Computational Methods for Electromagnetic and Optical Systems (Optical Engineering)

Copyright © 2000 Marcel Dekker, Inc.

Copyright © 2000 Marcel Dekker, Inc.

Copyright © 2000 Marcel Dekker, Inc.

Copyright © 2000 Marcel Dekker, Inc.

ISBN: 0-8247-7916-9 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http:/www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright # 2000 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, micro®lming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10 9 8 7 6 5 4 3 2 PRINTED IN THE UNITED STATES OF AMERICA

Copyright © 2000 Marcel Dekker, Inc.

John Jarem dedicates this book to his wife, Elizabeth A. Connell Jarem, and his children, Amy, Chrissy, and Sean. Partha Banerjee dedicates this book to his wife, Noriko Tsuchihashi Banerjee, and his sons, Hans and Neil.

Copyright © 2000 Marcel Dekker, Inc.

From the Series Editor

This volume is about neither mathematics for the sake of mathematics nor electromagnetic theory for the sake of electromagnetic theory. It is about the important and useful computational methods that need to be applied to the analysis and hence the design of electromagnetic and optical systems. Computational Methods for Electromagnetic and Optical Systems presents the best and most pertinent mathematical tools for the solution of current and future analysis and synthesis of systems applications without overgeneralization; that means using the best and most appropriate tools for the problem at hand. Optical design certainly proves that some problems can be evaluated by ray tracing; others need scalar wave theory; still others need electromagnetic wave analysis; and, ®nally, some systems require a quantum optics approach. Thus, rays, waves, and photons have coexisted in optical science and engineering, each with its own domain of validity and each with its own computational methods. Solutions of Maxwell's equations are described that can be applied to the analysis of diffraction gratings, radiation, and scattering from dielectric objects and holograms in photorefractive materials. Fundamentally it is necessary to understand how electromagnetic radiation is transmitted, re¯ected, and refracted through one- and two-dimensional isotropic and anistrophic materials. One- and two-dimensional Fourier transform theory allows for the study of how spectral components are propagated. The alternative method of split-step beam propagation can be applied to inhomogeneous media. Other computational methods covered in these pages include: coupledwave analysis of inhomogeneous cylindrical and spherical systems, state variable methods for the propagation of anisotropic waveguide systems, and rigorous coupled wave analysis for photorefractive devices and systems.

Copyright © 2000 Marcel Dekker, Inc.

The computational methods described here should be very valuable whether the reader needs to simulate, analyze, or design electromagnetic and optical systems. Brian J. Thompson

Copyright © 2000 Marcel Dekker, Inc.

Preface

Exact solutions of problems in electromagnetics and optics have become an increasingly important area of research. The analysis and design of modern applications in optics and those in traditional electromagnetics demand increasingly similar numerical computations due to reduction in feature sizes in optics. In electromagnetics a large amount of research concentrates on numerical analysis techniques such as the method of moments, ®nite element analysis, and the ®nite difference analysis technique. In the ®eld of optics (a part of electromagnetics), much research has been done on the analysis of thin and thick diffraction gratings for application to spectrometry and holography. From the late 1970s to the present, an extremely important technique for the analysis of planar diffraction gratings, developed by different researchers, has been a state variable technique called rigorous coupled wave analysis. This technique is based on expanding Maxwell's equations in periodic media in a set of Floquet harmonics and, from this expansion, arranging the unknown expansion variable in state variable form, from which all unknowns of the system can be solved. For planar diffraction gratings this technique has proved to be very effective, providing a fast, accurate solution and involving only a small matrix and eigenvalue equation for the solution. In control theory and applications, the state variable method has been widely applied and in fact forms a foundation for this area. In the electromagnetics area (including optics), the state variable method, although a powerful analysis tool, has seen much less application. When used, it is applied in conjunction with other methods (for example, the spectral domain method, transmission ladder techniques, K-space analysis techniques, and the spectral matrix method) and is rarely listed as a state variable method. The purpose of the present volume is to tie together different applications in electromagnetics and optics in which the state variable

Copyright © 2000 Marcel Dekker, Inc.

method is used. We place special emphasis on the analysis of planar diffraction gratings using the rigorous coupled wave theory method. This book introduces students and researchers to a variety of spectral computational techniques including K-space theory, Floquet theory, and the beam propagation technique, which are then used to analyze a variety of electromagnetic and optical systems. Examples include analysis of radiation through isotropic and anisotropic material slabs, planar diffraction gratings in isotropic and anisotropic media, propagation through nonlinear and inhomogeneous optical media, radiation and scattering from threedimensionally inhomogeneous cylindrical and spherical structures, and diffraction from photorefractive materials. The K-space and Floquet theory are applied in the form of a recently developed algorithm called rigorous coupled wave analysis. A full-®eld approach is used to solve Maxwell's equations in anistropic media in which standard wave equation approach is intractable. The spectral techniques are also used to analyze wave mixing and diffraction from dynamically induced nonlinear anisotropic gratings such as in photorefractive materials. This book should be particularly valuable for researchers interested in accurately solving electromagnetic and optical problems involving anisotropic materials. Ef®cient and current, rapidly convergent, numerical algorithms are presented. The organization of the book is as follows. In Chapter 1, mathematical preliminaries, including the Fourier series, Fourier integrals, Maxwell's equations, and a brief review of eigenanalysis, are presented. Chapter 2 deals with the K-space state variable formulation, including applications to anisotropic and bianisotropic planar systems. Chapter 3 covers the state variable method and the rigorous coupled wave analysis method as applied to planar diffraction gratings. Many types of gratings are analyzed, including thin and thick gratings, surface relief gratings, re¯ection gratings, and anistropic crossed diffraction gratings. In both Chapters 2 and 3, we apply the complex Poynting theorem to validate numerical solutions. Chapter 4 reviews the split-step beam propagation method for beam and pulse propagation. Chapter 5 applies the state variable method and rigorous coupled wave theory to the solution of cylindrical and spherical scattering problems. The interesting problem of scattering from a cylindrical diffraction is considered. Chapter 6 uses state variable and full-®eld analysis to study modal propagation in anisotropic, inhomogeneous waveguides and in anisotropic, transversely periodic media. Chapter 7 is concerned with the use of spectral techniques and rigorous coupled wave theory to study dynamic waves moving in photorefractive materials with emphasis on induced transmission and re¯ection gratings. The intended primary audience is seniors and graduate students in electrical and optical engineering and physics. The book should be useful for

Copyright © 2000 Marcel Dekker, Inc.

researchers in optics specializing in holography, gratings, nonlinear optics, and photorefractives, as well as researchers in electromagnetics working in antennas, propagation and scattering theory, or electromagnetic numerical methods. The book will also be of interest to the military, industry, and academia, and to all interested in solving various types of electromagnetic propagation problems. The book should be ideal for either classroom adoption or as an ancillary reference in graduate-level courses such as numerical methods in electromagnetics, diffractive optics, or electromagnetic scattering theory. We would like to acknowledge Dr. Brian J. Thompson for encouraging us to write this book and for his interest in the subject. We are also indebted to Linda Grubbs, who typed parts of the manuscript. We acknowledge all those who allowed us to reproduce part of their work. We also thank the ECE department at the University of Alabama for their long-term support, which made the writing possible. Finally, we acknowledge the support and encouragement of our wives, Elizabeth Jarem and Noriko Banerjee, and our parents and families, during the writing of the book. John M. Jarem Partha P. Banerjee

Copyright © 2000 Marcel Dekker, Inc.

Contents

From the Series Editor Preface

Brian J. Thompson

1.

Mathematical Preliminaries

2.

Spectral State Variable Formulation for Planar Systems

3.

Planar Diffraction Gratings

4.

The Split-Step Beam Propagation Method

5.

Rigorous Coupled Wave Analysis of Inhomogeneous Cylindrical and Spherical Systems

6.

Modal Propagation in an Anisotropic Inhomogeneous Waveguide and Periodic Media

7.

Application of Rigorous Coupled Wave Analysis to Analysis of Induced Photorefractive Gratings

Copyright © 2000 Marcel Dekker, Inc.

      





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2    
  

  

2.1

INTRODUCTION

A problem that is extremely important in optics, microwave theory, antenna theory, and electromagnetics in general [1±34] is the way radiation is transmitted, re¯ected, refracted, and propagates through two-dimensionally in®nite homogeneous material layer systems. This problem has been studied for a wide variety of different material layers, e.g., isotropic dielectric materials, isotropic permeable materials, anisotropic dielectric and permeable materials, and bi-anisotropic materials. It has also been studied when a wide variety of different types of electromagnetic (EM) source radiation is incident on, or is present in, a layer of the planar system, e.g., incident plane wave, dipole source, line source, Gaussian beam, antenna source, waveguide-¯ange system, microstrip line source strip. The synthesis and design of isotropic planar multilayer optical systems has also received considerable attention [11-13]. In carrying out EM studies of these types of systems, a very powerful tool for analysis [1±10] is provided by one- and two-dimensional Fourier transform theory (also called k-space theory). This theory is a powerful tool because it allows virtually any time-reduced EM source in any layer to be represented as a sum of plane waves whose propagation through the layers of the system can be analyzed in several manageable, tractable ways. Thus by using two-dimensional Fourier transform theory one can study (a) how individual plane spectral components propagate through the overall EM system, (b) the strength of the spectral components that are excited by the source in the system, and (c) the overall spatial response of the system at any given point in the system by adding up (using superposition) the different spectral components.

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The determination of the EM ®elds and their propagation, re¯ection, transmission, and scattering from isotropic, anisotropic, and bi-anisotropic planar layered media has received wide attention for a long time. References 2 and 8 give a complete review and description of re¯ection from planar isotropic single and multilayers. A topic that has received less attention but still has been studied by a number of researchers is the problem of determining the radiation and scattering when sources and external incident ®elds (plane waves, Gaussian beams, etc.) excite EM ®elds in an anisotropic or bianisotropic planar multilayer system. The anisotropic and bi-anisotropic EM scattering problem is considerably more dif®cult to analyze than the isotropic case because the anisotropic or bi-anisotropic constitutive material parameters couple the ®eld components together, creating from Maxwell's equations a much more complicated system than arises in the isotropic case. In most isotropic propagation problems the typical approach, based on Maxwell's equations, is to decouple one component from one another and then derive a second-order partial differential wave equation from which the solution to the EM problem can be obtained. For most anisotropic and bianisotropic scattering problems this procedure is quite intractable. Attempting this procedure for most anisotropic or bi-anisotropic systems would lead to fourth-, sixth-, or eighth-order partial differential equations that would be quite dif®cult to solve. For anisotropic and bi-anisotropic materials an alternate procedure that has been developed for transversely homogeneous planar layers is to Fourier transform all EM ®eld quantities with respect to (w.r.t.) the transverse coordinate(s) and then algebraically manipulate the reduced Fourier transformed ®eld variable equations into a standard state variable form. Eigenanalysis of these ®rst-order state variable equations yields the propagation constants and propagation modes of the system. In this procedure, the two longitudinal ®eld components are expressed in terms of the four transverse ®eld components and then substituted into Maxwell's equations to reduce the system to a 4  4 state variable form. Expressing the longitudinal ®elds in terms of the transverse ®elds is useful as it allows simple boundary matching of the tangential ®eld components from one layer interface to another. The eigenanalysis method is also known as the exponential matrix method [25,26] and was discussed in Chapter 1. The approach just mentioned [18±29] will be used in this chapter and consists of (1) replacing ®rst-order transverse derivative operators with terms proportional to their wavenumbers @=@   , @=@    , (2) writing out the six ®eld component equations (these equations will contain ®rst-order longitudinal derivative operator terms @=@ , (3) manipulating these equations so as to eliminate the longitudinal electric ®eld component  and the longitudinal magnetic ®eld component 

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(this reduces the number of curl equations from six to four), and ®nally (4) putting the four remaining equations into a standard 4  4 ®rst-order state variable matrix equation form. The four transverse components  ,  ,  , and  form the components of the 4  1 state variable column matrix. As shown in Section 2.4, this procedure provides a straightforward method of analyzing bi-anisotropic material layers whenever oblique and arbitrarily polarized plane wave radiation is incident on the material layers. This 4  4 state variable matrix procedure has been ®rst implemented by Teitler and Henvis [19], and perhaps others, who have reduced Maxwell's equations in an anisotropic layer to a set of four ®rst-order linear differential equations and then, assuming an exponential form of solution, have solved for the normal or eigen modes that describe propagation in the layer. The method is further developed by Berreman [20], who, starting from Maxwell's six component equations, puts the general anisotropic equations into a 4  4 form (where the 4  1 column vector contains the two tangential electric ®eld components and two tangential magnetic ®eld components), and then solves, using matrix techniques, for the four eigenvectors and eigenvalues of the system. Berreman [20] has studied several anisotropic material examples, including propagation in an orthorhombic crystal, propagation in an optically active material (described by the Drude model), and propagation involving Faraday rotation based on Born's model. Berreman [20] has also considered the state variable method as applied to determining propagation in media that are anisotropic and longitudinally periodic. LinChung and Teitler [21], Krowne [22], and Morgan et al. [23] have used the 4  4 matrix method of Berreman [20] to study propagation of plane waves in strati®ed or multilayer anisotropic media. Weiss and Gaylord [24] have used the Berreman method to study strati®ed multilayer resonators and optical ®lters (Fabry-Perot/Solc ®lter) composed of anisotropic materials. Two recent papers by Yang [25,26] study the important problem of formulating the EM state variable equations so that ef®cient numerical solution of the equations arises. This problem has also been studied by Moharam et al. (see Ref. 23 in Chapter 3 of this book). Dispersion in anisotropic and birefringent materials, and properties of the EM ®eld propagation in these materials, have been studied by many other researchers. Yeh [27] has studied EM propagation in layered birefringent media. Alexopoulos and Uslenghi [28] study re¯ection and transmission with arbitrarily graded parameters. Graglia et al. [29] study dispersion relations for bi-anisotropic materials and their symmetry properties. The book by Lindell et al. [6] also quotes many papers that have studied propagation in bi-anisotropic materials. Another area where the k-space state variable analysis is useful is in the problem of characterizing radiation from antennas, dipoles, and metallic

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structures in millimeter and microwave integrated circuits (MMICs). Several papers [30±34] have studied the problem of determining the radiation from arbitrarily oriented electric and magnetic dipoles embedded in anisotropic planar layers. Tsalamengas and Uzunoglu [32] have studied the problem of determining the EM ®elds of an electric dipole in the presence of a general anisotropic layer backed by a ground plane. Their method consists of Fourier transforming all EM ®elds in the transverse coordinates, casting the Fourier transformed differential equations into the form of a ®rstorder matrix differential equation, and, after solving this, matching EM boundary conditions at the half-space±anisotropic layer interface, to determine all ®elds of the system. An interesting feature of the Tsalamengas and Uzunoglu [32] method is that they have de®ned auxiliary vector components (the electric ®eld and magnetic ®eld were resolved into components parallel and perpendicular to the planar interfaces) that allow them to construct a matrix solution where the ground plane boundary condition is built into their matrix solution. This simpli®es the problem to matching of the boundary conditions at the half space±anisotropic layer boundary. Tsalamengas and Uzunoglu [32] have solved several numerical examples including radiation from a dipole when uniaxial materials, ferrites, or magnetoplasmas comprise the anisotropic layer. The method differs from other methods in that the fundamental matrix differential equation is for a 2  2 matrix rather than the usual column matrices used by almost all other researchers. Krowne [34] has used Fourier transform theory and the 4  4 matrix formalism of Berreman [20] to study propagation in layered, completely general bi-anisotropic media and to study Green's functions in bi-anisotropic media. Krowne's [34] analysis, in addition to determining the modes of propagation in all bi-anisotropic layers, includes the effect of arbitrary electric and magnetic surface currents located at the interfaces of the bi-anisotropic layers. The surface current sources are delta source functions in the spatial domain and therefore planar sources in the Fourier k-space transform domain. Tang [31] has studied the EM ®elds in anisotropic media due to dipole sources using Sommerfeld integrals and a transverse electric and transverse magnetic decomposition of the ®elds of the system. Ali and Mahmoud [30] have also studied dipole radiation in strati®ed anisotropic materials using a 3  3 state variable matrix technique. In addition to the state variable analysis, a second theme that will be developed in this chapter is the use of the complex Poynting theorem as an information aid to the computation of the EM ®elds of the system. First, the complex Poynting theorem will be used as a cross-check of the numerical calculations themselves. The use of this theorem over a given region of space, regardless of whether the region contains lossy (gain) material or

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not, must show equality between the power radiated out of the region and the power dissipated and energy stored in the region. This is a more stringent and useful test than the more standard test of checking conservation of power from one layer to another. Checking power conservation from one layer to another is a conclusive test as long as the materials inside the layers are nonlossy. It is inconclusive if the layers inside are lossy, since in this case the power transmitted out of a given region will necessarily be less than the power transmitted into the given region, since some power must be dissipated as heat in the lossy layer. The complex Poynting theorem on the other hand accounts for not only all power transmitted into and out of a given region but also all power dissipated and energy stored in the region. In a given computation, if the surface and volume integrals of the complex Poynting theorem do not agree precisely, some degree of numerical error has been made in the computation. If the agreement is too poor, most likely a signi®cant computational error has been made somewhere in the calculations, and it is most likely that the computations cannot be trusted. A second way that the complex Poynting theorem is an aid to EM ®eld analysis is that it can give insight into the way that energy is stored and power is dissipated in a given region of space. Often in making EM ®eld plots, the plots of the individual ®eld components, either electric or magnetic, can be deceptive, since, for example, the ®elds can appear large but in reality be standing waves, which are actually transmitting very little real power into a system. Plots of the energy stored and power dissipated then give great insight into how EM radiation is actually interacting with a material at a given place in space. In what follows, both the state variable method (in conjunction with kspace analysis) and the complex Poynting theorem will be applied to study a wide variety of different EM planar re¯ection and transmission problems. Section 2.2 will consider one of the simplest possible cases, namely when a normally incident plane wave impinges on an isotropic lossy material slab. Section 2.3 will study the case when an oblique incident plane wave impinges on an anisotropic layer. Section 2.4 will develop the general 4  4 state variable equations that apply to re¯ection and transmission through a general bi-anisotropic layer. The analysis will apply to the case when the incident radiation is an oblique arbitrarily polarized plane wave. The complex Poynting theorem will also be applied to this case. Section 2.5 will consider cases when EM sources that are not plane waves impinge on an anisotropic layer. In this section k-space theory is used to decompose the EM source into a plane wave Fourier spectrum from which a tractable analysis can be carried out. In particular, the cases of a waveguide±¯ange system that radiates into an anisotropic lossy layer are considered. The expression for the wave slot admittance is developed. In this

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section radiation of a Gaussian beam through the anisotropic layer is also considered. The complex Poynting theorem is applied to radiation in this section. Section 2.6 summarizes the work of Tsalamengas and Uzunoglu [32] who have considered the case of EM radiation from a dipole in the proximity of a general anisotropic grounded layer using k-space theory. Finally, Section 2.7 presents the work of Yang [25,26], which concerns ef®cient methods of solving the state variable equations when large evanescent plane wave components are present in the analysis. In this case, the presence of the large evanescent plane waves causes severe numerical singularity of the solutions. Yang presents a method of removing these singularities from the calculations, yielding a useful EM solution. Overall in this chapter only cases of homogeneous single-layer material slabs are considered. Only a single-layer analysis has been carried out in order to make the analysis as simple and clear as possible. Extension to multilayer analysis is straightforward. Later chapters use multilayer analyses extensively. The multilayer analysis is described thoroughly in these chapters.

2.2 2.2.1

STATE VARIABLE ANALYSIS OF AN ISOTROPIC LAYER Introduction

In this section we study one of the simplest EM state variable problems, namely the problem of determining the EM ®elds that result when a plane wave propagates with normal incidence in an isotropic lossy dielectric slab ~2  ~   ~  , ~ 2  ~   ~  ) (see Fig. 1). Three cases are studied: (1) a plane wave is normally incident on the slab, (2) a plane wave is normally incident on the slab backed by a perfect conductor, and (3) the EM ®elds are excited by an electric or magnetic current source. These cases are solved by the state variable method. Because all eigenvectors or eigenmodes of the state variable system can be solved in closed form, these examples show in a simple manner the principles and properties of the state variable formalism that apply to much more complicated problems (anisotropic planar slabs, diffraction gratings, etc.). 2.2.2

Analysis

To begin the analysis in this section we assume that all propagation is at normal incidence and that the EM ®elds of the system in Regions 1, 2, and 3 ~ ; ~ 

~ coordinate system are given by in an ;

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Figure 1 Geometry of a planar dielectric layer and a complex Poynting box. `  ~` =0 , `  ~ ` =0 , `  1; 2; 3, 0  8:85  10 12 (F/m), 0  4  10 7 (H/m).

 ~ ^  `   ` 

 ~ ^  `  ` 

`  1; 2; 3

2:2:1

where ` denotes the Region number. From Maxwell's equations assuming source free regions,   ~   `  !~ `  `

  ~   `  !~`  `

2:2:2

we ®nd substituting Eq. 2.2.1 that @ `  !~ ` ` @~ @`  !~`  ` @~

2:2:3

It is convenient to make the above equations dimensionless. We introduce   ~ 0  0 =0  377 , the state variables  `  ( ` , `  )` =0 ,   0 ,  0  ! 0 0  2  , where !  2 ,  is the frequency, and  is the freespace wavelength, `  ~ ` =0  ` ` , `  ~` =0  ` ` , and after substitution we ®nd

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@( `   ` )` @

2:2:4

@)`   ` ( ` @ Letting 

( ` 5 )`





0 + `

` 0



2:2:5

(and dropping the ` subscript for the moment) we may write Eq. 2.2.5 in the general state variable from @5 

 +5 

@

2:2:6

Equation 2.2.6 can be solved by determining the eigenvalues and eigenvectors of the matrix + according to the equation +5  5

2:2:7

From this, the general solution of Eq. 2.2.6 is then given by 5

 

* 5  

1

2:2:8

where   2, 5 and  ,  1; 2, are eigenvectors and eigenvalues of the matrix +, and * are general constants. We may demonstrate that 5   is a solution of Eq. 2.2.8 by direct substitution. We have for  1; 2, = 5    5 =     5   . But  5  +5 , hence  5    + 5  



2:2:9

which is the original equation. Superposition of the distinct modes of 5   then gives the full EM solution. The eigenvalues of  ,  1; 2; of + in Eq. 2.2.7 satisfy det+ 
   2  2    0

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or 2     0

2:2:10

Let      1 ,  > 0, > 0 ( and are real numbers), be the forward traveling mode in the `  1; 2; 3 regions. Substituting in Eq. 2.2.10 we have    2             0

2:2:11

After performing algebra it is found that 1=2 1  2 2 1=2         2    1=2 1  2 2 1=2        2 

2:2:12

where                            . Usually,   > 0. We note that      ,  1, corresponds to a forward traveling wave and that      ,  2, corresponds to a backward traveling wave in all regions of the system. We also note that these solutions obey proper boundary conditions in all regions. For example, in Region 3, we have for the forward traveling wave  1 , that for the exponential part of the EM wave,   exp    0 as    when  > 0, and for the oscillary part of the wave  ;   cos   ! , which indicates a wave traveling to the right, since the phase velocity +'  != < 0. A similar analysis in Region 1 shows that the second eigenvalue 2     corresponds to a backward traveling wave. The eigenvector 51  ( 1 ; )1  , 52  ( 2 ; )2  can be determined from Eq. 2.2.7 after substitution of the eigenvalue  ,  1; 2, into Eq. 2.2.10. For the forward traveling wave in any of the three regions we have 1      , 0



1 

 1





( 1 )1



2:2:13

Because 1 is an eigenvalue, the two equations of Eq. 2.2.13 are linearly dependent. We have 1 ( 1  )1  0 or )z1  1 = ( 1 . Letting (x1  1, the forward traveling eigenvector is 51  1;  = " , where " denotes the matrix transpose. Substituting the backward traveling wave with 2      , the backward traveling eigenvector corresponding to 2 is 52  1;  = " .

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The electric ®eld associated with the eigenmodes  ,  1; 2, is given in Regions `  1; 2; 3 as  `

  ( `  `  ^

2:2:14a

 `

1   ) `  `  ^  0 where ( `  1 )1` 

 ` `

 1; 2 )2` 

 ` `

22:14b

`  `   ` Since the medium is linear, a superposition over the modes in Eq. 2.2.14 gives the total ®eld in any region. The total electric and magnetic ®elds which can exist in Regions 1, 2, and 3 is given by 2  `   `

  * ` 

1

2  `

 `  * ` 

 

1

2:2:15a

2:2:15b

where * ` are general complex coef®cients that need to be determined from boundary conditions. As a cross-check of the solution we note that for any region (suppressing the ` subscript and superscript),   1 @ 1  @    !~ @~ 0  @ From Eq. 2.2.15a we note that   *1 exp   *2 exp 

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2:2:16

Substituting this in Eq. 2.2.16, we have   1    *1 exp  *2 exp   0 

2:2:17

which is the same solution as Eq. 2.2.15b when the eigenvectors of Eq. 2.2.14 are used. In addition to the ®eld amplitudes of the electric and magnetic ®elds, another important quantity to calculate is the time-averaged power that passes through any layer parallel to the material interface. This is explained in detail in the next subsection.

