is satisfied. For this band, the lower

From Figure 6a and b, one can conclude that the upper passband possesses typical behaviors in the k<sup>0</sup> � kx plane, whereas the width and position of the bottom passbands are defined by the corresponding resonant frequencies of effective bulk permeability μ<sup>v</sup> and effective bulk permittivity εv. In fact, these constitutive parameters are multipliers of the numerator of Eq. (35), whereas the denominator of this equation originates a singularity at the asymptotic line where 1 � εxzμxz=εxxμxx ! ∞. This asymptotic line splits the bottom passbands onto two separated sub-passbands which corresponds to the structures with a predominant impact of semiconductor and magnetic subsystems, respectively (Figure 6e and f). Note, the intermediate interaction with forming flattened region in dispersion curves between bulk modes from these two sub-bands is observed nearly the Morse critical points (H < 0). Such coupled waves

ropy, for which, as already mentioned, the first anisotropy axis is associated with the structure

), the superlattice is characterized by the biaxial anisot-

[42]: ε<sup>v</sup> > 0, μ<sup>v</sup> < 0 and εxzμxz=εxxμxx > 1 or ε<sup>v</sup> > 0, μ<sup>v</sup> > 0 and εxzμxz=εxxμxx < 1.

and upper limits are restricted by the lines at which μ<sup>v</sup> ¼ 0 and μ<sup>v</sup> ! ∞, respectively.

these waves are codirectional forward in the area nearly the critical point.

respectively.

they are described by condition (29).

are contradirectional since condition (29) is met.

! ∥z, M ! ∥ k !

In the Faraday geometry (M

Figure 6. Complete sets of dispersion curves of (a) ordinary (blue curves) and (b) extraordinary (red curves) bulk polaritons for different filling factor δ<sup>m</sup> for the structure being in the polar geometry. Manifestation of (c) crossing and (d–f) anti-crossing effects in dispersion curves of bulk polaritons at the particular value of filling factor δm, (c) δ<sup>m</sup> ¼ 0:267, (d) δ<sup>m</sup> ¼ 0:99, (e) δ<sup>m</sup> ¼ 0:05, and (f) δ<sup>m</sup> ¼ 0:99. All structure constitutive parameters are shown in Figure 2.

crossing effect (H < 0) between dispersion curves which restrict upper and bottom passbands of ordinary bulk polaritons can be observed in the composite structure with a predominant impact of the magnetic subsystem (i.e., δ<sup>m</sup> ≫ δs) as presented in Figure 6a and d. Coupled modes exhibit an intermediate interaction for which an appearance of the flattened branches in dispersion curves is peculiar (see, for instance, Figure 3b). Besides, condition (27) is met, so these waves are codirectional forward in the area nearly the critical point.

As shown in Figure 6b, there are two separated areas of existence of extraordinary bulk polaritons for each particular filling factor δm. Moreover, there are two possible combinations of conditions for their passbands, which depend strictly on the value of filling factor δ<sup>m</sup> [42]. Therefore, the particular critical filling factor is denoted here as δ<sup>c</sup> at which ε<sup>v</sup> ¼ μ<sup>v</sup> ¼ 0 at kx ¼ 0, and it is considered as a separation point between these two combinations. Thus, the overall range of values δ<sup>m</sup> can be separated onto two subranges 0 ≤ δ<sup>m</sup> < δ<sup>c</sup> and δ<sup>c</sup> < δ<sup>m</sup> ≤ 1, respectively.

The value of the critical filling factor δ<sup>c</sup> depends on the constitutive parameters of magnetic and semiconductor layers, and for the structure under study, it is δ<sup>с</sup> ¼ 0:267. Such critical state in the effective parameters of the superlattice leads to the absence of the forbidden band between the upper and bottom passbands as shown in Figure 6c. Remarkably, these dispersion branches appear to be coupled exactly at the frequency where ε<sup>v</sup> ¼ μ<sup>v</sup> ¼ 0, i.e., the crossing effect (H ¼ 0) is found to be at kx ! 0. Besides, such coupled waves are contradirectional, and they are described by condition (29).

At the same time, for all present values of filling factor δ<sup>m</sup> from the first subrange 0 ≤ δ<sup>m</sup> < δc, the lower limits of both the upper and bottom passbands are restricted by the lines at which ε<sup>v</sup> ¼ 0, as well as the bottom passband is bounded above by the asymptotic line where ε<sup>v</sup> ! ∞. The upper passband is outlined by the set of following conditions [40]: ε<sup>v</sup> > 0, μ<sup>v</sup> > 0, εxzμxz=εxxμxx < 1. The bottom passband exists when the set of following conditions holds [42]: ε<sup>v</sup> > 0, μ<sup>v</sup> < 0 and εxzμxz=εxxμxx > 1 or ε<sup>v</sup> > 0, μ<sup>v</sup> > 0 and εxzμxz=εxxμxx < 1.

For all other values of filling factor δ<sup>m</sup> (i.e., δ<sup>c</sup> < δ<sup>m</sup> ≤ 1), the conditions for the upper passband are the same (i.e., ε<sup>v</sup> > 0, μ<sup>v</sup> > 0, εxzμxz=εxxμxx < 1), whereas its lower limit is at the line where μ<sup>v</sup> ¼ 0. The bottom passband exists when either the set of conditions ε<sup>v</sup> < 0, μ<sup>v</sup> > 0 and εxzμxz=εxxμxx > 1 or ε<sup>v</sup> > 0, μ<sup>v</sup> > 0 and εxzμxz=εxxμxx < 1 is satisfied. For this band, the lower and upper limits are restricted by the lines at which μ<sup>v</sup> ¼ 0 and μ<sup>v</sup> ! ∞, respectively.

From Figure 6a and b, one can conclude that the upper passband possesses typical behaviors in the k<sup>0</sup> � kx plane, whereas the width and position of the bottom passbands are defined by the corresponding resonant frequencies of effective bulk permeability μ<sup>v</sup> and effective bulk permittivity εv. In fact, these constitutive parameters are multipliers of the numerator of Eq. (35), whereas the denominator of this equation originates a singularity at the asymptotic line where 1 � εxzμxz=εxxμxx ! ∞. This asymptotic line splits the bottom passbands onto two separated sub-passbands which corresponds to the structures with a predominant impact of semiconductor and magnetic subsystems, respectively (Figure 6e and f). Note, the intermediate interaction with forming flattened region in dispersion curves between bulk modes from these two sub-bands is observed nearly the Morse critical points (H < 0). Such coupled waves are contradirectional since condition (29) is met.

In the Faraday geometry (M ! ∥z, M ! ∥ k ! ), the superlattice is characterized by the biaxial anisotropy, for which, as already mentioned, the first anisotropy axis is associated with the structure

crossing effect (H < 0) between dispersion curves which restrict upper and bottom passbands of ordinary bulk polaritons can be observed in the composite structure with a predominant impact of the magnetic subsystem (i.e., δ<sup>m</sup> ≫ δs) as presented in Figure 6a and d. Coupled modes exhibit an intermediate interaction for which an appearance of the flattened branches in

Figure 6. Complete sets of dispersion curves of (a) ordinary (blue curves) and (b) extraordinary (red curves) bulk polaritons for different filling factor δ<sup>m</sup> for the structure being in the polar geometry. Manifestation of (c) crossing and (d–f) anti-crossing effects in dispersion curves of bulk polaritons at the particular value of filling factor δm, (c) δ<sup>m</sup> ¼ 0:267,

(d) δ<sup>m</sup> ¼ 0:99, (e) δ<sup>m</sup> ¼ 0:05, and (f) δ<sup>m</sup> ¼ 0:99. All structure constitutive parameters are shown in Figure 2.

116 Surface Waves - New Trends and Developments

periodicity (so it is directed along the y-axis), while the second anisotropy axis is a result of the external static magnetic field influence (so it is directed along the z-axis). For this geometry, the bulk polaritons in the structure under study appear as right-handed and left-handed elliptically polarized waves [10, 12, 44], and their passbands are outlined by curves governed by the following dispersion law [44]:

$$
\lambda \varkappa\_x^2 \varkappa\_y^2 - k\_0^4 \varepsilon\_{xy} \varepsilon\_{yx} = 0,\tag{37}
$$

where ϰ<sup>2</sup> <sup>ν</sup> ¼ k 2 <sup>z</sup> � k 2 <sup>0</sup>ςνν.

The surface polaritons are hybrid EH and HE waves, and their dispersion relation can be written in the form [44]:

$$\begin{aligned} \left(\kappa\_2 + \kappa\_0 \tilde{g}\_{zz}\right) \left(\kappa\_2^2 \varepsilon\_{yy} - \kappa\_y^2 \varsigma\_{zz}\right) \left\{\kappa\_1 \psi \left(\kappa\_0^2 - k\_z^2\right) + \kappa\_0 \left[\kappa\_1^2 \tilde{g}\_{xy} \varsigma\_{yy} - \varsigma\_{zz} \left(k\_0^2 \rho + k\_z^2 \tilde{g}\_{xy}\right)\right]\right\} \\ - \left(\kappa\_1 + \kappa\_0 \tilde{g}\_{zz}\right) \left(\kappa\_1^2 \varsigma\_{yy} - \kappa\_y^2 \varsigma\_{zz}\right) \left\{\kappa\_2 \psi \left(\kappa\_0^2 - k\_z^2\right) + \kappa\_0 \left[\kappa\_2^2 \tilde{g}\_{xy} \varsigma\_{yy} - \varsigma\_{zz} \left(k\_0^2 \rho + k\_z^2 \tilde{g}\_{xy}\right)\right]\right\} = 0. \end{aligned} \tag{38}$$

In Eq. (38), two distinct substitutions εvv<sup>0</sup> ! g~vv<sup>0</sup> , ε<sup>v</sup> ! gv and μvv<sup>0</sup> ! g~vv<sup>0</sup> , μ<sup>v</sup> ! gv correspond to the problem resolving with respect to the vector E ! and vector H ! , respectively; <sup>ε</sup><sup>v</sup> <sup>¼</sup> <sup>ε</sup>xx <sup>þ</sup> <sup>ε</sup><sup>2</sup> xy=εyy, <sup>μ</sup><sup>v</sup> <sup>¼</sup> <sup>μ</sup>xx <sup>þ</sup> <sup>μ</sup><sup>2</sup> xy=μyy, ψ ¼ gvg~yyςyx; r ¼ g~yyςyx � g~xyςyy; and the constant g<sup>0</sup> ¼ 1 is omitted.

For the semiconductor superlattice, the dispersion Eq. (38) agrees with Eq. (38) of Ref. [10], while in the case of magnetic superlattice, it coincides with Eq. (13) of Ref. [12], which verifies the obtained solution.

