Computational Methods for Electromagnetic and Optical Systems

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.

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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

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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

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1

Mathematical Preliminaries

1.1

INTRODUCTION

Popular T-shirts advertising Maxwell's equations do not go beyond merely stating them. In this book, we enter into a little more depth and solve these equations for analyzing various electromagnetic (EM) and optical problems, e.g., diffraction gratings, radiation and scattering from dielectric objects, and holograms in photorefractive materials. The emphasis on ®nding the solutions in our text concerns the use of Fourier and state variable analyses. In this chapter, we brie¯y restate Maxwell's equations and review mathematical techniques pertinent to the analyses presented in later chapters. Maxwell's equations in differential form are a set of four coupled ! partial differential equations relating the electric ®eld E , the magnetic ! ! ! ®eld H , the electric displacement, D , and the magnetic ¯ux density B : ! r
D ˆ i

…1:1:1†

! r
B ˆ0

…1:1:2†

! r E ˆ

! @B @t

! ! ! ƒ! @ D r  H ˆ Jc ‡ J i ‡ @t

…1:1:3† …1:1:4†

! ! In Eqs. 1.1.1±4, i denotes the impressed charge density, and J c , and J i are conduction and impressed current densities, respectively. In time-reduced ! ! form (i.e., assuming variations of the form A ˆ Re… A exp j!t†, Eqs. 1.1.1±4 read

Copyright © 2000 Marcel Dekker, Inc.

2

Chapter 1

! r
D ˆ i

…1:1:5†

! r
B ˆ0 ! r E ˆ

…1:1:6† ! j! B

…1:1:7†

! ! ! ! r  H ˆ Jc ‡ Ji ‡ j! D

…1:1:8†

In Eqs. 1.1.5±8, the electric and magnetic ®eld variables are related through the constitutive relations as ! ! D ˆ 0  E

! ! B ˆ 0 lH

! ! J c ˆ rE

…1:1:9†

where 0 and 0 are the free-space permittivity and permeability, respectively,  and l are the relative permittivity and permeability tensor, respectively, of the material, and r is the conductivity tensor. More general constitutive relations that apply to, for instance, chiral media, can be found later on in the book.

1.2

THE FOURIER SERIES AND ITS PROPERTIES

It is easy to show that a set of exponential functions fexp jnKxg, n ˆ 0, 1, p 2, etc., where j ˆ 1, is orthogonal over an interval …x0 ; x0 ‡ 2=K† for any value of x0 . The orthogonality can be demonstrated by considering the integral Iˆ

… x0 ‡2=K x0

exp…jnKx† exp… jmKx† dx ˆ

2  K m;n

…1:2:1†

where m;n is the Kronecker delta function:  m;n ˆ

1 0

mˆn m 6ˆ n

…1:2:2†

Using this, a function f …x† can be expanded in a Fourier series over an interval …x0 ; x0 ‡ 2=K† as f …x† ˆ

1 X nˆ 1

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Fn exp…jnKx†

…1:2:3†

Introduction

3

Multiplying Eq. 1.2.3 by exp… jmKx†, integrating over the interval …x0 , x0 ‡ 2=K†, interchanging the summation and the integral, and using Eqs. 1.2.1 and 1.2.2, we obtain … x0 ‡2=K x0

f …x† exp… jnKx† dx ˆ

1 X nˆ 1

Fn

… x0 ‡2=K x0

exp…jnKx†

exp… jmKx† dx   1 X 2 2 F m;n ˆ Fn ˆ K K m nˆ 1 Now, replacing m by n, Fn ˆ

K 2

… x0 ‡2=K x0

f …x† exp… jnKx† dx

…1:2:4†

Note that if a function fe …x† is de®ned as fe …x† ˆ f …x† exp… jx†, where  is a constant, then over the interval …x0 ; x0 ‡ 2=K†, it can be written as fe …x† ˆ

1 X nˆ 1

Fn exp… jkxn x†

kxn ˆ 

nK

…1:2:5†

Li refers to this expansion in Eq. 1.2.5 as a pseudo-Fourier series of fe …x† [1]. If two functions f …x† and g…x† having Fourier series expansions f …x† ˆ

1 X nˆ 1

Fn exp…jnKx†

g…x† ˆ

1 X nˆ 1

Gn exp…jnKx†

…1:2:6†

over the same interval are multiplied, the product function h…x† has a Fourier series expansion h…x† ˆ

1 X nˆ 1

Hn exp…jnKx†

…1:2:7†

over the same interval. We can ®nd the Fourier coef®cients of h…x† in the following way:

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4

Chapter 1 1 X

h…x† ˆ f …x†g…x† ˆ

nˆ 1

ˆ

1 X

1 X

nˆ 1 mˆ 1

ˆ

1 1 X X lˆ 1 mˆ 1



1 X lˆ 1

Fn exp…jnKx†

1 X mˆ 1

Gm exp…jmKx†

Fn Gm exp…j…n ‡ m†Kx† …1:2:8† Fl

m Gm

exp…jlKx†

Hl exp…jlKx†

The limits on l are 1 to ‡1 since l ˆ m ‡ n and m and n each have limits 1 to ‡1. Hence the Fourier coef®cients Hl of h…x† can be expressed as Hl ˆ

1 X mˆ 1

Fl

m Gm

…1:2:9†

Equation 1.2.7 is sometimes referred to as the Laurent rule [1]. To be more precise, Eqs. 1.2.8 and 1.2.9 should be understood in the following sense [1]: h…x† ˆ lim

N!1

ˆ lim

L!1

N X lˆ N L X lˆ L

Hl exp…jlKx† lim

M!1

M X mˆ M

…1:2:10†

! Fl

m Gm

exp…jlKx†

The above equation, in the way it is written, emphasizes two important points. First, the two limits L and M are independent of each other, and the inner limit has to be taken ®rst. Secondly, the upper and lower bounds in each sum should tend to in®nity simultaneously [1]. In solving a practical problem on a computer, the truncation of the in®nite series is inevitable. Later, in Chapter 3, we will show that there is a convergence problem resulting from application of the Laurent rule to ®nd the Fourier coef®cients of the product of two functions f …x† and g…x†, represented by ®nite or truncated Fourier series, which are pairwise discontinuous at x ˆ x0 , though their product h…x† is continuous at that point. We will further show that the convergence problem can be alleviated using the socalled inverse rule. Situations like this arise in the analysis of surface relief diffraction gratings in electromagnetics, when the permittivity is a discontinuous function of x. In this important case, the normal electric ®eld and

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Introduction

5

the permittivity must be pairwise discontinuous at x ˆ x0 because, from EM boundary conditions, the normal electric displacement (which is a product of the two) must be continuous.

1.3

THE FOURIER TRANSFORM

The one-dimensional spatial Fourier transform of a square-integrable function f …x† is given as [2] F…kx † ˆ

…1 1

f …x† exp…jkx x† dx

…1:3:1†

The inverse Fourier transform is 1 f …x† ˆ 2

…1 1

F…kx † exp… jkx x† dx

…1:3:2†

The de®nitions for the forward and backward transforms are consistent with the engineering convention for a traveling wave, as explained in [2]. If f …x† denotes a phasor EM ®eld quantity, multiplication by exp j!t gives a collection or spectrum of forward traveling plane waves. The two-dimensional extensions of Eqs. 1.3.1 and 1.3.2 are F…kx ; ky † ˆ

…1 …1 1

1 f …x; y† ˆ …2†2

1 …1

f …x; y† exp…jkx x ‡ jky y† dx dy

1

…1 1

F…kx ; ky † exp… jkx x

…1:3:3†

jky y† dx dy

…1:3:4†

In many EM applications, the function f …x; y† represents the transverse pro®le of an EM ®eld at a plane z. Hence in Eqs. 1.3.3 and 1.3.4, f …x; y† and F…kx ; ky † have z as a parameter. For instance, Eq. 1.3.4 becomes 1 F…x; y; z† ˆ …2†2

…1 …1 1

1

F…kx ; ky ; z† exp… jkx x

jky y† dx dy …1:3:5†

The usefulness of this transform lies in the fact that when substituted into Maxwell's equations, one can reduce the set of three-dimensional PDEs to a

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6

Chapter 1

set of one-dimensional differential equations (ODEs) for the spectral amplitudes F…kx ; ky ; z†. 1.4

THE DISCRETE FOURIER TRANSFORM

Given a discrete function f …n†; n ˆ 0; . . . N 1, a corresponding periodic function fp …n† with period N can be formed as [3] fp …n† ˆ

1 X

f …n ‡ rN†

…1:4:1†

rˆ 1

The discrete function f …n† may be formed by the discrete values of a continuous function f …x† evaluated at the points x ˆ n. The discrete Fourier Transform (DFT) of fp …n† is de®ned as Fp …mK† ˆ

N X1 nˆ0

fp …n† exp…jmnK†

Kˆ

2 N

…1:4:2†

The inverse DFT is de®ned as fp …n† ˆ

X1 1N F …mK† exp… jmnK† N nˆ0 p

…1:4:3†

For properties of the DFT, e.g., linearity, symmetry, periodicity, as well as relationship to the z-transform, the Fourier transform and the Fourier series, the readers are referred to any standard book on digital signal processing [3]. For the purposes of this book, the DFT is a way of numerically approximating the continuous Fourier transform of a function. The DFT is of interest because it can be ef®ciently and rapidly evaluated by using standard Fast Fourier Transform (FFT) packages. The direct connection between the continuous Fourier transform and the DFT is given below. For a function f …x† and its continuous Fourier transform F…kx †, Fp …mK† 

1 F…mK† 

jmKj <

 

…1:4:4†

In Eq. 1.4.4, Fp …mK† is de®ned, as in Eq. 1.4.2, to be the DFT of fp …n†. The equality holds for the ®ctitious case when the function is both space and spatial frequency limited.

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Introduction

1.5

7

REVIEW OF EIGENANALYSIS

Many of the computations in this book are based on determining the eigenvalues and eigenvectors of a matrix A. Therefore this section will brie¯y review the methods and techniques associated with numerically solving this problem [4,5]. The matrix A, which is a square matrix, in general transforms a column vector x that transform into themselves and satisfy Ax ˆ qx

…1:5:1†

These column vectors are called the eigenvectors of the system. The values q which satisfy are known as the eigenvalues, the characteristic values, or the latent roots of the matrix A. Equation 1.5.1 can be written as a linear set of equations as …a11

q†x1 ‡ a12 x2 ‡ a13 x3 ‡


‡ a1n xn ˆ 0

a21 x1 ‡ …a22

q†x2 ‡ a23 x3 ‡


‡ a2n xn ˆ 0




…1:5:2†



an1 x1 ‡ an2 x2 ‡ an3 x3 ‡


‡ …ann

q†xn ˆ 0

A nontrivial solution exists for the above equations if and only if P…q†  det…qI

A† ˆ 0

…1:5:3†

where I is the identity matrix. The result of Eq. 1.5.3 is an nth order polynomial called the characteristic equation or eigenvalue equation. The equation is given by P…q† ˆ qn ‡ a1 qn

1

‡ a2 qn

2

‡


‡ an 1 q ‡ an

…1:5:4†

The roots of this equation are the eigenvalues of the matrix A. When the roots are all unequal to one another, the roots or eigenvalues are called distinct. When the eigenvalue occurs m times, the eigenvalue is a repeated value of order m. When the root has a real and nonzero imaginary part, the roots occur in complex conjugate pairs. In factored form Eq. 1.5.4 can be written as

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8

Chapter 1

P…q† ˆ …q

q1 †…q

q2 †


…q

qn †

…1:5:5†

The coef®cients of the eigenvalue equation can be found directly from the matrix A. For instance, setting q to zero in Eq. 1.5.5, we ®nd P…0† ˆ an ˆ det… A† ˆ … 1†n det…A† ˆ … 1†n q1 q2


qn

…1:5:6†

and thus from Eq. 1.5.6 det…A† ˆ q1 q2


qn

…1:5:7†

The coef®cient a1 can be found by expanding the factor characteristic equation and comparing the polynomial coef®cients of the resulting equation. For example, if n ˆ 2, P…q† ˆ q2 ‡ a1 q ‡ a2 ˆ …q

q1 †…q

q2 † ˆ q2

…q1 ‡ q2 †q ‡ q1 q2 …1:5:8†

and thus a1 ˆ

…q1 ‡ q2 †

…1:5:9†

after equating coef®cients. For general n, a1 ˆ

…q1 ‡ q2 ‡


‡ qn †

…1:5:10†

If the determinant is expanded we also ®nd that the determinant is the negative sum of diagonal coef®cients, that is, a1 ˆ

…a11 ‡ a22 ‡


‡ ann †

…1:5:11†

The quantity in parentheses is an important quantity and is called the trace of A. Tr…A† ˆ a11 ‡ a22 ‡


‡ ann ˆ q1 ‡ q2 ‡


‡ qn

…1:5:12†

Let Tk ˆ Tr…Ak †. Then a useful formula for the coef®cient an of the characteristic equation is

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Introduction

a1 ˆ

9

T1 1 …a T ‡ T2 † 2 1 1 1 …a T ‡ a1 T2 ‡ T3 † 3 2 1

a2 ˆ a3 ˆ

…1:5:13†




1 …a T ‡ an 2 T2 ‡


‡ a1 Tn n n 1 1

an ˆ

1

‡ Tn †

For the case when the roots of P…q† are distinct, a nontrivial vector xi can be found for each root that satis®es …qi I

A†xi ˆ 0

i ˆ 1; 2; . . . n

…1:5:14†

The matrix formed of the columns of xi is called the modal matrix M. The name modal matrix comes from control theory where a dynamical system can be decomposed into dynamic modes of operation. For EM diffraction grating problems and also for EM problems which use k-space (spatial Fourier transform) techniques, the EM ®eld solutions associated with a state variable analysis can be decoupled into spatial mode solutions. These modes are analogous to the dynamical modes of operation encountered in control systems. If the eigenvalues are distinct, which is mainly the case under consideration in this text, the modal matrix is nonsingular and therefore its inverse exists. Letting M be the modal matrix, we may write MQ ˆ AM

…1:5:15†

where Q is a diagonal matrix holding the eigenvalues of qi on the diagonal. It can be shown that the inverse of M exists; hence, from Eq. 1.5.15 we obtain Q ˆ M 1A M

…1:5:16†

If Q is squared we have Q2 ˆ …M 1 A M†…M 1 A M† ˆ …M 1 A2 M†

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…1:5:17†

10

Chapter 1

and if further we pre- and post-multiply by M and M 1 , respectively, we have A2 ˆ M Q2 M

1

…1:5:18†

Similarly if A is raised to the pth power we have Ap ˆ M Qp M

1

…1:5:19†

where Qp is the diagonal matrix formed by raising each eigenvalue qi to the pth power. A matrix polynomial N…A† can be conveniently evaluated as N…A† ˆ M N…Q† M

1

…1:5:20†

where linear combinations of powers of A as given by Eq. 1.5.19 have been used. N…Q† is the diagonal matrix formed by placing in each diagonal matrix entry the polynomial N…qi †. Thus the modal matrix provides a convenient way to evaluate quickly and accurately the powers and polynomials of the matrix A. In this text we will be greatly concerned with calculating the exponential function of the matrix A. The exponential function of the matrix A, namely exp…A†, is de®ned as 1 1 exp…A† ˆ I ‡ A ‡ …A†2 ‡


‡ …A†k ‡


2 k!

…1:5:21†

which is the same in®nite series expansion as is used to de®ne the exponential function exp…a†. We now review two important aids that help in the solution and evaluation of an exponential matrix and in fact any function of the matrix A. These are called the Cayley±Hamilton theorem and the Cayley±Hamilton technique. These aids will be presented only for the cases of matrices with distinct eigenvalues. The ®rst theorem to be reviewed is the Cayley±Hamilton theorem. If we have a polynomial N…q† ˆ qn ‡ c1 qn 1 ‡


cn 1 q ‡ cn then using Eq. 1.5.20 we have 2

3 0 0 N…q1 † 6 0 7 N…q2 † 6 7 M N…A† ˆ M6 0 N…q3 † 7 4 5 .. .

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1

…1:5:22†

Introduction

11

where M is the modal matrix. If the polynomial N…q† is chosen to the characteristic equation, that is, N…q† ˆ P…q†, then N…qi † ˆ P…qi † ˆ 0, i ˆ 1; 2;


; n, and thus 2

0 60 P…A† ˆ M6 40

0 0 0

0 0 0

3 7 7M 5

1

ˆ0

…1:5:23†






We thus see that the matrix A satis®es its own characteristic equation. Another important aid in evaluating a function of a matrix, where the function is analytic over a given range of interest, is provided by the Cayley± Hamilton technique. We ®rst consider the case where the analytic function is a polynomial of higher degree than the characteristic polynomial P…q† of order n. Let the polynomial be N…q†. We consider the case where the roots (or eigenvalues) of P…q† are distinct. In this case, N…q† R…q† ˆ Q…q† ‡ P…q† P…q† where Q…q† is a polynomial and R…q† is a polynomial of order n Multiplying by P…q†, we have N…q† ˆ Q…q†P…q† ‡ R…q†

…1:5:24† 1 or less. …1:5:25†

If q is an eigenvalue or root of P…q†, then P…q† ˆ 0 and N…q† ˆ R…q†. If we substitute A for q in Eq. 1.5.25, we have N…A† ˆ Q…A†P…A† ‡ R…A†

…1:5:26†

Since by the Cayley±Hamilton theorem the matrix P…A† ˆ 0 we have N…A† ˆ R…A†

…1:5:27†

Thus a higher order polynomial matrix can be represented and evaluated using an n 1 polynomial expression. Consider next the case where the matrix function is a general analytic function over a region of interest, for example F…A† ˆ exp…A†. In this case F…q† can be expanded in an in®nite power series over the analytic region of interest. As in the case when F…q† was a polynomial, F…q† can be written as F…q† ˆ Q…q†P…q† ‡ R…q†

Copyright © 2000 Marcel Dekker, Inc.

…1:5:28†

12

Chapter 1

where R…q† is a polynomial of order n

1 given by

R…q† ˆ 0 ‡ 1 q ‡ 2 q2 ‡


‡ n 1 qn

1

…1:5:29†

Let q ˆ q1 ; q2 ; . . . ; qn be the distinct roots of P…qi † ˆ 0, i ˆ 1; . . . ; n. We have, after evaluating Eq. 1.5.28, F…q1 † ˆ R…q1 † F…q2 † ˆ R…q2 †



…1:5:30†


F…qn † ˆ R…qn † This de®nes a set of n  n linear equations from which the coef®cients i , i ˆ 1; . . . ; n can be determined. At this point we would like to show that the function Q…q† is analytic. To do this we write Q…q† as Q…q† ˆ

F…q† R…q† P…q†

…1:5:31†

In this expression we note that over the region of interest, the numerator and denominator of Eq. 1.5.31 have the same zeros. Since in Eq. 1.5.31 all functions F…q†, Q…q†, P…q†, and R…q† are analytic over the range where F…q† is, we may replace q by the matrix A. We have F…A† ˆ Q…A†P…A† ‡ R…A†

…1:5:32†

Since by the Cayley±Hamilton theorem P…A† ˆ 0 we have F…A† ˆ R…A†

…1:5:33†

Thus we have shown that the analytic matrix function F…A† can be evaluated by using a polynomial matrix expression of order n 1 as given by R…A† in Eq. 1.5.29.

Copyright © 2000 Marcel Dekker, Inc.

Introduction

13

PROBLEMS 1. Derive the wave equation for the electric and magnetic ®elds starting from Maxwell's equations in a homogeneous isotropic source free region. How does this change if the material is anisotropic? 2. Find from ®rst principles the Fourier series coef®cients for a periodic square wave s…x† of unit amplitude and 50% duty cycle. Now ®nd the Fourier series coef®cients of s2 …x† (a) from ®rst principles and (b) using the Laurent rule. Plot s2 …x† vs x by employing the Fourier series coef®cients you found using (b). Use 5, 10 and 100 Fourier coef®cients. Describe the general trend(s). 3. Find the two-dimensional Fourier transform of a rectangle (rect) function of unit height and width a in each dimension. 4. Show that the two-dimensional Fourier transform of a Gaussian function of width w is another Gaussian function. Functions like this are called self-Fourier transformable. Find its width in the spatial frequency domain. Can you think of any other functions that are self-Fourier transformable? 5. Find the DFT of a square wave function using a software of your choice. Comment on the nature of the spectrum numerically computed as the width of the square wave changes. 0 1 1 20 0 1 A using 6. Find sin A where A is a matrix given by @ 1 7 the Cayley Hamilton theorem [5]. 3 0 2 REFERENCES 1. L. Li, Use of Fourier series in the analysis of discontinuous periodic structures, J. Opt. Soc. Am. A, 15, 1808±1816 (1996). 2. P. P. Banerjee and T.-C. Poon, Principles of Applied Optics, Irwin, New York, 1991. 3. A. Antoniou, Digital Filters: Analysis and Design, McGraw-Hill, New York, 1979. 4. P. M. Deruso, R. J. Roy, and C. M. Close, State Variables for Engineers, John Wiley, New York, 1967. 5. L. A. Pipes and L. R. Harvill, Advanced Mathematics for Engineers and Scientists, McGraw-Hill, New York, 1970.

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2

Spectral State Variable Formulation for Planar Systems

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.

Copyright © 2000 Marcel Dekker, Inc.

16

Chapter 2

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 …@F=@x / jkx F, @F=@z / jkz F†, (2) writing out the six ®eld component equations (these equations will contain ®rst-order longitudinal derivative operator terms @F=@y†, (3) manipulating these equations so as to eliminate the longitudinal electric ®eld component Ey and the longitudinal magnetic ®eld component Hy

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Spectral State Variable Formulation

17

(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 Ex , Ez , Hx , and Hz 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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18

Chapter 2

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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Spectral State Variable Formulation

19

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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20

Chapter 2

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 ˆ ~ 0 j ~ 00 , ~ 2 ˆ ~ 0 j ~ 00 ) (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 ~ y; ~ z† ~ coordinate system are given by in an …x;

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Spectral State Variable Formulation

21

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

! ~ a^ x E ` ˆ Ex` …y† ! ~ a^ z H ` ˆ Hz` …y†

` ˆ 1; 2; 3

…2:2:1†

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

! j!~ ` H `

! ! r~  H ` ˆ j!~` E `

…2:2:2†

we ®nd substituting Eq. 2.2.1 that @Ex` ˆ j!~ ` Hz` @y~ @Hz` ˆ j!~` Ex` @y~

…2:2:3†

It is convenient to make the above equations dimensionless. We introduce p  ~ 0 ˆ 0 =0 ˆ 377 , the state variables Ex` ˆ Sx` , Hz` ˆ Uz` =0 , y ˆ k0 y, p 2 k0 ˆ ! 0 0 ˆ  , where ! ˆ 2f , f is the frequency, and  is the freespace wavelength, ` ˆ ~ ` =0 ˆ `0 j`00 , ` ˆ ~` =0 ˆ `0 j`00 , and after substitution we ®nd

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22

Chapter 2

@Sx` ˆ j ` Uz` @y

…2:2:4†

@Uz` ˆ j ` Sx` @y Letting 

Sx` Vˆ Uz`





0 Aˆ j`

j` 0

 …2:2:5†

(and dropping the ` subscript for the moment) we may write Eq. 2.2.5 in the general state variable from @V…y† ˆ AV…y† @y

…2:2:6†

Equation 2.2.6 can be solved by determining the eigenvalues and eigenvectors of the matrix A according to the equation AV ˆ qV

…2:2:7†

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

N X nˆ1

Cn Vn eqn y

…2:2:8†

where N ˆ 2, Vn and qn , n ˆ 1; 2, are eigenvectors and eigenvalues of the matrix A, and Cn are general constants. We may demonstrate that Vn eqn y is a solution of Eq. 2.2.8 by direct substitution. We have for n ˆ 1; 2, …d=dy†…Vn eqn y † ˆ Vn …d=dy†eqn y ˆ qn Vn eqn y . But qn Vn ˆ AVn , hence d …V eqn y † ˆ A…Vn eqn y † dy n

…2:2:9†

which is the original equation. Superposition of the distinct modes of Vn eqn y then gives the full EM solution. The eigenvalues of qn , n ˆ 1; 2; of A in Eq. 2.2.7 satisfy det‰A

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qIŠ ˆ … q†2

…j 2  † ˆ 0

Spectral State Variable Formulation

23

or q2 ‡   ˆ 0

…2:2:10†

Let   ‡ j  q1 ,  > 0, > 0 ( and are real numbers), be the forward traveling mode in the ` ˆ 1; 2; 3 regions. Substituting in Eq. 2.2.10 we have … ‡ j †2 ‡ ‰ 0

j 00 Š ‰ 0

j 00 Š ˆ 0

…2:2:11†

After performing algebra it is found that 

  1=2 1 0 r ‡ ‰r 02 ‡ r 002 Š1=2 2    1=2 1 0 02 002 1=2 r ‡ ‰r ‡ r Š ˆ 2

ˆ

…2:2:12†

where   ˆ ‰ 0  0  00  00 Š j‰ 00  0 ‡  0  00 Š ˆ r 0 jr 00 . Usually, r 00 > 0. We note that qn ˆ  ‡ j , n ˆ 1, corresponds to a forward traveling wave and that qn ˆ … ‡ j †, n ˆ 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 …n ˆ 1†, that for the exponential part of the EM wave, Ex / exp… y† ! 0 as y ! 1 when  > 0, and for the oscillary part of the wave Ex …y; t† cos… y ‡ !t†, which indicates a wave traveling to the right, since the phase velocity v' ˆ != < 0. A similar analysis in Region 1 shows that the second eigenvalue q2 ˆ … ‡ j † corresponds to a backward traveling wave. The eigenvector V1 ˆ ‰Sx1 ; Uz1 Št , V2 ˆ ‰Sx2 ; Uz2 Št can be determined from Eq. 2.2.7 after substitution of the eigenvalue qn , n ˆ 1; 2, into Eq. 2.2.10. For the forward traveling wave in any of the three regions we have q1 ˆ  ‡ j ˆ ,  0ˆ

q1 j

j q1





Sx1 Uz1

 …2:2:13†

Because q1 is an eigenvalue, the two equations of Eq. 2.2.13 are linearly dependent. We have q1 Sx1 ‡ jUz1 ˆ 0 or Uz1 ˆ … jq1 =†Sx1 . Letting Sx1 ˆ 1, the forward traveling eigenvector is V1 ˆ ‰1; … j =†ŠT , where T denotes the matrix transpose. Substituting the backward traveling wave with q2 ˆ … ‡ j † ˆ , the backward traveling eigenvector corresponding to q2 is V2 ˆ ‰1; …j =†ŠT .

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24

Chapter 2

The electric ®eld associated with the eigenmodes qn , n ˆ 1; 2, is given in Regions ` ˆ 1; 2; 3 as !…`† E n ˆ Sxn` eqn` y a^ x

…2:2:14a†

!…`† 1 H n ˆ Uzn` eqn` y a^ z 0 where Sxn` ˆ 1 Uz1` ˆ

n ˆ 1; 2

j ` `

Uz2` ˆ

j ` `

…22:14b†

` ˆ ` ‡ j ` 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 !…`† X !…`† E ˆ Cn` E n

…2:2:15a†

2 !…`† !…`† X Cn` H n H ˆ

…2:2:15b†

nˆ1

nˆ1

where Cn` 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), Hz ˆ

1 @Ex 1 ˆ j!~ @y~ 0



 j @Ex  @y

From Eq. 2.2.15a we note that Ex ˆ C1 exp… y† ‡ C2 exp… y†

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…2:2:16†

Spectral State Variable Formulation

25

Substituting this in Eq. 2.2.16, we have Hz ˆ

1 0



 j ‰C1 exp… y† 

C2 exp… y†Š

…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 P contained in a ! ! volume
V~ ! 0 by the electric and magnetic sources J i and M i should be balanced by the sum of (1) the time-averaged power Pf (meter 3 ) radiated ~ (2) the electric power PDE over the surface
S~ enclosing the volumes
V, 3 ~ and (3) and magnetic power PDM (meter ) dissipated over the volume
V, 2j! times the difference between the time-averaged magnetic energy W M ~ where stored in
V~ and the time-averaged electric energy W E stored in
V, ! ˆ 2f (radians) is the angular frequency and f is the frequency in Hertz. Mathematically the complex Poynting theorem for a general isotropic material is given by [3]

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26

Chapter 2

1 2

……

! ! 1 E  H
a^ n d S~ ‡ 2 S~

 ………   ! !t ! !t E
J ‡ H
M d V~ ˆ 0 ~ V

…2:2:18† !t where J is a general electric displacement, conduction and source current !t term and M represents the generalized magnetic current. Mathematically these currents are given by !t J ˆ j! ~ 0  !t M ˆ j! l~ 0


! !i j ~ 00 E ‡ J ! !i j l~ 00 H ‡ M

…2:2:19† …2:2:20†

!i !i where J and M 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, Ps ˆ Pf ‡ PDE ‡ PDM ‡ j… PWE ‡ PWM † where 1 2

Ps ˆ

 ………   ! !i ! !i E
J ‡ H
M d V~ ~ V

(source power) ‡‡ ! ! Pf ˆ 12 E  H ^
a^ n d S~ S~

PWE

(net outward power flow)  ………  ! h 0 !i ~ 1 ˆ 2!W E ˆ 2! 4 E
~ E d V

PWM

(proportional to stored electric energy)  ………  ! h 0 !i ~ ˆ 2!W M ˆ 2! 14 H
l~ H d V

~ V

~ V

(proportional to stored magnetic energy)

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…2:2:21†

Spectral State Variable Formulation

PDE ˆ

PDM

1 2

……… ~ V

27

! h 00 !i ~ E
~ E d V

(electric power dissipated) ……… ! h 00 !i ~ 1 ˆ2 H
l~ H d V

…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
S~ and are parallel to the interfaces of the slab. For this box we ®rst note that in the power ¯ow integral Pf , 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 x- and z-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. Pf ˆ

PIN ‡ POUT

…2:2:23†

where PIN ˆ

1 2

POUT ˆ

‡‡
S~

1 2

! ! E H

‡‡
S~

y~ ˆ~y‡

! ! E H

^
… a^ y † d S~

~ y~ yˆ

^
… a^ y † d S~

…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 y~ ‡ end cap is a^ y . Using Eqs. 2.2.21 and 2.2.23, we ®nd that the complex Poynting theorem for the present problem can be written as PIN ˆ

Ps ‡ POUT ‡ PDE ‡ PDM ‡ j… PWE ‡ PWM †

…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 x ˆ k0 x, 0 2 ~ theorem equations p by an amount of power PFS INC ˆ ‰
S=…20 †Š…E0 =1 † p (watts), where 0 ˆ ~ 0 =~0 ˆ 377 , 1 ˆ ~ 1 =~1 =0 (dimensionless), and 0 …E02 =1 † ˆ 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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28

Chapter 2

PIN ˆ PWE PWM

Ps ‡ POUT ‡ PDE ‡ PDM ‡ j… PWE ‡ PWM † …2:2:27† … … ! 0 ! ! 0 !  ˆ 10 2 E
 E dy ˆ E
 E dy (dimensionless) E0 `y `y … ! ! ˆ …0 H †
l 0 …0 H † dy (dimensionless) `y

… PDE ˆ

`y

… PDM ˆ POUT

PIN

`y

! 00 ! E
 E dy (dimensionless) ! ! …0 H †
l 00 …0 H † dy (dimensionless)

! i 1 h ! ˆ 0 2 E  …0 H †
… a^ y † E0 yˆy h! ! i ˆ E  …0 H †
… a^y † (dimensionless) h! ! i ˆ E  …0 H †

yˆy

yˆy‡


… a^ y † (dimensionless)

… h 1 ! ! ! ! i 1 k  E
J i ‡ H
M i dy 0 0 0 E02 `y … h ! ! ! ! i E
J i ‡ H
M i dy (dimensionless) ˆ k0 1 0

Ps ˆ

`y

where  ˆ  0 j  00 and l ˆ l 0 jl 00 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 y ˆ 1 is normally incident as a dielectric slab. In this case the C11 and C23 coef®cients

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Spectral State Variable Formulation

29

are known (see Eq. 2.2.15), with C11 ˆ E0 , where E0 is the incident amplitude (volts/m), and C23 ˆ 0 also, since there is no re¯ected wave from Region 3. As the coef®cient C21 represents the complex amplitude of the re¯ected ®eld in Region 1, we let C21 ˆ R, and since the coef®cient C13 represents the complex amplitude of the transmitted ®elds in Region 3, we let C13 ˆ T. Using these coef®cients, the ®elds in Regions 1, 2, and 3 are given by (see Fig. 1). Region 1 Ex…1† ˆ E0 exp… 1 y† ‡ R exp… 1 y† Hx…1† ˆ

j 1 ‰ E0 exp… 1 y† ‡ R exp… 1 y†Š  0 1

…2:2:28†

Region 2 Ex…2† ˆ C12 exp… 2 y† ‡ C22 exp… 2 y† Hx…2† ˆ

1 j 2 ‰ C12 exp… 2 y† ‡ C22 exp… 2 y†Š  0 2

…2:2:29†

Region 3 Ex…3† ˆ T exp… 3 …y ‡ L†† Hz…3† ˆ

1 j 3 ‰ T exp… 3 …y ‡ L††Š  0 3

…2:2:30†

The Ex…3† and Hz…3† ®elds have been written with a exp… 3 …y ‡ L†† in order to refer the phase of the T coef®cient to the y ˆ L boundary. The boundary conditions require that the tangential electric and magnetic ®elds match at y ˆ 0, L. Matching of the tangential electric and magnetic ®elds at y ˆ 0 and y ˆ L 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 E ˆ a11 C12 ‡ a12 C22 1 0 0 ˆ a21 C12 ‡ a22 C22

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…2:2:31†

30

Chapter 2

where

2 1

2 ‡ a12 ˆ 1  2 1  1 2    

2 3

exp… 2 L† ˆ ‡ a22 ˆ 2 ‡ 3 exp… 2 L† 2  3 2  3

a11 ˆ a21

…2:2:32† Also Rˆ

E0 ‡ C12 ‡ C22

T ˆ C12 exp… 2 L† ‡ C22 exp… 2 L†

…2:2:33†

Inversion of the 2  2 as given by Eqs. 2.2.31 then determines the unknown coef®cients C12 and C22 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., y~ ‡ ˆ y~ in ˆ 0:5, and has its right face at y~ ˆ y~ out , y~ out  0. For the present analysis there are no sources in the layer, so Ps ˆ 0. Substituting we ®nd that the complex Poynting theorem is given by PIN ˆ POUT ‡ PDE ‡ PDM ‡ j… PWE ‡ PWM †  PBOX where PDE ˆ PDE1 ‡ PDE2 ‡ PDE3 PDE1 ˆ PDE3 ˆ 0 …0 c12 exp… 2 y† ‡ c22 exp… 2 y† 2 dy PDE2 ˆ 200 y2

where  y2 ˆ

Copyright © 2000 Marcel Dekker, Inc.

yout

yout >

L

L

yout <

L

…2:2:34†

Spectral State Variable Formulation

31

PDM ˆ PDM1 ‡ PDM2 ‡ PDM3 PDM1 ˆ PDM3 ˆ 0 2 … 0 c exp… 2 y† PDM2 ˆ 200 2 2 y2 12

2 c22 exp… 2 y† dy

PWE ˆ PWE1 ‡ PWE2 ‡ PWE3 … yin E0 exp… 1 y† ‡ R exp… 1 y† 2 dy PWE1 ˆ 10 0

PWE2 ˆ 20 PWE3 ˆ 30

…0 y2

…

c12 exp… 2 y† ‡ c22 exp… 2 y† 2 dy

L

y3

T exp… 3 …y ‡ L† 2 dy

where (

yout >

L

y3 ˆ

yout

yout <

L L

PWM ˆ PWM1 ‡ PWM2 ‡ PWM3 2 … yin E0 exp… 1 y† PWM1 ˆ 10 1 1 0

PWM2 ˆ PWM3 PIN

POUT



2 … 0 c12 exp… 2 y†

20 2 2

y2

2 c22 exp… 2 y† dy

2 … L T exp… 3 …y ‡ L†† 2 dy ˆ 30 3 3 y3  

ˆ j 1 ‰E0 exp… 1 yin † ‡ R exp… 1 yin †Š‰E0 exp… 1 yin † 1 R exp… 1 yin †Š  

ˆ j 2 ‰c12 exp… 2 yout † ‡ c22 exp… 2 yout †Š‰c12 exp… 2 yout † 2 c22 exp… 2 yout †Š

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2 R exp… 1 y† dy

32

when

Chapter 2

yout >

L 

POUT ˆ j

3 3



T exp… 2 … yout ‡ L†† 2

when yout < L. In these equations R is the re¯ection coef®cient in Region 1, T is the transmission coef®cient in Region 3, and c12 and c22 are wave coef®cients in Region 2. The expressions for PWE3 and PWM3 have been chosen so that when yout > L (that is, yout is in Region 2) the lower limit y3 equals the upper limit and PWE3 and PWM3 are zero as they should be. The conservation theorem as given by Eq. 2.2.34 states (1) that the sum of Re…POUT † and PD ˆ PDE ‡ PDM …PD is real and nonnegative), which by de®nition equals Re…PBOX †, should equal Re…PIN † and (2) that the sum of Im…POUT † and the energy±power difference PWE ‡ PWM , which by de®nition equals Im…PBOX †, should equal the sum of Im…PIN †. As a numerical example for the normal incidence case, we assume that the layer thickness is L~ ˆ 0:6, that free space bounds the layer in Regions 1 and 3, and that the slab has a lossy permittivity given by 2 ˆ 3 j0:4 and relative permeability 2 ˆ 2:5 j0:2. Figs. 2, 3, and 4 show plots of the EM ®elds and different power terms associated with the present example. Figure 2 shows the Ex electric ®eld (magnitude, real and imaginary parts) plotted vs. the distance y~ ˆ y~ from the incident side interface. In observing the real and imaginary plots of Ex , one notices that the standing wave wavelength of Ex is greatly shortened in Region 2 as opposed to Region 1. This is due to the greater magnitude of the material constants j2 j ˆ j3 j0:4j and j2 j ˆ j2:5 j0:2j in Region 2 as opposed to Region 1. In observing the plots of Fig. 2 one also notices that the continuity of the Ex 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 ˆ jExMAX j=ExMIN j ˆ jE0 ‡ Rj= jE0 Rj  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 jEx j 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 PIN and PBOX are made as a function of the distance y~ out , 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 PIN (solid line) and PBOX (cross) agree very closely. One also observes that as the distance y~ OUT increases, the power dissipated PD increases, the Re…POUT † decreases, and both change so as to

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Spectral State Variable Formulation

33

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

leave the sum constant and equal to Re…PIN †. Also plotted in Fig. 3 is the Im…POUT † and the energy difference term PWE ‡ PWM . One observes from these plots that the Im…POUT † and PWE ‡ PWM 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…POUT † and PWE ‡ PWM is a constant equal to Im…PIN †. Thus the imaginary part of the power is exchanged periodically between Im…POUT † and PWE ‡ PWM so as to keep the Im…PIN † 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 y~OUT . As can be seen from Fig. 4, the electric and magnetic stored energy terms PWE and PWM 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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34

Chapter 2

Figure 3 Plots of the real and imaginary parts of PIN and PBOX as a function of the distance y~ OUT .

located at y ˆ L and that a plane wave from y ˆ 1 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 y~ ˆ 0 are the same as in the ®rst example. Thus 2 1 E ˆ a11 C12 ‡ a12 C22 1 0

…2:2:35†

where a11 and a12 have been de®ned previously. At y~ ˆ L~ the tangential component of the electric ®eld must vanish due to the presence of the metal. This leads to the equation 0 ˆ C12 exp… 2 L† ‡ C22 exp… 2 L†

…2:2:36†

From these equations C12 and C22 can be determined as well as all other coef®cients in the system. Figure 5 shows the Re…Ex †, Im…Ex †, and jEx j plotted versus the distance y~ ˆ y~ from the Region 1±2 interface, using the material parameter

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Spectral State Variable Formulation

35

Figure 4 Plots of the electric energy term, magnetic energy term, power stored, and power dissipated in the Poynting box, vs. the distance y~ 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 SWR pattern displayed by the jEx j 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 PIN (solid line) and POUT (cross) agree with each other very closely. We also notice from Fig. 6 that a higher oscillation of PWE ‡ PWM and Im…POUT † 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 ˆ 7

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36

Chapter 2

Figure 5

Plots of the Re…Ex †, Im…Ex †, and jEx j plotted versus the distance y~ .

j3:5 and 2 ˆ 2:5 j0:2. In this ®gure, the normalized re¯ected power is ~ As can be seen from Fig. 7, at a slab plotted versus the slab length L. ~ thickness of L ˆ 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), L~ ˆ 0:066 ˆ 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 J S ˆ Jsx a^ x ˆ J a^ x (Amp/m) is located in the interior of an isotropic two-layered medium.

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Spectral State Variable Formulation

37

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 Region 1 0 0

Ex…1 † ˆ C11 0 exp… 1 0 …y

L‡ †† ‡ C21 0 exp… 1 0 …y

L‡ ††

C11 0 ˆ 0 …2:2:37a†

0

0

Uz…1 † ˆ 0 Hz…1 † ˆ

Copyright © 2000 Marcel Dekker, Inc.

j 1 0 C 0 exp… 1 0 …y 1 0 21

L‡ †

…2:2:37b†

38

Chapter 2

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 ˆ 7 j3:5 and 2 ˆ 2:5 j0:2.

Region 1 Ex…1† ˆ C11 exp… 1 y† ‡ C21 exp… 1 y† Uz…1† ˆ 0 Hz…1† ˆ

j 1 ‰C11 exp… 1 y† 1

…2:2:38a† C21 exp… 1 y†Š

…2:2:38b†

Region 2 Ex…2† ˆ C12 exp… 2 y† ‡ C22 exp… 2 y† Uz…2† ˆ 0 Hz…2† ˆ

j 2 ‰C12 exp… 2 y† 2

…2:2:39a† C12 exp… 2 y†Š

…2:2:39b†

Region 3 Ex…3† ˆ C13 exp… 3 …y ‡ L †† Uz…3† ˆ 0 Hz…3† ˆ

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j 3 C exp… 3 …y ‡ L †† 3 13

…2:2:40a† …2:2:40b†

Spectral State Variable Formulation

Figure 8

39

Plots of the Re…Ex †, Im…Ex †, and jEx j plotted versus the distance y~ .

The total layer thickness is L ˆ L‡ ‡ L , where L‡  0 and L  0. Matching the tangential electric and magnetic ®elds at the Region 1±1 0 interface and eliminating the C21 0 coef®cient, it is found that C11 ˆ C21

ˆ

1 0 =1 0 ‡ 1 =1 exp… 2 1 L‡ †

1 0 =1 0 ‡ 1 =1

…2:2:41†

Matching the tangential electric and magnetic ®elds at the Region 2±3 interface and eliminating the C13 coef®cient it is found that C12 ˆ C22

ˆ

3 =3 ‡ 2 =2 exp…2 2 L †

3 =3 ‡ 2 =2

…2:2:42†

To proceed further we match EM boundary conditions at the Region 1±2 boundary y ˆ 0. These boundary conditions are given by 0 Hz…1† Ex…1†

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0 Hz…2† ˆ Uz…1† Ex…2† ˆ 0

Uz…2† ˆ 0 J

…2:2:43a† …2:2:43b†

40

Chapter 2

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 y ˆ 0. Performing algebra it is found that the following equations result, from which the unknown coef®cients of the system can be found. j 1 ‰ 1

1ŠC21 ‡

j 2 ‰ 2

‰ ‡ 1ŠC21

1ŠC22 ˆ 0 J

…2:2:44a†

‰ ‡ 1ŠC22 ˆ 0

…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 ˆ 2 j0:3, 1 ˆ 3 j0:5, 2 ˆ 3 j0:4, 2 ˆ 2:5 j0:2, L~ ˆ 0:4, L~ ‡ ˆ 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 y~ OUT‡ ˆ 0:25 to the left of the Region 1±2 interface (the source is located at the Region 1±2 interface at y~ ˆ 0), and its rightmost face is located at y~ ˆ y~ OUT ; y~OUT  0 from the Region 1±2 interface. (See Fig. 9 inset). For the present source problem, the complex Poynting theorem is given by PS ˆ POUT‡ ‡ POUT ‡ PDE ‡ PDM ‡ j… PWE ‡ PWM †  PBOX …2:2:45† where PS ˆ

! ! 0 E
J s

~ yˆ0

ˆ

0 Ex Js yˆ0 ~

…2:2:46†

ˆ …C11 ‡ C21 † is continuous at y~ ˆ 0. From Eq. The electric ®eld Ex jyˆ0 ~ 2.2.43a, 0 J ˆ

j 1 …C11 1

Ps ˆ

 j 1 …C11 ‡ C21 † …C11 1

C21 † ‡

j 2 …C 2 12

C22 †

…2:2:47†

Thus

Copyright © 2000 Marcel Dekker, Inc.

j C21 † ‡ 2 …C12 2

 C22 †

…2:2:48†

Spectral State Variable Formulation

41

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

The terms POUT ‡ and POUT are given by 

POUT ‡ ˆ 

j 1 1





C11 exp… 1 yOUT ‡ † ‡ C21 exp… 1 yOUT‡ †

C11 exp… 1 yOUT ‡ †

C22 exp… 1 yOUT‡ †



 …2:2:44a†



2 2

POUT ˆ j



‰C12 exp… 2 yOUT † ‡ C22 exp… 2 yOUT †Š

‰C12 exp… 2 yOUT † when

yOUT >

when

yOUT <

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C22 exp… 2 yOUT †Š

L 

POUT ˆ j

…2:2:44b†



3 3



T exp… 3 … yOUT ‡ L †† 2

…2:2:45†

L . The other terms in Eqs. 2.2.45 are given in Eq. 2.2.34.

42

Chapter 2

Figure 8 shows the Re…Ex †, Im…Ex †, and jEx j electric ®elds plotted versus the distance y 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…Ex †, Im…Ex †, and jEx j 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 y~ OUT . As can be seen from Fig. 9 the real and imaginary parts of PS  PSOURCE (cross) and PBOX (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 y~ OUT increases, the power dissipated PD increases, Re…POUT † decreases, and both change so as to leave the sum constant and equal to Re…PS †. Also plotted in Fig. 9 is the Im…POUT † and the energy±power difference PWE ‡ PWM . One observes from these plots that the Im…POUT † and the energy±power difference PWE ‡ PWM 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…POUT † and PWE ‡ PWM is a constant equal to IM…PS †. Thus the imaginary part of the power is exchanged periodically between Im…POUT † and PWE ‡ PWM so as to keep the Im…PS † 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 I . 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.

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Spectral State Variable Formulation

43

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 xx , xy , yx , yy , and zz 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  ˆ  0 j 00 . 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 Ex…1† ˆ Sx…1† …y† exp… jkx x† ˆ

ky1  E exp…jky1 y† 1 0

 R exp… jky1 y† exp… jkx x†

ˆ ‰Cx11 exp… 11 y† ‡ Cx21 exp… 21 y†Š exp… jkx x†

…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. Uz ˆ 0 Hz .

Copyright © 2000 Marcel Dekker, Inc.

44

Chapter 2

Ey…1† ˆ Sy…1† …y† exp… jkx x†  kx  E exp…jky1 y† ‡ R exp… jky1 y† exp… jkx x† 1 0   ˆ Cy11 exp… 11 y† ‡ Cy21 exp… 21 y† exp… jkx x†

ˆ

…2:3:2†

0 Hz…1† ˆ Uz…1† …y† exp… jkx x†   ˆ E0 exp…jky1 y† ‡ R exp… jky1 y† exp… jkx x† ˆ ‰Cz11 exp… 11 y† ‡ Cz21 exp… 21 y†Š exp… jkx x†

…2:3:3†

p p ~ y ˆ k0 y, ~ z ˆ k0 z, ~ k0 ˆ 2=, kx ˆ 1 sin…I †, ky1 ˆ 1 k2x , where x ˆ k0 x, and 0 ˆ 377 ; E0 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 Ex…3† ˆ Sx…3† …y† exp… jkx x† ˆ

 ky3  T exp…jky3 …y ‡ L†† exp… jkx x† 3

ˆ ‰Cx13 exp… 13 y† ‡ Cx23 exp… 23 y†Š exp… jkx x†

…2:3:4†

 kx  T exp…jky3 …y ‡ L†† exp… jkx x† 3   ˆ Cy13 exp… 13 y† ‡ Cy23 exp… 23 y† exp… jkx x† …2:3:5†   …3† ˆ Uz …y† exp… jkx x† ˆ T exp…jky3 …y ‡ L†† exp… jkx x†

Ey…3† ˆ Sy…3† …y† exp… jkx x† ˆ

0 Hz…3†

ˆ ‰Cz13 exp… 13 y† ‡ Cz23 exp… 23 y†Š exp… jkx x†

…2:3:6†

p where ky3 ˆ 3 k2x , T 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 ! r E ˆ

! jl…0 H †

! ! ! r  …0 H † ˆ j D ˆ j  … E †

…2:3:7†

where we assume that l is a diagonal matrix with xx ˆ yy ˆ zz . The x ! ! component of D ˆ  E is given by Dx ˆ xx Ex ‡ xy Ey ‡ xz Ez . The Dy and Dz 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 x, it is necessary that the EM ®elds of Region 2 all be proportional to exp… jkx x†. (This factor follows

Copyright © 2000 Marcel Dekker, Inc.

Spectral State Variable Formulation

45

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 ! E ˆ …Sx …y†a^ x ‡ Sy …y†a^ y ‡ Sz …y†a^z † exp… jkx x† ! 0 H ˆ …Ux …y†a^ x ‡ Uy …y†a^y ‡ Uz …y†a^z † exp… jkx x†

…2:3:8†

Using the fact that the only nonzero EM ®eld components in Region 1 are Ex , Ey , and Hz , a small amount of analysis shows that in Eqs. 2.3.7 a complete ®eld solution can be found taking only Sx , Sy , and Uz to be nonzero with Sz ˆ Ux ˆ Uy ˆ 0. Substituting Eqs. 2.3.8 in Eq. 2.3.7 and taking appropriate derivatives with respect to x, the following equations result: jkx Sy

@Sx ˆ @y

jzz Uz

…2:3:9†

@Uz ˆ jxx Sx ‡ jxy Sy @y

…2:3:10†

jkx Uz ˆ jyx Sx ‡ jyy Sy

…2:3:11†

To proceed further it is possible to eliminate the longitudinal electric ®eld component and express the equations in terms of the Sx and Uz components alone. Although other components could be eliminated, the Sy 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 Sy component is given by (from Eq. 2.3.11) Sy ˆ

yx k Sx ‡ x Uz yy yy

…2:3:12†

Substituting Eq. 2.3.12 into Eqs. 2.3.9, 2.3.10, " #   yx @Sx k2x ˆ j kx S ‡ j zz U @y yy x yy z     xy yx xy @Uz ˆ j xx Sx ‡ j kx U @y yy yy z

Copyright © 2000 Marcel Dekker, Inc.

…2:3:13† …2:3:14†

46

Chapter 2

The above equations are in state variable form and can be rewritten as @V ˆ AV @y

…2:3:15†

where a11 a21

" #   yx k2x a12 ˆ j zz ˆ j kx yy yy     xy yx xy ˆ j xx a22 ˆ j kx yy yy

…2:3:16† …2:3:17†

where V ˆ ‰Sx ; Uz Št . The basic solution method is to ®nd the eigenvalues and eigenvectors of the state variable matrix A, 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 V ˆ Vn exp…qn y†

…2:3:18†

where qn and Vn ˆ ‰Sxn ; Uzn Št are eigenvalues and eigenvectors of A and satisfy AVn ˆ qn Vn

n ˆ 1; 2

…2:3:19†

Because A is only a 2  2 matrix, it is possible to ®nd the eigenvalues and eigenvectors of the system in closed form. The quantities qn and Vn are given by "

a11

qn

a21

#"

a12 a22

qn

Sxn Uzn

# ˆ0

…2:3:20†

For this to have nontrivial solutions,  det

a11 qn a21

a12

a22

 qn

ˆ …a11

qn †…a22

qn †

a12 a21 ˆ 0 …2:3:21†

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Spectral State Variable Formulation

47

Using the quadratic equation to solve for qn we ®nd  qn ˆ 0:5…a11 ‡ a22 †  0:5 a211

2a11 a22 ‡ 4a12 a21 ‡ a222

1=2

n ˆ 1; 2 …2:3:22†

Letting Sxn ˆ 1, n ˆ 1; 2, it is found that the eigenvectors are given by  q Vn ˆ 1; n

a11

t

…2:3:23†

a12

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

yx k Sxn ‡ x Uzn yy yy

n ˆ 1; 2

…2:3:24†

Using these eigenvalues and eigenvectors it is found that the EM ®elds in Region 2 are given by Ex…2† ˆ Sx…2† …y† exp… jkx x† ˆ ‰C1 Sx1 exp…q1 y† ‡ C2 Sx2 exp…q2 y†Š exp… jkx x†  ‰Cx12 exp… 12 y† ‡ Cx22 exp… 22 y†Š exp… jkx x† Ey…2† ˆ Sy…2† …y† exp… jkx x†   ˆ C1 Sy1 exp…q1 y† ‡ C2 Sy2 exp…q2 y† exp… jkx x†    Cy12 exp… 12 y† ‡ Cy22 exp… 22 y† exp… jkx x†

…2:3:25†

…2:3:26†

0 Hz…2† ˆ Uz…2† …y† exp… jkx x† ˆ ‰C1 Uz1 exp…q1 y† ‡ C2 Uz2 exp…q2 y†Š exp… jkx x†  ‰Cz12 exp… 12 y† ‡ Cz22 exp… 22 y†Š exp… jkx x†

…2:3:27†

In these equations C1 and C2 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 R, T, C1 , and C2 . In the present problem the boundary conditions require that the tangential electric ®eld (the Ex ®eld) and the tangential magnetic ®eld (Hz ) 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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48

Chapter 2

ky1 ‰E 1 0

RŠ ˆ C1 Sx1 ‡ C2 Sx2

…2:3:28†

E0 ‡ R ˆ C1 Uz1 ‡ C2 Uz2

…2:3:29†

ky3 T ˆ C1 Sx1 exp… q1 L† ‡ C2 Sx2 exp… q2 L† 3

…2:3:30†

T ˆ C1 Uz1 exp… q1 L† ‡ C2 Uz2 exp… q2 L†

…2:3:31†

By substituting R and T 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     2ky1 ky1 ky1 E0 ˆ Uz1 ‡ Sx1 C1 ‡ Uz2 ‡ Sx2 C2 1 1 1  ky3 0 ˆ exp… q1 L† U ‡ Sx1 C1 ‡ exp… q2 L† 3 z1   ky3 Uz2 ‡ Sx2 C2  3

…2:3:32†



…2:3:33†

The C1 and C2 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 L~ ˆ 0:6 and material parameters xx ˆ yy ˆ 2:25 j0:3, yx ˆ 0:75 j0:1. We assume the permeability to be isotropic but lossy with zz ˆ 2 ˆ 2:5 j0:2. The incident plane wave (incident amplitude E0 ˆ 1 (V/m), electric ®eld polarization in the plane of incidence) is assume to have an angle of incidence I ˆ 25 . Figure 11 shows plots of the magnitudes of the Ex , Ey , and Uz ˆ 0 Hz EM ®elds in Regions 1±3 as a function of y ˆ y, 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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Spectral State Variable Formulation

49

Figure 11 Plots of the magnitudes of the Ex , Ey , and Uz ˆ 0 Hz electromagnetic ®elds in Regions 1±3 as a function of y ˆ y, 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 (Ex ) and tangential magnetic ®eld …Uz ˆ 0 Hz ) are continuous, and that the normal electric ®eld (Ey ) 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 y~ out (units of ) into Region 2 when y~ out  0:6 and into 3. In this ®gure „Pdexx , Pdexy , Regions 2 and 3 when y~ out > 0:6 into Region „ 00  00  Sx dy, Pdexy ˆ Sx xy Sy dy, etc. are given by the integrals Pdexx ˆ Sx xx etc. and PDE ˆ Pdexx ‡ Pdexy ‡ Pdeyx ‡ Pdeyy . Also PDM ˆ Pdmzz ˆ

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50

Chapter 2

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.

„

Uz zz00 Uz dy. As can be seen from Fig. 12, the dissipated electric and magnetic powers PDE and PDM are zero at y~ out ˆ 0 and increase in a monotonic fashion until y~ out ˆ 0:6 where they become constant for y~ out > 0:6. This is exactly to be expected since the only loss in the system is in Region 2 where 0  y~ out  0:6. We note also that the integrals Pdexy and Pdeyx are complex and satisfy Pdexy ˆ P deyx as expected. Thus Pdexy ‡ Pdeyx ˆ 2Re…Pdexy †. The integrals Pdexx and Pdeyy are purely real, and thus the electric dissipation integral PDE 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 Pdexy ‡ Pdeyx ˆ 2Re…Pdexy † 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 y~ out

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Spectral State Variable Formulation

51

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 y~ out  0:6 and into Regions 2 and 3 when y~ out > 0:6 into Region 3. In this ®gure P„ywe xx , Pywe xy , etc. are given by the integrals „ 0 0 Sx dy, Pwexy ˆ Sx xy S„y dy, etc. and PWE ˆ Pwexx ‡ Pwexy Pwexx ˆ Sx xx ‡ Pweyx ‡ Pweyy . Also PWM ˆ Pwmzz ˆ Uz zz0 Uz dy: As can be seen from Fig. 13, the stored electric and magnetic energy±powers PWE are nonzero at y~ out ˆ 0 and increase in a monotonic fashion thereafter. As in the case of the dissipation power integrals, we note that the integrals Pwexy and Pweyx are complex and satisfy Pwexy ˆ Pweyx . Thus Pwexy ‡ Pweyx ˆ 2Re…Pwexy †. The integrals Pwexx and Pweyy are purely real, so the electric energy±power integral PWE 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 Pwexy ‡ Pweyx ˆ 2Re…Pwexy † 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 PBOX ˆ POUT ‡ PDE ‡ PDM ‡ j… PWE ‡ PWM † and compare PIN and PBOX . As can be seen from Fig. 14, the real and imaginary parts of PIN (cross) and PBOX

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52

Chapter 2

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…POUT †, which decrease as y~ out increases, and PD ˆ PDE ‡ PDM (PD is purely real), which increase as y~ out increases. As can be seen from Fig. 14, the sum of these two quantities, namely Re…POUT † ‡ PD adds to Re…PIN †, which is constant as y~out increases. It makes sense that the Re…POUT † decreases as y~ out increases, due to increased power loss as y~out increases. Figure 14 shows plots of Im…POUT † and the energy difference term PWE ‡ PWM . 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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Spectral State Variable Formulation

2.4 2.4.1

53

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 I . 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 xx , xy , yx , yy , and zz 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  ˆ  0 j 00 . 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.

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54

Chapter 2

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 D and the magnetic ¯ux ! ! density vector B can be expressed in terms of the electric ®eld E and the magnetic ®eld H through the relations ! ! ! D ˆ ~
E ‡ m~
H

…2:4:1†

! ! ! B ˆ ~
E ‡ l~
H

…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 D and B of Eqs. 2.4.1 and 2.4.2 into Maxwell's equations, introducing the dimensionless dyadics

 ˆ ~ =0 ˆ  0 a ˆ a0

ja 00 ˆ

j  00

~ 0 ˆ l0 l ˆ l=

p ~ k0  0 0  ˆ  !

b ˆ b0

jl 00 jb 00 ˆ

p ~ k0 ~ 0 0 m ˆ m !

~ etc., we ®nd that and introducing normalized coordinates x ˆ k0 x, Maxwell's curl equations become h ! !i j a E ‡ lH h ! ! !i r  H ˆ j  E ‡ bH ! r E ˆ

…2:4:3† …2:4:4†

~ is the normalized curl operator. To proceed further where r ˆ …1=k0 †r we let all EM ®eld components in the material layer be proportional to the factor exp… j †, where  k~x x~ ‡ k~z z~ ˆ kx x ‡ kz 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

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Spectral State Variable Formulation

55

! ! ! j exp…j †r  … S exp… j †† ˆ l U ‡ a S

…2:4:5†

! ! ! j exp…j †r  … U exp… j †† ˆ  S ‡ b U

…2:4:6†

where the electric and magnetic ®elds are given by ! ! E ˆ S …y† exp… j †

…2:4:7†

! ! 0 H ˆ U …y† exp… j †

…2:4:8†

p where 0 ˆ 0 =0 ˆ 377 : If we carry out the differentiations as indicated by Eqs. 2.4.5 and 2.4.6, ! ! noting that S and U depend only on y, we ®nd after canceling the exponential factors that  @S a^ x j z @y  a^ x

 kz Sy ‡ a^ y ‰kz Sx

 kx Sz Š ‡ a^ z kx Sy



j

@Uz ‡ kz Uy ‡ a^ y ‰ kz Ux ‡ kx Uz Š ‡ a^ z @y ! ! ˆ  S ‡ bU

j

 ! ! @Sx ˆ lU ‡ a S @y

 kx Uy ‡ j

@Ux @y

…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 Sy and Uy can be eliminated, and equations for only the tangential components Sx , Sz , Ux and Uz 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 Sy and Uy components can be eliminated from Eqs. 2.4.9 and 2.4.10 in the following way. We equate the y components of Eqs. 2.4.9 and 2.4.10 and after transposing terms ®nd that ayy Sy ‡ yy Uy ˆ …kz

ayx †Sx ‡ … kx

ayz †Sz

yx Ux

yz Uz …2:4:10†

yy Sy ‡ byy Uy ˆ

yx Sx

yz Sz ‡ … kz

byx †Ux ‡ …kx

byz †Uz …2:4:11†

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56

Chapter 2

We can recast Eqs. 2.4.10 and 11 in the following matrix form: 2 " T22

Sy

#

Uy

Sx

3

7 6 6 Sz 7 7 6 ˆ R24 6 7 6 Ux 7 5 4

…2:4:12†

Uz so that after inverting T (we assume det…T† 6ˆ 0† we obtain 2 "

Sy

#

Sx

3

2

7 " 6 6 Sz 7 w11 7 6  T R6 7 6 Ux 7 w21 5 4 1

Uy

w12

w13

w22

w23

Uz

Sx

3

7 #6 7 w14 6 6 Sz 7 7 6 7 w24 6 4 Ux 5 Uz

…2:4:13†

Our next step is to substitute Sy and Uy as given by Eq. 2.4.13 into the x and z components of Eqs. 2.4.9 and 2.4.10. Doing so thus eliminates all longitudinal Sy and Uy terms from the equations. After performing considerable algebra it is found that the Sx , Sz , Ux , and Uz components can be placed in the following state variable form: 2

A11

6 @V 6 6 A21 ˆ6 @y 6 A31 4 A41

A12

A13

A22

A23

A32

A33

A42

A43

A14

3

7 A24 7 7 7V ˆ AV A34 7 5 A44

where  A11 ˆ j zy w21 ‡ azx ‡ …azy kx †w11  A12 ˆ j zy w22 ‡ …azy kx †w12 ‡ azz  A13 ˆ j zx ‡ zy w23 ‡ …azy kx †w13  A14 ˆ j zy w24 ‡ zz ‡ …azy kx †w14

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…2:4:14†

Spectral State Variable Formulation

A21 ˆ A22 ˆ A23 ˆ A24 ˆ

A31 ˆ A32 ˆ A33 ˆ A34 ˆ

57

 j xy w21 ‡ axx ‡ …axy ‡ kz †w11  j xy w22 ‡ …axy ‡ kz †w12 ‡ axz  j xx ‡ xy w23 ‡ …axy ‡ kz †w13  j xy w24 ‡ uxz ‡ …axy ‡ kz †w14

 j zx ‡ zy w11 ‡ …bzy ‡ kx †w21  j zy w12 ‡ zz ‡ …bzy ‡ kx †w22  j zy w13 ‡ bzx ‡ …bzy ‡ kx †w23  j zy w14 ‡ …bzy ‡ kx †w24 ‡ bzz

 A41 ˆ j xx ‡ xy w11 ‡ …bxy kz †w21  A42 ˆ j xy w12 ‡ xz ‡ …bxy kz †w22  A43 ˆ j xy w13 ‡ bxx ‡ …bxy kz †w23  A44 ˆ j xy w14 ‡ …bxy kz †w24 ‡ bxz

…2:4:15†

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

Vˆ

4 X nˆ1

Cn Vn exp…qn y†

3 Sxn 6 Szn 7 7 Vn ˆ 6 4 Uxn 5 Uzn

…2:4:16†

2

The EM ®elds in Region 2 are given by

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…2:4:17†

58

Chapter 2 4 ! X ! E ˆ Cn E n

…2:4:18†

4 ! ! X Cn H n H ˆ

…2:4:19†

nˆ1

nˆ1

where   ! E n ˆ Sxn a^ x ‡ Syn a^ y ‡ Szn a^ z exp…qn y

j †

 ! 1 Hn ˆ Uxn a^ x ‡ Uyn a^ y ‡ Uzn a^ z exp…qn y 0

…2:4:20† j †

n ˆ 1; 2; 3; 4 …2:4:21†

and 

Syn Uyn

 ˆ w24 Vn

…2:4:22†

Matching of the boundary conditions at the interfaces determines the ! ! ®nal Cn coef®cients and thus E and H .

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 ! ! E I ˆ S I exp… j

I†

! !! 0 H I ˆ UI HI exp… j I

! ˆ kI
! r ˆ kx x

…2:4:23† I†

ky1 y ‡ kz z

…2:4:24†

where ! k I ˆ kx a^ x

Copyright © 2000 Marcel Dekker, Inc.

ky1 a^ y ‡ kz a^ z

! r ˆ xa^ x ‡ ya^ y ‡ za^ z

…2:4:25†

Spectral State Variable Formulation

59

It is assumed for simplicity in this analysis that  ky1 ˆ 1 1

k2x

k2z

1=2

6ˆ 0

…2:4:26†

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

SyI ˆ

kx k S ‡ z S ˆ SxI ‡ SzI ky1 xI ky1 zI

…2:4:27†

UxI ˆ

1  ky1 SzI 1

…2:4:28†

kz SyI



1 ‰k S kx SzI Š 1 z xI  1  UzI ˆ k S ‡ ky1 SxI 1 x yI

UyI ˆ

…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 ! ! E R ˆ S R exp… j R

R †;

! ! 0 H R ˆ U R exp… j

! ˆ kR
! r ˆ kx x ‡ ky1 y ‡ kz z

R†

…2:4:31† …2:4:32†

where ! k R ˆ kx a^ x ‡ ky1 a^ y ‡ kz a^z

…2:4:33†

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

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60

Chapter 2

SyR ˆ

kx S ky1 xR

UxR ˆ

1  k S 1 y1 zR

1 ‰k S 1 z xR 1  ˆ k S 1 x yR

kz S ˆ ky1 zR kz SyR

SxR

SzR

…2:4:34†



UyR ˆ

kx SzR Š

UzR

ky1 SxR

…2:4:35† …2:4:36†



…2:4:37†

In Region 3 the EM ®elds are given by ! ! E T ˆ S T exp… j T

T†

! ! 0 H T ˆ U T exp… j

! ˆ k T
…! r ‡ La^ y † ˆ kx x

T†

ky3 …y ‡ L† ‡ kz z

…2:4:38† …2:4:39†

where ! k T ˆ kx a^ x ky3 a^y ‡ kz a^ z  1=2 6ˆ 0 ky3 ˆ 3 3 k2x k2z

…2:4:40† …2:4:41†

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

kx k S ‡ z S ˆ  0 SxT ‡ 0 SzT ky3 xT ky3 zT

…2:4:42†

UxT ˆ

1  ky3 SzT 3

…2:4:43†

kz SyT

1 ‰k S kx SzT Š 3 z xT  1  ˆ kx SyT ‡ ky3 SxT 3



UyT ˆ

…2:4:44†

UzT

…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

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Spectral State Variable Formulation

61

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 4 X ‰ExI ‡ ExR Š yˆ0‡ ˆ Cn Exn nˆ1 yˆ0 4 X ‰EzI ‡ EzR Š yˆ0‡ ˆ Cn Ezn nˆ1 yˆ0 4 X ‰HxI ‡ HxR Š yˆ0‡ ˆ Cn Hxn nˆ1 yˆ0 4 X ‰HzI ‡ HzR Š yˆ0‡ ˆ Cn Hzn nˆ1

…2:4:46†

yˆ0

P P P We letPSAx ˆ 4nˆ1 Cn Sxn , SAz ˆ 4nˆ1 Cn Szn , UAx ˆ 4nˆ1 Cn Uxn , and UAz ˆ 4nˆ1 Cn Uzn , evaluate the equations at y ˆ 0‡ and y ˆ 0 , cancel the exp… j…kx x ‡ kz z†† factor and express the unknowns of Eqs. 2.4.46, SxR and SzR , in terms of SAx , SAz , UAx , and UAz according to the relations SxR ˆ

SxI ‡ SAx

SzR ˆ

SzI ‡ SAz

…2:4:47†

After a small amount of algebra, it follows that VxI ˆ SAx ‰ kz Š ‡ SAz ‰ ky1 kz Š ‡ 1 UAx   VzI ˆ SAx kx  ‡ ky1 ‡ SAz ‰kx Š ‡ 1 UAz

…2:4:48†

where VxI ˆ SxI ‰ kz Š ‡ SyI ‰ kz Š ‡ SzI



2ky1

VzI ˆ SxI ‰2ky1 ‡ kx Š ‡ SyI ‰kx Š ‡ SzI ‰kx Š

kz

 …2:4:49†

The terms VxI , VzI represent the known source terms associated with the incident plane wave. If we further substitute the sums in SAx , SAz , UAx , and UAz and collect on the unknown coef®cients Cn in the sums, we ®nd

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62

Chapter 2 4 X

VxI ˆ

nˆ1 4 X

VzI ˆ

nˆ1

  Cn Sxn ‰ kz Š ‡ Szn ky1

 kz ‡ 1 Uxn …2:4:50†

   Cn Sxn kx  ‡ ky1 ‡ Szn ‰kx Š ‡ 1 Uzn

The boundary conditions at the Region 2±3 interface are ˆ Cn Exn nˆ1 yˆ 4 X ˆ Cn Ezn nˆ1 yˆ 4 X ˆ Cn Hxn nˆ1 yˆ 4 X ˆ Cn Hzn nˆ1 4 X

‰ExT Š

yˆ L‡

‰EzT Š yˆ

L‡

‰HxT Š

yˆ L‡

‰HzT Š yˆ

L‡

L

L

…2:4:51†

L

yˆ L

Substituting SxT ˆ SzT ˆ

4 X nˆ1 4 X nˆ1

Cn exp… qn L†Sxn …2:4:52† Cn exp… qn L†Szn

into Eqs. 2.4.51 and following a procedure very similar to the Region 1±2 interface we ®nd that 0ˆ

4 X nˆ1

0ˆ

4 X nˆ1

    Cn exp… qn L† Sxn kz  0 ‡ Szn ky3

kz 0

     Cn exp… qn L† Sxn kx  0 ‡ ky3 ‡ Szn kx 0



3 Uxn

3 Uzn



…2:4:53†

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Spectral State Variable Formulation

63

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 Cn 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 ˆ 1:3, 1 ˆ 1:8, and the incident plane wave of Eq. 2.4.27 has SxI ˆ 1 (V/m), SzI ˆ 0:9 (V/m), p p kx ˆ 1 1 sin…I † cos…I †, and kz ˆ 1 1 sin…I † sin…I †, where I ˆ   35 and I ˆ 65 . In Region 3 we assume that the material parameters are 3 ˆ 1:9 and 3 ˆ 2:7. In Region 2 we take the layer thickness L~ ˆ 0:6 and we consider a complicated numerical example where all material parameters of  , l, a, and b of Eqs. 2.4.5 are 2.4.6) are taken to be nonzero. The material parameters of Region 2 are taken to be 2

0:3 ‡ 0:2j

6 aˆ6 4 0:1 ‡ 0:05j

0:3

0:2 ‡ 0:2j

3

7 0:6 ‡ 0:65j 7 5

0:25 3 0:1 0:05 ‡ 0:05j 0:3 6 7 bˆ6 0:01 0:01 7 4 0:1 ‡ 0:1j 5 0:05 0:04 ‡ 0:08j 0:14 3 2 1:3 0:2j 0:3 0:1j 0:33 0:07j 7 6 7 ˆ6 0:1 2 0:01 5 4 0:02 0:01 3 2 3 0:1 :01 1:0 0:4j 6 7 lˆ6 0:013 7 4 0:15j 2:0 0:3j 5 0:011 0:012 1:3 0:2j 2

0:05

0:15 0:1 ‡ 0:1j

…2:4:54†

Figure 16 shows plots of the magnitude of the Ex , Ey , and Ez 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 Hx , Hy and Hz 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

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64

Chapter 2

Figure 16 Plots of the magnitudes of the Ex , Ey , and Ez 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 Ex , Ez , Hx , and Hz , 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 Ey and Hy , 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 J and M , 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.

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Spectral State Variable Formulation

65

Figure 17 Plots of the magnitudes of the Hx , Hy , and Hz 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 x- and y-directions and are constant in the z-direction.

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66

2.5.2

Chapter 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 x ˆ k0 x, y ˆ k0 y, ! E …x; y† ˆ ! 0 H …x; y† ˆ

…1 1 …1 1

! S …kx ; y† exp… j †dkx

…2:5:1†

! U …kx ; y† exp… j †dkx

…2:5:2†

where ˆ kx x. The subscript x 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 …r ˆ ~ and Fourier integrals we ®nd that …1=k0 †r† 0ˆ

…1 n 1

h! i r  S …kx ; y† exp… j †

h

! jl U

o !i ja S exp… j † dkx …2:5:3†

0ˆ

…1 n 1

h! i r  U …kx ; y† exp… j †

h ! o !i jb U ‡  S exp… j † dkx …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 ! ! ! j exp…j †r  … S exp… j †† ˆ l U ‡ a S ! ! ! j exp…j †r  … U exp… j †† ˆ b U ‡  S

…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 kz ˆ 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.