2.2.3

Complex Poynting Theorem

The previous subsection has presented the EM ®eld solution for a normally incident plane wave on a uniform, isotropic, lossy material layer. An important numerical consideration in all computations is the accuracy with which the numerical computations have been performed. A relatively simple test of the computation, which applies only when the slab is lossless, is provided by calculating the power incident on the slab, calculating the sum of the powers transmitted and re¯ected from the slab, and then calculating the difference of these two sums to compute the error in the numerical solution. As just mentioned, this test applies only when the layer is lossless. When the layer is lossy, the power re¯ected and transmitted does not equal the incident power, since some of the power is absorbed as heat inside the material layer. In the case when the layer is lossy, one can test numerical accuracy results by using the complex Poynting theorem. The purpose of this section will be to present the complex Poynting theorem (Harrington [3]) as it applies to the lossy material slab and also to test the numerical accuracy of the EM ®eld solutions that will be studied in Section 2.3.2. For an isotropic material, the complex Poynting theorem states that the time-averaged power delivered (meter 3 ) at a point ! contained in a   volume
,~  0 by the electric and magnetic sources
and 
should be balanced by the sum of (1) the time-averaged power ! (meter 3 ) radiated ~ (2) the electric power ! over the surface
(~ enclosing the volumes
,, 3 ~ and (3) and magnetic power ! (meter ) dissipated over the volume
,, 2! times the difference between the time-averaged magnetic energy -  ~ where stored in
,~ and the time-averaged electric energy -  stored in
,, !  2 (radians) is the angular frequency and  is the frequency in Hertz. Mathematically the complex Poynting theorem for a general isotropic material is given by [3]

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1 2



  1     ^  (~  2 S~

     t          ,~  0 ~ V

2:2:18

 where is a general electric displacement, conduction and source current  term and  represents the generalized magnetic current. Mathematically these currents are given by   
  ! ~   ~      
   ! l~   l~    

2:2:19

2:2:20



where and  are impressed source terms, and we have assumed that the permittivity and permeability are complex anisotropic quantities. After some algebra, we obtain from Eq. 2.2.18, !%  !  !  !   !-  !-

where

!% 

12

     
 
       ,~ ~ V

(source power)    !  12    ^ ^  (~ S~

!-

!-

(net outward power flow)        ~  2!-   2! 14   ~   , ~ V

(proportional to stored electric energy)        ~ 1  2!-   2! 4   l~   , ~ V

(proportional to stored magnetic energy)

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2:2:21

! 

!

1 2



~ V

    ~   ~   ,

(electric power dissipated)      ~ 1 2   l~   ,

2:2:22

V

(magnetic power dissipated)

For present applications we will consider a Poynting box as shown in Fig. 1. This box is assumed to have end faces that have the cross section
(~ and are parallel to the interfaces of the slab. For this box we ®rst note that in the power ¯ow integral ! , the integral over the lateral portion of the box (the portion between the end faces of the box) is zero. This follows since there is no variation in the EM ®elds or power ¯ow in the - and -directions. Thus the power ¯ow integral can be written as a sum of the power ¯ows as calculated over the two end faces of the box. !  !   !.)"

2:2:23

where !  

1 2

!.)" 




S~

1 2



   


S~

y~ ~y

   

^ ^   (~

~ ~ 

^ ^   (~

2:2:24

2:2:25

The minus sign in Eq. 2.2.25 is a result of the fact that the outward normal on the ~  end cap is ^  . Using Eqs. 2.2.21 and 2.2.23, we ®nd that the complex Poynting theorem for the present problem can be written as !   !%  !.)"  !  !   !-  !-

2:2:26

It is convenient to express the above power and energy integrals in dimen~ etc., and to normalize the complex Poynting sionless coordinates  0 ,  2 ~ theorem equations ! by an amount of power !(

*  
(= 20  0 =1

! (watts), where 0  ~ 0 =~0  377 , 1  ~ 1 =~1 =0 (dimensionless), and  02 =1  1 volt2 =m2 . With this normalization, and also carrying out all integrals in Eqs. 2.2.22, 24, 25, each term in Eq. 2.2.26 can be written as

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!   !%  !.)"  !  !   !-  !-

2:2:27

         !-  1 2            (dimensionless) 0 ` `    !-  0   l  0   (dimensionless) `

!  ! 



`



`

        (dimensionless)   0   l  0   (dimensionless)

  1     0 

 ^ 

2 0      ^ (dimensionless)    0 



!.)" 



! 

      0 



 ^  (dimensionless)

  1 1         
   
  0 0 02 `        1  
   
 (dimensionless)  0 0

!% 

`

where        and l  l  l  represent relative permittivity and permeability, respectively. Substitution of the ®eld solutions as obtained through the state variable technique into the above one-dimensional integrals gives the various power terms that make up the complex Poynting theorem. Because all permittivity and permeability tensor elements are constant, and because all EM ®eld solutions in the equations are exponentials, we note that all the one-dimensional power integrals can be carried out in closed form. For checking numerical error, this is important, since estimates of the error using these formulae do not depend on the accuracy of the numerical integration. 2.2.4

State Variable Analysis of an Isotropic Layer in Free Space

In this subsection we consider the case when a plane wave from    is normally incident as a dielectric slab. In this case the *11 and *23 coef®cients

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are known (see Eq. 2.2.15), with *11  0 , where 0 is the incident amplitude (volts/m), and *23  0 also, since there is no re¯ected wave from Region 3. As the coef®cient *21 represents the complex amplitude of the re¯ected ®eld in Region 1, we let *21  $, and since the coef®cient *13 represents the complex amplitude of the transmitted ®elds in Region 3, we let *13  ". Using these coef®cients, the ®elds in Regions 1, 2, and 3 are given by (see Fig. 1). ,%   1  0 exp 1   $ exp 1 

 1 

 1  0 exp 1   $ exp 1    0 1

2:2:28

,%   2  *12 exp 2   *22 exp 2 

 2 

1  2  *12 exp 2   *22 exp 2    0 2

2:2:29

,%  6  3  " exp 3   

 3 

1  3  " exp 3   

  0 3

2:2:30

The  3 and  3 ®elds have been written with a exp 3   

in order to refer the phase of the " coef®cient to the    boundary. The boundary conditions require that the tangential electric and magnetic ®elds match at   0, . Matching of the tangential electric and magnetic ®elds at   0 and    leads to four equations in four unknowns, from which the EM ®elds in all regions can be determined. It is convenient to use the electric ®eld equations at the boundaries to eliminate the unknowns in exterior Regions 1 and 3, thus reducing the number of equations from four to two. When we do so, we ®nd that 2 1   1 0

11 *12



12 *22

0

21 *12



22 *22

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2:2:31

where

11

21

2 1

1 2  12   2 1  1 2  

2 3 exp 2 

  2  3 

22





2 3 exp 2 

  2  3

2:2:32

Also $  0  *12  *22 "  *12 exp 2   *22 exp 2 

2:2:33

Inversion of the 2  2 as given by Eqs. 2.2.31 then determines the unknown coef®cients *12 and *22 of the system. We now apply the complex Poynting theorem of Eq. 2.2.27 to the normal incident plane wave case being studied in this section. We assume that the Poynting box has its left face 0:5 from the Region 1±2 interface, i.e., ~   ~
 0:5, and has its right face at ~  ~ /0 , ~ /0  0. For the present analysis there are no sources in the layer, so !%  0. Substituting we ®nd that the complex Poynting theorem is given by !   !.)"  !  !   !-  !-  !.1 where !  !1  !2  !3 !1  !3  0 0 2  12 exp 2   22 exp 2   !2  2 y2

where

2 

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/0 

/0 > 

/0 < 

2:2:34

!  !1  !2  !3 !1  !3  0 20 2  2  exp 2  22 exp 2   !2  2 2 2 12 !-  !-1  !-2  !-3  


2 !-1  1 0 exp 1   $ exp 1   0

!-2  2

0

12 exp 2   22 exp 2  

!-3  3

 

" exp 3    

"



2

3

2

2

where

3 

/0 >  /0 < 

/0

!-  !-1  !-2  !-3 

2 in 2 !-1  1 1 0 exp 1  $ exp 1   1 0

!-2 

2

!.)"

20

2

2  

$ exp 1 
  

  2 12 exp 2 /0  22 exp 2 /0 12 exp 2 /0

2 22 exp 2 /0 

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2

12 exp 2  22 exp 2  

3 2 " exp 3   

 3 3  

  1 0 exp 1 
 $ exp 1 
0 exp 1 


1

!-3  3 ! 

2 2

when /0 >  !.)"



 3 3



" exp 2 /0  

2

when /0 < . In these equations $ is the re¯ection coef®cient in Region 1, " is the transmission coef®cient in Region 3, and 12 and 22 are wave coef®cients in Region 2. The expressions for !-3 and !-3 have been chosen so that when /0 >  (that is, /0 is in Region 2) the lower limit 3 equals the upper limit and !-3 and !-3 are zero as they should be. The conservation theorem as given by Eq. 2.2.34 states (1) that the sum of Re !.)" and !  !  ! ! is real and nonnegative), which by de®nition equals Re !.1 , should equal Re !  and (2) that the sum of Im !.)" and the energy±power difference !-  !- , which by de®nition equals Im !.1 , should equal the sum of Im !  . As a numerical example for the normal incidence case, we assume that the layer thickness is ~  0:6, that free space bounds the layer in Regions 1 and 3, and that the slab has a lossy permittivity given by 2  3 0:4 and relative permeability 2  2:5 0:2. Figs. 2, 3, and 4 show plots of the EM ®elds and different power terms associated with the present example. Figure 2 shows the  electric ®eld (magnitude, real and imaginary parts) plotted vs. the distance ~  ~ from the incident side interface. In observing the real and imaginary plots of  , one notices that the standing wave wavelength of  is greatly shortened in Region 2 as opposed to Region 1. This is due to the greater magnitude of the material constants 2   3 0:4 and 2   2:5 0:2 in Region 2 as opposed to Region 1. In observing the plots of Fig. 2 one also notices that the continuity of the  is numerically obeyed as expected. In Fig. 2 one also notices that the presence of the lossy layer causes a standing wave in Region 1 with a standing wave ratio SWR   '1 =     0  R= 0 $  1:2. This means that the lossy layer represents a fairly matched load to the normally incident plane wave. In Region 2 of Fig. 2 it is observed that the   is attenuated to about 30% as the EM wave is multiply re¯ected in the lossy layer. In Fig. 3, plots of the real and imaginary parts of !  and !.1 are made as a function of the distance ~ /0 , the distance that the Poynting Box extends to the right of the Region 1±2 interface. As can be seen from Fig. 3, the complex Poynting theorem is obeyed to a high degree of accuracy as the real and imaginary parts of !  (solid line) and !.1 (cross) agree very closely. One also observes that as the distance ~ .)" increases, the power dissipated ! increases, the Re !.)" decreases, and both change so as to

Copyright © 2000 Marcel Dekker, Inc.

Figure 2 The  electric ®eld (magnitude, real and imaginary parts) plotted versus the distance $~ from the incident side interface is shown.

leave the sum constant and equal to Re !  . Also plotted in Fig. 3 is the Im !.)" and the energy difference term !-  !- . One observes from these plots that the Im !.)" and !-  !- vary sinusoidally in Region 2 and that the nonconstant portions of these curves are out of phase with one another by 180 . Thus the sum of Im !.)" and !-  !- is a constant equal to Im !  . Thus the imaginary part of the power is exchanged periodically between Im !.)" and !-  !- so as to keep the Im !  a constant throughout the system. Figure 4 shows plots of the electric and magnetic energy and power stored and dissipated in the Poynting box, again versus the distance ~.)" . As can be seen from Fig. 4, the electric and magnetic stored energy terms !- and !- are nearly equal to each other.

2.2.5

State Variable Analysis of a Radar Absorbing Layer (RAM)

As a second example, assume that a material similar to the one in the previous example is placed against an electric perfect conductor (EPC)

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Figure 3 Plots of the real and imaginary parts of !IN and !BOX as a function of the distance $~ OUT .

located at    and that a plane wave from    is incident on the layer. A practical application of this is in designing radar evading aircraft, where such a layer of appropriate thickness is pasted on the metal surface of the aircraft to minimize radar re¯ectivity. In this case the electric and magnetic ®eld equations at ~  0 are the same as in the ®rst example. Thus 2 1   1 0

11 *12



12 *22

2:2:35

where 11 and 12 have been de®ned previously. At ~  ~ the tangential component of the electric ®eld must vanish due to the presence of the metal. This leads to the equation 0  *12 exp 2   *22 exp 2 

2:2:36

From these equations *12 and *22 can be determined as well as all other coef®cients in the system. Figure 5 shows the Re  , Im  , and   plotted versus the distance ~  ~ from the Region 1±2 interface, using the material parameter

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Figure 4 Plots of the electric energy term, magnetic energy term, power stored, and power dissipated in the Poynting box, vs. the distance $~ OUT .

values of Section 2.2.4. As can be seen from Fig. 5, the presence of the EPC in Region 3 causes a larger standing wave (SWR) than was observed when a free space occupied Region 3. One also notices that the presence of the EPC causes more internal re¯ection within the slab layer, Region 2, as can be seen by the increased ripple or decaying (-$ pattern displayed by the   plot. Figure 6 shows the various normalized power terms associated with the complex Poynting theorem of Eq. 2.2.34. Figure 6 uses the same geometry as Fig. 3. The only difference between Fig. 3b and Fig. 6 is that an EPC is in Region 3 of Fig. 6, whereas free space was in Region 3 of Fig. 3. As can be seen in Fig. 6, as in Fig. 3, the complex Poynting theorem is obeyed to a high degree of accuracy since the real and imaginary part of !  (solid line) and !.)" (cross) agree with each other very closely. We also notice from Fig. 6 that a higher oscillation of !-  !- and Im !.)" occurs than in Fig. 2. This higher internal re¯ection in the slab is caused by the high re¯ectivity of the EPC at the Region 2±3 interface. Figure 7 shows the plot of normalized re¯ected power (re¯ected power/incident power, db) of a uniform slab that results when a plane wave is normal to the slab. Region 3 is an EPC, and in Region 2, 2  7

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Figure 5

Plots of the Re  , Im  , and   plotted versus the distance $~ .

3:5 and 2  2:5 0:2. In this ®gure, the normalized re¯ected power is ~ As can be seen from Fig. 7, at a slab plotted versus the slab length . ~ thickness of   0:066 the re¯ectivity of the layer drops sharply (about 21 db down from the re¯ection that would occur from a perfect conductor alone). At this slab thickness the layer has become what is called a ``radar absorbing layer'' (RAM), since at this slab thickness virtually all radiation illuminating a perfect conductor with this material will be absorbed as heat in the layer and very little will be re¯ected. Thus radar systems trying to detect a radar return from RAM-covered metal objects will be unable to detect signi®cant power. It is interesting to note that only a very thin layer of RAM material is needed for millimeter wave applications. For example, at millimeter wavelengths (95 GHz), ~  0:066  0:2088 mm. 2.2.6

State Variable Analysis of a Source in Isotropic Layered Media

In this subsection we consider the state variable analysis of the EM ®elds  that are excited when a planar sheet of electric surface current (  % ^  ^ (Amp/m) is located in the interior of an isotropic two-layered medium.

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Figure 6 Plots of the various normalized power terms associated with the complex Poynting theorem of Eq. 2.2.34. This ®gure uses the same geometry as Fig. 3.

The material slab, like the layer considered in Section 2.2.2, is assumed to be bounded on both sides by a uniform lossless dielectric material that extends to in®nity on each side. For this analysis we locate the origin of the coordinate system at the current source and label the different regions of the EM system as shown in Fig. 8. Following precisely the same state variable EM analysis as we followed in Section 2.2.2, we ®nd that the general EM ®eld solutions in each region are given by ,%   

 1  *11  exp 1   

 *21  exp 1   





) 1  0  1 

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 1  *  exp 1   

1  21

*11   0 2:2:37a

2:2:37b

Figure 7 Plots of normalized re¯ected power (re¯ected power/incident power, db) for the case where Region 3 is an EPC and Region 2 has 2  7 3:5 and 2  2:5 0:2.

,%   1  *11 exp 1   *21 exp 1 

2:2:38a

) 1  0  1 

2:2:38b

 1 *11 exp 1  *21 exp 1   1

,%   2  *12 exp 2   *22 exp 2 

2:2:39a

) 2  0  2 

2:2:39b

 2 *12 exp 2  *12 exp 2   2

,%  6  3  *13 exp 3   

2:2:40a

) 3  0  3 

2:2:40b

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 3 * exp 3   

3 13

Figure 8

Plots of the Re  , Im  , and   plotted versus the distance $~ .

The total layer thickness is      , where   0 and   0. Matching the tangential electric and magnetic ®elds at the Region 1±1  interface and eliminating the *21  coef®cient, it is found that *11  *21



1  =1   1 =1 exp 2 1 

1  =1   1 =1

2:2:41

Matching the tangential electric and magnetic ®elds at the Region 2±3 interface and eliminating the *13 coef®cient it is found that *12  *22



3 =3  2 =2 exp 2 2 

3 =3  2 =2

2:2:42

To proceed further we match EM boundary conditions at the Region 1±2 boundary   0. These boundary conditions are given by 0  1 0  2  ) 1 ) 2  0  1  2  0

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2:2:43a

2:2:43b

In the present problem, because an electric current source is present at the Region 1±2 boundary, the tangential magnetic ®eld given by Eq. 2.2.43a is discontinuous at   0. Performing algebra it is found that the following equations result, from which the unknown coef®cients of the system can be found.  1   1*21  2  1*22  0 1 2   1*21   1*22  0

2:2:44a

2:2:44b

To give a numerical example of the EM ®elds and complex Poynting results, we assume that the material slab (Region 2) has the parameters 1  2 0:3, 1  3 0:5, 2  3 0:4, 2  2:5 0:2, ~  0:4, ~   0:5 and that Regions 1 and 3 are free space. In this example we further assume that the Poynting box is the same one described in Section 2.2 except that its leftmost face is located ~ .)"  0:25 to the left of the Region 1±2 interface (the source is located at the Region 1±2 interface at ~  0), and its rightmost face is located at ~  ~ .)" ; ~.)"  0 from the Region 1±2 interface. (See Fig. 9 inset). For the present source problem, the complex Poynting theorem is given by !(  !.)"  !.)"  !  !   !-  !-  !.1 2:2:45

where   !(  0   s

~ 0

 0  s

~ 0

2:2:46

 *11  *21 is continuous at ~  0. From Eq. The electric ®eld  0 ~ 2.2.43a, 0 

 1  *11 *21  2 *12 *22

1 2

2:2:47

Thus    1  2 *11 *21  * *22

!s  *11  *21

1 2 12

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2:2:48

Figure 9 Plots of different power terms that make up the complex Poynting theorem of Eq. 2.2.45 plotted versus the distance $~ OUT .

The terms !.)"  and !.)" are given by !.)" 

!.)"

 

1     *11 exp 1 .)"   *21 exp 1 .)"

1   *11 exp 1 .)"  *22 exp 1 .)"





 2 2



*12 exp 2 .)"  *22 exp 2 .)"  

*12 exp 2 .)" *22 exp 2 .)" 

2:2:44a

2:2:44b

when .)" >  !.)"



 3 3



" exp 3 .)"  

2

2:2:45

when .)" <  . The other terms in Eqs. 2.2.45 are given in Eq. 2.2.34.

Copyright © 2000 Marcel Dekker, Inc.

Figure 8 shows the Re  , Im  , and   electric ®elds plotted versus the distance  from the Region 1±2 interface. As can be seen from Fig. 8, the presence of the electric current source in a lossy medium causes the electric ®eld to be greatest at the source location and attenuate as distance increases from the source. Because the regions are different to the left and right of the source, the ®elds are not symmetric about the source location. In observing Fig. 8 one notices that the Re  , Im  , and   are all continuous at the different interfaces as they must be to satisfy EM boundary conditions. Figure 9 shows different power terms that make up the complex Poynting theorem of Eq. 2.2.45 plotted versus the distance ~ .)" . As can be seen from Fig. 9 the real and imaginary parts of !(  !(.)$* (cross) and !.1 (solid line) agree with each other to a high degree of accuracy, thus showing that the complex Poynting theorem is being obeyed numerically for the present example. One also observes that as the distance ~ .)" increases, the power dissipated ! increases, Re !.)" decreases, and both change so as to leave the sum constant and equal to Re !( . Also plotted in Fig. 9 is the Im !.)" and the energy±power difference !-  !- . One observes from these plots that the Im !.)" and the energy±power difference !-  !- vary sinusoidally in Region 2 and that the nonconstant portions of these curves are out of phase with one another. Thus the sum of Im !.)" and !-  !- is a constant equal to IM !( . Thus the imaginary part of the power is exchanged periodically between Im !.)" and !-  !- so as to keep the Im !( a constant throughout the system. Although the EM ®elds were excited by an electric current source in Fig. 9 rather than a plane wave as in Fig. 3, the complex Poynting numerical results in the two ®gures are similar.

2.3 2.3.1

STATE VARIABLE ANALYSIS OF AN ANISOTROPIC LAYER Introduction

Thus far we have discussed several examples of EM scattering from isotropic layers. Another interesting problem is EM scattering from anisotropic media, such as crystals and the ionosphere. This section differs from the previous sections in two ways: namely, the media are anisotropic and couple the ®eld components into one another, and also the EM ®elds are obliquely incident on the dielectric slab at an angle  . The analysis [18±29] is a state variable analysis similar to that in the previous section and gives a reasonably straightforward and direct solution to the problem. We note that a traditional second-order wave equation analysis would lead to a fairly intractable equation set, due to the anisotropic coupling of the ®elds.

Copyright © 2000 Marcel Dekker, Inc.

We assume that the plane wave is polarized with its electric ®eld in the plane of incidence of the EM wave. The dielectric slab is assumed to be characterized by a lossy anisotropic relative dielectric permittivity tensor where  ,   ,  ,  , and  are nonzero and the other tensor elements are zero. The geometry is shown in Fig. 10. The slab's relative permeability is assumed to be isotropic and lossy and characterized by       . The basic analysis to be carried out is to solve Maxwell's equations on the incident side (Region 1), in the slab region (Region 2), and on the transmitted side (Region 3), and then from these solutions to match EM boundary conditions at the interfaces of the dielectric slab. 2.3.2

Basic Equations

A state variable analysis will be used to determine the EM ®elds in the dielectric slab region. We begin by specifying the EM ®elds in Regions 1 and 3 of the system. The EM ®elds in Region 1 are given by  1  ( 1  exp 



 1  0 exp 1  $ exp 1  exp 

1

 * 11 exp 11   * 21 exp 21   exp 

2:3:1

Figure 10 Geometry of a planar dielectric layer and a complex Poynting box is shown. A plane wave parallel polarization is obliquely incident on the layer. )  0  .

Copyright © 2000 Marcel Dekker, Inc.

 1  ( 1  exp 

   0 exp 1   $ exp 1  exp 

1    *11 exp 11   *21 exp 21  exp 



0  1  ) 1  exp 

   0 exp 1   $ exp 1  exp 

 *11 exp 11   *21 exp 21   exp 

2:3:2

2:3:3

!  ~   0 , ~   0 , ~ 0  2=,   1 sin  , 1  1 2 , where  0 , and 0  377 ; 0 is the incident plane wave amplitude,  is the free space wavelength in meters, and 1 is the relative permittivity of Region 1. The EM ®elds in Region 3 consist only of a transmitted wave and are given by  3  ( 3  exp  

 3  " exp 3   

exp 

3

 * 13 exp 13   * 23 exp 23   exp 

2:3:4

   " exp 3   

exp 

3    *13 exp 13   *23 exp 23  exp 

2:3:5

   ) 3  exp   " exp 3   

exp 

 3  ( 3  exp   0  3

 *13 exp 13   *23 exp 23   exp 

2:3:6

! where 3  3 2 , " is the transmitted plane wave amplitude, and 3 is the relative permittivity of Region 3. In the anisotropic dielectric slab region, Maxwell's equations are given by       l 0 

     0        

2:3:7

where we assume that l is a diagonal matrix with      . The   component of     is given by             . The  and  are similarly de®ned. In order that the EM ®elds of Region 1 and 3 phase match with the EM ®elds of Region 2 for all , it is necessary that the EM ®elds of Region 2 all be proportional to exp  . (This factor follows

Copyright © 2000 Marcel Dekker, Inc.

from application of the separation of variables method to Maxwell's equations.) Using this fact, the electric and magnetic ®elds in Region 2 can be expressed as    (  ^  (  ^   (  ^ exp 

 0   )  ^  )  ^  )  ^ exp 

2:3:8

Using the fact that the only nonzero EM ®eld components in Region 1 are  ,  , and  , a small amount of analysis shows that in Eqs. 2.3.7 a complete ®eld solution can be found taking only ( , ( , and ) to be nonzero with (  )  )  0. Substituting Eqs. 2.3.8 in Eq. 2.3.7 and taking appropriate derivatives with respect to , the following equations result: @(   ) @

2:3:9

@)   (    ( @

2:3:10

 )   (   (

2:3:11

 (

To proceed further it is possible to eliminate the longitudinal electric ®eld component and express the equations in terms of the ( and ) components alone. Although other components could be eliminated, the ( is the best, since the remaining equations involve variables that are transverse or parallel to the layer interfaces. These variables then may be used to match tangential EM boundary conditions directly. The ( component is given by (from Eq. 2.3.11) ( 

  (  )  

2:3:12

Substituting Eq. 2.3.12 into Eqs. 2.3.9, 2.3.10, # $    @( 2    ) (    @             @)    (    ) @   

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2:3:13

2:3:14

The above equations are in state variable form and can be rewritten as @5  +5 @

2:3:15

where

11

21

# $   2    12                  22      

2:3:16

2:3:17

where 5  ( ; )  . The basic solution method is to ®nd the eigenvalues and eigenvectors of the state variable matrix +, form a full ®eld solution from these eigensolutions, and then match boundary conditions to ®nd the ®nal solution. The general eigenvector solution is given by 5  5 exp  

2:3:18

where  and 5  ( ; )  are eigenvalues and eigenvectors of + and satisfy +5   5

 1; 2

2:3:19

Because + is only a 2  2 matrix, it is possible to ®nd the eigenvalues and eigenvectors of the system in closed form. The quantities  and 5 are given by #

11



21

12 22



$#

(

)

$

0

2:3:20

For this to have nontrivial solutions, det



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11



21

12 22







11



22



12 21

0 2:3:21

Using the quadratic equation to solve for  we ®nd   0:5

11



22

 0:5



2 11

2

11 22

4

12 21



2 1=2 22

 1; 2

2:3:22

Letting (  1,  1; 2, it is found that the eigenvectors are given by   5  1;

12

11



2:3:23

The longitudinal eigenvector components ( are given by, using Eq. 2.3.12, ( 

  (  )

 

 1; 2

2:3:24

Using these eigenvalues and eigenvectors it is found that the EM ®elds in Region 2 are given by  2  ( 2  exp 