Complete sets of dispersion curves that outline the bands of existence of the bulk polaritons (see Eq. (37)) as functions of filling factor δ<sup>m</sup> are presented in Figure 7a and b for right-handed (blue curves) and left-handed (red curves) elliptical polarizations. From these figures, one can conclude that there is a pair of corresponding sets of dispersion curves separated by a forbidden band for the bulk polaritons of each polarization. The dispersion curves of the righthanded elliptically polarized bulk waves demonstrate quite trivial behaviors, and they completely inherit characteristics of the right-handed circularly polarized waves of the corresponding reference semiconductor or magnetic medium (see, for instance [44]). Contrariwise, the dispersion features of the left-handed elliptically polarized waves are much more complicated being strongly dependent on filling factor δ<sup>m</sup> and resonant frequencies of constitutive parameters of both semiconductor and magnetic underlying materials. Therefore, in what follows we are interested only in the consideration of dispersion features of the bulk polaritons having the left-handed polarization.

One can conclude that the dispersion characteristics of the left-handed elliptically polarized bulk waves of the given gyroelectromagnetic structure are different from those ones of both convenient gyroelectric and gyromagnetic media. Indeed, in contrast to the characteristics of the left-handed circularly polarized waves of the corresponding reference medium whose passband has no discontinuity, the passband of the left-handed elliptically polarized bulk

polaritons of the superlattice is separated into two distinct areas. This separation appears nearly the frequency at which the resonances of the functions εxy and μxy occur for the composite structures with predominant impact of the magnetic and semiconductor subsystems,

Figure 7. (a, b) Complete sets of dispersion curves of both right-handed (blue curves) and left-handed (red curves) elliptically polarized bulk polaritons for different filling factor δ<sup>m</sup> for the structure being in the Faraday geometry. Manifestation of (c, e) anti-crossing and (d) crossing effects in dispersion curves of bulk polaritons at the particular value of filling factor δm; (c) δ<sup>m</sup> ¼ 0:15; (d) δ<sup>m</sup> ¼ 0:1; and (e) δ<sup>m</sup> ¼ 0:95. All structure constitutive parameters are shown in Figure 2.

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0 0

Modal Phenomena of Surface and Bulk Polaritons in Magnetic-Semiconductor Superlattices http://dx.doi.org/10.5772/intechopen.71837 119

periodicity (so it is directed along the y-axis), while the second anisotropy axis is a result of the external static magnetic field influence (so it is directed along the z-axis). For this geometry, the bulk polaritons in the structure under study appear as right-handed and left-handed elliptically polarized waves [10, 12, 44], and their passbands are outlined by curves governed by the

The surface polaritons are hybrid EH and HE waves, and their dispersion relation can be

In Eq. (38), two distinct substitutions εvv<sup>0</sup> ! g~vv<sup>0</sup> , ε<sup>v</sup> ! gv and μvv<sup>0</sup> ! g~vv<sup>0</sup> , μ<sup>v</sup> ! gv correspond

For the semiconductor superlattice, the dispersion Eq. (38) agrees with Eq. (38) of Ref. [10], while in the case of magnetic superlattice, it coincides with Eq. (13) of Ref. [12], which verifies

Complete sets of dispersion curves that outline the bands of existence of the bulk polaritons (see Eq. (37)) as functions of filling factor δ<sup>m</sup> are presented in Figure 7a and b for right-handed (blue curves) and left-handed (red curves) elliptical polarizations. From these figures, one can conclude that there is a pair of corresponding sets of dispersion curves separated by a forbidden band for the bulk polaritons of each polarization. The dispersion curves of the righthanded elliptically polarized bulk waves demonstrate quite trivial behaviors, and they completely inherit characteristics of the right-handed circularly polarized waves of the corresponding reference semiconductor or magnetic medium (see, for instance [44]). Contrariwise, the dispersion features of the left-handed elliptically polarized waves are much more complicated being strongly dependent on filling factor δ<sup>m</sup> and resonant frequencies of constitutive parameters of both semiconductor and magnetic underlying materials. Therefore, in what follows we are interested only in the consideration of dispersion features of the bulk

One can conclude that the dispersion characteristics of the left-handed elliptically polarized bulk waves of the given gyroelectromagnetic structure are different from those ones of both convenient gyroelectric and gyromagnetic media. Indeed, in contrast to the characteristics of the left-handed circularly polarized waves of the corresponding reference medium whose passband has no discontinuity, the passband of the left-handed elliptically polarized bulk

<sup>0</sup> � k 2 z � � <sup>þ</sup> <sup>κ</sup><sup>0</sup> <sup>κ</sup><sup>2</sup>

<sup>0</sup> � k 2 z � � <sup>þ</sup> <sup>κ</sup><sup>0</sup> <sup>κ</sup><sup>2</sup>

<sup>0</sup>ςxyςyx ¼ 0, (37)

2 <sup>0</sup>r þ k 2 <sup>z</sup>g~xy

<sup>0</sup>r þ k 2 <sup>z</sup>g~xy

and vector H

!

¼ 0:

, respectively;

(38)

<sup>1</sup>g~xyςyy � ςzz k

<sup>2</sup>g~xyςyy � <sup>ς</sup>zz <sup>k</sup><sup>2</sup>

!

xy=μyy, ψ ¼ gvg~yyςyx; r ¼ g~yyςyx � g~xyςyy; and the constant

n o h i � �

n o h i � �

ϰ2 xϰ<sup>2</sup> <sup>y</sup> � k 4

κ1ψ κ<sup>2</sup>

κ2ψ κ<sup>2</sup>

following dispersion law [44]:

118 Surface Waves - New Trends and Developments

where ϰ<sup>2</sup>

<sup>ν</sup> ¼ k 2 <sup>z</sup> � k 2 <sup>0</sup>ςνν.

written in the form [44]:

κ<sup>2</sup> þ κ0g~zz � � κ<sup>2</sup>

� κ<sup>1</sup> þ κ0g~zz � � κ<sup>2</sup>

<sup>ε</sup><sup>v</sup> <sup>¼</sup> <sup>ε</sup>xx <sup>þ</sup> <sup>ε</sup><sup>2</sup>

g<sup>0</sup> ¼ 1 is omitted.

the obtained solution.

<sup>2</sup>ςyy � <sup>ϰ</sup><sup>2</sup>

<sup>1</sup>ςyy � <sup>ϰ</sup><sup>2</sup>

xy=εyy, <sup>μ</sup><sup>v</sup> <sup>¼</sup> <sup>μ</sup>xx <sup>þ</sup> <sup>μ</sup><sup>2</sup>

polaritons having the left-handed polarization.

� �

� �

<sup>y</sup>ςzz

to the problem resolving with respect to the vector E

<sup>y</sup>ςzz

Figure 7. (a, b) Complete sets of dispersion curves of both right-handed (blue curves) and left-handed (red curves) elliptically polarized bulk polaritons for different filling factor δ<sup>m</sup> for the structure being in the Faraday geometry. Manifestation of (c, e) anti-crossing and (d) crossing effects in dispersion curves of bulk polaritons at the particular value of filling factor δm; (c) δ<sup>m</sup> ¼ 0:15; (d) δ<sup>m</sup> ¼ 0:1; and (e) δ<sup>m</sup> ¼ 0:95. All structure constitutive parameters are shown in Figure 2.

polaritons of the superlattice is separated into two distinct areas. This separation appears nearly the frequency at which the resonances of the functions εxy and μxy occur for the composite structures with predominant impact of the magnetic and semiconductor subsystems, respectively. Also, we should note that the exchange by the critical conditions for the asymptotic lines (at which kz ! ∞) between the bottom passbands of left-handed and right-handed elliptically polarized bulk polaritons appears at the particular frequency where εxy and μxy simultaneously tend to infinity.

the polar and Faraday geometries, pseudosurface waves (which attenuate only on one side of

Among all possible appearance of dispersion curves of the surface polaritons, we are only interested in those ones which manifest the crossing or anti-crossing effect. The search of their existence implies solving an optimization problem, where for the crossing effect the degeneracy point should be found. For the anti-crossing effect, the critical points are defined from

respect to k and k<sup>0</sup> for both EH and HE modes. During the solution of this problem, the period and constitutive parameters of the underlying materials of the superlattice are fixed, and the search for the effect manifestation is realized by altering the filling factors δ<sup>m</sup> and δ<sup>s</sup> within the period. The found crossing points are depicted in Figure 8a and b by green circles. We should note that in both configurations, the degeneracy points for EH (red line) and HE (blue line) surface polaritons can be obtained only for the case of composite structure with a predominant impact of the semiconductor subsystem. In particular, this state is found out to be at the values of filling factor δ<sup>m</sup> ¼ 0:27 and δ<sup>m</sup> ¼ 0:15 for the polar and Faraday configurations, respectively. Remarkable, in the Faraday geometry, both dispersion curves possess an anomalous dispersion line, namely, they start on the light line and fall just to the right of the light line, and then they flatten out and approach an asymptotic limit for large values of k. Contrariwise to the Faraday geometry, in the polar geometry, both dispersion curves demonstrate normal dispersion.

Figure 8. Manifestation of the crossing effect in dispersion curves of hybrid EH (red line) and HE (blue line) surface polaritons; (a) polar configuration (δ<sup>m</sup> ¼ 0:27) and (b) Faraday configuration (δ<sup>m</sup> ¼ 0:15). All structure constitutive

ð Þ <sup>k</sup>; <sup>k</sup><sup>0</sup> and second <sup>D</sup>00ð Þ <sup>k</sup>; <sup>k</sup><sup>0</sup> partial derivatives of Eq. (36) or (38) with

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the surface) can also be supported [10, 45].

calculation of the first D<sup>0</sup>

parameters are shown in Figure 2.

The dispersion curves of bulk waves, which have left-handed polarization, demonstrate significant variation of their slope having subsequent branches with normal and anomalous dispersion that possess approaching at some points (extreme states) as depicted in Figure 7c and e. The features of these dispersion curves in the vicinity of the critical points are different for the composite structure which has predominant impact either semiconductor (i.e., δ<sup>s</sup> ≫ δm) or magnetic (i.e., δ<sup>m</sup> ≫ δs) subsystem. The corresponding modes are contradirectional (i.e., condition (29) is met), and they demonstrate strong interaction nearly the Morse critical point (the anti-crossing effect with H < 0) in the case when δ<sup>s</sup> ≫ δm. The modes acquire an intermediate interaction in the superlattice with predominant impact of the magnetic subsystem as depicted in Figure 7e.

Moreover, a particular extreme state in dispersion curves is found out where the upper branches of the left-handed and right-handed elliptically polarized bulk polaritons merge with each other (see Figure 6d) and crossing point (H ¼ 0) occurs at kz ! 0. Likewise to the polar geometry, such extreme state corresponds to the particular superlattice configuration where ε<sup>v</sup> ¼ μ<sup>v</sup> ¼ 0 at kz ¼ 0. So, the interacting waves are degenerated at this point and possess the contradirectional propagation.