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Spectral State Variable Formulation

2.5.3

67

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 E0 (volt/m) and whose re¯ected amplitude is R0 (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 L and that the only nonzero, lossy, relative material parameters in the slab are xx , xy , yx , yy , and xx ˆ yy ˆ zz ˆ  ˆ  0 j 00 . All other material parameters in a, b,  , 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 ExI ˆ E0 exp… jk3 …y ‡ L†† 0 HzI ˆ

E0 exp… jk3 …y ‡ L†† 3

ExR ˆ R0 exp…jk3 …y ‡ L††

Figure 18

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

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…2:5:7† …2:5:8† …2:5:9†

68

Chapter 2

0 HzR ˆ

R0 exp…‡jk3 …y ‡ L†† 3

…2:5:10†

Ex…3† ˆ ExI ‡ ExR

…2:5:11†

Hz…3† ˆ HzI ‡ HzR

…2:5:12†

for jxj  2B and zero elsewhere in Region 3. In Eqs. 2.5.7±12, 0 ˆ 377 , p p ~ and B~ (meter) is the waveguide slot half 3 ˆ 3 =3 , k3 ˆ 3 3 , B ˆ k0 B, width. Since the EM ®elds are independent of the z-direction, it turns out that the only nonzero ®eld components in all regions of space are the Ex , Ey , and Hz components. The general state variable equations given by Eqs. 2.5.5 and 2.5.6 reduce to 

@V ˆ AV @y

a A ˆ 11 a21

a12 a22

 …2:5:13†

where

a11 a21

  yx ˆ j kx yy   xy yx ˆ j xx yy

" a12 ˆ j zz a22

 xy ˆ j kx yy

k2x yy 

# …2:5:14† …2:5:15†

and where V ˆ ‰Sx ; Uz Št . These are in fact the same exact equations as were studied in Section 2.3 except that here Sx and Uz 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 Ex…2† Ey…2† 0 Hz…2†

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ˆ ˆ ˆ

… 1 "X 2 1

nˆ1

1

nˆ1

1

nˆ1

… 1 "X 2 … 1 "X 2

# Cn Sxn exp…qn y† exp… jkx x† dkx

…2:5:16†

# Cn Syn exp…qn y† exp… jkx x†dkx

…2:5:17†

# Cn Uzn exp…qn y† exp… jkx x†dkx

…2:5:18†

Spectral State Variable Formulation

69

where Sxn ˆ 1 a Uzn ˆ 11

…2:5:19† qn

…2:5:20†

a12

yx k S ‡ xU yy yn yy zn

Syn ˆ

n ˆ 1; 2

…2:5:21†

and where  q1 ˆ 0:5‰a11 ‡ a22 Š ‡ 0:5 …a11  q2 ˆ 0:5‰a11 ‡ a22 Š 0:5 …a11

a22 †2 ‡ 4a12 a21 a22 †2 ‡ 4a12 a21

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  ky1 …1† Uz …kx † exp… jkx x jky1 y†dkx 1 1  …1  kx …1† ˆ Uz …kx † exp… jkx x jky1 y†dkx 1 1 …1 ˆ Uz…1† …kx † exp… jkx x jky1 y†dkx

Ex…1† ˆ Ey…1† 0 Hz…1†

…1 

1

…2:5:24† …2:5:25† …2:5:26†

where  ky1 ˆ

‰1 1 j‰k2x

k2x Š1=2 1 1 Š1=2

1  1 1  1

k2x  0 k2x < 0

…2:5:27†

The minus sign of ky1 (or branch of ky1 ) was chosen on the physical grounds p that the integrals converge as y ! 1 when the jkx j > 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 M s backed by an electrical perfect conductor. The boundary condition equation to determine ! the equivalent magnetic surface current M s backed by an in®nite ground plane is

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70

Chapter 2

 !…2† a^ y  E

yˆ



1



Aˆ

!…3† L‡

E

! Ms

…2:5:28†

yˆ L

where !…3† E

yˆ L

ˆ0

…2:5:29†

since the magnetic surface current is assumed to be backed by an in®nite ground plane. Also x !…2† E ˆ EA …x† rect …2:5:30† a^ 2B x yˆ L‡ where rect

x ˆ 2B

1

jxj < B

0

jxj > B

…2:5:31†

EA …x† represents the x-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 x ! ˆ a^ z M s ˆ a^ z EA …x† rect 2B

…1 1

M…kx † exp… jkx x†dkx

…2:5:32†

! The last part of Eq. 2.5.32 expresses M s in k-space. For the present problem the aperture electric ®eld is given by Eq. 2.5.30 evaluated at y ˆ L . Thus EA is a constant given by EA ˆ E0 ‡ R0 . Using this value of EA it is found from Fourier inversion that M…kx † ˆ

BEA sin…kx B† kx B 

…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 (Ex -component) and the tangential magnetic ®eld (Hz component) on the y ˆ 0‡ (in Region 1) and y ˆ 0 (in Region 2), and at the Region 2±3 interface, matching the tangential electric ®eld (Ex -compo! nent) at y ˆ L‡ (Region 2) to the magnetic surface current M s , and then recognizing that the Fourier amplitudes of all the k-space integrals must equal each other for all values of kx , we ®nd the following equations:

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Spectral State Variable Formulation

71

2 X ky1 …1† Uz …kx † ˆ Cn Sxn 1 nˆ1

Uz…1† …kx † ˆ 2 X nˆ1

2 X nˆ1

…2:5:34†

Cn Uzn

…2:5:35†

Cn Sxn exp… qn L† ˆ M…kx †

…2:5:36†

If we eliminate Uz…1† …kx † from Eqs. 2.5.34±36 we are left with a 2  2 set of equations from which to determine C1 and C2 in terms of M…kx †. We ®nd that C1 ˆ

T2 M…kx † T1 exp… q2 L† T2 exp… q1 L†

…2:5:37†

C2 ˆ

T1 M…kx † T1 exp… q2 L† T2 exp… q1 L†

…2:5:38†

where Tn ˆ a12

ky1 ‰a 1 11

qn Š

n ˆ 1; 2

…2:5:39†

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

yˆ

L‡

ˆ 0 Hz…3†

yˆ L

jxj  2B

…2:5:40†

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

…B  0 Hz…2† B

yˆ

 … 1 B  dx ˆ 0 Hz…3† ‡ L yˆ 2B B

 L

dx

…2:5:41†

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

…B  0 Hz…3† B

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yˆ L

 dx ˆ

1 ‰E 3 0

R0 Š

…2:5:42†

72

Chapter 2

Thus 1 ‰E 3 0

1 R0 Š ˆ 2B

…B  0 Hz…2† B

 yˆ L‡

dx

…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 y on the line y  L is de®ned as ~ Y…y† ˆ

Hz…3† …y†

…2:5:44†

Ex…3† …y†

and for the present case using Eqs. 2.5.42±44 is given by 1 E0 exp… jk3 …y ‡ L†† R0 exp…jk3 …y ‡ L†† ~ Y…y† ˆ 0 3 E0 exp… jk3 …y ‡ L†† ‡ R0 exp…jk3 …y ‡ L††

…2:5:45†

This quantity is useful for transmission lines because once a transmission line admittance load, call it Y~ LOAD , is speci®ed at a given point on the line it is possible to ®nd a relation between the incident wave amplitude E0 (assumed known) and the re¯ected wave amplitude R0 . With E0 assumed known and R0 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 y ˆ L. In this case we ®nd, calling the transmission line load admittance Y~ A` (in units of 1 (or mhos); the subscript A refers to aperture),

Y~ A` ˆ

Hz…3† yˆ …3† Ex

L

yˆ L

If we replace we ®nd that Y~ A` ˆ

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1=3 ‰E0

ˆ

Hz…3†

yˆ L

EA

R0 Š by …1=2B†

( … 1 1 1 B …2† 0 H z 0 EA 2B B yˆ

 ˆ „B

B

1 E0 R0 0 3 EA

0 Hz…2† jyˆ

L‡ dx

 …2:5:46†

using Eq. 2.5.43,

) L‡

dx

…2:5:47†

Spectral State Variable Formulation

73

De®ning a normalized aperture load admittance we have YA` ˆ 0 Y~ A` ˆ

…1=2B†

…2†  H B 0 z

„B

yˆ L‡

dx

…2:5:48†

EA

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

…1 1

Y…kx †dkx

…2:5:49†

where Y…kx † ˆ

 B T2 Uz1 exp… q1 L†  T1 exp… q2 L†

T1 Uz2 exp… q2 L† T2 exp… q1 L†



sin…kx B† kx B

2 …2:5:50†

We remind readers that in the above equation, the quantity in square brackets is a complicated function of kx , and the Uzn , n ˆ 1; 2, are eigenvector components associated with the magnetic ®eld in Region 2. Once the integral in Eq. 2.5.49 is carried out, YA` is known and then a relation between E0 and R0 can be found through the equation ~ YA` ˆ 0 Y…y†

yˆL

ˆ

1 E0 R0 3 E0 ‡ R0

…2:5:51†

If E0 is assumed known, then the normalized re¯ection coef®cient of the system is r

R0 1=3 YA` ˆ E0 1=3 ‡ YA`

…2:5:52†

In computing the integral as given in Eq. 2.5.50, care must be used in p carrying out the integral near the points where kx ˆ k1 , k1 ˆ 1 1 when k1   jkx j  k1 (this interval is in the visible region) and k1 ‡   jkx j  k1 (this interval is in the invisible region), where  is a small number say on the order of k1 =4 or possibly less. The reason for this is that the function in square brackets in the integrand of the YA` integral may be discontinuous

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74

Chapter 2

(or even singular) near the points kx ˆ k1 , 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 kx ˆ k1 cos…u†, 0  u  , in the visible region and kx ˆ k1 cosh…u†, 0  u  1, in the invisible region was employed to integrate the YA` integral. These formulas provide a very dense grid near kx ˆ k1 and thus provide an accurate integration of the YA` 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 P~  Y~ A ˆ jVj2

…2:5:53†

~ A , EA ˆ 1 (Volt/meter) and where where V ˆ 2BE P~ ˆ

…1 



Ex…2† ~ yˆ L~ ‡ 1

 Hz…2†

yˆ L~ ‡



d x~ ˆ

1 2

…1 1

 E x …k~x †H z …k~x †d k~x

…2:5:54† where E x …k~x † and H z …k~x † are the Fourier amplitudes (or k-space pattern space factors) of the Ex…2† electric ®eld and the Hz…2† magnetic ®eld, respectively. P~ has units of (watt/meter)=(volt amp/meter), so Y~ A 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 Y~ A as de®ned by Eq. 2.5.54 is very closely related to the transmission line load admittance expression Y~ A` . It is related by the equation Y~ Y~ A ˆ A` 2B~

…2:5:55†

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

  B1 sin kx B 2 dkx kx B 1 ky1

…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 Y~ A` as

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Spectral State Variable Formulation

75

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 x ˆ  cos…c †, y ˆ  sin…c †,  ! 1, is given by   ! 1 !…1† !…1† 1 1 …1† 2 S ˆ Re E  H ˆ U a^ r ho 2 2 0 z

…2:5:57†

where Uz…1† ˆ

…1 1

A…kx † exp… jkx x

jky1 y†dkx

…2:5:58†

and where A…kx † ˆ

1

    BEA ‰ T2 ‡ T1 Š sin…kx B† kx B  ky1 ‰T1 exp… q2 L† T2 exp… q1 L†Š …2:5:59†

We note in passing that Eq. 2.5.58 for Yz…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 2B~ 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  ! 1 the value Uz…1† ˆ F…k1 sin…'c ††



   2 1=2 j exp jk1  ‡ k1  4

…2:5:60†

where F…k1 sin…c ††  ‰k1 cos…c †ŠA…k1 sin…c †† ˆ     BEA ‰ T2 ‡ T1 Š sin …kx B† 1 kx B  ‰T1 exp… q2 L† T2 exp… q1 L†Š …2:5:61†

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76

Chapter 2

where k1 sin…c † and k1 cos…c † have been substituted for kx and ky1 , respectively, in Eq. 2.5.58. To describe the radiation from the waveguide aperture and material slab system in the far ®eld … ! 1† we plot the normalized radiation intensity, which here is de®ned as the radiation intensity,  ! 1, divided by the total radiation intensity integrated from c ˆ =2 to c ˆ =2. This quantity is called the directive gain D…c †. Applying this de®nition and using Eqs. 2.5.60 and 2.5.61 after cancelling common factors we ®nd D…c † ˆ  „ =2

jF…k1 sin…c ††j2

=2

2.5.4

…2:5:62†

jF…k1 sin…c ††j2 dc

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 ˆ 1 and 1 ˆ 1, 2

2 ˆ  ˆ 1:2

j2:6

xx  2 ˆ 4 yx 0

xy yy 0

3 0 0 5 zz

…2:5:63†

where xx ˆ 2, xy ˆ 0:3, yx ˆ 0:9 j0:2, and yy ˆ 2:1. The value of zz 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 2B~ ˆ 0:6. Figure 19 shows a plot of the Y…kx † aperture admittance integrand when the layer thickness has been taken to L~ ˆ 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 jkx j  5k1 ˆ 5. An inspection of Eq. 2.5.50 for Y…kx † shows that for kx large the integrand approaches 1=k3x and thus is guaranteed to converge. In an inspection of Fig. 19 one sees also that the integrand Y…kx † is not exactly symmetric with respect to the kx 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 kx ˆ k1 ˆ 1. One observes from Fig. 19 the effect that the discontinuous ky1 function of Eq. 2.5.27 has on the Y…kx † integrand in the kx regions near kx ˆ k1 ˆ 1. Figure 19 also lists values of the two lowest magnitude poles which were associated with the Y…kx † integrand. The two pole locations in

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Spectral State Variable Formulation

Figure 19

77

A plot of the Y…kx † aperture admittance integrand.

the complex kx plane …kxp1 ˆ 1:541 ‡ j0:218 and kxp2 ˆ 1:567 j0:146† were nonsymmetric because of the anisotropy of the material slab. The values of the poles were listed as they in¯uence the real kx integration when the kx integration variable passes close to the poles' location. Figure 20 shows a plot of the YA` aperture load admittance as a ~ At a value of L~ ˆ 0 the layer does not function of the layer thickness L. exist, and the waveguide aperture radiates into free space. As L~ increases, the real and imaginary parts of the aperture admittance are oscillatory up to a value of about L~ ˆ 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 ˆ 10 . The asymmetry is caused by the fact that the slot has radiated through an anisotropic material slab.

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78

Chapter 2

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

Figure 21

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

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Spectral State Variable Formulation

2.6 2.6.1

79

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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80

Chapter 2

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…j!t† time dependence] assume the form ! ~ ! r H a ˆ j!~ E a

…2:6:1†

~ ! r Ea ˆ

…2:6:2†

! ~ a j!lH

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

…1 …1 1

1

! ! ~ k~x d k~y ~ exp… j…k~x x~ ‡ k~y y††d f a … k ; z†

…2:6:3†

! ! where k ˆ k~x a~ x ‡ k~y a~ y ; F a represents, ! respectively, either the electric ®eld ! ! ! ~ represents, respectively, E a or magnetic ®eld H a , and where f a … k ; z† ! ! ~ either the spectral amplitude of ! the electric ®eld e … k ; z† or the spectral a ! ~ Substituting the Fourier transamplitude of the magnetic ®eld h a … k ; z†. forms integrals into Maxwell's equations and collecting coef®cients of the exponential in Eq. 2.6.3 we ®nd that ! ! ! ~ ˆ j!~ ! ~ D h a … k ; z† e a … k ; z†

…2:6:4†

! ~ ˆ D! e a … k ; z†

…2:6:5†

! ! ~ j!l~ h a … k ; z†

where 2

0 6 D ˆ 4 @=@z~ j k~y

@=@z~ 0 j k~x

3 j k~y 7 j k~x 5 0

…2:6:6†

De®ning the auxiliary ®eld column matrices " ~ ˆ ya …z†

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# ! ! ! ~ k
e a … k ; z† ! ! ~  k† a^z
…! e a … k ; z†

" xa …z† ˆ

# ! ! ! ~ k
h a … k ; z† ! ! ! ~  k† a^ z
… h a … k ; z† …2:6:7†

Spectral State Variable Formulation

81

We ®nd that Eqs. 2.6.4 and 2.6.5 can be put into the form d d z~



  ~ R xa …z† ˆ ~ V ya …z†

U W



~ xa …z† ~ ya …z†

 …2:6:8†

where the 2  2 matrices R, U, V, and W can be found in the Appendix of Ref. 32. The boundary conditions require that the tangential electric ®eld at ! z~ ˆ 0 must be zero. This requires at z~ ˆ 0 that a^ z  E a ˆ 0, which further ! requires, by the completeness of the Fourier transform, that a^ z  ! e a … k ; 0† ˆ ! ! 0 or eax … k ; 0† ˆ eay … k ; 0† ˆ 0. Thus the auxiliary column matrix ya …z† satis! ! ! ! ®es ya …0† ˆ 0, since ya1 …0†  k
! e a … k ; 0† ˆ kx eax … k ; 0† ‡ ky eay … k ; 0† ˆ 0 ! ! ! ! and ya2 …0†  a^z
…! e a … k ; 0†  k † ˆ eax … k ; 0†ky eay … k ; 0†kx ˆ 0. Consider the matrix differential equation    d X R ˆ V d z~ Y

U W



X Y

 …2:6:9†

where X and Y are 2  2 matrices with entries  Xˆ

~ x11 …z† ~ x21 …z†

~ x12 …z† ~ x22 …z†



 Yˆ

~ y11 …z† y12 …z† ~ y22 …z† y21 …z†



If X1 and Y1 are solutions of Eq. 2.6.9 that meet the boundary conditions X1 …0† ˆ I2 and Y1 …0† ˆ 0 (I2 is a 2  2 identity matrix), then the solution of Eq. 2.6.8 is given by ~ ~ ˆ X1 …z†…X ~ 1 1 …d††c xa …z†

…2:6:10†

~ ~ ˆ Y1 …z†…X ~ 1 1 …d††c ya …z†

…2:6:11†

where c ˆ ‰cx cy Št is a 2  1 constant column matrix. The matrices X1 and Y1 are given by the solution 

Y1 X1





0 ~ ˆ exp…Az† I2



and where the matrix A is given by  Aˆ

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W U

V R

 …2:6:12†

82

Chapter 2

and the 2  2 submatrices R, U, V, and W may be found in the Appendix of Ref. 32 as mentioned earlier. The matrix exp…Az† can be evaluated through the Cayley±Hamilton by the expression ~ ˆ C0 …z†I ~ 4 ‡ C1 …z†A ~ ‡ C2 …z†A ~ 2 ‡ C3 …z†A ~ 3 exp…Az†

…2:6:13†

~ i ˆ 0; 1; 2; 3; satisfy where Ci …z†, ~ ˆ exp…j z†

3 X

~ kj Ck …z†

j ˆ 1; 2; 3; 4

…2:6:14†

kˆ0

and j , j ˆ 1; 2; 3; 4, are the distinct roots of the characteristic equation det…I4

A† ˆ 4 ‡ a1 3 ‡ a2 2 ‡ a3 1 ‡ a4 ˆ 0

…2:6:15†

where a1 ˆ tr…A†, a2 ˆ ‰a1 tr…A† ‡ tr…A2 †Š=2, a3 ˆ ‰a2 tr…A† ‡ a1 tr…A2 † ‡tr…A3 †Š=3, and a4 ˆ det…A† and where tr…
† 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 y11 …z† ~ cedure one can determine the eight matrix elements x11 …z†; making up the 2  2 matrices X and Y 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 ! ! ~ the electric and magnetic Fourier amplitude ®eld components e … k ; z† and a ! ! ~ respectively. The algebraic form of these amplitudes is given in h a … k ; z†, 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 c ˆ ‰cx cy Št . Speci®cation of the general EM ®elds in the half space z~ > d~ (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 z~ > d~ 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 0 !the! ! dipole !current ! source ! ~ Letting ! ^ r~ ~ and h 0 … k ; z† ~ be the J … r~ † ˆ p… r~ †, where z~ 0 > d. e 0 … k ; z† two-dimensional Fourier amplitude of the electric ®eld and magnetic ®elds due to the dipole source [using the Fourier representation as given by Eq.

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Spectral State Variable Formulation

83

~ and y0 …z† ~ de®ned ana2.6.3, and using the auxiliary ®eld quantities x0 …z† logously to Eqs. 2.6.10 and 2.6.11, the free space dipole can be written as 1 ~ ˆ 2 x0 …z† 8

"

jsgn…z~ z~ 0 † ! 0 Q exp…j k
! 

~0 jz~

" 1 0 …!0 † ~ ˆ 2 y0 …z† 8 0

z~ 0 †

jsgn…z~

0

1

0 j k~2 ~0 1

0

sgn…z~

!0 ~0 1

~0 jz~

where z 6ˆ z 0 , k~2 ˆ k~2x ‡ k~2y , ~0 ˆ …k~2 3 ! j p^
k 7 6 7 6 Q ˆ 6 j a^
…p^  ! k †7 5 4 z 2

…2:6:15†

z~ 0 j†

0

! 0 Q exp…j k~
! 

#

z~ 0 †k~2 …!0 †

1

#

0

…2:6:16†

z~ 0 j† k20 †1=2 , k~20 ˆ !2 0 0 , 2

3 ! ! ! ~ k
h 0 … k ; z† 5 x0 …z† ˆ 4 ! ! ! ~  k† z^
… h 0 … k ; z†

p^
a^ z 3 ! ! ! ~ k
e 0 … k ; z† 5 y0 …z† ˆ 4 ! ! ! ~  k† z^
… e 0 … k ; z† 2

…2:6:17† and !0 !0 r~ ˆ ~ ‡ z~ 0 a^ z

…2:6:18†

For the ®eld re¯ected from the anisotropic layer (an outgoing wave moving away from the layer), " ~ ˆ xr …z† " ~ ˆ yr …z†

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#"

j ~0

0

0

!0 0

!0

j ~0 0

F

D #"

# exp… ~0 jz~ F D

~ dj†

…2:6:19†

# exp… ~0 jz~

~ dj†

…2:6:20†

84

Chapter 2

! ! ! ~ and yr …z† ~ are determined from ! ~ and h r … k ; z† ~ in a where xr …z† e r … k ; z† ~ ~ … z† and y … z† were determined manner similar to the way x 0 0 ! ! ! ! ! !from ! ~ and h 0 … k ; z† ~ or xa …z† ~ and ya …z† ~ from ! ~ and h a … k ; z†. ~ e a … k ; z† e 0 … k ; z† The ®nal step in obtaining the solution is to boundary match the ~ The total EM ®elds for z~  d~ is the sum of tangential EM ®elds at z~ ˆ d. the incident and re¯ected ®elds, and the total ®elds for z~  d~ is the anisotropic slab ®eld; thus equating these total ®elds (using the three sets of auxiliary vectors) we have ~ ˆ c ˆ x0 …d† ~ ‡ xr …d† ~ xa …d†

…2:6:21†

~ ˆ y …d† ~ ‡ y …d† ~ ya …d† 0 r

…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 " c " ~ ~ Y1 …d†X …d†c 1

j ~0

0

0

!0

0 !0

j ~0 0

#" #"

F D F D

# ~ ˆ x0 …d†

…2:6:23†

~ ˆ y0 …d†

…2:6:24†

#

Once the four constants c ˆ ‰cx cy Št , F, and D 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 r ! 1, ®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 N^ ˆ cos 0 a^ z ‡ sin 0 …cos 0 a^ x ‡ sin 0 a^ y †

Copyright © 2000 Marcel Dekker, Inc.

…2:6:25†

Spectral State Variable Formulation

85

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

Figure 23 Radiation patterns jE j, jE j versus  in the  ˆ 0 (180 ) plane for a uniaxial substrate with xx ˆ 10:70 , zz ˆ 10:40 , l ˆ 0 I3 , d ˆ 1 mm, and f ˆ 30 GHz. The primary source is an electric dipole located at the substrate surface ^ (# IEEE, 1985.) …z 0 ˆ d†, and its orientation is de®ned with the unit vector .

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86

Chapter 2

along the principal axes are ~xx ˆ ~yy ˆ 10:70 and ~zz ˆ 10:40 . In this case the  …0 ; 0 † is independent of the 0 angle and l ˆ 0 I 3 . The substrate thickness is taken to be d ˆ 1 mm. Both vertical ^ ˆ z^ and horizontal ^ ˆ x^ 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 z-axis. In treating ferrite substrates it is assumed that  …0 ; 0 † ˆ 150 I3 and that a strong magnetic type of anisotropy is used with ~ 11 ˆ 0:6750 , ~ 12 ˆ 0:494 0 , !0 =! ˆ 2:35 [32], ! ˆ M, 0 M ˆ 0:3Wb=m2 ( being the magnetomechanical ratio). Corresponding to various biasing static magnetic ®eld orientations, the computed radiation patterns on various  ˆ constant ^ planes are given in Figs. 24±26 for x-directed dipoles. The radiation frequency is taken f ˆ 30 GHz, and the ferrite layer thickness is d ˆ 1 mm. In general there is a strong dependence of the far ®eld to 0 orientation. When the  ˆ constant observation plane coincides with the  ˆ 0 plane (i.e.,  ˆ 0 ˆ 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 ˆ 20 ) to a rather directional diagram …0 ˆ 80 ).

Figure 24 Radiation patterns jE j, jE j versus  in the  ˆ 90 observation plane of a ferrite substrate for various 0 angles and 0 ˆ 0 . The material properties of the ferrite are ~ 11 ˆ 0:6750 , ~ 12 ˆ 0:4940 [32], and ~ …0 ; 0 † ˆ 0 I3 , d~ ˆ 1 mm. and f ˆ 30 GHz. The dipole axis is along the x-axis …a^ ˆ a^ x † and is located at the ~ (# 1985, IEEE.) substrate surface …z~ 0 ˆ d†.

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Spectral State Variable Formulation

87

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

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

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88

Chapter 2

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 ˆ 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 f ˆ 30 GHz and the plasma layer thickness is d ˆ 1 mm. The parameters characterizing the plasma are taken as l ˆ 0 I3 , while  …0 ˆ 0; 0 † is computed with !c = !p ˆ 1:8 and !=!p ˆ 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  ˆ 0 ˆ 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 jE j, jE j versus  in the  ˆ 0 plane for a grounded plasma layer with !c =!p ˆ 1:8, !=!p ˆ 2:4, d~ ˆ 1 mm, and f ˆ 30 GHz [32]. The ~ and ^ and it is located on the plasma surface …z~ 0 ˆ d† dipole is along the x-axis …^ ˆ x†,  0 ˆ 0 . (# 1985, IEEE.)

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Spectral State Variable Formulation

2.7

2.7.1

89

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 (x-y ~ shown in Fig. 28. All coorplane) generalized anisotropic layer …0 < z~ < d† 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 x~ and y~ spectral ®eld components in the anisotropic medium derived from Maxwell's equations with some algebraic manipulation

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90

Chapter 2

Figure 28 Re¯ection from an in-plane biased ferrite layer. Biased ®eld …H0 † 1000 Gauss in the a^x direction; magnetization 2500 Gauss. Transverse magnetic incidence i ˆ 30 and i ˆ 40 , ~f ˆ 12:80 , and d~ ˆ 3 cm. (Copyright 1995, IEEE [26].)

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

…2:7:1†

where 3 k~x H~ x ‡ k~y H~ y 6 k~ H~ k~x H~ y 7 7 6 ~ z† ~ ˆ6 y x w… 7 4 k~x E~ x ‡ k~y E~ y 5 k~y E~ x k~x E~ y 2

…2:7:2†

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

i   ~ ~ …0† ~ …d~ † ˆ T…d†

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…2:7:3†

Spectral State Variable Formulation

91

where 2 6 6 6 ~ ~ T…d† ˆ ‰/Š6 6 4

~ exp…1 d†

0

0

0

0

~ exp…2 d†

0

0

0

0

~ exp…3 d†

0

0

0

0

~ exp…4 d†

3 7 7 7 ~ 7‰/Š 7 5

1

…2:7:4† The electromagnetic ®elds in the air …z~  d~‡ and z~  0 ) can be derived from a set of transverse electric and transverse magnetic vector potential functions. This result can be shown to be q 3 ja 0 k~2 k20 7 6 7 6 0 7 6 b ! 0 6 ~ ‡ † ˆ 6 q 7 w…d 7 7 6 0 ~2 27 6 jb k k 05 4 2

2 6 6 6 6 ~ w…0 † ˆ 6 6 6 4

!0 a 0 q 3 jc 0 k~2 k20 7 7 0 7 !0 d 7 q 7 7 jd 0 k~2 k20 7 5

…2:7:5†

!0 c 0 The unknown a 0 , b 0 , c 0 , and d 0 are quantities to be determined from the equation ~ ‡† w…d

~ w…0 ~ T…d† † ˆ Qinc

…2:7:6†

q where k~ ˆ k~2x ‡ k~2y and where Qinc 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

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92

Chapter 2

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 ~ ~ T…d† ˆ exp…1 d†A 1 ‡ exp…2 d†A2

…2:7:7†

where the singular matrices A1 and A2 do not contain any terms that grow exponentially. We have 2

1

6 60 6 A1 ˆ r~ 6 60 4 0

0

0

0

0

0

0

0

0

0

3

7 07 7~ 7r 07 5 0

1

…2:7:8†

and 2

0

60 6 A2 ˆ r~ 6 40 0

0

0

1 0 ~ 0 exp‰…3 2 †dŠ 0

3

0

0

0 0 exp‰…4

7 7~ 7r 5

1

…2:7:9†

~ 2 †dŠ

Note that A1 is obtained from Eq. 2.7.4 by replacing the terms of exp…2 d†, exp…3 d†, and exp…4 d† with 0 and replacing exp…1 d† with 1. Since A1 is a singular matrix, it can be shown that 2

a1

3

6 7 6 a2 7 6 7 ~ A1 w…0 † ˆ …c ‡ d †6 7 6 a3 7 4 5 0

0

…2:7:10†

a4 where , and ai , i ˆ 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: ~ ˆu …c 0 ‡ d 0 † exp…1 d†

…2:7:11†

~ ˆv …c 0 ‡ d 0 † exp…2 d†

…2:7:12†

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Spectral State Variable Formulation

93

where u and v are the new variables replacing c 0 and d 0 . With the variable transformations, we have 2

a1

2

3

6 7 6 a2 7 u 6 7 ~ w…0 ~ T…d† † ˆ u6 7 ‡ A exp‰…2 6 a3 7  2 4 5 a4 2 ‡

v 

6 6 6 A2 6 6 6 4

q 3 j k~2 k20 7 7 !0 7 q 7 7 j k~2 k2 7 5

6 6 6 ~ 1 †dŠ6 6 6 4

q 3 j k~2 k20 7 7 !0 7 q 7 7 j k~2 k20 7 5 !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 a 0 , b 0 , u, and v (and therefore c 0 and d 0 ). 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-

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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 (xx ; xy ; yx ; yy ,

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Spectral State Variable Formulation

7. 8.

9. 10.

11.

95

and zz 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 ("; ; a; b scalar) layer immersed in air and for the case a ˆ b. A propagating transverse magnetic (TM) mode whose longitudihm i …x ‡ B† exp… y† is incinal electric ®eld is given Ey ˆ A sin 2B 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; "xx ˆ 2., "xy ˆ "yx ˆ :5, "yy ˆ 4: "zz ˆ 1:, "xz ˆ "zx ˆ "yz ˆ "zy ˆ 0:,  ˆ 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.

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Chapter 2

REFERENCES 1. D. R. Rhodes, Synthesis of Planar Aperture Antenna Sources, Oxford University Press, 1974. 2. H. A. MacLeod, Thin-Film Optical Filters, Macmillan, New York, 1986. 3. R. F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill, New York, 1961. 4. A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering, Prentice Hall, Englewood Cliffs, New Jersey, 1991. 5. L. B. Felson and N. Marcuvitz, Radiation and Scattering of Waves, Prentice Hall, Englewood Cliffs, New Jersey, 1973. 6. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media, Artech House, Boston, London, 1994. 7. J. Galejs, Antennas in Inhomogeneous Media, Pergamon Press, Oxford, 1969. 8. J. R. Wait, Electromagnetic Waves in Strati®ed Media, Pergamon Press, Oxford, 1970. 9. S. V. Marshall, R. E. DuBroff, and G. G. Skitek, Electromagnetic Concepts and Applications, 4th ed., Prentice Hall, Englewood Cliffs, New Jersey, 1996. 10. P. P. Banerjee and T. C. Poon, Principles of Applied Optics, Aksen Associates, Homewood, IL. 11. P. Baumeister, Utilization of Kard's equations to suppress the high frequency re¯ectance bands of periodic multilayers, Appl. Opt., 24, 2687±2689 (1985). 12. J. A. Dobrowolski and D. Lowe, Optical thin ®lm synthesis based on the use of Fourier transforms, Appl. Opt., 17, 3039±3050 (1978). 13. W. H. Southwell, Coating design using very thin high- and low-index layers, Appl. Opt., 24, 457±460 (1985). 14. J. M. Jarem, The minimum quality factor of a rectangular antenna aperture, Arab. J. Sci. Eng., 7(1), 27±32 (1982). 15. J. M. Jarem, Method-of-moments solution of a parallel-plate waveguide aperture system, J. Appl. Phys., 59(10), 3566±3570 (1986). 16. J. M. Jarem, The input impedance and antenna characteristics of a cavitybacked plasma covered ground plane antenna, IEEE Trans. Antennas Propagation, AP-34, 262±267 (1986). 17. 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, IEEE Trans. Antennas Propagation, AP-37, 187±193 (1989). 18. D. A. Holmes and D. L. Feucht, Electromagnetic wave propagation in birefringent multilayers, J. Opt. Soc. Am., 56, 1763±1769 (1966). 19. S. Teitler and B. W. Henvis, Refraction in strati®ed, anisotropic media, J. Opt. Soc. Am., 60, 830±834 (1970). 20. D. W. Berreman, Optics in strati®ed and anisotropic media: 4  4-matrix formulation, J. Opt. Soc. Am., 62, 502±510 (1972). 21. P. J. Lin-Chung and S. Teitler, 4  4 matrix formalisms for optics in strati®ed anisotropic media, J. Opt. Soc. Am. A, 1, 703±705 (1984).

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22. C. M. Krowne, Fourier transformed matrix method of ®nding propagation characteristics of complex anisotropic layer media, IEEE Trans. Microwave Theory Techniques, MTT-32, 1617±1625 (1984). 23. M. A. Morgan, D. L. Fisher, and E. A. Milne, Electromagnetic scattering by strati®ed inhomogeneous anisotropic media, IEEE Trans. Antennas Propagation, AP-35, 191±197 (1987). 24. R. S. Weiss and T. K. Gaylord, Electromagnetic transmission and re¯ection characteristics of anisotropic multilayered structures, J. Opt. Soc. Am. A, 4, 1720±1740 (1987). 25. H. Y. D. Yang, A spectral recursive transformation method for electromagnetic waves in generalized anisotropic layered media, IEEE Trans. Antennas Propagation, AP-45, 520±526 (1997). 26. H. Y. D. Yang, A numerical method of evaluating electromagnetic ®elds in a generalized anisotropic medium, IEEE Trans. Microwave Theory Techniques, MTT-43, 1626±1628 (1995). 27. P. Yeh, Electromagnetic propagation in birefringent layered media, J. Opt. Soc. Am., 69, 742±756 (1979). 28. N. G. Alexopoulos and P. L. E. Uslenghi, Re¯ection and transmission for materials with arbitrarily graded parameters, J. Opt. Soc. Am., 71, 1508±1512 (1981). 29. R. D. Graglia, P. L. E. Uslenghi, and R. E. Zich, Dispersion relation for bianisotropic materials and its symmetry properties, IEEE Trans. Antennas Propagation, AP-39, 83±90 (1991). 30. S. M. Ali and S. F. Mahmoud, Electromagnetic ®elds of buried sources in strati®ed anisotropic media, IEEE Trans. Antennas Propagation, AP-37, 671± 678 (1979). 31. C. M. Tang, Electromagnetic ®elds due to dipole antennas embedded in strati®ed anisotropic media, IEEE Trans. Antennas Propagation, AP-27, 665±670 (1979). 32. J. L. Tsalamengas and N. K. Uzunoglu, Radiation from a dipole in the proximity of a general anisotropic grounded layer, IEEE Trans. Antennas Propagation, AP-33 (2), 165±172 (1985). 33. J. L. Tsalamengas, Electromagnetic ®elds of elementary dipole antennas embedded in strati®ed general gyrotropic media, IEEE Trans. Antennas Propagation, AP-37, 399±403 (1989). 34. 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, IEEE Trans. Antennas Propagation, AP34, 247±253 (1986). 35. B. Lax and K. J. Button, Microwave Ferrites and Ferrimagnetics, McGraw-Hill, New York, 1962.

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3

Planar Diffraction Gratings

3.1

INTRODUCTION

In the past thirty years the study and use of periodic structures and diffraction gratings has become increasingly important. Diffraction gratings have been constructed for applications in the frequency ranges of microwaves, millimeter waves, far infrared, infrared, optics, and x-rays. Diffraction gratings occur in such applications as holography, memory storage, spectroscopy, phase conjugation, photorefractives, image reconstruction, optical computing, transducers, integrated optics, microwave phased arrays, acoustooptics, interdigitated, voltage controlled, liquid crystal displays, and many other areas. Petit [1], Gaylord and Moharam [2], Solymar and Cooke [3], and Maystre [4] give extensive reviews on the applications of diffraction gratings. Chapter 7 of this book cites many references on diffraction gratings in photorefractive materials. We will give a brief description and overview of the physical makeup of diffraction gratings. Diffraction gratings have been manufactured and constructed in many different forms and types. Two main classi®cations of diffraction gratings are those that are metallic and those that are dielectric. Metallic gratings have grooves that are etched or cut from a ¯at metal surface. These grooves may be rectangular or triangular in shape. Triangular grooves are referred to as blazed gratings. Metallic gratings are operated in the re¯ection mode, as the diffracted waves are re¯ected from the metal surface. Metallic gratings are also examples of surface relief gratings, as the rectangular or triangular groove shape of the grating is cut from the ¯at metal surface. Dielectric gratings are constructed of dielectric materials that are transparent to the electromagnetic radiation that impinges on it. Dielectric gratings can be classi®ed into two major types: dielectric gratings that are surface relief gratings and dielectric gratings that are volume grat-

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100

Chapter 3

ings. Surface relief dielectric gratings tend to have a large periodic modulation but small thickness, whereas volume dielectric gratings tend to have a small periodic modulation but a large thickness. The large modulation of the surface relief grating occurs because the grating material from which the grating is constructed has a large difference in index of refraction compared to the medium adjacent to the grating. Dielectric gratings may be operated in either the transmission mode or the re¯ection mode. Transmission gratings have periods on the order of a few wavelengths with the grating vector parallel to the grating surface, whereas re¯ection gratings have periods on the order of a half wavelength and grating vectors perpendicular to the grating surface. Gratings that are neither exactly parallel or not exactly perpendicular to the grating surface are referred to as slanted gratings. Scattering from dielectric diffraction gratings depends strongly on three main factors, namely the type and strength of the periodic variation of the index of refraction that exists in the grating, the type of material (anisotropic or isotropic, nonlossy or lossy) the grating is made from, and the type of EM wave that is incident on the grating. We will now brie¯y discuss these three factors. The periodic variation of the index of refraction that induces diffraction when a grating is illuminated may consist of many different forms. The periodic variation may be one-dimensional; it may be two-dimensional, in which case it is referred to as a crossed grating; or the grating may consist of two superimposed one-dimensional gratings. In addition to the index varying in one or two dimensions, the periodic variation of the index of refraction may vary longitudinally throughout the grating. A sinusoidal surface relief grating and a triangular blaze grating that has air as an interface are examples of this type of variation. A surface relief grating is longitudinally inhomogeneous because at a plane where the groove is deeper, more material will be included in the duty cycle of the grating that at a plane closer to the homogeneous air half space. The type of material that makes up the grating may be isotropic and nonlossy, like glass; it may be anisotropic, like calcite or LiNbO3 (lithium niobate); it may be either weakly lossy (e.g., BaTiO3 ) or strongly lossy. Lossy gratings attenuate the diffracted waves as they propagate through the system. In anisotropic materials the anisotropy tends to couple the polarization states of the incident wave in the medium and induce new polarization states in the system. In anisotropic systems the diffracted waves consist of ordinary and extraordinary waves coupled together through the grating vector. The type of EM radiation that is incident on the grating strongly in¯uences the diffraction that will result from the grating. The EM radiation may consist of either a plane wave or a collection or spectrum of plane

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101

waves (e.g., a Gaussian beam). Further, each of these types of waves may be incident on the grating at an oblique angle and possess an arbitrary polarization. Later in this chapter we will show examples of H-mode (magnetic ®eld in plane of incidence) and E-mode polarization (electric ®eld in plane of incidence) states that may be used to illuminate a diffraction grating. Particularly for anisotropic gratings, the type of incident wave and its polarization determine strongly how the EM wave will couple and diffract from the grating. Many mathematical analyses and numerical algorithms have been developed so that the diffraction that occurs from planar gratings can be predicted. Some of the main diffraction grating methods and algorithms are (1) coupled wave analysis [5±9], (2) rigorous coupled wave analysis (RCWA) [2,10±53], (3) coupled mode theory [54±61] (Refs. 57±59 have been referred to as the Australian method), (4) the differential method [1,62±65], (5) the integral method [66], (6) the ®nite difference method [67±69], (8) the boundary element method [70], (7) the unimoment method [71] and (9) other methods [72,73], which are either closely related to or variations of the methods listed above. References 74±76 list papers on energy and power conservation in electromagnetic and electromagnetic diffraction grating systems. Concerning the ®rst three methods, within the last ten years, several researchers have been concerned with the problem of improving the convergence or increasing the stability (that is, allowing analysis of thicker grating structures that have increased grating strength) of the coupled mode and coupled wave algorithms, and they also have been concerned with the problem of understanding in the ®rst place, for certain polarizations and material types, the coupled mode and coupled wave algorithms that are unstable and why they do not converge well. Just about all the above-mentioned algorithms solve the EM grating diffraction problem in three basic steps: one must (1) express the EM ®elds outside the diffraction grating region as Rayleigh series of propagating and evanescent planes waves whose amplitudes are unknown and are yet to be determined (the series is transversely periodic with the period equal to grating period of the periodic structure), (2) by an appropriate method, ®nd a general solution of Maxwell's equations in the diffraction grating region, and (3) match EM boundary conditions at the diffraction grating and homogeneous grating interfaces to determine all the unknown coef®cients of the diffraction grating system. Most of the methods differ in the way that Maxwell's equations are solved in the diffraction grating region. We will now give a brief description of all of these algorithms. This chapter will primarily focus on the rigorous coupled wave approach. Chapter 6, Section 6.2 will brie¯y describe the coupled mode algorithm and show its connection to anisotropic waveguide propagation theory as developed by

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Chapter 3

Gardiol [1, Chapter 6]. The reader may refer to the references for further details on the other methods. We will now give a brief description of the above-mentioned algorithms. The description here, in order to simplify the discussion and description, is assumed to apply only to longitudinally homogeneous gratings. When using coupled wave analysis [5±9] and RCWA [10±53], Maxwell's equations in the diffraction grating region are solved by expanding the periodic dielectric in the diffraction grating region in a Fourier series, expanding the EM ®elds in the diffraction grating region in a set of Floquet harmonics whose amplitudes are functions of the longitudinal coordinate, and after substituting these expansions in Maxwell's equations, organizing the resulting equations into state variable form where eigensolutions to the state variable system can be found. Coupled wave analysis [5±9] differs from RCWA [10±53] in that in coupled wave analysis only a very few Floquet harmonics are used in the analysis (two or three), whereas in RCWA the analysis is made nearly exact by including however many Floquet harmonics are necessary until convergence of the solution is obtained. Typical state variable matrix sizes in the rigorous coupled wave analysis method may range from 10  10 to 100  100. In coupled mode theory algorithms, the transverse periodic region of the gratings is divided into homogeneous subregions, and wave equation solutions in the homogeneous subregions [which are linear combinations of sinusoids proportional to a longitudinal propagation factor exp… z†, where z is the longitudinal coordinate] are EM boundary matched to the adjacent homogeneous subregions. After imposing the boundary condition that the overall EM solution across the grating period repeat itself every grating period, one derives a nonstandard eigenvalue equation, whose multiple roots thus determine the propagation constant of the modes that can propagate in the system. The propagation constant can of course be purely imaginary (nonevanescent), purely real (evanescent or attenuating), or complex if the medium is lossy, propagating with attenuation. By summing the forward and backward modes in the diffraction grating region, a complete solution of Maxwell's equations in the grating region is found. This method is particularly useful for lamellar gratings or step gratings, where there are just two or just a few uniform layers within one grating period. This method is called a coupled mode approach because it is based on determining the propagating modes of the system. In the special one-dimensional case when the grating period is bounded by perfect conductors and the overall grating region is uniform, the method reduces to the well-known problem of determining the propagating modes in a parallel plate waveguide. We would like to caution readers that the algorithm names, coupled mode analysis and

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Planar Diffraction Gratings

103

coupled wave analysis, have been sometimes used interchangeably in the literature. The differential method [1,62±65] is designed primarily for solving for diffraction from surface relief gratings. This method is based on solving Maxwell's equations in the diffraction grating region, by (1) de®ning a function y ˆ f …x† that speci®es the shape of the surface relief grating in the diffraction grating region (that is, over the range 0 < y < L, where L is the grating thickness), (2) expanding the dielectric permittivity function  …x; y† in a Fourier series over the grating period, (3) expanding the EM ®elds of the system in a Fourier series with the series amplitudes expressed as a function of the longitudinal coordinate y, (4) substituting this …x; y† and the expanded EM ®elds in either the wave equation resulting from Maxwell's equations or into Maxwell's equations directly, (5) organizing the system of Fourier series amplitudes into a state variable form (with ®rst-order derivatives of the system being taken with respect to the coordinate y), and (6) solving the state variable system using differential equation shooting methods. Petit [1] gives a detailed description and survey of this method and its application to metallic and dielectric surface relief gratings. The integral method [66], which is particularly useful for metallic surface relief gratings, is based on four basic steps: (1) deriving a periodic Green's function that describes the way that an electrical surface current radiates from one point on the metal surface to an arbitrary point in space, (2) using this Green's function, writing an electric ®eld integral that represents the way in which the grating current radiates to an arbitrary point in space, (3) summing the electric ®eld integral of Step (2) and the incident electric plane wave ®eld together, and (4) setting the total tangential electric ®eld at the grating surface to zero to form an integral equation from which the surface current of the grating can be determined. This formulation is similar to that used to solve for surface currents on an antenna or on a metallic scatterer. The ®nite difference method [67±69] of determining diffraction from a grating is based on solving Maxwell's equation in the diffraction grating region by dividing the diffraction grating region over one period into a large grid, and then approximating the spatial partial derivatives of Maxwell's equations by using ®nite differences. In the ®nite element method [70], the diffraction grating region is divided into cells, and the ®eld variables over a cell are expanded as boundary element functions. By substituting these as boundary element expansions in Maxwell's equations, a large system matrix is formed from which the ®elds of the system are determined. In both the ®nite difference and ®nite element methods, the solutions found in the diffraction grating region are matched to the plane wave Rayleigh expansion exterior to the diffraction grating system. The unimoment method [71] deter-

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Chapter 3

mines, by using either ®nite differences or ®nite elements, sets of special expansion functions in the diffraction grating that satisfy the wave equation and can be used to expand the unknown ®elds of the overall system. Other methods and applications of diffraction gratings (including diffraction analysis of interdigitated, voltage controlled, liquid crystal displays [77±82]) are listed in Refs. 77±95. In the previous paragraphs, we have given a brief overview of available methods for solving diffracting grating problems. In what follows, we will concentrate on analyzing several different diffraction grating structures using the RCWA method [2,10±53]. The RCWA technique is relatively simple and straightforward, provides rapid convergence in many cases, and can apply equally well to thick or surface relief gratings. Hence it has become a popular method for solving diffraction grating problems. In Section 3.2 we will study the full ®eld analysis when an H-mode polarized plane wave is incident on a one-dimensional anisotropic grating. In Section 3.3 we will study two formulations of the RCWA algorithm when an Emode polarized wave is incident on the grating. In Sections 3.2 and 3.3 the complex Poynting theorem will be used to check convergence of the solutions and to study evanescent power in the grating. Section 3.4 will be devoted to the study of slanted and re¯ection gratings. The special case of a pure re¯ection grating will also be studied. Section 3.5 will be concerned with multilayer diffraction theory. Section 3.6 will study diffraction from a crossed diffraction grating when a general conical wave is incident on the grating. Finally, in Sections 3.7 and 3.8 we will summarize recent work that has been performed to increase the ef®ciency and stability of the RCWA algorithm.

3.2

H-MODE PLANAR DIFFRACTION GRATING ANALYSIS

In this section we are interested in using the rigorous coupled wave analysis algorithm (RCWA) to study the diffraction case that occurs when a plane wave is incident on the planar grating shown in Fig. 1. The diffraction ! grating is assumed to have its grating vector speci®ed by K ˆ K~ a^ x , where ~ and  ~ is the grating period or grating wavelength. In this case the K~ ˆ 2= electric ®eld is assumed to be polarized perpendicular to the plane of inci! dence as E ˆ Ez a^ z . In this section two RCWA formulations will be presented. In the ®rst formulation (given in Section 3.2.1), the state variable equations will be derived directly from Maxwell's equations, whereas in the following section, Maxwell's equations will be reduced to a second-order wave equation and then placed in state variable form. The complex Poynting theorem using the solutions found from the full ®eld RCWA

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105

Figure 1 H-mode problem geometry assuming an arbitrary dielectric permittivity pro®le occupies Region 2. The inset shows one possible pro®le.

algorithm will be used to calculate the real and reactive power of the diffraction grating system and thus validate the overall analysis.

3.2.1

Full Field Formulation

The basic overall RCWA [16±24] approach that will be used to study diffraction in this section will be to solve Maxwell's equations in Regions 1, 2, and 3 and then using the general solutions to match the electromagnetic boundary to determine the speci®c EM ®elds in each region. The EM ®eld solutions in Regions 1 and 3, after we solve Maxwell's equations in homogeneous space, consist of an in®nite set of propagating and evasnescent re¯ected and transmitted plane waves. The EM ®eld solution in Region 2 is determined by expanding the electric and magnetic ®elds in a set of periodic or Floquet harmonics (the periodicity of the Floquet harmonics equals that of the diffraction grating), substituting these expansions into Maxwell's equations, and form the resulting equations, developing a set of state variable equations from whose solution the general EM ®elds of Region 2 are found. We begin the analysis by determining the general EM ®eld solution of Region 2, the diffractive grating region. Using normalized coordinates, ~ y ˆ k0 y, ~ and z ˆ k0 z, ~ where k0 ˆ 2= and  (meters) is where x ˆ k0 x,

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Chapter 3

the free space wavelength, we ®nd that Maxwell's normalized equations in Region 2 are given by ! j…0 H †

! r E ˆ

…3:2:1†

! ! r  …0 H † ˆ j E

…3:2:2†

p where 0 ˆ 0 =0 ˆ 377 is the intrinsic impedance of free space,  ~ 0 is the relative permeability of Region 2, 0 is the permeability of free ˆ = ~ 0 is the relative permittivity of Region 2, and 0 is the permitspace,  ˆ = tivity of free space. We expand the electric and magnetic ®eld as 1 X ! Szi …y† exp… jkxi x†a^ z E ˆ

…3:2:3†

iˆ 1

1  X  ! ! Uxi …y†a^ x ‡ Uyi …y†a^y exp… jkxi x† U  0 H ˆ

…3:2:4†

iˆ 1

kxi ˆ kx0

iKx

kx0 ˆ

p 1 1 sin…†

Kx ˆ 2=

~  ˆ k0 

Substituting we have  1  X ! @Szi r E ˆ a^ ‡ jkxi Szi a^y exp… jkxi x† ˆ @y x iˆ 1 ˆ

! j…0 H †

1  X  Uxi a^ x ‡ Uyi a^ y exp… jkxi x†

j

iˆ 1

1  X ! jkxi Uyi r U ˆ

 @Uxi exp… jkxi x†a^ z ˆ j…x†Ez a^ z @y

iˆ 1

1 X

ˆ j…x†

iˆ 1

…3:2:5†

Szi exp… jkxi x†a^z

…3:2:6†

The term …x†Ez can be written " …x†Ez ˆ

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1 X i 00 ˆ 1

#" i 00 e

ji 00 Kx x

1 X i 0ˆ 1

# Szi 0 e

j…kxo i 0 Kx †x

…3:2:7†

Planar Diffraction Gratings

107

or after combining sums we ®nd that …x†Ez ˆ

1 X

1 X

i 00 ˆ 1 i 0 ˆ 1

jkxo x j…i 0 ‡i 00 †Kx x

i 00 Szi 0 e

e

…3:2:8†

At this point we will make a substitution and let i ˆ i 0 ‡ i 00 , or i 00 ˆ i i 0 . We notice in the i 00 summation that when i 00 ˆ 1, i ˆ P1 for a ®xed 1 0 00 be ®nite i 0 . Thus in making P1the substitution of i ˆ i ‡ i , the i 0000ˆ 1 may replaced by the sum iˆ 1 . Carrying out the substitution i ˆ i i 0 we ®nd that …x†Ez ˆ

1 X

#

1 X

iˆ 1 i 0 ˆ 1

Using kxi ˆ kxo …x†Ez ˆ

"

i

i 0 Szi 0

e

j…kxo iKx †x

e

jkxi x

…3:2:9†

iKx we ®nd that 1 X

iˆ 1

"

#

1 X i 0ˆ

i

i 0 Szi 0

…3:2:10†

1

Substituting (3.2.10) in (3.2.5) and (3.2.6) we ®nd @Szi ˆ @y jkxi Uyi

jUxi

1 X @Uxi ˆj i @y i 0ˆ 1

jkxi Szi ˆ

jUyi …3:2:11†

i 0 Szi 0

It is useful to introduce column and square matrices and put the preceding equations into state variable form. Let Ux ˆ ‰Uxi Š, Uy ˆ ‰Uyi Š, Sz ˆ ‰Szi Š, i ˆ 1; . . . ; 1 and let  ˆ ‰i;i 0 Š ˆ ‰i i 0 Š, Kx ˆ ‰kxi i;i 0 Š, I ˆ ‰i;i 0 Š, …i; i 0 † ˆ 1; . . . ; 1, be square matrices. i;i 0 is the Kronecker delta and I is the identity matrix. We ®nd that @Sz ˆ @y jKx Uy

Copyright © 2000 Marcel Dekker, Inc.

jUx

@Ux ˆ jSz @y

jKx Sz ˆ

jUy …3:2:12†

108

Chapter 3

We eliminate the longitudinal vector component Uy and ®nd that jKx Uy

@Ux ˆ @y

 jKx

1 K S  x z



@Ux ˆ j Sz @y

…3:2:13†

Rearranging we ®nd the state variable form @Sz ˆ 0Sz jIUx @y   @Ux 1 ˆj K K  Sz ‡ 0Ux  x x @y

…3:2:14†

These equations may be put into state variable form if we introduce the super matrices Ve ˆ



Sz Ux



 Aˆj

0 …Kx Kx =

†

I 0

 …3:2:15†

we then have @Ve …y† ˆ A Ve …y† @y

…3:2:16†

These equations can be solved numerically by truncating the matrices A and Ve and using state variable techniques to solve the resulting equation. The truncation may be carried out by keeping mode orders whose magnitude is not greater than MT , that is, keeping modal terms where …i; i 0 † ˆ MT ; . . . ; 1; 0; 1; . . . ; MT . Making the truncation we ®nd Ve …y† is a column matrix of size NT ˆ 2…2MT ‡ 1† and A is a constant matrix of size NT  NT . Eq. 3.2.16, when truncated to size NT , can be solved by ®nding the eigenvector and eigenvalues of the constant coef®cient matrices A as was done in Chapter 2. Let qn and Vn be the eigenvalues and eigenvector of the matrix A. We have AVn ˆ qn Vn

…3:2:17†

The general solution for the electromagnetic ®eld in the grating region can be found from the state variable solution. The electric ®eld associated with the nth eigenvector mode is given by

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Planar Diffraction Gratings

!e En ˆ

(

MT X MT

109

) ‰Szin a^ z Š exp… jkxi x† exp…qn y†

…3:2:18†

where Szin , i ˆ MT ; . . . ; MT , correspond to the electric ®eld part of the eigenvector Vn . The magnetic ®eld associated with the nth eigenvector mode similarly is given by !e !e U n ˆ 0 H n ˆ

(

MT X  MT

)



Uxin a^ x ‡ Uyin a^ y exp… jkxi x† exp…qn y†

…3:2:19†

where Uxin , i ˆ MT ; . . . ; MT corresponds to the magnetic ®eld part of the eigenvector Vn and Uyin is found from Eq. 3.2.11. Summing over the individual eigenmodes we ®nd that NT MT X !e !…2† X Cn E n ˆ E ˆ

(

mˆ MT

nˆ1

NT X nˆ1

) Cn ‰Szin a^z Š exp…qn y† exp… jkxi x† …3:2:20†

T !…2† X !e !…2† U ˆ 0 H ˆ Cn U n

N

ˆ

MT X mˆ MT

(

nˆ1

NT X nˆ1

)

Cn ‰Uxin a^ x ‡ Uyin a^y Š exp…qn y† exp… jkxi x† …3:2:21†

Equations 3.2.20 and 3.2.21 represent NT ˆ 2…2MT ‡ 1† forward and backward traveling, propagating and nonpropagating eigenmodes, which when summed together give the general electromagnetic ®eld solution in Region 2, the grating region. An important problem that remains is to determine the NT coef®cients Cn of Eqs. 3.2.20 and 3.2.21. Up to this point we have speci®ed the general form of the diffracted ®elds in the grating region. The EM ®elds on the incident side of the diffraction grating (Region 1 of Fig. 1), and on the transmission side of the diffraction grating (Region 3 of Fig. 1), consist of an in®nite number of propagating and nonpropagating plane waves whose tangential wave vectors are given by kxi , i ˆ 1; . . . ; 1; 0; 1; . . . ; 1. The EM ®elds in Region 1 consist of a single incident H-mode polarized wave making an angle  with the y-axis and consist of an in®nite number of

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110

Chapter 3

re¯ected propagating and evanescent H-mode polarized plane waves. The tangential incident ®eld in Region 1 is given by …1† Ezinc ˆ Eo i0 e

ky1i e 0

…1† ˆ Hxinc

where n1 ˆ

jkxi x‡jky1i y

E0 i0

jkxi x‡jky1i y

…3:2:22†

E0 i0

…3:3:23†

p 1 1 is the index of refraction

…1† ˆ Ezref …1† Hxref ˆ

1 X iˆ 1

ri e

jkxi x jky1i y

1 1 X k re 0 iˆ 1 y1i i

…3:2:24†

jkxi x jky1i y

…3:2:25†

where ( ky1i ˆ

‰n21

k2xi Š1=2

j‰k2xi

n1 > kxi

n21 Š1=2

kxi > n1

…3:2:26†

as y ˆ ‡1, we note for kxi > n1 , e

j‰ j‰k2xi n21 Š1=2 Šy

ˆe

‰k2xi n21 Š1=2 Šy

…3:2:27†

! 0 as y ! 1 and thus the evanescent ®elds meet proper boundary conditions as y ! 1. The total tangential ®elds in Region 1 are given by …1† …1† Ez…1† ˆ Ezinc ‡ Ezref

…3:2:28†

…1† …1† Ux…1† ˆ 0 Hx…1† ˆ 0 …Hxinc ‡ Hxref †

…3:2:29†

In the transmitted region y < ®elds are given by Ez…3† ˆ

1 X iˆ 1

ti e

Ux…3† ˆ 0 Hx…3† ˆ

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L the tangential electric and magnetic

jkxi x‡jky3i …y‡L† 1 X iˆ 1

ky3i ti e

…3:2:30† jkxi x‡jky3i …y‡L†

…3:2:31†

Planar Diffraction Gratings

111

where  ky3i ˆ as y ! ej‰

‰n23 k2xi Š1=2 j‰k2xi n23 Š1=2

n3 > kxi kxi > n3

…3:2:32†

1, we note for kxi > n3 j‰k2xi n23 Š1=2 Š…y‡L†

2

ˆ e‰kxi

n23 Š1=2 Š…y‡L†

…3:2:33†

! 0 as y ! 1. We thus see that the evanescent ®elds in Region 3 meet the proper boundary conditions as y ! 1. Now that the EM ®elds have been de®ned in Regions 1, 2, and 3, the next step is to match boundary conditions at the interfaces y ˆ 0 and y ˆ L. At the y ˆ 0 interface we have Ez…1† Hx…1†

yˆ0‡ yˆ0‡

ˆ Ez…2†

…3:2:34†

yˆ0

ˆ Hx…2†

…3:2:35†

yˆ0

Substituting Eqs. 3.2.28 and 3.2.29 and keeping orders of jij  MT , we ®nd that MT X

fE0 i0 ‡ ri ge

jkxi x

f ky1i i0 E0 ‡ ky1i ri ge

jkxi x

iˆ MT MT X iˆ MT

At the y ˆ Ez…2† Hx…2†

iˆ MT

ˆ

MT X iˆ MT

( (

NT X nˆ1 NT X nˆ1

) Cn Szin e

jkxi x

…3:2:36†

) Cn Uxin e

jkxi x

…3:2:37†

L boundary we have

yˆ L‡ yˆ L‡

Copyright © 2000 Marcel Dekker, Inc.

ˆ

MT X

ˆ Ez…3†

yˆL

ˆ Hx…3†

yˆL

…3:2:38† …3:2:39†

112

Chapter 3

(

MT X

NT X

iˆ MT

nˆ1

MT X

Nt X

(

iˆ MT

nˆ1

) Cn Szin e

qn L

Cn Uxin e

qn L

e

jkxi x

ˆ

MT X iˆ MT

) e

jkxi x

ˆ

MT X iˆ MT

fti ge

jkxi x

…3:3:40†

f ky3i ti ge

jkxi x

…3:3:41†

In Eqs. 3.2.36 and 3.2.37 and Eqs. 3.2.40 and 3.2.41, in order for the leftand right-hand side of the equations to agree, it is necessary for the Fourier coef®cients of e jkxi x to agree for each Floquet harmonic e jkxi x . Thus for the unknown coef®cients ri , Cn , and ti we have the equations E0 i0 ‡ ri ˆ ky10 i0 E0 ‡ ky1i ri ˆ NT X

nˆ1

nˆ1 NT X nˆ1

Cn Szin e

qn L

ˆ ti

Cn Uxin e

qn L

ˆ

nˆ1 NT X

NT X

Cn Szin

…3:2:42†

Cn Uxin

…3:2:43† …3:2:44†

ky3i ti

…3:2:45†

for i ˆ MT ; . . . ; MT . We notice in Eqs. 3.2.42 and 3.2.44 that the ri and ti variables can be eliminated. These equations can be simpli®ed by substituting ri and ti of Eqs. 3.2.42 and 3.2.44, respectively, into Eqs. 3.2.43 and 3.2.45. We have " ky10 i0 E0 ‡ ky1i

E0 i0 ‡ NT X nˆ1

NT X nˆ1

# Cn Szin ˆ

Cn Uxin e

qn L

ˆ

NT X nˆ1

Cn Uxin

ky3i

"

NT X nˆ1

…3:2:46† #

Cn Szin e

qn L

…3:2:47†

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Planar Diffraction Gratings

113

or altogether NT X nˆ1 NT X nˆ1

 Cn e

Cn fky1i Szin

qn L

Uxin g ˆ 2E0 ky10 i0

‰Uxin ‡ ky3i Szin Š ˆ 0

…3:2:48† …3:2:49†

where i ˆ MT ; . . . ; MT . The above constitutes a set of NT ˆ 2…2MT ‡ 1† equations for the NT unknown coef®cients Cn . Power is excited in the diffraction grating system through the 2E0 ky10 i0 term on the right-hand side of Eq. 3.2.48. Once the Cn are determined, the ri and ti can be found form Eqs. 3.2.42 and 3.2.44.

3.2.2

RCWA Wave Equation Method

A different way of analyzing the diffraction from a grating in the H-mode case under consideration is to eliminate the magnetic ®eld from Maxwell's equations directly and then analyze the second-order partial differential for the electric ®eld that results. In the analysis to be presented it will be assumed that the dielectric permittivity is a sinusoidal one. In this section we will follow the formulation of Moharam and Gaylord's [16] original paper but use the geometry of Fig. 1. We refer to this formulation as a wave equation formulation as it is based on placing the wave equation in state variable form and proceeding with the solution from that point. ! To start the analysis we assume that E ˆ Ez …x; y†a^ z in all regions and that all ®elds are independent of z. In Region 2 in normalized coordinates ~ y ˆ k0 y, ~ and z ˆ k0 z~ we have x ˆ k0 x, ! r E ˆ

! j…0 H †

…3:2:50†

! ! r  …0 H † ˆ j…x† E ! rr E ˆ

! ! jr  …0 H † ˆ …x† E

! ! r  r  E ˆ rr
E ! @E r
E ˆ z ˆ0 @z

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…3:2:51† ! r2 E

…3:2:52† …3:2:53† …3:2:54†

114

Chapter 3

Therefore we have r2 Ez ‡ …x†Ez ˆ 0

…3:2:55†

or since @2 Ez =@z2 ˆ 0, we have @2 @2 Ez E ‡ ‡ …x†Ez ˆ 0 z @x2 @y2

…3:2:56†

For the present analysis we will let …x† ˆ 2 ‡
 cos Kx and take  ˆ 1. K is a normalized wave number Kˆ

2 2 K~ ˆ ˆ ~  ko ko 

…3:2:57†

~ is the normalized grating period of the system. To start the and  ˆ k0  analysis we expand the electric ®eld of Region 2, namely Ez , in the Floquet harmonic series Ez ˆ

1 X iˆ 1

Si …y† exp…j i †

k~2 p cos  0 ˆ 2 cos  0 ko p i ˆ 1 sin  iK

2 ˆ

…3:2:58†

i

ˆ

i x ‡ 2 y

…3:2:59† …3:2:60†

where i ˆ . . . ; 1; 0; 1; . . . and  0 is the angle of light refracted into the dielectric grating. Si …y† are Floquet modal amplitudes that need to be determined. Differentiating Ez with respect to y and x, we ®nd that " 1 X @2 @2 @ E ˆ Si …y† ‡ 2j2 Si …y† z 2 2 @y @y @y iˆ 1 X 2  @2 E ˆ i Si …y† exp…j i † z 2 @x i

Copyright © 2000 Marcel Dekker, Inc.

# 22 Si …y†

exp…j i †

…3:2:61† …3:2:62†

Planar Diffraction Gratings

115

The term  
 ‰exp… jKx† ‡ exp…jKx†Š …x†Ez ˆ 2 ‡ 2 1 X iˆ 1

…3:2:63†

Si …y† exp… j i x ‡ j2 y†

equals …x†Ez ˆ 2 ‡

1 X iˆ 1

Si …y† exp…j i † ‡

1
 X S …y† exp… j… i ‡ K†x ‡ j2 y† 2 iˆ 1 i

1
 X S …y† exp… j… i 2 iˆ 1 i

K†x ‡ j2 y† …3:2:64†

The terms in the exponential factors can be manipulated to give
p 1 sin  iK ‡ K p i ‡ K ˆ 1 sin  …i 1†K† ˆ i i ‡ K ˆ

…3:2:65† 1

…3:2:66†

Similarly i

K ˆ i‡1

…3:2:67†

The second term of Eq. 3.2.64 can be rewritten T2 

1
 X S …y† exp… j… i ‡ K†x ‡ j2 y† 2 iˆ 1 i

1
 X S …y† exp… j i 1 x ‡ j2 y† ˆ 2 iˆ 1 i

…3:2:68†

In this right-hand side summation we will made the substitution i 0 ˆ i Doing this we obtain T2 ˆ

1
 X S 0 …y† exp… j i 0 x ‡ j2 y† 2 i 0 ˆ 1 i ‡1

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1.

…3:2:69†

116

Chapter 3

Similarly the third term of Eq. 3.2.64 can be written T3 ˆ

1
 X S …y† exp… j… i 2 iˆ 1 i

K†x ‡ j2 y† …3:2:70†

1
 X S 0 …z† exp… j i 0 x ‡ j2 y† ˆ 2 i 0ˆ 1 i 1

Substituting T2 and T3 into Eq. 3.2.64 and using i instead of i 0 in summation we ®nd that  1  X

 S …y† ‡ S …y† exp…j i † 2 Si …y† ‡ …x†Ez ˆ 2 i‡1 2 i 1 iˆ 1

…3:2:71†

Substituting into the original differential equation for Ez we ®nd that 0ˆ

1 X iˆ 1

(

@2 @ Si …y† ‡ 2j2 Si …y† @y @y2

22 Si …y†

2i Si …y† …3:2:72†



 ‡2 Si …y† ‡ S …y† ‡ S …y† exp…j i † 2 i‡1 2 i 1

The only way that the above equation can be zero for all values of x and y is if the curly bracketed expression is zero. Thus Eq. 3.2.72 describes a series of coupled modal amplitude equations to determine the EM ®elds of the system. At this point it is useful to introduce scaled coordinates into analysis. We let j
 p y~ ˆ 2 2

uˆ

j
 p y ˆ 2 2 ko

j
 p y ˆ 4 2

jy

…3:2:73†

Substituting the above scaling into Eq. 3.2.72 we ®nd after algebra that
 d 2 Si dS ‡ cos  0 i du 82 du2

i‰i

BŠSi ‡ Si‡1 ‡ Si

1

ˆ0

…3:2:74†

where Bˆ

~ p ~ p 2 2 2 1 sin  0 ˆ sin   

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…3:2:75†

Planar Diffraction Gratings

117

and ˆ

22 ~ 2
 

…3:2:76†

and the last equation of B follows from Snell's law, p p 2 sin  0 ˆ 1 sin 

…3:2:77†

If we further let a ˆ 82 =
, bi ˆ the equation as

B†a, c ˆ a cos  0 , we can rewrite

i…i

1 d 2 Si c dSi bi ‡ ‡ Si ‡ Si‡1 ‡ Si a a du2 a du

1

ˆ0

…3:2:78†

Equation 3.2.78 is a second-order coupled differential equation. It can be put into the form of a ®rst-order state variable equation, if the following new variables are de®ned. Let S1i ˆ Si …u† S2i ˆ

…3:2:79†

dSi …u† du

…3:2:80†

Making these substitutions we ®nd that the second order Eq. 3.2.78 can be written as dS1i ˆ S2i du dS2i ˆ aS1i‡1 ‡ bi S1i ‡ aS1i du

…3:2:81† 1

‡ cS2i

…3:2:82†

If we differentiate Eq. 3.2.81 with respect to u, we ®nd that d 2 S1i dS2i ˆ aS1i‡1 ‡ bi S1i ‡ aS1i ˆ du du2

1

‡ cS2i

…3:2:83†

Dividing Eq. 3.2.83 by a, transferring the second derivative term to the right-hand side, and substituting the original de®nitions of S1i and S2i , we have

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118

Chapter 3

1 d2 b S ‡ Si‡1 ‡ i Si ‡ Si a du2 i a

1

‡

c dSi ˆ0 a du

…3:2:84†

This is identical to Eq. 3.2.78, thus showing that Eq. 3.2.83 is the correct ®rst-order state variable form of Eq. 3.2.78. The full matrix form for Eqs. 3.2.80 and 3.2.81 when written out for MT ˆ 2 is    d S1 A11 ˆ S A21 du 2

A12 A22



S1 S2

 …3:2:85†

where  t S1 ˆ S1; 2 S1; 1 S1;0 S1;1 S1;2  t S2 ˆ S2; 2 S2; 1 S2;0 S2;1 S2;2   A11 ˆ 0 55   A12 ˆ I ˆ ii 0 55 3 2 a 0 0 0 b 2 7 6 6 a b 1 a 0 07 7 6 7 6 6 a bo a 0 7 A21 ˆ 6 0 7 7 6 7 6 0 0 a b a 1 5 4 0 0 0 a b2 55   A22 ˆ cii 0 55 ( 1; i ˆ i0 i;i 0 ˆ 0; i 6ˆ i 0

…3:2:86† …3:2:87† …3:2:88† …3:2:89†

…3:2:90†

…3:2:91† …3:2:92†

If we let V ˆ ‰S1 S2 Št and  Aˆ

A11 A21

A12 A22

 1010

ˆ ‰aii 0 Š1010

…3:2:93†

aii 0 …i; i 0 † ˆ …1; . . . ; 10† represent the individual matrix elements of the overall matrix A. Using the just de®ned matrices, Eq. 3.2.85 can be written in full state variable form as

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Planar Diffraction Gratings

119

d V ˆ AV du

…3:2:94†

If we let Vn and qn be the eigenvalues and eigenvectors of the matrix A, then we ®nd that the solution for Si …u† and S1i …u† is given by Si …u† ˆ S1i …u† ˆ

NT X nˆ1

Cn win exp…qn u†

…3:2:95†

where NT ˆ 2…2MT ‡ 1† and where win represents the ith row of the nth eigenvector …S1 †n . The electric ®eld Ez is given by Eq. 3.2.58 with Si …u† substituted. We have Ez ˆ

MT X iˆ MT

( exp‰ j… i x

2 y†Š

NT X nˆ1

) Cn win exp‰ jqn yŠ

…3:2:96†

p where, as already de®ned,  ˆ
=…4 2 ). To proceed further it is necessary to ®nd the magnetic ®eld associated with Ez . Using Maxwell's equations, the tangential magnetic ®eld Hx is found from Hx ˆ

1 @Ez j0 @y

…3:2:97†

Altogether the tangential electromagnetic ®elds in Region 2, the diffraction grating region, are given by (including now the Region 2 subscript label) Ez2 ˆ

MT X NT X iˆ MT nˆ1

Ux2 ˆ 0 Hx2 ˆ

Cn win exp MT X Nt X

iˆ Mt nˆ1



j‰ i x

Cn win ‰…qn 

…2

qn †y

2 †Š exp





…3:2:98† j‰ i x

…2

qn †yŠ



…3:2:99† The differential equation method provides an alternate state variable representation from which to obtain the electromagnetic ®elds of Region 2. Although the state variable representations of Sections 3.2.2 and 3.2.3 are exactly equal as MT ! 1, the two representations give different solutions for ®nite MT . Thus a comparison of the two methods for different values of MT gives a good measure of how well both representations are converging.