 *1 ( 1 exp 1   *2 ( 2 exp 2   exp 

 * 12 exp 12   * 22 exp 22   exp 

 2  ( 2  exp 

   *1 (1 exp 1   *2 (2 exp 2  exp 

   *12 exp 12   *22 exp 22  exp 

2:3:25

2:3:26

0  2  ) 2  exp 

 *1 )1 exp 1   *2 )2 exp 2   exp 

 *12 exp 12   *22 exp 22   exp 

2:3:27

In these equations *1 and *2 are ®eld coef®cients yet to be determined. To proceed further it is necessary to determine the unknown coef®cients of the ®eld solution in Regions 1±3. In this case the unknown coef®cients are $, ", *1 , and *2 . In the present problem the boundary conditions require that the tangential electric ®eld (the  ®eld) and the tangential magnetic ®eld ( ) must be equal at the two slab interfaces. Thus in this analysis there are four boundary condition equations from which the four unknown constants of the system can be determined. Matching boundary conditions at the Region 1±2 interface we ®nd

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1  $  *1 ( 1  *2 ( 2 1 0

2:3:28

0  $  *1 )1  *2 )2

2:3:29

3 "  *1 ( 1 exp 1   *2 ( 2 exp 2 

3

2:3:30

"  *1 )1 exp 1   *2 )2 exp 2 

2:3:31

By substituting $ and " from Eqs. 2.3.28, 2.3.31 in Eqs. 2.3.29, 2.3.30, the 4  4 system may be reduced to the following 2  2 set of equations     21 1 1   )  ( 1 *1  )  ( 2 *2 1 0 1 1 1 2

2:3:32

  3 0  exp 1  )  ( 1 *1  exp 2 

3 1   3 )  ( 2 *2  3 2

2:3:33

The *1 and *2 can be found from the above in closed form. Using Eqs. 2.3.28, 31, the other coef®cients may be found. 2.3.3

Numerical Results

This section will be concerned with presenting a numerical example from an anisotropic layer when an obliquely incident plane wave impinges on the layer. In this example Regions 1 and 3 are free space, and Region 2 is a material slab with a thickness ~  0:6 and material parameters     2:25 0:3,   0:75 0:1. We assume the permeability to be isotropic but lossy with   2  2:5 0:2. The incident plane wave (incident amplitude 0  1 (V/m), electric ®eld polarization in the plane of incidence) is assume to have an angle of incidence   25 . Figure 11 shows plots of the magnitudes of the  ,  , and )  0  EM ®elds in Regions 1±3 as a function of   , which is the location of the ®eld relative to the incidence side of the Region 1±2 interface (see Fig. 10). As can be seen from Fig. 11, the material slab represents a mismatched medium to the incident wave and thus the incident and re¯ected waves interfere in Region 1 forming a standing wave pattern. In Region 2, because the layer is lossy, one also observes that all three EM ®eld magnitudes

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Figure 11 Plots of the magnitudes of the  ,  , and )  0  electromagnetic ®elds in Regions 1±3 as a function of $  , which is the location of the ®eld relative to the incidence side of the Region 1±2 interface (see Fig. 10), are shown.

attenuate as the distance from the incident side increases. In Region 2, an SWR pattern is also observed in addition to the attenuation, which has already been mentioned. The SWR pattern is caused by the multiple internal re¯ections that occur within the slab. In Region 3, only a forward traveling transmitted wave is excited; thus the EM ®eld amplitude is constant in this region. One also notices from Fig. 11 that the tangential electric ®eld ( ) and tangential magnetic ®eld )  0  ) are continuous, and that the normal electric ®eld ( ) is discontinuous, as should be the case. Figure 12 shows plots of normalized dissipated power that results when the complex Poynting theorem of Section 2.2 is used to study the example of this section. In this ®gure the Poynting box has been chosen to extend a half wavelength into Region 1 (see Fig. 12 inset) and to extend a variable distance ~ /0 (units of ) into Region 2 when ~ /0 0:6 and into 3. In this ®gure %! , !  , Regions 2 and 3 when ~ /0 > 0:6 into Region %     ( , !   (   ( , etc. are given by the integrals !d  (  etc. and !  !  !   !  ! . Also !  ! 

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Figure 12 Plots of normalized dissipated power as results when the complex Poynting theorem, as given by Eqs. 2.2.21±27 of section 2.2.3, is used to study the example of this section are shown.

%

)  ) . As can be seen from Fig. 12, the dissipated electric and magnetic powers ! and ! are zero at ~ /0  0 and increase in a monotonic fashion until ~ /0  0:6 where they become constant for ~ /0 > 0:6. This is exactly to be expected since the only loss in the system is in Region 2 where 0 ~ /0 0:6. We note also that the integrals !  and ! are complex and satisfy !   !  as expected. Thus !   !  2Re !  . The integrals ! and ! are purely real, and thus the electric dissipation integral ! is purely real. Note as can be seen from Fig. 12 that although the total electric dissipation integral is positive, the cross-term contribution given by !   !  2Re !  is negative. This is interesting as one would usually associate only positive values with typical power dissipation terms. Figure 13 shows plots of normalized energy±power terms as result from Eqs. 2.2.21±27 using the example of this section. In this ®gure as in the previous one, the Poynting box has been chosen to extend a half wavelength into Region 1 (see Fig. 13 inset) and to extend a variable distance ~ /0

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Figure 13 Plots of normalized energy±power terms as results from Eqs. 2.2.21±27 using the example of this section are shown.

into Region 2 when ~ /0 0:6 and into Regions 2 and 3 when ~ /0 > 0:6 into Region 3. In this ®gure !%& , !&  , etc. are given by the integrals %   !&  (  (  , !&   (   (% , etc. and !-  !&  !&   !&  !& . Also !-  !&  )  ) : As can be seen from Fig. 13, the stored electric and magnetic energy±powers !- are nonzero at ~ /0  0 and increase in a monotonic fashion thereafter. As in the case of the dissipation power integrals, we note that the integrals !&  and !we are complex and satisfy !we   !& . Thus !&   !&  2Re !&  . The integrals !& and !& are purely real, so the electric energy±power integral !- is purely real. Note that, as can be seen from Fig. 13, although the total electric energy±power integral is positive, the cross-term contribution given by !we   !we  2Re !we  is also negative. Figure 14 shows plots of the real and imaginary parts of the complex Poynting theorem terms as result from Eqs. 2.2.21±27 given the same Poynting box as was used in Figs. 12 and 13. In this ®gure, since we are testing the numerical accuracy of the computation formulae, we let !.1  !.)"  !  !   !-  !- and compare !  and !.1 . As can be seen from Fig. 14, the real and imaginary parts of !  (cross) and !.1

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Figure 14 Plots of the real and imaginary parts of the complex Poynting theorem terms as results from Eqs. 2.2.21±27 given the same Poynting box as was used in Figs. 12 and 13 are shown.

(solid line) are numerically indistinguishable from one another, showing that the numerical computations have been carried out accurately. Figure 14 also shows plots of Re !.)" , which decrease as ~ /0 increases, and !  !  ! (! is purely real), which increase as ~ /0 increases. As can be seen from Fig. 14, the sum of these two quantities, namely Re !.)"  ! adds to Re !  , which is constant as ~/0 increases. It makes sense that the Re !.)" decreases as ~ /0 increases, due to increased power loss as ~/0 increases. Figure 14 shows plots of Im !.)" and the energy difference term !-  !- . As can be seen from Fig. 14, within Region 2 the two terms are oscillatory, with the oscillatory terms out of phase with one another by 180 . The complex Poynting results of this section are similar to those of Section 2.2.

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2.4 2.4.1

STATE VARIABLE ANALYSIS OF A BI-ANISOTROPIC LAYER Introduction

In the previous section, we have discussed re¯ection and transmission from an anisotropic layer when an oblique incident plane wave impinges on the slab at an angle  . It was assumed that the plane wave was polarized with its electric ®eld in the plane of incidence of the EM wave, and the dielectric slab was assumed to be characterized by a lossy anisotropic relative dielectric permittivity tensor where  ,   ,  ,  , and  were nonzero and the other tensor elements were zero, and the slab was assumed have a permeability which was isotropic and lossy and characterized by       . A generalization of this problem that will be studied in this section is to calculate the EM ®elds that result when a plane wave of arbitrary polarization is obliquely incident on a uniform bi-anisotropic material layer. This problem has been studied by many authors. Lindell et al. [6] discuss scattering from bi-anisotropic layers extensively and include many references on this subject. The geometry is shown in Fig. 15. Again, the basic analysis to be carried out is to solve Maxwell's equations on the incident side (Region 1), in the slab region (Region 2), and in the transmitted side (Region 3) and then from these solutions to match EM boundary conditions at the interfaces of the dielectric slab. This solution method is similar to that of Section

Figure 15 Geometry of a planar bianisotropic layer and a complex Poynting box is shown. A general plane with arbitrary polarization is obliquely incident on the layer.

Copyright © 2000 Marcel Dekker, Inc.

2.3, except that the state variable analysis in Region 2 the slab region is more complicated than in Section 2.3. The analysis will be based on the general formulations of Refs. 18±29.

2.4.2

General Bi-Anisotropic State Variable Formulation

The following section covers the derivation of the state variable equations for a single bi-anisotropic layer. Following the analysis of Lindell et al. [Eqs.  1.10, 2.3, 2.4], the electric ¯ux density vector  and the magnetic ¯ux   density vector  can be expressed in terms of the electric ®eld  and the magnetic ®eld  through the relations      ~    m~  

2:4:1

     ~    l~  

2:4:2

~ ~ , and l~ in Eqs. It is assumed that each component of the four dyadics ~ , m, 2.4.1 and 2.4.2 are in general lossy nonzero complex constants. After sub  stituting  and  of Eqs. 2.4.1 and 2.4.2 into Maxwell's equations, introducing the dimensionless dyadics

  ~ =0             

~ 0  l  l  l  l=

 ~ 0  0 0    !

      

 ~ 0 ~ 0 0 m  m !

~ etc., we ®nd that and introducing normalized coordinates  0 , Maxwell's curl equations become             l             

2:4:3

2:4:4

~ is the normalized curl operator. To proceed further where   1=0  we let all EM ®eld components in the material layer be proportional to the factor exp  , where  ~ ~  ~ z~      (since an incident plane wave possessing this factor is incident on the layer and phase matching must occur at the interfaces of the slab), and substitute the resulting expressions into Maxwell's normalized equations. Carrying out the above operation we ®nd that Maxwell's equations become

Copyright © 2000 Marcel Dekker, Inc.

    exp    ( exp 

 l )   (

2:4:5

    exp    ) exp 

  (   )

2:4:6

where the electric and magnetic ®elds are given by     (  exp 

2:4:7

  0   )  exp 

2:4:8

 where 0  0 =0  377 : If we carry out the differentiations as indicated by Eqs. 2.4.5 and 2.4.6,   noting that ( and ) depend only on , we ®nd after canceling the exponential factors that       @( @( ^    (  ^   (  (   ^   (   l )   ( @ @ 2:4:9

    @) @) ^     )  ^    )   )   ^   )   @ @     (  ) 2:4:10

Useful relations may be found from Eqs. 2.4.9 and 2.4.10, if out of the six equations given, the longitudinal components ( and ) can be eliminated, and equations for only the tangential components ( , ( , ) and ) be used. This is highly useful because the tangential components can be matched with other tangential EM ®eld components at the parallel boundary interfaces. The longitudinal ( and ) components can be eliminated from Eqs. 2.4.9 and 2.4.10 in the following way. We equate the  components of Eqs. 2.4.9 and 2.4.10 and after transposing terms ®nd that  (

  )  

 (

 

 (

 )  ) 2:4:10

 (  2 )   (  (   2 )   2 ) 2:4:11

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We can recast Eqs. 2.4.10 and 11 in the following matrix form:

/22

#

( )

$



(

   (     ,24    )  

2:4:12

)

so that after inverting / (we assume det /  0 we obtain

#

( )

$



(

 #   (  &11    / 1 ,   )  &21 

&12

&13

&22

&23

)



(

 $  &14  (       &24  )  )

2:4:13

Our next step is to substitute ( and ) as given by Eq. 2.4.13 into the and  components of Eqs. 2.4.9 and 2.4.10. Doing so thus eliminates all longitudinal ( and ) terms from the equations. After performing considerable algebra it is found that the ( , ( , ) , and ) components can be placed in the following state variable form:



'11

'12

'13

'14

 @5   '21  @  '31 '41

'22

'23

'32

'33

'42

'43

 '24   5  +5 '34   '44

where & ' '11    &21      &11 & ' '12    &22    &12   & ' '13      &23    &13 & ' '14    &24      &14

Copyright © 2000 Marcel Dekker, Inc.

2:4:14

& ' '21     &21      &11 & ' '22     &22     &12   & ' '23       &23     &13 & ' '24     &24  0      &14 & ' '31      &11  2   &21 & ' '32    &12    2   &22 & ' '33    &13  2  2   &23 & ' '34    &14  2   &24  2 & ' '41       &11  2   &21 & ' '42     &12     2   &22 & ' '43     &13  2  2   &23 & ' '44     &14  2   &24  2 

2:4:15

Equation 2.4.14 is in state variable form and its solution can be determined from the eigenvector and eigenvalues of + as was done in Sections 2.3 and 2.2. The solution is given by

5

4 

* 5 exp  

1

(

 (   5   )  )



The EM ®elds in Region 2 are given by

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2:4:16

2:4:17

4      * 

2:4:18

1

4      * 

2:4:19

1

where      ( ^  ( ^   ( ^  exp   

  1   ) ^  ) ^   ) ^  exp   

0

2:4:20

 1; 2; 3; 4 2:4:21

and 

(

)



2:4:22

 724 5

Matching of the boundary conditions at the interfaces determines the   ®nal * coef®cients and thus  and  .

2.4.3

Incident, Reflected, and Transmitted Plane Wave Solutions

In Region 1 (see Fig. 15) we assume that an oblique incident plane wave with arbitrary polarization is incident on the bi-anisotropic material slab. We assume that the oblique incident plane wave is given mathematically by    I  ( exp 



  0  I  )  exp  I

2:4:23



  I    1    

2:4:24

where   I   ^ 1 ^    ^ 

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   ^   ^   ^

2:4:25

It is assumed for simplicity in this analysis that  1=2 1  1 1 2 2  0

2:4:26

It is further assumed that the wave vector values  ,  are known and given and that the incident plane wave polarization is speci®ed by known and given values of ( and ( . From Maxwell's equations and the assumed  known value of  I , the other ®eld components of the incident wave are given by

( 

  (   (  (  ( 1 1

2:4:27

) 

 1  1 (  ( 1

2:4:28

1  (  (  1   1  )   (  1 ( 1

) 

2:4:29

2:4:30

We note that Eqs. 2.4.27±30 represent an arbitrary oblique plane wave of arbitrary polarization. The re¯ected wave in Region 1 as results from Maxwell's equations is given by    $  ( $ exp  $

$ ;

  0  $  ) $ exp 

  $     1    

$

2:4:31

2:4:32

where   $   ^  1 ^    ^

2:4:33

If the tangential values of the electric ®eld ( $ and ($ can be found, it turns out from Maxwell's equations that the other ®eld components of the re¯ected wave are given by

Copyright © 2000 Marcel Dekker, Inc.

($  ) $ 

  ( $  ($  ( $ ($ 1 1

2:4:34

 1  1 ($  ($ 1

2:4:35

1  (  ($  1  $  1    ($ 1 ( $ 1

)$ 

2:4:36

)$

2:4:37

In Region 3 the EM ®elds are given by    T  ( T exp  T

T

  0  T  ) T exp 

T

   T      ^    3      

2:4:38

2:4:39

where   T   ^ 3 ^   ^   1=2 3  3 3 2 2  0

2:4:40

2:4:41

If the tangential values of the electric ®eld ( " and (" can be found, it turns out from Maxwell's equations that the other ®eld components of the transmitted wave are given by (" 

  ( "   ("    ( "   (" 3 3

2:4:42

) " 

 1  3 ("  (" 3

2:4:43

1  (  ("  3  "  1    ("  3 ( " 3

)" 

2:4:44

)"

2:4:45

Now that the general EM ®elds have been found in Regions 1±3 of space (see Fig. 15), as mentioned earlier, the next step is to match EM boundary conditions at the Region 1±2 and Region 2±3 interfaces. The boundary conditions for the present problem require that the tangential

Copyright © 2000 Marcel Dekker, Inc.

electric and magnetic ®elds at all interfaces be continuous. These boundary conditions follow from Maxwell's equations [3] using a small pillbox analysis. The boundary conditions for the present problem at the Region 1±2 interface are    $    $ 

0

0

   $ 

0

  $ 

0



4 



4 



4 



4 

* 

1

0

* 

1

0

* 

1

1

2:4:46

0

* 

0

( ( ( We let(('  4 1 * ( , ('  4 1 * ( , )'  4 1 * ) , and )'  4 1 * ) , evaluate the equations at   0 and   0 , cancel the exp     

factor and express the unknowns of Eqs. 2.4.46, ( $ and ($ , in terms of (' , (' , )' , and )' according to the relations ( $  (  (' ($  (  ('

2:4:47

After a small amount of algebra, it follows that

where

,  ('     ('z  1    1 )'   ,  ('    1  ('    1 )'   ,  (     (     ( 21  ,  ( 21     (    (  

2:4:48

2:4:49

The terms , , , represent the known source terms associated with the incident plane wave. If we further substitute the sums in (' , (' , )' , and )' and collect on the unknown coef®cients * in the sums, we ®nd

Copyright © 2000 Marcel Dekker, Inc.

,  , 

4 

1

4 

1

& '   * (     ( 1   1 )

&   ' * (    1  (    1 )

2:4:50

The boundary conditions at the Region 2±3 interface are

 "  "   "  " 

 

 

 

 



4 



4 



4 



4 

* 

1

 

* 

1

 

* 

1

1

2:4:51

 

* 

 

Substituting

( "  (" 

4 

* exp   (

1

4 

2:4:52

* exp   (

1

into Eqs. 2.4.51 and following a procedure very similar to the Region 1±2 interface we ®nd that

0 0

4 

1

4 

1

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&     ' * exp   (     ( 3   3 )

&     ' * exp   (     3  (   3 )

2:4:53

Altogether Eqs. 2.4.50 and 2.4.53 represent a set of 4  4 matrix equations from which the four unknown Region 2 coef®cients can be found. Once the * coef®cients are found, all coef®cients of the system can be found. 2.4.4

Numerical Example

In this section we present a numerical example of the theory presented in the previous subsections. In Region 1 we assume that 1  1:3, 1  1:8, and the incident plane wave of Eq. 2.4.27 has (  1 (V/m), (  0:9 (V/m),     1 1 sin I cos I , and   1 1 sin I sin I , where I  35 and I  65 . In Region 3 we assume that the material parameters are 3  1:9 and 3  2:7. In Region 2 we take the layer thickness ~  0:6 and we consider a complicated numerical example where all material parameters of  , l, , and  of Eqs. 2.4.5 are 2.4.6) are taken to be nonzero. The material parameters of Region 2 are taken to be

0:3  0:2

  0:1  0:05

0:3 0:1  0:1

0:2  0:2

 0:6  0:65  

0:25

0:1 0:05  0:05 0:3    0:01 0:01  0:1  0:1  0:05 0:04  0:08 0:14

1:3 0:2 0:3 0:1 0:33 0:07     0:1 2 0:01  0:02 0:01 3

0:1 :01 1:0 0:4    l 0:15 2:0 0:3 0:013  0:011 0:012 1:3 0:2

0:05

0:15

2:4:54

Figure 16 shows plots of the magnitude of the  ,  , and  electric ®eld components in Regions 1, 2, and 3 of the EM system under consideration, and Fig. 17 shows plots of the magnitude of the  ,  and  magnetic ®eld components in the same regions as Fig. 16. As can be seen from Figs. 16 and 17, the bi-anisotropic layer for the material values and layer thickness used represents a highly re¯ective layer. This is concluded from the large standing wave pattern observed in the re¯ected EM ®elds. It is also

Copyright © 2000 Marcel Dekker, Inc.

Figure 16 Plots of the magnitudes of the  ,  , and  electric ®eld components in Regions 1, 2, and 3 of the EM system of Fig. 15 are shown.

noticed from Figs. 16 and 17 that the tangential components of the EM ®elds, namely  ,  ,  , and  , are continuous at the interfaces, as they should be if correct EM boundary condition matching is occurring. It is also observed that the longitudinal or normal components to the interface, namely  and  , are discontinuous at the interfaces also as one would expect for the present problem. In Figs. 16 and 17 it is further observed that the magnitudes of the EM ®elds are constant in Region 3. This is expected since only a transmitted wave occurs in this region. In concluding this section, the authors would like to make the comment that the veri®cation of the complex Poynting theorem is a complicated but important calculation for the present problem. Using Eqs. 2.2.18±20 and t t generalizing the electric and magnetic currents and  , respectively, to include the additional contributions resulting from the bi-anisotropic material parameters of Region 2, one can verify the complex Poynting theorem by using the Poynting box shown in Fig. 15. We have veri®ed that the complex Poynting theorem is indeed obeyed to a high degree of accuracy.

Copyright © 2000 Marcel Dekker, Inc.

Figure 17 Plots of the magnitudes of the  ,  , and  magnetic ®eld components in the same regions as Fig. 16 are shown.

2.5 2.5.1

ONE-DIMENSIONAL k-SPACE STATE VARIABLE SOLUTION Introduction

In this section we apply the state variable method to solve problems where the EM ®eld pro®les vary in one transverse dimension and are incident on, in general, a bi-anisotropic slab. The bi-anisotropic slab is assumed to be bounded by either a homogeneous lossless half space or a perfect electric or magnetic conductor. Examples of this type of problem are a one-dimensional Gaussian beam incident on a material slab, an electric or magnetic line source incident on the slab (or located within the slab), and a slot radiating from a ground plane located adjacent to the material slab. In this section we assume that the EM ®elds vary in the - and -directions and are constant in the -direction.

Copyright © 2000 Marcel Dekker, Inc.

2.5.2

k-Space Formulation

To begin the analysis we expand the EM ®elds in Regions 1±3 in a onedimensional Fourier transform [1±8] (also called a k-space expansion) and substitute these ®elds in Maxwell's equations. As in other sections, all coor~ etc. We have dinates are normalized as  0 ,   0 ,   ;    0  ;  



  

 (  ;  exp  

2:5:1

 )  ;  exp  

2:5:2

where   . The subscript refers to the spatially varying EM ®elds, and Eqs. 2.5.1 and 2.5.2 apply to Regions 1±3. Our objective is to ®nd the EM ®eld solutions in Regions 1±3 of space and then to match appropriate EM boundary conditions at the Region 1±2 and Region 2±3 interfaces. In Region 2, we assume the same bi-anisotropic layer as was studied in Section 2.4. Substituting the electric and magnetic ®eld of Eqs. 2.5.1 and 2.5.2 into Maxwell's equations and interchanging the curl operators   ~ and Fourier integrals we ®nd that 1=0 

0

0

 )

   *     (  ;  exp  l )  ( exp  

 )

    *    )  ;  exp   )   ( exp  





2:5:3

2:5:4

Setting the quantities in the curly brackets of Eqs. 2.5.3 and 2.5.4 to zero and performing a small amount of algebra it is found that     exp    ( exp 

 l )   (

    exp    ) exp 

  )   (

2:5:5

2:5:6

These equations are of the same form as Eqs. 2.4.5 and 2.4.6 if we take   0. We thus ®nd in Region 2 that the variable equations given in Section 2.4 represent a general solution of the problem being studied here.

Copyright © 2000 Marcel Dekker, Inc.

2.5.3

Ground-Plane Slot-Waveguide System

As a speci®c example of the theory of this section we consider the problem of a slot parallel plate waveguide radiating from an in®nite ground plane through an anisotropic material slab into a homogeneous half space. Figure 18 shows the geometry of the system. We initially assume that the EM ®elds inside of the slot waveguide consist only of an incident and re¯ected TEM waveguide mode whose incident amplitude is 0 (volt/m) and whose re¯ected amplitude is $0 (volt/m) and to be determined. The material parameters in the slot are taken to be lossless, isotropic, and characterized by relative parameters 3 and 3 . We assume that the material layer (Region 2) has a ®nite thickness  and that the only nonzero, lossy, relative material parameters in the slab are  ,   ,  ,  , and             . All other material parameters in , ,  , and l tensors are zero. The in®nite half space is assumed to have lossless material parameters 1 and 1 . Assuming only a TEM wave in Region 3 we ®nd that the EM ®elds in the waveguide slot referring to Fig. 18 are given by   0 exp 3   

0  

0 exp 3   

3

 $  $0 exp 3   

Figure 18

The geometry of the ground-plane slot-waveguide system.

Copyright © 2000 Marcel Dekker, Inc.

2:5:7

2:5:8

2:5:9

0 $ 

$0 exp 3   

3

2:5:10

 3     $

2:5:11

 3    $

2:5:12

for   2 and zero elsewhere in Region 3. In Eqs. 2.5.7±12, 0  377 ,   ~ and ~ (meter) is the waveguide slot half 3  3 =3 , 3  3 3 ,   0 , width. Since the EM ®elds are independent of the -direction, it turns out that the only nonzero ®eld components in all regions of space are the  ,  , and  components. The general state variable equations given by Eqs. 2.5.5 and 2.5.6 reduce to @5  +5 @

+



11

12

21

22



2:5:13

where

11

21



              

#

2 12         22    

$

2:5:14

2:5:15

and where 5  ( ; )  . These are in fact the same exact equations as were studied in Section 2.3 except that here ( and ) represent k-space Fourier amplitudes rather than spatial EM ®eld components as they did in Section 2.3. The general solution to Eqs. 2.5.13 in Region 2 is

 2



 2  0  2 

Copyright © 2000 Marcel Dekker, Inc.