We should note that in [37] some unusual and counterintuitive consequences of such behaviors of the dispersion curves (e.g., backward wave propagation, reversed Doppler shift, reversed Cherenkov radiation, atypical singularities in the density of states, etc.) for the TE and TM modes of an axially uniform waveguide are discussed, and it is emphasized that these effects are of considerable significance for practical applications.

Finally, in order to obtain the dispersion curves k0ð Þk of surface polaritons in both the polar and Faraday configurations, characteristic Eqs. (36) and (38) are solved numerically [42, 44]. Here, as previously, the problem is decomposed into two particular solutions with respect to the vector H ! (EH waves) and vector E ! (HE waves) [10, 12]. Note, as evident from Eqs. (36) and (38), kx and kz appear only in even powers, so the dispersion curves of the surface waves appear to be reciprocal.

For both initial problem considerations, dispersion Eqs. (36) and (38) have four roots from which two physically correct ones (denoted here as κ<sup>1</sup> and κ2) are selected [10] ensuring they correspond to the attenuating waves. In papers [10] and [12], it was noted that depending upon the position in the k<sup>0</sup> � kw plane, in the non-dissipative system, the following combinations between two roots κ<sup>1</sup> and κ<sup>2</sup> may arise: (i) both roots are real and positive (bona fide surface modes); (ii) one root is real and the other is pure imaginary or vice versa (pseudosurface modes); (iii) both roots are complex in which case they are conjugate (generalized surface modes); and (iv) both roots are pure imaginary (the propagation is forbidden). In our study we consider only bona fide surface mode. Nevertheless, we should note that in both the polar and Faraday geometries, pseudosurface waves (which attenuate only on one side of the surface) can also be supported [10, 45].

respectively. Also, we should note that the exchange by the critical conditions for the asymptotic lines (at which kz ! ∞) between the bottom passbands of left-handed and right-handed elliptically polarized bulk polaritons appears at the particular frequency where εxy and μxy

The dispersion curves of bulk waves, which have left-handed polarization, demonstrate significant variation of their slope having subsequent branches with normal and anomalous dispersion that possess approaching at some points (extreme states) as depicted in Figure 7c and e. The features of these dispersion curves in the vicinity of the critical points are different for the composite structure which has predominant impact either semiconductor (i.e., δ<sup>s</sup> ≫ δm) or magnetic (i.e., δ<sup>m</sup> ≫ δs) subsystem. The corresponding modes are contradirectional (i.e., condition (29) is met), and they demonstrate strong interaction nearly the Morse critical point (the anti-crossing effect with H < 0) in the case when δ<sup>s</sup> ≫ δm. The modes acquire an intermediate interaction in the superlattice with predominant impact of the magnetic subsystem as

Moreover, a particular extreme state in dispersion curves is found out where the upper branches of the left-handed and right-handed elliptically polarized bulk polaritons merge with each other (see Figure 6d) and crossing point (H ¼ 0) occurs at kz ! 0. Likewise to the polar geometry, such extreme state corresponds to the particular superlattice configuration where ε<sup>v</sup> ¼ μ<sup>v</sup> ¼ 0 at kz ¼ 0. So, the interacting waves are degenerated at this point and possess the

We should note that in [37] some unusual and counterintuitive consequences of such behaviors of the dispersion curves (e.g., backward wave propagation, reversed Doppler shift, reversed Cherenkov radiation, atypical singularities in the density of states, etc.) for the TE and TM modes of an axially uniform waveguide are discussed, and it is emphasized that these effects

Finally, in order to obtain the dispersion curves k0ð Þk of surface polaritons in both the polar and Faraday configurations, characteristic Eqs. (36) and (38) are solved numerically [42, 44]. Here, as previously, the problem is decomposed into two particular solutions with respect to the

(38), kx and kz appear only in even powers, so the dispersion curves of the surface waves

For both initial problem considerations, dispersion Eqs. (36) and (38) have four roots from which two physically correct ones (denoted here as κ<sup>1</sup> and κ2) are selected [10] ensuring they correspond to the attenuating waves. In papers [10] and [12], it was noted that depending upon the position in the k<sup>0</sup> � kw plane, in the non-dissipative system, the following combinations between two roots κ<sup>1</sup> and κ<sup>2</sup> may arise: (i) both roots are real and positive (bona fide surface modes); (ii) one root is real and the other is pure imaginary or vice versa (pseudosurface modes); (iii) both roots are complex in which case they are conjugate (generalized surface modes); and (iv) both roots are pure imaginary (the propagation is forbidden). In our study we consider only bona fide surface mode. Nevertheless, we should note that in both

(HE waves) [10, 12]. Note, as evident from Eqs. (36) and

simultaneously tend to infinity.

120 Surface Waves - New Trends and Developments

depicted in Figure 7e.

vector H !

appear to be reciprocal.

contradirectional propagation.

are of considerable significance for practical applications.

!

(EH waves) and vector E

Among all possible appearance of dispersion curves of the surface polaritons, we are only interested in those ones which manifest the crossing or anti-crossing effect. The search of their existence implies solving an optimization problem, where for the crossing effect the degeneracy point should be found. For the anti-crossing effect, the critical points are defined from calculation of the first D<sup>0</sup> ð Þ <sup>k</sup>; <sup>k</sup><sup>0</sup> and second <sup>D</sup>00ð Þ <sup>k</sup>; <sup>k</sup><sup>0</sup> partial derivatives of Eq. (36) or (38) with respect to k and k<sup>0</sup> for both EH and HE modes. During the solution of this problem, the period and constitutive parameters of the underlying materials of the superlattice are fixed, and the search for the effect manifestation is realized by altering the filling factors δ<sup>m</sup> and δ<sup>s</sup> within the period. The found crossing points are depicted in Figure 8a and b by green circles. We should note that in both configurations, the degeneracy points for EH (red line) and HE (blue line) surface polaritons can be obtained only for the case of composite structure with a predominant impact of the semiconductor subsystem. In particular, this state is found out to be at the values of filling factor δ<sup>m</sup> ¼ 0:27 and δ<sup>m</sup> ¼ 0:15 for the polar and Faraday configurations, respectively. Remarkable, in the Faraday geometry, both dispersion curves possess an anomalous dispersion line, namely, they start on the light line and fall just to the right of the light line, and then they flatten out and approach an asymptotic limit for large values of k. Contrariwise to the Faraday geometry, in the polar geometry, both dispersion curves demonstrate normal dispersion.

Figure 8. Manifestation of the crossing effect in dispersion curves of hybrid EH (red line) and HE (blue line) surface polaritons; (a) polar configuration (δ<sup>m</sup> ¼ 0:27) and (b) Faraday configuration (δ<sup>m</sup> ¼ 0:15). All structure constitutive parameters are shown in Figure 2.

### 6. Conclusions

To conclude, in this chapter, we have studied dispersion features of both bulk and surface polaritons in a magnetic-semiconductor superlattice influenced by an external static magnetic field. The investigation was carried out under an assumption that all characteristic dimensions of the given superlattice satisfy the long-wavelength limit; thus, the homogenization procedures from the effective medium theory was involved, and the superlattice was represented as a gyroelectromagnetic uniform medium characterized by the tensors of effective permeability and effective permittivity.

[4] Anker JN, Hall WP, Lyandres O, Shah NC, Zhao J, Van Duyne RP. Biosensing with plasmonic nanosensors. Nature Materials. 2008;7:442-453. DOI: 10.1038/nmat2162 [5] Jun YC. Electrically-driven active plasmonic devices. In: Ki YK, editor. Plasmonics - Principles and Applications. InTech: Rijeka; 2012. pp. 383-400. DOI: 10.5772/50756 [6] Dicken MJ, Sweatlock LA, Pacific D, Lezec HJ, Bhattacharya K, Atwater HA. Electrooptic modulation in thin film barium titanate plasmonic interferometers. Nano Letters. 2008;

Modal Phenomena of Surface and Bulk Polaritons in Magnetic-Semiconductor Superlattices

http://dx.doi.org/10.5772/intechopen.71837

123

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009541

(85)90094-1

The general theory of polaritons in the gyroelectromagnetic medium whose permittivity and permeability simultaneously are tensor quantities was developed. Three particular configurations of the magnetization, namely, the Voigt, polar, and Faraday geometries, were discussed in detail.

The crossing and anti-crossing effects in the dispersion curves of both surface and bulk polaritons have been identified and investigated with an assistance of the analytical theory about the Morse critical points.

We argue that the discussed dispersion features of polaritons identified in the magneticsemiconductor superlattice under study have a fundamental nature and are common to different types of waves and waveguide systems.

### Author details

Vladimir R. Tuz1,2, Illia V. Fedorin3 and Volodymyr I. Fesenko1,2\*

\*Address all correspondence to: volodymyr.i.fesenko@gmail.com

1 International Center of Future Science, State Key Laboratory on Integrated Optoelectronics, College of Electronic Science and Engineering, Jilin University, China


#### References


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6. Conclusions

122 Surface Waves - New Trends and Developments

and effective permittivity.

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Author details

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Vladimir R. Tuz1,2, Illia V. Fedorin3 and Volodymyr I. Fesenko1,2\*

\*Address all correspondence to: volodymyr.i.fesenko@gmail.com

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College of Electronic Science and Engineering, Jilin University, China

3 National Technical University 'Kharkiv Polytechnical Institute', Ukraine

To conclude, in this chapter, we have studied dispersion features of both bulk and surface polaritons in a magnetic-semiconductor superlattice influenced by an external static magnetic field. The investigation was carried out under an assumption that all characteristic dimensions of the given superlattice satisfy the long-wavelength limit; thus, the homogenization procedures from the effective medium theory was involved, and the superlattice was represented as a gyroelectromagnetic uniform medium characterized by the tensors of effective permeability

The general theory of polaritons in the gyroelectromagnetic medium whose permittivity and permeability simultaneously are tensor quantities was developed. Three particular configurations of the magnetization, namely, the Voigt, polar, and Faraday geometries, were discussed in detail. The crossing and anti-crossing effects in the dispersion curves of both surface and bulk polaritons have been identified and investigated with an assistance of the analytical theory

We argue that the discussed dispersion features of polaritons identified in the magneticsemiconductor superlattice under study have a fundamental nature and are common to differ-

1 International Center of Future Science, State Key Laboratory on Integrated Optoelectronics,

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**Chapter 7**

Provisional chapter

) were obtained by analyzing

**Video Measurements and Analysis of Surface Gravity**

DOI: 10.5772/intechopen.73042

This paper discusses a shallow-water wave height measurement method that uses high definition video cameras to image a water surface wave patch. Wave height time series are extracted from water surface video sequences. Wave features such as the wavelength

the extracted wave height time series and expressing the wind-driven wave energy as a wave energy spectrum. A Weibull probability distribution was used as the mathematical form of the energy spectrum. Wave spectra are used as input to a wave patch simulation model that generates simulated wind-driven wave images. The measurement protocol is inexpensive, easy to implement, and useful to calibrate and validate wind-driven wave models. The protocol is used to understand resuspension of bottom muds due to wind waves in shallow waters. Scaled staff gauges made of polyvinyl chloride (PVC) materials are placed in shallow water and imaged at 30 Hz followed by frame based image analysis to extract wave height time series. Wave spectra calculated using the fast Fourier transform (FFT) results in a Weibull probability distribution function (WPDF) energy spectrum. The estimated wave spectrum is used to estimate wave energy in W/m<sup>2</sup> followed by generation of wave patch simulations of the water surface. Simulated wave patches are compared with the sensor-based wave patch video measurements. Sensitivity analysis of coefficients α and β in the model are used to adjust the synthetic wave images to measured wave patch images. The approach allows one to obtain an estimate of the energy