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120

Chapter 3

The ®nal matrix equations for Cn of this section can be found by matching the tangential electromagnetic ®elds as given in Section 3.2 by Eqs. 3.2.28±32 for Regions 1 and 3 with the EM ®eld solutions of Region 2 that have been just derived. The ®nal matrix equations from which that result for Cn are 2ky10 E0 io ˆ 0ˆ

NT X nˆ1 NT X nˆ1

Cn win ‰ky1i ‡ 2

qn Š

Cn en win ‰ ky3i ‡ 2

qn Š

…3:2:100† …3:2:101†

where en ˆ exp‰j…qn 

2 †LŠ

…3:2:102†

The re¯ection and transmission coef®cients are given by ri ˆ ti ˆ

NT X nˆ1 NT X nˆ1

Cn win

E0 i0

Cn win en

…3:2:103† …3:2:104†

These equations have been presented in Ref. 16. 3.2.3

Numerical Results

In this section we will present numerical examples of the diffraction ef®ciency and complex Poynting Theorem power balance as results from RCWA. The examples to be presented consist of an RCWA study of a cosine diffraction grating (lossless and lossy bulk dielectric cases) and an RCWA study of a square or step pro®le diffraction grating. Both gratings, consistent with the theory of this section, are assumed to be homogeneous in the longitudinal direction. These two gratings have been chosen because the cosine grating is relatively smooth, containing low spatial frequencies i ˆ 1; 0; 1, whereas the square wave or step pro®le contains sharp dielectric discontinuities at dielectric steps and thus possesses a high spectral content i ˆ 1; . . . ; 1; 0; 1; . . . ; 1. The complex Poynting energy balance is based on the formulation presented in Section 2.2.3, Eq. 2.2.26 with the source terms set to zero. The complex Poynting box taken to include a

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Planar Diffraction Gratings

121

~ (see Fig. 7.) Details of the calculation are given in transverse wave period  Section 3.3. We begin presenting results for the cosine grating. The solid line plots in Fig. 2 show the transmitted diffraction ef®ciencies DET …%† for ®ve orders i ˆ 2; 1; 0; 1; 2 as calculated by the full ®eld method (see Section 3.2.1, Eqs. 3.2.48 and 3.2.49) using the lossless cosine grating as speci®ed in Fig. 2 inset and heading. These plots show DET …%† versus the layer length L~ (in units of free space wavelength ). As can be seen from Fig. 2, as the layer length L~ increases from 0 to 9, because  is at the Bragg angle (implying Bragg incidence), power is primarily diffracted from the i ˆ 0 order into the i ˆ 1 order with a small amount of power being diffracted into the other ~ 9 to 18, power is diffracted orders i ˆ 2; 1; 2. For larger values of L, from the i ˆ 1 order into the i ˆ 0 order with a small amount of power being diffracted into the other orders i ˆ 2; 1; 2. This cycle is repeated over a long range of L~ values. Because the bulk regions had matched permittivities, the re¯ected diffractions were small and have not been plotted. Also shown in Fig. 2 is the DET …%† as calculated by a differential equation, the state variable method described in Section 3.2.2 and derived originally in Ref. 16 (dots, i ˆ 0). In this analysis, Maxwell's equations are reduced to a second-order differential equation for the electric ®eld, and this differential equation is put in state variable form. The state variable form that results is

Figure 2 The transmitted diffraction ef®ciencies DET …%† for ®ve orders i ˆ 2; 1; 0; 1; 2 as calculated by Eqs. 3.2.48 and 3.2.49 using a lossless cosine grating.

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122

Chapter 3

different from the present one, although as MT ! 1 the two methods are mathematically equivalent. As can be seen from Fig. 2, a comparison of the i ˆ 0 order plots (typical of all orders) shows that virtually identical results occur from the use of the two methods. Figures 3 and 4 show plots of the real and imaginary parts of the normalized complex power PIN (line) and PBOX (dot) of the complex Poynting theorem, ®rst introduced in Chapter 2 (using the Poynting box shown in Fig. 15). For more detail on the application of the Poynting theorem to gratings, see the next subsection. As mentioned earlier, this case represents a lossless diffraction grating, bulk dielectric case. In these ~ As can be seen plots the complex power is plotted versus the layer length L. from Figs. 3 and 4, excellent agreement in both plots is obtained from the calculation. Figure 5 shows a plot of the electric and magnetic energies PWE and PWM versus layer length L~ that results for the example under consideration. As can be seen from Fig. 5, the electric and magnetic energies are very nearly equal to one another, and in a L~ ˆ 1 size slab, the electric and magnetic energies PWE and PWM are much larger than the peak magnitude energy difference between the two energies. Figure 6 shows the Im…PBOX † versus layer length L~ when the electromagnetic ®elds are computed using MT ˆ 3 and MT ˆ 6. As can be seen

Figure 3 The real part of the normalized complex power PIN and PBOX of the complex Poynting theorem.

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Figure 4 The imaginary part of the normalized complex power PIN and PBOX of the complex Poynting theorem.

Figure 5 Plots of the electric and magnetic energies PWE and PWM versus layer length are shown.

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124

Chapter 3

Figure 6 The Im…PBOX † versus layer length L~ when the electromagnetic ®elds are computed using MT ˆ 3 and MT ˆ 6.

from Fig. 6, extremely good convergence is observed using the two different truncation sizes. Figures 7 and 8 show plots of the real and imaginary parts of the complex power PIN and PBOX versus layer length L~ when the diffraction grating bulk dielectric 2 is lossy rather than lossless and has a value of 2 ˆ 1 j0:02. In this ®gure one again observes extremely good agreement between the real and imaginary parts of PIN and PBOX , again showing that the complex Poynting theorem is obeyed to a high degree of accuracy. A comparison of Figs. 3 and 4 (lossless case) with Figs. 7 and 8 (lossy case) shows a very clear difference in the shapes of the real and imaginary parts of PIN and PBOX that is being computed in the four ®gures. In the lossy case, as L~ increases, the envelope of the oscillations of PIN and PBOX damps out, whereas in the lossless case the envelope maintains a longitudinal periodic shape. The damping of the envelope with increasing L~ in the lossy case is expected, since as the layer length increases, the EM ®elds in the system attenuate near the exit side of the diffraction grating due to the lossiness. When the diffraction grating becomes suf®ciently long, the EM ®elds at the exit side approach zero; therefore PIN and PBOX become ~ and thus there is no oscillation. independent of L,

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Figure 7 Plots of the real part of the complex power PIN and PBOX versus layer length L~ of the complex Poynting theorem when the diffraction grating bulk dielectric 2 is lossy rather than lossless and has a value of w ˆ 1 j0:02.

Figure 8 Plots of the imaginary part of the complex power PIN and PBOX , versus layer length L~ of the complex Poynting theorem when the diffraction grating bulk dielectric 2 is lossy rather than lossless and has a value of 2 ˆ 1 j0:02.

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126

Chapter 3

Figure 9 shows a plot of the power dissipated PD , Re…PIN †, and Re ~ The ripple in the Re…PIN † observed in Fig. 7 is …POUT † versus layer length L. not observed here because of the scale of the Fig. 9 plot. As can be seen from Fig. 9, the sum of power radiated out of the Poynting box and the power dissipated is balanced by the real power radiated into the box as one would physically expect. Figure 10 shows the transmitted diffraction ef®ciency (i ˆ 0 and i ˆ 1 orders) versus layer length L~ that arises when a plane wave is incident on a square wave or step pro®le dielectric grating. In the present ®gure, diffraction ef®ciency results are presented for two cases, namely when the diffraction grating region contains lossless dielectric material and the case when the grating contains lossy dielectric material. The square wave grating in both cases is taken to have a grating period of  ˆ  and a transverse groove width of =2 (or duty cycle of 50%). The bulk and groove dielectric values and their orientation in the diffraction grating and the angle of incidence are speci®ed in the Fig. 10 title and inset. The lossless case presented is the same case presented by Moharam and Gaylord [19]. As can be seen from Fig. 10, the presence of the lossy dielectric material in the diffraction grating for the lossy case causes a signi®cant drop in the size of the transmitted diffraction

Figure 9 Plot of the power dissipated PD , Re…PIN †, and Re…POUT † versus layer length L~ is shown. The ripple in the Re…PIN † observed in Fig. 7 is not observed here because of the scale of the Fig. 9 plot.

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Figure 10 The transmitted diffraction ef®ciency (i 0 and i ˆ 1 orders) versus layer length L~ that arises when a plane wave is incident on a square wave or step pro®le dielectric grating is shown. The diffraction ef®ciency results are presented for lossless and lossy cases. The square wave grating in both cases is taken to have a grating period of  ˆ  and a transverse groove width of =2 (or duty cycle of 50%).

ef®ciency as the layer length L~ increases, which is observed in the system as compared to the lossless case. Note that ef®cient coupling between 0 and 1 orders is possible in spite of the high spectral content and modulation depth of the grating, as long as incidence is at the Bragg angle. Higher diffracted orders in this case are all evanescent. Figure 11 shows the re¯ected diffraction ef®ciency results (i ˆ 0 and i ˆ 1 orders) versus layer length L~ which arises for the case under consideration. In these ®gures one observes a perceptible difference between the lossless and lossy diffraction grating cases. The reduction in the peak-to-peak envelopes in the lossy re¯ected diffraction ef®ciencies with increasing L~ is due to the fact that the EM ®elds near the transmit side of the diffraction grating are attenuating more strongly as L~ becomes larger, and thus re¯ected EM radiation is less sensitive to the layer length, which is then seen as a reduction in the ripple of the lossy diffraction ef®ciency results. Figures 12±13 show the real and imaginary parts of PIN and PBOX versus L~ for the lossless square wave diffraction case under study.

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Chapter 3

Figure 11 The re¯ected diffraction ef®ciency results (i ˆ 0 and i ˆ 1 orders) versus layer length L~ that arises for the case under consideration is shown.

Figure 12 The real part of PIN and PBOX versus L~ for the lossless square wave diffraction cases that were studied in Figs. 10 and 11.

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Figure 13 The imaginary part of PIN and PBOX versus L~ for the lossless square wave diffraction cases that were studied in Figs. 10 and 11.

3.2.4

Diffraction Grating±Mirror

Another important EM case that can be studied using an RCWA analysis consists of determining the EM ®elds that are diffracted when an H-mode polarized plane wave is incident on a diffraction grating backed by a mirror (also called a short circuit plate). Thus Region 3 is an electrical perfect conductor rather than a dielectric material (see the inset of Fig. 14). The analysis for this case is identical to that presented in Section 3.2.1 except that inside Region 3 (y < L) the EM ®elds are taken to be zero, and at the Region 2±3 interface …y ˆ L† the EM ®elds are required to meet the wellknown boundary condition that the tangential electric ®elds are zero. Mathematically for the present H-mode polarization case, this requires Ez…2† …x; y; z†

yˆ L‡

ˆ0

…3:2:105†

If this boundary condition is imposed, it is found that the overall matrix equations that must be solved to determine the EM ®eld of the grating mirror system are

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130

Chapter 3

Figure 14 The re¯ected diffraction ef®ciency (i ˆ 0 and i ˆ 1 orders) versus layer length L~ that arises when a plane wave is incident on a square wave or step pro®le dielectric grating that is backed by a mirror (or short circuit plate). The same lossless square grating that was studied in Fig. 10 is analyzed here.

NT X nˆ1

 Cn ky1i Szin NT X nˆ1

 Cn e

Uxin ˆ 2Eo ky1o io

…3:2:106†

Szin ˆ 0

…3:2:107†

qn L

where i ˆ MT ; . . . ; MT . Equation 3.2.106 is identical to Eq. 3.2.48 as it should be since it results from matching EM ®elds at the Region 1±2 interface, and the general form of the unknown EM ®elds in the two regions is the same whether the mirror is present or not. The second matrix equation, Eq. 3.2.107, is quite different from the second transmission grating matrix equation, Eq. 3.2.49. Equation 3.2.107 was determined by imposing the boundary condition that the tangential electric ®eld at a perfect conductor boundary is zero, whereas Eq. 3.2.49 was determined by matching the electric and magnetic ®elds at y ˆ L and then eliminating the electric ®eld unknown coef®cients. Substitution of the Cn coef®cients of Region 2 into Eq. 3.2.42 then allows for the determination of the ri re¯ection coef®cients

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131

of the system. Incident and re¯ected power are given by the same formulas as already given for the transmission grating analysis. Figure 14 shows the re¯ected diffraction ef®ciency …i ˆ 0 and i ˆ 1 orders) versus layer length L~ that arises when a plane wave is incident on a square wave or step pro®le dielectric grating backed by a mirror (or short circuit plate). In Fig. 14 the same lossless square grating that was studied in Fig. 10 is analyzed. The square wave grating was taken to have a grating ~ ˆ  and a transverse groove width of =2 (or duty cycle of period of  50%). The bulk and groove dielectric values and their orientation in the diffraction grating and the angle of incidence are speci®ed in the Fig. 14 caption and inset. For the present case, for the angle of incidence used, it turns out that the i ˆ 0; 1 orders are the only orders that are re¯ected, diffracted propagating plane waves. All the other orders are evanescent. the value of MT ˆ 6 was used to calculate the data of Fig. 14. As can be seen, power for a small grating thickness is diffracted from the i ˆ 0 order into the i ˆ 1 …0  L~  0:6†. As the thickness increases, however, power is transferred back to the i ˆ 0 order from the i ˆ 1, 1  L~  1:6. This cycle ~ In observing the i ˆ 0; 1 plots it is very is repeated for larger values of L. interesting to note that the transfer of power between the i ˆ 0; 1 orders is not periodic with increasing L~ but irregular and unpredictable. The nonperiodicity is undoubtedly due to interaction of the evanescent and propagating waves that resultP from the matrix solution. Conservation of incident and re¯ected power … i ‰DERi ‡ DETi Š ˆ 1, DETi ˆ 0† was observed to a high degree of accuracy.

3.3

APPLICATION OF RCWA AND THE COMPLEX POYNTING THEOREM TO E-MODE PLANAR DIFFRACTION GRATING ANALYSIS

In the previous section, RCWA was used to study H-mode polarization as it diffracts from isotropic diffraction gratings. In many real-life applications, it is necessary to study diffraction from anisotropic gratings, e.g., photorefractive materials (discussed in detail in Chapter 7). In this section RCWA and the complex Poynting theorem will be used to study, respectively, the EM ®elds and power ¯ow and energy storage when a plane wave (E-mode polarization) is scattered from an in general lossy and anisotropic diffraction grating. Full calculation of the diffraction ef®ciency, the electromagnetic energy (electric and magnetic), and the real, reactive, dissipative, and evanescent power of the grating will be made. In this section several numerical examples involving a step pro®le will be studied.

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132

Chapter 3

The grating in Fig. 15a is assumed to have its grating vector speci®ed ! ~ and  ~ is the grating period or grating waveby K ˆ K~ a^ x , where K~ ˆ 2= length. In this case the magnetic ®eld is assumed to be polarized perpendi! cular to the plane of incidence as H ˆ Hz a^ z . In the present study, the complex Poynting theorem will be applied to a Poynting box whose length ~ whose width extends over a grating period extends over the grating region L, ~ and whose thickness is
z~ (the electromagnetic ®elds do not vary in the z, direction, so the thickness of the Poynting box is immaterial to the Poynting power calculation). Figure 15b illustrates the Poynting box of this section and that of Sec. 3.2 as well. In Section 3.3.1, we will brie¯y summarize the Emode RCWA equations for anisotropic diffraction gratings. In Section 3.3.2, the pertinent equations for the power budget as results from the complex Poynting theorem will be presented. In Section 3.3.3, illustrative examples will be given for anisotropic media where the permittivity tensor is either Hermitian or arbitrary. In much of the existing diffraction grating literature [1±53], power conservation is veri®ed by calculating the time-averaged real power transmitted and re¯ected from a lossless grating and then verifying that the sum of these powers equals the power incident on the grating. Computing the

Figure 15 (a) The geometry of the E-mode diffraction grating system is shown. (b) The complex Poynting box used for calculations is shown.

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power budget using only the time-averaged real power has two large limitations associated with it. First, it cannot be used to verify power conservation for the very common case of lossy gratings, since in this case some power is dissipated as heat, and thus the transmitted and re¯ected powers will not equal the incident power. A second limitation of computing the power budget using only the time-averaged real power is that information about the reactive power, evanescent ®elds, electric energy, magnetic energy, and power dissipated within the grating is left undetermined and therefore unknown. All of these quantities contain important information about the nature and behavior of the grating. In the area of near ®eld optics, considerable attention has been paid to evanescent waves, since these carry information about the diffracting or scattering object. Speci®cally, evanescent wave monitoring has applications in the area of submicron microscopy. Evanescent waves may also be excited from sharp discontinuities in the grating, e.g., corners, blaze tips [95]. A power budget approach that can study energy and power, both real and reactive, during diffraction from such gratings, is incorporated in the framework of the complex Poynting theorem. Botten et al. [58] consider the problem of energy balance in isotropic lossy gratings when both E-mode and H-mode polarized incidence plane waves impinge on the grating. Energy ¯ow distributions, the generation of plasmon surface waves, and the absorption of EM energy by metallic sinusoidal gratings has been studied by Popov et al. [98±101] for shallow and deep gratings. The nature of the Poynting vector in a dielectric sinusoidal grating under total internal re¯ection has been studied by Shore et al. [102]. Our discussion of the Poynting vector is fundamentally different from that of Popov et al. [98±101] and Shore et al. [102]. In their work they were concerned with the problem of studying the spatial variation of the Poynting vector (and energy density) on a point-to-point basis over a region of space close to the diffraction grating surface. The point of their work was to relate local variation of the Poynting vector to the diffraction that occurred from the grating. They studied the physical mechanisms of blazing and antiblazing and its relation to Poynting vector. In this section, we focus on the Poynting vector power that has been averaged transversely over a diffraction grating period and relate this averaged Poynting power to the power dissipated, transmitted, and re¯ected from the grating [105]. We apply the complex Poynting theorem for EM incidence on periodic diffraction gratings of arbitrary pro®le and made of anisotropic lossy materials. We explicitly show also that the energy dissipated in the grating can result from both imaginary and real parts of the permittivity and permeability for the case of anisotropic nonreciprocal grating media.

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134

3.3.1

Chapter 3

E-Mode RCWA Formulation

We begin the analysis by determining the general EM ®eld solution of Region 2, the diffractive grating region. Using normalized coordinates ~ y ˆ k0 y, ~ and z ˆ k0 z, ~ and where k0 ˆ 2= and  (meters) where x ˆ k0 x, is the free space wavelength, we ®nd that Maxwell's normalized equations in Region 2 are given by ! r E ˆ

! j…0 H †

…3:3:1†

! ! r  …0 H † ˆ j E

…3:3:2†

p where 0 ˆ 0 =0 ˆ 377 is the intrinsic impedance of free space,  ˆ ~ 0 is the relative permeability of Region 2, 0 is the permeability of free = space,  ˆ ~ =0   0 j 00 is the relative tensor permittivity of Region 2, and 0 is the permittivity of free space. In this section we consider the important case when the relative permittivity tensor is anisotropic and has the speci®c form 2

xx  ˆ 4 yx 0

xy yy 0

3 0 0 5 zz

…3:3:3†

We expand the electric and magnetic ®eld as 1  X  ! E ˆ Sxi …y†a^ x ‡ Syi …y†a^ y exp… jkxi x†

…3:3:4†

iˆ 1

1 X ! ! Uzi …y† exp… jkxi x†a^z U  0 H ˆ

…3:3:5†

iˆ 1

kxi ˆ kx0

iKx

kx0 ˆ

p 1 1 sin…†

Kx ˆ 2=

~  ˆ k0  …3:3:6†

~ is the grating wavelength. Letting where  is the angle of incidence and  …x† represent any of the elements of the tensor  of Eq. 3.3.3, we also expand those permittivity elements as …x† ˆ

1 X iˆ 1

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i exp…jiKx x†

…3:3:7†

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135

where i represent the Fourier coef®cients of …x†. Substituting Eqs. 3.3.4±7 in Maxwell's equations; taking the relative permeability of Region 2 to be  ˆ 1; introducing column and square matrices, namely, Sx ˆ ‰Sxi Š, Uz ˆ ‰Uzi Š, i ˆ 1; . . . ; 1, xx ˆ ‰xxi;i 0 Š ˆ ‰xxi i 0 Š, Sy ˆ ‰Syi Š, xy ˆ ‰xyi;i 0 Š ˆ ‰xyi i 0 Š yx ˆ ‰yxi;i 0 Š ˆ ‰yxi i 0 Š, and yy ˆ ‰yyi;i 0 Š ˆ ‰yyi i 0 Š (here the underbar denotes a square …i; i 0 † matrix), Kx ˆ ‰kxi i;i 0 Š, I ˆ ‰i;i 0 Š, …i; i 0 † ˆ 1; . . . ; 1 square matrices; i;i 0 the Kronecker delta and I the identity matrix; eliminating Sy using the equation Sy ˆ yy1 …Kx Uz

yx Sx †

…3:3:8†

and rearranging terms, we ®nd the following state variable form @Ve …y† ˆ A Ve …y† @y

…3:3:9†

where Ve ˆ



Sx Uz



 Aˆ

a11 a21

a12 a22

 …3:3:10

where a11 ˆ j…Kx yy1 yx † a21 ˆ j…xx

a12 ˆ j… Kx yy1 Kx ‡ I†

xy yy1 yx †

…3:3:11†

a22 ˆ j…xy yy1 Kx †

…3:3:12†

where the superscript 1 in these equations denotes the matrix inverse. The above equations have been found by expressing each of the product terms xx …x†Ex …x; y†, xy …x†Ey …x; y†, etc. in a convolution form (see 3.2.7±10) when ! the Fourier series expansions of …x† and E …x; y† are substituted in each ! of the product terms making up …x† E and collecting coef®cients on common i orders. Let qn and Vn be the eigenvalues and eigenvectors of the matrix A after truncation. Summing over the individual eigenmodes we ®nd that the overall electric and magnetic ®elds in Region 2 are given by NT MT X !…2† X !e E ˆ Cn E n ˆ nˆ1

exp… jkxi x†

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iˆ MT

(

NT X nˆ1

) Cn ‰Sxin a^ x ‡ Syin a^ y Š exp…qn y† …3:3:13†

136

Chapter 3 NT MT X !…2† !…2† X !e U ˆ 0 H ˆ Cn U n ˆ nˆ1

mˆ MT

(

NT X nˆ1

) Cn ‰Uzin a^ z Š exp…qn y†

exp… jkxi x†

…3:3:14†

where NT ˆ 2…2MT ‡ 1†. Equations 3.3.13 and 3.3.14 represent the sum of NT ˆ 2…2MT ‡ 1† forward and backward traveling, propagating and non!e !e !e propagating eigenmodes E n and U n  0 H n , which gives the general electromagnetic ®eld solution in Region 2, the diffraction grating region. An important problem that remains is to determine the NT coef®cients Cn of Eqs. 3.3.13 and 3.3.14. Up to this point we have speci®ed the general form of the diffracted ®elds in the grating region. The EM ®elds on the incident side of the diffraction grating (Region 1 of Fig. 15a), on the transmission side of the diffraction grating (Region 3 of Fig. 15a), consist of an in®nite number of propagating and nonpropagating plane waves whose tangential wave numbers are given by kxi , i ˆ 1; . . . ; 1; 0; 1; . . . ; 1. The electromagnetic ®elds in Region 1 consist of the sum of a single E-mode polarized incident plane wave and an in®nite number of re¯ected propagating and evanescent plane waves. The total electric and magnetic ®elds in Regions 1 and 3 after summing the incident and re¯ected ®elds is given by Region 1 Uz…1† ˆ 0 Hz…1† ˆ

1  X iˆ 1

 Eo i;o exp…jky1i y† ‡ ri exp… jky1i y† exp… jkxi x† …3:3:15†

Ex…1† ˆ

1 X

1 1 iˆ

1

 ky1i Eo i;o exp…jky1i y†


 ri exp… jky1i y† exp… jkxi x† …3:3:16†

Ey…1† ˆ

1 X

1 1 iˆ

1


 kxi Eo i;o exp…jky1i y† ‡ ri exp… jky1i y† exp… jkxi x† …3:3:17†

Region 3 Uz…3† ˆ 0 Hz…3† ˆ

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1  X iˆ 1

 ti exp…jky3i …y ‡ L† exp… jkxi x†

…3:3:18†

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137

Ex…3† ˆ

1 
 1 X ky3i ti exp jky3i …y ‡ L† exp… jkxi x† 3 iˆ 1

…3:3:19†

Ey…3† ˆ

1 
 1 X k t exp jky3i …y ‡ L† exp… jkxi x† 3 iˆ 1 xi i

…3:3:20†

where ( kyri ˆ

‰r r k2xi Š1=2 j‰k2xi r r Š1=2

p r r > kxi p kxi > r r

r ˆ 1; 3

…3:3:21†

Now that the electromagnetic ®elds have been de®ned in Regions 1, 2, and 3, the next step is to match boundary conditions at the interfaces y ˆ 0 and y ˆ L. Matching the tangential components at the grating interfaces we have the ®nal matrix equation NT X nˆ1

NT X nˆ1

 Cn

 ky1i 2Eo ky1i Uzin ‡ Sxin ˆ io 1 1

 Cn exp… qn L†‰Sxin

 ky3i Uzin Š ˆ 0 3

…3:3:22† …3:3:23†

where i ˆ MT ; . . . ; MT : The above constitutes a set of NT ˆ 2…2MT ‡ 1† equations for the NT unknown coef®cients Cn . Power is excited in the diffraction grating system through the right-hand side (RHS) term of Eq. 3.3.22.

3.3.2

Complex Poynting Theorem

We will use the complex Poynting theorem [94,103] to study the power transmitted into and from the diffraction grating under consideration. In the present case we will choose the Poynting box to have a width  (normalized grating period length) in the x-direction, a thickness
z in the z-direction (the diffraction grating is z-independent, so the Poynting box dimension can be chosen to have an arbitrary value in this direction), and its back face located at y ˆ L (in Region 3, just behind the Region 2±3 interface). Figure 15b shows the Poynting box. We assume that no sources are present in Region 2. With these assumptions we ®nd the complex Poynting theorem is given by

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138

Chapter 3

PufIN ˆ PufOUT ‡ PuDE ‡ PuDM ‡ j… PuWE ‡ PuWM †

…3:3:24†

where PufIN PufOUT

 !…1† !…1† ˆ E U
… a^ y †dS
S yˆ0‡ ……  !…3† !…3† ˆ E U
… a^ y †dS

PuDE ˆ PuDM ˆ PuWE ˆ PuWM

……


S

……… ……… ……… ………

ˆ


V


V


V


V

yˆ L

…3:3:25† …3:3:26†

  !…2† !…2†  E
 00 E dV

…3:3:27†

  !…2† !…2†  U
 00 U dV

…3:3:28†

  !…2† !…2†  E
0 E dV

…3:3:29†

  …2†  !…2† 0! U
 U dV

…3:3:30†

In Eqs. 3.3.24±30, PufIN represents the complex power radiated into the diffraction grating power (it is the sum of the incident power, the re¯ected power, and the interaction power between the incident and re¯ected power), PuDE and PuDM represent the electric and magnetic dissipated power loss in the case when the grating material is isotropic, while PuWE and PuWM denote the reactive powers proportional to the electric and magnetic energies in the case when the grating material is isotropic. In the general anisotropic case, however, all four quantities can be complex. Hence, for instance, energy loss can result from both the imaginary and the real parts of  and l, as in a nonHermitian medium, to be discussed later. The superscript u in Eqs. 3.3.24± 30 means unnormalized. These power terms will be later normalized to the incident plane wave power. We will now be concerned with evaluating these equations for the E-mode plane wave polarization case under consideration. Sample Calculation of P uWE We illustrate the evaluation of the integrals in Eqs. 3.3.24±30 by calculating the PuWE integral of Eq. 3.3.29. Substituting the electric ®eld of Region 2 we ®nd that the dot product inside the integral is

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139



   !…2† 0 !…2† 0 0 0 E
 E ˆ Ex…2† xx …x†Ex…2† ‡ Ex…2† xy …x†Ey…2† ‡ Ey…2† yx …x†Ex…2†

0 ‡ Ey…2† yy …x†Ey…2†



…3:3:31† Each of the four terms in Eq. 3.3.31 must be substituted into Eq. 3.3.29 and the subsequent volume energy integrals must be evaluated. The analysis consists of substituting the Fourier series expansions of the electric ®eld quantities and dielectric tensor quantities into the energy volume integral, interchanging sum and integral expressions, carrying out all exponential integrals exactly and in closed form, and ®nally simplifying all summations. Letting V…x; y† ˆ

X i 0 ;n 0

Cn 0 Vi 0 ;n 0 exp…qn 0 y† exp… jkxi 0 x†

…3:3:32†

represent the electric ®eld Ex …x; y† or Ey …x; y†, letting …x† ˆ

X i

000

000

i0000 exp‰ji Kx xŠ

…3:3:33†

0 0 0 0 represent xx …x†, xy …x†, yx …x†, or yy …x†, and letting

W…x; y† ˆ

X i 00 ;n 00

Cn 00 Wi 00 ;n 00 exp…qn 00 y† exp… jkxi 00 x†

…3:3:34†

represent the electric ®eld Ex …x; y† or Ey …x; y†, we ®nd that any of the four terms of the unnormalized energy volume integral PuWE can be expressed in the general form ……… Pˆ


V

……… ˆ


V




8 <X :

i

000

V…x; y†…x†W…x; y† dV (

X i 0 ;n 0

) Cn 0 Vi 0 n 0 exp‰qn 0 yŠ exp‰ jkxi 0 xŠ

9( ) = X C 00 W 00 00 exp‰qn 00 yŠ exp‰ jkxi 00 xŠ i 000 exp‰ji Kx xŠ ; i 00 ;n 00 n i n 000

dx dy dz …3:3:35†

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140

Chapter 3

The integrals are independent of the z-coordinate. We ®nd after some algebra that ………
V

V…x; y†…x†W…x; y† dxdydz ˆ
z X i 0 ;i 00

X n 0 ;n 00

i 00

i

0

Cn 0 Cn 00 Iyn 0 ;n 00

V

i 0n 0

Wi00 n 00

…3:3:36†

where

Iyn 0 ;n 00 ˆ

… 0‡ L

exp‰…qn 0 ‡ qn 00 †yŠ dy

…3:3:37†

Substitution of the four terms of Eq. 3.3.31, with each term simpli®ed according to Eq. 3.3.36, produces a closed form expression (that is, all integrations have been carried out exactly) from which the normalized energy volume integral PuWE can be evaluated. The evaluation of the PuDE is identical to the analysis of the PuWE integral except that the lossy relative permittivity  00 …x† tensor is used rather than  0 …x† Other Poynting Theorem Integrals The evaluation of the magnetic volume integrals PuWM and PuDM is identical to that of PuWE and PuDE except that the magnetic ®eld is used rather than the electric ®eld. In the present case under analysis the permeability  ˆ  0 j 00 is uniform in the x-direction and its Fourier representation is …x†   ˆ

1 X iˆ 1

 i ejix

…3:3:38†

where  i ˆ … 0

j 00 †i;0

…3:3:39†

Following the analysis used to determine PuWE and evaluating the discrete Kronecker delta found, we ®nd that

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Planar Diffraction Gratings

PuWM ˆ
z 0 PuDM ˆ
z 00

X n 0 ;n 00

141

Cn 0 Cn 00 Iyn 0 ;n 00

X

Cn 0 Cn 00 Iyn 0 ;n 00

n 0 ;n 00

Xh i

…2† …2† Uzin 0 Uzin 00

Xh i

i

…2† …2† Uzin 0 Uzin 00

…3:3:40†

i

…3:3:41†

Simpli®cation of Results and Normalization Substituting the electromagnetic ®elds of Region 1 into Eq. 3.3.25, integrating over the cross section
z , and performing algebra we ®nd that the power radiated into the Poynting box at y ˆ 0‡ is given by PufIN ˆ


z X  k E  1 i yli 0 i;0

  ri E0 i;0 ‡ ri

…3:3:42†

It is convenient to normalize the power terms of Eq. 3.3.24 to the incident power radiated into the Poynting box. Integration of the incident power Puinc over the cross section
z  at y ˆ 0‡ for the present E mode polarization case, after noting that the integral is purely real, gives the incident power as Puinc ˆ


z jE0 j2 ky10 1

…3:3:43†

If we normalize the power of Eq. 3.3.42 to the incident power, we ®nd that P ri Š‰E0 i;0 ‡ ri Š Pu i kyli ‰E0 i;0 PIN  fIN ˆ …3:3:44† Puinc ky10 jE0 j2 Of interest are the powers re¯ected and transmitted from the diffracting grating at y ˆ 0‡ and y ˆ L , respectively, and the relation that these powers have to the power PfIN radiated into the Poynting box y ˆ 0‡ . The unnormalized re¯ected and transmitted powers are given by the expressions Puref ˆ

PufOUT  Putrans ˆ

……
S

……
S

!…1† !…1† E ref  U ref !…3† !…3† E U

yˆ0‡


a^ y dS ˆ


z X k r r 1 i y1i i i …3:3:45†

yˆ L


… a^ y †dS ˆ


z X k t t 3 i y3i i i …3:3:46†

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142

Chapter 3

To ®nd a relation between PufIN and Puref , we take E0 ˆ jE0 j exp…j0 † we note that kyli is purely real for i ˆ 0, and we analyze the summation in the numerator of the RHS of Eq. 3.3.44, namely, T

X i

kyli ‰E0 i;0

ri Š‰E0 i;0 ‡ ri Š

…3:3:47†

After expanding the product term of Eq. 3.3.47 in square brackets and after separating the i ˆ 0 term from the i 6ˆ 0 we ®nd that Tˆ

X i

 ky1i E02 i;0 ‡ … ri ‡ ri †E0 i;0 †

 T ˆ ky10 E02 ‡ … 2j Im…r0 †E0 T ˆ ky10 E02

2jky10 Im…r0 †E0

r0 r0 X i



ri ri X i;i6ˆ0



…3:3:48†

ky1i ri ri

…3:3:49†

ky1i ri ri

…3:3:50†

Thus PufIN

" #
z X  ˆ ky1i E0 E0 i;0 1 i


z X ‰k r r Š 1 i y1i i i

…3:3:51†


z  2jky10 Im…r0 †E0 1

The ®rst and second summation terms of Eq. 3.3.51 represent the unnormalized incident and re¯ected power at y ˆ 0‡ . The third term is an interaction term between the incident and re¯ected EM wave. We have PufIN ˆ Puinc

Puref


z  2jky10 Im…r0 †E0 1

…3:3:52†

We now substitute PufIN of Eq. 3.3.52 into the left-hand side (LHS) of Eq. 3.3.24. We ®nd that Puinc

Puref


z  2jky10 Im…r0 †E0 ˆ PufOUT ‡ PuDE ‡ PuDM 1 ‡ j…

PuWE

‡

PuWM †

…3:3:53†

Transposing the re¯ected power term and the interaction power term to the RHS of Eq. 3.3.53, we ®nd that

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Puinc ˆ Puref ‡ ‡ j…

143


z  2jky10 Im…r0 †E0 ‡ PufOUT ‡ PuDE ‡ PuDM 1

PuWE

‡

PuWM †

…3:3:54†

De®ning the normalized power terms (taking E0 ˆ jE0 j exp…j0 †† ˆ 1, Pinc 

Puinc ˆ1 Puinc

Pref ˆ

Puref 1 X k r r u ˆ Pinc ky10 i y1i i i

POUT  Ptrans ˆ

PufOUT 1 1 X ˆ k t t Puinc 3 ky10 i y3i i i

PDE 

PuDE Puinc

PDM 

PuDM Puinc

PWE 

PuWE Puinc

PWM 

PuWM Puinc

…3:3:55†

we now ®nd that the complex Poynting theorem of Eq. 3.3.24 after division of all terms by Puinc can be written in normalized form as PIN ˆ POUT ‡ PDE ‡ PDM ‡ j… PWE ‡ PWM †  PBOX

…3:3:56a†

or in terms of the re¯ected and transmitted powers as 1 ˆ Pinc ˆ Pref ‡ 2j Im…r0 † ‡ Ptrans ‡ PDE ‡ PDM ‡ j… PWE ‡ PWM † …3:3:56b† Equations 3.3.56a, b represent the main complex Poynting theorem conservation relation, which relates the input, incident, re¯ected, and transmitted powers to the dissipated power and stored energy of the system. By taking the real and imaginary parts of Eq. 3.3.56b we can derive useful relations from which the numerical accuracy of the diffraction analysis can be checked, and we can also derive useful expressions into which numerical insight of the diffraction process can be gained. Taking the real part of Eq. 3.3.56b we ®nd that 1 ˆ Pinc ˆ Re…Pref † ‡ Re…Ptrans † ‡ Re…PDE † ‡ Re…PDM † ‡ Re‰j… PWE ‡ PWM †Š

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…3:3:57†

144

Chapter 3

Taking the imaginary part of Eq. 3.3.56b we ®nd that 0 ˆ Im…Pref † ‡ Im…Ptrans † ‡ 2Im…r0 † ‡ Im…PDE † ‡ Im…PDM † ‡ Im‰j… PWE ‡ PWM †Š

…3:3:58†

We remind the reader that by de®nition the quantities PDE , PDM , PWE , and PWM in the general anisotropic case are not purely real, so that taking the real and imaginary parts as speci®ed in Eqs. 3.3.58 is necessary as shown. From Eqs. 3.3.57 and 3.3.58 we will now de®ne three useful relations from which numerical plots can be made and which give insight into the diffraction process. We will now give the ®rst relation. From Eq. 3.3.57 if we transpose the Re‰j… PWE ‡ PWM †Š term we have Re‰j… PWE ‡ PWM †Š ˆ Pinc

Re…Pref †

Re…PDE †

Re…Ptrans †

Re…PDM †

…3:3:59†

Letting the LHS of Eq. 3.3.59 be PWEM diffR  Re‰j… PWE ‡ PWM †Š

…3:3:60†

and letting the RHS of Eq. 3.3.59 be PdiffR  Pinc

Re…Pref †

Re…Ptrans †

Re…PDE †

Re…PDM †

…3:3:61†

we have from inspecting Eq. 3.3.59 PWEM diffR ˆ PdiffR

…3:3:62†

Equations for PWEM diffR and PdiffR are in general useful quantities to calculate. When the medium is reciprocal, PWE , PWM , PDE , and PDM are all purely real quantities; therefore from Eq. 3.3.61, PWEM diffR ˆ 0 and PdiffR ˆ 0, and thus Eq. 3.3.61 represents a conservation relation stating that the incident power should equal the sum of the transmitted, re¯ected, and dissipated powers. When the medium is anisotropic, PWE and PWM are in general complex, and WEM thus PWEM diffR is not necessarily zero. In this case PdiffR (which should equal PdiffR ) give a sense of how much the anisotropic nature of the medium is present in the EM ®eld calculation. The computation of PWEM diffR and PdiffR is also useful in this case as a cross-check of the numerical calculation. It is useful since both terms are computed from EM ®eld quantities located in different regions of space. Numerically if PWEM diffR and PdiffR are not equal or

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145

almost equal to each other, the numerical computation has to be in error. A second useful relation is given by Eq. 3.3.58. We de®ne PWEM diffI  Im‰ j… PWE ‡ PWM †Š PdiffI 

…3:3:63†

Im…Pref ‡ Ptrans ‡ PDE ‡ PDM †

2Im…r0 †

…3:3:64†

Proper balance of the complex Poynting theorem requires PWEM diffI ˆ PdiffI

…3:3:65†

We will now de®ne a third useful relation that results from Eqs. 3.3.56. Transposing the terms Im…Pref † and Im…Ptrans † to the left-hand side of Eq. 3.3.58 and multiplying by 1 we ®nd that Im…Pref † ‡ Im…Ptrans † ˆ

2Im…r0 † Im…PDM †

Im…PDE † Im‰j… PWE ‡ PWM †Š

…3:3:66†

Letting Pevan  Im…Pref † ‡ Im…Ptrans †

…3:3:67†

and letting Pdif evan 

2Im…r0 †

Im…PDE †

Im…PDM †

Im‰j… PWE ‡ PWM †Š …3:3:68†

we have from inspecting Eqs. 3.3.66±68 Pevan ˆ Pdiff evan

…3:3:69†

Equation 3.3.69 is a useful quantity to calculate as it gives a measure of the evanescent power stored in the diffraction system. That it gives the evanescent power can be seen from the equations for the re¯ected power in Region 1 and the transmitted power in Region 3. For the re¯ected power in Region 1 we have Pref ˆ

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Puref 1 X k r r u ˆ Pinc ky10 i y1i i i

…3:3:70†

146

Chapter 3

where from Eq. 3.3.21 ( ky1i ˆ

‰1 1 k2xi Š1=2 j‰k2xi 1 1 Š1=2

p 1 1 > kxi p kxi > 1 1

…3:3:71†

In these equations we note ky10 is real and positive. Because the term ri ri is purely real, the re¯ected power Pref is purely imaginary only when ky1i is purely imaginary. Thus we see that Im…Pref † is only nonzero when p ky1i ˆ j‰k2xi 1 1 Š1=2 , kxi > 1 1 , which occurs only for those space harmonics that are evanescent. The transmitted power is evanescent for Im…Ptrans † 6ˆ 0, just as for the re¯ected power. Relations 3.3.67±69 like Eqs. 3.3.60±62 are useful for two reasons. First, they give the evanescent power, thus they give a measure of how much power and energy is stored in nonpropagating EM waves near the diffraction grating interfaces. The larger the evanescent power and energy, the larger and rougher are the diffraction grating interfaces relative to the bounding regions. A second reason that they are useful is that they provide an excellent cross-check of the numerical diffraction solution. Pevan and Pdiff evan are computed from EM terms that exist in different regions of the EM system. Thus equality or very close numerical equality of Pevan and Pdiff evan helps show that the computations are being made correctly. We also note that the sum of evanescent powers tends to be small relative to the other diffraction power terms in the system. Thus the sum of evanescent powers tends to be a fairly sensitive test of the EM algorithm. The type of power budget analysis presented here can be applied to virtually any type of diffraction grating and any kind of polarization for the incident wave and may be extended to multilayer grating structures in a straightforward way. For example, results for H-mode incidence on an isotropic grating have already been presented in the previous section. 3.3.3

Numerical Results

In this section we will present numerical examples of the diffraction as results from RCWA and the complex Poynting theorem, which have been discussed previously for the E-mode polarization case under consideration. The examples to be presented consist of an RCWA study of a square or step pro®le diffraction grating when several different isotropic and anisotropic materials make up the step pro®le. The step grating is assumed to be homogeneous in the longitudinal direction and is assumed to contain sharp dielectric discontinuities at dielectric steps and thus possesses a high spectral content i ˆ 1; . . . ; 1; 0; 1; . . . ; 1. Thus studying isotropic and anisotro-

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147

~ ˆ  tests the RCWA pic materials with relatively short grating periods  and the complex Poynting theorem in a fairly severe way. We begin presenting results for the isotropic step grating. Figure 16a shows the transmitted diffraction ef®ciencies DET …%† for the i ˆ 0; 1 orders when a lossless and lossy grating is present. The parameters of the grating re given in the Fig. 16a inset. The plots show DET …%† versus the layer length L~ (in units of free space wavelength ). The lossless grating example of Fig. 16a was ®rst studied by Yamakita and Rokushima [54], and the lossless diffraction ef®ciency results of Fig. 16a are identical to their results [54]. As can be seen from Fig. 16a, as the layer length L~ increases from 0 to 2, because  is at the Bragg angle, power is diffracted from the i ˆ 0 order into ~ 2 to 4, power is diffracted from the the i ˆ 1 order. For larger values of L, i ˆ 1. This cycle of blazing and antiblazing (see [98±101] for an insightful discussion of blazing and antiblazing in grating analysis and its relation to the Poynting vector) is repeated over a long range of L~ values. One also observes that if 2b is lossy, the diffraction ef®ciency of both the i ˆ 0 and i ˆ 1 orders is attenuated as one would expect in a lossy material.

Figure 16 (a) The transmitted diffraction ef®ciency of a lossless and lossy step diffraction grating are shown. (b±e) Plots of the real and imaginary parts of the normalized complex power PIN and PBOX as computed by Eq. 56a of the complex Poynting theorem for the lossless (b and c) and lossy cases (d and e) are shown. (f±g) Plots of the evanescent power as computed by Eqs. 3.3.67±69 for the lossless (f) and lossy cases (g) are shown. OSA 1999 [103].

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148

Figure 16

Chapter 3

(continued)

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Figure 16

(continued)

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149

150

Figure 16

Chapter 3

(continued)

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151

Figures 16b±e show plots of the real and imaginary parts of the normalized complex power PIN and PBOX of the complex Poynting theorem for the lossless (Figs. 16b and 16c) and lossy (Figs. 16d and 16e) cases as speci®ed by Eq. 3.3.56a. As can be seen from these four plots, the complex Poynting theorem is obeyed to a high degree of accuracy as evidenced by the close ®t between the data for PIN (solid line) and the data for PBOX (dots). In comparing the lossless and lossy cases (using Eq. 3.3.56a), one also notices a signi®cant difference in the complex Poynting results for these cases. In the lossy case, as the layer becomes larger, the oscillations of the power results decrease. This is because in the lossy case, less EM power is re¯ected from the boundary and interferes with forward traveling waves than when the medium is lossless. Figures 16f and 16g show plots of the evanescent power as computed by Eqs. 3.3.67±69. As can be seen in these ®gures, excellent agreement with the complex Poynting theorem is observed in both lossy and nonlossy cases. As discussed earlier, because the grating width is on the order of a wavelength, a certain amount of energy is stored in the evanescent ®elds of Regions 1 and 3. The comparison of the lossy and nonlossy materials shows a de®nite difference in the evanescent ®eld quantities of the system. Figures 17a and 17b show plots of the real and imaginary parts, respectively, of the normalized complex power as computed by Eq. 3.3.56a for the lossless case when a general, nonreciprocal, anisotropic material occupies Region 2a of the step diffractive region as shown in the inset of Figs. 17a and 17b. Region 2b of the step was chosen to have 2b ˆ 2:5, and Region 2a of the step was chosen to have 2ayy ˆ 1:52axx , 2axx ˆ 1 j0:1, 2axy ˆ 0:22axx , and 2ayx ˆ 0. The example being considered is non-Hermitian since 2axy 6ˆ 2ayx . This situation may be encountered in, for instance, materials with stimulated Raman scattering, a process whose governing susceptibility does not exhibit overall permutation symmetry [96]. As can be seen from these ®gures, the complex Poynting theorem is obeyed to a high degree of accuracy. Figure 17c shows a plot of the evanescent power as calculated by Eqs. 3.3.67±69 for the nonreciprocal anisotropic case under consideration. Figures 17d and 17e show plots of PWEM diffR and PdiffR (calculated in Eqs. and P (calculated in Eqs. 3.3.63±65) for the 3.3.60±62) and plots of PWEM diffI diffI same anisotropic case as was considered in Fig. 17a±c (plots labeled 2ayx ˆ 0†. Also shown in these ®gures are plots made for the case when all the permittivity elements are the same as in Fig. 17a±c except that instead of taking 2ayx ˆ 0, 2axy has been taken to be 2axy ˆ 2ayx and thus the medium is Hermitian. These plots are labeled 2axy ˆ 2ayx in Figs. 17d and 17e. As can be seen from all the plots shown in Figs. 17d and 17e, WEM Eqs. 3.3.62 and 3.3.65 (namely, PWEM diffR ˆ PdiffR and PdiffI ˆ PdiffI ) are

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152

Chapter 3

Figure 17 (a,b) Plots of the real and imaginary parts, respectively, of the normalized complex power PIN and PBOX as computed by Eq. 3.3.56a of the complex Poynting theorem are shown. (c) A plot of the evanescent power as calculated by Eqs. 3.3.67±69 is shown. (d) Plots of PWEM diffR and PdiffR (calculated in Eqs. 3.3.60±62) for Hermitian and non-Hermitian step diffraction gratings are shown. (e) Plots of PWEM diffI and PdiffI as calculated in Eqs. 3.3.63±65 for Hermitian and non-Hermitian step diffraction gratings are shown.

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Planar Diffraction Gratings

Figure 17

(continued)

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153

154

Chapter 3

Figure 17

(continued)

obeyed to a very high degree of accuracy as can be seen by the close agreement between lines and dots as displayed in the ®gures. In Fig. 17d it is very interesting to compare the power results for the Hermitian and non-Hermitian cases. In Fig. 17d, for the plots labeled 2ayx ˆ 0, the material is non-Hermitian, and thus PWE and PWM are not necessarily purely real, and thus PWEM diffR of Eqs. 3.3.60±62 is not necessarily zero. The nonzero nature of PWEM diffR is clearly seen in the plot of Fig. 17d. On the other hand, for the plots labeled 2axy ˆ 2ayx (Hermitian case), it is noticed that PWEM diffR ˆ Re‰j… PWE ‡ PWM †Š ˆ PdiffR ˆ 0

…3:3:72†

or PdiffR  Pinc

Re…Pref †

Re…Ptrans †

Re…PDE †

Re…PDM † ˆ 0 …3:3:73†

or Pinc ˆ Re…Pref † ‡ Re…Ptrans † ‡ Re…PDE † ‡ Re…PDM †

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…3:3:74†

Planar Diffraction Gratings

155

It is expected in the Hermitian case that the medium should be completely conservative and should show the basic property that the incident power on the grating should equal the sum of the power dissipated in the grating and re¯ected and transmitted from the grating. This is exactly what is observed in Fig. 17d as can be seen from Eq. 3.3.74.

3.4

COUPLED WAVE ANALYSIS OF SLANTED AND PURE REFLECTION GRATINGS

In the previous two sections, transmission gratings have been analyzed by the RCWA method in the cases of E-mode and H-mode polarization. In both these cases the diffraction grating has been taken to have its grating vector directed in the x-direction (the dielectric modulation varies in this direction). Another important type of grating is the slanted or pure re¯ection grating where the grating vector will have a component along the ydirection. Coupled wave analysis of re¯ection gratings has been carried out by numerous researchers, see for instance Kogelnik [5], Raman-Nath, Kong [6], Marcuse [14], and Moharam and Gaylord [16,18,19]. The basic assumption in all of these is that the EM ®elds can be expanded in terms of a transverse space harmonic Fourier series, which upon substitution into Maxwell's equations can be reduced to state variable form for subsequent analysis. Rigorously speaking, this in turn implies that the analysis is only valid for ®nite transverse spatial periods of the grating vector. In a pure re¯ection grating, the transverse grating period is in®nite; the RCWA based on a transverse space harmonic expansion cannot strictly hold (even though it has been shown to hold in the limit). This does not mean however that a state variable analysis of this case cannot be performed. Zylberberg and Marom [25] have, in fact, employed a novel expansion of the EM ®elds in terms of harmonics of the phase of the grating to reduce the Maxwell's equations to state variable form. 3.4.1

Formulation

We consider the case of an EM wave (H-mode polarization) (see Fig. 18) incident on a sinusoidally modulated relative dielectric constant (relative permeability equal to unity in all space) …x; z† ˆ 2 ‡
 cos‰K…x sin  ‡ z cos † ‡ Š

…3:4:1†

where 2 is the bulk relative dielectric constant,
 is the modulation ampli~ is the normalized grating tude,  is the slant angle, K ˆ 2= and  ˆ k0 

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156

Chapter 3

Figure 18 Geometry for planar-grating diffraction. Used with permission of Optical Society of America (OSA), Ref. 25.

period.  is the phase angle of the modulation and plays a signi®cant role in the state variable analysis of the system. The grating is assumed to be bounded by lossless homogeneous regions on either side of the grating. The normalized electric ®eld in each region expressed in normalized coordi~ etc., is nates x ˆ k0 x,

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157

Region 1 (z < 0† E1 ˆ exp‰ j… 0 x ‡ 10 z†Š ‡

X i

R~ i exp‰ j… i x

1i z†Š

…3:4:2†

Region 2 (0 < z < d† E2 ˆ

X i

S~i exp‰ j… i x ‡ 2i z†Š

…3:4:3†

Region 3 …z > d† E3 ˆ

X i

T~i exp‰ j… i x ‡ 3i …z

d††Š

…3:4:4†

Here i ˆ k1 sin  2 mi ˆ k2m

2i

iK sin  km ˆ

p m

2i ˆ k2 cos  0 iK cos  "  # 1 1=2 0 1 sin   ˆ sin 2 i ˆ k1 sin 

iK sin 

m ˆ 1; 2; 3

i ˆ 0; 1; 2; . . .  is the wavelength in vacuum of the waves,  is the angle of incidence in Region 1, and  0 is the refraction angle inside the grating (Region 2); R~ i and T~i are the amplitudes of the re¯ected and transmitted diffraction orders, respectively. S~i is the amplitude of the ith order inside the modulated region. It will be shown that R~ i , T~i , and S~i depend on the modulation phase angle . When Eqs. 3.4.1 and 3.4.3 are substituted into the wave equation for the grating region, r2 E2 ‡ …x; z†E2 ˆ 0

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…3:4:5†

158

Chapter 3


 d 2 Si …u† dS …u† i…i B†S i …u† ˆ …cos  0 i cos † i 82 d 2 u du ‡ S i‡1 …u† exp… j† ‡ S i 1 …u† exp…j† ˆ 0

…3:4:6†

where S i …u†  S~i …z† u ˆ jz~ ˆ ˆ ˆ Bˆ

j
z~ 2…2 †1=2

 ~ 2 †1=2 … 22 22 2 ˆ ~ 2

  1=2 ~ 2 † cos… 2…

 0†



ˆ

2 cos… 

 0†

Solutions to Eq. 3.4.6 represent the EM ®elds in the diffracting grating regions. To ®nd an overall EM solution, the ®elds of Region 2 must be matched at the boundaries z ˆ 0 and z ˆ d to the EM ®eld solutions in Regions 1 and 3. Matching the tangential electric and magnetic ®elds, respectively, at z ˆ 0 we have R~ i ‡ i;0 ˆ S~i …0† 1i …R~ i

i;0 † ˆ

d S~i …z† j dz

…3:4:7† 2i S~i …0†

…3:4:8†

zˆ0

Matching the tangential electric and magnetic ®elds, respectively, at z ˆ d we have T~i ˆ S~i …d† exp… j2i d† " # ~i …z† d S 3i T~i ˆ j 2i S~i …d† exp… j2i d† dz

…3:4:9† …3:4:10†

zˆd

R~ i , S~i …z†, and T~i , the normalized amplitudes of the ith diffraction orders in regions 1, 2, and 3, respectively, can be written in a form that shows explicitly their dependence on the arbitrary phase modulation .

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159

Upon substituting Si …u† ˆ Si …u† exp…ji†

…3:4:11†

into Eq. 3.4.6, one obtains the differential equation
 d 2 Si …u† ˆ …cos  0 82 d 2 u

i cos †

dSi …u† du

i…i

B†Si …u†

…3:4:12†

‡ Si‡1 …u† ‡ Si 1 …u† ˆ 0 Equation 3.4.12 can also be obtained from Eq. 3.4.6 by letting  ˆ 0. Therefore Si …u† is the ®eld amplitude in the grating region for  ˆ 0. From Eq. 3.4.11 it follows that S~i …z† ˆ S^i …z† exp…ji†

…3:4:13†

where S^i …z† ˆ Si …u†

…3:4:14†

By substituting Eq. 3.4.13 into Eqs. 3.4.7±10 and de®ning R~ i ˆ Ri exp…ji†

…3:4:15†

T~i ˆ Ti exp…ji†

…3:4:16†

one obtains the following four equations: Ri ‡ i;0 ˆ S^i …0† 1i …Ri

i;0 † ˆ

d S^i …z† j dz

…3:4:17† 2i S^i …0†

…3:4:18†

zˆ0

Ti ˆ S^i …d† exp… j2i d† " # d S^i …z† j 2i S^i …d† exp… j2i d† 3i Ti ˆ dz

…3:4:19† …3:4:20†

zˆd

Equations 3.4.17±20 can also be obtained from Eqs. 3.4.7±10 by letting  ˆ 0. Therefore Ri and Ti are the amplitudes of the ith re¯ected and transmitted order for a grating with  ˆ 0.

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160

Chapter 3

Equation 3.4.12 and Eqs. 3.4.17±20, which correspond to the case of  ˆ 0, are identical with the coupled wave equations and boundary condition equations given by Gaylord [16] and can be solved by using the method described there to obtain the amplitudes S^i …z†, Ri , and Ti . After Ri and Ti are calculated, the amplitudes R~ i and T~i for an arbitrary modulation phase  can be determined by using Eqs. 3.4.15 and 16. From Eqs. 3.4.13±16, it is seen that a phase shift  of the grating's dielectric constant modulation causes a shift of i in the phases of the ith diffraction orders in the three regions. In retrospect, it is not hard to see why this should be true by examining the diffraction orders from a simple thin sinusoidal grating. The overall optical ®eld immediately behind a thin matched sinusoidal grating of thickness d in free space illuminated by a plane wave of wavenumber k0 can be expressed in the form E… † / exp…j
d cos †

…3:4:21†

where, with  ˆ =2, ˆ Kx ‡ 

…3:4:22†

Now the RHS of Eq. 3.4.21 can be expanded in a Fourier series by a Bessel identity of the form exp…j
d cos † ˆ

1 X iˆ 1

Ji …
d† exp…ji †

…3:4:23†

Using Eqs. 3.4.22 and 3.4.23, it readily follows that E/

1 X iˆ 1

Ji …k0
d† exp…ji † ˆ

1 X iˆ 1

‰Ji …k0
d† exp…ji†Š exp…jiKx† …3:4:24†

The quantity in square brackets is the amplitude Ti of the ith order diffracted ®eld at the exit plane of the grating (transmission type in this simple example), and it is readily seen that the phase associated with the ith diffraction order is always i. When the diffraction ef®ciencies are calculated using RCW analysis two distinct cases must be considered: 1. Slanted gratings … 6ˆ 0†. In this case the re¯ected (transmitted) orders in Region 1 (3) have different values of 1i …3i ) for different values

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of i. Therefore some of them propagate in different directions while others are evanescent. The diffraction ef®ciencies for the ith re¯ected order and for the ith transmitted orders, when the grating modulation has an arbitrary phase , are given, respectively, by 

 1i ~ 2 jRi j DERi ˆ Re 10    DETi ˆ Re 3i jT~i j2 10

…3:4:25† …3:4:26†

By substituting Eqs. 3.4.15 and 3.4.16 into Eqs. 3.4.25 and 3.4.26, respectively, we obtain 

 1i jRi j2 10    DETi ˆ Re 3i jTi j2 10

DERi ˆ Re

…3:4:25a† …3:4:26a†

Therefore for slanted gratings the diffraction ef®ciencies are independent of the modulation phase . This is also in agreement with our heuristic theory given above. 2. Pure re¯ection gratings ( ˆ 0). The dielectric constant of a pure re¯ection grating is constant along the boundaries, and on any plane parallel to the boundaries inside the grating region. It changes along the z-direction and is given by setting  ˆ 0: …z† ˆ 2 ‡
 cos…Kz ‡ †

…3:4:27†

Two important differences exist between the diffraction characteristics of purely re¯ection gratings and slanted gratings: 1. Only one re¯ected wave and one transmitted wave exist outside the pure re¯ection grating. 2. Different values of the modulation phase  correspond generally to different values of the dielectric constant at the boundaries. Therefore one would expect the diffraction ef®ciencies of a pure re¯ection grating to be affected, even if only the value of  is changed. A pure re¯ection grating can be viewed as a dielectric slab with plane boundaries that are perpendicular to the z-axis and with a dielectric constant that varies as a function of z only. A rigorous solution of the problem of

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162

Chapter 3

re¯ection and transmission of an incident TE plane wave by a pure re¯ection grating therefore has the following form: Region 1 (z < 0† E1 ˆ exp‰ j… 0 x ‡ 10 z†Š ‡ R~ exp‰ j… 0 x

10 z†Š

…3:4:28†

Region 2 (0 < z < d† E2 ˆ U…z† exp‰ j 0 xŠ

…3:4:29†

Region 3 (z > d) E3 ˆ T~ exp‰ j… 0 x ‡ 30 …z

d††Š

…3:4:30†

R~ and T~ are the electric ®eld amplitudes of the single re¯ected and transmitted plane waves, respectively, and U…z† is the z-dependent amplitude of the electric ®eld inside the pure re¯ection grating. In addition to satisfying the wave equation in the three regions, the expressions 3.4.28±30 must also satisfy the boundary conditions at z ˆ 0 and z ˆ d. The four boundary equations are tangential E and z ˆ 0: R~ ‡ 1 ˆ U…0†

…3:4:31†

tangential H at z ˆ 0: j10 …R~

dU 1† ˆ dz zˆ0

…3:4:32†

tangential E at z ˆ d: T~ ˆ U…d†

…3:4:33†

tangential H at z ˆ d: j30 T~ ˆ

dU dz zˆd

…3:4:34†

It is now shown that the RCWA equations can also be derived for  ˆ 0 and therefore are also valid for pure re¯ection gratings. Since changing the modulation phase  by an integral multiple of 2 ~ T, ~ and U…z† in Eqs. 3.4.28±30 results in the same grating, the amplitudes R,

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163

should be periodic functions of  with 2 periodicity and thus can be expanded in Fourier series in the forms R~ ˆ

X i

U…z† ˆ

X i

T~ ˆ

X i

Ri exp…ji†

…3:4:35†

Ui …z† exp…ji†

…3:4:36†

Ti exp…ji†

…3:4:37†

The Fourier coef®cients Ri , Ui …z†, and Ti are independent of . The Fourier coef®cients Ui …z† can be written without loss of generality in the form Ui …z† ˆ S^i …z† exp…j2i z†

…3:4:38†

where 2i ˆ k2 cos  0

iK

By setting  ˆ 0 in Eq. 3.4.1 and substituting Eqs. 3.4.1, 3.4.29, and 3.4.36, into the wave equation Eq. 3.4.5, the following equation is obtained: " X d 2 S^i i

dz2

d S^ 2j2i i dz

2 2i

‡

20

k22





  ^ Si S^i ‡ 2

1

‡ S^i‡1



#

 exp… j2i z† exp…ji† ˆ 0 …3:4:39† Since  can have any value, it follows that the coef®cient of each exponential (the term in square brackets in Eq. 3.4.39) must equal zero: d 2 S^i dz2

2j2i

d S^i dz

2 2i ‡ 20



  ^ Si k22 S^i ‡ 2

1

 ‡ S^i‡1 ˆ 0

…3:4:40†

It can easily be shown that 3.4.40 is identical to 3.4.12 with  ˆ 0. The boundary conditions are derived by substituting Eqs. 3.4.35±38 into Eqs. 3.4.31±34. For instance, after substituting Eqs. 3.4.35, 36, 38 into 3.4.31, one obtains

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164

Chapter 3

X i

Ri exp…ji† ‡ 1 ˆ

Noting that 1 ˆ

P

i i0

X i

S^i …0† exp…ji†

…3:4:41†

exp…ji†, it follows that

Ri ‡ i0 ˆ S^i …0†

…3:4:42†

which is identical to Eq. 3.4.17. Other boundary condition equations can be derived in a similar manner. By multiplying Eq. 3.4.40 and the corresponding boundary conditions by exp…ji† and de®ning S~i …z† ˆ S^i …z† exp…ji†

…3:4:43†

R~ i ˆ Ri exp…ji†

…3:4:44†

T~i ˆ Ti exp…ji†

…3:4:45†

a set of equations is obtained that is identical to Eqs. 3.4.6±10 with  ˆ 0. From Eqs. 3.4.35±38 it follows that the amplitudes of the re¯ected and transmitted waves are given by the sums of the amplitudes of the re¯ected and transmitted orders, respectively. The dependence on  is obtained by multiplying the amplitudes of the ith orders Ri , S^i …z†, Ti , which correspond to  ˆ 0, by exp…ji†. We have just shown that a state variable formulation in terms of the phase  of the grating can lead to a set of coupled wave equations that are equivalent to the RCWA equations for  6ˆ 0 but in the limit as  ! 0. When  ˆ 0, all the re¯ected orders have 1i ˆ 10 and propagate in the same direction in Region 1. The result is a single re¯ected wave whose amplitude R~ is given by the coherent addition of the amplitudes of the individual re¯ected orders R~ ˆ

X i

R~ i

…3:4:46†

Similarly, the single transmitted wave whose amplitude T~ is given by the coherent addition of the amplitudes of the individual transmitted orders T~ ˆ

X i

T~i

…3:4:47†

The re¯ection and transmission ef®ciencies of a pure re¯ection grating are given respectively by

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165

X 2 2 ~ ~ DER ˆ jRj ˆ Ri i

…3:4:48†

    X 2 30 ~ 2  jTj ˆ Re 30 T~ DET ˆ Re 10 10 i i

…3:4:49†

and

The conservation of power for a lossless pure re¯ection grating dictates that DER ‡ DET ˆ 1

…3:4:50†

Note that substituting 3.4.44 into 3.4.48, DER ˆ

X i

jRi j2 ‡

XX i

n;n6ˆi

Ri Rn exp‰j…i

n†Š

…3:4:51†

Averaging DER w.r.t.  over a period of 2 yields hDER i ˆ

X i

jRi j2

…3:4:52†

A similar operation on DET yields  X  jTi j2 hDET i ˆ Re 30 10 i

…3:4:53†

Averaging 3.4.50 w.r.t.  over a period of 2 and using 3.4.52, 53 therefore yields hDER i ‡ hDET i ˆ

X i

 X  jRi j ‡ Re 30 jTi j2 ˆ 1 10 i 2

…3:4:54†

It is interesting to note that Eq. 3.4.54 is identical to the conservation of power relationship for a slanted grating in the limit  ! 0. We would like to stress that the true power conservation relation for a pure re¯ection grating with a ®xed  is given by the relations 3.4.48±50.

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166

3.4.2

Chapter 3

Numerical Results

Slanted Gratings ( 6ˆ 0† In Section 3.2 above, we have given an example of a pure transmission grating … ˆ =2† [16, Fig. 1]. We now give an example of diffraction from slanted gratings. Figs. 19 and 20 show the diffraction ef®ciencies as a function of the grating strength parameter ˆ d= cos  0 (proportional to grating thickness d) for  ˆ =3 and =6, respectively. Note that as  increases, the grating behavior changes from a predominantly transmission type to a predominantly re¯ection type. Also shown in the ®gures is comparison with Kogelnik's two-mode theory. The case  ˆ 0 (pure re¯ection grating) has to be treated separately, and examples as given below.

Figure 19 The diffraction ef®ciencies of the transmitted waves for a  ˆ 60 slanted grating with p ˆ 2. The diffraction ef®ciencies of all re¯ected and transmitted waves not shown in the ®gure are less than 0.01. Used with permission of OSA, 1981 [16, Fig. 3].

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Figure 20 The diffraction ef®ciencies for a  ˆ 30 grating with  ˆ 10. The diffraction ef®ciencies of all re¯ected and transmitted waves not shown in the ®gure are less than 0.01. Used with permission of OSA, 1981 [16, Fig. 4].

Pure Re¯ection Gratings ( ˆ 0) Using RCWA we present examples of the re¯ection ef®ciencies of a pure re¯ection grating at ®rst-Bragg (B ˆ 1) and second-Bragg incidence. The data used in the calculations were as follows 1 ˆ 2 ˆ 3 ˆ 2:25


 ˆ 0:2;  ˆ 0:5 m

…z† ˆ 2 ‡
 cos‰…2z=† ‡ Š

…3:4:55†

In Figs. 21 and 22, the solid curves P are the rigorously correct re¯ection ef®ciencies of the PRG given by j i Ri j2 as a function of grating thickness, for B ˆ 1 and B ˆ 2, respectively. In both ®gures, the modulation phase  is

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168

Chapter 3

P zero. The dashed curves in Figs. 20 and 21 were obtained by using i jRi j2 as in Ref 16. P In the case of ®rst-Bragg incidence (Fig. 21), the curve P drawn using j i Ri j2 (solid line) ripples around the curve drawn using i jRi j2 (dashed curve). The ripple results from the fact that as the thickness of the grating changes the interference between the re¯ected, diffracted orders alternates from partially constructive to partially destructive interference. The effect of interference between re¯ected orders is even more pronounced in the case of second-Bragg incidence as shown in Fig. 22. Figures P 23 and 24 P show the deviations from the conservation of power given by j i Ri j2 ‡ j i Ti j2 1 computed using the same data as was used for Figs. 21 and 22. The deviations result from using a ®nite number of

Figure 21 Re¯ection ef®ciency of a PRG at a ®rst-Bragg incidence (solid line). The re¯ection ef®ciency line) is calculated by summing the re¯ection diffraction P(dashed 2 ef®ciency orders i jRi j . Used with permission of Optical Society of America (OSA), 1983 [25, Fig. 2].

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Figure 22 Re¯ection ef®ciency of a PRG at second-Bragg incidence (solid line). The re¯ection ef®ciency P(dashed line) is calculated by summing the re¯ection diffraction ef®ciency orders i jRi j2 . Used with permission of Optical Society of America (OSA), 1983 [25, Fig. 33].

diffraction orders in the numerical simulations. It is readily seen from Figs. 23 and 24 that the deviation decreases as the number of diffraction orders retained in the calculations is increased. The reader should note that the multipliers on the vertical scales decrease from top to bottom. In Figs. 25±28 we show the effect of  on the re¯ection ef®ciency for ®rst and second Bragg incidence. Figures 25 and 26 show the re¯ection ef®ciencies for  ˆ 90 and  ˆ 180 , respectively, for B ˆ 1, and Figs. 27 and 28 show the re¯ection ef®ciencies for  ˆ 90 and  ˆ 180 for B ˆ 2. As can be seen from the ®gures, a perceptible difference exists between the diffraction ef®ciencies for the case  ˆ 0 (dashed curves) and nonzero .

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170

Chapter 3

P P Figure 23 Deviations from conservation of power …j i jRi j2 ‡ j i Ti j2 1† obtained with the data of Fig. 21 when (top to bottom) 11, 17, and 23 diffraction orders were used in the numerical calculations. Used with permission of Optical Society of America (OSA), 1983 [25, Fig. 4].

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P P Figure 24 Deviations from conservation of power … i jRi j2 ‡ j i Ti j2 1† obtained with the data of Fig. 22 when (top to bottom) 11, 17, and 23 diffraction orders were used in the numerical calculations. Used with permission of Optical Society of America (OSA), 1983 [25, Fig. 5].

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172

Chapter 3

Figure 25 Re¯ection ef®ciency of a PRG with modulation phase  ˆ 908 at a ®rstBragg incidence (solid line). The re¯ection ef®ciency (dashed line) is when  ˆ 08 (same as solid curve in Fig. 21). Used with permission of Optical Society of America (OSA), 1983 [25, Fig. 6].