  # 2 

1

  # 2 

1

  # 2 

1

$

* ( exp   exp  

2:5:16

$

* ( exp   exp  

2:5:17

$

2:5:18

* ) exp   exp  

where (  1 11

) 



( 

2:5:19



2:5:20

12

  (  )    

 1; 2

2:5:21

and where   0:5  2  0:5 11  22  0:5 1  0:5

11



22 

11



2 22

4

12 21

11



2 22

4

12 21

1=2 1=2

2:5:22

2:5:23

From Maxwell's equations and including the boundary condition that only an outgoing wave can propagate away the material slab and waveguideslot, the EM ®elds in Region 1 are given by  

 1 1

 )  exp  1   1       1

 )  exp  1    1   ) 1  exp  1  

 1

 1

0  1



2:5:24

2:5:25

2:5:26

where 1 



1 1 2 1=2 2 1 1 1=2

1 1 2  0 1 1 2 < 0

2:5:27

The minus sign of 1 (or branch of 1 ) was chosen on the physical grounds  that the integrals converge as    when the   > 1 1 . To proceed it is necessary to match EM boundary conditions at the Region 1±2 and Region 2±3 interfaces. To facilitate the Region 2±3 EM boundary matching, it is convenient to represent and replace the waveguide  aperture slot with an equivalent magnetic surface current  s backed by an electrical perfect conductor. The boundary condition equation to determine  the equivalent magnetic surface current  s backed by an in®nite ground plane is

Copyright © 2000 Marcel Dekker, Inc.

  2

^  

 3

 



 



    s

2:5:28

where  3

  

0

2:5:29

since the magnetic surface current is assumed to be backed by an in®nite ground plane. Also  2

  

 ' rect

  ^ 2

2:5:30

where   rect  2



1

  < 

0

  > 

2:5:31

' represents the -component of the electric ®eld in the aperture. Using Eq. 2.5.30 it is found that the equivalent magnetic surface current is given by     ^  s  ^  ' rect 2





  exp  

2:5:32

 The last part of Eq. 2.5.32 expresses  s in k-space. For the present problem the aperture electric ®eld is given by Eq. 2.5.30 evaluated at    . Thus ' is a constant given by '  0  $0 . Using this value of ' it is found from Fourier inversion that   

' sin  

  

2:5:33

We will now present the boundary value equations at the Region 1±2 and Region 2±3 interfaces. At the Region 1±2 interface, matching the tangential electric ®eld ( -component) and the tangential magnetic ®eld ( component) on the   0 (in Region 1) and   0 (in Region 2), and at the Region 2±3 interface, matching the tangential electric ®eld ( -compo nent) at    (Region 2) to the magnetic surface current  s , and then recognizing that the Fourier amplitudes of all the k-space integrals must equal each other for all values of  , we ®nd the following equations:

Copyright © 2000 Marcel Dekker, Inc.



2  1 1

)   * (

1

1

) 1   2 

1

2 

2:5:34

2:5:35

* )

1

* ( exp     

2:5:36

If we eliminate ) 1  from Eqs. 2.5.34±36 we are left with a 2  2 set of equations from which to determine *1 and *2 in terms of   . We ®nd that *1  *2 

"2  

"1 exp 2  "2 exp 1 

2:5:37

"1  

"1 exp 2  "2 exp 1 

2:5:38

where " 

12



1  1

11

 

 1; 2

2:5:39

The last boundary condition to be imposed is that the tangential magnetic ®eld at    (Region 2) should match the tangential magnetic ®eld at    (Region 3, inside the waveguide aperture). We have 0  2

 

 0  3

 

 

2:5:40

2

In this section we will enforce this boundary condition by averaging Eq. 2.5.40 over the width of the waveguide slot   < . Integrating over    and dividing by 2 we have 1 2

  0  2



  1     3

    2  0 

 





2:5:41

The right-hand side of Eq. 2.5.41 integrates after using Eq. 2.5.12 to 1 2

  0  3



Copyright © 2000 Marcel Dekker, Inc.

 1   0 $0    3

2:5:42

Thus

1 1  $0   3 0 2

  0  2

 





2:5:43



When only TEM waves propagate in a parallel plate waveguide, the parallel plate waveguide forms a two-conductor transmission line system. An important quantity associated with the transmission line system is a quantity called the transmission line admittance, which for the present case at location  on the line   is de®ned as 3

 

~ 3 

 3

 

2:5:44

and for the present case using Eqs. 2.5.42±44 is given by 1 0 exp 3   

$0 exp 3   

~ 3 

 0 3 0 exp 3   

 $0 exp 3   

2:5:45

This quantity is useful for transmission lines because once a transmission line admittance load, call it 3~ .' , is speci®ed at a given point on the line it is possible to ®nd a relation between the incident wave amplitude 0 (assumed known) and the re¯ected wave amplitude $0 . With 0 assumed known and $0 known from Eq. 2.5.45, the ®elds everywhere on the line can then be determined using Eqs. 2.5.7±12. In the present problem we de®ne a transmission line load admittance to be located at the waveguide aperture at   . In this case we ®nd, calling the transmission line load admittance 3~ '` (in units of 1 (or mhos); the subscript ' refers to aperture),

3~ '` 

 3

 3

 

 3



 

'

"  1 1 1     2

0 ' 2  0 

Copyright © 2000 Marcel Dekker, Inc.

1 0 $0  0 3 '

 

If we replace 1=3 0 $0  by 1=2

we ®nd that 3~ '`



%



2:5:46

0  2    using Eq. 2.5.43,

  



+

2:5:47

De®ning a normalized aperture load admittance we have

3'`

 0 3~ '` 

1=2

%



0  2

 



'

2:5:48

If we substitute the EM ®eld solution for the magnetic ®eld in Region 2 into Eq. 2.5.48, interchange the  and  integrals in the numerator of Eq. 2.5.48, and cancel the common constant ' in the numerator and denominator of Eq. 2.5.48, we ®nd the following expression for the normalized aperture load admittance: 3'` 





3  

2:5:49

where     "2 )1 exp 1  "1 )2 exp 2  sin   2 3    "1 exp 2  "2 exp 1 

 

2:5:50

We remind readers that in the above equation, the quantity in square brackets is a complicated function of  , and the ) ,  1; 2, are eigenvector components associated with the magnetic ®eld in Region 2. Once the integral in Eq. 2.5.49 is carried out, 3'` is known and then a relation between 0 and $0 can be found through the equation ~ 3'`  0 3 





1 0 $0 3 0  $0

2:5:51

If 0 is assumed known, then the normalized re¯ection coef®cient of the system is 

$0 1=3 3'`  0 1=3  3'`

2:5:52

In computing the integral as given in Eq. 2.5.50, care must be used in  carrying out the integral near the points where   1 , 1  1 1 when 1    1 (this interval is in the visible region) and 1       1 (this interval is in the invisible region), where  is a small number say on the order of 1 =4 or possibly less. The reason for this is that the function in square brackets in the integrand of the 3'` integral may be discontinuous

Copyright © 2000 Marcel Dekker, Inc.

(or even singular) near the points   1 , and thus signi®cant numerical error can occur if a very ®ne numerical integration grid is not used around these points. In the present section using the quadrature formulas   1 cos 0 , 0 0 , in the visible region and   1 cosh 0 , 0 0 , in the invisible region was employed to integrate the 3'` integral. These formulas provide a very dense grid near   1 and thus provide an accurate integration of the 3'` integral. Harrington [3, p. 183, Eqs. 4-104, 4-105] de®nes an aperture admittance for the present slot radiator problem through the Parseval power relation !~  3~ '  ,2

2:5:53

~ ' , '  1 (Volt/meter) and where where ,  2 !~ 

 

 2



~  ~

  2



 ~ 



1  ~  2





  ~   ~  ~

2:5:54

where  ~ and   ~ are the Fourier amplitudes (or k-space pattern space factors) of the  2 electric ®eld and the  2 magnetic ®eld, respectively. !~ has units of (watt/meter)=(volt amp/meter), so 3~ ' has units of ( meter 1 (or mho/meter). Substituting the EM ®eld solutions derived earlier in Eq. 2.5.54, it is found that the aperture admittance 3~ ' as de®ned by Eq. 2.5.54 is very closely related to the transmission line load admittance expression 3~ '` . It is related by the equation 3~ 3~ '  '` 2~

2:5:55

where 2~ is the width of the slot. We note that, in calculating the 3'` integral using Eq. 2.5.49 in the limits as   0, the exponential terms in Eq. 2.5.50 approach unity, and it is found after a small amount of algebra that 3'`  0 3~ '`

  1 sin   2      1 

2:5:56

which is an expression for the aperture load admittance of a slot waveguide radiating into a homogeneous lossless half space. If one substitutes 3~ '` as

Copyright © 2000 Marcel Dekker, Inc.

given by Eq. 2.5.56 in the aperture admittance expression as given by Eq. 2.5.55, one derives the same expression as derived by Harrington [3, p. 183, Eqs. 4-104, 4-105] for a ground plane slot radiating into a lossless half space. Another quantity of interest is the power that is radiated as one moves in®nitely far away from the radiating slot. The Poynting vector at a location   cos c ,    sin c ,   , is given by    1  1  1

1 1 1 2 ^  /  (  Re    ) 2 2 0 

2:5:57

where ) 1







'  exp  1  

2:5:58

and where     '  "2  "1  sin  

'   1    1 "1 exp 2  "2 exp 1  

2:5:59

We note in passing that Eq. 2.5.58 for 3 1 is identical to that given by Ishimaru [4, Chapter 14] when one (1) lets the dielectric layer be isotropic, (2) lets the slot waveguide width 2~ approach zero while holding the voltage potential difference between the parallel plate conductors constant, and (3) makes the correct geometry association between Ishimaru's analysis and the present one. Ishimaru [4] shows, by using the method of steepest descent, that the integral in Eq. 2.5.58 asymptotically approaches as    the value ) 1

 1=2   2    1 sin 'c

exp 1   1  4

2:5:60

where  1 sin c

 1 cos c ' 1 sin c

     '  "2  "1  sin  

1    "1 exp 2  "2 exp 1  

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2:5:61

where 1 sin c and 1 cos c have been substituted for  and 1 , respectively, in Eq. 2.5.58. To describe the radiation from the waveguide aperture and material slab system in the far ®eld    we plot the normalized radiation intensity, which here is de®ned as the radiation intensity,   , divided by the total radiation intensity integrated from c  =2 to c  =2. This quantity is called the directive gain  c . Applying this de®nition and using Eqs. 2.5.60 and 2.5.61 after cancelling common factors we ®nd  c   % =2

 1 sin c

2

2:5:62

2 =2  1 sin c

 c

2.5.4

Ground-Plane Slot-Waveguide System, Numerical Results

As a numerical example of the radiation through a waveguide slot radiating through the anisotropic layer under study we consider the layer formed when 1  1 and 1  1, 2    1:2 2:6



  2   0

   0

0 0  

2:5:63

where   2,    0:3,   0:9 0:2, and   2:1. The value of  is immaterial to the present analysis and is not speci®ed here. For all calculations in this section the slot width has been taken to be 2~  0:6. Figure 19 shows a plot of the 3  aperture admittance integrand when the layer thickness has been taken to ~  0:6. As can be seen from Fig. 19 for the values used in the present example, the integrand converges fairly rapidly for values of    51  5. An inspection of Eq. 2.5.50 for 3  shows that for  large the integrand approaches 1=3 and thus is guaranteed to converge. In an inspection of Fig. 19 one sees also that the integrand 3  is not exactly symmetric with respect to the  variable. This is a result of the slot radiating through an anisotropic rather than an isotropic medium. For the present example, the boundary of the visible and invisible [1] (i.e., propagating and evanescent) radiation range is at   1  1. One observes from Fig. 19 the effect that the discontinuous 1 function of Eq. 2.5.27 has on the 3  integrand in the  regions near   1  1. Figure 19 also lists values of the two lowest magnitude poles which were associated with the 3  integrand. The two pole locations in

Copyright © 2000 Marcel Dekker, Inc.

Figure 19

A plot of the 3  aperture admittance integrand.

the complex  plane  1  1:541  0:218 and  2  1:567 0:146

were nonsymmetric because of the anisotropy of the material slab. The values of the poles were listed as they in¯uence the real  integration when the  integration variable passes close to the poles' location. Figure 20 shows a plot of the 3'` aperture load admittance as a ~ At a value of ~  0 the layer does not function of the layer thickness . exist, and the waveguide aperture radiates into free space. As ~ increases, the real and imaginary parts of the aperture admittance are oscillatory up to a value of about ~  1, where it starts to approach a constant value. Figure 21 shows a plot of the directive gain as a function of the angle c . One observes from this ®gure that the radiation pattern is concentrated in a 90 angle around the broadside direction and one also observes that the radiation pattern is asymmetric in the angle c , with the peak radiation value occurring at about angle c  10 . The asymmetry is caused by the fact that the slot has radiated through an anisotropic material slab.

Copyright © 2000 Marcel Dekker, Inc.

Figure 20 A plot of the 3A` aperture load admittance as a function of the layer ~ thickness .

Figure 21

A plot of the directive gain as a function of the angle  .

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2.6 2.6.1

RADIATION FROM A DIPOLE IN THE PROXIMITY OF A GENERAL ANISOTROPIC GROUNDED LAYER [32] Introduction

In the previous sections we have studied general plane-wave incidence on an anisotropic material slab and have used one-dimensional k-space theory to study radiation from a waveguide slot aperture into an anisotropic material. In this section we will study the problem of determining the EM ®elds when an electric dipole is in the presence of a slab of anisotropic material that is backed by an electrical ground plane (see Fig. 22). As is well known, the radiation from a dipole varies in all three dimensions in space. The solution to this problem is one level of complexity higher than the previous example and thus requires two-dimensional k-space theory rather than one-dimensional k-space theory. Furthermore, the presence of the anisotropic layer near the radiating dipole makes this a formidable problem to tackle. This follows because the anisotropic material couples all of the EM ®eld components in a very complicated way. Two-dimensional k-space theory in conjunction with state variable techniques is probably the only tractable way to approach this problem. We will summarize the basic formulation and numerical solution as presented by Tsalamengas and Uzunoglu [32], who have developed a useful and interesting formulation to this problem that we will brie¯y summarize in the following section. The formulation of Ref. 32 is useful because it constructs an EM ®eld solution that, despite the complexity of the general anisotropic layer, builds the ground plane boundary condition (tangential electric ®eld zero at the surface of the ground plane) into the form of the EM ®eld solution. In the following we follow the coordinate system and notation of Ref. 32.

Figure 22

General anisotropic grounded layer geometry. (# 1985, IEEE.)

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2.6.2

The Field Inside the Anisotropic Layer

Following Ref. 32 we assume that the permittivity and permeability tensor components of the anisotropic layer are characterized by the general com~ Using the notation in Ref. 32, Maxwell's equations in plex values ~ and l. the anisotropic region [assuming exp ! time dependence] assume the form  ~     !~ 

2:6:1

 ~  ~    !l

2:6:2

where the subscript ``a'' stands for anisotropic. We express the spatial electric and magnetic ®elds in a two-dimensional k-space Fourier transform as   

 

 

  ~ ~  ~ ~ exp  ~ ~  ~ 

   ; 

2:6:3

  respectively, either the electric ®eld where   ~ ~  ~ ~  ;  represents,     ~ represents,  or magnetic ®eld  , and where   ; 

respectively,   ~ either the spectral amplitude of  the electric ®eld   ; 

or the spectral  ~ Substituting the Fourier transamplitude of the magnetic ®eld   ;  . forms integrals into Maxwell's equations and collecting coef®cients of the exponential in Eq. 2.6.3 we ®nd that    ~  !~  ~    ; 

  ; 

2:6:4

   ~  !l~   ; 

~    ; 

2:6:5

where

0    @=@~  ~

@=@~  ~  0  ~   ~ 0

2:6:6

De®ning the auxiliary ®eld column matrices

~  $ 

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#

$    ~     ; 

  ^    ~  

  ; 

3  

#

$    ~     ; 

   ^     ; 

~  

2:6:7

We ®nd that Eqs. 2.6.4 and 2.6.5 can be put into the form     3 

~ ,  ~ 5  ~ $ 

0 !



~ 3 

~ $ 



2:6:8

where the 2  2 matrices ,, 0, 5, and ! can be found in the Appendix of Ref. 32. The boundary conditions require that the tangential electric ®eld at  ~  0 must be zero. This requires at ~  0 that ^     0, which further  requires, by the completeness of the Fourier transform, that ^      ; 0    0 or   ; 0     ; 0  0. Thus the auxiliary column matrix $  satis    ®es $ 0  0, since  1 0       ; 0     ; 0      ; 0  0       ; 0      ; 0     ; 0   0. and  2 0  ^   Consider the matrix differential equation     8 ,  5  ~ 9

0 !



8 9



2:6:9

where 8 and 9 are 2  2 matrices with entries 

~ 11 

8 ~ 21 

~ 12 

~ 22 





~   12 

9  11 ~ 22 

21 



If 81 and 91 are solutions of Eq. 2.6.9 that meet the boundary conditions 81 0 
2 and 91 0  0 (
2 is a 2  2 identity matrix), then the solution of Eq. 2.6.8 is given by ~ ~  81  8 ~ 1 3 

1 

'

2:6:10

~ ~  91  8 ~ 1 $ 

1 

'

2:6:11

where '    t is a 2  1 constant column matrix. The matrices 81 and 91 are given by the solution 

91 81





 ~  exp +


2



and where the matrix + is given by 

! 5 + 0 ,

Copyright © 2000 Marcel Dekker, Inc.



2:6:12

and the 2  2 submatrices ,, 0, 5, and ! may be found in the Appendix of Ref. 32 as mentioned earlier. The matrix exp + can be evaluated through the Cayley±Hamilton by the expression ~  *0 
~ 4  *1  + ~  *2  + ~ 2  *3  + ~ 3 exp +

2:6:13

~
 0; 1; 2; 3; satisfy where *i  , ~  exp  

3  0

~ k *k 

  1; 2; 3; 4

2:6:14

and  ,   1; 2; 3; 4, are the distinct roots of the characteristic equation det 
4 +  4 

1

3



2

2



3

1



4

0

2:6:15

where 1   + , 2   1  +   +2 =2, 3   2  +  1  +2

 +3 =3, and 4  det + and where   is the trace operator. In this analysis, only the case of distinct roots is treated. When repeated roots are present a more general analysis is required. After a lengthy algebraic pro~ . . . ; and 11 

~ cedure one can determine the eight matrix elements 11  ; making up the 2  2 matrices 8 and 9 respectively. A full listing these matrix elements is given in Ref. 32, Eqs. 16a±d and 17a±d. Using Eqs. 2.6.4±15 one can ®nally ®nd full algebraic expressions for   ~ and the electric and magnetic Fourier amplitude ®eld components   ; 

  ~ respectively. The algebraic form of these amplitudes is given in   ;  , Ref. 32. We remind the reader that these ®eld components at this stage of the analysis are speci®ed in terms of the still unknown '    t . Speci®cation of the general EM ®elds in the half space ~ > ~ (which contains the electric dipole source) and boundary matching of these ®elds to ®elds of the anisotropic layer must be performed in order to determine all ®elds of the EM system. 2.6.3

Solution of the Boundary Value Problem

The ®eld in the region ~ > ~ is the superposition of the EM ®elds due to the dipole source and the ®elds re¯ected from the anisotropic layer. The primary EM ®eld due to the dipole source is assumed to be excited in free space (in the  absence of anisotropic slab) and to the electric  the  dipole current  source    ~ ^ ~ ~ , where ~ > . Letting  0  ; 

~ and  0  ; 

~ be the ~   two-dimensional Fourier amplitude of the electric ®eld and magnetic ®elds due to the dipole source [using the Fourier representation as given by Eq.

Copyright © 2000 Marcel Dekker, Inc.

~ and $0 

~ de®ned ana2.6.3, and using the auxiliary ®eld quantities 30 

logously to Eqs. 2.6.10 and 2.6.11, the free space dipole can be written as # 0 sgn ~ ~ 

1 ~  2 30 

8 sgn ~ ~ 

0   4 exp      ~0 ~ ~  

# 1 1 0 !0

~  2 $0 

8 0

0 !0 ~0 1

0  ~2 ~0 1

$

sgn ~ ~  ~2 !0 1 0

2:6:15

$

  4 exp  ~    ~0 ~ ~  

2:6:16

where     , ~2  ~2  ~2 , ~0  ~2 20 1=2 , ~20  !2 0 0 ,

  ^       4    ^  ^      



   ~    0  ; 

 30      ~  

^   0  ; 



^  ^ 

   ~    0  ; 

 $0      ~  

^   0  ; 



2:6:17

and   ~  ~  ~  ^ 

2:6:18

For the ®eld re¯ected from the anisotropic layer (an outgoing wave moving away from the layer), ~  3 

#

~  $ 

#

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 ~0

0

0

!0

$#

0

 ~0

!0

0



 $#

$

~ exp ~0 ~ 

2:6:19

$

2:6:20

 

~ exp ~0 ~ 

   ~ and $ 

~ are determined from  ~ and    ; 

~ in a where 3 

   ; 

~ and $0 

~ were manner  similar  to the way 30 

 determined  from  ~ and  0  ; 

~ or 3 

~ and $ 

~ from  ~ and   ;  . ~  0  ; 

  ; 

The ®nal step in obtaining the solution is to boundary match the ~ The total EM ®elds for ~  ~ is the sum of tangential EM ®elds at ~  . the incident and re¯ected ®elds, and the total ®elds for ~ ~ is the anisotropic slab ®eld; thus equating these total ®elds (using the three sets of auxiliary vectors) we have ~  '  30 

~  3 

~ 3 

2:6:21

~  $ 

~  $ 

~ $ 

0 

2:6:22

On substituting Eqs. 2.6.15±20 into Eqs. 2.6.21 and 2.6.22, the following set of 4  4 equations is obtained, from which all unknown constants of the system can be found. The 4  4 equations are ' ~ ~ 91  8  ' 1

#

#

 ~0

0

0

!0

0

 ~0

!0

0

$# $#

   

$

~  30 

2:6:23

$

~  $0 

2:6:24

Once the four constants '    t , , and  are known, the EM ®elds in the anisotropic and isotropic regions can be speci®ed. Reference 32 gives a complete speci®cation of these ®elds both in the anisotropic region and in the isotropic region. Reference 32, further, by letting   , ®nds, from an asymptotic approximation of the Fourier integrals, expressions for the electric far ®eld. From these far ®eld expressions, Ref. 32 is able to compute the far ®eld radiation patterns of the dipole anisotropic slab.

2.6.4

Numerical Results and Discussion

Numerical computations [32] have been carried out for the far ®eld structure related to several anisotropic substrates. The anisotropic cases considered are uniaxial crystals, ferrites, and plasmas. For the ferrite and plasma layers, the orientation of the static magnetic ®eld is taken as ^  cos 0 ^   sin 0 cos 0 ^  sin 0 ^ 

Copyright © 2000 Marcel Dekker, Inc.

2:6:25

The general ferrite tensor l 0 ; 0 and the plasma tensor  0 ; 0 are computed by applying unitary transformations to l 0  0; 0 , and  0  0; 0 , respectively. The expressions for these tensor are referred to in [32]. For uniaxial media the ^ vector represents the orientation of the optical axis. The direction of the radiating dipole is determined by the unit vector ^ and is parallel to one of the unit vectors ^ , ^ , ^  . Figure 23 (kindly supplied to us in corrected form by the authors of Ref. 32), gives results for  and  relative far ®eld amplitudes for a ceramic Polytetra¯uoroethylene (PTFE) uniaxial substrate for various optical axis orientations 0  20 , 40 , 60 , and 80 ). The dielectric constants

Figure 23 Radiation patterns  ,   versus  in the   0 (180 ) plane for a uniaxial substrate with   10:70 ,   10:40 , l  0 3 ,   1 mm, and   30 GHz. The primary source is an electric dipole located at the substrate surface ^ (# IEEE, 1985.)     , and its orientation is de®ned with the unit vector .

Copyright © 2000 Marcel Dekker, Inc.

along the principal axes are ~  ~  10:70 and ~  10:40 . In this case the  0 ; 0 is independent of the 0 angle and l  0 I 3 . The substrate thickness is taken to be   1 mm. Both vertical ^  ^ and horizontal ^  ^ dipoles are considered assuming the same excitation. The variation of the radiation diagrams is noticeable only for the horizontal dipoles, while for the vertical dipoles there is almost no effect of the optical axis orientation. The radiation diagrams, as in the case of isotropic substrates, retain their symmetry with respect to the -axis. In treating ferrite substrates it is assumed that  0 ; 0  150 3 and that a strong magnetic type of anisotropy is used with ~ 11  0:6750 , ~ 12  0:494 0 , !0 =!  2:35 [32], !  , 0   0:3-2=2 ( being the magnetomechanical ratio). Corresponding to various biasing static magnetic ®eld orientations, the computed radiation patterns on various   constant ^ planes are given in Figs. 24±26 for -directed dipoles. The radiation frequency is taken   30 GHz, and the ferrite layer thickness is   1 mm. In general there is a strong dependence of the far ®eld to 0 orientation. When the   constant observation plane coincides with the   0 plane (i.e.,   0  0) and the dipole axis is also parallel to this plane, the patterns are axisymmetric. This symmetry is not exhibited for other observation planes such as in Fig. 25, where patterns are varying from an almost omnidirectional coverage (0  20 ) to a rather directional diagram 0  80 ).

Figure 24 Radiation patterns  ,   versus  in the   90 observation plane of a ferrite substrate for various 0 angles and 0  0 . The material properties of the ferrite are ~ 11  0:6750 , ~ 12  0:4940 [32], and ~ 0 ; 0  0 3 , ~  1 mm. and   30 GHz. The dipole axis is along the -axis ^  ^ and is located at the ~ (# 1985, IEEE.) substrate surface ~    .

Copyright © 2000 Marcel Dekker, Inc.

Figure 25 Radiation patterns  ,   versus  for the same parameters as Fig. 24 except the observation plane is   0 . The magnetostatic ®eld is inside the 0  0 plane. (# 1985, IEEE.)

Figure 26 Radiation patterns  ,   versus  for the same parameters as Fig. 25 except the observation plane is   0 and 0  45 . (# 1985, IEEE.)

Copyright © 2000 Marcel Dekker, Inc.