) transferred from the local wind field to a water surface gravity wave patch.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Keywords: water surface waves, gravity waves, wind driven waves, wave patches, wave patch energy, remote sensing, staff gauges, video imaging, video analysis,

energy spectra, Weibull distributions, synthetic imaging, wind energy

Video Measurements and Analysis of Surface Gravity

**Waves in Shallow Water**

Waves in Shallow Water

http://dx.doi.org/10.5772/intechopen.73042

Tyler Rotkiske

Abstract

(W/m<sup>2</sup>

Tyler Rotkiske

Charles R. Bostater Jr, Bingyu Yang and

Charles R. Bostater Jr, Bingyu Yang and

Additional information is available at the end of the chapter

distribution and energy contained in a wave patch (W/m<sup>2</sup>

Additional information is available at the end of the chapter

#### **Video Measurements and Analysis of Surface Gravity Waves in Shallow Water** Video Measurements and Analysis of Surface Gravity Waves in Shallow Water

DOI: 10.5772/intechopen.73042

Charles R. Bostater Jr, Bingyu Yang and Tyler Rotkiske Charles R. Bostater Jr, Bingyu Yang and Tyler Rotkiske

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.73042

#### Abstract

This paper discusses a shallow-water wave height measurement method that uses high definition video cameras to image a water surface wave patch. Wave height time series are extracted from water surface video sequences. Wave features such as the wavelength distribution and energy contained in a wave patch (W/m<sup>2</sup> ) were obtained by analyzing the extracted wave height time series and expressing the wind-driven wave energy as a wave energy spectrum. A Weibull probability distribution was used as the mathematical form of the energy spectrum. Wave spectra are used as input to a wave patch simulation model that generates simulated wind-driven wave images. The measurement protocol is inexpensive, easy to implement, and useful to calibrate and validate wind-driven wave models. The protocol is used to understand resuspension of bottom muds due to wind waves in shallow waters. Scaled staff gauges made of polyvinyl chloride (PVC) materials are placed in shallow water and imaged at 30 Hz followed by frame based image analysis to extract wave height time series. Wave spectra calculated using the fast Fourier transform (FFT) results in a Weibull probability distribution function (WPDF) energy spectrum. The estimated wave spectrum is used to estimate wave energy in W/m<sup>2</sup> followed by generation of wave patch simulations of the water surface. Simulated wave patches are compared with the sensor-based wave patch video measurements. Sensitivity analysis of coefficients α and β in the model are used to adjust the synthetic wave images to measured wave patch images. The approach allows one to obtain an estimate of the energy (W/m<sup>2</sup> ) transferred from the local wind field to a water surface gravity wave patch.

Keywords: water surface waves, gravity waves, wind driven waves, wave patches, wave patch energy, remote sensing, staff gauges, video imaging, video analysis, energy spectra, Weibull distributions, synthetic imaging, wind energy

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 1. Introduction

Differential heating of the earth causes large-scale atmospheric pressure gradients. These gradients and the acceleration due to rotation of the earth result in the major wind fields around the globe. The winds cause friction on the water surface and result in the generation of wind-driven water surface gravity waves. These wind-driven gravity waves are a major source of transfer of energy from the sun and atmosphere to the earth's water system on a global scale as well as in small water regions. Measurements of these wind-driven shallowwater surface gravity wave characteristics such as wave height, period, and direction of propagation are key to understanding the magnitude of energy in a small gravity wave field or "wave patch." These surface wind-driven waves cause orbital motion of water parcels [1]. In water the wind stress or friction causes the downward movement or transport of momentum from the water surface to the water column. The downward transport of momentum causes internal friction in the water in the form of turbulent friction or viscosity, and the associated circular eddy-type motions influence the bottom lutocline within the bottom boundary layer [2, 3]. The resulting shearing forces at the bottom in shallow water cause resuspension of and fluidization of bottom sediments and associated nutrients and/or trace metals. The fluidization can result in fluidized mud, muck, high-bottom water turbidity, and nephelometric wave motions (internal wavelike motions that are observed in acoustic imaging of the bottom water column). Thus, water quality variables are influenced by the surface water wave field driven by surface winds. Bottom orbital or elliptical velocities due to these wind-driven gravity waves in shallow water can be calculated using surface wind-driven gravity wave information, such as significant wave height and wave spectra [3, 4].

In this chapter, techniques to describe the protocol are demonstrated for shallow water near the Atlantic Ocean and Space Coast Region of the Banana River estuary and watershed, near Melbourne, Florida. Simulated wave patch synthetic images are then generated using estimated wave spectrum derived from video imagery and compared to the video images.

Video Measurements and Analysis of Surface Gravity Waves in Shallow Water

http://dx.doi.org/10.5772/intechopen.73042

129

Staff gauges and HD video cameras (JVC Everio HD cameras) are used to measure the surface wave heights in shallow water (1–2 m) as shown in Figure 1(a). Four staff gauges are mounted in the water by inserting them into the bottom sediment. Staff gauges are constructed from white PVC pipes. Waterproof paper with 0.5 spaced horizontal lines is

Images recorded at 1920 1080 pixels (30 per second of recording time) obtained from the video sequence yield spatial resolution (using lens zooming) is 0.04 cm/pixel. Each line is either 1 mm (≈2 pixels) or 0.5 mm (≈1 pixel) as shown in Figure 1(b). The HD cameras are mounted on a tripod on the shore or from a fixed platform or mount in the water. The video staff gauges

Figure 1. Imaging system deployment and video image of the wave patch (a), staff gauges (b), and example camera

constructed using PVC pipes and striped paper with 0.5 cm line (alternating two colors and thickness) separation. Two video cameras can be used for binocular stereo imaging, and the third camera can be mounted on the vertical bar in the middle or

). Staff gauges were

deployment platform (c). Target wave patch defined by four staff gauges is 4.5 4.5 m (20.25 m2

either side in order to conduct a stereo or trinocular imaging of a water patch or zoomed to image staff gauges.

allow one to define a wave patch of ~4.5 4.5 m (Wx = Wy = 4.5 m).

2. Techniques and methods

taped to each gauge pipe.

2.1. Video imaging and staff gauge system

Wind-driven gravity waves in water can be measured using numerous techniques and descriptions, including satellite altimeters, video cameras, wave buoys, and many types of wave gauges [5]. Altimeters flown on satellites such as Seasat in 1978, Geosat from 1985 to 1988, ERS-1 and ERS-2 from 1991, TOPEX/Poseidon from 1992, Jason from 2001, and Envisat can provide data for monthly mean maps of wave heights and the variability of wave energy density in time and space [5]. Wave height estimations based on binocular or trinocular imaging also allow acquisition of both spatial and temporal wave information for surface wave patches [6, 7, 14]. Many types of wave gauges including resistance-type, capacitance-type, and wave pressure gauges can also provide data for quantifiable wind-driven gravity wave height estimations [8]. However, there is a need to develop measurement system methods [17, 18] for shallow-water areas since most systems developed to date can only be used in deep waters (>3–5 m).

In this research, a technique or protocol that combines gravity wave staff gauges, HD video cameras [14], and image analyses is used to measure wind-driven gravity waves in shallow-water environments that receive energy from the atmosphere. Wave height time series are extracted from video sequence taken using high-definition (HD) video cameras at a 30 Hz frame rate. Significant wave height, wave spectrum, and wave energy (W m<sup>2</sup> ) are estimated from the video-based wave height time series.

In this chapter, techniques to describe the protocol are demonstrated for shallow water near the Atlantic Ocean and Space Coast Region of the Banana River estuary and watershed, near Melbourne, Florida. Simulated wave patch synthetic images are then generated using estimated wave spectrum derived from video imagery and compared to the video images.

#### 2. Techniques and methods

1. Introduction

128 Surface Waves - New Trends and Developments

as significant wave height and wave spectra [3, 4].

estimated from the video-based wave height time series.

Differential heating of the earth causes large-scale atmospheric pressure gradients. These gradients and the acceleration due to rotation of the earth result in the major wind fields around the globe. The winds cause friction on the water surface and result in the generation of wind-driven water surface gravity waves. These wind-driven gravity waves are a major source of transfer of energy from the sun and atmosphere to the earth's water system on a global scale as well as in small water regions. Measurements of these wind-driven shallowwater surface gravity wave characteristics such as wave height, period, and direction of propagation are key to understanding the magnitude of energy in a small gravity wave field or "wave patch." These surface wind-driven waves cause orbital motion of water parcels [1]. In water the wind stress or friction causes the downward movement or transport of momentum from the water surface to the water column. The downward transport of momentum causes internal friction in the water in the form of turbulent friction or viscosity, and the associated circular eddy-type motions influence the bottom lutocline within the bottom boundary layer [2, 3]. The resulting shearing forces at the bottom in shallow water cause resuspension of and fluidization of bottom sediments and associated nutrients and/or trace metals. The fluidization can result in fluidized mud, muck, high-bottom water turbidity, and nephelometric wave motions (internal wavelike motions that are observed in acoustic imaging of the bottom water column). Thus, water quality variables are influenced by the surface water wave field driven by surface winds. Bottom orbital or elliptical velocities due to these wind-driven gravity waves in shallow water can be calculated using surface wind-driven gravity wave information, such

Wind-driven gravity waves in water can be measured using numerous techniques and descriptions, including satellite altimeters, video cameras, wave buoys, and many types of wave gauges [5]. Altimeters flown on satellites such as Seasat in 1978, Geosat from 1985 to 1988, ERS-1 and ERS-2 from 1991, TOPEX/Poseidon from 1992, Jason from 2001, and Envisat can provide data for monthly mean maps of wave heights and the variability of wave energy density in time and space [5]. Wave height estimations based on binocular or trinocular imaging also allow acquisition of both spatial and temporal wave information for surface wave patches [6, 7, 14]. Many types of wave gauges including resistance-type, capacitance-type, and wave pressure gauges can also provide data for quantifiable wind-driven gravity wave height estimations [8]. However, there is a need to develop measurement system methods [17, 18] for shallow-water areas

In this research, a technique or protocol that combines gravity wave staff gauges, HD video cameras [14], and image analyses is used to measure wind-driven gravity waves in shallow-water environments that receive energy from the atmosphere. Wave height time series are extracted from video sequence taken using high-definition (HD) video cameras at a 30 Hz frame rate. Significant wave height, wave spectrum, and wave energy (W m<sup>2</sup>

) are

since most systems developed to date can only be used in deep waters (>3–5 m).