3.5

MULTILAYER ANALYSIS OF E-MODE DIFFRACTION GRATINGS

Up to this point in this book we have mainly considered the diffraction problem when the diffracting layer is a uniform, homogeneous layer in the longitudinal direction. A very important problem that remains is when the diffracting layer is not constant or homogeneous in the y-direction but varies (is inhomogeneous) in the y-direction. All surface relief gratings are examples of gratings that are inhomogeneous in the y-direction. Figure 29a shows an example of a symmetric blaze surface relief grating. As can be seen from this ®gure, taking 2a ˆ 1 and 2b ˆ 2 , the surface of Region 3 is cut or grooved with the triangular blazes. Any grating that has a longitudinal variation in the bulk index or possesses a variation in its modulation is

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Figure 26 Re¯ection ef®ciency of a PRG with modulation phase  ˆ 1808 at a ®rst-Bragg incidence (solid line). The re¯ection ef®ciency (dashed line) is when  ˆ 08 (same as solid curve in Fig. 21). Used with permission of Optical Society of America (OSA), 1983 [25, Fig. 7].

an example of a longitudinal inhomogeneous dielectric grating. An interdigitated electrode device that induces a nonuniform periodic index of refraction change is a second example of a nonuniform nonhomogeneous periodic grating [77±82]. The basic theory that will be used in this section to analyze longitudinally inhomogeneous gratings will be to divide the grating into a set of thin layers (suf®ciently thin that very little longitudinal variation occurs inside the grating thin layer), in each thin layer to ®nd the Fourier coef®cients that correspond to each thin layer, to formulate a set of state variable equations

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174

Chapter 3

Figure 27 Re¯ection ef®ciency of a PRG with modulation phase  ˆ 908 at second-Bragg incidence (solid line). The re¯ection ef®ciency (dashed line) is when  ˆ 08 (same as solid curve in Fig. 22). Used with permission of Optical Society of America (OSA), 1983 [25, Fig. 8].

within the thin layer as has been done in earlier sections, and to boundary match the state variable solutions at all thin layer interfaces and at all regions exterior to the grating to ®nd an EM solution to the inhomogenous grating problem. This procedure has been used by numerous researchers to study longitudinally inhomogeneous diffraction gratings [18,19,21±23, 28,31,33,55,65] and other EM systems. In observing Fig. 29a one also notices why the grating must be considered inhomogeneous in the y-direction. The dotted lines of Fig. 29a show examples of the thin layers that can be used to analyze the grating. In observing these dotted lines, for the example presented, one notices that if the thin layer is chosen close to the incident side of the grating, little material of Region 3 is included in the thin layer, and the Fourier series representing that thin layer will nearly be that of Region 1. On the other hand, if the thin layer is chosen close to the transmit side of the grating, then in this case most

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Figure 28 Re¯ection ef®ciency of a PRG with modulation phase  ˆ 1808 at second-Bragg incidence (solid line). The re¯ection ef®ciency (dashed line) is when  ˆ 08 (same as solid curve in Fig. 22). Used with permission of Optical Society of America (OSA), 1983 [25, Fig. 9].

of the material in the thin layer is that of Region 3, and the Fourier series representing this thin layer will nearly be that of Region 3. Clearly, from this discussion and observing the thin layer in Fig. 29a one can see that the Fourier series in the thin layer near the incident side are signi®cantly different from those at the transmit side. Thus from this discussion, the RCWA method must re¯ect the change in Fourier series coef®cients as one changes the y position in the grating. In the following sections we will carry out the multilayer analysis for the E-mode case studied in Section 3.3. The analysis for the H-mode case is similar to the E-mode case.

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176

Chapter 3

Figure 29

3.5.1

Symmetric blaze grating with multilayers shown.

E-Mode Formulation

To begin the analysis of longitudinal inhomogeneous gratings we divide the inhomogeneous region into N` regions as shown in Fig. 29b. If the P width of ` the lth layer is S` , then the overall layer thickness is given by L ˆ N `ˆ1 S` . Each layer is assigned a local coordinate system y` with its local origin as shown in Fig. 28. The ®rst layer has the coordinate y1 and the last layer to the right is yN . Enough layers N` are used so that the grating inhomogeneity `

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is nearly constant in each layer. The periodic dielectric permittivity tensor element in each layer is labeled ~ss 0 ` and is expanded the same way as in Sections 3.2 and 3.3, namely as ~ss 0 ` ˆ 0

1 X iˆ 1

ss 0 i` ejiKx x

…s; s 0 † ˆ …x; y; z†

…3:5:1†

where x; y and z are normalized space coordinates, Kx ˆ 2= is the normalized magnitude of the grating vector,  is the grating wavelength, ss 0 i` is the ith Fourier expansion coef®cient of ~ss 0 ` in the `th layer, and 0 is the permittivity of free space. Using a local coordinate system is an important feature of the analysis. The ®rst step is to determine the full EM ®elds in the `th layer of the grating. This analysis has been seen already in Section 3.2, since the grating is assumed uniform in each layer. Thus the EM ®elds from Maxwell's equa! tions in Region ` after truncation to order jij  MT are given by E ˆ Ez a^ z (all regions) ( ) MT NT X X qn` y Uz` ˆ 0 Hz` ˆ Cn` Uzin` e ` e jkxi x …3:5:2† Ex` ˆ

MT X

(

iˆ MT

iˆ MT

NT X nˆ1

nˆ1

)

Cn` Sxin` eqn` y` e

jkxi x

…3:5:3†

In these equations kxi ˆ kx0 iKx , NT ˆ 2…2MT ‡ 1†, qn` and V` ˆ ‰Stx` ; Utz` Št …Sx` ˆ ‰Sxi` Š, Uz` ˆ ‰Uzi` Š† are the eigenvalues and eigenvectors of the `th region. The eigenvector Vn` …y† ˆ Vn eqn `y is assumed to satisfy the eigenvalue equation @Vn` ˆ A` Vn` @y

…3:5:4†

where the matrix A` is given by Eq. 3.2.15 with the ss 0 i` of Eq. 3.5.1 used to de®ne A` . The EM ®elds in Regions 1 and 3 are the same as in the uniform case and are given in Section 3.2. The analysis proceeds by matching the tangential electric ®eld (Ex in this case) and tangential magnetic ®eld (Hz in this case) at every boundary interface. The Ex and Hz ®elds at Region 1: Layer ` ˆ 1 interface are Ex…1† ‡ ˆ Ex1 y ˆ0 Hz…1† ‡ ˆ Hz1 y ˆ0 …3:5:5† yˆ0

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1

yˆ0

1

178

Chapter 3

Substituting we have MT X ky1i …Eo io 1 iˆ M

jkxi x

Ri †e

(

MT X

ˆ

NT X

iˆ MT

T

MT 1 X …E  ‡ Ri †e 0 iˆ M o io

jkxi x

ˆ

T

nˆ1

MT 1 X 0 iˆ M

(

) Cn1 Sxin1 e

1

e



jkxi x y ˆ0 1

NT X nˆ1

T

qn1 y

) Cn1 Uzin1 eqn1 y1

…3:5:6a† jkxi x e y ˆ0 1






3:5:6b†

At the Layer `: Layer ` ‡ 1 interface we have Ex;` y ˆ `

(

MT X

ˆ Ex;`‡1 y

S‡ `

NT X

iˆ MT

nˆ1

Cn` Sxin` e (

(

NT X nˆ1

NT X nˆ1

T

e

Cn;`‡1 Sxin;`‡1 e )

Cn` Uzin` eqn` y`

MT X

1 ˆ 0 iˆ

`



MT

(

y ˆ S‡ `

qn;`‡1 y

jkxi x e

) e

`‡1

nˆ1

`‡1

Cn;`‡1 Uzin;`‡1 e



jkxi x

ˆ0

…3:5:7†

…3:5:8a† y

`‡1

y ˆ S`‡ `

NT X

ˆ Hz;`‡1 y

jkxi x `

iˆ MT MT 1 X 0 iˆ M

qn` y

S‡ `

`

)

MT X

ˆ

`‡1

Hz;` y ˆ

ˆ0

qn;`‡1 y

) e

`‡1

jkxi x



ˆ0

…3:5:8b† y

`‡1

ˆ0

At the Layer N` Region 3 interface we ®nd MT X

(

iˆ MT

NT X nˆ1

ˆ

) CnN` SxinN` exp…qnN` yN † e `

MT X ky3i Te 3 i iˆ M T

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jkxi x



jkxi x y

N`

ˆ SN`

…3:5:9a†

Planar Diffraction Gratings MT 1 X 0 iˆ M

( T

NT X nˆ1

ˆ

179

) CnN` UzinN` exp…qnN` yN † e `

MT 1 X Te 0 iˆ M i



jkxi x y

N`

ˆ SN`

jkxi x

…3:5:9b†

T

Each of the Fourier coef®cients of each exponential e jkxi x on the left and right sides of the above equations must be equal in order that the equations be all satis®ed. Equating Fourier coef®cients after evaluation of the y-dependent terms we ®nd ky1i …Eo io 1

Ri † ˆ

…Eo io ‡ Ri † ˆ

NT X nˆ1

NT X nˆ1 NT X nˆ1

Cn1 Sxin1

…3:5:10a†

Cn1 Uzin1

…3:5:10b†

Cn` Sxin` exp… qn` S` † ˆ

NT X nˆ1

Cn;`‡1 Sxin;`‡1

` ˆ 1; . . . ; N` ˆ 1 …3:5:11a†

NT X nˆ1

Cn` Uzin` exp… qn` S` † ˆ

NT X nˆ1

Cn;`‡1 Uzin;`‡1

` ˆ 1; . . . ; N`

1

…3:5:11b† NT X nˆ1 NT X nˆ1

Cn;N` SxinN` exp… qn;N` SN` † ˆ

ky3i T 3 i

Cn;N` UzinN` exp… qn;N` SN` † ˆ Ti

…3:5:12a† …3:5:12b†

Each equation assumes a range of i values from i ˆ MT ; . . . ; MT . NT ˆ 2…2MT ‡ 1†. The above equations can be simpli®ed. Equations 3.5.11a and 11b for ` ˆ 1; . . . ; N` 1 can be written, with i ˆ MT ; . . . ; MT , as

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Chapter 3



 ‰Sxin` Š K` ˆ e ‰Uzin` Š

qn` S`



K`‡1

‰Sxin;`‡1 Š ˆ ‰Uzin;`‡1 Š

 …3:5:13†

K1 C1 ˆ K‡ 2 C2 K2 C2 ˆ K‡ 3 C3 .. . KN` 1 CN`

1

ˆ K‡ N` CN`

…3:5:14†

where C` ˆ ‰C‰ 1`; C2` ; . . . CNT ` Št , ` ˆ 1; . . . ; N` . If we invert K` , ` ˆ 1; . . . ; N` 1 in each of the above equations, we have C1 ˆ …K1 † 1 …K‡ 2 †C2 C2 ˆ …K2 † 1 …K‡ 3 †C3 .. . CN` CN`

2 1

ˆ …KN` 2 † 1 …K‡ N` †CN` ˆ …KN` 1 †

1

…3:5:15† 1

…K‡ N` †CN`

Substituting we ®nd that    1 ‡ …K2 † 1 …K‡ C1 ˆ …K1 † 1 …K‡ 2† 3 †


…KN` 1 † …KN` † CN`

…3:5:16†

Letting the cascaded matrix be MNT NT , we have C1 ˆ M CN`

…3:5:17†

We can simplify Eq. 3.5.10a by solving for Ri in Eq. 3.5.10b and substituting in Eq. 3.5.10a. We can also substitute Ti from Eq. 3.5.12b into Eq. 3.5.12a. Doing both operations we ®nd that ! NT NT X X ky1i Eo io ‡ Eo io Cn1 Uzin1 ˆ Cn1 Sxin1 1 nˆ1 nˆ1   NT X 2Eo ky1i ky1i Vi ˆ io ˆ Cn1 Uzin1 ‡ Sxin1 1 1 nˆ1   N T X ky3i 0ˆ CnN` exp… qn;N` SN` † UzinN` ‡ SxinN` 3 nˆ1 …3:5:18†

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181

We also have ky10 ˆ n1 cos 1 so that, n1 ˆ

p 1 1 ,

ky10 n1 cos 1 cos 1 ˆ ˆ 1 n1 n21

…3:5:19†

so that Vi ˆ

2Eo cos 1 i;o n1

…3:5:20†

Letting …1†

R

  ky1i ˆ Uzin1 ‡ Sxin1 1 …NT =2†NT

I ˆ ‰ii 0 ŠNT NT 0 ˆ ‰0ŠNT =2NT    ky3i …N` † R ˆ exp… qnN` SN` † U ‡ SxinN` 3 zinN` …NT =2†NT iˆ

MT ; . . . ; 0; MT

n ˆ 1; . . . ; NT V ˆ ‰Vi i;o Š 2 3 V 6 7 607 6 7 Vext ˆ 6 7 607 4 5 2

0 R…1†

6 Aˆ6 4 M 0

iˆ

MT ; . . . ; 0; . . . ; MT "

Cˆ

2NT

0

C1 CN`

…3:5:21† 2NT

3

7 I 7 5

R…N` †

#

…3:5:22† 2NT 2NT

We ®nd the ®nal matrix equation from which C1 and CN` can be found. It is given by Vext ˆ A C

…3:5:23†

where Ct ˆ ‰Ct1 ; CtN` ] and t refers to the matrix transpose. Inversion of this equation gives C1 and CN` . Power may be analyzed in the same way as in the

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182

Chapter 3

single layer E-mode case. The analysis of the H-mode multilayer case is very similar to that of the E-mode case. The analysis of the H-mode multilayer case is very similar to that of the E-mode case. 3.5.3

Numerical Results

This section will present some numerical examples of H-mode and E-mode diffraction ef®ciency as results from the multilayer analysis described in this section. Figure 30 shows transmitted and re¯ected diffraction ef®ciencies of a sinusoidal surface relief grating when an H-mode polarized plane wave is incident on the grating. The layer thickness is taken to be L ˆ k0 L~ and extend from the peak to trough of the grating. In this example ~ ˆ , N` ˆ 5, and  ˆ 30 . In this example 2a ˆ 1 ˆ 1, 2b ˆ 3 ˆ 2:5,  the full ®eld formulations of Section 3.2 were used to calculate the EM ®elds of the diffraction grating system. As can be seen from the plots of Fig. 30, diffraction power is mainly transferred from the zero-order diffracted power into the ®rst-order with a small amount of diffracted power being re¯ected and transferred into higher order modes. This example has been analyzed by Moharam and Gaylord [19, Fig. 4, p. 1388] using the multilayer RCWA method to determine the EM ®elds of the system. In comparing their ®gure

Figure 30 The transmitted and re¯ected diffraction ef®ciencies of a sinusoidal surface relief grating when an H-mode polarized plane wave is incident on the grating.

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Planar Diffraction Gratings

183

to the one we present here, almost identical results cam from the two formulations. In Fig. 30 we mention that conservation of the real power was observed to a high degree of accuracy. Figure 31 shows transmitted (i ˆ 1; 0; 1; 2† and re¯ected (i ˆ 0; 1) diffraction ef®ciencies of a sinusoidal surface relief grating when an Emode polarized plane wave is incident on the grating rather an H-mode polarization. The layer thickness is taken to be L ˆ k0 L~ and extend from peak to trough of the grating as in the previous ®gure. In this example ~ ˆ , N` ˆ 5, and  ˆ 30 . In this example 2a ˆ 1 ˆ 1, 2b ˆ 3 ˆ 4,  the full ®eld formulation of Section 3.3 was used to calculate the EM ®elds of the diffraction grating system in each thin layer. As can be seen from the plots of Fig. 31, diffraction power is mainly transferred from the zero-order diffracted power to the ®rst-order with a small amount of diffracted power being re¯ected and transferred into higher order modes. This example has been analyzed by Yamakita et al. [55, Fig. 6, p. 156] who used a multilayer coupled mode method to determine the EM ®eld of the system. In comparing their ®gure to the present one, almost identical results came from the two formulations. In this paper [55] the numerical value of the diffraction orders was opposite of that used in Ref. 19 and that used in this section. (That is,

Figure 31 The transmitted …i ˆ 1; 0; 1; 2† and re¯ected …i ˆ 0; 1† diffraction ef®ciencies of a sinusoidal surface relief grating when an E-mode polarized plane wave is incident on the grating rather than an H-mode polarization is shown.

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184

Chapter 3

Yamakita's i ˆ 1 order is the same as the i ˆ 1 order in this section.) In Fig. 31 we mention that conservation of the real power was observed to a high degree of accuracy. Figure 32 shows transmitted and re¯ected diffraction ef®ciencies of a symmetric blaze surface relief grating when an H-mode polarized plane wave is incident on the grating. The layer thickness is taken to be L ˆ k0 L~ and extend from peak to trough of the grating. In this example ~ ˆ , N` ˆ 5, and  ˆ 30 . In this example the 2a ˆ 1 ˆ 1, 2b ˆ 3 ˆ 2:5,  full ®eld formulation of Sections 3.2 and 3.5.2 was used to calculate the EM ®elds of the diffraction grating system. As can be seen from the plots of Fig. 32, diffraction power is mainly transferred from the zero-order diffracted power into the ®rst-order with a small amount of diffracted power being re¯ected and transferred into higher order modes. This example has been analyzed by Moharam and Gaylord [19, Fig. 6, p. 1389], who used the same multilayer wave equation RCWA method as was used for Fig. 30 to determine the EM ®elds of the system. In comparing their ®gure to the present one, almost identical results came from the two formulations. In Fig. 32 we mention that conservation of the real power was observed to a high degree

Figure 32 The transmitted and re¯ected diffraction ef®ciencies of a symmetric blaze surface relief grating when an H-mode polarized plane wave is incident on the grating.

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Planar Diffraction Gratings

185

of accuracy. We mention that the blaze represents a dif®cult scattering case because of the presence of the sharp discontinuities caused by the blaze.

3.6 3.6.1

CROSSED DIFFRACTION GRATINGS Crossed-Diffraction Grating Formulation

In this section we are interested in studying diffraction from a crossed twodimensional grating using RCWA based on Floquet theory and state variable analysis. Figure 33 shows, as an example, the geometry of a two-dimensional crossed pyramidal grating (front and side view). We are interested in the case when a plane wave of oblique incidence and general polarization is incident on a crossed diffraction grating or a grating that is periodic in two

Figure 33 The geometry of a two-dimensional crossed pyramidal grating: (a) front view; (b) side view.

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186

Chapter 3

independent directions. We assume that the permittivity of the grating is described by an anisotropic dielectric permittivity tensor and that the permeability of the grating is that of homogeneous space, and that the permittivity tensor elements pq , …p; q† ˆ …x; y; z† can be expressed as a sum of Floquet harmonics as ~pq ˆ 0

1 X 1 X iˆ1 iˆ 1

pq;ii exp…j…iKx x ‡ iKz z††

…3:6:1†

where Kx ˆ 2=x , Kz ˆ 2=z , x and z are normalized, and x and z are ~ x , z ˆ k o  ~ z. normalized grating periods given by x ˆ ko  To start the analysis we expand the electric ®elds in Region 2 in a set of Floquet and space harmonics as !…2† X ! E ˆ S i;i …y† exp… jkxi x

jkzi z†

…3:6:2†

i;i

where kxi ˆ kx0

iKx

iˆ

MT ; . . . ; 0; . . . ; MT

kzi ˆ kz0

iKz

iˆ

MT ; . . . ; 0; . . . ; MT

…3:6:3†

where kx0 ˆ n1 sin  sin

…3:6:4†

kz0 ˆ n1 sin  cos

p where k~xo ˆ ko kxo , k~zo ˆ ko kzo , n1 ˆ 1 1 , and , are the incident angles of the incoming plane wave. The quantity n1 is the index of refraction of !…2† Region 1. The magnetic ®eld H in Region 2 can also be expanded in Floquet harmonics, and it is given by ! !…2† U 1 X! H ˆ ˆ U i;i …y† exp… jkxi x o o i;i

jkzi z†

…3:6:5†

p !…2† !…2† where 0 ˆ ~ 0 =~0 . If E and H are substituted in Maxwell's equations we ®nd that

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Planar Diffraction Gratings

! r S ˆ

187

! jU

…3:6:5a†

! ! r  U ˆ j
S

…3:6:5b†

where the curl r is expressed in normalized coordinates and ! ! r  F ˆ ko r~  F , where r is the curl in unnormalized coordinates. If ! ! ! we let F i;i represent either S i;i e j i;i or U i;i e j i;i , where i;i ˆ kxi x ‡ kzi z, we ®nd that    @Fzii ! ‡ jkzi Fyii r  F ii ˆ a^ x @y h i ‡a^ y jkzi Fxii ‡ jkxi Fzii   @Fxii ‡a^ z jkxi Fyii e @y

…3:6:6† j

i;i

Further, if we let f represent any of the dyadic components xx , xy ; . . ., with fi;i representing the Fourier amplitudes, and let F represent Sx , Sy , Sz , Ux , Uy , Uz , we ®nd that 1 1 X X

fFˆ

iˆ1 iˆ 1

2 4

1 X

1 X

i 0 ˆ1 i 0 ˆ 1

3 …fi

i 0 ;i i 0 Fi 0 ;i 0 †

5 exp… jkxi x

jkzi z†

…3:6:7†

The two-dimensional discrete convolution form of Eq. 3.6.7 is derived by the same manipulations as were used for the one-dimensional diffraction ! ! grating studied in Section 3.2. When the Floquet expansions of S , U , and  are substituted in Maxwell's equations and common modal amplitudes of exp… jkxi x jkzi z† are equated, the following set of coupled equations is found: @Szii ‡ jkzi Syii ˆ jUxi;i @y

…3:6:8a†

jkzi Sxii ‡ jkxi Szii ˆ

jUyii

…3:6:8b†

@Sxii ˆ @y

jUzii

…3:6:8c†

jkxi Syii

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188

Chapter 3

X @Uzii ‡ jkzi Uyii ˆ j @y i;i 0 h xx;i

i 0 ;i

i0

Sxi 0 i 0 ‡ xx;i

i 0 ;i

i0

Syi 0 i 0 ‡ xz;i

i 0 ;i

i0

Szi 0 i 0

i

…3:6:8d† jkzi Uxii ‡ jkxi Uzii ˆ j h

X i 0 ;i 0

yx;i

i 0 ;i i 0 Sxi 0 i 0

‡ yy;i

i 0 ;i i 0 Syi 0 i 0

‡ yz;i

i 0 ;i i 0 Szi 0 i 0

i

…3:6:8e†

jkxi Uyii

X @Uzii ˆj @y i 0 ;i 0 h zx;i i 0 ;i

i0

Sxi 0 i 0 ‡ zy;i

i 0 ;i

i0

Syi 0 i 0 ‡ zz;i

i 0 ;i

i0

Szi 0 i 0

i

…3:6:8f† To make further progress we eliminate the longitudinal electric and magnetic ®eld Floquet harmonic amplitudes Syii and Uyii , which occur in Eqs. 3.6.8a±f, by expressing these amplitudes in terms of the tangential ®eld components Sxii , Szii , Uxii , and Uzii . To do this it is necessary to express Eqs. 3.6.8a±f in matrix form. We now de®ne the column matrices Sq , Uq …q ˆ x; y; z† de®ned by Sq ˆ ‰Sqii Š and Uq ˆ ‰Uqii Š. Each of ordered pair …i; i† de®nes a position in the column matrix. We also de®ne the square diagonal matrices Kx and Kz , where these matrices represent the multipliers kxi and kzi , respectively. These square matrices are de®ned by Kx ˆ ‰kxi ii;i 0 i 0 Š

…3:6:9†

and Kz ˆ ‰kzi ii;i 0 i 0 Š ……ii†; …i 0 i 0 †† ˆ …… MT ; MT †; . . . ; …0; 0†; . . . …MT ; MT †† …3:6:10† We also de®ne the square matrices pq ,…p; q† ˆ …x; y; z†: pq ˆ ‰pq…i;i†;…i 0 ;i 0 † Š ˆ ‰pqi

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i 0 ;i i 0 Š

…3:6:11†

Planar Diffraction Gratings

189

where pqii;i 0 i 0 is the …ii†; …i 0 i 0 † matrix element of matrix pq and pqi i 0 ;i i 0 is the Floquet harmonic, which de®nes pqii;i 0 i 0 . With these matrix de®nitions it is possible to express Eqs. 3.6.8a±f in matrix form. Doing so we ®nd that Eqs. 3.6.8a±f satisfy @Sz ‡ jKz Sy ˆ @y

jUx

…3:6:12a†

Kz Sx ‡ Kx Sz ˆ

Uy

…3:6:12b†

@Sz ˆ @y

jUz

…3:6:12c†

  @Uz ‡ jKz Uy ˆ j xx Sx ‡ xy Sy ‡ xz Sz @y

…3:6:12d†

  jKz Ux ‡ jKx Uz ˆ j yx Sx ‡ yy Sy ‡ yz Sz

…3:6:12e†

  @Ux ˆ j zx Sx ‡ zy Sy ‡ zz Sz @y

…3:6:12f†

jKx Sy

jKx Uy

We may express Sy in terms of the tangential mode amplitudes Sx , Sz , Ux , and Uz if we express yy Sy in terms of Sx , Sz , Ux , and Uz and then multiply the resulting equation by yy1 . Doing so we ®nd that the longitudinal modal amplitude Sy and Uy are given by yy1 Kz Ux ‡ yy1 Kx Uz

Sy ˆ

Uy ˆ Kz Sx

yy1 yx Sx

yy1 yz Sz

Kx Sz

…3:6:13a† …3:6:13b†

If we substitute Sy and Uy into Eqs. 3.6.13a and perform algebra we ®nd that 2

Sx

3

2

A11

7 6 6 7 6 @ 6 6 Sz 7 6 A21 7ˆ6 6 @y 6 Ux 7 6 A31 5 4 4 Uz A41

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A12

A13

A22

A23

A32

A33

A42

A43

A14

32

7 A24 7 7 7 A34 7 5 A44

Sx

3

7 6 6 Sz 7 7 6 7 6 6 Ux 7 5 4 Uz

…3:6:14†

190

Chapter 3

where A11 ˆ jKx yy1 yx A12 ˆ jKx yy1 yz A13 ˆ jKx yy1 Kz A14 ˆ j‰I

Kx yy1 Kx Š

A21 ˆ j‰Kz yy1 yx Š A22 ˆ j‰Kz yy1 yz Š A23 ˆ j‰Kz yy1 Kz A24 ˆ j‰

IŠ

Kz yy1 Kx Š zx ‡ zy yy1 yx Š

A31 ˆ j‰ Kx Kz

A32 ˆ j‰K2x ‡ zy yy1 yz

zz Š

A33 ˆ j‰zy yy1 Kz Š A34 ˆ j‰ zy yy1 Kx Š A41 ˆ j‰ K2z ‡ xx A42 ˆ j‰Kx Kz

xy yy1 yx Š

xy yy1 yz ‡ xz Š

A43 ˆ j‰ xy yy1 Kz Š

…3:6:15†

A44 ˆ j‰xy yy1 Kx Š where I is the identity matrix. If we introduce the column matrix (t is the matrix transpose) V ˆ ‰Stx ; Stz ; Utx ; Utz Št

…3:6:16†

and the square matrix A ˆ ‰Ars Š

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…r; s† ˆ …1; 2; 3; 4†

…3:6:17†

Planar Diffraction Gratings

191

we can express Eq. 3.6.14 as @V ˆ AV @y

…3:6:18†

The solution of this equation by the state variable method, as has been discussed previously, is given by Vn …y† ˆ

NT X nˆ1

Vn eqn y

…3:6:20†

where AVn ˆ qn Vn

…3:6:21†

The quantities Vn and qn are the eigenvectors and eigenvalues of the matrix A and have dimension NT ˆ 4…2MT ‡ 1†…2MT ‡ 1†. The electromagnetic ®elds in Region 2 are given by Eq…2† …x; y; z†

ˆ

NT X X ii

nˆ1

! Cn Sqiin e

qn y

exp‰ j…kxi x ‡ kzi z†Š

and Hq…2† …x; y; z†

! NT 1X X qn y ˆ C U e e o i;i nˆ1 n qiin

j…kxi x‡kzi z†

q ˆ …x; y; z† …3:6:22†

where Sxiin , Ssiin , Uxiin , and Uxiin are eigenvectors obtained from the eigenvector Vn . The quantities Syiin and Uyiin are obtained from Eqs. 3.6.13a,b, using the known eigenvectors Sxiin , Sziin , Uxiin , and Uziin . Now that the EM ®elds have been determined in Region 2, the next step is to determine the EM ®elds in Region 1 (incident side) and Region 3 (transmit side) of the grating. The ®elds in Region 1 consist of an obliquely incident plane wave and consist of an in®nite number of Floquet harmonic re¯ected waves. Using the coordinates shown in Fig. 32, and assuming that the incident plane wave has polarization ( and are the incident angles of the incoming wave),

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192

Chapter 3

 !I  I E ˆ E a^ ‡ EI a^  exp… jkxi x ‡ jky1ii y

jkzi z†

iˆiˆ0 …3:6:23†

where ( ky1ii ˆ

‰1 k2xi k2zi Š1=2 j‰k2xi ‡ k2zi 1 Š1=2

k2xi ‡ k2zi  1 k2xi ‡ k2zi > 1

…3:6:24†

and noting that a^ ˆ

sin a^ z ‡ cos a^ x

a^  ˆ cos  cos a^z ‡ cos  sin a^ x and letting

I ii

ˆ kxi x

…3:6:25a† sin a^y

…3:6:25b†

kyii y ‡ kzi z, and substituting a^ and a^  , we ®nd that

   !I  sin EI a^y E ˆ cos E I ‡ cos  sin EI a^x ‡   ‡ sin EI ‡ cos  cos EI a^ z exp… j I00 † i !I Xh I I I E ˆ Exii a^ x ‡ Eyii a^ y ‡ Ezii a^ z exp… j Iii †ii;00

…3:6:26a† …3:6:26b†

i;i

where ii;00  i;0 io and ; is the Kronecker delta. The incident magnetic ®eld can be determined from the second Maxwell curl equation. We have !I H ˆ

!I 1 r E j0 i h i !I 1 nh I I I I kylii Ezi;i kzi Eyii kxi Ezii H ˆ a^ x ‡ kzi Exii a^ y 0 h i o I I ‡ kxi Eyii ‡ ky1ii Exi;i a^ z exp… j Iii †ii;00

…3:6:27b†

i !I Xh I I I H ˆ Hxii a^ x ‡ Hyii a^ y ‡ Hzii a^ z exp… j

…3:6:27c†

i;i

I ii †ii;00

…3:6:27a†

The re¯ected EM ®eld consists of an in®nite number of propagating Floquet harmonics, and evanescent backward traveling plane waves. The re¯ected EM electric ®eld is given by

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Planar Diffraction Gratings

193

!R X !R E ˆ E i;i exp… j

R ii †

…3:6:28a†

!R X !R H ˆ H ii exp… j

R ii †

…3:6:28b†

i;i

i;i

where i !R h E ii ˆ Rxii a^ x ‡ Ryii a^ y ‡ Rzii a^ z R ii

…3:6:28c†

ˆ kxi x ‡ ky1ii y ‡ kzi z

…3:6:28d†

Notice that, in Eq. 3.6.24 for the case that ky1ii is evanescent, e j… jjky1ii jy†ˆ e jky1ii jy ! 0 as y ! 0. We thus see that for the evanescent plane wave wavenumber h ky1ii ˆ j k2xi ‡ k2zi

1

i1=2   ; k2xi ‡ k2zi > 1

…3:6:29†

the minus is the correct root. This is the one used in Eq. 3.6.24. The re¯ected magnetic ®eld in Region 1 is given by i !R h R R R H ii ˆ Hxii a^ x ‡ Hyii a^ y ‡ Hzii a^ z ˆ

i 1 nh ky1ii Rzii kzi Ryii a^x 0 h i ‡ kzi Rxii kxi Rzii a^ y h i o ‡ kxi Ryii ky1ii Rxii a^ z

…3:6:30†

!I !R In Eq. 3.6.26 for E and Eq. 3.6.28a for E the longitudinal y-electric ®eld component Ey can be expressed in terms of the tangential electric ®elds Ex and Ez . Using the electric ¯ux density equation in Region 1, ! ! ! r
D ˆ r
1 E ˆ 1 r
E ˆ 0

! r
E ˆ0

…3:6:31†

!R ! !I where E represents either E or E . Using this equation we have

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194

Chapter 3

" X !I !I r
E ˆr
E ii e

I ii

ii;00 ˆ 0

…3:6:32a†

jkz0 Ez00 ˆ 0

…3:6:32b†

kx0 I k I Ex00 ‡ z0 Ez00 ky100 ky100

…3:6:32c†

ii

I I jkx0 Ex00 ‡ jky100 Ey00 I Ey00 ˆ

# j

!R and for E " X !R r
E ˆ0ˆr
Rii e

# j

R ii

…3:6:33a†

ii

jkxi Rxii

jky1ii Ryii

jkzi Rzii ˆ 0 Ryii ˆ

…3:6:33b† kxi R ky1ii xii

kzi R ky1ii zii

…3:6:33c†

!R !I We thus see that E and E are known once they can be expressed entirely I I , Ez00 , Rxii , and Rzii . in terms of the known coef®cients of Ex00 !I !R The incident and re¯ected magnetic ®elds H and H can be !I !R expressed in terms of the tangential E and E ®elds. After substitution I I I , Ey00 , Ez00 in Eq. 3.6.27b, we ®nd that the tangential incident magof Ex00 I I and Hz00 are given by netic ®eld amplitudes Hx00  1 I I I I Yxx00 Ex00 ‡ Yxz00 Ez00 0  1 I I I I ˆ Yzx00 Ex00 ‡ Yzz00 Ez00 0

I Hx00 ˆ

…3:6:34a†

I Hz00

…3:6:34b†

where I Yxx00 ˆ

kz0 kx0 ky100

I Yxz00 ˆ

ky100

I Yzx00 I Yzz00

k2z0 ky100

k2 ˆ ky100 ‡ x0 ky100 k k ˆ x0 z0 ky100

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…3:6:35†

Planar Diffraction Gratings

195

I The quantities Ypq00 , …p; q† ˆ …x; z† may be considered the normalized surface admittances of the system. They are analogous to the surface aperture admittances used in k-space theory to analyze radiation from inhomogeneously cover surface aperture antennas [Chapter 2, this book, [1], Rhodes and Galejs[7]]. The tangential magnetic ®eld re¯ected modal amplitudes can also be expressed in terms of the tangential re¯ected electric ®eld modal amplitudes Rxii and Rzii using Eq. 3.6.30. We have

i 1h R R Yxxii Rxii ‡ Yxzii Rzii 0 i 1h R R ˆ Yzxii Rxii ‡ Yzzii Rzii 0 kzi kxi ˆ kylii

R ˆ Hxii

…3:6:36a†

R Hzii

…3:6:36b†

R Yxxii

R Yxzii ˆ kylii ‡

k2zi ky1ii

R Yzxii ˆ

ky1ii

R Yzzii ˆ

kxi kzi ky1ii

k2xi ky1ii …3:6:37†

Overall the EM ®elds in Region 1 are given by   !…1† X !I j Iii !R j R E ii e ii;00 ‡ E ii e ii E ˆ i;i

 !…1† X !I H ˆ H ii e i;i

j

I ii

!R i;i;00 ‡ H ii e

j

R ii

…3:6:38a†

 …3:6:38b†

The analysis for the EM ®elds in Region 3 on the transmit side is very similar to the analysis made in Region 1. In Region 3 the electric and magnetic ®elds consist of an in®nite number of Floquet harmonic diffracted plane waves. The electric ®eld in Region 3 is given by i !…3† !T Xh E  E ˆ Txii a^ x ‡ Tyii a^ y ‡ Tzii a^ z e

j

T ii

…3:6:39†

ii

where T ii

ˆ kxi x

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ky3ii …y ‡ L† ‡ kzi z

…3:6:40†

196

Chapter 3

where

ky3ii

8h i1=2 > < 3 k2xi k2zi ˆ h i1=2 > : j k2 ‡ k2  3 xi zi

k2xi ‡ k2zi  3

…3:6:41†

k2xi ‡ k2zi > 3

Note that when the plane wave is evanescent …k2xi ‡ k2zi > 3 †, the exponent in 3.6.39 tends to zero as y ! 1. Note that in 3.6.40, Tii has been chosen so that Tiijyˆ L ˆ kxi x ‡ kzi z, which simpli®es boundary matching. Using the fact that "  X !T !T !T 0 ˆ r
D ˆ 3 r
E ˆ 3 r
E ii e

j

T ii

# …3:6:42†

ii

and differentiating Eq. 3.6.42, we ®nd that 0ˆ Tyii ˆ

jkxi Txii ‡ jky3ii Tyii

jkzi Tzii

…3:6:43†

kzi kxi T ‡ T ky3ii xii ky3ii zii

…3:6:44†

The magnetic ®eld in Region 3 can be found from Maxwell's ®rst curl equation. We have " i Xh !…3† !…T† T T T H H ˆ Hxii a^ x ‡ Hyii a^ y ‡ Hzii a^ z e

j

T ii

…3:6:45†

ii

i !…3† 1 Xnh ky3ii Tzii kzi Tyii a^ x H ˆ 0 ii h i ‡ kzi Txii kxi Tzii a^y h i o j T ‡ kxi Tyii ‡ ky3ii Txii a^ z e ii

…3:6:46†

Using Eq. 3.6.44, Tyii can be expressed in terms of Txii and Tzii . Thus it is possible to express all the magnetic ®eld components in terms of Txii and T T Tzii . The tangential magnetic ®eld modal amplitudes Hxii and Hzii are given by

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Planar Diffraction Gratings

197

i 1h T T Yxxii Txii ‡ Yxzii Tzii 0 i 1h T T ˆ Yzxii Txii ‡ Yzzii Tzii 0

T Hxii ˆ T Hzii

…3:6:47†

where T Yxxii ˆ

kzi kxi ky3ii

T Yxzii ˆ

ky3ii

T Yzxii

k2zi ky3ii

…3:6:48†

k2xi ˆ ‡ ky3ii ky3ii

T Yzzii ˆ

kxi kzi ky3ii

The next step in the analysis is to match the EM ®eld solutions at the y ˆ 0 and y ˆ L interfaces and determine all the unknown constants of the system. Now that the EM ®elds have been de®ned in Regions 1, 2, and 3, the next step in the analysis is to match the tangential electric and magnetic ®elds at boundary plane y ˆ 0 and y ˆ L. At y ˆ 0 we have …1† Ex;z …1† Hx;z

yˆ0‡ yˆ0‡

…2† ˆ Ex;z

yˆ0

…2† ˆ Hx;z

…3:6:49†

yˆ0

Substituting the previous equations for the EM ®elds and evaluating at y ˆ 0, we have Xh

I Exii ii;00

i ‡ Rxii e

jkxi x jkzi z

ˆ

i;i

i;i

i Xh I Ezii ii;00 ‡ Rzii e i;i

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" NT X X

jkxi x jkzi z

ˆ

nˆ1

" NT X X ii

nˆ1

# Cn Sxiin e

jkxi x jkzi z

# Cn Sziin e

jkxi x jkzi z

198

Chapter 3

 1 Xh I I I I ii;oo Yxxii Exii ‡ Yxzii Ezii o ii  i R R Rxii ‡ Yxzii Rzii e ‡ Yxxii

jkxi x jkzi z

 1 Xh I I I I ii;oo Yzxii Exii ‡ Yzzii Ezii o ii  i R R Rxii ‡ Yxzii Rzii e ‡ Yzxii

jkxi x jkzi z

" # NT 1X X ˆ C U e o ii nˆ1 n xiin

jkxi x jkzi z

" # NT 1X X ˆ C U e o ii nˆ1 n ziin

jkxi x jkzi z

…3:6:50† At the y ˆ …3† Ex;z …3† Hx;z

L interface we have

yˆ L yˆ L

…2† ˆ Ex;z

yˆ L‡

…2† ˆ Hx;z

yˆ L‡

X ii

X ii

Txii e Tzii e

jkxi x jkzi z

ˆ

" NT X X ii

jkxi x jkzi z

ˆ

ii

i 1 Xh T T Yxxii Txii ‡ Yxzii Tzii e o ii

jkxi x jkzi z

i 1 Xh T T Yzxii Txii ‡ Yzzii Tzii e o ii

jkxi x jkzi z

nˆ1

" NT X X nˆ1

# Cn Sxiin e

qn L

Cn Sziin e

qn L

e

jkxi x jkzi z

e

jkxi x kzi z

#

" # NT 1X X qn L ˆ C U e e jkxi x o ii nˆ1 n xiin " # NT 1X X ˆ C U e jkxi x jkzi z o ii nˆ1 n ziin …3:6:51†

Equating modal coef®cients we obtain the following set of equations: I Exii ii;oo ‡ Rxii ˆ

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NT X nˆ1

Cn Sxiin

…3:6:52†

jkzi z

Planar Diffraction Gratings

199 I Ezii ii;oo ‡ Rzii ˆ

NT X

Cn Sziin

…3:6:53†

NT     X I I I I R R Yxxii ii;oo ‡ Yxxii Exii ‡ Yxzii Ezii Rxii ‡ Yxzii Rzii ˆ Cn Uxiin

…3:6:54†

NT     X I I I I R R Yzxii ii;oo ‡ Yzxii Exii ‡ Yzzii Ezii Rxii ‡ Yzzii Rzii ˆ Cn Uziin

…3:6:55†

nˆ1

nˆ1

nˆ1

We can eliminate Rxii and Rzii and determine equations for Cn alone. We have   I I I I R Yxxii ‡Yxzii ii;oo ‡ Yxxii Exii Ezii " ‡

R Yxzii

I Ezii ii;oo

‡

"

NT X nˆ1

I Exii ii;oo

‡

NT X nˆ1

# Cn Sziin ˆ

NT X nˆ1

# Cn Sxiin …3:6:56† Cn Uxiin

Collecting common terms, we have h

I …Yxxii

R I I Yxxii †Exii ‡ …Yxzii NT X nˆ1

Cn

h

R Yxxii Sxiin

i R I ii;oo ˆ Yxzii †Ezii R Yxzii Sziin ‡ Uxiin

i

…3:6:57†

A similar analysis for the Hz , y ˆ 0 equation shows that h

I …Yzxii

R I I Yzxii †Exii ‡ …Yxzii NT X nˆ1

Cn

h

R Yzxii Sxiin

i R I ii;oo ˆ Yxzii †Ezii R Yzzii Sziin ‡ Uziin

If the modal coef®cients of the Hx and Hz , y ˆ we ®nd that

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i

…3:6:58†

L, equations are computed,

200

Chapter 3

Txii ˆ Tzii ˆ T T Yxxii Txii ‡ Yxzii Tzii ˆ

T T Yzxii Txii ‡ Yzzii Tzii ˆ

NT X nˆ1 NT X nˆ1 NT X nˆ1 NT X nˆ1

Cn Sxiin e

qn L

Cn Sziin e

qn L

Cn Uxiin e

qn L

…3:6:59a†

Cn Uziin e

qn L

…3:6:59b†

If we substitute Txiin and Tziin into Eqs. 3.3.59a,b, we ®nd that 0ˆ

NT X nˆ1

0ˆ

NT X nˆ1

Cn e

qn L

Cn e

qn L

n n

T Yxxii Sxiin

T Yxzii Sziin ‡ Uxiin

T Yzxii Sxiin

T Yzzii Sziin ‡ Uziin

o

o

…3:6:60a† …3:6:60b†

Equations 3.6.60a,b form a set of NT  NT equations from which all the modal coef®cients can be determined. Another important quantity that needs to be studied is the power incident on the cross-grating and the power re¯ected, diffracted, and transmitted from the grating. The power incident on the grating over one grating cell in the … a^y † direction is given by PIR ˆ 12 Re…PIc †

…3:6:61†

where PIc is given by PIc

ˆ

… ~ x … ~ z =2 2

~x  2

~ z =2 



!I !I ~ z† ~ E  H
… a^ y d xd

or after being put in normalized form and carrying out the PIc ˆ

1 k2o

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… z =2 … x =2 h z =2

x =2

i I I I I Ez00 Hx00 ‡ Ex00 Hz00 dx dy

…3:3:62† a^ y dot product, …3:6:63†

Planar Diffraction Gratings

201

The quantity in square brackets is a constant. After integrating Eq. 3.6.63 I ; . . ., it is found that and substituting the incident modal admittances Yxx00 I Pc is given by PIc ˆ

i x z n I h I I I I E Y E ‡ Y E z00 xx00 x00 xz00 z00 k20 0

h  io I I I I I Yzx00 Ex00 ‡ Yzz00 Ez00 Ex00 …3:6:64†

I I The quantities Ex00 and Ez00 are given in terms of the incident angles and polarizations by Eq. 3.6.26a. The re¯ected power from the crossed grating is given by

PR ˆ 12 Re…PR c †

…3:6:65†

where PR c

1 ˆ 2 ko

PR c

1 ˆ 2 ko

PR c ˆ

… ~ z =2

… ~ x =2

z =2

x =2

… z =2



!R !R E  H
…a^ y dx dz†

… x =2 h  EzR HxR

z =2

x =2

1 R Izx k2o

R Ixz

…3:6:66†

i  ExR HzR dx dz

…3:6:67†




…3:6:68†

R R and Ixz refer to the ®rst and second terms in Eq. 3.6.67. If we where Izx R substitute Ex and EzR into Izx , we ®nd after interchanging summation and integration that

Izx ˆ

XX ii




i 0i 0

h  i R  R  0 i 0 Rxi 0 i 0 ‡ Yxzi 0 i 0 Rzi 0 i 0 Rzii Yxxi

… x =2 x =2

e

j…kxi kxi 0 †x

dx

 … z =2 z =2

e

j…kzi kzi 0 †z



…3:6:69†

dz

The ®rst integral (x-integral) equals x ii 0 and the second integral equals z ii 0 , where ii 0 is the Kronecker delta. Substituting these values in Eq. 3.6.69 we ®nd that Izx ˆ x z

X ii

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h  i R R  0 i 0 Rzii Rzii Yxxii Rxii ‡ Yxzi

…3:6:70†

202

Chapter 3

R Carrying out a similar analysis for Ixz and substituting the expressions into Eq. 3.6.68, we ®nd that

X

PR c ˆ

ii

PR cii

i x z Xnh  R  R  R Y R ‡ Y R zii xxii xii xzii zii k2o o ii h   io R R  Rxii Yzxii Rxii ‡ Yzzii Rzii

ˆ

The power transmitted in the crossed x z grating cell is given by

a^ y direction at y ˆ

…3:6:71†

L

PT ˆ 12 Re…PTc †

over one …3:6:72†

where PTc

ˆ

PTc ˆ

… z =2

… x =2

z =2

1 k2o

… z =2



x =2

z =2

!T !T E H


… x =2 h x =2


a^y d x~ d z~

i   EzT HzT ‡ ExT HzT dx dz

…3:6:73† …3:6:74†

Substituting the transmitted electric and magnetic ®elds into Eq. 3.6.74 for PTc , and carrying out an analysis similar to that used to determine PR c , we ®nd that PTc ˆ

i x z Xn h  T  T  T Y T ‡ Y T zii xxii xzii xii zii k2o o ii ii h   io T  T  ‡ Txii Yzxii Txii ‡ Yzzii Tzii X

PTcii ˆ

…3:6:75†

An important quantity associated with the transmitted and re¯ected differential power is the diffraction ef®ciency of the iith order. The diffraction ef®ciency of the re¯ected iith is given by and de®ned by Re …PR † cii 2 DR ˆ ii Re …PI † c 2 The diffraction ef®ciency of iith order is given by and de®ned by

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…3:6:76†

Planar Diffraction Gratings

Re …PT † cii DTii ˆ 2 Re …PI † c 2

203

…3:6:77†

For a lossless crossed grating, the re¯ected and transmitted diffraction ef®ciencies obey the conservation of power relation X T …DR ii ‡ Dii † ˆ 1

…3:6:78†

ii

3.6.2

Numerical Results

This section will present some numerical examples of the diffraction ef®ciency that results when an oblique plane wave is scattered or diffracted from a crossed or two-dimensional diffraction grating. The example to be presented involves scattering from a one-dimen~ x ˆ , sional square wave grating where 1 ˆ 3 ˆ 1, 2 ˆ 2:5,  ˆ 30 ,  ~ z ˆ 1. This example has been previously studied for the H-mode case and  in Section 3.2 and the E-mode case in Section 3.3. In the literature it has been ®rst presented by Yamakita and Rokushima [54]. The purpose of using the more general crossed grating algorithm to study a one-dimensional case is to validate that in the limiting case the operation of the RCWA crossed grating algorithm presented in this section can produce the same results as the one-dimensional RCWA algorithm. The H-mode square wave case was ~ z to have a large but not in®nite value.  ~ z in numerically studied by taking  ~ z ˆ 15, MT ˆ 6, MT ˆ 0, ˆ 270 ,  ˆ 30 , the algorithm was set to  and E I ˆ E0 , EI ˆ 0. The nonzero relativePdielectric permittivities were T taken to be xx …x; z† ˆ yy …x; z† ˆ zz …x; z† ˆ M iˆ MT i exp…jiKx x†, where i are the Fourier coef®cients of the square pro®le used in the square wave example of Section 3.2. The E-mode square wave case was studied using the same parameters as the H-mode square wave case except that the polarization was taken to be E I ˆ 0, EI ˆ E0 . Figure 34 shows the diffraction ef®ciency results using the one-dimensional theory of Sections 3.2 and 3.3 and using the crossed diffraction grating theory of this section. As can be seen from Fig. 34, nearly identical diffraction ef®ciency results from the two algorithms. The crossed diffraction grating theory has been also used to calculate the scattering from the H-mode cosine grating (Figure 2) (Gaylord [16]) described in Section 3.2. After setting the parameters of the crossed grating algorithm to match those of the H-mode cosine grating, identical diffraction

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204

Chapter 3

Figure 34 The diffraction ef®ciency results using the one-dimensional theory of Sections 3.2 and 3.3 are used to validate the two-dimensional crossed diffraction grating theory of this section. Results here are identical to those of Yamakita and Rokushima [54, Fig. 5, p. 242], who ®rst calculated this example by coupled mode theory.

ef®ciency results were obtained for the one- and two-dimensional RCWA algorithms for the case also. Figure 35 shows the diffraction ef®ciency data that results when the crossed grating theory of this section is applied to study scattering from a two-dimensional crossed cosine wave grating where 1 ˆ 2 ˆ 3 ˆ 1, ~ x ˆ 2:8747, and  ~ z ˆ 1:5 ~ x , L~ ˆ 9, MT ˆ 3, MT ˆ 3,  ˆ 10 ,  ˆ 270 , E I ˆ 1, and EI ˆ 0. The nonzero relative dielectric permittivities were taken to be

xx …x; z† ˆ yy …x; z† ˆ zz …x; z† ˆ rs …x; z† ˆ 0

r 6ˆ s

MT X

MT X

iˆ MT iˆ MT

ii exp‰j…iKx x ‡ iKz z†Š

…r; s† ˆ …x; y; z† …3:6:79†

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Planar Diffraction Gratings

205

Figure 35 The diffraction ef®ciency data that results when crossed grating theory is applied to study scattering from a two-dimensional crossed cosine wave grating is shown.

In Eq. 3.6.79, 0;0 ˆ 1 ; 1;1 ˆ  1;1 ˆ 1; 1 ˆ  1; 1 ˆ 1=4, and all other Fourier coef®cients i;i in Eq. 3.6.79 are zero. In Fig. 35 transmitted diffraction ef®ciencies (denoted by Ti;i ) of the T00 , T01 , T10 , and T11 orders was plotted versus the azimuthal angle , which was varied over the range 180   270 . As can be seen from the Fig. 35 plot, changing the angle of incidence causes a perceptible change in the diffraction ef®ciency observed from the grating. In making the Fig. 35 plot conservation of real power, Eq. 3.6.78 was observed to a high degree of accuracy. Table 1 shows the transmitted diffraction ef®ciency for the crossed cosine diffraction grating studied in Fig. 35 (taking ˆ 270 ) that results for ®fteen orders (taking all pair combinations of i ˆ 2; 1; 0; 1; 2 and i ˆ 1; 0; 1) when ®ve different matrix truncations MT ˆ MT ˆ 1; 2; 3; 4, 5 are used. (For those truncations where the i; i) pair order exceeds the truncation order [for example, when the pair …i; i† ˆ … 2; 1† exceeds the truncation order, MT ˆ MT ˆ 1Š the diffraction ef®ciency is set to zero.) A striking and reassuring feature of the diffraction ef®ciencies displayed in Table 1 is how rapidly the diffraction ef®ciency converges to a ®nal value that does not change with increasing order. After the value of MT ˆ

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206

Chapter 3

Table 1 Transmitted diffraction ef®ciency for the crossed cosine diffraction grating studied in Fig. 35 ( ˆ 270 ) that results for ®fteen orders (Taking All Pair Combinations of i ˆ 2; 1; 0; 1; 2 and i ˆ 1; 0; 1) When Five Different Matrix Truncations MT ˆ MT ˆ 1; 2; 3; 4; 5 Are Used

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Planar Diffraction Gratings

207

Figure 36 The transmitted diffraction ef®ciencies (solid line) of the Ti;i orders when i and i rnge from … 1; 0; 1†, when the grating thickness is varied from L~ ˆ 0 to L~ ˆ 4, are shown.

MT ˆ 3 there is virtually no change in diffraction ef®ciency for any of the ®fteen orders displayed. The second diffraction ef®ciency example to be presented consists of the diffraction ef®ciency data that results when the crossed grating theory of this section is applied to study scattering from a two-dimensional rectangular surface relief grating composed of isotropic dielectric material. The rectangular dielectric (shown in Fig. 36 inset) making up the surface relief grating in Region 2 was assumed to be centered in each two-dimensional ~ x =2, a length of 2z1 ˆ  ~ z =2, a grating period and to have a width of 2x1 ˆ  ~ thickness of L, and a relative permittivity value of 3 . The region surrounding the rectangular dielectric was assumed to have a dielectric value of 1 . In Region 2, mathematically the permittivity tensor of the surface relief grating is given by xx …x; y; z† ˆ yy …x; y; z† ˆ zz …x; y; z† ˆ …x; y; z† rs …x; y; z† ˆ 0 where

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r 6ˆ s

…r; s† ˆ …x; y; z†

…3:6:80†

208

Chapter 3

 …x; y; z† ˆ

3 1

~  x~ 1 ; jzj ~  z~1 jxj ~ x ; jzj ~z ~  ~  elsewhere in the cell jxj …3:6:81†

~ x =2 and z~1 ˆ  ~ z =2. Fourier inversion of Eq. 3.6.1 using and where x~ 1 ˆ  the speci®ed permittivity value given by Eq. 3.6.81 speci®es the two-dimensional Fourier coef®cients of Eq. 3.6.80. Figure 36 shows the transmitted diffraction ef®ciencies (solid line) of the Ti;i orders where i and i range from … 1; 0; 1†; when the grating thickness is varied from L~ ˆ 0 to L~ ˆ 4; when ~ x ˆ ,  ~ z ˆ 1:5 ~ x , ˆ 270 , E I ˆ 1, and 1 ˆ 1, 3 ˆ 2:5,  ˆ 30 ,  I E ˆ 0; and when MT ˆ MT ˆ 3. As can be seen from Fig. 36, EM power is diffracted out of the T0;0 order (pump wave or incident wave) and is subsequently diffracted into the higher orders. Because of symmetry, the diffraction ef®ciencies of the T1;1 and T1; 1 orders were the same and the diffraction ef®ciencies of the T0;1 and T0; 1 orders were the same. By the same token, for ˆ 0 , 90 , and 180 , we should observe similar symmetry in the diffracted orders. Figure 36 also shows the diffraction ef®ciency of the T1;0 order (dotted) when the truncation was taken to be MT ˆ MT ˆ 2 rather than MT ˆ MT ˆ 3 as was done for the curves discussed earlier. As can be seen from the ®gure, very little diffraction ef®ciency difference exists between the two truncations. Table 2 shows the transmitted diffraction ef®ciency for the crossed rectangular diffraction grating studied in Fig. 36 (taking ˆ 270 and L~ ˆ 1:7) that results for ®fteen orders (taking all pair combinations of i ˆ 2; 1; 0; 1; 2 and i ˆ 1; 0; 1) when ®ve different matrix truncations MT ˆ MT ˆ 1; 2; 3; 4; 5 are used. (For those truncations where the …i; i† pair order exceeds the truncation order [for example, when the pair …i; i† ˆ … 2; 1† exceeds the truncation order, MT ˆ MT ˆ 1] the diffraction ef®ciency is set to zero.) A striking and reassuring feature of the diffraction ef®ciencies displayed in Table 2, like those of Table 1, is that the diffraction ef®ciency converges fairly rapidly to a ®nal value that does not change with increasing order. In comparing Table 2 to Table 1 it is interesting to note that the convergence to a ®nal value is slightly slower in Table 2 than in Table 1. This is believable since the grating studied in Table 2 is much smaller in size than the grating studied in Table 1 and also the grating studied in Table 2 has a much higher spatial spectral content than the grating studied in Table 1 (cosine grating). Both these factors would cause a slower convergence with truncation order. The third diffraction ef®ciency example to be presented consists of the diffraction ef®ciency data that results when the crossed grating theory of this section is applied to study scattering from a two-dimensional rectangu-

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Planar Diffraction Gratings

209

Table 2 Transmitted Diffraction Ef®ciency for the Crossed Rectangular Diffraction Grating Studied in Fig. 35 ( ˆ 270 and L~ ˆ 1:7) That Results for Fifteen Orders (Taking All Pair Combinations of i ˆ 2; 1; 0; 1; 2 and i ˆ 1; 0; 1) When Five Different Matrix Truncations MT ˆ MT ˆ 1; 2; 3; 4; 5 are used

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210

Chapter 3

lar surface relief grating that contains anisotropic dielectric material. In Region 2, we ®rst need to express the permittivity tensor of the surface relief grating cell for this example. Let  f …x; z† ˆ

~  x~ 1 ; jzj ~  z~1 jxj ~ x ; jzj ~z ~  ~  elsewhere in the cell jxj

b a

…3:6:82†

where b ˆ

n2o ‡ n2e 2

…3:6:83†

and also let xxC ˆ

n2o ‡ …n2e n2o †Cx2 b

yyC ˆ

n2o ‡ …n2e n2o †Cy2 b

zzC ˆ

n2o ‡ …n2e n2o †Cz2 b

xyC ˆ yxC ˆ xzC ˆ zxC ˆ

…n2e

…3:6:83a†

n2o †Cx Cy b

…n2e

n2o †Cx Cz b

yzC ˆ zyC ˆ zxC

Cx ˆ sin…c † sin…c † Cy ˆ cos…c † sin…c † Cz ˆ cos…c † ~ x =2, z~1 ˆ  ~ z =2, a ˆ 1,  ~ x ˆ , and where C ˆ C ˆ 45 , x~ 1 ˆ  ~ ~ z ˆ 1:5x . Using these parameters and functions we de®ne the relative dielectric permittivity to be

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Planar Diffraction Gratings

2

xxC …x; z† ˆ 4 yxC zxC

xyC yyC zyC

211

3 xzC yzC 5f …x; z† zzC

…3:6:84†

Note that the diffraction grating in Region 2 is made up of two different anisotropic materials, as speci®ed above. Figure 37 shows plots of the diffraction ef®ciency of, say, the T10 order …i ˆ 1; i ˆ 0†; when the grating thickness is varied from L~ ˆ 0 to L~ ˆ 4; when 3 ˆ 2:5,  ˆ 30 , ~ x ˆ ,  ~ z ˆ 1:5 ~ x , ˆ 270 , E I ˆ 1, and EI ˆ 0; when  MT ˆ MT ˆ 2; and when the values of the parameters n2o , n2e were taken to be n2o ˆ 2, n2e ˆ 3 (curve marked T10a ), n2o ˆ 2:4, n2e ˆ 2:6 (curve marked T10b ), n2o ˆ 2:5, n2e ˆ 2:5 (curve marked T10c ). As can be seen from Fig. 37, considerable power is diffracted into the T10 order. Figure 37 also shows that as the grating is made more anisotropic (that is, by increasing the magnitude of the difference between n2o and n2e ), a more perceptible difference between the isotropic and anisotropic diffraction ef®ciencies occurs. It is interesting to note that even with severe anisotropy there is not too much difference in the behavior of the diffraction ef®ciency as compared to the isotropic case. The fourth diffraction ef®ciency example to be presented consists of the diffraction ef®ciency data that results when the crossed grating theory of

Figure 37 Plots of the diffraction ef®ciency of the T10 order …i ˆ 1; ˆ 0† when the grating thickness is varied from L~ ˆ 0 to L~ ˆ 4, are shown.

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212

Chapter 3

Table 3 Transmitted Diffraction Ef®ciency for the Crossed Rectangular Anisotropic Diffraction Grating Studied in Fig. 37 (Taking n2o ˆ 2, n2e ˆ 3, L~ ˆ 1:7 and All Other Parameters the Same as Fig. 37) That Results for Fifteen Orders (Taking All Pair Combinations of i ˆ 2; 1; 0; 1; 2 and i ˆ 1; 0; 1) When Five Different Matrix Truncations MT ˆ MT ˆ 1; 2; 3; 4; 5 are used

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Planar Diffraction Gratings

213

this section is applied to study scattering from a rectangular pyramidal surface relief grating that contains anisotropic dielectric material. Let the function f …x; y; z† at any given value of y in the interval L  y  0 be de®ned as  f …x; y; z† ˆ

b a

jxj  x1 …y†; jzj  z1 …y† elsewhere in jxj  x=2 ; jzj  z=2

…3:6:85†

where p 3jyj x1 …y† ˆ 4x p 3jyj z1 …y† ˆ 4z then the permittivity in the grating cell in Region 2 is given by 2

xxC …x; y; z† ˆ 4 yxC zxC

xyC yyC zyC

3 xzC yzC 5f …x; y; z† zzC

…3:6:86†

where the parameters of Eq. 3.6.86 are already given in Eq. 3.6.83. Because the grating is longitudinally inhomogeneous, a two-dimensional multilayer analysis based on the theory of Section 3.5 was used to calculate the diffraction ef®ciency. Figure 38 shows plots of the diffraction ef®ciency of the T00 , T10 , T01 , and T11 orders; when the grating thickness is varied from L~ ˆ 0 to ~ x ˆ , and  ~ z ˆ 1:5 ~ x ;  ˆ 30 , ˆ 270 , E I ˆ 1, L~ ˆ 2:5; 3 ˆ 2:5,  I and E ˆ 0; when MT ˆ MT ˆ 2; when the values of the parameters n2o , n2e were taken to be n2o ˆ 2, n2e ˆ 3; and when ten layers …N` ˆ 10) were used to carry out the two-dimensional multilayer analysis. As can be seen from Fig. 38, for the grating under study, power is diffracted out of the T00 order into higher orders. Conservation of power as speci®ed by equations was observed to a high degree of accuracy. In the ®nal example, a crossed pyramidal diffraction grating is again studied (same pyramid geometry as Fig. 37), but with a mirror (or a perfectly conducting short circuit plate) placed at the Region 2±Region 3 interface at y~ ˆ L~ (see Fig. 39). In this case just the re¯ected diffraction ef®ciency was studied (the transmitted diffraction ef®ciency in Region 3 is zero). The overall EM analysis in this case requires that the tangential EM ®elds at y~ ˆ L~ be zero. Imposing this condition (see Section 3.2.4 for an H-

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214

Chapter 3

Figure 38 Plots of the diffraction ef®ciency of the T00 , T10 , T01 , and T11 orders, when the grating thickness is varied from L~ ˆ 0 to L~ ˆ 2:5, are shown.

mode one-dimensional analysis) leads to a set of multilayer matrix equations from which the EM ®elds in the diffraction grating system can be found. Fig. 39 shows plots of the diffraction ef®ciency of the R00 , R10 , R01 , and R11 orders of the mirror±grating system; when the grating thickness is varied ~ z ˆ 1:5 ~ x ;  ˆ 30 , ˆ 270 , ~ x ˆ , and  from L~ ˆ 0 to L~ ˆ 2:5;  I I E ˆ 1, and E ˆ 0; when MT ˆ MT ˆ 2; when the values of the parameters n2o , n2e were taken to be n2o ˆ 2, n2e ˆ 3; and when ten layers …N` ˆ 10† were used to carry out the two-dimensional multilayer analysis. As can be seen from Fig. 39, for the grating under study, power is again diffracted out of the T00 pump order and into higher orders. Conservation of power was observed to a high degree of accuracy.

3.7

STABLE IMPLEMENTATION OF RCWA FOR MULTILAYER DIFFRACTION GRATINGS: AN ENHANCED TRANSMITTANCE APPROACH

In Section 3.4 and later sections a multilayer analysis was used to solve for the diffraction from both one-dimensional and two-dimensional diffraction gratings. The method of analysis was to divide the longitudinally inhomo-

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Planar Diffraction Gratings

215

Figure 39 Pyramidal, anisotropic surface relief grating re¯ected diffraction ef®ciency mirror, multilayer analysis (10 layer).

geneous grating into a series of thin layers, solve Maxwell's equations in each thin layer region using state variable techniques, and then match EM boundary conditions from layer to layer to determine all the unknown coef®cients of the system. The technique is an effective one, and it is able to solve a wide range of diffraction problems for both isotropic and anisotropic gratings. A major limitation of the method, however, is that when the grating is too thick, the dielectric modulation is too large, and ill conditioned matrices can result, and thus unstable and very inaccurate EM ®eld solutions can result. The source of the problem is that the state variable matrix method gives rise in its solution to large exponential arguments (eigenvalue times thickness values) due to large grating modulation or large grating thickness. Thus when these exponential eigensolutions are evaluated and used to compute the boundary matching matrix of the overall system, exponentially large and small matrix elements result. These exponentially large and small matrix elements cause the overall system matrices to be singular or ill conditioned, thus causing either no solution or inaccurate solutions. Exactly this same type of problem was observed in Section 2.8 for performing k-space analysis by Yang, Section 2.7.

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216

Chapter 3

Moharam et al. [22,23] have presented an elegant and powerful algorithm that can overcome this problem for the case of propagation in an isotropic longitudinally inhomogeneous diffraction grating system. This method is based on formulating Maxwell's equations into a second-order matrix state variable form and then transforming the unknown transmittance variables into a form where the exponentially large and small terms are summed together into the same matrix element. In this way when the matrix terms are exponentially small, the exponentially small terms drop out (or appear as small numerical noise relative to the large terms), and thus the exponentially small terms do not affect the stability of the calculation. We will now summarize the method presented by Moharam et al. [22,23].

3.7.1

Second-Order RCWA

In this section we present the second-order RCWA formulation as given by Moharam et al. [22,23]. In the following we will use the geometry in Fig. 40 and most of the multilayer notation as given in Refs. 22 and 23. We consider the RCWA formulation for E-mode polarization (the polarization used in Section 3.3. of the present text) for the case when Region 2 is isotropic. We assume that Region 2 is divided into L thin layers each with a thickness d` and that P the distance to the right thin layer interface from z ˆ 0 is Di ˆ i`ˆ1 d` , i ˆ 1; . . . ; L. In the `th layer of the diffraction grating in Region 2 of Fig. 40, we assume that the magnetic and electric ®elds (using ~ y ˆ k0 y, ~ etc.) are expanded in the space normalized coordinates x ˆ k0 x; harmonics X ! H g` ˆ Hy;g` a^ y ˆ Uyi` …z† exp… jkxi x† i

X ! E g` ˆ Sx;g` a^ x ‡ Sz;g` a^z ˆ j0 ‰Szi` …z†a^ x ‡ Szi` …z†a^ z Š exp… jkxi x† 2` …x† ˆ

X i

i

i` exp…i j  x† …3:7:1†

~  ~ where kxi ˆ kx0 i, i ˆ . . . 2; 1; 0; 1; 2; . . . ;  ˆ 2=, and  ˆ k0 , is the period of the diffraction grating. If these space harmonic expansions are substituted in Maxwell's equations, appropriate derivatives are taken, and the coef®cients of the exponential terms are equated, it is found that the following coupled equations result:

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Planar Diffraction Gratings

217

Figure 40 Geometry and coordinate system for the diffraction grating problem under consideration. The column matrices C` and C‡ ` represent the coef®cients of the forward and backward traveling waves in each thin layer, and 2` …x; z† represents the periodic dielectric function in each thin layer. Used with permission of OSA, 1995 [23].

@Uyi` X ˆ `;i @z i0 X jkxi Uyi` ˆ `;i i0

@Sxi` ˆ @z

i 0 Sxi 0 ` i 0 Szi 0 `

jkxi Szi`

…3:7:2†

Uyi` ^

where i ˆ . . . ; 2; 1; 0; 1; 2; . . .. It is convenient to introduce row and column matrices in the analysis. Letting Sx` ˆ ‰Sx` Š, Sz` ˆ ‰Szi` Š, and Uy` ˆ bUyi` c be column matrices and letting Kx ˆ kxi i;i 0 and  ˆ i;i 0 ˆ i i 0 be square matrices, Eqs. 3.7.2 can be put in matrix form as @Uy` ˆ ` Sx` @z

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…3:7:3a†

218

Chapter 3

jKx Uy` ˆ ` Sz` @Sx` ˆ @z

jKx Sz`

…3:7:3b† Uy`

…3:7:3c†

If we invert ` in Eq. 3.7.3b and thus express Sz` ˆ j…` 1 †Kx Uy` and further substitute Sz` in Eq. 3.7.3c we ®nd that @Sx` ˆ @z

jKx …j` 1 Kx Uy` †

Uy`

…3:7:4†

Taking the derivative of Eq. 3.7.3a and substituting Eq. 3.7.4 we ®nd that h @2 Uy` ˆ  Kx ` 1 Kx ` @z2

i I Uy`

…3:7:5†

where I ˆ ‰i;i 0 Š is the identity matrix. Further letting B`  Kx ` 1 Kx and E`  ` we ®nd Eq. 3.7.5 becomes @2 Uy` ˆ E` B` Uy` ˆ A` Uy` @z2

I

…3:7:6†

where A` ˆ E` B` . We further note that the tangential electric ®eld space harmonic amplitude is given by Sx` ˆ E` 1

@Uy` @z

…3:7:7†

Equation 3.7.5, unlike those in Chapter 2 and the earlier sections of Chapter 3, is a second-order matrix differential equation rather than a ®rstorder matrix equation. Its solution for the eigenmodes of the system, however, is similar to the ®rst-order matrix differential equation analysis. We will now proceed with the solution. Let q2n` represent the eigenvalues of the matrix A` ˆ E` B` and also let the sign of the square root of q2n` be chosen so that Re…qn` †  0. Further let Wn` represent the nth eigenvector of the matrix A` . The eigenvalues q2n` and eigenvectors Wn` satisfy the eigen matrix equation A` Wn` ˆ q2n` Wn` One can show that the exponential solution

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…3:7:8†

Planar Diffraction Gratings

219

U‡ yn` …z† ˆ W`n exp… qn` z†

…3:7:9†

is an eigensolution of the second-order matrix differential equation Eq. 3.7.6. Differentiating Eq. 3.7.9 twice with respect to z we ®nd that @2 @2 Uy`n …z† ˆ W`n 2 exp… qn` z† ˆ q2n` W`n exp… qn` z† 2 @z @z

…3:7:10†

ˆ A` Wn` exp… qn` z† ˆ A` Uyn` …z† thus showing that Uyn` …z† is a solution as stated. Using the matrix eigensolutions of Eq. 3.7.9, we can now form a general expression for the tangential magnetic and electric ®elds in the `th layer of the diffraction grating region. The tangential magnetic ®elds associated with the nth mode in the `th layer associated with the eigensolution U‡ yn` …z† ˆ Wn` exp… qn` z† is given by ‡EV Uyn` …x; z† ˆ

X i

Win` exp‰ qn` …z

D` ‡ d` †Š exp… jkxi x†

…3:7:11†

where the superscript EV stands for eigenvector solution. From Maxwell's equation, it is found that @ ‡EV ‡EV U …x; z† ˆ 2` …x†Sxn` …x; z† @z yn`

…3:7:12†

‡EV After expressing 2` …x† and Sxn` …x; z† in an exponential Fourier series sum, ‡EV combining the Fourier sums in the product term ` …x†Sxn` …x; z† into a convolution summation term, differentiating Eq. 3.7.11, and equating coef®cients of exp… jkxi x†, it is found that

… qn` †Win` ˆ

X i0

i

‡ i 0 ;` Sxi 0 n`

…3:7:13†

‡ 0 Using matrix inversion one ®nds the amplitude Sxi 0 n` . It is given by the i th row component of the column vector 1 S‡ xn` ˆ … qn` †E` Wn` ˆ

Vn`

…3:7:14†

where we have let Vn` ˆ qn` E` 1 Wn`

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…3:7:15†

220

Chapter 3

Using this value it is found that ‡EV …x; z† ˆ Sxn`

X … Vin` † exp‰ qn` …z i

D` ‡ d` †Š exp… jkxi x†

…3:7:16†

The tangential magnetic ®eld associated with the nth mode in the `th layer associated with the eigensolution Uyn` …z† ˆ Wn` exp…qn` z† is given by Uyn`EV …x; z† ˆ

X i

Win` exp‰qn` …z

D` †Š exp… jkxi x†

…3:7:17†

From Maxwell's equation, it is found that @ U EV …x; z† ˆ 2` …x†Sxn`EV …x; z† @z yn`

…3:7:18†

After carrying out the differentiation in Eq. 3.7.18 and equating coef®cients of exp… jkxi x†, it is found that …qn` †Win` ˆ

X i0

i

i 0 ;` Sxi 0 n`

…3:7:19†

Using matrix inversion one ®nds the amplitude Sxi 0 n` . It is given by the i0 th component of the column vector Sxn` ˆ Vn` , where Vn` has been de®ned previously. Using this value it is found that Sxn`EV …x; z† ˆ

X i

Vin` exp‰qn` …z

D` †Š exp… jkxi x†

…3:7:20†

If we sum the forward and backward tangential magnetic and electric ®elds as given in Eqs. 3.7.11, 3.7.16, 3.7.17, and 3.7.20 we ®nd that a total expansion of these ®elds is given by Tot …x; z† ˆ Uyn` Tot Sxn` …x; z†

or

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X ‡ ‡EV Cn` Uy`n …x; z† ‡ Cn` Uy`nEV …x; z† n

X ‡ ‡EV ˆ Cn` Sx`n …x; z† ‡ Cn` Sx`nEV …x; z† n

…3:7:21†

Planar Diffraction Gratings Tot Uy` …x; z† ˆ

XX i

n

221

 ‡ Win` Cn` exp‰ qn` …z

D` ‡ d` †Š

‡Cn` exp‰qn` …z D` †Š exp… jkxi x† XX  Tot ‡ …x; z† ˆ Vin` Cn` exp‰ qn` …z D` ‡ d` †Š Sx` i

n

‡Cn` exp‰qn` …z

…3:7:22†

D` †Š exp… jkxi x†

i and n range over the space harmonics of the system. 3.7.2

Electromagnetic Fields in Regions 1 and 3

Up to now we have speci®ed the EM ®elds in the `th thin layer region of Region 2. We will now specify the EM ®elds in Regions 1 and 3. After solving Maxwell's equations in Region 1 we ®nd the magnetic ®eld is given by Hy…1† ˆ

X ‰exp… jkzi1 z†i0 ‡ Ri exp…jkzi1 z†Š exp… jkxi x†

…3:7:23†

i

where kxi has been previously de®ned and  kzi1 ˆ

‰1 1 k2xi Š1=2 j‰k2xi 1 1 Š1=2

1 1 k2xi  0 k2xi 1 1  0

…3:7:24†

It is assumed that the incident plane wave amplitude is 1 (V/m). The coef®cients Ri represent the amplitudes of the re¯ected, diffracted ®elds in Region 1. The tangential electric ®eld in Region 1 is given by Ex…1† ˆ

j0 X ‰ jkzi1 exp… jkzi1 z†i0 ‡ jkzi1 Ri exp…jkzi1 z†Š exp… jkxi x† 1 i …3:7:25†

After solving Maxwell's equations in Region 3 we ®nd that the magnetic ®eld is given by Hy…3† ˆ

X i

Ti exp‰ jkzi3 …z

DL †Š exp… jkxi x†

where kxi has been previously de®ned and

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…3:7:26†

222

Chapter 3

 kzi3 ˆ

‰3 3 k2xi Š1=2 j‰k2xi 3 3 Š1=2

3 3 k2xi  0 k2xi 3 3  0

…3:7:27†

The tangential electric ®eld in Region 3 is given by Ex…3† ˆ

j0 X ‰ jkzi3 Ti exp‰ jkzi3 …z 3 i

DL †ŠŠ exp… jkxi x†

…3:7:28†

The coef®cients Ti represent the amplitudes of the transmitted, diffracted ®elds in Region 3. Now that the general form of the EM ®elds has been speci®ed in all regions of space, an important problem that remains is to match EM boundary conditions at the different interfaces of the system. Matching the tangential magnetic ®eld at the Region 1: Region 2, ` ˆ 1, thin layer interface located at z ˆ 0 we ®nd that i0 ‡ Ri ˆ

X n

 ‡  Win1 Cn1 ‡ Cn1 exp… qn1 d1 †

…3:7:29†

Matching the tangential electric ®eld at z ˆ 0 we ®nd similarly, after canceling the j0 factor common to both terms, 1 ‰ jk  1 zi1 i0

jkzi1 Ri Š ˆ

X n

 ‡ Vin1 Cn1

 Cn1 exp… qn1 d1 †

…3:7:30†

These two equations can be written in matrix form as 

i0 jkzi1 1 i0



 ‡

  I W1 Rˆ V1 jZ1

W1 X1 V1 X1



C‡ 1 C1

 …3:7:31†

where Z1 ˆ ‰…kzi1 =1 †i;i 0 Š, W1 ˆ ‰Win1 Š, V1 ˆ ‰Vin1 Š, and X1 ˆ ‰i;n exp… qn1 d1 †Š where i and n range over the number of space harmonics in the system. Matching the tangential magnetic ®eld at the Region 2, …` 1†th thin layer interface to the Region 2, `th thin layer located at z ˆ D` 1 , where ` ˆ 2; . . . ; L, we ®nd that X n

ˆ

Win;`

X n

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1



‡ Cn;` 1 exp… qn;` 1 d` 1 † ‡ Cn;`

 ‡ Win;` Cn;` ‡ Cn;`

1

exp… qn;` d` †



1

 …3:7:32†

Planar Diffraction Gratings

223

Matching the tangential electric ®eld at z ˆ D` 1 ` ˆ 2; . . . ; L we ®nd similarly, after canceling the j0 factor common to both terms, that X n

Vin;`

1

 ˆ

‡ Cn;`

X n

1

exp… qn;` 1 d` 1 † ‡ Cn;`

Vin;`



‡ Cn;` ‡ Cn;`

1

1



exp… qn;` d` †

…3:7:33†



As before, these two equations can be written in matrix form as 

W` V`

1 1

X` X`

1 1

W` 1 V` 1



C‡ ` C`

 1 1

 ˆ

W` V`

W` X ` V` X`



C‡ ` C`

 …3:7:34†

where X` is a diagonal matrix with center diagonal matrix elements given by exp… qn` d` †. Matching the tangential magnetic ®eld at the Region 2, ` ˆ L thin layer interface located at z ˆ DL to the Region 3 interface we ®nd that X n

 ‡  WinL CnL exp… qnL dL † ‡ CnL ˆ Ti

…3:7:35†

Matching the tangential electric ®eld at z ˆ 0 we ®nd similarly, after canceling the j0 factor common to both terms, that X n

VinL



 ‡ CnL exp… qnL dL † ‡ CnL ˆ

jkzi3 Ti 3

…3:7:36†

These two equations can be written in matrix form as 

W L XL VL XL

WL VL



C‡ L CL





 I ˆ T jZ3

…3:7:37†

where Z3 ˆ ‰…kzi3 =3 †i;i 0 Š. 3.7.3

Enhanced Transmittance Matrix Analysis

We will now be concerned with reducing the cascaded set of matrix equations that have been presented in the previous subsection. The matrix cascade will be done so that matrix singularities do not occur as a result of exponentially small matrix elements.