There is also high cross-polarization due to the anisotropic layer. Numerical computations have shown that the nonsymmetry in the lobe structures is considerably smaller for weaker anisotropies 11  0:90 , 12 0:2 0 ). With this, however, strong depolarization phenomena have been observed with a strong dependence on the 0 angle. Finally we consider the excitation of a grounded plasma layer with a horizontal dipole excitation. Again the radiation frequency is   30 GHz and the plasma layer thickness is   1 mm. The parameters characterizing the plasma are taken as l  0 3 , while  0  0; 0 is computed with !c = !p  1:8 and !=!p  2:4. In Fig. 27 computed radiation patterns are given. For this particular set of plasma parameters the variation in the radiation pattern is weak. However when   0  0, strong variation in the sidelobes is observed.

2.6.5

Conclusion

In conclusion of this section a general formulation is presented for the analysis of an EM ®eld originating from an arbitrary oriented dipole source in the presence of a grounded general anisotropic layer. The Green's function is determined by using linear algebra techniques without restriction on the anisotropic permittivity or permeability. Several numerical examples have been presented.

Figure 27 Radiation patterns  ,   versus  in the   0 plane for a grounded plasma layer with ! =!  1:8, !=!  2:4, ~  1 mm, and   30 GHz [32]. The ~ and ^ and it is located on the plasma surface ~   

dipole is along the -axis ^  ,  0  0 . (# 1985, IEEE.)

Copyright © 2000 Marcel Dekker, Inc.

2.7

2.7.1

A NUMERICAL METHOD OF EVALUATING ELECTROMAGNETIC FIELDS IN A GENERALIZED ANISOTROPIC MEDIUM [25, 26] Introduction

In the previous sections a 4  4 matrix formulation has been presented to study EM ®elds in an anisotropic or bi-anistropic medium. As mentioned previously, for anisotropic or bi-anisotropic media, the full ®eld method is the only tractable method, because of the analytic complexity of dealing with the complicated coupled tensor equations that result. A critical step in the state variable or exponential matrix method is to develop the transition matrices, which relate the EM ®elds at one planar interface to others. This method, although ef®cient at handling the formulation, has problems in the actual numerical computation. Problems arise when the wave numbers in the direction of the inhomogeneity are complex valued. If the layers are electrically thick enough, the transition matrices become numerically singular due to some exponentially large matrix elements. The problem of singularity of the transition matrix is particularly severe in systems that have sharp discontinuities such as antennas and circuits, as these systems generate signi®cant evanescent ®elds; thus generating the correct numerical solution in the evanescent wave number range is dif®cult. In this section a scheme utilizing variable transformation is developed. The idea is to extract the large exponential terms in the formulation and transform them into variables that are then used to represent the ®elds at each interface. In the following section only a single layer analysis is performed. A detailed review of this algorithm as applied to multilayer analysis is given in Ref. 25. In the following we use the coordinate system and notation of Yang [26] to describe the ®eld problem. Yang refers to this as the spectral recursive transformation method [25].

2.7.2

Variable Transformation in the Matrix Exponential Method

We consider the problem of a plane wave scattering from a planar ( - ~ shown in Fig. 28. All coorplane) generalized anisotropic layer 0 < ~ < 

dinates and ®eld quantities are in unnormalized coordinates. The approach using ®eld excitation by current sources is similar in principle to plane wave analysis under consideration. The extension of the method to multilayer systems is discussed elsewhere [25]. In the spectral exponential matrix method the ~ and ~ spectral ®eld components in the anisotropic medium derived from Maxwell's equations with some algebraic manipulation

Copyright © 2000 Marcel Dekker, Inc.

Figure 28 Re¯ection from an in-plane biased ferrite layer. Biased ®eld 0 1000 Gauss in the ^ direction; magnetization 2500 Gauss. Transverse magnetic incidence i  30 and i  40 , ~  12:80 , and ~  3 cm. (Copyright 1995, IEEE [+3].)

become four coupled ®rst-order differential equations of Berreman [20] or Tsalamengas and Uzunoglu [32]. In matrix form the equations are @ ~ ~ w   +w 

@~

2:7:1

where

~ ~  ~ ~   ~ ~ ~ ~    ~ 

~   w  ~ ~  ~ ~   ~ ~ ~ ~ 

2:7:2

~ , ~  , ~ , and ~  are the Fourier transforms of the tangential components, and + is a 4  4 matrix where the elements are functions of spectral variables ~ and ~ and material parameters. If one de®nes the 4  4 matrix r~ as the eigenvector matrix with the eigenvalues i ,
 1; 2; 3; 4, of +, the solution of Eq. 2.7.1 is 

   ~ ~ 0

~ ~  / 

Copyright © 2000 Marcel Dekker, Inc.

2:7:3

where

~ exp 1 

0

0

0

0

~ exp 2 

0

0

0

0

~ exp 3 

0

0

0

0

~ exp 4 

  ~  / ~  / 

 

   ~ 1 /  

2:7:4

The electromagnetic ®elds in the air ~  ~ and ~ 0 ) can be derived from a set of transverse electric and transverse magnetic vector potential functions. This result can be shown to be , ~2 20        ! 2 0  :    ,  w      ~2  2  20  





!0  ,

  ~2 20        !0    : w 0

 ,       ~2 20  

2:7:5

!0  

The unknown equation



, 2  ,   , and   are quantities to be determined from the

~ w 0 :  / 

:  4inc w 

2:7:6

, where ~  ~2  ~2 and where 4inc is related to the incident plane wave. For the problem with a current source the right-hand side should be the corresponding spectral current component. The state variable exponential matrix method described above is rigorously correct. However in numerical implementation this method may break down. Without loss of generality it is assumed that Re 1  Re 2  Re 3  Re 4 . In many practical applications when Re 1 ! 1, the transition matrix de®ned in Eq. 2.7.4

Copyright © 2000 Marcel Dekker, Inc.

becomes numerically singular. As a result the numerical inversion of Eq. 2.7.6 provides erroneous results. In Eq. 2.7.4 the transition matrix can be written ~ ~ /   exp 1  + 1  exp 2  +2

2:7:7

where the singular matrices +1 and +2 do not contain any terms that grow exponentially. We have



1

0

0

0

 0  +1  r~  0

0

0

0

0

0

0

 0  ~ 1 r 0  0

0

2:7:8

and

0

0  ~ +2  r  0

0

0

0

0

1 0 ~ 0 exp 3 2  0

0

0 0 ~ exp 4 2 

  ~ 1 r 

2:7:9

Note that +1 is obtained from Eq. 2.7.4 by replacing the terms of exp 2  , exp 3  , and exp 4  with 0 and replacing exp 1  with 1. Since +1 is a singular matrix, it can be shown that

   ~ +1 w 0

     





1

     3 2

2:7:10

4

where , and i ,
 1; 2; 3; 4, are associated with the eigenvectors and found from Eqs. 2.7.6, 2.7.7, and 2.7.8. In order to overcome the over¯ow problem, the following variable transformations are de®ned: ~ 0      exp 1 

2:7:11

~ +      exp 2 

2:7:12

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where 0 and + are the new variables replacing   and   . With the variable transformations, we have

   ~ w 0 ~ / 

 0 

,  ~2 20       !0   2 0 ~  ,  +2 exp 2 1         ~2 2  3 0  1



4

,  ~2 20     !   + 0   +2  ,       ~2 2 

!0

0

!0

2:7:13

Upon inspecting Eq. 2.7.13, one observes why the transformation provides a stable invertible matrix equation from which to determine the unknown coef®cients  , 2  , 0, and + (and therefore   and   ). The right-hand side of Eq. 2.7.13 is a sum of an exponential and two nonexponential terms. When 1 ! 2 , the exponential term becomes much smaller than the nonexponential terms. In this case, when the left-hand side is then numerically computed, the exponential term will make a negligible contribution to the matrix elements of Eq. 2.7.6, and the nonexponential terms alone will provide a ®nite and numerically correct value for the matrix elements of the system. As mentioned earlier, without using this transformation, a row of exponentially small matrix elements exists, leading to numerical singularity of the matrix equation.

2.7.3

An Example: Scattering from a Biased Ferrite Layer

A practical example of the case of scattering from a biased ferrite layer is shown in Fig. 28. It is known that the (magnetically) biased ferrites may couple ordinary and extraordinary waves due to the presence of magnetic®eld-dependent off-diagonal terms in the permeability tensor. Hence an incident ordinary wave could excite extraordinary waves inside the material. The extraordinary wave is evanescent [35]. When the decay factor of this extraordinary wave is large, the matrix equation that directly results from boundary matching is no longer numerically invertible, for reasons dis-

Copyright © 2000 Marcel Dekker, Inc.

cussed above, and therefore the variable transformation technique should be used. The result for the re¯ection from a biased ferrite layer is shown in Fig. 28 for both methods. It is seen that there exists a frequency band where the ordinary transition matrix method provides nonphysical results. Outside this frequency band the two methods provide identical results. Further examples of the variable transformation technique can be found from [25]. 2.7.4

Conclusion

A numerical algorithm was developed for the computation of EM ®elds in a generalized anisotropic structure. The proposed method using variable transformation overcomes the dif®culty frequently encountered in the transition cascade method, without increasing computational time or memory. The extension of this technique to multilayer structures is given in detail by Yang [25].

PROBLEMS 1.

2.

3.

4.

5. 6.

Using the wave equation for the electric ®eld, write down the EM ®eld solutions in the three regions in Fig. 1. Assume normal incidence from Region 1. Show that your results are the same as the state variable solutions of Section 2.2. If the interface between Regions 2 and 3 in Fig. 2 has a perfectly electrically conducting surface, write down the state variable solutions in each of the three regions for normal incidence from Region 1. Using these solutions and the EM boundary conditions, solve for all the EM ®elds. Extend the state variable solutions developed in Sec. 2.2 to the case of normal incidence onto 2 layers sandwiched in air. Assume that the permeabilities of the layers are equal to that of free space, and that the layer relative permittivities are 2 and 4. Determine the condition on layer thicknesses to achieve maximum re¯ection from the sandwich. Verify the complex Poynting theorem for the solutions to the two-layer sandwich in Problem 3. Assume the Poynting box to be of unit cross-sectional area and of suf®cient thickness to enclose both layers. If the electric current source in Fig. 8 is replaced with a magnetic current source, ®nd the ®eld solutions for the system. Starting from Eq. (2.3.7), develop the state variable solution for the case where the permeability is anisotropic ( ;   ;  ;  ,

Copyright © 2000 Marcel Dekker, Inc.

7. 8.

9.

10.

11.

and  are nonzero) and the permittivity is isotropic. Assume that plane wave which is polarized with its electric ®eld perpendicular to the plane of incidence impinges on the layer. Develop the EM ®eld solutions within a bi-isotropic ("; ; ; 2 scalar) layer immersed in air and for the case  2. A propagating transverse magnetic (TM) mode whose longitudi  nal electric ®eld is given   ' sin   exp  is inci2 dent on the anisotropic layer shown in Fig. 18. Assume only a single TM mode is re¯ected from the layer. a) Determine the EM ®elds associated with the incident TM mode. b) Determine the EM ®elds associated with the re¯ected TM mode. c) Determine the state variable equations and solutions which electromagnetically couple to the incident and re¯ected ®elds from the slot waveguide. d) Determine the EM ®eld solution which exists in Region 1 of Fig. 18 (Sec. 2.5). e) Follow the procedure outlined in Sec. 2.5 to determine the re¯ection coef®cient of the incident TM mode. f) Find the far ®eld radiation pattern associated with the system. Repeat Problem 8 assuming a transverse electric (TE) mode is incident in the waveguide. How does this mode couple to the anisotropic layer? Solve Problem 8 exactly by including in your solution all propagating and evanescent TEM, TE, and TM modes which may be re¯ected from the anisotropic layer system. What is the coupling that occurs between the TEM, TE, and TM modes? a) Considering the slot-waveguide, anaisotopic layer system displayed in Fig. 18, using the parameters; "  2., "   "  :5, "  4: "  1:, "   "  "  "  0:,   1. (all regions), waveguide width equal to :9; and using the numerical method described in Sec. 2.5, determine the EM ®elds of the system if the layer thickness is :2: b) Using the numerical algorithm and parameters of Part a), investigate the largest thickness that may be used before numerical instability of the solution becomes evident. c) Use the spectral recursive transformation method of Yang [25, 26] described in Sec. 2.7, to obtain numerically stable EM solution for layer thickness which were equal to or greater than those determined in Part b) to lead to numerical instability.

Copyright © 2000 Marcel Dekker, Inc.

REFERENCES 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

17.

18. 19. 20. 21.

D. R. Rhodes,  

  "      , Oxford University Press, 1974. H. A. MacLeod,  4&  %"  &   , Macmillan, New York, 1986. R. F. Harrington,  4!   $  &  , McGraw-Hill, New York, 1961. A. Ishimaru,  $  5' "$  6      $, Prentice Hall, Englewood Cliffs, New Jersey, 1991. L. B. Felson and N. Marcuvitz, 6      $  5' , Prentice Hall, Englewood Cliffs, New Jersey, 1973. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen,  $  5'      4(  "  # , Artech House, Boston, London, 1994. J. Galejs,    ($ # , Pergamon Press, Oxford, 1969. J. R. Wait,  $  5'    7 # , Pergamon Press, Oxford, 1970. S. V. Marshall, R. E. DuBroff, and G. G. Skitek,  $  "  ""   , 4th ed., Prentice Hall, Englewood Cliffs, New Jersey, 1996. P. P. Banerjee and T. C. Poon,   "  ""  %"  , Aksen Associates, Homewood, IL. P. Baumeister, Utilization of Kard's equations to suppress the high frequency re¯ectance bands of periodic multilayers, "" %"  +* 2687±2689 (1985). J. A. Dobrowolski and D. Lowe, Optical thin ®lm synthesis based on the use of Fourier transforms, "" %"  .8 3039±3050 (1978). W. H. Southwell, Coating design using very thin high- and low-index layers, "" %"  +* 457±460 (1985). J. M. Jarem, The minimum quality factor of a rectangular antenna aperture, 
   $ 8(1), 27±32 (1982). J. M. Jarem, Method-of-moments solution of a parallel-plate waveguide aperture system,
 ""   /,(10), 3566±3570 (1986). J. M. Jarem, The input impedance and antenna characteristics of a cavitybacked plasma covered ground plane antenna, (     "$  49*, 262±267 (1986). J. M. Jarem and F. To, A K-space methods of moments solution for the aperture electromagnetic ®elds of a circular cylindrical waveguide radiating into an anisotropic dielectric half space, (     "$  498, 187±193 (1989). D. A. Holmes and D. L. Feucht, Electromagnetic wave propagation in birefringent multilayers,
 %"    /3, 1763±1769 (1966). S. Teitler and B. W. Henvis, Refraction in strati®ed, anisotropic media,
 %"    3:, 830±834 (1970). D. W. Berreman, Optics in strati®ed and anisotropic media: 4  4-matrix formulation,
 %"    3+, 502±510 (1972). P. J. Lin-Chung and S. Teitler, 4  4 matrix formalisms for optics in strati®ed anisotropic media,
 %"     ., 703±705 (1984).

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23.

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C. M. Krowne, Fourier transformed matrix method of ®nding propagation characteristics of complex anisotropic layer media, (   # '   ;   #49+, 1617±1625 (1984). M. A. Morgan, D. L. Fisher, and E. A. Milne, Electromagnetic scattering by strati®ed inhomogeneous anisotropic media, (     "$  49/, 191±197 (1987). R. S. Weiss and T. K. Gaylord, Electromagnetic transmission and re¯ection characteristics of anisotropic multilayered structures,
 %"     *, 1720±1740 (1987). H. Y. D. Yang, A spectral recursive transformation method for electromagnetic waves in generalized anisotropic layered media, (     "$  4*/, 520±526 (1997). H. Y. D. Yang, A numerical method of evaluating electromagnetic ®elds in a generalized anisotropic medium, (   # '   ;   #4*9, 1626±1628 (1995). P. Yeh, Electromagnetic propagation in birefringent layered media,
 %"    3,, 742±756 (1979). N. G. Alexopoulos and P. L. E. Uslenghi, Re¯ection and transmission for materials with arbitrarily graded parameters,
 %"    8., 1508±1512 (1981). R. D. Graglia, P. L. E. Uslenghi, and R. E. Zich, Dispersion relation for bianisotropic materials and its symmetry properties, (     "$  49,, 83±90 (1991). S. M. Ali and S. F. Mahmoud, Electromagnetic ®elds of buried sources in strati®ed anisotropic media, (     "$  498, 671± 678 (1979). C. M. Tang, Electromagnetic ®elds due to dipole antennas embedded in strati®ed anisotropic media, (     "$  4+8, 665±670 (1979). J. L. Tsalamengas and N. K. Uzunoglu, Radiation from a dipole in the proximity of a general anisotropic grounded layer, (     "$  499 (2), 165±172 (1985). J. L. Tsalamengas, Electromagnetic ®elds of elementary dipole antennas embedded in strati®ed general gyrotropic media, (     "$  498, 399±403 (1989). C. M. Krowne, Determination of the Green's function in the spectral domain using a matrix method: application to radiators or resonators immersed in a complex anisotropic layered medium, (     "$  4 9*, 247±253 (1986). B. Lax and K. J. Button, # ' &   & $  , McGraw-Hill, New York, 1962.

Copyright © 2000 Marcel Dekker, Inc.

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,    "   + ,  ,   

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   ( % +

Since its discovery by Ashkin et al. [1] in the mid 1960s, a tremendous amount of research has been carried out to study the photorefractive (PR) effect and apply it to real-time image processing [2], beam ampli®cation [3], self-pumped phase conjugation [4], four-wave mixing [5], and optical computing [6], to name a few applications. A preliminary discussion of beam propagation through PR materials was presented in Chapter 4. When two coherent plane waves of light intersect in a PR material, they form an intensity interference pattern comprising bright and dark regions (see Fig. 1). Assuming that the PR material is predominantly n-doped, the electrons migrate from bright to dark regions, thus creating an approximately sinusoidal charge distribution. This diffusion-controlled PR effect in turn creates an electrostatic ®eld that is ideally 90 phase shifted from the intensity pattern and modulates the refractive index of the crystal via the electrooptic effect (see Fig. 1). The incident plane waves are, in turn, scattered by the grating so that one wave may have constructive recombination while the other may encounter a destructive recombination. This effect leads to energy coupling between the beams through what is commonly referred to as the two-beam coupling effect [3]. A steady-state nonlinear coupling theory of the two-beam coupling phenomenon using participating plane waves has been derived and the results reconciled with numerical simulations of the coupling between two, in general, focused Gaussian beams [7]. The results indicate that the twobeam coupling parameter  [8] strongly depends on the initial intensity ratio of the plane waves (or power ratio of the input beams), an effect that has

Copyright © 2000 Marcel Dekker, Inc.

5" 

Geometry of the PR diffraction grating.

been experimentally observed before [9,10] but for which only an empirical theory existed [11]. The empirical theory, however, was an improvement over linearized time-dependent theories that predict that  is independent of the initial intensity ratio [8]. The dif®culty in providing an analytical nonlinear time-dependent theory stems from the impossibility of exactly decoupling the Kukhtarev equations [9] except in the steady state, and numerical simulations of these coupled equations are also rather formidable. Simpli®ed beam propagation results, based on the approximate steady state solutions, was presented in Chapter 4. Over and above the computational dif®culty with the Kukhtarev equations is that in a typical experimental setup, the PR crystal is ®nite and often has a large linear refractive index (typically greater than 2), prompting re¯ections from boundaries. This introduces the added complication of taking into account forward and backward traveling waves during the twobeam coupling process and in fact extends the two-beam problem to a degenerate four-wave mixing problem in a PR material. Recently, a ``uni®ed'' method for solution of four-wave mixing problems in PR crystals in both transmission and re¯ection geometries has been proposed and applied to problems of double phase conjugation and two-wave mixing with crossed polarization in cubic crystals [12]. However, the methodology implies linearized interaction equations between the PR grating and the intensity grating and furthermore has been developed for the steady state. A rigorous analysis of wave interactions, particularly between ordinary and extraordinary polarizations, coupled through induced material anisotropy, has been developed by Wilson et al. [13]. The material analyzed is LiNbO3 : Fe where the PR effect is primarily photovoltaic. Arbitrary-direction two-wave coupling is studied and extended to multiwave coupling. The analysis is rigorous from an electromagnetics standpoint since it employs the

Copyright © 2000 Marcel Dekker, Inc.

rigorous coupled wave approach [14±16]. Once again, the analysis is restricted to the steady state. In our approach, we assume the PR effect to be diffusion dominated, and we ®rst write down the expression for the electrostatic ®eld (and hence the induced refractive index pro®le) as a nonlinear function of the optical intensity in the PR material in the steady state. The time-dependent Kukhtarev equations are then decoupled, ®rst under a set of approximations, to yield a differential equation in time for the induced refractive index pro®le. Rigorous coupled wave analysis (RCWA) is then used to analyze the diffraction of each of the incident plane waves from the slowly induced grating, including the effect of multiple re¯ections from the interfaces as well as from dielectric mismatches along the direction of propagation created through the changing intensity pattern due to energy transfer. We have performed our simulations ®rst for the case of two coherent incident plane waves to check our results with those in Ref. 7 (derived for the steady state and unidirectional propagation) and then extended our approach to include the case(s) when the incident waves are mutually partially incoherent. We have also investigated the difference in the time responses of two-wave coupling between taking the time constant as intensity dependent and taking it to be independent of the intensity. Furthermore, we have repeated our calculations with different incident polarizations for different materials with varying photorefractive gains using the exact Kukhtarev model. The relevance of our generalized approach using a nonlinear timedependent theory and rigorous coupled wave analysis can be summarized in the context of the extensive amount of work on the nature of wave mixing in PR materials. It has been pointed out that spatiotemporal instabilities, possibly due to internal re¯ections, may cause self-pulsations, irregular ¯uctuations in time, and optical chaos in self-pumped BaTiO3 phase conjugate mirrors and during naturally pumped phase conjugation [17±20]. Light waves at the interface of linear and PR media have also been analyzed [21], and it has been experimentally shown that optical bistability can be realized through optical feedback of the signal beam in a two-beam coupling geometry [22]. In the latter paper, a simple analysis has also been advanced to explain the hysteresis effect. The effect of a position-dependent time constant during two-wave coupling in both nominally copropagating and counterpropagating geometries has also been analyzed [23]. The rationale for this calculation is that the PR time constant is, in general, dependent on intensity. Furthermore, PR two-beam coupling with beams of partial spatiotemporal coherence has been theoretically and experimentally investigated [24,25]. The results indicate weaker coupling and beam-pro®le deformation along with spatial coherence enhancement and deterioration for the ampli®ed and deampli®ed beams, respectively. Finally, recording

Copyright © 2000 Marcel Dekker, Inc.

kinetics in PR materials in a two-beam coupling geometry, resulting in multiple two-wave mixing and fanning, and ``kinky'' beam paths have been theoretically predicted [26±28]. The results show that at least one higher order may develop with a high diffraction ef®ciency. As shown in this chapter, our algorithm can be effectively used to analyze diffraction from induced nonlinear transient transmission and re¯ection gratings and in materials with arbitrary complex permittivities (including absorption) at optical frequencies. The rigorous coupled wave algorithm is a straightforward, easy-to-understand diffraction analysis method, which allows in a simple, complete, and elegant way the electromagnetic ®elds of the diffraction grating and the surrounding medium to be accurately determined. RCWA techniques are routinely used in a wide variety of diffraction grating problems arising in optics and electromagnetics. The electromagnetic solution includes in its analysis the optical of the anisotropy medium as well as all multiply re¯ected waves that occur from the boundary interfaces. The RCWA method is a generalization of the beam propagation method, and it reduces to the beam propagation method when only forward traveling waves are allowed in the analysis. To the best of our knowledge there exist no other optical analysis method that could even approximately analyze the mismatched boundary case that we present in this chapter (see Fig. 1). The beam propagation method cannot be applied in the present case because it does not account for multiply re¯ected waves form the boundary interfaces. We believe that it is useful to use a rigorous electromagnetic ®eld theory even if the material model is simple, because the use of a rigorous ®eld theory at least removes the uncertainty that an inaccurate electromagnetic solution is the source of discrepancy between the overall model and experiment. All discrepancies between theory and experiment can then be ascribed to the materials model.

-

,% +  % +   9 ++   % +

The time-dependent Kukhtarev equations [8] for a PR material for nominal propagation of the optical ®eld along  and assuming transverse ®eld components     s  s  and transverse coordinate are   D

1     

Copyright © 2000 Marcel Dekker, Inc.

721

 D    D D

%   R D



 s s



  s 

s  A

  D 

722

723

724

where

  

 D   

  

s   

  

D  %  R s s  A  

= free electron density = ionized donor concentration = current density =electrostatic space-charge ®eld = intensity distribution =donor concentration =electronic charge =ionization cross section = thermal excitation rate (proportional to dark current) = recombination rate =diffusion constant = effective quasi-static permittivity    s    2 = effective static mobility        2 =acceptor concentration =  x   0 0 is the grating vector,   2  = grating period

The effective values of s and  are given by Yeh [8, p. 88, Eqs. 3.1-32, 33]. In what follows we will determine how the dielectric permittivity modulation     is related to the optical intensity .

-#

;% +,  </:#0=

To start the analysis, we differentiate Eq. 7.2.4 with respect to time, and after noting that A   0, we ®nd that   D

s 2  1         

Copyright © 2000 Marcel Dekker, Inc.