#### 2.1. Video imaging and staff gauge system

Staff gauges and HD video cameras (JVC Everio HD cameras) are used to measure the surface wave heights in shallow water (1–2 m) as shown in Figure 1(a). Four staff gauges are mounted in the water by inserting them into the bottom sediment. Staff gauges are constructed from white PVC pipes. Waterproof paper with 0.5 spaced horizontal lines is taped to each gauge pipe.

Images recorded at 1920 1080 pixels (30 per second of recording time) obtained from the video sequence yield spatial resolution (using lens zooming) is 0.04 cm/pixel. Each line is either 1 mm (≈2 pixels) or 0.5 mm (≈1 pixel) as shown in Figure 1(b). The HD cameras are mounted on a tripod on the shore or from a fixed platform or mount in the water. The video staff gauges allow one to define a wave patch of ~4.5 4.5 m (Wx = Wy = 4.5 m).

Figure 1. Imaging system deployment and video image of the wave patch (a), staff gauges (b), and example camera deployment platform (c). Target wave patch defined by four staff gauges is 4.5 4.5 m (20.25 m2 ). Staff gauges were constructed using PVC pipes and striped paper with 0.5 cm line (alternating two colors and thickness) separation. Two video cameras can be used for binocular stereo imaging, and the third camera can be mounted on the vertical bar in the middle or either side in order to conduct a stereo or trinocular imaging of a water patch or zoomed to image staff gauges.

An inverted T-shaped (Figure 1(c)) aluminum and steel camera optical mount is used to mount one, two, or three HD cameras. Cameras can be time synchronized, using a known or fixed "digital zoom" mode. During measurement experiments, one camera is focused upon and zoomed (18) on a staff gauge, and the other two cameras image the entire wave patch and record at 30 Hz or frames per second (fps) (Figure 2).

#### 2.2. Image processing

In order to determine the water surface level based on video sequences, three steps are needed as summarized below:

#### Step 1: Determine the region of interest (ROI)

The water surface level at the staff gauges was clearly observed from the 18 zoomed video frame images. A representative image is shown in Figure 3(a). A ROI was selected to be the area between the top horizontal scale line on the gauge and the bottom margin of the image and was slightly less in width than the actual gauge as defined by the rectangular area in Figure 3(b). A thinner ROI than the actual gauge prevents pixels outside of the staff gauge influencing the results.

Step 3: Locate the air-water level on the gauge

energy in the patch in W=m<sup>2</sup> is 1.1 105

height is 0.1 m.

wave crests.

automatically locate the interface within each video frame.

The shape of the pixel value curve changes dramatically at the interface of water and gauge as shown in Figure 3(c), which is the water surface level. A threshold value is selected in order to

Figure 3. Measured shallow-water surface oscillation series from the video expressed in "wave coordinate." The sequence shown is 347 s with 30 Hz frame rate (10,388 observations). The average water depth during the experiment was 35 cm. The average water surface level during the experiment ranged from 30 cm at the wave troughs and 40 cm at

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The threshold value was determined based on the average of part of the "representative line" (first 500 pixels from the top in the example shown in Figure 3 (b)) and it slightly varies from frame to frame. The key is to semiautomatically select a threshold value and apply this value to each frame. The location of the start of values lower than the threshold value is considered as

Figure 4. Estimated wave energy spectrum with simulated wave patch (a) and wave energy spectrum with simulated wave patch image using adjusted model coefficients α and β. The peak frequency of the spectrum is 0.39 Hz, and the wave

wave spectrum from field measurements. The simulated image size is 1024 1024 pixels and represents the 4.5 4.5 m<sup>2</sup> wave patch defined by the staff gauges shown in Figure 1. Simulated mean wave height is 7.6 cm, and significant wave

. The simulated wave patch synthetic images are generated using the estimated

#### Step 2: Average the digital counts within the ROI

Within the ROI, the mean pixel value across each horizontal line is calculated to compress the ROI into a "representative line" shown in Figure 3(b). The average of the three color bands (R, G, B) is calculated in this step as well. The oscillating line in Figure 3(c) is the average pixel value of the example frame, and the air-water interface is obvious.

Figure 2. Three steps are used to determine the water surface level from the video. (a) Example of a video frame from the zoomed video (18). (b) Then, a region of interest (ROI) is defined by a rectangular area, and then a representative line at the center is calculated by averaging the pixel value on each horizontal level. (c) Pixel values along the averaged representative line are shown, and a selected threshold value is selected to locate the interface of water and gauge shown by the solid line.

Figure 3. Measured shallow-water surface oscillation series from the video expressed in "wave coordinate." The sequence shown is 347 s with 30 Hz frame rate (10,388 observations). The average water depth during the experiment was 35 cm. The average water surface level during the experiment ranged from 30 cm at the wave troughs and 40 cm at wave crests.

#### Step 3: Locate the air-water level on the gauge

An inverted T-shaped (Figure 1(c)) aluminum and steel camera optical mount is used to mount one, two, or three HD cameras. Cameras can be time synchronized, using a known or fixed "digital zoom" mode. During measurement experiments, one camera is focused upon and zoomed (18) on a staff gauge, and the other two cameras image the entire wave patch and

In order to determine the water surface level based on video sequences, three steps are needed

The water surface level at the staff gauges was clearly observed from the 18 zoomed video frame images. A representative image is shown in Figure 3(a). A ROI was selected to be the area between the top horizontal scale line on the gauge and the bottom margin of the image and was slightly less in width than the actual gauge as defined by the rectangular area in Figure 3(b). A thinner ROI than the actual gauge prevents pixels outside of the staff gauge

Within the ROI, the mean pixel value across each horizontal line is calculated to compress the ROI into a "representative line" shown in Figure 3(b). The average of the three color bands (R, G, B) is calculated in this step as well. The oscillating line in Figure 3(c) is the average pixel

Figure 2. Three steps are used to determine the water surface level from the video. (a) Example of a video frame from the zoomed video (18). (b) Then, a region of interest (ROI) is defined by a rectangular area, and then a representative line at the center is calculated by averaging the pixel value on each horizontal level. (c) Pixel values along the averaged representative line are shown, and a selected threshold value is selected to locate the interface of water and gauge shown

record at 30 Hz or frames per second (fps) (Figure 2).

Step 1: Determine the region of interest (ROI)

Step 2: Average the digital counts within the ROI

value of the example frame, and the air-water interface is obvious.

2.2. Image processing

130 Surface Waves - New Trends and Developments

as summarized below:

influencing the results.

by the solid line.

The shape of the pixel value curve changes dramatically at the interface of water and gauge as shown in Figure 3(c), which is the water surface level. A threshold value is selected in order to automatically locate the interface within each video frame.

The threshold value was determined based on the average of part of the "representative line" (first 500 pixels from the top in the example shown in Figure 3 (b)) and it slightly varies from frame to frame. The key is to semiautomatically select a threshold value and apply this value to each frame. The location of the start of values lower than the threshold value is considered as

Figure 4. Estimated wave energy spectrum with simulated wave patch (a) and wave energy spectrum with simulated wave patch image using adjusted model coefficients α and β. The peak frequency of the spectrum is 0.39 Hz, and the wave energy in the patch in W=m<sup>2</sup> is 1.1 105 . The simulated wave patch synthetic images are generated using the estimated wave spectrum from field measurements. The simulated image size is 1024 1024 pixels and represents the 4.5 4.5 m<sup>2</sup> wave patch defined by the staff gauges shown in Figure 1. Simulated mean wave height is 7.6 cm, and significant wave height is 0.1 m.

the water surface. Figure 4 shows the result of 30 Hz time series of the water wave surface oscillations calculated from the example video sequence.

#### 3. Wave spectrum development

A wave energy density spectrum of a wind-driven surface gravity wave patch E can be expressed in units of W=m<sup>2</sup> and is related to the wave height or the variance of the wave displacement, given as <sup>E</sup> <sup>¼</sup> <sup>1</sup> <sup>8</sup> <sup>ρ</sup>gH<sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> ρga2, where a is the wave displacement and the wave height (H) is twice the wave displacement or H ¼ 2a [1]. Given a measured wave time series, the wave spectrum can be calculated using the fast Fourier transform (FFT) [5]:

$$S(f) = Z\_n Z\_{n'}^\* \tag{1}$$

number of the data point in the measured sequence (dimensionless), t is the instant time corresponding to the data point in the unit of second (s), and T is the length of measurement time in the unit of second (s). A two-parameter Weibull distribution function was used as a test model for mathematical expression of the spectrum [2]. The Weibull probability density func-

> αβ � �Y<sup>β</sup>�<sup>1</sup>

where Y is a variable that follows the Weibull distribution. In this research Y has the unit of 1/s (temporal frequency) or 1/m (wavenumber (spatial frequency); see Figure 5(a)), and Table 1 on x-axis, α = scale parameter has the units of the variable Y, which is 1/s or 1/m, β = non-

By using the three steps described in Section 2.2, the water surface-level change throughout the video sequence was obtained as shown in Table 1 and Figure 4. Mean wave height of the measured series was 7.9 cm, and significant wave height Hs was 13 cm. The significant wave height Hs is defined as average of the highest one-third of the wave heights [10]. The speed of

The wave spectrum coefficients estimated using the video-based wave series were α ¼ 0:55 and β ¼ 1:31. The calculated wave energy spectrum of the measured wave height time series

α = 0.55 is the Weibull scale parameter (Hz), and β = 1.31 is the Weibull shape parameter (dimensionless). The estimated wave spectrum can then be used in the spectral wave patch model [11, 12]. Since the wave simulation program by Bostater et al. [11, 12] was designed to generate a shallow-water gravity wave patch that was independent of time, a dispersion relation was applied to transfer the energy spectrum from temporal frequency domain (Hz)

exp � <sup>ω</sup>

α

� �<sup>β</sup> � �, (4)

/Hz), ω is the temporal frequency (Hz),

CDF � cumulative : ð Þ¼ <sup>Y</sup> <sup>1</sup> � exp � <sup>Y</sup>

4.1. Image-based wave height time series and statistical characteristics

this wave series extracting process is about two frames per second.

on temporal frequency domain (Hz) is calculated from Eq. (4):

where <sup>S</sup>ð Þ <sup>ω</sup> is the wave energy spectral density (m<sup>2</sup>

to spatial frequency domain (1/m):

<sup>S</sup>ð Þ¼ <sup>ω</sup> <sup>β</sup>

αβ � �ω<sup>β</sup>�<sup>1</sup>

4.2. Wave spectrum, wave energy, and simulation for the Banana River site

exp � <sup>Y</sup>

α

α

Video Measurements and Analysis of Surface Gravity Waves in Shallow Water

� �<sup>β</sup> ! <sup>Y</sup> <sup>&</sup>gt; <sup>0</sup>, (2)

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� �<sup>β</sup> ! <sup>Y</sup> <sup>&</sup>gt; <sup>0</sup>, (3)

tion (PDF) and cumulative density function (CDF) are given by [9]:

PDF � probability : ð Þ¼ <sup>Y</sup> <sup>β</sup>

dimensional shape parameter.