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224

Chapter 3

We start by writing out the matrix equations and presenting a cascade analysis of the system. Writing out the equations for ` ˆ 1; 2; . . . ; L we ®nd that 2

i0

3

4 jkzi1 5 ‡  1 i0



  I W1 Rˆ V1 jZ1



W1 X1 V1 X1

C‡ 1 C1

 …3:7:38†

From Eq. 3.7.34, setting ` ˆ 2, we have 

C‡ 1 C1





W1 X1 ˆ V1 X1

W1 V1

 1

W2 V2

W2 X 2 V2 X2

W1

W1 X1

V1

V1 X1

W2

W2 X2



C‡ 2 C2

 …3:7:39†

so that 2

i0

3

7 6 4 jkzi1 5 ‡ i0 1

"

I

#

" Rˆ

jZ1

"

V2

#"

W1 X 1

W1

V1 X1

V1

#"

V2 X2

#

1

#

C‡ 2 C2

…3:7:40† If we repeat this process L 2

i0

3

6 7 4 jkzi1 5 ‡ i0 1

"

I jZ1

1 times we have #

("

Rˆ "

L Y1

W`

W` X `

`ˆ1

V`

V` X`

W` X ` V` X`

"

#

# 1 91 = A ; V`

W`

WL

WL XL

VL

VL XL

#"

C‡ L

…3:7:41† #

CL

Inverting Eq. 3.7.37 and substituting in the above equation we ®nd that

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Planar Diffraction Gratings

2

i0

3

7 6 4 jkzi1 5 ‡ i0 1

"

225

#

I

Rˆ

jZ1

(" L W` Y

W` X `

V`

V` X`

`ˆ1

"

#

# 1 91" # = f L‡1 A T ; gL‡1 V`

W` X `

W`

V` X`

…3:7:42† where fL‡1 ˆ I and gL‡1 ˆ jZ3 . At this point we will rearrange the matrix products in the preceding equation and proceed with the enhanced matrix method. We ®rst note by direct multiplication that "

W` X`

W`

V` X`

V`

#

" ˆ

W`

V`

V`

Using the matrix property …A B† "

W` X`

W`

V` X`

V`

#

"

1

#"

W`

ˆ

1

X`

0

0

I

ˆ B 1A

X`

0

0

I

# 1"

1

# …3:7:43†

we ®nd that

W`

W`

V`

V`

#

1

…3:7:44†

The last factor of Eq. 3.7.42 can now be written as "

WL VL

WL XL

#"

VL XL

WL X L

WL

# 1"

VL XL "

XL 0

VL # 1" WL 0 VL

I

f L‡1

#

"

WL

Tˆ gL‡1 VL # 1" # f L‡1 WL T gL‡1 VL

WL X L

#

VL XL

…3:7:45† To impose an enhanced transmittance approach, we let 

aL bL



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WL ˆ VL

WL VL

 1

f L‡1 gL‡1

 …3:7:46†

226

Chapter 3

and also let T ˆ aL 1 XL TL . Thus "

#"

WL

WL X L

VL

VL XL

XL

0

0 "

I

ˆ " ˆ

# 1"

aL bL

# aL 1 X L T L

WL

WL X L

VL

VL XL

WL

WL X L

#"

1

XL

0

0

#"

#"

I I

b L aL 1 X L 2 3 WL …I ‡ XL bL aL 1 XL † 5TL ˆ4 VL …I XL bL aL 1 XL † VL

VL XL

#

XL

#

bL aL 1 XL

TL

TL

…3:7:47† De®ning 

fL gL

"

 ˆ

# WL …I ‡ XL bL aL 1 XL † TL VL …I XL bL aL 1 XL †

…3:7:48†

we ®nd that Eq. 3.5.42 can be written as 2

i0

3

6 7 4 jkzi1 5 ‡ i0 1

"

#

I

("

Rˆ

jZ1

"

L Y1

W`

W` X `

`ˆ1

V`

V` X`

W` X ` V` X`

#

# 1 91" # = fL A TL ; gL V`

…3:7:49†

W`

We next let TL ˆ aL 1 1 XL

1

TL

…3:7:50†

1

and let 

aL bL

1 1

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WL ˆ VL

1 1

WL 1 VL 1

 1

fL gL

 …3:7:51†

Planar Diffraction Gratings

227

From this we now ®nd that 2

i0

3

7 6 4 jkzi1 5 ‡ i0 1

"

I jZ1

#

("

Rˆ "

L Y2

W`

W` X `

`ˆ1

V`

V` X`

W` X ` V` X`

#

# 1 91" = fL A ; gL V`

W`

1 1

# TL

1

…3:7:52† where 

fL gL

1

"



1

ˆ

WL 1 …I ‡ XL VL 1 …I XL

1 1

bL bL

1 1

# aL 1 1 XL 1 † TL aL 1 1 XL 1 †

1

…3:7:53†

Repeating this cycle and process until the last layer we ®nd that 2

i0

3

4 jkzi1 5 ‡  1 i0



   f I R ˆ 1 T1 g1 jZ1

…3:7:54†

where f 1 and g1 are found from repeated calculations of Eqs. 3.7.46±53. T1 is found from the matrix solution of Eq. 3.7.54 and back substitution shows that the T matrix is given by T ˆ …aL 1 XL †…aL 1 1 XL 1 † . . . …a1 1 X1 †T1 :

…3:7:55†

In inspecting Eqs. 3.7.53 and Eqs. 3.7.54 one observes why the present algorithm is extremely ef®cient and stable. In Eq. 3.7.53, the X` matrix is diagonal and contains the exponential term exp… qn` d` ). When qn` d` is large, the exponential term exp… qn` d` † is very small, and thus the matrix X` in this case is near zero. The matrix terms f ` ˆ W` …I ‡ X` b` a` 1 X` † and g` ˆ V` …I X` b` a` 1 X` †, which form an important part of the algorithm, are the only terms that contain exponential terms. Further, the terms in the matrices making up f ` and g` appear as the sum of matrix element terms near unity (coming form the identity matrix I) and the exponential terms (coming from matrix X` b` a` 1 X` ). Thus when the matrix elements of the matrices X` b` a` 1 X` are exponentially small, the matrix elements making up f ` and g` are not all near zero (because of the presence of the identity matrix I). Thus when this procedure is repeated for each layer starting at ` ˆ

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228

Chapter 3

L with the computation of f L and gL and repeated until ` ˆ 1 and the last f 1 and g1 is produced, one sees that the ®nal matrix equation for R and T1 , which uses the matrices f 1 and g1 , will thus not have ill-conditioned or nearsingular matrices, since f 1 and g1 , which make up this equation, do not possess all exponentially small terms. It is interesting to note that the principle used here is similar to the method used by Yang, Section 2.7, discussed in the previous chapter. It would be useful to extend this technique to the case of diffraction from gratings in anisotropic materials. In this section the enhanced transmittance method has been applied to the case when the electric ®eld polarization was in the plane of incidence. In a companion paper [23] written with the paper the present analysis was based on, the authors present an enhanced transmittance method for Hmode incidence which deals with the conical plane wave diffraction case. 3.7.4

Numerical Stability and Convergence [23]

To illustrate the stability of the algorithm, Fig. 41 shows the diffraction ef®ciency of the ®rst diffracted order plotted versus the normalized grating depth for a 16-level asymmetric dielectric grating as shown in Fig. 42 …n1a ˆ n1 ˆ 1 (n2a is the bulk index value in Region 2 outside the step portion of the grating) and n2b ˆ n3 ˆ 2:04 (n2b is the bulk index value in Region 2 inside the step portion of the grating)] up to excessive depths of 50 wavelengths for two grating periods of 1 and 10 wavelengths, respectively. The asymmetric grating is a sawtooth 15-layer stairstep with a step width of 1/16 of the grating period and a layer depth of 1/15 of the total depth of the structure. The diffraction ef®ciency is shown for TE and TM polarizations and for conical diffraction with  ˆ 30 (azimuth angle) and ˆ 45 (polarization angle between the incident electric ®eld and the plane of incidence). A suf®cient number of terms are retained in the space harmonic expansions to ensure accuracy to four places past the decimal. Conservation of energy has been observed to one part in 1010 . Conservation of energy is a necessary condition for numerical stability of the algorithm. Figure 43 illustrates the convergence of the diffraction ef®ciency of the 16-level, asymmetric dielectric grating shown in Fig. 42 as the number of ®eld harmonics is increased. Results are given for two grating depths (1 and 49 wavelengths) and for two grating periods, respectively, for both TE and TM polarization and for conical mounting. It is clear that, in all cases, the diffraction ef®ciency converges to the proper values when a suf®cient number of harmonics are included in the formulation. Note that the TE polarization requires fewer harmonics than are required by the conical diffraction and by the TM polarization. Also more harmonics are required for deeper gratings with larger grating periods.

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Figure 41 Diffraction ef®ciency dependency on the normalized grating of a 16level (15 layer) asymmetric binary dielectric grating (ng ˆ n3 ; n1 ˆ 1†. The angle of incidence is 108. TE-polarization, TM-polarization, and conical-mount polarization results are shown for two grating periods of 1 and 10 wavelengths, respectively. Used with permission of OSA, 1995 [23, Fig. 4].

Figure 42 Geometry for the surface relief grating diffraction problem analyzed herein. Used with permission of OSA [23, Fig. 2].

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Figure 43 Diffraction ef®ciency dependency on the number of space harmonics for the grating shown in Fig. 40 for two values of the grating depth (1 and 49 wavelengths, respectively) and for two grating periods of 1 and 10 wavelengths, respectively. Used with permission of OSA, 1995 [23, Fig. 3].

3.8

HIGHLY IMPROVED CONVERGENCE OF THE COUPLED WAVE METHOD FOR E-MODE INCIDENCE

In this section, we revisit diffraction of an E-mode polarized ®eld obliquely incident onto a grating formed from an arbitrary permittivity pro®le …x† and having a ®nite depth along the longitudinal, y-direction. This case has previously been studied in Section 3.3. The grating, in this case, is assumed to be one-dimensional, i.e., there is no variation along y except at the grating boundaries. We show, following Lalanne and Morris [29], Peng and Morris [103], and Li [53], that by reformulating the eigenvalue problem of the coupled wave method, highly improved convergence rates can be obtained. All variables in this section are normalized as in previous sections. The analysis starts from the x and y components of the E ®eld and the z component of the H ®eld for the case of E-mode (or TM) incidence on the gratings (see Fig. 44).

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Figure 44 Geometry for the nonconical grating diffraction problem for E-mode TM polarization. The parameters of the grating are: grating thickness L~ ˆ :36:36; ~ ˆ :4545; relative permittivity of Region 1, "1 ˆ 1:0; relative grating wavelength  permittivity of Region 2a (nonmetallic portion of grating), "2a ˆ 2:25; relative permittivity of Region 2b (metallic portion of grating), "2b ˆ …3:18 j4:41†2 ; relative permittivity of Region 3, "3 ˆ 2:25; and grating duty cycle, 30%. Used with permission of OSA, 1996 [29, Fig. 1].

The ®eld expansions for this case are given by Eqs. 3.3.4.5. We assume an isotropic grating in Reg. 2 and in Eq. 3.4.3 we take "xx …x† ˆ "yy …x† ˆ "zz …x†; " ˆ 0;  6ˆ . With these assumptions we ®nd that Maxwell's equations in Reg. 2 are given by @…0 Hz † ˆ j"…x† Ex @y @…0 Hz † ˆ j"…x† Ey @x @Ey @Ex ˆ j…0 Hz † @x @y

…3:8:1† …3:8:2† …3:8:3†

If we substitute Ey of Eq. 3.8.2 in Eq. 3.8.3; if we solve for Ex of Eq. 3.8.1, differentiate with respect to y, and substitute this result in Eq. 3.8.3; and if we perform a small amount of additional algebra we ®nd after letting Uz ˆ 0 Hz @2 U z ˆ @y2

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  @ 1 @Uz "…x† @x "…x† @x

"…x†Uz

…3:8:4†

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Substitution of Eq. 3.3.5 into Eq. 3.8.4 and after collection of coef®cients for the Fourier exponential term exp( jkxi x†, for i ˆ MT ; . . . ; 0; . . . ; MT , Mt ! 1, the following matrix equation results @2 U z @y2

ˆ E…Kx A Kx

I†U z

…3:8:5†

where E ˆ ‰"i;i0 Š, "i;i0 ˆ "i i0 ; A ˆ ‰ai;i0 Š, ai;i0 ˆ a i i0 and a i are the Fourier coef®cients of the reciprocal permittivity function M X 1 ˆ "…x† T

a i exp…ji x† MT

The other terms are de®ned in Section 3.3. We note in the limit Mt ! 1 that the matrices E and A are inverses of each other and we thus have E ˆ A 1 and A ˆ E 1 . Using this inverse relation we may also express Eq. 3.8.5 as Mt ! 1 @2 U z @y2

ˆ A 1 …Kx E 1 Kx

I†Uz

…3:8:6†

It turns out that the above eigenvalue formulation (Eq. 3.8.6) is superior to the one in Eq. 3.8.5 as far as convergence rates are concerned. As an example we quote from Lalanne and Morris [29] the case of diffraction from a chrome (refractive index 3.18-j4.41) lamellar grating deposited on a glass substrate (see Fig. 44). The diffraction ef®ciency of the zeroth order is shown in Fig. 45 along with the convergence rates, using Eqs. 3.8.5 (line) and 3.8.6, (circle) respectively. The results clearly show the superiority of the convergence from the second formulation. The reason for the improvement in convergence using the second method above has been shown rigorously by Li [53]. The dif®culty arises in the convergence because 1. The EM ®elds and the periodic dielectric permittivity are discontinuous at the points in the grating where  changes from 2a to 2b , and vice versa yet their product must be a continuous function. 2. Expressing products of periodic functions as in a Fourier series (the Fourier coef®cients of the product function are a convolution of the Fourier coef®cients of each seriesÐthis is called Laurent's rule) involves a ®nite truncation of the convolution series.

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Figure 45 Diffraction ef®ciency of the transmitted zeroth order of a metallic grating with E-mode polarized light. The solid curve is obtained by using the conventional eigenproblem formulation. The circles are provided by the new eigenproblem formulation.  ˆ 08 (normal incidence). Used with permission of OSA, 1996 [Fig 2, 29].

To illustrate Statement 1 we will now return to Eq. 3.8.1. In this equation we ®rst note that on the right-hand side, the Ex ®eld is discontinuous at the points where …x† changes discontinuously from 2a to 2b or vice versa. This follows because the electric ¯ux density Dx ˆ …x†Ex is continuous at all points in x. The fact that Dx is continuous on the right-hand side z also forces dH dy to be continuous on the left-hand side. Thus we see from boundary conditions that Eq. 3.8.1 is a product of two discontinuous functions that produce a continuous product. Li [53] explains that the source of the convergence problem has to do with the fact that the ®nite Fourier sum of a periodic product function h…x† ˆ f …x†g…x†, formed by convolving the ®nite Fourier sums of each periodic function f …x† and g…x†, converges very slowly when f …x† and g…x† are piecewise, pairwise discontinuous at a point x yet their product h…x† is continuous at that point. Li goes on to show that the ®nite Fourier sum of a periodic product function h…x† ˆ f …x†g…x†, formed by (1) ®nding the ®nite Fourier sum of the periodic function 1=f …x†, call it f REC …x†, (2) taking the recriprocal of f REC …x†, and (3) convolving this periodic function with the periodic function g…x†, converges very rapidly at the point x where the pairwise discontinuity of f …x† and g…x† occurred. Li [53] calls the ®rst method the Laurent rule and the second method the inverse rule. Li [53] illustrates

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numerically the convergence problems and different convergence rates that result by using the two ways of convolving the periodic functions f …x† and g…x† that have just been described. Although Li [53] shows in great detail and rigor, both numerically and theoretically, the improved convergence that occurs by using the inverse rule, we feel that a physical or geometric explanation of why the convergence is better would be bene®cial to the readers. As Li [53] has shown, periodic functions expressed using the Laurent rule can have severe convergence problems when represented by a truncated convolution series. We illustrate the problem with the following example. Consider two periodic square wave functions f and g de®ned over a period  as ( f …x† ˆ ( g…x† ˆ

a b b a

=2 < x < 0 0 < x < =2 =2 < x < 0 0 < x < =2

…3:8:7† …3:8:8†

The product h…x† ˆ f …x†g…x† ˆ ab is, of course, a continuous function. Note that the ®nite Fourier series expansion of f …x† and g…x†, call them fT …x† and gT …x†, have the value …a ‡ b†=2 at x ˆ 0. This follows from a well-known theorem in Fourier series: If f …x† is a piecewise smooth function and/or satis®es the Dirichlet conditions, then its Fourier series converges to …1=2† ‰f …x 0† ‡ f …x ‡ 0†Š where x ˆ 0 is the point of discontinuity (Butkov [104, Chapter 1]). Hence, in general, hT…1† …0† ˆ fT …0†gT …0† ˆ …a ‡ b†2 =4 6ˆ h…0† ˆ ab, and thus the product of the ®nite Fourier series gives, in general, an erroneous value at the discontinuity point even though each of the ®nite Fourier series give the value (a ‡ b)/2 at the point of discontinuity. Around this point the error decreases as the number of harmonics retained in the Fourier expansion is increased. The inverse method used by Li [53] and Lalanne and Morris [29] involves taking the ®nite Fourier series expansion of 1=f …x† (call it fTREC ), inverting it, and multiplying with the ®nite Fourier series expansion of g…x† to approximate the product function h…x†. Let hT…2† …x† ˆ …1=fTREC † gT …x†. Using the same Fourier theorem stated above, h…2† T …0† ˆ

1 a‡b
ˆ ab ˆ h…0† …1=a ‡ 1=b†=2 2

…3:89†

Thus, this shows that by (a) taking the ®nite Fourier series fTREC …x†; of the reciprocal function 1=f …x†, (b) then inverting fTREC …x† to obtain the function

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1=fTREC …x†, (c) ®nding the ®nite Fourier series expansion of 1=fTREC x, and (d) multiplying this with the ®nnite Fourier series gT …x†, gives the accurate value of the product h…x† at the point of discontinuity. Note that since fTREC …x† is a ®nite Fourier series, it is ®nite and continuous at the point of discontinuity of f …x†, and hence the Fourier series of its reciprocal, namely 1=fTREC …x†, is continuous at this point and has the value 1/[(1/a+1/b)/2]. Note further that we could achieve the same result by interchanging f and g. Summarizing, the Laurent rule gives an, in general, incorrect value at the point of discontinuity, whereas the inverse method yields the correct result. When these types of computations are encountered in grating problems, it is easy to see why numerical dif®culties encountered with the Laurent method can be alleviated using the inverse rule. It is instructive to compute the ®rst and second derivatives of the product functions h…i† T …x†; i ˆ 1; 2 around the point of discontinuity x ˆ 0. This gives insight into the way the two approximations vary around the point of discontinuity (by using these derivatives in a Taylor series). Note that h…1† T …x† ˆ fT …x†gT …x† ˆ …c0 ‡ DN …x††…c0

DN …x††

…3:8:10†

where DN …x† ˆ

N X nˆ1

cn sin nKx;

n odd and cn ˆ 0; even;

c0 ˆ …b ‡ a†=2;

cn ˆ …2=n†…b

a†; …3:8:11†

and where we have assumed K ˆ 1 for simplicity. Then hT…1† 0 …x† ˆ 2 DN D0N , 0 …1† 00 …1† 00 0 2 hT …x† ˆ 2…DN ‡ DN D00N †: Hence h…1† …8=†2 T …0† ˆ 0, and hT …0† ˆ 2 2 …b a† ‰N=2Š , where [y] refers to the highest integer less than y. Thus we see that when expanding the h…1† T …x† the Taylor series about x ˆ 0 has a parabolic shape with narrower and narrower width around the peak as N increases. This follows since the ®rst derivative is zero and the second derivative increases as the square of the number of harmonics. Thus we can clearly see the nature of the error in using the Laurent approximation. As the number of harmonics increase, the value of the approximation to the left and right of the discontinuity approaches the correct product value. Further, as one approaches the discontinuity, the approximation deviates from the true value ab in the form of a parabolic function whose value at the discontinuity still remains at …a ‡ b†2 =4. This nature is evident from Figs. 2-4 of Li [3]. In the inverse method described earlier, recall that h…2† T …x† ˆ REC …1=fT †gT …x†. It is crucial to note that 1=f ˆ f REC ˆ …1=ab†g. Hence it

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follows that fTREC ˆ …1=ab†gT , provided the same number of harmonics are used to expand both functions. Therefore REC h…2† †gT …x† ˆ T …x† ˆ …1=fT

1 g ˆ ab; …1=ab†gT T

for all x:

…3:8:12†

Furthermore, this is true irrespective of the number of harmonics used! Thus using this method, the discontinuity of f and g does not affect the product h at any point. The equations above hold exactly for the square wave example considered. For an arbitrary pairwise discontinuous set of functions f and g whose product is continuous, we remark that f and g can each be decomposed into the sum of a square wave, as in the example above, and a continuous function whose value at the point of discontinuity is zero. Thus the product of these f and g's behave exactly as predicted above in a small neighborhood around the point of discontinuity. This follows since the continuous parts of the functions are zero at the point of discontinuity. Any truncation in the Fourier series representation of the functions f and g is thus not going to be re¯ected at the point of discontinuity.

PROBLEMS 1. A thin sinusoidal phase grating with spatial period  and thickness L is given by   2 x "…x† ˆ "2 ‡
" cos  is illuminated by a normally incident linearly polarized optical ®eld. Find the angles between the diffracted orders and the respective diffraction ef®ciencies of each order. 2. A circularly polarized optical ®eld is incident on the grating described in Fig. 2. Find the polarized state of each diffracted order, and the corresponding diffraction ef®ciency. 3. A plane wave (see Fig. 1) is obliquely incident (angle  to the grating normal) on a lamellar diffraction grating of spatial period  and thickness L whose permitivity pro®le is given by  "…x† ˆ

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"2a ‡
"…1: jxj=d†; jxj  d "2b ; d < jxj  =2

Planar Diffraction Gratings

4. 5.

6.

7.

237

~ where  ˆ 108, "1 ˆ 1:; "3 ˆ 2:25; "2a ˆ 2:25; "2b ˆ 1:;
" ˆ :5;  ˆ k0  ˆ 5; d ˆ =5:; L~ ˆ :75; k0 ˆ 2= and is the free space wavelength. Determine the diffraction ef®ciency of the transmitted and re¯ected orders of the system if the plane wave is a) H-mode polarized. b) E-mode polarized. c) Verify in Parts a) and b) that conservation of real power is conserved. d) What is the approximate number of orders MT needed to ensure convergence of the EM solution in this problem? In Problem 3, verify the complex Poynting theorem as developed in Sec. 3.3 for the Poynting box shown in Fig. 15. Determine the diffraction ef®ciency of the transmitted and re¯ected orders of the system in Problem 3 if an in®nitely thin, perfectly conducting strips are placed at the interfaces between "2a and "2b . a) Use the method of [25] to determine the EM ®elds of a pure re¯ection grating when the permitivity is given by Eq. (3.4.5) and for the data of Fig. 21 except that
" ˆ :4 rather than
" ˆ :2. b) Find and plot the re¯ection ef®ciencies for data of Problem 5a) as was done in Fig. 21. Determine the re¯ected and transmitted diffraction ef®ciencies of an asymmetric diffraction grating when the relative dielectric permitivity is given by  "2a , 0 < x < (=L) y "…x; y† ˆ "2b , …=L†y < x < 

where L  y  0, where  ˆ 108, "1 ˆ 1., "3 ˆ 2:25, "2a ˆ 2:25, ~ k0 ˆ 2= ~ ˆ k0  ˆ 5, d ˆ =5., L~ ˆ :75, L ˆ :k0 L, "2b ˆ 1.,   and is the free space wavelength and x; y are in normalized coordinates. Be sure to include suf®cient diffraction orders and multilayers to ensure proper convergence of your solution. 8. Use RCWA and the theory of Sec. 3.6 to determine the EM ®elds in a lamellar, crossed diffraction (see Fig. 33) where the relative permitivity is given by "…x; z† ˆ "2 ‡
"…1:

…2x=x †2 †…1:

…2z=z †2 †

where I ˆ 208, I ˆ 108, "1 ˆ 1., "3 ˆ 2:25, "2 ˆ 2:25,
" ˆ :2, ~ x ; x ˆ 5:, L~ ˆ :75, and k0 ˆ 2= and is the z ˆ x ˆ k0  free space wavelength. Assume the incident plane wave is circularly

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polarized. Determine the diffraction ef®ciency of the transmitted and re¯ected orders of the system. 9. a) Considering the diffraction grating described in Problem 3 and your solution determined therein, determine the maximum grating thickness before your numerical solution becomes unstable and illconditioned. b) Apply the enhanced transmittance approach described in Sec. 3.7 [22,23] to provide a numerically stable solution to Part a) for those layer thickness for which numerically unstable solutions occurred.

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50. A. Vasara, E. Noponen, J. Turunen, J. M. Miller, and M. R. Taghizadeh, Rigorous diffraction analysis of Dammann gratings, Optical Comm., 81, 337± 342 (1991). 51. G. Granet, and G. Guizal, Ef®cient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization, J. Opt. Soc. Am. A, 13, 1019±1023 (1996). 52. M. G. Moharam, T. K. Gaylord, and R. Magnusson, Criteria for Bragg regime diffraction by phase gratings, Optical Comm., 32, 14Ð18 (1980). 53. L. Li, Use of Fourier series in the analysis of discontinuous periodic structures, J. Opt. Soc. Am. A., 13, 1870±1876 (1996). 54. J. Yamakita and K. Rokushima, Modal expansion method for dielectric gratings with rectangular grooves, Proc. SPIE, 503, 239±243 (1984). 55. J. Yamakita, K. Rokushima, and S. Mori, Numerical analysis of multistep dielectric gratings, SPIE, 815, Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, 153±157 (1987). 56. J. Yamakita and K. Rokushima, Scattering of plane waves from dielectric gratings with deep grooves, IECE Japan, J66-B, 375 (1983). 57. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewarthar, The dielectric lamellar diffraction grating, Optica Acta, 28 (3), 413±428 (1981). 58. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewarthar, The ®nitely conducting lamellar diffraction grating, Optica Acta, 28(8) 1087±1102 (1981). 59. L. C. Botten, M. S. Craig, and R. C. McPhedran, Highly conducting lamellar diffraction grating, Optica Acta, 28(8) 1103±1106 (1981). 60. L. Li, Multilayer modal method for diffraction gratings of arbitrary pro®le, depth, and permittivity, J. Opt. Soc. Am. A, 10, 2581±2591 (1993). 61. D. M. Pai and K. A. Awada, Analysis of dielectric gratings of arbitrary pro®les and thickness, J. Opt. Soc. Am. A, 8, 755±762 (1991). 62. M. Nevievre, Diffraction of light by gratings studied with the differential method, SPIE Periodic Structures, Grating, Moire Patterns and Diffraction Phenomena, 240, 90±96 (1980). 63. P. Vincent, New improvement of the differential formalism for high-modulated gratings, SPIE Periodic Structures, Grating, Moire Patterns and Diffraction Phenomena, 240, 147±154 (1980). 64. R. Petit and G. Tayeb, Theoretical and numerical study of gratings consisting of periodic arrays of thin and lossy strips, J. Opt. Soc. Am. A, 7, 1686±1692 (1990). 65. J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, Multicoated gratings: a differential formalism applicable in the entire optical region, J. Opt. Soc. Am., 68, 838±846 (1982). 66. D. Maystre, A new general integral theory for dielectric coated gratings, J. Opt. Soc. Am., 68, 490±495 (1978). 67. M. K. Moaveni, Plane wave diffraction by dielectric gratings, ®nite-difference formulation, IEEE Trans. Antennas Propagation, 37(8), 1026±1031 (1989).

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68. H. A. Kalhor and M. K. Moaveni, Analysis of diffraction gratings by ®nitedifference coupling technique, J. Opt. Soc. Am., 63, 1584±1588 (1973). 69. M. K. Moaveni, Application of ®nite differences to the analysis of diffraction gratings embedded in an inhomogeneous and lossy dielectric, Int. J. Electronics, 61(4), 465±476 (1986). 70. Y. Nakata and M. Koshiba, Boundary-element analysis of plane-wave diffraction from groove-type dielectric and metallic gratings, J. Opt. Soc. Am. A, 7, 1494±1502 (1990). 71. D. E. Tremain and K. K. Mei, Application of the unimoment method to scattering from periodic dielectric structures, J. Opt. Soc. Am., 68, 775±783 (1978). 72. A. K. Cousins and S. C. Gottschalk, Application of the impedance formalism to diffraction gratings with multiple coating layers, Appl. Opt., 29, 4268±4271 (1990). 73. R. A. Depine, C. E. Gerber, and V. Brundy, Lossy gratings with a ®nite number of grooves: a canonical model, J. Opt. Soc. Am. A, 9, 573±577 (1992). 74. P. St. J. Russell, Power conservation and ®eld structures in uniform dielectric gratings, J. Opt. Soc. Am., A, 1(3), 293±299 (1984). 75. R. Petit and G. Tayeb, On the use of the energy balance criteria as a check of validity of computations in grating theory, SPIE Application and Theory of Periodic Structures, Grating, Moire Patterns and Diffraction Phenomena III 815, 2±10 (1987). 76. R. F. Harrington, Time Harmonic Electromagnetic Fields, Section 1±10, ``Complex Power,'' McGraw-Hill, New York, 1961. 77. E. N. Glytsis and T. K. Gaylord, Anisotropic guided-wave diffraction by interdigitated electrode-induced phase gratings, Appl. Opt., 27, 5035±5050 (1988). 78. P. M. Van Den Berg, W. J. Ghijsen, and A. Venema, The electric-®eld problem of an interdigital transducer in a multilayer structure, IEEE Trans. Microwave Theory Techniques, MTT±33(2) (1985). 79. D. Quak and G. Den Boon, Electric input admittance of an interdigital transducer in a layered, anisotropic, semiconducting structure, IEEE Trans. Sonic Ultrasonics, SU-25(1) (1978). 80. E. N. Glytsis, T. K. Gaylord, and M. G. Moharam, Electric ®eld, permittivity, and strain distributions induced by interdigitated electrodes on electrooptic waveguides, IEEE J. Lightwave Technology, LT-5, 668±683 (1987). 81. K. Rokushima and J. Yamakita, Analysis of diffraction in periodic liquid crystals: the optics of the chiral smetic C phase, J. Opt. Soc. Am. A, 4(1), 27±33 (1987). 82. D. Y. K. Ko and J. R. Sambles, Scattering matrix method for propagation of radiation in strati®ed media: attenuated total re¯ection studies of liquid crystals, J. Opt. Soc. Am. A, 5, 1863±1866 (1988). 83. C. Schwartz and L. F. DeSandre, New calculational technique for multilayer stacks, Appl. Opt., 26, 3140±3144 (1987).

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84. S. L. Chuang and J. A. Kong, Wave scattering from periodic dielectric surface for a general angle of incidence, Radio Sci. 17, 545±557 (1982). 85. T. K. Gaylord and M. G. Moharam, Thin and thick gratings: terminology clari®cation, Symbols, Units, Nomenclature, Applied Optics, 20(19), 3271±3273 (1981). 86. T. Jaaskelainen and M. Kuittinen, Inverse grating diffraction problems, SPIE International Colloquium on Diffractive Optical Elements, 1574, 272±281 (1991). 87. E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, and N. Streibl, Form birefringence of surface-relief gratings and its angular dependence, Optics Commun., 89, 173±177 (1992). 88. J. M. Elson, L. F. DeSandre, and J. L. Stanford, Analysis of anomalous resonance effects in multilayer-overcoated, low-ef®ciency gratings, J. Opt. Soc. Am. A, 5, 74±88 (1988). 89. T. Tamir and H. C. Wang, Scattering of electromagnetic waves by a sinusoidally strati®ed half space: I. Formal solution and analysis approximations, Can. J. Phys., 44, 2073±2094 (1966). 90. T. Tamir, Scattering of electromagnetic waves by a sinusoidally strati®ed half space: II. Diffraction aspects at the Rayleigh and Bragg wavelengths, Can. J. Phys., 44, 2461±2494 (1966). 91. S. T. Peng, T. Tamir, and H. L. Bertoni, Theory of periodic dielectric waveguides, IEEE Trans. Microwave Theory Tech., MTT-23, 123±133 (1975). 92. T. Tamir, H. C. Wang, and A. A. Oliner, Wave propagation in sinusoidally strati®ed dielectric media, IEEE Trans. Microwave Theory Tech., MTT-12, 323±335 (1964). 93. R. V. Johnson and R. Tanguay, Optical beam propagation method for birefringent phase grating diffraction, Opt. Eng., 25, 235±249 (1986). 94. H. Haus, Waves and Fields in Optoelectronics, Prentice-Hall, New York, 1984, Section 11.1. 95. S. Samson, A. Korpel, and H. S. Snyder, Conversion of evanescent waves into propagating waves by vibrating knife edge, Internat. J. Imaging Systems Technology, 7, 48±53 (1996). 96. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics, Cambridge, UK, 1990, p. 127. 97. T. L. Zinenko, A. I. Nosich, and Y. Okuno, Plane wave scattering and absorption by resistive-strip and dielectric strip periodic gratings, IEEE Trans. Antennas Propagation, 46(10), 1498±1505 (1998). 98. E. Popov, L. Tsonev, and D. Maystre, GratingsÐgeneral properties of the Littrow mounting and energy ¯ow distribution, J. Mod. Optics, 37(3), 367±377 (1990). 99. E. Popov, L. Tsonev, and D. Maystre, Losses of plasmon surface waves on metallic grating, J. Mod. Optics, 37(3), 379±387 (1990). 100. E. Popov, and L. Tsonev. Total absorption of light by metallic gratings and energy ¯ow distribution, Surface Science, 230, 290±294 (1990).

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101. E. Popov, Light diffraction by relief gratings: a macroscopic and microscopic view, Progress in Optics, XXXI, Elsevier Science publishers B.V., 139±294 (1993). 102. B. W. Shore, L. Li, and M. D. Feit, Poynting vectors and electric ®eld distributions in simple dielectric gratings. J. Mod. Optics, 44(1), 69±81 (1997). 103. S. Peng and G. M. Morris, Ef®cient implementation of rigorous coupled-wave analysis for surface-relief gratings, J. Opt. Soc. Am. A, 12, 1087±1096 (1995). 104. E. Butkov, Mathematical Physics, Addison-Wesley, 1968, Chapter 1. 105. J. Jarem and P. P. Banerjee, Application of the complex Poynting theorem to diffraction gratings, J. Opt. Soc. Amer. A, 16(5), 819±831 (1999).

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4

The Split-Step Beam Propagation Method In the previous chapter, we have discussed the RCWA in detail and also shown applications where plane waves are transmitted and re¯ected upon incidence on a layer of arbitrary permittivity and from ®xed gratings. In this chapter we discuss an alternative method to determine the propagation of a beam in a semi-in®nite region that may contain a certain optical inhomogeneity, whether ®xed (such as in a grating) or induced (due to the nonlinear change in refractive index). The extension of this method to analyze pulse propagation as well has been performed but will not be treated here for the sake of simplicity.

4.1

TRANSFER FUNCTION FOR PROPAGATION

For simplicity, consider the scalar wave equation 1 @2 E v2 @t2

r2 E ˆ 0

…4:1:1†

and substitute E…x; y; z; t† ˆ Re fEe …x; y; z† exp‰j…!0 t

k0 z†Šg

…4:1:2†

with !0 =k0 ˆ v. The quantity Ee is related to the phasor Ep according to Ep …x; y; z† ˆ Ee …x; y; z† exp… jk0 z†

…4:1:3†

and we will use one or the other notation according to convenience. Substituting Eq. 4.1.2 into 4.1.1 and assuming that Ee is a slowly varying

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function of z (the direction of propagation) in the sense that j@2 Ee =@z2 j= j@Ee =@zj  k0 we obtain the paraxial wave equation [1] 2jk0

@Ee 2 ˆ r? Ee @z

…4:1:4†

2 denotes the transverse Laplacian. Equation 4.1.4 describes the where r? propagation of the envelope Ee …x; y; z† starting from the initial pro®le Ee jzˆ0 ˆ Ee0 …x; y†. Equation 4.1.4 can be solved readily using Fourier transform techniques. Assuming Ee to be Fourier transformable, we can employ the de®nition of the Fourier transform

E~ e …kx ; ky ; z† ˆ F x;y fEe …x; y; z†g ˆ

…1 1

Ee …x; y; z† exp…jkx x ‡ jky y† dx dy …4:1:5†

and its properties to transform Eq. 4.1.4 into the ODE
d E~ e j ˆ k2 ‡ k2y E~ e 2k0 x dz

…4:1:6†

We can easily solve Eq. 4.1.6 to give  
z E~ e …kx ; ky ; z† ˆ E~ e0 …kx ; ky † exp j k2x ‡ k2y 2k0

…4:1:7†

where E~ e0 …kx ; ky † is the Fourier transform of Ee0 …x; y†. We can interpret Eq. 4.1.7 in the following way: Consider a linear system with an input spectrum of E~ e0 …kx ; ky † at z ˆ 0 where the output spectrum at z is given by E~ e …kx ; ky ; z†. The spatial frequency response of the system, which we will call the paraxial transfer function for propagation, is then given by  
z E~ e 2 2  H…kx ; ky ; z† ˆ exp j kx ‡ ky 2k0 E~ e0

…4:1:8†

As we will show later, in the split-step beam propagation method we model propagational diffraction by means of the transfer function or propagation derived above. For more exact calculations, the nonparaxial transfer function can be used. This can be derived starting from the nonparaxial wave equation, but it will not be presented here for the sake of simplicity.

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Incidentally, the inverse Fourier transform of the transfer function for propagation yields the impulse response for propagation. Starting from the paraxial transfer function for propagation, which resembles a complex Gaussian, the inverse Fourier transform is a complex Gaussian as well and has the form jk h…x; y; z† ˆ 0 exp 2z



k j…x ‡ y † 0 2z 2

2

 …4:1:9†

This, when convolved with the initial beam pro®le, yields the pro®le of the diffracted beam in the spatial domain directly. This convolution integral is in fact the Fresnel diffraction formula. 4.2

SPLIT-STEP BEAM PROPAGATION ALGORITHM

If we wish to consider propagation in a material where the propagation constant or equivalently the refractive index is a function of position, either due to pro®ling of the material itself (such as a graded index ®ber or a grating) or due to induced effects such as third-order nonlinearities, the paraxial wave equation changes to @Ee 1 2 ˆ r E 2jk0 ? e @z

j
nk0 Ee

…4:2:1†

The quantity
n is the change in the refractive index over the ambient refractive index n0 ˆ c=v, where c is the velocity of light in vacuum. Equation 4.2.1 is a modi®cation of Eq. 4.1.4 and can be derived from the scalar wave equation when the propagation constant, or equivalently the velocity of the wave, is a function of …x; y; z† explicitly, as in gratings or ®bers, or implicitly, such as through the intensity-dependent refractive index. The paraxial propagation equation (4.2.1) is a partial differential equation that does not always lend itself to analytical solutions, except for some very special cases involving special spatial variations of
n, or when, as in nonlinear optics, one looks for a particular soliton solution of the resulting nonlinear PDE using exact integration or inverse scattering methods. Numerical approaches are often sought to analyze beam (and pulse) propagation in a complex system such as optical ®bers, volume diffraction gratings, Kerr and photorefractive media, etc. A large number of numerical methods can be used for this purpose. The pseudospectral methods are often favored over ®nite difference methods due to their speed advantage. The

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Chapter 4

split-step beam propagation method (BPM) is an example of a pseudospectral method. To understand the philosophy behind the BPM, it is useful to rewrite Eq. 4.2.1 in the form [2,3] @Ee ^ e ˆ …D^ ‡ S†E @z

…4:2:2†

where D^ and S^ are a linear differential operator and a space-dependent or nonlinear operator, respectively (see, for instance, the structure of Eq. 4.2.1). Thus, in general, the solution of Eq. 4.2.2 can be symbolically written as ^ Ee …x; y; z ‡
z† ˆ exp‰…D^ ‡ S†
zŠE e …x; y; z†

…4:2:3†

If D^ and S^ are assumed to be z-independent. Now for two noncommuting ^ operators D^ and S, 2 ^ ^ SŠ…
z† ‡


exp…D^
z† exp…S^
z† ˆ exp…D^
z ‡ S^
z ‡ 12 ‰D;

…4:2:4† ^ represents the com^ SŠ according to the Baker-Hausdorff formula, where ‰D; ^ ^ mutation of D; S. Thus up to second order in
z, ^ ^ ^ ^ exp…D
z ‡ S
z†  exp…D
z† exp…S
z†

…4:2:5†

which implies that in Eq. 4.2.4 the diffraction and the inhomogeneous operators can be treated independently of each other. The action of the ®rst operator on the RHS of Eq. 4.2.5 is better understood in the spectral domain. Note that this is the propagation operator that takes into account the effect of diffraction between planes z and z ‡
z. Propagation is readily handled in the spectral or spatial frequency domain using the transfer function for propagation written in Eq. 4.1.8 with z replaced by
z. The second operator describes the effect of propagation in the absence of diffraction and in the presence of medium inhomogeneities, either intrinsic or induced, and is incorporated in the spatial domain. A schematic block diagram of the BPM method in its simplest form is shown in Fig. 1. There are other modi®cations to the simple scheme, viz., the symmetrized split-step Fourier method and the leap-frog techniques. These are discussed in detail elsewhere [2].

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The Split-Step Beam Propagation Method

Figure 1

4.3

249

Flow diagram for the BPM split-step method.

BEAM PROPAGATION IN THE LINEAR MEDIA

In this section we will illustrate various cases where the BPM can be used to analyze propagation in inhomogeneous media. While most of the examples will be connected with beam propagation, we must point out to readers that the method can be used to analyze pulse propagation as well, simply by replacing z in Eq. 4.2.2 with t (time) and making the linear spatial transverse differential operator a similar differential operator in z. With this modi®cation, Eq. 4.2.2 can model the propagation of one-dimensional longitudinal pulse through an optical ®ber with arbitrary group velocity dispersion. For details, we refer the readers to Agrawal [2].

4.3.1

Linear Free-Space Propagation

In this case, the inhomogeneous operator is zero, and we can solve Fresnel diffraction of beams using the BPM method. Of course, propagation from a plane z ˆ 0 to arbitrary z can be performed in one step in this case, but in the example we provide we use the split-step method to convince readers that the result is identical to what one would obtain if the propagation were covered in one step. In Fig. 2, we show the pro®le of a diffracted Gaussian beam after propagation through free space, and the results agree with the

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Chapter 4

Figure 2 Diffraction of a Gaussian beam during free space propagation. (a) Pro®le at z ˆ 0 (plane wave fronts assumed); (b) pro®le at z ˆ zR , where zR is the Rayleigh length of the original Gaussian beam.

physical intuition of increased width and decreased on-axis amplitude during propagation. 4.3.2

Propagation of Gaussian Beam Through a Graded Index Medium

A graded index medium has a refractive index variation of the form n ˆ n0 ‡ n…2† …x2 ‡ y2 †

…4:3:1†

where n0 denotes the intrinsic refractive index of the medium and n…2† is a measure of the gradation in the refractive index. In this case, the operator S^ becomes S^ ˆ

jk0 n…2† …x2 ‡ y2 †

…4:3:2†

Propagation of a Gaussian beam in a medium with a graded index pro®le is shown in Fig. 3. The contour plots show the initial (Gaussian) beam pro®le, the beam pro®le where the initial Gaussian attains its minimum waist during propagation before returning back to its original shape again, due to periodic focusing by the graded index distribution. Note that there exists a speci®c eigenmode (a Gaussian of a speci®c width, related to the refractive index gradient) for which the beam propagates through the material without a change in shape as a result of a balance between the diffraction of the beam and the guiding due to the parabolic gradient index pro®le. The contour plot of such a beam is shown in Fig. 4.

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The Split-Step Beam Propagation Method

Figure 3

4.3.3

251

Contour plots showing periodic focusing of an initial Gaussian pro®le.

Beam Propagation Through Diffraction Gratings: Acousto-optic Diffraction

The beam propagation algorithm has been applied to the propagation of a beam through a grating and can be also used to analyze the case where the grating is a sound ®eld. In what follows, we give an example of the use of the beam propagation method to analyze the diffraction of light by an acoustooptic cell in which a traveling wave of sound causes a change in the refractive index using a modi®ed split-step technique. The modi®cation is that the inhomogeneity due to the refractive index grating is accommodated for in the spatial frequency domain as well.

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Chapter 4

Figure 4

Fundamental mode in a graded index ®ber.

The perturbation
n in the case of sound-induced gratings is a function of time and space:
n…x; z; t† ˆ Cs…x; z; t†

…4:3:3†

where C is an interaction constant (for details, see Korpel [4]) and s…x; z; t† is the real sound amplitude given by s…x; z; t† ˆ 12 ‰Se …x; z† exp… jKx† exp…j t† ‡ c:cŠ

…4:3:4†

where Se is the complex amplitude of the sound ®eld that interacts with the light beam and is traveling in the x-direction and c:c: denotes the complex conjugate. The quantities K and are the propagation constant and the angular frequency of the sound ®eld. Following Korpel [4,5], a snapshot of the sound ®eld is used at t ˆ 0, so that using Eqs. 4.3.3 and 4.3.4, ^ exp…S
z† ˆ exp… jk0
n
z†  1 ˆ1

jk0
n
z

1 2 jk0
zC‰Se …x; z† exp…

jKx† ‡ Se …x; z† exp…‡jKx†Š …4:3:5†

In the modi®ed split-step technique, we take the Fourier transform of the above operator on the optical ®eld Ee …x; z†, taking care to note from the property of Fourier transforms that

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The Split-Step Beam Propagation Method

F x ‰f …x† exp…jKx†Š ˆ F…kx  K†

253

…4:3:6†

The main propagation loop of the algorithm is modi®ed from Fig. 1 and is shown in Fig. 5. The boxes marked ``Shift K'' are used to facilitate the operation shown in Eq. 4.3.6 in the spatial frequency domain. Figure 6 shows problem geometry of a Gaussian beam incident nominally at Bragg angle on a sound column of width z ˆ L. The simulated evolution of the Gaussian beam is shown in Fig. 7 and is taken from Ref. [4]. The peak phase delay  of the light traveling through the acousto-optic cell is taken equal to , and the Klein±Cook parameter Q ˆ K 2 L=k0 ˆ 13:1. We would like to point out that the same answers could be derived by using the transfer function for acousto-optic interaction, as given in Refs. 4 and 6.

Figure 5 Flow diagram for the modi®ed split-step technique to analyze acoustooptic interaction.

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Chapter 4

Figure 6 Geometry of acousto-optic interaction with a Gaussian beam at nominally Bragg incidence.

Figure 7 Simulation plot of the intensity of the angular spectrum of the total ®eld at different positions along interaction length [Ref. 4].

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The Split-Step Beam Propagation Method

4.4 4.4.1

255

BEAM PROPAGATION IN NONLINEAR MEDIA Nonlinear Self-focusing and Defocusing of Beams

The nonlinear propagation of beams through a cubically nonlinear material is modeled by the nonlinear PDE, also called the nonlinear SchroÈdinger (NLS) equation [4] 2j k0

@Ee 2 ˆ r? Ee ‡ 2n2 k20 jEe j2 Ee @z

…4:4:1†

where n2 is the nonlinear refractive index coef®cient de®ned by the functional dependence of the total refractive index n on the intensity [4]:

Figure 8 Gaussian beam propagation in a self-focusing medium showing periodic focusing. Used with permission OSA, 1986 [7].

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Chapter 4

Figure 9 Gaussian beam propagation in a self-defocusing medium. Used with permission OSA, 1986 [7].

n ˆ n0 ‡ n2 jEe j2

…4:4:2†

In writing Eq. 4.4.1, we have taken the linear refractive index n0 equal to unity for the sake of simplicity. For a medium with n2 > 0, one can observe self-focusing of a Gaussian beam traveling through a medium, while selfdefocusing is observed fora medium with n2 < 0. The nonlinear operator ^ expfS
zg ˆ exp jk0 n2 jEe j2
z. Typical plots showing self-focusing and self-defocusing of initial Gaussian pro®les in one transverse dimension are shown in Figs. 8 and 9, respectively. Note that in Fig. 8, the initial power in the Gaussian beam is taken to be higher than the so-called critical power required for self-focusing. For this reason, one observes periodic focusing during propagation through the medium. The physical reasoning behind self-focusing is as follows. The Gaussian beam induces a positive lens in the nonlinear material for n2 > 0 because where the beam intensity is high

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257

(e.g., on-axis), the induced refractive index is higher as well, amounting to larger slowing down of the wave fronts. The wave fronts are therefore bent similar to the action of a positive lens, resulting in initial focusing of the beam. This process continues till the beam width is small enough for the diffraction effects to take over, leading to an increase in the beam width. The converse is true for the case of n2 < 0. In this case, the beam spreads more than in the linear diffraction limited case. A stable nonspreading solution in one transverse dimension can be analytically found from the NLS equation for n2 > 0 and has the form  Ee …x† ˆ

8 n2 k0

1=2

sec h

x …2k0 †1=2

…4:4:3†

where  is a free parameter. With such a beam pro®le as an initial condition, the propagation has been plotted using the split-step method, as shown in Fig. 10. This pro®le is called a spatial soliton and can be regarded as a nonlinear eigenmode of the NLS system. Higher order spatial soliton solutions can also be derived. We would like to point out that the above simulations can be easily modi®ed to analyze pulse propagation through a nonlinear ®ber and in the presence of group velocity dispersion. This is possible because the interchange z ! t and x ! z in the NLS equation with a suitable coef®cient in front of the second-order derivative term transforms the equation to one that can model the propagation of pulses in time t along a ®ber. The split-step method has also been used to analyze propagation of pulses through ®bers having higher order dispersion, and other kinds of nonlinearities, such as that stemming from Raman scattering, etc. An excellent reference on nonlinear propagation through ®bers is Agrawal [4]. The split-step technique has also been applied to analyze propagation of pro®les in two transverse dimensions [7], and also to analyze propagation of optical ®elds that are pulsed in time and have a spatial pro®le in the transverse dimension [8].

4.4.2

Beam Fanning and Distortion in Photorefractive Materials

In this section, a model for beam propagation through a nonlinear photorefractive material that takes into account inhomogeneous induced refractive index changes due to the nonlinearity is ®rst developed. In some cases a focused Gaussian beam asymmetrically distorts due to passage through the nonlinear material.

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Chapter 4

Figure 10 Spatial soliton propagation in a self-focusing medium. Used with permission OSA, 1986 [7].

The photorefractive (PR) effect, discussed in detail in Chapter 7, has been used in a wide variety of applications, viz., image processing, optical interconnections, optical data storage, optical limiters, and self-pumped phase conjugators [9]. When a PR material is illuminated by a light beam or by a fringe pattern generated by the interference of two light beams, photoexcited carriers are redistributed in the volume of the crystal [9]. This sets up a space charge ®eld which, through the linear electro-optic effect, gives rise to a refractive index pro®le and hence a phase hologram. The phenomenon of PR beam fanning, where the incident light beam is de¯ected and/or distorted when it passes through a high-gain PR crystal, has been observed in BaTiO3 , LiNbO3 , and SBN [10±12]. One of the ways this has been explained is through the fact that a symmetric beam may create an asymmetric refractive index pro®le, leading to beam distortion, or what we will call deterministic beam fanning (DBF) in the far ®eld [13].

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This analysis has been done for a thin sample, meaning one where diffraction of the beam is neglected during its travel through the PR material, and by using a linearized theory to determine the induced refractive index pro®le. We have recently extended the linearized approach to the case of a thick sample, and have included the transient effects, and are in the process of determining the effects of transient DBF when a reading beam is used to illuminate a previously stored hologram in the PR material [14]. Another school of thought is that beam ``fanning'' results from light scattering from the random distribution of space charges in the PR material. However, a larger contribution to random beam fanning (RBF) is the socalled ampli®ed noise [15] that may arise from the couplings between the plane wave components scattered from crystal defects. In this section, we examine steady state DBF in a diffusion dominated PR material by deriving a closed form expression for the induced refractive index change from the nonlinearly coupled Kukhtarev equations. We also assess the role of propagational diffraction in DBF by determining the similarities and differences between the thin and thick sample models. It can be shown that the coupled set of simpli®ed Kukhtarev equations [9] (see Chapter 7 for details) for a diffusion dominated PR material can be decoupled in the steady state to yield an ordinary differential equation for the space charge electric ®eld [13]. In denormalized form, we can express this electric ®eld Es …x; y; z† as 

eNA r Es ˆ s r
Es 2



e E kB T s

rI =s ‡ I

 …4:4:4†

where e is the electronic change, NA is the (ionized) acceptor concentration, s is the static permittivity, kB is the Boltzmann constant, T is the temperature, s is the ionization cross section per unit photon energy, and is the thermal generation rate. The last coef®cient is important if the beam pro®le decays to zero for large x; y, which represent directions transverse to propagation (z) of the beam in the PR material. I…x; y; z† denotes the intensity distribution along x; y at a position z in the PR material. We have numerically checked that a good approximation to the solution of Eq. 4.4.4 is Es 

kB T rI ˆ Esx a^ x ‡ Esy a^ y e =s ‡ I

…4:4:5†

if …kB =T=e†=W2  eNA =s , where W is the characteristic width of the complex envelope Ee …x; y; z† of the optical ®eld. The quantities a^ x and a^ y refer to unit vectors in the x- and y-directions, respectively. Now this electrostatic ®eld induces a refractive index change
next …x; y; z† for extraordinary polar-

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ized (say along x, see Fig. 11) plane waves of light in the PR material, assumed BaTiO3 from now on, through the linear electro-optic effect, given by
next …x; y; z; y† ˆ Ex …x; y; z† f …y† f …y† ˆ

1 3 2 ne …y† cos y

 …r13 sin y ‡ r33 cos2 y ‡ 2r42 sin2 y

n2e …y† ˆ …sin2 y=n2o ‡ cos2 y=n2e †

1

…4:4:6†

where no and ne are the linear ordinary and extraordinary refractive indices and the rij are the linear contracted electro-optic coef®cients [9]. The angle  in Eq. 4.4.6 is de®ned in Fig. 11. Note that f …y† is a slowly varying function of  over the spectral content of the optical ®eld. It can be readily shown that, in general, propagation through the PR material under the slowly varying envelope approximation can be modeled by means of the PDE [13] @Ee ˆ @z

2 jk0
next Ee j‰1=2ne …†k0 r? Ee    kx
next …x; † ˆ F x 1 ‰F x ‰Ee …x†Šf ‡0 ne …†k0

Esx …x† ˆ

kB T @jEe …x†j2 =@x e  =s ‡ jEe …x†j2

…4:4:7†

where  is the characteristic impedance of the medium. For values of  around 40 , a symmetric beam could induce an asymmetric refractive index pro®le, leading to beam bending and DBF in the far ®eld. However, for some other value of , for instance 90 , our theory predicts symmetric beam shaping, in agreement with the ®ndings of Segev et al. [15]. In this respect, the nature of the optical nonlinearity in a PR material is more involved as compared to that in a nonlinear Kerr-type material. We

Figure 11 1993 [13].

Geometry used to study DBF. Used with permission of North-Holland,

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point out that in a Kerr-type material for instance, only an asymmetric beam pro®le can cause beam bending, as reported in [16], while a symmetric beam undergoes self-focusing or defocusing. In what follows, we ®rst provide results for the far ®eld beam pro®les by assuming the PR material to be a thin sample, in the sense that we neglect the effects of propagational diffraction through the material. A Gaussian input Ee …x; y; 0† ˆ …I0 †1=2 exp

…x2 ‡ y2 † W2

…4:4:8†

with I0 ˆ 2P=W 2 , where I0 denotes the on-axis intensity and P is the beam power, is phase modulated owing to the induced refractive index pro®le. The resulting output ®eld is Ee …x; y; L† ˆ Ee …x; y; 0† exp… jk0
next …x†L†, where L is the thickness of the PR material. Such a phase modulation results in a shift of the far ®eld pattern with respect to the axis …z† of propagation of the optical beam, and in the appearance of asymmetric side lobes, the so-called fanning of the beam. Numerical simulations for BaTiO3 with parameters n0 ˆ 2:488, ne ˆ 2:434, r42 ˆ 1640 pm/V, r13 ˆ 8 pm/V, r33 ˆ 28 pm/V, NA ˆ 2  1022 m 3 , s ˆ 3:28  10 8 F/m, s ˆ 2:6  10 5 m2 =J, b ˆ 2 s 1 , T ˆ 298 K [13], and L ˆ 1 cm and using an incident wavelength of 514.5 nm show a monotonic increase in the shift of the far ®eld main lobe from the z axis with increase in I0 (implying either an increase in power P or a decrease in width W). In Figs. 12a and b, kx is the spatial frequency variable corresponding to x and is related to the far ®eld coordinate xf by kx ˆ k0 xf =d, d being the distance of propagation from the exit of the crystal to the far ®eld. However, the amount of DBF (de®ned by the relative amount of power in the side lobes) varies nonmonotonically with intensity, initially increasing as the intensity is increased from low levels to attain a maximum, and then decreasing with further increase in intensity. Note that our results are different from those of Feinberg [10], in that the latter, based on a linearized two-beam coupling theory that neglects coupling of the angular plane wave components of the Gaussian with any ! background illumination, yields E s / rI=I0 , where I0 is the quiescent intensity (to be compared with our Eq. 4.4.7). For a Gaussian intensity pro®le, the locations of the extrema of E in Feinberg's formulation are ®xed w.r.t. to the incident pro®le and hence can be shown to predict a monotonic increase in DBF with a decrease in W. In our nonlinear formulation, however, for decreasing W, the extrema of E move out with respect to the incident pro®le, so that the pro®le essentially sees a linear induced refractive index for suf®ciently small W, resulting in reduced DBF.

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Chapter 4

Figure 12 Normalized far ®eld intensity pro®les for the thin sample model. (a) P ˆ 1:5 mW; (b) W ˆ 40 microns. Used with permission of North-Holland, 1993 [13].

The Split-Step Beam Propagation Method

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Before comparing the thin sample results with the ®ndings for the thick sample case, we will, at this point, provide a simple alternate explanation for the observed behavior of DBF when monitored as a function of the intensity. Our explanation is based on the examination of the spectrum of the phase modulation exp… ik0
next …x†L†. The far ®eld pattern is the convolution of the above spectrum with that of the input pro®le. Since
next …x† is an odd function of x (see Eq. 4.4.6), it can be expanded in a power series of the form ax3 bx, where a and b are given by bˆ

4f …y†kB T=eW 2 =sI0 ‡ 1

aˆ

…2b=W 2 †… =sI0 † =sI0 ‡ 1

…4:4:9†

Note that the coef®cients of this expansion hold for all values of the ratio =sI0 . The spectrum H…kx † of exp… ik0
next …x†L† is then H…kx † ˆ

2 k bk0 L Ai x 1=3 …3a† …3a†1=3

…4:4:10†

Once again, kx above has the same implication as in the discussion on Fig. 12. We comment that if d is replaced by f , where f is the focal length of a lens at the exit plane of the crystal, kx , and hence xf , would be representative of the spatial coordinate on the back focal plane of the lens. Ai‰
Š is the Airy function [17]. The ith zero, i , of H…kx † is related to the ith zero, i …< 0†, of Ai‰
Š by i ˆ bk0 L ‡ …3a†1=3 i . It then follows that the spatial extent of the Airy pattern for kx < bk0 L, up to say the ith zero, and normalized by the spectral width 2=W of the incident Gaussian pro®le, varies nonmonotonically with I0 . Figs. 13a and b show, for instance, the variations of i ˆ bk0 L ‡ …3a†1=3 i =…2=W† for i ˆ 1 with W and P, respectively. The shift in the Airy pattern, bk0 L, however, increases with an increase in I0 . For large I0 , it can be shown that the shift is proportional to 1=W 2 , in agreement with the trend in Fig. 12a. The resulting far ®eld pattern, which is the convolution of the Gaussian spectrum and the Airy pattern, generally exhibits decreased DBF when the Airy pattern has a (denormalized) width much smaller than that of the Gaussian spectrum (which may occur, for instance, for both small and large W). This is in agreement with our numerical simulations in Fig. 12. Appreciable DBF occurs in the region where the normalized bandwidth (see Figs. 13a and b) is greater than unity. As an example, for P ˆ 1:5 mW, maximum beam fanning, de®ned by the maximum of the ratio of the peak value of the side lobe and that of the main lobe, occurs when W ˆ 30 microns.

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Figure 13 Normalized bandwidth of the induced PR phase modulation (a) for P ˆ 1:5 mW, plotted as a function of W, and (b) for W ˆ 40 microns, plotted as a function of P. Used with permission of North-Holland, 1993 [13].

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In the remainder of this section, we will present the results for the far ®eld beam pro®les using a thick sample model for the PR material and point out the similarities and differences with the thin sample approach. Numerical simulations for the thick sample model were performed on the basis of Eq. 4.4.7 by employing a split-step beam propagation technique [7]. In this simulation, we track both the phase and the amplitude modulation of the beam within the crystal due to the combined effects of propagational diffraction (along x; y) and induced refractive index (along x) arising from the PR effect. Figs. 14a and b show the normalized far ®eld intensity patterns with W and P as parameters. By W we now mean the beam waist that would be expected at z ˆ L=2 (i.e., the location of the center of the sample) in the absence of any electro-optic effect (rij ˆ 0) (see inset in Fig. 14a). The results are qualitatively similar: DBF is seen to reduce at very low (high) and very high (low) values of P …W†. Quantitatively, for a ®xed power P (viz., 1.5 mW), we can predict the absence of DBF for suf®ciently large values for W (viz., 70 microns) which are independent of the model (thin or thick sample) used for simulation. Physically, this makes sense, since the thin and thick sample models must agree if the diffraction effects in the crystal are suf®ciently small. On the other hand, the reason for the absence of DBF for a suf®ciently small value of W in the thick sample approach is that effectively, the beam width, if monitored over most of the sample, is large (due to a large diffraction angle), implying a reduced PR effect. This in turn implies that propagation through the crystal is predominantly diffraction limited. For small W, the thick sample model therefore is more accurate than the corresponding thin sample model for the same value of W, since the latter model overestimates the amount of cumulative PR effect. For the thick sample model, for the same value of P as above, we see negligible DBF for W less than 25 microns. On the other hand, the thin sample model predicts a value of W less than 5 microns for negligible beam fanning. The reason for the disappearance of DBF in the thin sample approach has been presented above using the Airy function argument and the movement of the extrema of E w.r.t. the incident optical ®eld. Maximum DBF for P ˆ 1:5 mW occurs for W ˆ 40 microns, in close agreement with the thin sample computations and the Airy function approach. However, the shift in the position of the main lobe in the thick sample model is much smaller as compared to the thin sample case due to the effective decrease in the PR effect for a small waist size, as explained above. Referring to Fig. 14a, we note that for W ˆ 40 microns, P ˆ 1:5 mW and f ˆ 10 cm, and the spatial shift in the back focal plane of a lens of focal length f located at the exit plane of the PR material is about 0.2 mm. We would like to comment that for the above parameters, DBF was also numerically observed at the exit face of the thick PR sample.

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Figure 14 Normalized far ®eld pro®les for the thick sample model. (a) P ˆ 1:5 mW; (b) W ˆ 40 microns. Used with permission of North-Holland, 1993 [13].

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4.4.3

267

Two-Beam Coupling in Photorefractive Materials [18]

As seen in the previous section, in diffusion dominated photorefractive (PR) materials, the induced refractive index can be written as
next /

kB T rI e =s ‡ I

…4:4:11†

The interaction between two focused Gaussian beams incident on the material can be effectively studied numerically using the split-step method. In this case, two-beam coupling results in energy exchange between the two beams after interaction through the photorefractive material. The problem geometry is shown in Fig. 15. The two Gaussian beams are focused in the center of the photorefractive material, and the angle between them is 2. The Gaussian beams are expressed in terms of their q-parameters at the entry face of the material. The split-step algorithm is used to determine the interaction and energy exchange between the two beams. The induced refractive index
n written above is used to construct the operator representing the induced inhomogeneity in the material. The results on two-wave mixing are shown in Fig. 16. The dot-dashed lines show the far ®eld intensity pro®les of the two Gaussian beams in the absence of the photorefractive material. The dashed lines show the beams after energy transfer due to the induced refractive index. The initial pump-to-signal power is 3. The peak intensity of the

Figure 15 Geometry for photorefractive two-beam coupling. Used with permission of North-Holland, 1994 [18].

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Chapter 4

Figure 16 (a) Dotted and dashed lines are respectively the far ®eld signal and pump intensities with the absence of any PR material, and chain dots and chain dashes represent the resulting far ®eld intensities after the beams have propagated through a 5 mm BaTiO3 sample. Incident beams are focused to the center (z ˆ L=2) of the Pr crystal, and the waist of each beam at wavelength 0.632 microns is 100 microns. Signal-to-pump ratio is 3, and semi-angle of crossing  is 0.5 degrees. Note that w1 ˆ w2 ˆ 1:0  10 4 . (b) Interference pattern at center (z ˆ L=2) of the PR crystal for the beams described in (a). (c) Space charge ®eld (V/m) at the center (z ˆ L=2) of the crystal for the beams of (a). Used with permission of NorthHolland, 1994 [18].

pump and signal beams are 63 and 21 W/cm2 , respectively, before the interaction. The beams are coupled by a 5 mm BaTiO3 photorefractive material. The output beams do not show any effect of beam fanning at this power, but with larger beam powers, distortion of the beams due to beam fanning is observed. The results have been used to ®nd the two-beam coupling strength

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and their dependence on the intensities of the two participating beams. The results, shown in Ref. 18, depict that the coupling strength depends on the power ratio between the two beams, a fact that is ignored in perturbation calculations of two-wave mixing in photorefractive materials. Later, in Chapter 7, we will analyze this effect in more detail with participating plane waves and using rigorous coupled wave theory.

4.5

q-TRANSFORMATION OF GAUSSIAN BEAMS THROUGH NONLINEAR MATERIALS: z-SCAN AND PSCAN TECHNIQUES

The previous examples illustrated the use of the split-step method in calculating the beam pro®les during diffraction in space or during propagation through a guided (externally or internally induced) medium. If a Gaussian beam is assumed, however, the split-step method can be reformulated in terms of a differential equation that shows the evolution of the Gaussian beam's parameters, e.g., width, during propagation. The ensuing equation can be exactly solved in some cases, e.g., for a Kerr-type material, and is therefore physically more transparent than the results obtained using the split-step method. The differential equation for the parameter(s) may not be simpler to solve than the split-step method, but having an analytical solution (Gaussian beam) adds a tremendous insight into the actual propagation of the wave through the material, whereas the split-step method only presents simulation results. When a Gaussian beam travels a distance
z in an n2 medium, the q-parameter [19] change using the split-step method can be written as
q ˆ
z ‡

q2 find …
z†

…4:5:1†

where find is the nonlinearly induced focal length of the slice
z [19]. The above equation shows that the q of a Gaussian beam changes due to propagational diffraction and due to the induced nonlinearity of the material. In LiNbO3 the photovoltaic effect is responsible for breaking the circular symmetry of an incident focused extraordinarily polarized Gaussian beam. Therefore the propagation model is based on the propagation of an elliptical Gaussian beam. As discussed in the last section, beam fanning in photorefractive crystals has received considerable attention for its possible implications in holographic information recording [10,20±23]. Light-induced scattering resulting

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Chapter 4

in deterministic beam fanning (DBF) has been observed in PR LiNbO3 and can be explained on the basis of an induced nonlinear refractive index primarily due to the photovoltaic and thermal effects [11]. This type of beam fanning is distinct from random beam fanning (RBF) due to light scattering from the randomly distributed space charges or crystal defects [9,15]. In LiNbO3 the photovoltaic effect is responsible for breaking the circular symmetry of an incident focused extraordinarily polarized Gaussian beam in the far ®eld, while the thermal effect manifests itself in circularly symmetric far ®eld patterns [11]. Over a range of input powers the photovoltaic effect dominates, resulting in an elongated far ®eld pattern with the spreading dominant along the c-axis of the crystal. An interesting consequence of monitoring the q-parameter variation of a Gaussian beam as it propagates through a nonlinear material is that one can thereby estimate the amount of nonlinearity in the material. Conventional methods of estimating the sign and magnitude of the optical nonlinearity in materials include the z-scan technique where the far ®eld onaxis transmittance is monitored as a function of the scan distance about the back focal plane of an external lens. The z-scan method, however, may be rather cumbersome, since it involves scanning the material; so we developed a simpler technique in which the longitudinal position of the sample is not changed. Instead the beam ellipticity is monitored as a function of the incident beam power P while testing materials with induced inhomogeneous nonlinearities, e.g., photorefractive (PR) LiNbO3 . Another disadvantage of the z-scan is that monitoring the on-axis intensity may be dif®cult owing to aberrations, optical misalignments, sample imperfections, refractive index mismatch, and nonparallelism of the entry and exit faces of the material. The imperfections can give rise to ®ne interference patterns within the far ®eld intensity pro®le. These problems have been observed during z-scan measurements of LiNbO3 , which led us to develop the P-scan technique as an attractive and simple alternative. In what follows, we present a new technique for determining the nonlinear refractive index of PR LiNbO3 that uses an appropriate model for beam propagation through a nonlinear material. The model takes into account inhomogeneous induced refractive index changes due to the optical nonlinearity. For the case of LiNbO3 , induced refractive index changes are primarily due to photovoltaic contributions over the range of powers used. The model is based on the evolution of beam widths of an incident circularly symmetric Gaussian beam focused by a lens onto the material in order to reduce RBF. The calculations closely follow the analysis for the z-scan determination of nonlinearities in a thick sample of a nonlinear material previously derived by several groups [13,24]. Under certain approximations, the model reduces to that used by Song et al. to study anisotropic light-

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induced scattering and ``position dispersion'' in PR materials [25]. Since we consider a ``thick'' sample, i.e., a sample whose thickness is much larger than the Rayleigh range of the focused Gaussian beam, diffraction effects become important and cannot be neglected. Therefore we determine the beam shape as it leaves the nonlinear sample and then calculate the beam pro®le after it has propagated some distance outside the medium. The information about the effective n2 is contained in the nature of this pro®le. In general, the magnitude and sign of the nonlinearity can be determine from the beam pro®le variation as the sample position is varied about the back focal length of the external lens. The nonlinearity depends on the acceptor-to-donor concentration ratio NA =ND , which in turn determines the far ®eld diffraction pattern. Conversely, measurements of the far ®eld pattern can be used to calculate NA =ND and used as a tool for characterizing different LiNbO3 samples.