731

If the last two equation statements of Eq. 7.3.1 are integrated with respect to (the constant of integration can be shown to be zero) and of Eq. 7.2.3 is substituted, we ®nd that s s 

  s s   

732

For a particular choice of c-axis of the PR crystal, the index of refraction modulation may be taken to be    30 s , where 0 is the pertinent index of refraction of the bulk dielectric of the PR material and where  is the effective electro-optic coef®cient. The dielectric permittivity modulation  is related to  by    2 0 . Thus      s  40

733

Substituting s from Eq. 7.3.3 into Eq. 7.3.2, we ®nd the following equation for :  

 

   s  40  s s 

734

This equation shows that the dielectric modulation satis®es a ®rst-order reaction type of differential equation, which is driven by the spatial gradient of the electron density of . The equation is nonlinear since the electron density depends on  through the optical diffraction, re¯ection, and refraction which occur in the PR media. At this point no approximations of Kukhtarev's equations have been made. In order to derive an expression that explicitly shows the dependence of on in Eq. 7.3.4, in this chapter we will study the commonly occurring   case when A & D and ) D ) A . In this case the D term in the ®rst term of the right-hand side of Eq. 7.2.2 can be dropped, and we can solve for R , so that R 

 %   D 1 D    D D

735

The second term on the right-hand side (RHS) of Eq. 7.3.5 is the derivative of a logarithmic term    1 D    ln D    A D

Copyright © 2000 Marcel Dekker, Inc.

and can be shown to be small relative to the ®rst term on the RHS of Eq.  7.3.5. It has been ignored for now in this approximate analysis. If D is approximated by A in the denominator of Eq. 7.3.5, we can ®nally relate the electron density to the optical intensity through the linear relations



%   D R  A



%D    R A 

736

Substituting Eq. 7.3.6 in Eq. 7.3.5 we ®nd that   40 s  

      % 

737

where  

R A s D %  

738

The dielectric perturbation  can be put in the normalized form  

         *



739

where   0

 

  0

*

 %



 40 s 0 

  



 0

7310

and where 0 is the peak intensity of the incident optical ®eld. In Eq. 7.3.10,  0 denotes a normalizing constant that makes  and  dimensionless. A typical range for values of  is from * 10 5 (e.g., in BSO) to * 10 2 (BaTiO3 ). Also, the constant * is of the order of 104 ±106 (W/m2 ) in most PR materials. We remark also that for small  * 10 5 ), we have found that the intensity pro®le is nearly constant in time; thus Eq. 7.3.9 can be directly integrated w.r.t.  to yield an explicit expression for :   

Copyright © 2000 Marcel Dekker, Inc.

      

 1   *

7311

The dielectric tensor of a BaTiO3 crystal can be shown to be given by   2CO cos2   2CE sin2           2CO 2CE sin  cos               2CO sin2   2CE cos2       

7312

where   OE

"

13 2CO 33 2CE 3 2 2 sin  cos   2 sin  cos   sin      42 2CE 42 2CO

   OE

"

13 2CO 2 sin  cos   cos   cos2   sin2   42 2CE +

33 2CE 2 sin  cos  42 2CO " + 13 2CO 3 33 2CE 2 2  OE sin  2 sin  cos   cos  sin  42 2CE 42 2CO



+

OE 

2CO 2CE

2O 2E

2CO  2O O

2CE  2E E

7313

 in Eq. 7.3.12 is found by solving  

        * 



% 2 2

   O E 0 42

7314

 

is given in Eq. 7.3.10. The optical power intensity    in this problem is assumed proportional to  2   2 . The terms 2CO and 2CE are the relative complex dielectric permittivities of the bulk crystal. The terms 2O and 2E are real parts of the relative bulk dielectric permittivity and O and E are the lossy parts. Note that xsc , electric space charge ®eld, is linearly related to the dielectric perturbation or modulation function by the equation

Copyright © 2000 Marcel Dekker, Inc.

     2O 2E 42 xsc   

7315

The electro-optic constants 13 , 33 , and 42 are speci®ed in Ref. 8, Table 1.2, pp. 26±29.

-#

" 2 25

We will now describe an algorithm, based on the RCWA of Chapter 3, that can be used to determine mode coupling and diffraction from a slowly varying PR medium. The temporal variation of the PR medium is speci®ed by Eq. 7.3.14. The algorithm proceeds temporally as follows. At   0 (®rst time step) a signal (  
and pump wave (  
) are incident on a dark, uniform PR slab of material. The pump and signal wave interfere, creating a periodic optical intensity pattern in the PR slab. The periodic optical intensity modi®es the PR slab and produces a small periodic modulation . During this time step, no diffraction has occurred, as the dielectric modulation was zero at the beginning of the time step. At the second time step, the  generated by the optical interference of the ®rst time step will begin to diffract light. The algorithm calculates the total optical intensity in the medium (1) by calculating the EM ®elds and diffraction from  in the PR slab from the pump wave by itself, (2) by calculating the EM ®elds and diffraction from the signal wave from  in the PR slab by itself, and (3) by adding these two EM ®elds together to ®nd the overall EM ®eld in the PR medium and to ®nd the overall transmitted and re¯ected EM ®elds in free space. The algorithm proceeds by substituting the just-described total optical intensity into Eq. 7.3.14 to ®nd a new  modulation that can be used for the third time step. The algorithm repeats the above process for as many time steps as is required for a solution. The new value of  for any time step is calculated approximately using the ®nite difference formula       

   

 

7316

          

7317

or

where   is the right-hand side of Eq. 7.3.14 and is the optical intensity of the light at time .

Copyright © 2000 Marcel Dekker, Inc.

The time steps are chosen to be suf®ciently small so that only a very small change in  is observed at any given time step. A numerical simulation of the nonlinear photorefractive dielectric equations of Section 7.2 was made in order to study the interaction of the TE and TM optical electric ®elds with the photorefractive material. The simulation was performed for a photorefractive crystal located in free space and was illuminated by two incident plane waves whose separation angles with the normal to the crystal were 571 and 571 , respectively, and whose complex amplitudes were respectively 0 0 and 1 rand . These two plane waves caused an electric ®eld interference pattern whose nulls were   5 apart. -#

 " 2 "2!  "2

This section will describe the TE simulation results that arise from the theory of the previous section. For the TE case of this section, the photorefractive crystal was assumed to have an isotropic bulk dielectric of   2  2 2 , a thickness of 1.5 cm, and a wavelength   05 m. For the TE case, the ratio of the magnitudes of the incident plane wave   1  0  was taken to have a large range of values that extended from 0.1 to 10. The parameters were varied so as to keep the total power from both waves inc transmitted into the photorefractive material constant and equal to 9949  104 W/m2 . Concerning the phases of the incident waves, in our simulation we studied wave coupling when the phase difference rand between the interfering waves was equal to zero, which corresponds to the case of coherent illumination; and we studied the case where rand was uniformly distributed random phase (the limits of the random phase were taken to be 15 and 30 ), which corresponds to a partially incoherent illumination. Numerically the random phase was generated by a random number generator. Figure 2 shows the transmitted and re¯ected normalized power in the directions "1  "0  $1  $0 as a function of time that results when equal amplitude plane waves (  10) are incident on an initially dark photorefractive material bulk dielectric 2  8. The smooth line shows the result in the case of coherent illumination (rand  0), while the nonsmooth line depicts the response for partially incoherent illumination when the uniform probability density function, generating the random phase data, is bounded to 30 ( 30 rand 30 ). In our case, as is evident from the ®gure, there is energy transfer between the beams even though the amplitude incident beams are equal. For partially incoherent illumination, despite a moderate level of phase incoherence, steady-state results similar to those of the coherent

Copyright © 2000 Marcel Dekker, Inc.

5"  Plots of the normalized power in the directions $0 , $1 , "0 , and "1 (see inset) that is transmitted and re¯ected from a slab of PR material when the incident waves are coherent (rand  0 ) and partially coherent ( 30 rand 30 ). The angle of incidence of the pump wave is 57 , and the angle of incidence of the signal wave is 57 . These values are used for all ®gures.   5 inc  9949  104 (W/ m2 ),   05 m, coherent rand  0 , 1  0 exp rand , incoherent 30 rand 30 ,   2  10 5 , *  104 (W/m2 ),   1. Used with permission of Opt. Comm, 1996 [29].

case are achieved. It is interesting to note that a lower level of energy transfer occurs in the incoherent case as compared to the coherent case, in agreement with the ®ndings in Ref. 24. Figure 3 shows the dielectric constant perturbation  (total dielectric minus average bulk dielectric) that results from the simulation at time step 100 of Fig. 2. This case corresponds to coherent illumination by two equalamplitude interfering plane waves. The -axis of the ®gure extends over one grating period  (or interference period) of the photorefractive material. The -axis extends over the length of the crystal from the input plane   0 to the exit plane at   15 cm. The -coordinate used to plot numerical results is opposite in direction to that used in Section 7.3 and in Fig. 1 (RCW coordinate system). The coordinate system of Section 7.3 was chosen to coincide with the coordinate system used by Gaylord [16]. As can be seen from the ®gure, the dielectric permittivity perturbation is largely uniform in the direction of propagation () and blaze like in the transverse grating vector direction ( ). Thirty layers were used to model the possible longitudinal inhomogeneity of the grating. The  variation seen in Fig. 3 can be understood in the following way. At time step 100 of Fig. 2,  of the PR material has reached a steady state. Thus in Eq. 7.3.14,    0 and

Copyright © 2000 Marcel Dekker, Inc.

5" # Plots of the relative dielectric perturbation  when   1,   2  10 5 ,  2  8,   05 m, at time step=100. The  shown correspond to the random case of Fig. 4.   0 is the incident plane, and   15 cm is the exit plane. All other parameters are the same as in Fig. 2. min  358  10 6 , max  351  10 6 . Used with permission of Opt. Comm., 1996 [29].

  

   *

7318

Because the PR material is weakly modulated, the electric ®eld intensity inside the PR material is very little different from that of the unmodulated sample, and thus the intensity of Eq. 7.3.14 is

  0 cos2  

7319

where 0  4443  104 W m2 , *  104 W m2 ,   2  10 5 ,   01. Substitution of Eq. 7.3.19 into Eq. 7.3.18 shows  

 0  sin 2  *  0 cos2  

and

7320

A numerical comparison of Eq. 7.3.20 with the numerical data of Fig. 3 shows almost exact equality between the two sets of data. Figure 4a shows a plot of the position-dependent [23] nonlinear  1 function, which results from using the more exact space charge dielectric constant equation, Eq. 7.3.14. The nonlinear  1 function was chosen to be unity at the center of the grating fringe. Figure 4b shows the normalized power in, for instance, the $1 direction (see Fig. 1) when   1 for coherent and incoherent illumination ( 30 rand 30 ) and compares with the case when the nonlinear  function of Fig. 4a is used in the numerical

Copyright © 2000 Marcel Dekker, Inc.

5" $ (a) Plot of the nonlinear time factor  1 as a function of transverse distance and longitudinal distance  when   1,   2  10 5 , 2  8,   05 m, and rand  0 (coherent case). (b) Comparison of the re¯ected normalized power $1 for the case of Fig. 2 when the  factor was taken to be constant (  1) and nonlinear (a) in the cases when the incident waves were coherent (rand  0 ) and incoherent ( 30 rand 30 ). As can be seen, the nonlinear  factor increases the time necessary for the system to reach a steady state. Used with permission of Opt. Comm., 1996 [29].

simulation. As can be predicted from Fig. 4b, the net effect of the nonlinear  function is, in general, to reduce and slow down the transfer of modal energy in the $1 direction. Similar results occur for the normalized power in the $0 , "1 , and "0 directions. It appears that the nonlinear  does not have a strong effect on the mode coupling as regards the ®nal steady state (although the position-dependent case yields a slightly lower steady-state value [23]); for this reason, a constant  has been chosen for all subsequent calculations in the interest of computation time.

Copyright © 2000 Marcel Dekker, Inc.

Figures 5 a±c show plots of the dielectric modulation  as a function of time  (time step) and transverse dimension (fringing ®eld direction ) when sampled at the center of the photorefractive medium   075 cm. Because of the longitudinal uniformity of the  plots (see Fig. 3), the   075 cm plane is representative of  at other longitudinal planes. Figure 5a shows the  growth when   1 and coherent illumination was used; Fig. 5b shows the  growth when  was nonlinear (see Fig. 4a) and coherent illumination was used; and Fig. 5c shows the  growth when  was nonlinear (see Fig. 4a) and incoherent illumination was used. As can be seen from a comparison of Figs. 5a and 5b, the nonlinear  function reduces the rate of the  growth relative to the   1 case. A comparison of Figs. 5b and 5c shows that for the same  function (the nonlinear  was used) incoherent illumination produces a lower  modulation (max  338  10 6 , min  336  10 6 ) than does coherent illumination (max  377  10 6 , min  377  10 6 ) and hence reduced energy transfer. Figure 6 shows the plot of the two-wave coupling coef®cient  in the study state that results when a coherent wave rand  0 and when two incoherent waves 15 rand 15 ,  30 rand 30 are incident on the photorefractive slab. The  [7] is de®ned by 3  t 

3 3 . t1 1inc 0

t  10 log 3 3  t 

3 / t0 0inc 0

2

0 3 3 3 1

23

3 3 2

3  r 0

3 3 .  r 1 0

1inc r  10 log 3 3  r 0

3 / r0 0inc 0

2

0 3 3 3 1

23

3 3 2

7321

 In these expressions 
inc 0
 0 1 is the incident electric ®eld in the
th  order direction, 

 0 1 is the transmitted electric ®eld at the ®nal  time step transmitted out of the slab in the
th order direction, 
inc 0
 0 1 is the re¯ected incident electric ®eld in the
th order direction, and 
 0


 0 1 is the re¯ected electric ®eld at the ®nal time step re¯ected out of the slab in the
th order direction. The gamma mode ratio is a measure that shows to what degree the photorefractive medium has been able to convert power from one mode to another. For the plots of Fig. 6, the bulk dielectric was taken to be 2  8,   2  10 5 , and the incident wave's refraction ratio   1  0  was varied from   01 10 db) to   10 10 db . The values of 0 and 1 were adjusted to keep the incident power inc  9949  10 4 (W/m2 ) the same for all values of  used. The results of Fig. 6 show that the PR medium transfers the most energy when the incident waves are coherent, and that a gradual decrease in coupling ef®ciency occurs as the waves become more incoherent as seen

Copyright © 2000 Marcel Dekker, Inc.

5" & Plots of  versus transverse distance and time step as measured at a plane midpoint in the PR slab (  075 cm, slab length=1.5 cm) for the parameters of Figs. 2±4. (a)  when coherent illumination was used and   1. (b)  when coherent illumination was used and the nonlinear  factor of Fig. 4a was used. (c)  when the nonlinear  factor of Fig. 4a was used and when incoherent illumination was used.   2  10 5 . Used with permission of Opt. Comm., 1996 [29].

Copyright © 2000 Marcel Dekker, Inc.

5" ' Plots of the r and t ratio of Eq. 31 for   2  10 5 , 2  8,   05 m, slab length =1.5 cm (Figs. 2±5) in the case when rand  0 (coherent), 15 rand 15 , and 30 rand 30 . The nonstarred lines refer to the re¯ected r ratio, and the starred lines refer to the transmitted t ratio. As can be seen, as the incoherence of the interfering waves increases, and the r and t ratios  decrease. inc  9949  104 (W/m2 ), *  104 (W/m2 ),   05 m, 2  8, and 5   2  10 . Used with permission of Opt. Comm., 1996 [29].

by the drop in  and  when 15 rand 15 and 30 rand 30 . The results of Fig. 6 also show that the maximum mode coupling occurs when the wave amplitudes are equal,   1  1  0 . This is to be expected since the equal amplitude waves cause the greatest interference pattern in the PR medium and thus cause the largest change in the medium, which causes the most change in the mode power. The decrease in mode coupling ef®ciency with increasing incoherence is expected since increasing incoherence implies less interference of the waves and therefore less change in the PR medium and thus less mode coupling. Note also that the ¯uctuations in  increase with rand indicating possibly reduced coherence of the signal at the output, in agreement with Ref. 25. A time averaged , which can be found by numerically repeating the experiment several times, will also

Copyright © 2000 Marcel Dekker, Inc.

show a decrease in the coupling coef®cient due to increased incoherence (see Problem 8). !  Figure 7 shows the power ef®ciency that results when the  2 ratio is signi®cantly increased over the value used in Figs. 2±6 in the case when the bulk index nearly matched the free space 2  1 10 6 and in the case  8 282  10 7 ). when the PR medium !  is mismatched to free space 2 4 In Fig. 7 the  2 has been taken to be 318  10 , 388  10 4 , and

5" - (a) Plot of the power transmitted in the "0 and "1 directions as functions of time step in the nearly matched case when 2  1 1  10 6 and   001 for the values of   318  10 4 , 388  10 4 , and 459  10 4 when the PR slab length=1.5 cm. Also *  2124  104 (W/m2 ). As  increases, the speed and completeness of mode conversion increases. (b) Plot of the normalized ! power re¯ected and transmitted in the directions $0 , $1 , "0 , "1 when  2  459  10 4 , 7 4 2 2  8 282  10 , *  2124  10 (W/m ), and   001. Because of the dielectric mismatch, aperiodic variation of the normalized power results. Used with permission of Opt. Engr., 1995 [30].

Copyright © 2000 Marcel Dekker, Inc.

459  10 4 , and fractional ratio of the incident beams has been taken to be   1  0   001. The waves marked "0 and "1 , "02 and "12 , and ! "0 and "1 show the as indicated in the diffraction ef®ciencies for the three values of  2 ®gure insets. Because of index matching (for the case of Fig. 7a) the re¯ected waves from the slab were nearly zero and thus are ! not  shown. The main effect that is observed in Fig. 7a using the values of  2 that were given is that a great deal of energy is transferred from the (order 0) transmitted 0 wave (which is large at   0) to the (order 1) transmitted 1 wave (which is small at   0). As can be seen from Fig. 7a,! the conversion of the modal  used and affects the value energy depends very strongly on the value  2 of power diffracted in the "1 direction and the speed with which the mode power transfer reaches the steady state. In Fig. 7a the presence of absorption  (2  1  10 6 ) seemed to have a minimal effect on the diffraction except, of course, to attenuate the "0 and "1 propagating waves. Figure 7b shows the diffraction the real part of!the   !  that occurs when  bulk dielectric is 2  8 and  2  459  10 4 . The value of  2 used in Fig. 7b is the same as in the ``c'' case of Fig. 7a. In the simulation case of Fig. 7b, because there is a large bulk dielectric mismatch between free space and the PR medium, the incident and diffracted optical energy is multiply re¯ected from the dielectric boundaries on the incident and transmitted sides in a complicated way, causing a standing wave pattern to arise in the PR medium. The intensity of this standing wave pattern changes slowly in time, modi®es the PR medium according to Eq. 7.3.14 and thus further diffracts the incident and re¯ected light. The net effect over time is a highly inhomogeneous PR medium. The simulations, shown in Fig. 7b, predict that the optical energy may oscillate or self-pulsate quasi-periodically between the $0  "0  "1 , and $1 directions, transferring energy back and forth between the different orders. This is due to the temporal longitudinal inhomogeneous PR medium formed by the interfering waves. Similar results have been reported during self-pumped and mutually pumped phase conjugation in PR materials [17±20]. Figure 8 shows a plot of the dielectric perturbation  that occurs at the line   075 cm (this line is midway between the PR medium boundaries) as a function of time step and transverse distance (wavelength) in  the case!(1) dielectric is matched (2  1) to free space (Fig.   when the bulk 8a,  2  388  10 4 ), (2) when the bulk dielectric is mismatched !    459  10 4 ) and (3) when the  8) to free space (Fig. 8b,  2 (2  bulk ! dielectric is 4 mismatched (2  8) to free space (Fig. 8c,  2  530  10 ). As can be seen from these ®gures, when the boundary is matched, the dielectric modulation  approaches the steady state quickly (Fig. 8a), whereas when the bulk dielectric is mismatched, the

Copyright © 2000 Marcel Dekker, Inc.

5" of  versus time step and transverse distance when ! . (a) Plots 2  1 1  10 6 ,   001. (b)!  when  ! 2  388  10 4 , 4 7    !  459  10 has increased ,   001, and   8 282  10 . (c)   2 2  2   530  10 4 . As can be seen, the dielectric mismatch cases have a great to  2 effect on the  that forms in the PR medium. Used with permission of Opt. Engr., 1995 [30].

Copyright © 2000 Marcel Dekker, Inc.

dielectric modulation does not approach a ®nal steady state but oscillates in the diagonal ripple pattern shown in Fig. 8b. The  temporal pattern of Fig. 8b, as mentioned earlier is a result of the optical standing wave pattern slowly changing the PR medium, thus further causing a time change in the PR medium. Figure 8c ! shows a mismatched case)   the same case as in Fig. 8b (this is  4 except that  2  530  10 . has been increased to a value   2 !  Because of the higher  2 ratio (stronger PR medium), the diffracted waves more strongly in¯uence the standing wave pattern of the PR medium and thus more severely change the PR medium. As can be seen from Fig. 8c, the diffracted waves build up to such a point that almost chaotic behavior occurs as time proceeds [17]. A simulation was run for the same parameters  as given in Fig. 8 except that 2  0 (no absorption). The resulting  was nearly identical to that seen in Fig. 8c. For this case conservation of power was observed numerically almost exactly. Figure 9 shows the PR medium as a function of and  that results in the matched at the time step 200 (last time step !case !  of Figs. 7 4and 8)  4  (Fig. 9a), (2)   (Fig.  318  10 when (1)  2 2  388  10 !  !  4 9b), and (3)  2  459  10 (Fig. 9c). The three values of  2 !  used in Fig. 7 used in Fig. 9a, b, and c correspond to values of  2 (cases a, b, and c respectively). As can be seen from these plots, the dielectric modulation  is highly inhomogeneous longitudinally, with a peak maximum and minimum value occurring in the PR medium. The position !  of the peak maximum and minimum values is dependent on the  2 value used and tends to occur closer to the incident side as!the !     value of  2 increases. This occurs because the larger value of  2 causes the weak signal to be ampli®ed over a shorter distance in the PR medium, thus causing the peaks in a maximum and minimum to form !   caused a much more quickly. Notice that the higher values of  2 more complete transfer of optical energy as can be seen in Fig. 7a (cases a, b, c). Note that the diffraction numerical calculation can only be made correctly using a longitudinally cascaded diffraction analysis algorithm. The plots shown in Fig. 9 are steady state plots. This was veri®ed by noting that there was no change in the plots shown with plots recorded at time step 100. In all our numerical simulations for the TE case, thirty layers were used to model the inhomogeneity of the PR medium. Because each individual layer consisted of many wavelengths (  15 cm/30,   05 m), it was necessary when calculating the intensity   to average this over a number of  points distributed over a one-wavelength interval in the center of each layer. If this were not performed, random widely varying samples of the standing wave pattern would be obtained, leading to samples that would

Copyright © 2000 Marcel Dekker, Inc.

5" /  pro®les at !time  1 1  10 6 , slab   step=200, 4   001, !2   388  10 4 . 10 . (b)  2 (c) length=1.5 cm. (a)  2  318 !  4  interfer 2  459  10 . Higher values of  2 cause mode conversion and ! ence in the PR medium to occur closer to the incident side as the  2 value increases. Used with permission of Opt. Engr., 1995 [30].

not be representative of the average intensity. Three points provided suf®cient averaging to produce smooth intensity results. -##

% " 2 "2!  "2

As a TM example we consider the photorefractive grating that occurs when two equal-amplitude in-phase TM plane waves   0633 m impinge on BaTiO3 crystal at an angle of incidence i  571 as shown in Fig. 1. The grating period formed in the crystal at this angle of incidence is   5.

Copyright © 2000 Marcel Dekker, Inc.

The BaTiO3 crystal complex relative dielectric permittivity matrix is assumed to be described by Eqs. 7.3.12 and 7.3.13. For the present simulation we choose 42  1640  10 12 m/V, 13  8  10 12 m/V, 33  28  10 12 m V, O  2437, E  2365, and E  O  242  10 6 (these values of O and E correspond to an absorption coef®cient   1 cm 1 ), which are the values given for BaTiO3 in Ref. 8 at   0633 m. The electro-optic coupling constant   2O 2E 0 42 s  for the values given is   00139. In the simulation we have assumed that the crystal length   1500  094 mm and that the BaTiO3 c-axis makes a 135 angle with the -axis of Fig. 1 (  135 in Eqs. 7.3.12 and 7.3.13). Free space was assumed to occupy the regions outside the crystal. We further assume [31] R  5  10 14 m3 /s,   05  10 4 m2 V s A D  001, and [32] %  1  10 5 m2 J s and   2 1 s. With these values we ®nd that the dark current *   s  2  105 W/m2 . For BaTiO3 and for the geometry shown in Fig. 1, the effective value of      %  %2 cos2   %3 sin2  , where %1  %2  36000 , %3  1350 [8, Table    1.2, p. 28], and where %  %

  ,

 1 2 3 is the static dielectric permittivity tensor when the BaTiO3 PR crystal has its c-axis aligned along the   -axis. For   135 , %  18670 . We assume that the total incident power in the  direction of the interfering incident waves is !T INC  17 9 107 watts/m2 . With this value, it is found that the maximum power intensity inside the crystal is MAX  160  107 watts/m2 . Using this value, we ®nd the maximum value of  %  MAX  162  107 watts/m2 . Using this value, we ®nd that the approximate time constant  0 is  0  638 ms. The  time constant was chosen to be    0 5  127 ms. This time step was suf®ciently small to cause a smooth change in the dielectric modulation with time. An additional run at    0 15  423 ms was also made, to check the time step size. It was found that the two time step sizes gave similar numerical results. We will now make an approximate numerical estimate of the ratio of the two terms on the RHS of Eq. 7.3.9 for the present BaTiO3 example. We  assume that the ionized donor density D varies on a time scale of about  0 , which equals 6.38 ms. We approximate    1 D 1 D 1 )   156  102 s 1    0

D  D  0

 For the ®rst term of the RHS of Eq. 7.3.5 we have %   D D  5 7 2 1 4 1 % * D A  10 162  10 10 s  162  10 s . The ratio of

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the second term to the ®rst term of the RHS of Eq. 7.3.9 is 096  10 2 . We see that the second term is roughly a hundred times smaller than the ®rst, which justi®es the approximation used. The numerical simulation for this case was performed by using Eq. 7.3.14 to calculate  at every time step and using RCWA to calculate the grating diffraction at every time step. The RCWA calculation was performed using L  160 layers, which gave each layer a length of   9375. The optical power intensity point          

( and  are the optical electric ®elds) was calculated at 10 equally spaced points over the layer length    L  9375 ( point   

was sampled every 09375). The values of point    were then averaged. This value was used as the average optical power intensity in Eq. 7.3.18. Averaging the intensity over the layer length  represents an important part of the interaction of the incident optical light with the PR medium. Physically averaging the optical intensity over  represents the way that partially coherent light or light with a ®nite frequency spectrum (or both) would interact with the PR medium. When perfectly coherent, monochromatic light enters the PR slab, the energy is multiply re¯ected at the interfaces, and a strong standing wave pattern is formed in the PR slab. The distance from peak to peak longitudinally (-direction) is a few free space wavelengths . When the frequency of the light is changed by a small amount, the peaks and nulls change position by a few wavelength, and thus in the crystal a standing wave pattern is formed whose peaks are in an entirely different position from the original monochromatic wave from which the frequency was changed. When many different frequency waves are added together, the peaks and nulls will tend to average out, and a longitudinally average ®eld will result. Thus spatial averaging simulates the frequency spread of real optical energy. Figure 10a shows the dielectric modulation function     that results for the BaTiO3 simulation at time step  100. At this time (see Figs. 10b and 10c) the photorefractive crystal is nearly in a steady state. The minimum and maximum dielectric modulations that occur are min  000605 and max  000565. Using the relation   42 2O 2E xcs , this min corresponds to a minimum and maximum space charge electric ®eld of xsc 4 4  111  10 V/m and 926  10 V/m. These ®gures are roughly in line (to a factor of 2) with typical values given in [8, Fig. 3.3b, p. 91]. The simulation shown in Fig. 10a predicts an interesting feature of the dielectric modulation: two different dielectric slanted gratings have formed as a result of the photorefractive±optical interaction. The ®rst grating extends from about

Copyright © 2000 Marcel Dekker, Inc.