4. Experimental results

where S fð Þ is the wave spectrum density in the unit of cm2 /Hz, f is temporal frequency in the unit of Hz (1/s), Zn <sup>¼</sup> <sup>1</sup> T Ð <sup>T</sup>=<sup>2</sup> �T=<sup>2</sup> <sup>ζ</sup>ð Þ<sup>t</sup> exp ð Þ <sup>i</sup>2πnft is the Fourier transform of the original signal <sup>ζ</sup>ð Þ<sup>t</sup> (dimensionless), Z<sup>∗</sup> <sup>n</sup> <sup>¼</sup> <sup>1</sup> T Ð <sup>T</sup>=<sup>2</sup> �T=<sup>2</sup> <sup>ζ</sup>ð Þ<sup>t</sup> exp ð Þ �i2πnft is the conjugate of Zn (dimensionless), <sup>n</sup> is the


Figure 5. Significant wave height Hs (m) of simulated waves using the wave model (see Eqs. (2) and (3)) parameters α (scale parameter) and β (nondimensional shape parameter) and selected example simulated wave patch synthetic images.

number of the data point in the measured sequence (dimensionless), t is the instant time corresponding to the data point in the unit of second (s), and T is the length of measurement time in the unit of second (s). A two-parameter Weibull distribution function was used as a test model for mathematical expression of the spectrum [2]. The Weibull probability density function (PDF) and cumulative density function (CDF) are given by [9]:

$$\text{PDF} - \text{probability} : (Y) = \left(\frac{\beta}{a^{\beta}}\right) Y^{\beta - 1} \exp\left(-\left(\frac{Y}{a}\right)^{\beta}\right) \quad Y > 0,\tag{2}$$

$$\text{CDF} - \text{cumulative} : (Y) = 1 - \exp\left(-\left(\frac{Y}{a}\right)^{\beta}\right) \quad Y > 0,\tag{3}$$

where Y is a variable that follows the Weibull distribution. In this research Y has the unit of 1/s (temporal frequency) or 1/m (wavenumber (spatial frequency); see Figure 5(a)), and Table 1 on x-axis, α = scale parameter has the units of the variable Y, which is 1/s or 1/m, β = nondimensional shape parameter.

#### 4. Experimental results

the water surface. Figure 4 shows the result of 30 Hz time series of the water wave surface

A wave energy density spectrum of a wind-driven surface gravity wave patch E can be expressed in units of W=m<sup>2</sup> and is related to the wave height or the variance of the wave

height (H) is twice the wave displacement or H ¼ 2a [1]. Given a measured wave time series,

S fð Þ¼ ZnZ<sup>∗</sup>

Figure 5. Significant wave height Hs (m) of simulated waves using the wave model (see Eqs. (2) and (3)) parameters α (scale parameter) and β (nondimensional shape parameter) and selected example simulated wave patch synthetic images.

<sup>2</sup> ρga2, where a is the wave displacement and the wave

�T=<sup>2</sup> <sup>ζ</sup>ð Þ<sup>t</sup> exp ð Þ <sup>i</sup>2πnft is the Fourier transform of the original signal <sup>ζ</sup>ð Þ<sup>t</sup>

�T=<sup>2</sup> <sup>ζ</sup>ð Þ<sup>t</sup> exp ð Þ �i2πnft is the conjugate of Zn (dimensionless), <sup>n</sup> is the

<sup>n</sup>, (1)

/Hz, f is temporal frequency in the

oscillations calculated from the example video sequence.

<sup>8</sup> <sup>ρ</sup>gH<sup>2</sup> <sup>¼</sup> <sup>1</sup>

where S fð Þ is the wave spectrum density in the unit of cm2

T Ð <sup>T</sup>=<sup>2</sup>

<sup>n</sup> <sup>¼</sup> <sup>1</sup> T Ð <sup>T</sup>=<sup>2</sup>

the wave spectrum can be calculated using the fast Fourier transform (FFT) [5]:

3. Wave spectrum development

132 Surface Waves - New Trends and Developments

displacement, given as <sup>E</sup> <sup>¼</sup> <sup>1</sup>

unit of Hz (1/s), Zn <sup>¼</sup> <sup>1</sup>

(dimensionless), Z<sup>∗</sup>

#### 4.1. Image-based wave height time series and statistical characteristics

By using the three steps described in Section 2.2, the water surface-level change throughout the video sequence was obtained as shown in Table 1 and Figure 4. Mean wave height of the measured series was 7.9 cm, and significant wave height Hs was 13 cm. The significant wave height Hs is defined as average of the highest one-third of the wave heights [10]. The speed of this wave series extracting process is about two frames per second.

#### 4.2. Wave spectrum, wave energy, and simulation for the Banana River site

The wave spectrum coefficients estimated using the video-based wave series were α ¼ 0:55 and β ¼ 1:31. The calculated wave energy spectrum of the measured wave height time series on temporal frequency domain (Hz) is calculated from Eq. (4):

$$S(\omega) = \left(\frac{\beta}{a^{\beta}}\right) \omega^{\beta - 1} \exp\left(-\left(\frac{\omega}{a}\right)^{\beta}\right),\tag{4}$$

where <sup>S</sup>ð Þ <sup>ω</sup> is the wave energy spectral density (m<sup>2</sup> /Hz), ω is the temporal frequency (Hz), α = 0.55 is the Weibull scale parameter (Hz), and β = 1.31 is the Weibull shape parameter (dimensionless). The estimated wave spectrum can then be used in the spectral wave patch model [11, 12]. Since the wave simulation program by Bostater et al. [11, 12] was designed to generate a shallow-water gravity wave patch that was independent of time, a dispersion relation was applied to transfer the energy spectrum from temporal frequency domain (Hz) to spatial frequency domain (1/m):

$$
\omega(k) = \sqrt{\text{gk\,tanh}(kd)},\tag{5}
$$

where ωð Þk is the temporal frequency (Hz), k is the spatial frequency wavenumber (1/m), g is the gravity acceleration (m/s<sup>2</sup> ), and d is the water depth (m).

The wave energy in the units of W=m<sup>2</sup> can now be calculated using the estimated wave spectrum, based on the area under the spectrum curve [13]:

$$E = \rho g \int \mathcal{S}(\omega) d\omega,\tag{6}$$

where <sup>E</sup> is the wave energy in units of W=m<sup>2</sup> (Joules=s m2), <sup>S</sup>ð Þ <sup>ω</sup> is the wave energy spectrum in units of m2, <sup>ω</sup> is the temporal frequency in units of Hz, <sup>ρ</sup> <sup>¼</sup> 1000 kg=m3 is the density of the water used in this research, and <sup>g</sup> <sup>¼</sup> <sup>9</sup>:8 m=s2 is the acceleration due to gravity. The discrete approximation of wave energy in this research is calculated by summing the area under the spectral curve:

$$E = \rho \text{g} \sum\_{0}^{N} \mathcal{S}(\omega\_n) \Delta \omega\_{n\nu} \tag{7}$$

Figure 6. Example shallow-water wave displacement (m) and wavelength (m) sequences (a–f) extracted from synthetic imagery (see Figure 5) used to demonstrate the sensitivity of the energy in a simulated wave patch as shown in Figure 7.

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The simulated image results are shown in Figure 5.

where N is the total number of discrete frequencies integral, Sð Þ ω<sup>n</sup> is the magnitude of the energy spectrum at each frequency ωn in units of m2, and Δω<sup>n</sup> is the discrete frequency interval in units of Hz. A representative shallow-water wind-driven gravity wave can be simulated using the

Table 1. (right) Input equations and parameters for running the wave simulation program (Bostater et al. [11, 12]) for simulating the wave patch shown in Figure 4 and the measured and simulation distributions (left image). Comparison of the calculated Weibull cumulative curve (solid line) and the cumulative curve of the wave energy spectrum extracted from the measured wave displacements with estimated Weibull parameters of α = 0.55 (Hz) and β = 1.31 (dimensionless) as shown in the table.

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ωð Þ¼ k

spectrum, based on the area under the spectrum curve [13]:

the gravity acceleration (m/s<sup>2</sup>

134 Surface Waves - New Trends and Developments

as shown in the table.

q

), and d is the water depth (m).

E ¼ ρg

E ¼ ρg

where ωð Þk is the temporal frequency (Hz), k is the spatial frequency wavenumber (1/m), g is

The wave energy in the units of W=m<sup>2</sup> can now be calculated using the estimated wave

ð

where <sup>E</sup> is the wave energy in units of W=m<sup>2</sup> (Joules=s m2), <sup>S</sup>ð Þ <sup>ω</sup> is the wave energy spectrum in units of m2, <sup>ω</sup> is the temporal frequency in units of Hz, <sup>ρ</sup> <sup>¼</sup> 1000 kg=m3 is the density of the water used in this research, and <sup>g</sup> <sup>¼</sup> <sup>9</sup>:8 m=s2 is the acceleration due to gravity. The discrete approximation of wave energy in this research is calculated by summing the area under the spectral curve:

> X N

> > 0

where N is the total number of discrete frequencies integral, Sð Þ ω<sup>n</sup> is the magnitude of the energy spectrum at each frequency ωn in units of m2, and Δω<sup>n</sup> is the discrete frequency interval in units of Hz. A representative shallow-water wind-driven gravity wave can be simulated using the

Table 1. (right) Input equations and parameters for running the wave simulation program (Bostater et al. [11, 12]) for simulating the wave patch shown in Figure 4 and the measured and simulation distributions (left image). Comparison of the calculated Weibull cumulative curve (solid line) and the cumulative curve of the wave energy spectrum extracted from the measured wave displacements with estimated Weibull parameters of α = 0.55 (Hz) and β = 1.31 (dimensionless)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gk tanh kd ð Þ

, (5)

Sð Þ ω dω, (6)

Sð Þ ω<sup>n</sup> Δωn, (7)

Figure 6. Example shallow-water wave displacement (m) and wavelength (m) sequences (a–f) extracted from synthetic imagery (see Figure 5) used to demonstrate the sensitivity of the energy in a simulated wave patch as shown in Figure 7. The simulated image results are shown in Figure 5.