4.5.1

Model for Beam Propagation Through a PR LiNbO3 Single Crystal

Assume an incident Gaussian beam in the form Ee …x; y; z† ˆ a…z† exp

! x2 exp w2x

y2 w2y

! …4:5:2†

For an elliptical Gaussian beam the following relationships hold:
qx ˆ
z ‡

q2x

findx


qy ˆ
z ‡

q2y

…4:5:3†

findy

Since 2

n ˆ ne ‡ n2 jEe j  ne

x2 y2 2n2 a …z† 2 ‡ 2 wx wy 2

! …4:5:4†

where n2 is the effective nonlinear refractive index coef®cient, ne is the linear refractive index, and Ee is the optical ®eld, we can compute the phase change upon nonlinear propagation through a section
z of the sample and thereby determine the induced focal length. As expected, these focal lengths are inversely proportional to
z and can be expressed as

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Chapter 4

findx ˆ

ne w2x 4n2x a2 …z†
z

findy ˆ

ne w2y 4n2y a2 …z†
z

…4:5:5†

Substituting Eq. 4.5.5 into 4.5.3 and taking the limit as
z ! 0 we obtain the system of equations 4n2x a…z†q2x dqx ˆ1‡ dz ne w2x 4n2y a…z†q2y dqy ˆ1‡ dz ne w2y

…4:5:6†

Using the well known relationship 1=q ˆ 1=R ‡ j=ne w2 , where R is a radius of Gaussian beam curvature, 1=R ˆ …1=w†…dw=dz†, and  is the wavelength in vacuum, we obtain 1 dRx n2e 2 w4x 2 R2x ˆ R2x dz …ne w2x Rx †2

4n2x a2 ne w2x

2 2 4 2 2 1 dRy ne  wy  Ry ˆ R2y dz …ne w2y Ry †2

4n2y a2

d 2 wx 2 ˆ dz2 n2e 2 w3x

4n2x a2 ne w x

d 2 wy 2 ˆ dz2 n2e 2 w3y

4n2y a2 ne wy

ne w2y

…4:5:7†

…4:5:8†

Taking into account the relationship for the beam's power, P ˆ …=2†a2 …z†wx …z†wy …z†, where  is the characteristic impedance of the material, which is conserved, we ®nally have the system of equations describing the Gaussian beam propagation in a thick LiNbO3 crystal: d 2 wx 2 ˆ 2 2 3 2 dz ne  wx

8n2x P ne w2x wy

d 2 wy 2 ˆ dz2 n2e 2 w3y

8n2y P ne w2y wx

…4:5:9†

Assuming n2x  n2y (true for photorefractive lithium niobate), the variation of the widths wx and wy of an elliptic Gaussian beam propagating through a

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thick LiNbO3 sample as shown in Fig. 17 can be modeled by the coupled differential equations [26] d 2 wx 2 ˆ dz2 n2e 2 w3x

8n2 P ne w2x wy

d 2 wy 2 ˆ 2 2 3 2 dz ne  wy

…4:5:10†

The case when n2x ˆ n2y has been studied [13] by employing the q-transformation approach to ®nd the widths of an elliptic Gaussian beam in a nonlinear medium in the presence of diffraction. Equation 4.5.10 assumes that the nonlinearity is highly inhomogeneous and only affects the width along the x-axis (which coincides with the c-axis of our crystals) due to the large electron mobility along that axis [25]. The effective n2 can be written as [11] nw 

1 3 k R NA n r 2 e 33 mebND

…4:5:11†

where r33 is the electro-optic coef®cient, k is the photovoltaic constant,  is the absorption coef®cient, R is the recombination constant, m is the mobility, e is the electron charge, and b is the thermal generation rate. In the

Figure 17 z-scan setup for a thick sample. The thick lines represent the path of the rays, described as the locus of the 1=e points of the Gaussian beam. The thin lines show the ray path in the absence of the medium. Circular symmetry of the Gaussian beam is assumed throughout the sample. Used with permission of OSA, 1998 [26].

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Chapter 4

above equation, we have made the assumption b  sI, where s is the ionization cross section per quantum of light and I is the optical intensity. 4.5.2

z-SCAN: ANALYTICAL RESULTS, SIMULATIONS, AND COMPARISON WITH SAMPLE EXPERIMENTS

In this section we present analytical and numerical simulation results using equations (1) and compare them with sample experiments using PR LiNbO3 . If the Gaussian beam incident on the sample is assumed to have planar wave fronts and waist w0 (approximately at the back focus of the lens), then w2y …z†

ˆ

w20

z2 1‡ 2 zR y

! and

zRy ˆ

ne w20 0

…4:5:12†

For a sample length L assumed to be much larger than the Rayleigh ranges zRy and zRx along z for the elliptic beam, the evolution of wx can be approximated as w2x …z†

ˆ

w20

z2 1‡ 2 zR x

! where

zRx

  ne w20 4ne n2 P ˆ 1‡ 0 20 …4:5:13†

It is clear that in the x-direction, the beam spread is more than that in the linear diffraction-limited case when n2 < 0 and less when n2 > 0. As seen from Eq. 4.5.12, the nonlinearity does not affect the beam width along the y direction, which leads to elliptic beam cross section pro®le at the exit of the crystal and, in general, in the far ®eld. For more general geometry, where the incident beam does not have a planar wave front, we have solved Eqs. 4.5.10 numerically. Figure 18 shows typical z-scan graphs plotted for four different values of power for the initially circularly symmetric Gaussian beam. In the calculations we have used the following parameters: crystal thickness L ˆ 10 mm, lens focal length f0 ˆ 10 cm, 0 ˆ 514 nm, initial beam width w0 ˆ 1:0 mm, ne ˆ 2:20, n2 ˆ 1:4  10 12 m2 =V2 , P ˆ 1 mW, crystal exit plane to observation plane distance D ˆ 1 m. A simple explanation of the behavior in the limiting case (s much smaller or larger than f0 ) seen in Fig. 18 can be given by referring to Fig. 17. When the distance s, the lens-to-sample separation, is much smaller than the lens focal length f0 , the incident beam is weakly focused and therefore the beam widths lie close to their linear values leading to semilinear diffrac-

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Figure 18 Typical z-scan graph drawn by solving Eq. 4.5.10 and propagating the Gaussian beam a distance D behind the sample. Used with permission of OSA, 1998 [26].

tion-limited propagation. When s is much larger than f0 , the incident beam is weakly diverging and the overall nonlinear effect is small, which in turn leads to semilinear diffraction-limited propagation. If s  f0 , the incident beam is highly focused and therefore the nonlinear effect is large. In this region, as s decreases, the normalized intensity decreases from its linear value, passes through a minimum, and then reaches its maximum before approaching its linear value again. The overall negative slope (between the peak and the valley) of the z-scan con®rms the net negative nonlinearity of the sample. Figure 19 depicts ellipticity wx =wy in the far ®eld versus displacement s drawn for the same set of parameters as that used to draw Fig. 17 but for P ˆ 0:2 mW. We have done a series of sample experiments and compared results. It turns out that the on-axis intensity measurement of far ®eld patterns may lead to signi®cant errors due to ®ne structures in the pattern as seen on Fig. 20 (obtained using a LiNbO3 crystal doped with Fe). We have used this crystal for all experimentation to validate our theory, unless otherwise stated. As stated in the Introduction, possible reasons for this include

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276

Chapter 4

Figure 19 Plot of ellipticity as a function of displacement s for parameters the same as in Fig. 1 but for P ˆ 0:2 mW. Used with permission of OSA, 1998 [26].

1. 2. 3. 4.

Interference patterns stemming from single-beam holography [27] Interference patterns from optical misalignment Light diffraction and scattering on crystal defects Interference patterns from nonparallel crystal edges

Note that the pattern is approximately symmetric (along x and y). This symmetry arises because the refractive index changes that are due to photovoltaic (and thermal) effects are symmetric and because there is little contribution from diffusion. Experimental results based on the measurement of ellipticity, as shown in Fig. 21, show the same trend as the theoretical predictions superposed on the same ®gure. The ellipticity was calculated from experimental observations by ®rst determining the extent wx , wy of the bright or gray region along x and y, respectively, from pictures such as Fig. 20 and taking the ratio of the two. Note that Fig. 21 is in fact a blow-up of Fig. 19 over the interval 9.5±10.5 cm. The theoretical graph in Fig. 21 was drawn after examining the experimental results shown in the same ®gure and choosing that value n2 for the analytical graph that minimizes the sum of the differences between the experimental points and the corresponding theoretical data.

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Figure 20 Typical beam pattern at D ˆ 0:5 m for P ˆ 0:05 mW, f0 ˆ 20 cm, and s ˆ 19:5 cm for Fe doped LiNbO3 crystal. Used with permission of OSA, 1998 [26].

Figure 21 Experimental (points) and theoretical (line) variation of the beam ellipticity on the observation plane as a function of scan distance. here, P ˆ 0:2 mW, D ˆ 0:5 m, f0 ˆ 10 cm. Upon comparison, n2 ˆ 1:4  10 12 m2 =V2 . Used with permission of OSA, 1998 [26].

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As a ®nal note, we would like to point out that each time the crystal was displaced along the longitudinal direction for a fresh z-scan ellipticity measurement, we also made a transverse movement of the crystal in order to make sure that we were starting out from a virgin location in the crystal for each data point. In other words, we always started out from an initially unexposed region of the crystal and exposed it to the incident illumination until the steady state was achieved.

4.5.3

P-SCAN SIMULATIONS AND COMPARISON OF THEORY WITH EXPERIMENT

In order to make the data acquisition less cumbersome and still get an accurate value for the effective n2 , we have developed and used a new technique that we call P-scan, where we simply vary the power of a Gaussian beam focused at a ®xed longitudinal position within the sample and monitor the beam ellipticity. Each data point for the ellipticity is however the steady state value for a different transverse location in the crystal. Figure 22a shows a typical theoretically obtained P-scan graph. From sample experiments we have found that the value of the nonlinearity coef®cient of PR LiNbO3 derived with z-scan compares favorably with that found using P-scan. To determine the effective n2 we have matched experimental data (Fig. 22b) with those obtained numerically. In the experiment we de®ne the extent of DBF by measuring the ellipticity of the distinct bright spot. The 10  10  10 mm crystals have been illuminated with extraordinary polarized light at 0 ˆ 514 nm. The 0.8 mm wide beam has been focused by a lens with a focal length of f0 ˆ 20 cm into the crystal (see Fig. 17). The observation plane is 0.5 m from the crystal. It took between 10 and 45 min (depending on the dopant and its concentration) to achieve the steady state pattern. As the beam power increases, both DBF and RBF increase. However, above a particular value of power Ps the photovoltaic effect saturates and thermal effects become more appreciable. Taking this into account we have used low power, i.e., P < Ps . We stress that our model does not take into account saturation or thermal effects. Experimental data have been used to build a cubic spline, and its slope has been compared and matched with slopes of those obtained numerically for a series of different n2 . Once the slopes are matched, the nonlinearity of the crystal is known. Given n2 , the acceptor-to-donor ratio can be readily determined using (2). Using Pscan we have evaluated effective nonlinearities (resulting from the photovoltaic effect) of 12 LiNbO3 samples doped with various materials [28]. Our results for the nonlinearity for most crystals, as shown in Table 1, are in general agreement with the trends from hologram peak diffraction ef®ciency

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Figure 22 (a) Theoretically and (b) experimentally obtained P-scan graph for D ˆ 0:5 m for f0 ˆ 20 cm and s ˆ 19:5 cm for the Fe doped LiNbO3 crystal. Fig. (a) was drawn with n2 ˆ 4  10 11 m2 =V2 to provide the best match with the experimental results in (b) for lower powers. Used with permission of OSA, 1998 [26].

from these crystals when holograms were stored in them with a 90 s exposure time at 514 nm using an incident power of approximately 200 mW. Given a crystal with unknown nonlinearity, the signal can be determined as follows. We place the crystal around the back focal plane of the lens. Upon moving it toward the lens, if the ellipticity reduces to less than unity, nw < 0. If, upon moving the crystal away from the lens, the ellipticity

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Table 1. Experimental results for effective nonlinearities (n2 ) of the 12 crystals. The last column shows the time needed to achieve the steady state. Crystal name

Dopant

Pat Rob Bob Sam Ned Hal Ted Eve Flo Liz Ian Moe

Fe Rh Cr Fe:Ce Cu Co Tb Ce Fe:Cr Fe:Mn Mn Ni

Nonlinearity 4:00E-11 3:00E-13 2:00E-13 1:00E-10 5:00E-11 4:00E-11 4:00E-13 3:00E-11 7:00E-11 9:00E-10 5:00E-13 4:00E-13

Time 10 25 20 5 10 20 25 15 10 5 25 20

min min min min min min min min min min min min

@ @ @ @ @ @ @ @ @ @ @ @

0.01 mW 1 mW 1 mW 0.005 mW 0.01 mW 0.01 mW 1 mW 0.01 mW 0.01 mW 0.005 mW 1 mW 1 mW

is less than unity, n2 > 0. The magnitude of n2 can now be evaluated using the P-scan technique. Details of this can be found in Ref. 26. In summary, a model for beam propagation through a nonlinear material that takes into account inhomogeneous induced refractive index changes due to the nonlinearity was developed. The theory based on this model can be used to analyze the propagation of Gaussian beams through PR LiNbO3 . A focused Gaussian beam of circular cross section incident on the sample emerges as an elliptic Gaussian after interaction in this material. The P-scan method can be used to evaluate the effective nonlinearities (resulting from the photovoltaic effect) of lithium niobate samples doped with different materials such as Fe, Co, Cr, Rh, Mn, etc. The value of the nonlinear coef®cient can then be used to determine the acceptor-to-donor ratio of dopants in the photorefractive samples. This method can be used to characterize any optically nonlinear material that has an induced intensity-dependent refractive index. We would like to point out that this method is very general and in principle can be applied to any nonlinear electromagnetic material and at any frequency.

PROBLEMS 1. Analyze the propagation of a Gaussian beam through free space using the transfer function for propagation de®ned in Section 4.1. Assume an initial Gaussian of waist W0 and having plane wave-

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2.

3.

4.

5.

6.

7.

281

fronts at z ˆ 0. Find expressions for the width W…z† and the radius of curvature of the wavefronts R…z† after an arbitrary distance z of propagation. A Gaussian beam of width W and having wavefront with a radius of curvature R is normally incident on the interface between air and glass of refractive index n. Find the width and radius of curvature: (a) immediately after transmission through the interface, and (b) immediately upon re¯ection at the interface. [Hint: Calculate the off-normal ray re¯ection and transmission angles.] A one-dimensional Gaussian beam symmetric along the x-direction of waist W0 is incident on a thin slice of dielectric material of thickness
z with a graded refractive index n…x† ˆ n0 n…2† x2 for small x. Find the effective focal length of the induced lens as the Gaussian beam propagates through the material. Hence ®nd the approximate location of the focal point beyond the thin sample where the Gaussian beam would have minimum waist. A one-dimensional Gaussian beam symmetric along the x-direction of waist W0 is incident on a thin slice of dielectric material of thickness
z with a graded refractive index n…x† ˆ n0 ‡ n1 cos Kx; W0  2=K. Calculate the far-®eld diffraction pattern. Use the split step beam propagation technique to analyze propagation along z of a one-dimensional Gaussian beam of W0 ˆ 100 ( is the free-space wavelength) incident onto a lamellar grating bounded by a material with refractive index n0 . The lamellar grating has a thickness of 100 with a refractive index pro®le n…x† ˆ n0 ‡ n1 sgn(cosKx), K ˆ 2=; ˆ 5, where sgn(y† ˆ ‡1 if y > 0 and 1 if y < 0. Take n0 ˆ 1:5, n1 ˆ 0:15. Calculate the pro®le at the exit plane of the grating and in the far ®eld. Repeat the problem for the case where the thickness of the lamellar grating is 1000 and characterize the differences between the two cases. Use the split step method to analyze the propagation of a Gaussian beam of waist W0 ˆ 100 through a material of thickness 100 having a refractive index pro®le n…x† ˆ n0 ‡ …x=W0 †, jxj < 5W0 . Let n0 ˆ 1:5,  ˆ 0:015. Assume that the material is bounded by a material of refractive index n0 . Determine the far®eld intensity pro®le. A Gaussian beam of waist W0 ˆ 100 symmetric about x ˆ 0 is incident from air onto a nonlinear material slab of thickness 100 and of refractive index n…x† ˆ 1 ‡ n2 I…x† where I…x† is the intensity

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of the Gaussian beam. Assume that a knife edge is present at z ˆ 0, x < 0. Use the split step method to determine the far ®eld pro®le. At z ˆ 0, take n2 I…0† ˆ 10 4 . REFERENCES 1. P. P. Banerjee and T. C. Poon, Principles of Applied Optics, CRC Press, 1991. 2. G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, London, 1989. 3. R. H. Hardin and F. D. Tappert, Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coef®cient wave equations, SIAM, 15, 423 (1973). 4. A. Korpel, Acousto-optics, 2d ed., Marcel Dekker, Amsterdam, 1997. 5. C. Venzke, A. Korpel, and D. Mehrl, Appl. Opt., 31, 656 (1992). 6. P. P. Banerjee and C.-W. Tarn, A Fourier transform approach to acoustooptic interactions in the presence of propagational diffraction, Acustica, 74, 181±191 (1991). 7. A. Korpel, K. E. Lonngren, P. P. Banerjee, H. K. Sim, and M. R. Chatterjee, J. Opt. Soc. Am., 3, 885 (1986). 8. H. K. Sim, A. Korpel, K. E. Lonngren, and P. P. Banerjee, Simulation of twodimensional nonlinear envelope pulse dynamics by a two-step spatiotemporal angular spectrum method, J. Opt. Soc. B, 5, 1900±1909 (1988). 9. P. Yeh, Introduction to Photorefractive Nonlinear Optics, John Wiley, New York, 1993. 10. J. Feinberg, Asymmetric self-defocusing of an optical beam from the photorefractive effect, J. Opt. Soc. Am., 72, 46±51 (1982). 11. J. J. Liu, P. P. Banerjee, and Q. W. Song, Role of diffusive, photovoltaic, and thermal effects in beam fanning in LiNbO3 , J. Opt. Soc. Am. B, 11, 1688±1693 (1994). 12. V. V. Voronov, I. R. Dorosh, Yu. S. Kuz'minov, and N. V. Tkachenko, Photoinduced light scattering in cerium-doped barium strontium niobate crystals, Sov. J. Quantum Electron, 10, 1346±1349 (1980). 13. P. P. Banerjee and R. M. Misra, Dependence of photorefractive beam fanning on beam parameters, Opt. Comm., 100, 166±172 (1993). 14. P. P. Banerjee and J.-J. Liu, Perturbational analysis of steady state and transient beam fanning in thin and thick photorefractive media, J. Opt. Soc. Am. B, 10, 1417±1423 (1993). 15. M. Segev, Y. Ophir, and B. Fischer, Nonlinear multi two-wave mixing. The fanning process and its bleaching in photorefractive media, Opt. Comm., 77, 265±274 (1990). 16. G. A. Swartzlander and A. E. Kaplan, J. Opt. Soc. Am. B, 5, 765 (1988). 17. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970. 18. K. Ratnam and P. P. Banerjee, Nonlinear theory of two-beam coupling in a photorefractive material, Opt. Comm., 107, 522±530 (1994).

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19. P. P. Banerjee, R. M. Misra, and M. Maghraoui, Theoretical and experimental studies of propagation of beams through a ®nite sample of a cubically nonlinear material, JOSA B, 8, 1072±1080 (1991). 20. M. Cronin-Golomb and A. Yariv, Optical limiter using photorefractive nonlinearities, J. Appl. Phys., 57, 4906±4910 (1985). 21. S. E. Bialkowski, Application of BaTiO3 beam-fanning optical limiter as an adaptive spatial ®lter for signal enhancement in pulsed infrared laser-excited photothermal spectroscopy, Opt. Lett., 14, 1020±1022 (1989). 22. J. Feinberg, Self-pumped, continuous-wave phase conjugator using internal re¯ection, Opt. Lett., 7, 486±488 (1982). 23. J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3 , Appl. Phys. Lett., 40, 450±452 (1982). 24. M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, High-sensitivity, singlebeam n2 measurements, Opt. Lett., 14, 955±957 (1989). 25. Q. W. Song, C.-P. Zhang, and P. J. Talbot, Anisotropic light-induced scattering and ``position dispersion'' in KNbO3 :Fe crystal, Opt. Commun., 98, 269±273 (1993). 26. P. P. Banerjee, A. Danilieko, T. Hudson, and D. McMillen, P-scan analysis of induced inhomogeneous optical nonlinearities, J. Opt. Soc. Am., 15, 2446±2456 (1998). 27. N. Kukhtarev, G. Dovgalenko, G. Duree, G. Salamo, E. Sharp, B. Wechler, and M. Klein, Single beam polarization holographic grating recording, Phys. Rev. Lett., 71, 4330±4332 (1993). 28. T. Hudson and D. McMillen, personal communication.

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5

Rigorous Coupled Wave Analysis of Inhomogeneous Cylindrical and Spherical Systems 5.1 5.1.1

INTRODUCTION Background

In Chapters 2, 3, and 4 the RCWA method and spectral domain techniques were used extensively to treat the solution of Maxwell's equations for planar dielectric systems, which were isotropic, anisotropic, or bianisotropic. Chapter 2 concentrated on the case when the dielectric layers were transversely homogeneous and the source and EM ®elds of the system could be effectively represented as a Fourier or K-space integral (i.e., waveguide slot, dipole antenna, etc.). Chapter 3 concentrated on the case when the dielectric layers were periodic diffraction gratings and the source of the system were Rayleigh plane waves. In Chapter 4 the split-step beam propagation method was used to solve the dif®cult problem that occurs when the dielectric layer is a diffraction grating and the source (i.e., Gaussian beam) is represented by a Fourier or K-space integral. In this chapter we will deal with the problem of using the RCWA or exponential matrix method to solve Maxwell's equations in circular and spherical systems that may be inhomogenous in the radial and angular coordinates and which may be isotropic or anisotropic. In cylindrical or spherical systems the RCWA method is applied by expanding all EM ®eld and source quantities in Floquet harmonics that are periodic in the angles  or . The method is similar to that used when applying the RCWA method to planar diffraction gratings systems where all EM ®eld and source quantities are expanded in Floquet harmonics, which are periodic in the grating periods x or z .

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Two major differences exist between the cylindrical and spherical RCWA formulations presented in this chapter and the RCWA formulation presented in Chapter 3. The ®rst difference is that, in cylindrical or spherical systems, when Maxwell's equations are being reduced to state variable form (a set of ®rst order, partial differential equations), the scale factors associated with Maxwell's equations (for example, 1=r, 1=…r sin…††, etc.) vary with radial coordinate r, whereas in homogeneous planar diffraction grating systems, this variation does not occur. This difference does not cause signi®cant trouble, however, and may be overcome by simply dividing the cylindrical or spherical system into a set of thin layers where the scale factors are nearly homogeneous, and then using a multilayer analysis as was performed in Chapters 3 and 4 to solve the cascaded system. The second major difference is that the ®eld solutions that exist in the uniform space regions that bound the inhomogeneous scatterer have Maxwell equation solutions that consist of Hankel and Bessel functions for cylindrical systems and Tesseral harmonics (that is, half-order Bessel and Legendre polynomial solutions) for spherical systems. When using the RCWA planar diffraaction grating method, the Maxwell equation solutions consist of Rayleigh plane waves. Having to use Hankel, and Bessel and Tesseral harmonic functions in the boundary matching procedure causes the overall solution to be more complicated than in the planar diffraction grating case. A detailed description of the differences of the RCWA cylindrical and planar diffraction grating formulations will be given, along with several numerical examples. An important and well-known problem in electromagnetics is the problem of determining the scattering that occurs when an electromagnetic wave is incident on a circular cylindrical object. This problem has been extensively studied in the cases where (1) the EM incident wave is an oblique or nonoblique plane wave, (2) the incident EM wave has been generated by a line source, (3) the circular cylindrical scattering object is an inhomogeneous dielectric, and (4) the circular cylindrical object is a dielectric-coated metallic object [1±7]. The problem of determining plane wave and line source scattering from eccentric circular dielectric systems (circular dielectric cylinders of varying dielectric value whose axes are not centered on a single line) has also been studied. Recently Kishk et al. [8] have obtained a complete solution to this problem and give a complete literature survey of scattering from eccentric and centered circular cylindrical dielectric systems. A problem concerning circular cylindrical object scattering that has not, to the authors' knowledge, received a great deal of attention is the problem of determining the scattering and radiation that occurs when the circular cylindrical dielectric system contains a region whose permittivity is inhomogeneous and periodic in the phi (') direction. Figure 1 shows two

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287

Figure 1 (a) The geometry of a phi-periodic system when Region 2 is a semicylindrical half shell and a plane wave is incident on the cylindrical system. (b) An eight-section (four grating period) shell system excited by a line source located at  s ˆ s a^ x , 0  s < a. Used with permission of VSP BV, 1997 [15, Fig. 1].

examples of such a system. In this section we treat the cases (1) where the 'inhomogeneity and its excitation are only periodic over the '-period of ' ˆ 2 (for example, plane wave scattering off a '-inhomogeneous dielectric cylinder, see Fig 1a) and (2) where the '-inhomogeneity and its excitation possesses a higher symmetry than the ®rst case (for example, Fig. 1b with a centered line source) in which the '-period can be taken to be ' ˆ 2=p, p  2, where p is an integer. The '-inhomogeneous cylindrical dielectric system that is being studied in this paper can also be viewed as a circular diffraction grating that has been placed in a circular region. The solution of the problem just stated may have applications, for example, to scattering from circular frequency-selective surfaces and scattering from circular surfaces covered with periodically spaced radar absorbing material (RAM). It can also be used as a numerical cross-check of other numerical algorithms that concern scattering from dielectric systems. The solution method to be proposed in this chapter to solve the circular diffraction grating problem will be based on a recently developed algorithm called rigorous coupled wave theory (RCWA), which has been

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Chapter 5

used extensively to determine diffraction from a planar dielectric diffraction grating [9±14]. This algorithm calculates the diffraction from the grating in four basic steps: (1) expands all electric and magnetic ®eld and dielectric permittivity tensor components in a set of Floquet harmonics (this is an exponential Fourier series whose period is the diffraction grating period), (2) solves Maxwell's equations in the nondiffractive regions on the incident and transmit sides of the diffraction grating, (3) solves Maxwell's equations in the diffractive region of space using a state variable approach (the EM ®elds in the diffraction gratings consist of an in®nite number of forward and backward propagating and nonpropagating state variable eigenmodes), and (4) uses the solutions of Steps 2 and 3 to match EM boundary conditions at the front and back boundaries of the diffraction grating. The solution of Maxwell's equations in the incident and transmit sides of the diffraction grating consists of an in®nite number of propagating and eva~ nescent plane waves whose x-propagation factors are e jkxi x~ (x~ is a coordinate along the grating interface) where k~xi ˆ k~x0 iK~ x , where p k~x0 ˆ k0 I sin I , k0 ˆ 2=, where  is the free space wavelength, ~ x,  ~ x is the grating period, 1 is i ˆ 1; . . . ; 1; 0; 1 . . . ; 1, K~ x ˆ 2= the relative permittivity on the incident side, and I is incident angle of the plane wave incident on the diffraction grating. Chapter 3 gives a detailed description of the RCWA method and its application to planar diffraction gratings. The solution method described previously for planar diffraction gratings will be used to determine the EM ®elds of the circular cylindrical phiperiodic problem. In the regions bounding the phi-periodic region (Regions 1 and 3 of Fig. 1), the Maxwell equation solution consist of an in®nite sum of Bessel and Hankel function solutions Ji eji' , Hi…1† eji' , and Hi…2† eji' , where i ˆ 1; . . . ; 1; 0; 1; . . . ; 1. The eji' factor that makes up the cylindrical ~ solutions is analogous to the e j kxi x~ Floquet harmonic x-propagation factor used in the planar diffraction analysis. In the phi-periodic cylindrical region, all electric ®eld, magnetic ®eld, and dielectric permittivity tensor components are expanded in a set of eji' exponential Fourier series harmonics (Floquet harmonics), and Maxwell's equations are then cast in state variable form and then solved numerically. Because of the radial inhomogeneous nature of the state variable equations in cylindrical coordinates, the state variable equations are solved by dividing the phi-periodic cylindrical region into a series of thin layers (thin enough so that the radial coordinate is approximately constant in the layer), solving Maxwell state variable equations in each layer, and matching EM boundary conditions form one layer to the next to obtain an overall solution in the phi-periodic cylindrical region. This ladder approach is identical to the approach used by

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Inhomogeneous Cylindrical and Spherical Systems

289

Moharam and Gaylord [12] to solve planar surface relief diffraction gratings. We next use RCWA to determine the radiation and scattering that arises from inhomogeneous anisotropic cylindrical dielectric and permeable systems that have arbitrary radial and azimuthal spatial variation. The RCWA state matrix equations and the associated boundary matrix equations (derived from a multilayer ladder analysis) are presented and solved for the ®rst time for the cases when a plane wave (TM polarization, electric ®eld parallel to the cylinder axis) or electric line source is incident on a cylinder that possesses an inhomogeneous permittivity pro®le …; '† and possesses inhomogeneous anisotropic permeability pro®les  …; '†, ' …; '†, ' …; '†, and '' …; '†. In this chapter, radiation and scattering from three inhomogeneous examples were studied using the cylindrical RCWA method. Finally, we present a rigorous coupled wave analysis of the electromagnetic radiation that occurs when a centered electric dipole excites power and energy in a general three-dimensional inhomogeneous spherical system. The formulation consists of a multilayer state variable (SV) analysis of Maxwell's equations in spherical coordinates (the SV analysis used transverse-to-r spherical EM ®eld components) as well as a presentation of the EM ®elds that exist in the interior and exterior regions that bound the inhomogeneous spherical system. A detail description of the matrix processing that is involved with ®nding the ®nal EM ®elds of the overall system is given and three numerical examples of the RCWA method are studied

5.2 5.2.1

RIGOROUS COUPLED WAVE ANALYSIS CYLINDRICAL FORMULATION [15] Introduction

This section is concerned with the problem of determining the EM ®elds that arise when a plane wave (see Fig. 1a) and an off-center interior line source (see Fig. 1b) excite EM ®elds in a circular cylindrical dielectric system as shown in Fig. 1. The EM analysis will be carried out by (1) solving Maxwell's equation in the interior and exterior regions of Fig. 1 in terms of cylindrical Bessel functions, (2) solving Maxwell's equation in the dielectric shell region by using a multilayer state variable approach, and (3) matching EM boundary conditions at the interfaces. It is convenient to ~  ˆ k0 , ~ b ˆ k0 b, ~ etc., introduce normalized coordinates. We let a ˆ k0 a, where unnormalized coordinates are in meters and k0 ˆ 2= is the free space wave number (1/meter).

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290

5.2.2

Chapter 5

Basic Equations

It is assumed that all ®elds and the medium are z-independent and that the relative dielectric permittivity in Region 2 is given by 2 …; '† ˆ

1 X iˆ 1

i …†eji'

0  '  2

ab

…5:2:1†

where i …† represents '-exponential Fourier coef®cients in Region 2. The permeability is assumed to be that of free space,  ˆ 0 . To begin the analysis we determine the EM ®elds in the regions interior and exterior to the Region 2 dielectric shell. In the interior region, the EM ®elds of an off-center line source and the general electric scattered ®elds are given by [5,6] Ez…1† ˆ H'…1†

ˆ

1 X iˆ 1

I …2† p ~ c…1† i Ji …X1 † ‡ c0 H0 … 1 j

~s j†

p j 1 @Ez…1† @X1 0

…5:2:2a† …5:2:2b†

p p p where cI0 ˆ !0 I=4, ~s ˆ s a^ x , 0 ˆ 0 =0 , X1 ˆ 1 k0 ~ ˆ 1 , 0    a, and I is the electric current line source. The EM ®elds in the region s <   a can be expressed [5,6] as Ez…1† ˆ ˆ

1 h i X ci…1† Ji …X1 † ‡ cI0 Ji …X1s †Hi…2† …X1 † eji'

iˆ 1 1 X iˆ 1

H'…1† ˆ ˆ

szi…1† …X1 †eji'

…5:2:3a†

1 i 0 j X ph …1† 0 1 ci Ji …X1 † ‡ cI0 Ji …X1s †Hi…2† …X1 † eji'

0

iˆ 1

1 j X

0

iˆ 1

ji' u…1† 'i …X1 †e

…5:2:3b†

p p where J 0 …X† ˆ dJ…X†=dX, etc., X1s ˆ 1 k0 ~ s ˆ 1 s , and 1 is the relative permittivity of Region 1. In the exterior region, the EM ®elds arepa sum of an incident plane wave (electric ®eld given by E I ˆ E0I e j 3  x a^ z , ~ and a general EM scattered wave. The exterior EM ®elds in x ˆ k0 x) Region 3 are given by

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Inhomogeneous Cylindrical and Spherical Systems

Ez…3† ˆ H'…3† ˆ ˆ

1 h 1 i X X ci…3† Hi…2† …X3 † ‡ EiI Ji …X3 † eji' ˆ szi…3† …X3 †eji'

iˆ 1 1 X

j

0

iˆ 1 1 X

j 0 iˆ

1

291

…5:2:4a†

iˆ 1

i ph …3† …2† 0 3 ci Hi …X3 † ‡ EiI Ji0 …X3 † eji' ji' u…3† 'i …X3 †e

…5:2:4b†

p p where EiI ˆ E0I j i , X3 ˆ 3 k0 ~ ˆ 3 , and 3 is the relative permittivity in Region 3. In Region 2, the middle cylindrical dielectric region, we divide P the dielectric region into L thin shell layers of thickness d` , b a ˆ L`ˆ1 d` , and solve Maxwell's equations in cylindrical coordinates by a state variable approach in each thin layer. The layers are assumed to be thin enough so that the -dependence of 2 …; '† and the  scale factors can be treated as a ~ we ®nd that Maxwell's equations in constant in each layer. Letting  ˆ k0 , a cylindrical shell of radius  are given by @Ez ˆ @

j0 H'

…5:2:5a†

@‰0 H' Š 1 @2 Ez ˆ j2 Ez ‡ j @'2 @

…5:2:5b†

To solve Eq. 5.2.5, we expand Ez , 2 …; '†, and 0 H' in the Floquet harmonics: Ez ˆ 0 H' ˆ 2 …; '†Ez ˆ

1 X iˆ 1 1 X iˆ 1 1 X

szi …†eji'

…5:2:6a†

u'i …†eji'

…5:2:6b†

"

1 X 0

iˆ 1 i ˆ 1

# i

i 0 szi 0

eji'

…5:2:6c†

These expansions are substituted into Eqs. 5.2.5a and 5.2.5b and we let sz …† ˆ ‰szi …†Š and uu …† ˆ ‰u'i …†Š be column matrices and  ˆ ‰i i 0 Š, K ˆ ‰Ki;j 0 Š, K ˆ 2= 0 (0 is the circular grating period and i;i 0 is the Kronecker delta) be square matrices. We ®nd then after a small amount of manipulation that

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@V ˆ A V; @ " Aˆj

" Vˆ

sz

#

uu

0

…I†=

…K K†= ‡  

#

0

…5:2:7†

In the `th cylindrical shell it is convenient to introduce the local coordinates s1 ˆ  b for b d1    b, s2 ˆ  …b d1 † for b d1 d2    b d1 ; . . . ; sL ˆ  ‰b d1 d2


dL 1 Š for b d1 d2


dL    b d1 d2


dL 1 . The state variable equation Eq. 5.2.7 in each cylindrical shell can be expressed in the local coordinates. Further, if the thickness of each cylindrical shell is chosen to be suf®ciently thin so that the  variation in A is negligible, the A…† matrix of Eq. 5.2.7 can be approximated by the thin ˆ b d1 =2, mid ˆ b d1 d2 =2; . . . ; shell's midpoint value where mid 1 2 mid L ˆ b d1


dL 1 dL =2: Letting A` ˆ Ajmid , we have the approximate state variable equation ` in each thin shell given by @V` …s` † ˆ A` V` …s` † @s`

` ˆ 1; . . . ; L

…5:2:8†

If Eq. 5.2.8 is truncated at order MT (i ˆ MT ; . . . ; 1; 0; 1; . . . ; MT ), it represents a NT ˆ 2…2MT ‡ 1† state variable equation (with matrix …A` †NT NT ). The solution of this equation is given by V`n …s` † ˆ V`n exp…qne` s` †

…5:2:9†

where qn` and V`n are the nth eigenvalue and eigenvector of the constant matrix A` …A` V`n ˆ qn` V`n †. Using the V`n …s` † eigenvector solution, the general EM ®elds in the `th thin shell region are given by Ez` ˆ 0 H'` ˆ

MT X NT X iˆ MT nˆ1 MT X NT X iˆ MT nˆ1

cn` szin` exp…qn` s` †

…5:2:10a†

cn` u'in` exp…qn` s` †

…5:2:10b†

h i V t`n ˆ stzn` ut'n`

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…5:2:10c†

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293

where t represents matrix transpose. For MT ˆ 1 for example, suppressing the n` subscripts, we have V t ˆ ‰V1 ; V2 ; V3 ; V4 ; V5 ; V6 Š stz ˆ ‰sz; 1 ; sz;0 ; sz;1 Š ˆ ‰V1 ; V2 ; V3 Š ut' ˆ ‰u'; 1 ; u';0 ; u';1 Š ˆ ‰V4 ; V5 ; V6 Š

…5:2:11†

To proceed further it is necessary to match EM boundary conditions at all interfaces to determine all the unknowns of the system. If the series in Regions 1 and 3 are truncated for jij  MT , then there are 2MT ‡ 1 unknowns in Region 1, 2MT ‡ 1 unknowns in Regions 3, and LNT unknowns in Regions ` ˆ 1; . . . ; L. There are L ‡ 1 interfaces, and 2…2MT ‡ 1† equations to be matched at each interface …2MT ‡ 1 equations for each eji' coef®cient of Ez and 2MT ‡ 1 equations for each eji' coef®cient of H' ). Thus for every truncation order MT , there are an equal number of equations and unknowns from which the EM solution of the overall system can be obtained. Although a large matrix equation exists from which the overall solution of the problem can be obtained, a more ef®cient solution method is to use a ladder approach [12] (that is, successively relate unknown coef®cients from one layer to the next) to express the cnL coef®cients of the Lth last layer in terms of the cn1 coef®cients of the ®rst layer, and then match boundary conditions at  ˆ a and  ˆ b interfaces to obtain the ®nal unknowns of the system. At the `th and …` ‡ 1†th interface, matching the Ez` and 0 H'` ®elds to the Ez; `‡1 and 0 H' `‡1 ®elds, we have NT X nˆ1 NT X nˆ1

cn` szin` exp… qn` d` † ˆ cn` u'in` exp… qn` d`† ˆ

NT X nˆ1 NT X nˆ1

cn;`‡1 szin;`‡1 cn;`‡1 u'in;`‡1

…5:2:12a† i ˆ MT ; . . . ; Mt ` ˆ 1; . . . ; L 1

…5:2:12b†

Letting Ct` ˆ ‰c1` ; . . . ; cN;` Š, these equations can be written D` C` ˆ E` C`‡1

…5:2:13a†

C`‡1 ˆ E` 1 D` C` ˆ F` C`

…5:2:13b†

or

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where the we have

1 superscript denotes matrix inverse. Substituting successively

CL ˆ FL 1 …FL

2




…F1 C1 †† ˆ M C1

…5:2:14†

At the  ˆ a boundary, if we match the Ez…1† solution with the EzL solution (the ` ˆ L thin layer is assumed adjacent to Region 1, and the ` ˆ 1 thin layer is assumed adjacent to Region 3) and solve for the Region 1 ci…1† coef®cient, we ®nd c…1† i ˆ

cI0 Ji …X1s †Hi…2† …X1a † ‡

NT X nˆ1

cnL szinL exp… qnL dL † =Ji …X1a †

…5:2:15†

p where X1a ˆ 1 a: If the 0 aH'…1† solution is matched with the 0 aH'L solution the ci…1† coef®cient is substituted, and the well-known Wronskian equation for Bessel functions is used, it is found that    Ji …X1a † p Ji0 …X1a † s ja 1 ˆ cnL exp… qnL dL † ‡ u'inL 2 Ji …X1a † zinL nˆ1 NT X

cI0 Ji …X1s †

iˆ

MT ; . . . ; 0; . . . ; MT

…5:2:16†

At the  ˆ b boundary, after matching the tangential electric ®eld Ez…3† from Region (3) to the electric ®eld Ez1 from the ` ˆ 1 layer and solving for the Region 3 c…3† i coef®cient, we have " c…3† i

EiI Ji …X3b †

ˆ

iˆ

‡

NT X nˆ1

# cn1 szin1 =Hi…2† …X3b †

…5:2:17†

MT ; . . . ; 0; . . . ; MT

p where X3b ˆ 3 b. If the 0 bH'…3† is matched with the 0 bH'1 ®eld solution of the ®rst layer, the ci…3† coef®cient is substituted, and again a Wronskian Bessel function relation is used, it is found that EiI

ˆ

NT X nˆ1

iˆ

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" cn1

) #( …2† 0 Hi…2† …X3b † p Hi …X3b † szin1 ‡ u'in1 jb 3 …2† 2 Hi …X3b †

MT ; . . . ; 0; . . . ; MT

…5:2:18†

Inhomogeneous Cylindrical and Spherical Systems

295

Equation 5.2.16 represents a set of 2MT ‡ 1 equations, Eq. 5.2.18 represents a set of 2MT ‡ 1 equations, and the matrix equation Eq. 5.2.14 represents a set of NT ˆ 2…2MT ‡ 1† equations. Thus Eqs. 5.2.14, 5.2.16, and 5.2.18 represent a set of 2NT ˆ 4…2MT ‡ 1† equations to calculate the 2NT set of unknowns represented by C1 and CL . Once these quantities are known all other unknown coef®cients in the system can be found. An important quantity to calculate is the normalized power of each order. We consider the important cases when either the power is radiated from the line source in Region 1 …cI0 6ˆ 0, E0I ˆ 0† or the power is scattered by a plane wave from Regions 3 …cI0 ˆ 0, E0I 6ˆ 0†. In the case when cI0 6ˆ 0 and E0I ˆ 0, it is useful to calculate the normalized power of each order radiated at three different places, namely at  ˆ a,  ˆ b, and  ˆ 1. These are useful points because, according to the laws of conservation of power, the sum of the powers over all orders at each interface should be equal (or conserved) if there are no losses in the system. Thus a check of the power conservation is a check of the numerical consistency of the solution. It is useful secondly because it gives information on how the different power levels of the orders change as the EM waves propagate through the system. After substitution of the electric and magnetic ®elds into the Poynting real power formula, and carrying out the '-integrals, the normalized power in each order …PNi ˆ PRAD =PINC , i is the order) is given at  ˆ a by i PNi …a† ˆ

p o a 1 n …1† …1† Re s …X †u …X † 1a 1a 'i zi 2jcI0 j2

…5:2:19†

…1† where s…1† zi …X1a † and u'i …X1a ) are de®ned in Eq. 5.2.3 and the  represents the complex conjugate. At  ˆ b the normalized power radiated by the ith order is given by

PNi …b† ˆ

p o b 3 n …3† 2 …2† …2† 0  Re jjc j H …X †H …X † 3b 3b i i i 2jcI0 j2

…5:2:20†

At  ˆ 1 the normalized power radiated by the ith order is given by PNi …1† ˆ

2 jc…3† i j jcI0 j2

…5:2:21†

For the plane wave scattering case …cI0 ˆ 0, E0I 6ˆ 0), it is useful to calculate the normalized scattered power ({(power/meter)/wavelength)}/ {Poynting power intensity (watts/m2 )}) in each order. The normalized scattered power at  ˆ b is given by

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296

Chapter 5

PScat Ni …b†

PScat …b†= p ˆ i ˆ b 3 Re SINC

(

) 2 0 jc…3† j …2† …2†  j i I 2 Hi …X3b †Hi …X3b † jE0 j

…5:2:22†

The normalized scattered power at  ˆ 1 is given by PScat Ni …1† ˆ

2 PScat …1†= 2 jc…3† i i j ˆ SINC  jE0I j2

…5:2:23†

In Eqs. 5.2.22 and 5.2.23, SINC is the power per unit area (watts/meter2 ) of is the scattered power per unit length the incident plane wave, and PScat i (watts/m) of the ith order. 5.2.3

NUMERICAL RESULTS [15]

In this section two numerical examples of scattering from phi-periodic cylindrical systems are presented. In the ®rst example we study the scattering that occurs when a plane wave is incident on the phi-periodic dielectric system shown in Fig. 1a. In this cylindrical system the inner cylinder has a relative dielectric value of 1 ˆ 1:5, Region 2 consists of two semicircular dielectric regions where 20 ˆ 2:8 for the right half dielectric region … 90    90 † and 200 ˆ 2:3 for the left half dielectric region, and Region 3 consists of free space 3 ˆ 1. The phi-Fourier coef®cients (square wave Fourier coef®cients) are given by 0 ˆ …20

200 †

1

i ˆ …20

200 †

1

2

2

‡ 200 sin i i

jij  1

…5:2:24†

where 0 ˆ 2 and 1 ˆ . Figure 2 shows a comparison of the '-periodic semicylindrical shell ~ normalized plane wave scattered power in each order (calculated at ~ ˆ b, according to Eq. 5.2.22, using MT ˆ 20 and L ˆ 30 layers) with that of a uniform dielectric shell as a function of order i when b ˆ k0 b~ ˆ 10. In Region 2 the uniform shell dielectric value was taken to be 2 ˆ 2:5 (this value is the average or bulk dielectric value used for the semicylindrical shell). The scattering from the uniform dielectric shell (dashed curves in Fig. 2) was calculated both by the current state variable algorithm (using MT ˆ 20 and L ˆ 30 layers) and by solving Maxwell's equations in Regions 1, 2, and 3 in terms of Bessel and Hankel functions and then

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297

Figure 2 A comparison of the '-periodic semicylindrical shell plane wave scattered ~ normalized according to Eq. power in each order (see Fig. 1a) (calculated at ~ ˆ b, 5.2.22) with that of a uniform dielectric shell as a function of order i when b ˆ k0 b~ ˆ 10. In Region 2 the uniform shell dielectric value was taken to be 2 ˆ 2:5. Used with permission of VSP BV, 1997 [15, Fig. 2].

matching electromagnetic boundary conditions at the interfaces. The numerical results obtained by both methods were so close that the two dashed curves shown in Fig. 2 (both labelled uniform) cannot be distinguished. As can be seen from Fig. 2, the half cylinder shell causes an increased oscillation of the order power over that of the uniform shell. We note that the order power is symmetric for positive and negative values of the order as is expected. We note that the order m ˆ 5 produces the largest change in diffracted power from the uniform shell case for the value of b ˆ k0 b~ ˆ 10 which was used. Figure 3 shows the total plane wave scattered power from the semicylindrical half shell described at the beginning of this section (see Fig. 1a) ~ solid line curve, normalized, see Eq. 5.2.22) when a ˆ (calculated at ~ ˆ b, k0 a~ ˆ 5 and where b~ ranges from b~ ˆ 0:8 to b~ ˆ 1:6. The values of MT ˆ 10 and L ˆ 30 layers (for each Region 2 shell of inner radius a and outer radius b) were used to make this plot. For comparison, Fig. 3 also shows the total plane wave scattered power (dashed curve) that results when a plane wave is incident on the same dielectric system that has already been described, except that Region 2 is taken to be a uniform dielectric shell whose dielectric value is 2 ˆ 2:5. The scattering from the uniform shell (dashed curve, Fig. 3) was calculated both by the current algorithm and by solving Maxwell's equations in Regions 1, 2, and 3 in terms of Bessel

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Chapter 5

Figure 3 The total plane wave scattered power from the semicylindrical half shell ~ solid line curve, normalized, see Eq. 5.2.23, labeled (see Fig. 1a) (calculated at ~ ˆ b, periodic) when a ˆ k0 a~ ˆ 5 and where b~ ranges from b~ ˆ 0:8, to b~ ˆ 1:6. The value of MT ˆ 10, was used to make this plot. For comparison Fig. 1 also shows the total plane wave scattered power (dashed curve, labeled Uniform) that results when a plane wave is incident on the same dielectric system that has already been described except that Region 2 is taken to be a uniform dielectric shell whose dielectric value is 2 ˆ 2:5. Used with permission of VSP BV, [15, Fig. 3].

and Hankel functions and then matching electromagnetic boundary conditions at the interfaces. Nearly identical numerical results were obtained by the two methods. The dashed curve shown in Fig. 3 was calculated by using Bessel and Hankel functions in all three regions and then matching boundary conditions at the interfaces. As can be seen from Fig. 3, the presence of the semicylindrical shell results in a small but perceptible difference in the power scattered when compared to the scattered power from the uniform shell system. Figure 4 shows a comparison of the uniform and phi-periodic half shell dielectric systems (same case as described in Figs. 2 and 3, normalized by Eq. 5.2.22) for the orders of i ˆ 0 and i ˆ 7 when a ˆ k0 a~ ˆ 5 and b ˆ k0 b~ is varied from values of b ˆ 5 to b ˆ 10 (MT ˆ 20 and L ˆ 30 layers). As can be seen from Fig. 4 for the i ˆ 0 order, the uniform and half shell cases show only a small difference in the scattered power, whereas for the i ˆ 7 order a large difference in the scattered order power occurs. Figure 5 shows the radiated power (normalized) that results when a line source located at the origin radiates into the quarter phi-periodic shell shown in Fig. 1b. The values of the relative dielectric in the Region 2 quarter

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299

Figure 4 A comparison of the uniform and phi-periodic half shell dielectric systems (same case as described in Figs. 2 and 3) for the i ˆ 0 and i ˆ 7 orders when a ˆ k0 a~ ˆ 5 and b ˆ k0 b~ is varied from values of b ˆ 5 to b ˆ 10. Used with permission of VSP BV, 1997 [15, Fig. 4].

shell centered at ' ˆ 0 …j'j  45 † are 20 ˆ 3:25 for 22:5    22:5 and 200 ˆ 1:75 for 22:5 < j'j  45 . The Region 2 quarter shell regions centered at ' ˆ 90 , 180 , and 270 repeat the ' ˆ 0 centered pattern. The Regions 1 and 3 dielectric values are the same as those used in the ®rst example. In this case because of the centered location of the line source, the grating period of the system can be taken to be 0 ˆ =2. Using this grating period and the Region 2 dielectric values already given, the Fourier coef®cients are given by Eq. 5.2.24 with 1 ˆ 45 . Also shown in Fig. 5, for comparison, is the radiation that results when a line source radiates through a uniform dielectric shell (Regions 1 and 3 have the same values as previous examples) and Region 2 has a bulk dielectric value of 2 ˆ 2:5. In Fig. 5, a ˆ k0 a~ ˆ 5 and the radiated power (normalized in Eq. 5.2.20) is plotted versus the outer radius b ˆ k0 b~ with b ˆ k0 b~ varying from 5 to 10 (MT ˆ 4 and L ˆ 50 layers). The solid lines of Fig. 5 show a comparison of the total power (normalized) radiated by the uniform and quarter shell periodic dielectric systems. For the uniform shell, the m ˆ 0 order also represents the total power radiated by the system, since the line is centered on the cylinder axis. Figure 5 also shows the i ˆ 0 and i ˆ 4 order powers. The i ˆ 4 order is based on a 0 ˆ 2 full circle grating period. It is the i ˆ 1 order if based on a 0 ˆ =2 quarter circle grating period. The uniform shell radiated power was calculated both by the current algorithm and by solving Maxwell's equations in Regions 1, 2, and 3 in terms of Bessel and Hankel functions and matching EM boundary conditions at the interfaces. The methods gave

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300

Chapter 5

Figure 5 The radiated power that results when a line source located at the origin radiates into the quarter phi-periodic shell shown in Fig. 1b. Here a ˆ k0 a~ ˆ 5 and the radiated power is plotted versus the outer radius b ˆ k0 b~ with b ˆ k0 b~ varying from 5 to 10. Used with permission of VSP BV, 1997 [15, Fig. 5].

nearly identical results. The Bessel function matching method was used to make the plot of Fig. 4. As can be seen, the quarter shell dielectric system causes signi®cantly different radiation than did the uniform shell system, although both systems had the same bulk dielectric value in Region 2. The total radiated power shown in Fig. 5 was determined by calculating the ith order power at ~ ˆ b~ (Eq. 5.2.20) and summing these order powers to obtain the total scattered power. The power was calculated at ~ ˆ ~ The ith 1 and found to be almost exactly equal to that found at ~ ˆ b. order power was also calculated at ~ ˆ a~ (Eq. 5.2.19). It was found that the order power of the higher orders at ~ ˆ a~ was almost exactly zero for jij  1 and that the i ˆ 0 power at ~ ˆ a~ almost exactly equaled the total radiated ~ We thus see that conservation of power was power calculated at ~ ˆ b. obeyed to a high degree of accuracy. It is interesting that almost no power was radiated and diffracted into higher orders in the interior region of the cylindrical system.

5.3 5.3.1

ANISOTROPIC CYLINDRICAL SCATTERING Introduction

A problem concerning circular cylindrical object scattering that has been studied is the problem of determining the scattering and radiation that occurs when a circular cylindrical dielectric system contains a region whose permittivity is inhomogeneous and periodic in the phi (') azimuthal

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Inhomogeneous Cylindrical and Spherical Systems

301

direction [16±18]. Elsherbeni and Hamid [16] studied EM transverse magnetic (TM, electric ®eld parallel to the cylinder axis) scattering from the inhomogeneous radial dielectric shell permittivity pro®le …; † ˆ a …0 =†2 …  cos‰2'Š†, where a , 0 , , and  are constants de®ned in [16] and  and ' are cylindrical coordinates. Mathieu functions are used to solve for the EM ®elds in the inhomogeneous shell region. The choice of …; '† used in Refs. 16±18 was necessary in order that the Region 2 solution could be expressed in terms of Mathieu functions. A limitation of the solution in Refs. 16±18 is that their solution does not apply to an arbitrary …; '† pro®le but only to one to which a Mathieu function solution can be found. In the previous section, we generalized the work in Refs. 16±18 and presented an EM cylindrical solution algorithm to analyze radiation and scattering from isotropic dielectric cylindrical systems that have an arbitrary radial and azimuthal …; '† pro®le rather than the …; '† pro®le used in Refs. 16±18. The solution algorithm above and in Ref. 15 was based on a recently developed EM planar diffraction grating algorithm called rigorous coupled wave (RCW) analysis [5,19,20]. The purpose of this section will be to extend the RCWA cylindrical algorithm of Ref. 15 to handle the analysis of anistropic inhomogeneous dielectric and permeable material cylinders. Other research on uniform anistropic cylinder scattering may be found in Refs. 21 and 22. Speci®cally the algorithm of this chapter will study the case when (1) the electric ®eld is polarized parallel to the material cylindrical axis (TM case), (2) the cylindrical scattering object has an arbitrary isotropic inhomogeneous dielectric permittivity pro®le …; '†, and (3) the cylindrical scattering object has arbitrary anisotropic inhomogeneous relative permeability tensor pro®les xx …x; y†, xy …x; y†, yx …x; y†, and yy …x; y† …xz ; zx ; zy , and yz are taken to be zero). Equations 5.3.2 and 5.3.3 of this chapter and Refs. 21 and 22 express the tensor elements in cylindrical components. The analysis of this chapter also applies to the case when (1) the magnetic ®eld is polarized parallel to the cylindrical axis (TE case), (2) the cylindrical scattering object has an arbitrary isotropic inhomogeneous permeable pro®le …; '†, and (3) the cylindrical scattering object has arbitrary anisotropic inhomogeneous relative permittivity tensor pro®les xx …x; y†, xy …x; y†, yx …x; y† and yy …x; y† …xz ; zx ; zy , and yz are taken to be zero). This follows since the TE and TM cases just described are dual to one another. The solution of this problem is of great interest in several areas of EM research. In the area of cylindrical aperture antenna theory, radial and azimuthal dielectric loading in front of a cylindrical aperture antenna can greatly alter, and therefore possibly enhance, the radiation characteristics of cylindrical aperture antennas [16±18]. Other EM applications include (1)

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302

Chapter 5

scattering from circular frequency-selective surfaces, (2) scattering from cylindrical surfaces covered with periodically spaced inhomogeneous anisotropic radar absorbing material (RAM), (3) scattering from irregularly shaped inhomogeneous mounting struts in an anechoic chamber, and (4) use as a cross-check of other numerical algorithms (FD-TD or FE) which concern scattering from inhomogeneous anisotropic systems.

5.3.2

State Variable Analysis [23]

This section is concerned with the problem of determining the EM ®elds that arise when a plane wave and an off-center interior line source excite EM ®elds in a circular cylindrical dielectric anistropic permeability system as shown in Figs. 6±8 by using the RCWA method. The EM analysis will be carried out by (1) solving Maxwell's equation in the interior and exterior regions of Figs. 6±8 in terms of cylindrical Bessel functions, (2) solving Maxwell's equation in the shell region by using a multilayer state variable approach, and (3) matching EM boundary conditions at the interfaces. It is ~ ~ b ˆ k0 b, convenient to introduce normalized coordinates. We let a ˆ k0 a, ~ etc., where unnormalized coordinates are in meters, k0 ˆ 2= is  ˆ k0 , the free space wave number (1/meter), and  is the free space wavelength.

Figure 6 The geometry of a uniform cylindrical shell system when a plane wave is incident on the cylindrical system and when an electric line source excites EM ®elds in the system is shown. The polarization of the electric ®eld of the plane wave is parallel to the cylinder axis. Used with permission of EMW Publishing 1998, [23, Fig. 1].

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Inhomogeneous Cylindrical and Spherical Systems

303

Figure 7 The geometry of an anisotropic permeable cylindrical half shell is shown along with a plane wave (electric ®eld polarized parallel to the cylinder axis) and an electric line source excitation. Used with permission of EMW Publishing 1998, [23, Fig. 1].

It is assumed that all ®elds and the medium are z-independent and that the relative dielectric permittivity in an inhomogeneous region of the material system is given by …; '† ˆ

1 X iˆ 1

i …†eji'

0  '  2

…5:3:1†

 where …† represent '-exponential Fourier coef®cients. The anisotropic permeability tensor is assumed to be given in rectangular and cylindrical coordinates by [21,22]

Figure 8 The geometry of an isotropic dielectric square cylinder embedded in an anisotropic permeable cylindrical half shell is shown along with an electric line source excitation. Used with permission of EMW Publishing 1998, [23, Fig. 3].

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304

Chapter 5

2

xx

6 ˆ6 4 yx 0

xy yy 0

0

3

2

7 0 7 5 zz



6 ˆ6 4 '

'

0

7 0 7 5 zz

''

0

3

0

…5:3:2†

where  ˆ xx cos2 …'† ‡ …xy ‡ yx † sin…'† cos…'† ‡ yy sin2 …'† ' ˆ xy cos2 …'† ‡ … xx ‡ yy † sin…'† cos…'†

yx sin2 …'†

' ˆ yx cos2 …'† ‡ … xx ‡ yy † sin…'† cos…'†

xy sin2 …'†

'' ˆ yy cos2 …'† ‡ … xy

yx † sin…'† cos…'† ‡ xx sin2 …'†

…5:3:3†

The cylindrical permeability tensor components are assumed to be expanded in the exponential Fourier series rs …; '† ˆ

1 X iˆ 1

 rs …†eji'

0  '  2

…r; s† ˆ …; '†

…5:3:4†

where  rs …† represents '-exponential Fourier coef®cients. The EM ®elds interior and exterior (Regions 1 and 3 of Figs. 6 and 7) when a line source (Region 1) and a plane wave (Region 3) excite EM radiation in a cylindrical system are well known to be an in®nite expansion of the Fourier±Bessel functions Hn…2† ejn' ;

Jn ejn' ;

Yn ejn'

In Region 2, the middle cylindrical dielectric region, we divide the dielectric region into L thin shell layers of thickness d` , b a ˆ L`ˆ1 d` (` ˆ 1 is adjacent to  ˆ b and ` ˆ L is adjacent to  ˆ a) and solve Maxwell's equations in cylindrical coordinates by a state variable approach in each thin layer. The layers are assumed to be thin enough so that the  dependence of …; '†,  …; '†, ' …; '†, ' …; '†, and '' …; '† and the  scale factors can be treated as a constant in each layer. Making the substitutions Sz ˆ Ez , U ˆ 0 H , and U' ˆ 0 H' , where Ez , H , and H' represent the electric and magnetic ®elds in the thin shell region and 0 ˆ 377 is the intrinsic impedance of free space, we ®nd that Maxwell's equation in a cylindrical shell of radius  are given by

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Inhomogeneous Cylindrical and Spherical Systems

@Sz ˆ @'

j U

305

j' U'

…5:3:5†

'' @Sz ˆ j' U ‡ j U @  '

…5:3:6†

@U' @

…5:3:7†

@U ˆ jSz @'

To solve Eqs. 5.3.5±5.3.7, we expand Sz …; '†, U …; '†, U' …; '†, …; '†, and rs …; '†, …r; s† ˆ …; '†, in the Floquet harmonics: Sz …; '† ˆ U' …; '† ˆ

1 X iˆ 1

szi …†eji'

1 X iˆ 1

U …; '† ˆ

u'i …†eji'

rs …; '† F…; '† ˆ

1 X

iˆ 1

…; '†Ez ˆ

"

iˆ 1

1 X

#

1 X i 0ˆ

 rs ;i

i 0 fi 0

ui …†eji'

1 X

"

1 X

iˆ 1 i 0 ˆ 1

eji'

# i

i 0 Szi 0

eji'

…r; s† ˆ …; '† …5:3:8†

1

where F…; '† represents either U …; '† or U' …; '† in Eq. 5.5.8. If these expansions are substituted in Eqs. 5.3.5±5.3.7, and after letting sz …† ˆ ‰szi …†Š, u …† ˆ ‰ui …†Š, and u' …† ˆ ‰u'i …†Š be column matrices and …† ˆ ‰i i 0 …†Š, lrs …† ˆ ‰ rsi i 0 …†Š, …r; s† ˆ …; '†, K ˆ ‰Ki;i 0 Š, K ˆ 2= ' (' is the circular grating period and i;i 0 is the Kronecker delta) be square matrices, we ®nd after manipulation that @V ˆAV @



s Vˆ z uu





A11 Aˆ A21

A12 A22

 …5:3:9†

where  j j luq lqq 1 K A12 ˆ luq lqq 1 lq' ‡ l''     1 j ˆj K lqq 1 K ‡  A22 ˆ K lqq 1 lq'  

A11 ˆ A21

In these equations uq was eliminated by ®nding the matrix inverse of lqq , namely lqq 1 , and then carrying out appropriate matrix multiplications. If Eq. 5.5.9 is truncated at order MT (i ˆ MT ; . . . ; 1; 0; 1; . . . ; MT ), Eq. 5.5.9 represents a NT ˆ 2…2MT ‡ 1† state variable equation [with matrix

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(A` †NT NT ]. The solution of this equation is given by Vn …† ˆ Vn exp… qn †, where qn and Vn are the nth eigenvalue and eigenvector of the constant matrix An . The quantities An , Vn , and qn satisfy An Vn ˆ qn Vn . The general EM ®elds in the `th thin shell region are given by Ez ˆ

MT X NT X iˆ MT nˆ1

cn szin exp…qn †

0 H' ˆ

MT X NT X iˆ MT nˆ1

cn u'in exp qn  …5:3:10†

where Vtn ˆ ‰stn ; utn Š and where t represents the matrix response. Although a large matrix equation exists from which the overall solution of the problem can be obtained, a more ef®cient solution method is to use a ladder approach [19] (that is, successively relate unknown coef®cients from one layer to the next) to express the cnL coef®cients of the Lth last layer in terms of the cn1 coef®cients of the ®rst layer, and then match boundary conditions at the  ˆ a and  ˆ b interfaces to obtain the ®nal unknowns of the system. From [19] we obtain the following overall matrix equation CL ˆ FL 1 …FL




…F1 C1 †† ˆ M C1 …5:3:11†     NT X Ji …X1a † p Ji0 …X1a † I s c0 Ji …X1s † ˆ ja 1 cnL exp… qnL dL † ‡ u'inL 2 Ji …X1a † zinL nˆ1

EiI ˆ

NT X nˆ1

" cn1

2

) #( …2† 0 Hi…2† …X3b † p Hi …X3b † szin1 ‡ u'in1 jb 3 …2† 2 Hi …X3b †

…5:3:12† …5:3:13†

p where i ˆ MT ; . . . ; 0; . . . ; MT , J 0 …X† ˆ dJ…X†=dX, etc., X1s ˆ 1 s , p p X1a ˆ 1 a, and X3b ˆ 3 b. Equation 5.3.12 represents a set of 2MT ‡ 1 equations, Eq. 5.3.13 represents a set of 2MT ‡ 1 equations, and the matrix equation Eq. 5.3.11 represents a set of NT ˆ 2…2MT ‡ 1† equation. Thus Eqs. 5.3.11±5.3.13 represent a set of 2NT ˆ 4…2MT ‡ 1† equations to calculate the 2NT set of unknowns represented by C 1 and C L . Once these quantities are known, all other unknown coef®cients in the system can be found. An important quantity to calculate is the normalized power of each order. We consider the important cases when either the power is radiated from the line source in Region 1 …cI0 6ˆ 0, E0I ˆ 0) or the power is scattered by a plane wave from Region 3 (cI0 ˆ 0, E0I 6ˆ 0). In the case when cI0 6ˆ 0 and E0I ˆ 0, the normalized power in each order is given by PNi ˆ PRAD =PINC , i INC RAD where P is the incident power of the line source and Pi is the radiation

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at a radial distance . For the plane wave scattering case (cI0 ˆ 0, E0I 6ˆ 0), it is useful to calculate the normalized scattered at p ˆ 1: PScat Ni …1† ˆ

2 PScat …1†= 2 jc…3† i i j ˆ SINC  jE0I j2

…5:3:14†

In Eq. 5.5.14, SINC is the power per unit area (watts/meter2 ) of the incident is the scattered power per unit length (watts/m) of the plane wave, and PScat i ith order. 5.3.3

Numerical Results [23]

In this section we will study line source radiation and plane wave scattering using the RCWA method for three different material system examples. The ®rst example consists of line source radiation and plane wave scattering from a uniform dielectric shell. In this example all of space was taken to have a permeability  ˆ 1 and the permittivity in Regions 1, 2, 3 was taken to be respectively 1 ˆ 1:5, 2 ˆ 2:5, and 3 ˆ 1. The inner radius was taken to be a ˆ k0 a~ ˆ 5…a~ ˆ 0:795†, and the outer shell radius was taken to range from b ˆ a ˆ 5 to b ˆ 10. Using a centered line source excitation only (see Fig. 6), Fig. 9 shows a comparison of the normalized radiated power (all normalized powers in this section are assumed normalized either to the incident dipole or to the incident plane wave amplitude) as determined by the RCWA method (using L ˆ 10 layers MT ˆ 1) with that determined by a Bessel function matching solution method (based on matching Bessel function solutions in Regions 1, 2, 3) when the outer radius was varied from b ˆ a ˆ 5 to b ˆ 10. As can be seen from Fig. 9, excellent agreement exists between the Bessel function matching algorithm and the RCWA method. Figure 10 shows the total radiated power that results when a centered line source radiates through an anisotropic permeable half shell (see Fig. 7, 1 ˆ 1:5, 3 ˆ 1, 200 ˆ 1:75, 200 ˆ 1:5, 20 ˆ 3:25, xx ˆ 1:5, xy ˆ 0:3, yx ˆ 0:3, yy ˆ 1:7, MT ˆ 10, L ˆ 10 layers) when the inner radius is a ˆ k0 a~ ˆ 5 and when the outer radius is varied from b ˆ a ˆ 5 to b ˆ 10. As can be seen, almost exact conservation of power at the inner and outer radius is observed. At  ˆ a (inner radius) no power was calculated to be diffracted into higher orders. This is why the total power at  ˆ a also equals the i ˆ 0 power at  ˆ a. Also shown in Fig. 10 are the i ˆ 1, i ˆ 0, and i ˆ 1 orders radiated at  ˆ b (outer radius) and the higher orders i ˆ 3; 2; 2; 3. As b~ is increased from b~ ˆ 0:8 to b~ ˆ 1:6 in Fig. 10, one clearly observes that as the outer radius is increased, power is depleted out

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Chapter 5

Figure 9 The normalized radiated power that results when a centered line source excites a uniform dielectric shell (see Fig. 6, 1 ˆ 1:5, 2 ˆ 2:5, 3 ˆ 1,  ˆ 1) is shown when determined by RCWA and when determined by a Bessel function matching solution. Used with permission of EMW Publishing, 1998 [23, Fig. 4].

of the i ˆ 0 order and is diffracted into higher orders. One also observes that unequal order power is radiated into the i ˆ 1 and the i ˆ 1 orders. This is to be expected and is a result of the anisotropy of the permeable half shell.

Figure 10 The total radiated power that results when a centered line source radiates through an anisotropic permeable half shell (see Fig. 7, 1 ˆ 1:5, 3 ˆ 1, 200 ˆ 1:75, 200 ˆ 1:5, 20 ˆ 3:25, xx ˆ 1:5, xy ˆ 0:3, yx ˆ 0:3, yy ˆ 1:7, MT ˆ 10, L ˆ 10 layers) when the inner radius is a ˆ k0 a~ ˆ 5 and when the outer radius is varied from b ˆ a ˆ 5 to b ˆ 10 is shown. Used with permission of EMW Publishing, 1998 [23, Fig. 5].

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Figures 11±13 display scattering results when a plane wave is incident on the cylindrical system. Figure 11 (solid line and squares) shows a comparison of the total plane wave power scattered by the same uniform dielectric shell example as considered in Fig. 10 (see Fig. 6, centered line source not present) as determined by the Bessel function matching solution (squares, MT ˆ 15) and as determined by the RCWA method (solid line, using L ˆ 15 layers, MT ˆ 15). As can be seen from Fig. 11, excellent agreement was obtained between the two methods. As can be seen from Fig. 11, the RCWA method was able to reproduce accurately even the small resonance peaks that arise in the scattering solution. Figure 11 (solid line labeled RCWA [anisotropic, half shell]) shows the total plane wave scattered power ~ that results when a plane wave is (as a function of the outer radius b) incident on an anisotropic permeable cylindrical half shell (see Fig. 7, 1 ˆ 1:5, 3 ˆ 1, 200 ˆ 1:75, 200 ˆ 1:5, 20 ˆ 3:25, xx ˆ 1:5, xy ˆ 0:3, yx ˆ 0:3, yy ˆ 1:7, MT ˆ 15, L ˆ 15 layers). As can be seen from Fig. 11, the presence of the anisotropic half shell causes a signi®cantly different scattering pro®le from that of the isotropic uniform shell cylinder. Figure 12 shows a three-dimensional plot of the plane wave scattered order power versus order i when i is varied from i ˆ 15 to i ˆ 15 and versus the outer radius b~ when b~ is varied from b~ ˆ 0:8 to b~ ˆ 1:6.

Figure 11 A comparison of the total plane wave power scattered by the same uniform dielectric shell example as considered in Fig. 9 (see Fig. 1, centered line source not present) as determined by the Bessel function matching solution (MT ˆ 15) and as determined by the RCWA method (using L ˆ 15 layers, MT ˆ 15) is shown. Plane wave scattering from an anisotropic cylinder is also shown. Used with permission of EMW Publishing, 1998 [23, Fig. 6].