5" 0 The dielectric modulation and diffracted power that results when TM optical energy illuminates BaTiO3 are shown.
  179  107 W/m2 ,  0  638 ms,   02 0  127 ms,   100, and   0633 m. Used with permission of Opt. Engr., 1995 [30].

zero to 300 and has a grating period of 1  120 with grating vector given by 2 sin 1   cos 1     1  1

1  15

7322

in the geometry of Fig. 1. The second grating extends from about   300 to 1500. This grating is more clearly de®ned and has a grating period of about 2  75. 2  2   sin 2   cos 2    2

Copyright © 2000 Marcel Dekker, Inc.

2  45

7323

Figures 10b and 10c show, respectively, the power transmitted and re¯ected in different diffraction directions as a function of time. As can be seen from the power transmission curves of Fig. 10b, the BaTiO3 PR crystal at the ®rst time step is uniform, and power is transmitted equally in the "0 and "1 directions. Within a few time steps, a modulation grating forms and power is rapidly depleted from the "1 direction and transferred to the "0 direction. As can be seen from Fig. 10b, the "0 modal direction is completely depleted of power. As time progresses over a period of approximately 30 time steps, diffracted waves in the " 1 direction build up, and ®nally a strong mode conversion from the "0 to the " 1 order occurs and the grating goes into a quasi±steady-state form. The "0 mode drops to about 10% of the total diffracted power, and the " 1 modal power builds from 0 to 60%. The ®gures are similar to those in Ref. 26, which also illustrates the growth of a higher order. The re¯ected power in the $0 , $1 , and $ 1 directions follows a similar time history as did the transmitted except that the $1 order is not depleted to a zero value by drops from 18% to above 10% as time increases. In contrast to the results in Refs. 26 and 28, we predict that the 1 order in the transmitted case virtually decays to zero. The reason may be that in the unmatched case (a true experimental possibility), the re¯ected orders also interact. This example can also be viewed as a case of six-wave coupling in PR materials, such as in KNbO3 , which leads to phase conjugation in a two-beam coupling geometry [33]. The BaTiO3 simulation was performed for four other PR crystal lengths, which were 
  1500 
 1875,
  1 2 3 4. An interesting result of the analysis was that for the lengths of 151875
  1 and 155625
  3 simulation showed that diffraction in the " 1 and $1 directions did not occur at all where they did occur for the values of
L  2 and
L  4. Evidently the growth of the " 1 and $1 mode perturbations depends on a resonant length of the crystal. -#$

"!   "2  

 !24

By using rigorous coupled wave diffraction theory along with a time-dependent nonlinear formulation, we have analyzed two-wave and multiwave coupling in a PR material. The two-beam coupling gain has been plotted as a function of the incident intensity ratio for both transmitted and re¯ected beams. Coherent and partially incoherent cases have been analyzed, and the results show that reasonable beam coupling occurs up to a certain degree of incoherence. Computations performed with both uniform and intensity-dependent time constants show that although the overall times needed to achieve a steady state somewhat differ, the ®nal steady-

Copyright © 2000 Marcel Dekker, Inc.

state values are unaffected. We have also examined wave coupling in PR media with different gain coef®cients and, for the ®rst time, cases where there exists signi®cant linear refractive index mismatch between the material and the surrounding medium. The analysis has been extended to an inhomogeneous anisotropic PR material, e.g., BaTiO3 , that includes a nearly exact solution of the Kukhtarev equations. Our simulations thus far predict periodic and nonperiodic oscillations during two-beam coupling, as well as generation of higher transmitted and re¯ected orders. We believe that our analysis is the ®rst on the anisotropy of the diffracting region, the multiple re¯ections, and the time dependence of beam coupling simultaneously in a rigorous way. In our work we have studied the following new phenomena. We have studied the effect that an intensity-dependent time scale has on beam coupling in a PR medium (Fig. 4b) and the effect of partial coherence on beam coupling (Figs. 5 and 6). We have also studied in detail six wave coupling examples in BaTiO3 , which are shown in Figs. 10 a±c. In the BaTiO3 example we have shown numerically that when mismatched boundaries occur the PR coupling seems to be resonant and very sensitive to the overall length  of the grating. For BaTiO3 we have not carried out experimental veri®cation because we do not have the resources to do so at the present time. However we would like to point out that the BaTiO3 case that we have analyzed numerically is a realistic one and one for which experiments could readily be performed. We feel that the case we have analyzed numerically would be interesting to do experimentally. Any discrepancies between theory and experiment would be due to the PR model and not the optical diffraction analysis. The mismatched boundaries could serve as partial Fabry±Perot effects and perhaps make it possible to determine from the optical diffraction data some of the Kukhtarev PR model parameters (mobility constant, A and D value, etc.) that pertain to the BaTiO3 crystal under consideration. Our developed model can thus be used to study wave mixing in PR materials with induced transmission and re¯ection gratings.

-$

; +,  <#$=

In Section 7.3, two signi®cant approximations were made to the Kukhtarev analysis. Both approximations were made to the electron rate production equation [29,30]. The ®rst approximation consists of assuming that the donor density D was much greater than the acceptor density A , the  electron density , and the ion density D . The effect of the approximation is that the production of electrons is not limited as the electron donors D are depleted. The second approximation consists of assuming that the term

Copyright © 2000 Marcel Dekker, Inc.

 D  is small compared to the source term in Eq. 7.2.2, which depends on the optical intensity. The effect of this approximation is that the full temporal behavior of the system is not included in the analysis. An important impact that the above approximations have, over and above the one already mentioned, is that indirectly both approximations lead to the omission of second-order transverse spatial derivative terms in the analysis. This is important for a few reasons. First, any analysis for which a higher order derivative is ignored is limited to cases where changes in the electric ®eld, the electron density, etc. in the transverse direction are small. Thus, for example, if higher order diffraction (which varies rapidly in the transverse direction) should be excited by the PR system, the effect of this cannot be in general studied for arbitrary hologram wave numbers because the second-order derivative terms may contribute more to the analysis than the ®rst-order derivatives. Thus the analysis could at best be valid only over a range of hologram wave numbers [29,30]. The purpose of this section is to study an exact nonlinear time-dependent solution of the Kukhtarev equations in conjunction with RCWA. To start the analysis, we substitute Eq. 7.2.3 into Eq. 7.2.1 and ®nd that   D  s 2

   s 2    

741

Using Eq. 7.2.4 we ®nd that  D 

s s  A 

 

742

or  D  2 s 

 s      

743

 where A   0 has been used in Eq. 7.4.3. If the D  of Eq. 7.4.3 is substituted into Eq. 7.4.1 and   is cancelled on the right- and left-hand sides of the resulting equation, we ®nd that

s 2 s  s

2   s 2  0      or, after integration with respect to ,

Copyright © 2000 Marcel Dekker, Inc.

744

s   



s  s s s  

745

In Eq. 7.4.5 we have set the integration constant to zero, assuming all dependent variables and their derivatives tend to zero as    . For a particular choice of c-axis of the PR crystal, the dielectric permittivity modulation  is related to the electrostatic electric ®eld by      2O 2E 42 s   

746

Substituting s from Eq. 7.4.6, we ®nd for :    

   s 42 20 2  s s 

747

Up to this point no use has been made of Eq. 7.4.2, the electron rate  production equation. If D  of Eq. 7.4.3 is substituted into Eq. 7.4.2, we ®nd that s 2 s 

    %   D D

R D    

748

If Eq. 7.4.4 is used to eliminate the s  2 s   and d of Eq. 7.4.2 is substituted, we ®nd that  

s 2 

s s s 2   %   D A

         s  A 

R s  

749

At this point it is useful to introduce normalized coordinates and variables and also to perform a small amount of algebraic manipulation of Eq. 7.4.9. Letting *   %, %    %* 1  * ,   0 ,      A , and using Eq. 7.4.6 to express s in terms of , we ®nd that     2   

 

3 4 1 

2 2  1   3 1   1 4

  *          

D 1   3 1 7410

* A 3  

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where 1 

A s

4 

R  A 

2 

0 s A 20 2 42 s

3 

A 20 2 42 s 0 7411

Also Eq. 7.4.7 in normalized form can be written as      1   2   

7412

Equations 7.4.10 and 7.4.12 are a pair of coupled nonlinear equations for the electron density  and dielectric modulation function . The form of both these equations for  and  at any given time and at any given point in the PR medium depends on the value of the optical intensity at that point in space and time. The value of itself in the PR medium depends on the incident optical ®eld and on the optical energy transmitted, re¯ected, and diffracted by the dielectric modulation function  in the PR medium at a given time. These transmitted, re¯ected, and diffracted ®elds can be found, as mentioned in the introduction, through the use of a diffraction algorithm called rigorous couple wave theory, which is the subject of the next section.

-$

!   ! D"C
!24

We will now describe an algorithm based on the exact Kukhtarev equations and on RCWA that can be used to determine mode coupling and diffraction from a slowly varying PR medium. The temporal variation of the PR medium is speci®ed by Eqs. 7.4.10±7.4.12. The algorithm proceeds temporally as follows. At   0 (®rst time step) a signal (  i ) and pump wave (  i ) is incident on a dark uniform PR slab of material. The pump and signal waves interfere, creating a periodic optical intensity pattern in the PR slab. The periodic optical intensity modi®es the PR slab and produces a small periodic dielectric modulation . During this time step, no diffraction has occurred, as the dielectric modulation  was zero at the beginning of the time step. At the second time step, the  generated by the optical interference of the ®rst time step will begin to diffract light. The algorithm calculates the total optical intensity in the medium (1) by calculating the EM ®elds and diffraction from  in the PR slab from the pump wave by itself, (2) by calculating the EM ®elds and diffraction from the signal wave

Copyright © 2000 Marcel Dekker, Inc.

from  in the PR slab by itself, and (3) by adding these two EM ®elds together to ®nd the overall EM ®eld in the PR medium and thus ®nd the overall transmitted and re¯ected EM ®elds in free space. The algorithm proceeds as follows. The time derivative   is approximated as          

    

 and after substitution in Eq. 7.4.12 and cross multiplication by  we ®nd that                 1   2    7413

  This equation is used to advance the  dielectric modulation function in time. Equation 7.4.10 is used to ®nd the electron density  for Eq. 7.4.13. Its  determination from Eq. 7.4.10 proceeds as follows. Forming an -grid  system of  divisions,     ,   0 , we let     05

,   0 1      ,   1 be sampled values of  over the grating period of the PR material at a longitudinal distance . The points   0 and   1 extend one point outside the grating period. These need to be included in order to specify periodic boundary conditions. We also let ,   ,     

        , be any sampled dependent variable (for example 

         , etc.) of Eqs. 7.4.10 or 7.4.12 in the grating period of the PR material at a longitudinal distance . Using these de®nitions we can approximate the ®rst and second spatial derivatives of Eqs. 7.4.10 and 7.4.12 by the well-known ®nite difference formulas. 2    2

 

Letting



  1 2 

    1

 2 

   

 



  1   1

 2  7414

!   1  

 



  3 4 1  

   3 1   1 4

   *          

 D 1 (   3 1 7415

 * A

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we ®nd for   1      2



     1 2 

    1

  1   1



  ! 

2  2

  ( 

  



7416

  1 ,  ,   1 are collected, Eq. 7.4.16 can be put If coef®cients of 

 in the following form   1  $ 



  $ 

  1  ( 

$ 



7417

where $  

!  2  2   2

$    

22

 2

$  

!  2  2   2

and where   1 2      . Because the diffraction grating is periodic, we have the important      and boundary conditions on the variable that 0

   1  1 . 

 These equations can be used to eliminate the variables     1 in Eq. 7.4.16 and thus give an equation that depends

0

and   only on the unknowns  ,   1      . We thus observe that Eq. 7.4.17    1      . represents a system of  equations in  unknowns  , The system of ®nite difference equations given by Eq. 7.4.17 can be conveniently expressed in terms of a matrix equation:

$ 1

$ 1

0 0   $ 2

$ 2 $ 2

0    0 $ 3 $ 3 $ 3

   0 0      0 0 0     0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0 0 0 0 $ 

 " (  ( 1 ( 2    ( 

   (

0 0

0

0

0

$ 1



     0 0 0 0 0 0     0 0 0 0 0      0 0 0 0       0 0 0    0    0 0    0 0 0 $  2 $  2 $  2

  0 0 0 $  1 $  1 $  1   0 0 0 0 $ 

$ 

 "

 1 2    

0 0

0

0

0

0

7418

Inverting this matrix equation gives

   1 (

Copyright © 2000 Marcel Dekker, Inc.

7419

which thus determines the electron density pro®le for a given value of  in the PR medium.      is determined [ 

 is The algorithm proceeds as follows. Once       ] and its deridetermined from matrix inversion and 

speci®es        is calculated,      and         are substituted vative       is found from Eq. back into Eq. 7.4.13. Once a new value of        is calculated for all values of  (all 7.4.13, this new value of   discrete layers of the PR slab). Once this step is completed, RCWA is used      , and thus a new to study diffraction from the new value of        is found. The new intensity      , optical intensity value       and its derivative, is substituted into Eqs. 7.4.17 along with          is found. By repeating the above and 18 and a new value of steps for many iterations, the time evolution of the PR material and the optical diffracted intensity can be found. -$

% " 2 "2!  "2

We consider the photorefractive grating that occurs when two in-phase TM plane waves (  0633 m) of amplitudes 0 and 1 impinge on BaTiO3 crystal at an angle of incidence 
as shown in Fig. 11. The BaTiO3 crystal complex relative dielectric permittivity matrix is assumed to be described by Eqs. 7.3.12 and 13. For the present simulation, we choose 42  1640 10 12 m/V, 13  8  10 12 m/V, 33  28  10 12 m/V, O  2437, E  2365, and E  O  242  10 6 (these values of O and E correspond to an absorption coef®cient   1 cm 1 ), which are the values given for BaTiO3 by [8] at   0633 m. In the simulation we have assumed that the BaTiO3 c-axis makes a 45 angle with the -axis of Fig. 1. We further assume [31,32] R  5  10 14 m3 /s,   05  10 4 m2 V s, A  3  1022 , D  200A , %  1  10 5 m2 J s, and   2 1 s. With these values we ®nd that the dark current *   %  2  105 W/m2 . For BaTiO3 and for the geometry shown in Fig. 1, the effective value of    %  %2 cos2   %3 sin2  , where %1  %2  36000  %3  1350 [8, Table    1.2, p. 28], and %  %

  ,

 1 2 3 is the static dielectric permittivity tensor when the BaTiO3 PR crystal has its c-axis aligned along the   -axis. For   45 , %  18670 . Figure 11 shows the intensity pro®le and steady-state dielectric modulation function  that results from that pro®le when two equal-amplitude plane waves impinge on a BaTiO3 grating whose thickness is   15. The grating thickness has been chosen to be so thin that no appreciable diffraction occurs within the grating. Figure 11a shows the normalized intensity pro®le for   1 2 5 and 10 at    2. The power intensity has the approximate intensity pro®le of a squared sinusoidal wave. The intensity

Copyright © 2000 Marcel Dekker, Inc.

5"  (a) The optical power intensity (normalized to the dark current *) in the grating when the grating period is   1, 2, 5, and 10 as a function of the normalized grating distance N  . The incident power (evaluated at    2) was adjusted in order that the intensity pro®le for each different size grating period would have the same peak intensity. (b) The steady-state dielectric modulation function  (also evaluated at    2) that results when the intensity pro®les of (a) were used to determine . Because the PR grating was so thin, the intensity pro®les of (a) were not assumed to change with time as the  pro®les reached a steady state. All grating parameters used in the simulation not listed on the ®gure are given in Section 7.3. Used with permission of JOSA-A, 1996 [34].

Copyright © 2000 Marcel Dekker, Inc.

pro®le peak is shifted to the left as the grating period becomes smaller. This is an effect of the anisotropy of the PR crystal. The incident amplitudes of the interfering waves have been chosen to keep the peak intensity the same for the different grating periods shown in Fig. 11a. The dielectric modulation function  that results from the intensity pro®les of Fig. 12a is shown in Fig. 11b. In Fig. 11b, the curve marked ``1'' corresponds to a grating period   1, the curve marked ``2'' corresponds to a grating period   2, and so on. The  of each of the curves has been plotted as a function of the normalized variable    as was the power intensity. As can be seen in Fig. 11b, the size of the grating period  as a large effect on the value of  that results. When the grating period is small (on the order of   1) the magnitude of  is small. As the grating period increases in size, the maximum magnitude value of the  pro®le increases in size until the grating period is about   5. During this range of 1  5, the  pro®le gradually changes shape with the  peak (maximum and minimum) rising more sharply as the grating period increases. When  is increased to the range 5  10, the maximum magnitude value of the  pro®le decreases in size. The  pro®le further gradually changes shape. The increase and then decrease of the maximum magnitude value of the  pro®le results because the equilibrium equation for and , namely Eq. 7.4.10, contains zero-, ®rst-, and second-order derivatives. Thus when  is small   2  is large) the second-order derivatives terms are large, and when  is large the zero-order derivative term tends to be large. Intermediate to this (  4 to 5), the largest magnitude  pro®les are reminiscent of the dependence of the linearized two-beam coupling coef®cient on . We note that for all values of the grating size  the  function is shifted to the right of the intensity pro®le, as can be seen by inspecting Figs. 12a and 12b. Figures 12a±d display the numerical PR mode diffraction, coupling, and conversion that occurs (using the Kukhtarev equation and RCWA) when two interfering plane waves (see Fig. 12a) whose amplitudes are 0 and 1  010 are incident on an index matched PR crystal of length   1530,   0633 m. The angle of incidence is such as to make the grating period   5. Figure 12c shows the normalized power transmitted in the "0 and "1 directions as a function of time step, and Fig. 12d shows the normalized power re¯ected in the $0 and $1 directions as a function of time step (   1 ms). As can be seen from Fig. 12d, because of the index matching, almost zero power ( 10 6 ) is re¯ected from the PR grating in the $0 and $1 directions. Figure 12c shows, for the geometry and material parameters of the case under consideration, that a large amount of energy is transferred from the "0 to the "1 in a period of about 35 ms, at which time the grating dynamics rapidly approaches the steady state. The power in the

Copyright © 2000 Marcel Dekker, Inc.

5"  The dielectric perturbation function  (b) and the power transmitted in the "1 , "0 , (c) $1 , and (d) $0 directions is shown when the regions (Region 1 and 3) bounding the PR crystal are indexed matched to the PR crystal. All grating parameters used in the simulation not listed on the ®gure are given in Section 7.3. Used with permission of JOSA-A, 1996 [34].

"0 and "1 directions adds to about 90% of the incident power. Because the grating is assumed to be lossy, the other 10% of the incident optical power is absorbed as heat in the grating. Figure 12b shows the dielectric modulation function  that results at   113 ms when the grating has been in a steady state for a long time. The pro®le has the form of a slanted sinusoidal grating that grows steadily from a small value at   0 to a peak-to-peak value of max min  000642, which occurs at about   1000. It may be noticed that the grating pro®le is slightly skewed at   1000. This may be a slight nonlinearity effect. The simulation grid used 34 divisions to solve Kukhtarev's equations, Eq. 7.4.10, for each layer (thus the matrix equation that was inverted in Eq. 7.4.18 was 34  34 size), and the grid used   160 layers to describe and simulate the optical wave (RCWA was used to determine the electromagnetic or optical ®elds of the system) as it propagated and diffracted through the diffraction grating. The optical intensity was averaged

Copyright © 2000 Marcel Dekker, Inc.

over 10 points in each     to ensure that a smooth and physically realistic optical intensity pro®le    was used in Kukhtarev's equations. A detailed discussion of the averaging is given in [29,30]. The temporal analysis was marched forward in time with a time step of   1 ms. Very stable numerical results were obtained for the grid parameters and time step used. The RCWA analysis was carried out for "  2 (
 2, 1 0, 1, 2) for the simulation in Fig. 12 and for all simulations in this chapter. For the matched case of Fig. 12, only an extremely small amount of diffracted power was found in the higher diffraction orders (
 2, 1, 2) in agreement with [29]. We would like to mention at this point that because an exact Kukhtarev analysis (exactly including all higher order spatial x derivatives) of the PR material grating was performed, a higher order RCWA analysis is also valid, since rapid spatial variation of the material equations has been accounted for. In other words, the Kukhtarev analysis is not limited to the D limit (D represents Debye) [8, p. 89] (a limit that arises when approximations to Kukhtarev's equations are made that eliminate and drop higher order derivative terms) because the Kukhtarev analysis is an exact one. The comments of this paragraph concerning the applicability of the higher order RCWA analysis apply not only to the results of Fig. 13 but also to all the computations to be presented in the rest of the ®gures. Figures 13 and 14 (see Fig. 12a, for the geometry) display the numerical PR mode coupling and conversion that occurs (using a Kukhtarev and RCWA analysis) when two interfering plane waves (see Fig. 12a) whose amplitudes are 0 and 1  010 are incident on a PR crystal,   0633 m. Unlike the numerical case of Fig. 13, the PR crystal is not now indexed matched to the surrounding medium (free space is assumed to surround the PR crystal, 1  3  1). In this case optical energy is multiply re¯ected and subsequently diffracted from the   0 and    indexed mismatched interfaces. The combination of multiple re¯ection, diffraction, and the anisotropy of the PR crystal leads to the possibility of a quite complicated dielectric modulation pro®le occurring in the PR grating region. For this reason, the numerical simulation of the mismatched grating case was made for several closely spaced crystal lengths  in order that the effect of crystal length on grating formation could be fully studied. For the values of   14821875   ,   95625,   1 2 3 4 5 6, Figs. 13a and 13b show the power transmitted in the "0 and "1 directions, respectively, of Fig. 12a, and Figs. 13c and 13d show the power re¯ected in the $0 and $1 directions, respectively, of Fig. 12a. Again, no appreciable higher order diffraction was observed. There are several interesting features of the plots for this case. The ®rst is that a relatively small change in the overall length  of the PR crystal can

Copyright © 2000 Marcel Dekker, Inc.

5" # The power transmitted in the "0 , "1 , $0 , and $1 directions is shown when the regions (Region 1 and 3) bounding the PR crystal are not indexed matched to the PR crystal (see Fig. 12) (air was assumed 1  3  1) in (a), (b), (c), and (d), respectively, for six slightly different PR crystal lengths . All grating parameters used in the simulation not listed on the ®gure are given in Section 7.3. The starred line on the 5 curve of (a) used 320 longitudinal layers, whereas all other simulation runs used 160 layers. Note that the number of longitudinal divisions made virtually no difference in the simulation. Used with permission of JOSA-A, 1996 [34].

make a very large change in the power transmitted and re¯ected in the different directions from the crystal. For example in Fig. 13b, the power transmitted in the "1 direction for the 4 length is small (about 12% in steady state), whereas when the length is increased to 6 the transmitted power jumps to a large value (about 50%). A length change of only about 2   19 has occurred. A second interesting feature of the plots is that depending on the length , the power transmitted ("0 and "1 directions) or re¯ected ($0 and $1 directions) may go into an oscillatory steady state or a nonoscillatory steady state. In Fig. 13b, it is observed that the 1 and 5 lengths form oscillatory steady states, whereas the lengths 2 , 3 , 4 and 6

Copyright © 2000 Marcel Dekker, Inc.