estimated wave spectrum from measurements as shown in Figure 4(a). A wave patch simulation model specific for shallow water such as Banana River and Indian River Lagoon was used [8]. The simulated mean wave height is 7.6 cm, and significant wave height is 0.1 m, which is consistent with the video-based measurements. Sensitivity analysis of the simulation model can also be conducted in order to determine the effects of the model coefficients α and β on the simulated wave patches. The results suggest that the model coefficients α and β can affect wave height scale and wave pattern of the simulated wave patches. Therefore, it is reasonable to assume that α and β are related to physical or environmental variables that affect the wave conditions, such as wind speed, wind duration, wind direction, fetch, water depth, bottom slope, etc. When the wave patch model coefficients are adjusted, realistic random water waves are produced as shown in Figure 4(b). The wave energy spectrum (Eq. (4)) can also be introduced or multiplied by a nondimensional scaling coefficient A = 15 in this example for sensitivity analysis

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A sensitivity analysis of the wave model parameter α (scale parameter) and the β (nondimensional shape parameter) allows one to clearly see how the water surface wave field would look for different parameter values. The sensitivity simulation results are shown in Figure 5 with selected wave patch synthetic images for selected model parameters. Figure 6 shows example wave height displacements, wavenumbers, and significant wave height, and the resulting wave patch energy is simulated wave patches. Figure 7 shows the associated (a–f) wave energy and significant wave heights obtained from synthetic image model runs. The results demonstrate the utility of making observations, followed by sensitivity analysis and resulting wave patch energy, and associated significant wave heights derived from the synthetic images which are represen-

This research developed an imaging methodology to measure wind-driven gravity waves in shallow water using high-definition (HD) video cameras and specially constructed staff gauges. Wave spectrum, wave energy, and significant wave height are estimated from videobased wave height measurement, which can be used to estimate bottom velocity and sediment resuspension. Simulated wave field images are generated using estimated wave spectrum; wave characteristics (significant wave heights) based on the simulation images agree with the

Coefficients α and β in the spectral wave simulation model can affect the wave height and pattern of the simulated wave patches, and they are very likely to be related with some physical and environmental variables. Ongoing research is investigating these variables for these small amplitude water surface waves. The approach developed and described above utilizes the conceptualization shown in Figure 8. The energy stored in a surface water gravity wave field drives the sediment resuspension process in the bottom boundary layer and lutocline consisting

in situ video-based measurement protocol and methods described in Section 3 above.

related to wave patch simulations shown Figures 4 and 5 (Table 1).

tative measured wind-driven surface gravity wave field patches.

5. Summary

Figure 7. Calculated wave energy in a simulated wave patch for different significant wave heights, wave displacement and wavenumber (spatial frequency) time series (a–f), and synthetic wave patch images shown in Figure 6. The representative synthetic mages are shown in Figure 5.

estimated wave spectrum from measurements as shown in Figure 4(a). A wave patch simulation model specific for shallow water such as Banana River and Indian River Lagoon was used [8]. The simulated mean wave height is 7.6 cm, and significant wave height is 0.1 m, which is consistent with the video-based measurements. Sensitivity analysis of the simulation model can also be conducted in order to determine the effects of the model coefficients α and β on the simulated wave patches. The results suggest that the model coefficients α and β can affect wave height scale and wave pattern of the simulated wave patches. Therefore, it is reasonable to assume that α and β are related to physical or environmental variables that affect the wave conditions, such as wind speed, wind duration, wind direction, fetch, water depth, bottom slope, etc. When the wave patch model coefficients are adjusted, realistic random water waves are produced as shown in Figure 4(b). The wave energy spectrum (Eq. (4)) can also be introduced or multiplied by a nondimensional scaling coefficient A = 15 in this example for sensitivity analysis related to wave patch simulations shown Figures 4 and 5 (Table 1).

A sensitivity analysis of the wave model parameter α (scale parameter) and the β (nondimensional shape parameter) allows one to clearly see how the water surface wave field would look for different parameter values. The sensitivity simulation results are shown in Figure 5 with selected wave patch synthetic images for selected model parameters. Figure 6 shows example wave height displacements, wavenumbers, and significant wave height, and the resulting wave patch energy is simulated wave patches. Figure 7 shows the associated (a–f) wave energy and significant wave heights obtained from synthetic image model runs. The results demonstrate the utility of making observations, followed by sensitivity analysis and resulting wave patch energy, and associated significant wave heights derived from the synthetic images which are representative measured wind-driven surface gravity wave field patches.

#### 5. Summary

Figure 7. Calculated wave energy in a simulated wave patch for different significant wave heights, wave displacement and wavenumber (spatial frequency) time series (a–f), and synthetic wave patch images shown in Figure 6. The represen-

tative synthetic mages are shown in Figure 5.

136 Surface Waves - New Trends and Developments

This research developed an imaging methodology to measure wind-driven gravity waves in shallow water using high-definition (HD) video cameras and specially constructed staff gauges. Wave spectrum, wave energy, and significant wave height are estimated from videobased wave height measurement, which can be used to estimate bottom velocity and sediment resuspension. Simulated wave field images are generated using estimated wave spectrum; wave characteristics (significant wave heights) based on the simulation images agree with the in situ video-based measurement protocol and methods described in Section 3 above.

Coefficients α and β in the spectral wave simulation model can affect the wave height and pattern of the simulated wave patches, and they are very likely to be related with some physical and environmental variables. Ongoing research is investigating these variables for these small amplitude water surface waves. The approach developed and described above utilizes the conceptualization shown in Figure 8. The energy stored in a surface water gravity wave field drives the sediment resuspension process in the bottom boundary layer and lutocline consisting

Figure 8. Conceptual model of energy stored in a measured or simulated wave patch and the downward transport of momentum to the bottom boundary layer that can cause resuspension and liquefaction of bottom mud and muck within the lutocline.

of fluid mud and muck. The flowchart shown in Figure 9 summarizes the protocol and methods described above.

Each time series of water surface displacements is linearly detrended before spectral analysis and calculation of the wind wave energy in the wave field. The mono imaging used in the protocol is the simplest method because water surface elevations can be easily obtained using in situ staff gauges. Algorithms can then be used to calculate wave displacement and wave heights using image processing techniques that analyze each video frame as an instantaneous realization of the water surface elevation or displacement at 0.033 s time intervals. The key for successful mono imaging is the bottom mounted staff gauge used for automatic water surface elevation detection. The staff gauge used in this research was scaled to the appropriate measured wave height field and the gauge and inserted into the bottom sediments.

Simulation of wind-driven water wave patch results shown in Figure 5 indicates that as α increases, the wave displacement increases and smaller wavelength waves appear on top of the larger waves. The effect of β is just the opposite of α. Increasing β causes decrease in wave displacements with coincident disappearance of smaller waves on top of the larger waves.

(4.5 4.5 m) using the energy distribution from the sum of the variance in the temporal frequency domain [13]. The difference of energy stored in the simulated surface wave patches results in differences in the surface wave heights. In this research, significant wave height was

Figure 9. Summary of the general procedures used to extract wind-driven gravity wave energy in a wave patch and

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The significant wave height is defined as the average of the highest one-third of the wave heights [10]. It can be used as a representation of wave height of an observed or simulated wave patch. The tendency of simulated significant wave heights changes with α and β as shown in Figures 5–7 and suggests that simulated water surfaces are smooth with a small α and large β using the Weibull wave spectrum. A larger α and smaller β tend to produce a rougher surface with more energy. However, considering the shallow-water circumstances of this research, α and β should be limited to a range appropriate to the experimental conditions from field measurements. The authors believe that the techniques, method, and the resulting protocol for the measurement of wind-driven surface water gravity waves described and

used to represent the simulated surface state.

resulting wave energy spectrum from measured wind wave videos.

The wave energy spectrum graphs (obtained from the synthetic images) shown in Figures 6 and 7 also provide a visual demonstration of the effect of α and β on wind-driven waves using the Weibull model. Increasing α causes the energy distribution to spread into larger wavenumber in the spatial frequency domain (1/m). This reveals that a simulated wave patch with larger α is consistent with waves with a greater variety of wavelengths. However, increasing β causes the narrowing of the energy distribution to smaller wavenumber (1/m) and suppresses the frequency of larger wavelength waves in a simulated wave patch.

From an energy point of view, wave energy in units of W/m<sup>2</sup> increases with increasing α and decreasing β. The wave energy can be calculated within the simulated wave patch area

of fluid mud and muck. The flowchart shown in Figure 9 summarizes the protocol and methods

Figure 8. Conceptual model of energy stored in a measured or simulated wave patch and the downward transport of momentum to the bottom boundary layer that can cause resuspension and liquefaction of bottom mud and muck within

Each time series of water surface displacements is linearly detrended before spectral analysis and calculation of the wind wave energy in the wave field. The mono imaging used in the protocol is the simplest method because water surface elevations can be easily obtained using in situ staff gauges. Algorithms can then be used to calculate wave displacement and wave heights using image processing techniques that analyze each video frame as an instantaneous realization of the water surface elevation or displacement at 0.033 s time intervals. The key for successful mono imaging is the bottom mounted staff gauge used for automatic water surface elevation detection. The staff gauge used in this research was scaled to the appropriate mea-

Simulation of wind-driven water wave patch results shown in Figure 5 indicates that as α increases, the wave displacement increases and smaller wavelength waves appear on top of the larger waves. The effect of β is just the opposite of α. Increasing β causes decrease in wave displacements with coincident disappearance of smaller waves on top of the larger waves.

The wave energy spectrum graphs (obtained from the synthetic images) shown in Figures 6 and 7 also provide a visual demonstration of the effect of α and β on wind-driven waves using the Weibull model. Increasing α causes the energy distribution to spread into larger wavenumber in the spatial frequency domain (1/m). This reveals that a simulated wave patch with larger α is consistent with waves with a greater variety of wavelengths. However, increasing β causes the narrowing of the energy distribution to smaller wavenumber (1/m)

From an energy point of view, wave energy in units of W/m<sup>2</sup> increases with increasing α and decreasing β. The wave energy can be calculated within the simulated wave patch area

and suppresses the frequency of larger wavelength waves in a simulated wave patch.

sured wave height field and the gauge and inserted into the bottom sediments.

described above.

138 Surface Waves - New Trends and Developments

the lutocline.

Figure 9. Summary of the general procedures used to extract wind-driven gravity wave energy in a wave patch and resulting wave energy spectrum from measured wind wave videos.

(4.5 4.5 m) using the energy distribution from the sum of the variance in the temporal frequency domain [13]. The difference of energy stored in the simulated surface wave patches results in differences in the surface wave heights. In this research, significant wave height was used to represent the simulated surface state.

The significant wave height is defined as the average of the highest one-third of the wave heights [10]. It can be used as a representation of wave height of an observed or simulated wave patch. The tendency of simulated significant wave heights changes with α and β as shown in Figures 5–7 and suggests that simulated water surfaces are smooth with a small α and large β using the Weibull wave spectrum. A larger α and smaller β tend to produce a rougher surface with more energy. However, considering the shallow-water circumstances of this research, α and β should be limited to a range appropriate to the experimental conditions from field measurements. The authors believe that the techniques, method, and the resulting protocol for the measurement of wind-driven surface water gravity waves described and

Figure 10. Steps performed to extract wind-driven gravity wave energy and spectra using the protocol methodology described above.

summarized in Figures 9 and 10 will allow other scientists, engineers, and meteorologists access to an inexpensive and rapidly deployable instrumental approach for shallow waters.