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Chapter 5

Figure 12 A three-dimensional plot of the plane wave scattered order power versus order i when i is varied from i ˆ 15 to i ˆ 15 and versus the outer radius b~ when b~ is varied from b~ ˆ 0:8 to b~ ˆ 1:6 is shown. This is part of the same numerical case as was studied in Fig. 11. Used with permission of EMW Publishing, 1998 [23, Fig. 7].

Figure 13 The scattered order power that occurs when a plane wave impinges on a uniform dielectric shell (see Fig. 6,, 1 ˆ 1:5, 2 ˆ 2:5, 3 ˆ 1,  ˆ 1) rather than an anisotropic half shell is shown. Used with permission of EMW Publishing, 1998 [23, Fig. 8].

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Figure 12 is part of the same numerical case as was studied in Fig. 11. As can be see from Fig. 12, one clearly observes asymmetry of the order power as the size of the outer radius b~ is increased. The sum of the plane wave order power at any given b~ gives the total scattered power, which is displayed in Fig. 11. We again note that this total plane wave scattered power obeys conservation of power as expected. Figure 13 in comparison with Fig. 12 shows the scattered order power that occurs when a plane wave impinges on a uniform dielectric shell (1 ˆ 1:5, 2 ˆ 2:5, 3 ˆ 1,  ˆ 1) rather than an anisotropic half shell. The uniform dielectric shell has dielectric permittivity values roughly the same size as that of the anisotropic half shell. As can be seen from Fig. 13, the three-dimensional shape of the Fig. 13 plot from the uniform shell is symmetric in the order parameter i and in general has quite a different shape from that of the anisotropic half shell in Fig. 13. Figure 14 shows the total radiated power (normalized to the dipole power of the centered line source) when a line source radiates from an isotropic square cylinder embedded in an anisotropic permeable half shell (see Fig. 8, 1 ˆ 1:5, 3 ˆ 1, 200 ˆ 3:5, 200 ˆ 1, 20 ˆ 3:25, xx ˆ 1:5, xy ˆ 0:3, yx ˆ 0:3, yy ˆ 1:7, MT ˆ 20, L ˆ 25 layers). The radiated power was calculated at  ˆ a …a~ ˆ 1† which is a circle inscribed in the square cylinder of Region 1 and was calculated at  ˆ b (b~ ˆ 2:5†, which ~ is the outer radius pof  the anisotropic half cylinder. The outer radius b was varied from b~ ˆ 2a~ ˆ 1:414 to b~ ˆ 2:5. As can be seen from Fig. 14, extremely good power conservation was observed at  ˆ a …a~ ˆ 1) and at ~ Despite the square shape of the cylinder, no power was observed to ~ ˆ b. be diffracted into higher orders at  ˆ a (a~ ˆ 1†. Also show in Fig. 14 are the i ˆ 1, i ˆ 0, and i ˆ 1 orders radiated at  ˆ b (outer radius). As in Fig. 12, one observes that power is depleted from the i ˆ 0 order and radiated into higher orders. Figure 14 shows the increase in the i ˆ 1 and i ˆ 1 orders, for example, that occurs when b~ is increased. One also observes in Fig. 14 that the order power is radiated asymmetrically into the i ˆ 1 and i ˆ 1 orders. As in Fig. 14 this is expected and is due to the anisotropy of the permeable half shell. Figure 15 shows a plot (dotted line) of the relative dielectric permittivity function …; '† when ~ ˆ 1:241 for the square cylinder anisotropic half shell case displayed in Fig. 8. The circular dashed line of Fig. 8 represents the approximate placement of the ~ ˆ 1:241 parameter used to make the Fig. 15 …; '† plots. Also shown in Fig. 15 (solid line) is the Fourier series representation of the …; '† pro®le when ~ ˆ 1:241 and MT ˆ 20 (MT ˆ 20 was used to make the RCWA method of Fig. 14). As can be seen from Fig. 14, enough Fourier terms ( 40 ˆ 2MT  i  2MT ˆ 40) were used to model correctly the inhomogeneous region as de®ned by the square cylinder. (Note: The convolution matrix of Eq. 5.5.8 requires 2MT ˆ

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Chapter 5

Figure 14 The total radiated power (normalized to the dipole power of the centered line source) when a line source radiates from an isotropic square cylinder embedded in an anisotropic permeable half shell (see Fig. 8, 1 ˆ 1:5, 3 ˆ 1, 200 ˆ 3:5, 200 ˆ 1, 20 ˆ 3:25, xx ˆ 1:5, xy ˆ 0:3, yx ˆ 0:3, yy ˆ 1:7, MT ˆ 20, L ˆ 25 layers) is shown. Used with permission of EMW Publishing, 1998 [23, Fig. 9].

Figure 15 A plot (dotted line) of the relative dielectric permittivity function …; '† when ~ ˆ 1:241 for the square cylinder anisotropic half shell case displayed in Fig. 8 is shown. The circular dashed line of Fig. 8 represents the approximate placement of ~ ˆ 1:241 parameter used to make …; ') plots here. Also shown (solid line) is the Fourier series representation of the …; '† pro®le when ~ ˆ 1:241 and MT ˆ 20. Used with permission of EMW Publishing, 1998 [23, Fig. 10].

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40 terms.) Figures 16±18 show the relative permeability tensor pro®les  …; '†, ' …; '†, ' …; '†, and '' …; '† for the same case and parameters as shown in Fig. 15. The numerical example of Fig. 14 was chosen so that ' …; '† ˆ ' …; '†.

5.4 5.4.1

SPHERICAL INHOMOGENEOUS ANALYSIS Introduction

Another important and well-known problem in electromagnetic theory is to determine the scattering that occurs when an electromagnetic wave is incident on a spherical object. These problems have been extensively studied in the cases where (1) the EM incident wave is an oblique or nonoblique plane wave, (2) the incident EM wave has been generated by a line source or dipole source, and (3) the circular or spherical object is a dielectric coated metallic object [1±5]. Ren [24] studied scattering from anisotropic homogeneous spherical systems and also studied Green's functions associated with anisotropic homogeneous spherical systems. Ren gives a complete literature survey of scattering from isotropic and anistropic spherical systems. Concerning the problem of EM scattering from inhomogeneous material spherical systems, the RCWA algorithm can be applied to the analysis

Figure 16 Plots of the relative permeability function  …; '†. Exact (dotted line) and Fourier series representations (solid line) for the same case as described in Fig. 15 are shown. Used with permission of EMW Publishing, 1998 [23, Fig. 11].

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Chapter 5

Figure 17 Plots of the relative permeability function ' …; '† ˆ ' …; '†. Exact (dotted line) and Fourier series representations (solid line) for the same case as described in Fig. 15 are shown. Used with permission of EMW Publishing, 1998 [23, Fig. 12].

Figure 18 Plots of the relative permeability function '' …; '†. Exact (dotted line) and Fourier series representations (solid line) for the same case as described in Fig. 15 are shown. Used with permission of EMW Publishing, 1998 [23, Fig. 13].

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315

of radiation and scattering from a spherical inhomogeneous object. Ref. 25 presents the basic spherical equations necessary to analyze an arbitrary 3D inhomogeneous scatterer by the RCWA method and also presents a simple example of dipole radiation from an inhomogeneous dielectric object that was azimuthally homogeneous (it varied in the  direction but had no dependence in the ' direction). The spherical RCWA method of this section will extend the results of Ref. 25 in the following ways. First the analysis of this section (following Ref. 26) will consider examples in which the inhomogeneous scatterer has an inhomogeneous permittivity and permeability pro®le, which in addition to varying arbitrarily in the radial direction also varies arbitrarily in the  and ' directions. In Ref. 25 the inhomogeneity variation is only in the  direction. This case is numerically much more challenging than that in Ref. 25 because matrix equations for all orders of m and n must be solved rather than a matrix equation for just m ˆ 0 and all n. The second way that the results of Ref. 25 are extended in this section is that the basic spherical state variable equations and the interior exterior Bessel function equation of Ref. 25 will be modi®ed to the general case when the inhomogeneous scatterer and EM source excitation is periodic in the ' coordinate over a region 2=, where  is an integer ( ˆ 1; 2; 3; . . .† rather than being periodic over just 2 as was presented in Ref. 25. This is particularly useful as only a centered line excitation is considered in this chapter. The third way that the results of Ref. 25 are extended is that full radiated power results for radiation from higher order m and n spherical Bessel±Legendre modes are given, whereas in Ref. 25 power results were given only for m ˆ 0 modes. This chapter will be concerned with determining the EM ®elds that result when a centered electric dipole radiates inside a three-dimensionally inhomogeneous material system (see Fig. 19). This problem can be viewed as either a material shielded antenna source problem or as a material microwave cavity problem in which the material cavity is formed from the inhomogeneous dielectric and permeable material that surrounds the electric dipole source.

5.4.2

Rigorous Coupled Wave Analysis Formulation [26]

This section will be concerned with putting Maxwell's equations in spherical coordinates into a form for which the RCWA formulation can be implemented. We consider the spherical system shown in Fig. 19. All coordinates ~ etc. where r ˆ k0 r~, will be assumed normalized as r ˆ k0 r~, a ˆ k0 a, k0 ˆ 2=,  being the free space wavelength in meters. In this ®gure, Region 1 (0  r  a) is assumed to be a uniform material with the relative

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Chapter 5

Figure 19 Geometry of the three-dimensionally inhomogeneous spherical system. Used with permission of IEEE, 1997, [26, Fig. 1].

permittivity 1 and relative permeability 1 . Region 3 (b  r) is assumed to be a uniform material with the relative permittivity 3 and relative permeability 3 . Region 2 with  ˆ cos…† is assumed to have an arbitrary inhomogeneous lossy relative permittivity …r; ; '† and an inhomogeneous lossy relative permeability …r; ; '†. For generality we assume that electromagnetic radiation may impinge on the 3-D object from Region 3 (a plane wave, for example) or from Region 1 (a dipole source, for example). We will now put Maxwell's equations of Region 2 into state variable form. If we substitute …r; ; '† and …r; ; '† into the Maxwell curl equations of Region 2 and expand the two Maxwell curl equations to their r; , and ' ®eld components, we ®nd that the longitudinal radial electric and magnetic ®eld components can be expressed in terms of the transverse , ' electric and magnetic ®eld components as Er ˆ

  @U' 1 @U 1 ‡

@' j…r; ; '†r2 @

  @S' 1 @S 1 ‡ Hr ˆ

@' j…r; ; '†r2 @ …5:4:1†

S' ˆ r sin E' , U ˆ 0 rH , where

ˆ …1 2 †1=2 , pS  ˆ rE ,  U' ˆ 0 r sin H' , and 0 ˆ 0 =0 ˆ 377 . Substituting these equations into the remaining Maxwell curl equations, we ®nd that

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Inhomogeneous Cylindrical and Spherical Systems

   @S

@ 1 @ …r; ; '† ˆj 2 U ‡ j @ …r; ; '† @'

@r r

317



@ 1 @ U' r2 @ …r; ; '† @ …5:4:2†

    @S' 1 @ 1 @ 1 @ 1 @ ˆ j …r; ; '† ‡ 2 U ‡ j 2 U' @r r @' …r; ; '† @' r @' …r; ; '† @ …5:4:3†     @U

@ 1 @ …r; ; '† @ 1 @ ‡ 2 ˆj 2 S ‡ j S' @r

r @ …r; ; '† @' r @ …r; ; '† @ …5:4:4† @U' ˆj @r



 ‡j

…r; ; '†

 1 @ 1 @ S r2 @' …r; ; '† @'

 1 @ 1 @ S' r2 @' …r; ; '† @

…5:4:5†

We will now be concerned with developing a multilayer RCWA method that can be used to solve Eqs. 5.4.2±5.4.5 in Region 2. To proceed, we divide Region 2, a  r  b, into L thin layers of width d` where b a ˆ L`ˆ1 d` . We assume that each layer has been made thin enough so that all inhomogeneous functions in the radial coordinate r on the righthand sides of Eqs. 5.4.2±5.4.5 can be considered constant in the thin shell region and approximated by the midpoint value of r in the thin layer. In each thin spherical shell it is convenient to introduce the local coordinates s1 ˆ r b for b d1  r  b, s2 ˆ r …b d1 † for b d1 d2  r  b d1 , and so on. These local coordinates will be used to express the ®nal state variable equations in each cylindrical shell. In the `th thin shell layer Eqs. 5.4.2±5.4.5 are put into state variable form in the local coordinates s` by expanding all ®eld variables and inhomogeneous factors …rmid ` ; ; '†, …†; , ; '†; . . . ; etc: (these functions are assumed sampled at the `th 1=…rmid ` ) in a two-dimensional exponential Fourier series, colradial midpoint rmid ` lecting terms together that have the same exponential coef®cient factors and forming a set of ®rst-order differential equations for the mode amplitudes …`† …`† …`† …`† Sim , S'im , Uim , and U'im . The mode amplitude expansion for S…`† …s` ; ; '†, …`† for example, is given by S…`† …s` ; ; '† ˆ i;m Sim …s` † exp‰j…i ‡ m'†Š, where ' =2  '  ' =2, 1    1,  ˆ 2=' ˆ 1; 2; 3; . . . may be called the

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Chapter 5

azimuthal grating wave vector and ' the azimuthal grating period. The matrix for a general inhomogeneous factor, say …rmid ` ; ; '†, for example, is …`† ˆ ‰…`† …i;m†;…i 0 ;m 0 † Š ˆ ‰i i 0 ;m m 0 Š, where i i 0 ;m m 0 are the two-dimensional …`† Fourier coef®cients of …rmid ` ; ; '†, and  …i;m†;…i 0 ;m 0 † represents a typical matrix element of the overall matrix …`† (note that …i; m† is an ordered pair representing a single integer in the …` matrix [same for …i 0 ; m 0 †]). The matrices for the differential operators @=@ and @=@' are given by the diagonal matrices D ˆ ‰jii;j 0 m;m 0 Š and D' ˆ ‰jmi;i 0 m;m 0 Š, respectively, where i;i 0 is the Kronecker delta, and the matrices describing the modal ®eld …`† t amplitudes are given by column matrices (for example, S…`†  ˆ ‰S …i;m† Š (t is transpose). Replacing each inhomogeneous factor, derivative operator, and ®eld amplitude by the appropriate matrix, the overall system state variable matrix can be found. The ®rst right-hand term of Eq. 5.4.2, for 2 …`† †…c …D …K1=c …Du U †††† ˆ A…`† example, is given by …j=rmid ` 1;3 Uh , where c …`† mid and K1=c matrices represent the factors …† and 1=……r` ; ; '† …††, respec-

tively. The matrix A…`† 1;3 , which was just formed, represents a square component submatrix of the overall state matrix A…`† . All component submatrices …`† Aa;b , …; † ˆ …1; 4† of the overall state matrix A…`† are de®ned in the same …`† way as was A1;3 . (Since the component submatrices can be de®ned by inspec…`† , …; † ˆ …1; 4† tion of Eqs. 5.8.2±5.8.5, it is not necessary to list the Aa;b matrices speci®cally.) The overall state variable equations, determined from Eqs. 5.8.2±5.8.5 in the `th thin shell layer are given by @V…`† ˆ A…`† V…`† @s`

` ˆ 1; 2; 3; . . . ; L

…5:4:6†

where 2

A…`†

0

6 6 6 0 6 ˆ6 6 …`† 6 A3;1 6 4 …`† A4;1

0

…`† A1;3

0

…`† A2;3

…`† A3;2

0

…`† A4;2

0

…`† A1;4

3

7 …`† 7 A2;4 7 7 7 7 0 7 7 5 0

2 V…`†

S…`†

3

7 6 6 S…`† 7 6 ' 7 7 ˆ6 6 …`† 7 6 U 7 5 4 U…`† '

…5:4:7†

If the overall state variable equation is truncated with jij  IT and jmj  MT , then A…`† is a PT  PT square matrix with PT ˆ 4…2IT ‡ 1†…2MT ‡ 1†. The solution of the overall state variable matrix …`† solution is given by V…`† …s` †p ˆ V…`† p exp…qp…`† s` †, where q…`† p and V p …p ˆ 1;

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. . . ; PT † are the eigenvalues and eigenvectors, respectively, of the state variable matrix A…`† . The overall EM ®eld solution in each thin shell region can be found by adding a linear combination of the PT eigensolutions. For example, if jij  IT and jmj  MT …p ˆ 1; . . . ; PT †, then the S…2;`† …s` ; ; '† …`† ®eld is given by S…2;`† …s` ; ; '† ˆ i;m;p Cp…`† Simp exp‰q…`† p s` ‡ j…i ‡ m'†Š, …`† where Simp is the pth eigenvector component of S…`† p in the overall eigenvector V…`† p and Cp…`† are unknown EM ®eld expansion coef®cients. Although a large matrix equation could be formed from matching EM boundary conditions at r ˆ a, r ˆ b, and at each thin shell layer interface in the inhomogeneous region, a more ef®cient solution method is to use a ladder approach [12] (that is, successively relate unknown coef®cients from one layer to the next) to express the Cp…L† coef®cients of the Lth thin shell layer (located in the layer adjacent to r ˆ a) in terms of the Cp…1† coef®cients (located in the layer adjacent to r ˆ b) and then match boundary conditions at r ˆ a and r ˆ b interfaces to obtain the ®nal unknowns of the system. At the `th and …` ‡ 1†th interface, matching the tangential magnetic and electric ®elds we have PT X pˆ1 PT X pˆ1 PT X pˆ1 PT X pˆ1

…`† Cp…`† Simp exp… qp…`† d` † ˆ

…`† Cp…`† S'imp exp… qp…`† d` † ˆ

…`† Cp…`† Uimp exp… qp…`† d` † ˆ

…`† Cp…`† U'imp exp… qp…`† d` † ˆ

PT X pˆ1 PT X pˆ1

…`‡1† Cp…`‡1† Simp

…5:4:8†

…`‡1† Cp…`‡1† S'imp

…5:4:9†

PT X pˆ1 PT X pˆ1

…`‡1† Cp…`‡1† Uimp

…5:4:10†

…`‡1† Cp…`‡1† U'imp

…5:4:11†

Letting C…`† ˆ ‰C1…`† ; . . . ; CP…`†T Št , these equations can be written …`† D…`† C…`† ˆ D‡ C…`‡1†

…5:4:12†

or h i …`† C…`‡1† ˆ D‡

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1

D…`† C…`† ˆ F…`† C…`†

` ˆ 1; . . . ; L

1

…5:4:13†

320

Chapter 5

where the we have

1 superscript denotes matrix inverse. Substituting successively

C…L† ˆ F…L

1†

…F…L

2†

…


…F…1† C…1† †


† ˆ M C…1†

…5:8:14†

Another important problem is to relate the ®elds of Region 1 (interior region) and Region 3 (exterior region) to the ®elds of Region 2 (inhomogeneous region). The ®elds in Regions 1 and 3, as is well known, can be expressed in terms of an in®nite number of transverse to r electric (TEr ) and transverse to r magnetic (TMr ) Schelkunoff spherical vector potential modes [3, Chap. 6]. These vector potential modes consist of half-order radial Bessel and Hankel functions and consist of Tesseral harmonics (products of Legendre polynomials and ' exponential functions). The scattered ®eld portions of the Regions 1 and 3 Bessel and Hankel function solution are chosen to satisfy the usual spherical boundary conditions of ®niteness at the origin and being an outgoing wave at in®nity. In this chapter the incident ®eld in Region 1 is the EM ®eld of an in®nitesimal dipole. The basic EM boundary matching procedure to be followed in this chapter is to equate the tangential electric ®elds at the interfaces r ˆ a and r ˆ b, eliminate unknown ®eld constants in Regions 1 and 3 in favor of the ®eld constants in Region 2 from these equations, equate the tangential magnetic ®elds at the interfaces r ˆ a and r ˆ b, and substitute the electric ®eld matching Region 2 constants into the magnetic ®eld matching equations. This general procedure is precisely the one followed in Refs. 9±12 in the analysis of diffraction from planar diffraction gratings. Equating the common terms of exp…jm'† of the Sm …† and S'm …† ®eld components at r ˆ a from Regions 1 and 2 we have …1† …1;Scat† …1;INC† …2;a† …† ˆ Sm …† ‡ Sm …† ˆ Sm …† Sm

…5:4:15†

…1† …1;Scat† …1;INC† …2;a† …† ˆ S'm …† ‡ S'm …† ˆ S'm …† S'm

…5:4:16†

where 8 9 IT <2IT ‡jmj‡ m;0 h i= X X …1;Scat† …1† …1† …1† …1† Sm …† ˆ EAimn Fmn ‡ EBimn AA exp…ji† mn ; : iˆ I nˆjmj‡ T

m;0

…5:4:17† 8 9 IT <2IT ‡jmj‡ i= X m;0 h …1† …1† X …1† …1;Scat† …† ˆ ECimn Fmn ‡ EDimn A…1† exp…ji† S'm mn ; : nˆjmj‡ iˆ I T

m;0

…5:4:18†

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Inhomogeneous Cylindrical and Spherical Systems

321

where …1† ˆ EAim …1† ECim ˆ

jm ^ J … a†gA 1 n 1 imn 1 ^ J … a†gC 1 n 1 imn

…1† EBim ˆ …1† EDim ˆ

j ^0 J … a†gB 1 n 1 imn

m ^ 0 J … a†gD 1 n 1 imn

Letting gAmn ˆ …1 2 † 1=2 Pnjmj …†, gBmn ˆ …1 2 †1=2 @=@…Pjmj n …†), gCmn ˆ jmj jmj …1 2 †@=@…Pjmj n …†), and gDmn ˆ Pn …†, where Pn …† are associated B C Legendre functions of order n and jmj, the coef®cients gA imn , gimn , gimn , and D gimn represent the exponential Fourier series expansion coef®cients for the terms gAmn …†, gBmn …†, gCmn …†, and gDmn …†, respectively, on the interval 1    1 …gAmn …† ˆ i gA imn exp…ji†). In the present analysis the Fourier coefB C ®cients gA , g , g , and gD imn imn imn imn have been determined exactly by calculating higher order derivatives of the Bessel function integral representation given in Ref. 27, Eq. 9.1.20, p. 360. The exact calculation of these Fourier coef®cients is an important step in order to ensure overall accuracy of the entire …1;INC† …1;INC† …1;INC† …† ˆ i Sim exp…ji† and S'm RCWA algorithm. The terms Sm …1;INC† …† ˆ i S'im exp…ji† represent electric ®eld EM incident waves evaluated at r ˆ a that emanate from Region 1. In this chapter, it is assumed that a centered electric dipole excites EM radiation in the overall system, and for …1;INC† this source it is found that Sm …† ˆ …j= 1 †AI0;1 H^ 10 … 1 a†‰1 2 Š1=2 m;0 , …1;INC† I …† ˆ 0, and A0;1 is the strength of the electric dipole source. In S'm p Eqs. 5.4.15±5.4.18, 1 ˆ 1 1 and J^n … 1 a† are spherical Schelkunoff Bessel functions [5]; the prime in Eqs. 5.4.15±5.4.18 represent differentiation w.r.t. the argument. For m 6ˆ 0, the lower n limits start at jmj, since the …2;a† …2;a† Legendre polynomials are zero when jmj > n. The terms Sm …† and S'm …† represent the state variable solution in Region 2 in the Lth thin shell layer region evaluated at r ˆ a and are given by …2;a† …† Sm

ˆ

( IT PT X X iˆ IT

ˆ

IT X iˆ IT

…2;a† S'm …†

ˆ

IT X iˆ IT

Copyright © 2000 Marcel Dekker, Inc.

exp…

q…L† p sL †

exp…ji†

…2;a† Sim exp…ji†

( IT PT X X

iˆ It

ˆ

pˆ1

) …L† Cp…L† Simp

pˆ1

…5:4:19† )

…L† Cp…L† S'imp

exp…

…2;a† S'im exp…ji†

q…L† p sL †

exp…ji† …5:4:20†

322

Chapter 5

When IT is in®nite, the boundary matching equations given by Eqs. 5.4.15± 5.4.20 are exact. When IT is truncated at a ®nite value, and common coef®cients of exp…ji† in Eq. 5.4.15±5.4.20 are collected, Eqs. 5.4.15±5.4.20 give a …1† ‰2…2IT ‡ 1†Š  ‰2…2IT ‡ 1†Š set of equations from which the A…1† mn and Fmn can …1† …L† , be expressed in terms of the Region 2 unknown coef®cients Cp . Letting EAm …1† …1† …1† EBm , ECm , and EDm be matrices representing Eqs. 5.8.17 and 5.8.18 (for …1† example, E…1† Am ˆ ‰EAimn Š, where i ˆ

IT ; . . . ; IT and n ˆ jmj ‡ m;0 ; . . . ;

…1;INC† …1;INC† …1;INC† …1;INC† 2IT ‡ jmj ‡ m;0 ), letting Sm ˆ ‰Sim Š and S'm ˆ ‰S'im Š for i ˆ …2;a† …L† …L† …2;a† …L† IT ; . . . IT and letting Sm ˆ ‰Simp exp… q…L† p sL †Š, S'm ˆ ‰S'imp exp… qp sL †Š

for i ˆ

IT ; . . . ; IT and p ˆ 1; . . . ; PT we ®nd the matrix equation 2

S…2;a† m

…L†

C

ˆ4

…2;a† Sm

3

2

…1† EAm

5C ˆ 4 …2;a† …1† S'm ECm " …1† # …1† Fm ˆ Em ‡ S…1;INC† m …1† Am …L†

E…1† Bm …1† EDm

3" 5

…1† Fm …1† Am

#

" ‡

S…1;INC† 

#

S…1;INC† ' …5:4:21†

where S…2;a† is a ‰2…2IT ‡ 1†Š  PT matrix, E…1† m m is a ‰2…2IT ‡ 1†  ‰2…2IT ‡ 1†Š is a 2…2IT ‡ 1† column matrix. square matrix, and S…1;INC† m To proceed further we match the terms common to exp…jm'† of the tangential magnetic ®eld at the Region 1, Region 2 interface at r ˆ a and ®nd that …1† …1;Scat† …1;INC† …2;a† …† ˆ Um …† ‡ Um …† ˆ Um …† Um

…5:4:22†

…1† U'm …†

…5:4:23†

ˆ

…1;Scat† U'm …†

‡

…1;INC† U'm …†

ˆ

…2;a† U'm …†

where 8 9 IT <2IT ‡jmj‡ m;0 h i= X X …1;Scat† …1† …1† …1† …1† …† ˆ HBimn Fmn ‡ HAimn Amn Um exp…ji† : ; iˆ I nˆjmj‡ T

m;0

…5:4:24† 8 9 IT <2IT ‡jmj‡ i= X m;0 h …1† …1† X …1† …1;Scat† exp…ji† …† ˆ HDimn Fmn ‡ HCimn A…1† U'm mn : nˆjmj‡ ; iˆ I T

m;0

…5:4:25†

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Inhomogeneous Cylindrical and Spherical Systems

323

where …1† HAim ˆ

jm ^ J … a†gA 1 n 1 imn

…1† HCim ˆ

1 ^ J … a†gC 1 n 1 imn

…1† HBim ˆ …1† HDim ˆ

j ^0 J … a†gB 1 n 1 imn m ^ 0 J … a†gD 1 n 1 imn

…1;INC† …1;INC† …1;INC† …1;INC† The terms Um …† ˆ i Uim exp…ji† and U'm …† ˆ i U'im exp …ji† represent the magnetic ®eld incident wave that may emanate from …1;INC† …† ˆ …1=1 †AI0;1 H^ 1 … 1 a†‰1 2 Šm;0 and Region 1. In this chapter, U'm …1;INC† …2;a† …† ˆ 0 for a centered electric dipole source. The terms Um …† Um …2;a† and U'm …† represent the state variable solution in Region 2 in the Lth thin shell layer region evaluated at r ˆ a and are given by

…2;a† Um …†

ˆ

( IT PT X X iˆ IT

ˆ

IT X iˆ IT

…2;a† U'm …† ˆ

ˆ

pˆ1

iˆ IT

iˆ IT

exp…

q…L† p sL †

exp…ji†

…2;a† Uim exp…ji†

( IT PT X X

IT X

) …L† Cp…L† Uimp

pˆ1

…5:4:26†

) …L† Cp…L† U'imp

exp… qp…L† sL † exp…ji†

…2;a† U'im exp…ji†

…5:4:27†

Equating common coef®cients of exp…ji† in Eqs. 5.4.22±5.4.27, a similar matrix equation as was formed for the tangential electric ®eld components can be formed for the tangential magnetic ®eld components. We have then 2 U…2;a† m

C

…L†

ˆ4

U…2;a† m

3

2

5C U…2;a† 'm " …1† # Fm …1†

ˆ Hm

A…1† m

…L†

ˆ4

…1† HBm

H…1† Am

H…1† Dm

H…1† Cm

…1;INC† ‡ Um

3" 5

F…1† m A…1† m

#

" ‡

U…1;INC†

#

U'…1;INC† …5:4:28†

To proceed further, our objective now is to eliminate the column matrix …1† t ‰F…1† m ; Am Š from Eq. 5.4.21 and therefore form a single matrix equation for

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324

Chapter 5

the C…L† coef®cients alone. By inspecting the matrix equation Eq. 5.4.21 and their de®nitions, we notice that two distinct cases arise, namely the cases when m 6ˆ 0 and the case when m ˆ 0. In the case of m 6ˆ 0, it turns and H…1† out that the matrices E…1† m are nonsingular; therefore it is straightm …1† …1† t forward to invert E…1† m and solve for ‰Fm ; Am Š . For m 6ˆ 0 we ®nd that



F…1† m A…1† m



1

…1† ˆ Em …S…2;a† C…L† †

1

…1;INC† …L† E…1† Sm ˆ Z…1† m m C

E…1;INC† m …5:4:29†

…1† …1† The determination of F0:n and A0;n (n ˆ 1; 2; 3; . . .† coef®cients for the m ˆ 0 case requires special matrix processing. We ®rst note for the m …1† …1† ˆ EDim ˆ 0 in Eqs. 5.4.17 and 5.4.18 and thus the ˆ 0 case that EAim …1† matrix equations for F0;n and A…1† 0;n are decoupled from one another. One …1† also observes from Eqs. 5.4.17 and 5.4.18 that when solving for either F0;n …1† …1† …1† …1† …1† …1† …1† and A0;n , the coef®cients of F0;1 ; F0;3 ; F0;5 ; . . . and A0;1 ; A0;3 ; A0;5 ; . . . are multiplied by the ®rst-derivative Legendre polynomials @=@…P01 …††, …1† …1† ; F0;4 ; @=@…P03 …††; . . ., which are even in , whereas the coef®cients of F0;2 …1† …1† …1† …1† F0;6 ; . . . and A0;2 ; A0;4 ; A0;6 ; . . . are multiplied by ®rst-derivative Legendre polynomials @=@…P02 …†), @=@…P04 …††; . . . which are odd in . This means …1† …1† and A0;n , the that when determining the m ˆ 0, Region 1 coef®cients F0;n …1;INC† …†, best numerical processing in Eq. 5.4.21 is to decompose Sm …2;a† …1;INC† …2;a† …†, Sm …†, S'm …† for m ˆ 0 into a sum of even and odd funcS'm …1† tions, and from the even functions in Eqs. 5.4.17 and 5.4.18 determine F0;1 …1† …1† …1† …1† …1† ; F0;3 ; F0;5 ; . . . and A0;1 ; A0;3 ; A0;5 ; . . ., and from the odd functions in Eqs. …1† …1† …1† …1† …1† 5.4.17 and 5.4.18 determine F0;2 ; F0;4 ; F0;6 ; . . . and A0;2 ; A…1† 0;4 ; A0;6 ; . . .. The …1† …1† …1† ; F0;5 ;... speci®c matrix processing that is carried out for say the F0;1 ; F0;3 …1;INC† …2a† …† and S'm …† for m coef®cients is as follows. After decomposing S'm …1† …1† …1† ; F0;3 ; F0;5 ; . . . is determined by ˆ 0 into even and odd functions of , F0;1 …1;INC† …2;a† …† and S'm …† for m ˆ 0 (1) expanding the even function part of S'm IT IT …2;a† …v† in a fcos…iv†giˆ0 cosine series (the fcos…i†giˆ0 series expansion of S'm …L† for m ˆ 0 depends on the Cp , p ˆ 1; . . . ; PT coef®cients in Region 2, and …1;INC† IT series expansion of S'm …† for m ˆ 0 depends on the the fcos…i†g iˆ0 incident EM source waves that emanate from Region 1); (2) expanding the ®rst-derivative Legendre polynomial @=@…P01 …††, @=@…P03 …††; . . . in a fcos… IT series; (3) equating common coef®cients of the cosine series i†g iˆ0 IT ; and (4) from these equations, developing an …IT ‡ 1†  …IT ‡ fcos…i†g iˆ0 …1† …1† …1† 1† matrix equation that upon matrix inversion expresses the F0;1 ; F0;3 ; F0;5 …L† ; . . . coef®cients in terms of the Cp , p ˆ 1; . . . ; PT coef®cients of Region 2 and incident EM wave coef®cients of Region 1. The determination of the

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Inhomogeneous Cylindrical and Spherical Systems

325

…1† …1† …1† F0;2 ; F0;4 ; F0;6 ; . . . coef®cients is found by (1) expanding the odd function …1;INC† …2;a† …† and S'm …† for m ˆ 0 and the odd derivative Legendre part of S'm 0 IT polynomials @=@…P2 …††; @=@…P04 …†† functions in a fsin…i†g iˆ1 series; (2) IT ; and (3) then forming an IT equating common coef®cients of fsin…i†g iˆ1 …1† …1† IT matrix equation which, upon matrix inversion, expresses the F0;2 ; F0;4 …1† ; F0;6 ; . . . coef®cients in terms of the Cp…L† , p ˆ 1; . . . ; PT coef®cients of Region 2 and incident EM wave coef®cients of Region 1. After following the above procedure, and combining the even and odd matrix expressions …1† …1† …1† …1† …1† …1† ; F0;3 ; F0;5 ; . . . and F0;2 ; F0;4 ; F0;6 ; . . ., a 2IT ‡ 1†x…2IT ‡ 1† matrix for F0;1 …1† …1† …1† …1† ; F0;2 ; F0;2 ; F0;4 ; . . . coef®cients relation is found between the overall F0;1 and the Cp…L† , p ˆ 1; . . . ; PT coef®cients and EM incident wave coef®cients of Region 1. A similar even and odd analysis allows a …2IT ‡ 1†  …2IT ‡ 1† …L† matrix relation between the A…1† 0;n coef®cients and the Cp , p ˆ 1; . . . ; PT coef®cients of Region 2 and EM incident wave coef®cients of Region 1. …1† and A…1† Altogether the F0;n 0;n coef®cients for m ˆ 0 in matrix form can be expressed as

"

F…1† 0 A…1† 0

#

" ˆ

Z0…1;F† 0

0

…1;A†

Z0

# C…L†

…L† E…1;INC† ˆ Z…1† 0 0 C

E…1;INC† 0 …5:4:30†

where the matrix Z…1† 0 is size ‰2…2IT ‡ 1†Š  ‰2…2IT ‡ 1†Š. …1† …1† t …1† t Substituting ‰F…1† m Am Š from Eq. 5.4.29 …m 6ˆ 0) and ‰F0 A0 Š of Eq. 5.4.30 …m ˆ 0† into Eq. 5.4.28, we ®nd that

h 2;a† Um

i …1† …L† …1;INC† H…1† Z ˆ Cm m m C

…1;INC† H…1† m Em

…5:4:31†

where m ˆ MT ; . . . ; 1; 0; 1; . . . ; MT . For each m, Eq. 5.8.31 has 2…2IT ‡ 1† rows. The boundary matching analysis at the r ˆ b interface is identical to that at r ˆ a, and the boundary matching equations at the r ˆ b interface are given by Eqs. 5.4.15±5.4.31 if (1) one replaces Region ``(1)'' superscripts with Region ``(3)'' superscripts; (2) one replaces the spherical Schelkunoff Bessel function J^n … 1 a† with the outgoing spherical Schelkunoff Hankel …2;a† …† function of the second kind, namely, H^ n … 3 b†; (3) one replaces Sm …2;a† …† in Eqs. 5.4.19 and 5.4.20 with the ` ˆ 1 thin shell state variable and S'm solution evaluated at r ˆ b, namely,

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326

Chapter 5 …2;b† Sm …†

ˆ

( IT PT X X iˆ IT

pˆ1

(

…2;b† S'm …† ˆ

IT PT X X iˆ IT

pˆ1

) …1† Cp…1† Simp

exp…ji† ˆ

IT X iˆ IT

…2;b† Sim exp…ji†

…5:4:32†

) …1† Cp…1† S'imp exp…ji† ˆ

IT X iˆ IT

…2;b† S'im exp…ji†

…5:4:33† and (4) one sets all Region 3 incident source terms to zero, since EM energy in this chapter is assumed to emanate only from a Region 1±centered dipole source. After algebraic manipulation it is found that the Region 3 boundary equations are h 2;b† Um

i …3† …1† H…3† ˆ U…3;INC† m Zm C m

…3;INC† H…3† m Em

…5:4:34†

where m ˆ MT . . . ; 1; 0; 1; . . . ; MT and where the right-hand side of Eq. 5.4.34 for the present chapter is zero. For each m, Eq. 5.4.34 has 2…2IT ‡ 1† rows. Using C…L† ˆ M C…1† from Eq. 5.4.14, we eliminate the C…L† column matrix and ®nd that h …2;a† Um

i …1† …1† …1;INC† H…1† ˆ Um m Zm M C

…1;INC† H…1† m Em

…5:4:35†

where m ˆ MT ; . . . ; 1; 0; 1; . . . ; MT . Including all values of m, Eq. 5.4.34 and Eq. 5.4.35 each represent ‰2MT ‡ 1Š  ‰2…wIT ‡ 1†Š ˆ PT =2 equations for C…1† . Thus Eqs. 5.4.34 and 5.4.35 represent together a PT  PT matrix equation from which C…1† can be determined. From knowledge of C…1† , all other unknown constants of the system can be determined. Once all the EM ®eld coef®cients are determined, one can calculate the power radiated in Regions 1 and 3 of the system. The radiated power associated with a given m and n spherical mode is well known, and the speci®c formulas can be found in Ref. 5, Chap. 6. In this chapter we will give numerical results in terms of normalized power. The normalized power of a given m and n spherical mode at radial distance r is de®ned here as the power radiated by the m and n spherical mode into a sphere located at a radius r divided by the total power radiated by the centered dipole when the centered dipole is in an in®nite region whose material parameters are those of Region 1, namely 1 and 1 .

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Inhomogeneous Cylindrical and Spherical Systems

5.4.3

327

Numerical Results [26]

In this section we illustrate the RCWA method of Section 5.4 by solving for the radiated and scattered EM ®elds that result when the Region 2 inhomogeneous material shell is assumed to have, as a speci®c example, the form …r; ; '† ˆ 2 ‡
 sin…† sgn…cos…††‰1 ‡
' …r† sgn…'†Š ˆ 2 ‡
 ‰1 …r; ; '† ˆ 2

2 Š1=2 sgn…†‰1 ‡
' …r† sgn…'†Š …5:4:36†

where sgn…X† ˆ 1 for X > 0 and sgn…X† ˆ 1 for X < 0. For this pro®le  ˆ 1. This inhomogeneity pro®le is convenient because if it is integrated over a spherical surface of radius r, its average or bulk value is always 2 , regardless of the value of
 or
' used. Using this dielectric inhomogeneity pro®le, three cases will be studied, namely, Case 1,
 ˆ 0:001,
' …r† ˆ 0:001; Case 2,
 ˆ 2:8,
' …r† ˆ 0:4; Case 3,
 ˆ 2:8,
' …r† ˆ 1 r ‡ 2 , where
' …r†jrˆ5 ˆ 0:6 and
' …4†jrˆ5:5 ˆ 1:5 for a  r  b, a ˆ 5, b ˆ 5:5. For all numerical examples of this chapter, the bulk material parameters will be taken to be 1 ˆ 1:5, 1 ˆ 1, 2 ˆ 7, 2 ˆ 1:3, 3 ˆ 1, and 3 ˆ 1. The ®rst case, which because of the small values of
 and
' can be called a homogeneous pro®le case, represents the application of the RCWA method to the solution of the problem of determining the EM radiation that occurs when a centered dipole radiates through a uniform dielectric shell. Since this problem of EM radiation through a homogeneous dielectric shell can be solved exactly by matching Bessel function solutions in Regions 1, 2, and 3, comparison of the RCWA method with the exact Bessel function matching solution represents a numerical validation of the RCWA method if close numerical results from the two methods occur. The second case, which may be designated a …; '†-inhomogeneity pro®le case, represents an inhomogeneous example in which the dielectric shell is homogeneous in the radial r direction but is inhomogeneous in the  and ' coordinates. This case will be solved by both a single layer RCWA algorithm and a multilayer RCWA algorithm. The purpose of solving this second case is to observe in general how much diffraction occurs in higher order spherical modes when a reasonably large  and ' inhomogeneity material pro®le is present in the dielectric shell. The purpose of comparing single-layer and multilayer RCWA results is to observe the importance that the scale factors of Eqs. 5.8.2±5.8.5 have on the overall scattering solution. The purpose is also to study how well the power conservation law is obeyed numerically. Power conservation at different radial distances is a good indi-

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328

Chapter 5

cation of the accuracy of the numerical solution in a lossless system such as the present one. The third case, which may be designated a …r; ; '†-inhomogeneity pro®le case, represents a solution of the RCWA method under the most general conditions, namely when the inhomogeneity variation occurs in the r, , and ' coordinates. The radial inhomogeneity variation has been chosen to vary in such a way that the magnitude of the '-variation in the inhomogeneity pro®le changes linearly. Single and multilayer analyses have been studied in order to gage the effect of the radial inhomogeneity variation of the …r; ; '†-inhomogeneity pro®le case relative to that of the (; '†-inhomogeneity pro®le case. The purpose of studying this case is to see the effect in general that a fully three-dimensional inhomogeneity variation has on diffraction and scattering into higher orders. The purpose also of Case 3, as in Case 2, is to study how well the power conservation law is obeyed numerically. Figure 20 shows a comparison of the normalized power radiated through a uniform material shell when calculated by a Bessel function matching algorithm (exact solution), when calculated by a single layer RCWA method, and when calculated by a multilayer RCWA method. Ten layers (L ˆ 10) were used to make all multilayer calculations in this section. In Fig. 20 the normalized power by all methods has been calculated at both r ˆ a and r ˆ b (the label r ˆ a; b means calculated at r ˆ a and calculated at r ˆ b). In Fig. 20, the outer radius is ®xed at b ˆ 5:5 (radians or rad) and the inner radius a is varied from 5 rad to 5.45 rad. As can be seen

Figure 20 Normalized total power as obtained by the RCWA method compared to the total normalized power as obtained by matching Bessel function solutions at the interfaces r ˆ a and r ˆ b. Used with permission of IEEE, 1997 [26, Fig. 2].

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Inhomogeneous Cylindrical and Spherical Systems

329

from Fig. 20, there is excellent numerical agreement between the three methods used. We also notice that the r ˆ a; b power results for each of the three methods are so close at r ˆ a and r ˆ b that the two power curves for each method cannot be distinguished from one another. In Fig. 20 the RCWA algorithm was calculated using MT ˆ 1 and IT ˆ 5. Because the inhomogeneity factor in this case was very close to that of a perfectly homogeneous shell, the RCWA algorithm could have calculated the power of this case using a value of MT ˆ 0 and IT ˆ 5, which would have meant a signi®cantly smaller matrix equation than would have resulted from MT ˆ 1. A larger matrix equation than necessary was solved for this case in order to test the numerical stability of the algorithm and also to test the sensitivity of the RCWA solution to error in the Fourier coef®cients. (Error in the Fourier coef®cients would arise for MT ˆ 1 because numerical integration is used to calculate the exp…jm'†, m ˆ 1; 0; 1 Fourier coef®cients, and thus instead of the m ˆ 1 coef®cients being exactly zero, they would have some small value.) The MT ˆ 1 matrix solution showed no ill-conditioned effects from using a matrix size larger than needed and showed no sensitivity to error in the Fourier coef®cients. An RCWA method was also carried out using MT ˆ 1 and IT ˆ 2. In this case the RCWA algorithm differed perceptibly from the Bessel matching solution. This indicates that for accurate results, enough Fourier harmonic terms must be included to calculate the state variable solution of Eqs. 5.4.2±5.4.5. We note ®nally that a case nearly the same for the purpose of validation, was also studied in Ref. 25 using MT ˆ 0 and IT ˆ 5. The numerical results between Fig. 20 and Ref. 25 were almost identical. Figure 21 shows a comparison of the total normalized powers that occur when the dielectric shell is taken to be a uniform layer (the power here is calculated at r ˆ b by Bessel function matching) and when the dielectric shell is taken to be a (; ')-inhomogeneity pro®le with
 ˆ 2:8,
' …r† ˆ 0:4 (Case 2). The power here is calculated at r ˆ a; b by a single-layer analysis and a multilayer analysis. As can be seen from Fig. 21, the presence of the (; ')-inhomogeneity pro®le causes a marked difference in the total scattered power of the inhomogeneous shell, although the bulk dielectric inhomogeneity pro®le was exactly the same as that of the uniform homogeneous shell. It is also noticed from Fig. 21 that for both the single- and multilayer analyses the law of power conservation at r ˆ a and r ˆ b is obeyed to a reasonable degree of accuracy. Also plotted in Fig. 21 is the m ˆ 0, n ˆ 1 power at r ˆ a. It is noticed that the m ˆ 0, n ˆ 1 (; ')-inhomogeneity pro®le power at r ˆ a almost exactly equals that of the total power at r ˆ a; b. This indicates that at r ˆ a no power has been diffracted into higher order modes at the r ˆ a interior boundary shell interface of the system. Figure 21 also shows the m ˆ 0, n ˆ 1 power as calculated at r ˆ b.

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Figure 21 A comparison of the total normalized powers that occur when the dielectric shell is taken to be a uniform layer (the power here is calculated at r ˆ a; b by Bessel function matching) and when the dielectric shell is taken to be a (; ')inhomogeneity pro®le with
 ˆ 2:8,
' …r† ˆ 0:4 (Case 2). The power here is calculated at r ˆ a; b by a single-layer analysis and a multilayer analysis. The m ˆ 0, n ˆ 1 order power is also shown. Used with permission of IEEE, 1997 [26, Fig. 3].

From this plot, one observes that the m ˆ 0, n ˆ 1 power is signi®cantly lower than the r ˆ a; b total power plots. This clearly indicates that as the EM waves have radiated through the dielectric shell, power has been diffracted into the m, n higher order modes of the system. The dielectric shell is acting very much like a planar diffraction grating operating in a transmission mode. Figure 22 shows plots of the n ˆ 2, m ˆ 0 and n ˆ 4, m ˆ 0 mode order power at r ˆ b for the same (; ')-inhomogeneity pro®le as was studied in Fig. 21. The n ˆ 3; 5 …m ˆ 0† orders were very small and not plotted. As can be seen from Fig. 22, as the inhomogeneous shell thickness b a is increased, the diffracted power is transferred from the n ˆ 1, m ˆ 0 lowest order mode (see Fig. 22) to the n ˆ 2; m ˆ 0 mode, to the n ˆ 4, m ˆ 0 mode, and to other higher order modes. One also notices the interesting behavior that at about a ˆ 5:1 rad, b a ˆ 0:4 rad, the n ˆ 2, m ˆ 0 mode power has reached a maximum value and decreases with further increase of the inhomogeneity shell thickness b a. Evidently the n ˆ 2, m ˆ 0 higher order mode is itself transferring energy to other higher order modes. This behavior is very common in planar diffraction gratings [28]. Figure 23 shows plots of the m ˆ 1 total power (formed by summing all m ˆ 1, n ˆ 1; 2; 3; . . . mode powers) as computed by using a multilayer analysis (dotted line-triangle) and using a single-layer analysis (solid line). One notices that the single- and multilayer analyses give approximately the

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Figure 22 Plots of the n ˆ 2, m ˆ 0 and n ˆ 4, m ˆ 0 mode order power for the same (; ')-inhomogeneity pro®le as was studied in Fig. 3 are shown. The n ˆ 3; 5 (m ˆ 0) orders were very small and not plotted. Used with permission of IEEE, 1997 [26, Fig. 4].

same result up to about a shell thickness of b a ˆ 0:4 rad, but after this value the multilayer analysis is needed for more accurate results. Figure 23 also shows the m ˆ 1, n ˆ 1; 2; 3 order power as calculated by a single-layer analysis. One observes that as the shell thickness b a increases, the m ˆ 1, n ˆ 1; 2; 3 order power increases. Figure 24 shows a plot of the total normalized power that results when the (r; ; ')-inhomogeneity pro®le of Case 3 was solved using a multilayer

Figure 23 Plots of the m ˆ 1 total power (formed by summing all m ˆ 1, n ˆ 1; 2; 3; . . . mode powers) as computed by using a multilayer analysis (dotted line-triangle) and using a single-layer analysis (solid line) are shown. Used with permission of IEEE, 1997 [26, Fig. 5].

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Figure 24 Plot of the total normalized power that results when the (r; ; '†-inhomogeneity pro®le of Case 3 was solved using a multilayer RCWA method and using MT ˆ 4 and IT ˆ 5 is shown. Also shown for comparison is the total power of a uniform shell system (Case 1 parameters) and the total power when a (; ')-inhomogeneity pro®le was used with
' …r† set to a constant value of
' …r† ˆ 0:375. Used with permission of IEEE, 1997 [26, Fig. 6].

RCWA method and using MT ˆ 4 and IT ˆ 5. Also shown for comparison is the total power of a uniform shell system (Case 1 parameters) and the total power that results when a (; ')-inhomogeneity pro®le was used with
' …r† set to a constant value of
' …r† ˆ 0:375. This value of
' …r† exactly equaled the average radial value of the
' …r† function of Case 3 over the interval a  r  b, a ˆ 5 rad and b ˆ 5:5 rad. As can be seen from Fig. 24, the
' …r† linear taper causes little difference in total power to be seen between the total power of the (r; ; ')-inhomogeneity pro®le of Case 3 and the total power of the (; ')-inhomogeneity pro®le that used a constant value of
' …r† ˆ 0:375. We notice from Fig. 24 that power conservation was observed to hold to a reasonable degree of accuracy. Figure 24 also shows a plot of the m ˆ 0, n ˆ 1 order power calculated at r ˆ b rad for the two inhomogeneity pro®les for which the total power was just described. As can be seen from Fig. 24, there is a difference in the plots due to the different inhomogeneity pro®les. Figure 25 shows an m ˆ 1 (and m ˆ 3) total order power comparison between the two inhomogeneity pro®les. At a ˆ 5 rad (shell thickness b a ˆ 0:5 rad), the presence of the
' …r† linear taper for the (r; ; ')-inhomogeneity pro®le of Case 3 causes an observable difference with the m ˆ 1 total order power of the (; ')-inhomogeneity pro®le that used a constant value of
' …r† ˆ 0:375. As the same multilayer algorithm was used to calculate the a ˆ 5 rad, m ˆ 1 total power plots, the only difference being that a linear and constant
' …r† function was used, we conclude that completely correct

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333

Figure 25 m ˆ 1 (and m ˆ 3) total order power comparison between the two inhomogeneity pro®les discussed in Fig. 24. Used with permission of IEEE, 1997 [26, Fig. 7].

results can only be achieved in the general (r; ; ')-inhomogeneity case by using a multilayer analysis. It is interesting to compare the unit periodic cell formed by the spherical dielectric shell of the present section with the unit cell of a planar ~ x and crossed dielectric diffraction grating whose grating dimensions are  ~ y . For the spherical system using the inner radius value of r~ ˆ a, ~ the area  p ~x ˆ ~ y ˆ 2 a, ~ the grating of the spherical unit cell is 4a~ 2 . If we chose  cell areas of the planar system and the spherical one are equal. For the present example, the inner surface of the dielectric shell had a ˆ k0 a~ ˆ …2 ~ y ˆ …5=p ~x ˆ † ˆ 2:82. It is inter=†a~ ˆ 5 rad, which thus leads to  esting to note that this grating cell size is typical of many analyses of planar diffraction gratings. For example, in holographic applications, if two interfering plane waves make an angle of 10:21 on opposite sides of a normal to ~x the holographic surface, a one-dimensional diffraction grating of width  ˆ 2:82 is formed. We thus see that diffraction from the spherical shell system studied in this section is on the same scale size as diffraction that occurs in many planar diffraction analyses. It is also interesting to note that the spherical shell scattering analysis studied in this section would, in the area of diffraction grating theory, be classi®ed as a thin grating diffraction analysis. This follows as the spherical shell thickness is less than the grating period and the percent of modulation for the spherical shell
 =2  2:8=7 ˆ 40%, which for planar diffraction grating analysis is large. (Holograms have a depth of modulation on the order of  0:03%.)

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334

Chapter 5

One of the best areas of research concerning the present chapter would be to implement the numerical stability algorithms for RCWA [19,20], which were described in the last section of Chapter 3. This would greatly increase the size of the spherical or cylindrical scatterer which could be analyzed by the RCWA algorithm. Problem 3 below suggests a cylindrical problem for which the RCWA enhanced transmittance matrix method may be implemented and numerically studied. Problem 10 suggests another application of this method which applies to an inhomogeneous spherical scattering system.

PROBLEMS 1. Referring to Fig. 1b assume that an electric line source [strength I ~  ˆ 0; a~ ˆ 1:5; b~ ˆ 2a~ and that (amps)] is located at ~ ˆ a=2;  "2 …† ˆ

"02 ; 0   < 1808 "200 ; 180   < 3608

where "02 ˆ 1:5; "02 ˆ 3:5: (a) Use the RCWA method to ®nd the EM ®elds that exist in all regions of space. (b) Find the far ®eld radiation pattern of the system. 2. Repeat Problem 1 assuming that a plane wave E~ ˆ E0 exp ~ a^z is incident on the cylinder rather than a line source. … jk0 x† 3. Referring to Problem 1: (a) numerically investigate the largest values of a~ that may be used until numerical instability and ill-conditioned results occur; (b) use the RCWA enhanced transmittance matrix method discussed in Chapter 3 (see references [19,20] this chapter) to signi®cantly improve convergence for the case found in (a). 4. Assume that a centered electric line source [strength I (amps)] excites EM ®elds in an anisotropic cylinder shell (homogeneous in the direction parallel to the cylinder axis) where  e…† ˆ  l2 …† ˆ

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"02 ; 0   < 0 ; a    b e2 ; 0   < 3608; a    b 02 ; 0   < 0 ; a    b l002 ; 0   < 3608; a    b

Inhomogeneous Cylindrical and Spherical Systems

335

where e2 00 and l2 00 are arbitrary 3  3 anisotropic material matrices. The shell radii a and b are close together in value and thus form a thin shell cylinder system. Assume isotrophic material occupies the interior ("1 ; 1 ) and exterior regions ("3 ; 3 ) of the shell. (a) Find from Maxwell's equations in Region 2 the 4  4 state variable equations that describe the EM ®elds in the system. The state variable equations should relate the nonradial components of the electric and magnetic ®elds. (b) Find the general EM ®elds of the system in Regions 1 and 3. (c) Match EM boundary conditions at the inner and outer cylindrical shell interfaces and determine the EM ®elds of the overall system. (d) Provide and solve a numerical example of your choice. The example should be suf®ciently complicated to require that all ®eld components are coupled together. (e) For your numerical example, ®nd the far ®eld radiatiotn pattern of the system. (f) Repeat (a±d) assuming that a plane wave is incident on the cylinder rather than an electric line source. (g) Provide and solve a numerical example of the present problem for values of a and b which are not close to each other value. In this case a multilayer analysis is required in order to ®nd a valid solution. 5. Referring to Fig. 1b, assume that a traveling wave-centered, electric line source I…z† ˆ I0 exp… j z† excites EM ®elds in a cylindri~ ˆ k0 ~ ˆ 5. Also, cal shell system. Assume a~ ˆ 1:; b~ ˆ 3a;  "2 …† ˆ

"02 ; 0   < 0 ; a    b "002 ; 0   < 3608; a    b

where 0 ˆ 1208; "02 ˆ 1:5; "002 ˆ 3:5. Assume that free space occupies Regions 1 and 3, which are outside the shell. (a) From Maxwell's equations in Region 2, ®nd the state variable equations that describe the EM ®elds in the system. The state variable equations should relate the nonradial components of the electric and magnetic ®elds. Be sure to include and account for all z derivatives in your solution. (Hint: Reduce z derivatives using @z@ / j :†

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Chapter 5

6.

7.

8.

9.

(b) Find the general EM ®elds of the system in Regions 1 and 3. The ®elds in these regions will be Bessel and Hankel functions q ~ ~ whose arguments have the form 2 k20 z: (c) Match EM boundary conditions at the inner and outer cylindrical shell interfaces and determine the EM ®elds of the overall system. (d) Find the far ®eld radiation pattern of the system. Repeat Problem 5 assuming that a general oblique incident plane wave impinges on the system. Assume that the incident angle of the plane with the cylinder axis is 308. For numerical calculations, use a polarization state of your choice. Referring to Fig. 19, assume that an off-center, vertical, electric dipole excites EM ®elds in the system rather than the centeredelectric dipole shown. (a) Derive the incident ®elds in Region 1 from this source. (b) Using this incident ®eld and the inhomogeneous spherical shell of Sec. 5.4.3, Eq. 4.3.6, Case 2, and the RCWA method of Sec. 5.4, determine the EM ®elds of the system. For numerical example purposes, choose the dipole to have any off-center position in Region 1. (c) Find the far ®eld radiation pattern of the system for your numerical example. Referring to Fig. 19 and the inhomogeneous spherical shell example of Sec. 5.4.3, Eq. 4.3.6, Case 2, assume that a general plane wave travelling in the a^ x direction and polarized with its electric ®eld in the a^ z is incident on the scattering system. (a) Derive the incident ®elds in Region 3 for this plane wave source. (b) Use the spherical RCWA method to determine the EM ®elds of the system in all regions. (c) Determine the bistatic cross section of the system. (See Ref. 6 for a de®nition of the bistatic cross section.) An anisotropic three dimensionally inhomogeneous spherical thin ~ is characterized by a shell, of inner radius a~ and outer radius b, general relative dielectric permittivity tensor e and relative permeability tensor l. (a) Starting from Maxwell's equations in spherical coordinates, and assuming that the shell is thin enough that the scale factors do not vary signi®cantly in the radial direction, 1. expand the EM ®elds and material tensors of the system in Floquet harmonics; and 2. manipulate the resulting equations in such a way that a set

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Inhomogeneous Cylindrical and Spherical Systems

337

of state variable equations for the EM ®eld variables arise. The state variable equations should have the form @V ˆ AV @r  t where V ˆ Sh t Sr t Uh t Ur t and Sh t Sr t Uh t Ur t are column matrices holding Floquet harmonics of the EM ®eld variables. (b) Assuming a centered dipole, solve a numerical example of your choice using the RCWA method. 10. Apply the RCWA enhanced transmittance matrix method [19,20] to Problem 7 for cases where numerical instabilities would otherwise arise.

REFERENCES 1. M. Le Blanc and G. Y. Delisle, Comments on ``Plane wave excitation of an in®nite dielectric rod'', IEEE Microwave Guided Wave Lett., 5, 233±234 (1995). 2. R. B. Keam, Plane wave excitation of an in®nite dielectric rod, IEEE Microwave Guided Wave Lett., 4(10), 326±328 (1994). 3. J. R. Wait, Electromagnetic Radiation from Cylindrical Structures, New York, Pergamon, 1959. 4. H. Y. Yee, Scattering of electromagnetic waves by circular dielectric coated conducting cylinders with arbitrary cross sections, IEEE Trans. Antennas Propagation, AP-13, 822±823 (1965). 5. R. F. Harrington, Time-Harmonic Electromagnetic Fields, New York, McGraw-Hill, 1961. 6. C. A. Balanis, Advanced Engineering Electromagnetics, New York, John Wiley, 1989. 7. P. Bhartia, L. Shafai, and M. Hamid, Scattering by an imperfectly conducting conductor with a radially inhomogeneous dielectric coating, Int. J. Electron., 31, 531±535 (1971). 8. A. A. Kishk, R. P. Parrikar, and A. Z. Elsherbeni, Electromagnetic scattering from an eccentric multilayered circular cylinder, IEEE Trans. Antennas Propagation, 40(3), 295±303 (1992). 9. M. G. Moharam and T. K. Gaylord, Rigorous coupled-wave analysis of planar grating diffraction, J. Opt. Soc. Am., 71, 811±818 (1981). 10. K. Rokushima and J. Yamakita, Analysis of anisotropic dielectric gratings, J. Opt. Soc. Am., 73, 901±908 (1983). 11. M. G. Moharam and T. K. Gaylord, Three-dimensional vector coupled-wave analysis of planar-grating diffraction, J. Opt. Soc. Am., 73, 1105±1112 (1983). 12. M. G. Moharam and T. K. Gaylord, Diffraction analysis of dielectric surfacerelief gratings, J. Opt. Soc. Am., 72, 1385±1392 (1982).

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13. K. Rokushima, J. Yamakita, S. Mori, and K. Tominaga, Uni®ed approach to wave diffraction by space-time periodic anisotropic media, IEEE Trans. Microwave Theory Techniques, 35 937±945 (1987). 14. E. N. Glytsis and T. K. Gaylord, Rigorous three-dimensional coupled-wave diffraction analysis of single cascaded anisotropic gratings, J. Opt. Soc. Am. A., 4 2061±2080 (1987). 15. J. M. Jarem, Rigorous coupled wave theory solution of phi-periodic circular cylindrical dielectric systems, Journal Electromagnetic Waves Appl., 11, 197±213 (1997). 16. A. Z. Elsherbeni and M. Hamid, Scattering by a cylindrical dielectric shell with inhomogeneous permittivity pro®le, Int. J. Electronics, 58(6), 949±962 (1985). 17. A. Z. Elsherbeni and M. Hamid, Scattering by a cylindrical dielectric shell with radial and azimuthal permittivity pro®les. Proceedings of the 1985 Symposium of Microwave Technology in Industrial Development, Brazil, July 22±25, 77±80 (1985), (invited). 18. A. Z. Elsherbeni and M. Tew, Electromagnetic scattering from a circular cylinder of homogeneous dielectric coated by a dielectric shell with a permittivity pro®le in the radial and azimuthal directionsÐeven TM case, IEEE Proceedings 1990, Southeastcon, Session 11A1, pp. 996±1001. 19. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, Formulation for stable and ef®cient implementation of rigorous coupledwave analysis of bindary gratings, J. Opt. Soc. Am. A, 12, 5, 1068±1076 (1995). 20. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord. Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach, J. Opt. Soc. Am. A, 12, 1077± 1086 (1995). 21. X. B. Wu, Scattering from an anisotropic cylindrical dielectric shell, Internat. J. Infrared Millimeter Waves, 15(10), 1733±1743 (1994). 22. X. B. Wu, An alternative solution of the scattering from an anisotropic cylindrical dielectric shell, Internat. J. Infrared Millimeter Waves, 15(10), 1745±1754 (1994). 23. J. M. Jarem, Rigorous coupled wave theory of anisotropic, azimuthally-inhomogeneous cylindrical systems, Progress in Electromagnetics Research, Pier 19, 109±127 (1998). 24. W. Ren, Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media, Physical Review E, 47(1), 664±673 (Jan. 1993). 25. J. M. Jarem, A rigorous coupled wave theory and crossed diffraction grating analysis of radiation and scattering from three-dimensional inhomogeneous objects, letter submitted to IEEE Antennas Propagation, 46(5) (1998). 26. J. M. Jarem, Rigorous coupled-wave-theory analysis of dipole scattering from a three-dimensional, inhomogenous, spherical dielectric and permeable system, IEEE Trans. Microwave Theory Techniques, 45(8), 1193±1203 (1997). 27. M. Abramowitz and I. Stegum, Handbook of Mathematical Functions, Dover, New York, 1972, Eq. 9.1.20, p. 360.

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28. J. M. Jarem and P. Banerjee, An exact, dynamical analysis of the Kukhtarev equations in photorefractive barium titanate using rigorous wave coupled wave diffraction theory, J. Opt. Soc. Am. A, 13(4), 819±831 (1996).

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6

Modal Propagation in an Anisotropic Inhomogeneous Waveguide and Periodic Media 6.1 6.1.1

PROPAGATION IN AN ANISOTROPIC INHOMOGENEOUS WAVEGUIDE Introduction

Previous chapters of this book have concentrated on determining the EM ®elds of an optical system using state variable theory and systems theory in domains that were unbounded or in®nite in the transverse directions; for example, scattering of plane waves or Gaussian beams from an inhomogeneous layer, scattering of plane waves from a diffraction grating, scattering of cylindrical and spherical waves. Another area where state variable theory and systems theory is an effective technique is in the study of EM propagation in an anisotropic inhomogeneous waveguide. State variable theory is effective in the study of these types of problems because all six of the EM ®eld components in an anisotropic waveguide, because of the anisotropy of the materials (relative permittivity  and permeability l), tend to be coupled to one another in a very complicated way. Reduction of Maxwell's equations to a higher order partial differential equation for a single component leads to an almost intractable separation-of-variables problem. On the other hand, state variable theory in some situations can provide a tractable method of dealing with multiple sets of ®eld components. In Section 2 of this chapter the mathematical formulation developed by Gardiol will be used to study diffraction from a one-dimensional grating composed of the cross-tensor materials described in the previous section. This chapter, using the paper by Gardiol [1], applies the state variable method to the study of propagation in a fairly general anisotropic waveguide system.

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6.1.2

Chapter 6

Waveguide Description

Waveguides containing gyrotropic materials, in particular magnetized ferrites, exhibit nonreciprocal properties and permit the realization of isolators, circulators, and phase shifters [2]. Similar properties exist in semiconductors and plasmas [3,4]. The problem of propagation in a rectangular waveguide containing Eplane slabs of gyrotropic materials has been considered by several authors. Most of the publications deal only with particular modes …TEm0 ) [5,6] and therefore do not include limitations to higher order modes or consider only the simple lossless cases [3,4,6]. This section will be concerned with developing a complete model of the mode propagation in a rectangular waveguide containing inhomogeneous anisotropic lossy cross-tensor material media. The general problem considered here is depicted in Fig. 1: it consists of an arbitrary number L of sections that can be ®lled with an anisotropic medium (ferrite, plasma, semiconductor) or isotropic media (air). All these regions extend uniformly across the waveguide in the y-direction. The system is uniform in the y-direction and in the z-direction. In this chapter we will express all coordinates, distances, and wave numbers in ~ y ˆ k0 y, ~ ˆ k0 ~ (propagation constant), dimensionless form using x ˆ k0 x, etc. where k0 ˆ 2= ( (in meters) is the free space wavelength, and x~ (meters), y~ (meters), etc. Each inhomogeneous material slab region interior to the waveguide is assumed to have a width s` and to be homogeneous in the vertical y-direction. The waveguide is assumed to have an overall width ` N `ˆ1 s` ˆ a and a height b. The waveguide walls are taken to be perfect electrical conductors (see Fig. 1).

Figure 1 The geometry of the anisotropic transversely inhomogeneous is shown. The local coordinates x1 ; x2 ; . . . ; x` ; . . . ; xN` that represent the EM ®elds inside each slab are also shown. The waveguide walls at x ˆ 0, a and y ˆ 0, b are assumed to be perfect electrical conductors. Used with permission of IEEE-MTT, 1970 [1, Fig. 1].

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Anisotropic Inhomogeneous Waveguide and Periodic Media

6.1.3

343

Cross-Tensor Media

In a rectangular waveguide coordinate system, Maxwell's equations, in uniform anisotropic material slab regions (see Fig. 1), are separable [7]. For a wave traveling toward values of increasing z, the ®eld components of a waveguide mode have the general form 1 h X ny nyi ‡ cjn …x† cos exp… z† sjn …x† sin b b nˆ0

…6:1:1†

with j ˆ 1 referring to Ex , j ˆ 2 referring to Ey , and j ˆ 3 referring to Ez . Equation 6.1.1 could be substituted in Maxwell's equations and a very complicated set of in®nite series equations could be derived from which a solution could be obtained. No attempt will be made here to study this most general case. However, when certain restrictions are imposed on the properties of the material ®lling the uniform region, an exact derivation of the dispersion relation can be made. We shall limit from now on the treatment to a particular case in which each term of Eq. 6.1.1 corresponds to a mode of the loaded waveguide. The constraints imposed on ` and ` will then be determined. The ®eld components of a waveguide mode then have the form h ny nyi ‡ cjn …x† cos sjn …x† sin exp… z† b b

…6:1:2†

We consider ®rst the case n 6ˆ 0; the case of TEm0 will be considered separately. The boundary conditions at y ˆ 0 and y ˆ b (see Fig. 1) specify the Ex and Ez components: Ex ˆ ‰s1n …x† sin hyŠ exp… z†

c1n …x† ˆ 0

…6:1:3a†

Ez ˆ ‰s3n …x† sin hyŠ exp… z†

c3n …x† ˆ 0

…6:1:3b†

where h ˆ n=b. These components correspond to the sum of two waves traveling with the same velocity in the positive and negative y direction. This can only occur in media presenting symmetry with respect to the y axis, namely media in which [8] ` ˆ a ` a

1

…6:1:4a†

l` ˆ a l` a

1

…6:1:4b†

where a is the tensor expressing the inversion of the y axis.

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Chapter 6

2

1 a ˆ 40 0

0 1 0

3 0 05 1

…6:1:5†

Equations 6.1.4a,b are satis®ed in media having the material properties 2

xx ` ˆ 4 0 zx

0 yy 0

3 xz 0 5 zz

2

xx l` ˆ 4 0 zx

0 yy 0

3 xz 0 5 zx

…6:1:6†

Only in such media, which are called cross-tensor media [1], are the ®eld components obtained by Eq. 6.1.6. The most interesting examples are the following. 1. Ferrite slabs magnetized in the a^ y direction (the case of intrinsic anisotropy in which xx 6ˆ zz is included). In this case we have a tensor l` and scalar ` . 2. Plasma or semiconductor (tensor i ; scalar i ) magnetized along the a^ y direction (tensor ` and scalar ` ). In this case we also have xx 6ˆ zz . 3. Uniaxial crystals (tetragonal, hexagonal, trigonal) and certain types of biaxial crystals (orthorhombic) having one of their principal dielectric axes along a^ y . 4. ``Simple anisotropic'' and isotropic materials: these are degenerate cases of cross-tensor media (diagonal tensors). It should be noted that longitudinally magnetized ferrites, plasmas, and semiconductors do not meet these conditions and are not covered by the present treatment. We mention at this point that the media studied in Chapter 2, Section 2.4 and Chapter 3, Sections 3.2 and 3.3 are classi®ed as cross-tensor media as is shown in Section 6.2.4 of this chapter.