5" $ The dielectric perturbation function  that results in the index mismatched case of Fig. 13 when the PR crystal length is 5  1530 and (a)   56 ms, (b)   90 ms (oscillatory steady state, see Fig. 13, 5  1530). (c) The dielectric perturbation function  that results in the index mismatched case of Fig. 14 when the PR crystal length is 3  1510875 and   90 ms (nonoscillatory steady state, see Fig. 14, 3  1510875). Parts (b±d) are drawn to the same scale. (d) The RMS  [ rms 2 is also proportional to the electrostatic energy in a stored a grating period] as a function of time step when the crystal length is 3  1510875 and when the crystal length is 5  1530. Used with permission of JOSA-A, 1996 [34].

form nonoscillatory steady states. It is also interesting to note that the oscillatory steady state periods depend on the length . For example the period of the length 1 is about 50 ms, whereas the length 5 is about 65 ms. We would like to point out that the results of Fig. 13 did not change when the number of layers was changed from 160 to 320. The starred line of length 5 in Fig. 14a was calculated using 320 layers (the power intensity was averaged over ten points for each of the 320 longitudinal divisions), and all other plots were made using 160 layers (the power intensity was averaged over ten points for each of the 160 longitudinal divisions). As can be seen from Fig. 13a, there was no difference in the numerical results obtained. Figures 14a and 14b show, respectively, the dielectric modulation function  that results when   5  1530 (the grating system for this

Copyright © 2000 Marcel Dekker, Inc.

length is an oscillatory steady state, see Figs. 13a±d) at times   56 ms and   90 ms. Figure 14c shows the  dielectric modulation function that results when   3  1510875 at a time   90 ms (the grating system for this length is in a nonoscillatory steady state, see Figs. 13a±d). From Figs. 14a and 14b one notices an interesting property of the oscillatory steady state: the shape of the dielectric modulation function pro®le  does not change with time, but the peak-to-peak amplitude of the  pro®le changes periodically in time with the same period as the diffracted powers "0 , "1 , $0 , $1 . In Fig. 14a at time   56 ms (  5  1530) we notice max min  157  10 3 , whereas in Fig. 14b at   90 ms (  5  1530 max min  531  10 3 . Figures 14a and 14b (and Fig. 14c also) are drawn on the same scale. They have almost an identical shape, but the peak-to-peak amplitude of Fig. 14a is about 25% that of Fig. 14b. Fig. 14d shows a plot of the root mean square (RMS) amplitude of the dielectric modulation function pro®le  (the RMS formula is given in Fig. 14d) as a function of time step (   1 ms) when the PR crystal length is   3  1510875 and when the PR crystal length is   5  1530. As can be seen from the plots of Fig. 14d, the   5  1530 curve shows that in the nonoscillatory steady state case the RMS value (and therefore peak-to-peak value) of the dielectric modulation function pro®le  does go through maximum and minimum values, whereas the   3  1510875 curve in the nonoscillatory steady state reaches a steady state RMS value of . It is interesting and reasonable that the nonoscillatory RMS value of  is almost exactly the average of the oscillatory RMS  value. We would like to note that the square of RMS  values shown in Fig. 14d is proportional to the electrostatic energy stored over one grating width  and length  of the PR crystal. Thus another way of viewing the nonoscillatory steady state is that the electrostatic (or quasi-static) energy of the PR crystal is in a continuous state of gaining and losing electrostatic energy as time progresses. We also note from Fig. 14d that the rise and fall of the RMS value of  (also electrostatic energy) is not symmetric in time but builds up from a minimum to a maximum in about 30 ms and falls from a maximum to a minimum in about 40 ms. Figure 15 displays the numerical PR mode coupling and diffraction that occurs (using the Kukhtarev and RCWA analysis) when two interfering plane waves whose amplitudes are 0 and 1  040 are incident on a PR crystal (  0633 m,   1530  5 ) that is not index matched to the surrounding medium (free space is assumed to surround the PR crystal, 1  3  1). The peak-to-peak value of  is approximately 0.00654 as is evident from Fig. 15b. The diffraction response in this case is very different from the Fig. 13 and 14 case, although that case differs from the case of Figs. 13 and 16 (  5  1530 curves) only in that the signal amplitude

Copyright © 2000 Marcel Dekker, Inc.

5" & Numerical PR mode coupling and diffraction that occur (with the use of the Kukhtarev equations and RCWT) when two interfering plane waves whose amplitudes are 0 and 1  040 are incident on a PR crystal (  0633 m,   1530  5 ) that is not index matched to the surrounding medium (free space is assumed to surround the PR crystal, 1  3  1). (a) The dielectric modulation function  that results when L  160 layers is used. (b) The results when L  640 layers is used. (c) The transmitted and re¯ected power diffracted in the zero and ®rst orders when L  640. (d) The power transmitted in the second order when L  160, 320, and 640 layers. The dashed line shown in (d) shows the transmitted power that is diffracted in the second order when L  160 layers and T  3. Used with permission of JOSA-A, 1996 [34].

1 is 1  040 rather 1  010 . In this case, as can be seen from Figs. 15c and 15d, power is initially diffracted into the ®rst-order diffraction mode directions "1 and $1 and then later at about   100 ms diffracted into the "2 and $2 directions. In Figs. 13 and 14 diffraction into higher orders was not observed at all. Thus the diffraction in Figs. 13 and 14 constitutes a case of four-wave coupling, whereas that of Fig. 15 represents six-wave coupling. Another large difference with the results of Figs. 13 and 14 is that in the case of Fig. 15 the pronounced oscillation that occurred in Figs. 13 and 14 did not occur in Figs. 15c and 15d. The response of Fig. 15c and 15d seems to indicate that "0 attains a quasi±steady state while there is power exchange

Copyright © 2000 Marcel Dekker, Inc.

between "1 and "2 . A third major difference in the PR response of Figs. 13 and 14 and Fig. 15 was in the shape of the dielectric modulation function  that was formed. In Figs. 14a,b,c the  pro®le was small at the incidence side and increased in value toward the transmit side. In Fig. 15b the  pro®le is nearly uniform in its peak-to-peak value throughout. The generation of higher order diffracted waves can be understood in the following way. In volume holograms, appreciable power transfer is possible not only for Bragg incidence but also for incidence at multiple Bragg angles. Alferness [35] has analyzed the diffraction ef®ciency for thick holograms operating in the second-order Bragg regime and concluded that 100% diffraction ef®ciency was possible. The second-order diffraction ef®ciency can be expressed as [36]  2 

2  sin2  #

7420

where #   2  0 and   0 

pp eff 4. In the above expression  is the hologram wave number, eff is the effective grating thickness, and 

pp denotes the peak-to-peak change in induced refractive index. In our case, with the hologram spacing   5 and eff )  2  1530 2,   0633 m and #  200. Also from the plots (Fig. 16b) the peak value of 

pp  000134 (since (

pp   pp 2 0 , pp  max min  000654 (see Fig. 16b), 0  2437) at   200 ms implying   16. Note that this approximately corresponds to the condition for maximum diffracted power in the second order. Equation 7.4.20 shows that as # increases, a larger value of  is required for enhanced power transfer to the second order. A numerical simulation was also performed for the mismatched case (see Fig. 12a, 1  3  1) when the incident amplitudes were 0 and 1  0 for the cases when   1500, In these cases very interesting results occurred. In the case when   1500 the power in the "0 order was diffracted and the mode converted into the "1 order, and no other appreciable diffraction occurred. The diffraction in the "0 and "1 directions was observed to be in a nonoscillatory steady state as time increased. For   1500, the peak-to-peak    max min  386  10 3 which made (

  079  10 3 . Further, it was observed that the peak-to-peak dielectric modulation function (  decreased in value, nearly linearly, from the incidence side to the transmission side, assuming a very small value at the transmission side. Thus the effective length of the grating was about    2  1500 2. In the case when   1530, power was initially diffracted from the "0 order to the "1 order and then subsequently at

Copyright © 2000 Marcel Dekker, Inc.

about   75mg was diffracted from the "1 order to the "2 second order. For this case, the peak-to-peak dielectric modulation function (   max min  533  10 3 , which made 

  109  10 3 . From this simulation we ®nd the interesting results that the index mismatched PR crystal for certain lengths appears to be resonant in the sense that for certain lengths (in this case   1530), the optical energy in the PR crystal can interact with the crystal in such a way that the dielectric modulation function  can build up to relatively large values in the crystal. Figure 16a shows plots of the normalized electron density   A that result from the simulation shown in the matched case of Fig. 13 (  113 ms,    2,   1530,   0633 m,   5) and the mismatched case of Figs. 14 and 15 (  200 ms,   2,   5  1530,   0633 m,   5). As can be seen from these plots, we ®rst notice that the results of the simulation show that the normalized electron density  

A assumes a very small value of  ) 10 5 or less. Secondly we note that the presence of matched or mismatched boundaries makes a signi®cant difference as to where the peaks of the electron density occur and the peak-to-peak size of the electron density. Figure 17b shows a plot of the dielectric modulation function  obtained at the same location where the electron densities of Fig. 17a were obtained. As we can see from these plots, the presence of matched or mismatched boundaries makes a signi®cant difference as to position of the peaks and the peak-to-peak amplitude of the  pro®les. We ®nally note that a comparison of Figs. 17a and 17b shows that in both the matched and the mismatched cases the electron density is always displaced from, and out of phase with, the  pro®le. This is a result of Eq. 7.4.12 and the fact that  and  are related by a ®rst space and time derivative equation.

-&

 +  ** <#-:#.=

So far we have examined wave mixing in diffusion-dominated PR materials assuming transmission gratings. However, as stated in the introduction, re¯ection gratings can also be ef®ciently induced and stored in PR media and have practical applications such as in the construction of tunable ®lters of very low spectral width. -&

)  2  2 !24

This section will be concerned with determining the EM ®elds that exist inside of a PR re¯ection grating using a RCWA analysis. In this case, because the PR grating depends on the optical intensity, the magnitude of

Copyright © 2000 Marcel Dekker, Inc.

5" ' (a) Plot of the normalized electron density   A that results from the simulation shown in the matched case (  113 ms,    2,   1530,   0633 m,   5) and the mismatched case (  200 ms,    2,   5  1530,   0633 m,   5). (b) Plot of the dielectric modulation function  obtained in the same location as that of the electron densities of (a). Used with permission of JOSA-A, 1996 [34].

Copyright © 2000 Marcel Dekker, Inc.

the optical modulation of the grating is longitudinally inhomogeneous. The RCWA analysis is carried out by (1) dividing the RG into a number of discrete layers, (2) expanding the EM ®elds in each layer region and expanding the EM ®elds in the incident and transmit sides of the RG in a set of Floquet harmonics, (3) solving Maxwell's equations in all regions in terms of the Floquet harmonics, and (4) matching EM ®eld solutions at all boundaries to determine the EM ®elds of the overall system. The RCWA optical analysis of this chapter closely follows the RCWA re¯ection grating analyses of Refs. 39±41. The RCWA re¯ection analysis of this section differs from that of Refs. 39±41 in two ways: (1) the re¯ection grating here is anisotropic rather than isotropic, and (2) the EM incident electric ®eld here is parallel to the plane of incidence rather than the perpendicular to it. For the convenience of readers, we will now summarize the RCWA optical ®eld equations in the RG and space surrounding the grating.    0 ,  etc., Normalizing all space coordinates according to  0 , where 0  2  and  in meters is the free space wavelength, the EM ®elds  and  in the th layer (the index is suppressed) of the RG are given by ( 

 (
  (
   exp 
 





 )
  exp 
 


) 

751




0  377 , 
   sin  , 1 sin 
 

where (  , )  0    

  cos  , and   0 . The angle  [39±41] is the tilt angle of the DG with respect to the planar interfaces and can be taken to be zero Maxwell's equations are for a pure RG analysis [41]. In terms of ( and ),

  (   )

  )  (



 where   % 0

  0

0 0  

752

In this chapter the anisotropic permittivity tensor is assumed to have its caxis at *  45 to the crystal interfaces as shown in Fig. 17. The permittivity tensor elements, including the RG modulation due to the presence of a nonzero optical intensity inside of photorefractive BaTiO3 , and a nonzero longitudinal electrostatic ®eld for the c-axis shown in Fig. 1 have been derived and speci®ed in Ref. 5. After substituting Eq. 7.5.1 in Maxwell's Eq. 7.5.2 following Refs. 39±41, the state variable equations arise

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5" -

Problem geometry.

 5   11 21 

 12 5 22

753

where 5   0  ,   (
, 0  )
,
 T      T , and   11   <  1   m   21       1     <  


   m  




  12   <  1  < 
  22   m     1  <     

     

" 1 for





  0 for



Solution of the state variable equations gives the propagating and nonpropagating eigenmode ®elds in each thin layer. Summing over these eigenmodes and matching boundary conditions at each thin layer and in Regions 1 and 3, the full EM solution in all regions can be found. -&

% 2 !24

In this section we will present a material analysis using Kukhtarev's equations and applicable to the re¯ection grating geometry. For the present case it is assumed that no variation occurs in the transverse -direction and that a PR re¯ection grating is formed in the -longitudinal direction. Following the analysis of transmission gratings above, but taking care to use the longitudinal spatial variable  (the direction of the re¯ection grating vector), we

Copyright © 2000 Marcel Dekker, Inc.

®nd that Kukhtarev's material equations can be reduced to differential equations     2





 3 4 1  

2 2  1   3 1   1 4   *         

 

  1 754

1  1  3  * *  '  

 1  2  

755

In these equations  is the optical intensity W/m2 and   20 2 42 % 

is a normalized dielectric modulation function linearly related to the longitudinal electrostatic ®eld %  (V/m). All other parameters have been de®ned in the previous sections. Because a re¯ection grating is being studied, it is useful to expand the optical ®eld  and the material variables  and   in a spatial Fourier series (the period of the Fourier series is the grating wavelength  where the Fourier amplitudes are all spatially varying functions   0 ) of the longitudinal coordinate . We have

 

    


 


 exp 


756


 


 exp 


757


 


 exp 


758


 


 


 

where   2  and  equals the number of Fourier components. If Eqs. 7.5.6±7.5.8 are substituted into Eq. 7.5.4 and the coef®cients of exp 
 are equated, we ®nally obtain  2


    2

2 
 2
  


2   where

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 *      * 759

 1 2 3  1  4

2 and 





"


  


7510

and where "
 


    1   4 
   

 3 4

 3
1 2 2 * 2  2

    1  
 3 



 
0  
 

 1 2
  2  * '    1  




    




 0  2
  *

7511

7512

  (in meters being Equation 7.5.12 for the interval   0   0 , the crystal layer thickness) represents a set of 2*  1 spatially varying equations from whose solution all material variables can be determined. In this chapter we impose the boundary conditions that the normalized electron density  vanishes at the crystal interfaces   0 and   . This boundary condition further imposes the boundary conditions on Eq. 7.5.12 that


0 
  0


 *      *

7513

To proceed further we now for the moment regard the RHS of Eq. 7.5.12 as a known function . Equation 7.5.12 along with its boundary conditions, for each
, is classi®ed as a linear second-order nonhomogeneous differential equation. The solution to this type of equation is well known and can be found by using a Green's function approach: (1) setting the RHS of Eq. 7.5.12 to a Dirac delta function    ; (2) solving the resulting differential equation  2 
 


   2 2   2
 



     

2  

Copyright © 2000 Marcel Dekker, Inc.

7514

with the boundary conditions and continuity condition, respectively 
0   
   0 


 

 

 
 





7515



where  and   represent locations an in®nitesimal to the left and right of   ; (3) superposing the Green's function solutions times the nonhomogeneous RHS 
to ®nd the overall response of the system. Regarding 


as a known function, the solution for
 is given by


 

0




  
   

7516

Although Eq. 7.5.16 is an exact integral for the differential equation Eq. 7.5.12, it is an unnecessarily complicated one for the current analysis. For typical PR parameters used in this chapter, the constant  is on the same order of magnitude as   2 . Investigation of the Green's function 
  for the just-described values of  and  shows that this Green's function has a signi®cant nonzero value only within a few  of the point     in the interval. Investigation has further shown that most of the exponential terms are exponentially small. After analysis it is found that 
  is well approximated by [37]  1 exp%1     
  %1 %2 exp%2     

     0

7517

where %12   
. 
  is signi®cantly nonzero only when     5. A basic assumption that has been made in this chapter is that all of the spatially varying amplitudes of Eqs. 7.5.6±7.5.8 vary over a much greater length than the exp 
 functions. Using this assumption one notices that in the integral of Eq. 7.5.16 the 
  function can be approximately taken as constant over the range where 
  of Eq. 7.5.17 is nonzero. With the approximations that (1) 
  is taken as constant and evaluated at     where the Green's function is maximum, (2) the approximate Green's function of Eq. 7.5.17 is used, and (3) the limits of the integral of Eq. 7.5.16 are taken to be     to    , it turns out that the integral of Eq. 7.5.16 can be evaluated in closed form. The result of the integration is


  



where

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7518




2

1 
 2

7519

Substituting Eq. 7.5.10 into Eq. 7.5.18 and collecting terms on
 , we derive a matrix equation for the normalized electron density
 : 


 



  
"


   


   




7520

To proceed further we substitute the spatially varying approximations Eqs. 7.5.6±7.5.8 for  and   into Eq. 7.5.5 and ®nd after equating like coef®cients of exp 
) to each other that    

 1  



 
  2  


7521

If Eq. 7.5.20 is substituted for
 , and matrix terms common to 
are collected in Eq. 7.5.21, it can be placed in the usual state variable form:  
  '

 
  



7522

where '

 and 
are de®ned in Ref. 37, where the complete temporal numerical solution is detailed. -&#

" 2  "2

This section will be concerned with presenting a numerical simulation of a re¯ection grating based on the above model. For the present simulation we assume a lossless grating and that 42  1640  10 12 (m/V), 13  8  10 12 m V , o  2437 and e  2365, which are the given crystal values for BaTiO3 by Ref. 8 at   0633 m. For these values the   0132 m. In the simulation we grating wavelength had the value  have assumed that the BaTiO3 c-axis makes a 45 and with the -axis of Fig. 18. We further assume R  5  10 14 (m3 /s),   025  10 4 (m2 /V s), A  3  1022 (1/m3 ), D  200A , %  1  10 5 (m2 /J s) and   2 (1/ s). With these values we ®nd that the dark current is *  2  105 (W/m2 ). For BaTiO3 and for the geometry shown in Fig. 18, the effective value of      %  %2 cos2   %3 sin2  , where %1  %2  36000 %3  1350 [8, Table 1.2, p. 28] and where %  %


 ,

  1 2 3, is the static dielectric permittivity tensor when BaTiO3 PR crystal has its c-axis aligned along

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the   -axis. The optical and material analysis was carried out using values of   0132 mm,      0  4795, T  5 and C  10. Also   1000    4858, and   6 . In Ref. 37, we have shown the plots of the dielectric modulation function    [proportional to the electrostatic space charge %   ] as   in meters,   0 , calculated over a range of two grating wavelengths    from a position 123 to 125 from the incident side of the grating at the times of   0059, 0.477, 4.97, 30.1, and 69.0 ms. Also plotted was the re¯ected diffraction ef®ciency R  as a function of time for the simulation under consideration. We have also shown plots of the terms of the electron density balance equation Eq. 7.5.20 at a given time   301 ms. These plots were all done  for a tuned layer and  for a thickness of 1000. Also in Ref. 37, no study was done of the convergence of the numerical solution with different spatial harmonics C (number of harmonics in the material equations) and t (the number of harmonics used in RCWA analysis). We will now present additional numerical simulations for the diffraction ef®ciency and the various terms of the electron density balance equation for different lengths of the photorefractive material, and for tuned and detuned cases, respectively. In Fig. 18, we study the convergence of the solution (diffraction ef®ciency) for a tuned layer of thickness 3000 by varying the values of C and t from C  1 and t  1 to C  14

5" . Transient re¯ected diffraction ef®ciency for different optical and material harmonics. Used with permission of JOSA-B, 1998 [37].

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and t  17. As can be seen from Fig. 18, reasonable convergence is achieved for values of C  6 and t  3 or higher. A nominal combination of C  10 and t  5 was chosen for all following calculations. Figure 19 shows the re¯ected diffraction ef®ciency results for tuned layers of different thicknesses as a function of time. As expected, the diffraction ef®ciency increases with layer thickness. Note that the time evolution is nonexponential and that the initial buildup of the diffraction ef®ciency is more rapid as the thickness increases. Conservation of transmitted and re¯ected power was observed for all computations. In Fig. 20, we show the time evolution of the diffraction ef®ciencies for a tuned layer of thickness 3000 and several other layers detuned from this value by 0.075, 025, 0375, and 05. As can be seen, the amount of detuning has a profound effect on the diffraction ef®ciency. For a slightly detuned layer, the initial re¯ection is small, but it increases in time rapidly as does the tuned case. On the other hand, a severely detuned layer has an initially large re¯ection, due to boundary mismatch, and the rate of increase of the diffraction ef®ciency is much slower because a smaller percentage of the incident illumination actually enters into and interacts within the photorefractive material to form the re¯ections grating.

5" / Transient re¯ected diffraction ef®ciency for different tuned layer thicknesses. Used with permission of JOSA-B, 1998 [37].

Copyright © 2000 Marcel Dekker, Inc.

5" 0 Transient re¯ected diffraction ef®ciency for different detuning lengths. Used with permission of JOSA-B 1998 [37].

Figures 21a and 21b show plots of several terms that make up the electron density matrix equation as given by Eq. 7.5.20 (the terms making up Eq. 7.5.20 are speci®ed in Eqs. 7.5.9±7.5.12) when a tuned layer re¯ection grating is formed (Fig. 21a,   0) and when a nontuned layer re¯ection grating is formed (Fig. 21b,   05). These ®gures were formed by Fourier summing the slowly varying amplitudes according to Eqs. 7.5.6± 7.5.8 and then plotting the sums as a function of loc as shown in Figs. 21a and 21b. As can be seen when comparing them, a nontuned layer re¯ection grating causes a signi®cantly different electron density pro®le from when a tuned layer re¯ection grating is present. Figure 22 shows plots of the dielectric modulation function (proportional to the electrostatic ®eld inside the re¯ection grating) when a tuned layer re¯ection grating is formed (same case as Fig. 21a,   0) and when a nontuned layer re¯ection grating is formed (same case as Fig. 21b,   05). As can be seen from Fig. 21, detuning of the layer causes a shift in the dielectric modulation function . -&$

!2"!

In this section, we have, after appropriate reformulation of the multilayer RCWA, analyzed the time evolution of re¯ection gratings in a photorefractive material of an arbitrary thickness. This has been accomplished by letting all but the last layer be an integral number of wavelengths and then

Copyright © 2000 Marcel Dekker, Inc.

5"  Spatial distribution of different terms in Eq. 7.5.20 for a tuned layer (a) and for a detuned layer (b). Used with permission of JOSA-B, 1998 [37].

letting the last layer have an arbitrary length. The re¯ected diffraction ef®ciency has been plotted as a function of time for various layer thicknesses, viz., from 100 to 6000. The evolution of the diffraction ef®ciency has also been shown for various detunings  ranging from 0 to 05. Numerical convergence was also analyzed. We have found that a minimum number of spatial harmonics (viz., *  6 and   3) should be used for reasonable accuracy. Examples of the spatial distribution of various physical quantities such as the electron density, induced permittivity etc., have been provided.

Copyright © 2000 Marcel Dekker, Inc.

5"  Typical induced permittivity distributions for a tuned and detuned layer. Used with permission JOSA-B, 1998 [37].

1+ %

1.

In many photorefractive materials such as lithium niobate, an important contribution to the photorefractive effect is the photocurrent generated (photovoltaic effect) when light is incident on the material (see Chapter 4). Using suitable approximations, decouple the Kukhtarev equations in the steady state when the current in Eq. 7.2.3 contains an additional term  on the righthand side, where  is the photovoltaic tensor element and is the optical intensity. 2. Assume that a thin photorefractive layer has incident on it two plane waves of amplitude ratio . (a) Use a ®nite difference scheme to discretize the original set of Kukhtarev equations (Eqs 7.2.1±7.2.4). (b) By marching forward in time, solve these equations using periodic boundary conditions along over a spatial period  formed by the interfering plane waves. Plot the electrostatic ®eld % over a spatial period for different times. Assume BaTiO3 as the photorefractive material. (The procedure is similar to that used in Sec. 7.4.) (c) Show that the temporal evolution of % is well approximated by means of the intensity dependent time constant  de®ned in Eq. (7.3.8). (d) Compare your results with the ®ndings of Ref. [42].

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3.

4. 5.

6.

7.

8.

(a) Decouple and linearize the Kukhtarev equations (7.2.1±7.2.4) in the small contrast approximation (small ). (b) Derive the set of differential equations coupling the two incident electric ®elds during interaction in the photorefractive material. (c) Exactly solve these equations in the steady state to determine the longitudinal variation of the two electric ®elds [8]. (e) Solve the differential equations derived in (b) in the transient case. Take the limit as    and compare your answer with part (c). Repeat Problem 3 but now including a photovoltaic term in the Kukhtarev equations (see Problem 1). Assume that two optical waves which are incident on a photorefractive material are frequency offset from one another by . This sets up a running interference pattern proportional to cos (Kx-t). Decouple and linearize the Kukhtarev equations (7.2.1±7.2.4) in the small contrast approximation (small ). Analyze the resulting equations and ®nd an expression for the space charge ®eld in terms of the intensity and the frequency offset [8]. Assume that two optical waves which are frequency offset from one another by , as in Problem 5, are incident on a photorefractive material at   0. (a) Reformulate the numerical algorithm for the exact Kuktarev equations to account for the traveling wave nature of the intensity grating. (b) Obtain a numerical solution for the photorefractive and optical response of BaTiO3 . What is the effect of varying ? Investigate the re¯ection grating problem of Section 7.5, and plot the steady state transmitted and re¯ected intensities as a function of the incident intensity. Can this effect be used for optical limiting for the transmitted ®eld? Fig. 5 in this chapter represents one time experiment each for the case where the phase difference between the two participating plane waves incident on the photorefractive crystal is rand  158 and 308, respectively. Make several independent computations for these values of rand , and average these values for each 1 0 ratio as shown in Figure 5. Compare your results with the coherent case.

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1.

2. 3.

4. 5. 6. 7. 8. 9.

10.

11.

12. 13.

14. 15.

16.

17.

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 %"     *, 2061±2080 (1987). P. Gunter, E. Voit, M. Z. Zha, and J. Albers, Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3 , %"   //, 210±214 (1985).

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19.

20. 21. 22. 23. 24. 25.

26.

27.

28. 29.

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 %"     ,, 1689±1692 (1992). L.-K. Dai, C. Gu, and P. Yeh, Effective of position-dependent time-constant on photorefractive two-wave mixing,
 %"     ,, 1693±1697 (1992). H. Kong, C. Wu, and M. Cronin-Golomb, Photorefractive two-beam coupling with reduced spatial coherence, %"  -  .3, 1183±1185 (1991). M. Cronin-Golomb, H. Kong, and W. Krolikowski, Photorefractive two-beam coupling with light of partial spatio-temporal coherence,
 %"     ,, 1698±1703 (1992). M. Snowbell, M. Horowitz, and B. Fischer, Dynamics of multiple two-wave mixing and fanning in photorefractive materials,
 %"     ,, 1972± 1982 (1994). E. Serrano, V. Lopez, M. Carrascosa, and F. Aguillo-Lopez, Recording and erasure kinetics in photorefractive materials at large modulation depths,
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35. 36. 37. 38.

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