Practical use and applications of water surface video imaging include studying of coastal nearshore processes [14]. The use of fixed sensor video platforms such as bridges, piers, and building which view the littoral zone for water quality constituent determinations has also been demonstrated [15] including the mathematical methods to correct video hyperspectral push broom imagery taken at high oblique angles. Video imaging of bottom features such as seagrasses and sand bottom features has also been demonstrated using practical imaging techniques [16]. The imaging technique described above is being used to assist in assessing moving fluid mud and muck in Indian River Lagoon and Banana River estuarine areas and tributaries [2]. In many shallow-water estuarine regions, wind-driven gravity waves are responsible for resuspension and transport of muds and decaying organic matter within the lutocline [17, 18]. Figure 11 shows the results of the vertical profile of horizontal mass transport (fluxes) using the water wave video imaging technique in conjunction with simultaneous deployment of a vertical array of six passive Sondes [20] during October 2017. Wind speed during the video wave gauge measurements were 16-17 knots.

zones. In essence, the novel approach [1] requires no expensive equipment [2], can be easily constructed, and [3] is self-calibrated to video imagery. The method can be used in conjunction with sophisticated wave patch imaging models and when used as a system can be used to improve scientific and engineering understanding of sediment transport due to wind-driven waves. The imaging techniques are currently being used in the field of transportation construction engineering concerned with pile driving rebound imaging and related soil engineering

Figure 11. Result (left) from deployment of a vertical array of fluid mud Sondes [20] (right images) in conjunction with

water littoral zone [2]. The figure demonstrates that water surface gravity waves in shallow coastal areas are responsible for suspending and transporting fluid mud and muck under steady winds of ≈16–17 knots with dry weight mass fluxes

) at the bottom greater than 450 times the surface transport within surface wave field.

33.0200N, 80�38<sup>0</sup>

Video Measurements and Analysis of Surface Gravity Waves in Shallow Water

http://dx.doi.org/10.5772/intechopen.73042

141

14.7200W)) in a shallow-

video imaging of wind-driven water surface gravity waves (October 2017 (28�12<sup>0</sup>

The work presented in this paper has been supported in part by the Northrop Grumman Corporation, the NASA, the Kennedy Space Center, the KB Science, the National Science Foundation, the US-Canadian Fulbright Program, and the US Department of Education, the FIPSE and European Union's grant Atlantis STARS (Sensing Technology and Robotic Systems) to Florida Institute of Technology, the Budapest University of Technology and Economics (BME), and the Belgium Royal Military Academy, Brussels, in order to support the involvement of students in obtaining international dual US-EU undergraduate engineering degrees.

problems.

(g m�<sup>2</sup> day�<sup>1</sup>

Acknowledgements

Sonde array [20] deployment was 16–17 knots, with east winds in a shallow ≈0.7 m water column and 20–30 cm amplitude wind-driven gravity waves at the location in Banana River near Pineda Causeway [2, 16] near the Atlantic Ocean in Florida (28�12<sup>0</sup> 33.0200N, 80�38<sup>0</sup> 14.7200W). The use of the staff gauge video imaging and Sondes shown in Figure 11 showed that the shallow-water gravity waves increase the bottom water fine particulate transport or flux of 15,071 g m�<sup>2</sup> day�<sup>1</sup> dry weight—nearly 450 times greater than the surface values of 13.1 g m�<sup>2</sup> day�<sup>1</sup> . The vertical profile of sediment and particulate mass flux or movement under the waves follows the same profile found in previous studies in tributaries during high flow conditions [19]. The above recent result describes the application of the novel approach to water wave imaging in shallow littoral

Figure 11. Result (left) from deployment of a vertical array of fluid mud Sondes [20] (right images) in conjunction with video imaging of wind-driven water surface gravity waves (October 2017 (28�12<sup>0</sup> 33.0200N, 80�38<sup>0</sup> 14.7200W)) in a shallowwater littoral zone [2]. The figure demonstrates that water surface gravity waves in shallow coastal areas are responsible for suspending and transporting fluid mud and muck under steady winds of ≈16–17 knots with dry weight mass fluxes (g m�<sup>2</sup> day�<sup>1</sup> ) at the bottom greater than 450 times the surface transport within surface wave field.

zones. In essence, the novel approach [1] requires no expensive equipment [2], can be easily constructed, and [3] is self-calibrated to video imagery. The method can be used in conjunction with sophisticated wave patch imaging models and when used as a system can be used to improve scientific and engineering understanding of sediment transport due to wind-driven waves. The imaging techniques are currently being used in the field of transportation construction engineering concerned with pile driving rebound imaging and related soil engineering problems.

#### Acknowledgements

summarized in Figures 9 and 10 will allow other scientists, engineers, and meteorologists access to an inexpensive and rapidly deployable instrumental approach for shallow waters.

Figure 10. Steps performed to extract wind-driven gravity wave energy and spectra using the protocol methodology

Practical use and applications of water surface video imaging include studying of coastal nearshore processes [14]. The use of fixed sensor video platforms such as bridges, piers, and building which view the littoral zone for water quality constituent determinations has also been demonstrated [15] including the mathematical methods to correct video hyperspectral push broom imagery taken at high oblique angles. Video imaging of bottom features such as seagrasses and sand bottom features has also been demonstrated using practical imaging techniques [16]. The imaging technique described above is being used to assist in assessing moving fluid mud and muck in Indian River Lagoon and Banana River estuarine areas and tributaries [2]. In many shallow-water estuarine regions, wind-driven gravity waves are responsible for resuspension and transport of muds and decaying organic matter within the lutocline [17, 18]. Figure 11 shows the results of the vertical profile of horizontal mass transport (fluxes) using the water wave video imaging technique in conjunction with simultaneous deployment of a vertical array of six passive Sondes [20] during October 2017. Wind speed

Sonde array [20] deployment was 16–17 knots, with east winds in a shallow ≈0.7 m water column and 20–30 cm amplitude wind-driven gravity waves at the location in Banana River near Pineda

the staff gauge video imaging and Sondes shown in Figure 11 showed that the shallow-water gravity waves increase the bottom water fine particulate transport or flux of 15,071 g m�<sup>2</sup> day�<sup>1</sup>

profile of sediment and particulate mass flux or movement under the waves follows the same profile found in previous studies in tributaries during high flow conditions [19]. The above recent result describes the application of the novel approach to water wave imaging in shallow littoral

dry weight—nearly 450 times greater than the surface values of 13.1 g m�<sup>2</sup> day�<sup>1</sup>

33.0200N, 80�38<sup>0</sup>

14.7200W). The use of

. The vertical

during the video wave gauge measurements were 16-17 knots.

described above.

140 Surface Waves - New Trends and Developments

Causeway [2, 16] near the Atlantic Ocean in Florida (28�12<sup>0</sup>

The work presented in this paper has been supported in part by the Northrop Grumman Corporation, the NASA, the Kennedy Space Center, the KB Science, the National Science Foundation, the US-Canadian Fulbright Program, and the US Department of Education, the FIPSE and European Union's grant Atlantis STARS (Sensing Technology and Robotic Systems) to Florida Institute of Technology, the Budapest University of Technology and Economics (BME), and the Belgium Royal Military Academy, Brussels, in order to support the involvement of students in obtaining international dual US-EU undergraduate engineering degrees.

#### Author details

Charles R. Bostater Jr\*, Bingyu Yang and Tyler Rotkiske

\*Address all correspondence to: bostater@fit.edu

Marine Environmental Optics Laboratory and Remote Sensing Center, Ocean Engineering and Sciences, College of Engineering, Florida Institute of Technology, Melbourne, Florida, USA

[14] Holland KT, Holman RA, Lippmann TC, Stanley J, Plant N. Practical use of video imagery in nearshore oceanographic field studies. IEEE Journal of Oceanic Engineering. 1997;

Video Measurements and Analysis of Surface Gravity Waves in Shallow Water

http://dx.doi.org/10.5772/intechopen.73042

143

[15] Bostater CR, Oney TS. Collection and corrections of oblique multiangle hyperspectral bidirectional reflectance imagery of the water surface. SPIE. 2017;10422:1042209-1-1042209-22

[16] Bostater CR, Oney TS, Rotkiske T, Aziz S, Morrisette C, Callahan K, Mcallister D. Hyperspectral signatures and WorldView-3 imagery of Indian River Lagoon and Banana

[17] Mehta A, Lee A, Li Y. Fluid Mud and Water Waves: A Brief Review of Processes and Simple Modeling Approaches. US Army Corps of Engineers, CR DRP-94-4. 1994. 79 pp

[18] McAnally W et al. Management of fluid mud in estuaries, bays, and lakes, II: Present state of understanding on character and behavior. Journal of Hydraulic Engineering, ASCE.

[19] Vanoni VA, editor. Sedimentation Engineering. American Society of Civil Engineering, ASCE Hydraulic Division Task Committee. 1975. p. 745. ISBN: 0-87262-001-8

[20] Bostater C, Rotkiske T, Oney T, Obot E. "Design and Operation of Sonde Arrays to Measure Fluid Mud in the Marine Environment", Proceedings of ISOPE. 2017. pp. 1511-

River Estuarine Water and bottom types. SPIE. 2017;10422:104220E-5-104220E-13

22(1):81-92

2007;133:23-38

1514. ISBN: 978-1-880653-97-5

#### References


[14] Holland KT, Holman RA, Lippmann TC, Stanley J, Plant N. Practical use of video imagery in nearshore oceanographic field studies. IEEE Journal of Oceanic Engineering. 1997; 22(1):81-92

Author details

142 Surface Waves - New Trends and Developments

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Charles R. Bostater Jr\*, Bingyu Yang and Tyler Rotkiske

Marine Environmental Optics Laboratory and Remote Sensing Center, Ocean Engineering and Sciences, College of Engineering, Florida Institute of Technology, Melbourne, Florida, USA

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\*Address all correspondence to: bostater@fit.edu


## *Edited by Farzad Ebrahimi*

Surface waves have drawn a significant attention and interest in the recent years in a broad range of commercial applications, while their commercial developments have been supported by fundamental and applied research studies. This book is a result of contributions of experts from international scientific community working in different aspects of *surface waves* and reports on the state-of-the-art research and development findings on this topic through original and innovative research studies. It contains up-to-date publications of leading experts, and the edition is intended to furnish valuable recent information to the professionals involved in surface wave analysis and applications. The text is addressed not only to researchers but also to professional engineers, students, and other experts in various disciplines, both academic and industrial, seeking to gain a better understanding of what has been done in the field recently and what kind of open problems are in this area.

Published in London, UK © 2018 IntechOpen © Meriç Dağlı / unsplash

Surface Waves - New Trends and Developments

Surface Waves

New Trends and Developments