6.1.4

Formulation

To begin the analysis we write Maxwell's equation in the `th slab (` ˆ 1; . . . ; L; ` index suppressed in the following equations) in the symmetric form [9] ~ ˆ0 r  E~ ‡ …H†

…6:1:7a†

E~

…6:1:7b†

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~ ˆ0 r  …H†

Anisotropic Inhomogeneous Waveguide and Periodic Media

345

p where  ˆ j 0 =0 ˆ j0 and  and l are dimensionless relative permeability and permittivity tensors whose nonzero values have been listed earlier. To ®nd a solution to Eqs. 6.1.7a,b we express the electric ®eld E and magnetic ®eld H as h i E~ ˆ E^ x …x† sin hya^ x ‡ E^ y …x† cos hya^ y ‡ E^ z …x† sin hya^ z e z …6:1:8a† h i H~ ˆ …H~ x …x†† cos hya^ x ‡ …H~ y …x†† sin hya^ y ‡ …H~ z …x†† cos hya^ z e z …6:1:8b† where h ˆ n=b. The quantities E^ r …x† and H^ r …x†, r ˆ 1; 2; 3, may be called modal amplitudes of the EM ®eld components and as yet are undetermined. As we mentioned earlier, the electric and magnetic ®elds in the above expression are all proportional to a single model function, either cos hy or sin hy. ~ ®elds are assumed to be propagating in the z-direction with The E~ and …H† an exponential propagation factor (determined from separation of variables) of e z . We also note that the modal functions cos hy and sin hy are all chosen to meet the proper boundary conditions for E~ and H~ on the y ˆ 0 and y ˆ b walls. Note that the choice of sin hy causes the Ex , Ez , and Hy ®elds to vanish at y ˆ 0; b, and the choice of cos hy allows Ex , Ez , and Hy to be nonzero at y ˆ 0; b. These are the proper boundary conditions for E~ and H~ on the bottom and top walls. To proceed further we substitute E and H of Eqs. 6.1.8a,b into Maxwell's eqs. 6.1.7a,b and ®nd that r  E ˆ a^ x ‰hE^ z cos hye

z

… †E^ y cos hye

‡ a^ y ‰ E^ x sin hye

z

rx E^ z sin hye

‡ a^ z ‰rx E^ y cos hye

z

hE^ x cos hye

z

Š

z

z

Š

Š

…6:1:9a†

^ ˆ a^ x ‰ h…H^ z † sin hye r  …H†

z

… †…H^ y † sin hye

z

‡ a^ y ‰ H^ x cos hye

z

rx …H^ z † cos hye

Š

‡ a^ z ‰rx H^ y sin hye

z

… h†…H^ x † sin hye

z

z

Š

Š

…6:1:9b†

where rx ˆ @=@x. Substituting into the lH and E terms of Eqs. 6.1.7a,b, we ®nd that

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346

Chapter 6

0 ˆ ‰xx E^ x ‡ xz E^ z ‡ h…H^ z †

…H^ y †Š sin hye

0 ˆ ‰yy E^ y ‡ …H^ x † ‡ rx …H^ z †Š cos hye 0 ˆ ‰zx E^ x ‡ zz E^ z ‰ hE^ z

rx …H^ y †

z

…6:1:10a† …6:1:10b†

h…H^ x †Š sin hye

z

Š

Š

E^ y ‡ xx …H^ x † ‡ xz …H^ z †Š cos hye

‰ E^ x ‡ rx E^ z ‡ yy …H^ y †Š sin hye

z

z

z

Š

…6:1:10c†

Šˆ0

…6:1:10d†

Šˆ0

‰ rx E^ y ‡ hE^ x ‡ zx …H^ x † ‡ zz …H^ z †Š cos hye

…6:1:10e†

z

Šˆ0

…6:1:10f†

As can be seen, these six equations can only be satis®ed if the quantity in square brackets multiplying the sin hye z and cos hye z terms are set to zero. Setting these square-bracketed terms to zero, one ®nds the following set of six ®rst-order differential equations in x for the modal amplitudes E~^ and H^~. Equations 6.10a±f can be put into the matrix form 2

0

6 6 6 6 h 6 6 6 xx 6 4 0 zx

h

xx

0

0 rx

rx 0

0 zx

yy 0

0

xz

0

yy 0

0 zz

h

0 rx

32 ^ 3 Ex 7 6 E^ y 7 0 7 76 7 6 7 zz 76 E^ z 7 7 76 7ˆ0 6 ^x 7 h 7  H 76 7 6 7 7 rx 5 6 4 H^ y 5 0 H^ z

xz

…6:1:11†

The above matrix equation is still in an awkward form for numerical computation. This analysis will now concentrate on eliminating variables and, in fact, putting it into a standard state variable form for which a solution can be found. When reducing the number of ®eld variables in Eq. 6.1.11, the most logical ones to eliminate are the x-directed variables E^ x ad H^ x . These variables are normal to the interface of the different anisotropic slabs and thus are in general discontinuous at the boundary interfaces. On the other hand, Maxwell boundary conditions require that the EM ®elds tangential to the interface boundaries should be equal at each boundary interface, and this implies that the modal amplitudes E^ y , E^ z , H^ y , and H^ z should be continuous at the boundary interfaces, thus greatly facilitating the boundary matching operation from layer to layer. The objective now is to express E^ x and H^ x in terms of E^ y , E^ z , H^ y , and H^ z and thus put Eq. 6.1.11 into a standard state variable form. Using Eq. 6.1.10e and 6.1.10b, respectively, we may write

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Anisotropic Inhomogeneous Waveguide and Periodic Media

E^ x ˆ H^ x ˆ

1 r E^

x z yy ^ E

y

1  …H^ y †

yy

347

…6:1:12a†

1 r …H^ z †

x

…6:1:12b†

Substituting in Eqs. 6.1.10a,c,d,f we ®nd that  1 ^ r …Hz † ‡ xz …H^ z † ˆ 0 …6:1:13a†

x   1 1 rx E^ z yy …H^ y † rx E^ y ‡ h

  yy 1 rx …H^ z † ‡ zz …H^ z † ˆ 0 ‡ zx …6:1:13b† E^ y

  1 1 ^ ^ r E  …Hy † ‡ xz E^ z ‡ h…H^ z † …H^ y † ˆ 0 xx

x z yy hE^ z

 zx

E^ y ‡ xx



yy E^

y

…6:1:13c†



1 1 r E^  …H^ y † ‡ zz E^ z

x z yy   yy 1 ^ ^ r …Hz † ˆ 0 h E

y x

rx …H^ y † …6:1:13d†

We notice from Eq. 6.1.13c that the only rx term is rE^ z , and thus we may solve for it alone. Doing so we ®nd that rx E^ z ˆ

1 h ^ 0Ez ‡ xz xx E^ z xx xx

xx k2y …H^ y † ‡ hxx …H^ z †

i …6:1:14a†

where k2y ˆ 2 ‡ yy xx . We also notice from Eq. 6.1.13a that rx H^ z is the only rx term in Eq. 6.1.13a. Solving for it we ®nd that rx H^ z ˆ

1 h xx k2x E^ y xx xx

h xx E^ z ‡ 0…H^ y † ‡ xz xx …H^ z †

i

…6:1:14b† We notice from Eq. 6.1.13b that it involves the rx E^ y , rx E^ z , and rx H^ z . If Eqs. 6.1.14a and 6.1.14b are used to eliminate rx E^ z and rx H^ z , it is then possible to solve for rx E^ y alone in terms of E^ y , E^ z , H^ y , and H^ z . Equation

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348

Chapter 6

6.1.13d involves rx …H^ y †, rx E^ z , and rx …H^ z †. Use of Eqs. 6.1.14a,b in Eq. 6.1.13d also allows rx …H^ y † to be expressed in terms of E^ y , E^ z , H^ y , and H^ z . After making the appropriate substitutions the standard state variable form is 2

E^ y

3

7 6 6 ^ 7 6 Ez 7 7 rx 6 6 ^ 7 6 H y 7 5 4 ^ H z

2

6 1 6 6 ˆ 6 xx xx 6 4

xx zx 0

hxx

xx k2x 3 E^ y 7 6 6 ^ 7 6 Ez 7 7 6 6 ^ 7 6 H y 7 5 4 ^ Hz 2

h…xx zx

xz xx †

xx k2y

xz xx xx …e xx

hxx

hyxx

h2 †

zx xx 0

xx …xx e

h2 †

7 7 7 7 zx xx † 7 5

hxx h…xx xz

xx xz

…6:1:14c†

where e ˆ zz e ˆ zz

xz zx xx xz zx xx

…6:1:15a† …6:1:15b†

k2x ˆ yy xx ‡ 2

…6:1:15c†

k2y ˆ xx yy ‡ 2

…6:1:15d†

The state variable form for the EM ®eld components parallel to each slab interface can be used to determine the ®nal propagation constant of modes that exist in the waveguide. We ®rst note that the 4  4 set of ®rstorder differential equations, because the A matrix in the `th slab is a constant matrix, can be solved by ®nding the eigenvalues and eigenvectors of the constant matrix A. If we let Ve …x† ˆ ‰E^ y …x†; E^ z …x†; H^ y …x†, H^ z …x†Št and

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3

Anisotropic Inhomogeneous Waveguide and Periodic Media

349

qn and Vn ˆ ‰V1n ; V2n ; V3n ; V4n Št …n ˆ 1; 2; 3; 4) be the nth eigenvalue and eigenvector of the matrix A, respectively, that is, let Vn and qn satisfy A Vn ˆ qn Vn

…6:1:16†

then it is found that @ e V …x† ˆ A Ve …x† @x

…6:1:17†

where Ve n …x† ˆ Vn exp…qn x†

…6:1:18†

Linear combinations of theVen …x† represent full EM solutions to Maxwell's equations in each slab region. The next section will be concerned with imposing boundary conditions from which a ®nal equation for the propagation constants of the system can be determined.

6.1.5

Multilayer Analysis

In this section we implement a multilayer analysis from which we can determine the propagation constants of the system. Using the local coordinate system shown in Fig. 1, and taking linear combinations of the general state variable solution given by Eq. 6.1.18, we ®nd that the EM ®elds in the `th slab are given by Ve ` …x` † ˆ

4 X n 0 ˆ1

Cn 0 exp…qn…`†0 x` †Vn…`†0

…6:1:19†

where Vn…`†0 are the state variable eigenvectors of the `th slab region (Eq. 6.1.16). The ®eld components of the `th slab region contained in the matrix Ve ` …x` † are h it Ve ` …x` † ˆ E^ y` …x` †; E^ z` …x` †; H^ y` …x` †; H^ z` …x` †

…6:1:20†

We now begin ®eld matching at the different interfaces of the system. We have at x ˆ 0, x1 ˆ 0, E^ y1 …x1 † ˆ 0; E^ z1 …x1 † ˆ 0, thus

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350

Chapter 6

2

0 0

3

6 7 …1† …1† 7 Ve 1 …0† ˆ 6 4 H^ y1 …0† 5 ˆ ‰Vnn 0 Š‰Cn 0 Š ˆ V1 C1 H^ z1 …0†

…6:1:21†

where (n 0 ; n† ˆ …1; 2; 3; 4†. Matrix inversion of this equation gives 2

0 0

3

7 6 1 e 7 C1 ˆ V1 1 6 4 H^ y1 …0† 5 ˆ V1 V 1 …0† H^ z1 …0†

…6:1:22†

At x ˆ s1 ‡


‡ s` 1 ; x` ˆ 0 for ` ˆ 2; . . . ; N` V e ` 1 …s` 1 † ˆ V e ` …0†

…6:1:23†

or h

ih i h ih i …` 1† …`† Cn…`†0 Vnn exp…qn…`0 1† s` 1 † Cn…`0 1† ˆ Vnn 0 0 h ih ih i h ih i …` 1† 1† …`† Vnn exp…q…` Cn…`†0 s` 1 †n 0 n Cn…`0 1† ˆ Vnn 0 0 n0 Letting D` 1 ˆ ‰exp…qn…`0 we then have V` 1 D` 1 C`

1

1†

…6:1:24† …6:1:25†

…`† …`† …`† s` 1 †n 0 n Š, V` ˆ ‰Vnn 0 Š, and V` ˆ ‰Vnn 0 Š, C` ˆ ‰Cn 0 Š,

ˆ V` C`

…6:1:26†

or C` ˆ V` 1 V` 1 D` 1 C`

…6:1:27†

1

At x ˆ a, xN` ˆ sN` , E^ yN` …sN` † ˆ 0, E^ z1 …sN` † ˆ 0; 2

3

0 0

7 6 7 Ve N` …sN` † ˆ 6 4 H^ yN` …sN` † 5 ˆ VN` DN` CN` H^ zN …sN † `

…n 0 ; n† ˆ …1; 2; 3; 4†

`

…6:1:28† Substituting CN` in terms of CN` 1 , substituting CN` substituting C2 in terms of C1 , we ®nd that

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1

in terms of CN` 2 ; . . .,

Anisotropic Inhomogeneous Waveguide and Periodic Media

V e N` …sN` † ˆ VN` DN` CN` CN`

1

CN` ˆ VN1` VN` 1 DN` 1 CN`

1

ˆ VN` 1 VN` 2 DN` 2 CN` 1

2

1

...

1

C1 ˆ V1 V e 1 …0†

C2 ˆ V2 V1 D1 C1

351

…6:1:29†

or Ve N` …sL † ˆ …VN` DN` VN1` †…VN` 1 DN` 1 VN1` 1 †


…V1 D1 V1 1 †Ve 1 …0† …6:1:30† If we let T` ˆ V` D` V` 1 for ` ˆ 1; . . . ; N` , then Eq. 6.1.30 may be written as Ve N` …sL † ˆ TN` TN` or letting T  TN` TN` 2

3

1

1




T2 T1 Ve 1 …0†




T2 T1 , then

2 T11 6 7 6 0 6 7 6 T21 6 7 6 H^ yN …sN † 7 ˆ 6 4 T31 ` ` 5 4 ^ T41 HzN` …sN` † 0

…6:1:31†

T12

T13

T22 T32

T23 T33

T42

T43

T14

32

0 0

3

6 7 T24 7 76 7 76 ^ 7 T34 54 Hy1 …0† 5 T44 H^ z1 …0†

…6:1:32†

For this equation to hold we must have T13 …H^ y1 …0†† ‡ T14 …H^ z1 …0†† ˆ 0

…6:1:33a†

T23 …H^ y1 …0†† ‡ T24 …H^ z1 …0†† ˆ 0

…6:1:33b†

Since the right-hand side of both these equations is zero, the equations must be linearly dependent for a nontrivial solution to correspond to them. Setting the determinant to zero we ®nd that the ®nal propagation equation for the n 6ˆ 0 mode is T13 … †T24 … †

T23 … †T14 … † ˆ 0

…6:1:34†

Reference 1 describes in detail a root ®nding technique that can be used to solve for the propagation constants of this equation.

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352

Chapter 6

TEn0 Modes To determine the modes of the n ˆ 0 case, a separate EM ®eld analysis of Maxwell's equations must be carried out. In this case we take Ex ˆ Ez ˆ 0 and Hy ˆ 0. After a reduction of equations similar to that carried out for the n 6ˆ 0 case, it is found that the E^ y and H^ z ®eld variables can be put into the state variable form " #  @ 1 zx E^ y ˆ k2x @x H^ z xx

xx e xz

"

E^ y H^ z

# …6:1:35†

By carrying out a multilayer matrix analysis based on eigenvalues and eigenvectors identical to that which has already been presented, we ®nd that "

E^ yN` …sN` † H^ zN` …sN` †

"

# ˆ T N` T N`

1




T 2T 1

E^ y1 …0† H^ z1 …0†

#

"

E^ y1 …0† ˆT H^ z1 …0†

#

…6:1:36† or "

0

H^ zN` …sN` †

#

 ˆ

T11

T12

T21

T22



 ˆ

0

H^ z1 …0†

 …6:1:37†

or T11 … † ˆ 0

…6:1:38†

for a nontrivial solution to this equation to hold. Using the same numerical techniques described earlier, the roots of this equation can be found. 6.1.6

Comparison with Experimental Data

The case of a transversely magnetized slab of ferrite on the broad wall of a rectangular waveguide, Fig. 2, was studied by Bernardi [10]. On the basis of the dispersion relation, calculated by assuming a lossless ferrite, he quite accurately predicted the frequency at which losses are likely to occur. However, the determination of the actual shape of the attenuation curve is beyond the scope of his approach. The algorithm described in this chapter was applied to this particular problem yielding results presented in Fig. 3 and Fig. 4, which show, respectively, the attenuation of the structure in the

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Anisotropic Inhomogeneous Waveguide and Periodic Media

353

Figure 2 Rectangular waveguide with H-plane ferrite slab. Used with permission of IEEE-MTT, 1970 [1, Fig. 5].

Figure 3 Attenuation of ferrite-loaded waveguide of Fig. 2. Solid lines: theoretical values. Dots: experimental results published by Bernardi [10]. Used with permission of IEEE-MTT, 1970 [1, Fig. 6].

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354

Chapter 6

Figure 4 Propagation characteristics of ferrite-loaded waveguide of Fig. 2. Plot of b against frequency, calculated for the material parameters of Fig. 3. Used with permission of IEEE-MTT, 1970 [1, Fig. 8].

reverse direction (curves 1 and 2 correspond to the two lowest modes) and the normalized phase coef®cient k (two modes in both directions) as a function of frequency. Comparison with values of the attenuation measured by Bernardi [10] and presented in Fig. 3 show good agreement between calculations and experiment. It is worthwhile noting that the particular hump on the right-hand side of the attenuation curve, which corresponds to the resonance of e (Fig. 5), is quite apparent from the oscillograms of Bernardi [10]. Comparing Figs. 3 and 4 it appears that coupling takes place between the empty waveguide and curve 1 in the dispersion diagram (Fig. 4) when the frequency is lower than the resonant; at higher frequencies coupling occurs to curve 2 of Fig. 4. It is also worth noting that in ferrite-loaded waveguides, large values of attenuation can occur with large values of phase shift. This corresponds

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Anisotropic Inhomogeneous Waveguide and Periodic Media

355

Figure 5 Effective ferrite permeability e against frequency, calculated for the material parameters of Fig. 3. Used with permission of IEEE-MTT, 1970 [1, Fig. 7].

to the evanescent modes reported by Tai [11]. Experimental veri®cation for other ferrite-loaded waveguides were made in Ref. 12.

6.1.7

Conclusion

The study of a rectangular waveguide containing slabs extending across it has shown that an exact derivation of the dispersion equation can be made when the material properties meet certain requirements. This is the case for cross-tensor media, which include in particular transversely magnetized ferrites, plasmas and semiconductors, certain crystal orientations of crystals, and also dielectrics and resistive materials. A state variable analysis of Maxwell's equations was made, the dispersion equations of the modes of the loaded structure were derived, and a numerical solution of the dispersion

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356

Chapter 6

equations was obtained. A veri®cation with published experimental data shows good agreement. 6.2 6.2.1

DIFFRACTION GRATING ANALYSIS USING THE GARDIOL WAVEGUIDE FORMULATION Introduction

In this section we use the Gardiol state variable waveguide formulation [1] to study diffraction from a one-dimensional grating composed of the crosstensor materials described in the previous section. The diffraction grating geometry is shown in Fig. 6. Regions 1 and 3 contain uniform lossless materials characterized by permittivity and permeability (1 ; 1 ) and (3 ; 3 ), respectively. The diffraction grating region is assumed to contain inhomogeneous cross-tensor material characterized by relative permittivity  and permeability , positioned in the uniform slabs as shown in Fig. 6. The grating is assumed to be uniform in the y- and z-directions. It is assumed that an H-mode plane wave (see Fig. 6) impinges on the grating. The diffraction grating analysis presented in this section combines elements of the diffraction grating analysis in Chapter 3 and the waveguide analysis of Gardiol in Section 6.1. The basic procedure to be followed will be (1) to use the Gardiol method to obtain the forward and backward propagating modes in the z-direction of the periodic diffraction grating, (2) to expand the EM ®elds of Regions 1 and 3 in a set of Floquet harmonics (as is done in applying RCWA to the study of gratings), and (3) to determine all unknown coef®cients of the system by matching EM boundary conditions at the Region 1±2 and Region 2±3 interfaces of the system. The procedure to be used to obtain the modes of the diffraction grating system is identical to that of the Gardiol analysis except that periodic boundary conditions will be imposed at the boundaries of each diffraction grating cell instead of requiring that the tangential electric ®eld be zero at waveguide boundaries. This difference will lead to a signi®cantly different eigenvalue equation from the one used by Gardiol. We mention at this point that the method to be presented here is identical to that already presented by Yamakita et al. [13±15] when E-mode and H-mode plane wave scattering from diffraction gratings composed of isotropic dielectric materials is studied. In the work of Yamakita et al. [13±15] the sets of propagating plane waves were termed Bloch wave modal functions. There are at least two reasons why determination of diffraction by the present method is useful. First, it represents an entirely different formulation of the diffraction grating problem than that of rigorous coupled wave analysis. Therefore close agreement of the two methods strongly recommends

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Anisotropic Inhomogeneous Waveguide and Periodic Media

357

Figure 6 The geometry of the anisotropic transversely inhomogeneous diffraction grating system is shown. One unit cell of the diffraction grating over the interval 0  x   is shown in detail. The other cells are assumed to repeat inde®nitely in the positive and negative x-directions. The local coordinates x1 ; x2 ; . . . x` ; . . . ; xN` that represent the EM ®elds inside each slab are also shown. Region 1 (incident side) and Region 3 (transmit side) are assumed to consist of homogeneous lossless media. The incident EM wave is assumed to have either E-mode or H-mode polarization.

the validity of both methods. Second, the present method, because it boundary matches the EM ®elds at different boundaries, should be an ideal method for describing the internal EM ®elds. In comparison, the RCWA method uses a large number of Fourier terms to represent a step discontinuity, whereas the present modal method builds step continuities into solution, therefore giving a very accurate representation of the EM ®elds internal to the grating system. 6.2.2

H-mode Formulation

Section 6.2.1 has given a brief introduction to the problem at hand. Because an H-mode wave is impinging on the grating, and because the grating is assumed uniform in the y-direction, it turns out that the Ex , Ez , and Hy EM ®eld components can be taken to be zero, Ex ˆ Ez ˆ Hy ˆ 0. In this sec-

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358

Chapter 6

~ y ˆ k0 y, ~ etc., and tion we will use normalized coordinates in which x ˆ k0 x, p k0 ˆ ! 0 0 , x~ in meters. In Gardiol's n ˆ 0 waveguide formulation (denoted in this section with a prime in order that the waveguide ®eld solutions will not be confused with the grading solution), the EM ®elds are expressed in the form E~ ˆ ‰E^ y0 …x†a^y Še

z

…6:2:1a†

  H~ ˆ …H x0 …x††a^ x ‡ …H z0 …x††a^ z e

z

…6:2:1b†

Substituting Eqs. 6.2.1a,b into Maxwell's equations using normalized coordinates led to the matrix equation " #  " 0 # 0 0 @ E^ y0 E^ y a12 a11 …6:2:2† ˆ ˆ A0 V0 0 0 0 ^ a21 a22 @x Hz H^ z0 where A0 ˆ

 1 zx k2x xx

xx e

zx



~ 0 is the normalized propagation constant, ~ in 1/meter, where ˆ =k e ˆ zz xz zx =xx , and k2x ˆ yy xx ‡ 2 , and where the other terms are de®ned as in Section 6.1. In order to facilitate boundary matching to the EM ®elds in Regions 1 and 3 of the present diffraction grating problem, it is useful to seek a solution of Maxwell's equations in the form E ˆ ‰E^ y …x†a^y Š exp… jkx0 x† exp… z†   H ˆ …H x …x††a^ x ‡ …H z …x††a^z exp… jkx0 x† exp… z†

…6:2:3a† …6:2:3b†

Letting V ˆ ‰E^ y …x† H^ z …x†Št , a comparison with Eqs. 6.2.1a,b shows that V 0 ˆ V exp… jkx0 x†. If V 0 ˆ V exp… jkx0 x† is substituted into Eq. 6.2.2, we ®nd after algebra that " #  # " @ a11 a12 E^ y E^ y …6:2:4† ˆ ˆAV a21 a22 @x H^ z H^ z where A ˆ …jkx0 †I ‡ A 0

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…6:2:5†

Anisotropic Inhomogeneous Waveguide and Periodic Media

359

The matrix A refers to any slab layer `, and the subscript ` has been suppressed in the above equations. By carrying out a multilayer matrix analysis based on eigenvalues and eigenvectors identical to that which has already been presented, we ®nd that "

E^ yL …sL † H^ zL …sL †

#

" ˆ TN` TN`

1




T2 T1

E^ y1 …0† H^ z1 …0†

#

"

E^ y1 …0† ˆT H^ z1 …0†

# …6:2:6†

where  T` ˆ Q

`

q1` 0

 0 Q 1 q2` `

where Q is the eigenvector or modal matrix corresponding to the matrix A` ` and q1` and q2` are the eigenvalue of the matrix A` . In the present diffraction grating problem we require that the modal amplitudes E^ y …x† and H^ z …x† be periodic functions of the grating period . This requirement leads to the equations E^ yL …sN` † ˆ E^ y1 …0†

…6:2:7a†

H^ zL …sN` † ˆ H^ z1 …0†

…6:2:7b†

An inspection of Eq. 6.1.3.6 shows that only nontrivial solutions will result for this case when det…T† ˆ T11 … †T22 … †

T12 … †T21 … † ˆ 0

…6:2:8†

This equation then forms an equation from which the propagation constants in the grating can be determined. Once the propagation constant has been determined, it is possible (up to a normalization constant) to determine the full form of each electromagnetic mode that can propagate in the grating system. The ®eld coef®cients E^ y`n …x` † and H^ z`n …x` † of the nth mode in the `th layer are found, after assuming an appropriate normalization constant, by using a matrix cascade to relate the ®eld amplitudes at one grating boundary to the `th layer. A small amount of algebra shows that in the `th layer, H^ x`n …x` † ˆ

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o 1n ‰ jkx0 ‡ a22`n ŠH^ z`n …x` † ‡ a21`n E^ y`n …x` †

n`

…6:2:9†

360

Chapter 6

We mention at this point that in order to determine the propagation of waves in the negative z-direction, that is, waves proportional to exp… z†, we use the analysis as presented in Eqs. 6.2.1±6.2.9 and replace in all equations

by . Because the material is not reciprocal it does not necessarily follow that the positive and negative traveling propagation constants and the associated modal ®elds will be the same. We represent the EM ®elds in Region 2 by expanding those ®elds in a set (multiplied by as yet unknown coef®cients) of 2MT ‡ 1 forward traveling modes … n‡‡ , n‡ ˆ 1; . . . ; 2MT ‡ 1 denotes the propagation constant of the forward traveling wave) and 2MT ‡ 1 backward traveling modes ( n , n ˆ 1; ; . . . ; 2MT ‡ 1 denotes the propagation constant of the backward traveling modes). The full expansion of the Ey…2† and H^ x…2† EM ®elds in Region 2 is given by Ey…2† ˆ exp… jkx0 x† Hx…2† ˆ exp… jkx0 x†

NT X nˆ1 NT X nˆ1

Cn E^ yn …x† exp… n z†

…6:2:10a†

Cn H^ xn …x† exp… n z†

…6:2:10b†

where (

n ˆ E^ yn …x† ˆ H^ xn …x† ˆ

n‡‡

n

8 ‡ < E^ yn ‡ …x† : E^ …x† yn ( ‡ Hxn‡ …x† Hxn …x†

…6:2:11†

…6:2:12a†

…6:2:12b†

where NT ˆ 2…2MT ‡ 1†, where plus and minus denote forward and backward traveling modes, respectively and where n‡ ˆ n; n ˆ 1;


; 2MT ‡ 1 and n ˆ n …2MT ‡ 1†; n ˆ 2MT ‡ 2;


; NT : In writing Eqs. 6.2.10± 6.2.12, it is assumed that EM ®eld solutions, which were found in terms of the local coordinates x` of each thin slab, have been properly translated into the overall transverse variable x, which extends over the periodic grating interval 0  x  . In Regions 1 and 3 the EM ®elds are given by

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Anisotropic Inhomogeneous Waveguide and Periodic Media

Ey…1† ˆ

MT X iˆ MT

‰E0 i0 exp… jkxi x

361

jkz1i z† ‡ ri exp… jkxi x ‡ jkz1i z†Š …6:2:13a†

Hx…1† ˆ

MT 1 X kz1i ‰ E0 i0 exp… jkxi x 0 iˆ M 1

jkz1i z†

T

‡ri exp… jkxi x ‡ jkz1i z†Š Ey…3† ˆ

MT X iˆ MT

‰ti exp… jkxi x

jkz3i …z

…6:2:13b† L††Š

MT 1 X ‰t exp… jkxi x jkz3i …z L††Š 0 iˆ M i T 8 p < ‰r r k2xi Š1=2 r r > kxi ˆ p : j‰k2 r r Š1=2 kxi > r r xi

…6:2:13c†

Hx…3† ˆ

kzri

…6:2:13d†

r ˆ 1; 3

…6:2:13e†

and kxi ˆ kx0

iKx

…6:2:13f†

where Kx ˆ 2x , x is the normalized grating period, and i ˆ MT ; . . . ; MT . To impose boundary conditions we equate the tangential electrical and magnetic ®elds at the Region 1±2 and Region 2±3 interfaces. Equating the ycomponent of the electric ®eld at z ˆ 0 we have Ey…1† jzˆ0 ˆ Ey…2† jzˆ0‡ Ey…1† jzˆ0 ˆ exp… jkx0 x†

Ey…2† jzˆ0‡ ˆ exp… jkx0 x† or

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…6:2:14a† MT X iˆ MT NT X nˆ1

‰…E0 i0 ‡ ri † exp…jiKx x†Š

Cn E^ yn …x†

…6:2:14b†

…6:2:14c†

362

Chapter 6 MT X iˆ MT

‰…E0 i0 ‡ ri † exp…jiKx x†Š ˆ

NT X nˆ1

Cn E^ yn …x†

…6:2:14d†

Using the orthogonality of Floquet harmonics (or Fourier series) (that is, after multiplying Eq. 6.2.14d by …1=x † exp… ji 0 Kx x†, integrating across a grating period, and replacing i 0 by i in the notation) we ®nd that E0 i0 ‡ ri ˆ

NT X nˆ1

Cn

1 x

… x =2 x =2

E^ yn …x† exp… jiKx x†dx ˆ

NT X nˆ1

Cn Eyin …6:2:15†

where Eyin ˆ

1 x

… x =2 x =2

E^ yn …x† exp… jiKx x†dx

…6:2:16†

Applying the same matching procedure to the tangential magnetic ®eld at z ˆ 0 and the tangential electric and magnetic ®eld at z ˆ L, we ®nd the additional equations … NT X kz1i 1 x =2 ^ … E0 i 0 0 ‡ ri † ˆ j Cn H …x† exp… jiKx x†dx 1 x x =2 xn nˆ1 ˆj

NT X nˆ1

ti ˆ

NT X nˆ1

Cn H^ xin

…6:2:17a†

Cn Eyin

…6:2:17b†

NT X kz3i … ti † ˆ j Cn H^ xin 1 nˆ1

…6:2:17c†

Elimination of ri and ti from these equations leads to the ®nal matrix equations for Cn . We have T X 2kz1i …Eo i0 † ˆ Cn 1 nˆ1

N

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k j H^ xin ‡ z1i Eyin 1

 …6:2:18a†

Anisotropic Inhomogeneous Waveguide and Periodic Media NT X



k 0ˆ Cn exp… n L† j H^ xin ‡ z3i Eyin 3 nˆ1

363

 …6:2:18b†

where i ˆ MT ; . . . ; MT in each equation. We notice that Eqs. 6.2.18a and 6.2.18b with NT ˆ 2…2MT ‡ 1† represent an NT  NT matrix equation from which all the Cn coef®cients of the system can be found. The E-mode case, which is the case dual to the case considered in Section 6.2.1, occurs when the EM ®eld components Hx ˆ Hz ˆ Ey ˆ 0 rather than when the ®eld components Ex ˆ Ez ˆ Hy ˆ 0 as was considered earlier. In conclusion of this subsection we mention that the present analysis can be extended to include longitudinally inhomogeneous diffraction gratings (for example a sinusoidal surface diffraction grating or a triangular blaze grating) by applying a multilayer analysis as described in Section 3.5 or as described in Ref. 15. This analysis consists of dividing the diffraction grating into electrically thin layers, determining the eigenmodes of each thin layer, and then boundary matching the EM ®elds of each thin layer and the bounding regions to determine the EM ®elds of the whole system.

6.2.3

Coordinate Transformations

In Section 6.1 the coordinate system was chosen with the z-coordinate parallel to the waveguide axis as is typical of most waveguide descriptions. This coordinate system corresponds exactly to Gardiol's waveguide analysis. In order that the Gardiol analysis could be extended to diffraction gratings in a simple way, the waveguide system of Section 6.1 was also used in Section 6.2 for diffraction gratings. We mention at this point that by using a simple coordinate transformation matrix, we can relate the coordinates and the permittivity and permeability tensors of the present chapter to the permittivity and permeability tensors expressed in the coordinate system used in Chapter 3. Letting F~s ˆ ‰Fxs ; Fys ; Fzs Št represent a vector in the coordinate system of Chapter 3 (See Fig. 3.2.1) we ®nd that a vector F~ in the coordinate system of this chapter is related to Fs by the relation 2

3 2 Fx 1 F~ ˆ 4 Fy 5 ˆ 4 0 Fz 0

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0 0 1

32 3 0 Fxs 1 54 Fys 5 ˆ MFs Fzs 0

…6:2:19†

364

Chapter 6

Also 2

3 2 1 Fxs F~s ˆ 4 Fys 5 ˆ 4 0 0 Fzs

0 0 1

32 3 0 Fx 1 54 Fy 5 ˆ M 1 F~ 0 Fz

…6:2:20†

In Chapter 3 the dielectric permittivity tensor satis®es 2 32 3 Exs xxs xys 0 D~ s ˆ s E~s ˆ 4 yxs yys 0 54 Eys 5 0 0 zzs Ezs

…6:2:21†

We have D~ ˆ MD~ s

E~ ˆ ME~s

D~ ˆ E~

MD~ s ˆ  M E~s

…6:2:22†

or D~ s ˆ …M 1  M†E~s ˆ s E~s

…6:2:23†

Thus s ˆ …M 1  M† and also  ˆ …M s M 1 ). Written out we have 2

xxs

xys

0

3

2

xx

xz

7 6 6 1 6 s ˆ 6 0 7 5 ˆ M  M ˆ 4 zx zz 4 yxs yys 0 0 0 0 zzs 3 2 2 xxs 0 xx 0 xz 7 6 6 1 7 6 zzs ˆ6 4 0 yy 0 5 ˆ M s M ˆ 4 0 yxs 0 zx 0 zz

0

3

7 0 7 5 …6:2:24a† yy 3 xys 7 0 7 5 …6:2:24b† yys

The permeability tensors ls and l obey the same relations as the permittivity tensors above. Thus the formulation presented in this section can be used to solve the diffraction grating problems considered in Sections 3.2 and 3.3, which were solved by rigorous coupled wave theory. 6.2.5

Numerical Results

The method presented here is identical to that presented by Yamakita et al. [13±15] when E-mode and H-mode plane wave scattering from diffraction gratings composed of isotropic dielectric materials is studied. When applying the analysis presented here to isotropic materials, the waves in each lateral

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Anisotropic Inhomogeneous Waveguide and Periodic Media

365

slab reduce to multiply re¯ected plane waves traveling in the positive and negative x-directions. Yamakita et al. [13±15] boundary matched expansions of these plane waves at the slab interfaces, and after imposing periodic boundary conditions at the grating cell boundaries they presented equations that determined which sets of plane waves can propagate in the diffraction grating system. The sets of propagating plane waves were termed Bloch wave modal functions. Once the Bloch wave modal functions were determined, boundary matching of the incident, re¯ected, and transmitted diffracted waves with expansions of the Bloch wave modal functions at the Region 1±2 and Region 2±3 interfaces provided the ®nal matrix equation from which all EM ®elds in the diffraction grating system could be uniquely determined. In addition to uniform rectangular groove gratings, Yamakita et al. [15] have, based on the Bloch wave modal function theory, developed a multilayer algorithm to study longitudinally inhomogeneous surface relief gratings. Figure 7 (Yamakita et al. [14], Fig. 5) shows examples of plane wave E-mode and H-mode diffraction ef®ciency versus groove depth as occurs from a rectangular groove grating when Bloch wave modal function theory (and therefore the theory of this section applied to isotropic materials) is used to determine the EM ®elds of the system. Yamakita et al. [13±15] number the diffraction ef®ciency orders in the number sequence opposite to that used in Chapter 3 of the present text. The Yamakita ordering is shown in the inset of Fig. 7.

PROBLEMS 1. Assume that the waveguide shown in Figure 1 has an isotropic permittivity given by  "…x† ˆ

"1 ; 0  x  x 1 "2 ; x1 < x  a

 …x† ˆ

and permeability given by

1 ; 0  x  x1 2 ; x1 < x  a

This type of waveguide is referred to as a loaded waveguide or partially ®lled waveguide and its modes may be solved exactly by using vector potential theory. Please see Harrington [16]. The modal solutions are found using the x component of the electric or magnetic vector potential and the modal solutions which result

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Figure 7 Diffraction ef®ciencies as a function of groove depth for rectangular surface relief grating. Used with permission of SPIE, 1987 [14, Fig. 5].

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Anisotropic Inhomogeneous Waveguide and Periodic Media

367

are termed transverse electric to x (TEx ) or transverse magnetic to x (TMx ) modal solutions. (a) Use the method of Gardiol [1] to solve for the EM ®elds and modes that propagate in the waveguide for the permittivity and permeability that are given at the start of the problem. (b) Compare this solution with the loaded waveguide solutions as found in Harrington [16]. 2. A partially loaded waveguide contains an inhomogeneous dielectric whose relative permittivity is given by  e…x† ˆ

"1 I; 0  x  x1 ; L~  z~  L~ "2 ; x1 < x  a; L~  z~  L~

where I is a 3  3 identity matrix, "1 ˆ 1:5.; the relative permeability  ˆ 1; and the nonzero elements of e2 are "2xx ˆ "2yy ˆ "2zz ˆ 2. and "2xz ˆ "2zx ˆ 3.; and where the region outside L~  z~  L~ is free space. Let a~ ˆ 2b~ ˆ 2 cm; ~ ~ ~ The guide is operated at a frequency f , which is x~ 1 ˆ b; L ˆ 5a. 25% higher than the cutoff frequency of the lowest order mode which propagates in the unloaded portion of the guide. (a) Find the propagation constants and EM ®elds in the wave~ guide of the lowest four modes in the region L~  z~  L. (b) If a TE10 (the lowest order dominant mode) is incident on the loaded portion of the waveguide from z~ ˆ 1, determine the amplitude of the TE10 mode which is re¯ected from the loaded portion of the guide and which is transmitted through the loaded portion of the guide. Solve this problem by expanding the unknown ®elds in each region in the propagating and evanescent modes that exist in eaach region (TEmn and TMmn modes outside in the region L~  z~  L~ and the modes of Part (a) inside ~ and by matching boundary conditions. L~  z~  L) (c) Repeat (b) assuming that the TE10 mode is incident from z~ ˆ 1. Are the re¯ection and transmission coef®cients the same as in part (b)? 3. A waveguide contains an inhomogeneous material whose relative permeability is given by l…x† ˆ 2:I ‡ : :5l0 exp‰ …x~

~ 2Š a=2†

where I is a 3  3 identity matrix; the relative permittivity " ˆ 1:5.; and the nonzero elements of l0 are 0xx ˆ yy ˆ

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368

Chapter 6

0zz ˆ 3., 0xz ˆ 1:5, and 0zx ˆ 2:5: Let a~ ˆ 2b~ ˆ 2 cm. The guide is operated at a frequency f , which is 15% higher than the cutoff frequency of the lowest order mode which propagates when free space occupies the guide. (a) Find the propagation constants and EM ®elds in the waveguide of the lowest four modes. In the numerical analysis be sure to include a suf®cient number of multilayers to assure proper convergence of the solution. (b) Calculate each mode's amplitude coef®cient so that each mode at z ˆ 0 transmits in magnitude 1. watt of power across cross section of the guide in the z direction. 4. Referring to the lamellar diffraction grating studied by Yamakita and Rokushima [14, Fig. 5] (and reproduced in Fig. 7 of Sec. 6.2): (a) Use the method described in Sec. 6.2 (based on the Gardiol waveguide formulation [1], described in Sec. 6.1) to determine the four lowest propagating (or possibly evanescent) transverse electric modes (Ez ˆ 0), which are excited in the lamellar diffraction grating. (Transverse electric is also called H-mode.) (b) Use rigorous coupled wave analysis as described in Chapter 3 to determine the four lowest transverse electric modes for the case studied in (a). (c) Plot the modal ®elds from both methods and compare the results. Which method does the best job in meeting EM boundary conditions of the system? (d) Using the modal ®elds from (a), calculate the diffraction ef®ciency of the grating diffraction results shown in Fig. 7, Sec. 6.2 [14, Fig. 5]. Compare the results with that found by RCWA. (e) Repeat (a) through (d) for transverse magnetic modes. (Transverse magnetic is also called E-mode.) 5. An anisotropic, lambellar diffraction grating is bounded by free space on the incident side (Region 1) and is bounded by a homogeneous, isotropic dielectric on the transmit side (Region 3), whose relative dielectric permittivity value is "3 ˆ 2:5. The permittivity in the diffraction grating region is given by ( "2 …x† ˆ

"2a ; 0  x~  x~ 1 ; 0  z~  L~ ~ 0  z~  L~ "2b I; x~ 1 < x~  a;

where I is a 3  3 identity matrix,  ˆ 1; the nonzero elements of "2a "2axx ˆ "2azz ˆ 1: j:1; "2ayy ˆ 1:5"2axx , and "2axz ˆ :2"2axx ; the relative permeability  ˆ 1; and "2b ˆ 2:5. Let a~ ˆ , x~ 1 ˆ =2.

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Anisotropic Inhomogeneous Waveguide and Periodic Media

369

(Note that the permittivity is nonreciprocal since "2axz ˆ :2"2axx and "2azx ˆ 0.) Use the method described in Sec. 6.2 (based on the Gardiol waveguide formulation [1], described in Sec. 6.1) to (a) Determine the four lowest propagating (or possibly evanescent) transverse magnetic modes (Hz ˆ 0), which may be excited in the anisotropic, lamellar diffraction grating in Region 2. (Transverse magnetic is also called E-mode. Hz ˆ 0† (b) If an E-mode polarized plane wave is incident on the grating at an angle of 308, use rigorous coupled wave analysis as described in Chapter 3 to determine the four lowest transverse magnetic modes for the case studied in (a). (c) Plot the modal ®elds from both methods to compare the results. (d) Using the modal ®elds from (a), calculate the diffraction ef®ciency of the grating. Be sure to include a suf®cient number of expansion modes in all regions to ensure proper convergence of the solution. Compare the results with that found by the RCWA method.

REFERENCES 1. F. E. Gardiol, Anisotropic slabs in rectangular waveguides, IEEE Trans. Microwave Theory Techniques, MTT-18(8), 461±467 (1970). 2. B. Lax and K. J. Button, Microwave Ferrimagnetics, McGraw-Hill, New York, 1962, Chap. 12. 3. M. H. Engineer and B. R. Nag, Propagation of electromagnetic waves in rectangular waveguides ®lled with semiconductor in the presence of a transverse magnetic ®eld, IEEE Trans. Microwave Theory Techniques, MTT-13, 641±645 (Sept. 1965). 4. W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas. Cambridge, Mass, MIT Press, 1963, Chap. 9. 5. H. Suhl and R. L. Walker, Topics in guided wave propagation through gyromagnetic media py2 transverse magnetization and non-reciprocal helix, Bell System Tech. J., 33, 939±986 (July 1954). 6. H. Seidel, Ferrite slabs in transverse electric waveguide, J. Appl. Phys., 28, 218± 226 (Feb. 1957). 7. P. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953, Sec. 5.1. 8. A. Angot, CompleÂments matheÂmatiques de l'usage des ingeÂnieurs de l'eÂlectrotechnique et des teÂleÂcommunications. Collection Techniques et Scienti®que, C.N.E.T., Paris, 1961, p. 274.

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370

Chapter 6

9. A. D. Bresler, G. H. Joshi, and M. Maecuvitz, Orthogonality properties for modes in passive and active uniform waveguides, J. Appl. Phys., 29, 794±799 (May 1958). 10. P. Bernardi, A tunable absorbing band-stop ®lter: the ®eld rotation ®lter, IEEE Microwave Theory Techniques, MTT-17, 62±66 (Feb. 1969). 11. C. T. Tai, Evanescent modes in a partially ®lled gyromagnetic rectangular waveguide, J. Appl. Phys., 31, 220±221 (1960). 12. F. E. Gardiol, Propagation in rectangular waveguides loaded with slabs of anisotropic materials, Ph.D. thesis, Faculty of Applied Sciences, Louvain University, Heverlee, Belgium, June 1969. 13. J. Yamakita and K. Rokushima, Scattering of plane waves from dielectric gratings with deep grooves, IECE Japan, J66-B, 375 (1983). 14. J. Yamakita and K. Rokushima, Modal expansion method for dielectric gratings with rectangular grooves, Proc. SPIE, 503, 239 (1984). 15. J. Yamakita, K. Rokushima, and S. Mori, Numerical analysis of multistep dielectric gratings, SPIE, 815. Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, 153±157 (1987). 16. R. F. Harrington, Time Harmonic Electromagnetic Fields, McGraw-Hill, New York, (1961).

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7

Application of Rigorous Coupled Wave Analysis to Analysis of Induced Photorefractive Gratings 7.1

INTRODUCTION TO PHOTOREFRACTIVE MATERIALS

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

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372

Figure 1

Chapter 7

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

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Induced Photorefractive Gratings

373

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

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374

Chapter 7

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.

7.2

DYNAMIC NONLINEAR MODEL FOR DIFFUSIONCONTROLLED PR MATERIALS

The time-dependent Kukhtarev equations [8] for a PR material for nominal propagation of the optical ®eld along y and assuming transverse ®eld components …J ˆ Ja^ x ; E s ˆ Es a^ x † and transverse coordinate x are ‡ @…ND n† ˆ @t

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1 @J e @x

…7:2:1†

Induced Photorefractive Gratings ‡ @ND ˆ …ND @t

J ˆ eDs s

‡ ND † …sI ‡ †

375 ‡

R N D n

@n ‡ enEs @x

@Es ‡ ˆ e…ND @x

NA

n†

…7:2:2† …7:2:3† …7:2:4†

where n…x; y; t† ‡ …x; y; t† ND J…x; y; t† Es …x; y; t† I…x; y; t† ND e s

R Ds s  NA K 

= 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 ˆ K
s
K=K 2 = effective static mobility ˆ K

K=K 2 =acceptor concentration =K a^ x ˆ ‰K; 0; 0Š is the grating vector, K ˆ 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
…x; y; t† is related to the optical intensity I.

7.3

APPROXIMATE ANALYSIS [29,30]

To start the analysis, we differentiate Eq. 7.2.4 with respect to time, and after noting that @NA =@t ˆ 0, we ®nd that ‡ @…ND n† s @2 E ˆ ˆ @t e @x@t

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1 @J e @x

…7:3:1†

376

Chapter 7

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

nEs

Ds

@n @x

…7:3:2†

For a particular choice of c-axis of the PR crystal, the index of refraction modulation may be taken to be
n ˆ rn30 Es , where n0 is the pertinent index of refraction of the bulk dielectric of the PR material and where r is the effective electro-optic coef®cient. The dielectric permittivity modulation
 is related to
n by
n ˆ
=2n0 . Thus
…x; y; t† ˆ

Es rn40

…7:3:3†

Substituting Es from Eq. 7.3.3 into Eq. 7.3.2, we ®nd the following equation for
: @
 en De @n ‡
 ˆ s rn40 @t s s @x

…7:3:4†

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 n. The equation is nonlinear since the electron density n 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 n on I in Eq. 7.3.4, in this chapter we will study the commonly occurring ‡ ‡ case when NA  ND and n  ND  NA . In this case the ND term in the ®rst term of the right-hand side of Eq. 7.2.2 can be dropped, and we can solve for

R n, so that

R n ˆ

…sI ‡ †ND ‡ ND

‡ 1 @ND ‡ ND @t

…7:3:5†

The second term on the right-hand side (RHS) of Eq. 7.3.5 is the derivative of a logarithmic term  ‡ ‡ 1 @ND @ N ˆ ln D ‡ @t @t NA ND

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377

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

…sI ‡ †ND

R N A

@n sND @I ˆ @x R NA @x

…7:3:6†

Substituting Eq. 7.3.6 in Eq. 7.3.5 we ®nd that …I†

@
 ˆ @t




rn40 Ds @I …I ‡ =s† @x

…7:3:7†

where …I† ˆ

R NA s eND …sI ‡ †

…7:3:8†

The dielectric perturbation
 can be put in the normalized form ~ …I†

@
 ˆ @t~


 ‡ 

@I=@x~ …I ‡ C†

…7:3:9†

where x~ ˆ k0 x

t~ ˆ

t …I0 †

Cˆ

s

ˆ

rn40 Ds k0 

~ ˆ …I†

…I† …I0 † …7:3:10†

and where I0 is the peak intensity of the incident optical ®eld. In Eq. 7.3.10, …I0 † denotes a normalizing constant that makes …I† and t dimensionless. A typical range for values of  is from  10 5 (e.g., in BSO) to  10 2 (BaTiO3 ). Also, the constant C 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 I is nearly constant in time; thus Eq. 7.3.9 can be directly integrated w.r.t. t~ to yield an explicit expression for
:
 ˆ 

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@I=@x~ h 1 C‡I

e

~ …I† ~ =

i

…7:3:11†

378

Chapter 7

The dielectric tensor of a BaTiO3 crystal can be shown to be given by xx ˆ n2CO cos2 c ‡ n2CE sin2 c ‡
…x; y; t†Fxx xy ˆ …n2CO

n2CE † sin c cos c ‡
…x; y; t†Fxy

yx ˆ xy yy ˆ n2CO sin2 c ‡ n2CE cos2 c ‡
…x; y; t†Fyy

…7:3:12†

where ( Fxx ˆ

FOE (

Fxy ˆ

Fyy ˆ FOE ˆ

FOE

r13 n2CO r33 n2CE 3 2 2 sin  cos  ‡ 2 sin  cos  ‡ sin c c c c c r42 n2CE r42 n2CO

)

r13 n2CO 2 sin c cos c ‡ cos c ‰ cos2 c ‡ sin2 c Š r42 n2CE )

r33 n2CE 2 sin c cos c r42 n2CO ( ) r13 n2CO 3 r33 n2CE 2 2 FOE sin c 2 sin c cos c ‡ cos c sin c r42 n2CE r42 n2CO n2CO n2CE n2O n2E

n2CO ˆ n2O

jO00

n2CE ˆ n2E

jE00

…7:3:13†


 in Eq. 7.3.12 is found by solving ~ …I†

@
 ˆ @t~


 ‡ 

@I=@x~ C‡I

ˆ

Ds 2 2 n n k r  O E 0 42

…7:3:14†

~ …I† is given in Eq. 7.3.10. The optical power intensity I…x; y; t† in this problem is assumed proportional to jEx j2 ‡ jEy j2 . The terms n2CO and n2CE are the relative complex dielectric permittivities of the bulk crystal. The terms n2O and n2E are real parts of the relative bulk dielectric permittivity and O00 and E00 are the lossy parts. Note that Exsc , electric space charge ®eld, is linearly related to the dielectric perturbation or modulation function by the equation

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Induced Photorefractive Gratings


…x; y; t† ˆ n2O n2E r42 Exsc …x; y; t†

379

…7:3:15†

The electro-optic constants r13 , r33 , and r42 are speci®ed in Ref. 8, Table 1.2, pp. 26±29. 7.3.1

Numerical Algorithm

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 t ˆ 0 (®rst time step) a signal ( ˆ i † and pump wave ( ˆ i ) 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 @
 @…t ‡
t†  @t
t


…t†

ˆ F…t; I†

…7:3:16†

or
…t ‡
t† ˆ
…t† ‡ F…t; I†
t

…7:3:17†

where F…t; I† is the right-hand side of Eq. 7.3.14 and I is the optical intensity of the light at time t.

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380

Chapter 7

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 5:71 and 5:71 , respectively, and whose complex amplitudes were respectively E0 ej0 and E1 ej rand . These two plane waves caused an electric ®eld interference pattern whose nulls were  ˆ 5 apart. 7.3.2

TE Numerical Simulation Results

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 0 00 2r ˆ 2r j2r , a thickness of 1.5 cm, and a wavelength  ˆ 0:5 m. For the TE case, the ratio of the magnitudes of the incident plane wave p ˆ jE1 j=jE0 j 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 Iinc transmitted into the photorefractive material constant and equal to 9:949  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 T1 ; T0 ; R1 ; R0 as a function of time that results when equal amplitude plane waves (p ˆ 1:0) are incident on an initially dark photorefractive material bulk dielectric 2r ˆ 8. The smooth line shows the result in the case of coherent illumination ( rand ˆ 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

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381

Figure 2 Plots of the normalized power in the directions R0 , R1 , T0 , and T1 (see inset) that is transmitted and re¯ected from a slab of PR material when the incident waves are coherent (rand ˆ 0 ) and partially coherent ( 30  rand  30 ). The angle of incidence of the pump wave is 5:7 , and the angle of incidence of the signal wave is 5:7 . These values are used for all ®gures.  ˆ 5 Iinc ˆ 9:949  104 (W/ m2 ),  ˆ 0:5 m, coherent rand ˆ 0 , E1 ˆ pE0 exp… jrand †, incoherent 30  rand  30 ,  ˆ 2  10 5 , C ˆ 104 (W/m2 ), p ˆ 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 x-axis of the ®gure extends over one grating period  (or interference period) of the photorefractive material. The y-axis extends over the length of the crystal from the input plane y ˆ 0 to the exit plane at y ˆ 1:5 cm. The y-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 (y) and blaze like in the transverse grating vector direction (x). 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, @
=@t ˆ 0 and

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382

Chapter 7

Figure 3 Plots of the relative dielectric perturbation
 when p ˆ 1,  ˆ 2  10 5 , 0 2r ˆ 8,  ˆ 0:5 m, at time step=100. The
 shown correspond to the random case of Fig. 4. y ˆ 0 is the incident plane, and y ˆ 1:5 cm is the exit plane. All other parameters are the same as in Fig. 2.
min ˆ 3:58  10 6 ,
max ˆ 3:51  10 6 . Used with permission of Opt. Comm., 1996 [29].


 ˆ 

@I=@x~ C‡I

…7:3:18†

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 I of Eq. 7.3.14 is I…z† ˆ I0 cos2 x~

…7:3:19†

where I0 ˆ 4:443  104 …W=m2 †, C ˆ 104 …W=m2 †,  ˆ 2  10 5 ,  ˆ 0:1. Substitution of Eq. 7.3.19 into Eq. 7.3.18 shows
 ˆ

I0  sin 2x~ C ‡ I0 cos2 x~

and

…7:3:20†

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 R1 direction (see Fig. 1) when  ˆ 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

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383

Figure 4 (a) Plot of the nonlinear time factor  1 as a function of transverse distance x and longitudinal distance y when p ˆ 1,  ˆ 2  10 5 , 2r ˆ 8,  ˆ 0:5 m, and rand ˆ 0 (coherent case). (b) Comparison of the re¯ected normalized power R1 for the case of Fig. 2 when the  factor was taken to be constant ( ˆ 1) and nonlinear (a) in the cases when the incident waves were coherent (rand ˆ 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 R1 direction. Similar results occur for the normalized power in the R0 , T1 , and T0 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.

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384

Chapter 7

Figures 5 a±c show plots of the dielectric modulation
 as a function of time t (time step) and transverse dimension (fringing ®eld direction ) x when sampled at the center of the photorefractive medium y ˆ 0:75 cm. Because of the longitudinal uniformity of the
 plots (see Fig. 3), the y ˆ 0:75 cm plane is representative of
 at other longitudinal planes. Figure 5a shows the
 growth when  ˆ 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  ˆ 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  3:38  10 6 ,
min  3:36  10 6 ) than does coherent illumination (
max ˆ 3:77  10 6 ,
min ˆ 3:77  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 ˆ 0 † and when two incoherent waves … 15  rand  15 , ˆ 30  rand  30 † are incident on the photorefractive slab. The [7] is de®ned by

t

8 9 E t …L† 2 > > > 1 > > > < t = E1inc …0† ˆ 10 log > E t …L† 2 > > > > > ; : t 0 E …0† 0inc

r

8 9 E r …0† 2 > > > > 1 > < E r …0† > = 1inc ˆ 10 log > E r …0† 2 > > > > > ; : r 0 E0inc …0†

…7:3:21†

t In these expressions Ei;inc …0† …i ˆ 0; 1† is the incident electric ®eld in the ith t order direction, Ei …L† …i ˆ 0; 1† is the transmitted electric ®eld at the ®nal r …0† …i ˆ time step transmitted out of the slab in the ith order direction, Ei;inc 0; 1† is the re¯ected incident electric ®eld in the ith order direction, and Eir …0† …i ˆ 0; 1† is the re¯ected electric ®eld at the ®nal time step re¯ected out of the slab in the ith 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 2r ˆ 8,  ˆ 2  10 5 , and the incident wave's refraction ratio p ˆ jE1 j=jE0 j was varied from p ˆ 0:1 … 10 db) to p ˆ 10 …10 db†. The values of E0 and E1 were adjusted to keep the incident power Iinc ˆ 9:949  10 4 (W/m2 ) the same for all values of p 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

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Induced Photorefractive Gratings

385

Figure 5 Plots of
 versus transverse distance x and time step as measured at a plane midpoint in the PR slab (y ˆ 0:75 cm, slab length=1.5 cm) for the parameters of Figs. 2±4. (a)
 when coherent illumination was used and  ˆ 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.  ˆ 2  10 5 . Used with permission of Opt. Comm., 1996 [29].

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386

Chapter 7

Figure 6 Plots of the r and t ratio of Eq. 31 for  ˆ 2  10 5 , 2r ˆ 8,  ˆ 0:5 m, slab length =1.5 cm (Figs. 2±5) in the case when rand ˆ 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 0 decrease. Iinc ˆ 9:949  104 (W/m2 ), C ˆ 104 (W/m2 ),  ˆ 0:5 m, 2r ˆ 8, and 5  ˆ 2  10 . Used with permission of Opt. Comm., 1996 [29].

by the drop in t and r 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, p ˆ 1 ˆ jE1 j=jE0 j. 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

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Induced Photorefractive Gratings

387

show a decrease in the coupling coef®cient due to increased incoherence (see Problem 8). p0  ratio Figure 7 shows the power ef®ciency that results when the = 2r is signi®cantly increased over the value used in Figs. 2±6 in the case when the bulk index nearly matched the free space …2r ˆ 1 j10 6† and in the case 8 j2:82  10 7 ). when the PR medium p0  is mismatched to free space …2r ˆ 4 In Fig. 7 the = 2r has been taken to be 3:18  10 , 3:88  10 4 , and

Figure 7 (a) Plot of the power transmitted in the T0 and T1 directions as functions of time step in the nearly matched case when 2r ˆ 1 j1  10 6 and p ˆ 0:01 for the values of  ˆ 3:18  10 4 , 3:88  10 4 , and 4:59  10 4 when the PR slab length=1.5 cm. Also C ˆ 2:124  104 (W/m2 ). As  increases, the speed and completeness of mode conversion increases. (b) Plot of the normalized p0 power re¯ected and transmitted in the directions R0 , R1 , T0 , T1 when = 2r ˆ 4:59  10 4 , 7 4 2 2r ˆ 8 j2:82  10 , C ˆ 2:124  10 (W/m ), and p ˆ 0:01. Because of the dielectric mismatch, aperiodic variation of the normalized power results. Used with permission of Opt. Engr., 1995 [30].

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388

Chapter 7

4:59  10 4 , and fractional ratio of the incident beams has been taken to be p ˆ jE1 j=jE0 j ˆ 0:01. The waves marked T0a and T1a , T0b and T1b , and p0 T0c and T1c show the diffraction ef®ciencies for the three values of = 2r as indicated in the ®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 p not 0 shown. The main effect that is observed in Fig. 7a using the values of = 2r that were given is that a great deal of energy is transferred from the (order 0) transmitted E0 wave (which is large at t ˆ 0) to the (order 1) transmitted E1 wave (which is small at t ˆ 0). As can be seen from Fig. 7a,p the conversion of the modal 0 used and affects the value energy depends very strongly on the value = 2r of power diffracted in the T1 direction and the speed with which the mode power transfer reaches the steady state. In Fig. 7a the presence of absorption 00 ˆ 1  10 6 ) seemed to have a minimal effect on the diffraction except, (2r of course, to attenuate the T0 and T1 propagating waves. Figure 7b shows the diffraction the real part ofpthe 0  p0  that occurs when 0 ˆ 8 and = 2r bulk dielectric is r2 ˆ 4:59  10 4 . The value of = 2r 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 R0 ; T0 ; T1 , and R1 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 y ˆ 0:75 cm (this line is midway between the PR medium boundaries) as a function of time step and transverse distance x (wavelength) in 0 the casep(1) dielectric is matched (2r ˆ 1) to free space (Fig. 0  when the bulk 4 8a, = 2r ˆ 3:88  10 ), (2) when the bulk dielectric is mismatched p0  0 ˆ 4:59  10 4 ) and (3) when the ˆ 8) to free space (Fig. 8b, = 2r (2r 0 bulk p0 dielectric is 4 mismatched (r2 ˆ 8) to free space (Fig. 8c, = 2r ˆ 5:30  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

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389

Figure of
 versus time step and transverse distance x when p0 8 (a) Plots =p 2 ˆ 3:88  10 4 , 2r ˆ 1 j1  10 6 , p ˆ 0:01. (b)p
 when 4 7 0 0 = p ˆ 4:59  10 has increased , p ˆ 0:01, and  ˆ 8 j2:82  10 . (c) =  2r 2r  2r 0 ˆ 5:30  10 4 . As can be seen, the dielectric mismatch cases have a great to = 2r effect on the
 that forms in the PR medium. Used with permission of Opt. Engr., 1995 [30].

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390

Chapter 7

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 p shows a mismatched case) 0  the same case as in Fig. 8b (this isp 0 4 except that = 2r ˆ 5:30  10 . has been increased to a value =  2r p0  Because of the higher = 2r 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 00 ˆ 0 (no absorption). The resulting
 was as given in Fig. 8 except that 2r 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 x and y that results in the matched at the time step 200 (last time step  p0case p  of Figs. 7 4and 8) 4 0 (Fig. 9a), (2) =  (Fig. when (1) = 2r ˆ 3:18  10 2r ˆ 3:88  10 p0  p0  ˆ 4:59  10 4 (Fig. 9c). The three 9b), and (3) = 2r values of = 2r p0  used in Fig. 9a, b, and c correspond to values of = 2r used in Fig. 7 (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 p0  of the peak maximum and minimum values is dependent on the = 2r value used and tends to occur closer to the incident side aspthe p 0   0 increases. This occurs because the larger value of = 2r value of = 2r 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 p  0 caused a much more quickly. Notice that the higher values of = 2r 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 (L ˆ 1:5 cm/30,  ˆ 0:5 m), it was necessary when calculating the intensity I…x; y† to average this over a number of y 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

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Induced Photorefractive Gratings

391

Figure 9
 pro®les at ptime ˆ 1 j1  10 6 , slab 0  step=200, 4 p ˆ 0:01, p2r 0 ˆ 3:88  10 4 . length=1.5 cm. (a) = 2r ˆ 3:18 p (c) 10 . (b) = 2r p 4 0 interfer= 2r ˆ 4:59  10 . Higher values of = 2r cause mode conversion and  p ence in the PR medium to occur closer to the incident side as the = 2r0 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. 7.3.3

TM Numerical Simulation Results

As a TM example we consider the photorefractive grating that occurs when two equal-amplitude in-phase TM plane waves … ˆ 0:633 m† impinge on BaTiO3 crystal at an angle of incidence i ˆ 5:71 as shown in Fig. 1. The grating period formed in the crystal at this angle of incidence is  ˆ 5.

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392

Chapter 7

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 r42 ˆ 1640  10 12 m/V, r13 ˆ 8  10 12 m/V, r33 ˆ 28  10 12 m=V, nO ˆ 2:437, nE ˆ 2:365, and E00 ˆ O00 ˆ 2:42  10 6 (these values of O00 and E00 correspond to an absorption coef®cient p ˆ 1 cm 1 ), which are the values given for BaTiO3 in Ref. 8 at  ˆ 0:633 m. The electro-optic coupling constant  ˆ n2O n2E k0 r42 Ds = for the values given is  ˆ 0:0139. In the simulation we have assumed that the crystal length L ˆ 1500 ˆ 0:94 mm and that the BaTiO3 c-axis makes a 135 angle with the y-axis of Fig. 1 (c ˆ 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 ˆ 5  10 14 m3 /s,  ˆ 0:5  10 4 m2 =V s NA =ND ˆ 0:01, and [32] s ˆ 1  10 5 m2 =J s and ˆ 2 1=s. With these values we ®nd that the dark current C ˆ =s ˆ 2  105 W/m2 . For BaTiO3 and for the geometry shown in Fig. 1, the effective value of 0 0 0 0 0 s ˆ s2 cos2 c ‡ s3 sin2 c , where s1 ˆ s2 ˆ 36000 , s3 ˆ 1350 [8, Table 0 0 0 1.2, p. 28], and where s ˆ ‰si ij 0 Š, …i; i † ˆ 1; 2; 3 is the static dielectric permittivity tensor when the BaTiO3 PR crystal has its c-axis aligned along the z 0 -axis. For c ˆ 135 , s ˆ 18670 . We assume that the total incident power in the y direction of the interfering incident waves is PT INC ˆ 1:7 9 107 watts/m2 . With this value, it is found that the maximum power intensity inside the crystal is IMAX ˆ 1:60  107 watts/m2 . Using this value, we ®nd the maximum value of =s ‡ IMAX ˆ 1:62  107 watts/m2 . Using this value, we ®nd that the approximate time constant …I0 † is …I0 † ˆ 6:38 ms. The
t time constant was chosen to be
t ˆ …I0 †=5 ˆ 1:27 ms. This time step was suf®ciently small to cause a smooth change in the dielectric modulation with time. An additional run at
t ˆ …I0 †=15 ˆ 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 ‡ varies on a time scale of about assume that the ionized donor density ND …I0 †, which equals 6.38 ms. We approximate  ‡ ‡ 1 @ND 1 ND 1  ‡ ˆ 1:56  102 s ˆ ‡ …I0 † ND @t ND …I0 †

1

‡ For the ®rst term of the RHS of Eq. 7.3.5 we have …sI ‡ †ND =ND  5 7 2 1 4 1 s…I‡C †ND =NA ˆ …10 †…1:62  10 †…10 † s ˆ 1:62  10 s . The ratio of

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Induced Photorefractive Gratings

393

the second term to the ®rst term of the RHS of Eq. 7.3.9 is 0:96  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 NL ˆ 160 layers, which gave each layer a length of
L ˆ 9:375. The optical power intensity Ipoint …x; y; t† / …Ex Ex ‡ Ey Ey † (Ex and Ey are the optical electric ®elds) was calculated at 10 equally spaced points over the layer length
L ˆ L=NL ˆ 9:375 (Ipoint …x; y; t† was sampled every 0:9375). The values of Ipoint …x; y; t† 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
L represents an important part of the interaction of the incident optical light with the PR medium. Physically averaging the optical intensity over
L 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 (y-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
…x; y; t† that results for the BaTiO3 simulation at time step ˆ 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 ˆ 0:00605 and
max ˆ 0:00565. Using the relation
 ˆ r42 n2O n2E Excs , this min corresponds to a minimum and maximum space charge electric ®eld of Exsc 4 4 ˆ 11:1  10 V/m and 9:26  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

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394

Chapter 7

Figure 10 The dielectric modulation and diffracted power that results when TM optical energy illuminates BaTiO3 are shown. I inc ˆ 1:79  107 W/m2 , …I0 † ˆ 6:38 ms,
t ˆ 0:2…I0 † ˆ 1:27 ms, t ˆ 100
t, and  ˆ 0:633 m. Used with permission of Opt. Engr., 1995 [30].

zero to 300 and has a grating period of 1 ˆ 120 with grating vector given by 2 K 1 ˆ ‰sin 1 a^ x ‡ cos 1 a^ y Š 1

1 ˆ 1:5

…7:3:22†

in the geometry of Fig. 1. The second grating extends from about y ˆ 300 to 1500. This grating is more clearly de®ned and has a grating period of about 2 ˆ 75. 2 K 2 ˆ ‰ sin 2 a^ x ‡ cos 2 a^ y Š 2

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2 ˆ 4:5

…7:3:23†

Induced Photorefractive Gratings

395

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 T0 and T1 directions. Within a few time steps, a modulation grating forms and power is rapidly depleted from the T1 direction and transferred to the T0 direction. As can be seen from Fig. 10b, the T0 modal direction is completely depleted of power. As time progresses over a period of approximately 30 time steps, diffracted waves in the T 1 direction build up, and ®nally a strong mode conversion from the T0 to the T 1 order occurs and the grating goes into a quasi±steady-state form. The T0 mode drops to about 10% of the total diffracted power, and the T 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 R0 , R1 , and R 1 directions follows a similar time history as did the transmitted except that the R1 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 LiL ˆ 1500 ‡ iL 18:75, iL ˆ 1; 2; 3; 4. An interesting result of the analysis was that for the lengths of 1518:75 …iL ˆ 1† and 1556:25 …iL ˆ 3† simulation showed that diffraction in the T 1 and R1 directions did not occur at all where they did occur for the values of iL ˆ 2 and iL ˆ 4. Evidently the growth of the T 1 and R1 mode perturbations depends on a resonant length of the crystal. 7.3.4

Discussion of Results from Approximate Analysis

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-

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396

Chapter 7

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 L 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, NA and ND 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. 7.4

EXACT ANALYSIS [34]

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 ND was much greater than the acceptor density NA , the ‡ electron density n, and the ion density ND . The effect of the approximation is that the production of electrons is not limited as the electron donors ND are depleted. The second approximation consists of assuming that the term

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Induced Photorefractive Gratings

397

‡ @ND =@t 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 ‡ @n @ND @nEs @2 n ˆ ‡ ‡ Ds 2 @t @x @t @x

…7:4:1†

Using Eq. 7.2.4 we ®nd that ‡ ND ˆ

s @Es ‡ NA ‡ n e @x

…7:4:2†

or ‡ @ND  @2 Es @n ˆ s ‡ @t e @x@t @t

…7:4:3†

‡ where @NA =@t ˆ 0 has been used in Eq. 7.4.3. If the @ND =@t of Eq. 7.4.3 is substituted into Eq. 7.4.1 and @n=@t is cancelled on the right- and left-hand sides of the resulting equation, we ®nd that

s @2 Es @…nEs † @2 ‡ Ds n2 ˆ 0 ‡ @x e @x@t @x or, after integration with respect to x,

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…7:4:4†

398

Chapter 7

@Es e ‡ nEs ˆ s @t

Ds

e @n s @x

…7:4:5†

In Eq. 7.4.5 we have set the integration constant to zero, assuming all dependent variables and their derivatives tend to zero as jxj ! 1. For a particular choice of c-axis of the PR crystal, the dielectric permittivity modulation
 is related to the electrostatic electric ®eld by
…x; y; t† ˆ n2O n2E r42 Es …x; y; t†

…7:4:6†

Substituting Es from Eq. 7.4.6, we ®nd for
: @
 e ‡ …n
† ˆ @t s

Ds e @n r42 n20 n2e s @x

…7:4:7†

Up to this point no use has been made of Eq. 7.4.2, the electron rate ‡ production equation. If @ND =@t of Eq. 7.4.3 is substituted into Eq. 7.4.2, we ®nd that s @2 Es @n ‡ ˆ …sI ‡ † …ND e @x@t @t

‡ ND †

‡

R nND

…7:4:8†

If Eq. 7.4.4 is used to eliminate the …s =e† @2 Es =@x@t and Nd‡ of Eq. 7.4.2 is substituted, we ®nd that nE  s @x

 @2 n @n s @Es Ds 2 ‡ ˆ …sI ‡ † ND NA e @x @t @x   s @Es ‡ NA ‡ n

R n e @x

 n …7:4:9†

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 C ˆ =s, sI ‡ ˆ sC…1 ‡ I=C†, x~ ˆ k0 x, t~ ˆ t n~ ˆ n=NA , and using Eq. 7.4.6 to express Es in terms of
, we ®nd that @2 n~ ‡ 2 @x~ 2

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   @n~ I @
 ‡ ‡… 1 1
 3 1‡ 4† @x~ C @x~     @n~ I ND 1 @
 ˆ 3 1‡ C NA @t~ 3 @x

 3 4 …1

~ n ‡ n† …7:4:10†

Induced Photorefractive Gratings

399

where 1

ˆ

eNA s

4

ˆ

R N A

2

ˆ

ek0 Ds NA n20 n2e r42 s

3

ˆ

eNA n20 n2e r42 s k0 …7:4:11†

Also Eq. 7.4.7 in normalized form can be written as @
 ‡ @t~

~ 1 n


ˆ

2

@n~ @x~

…7:4:12†

Equations 7.4.10 and 7.4.12 are a pair of coupled nonlinear equations for the electron density n~ and dielectric modulation function
. The form of both these equations for n~ and
 at any given time and at any given point in the PR medium depends on the value of the optical intensity I at that point in space and time. The value of I 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.

7.4.1

Finite Difference Kukhtarev Analysis

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 t ˆ 0 (®rst time step) a signal ( ˆ i ) and pump wave ( ˆ 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

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400

Chapter 7

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 @
=@t~ is approximated as ~ y; t~ ‡ t~† @

…x;  ~ @t t~

~ y; t~†
…x;

and after substitution in Eq. 7.4.12 and cross multiplication by t~ we ®nd that   @n~ ~ ~ ~ ~ y; t ‡
t† ˆ
…x; ~ y; t† ‡ ~
…x; @t~ …7:4:13† 1 n
 2 @x~ This equation is used to advance the
 dielectric modulation function in time. Equation 7.4.10 is used to ®nd the electron density n~ for Eq. 7.4.13. Its ~ determination from Eq. 7.4.10 proceeds as follows. Forming an x-grid  p,   ˆ k0 , we let x~ p ˆ …p 0:5†x, ~ system of Np divisions, x~ ˆ =N p ˆ 0; 1; . . . ; Np , Np ‡ 1 be sampled values of x~ over the grating period of the PR material at a longitudinal distance y. The points p ˆ 0 and Np ‡ 1 extend one point outside the grating period. These need to be included in order to specify periodic boundary conditions. We also let V…p† ˆ V…x~ p ; y; t~† ~ ˆ n… ~ x~ p ; y; t~†, be any sampled dependent variable (for example n…p†
…p† ˆ
…x~ p ; y; t~†, etc.) of Eqs. 7.4.10 or 7.4.12 in the grating period of the PR material at a longitudinal distance y. 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 n~ ~ ‡ 1† 2n…p† ~ ‡ n…p ~ ~ ‡ 1† n…p ~ @n~ n…p 1† n…p 1† x~ p  @x~ 2 x~ p  2 ~ ~ @ x 2 x ~ …x† …7:4:14† Letting P…p† ˆ

1
…p†

  I…p† @
 1 ‡ † ‡ … 3 1 4 ~ x~ p C @x~ xˆ     ~ @n…p† I…p† ND 1‡ S…p† ˆ 3 1 @t C NA

H…p† ˆ

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3 4 …1

~ ‡ n…p†† …7:4:15†

Induced Photorefractive Gratings

401

we ®nd for p ˆ 1; . . . ; Np  2

~ ‡ 1† n…p

~ ‡ n…p ~ 2n…p† 2 ~ …x†

1†





 1†

~ ‡ 1† n…p ~ n…p ‡ P…p† ~ 2x ~ ˆ S…p† ‡ H…p†n…p†

~ ‡ 1†, n…p†, ~ ~ If coef®cients of n…p n…p in the following form

…7:4:16†

1† are collected, Eq. 7.4.16 can be put

~ ‡ 1† ‡ R…p†n…p† ~ ‡ R …p†n…p ~ R‡ …p†n…p

1† ˆ S…p†

…7:4:17†

where R‡ …p† ˆ

P…p† ‡ 2 2x~ x~ 2

R…p† ˆ H…p†

2 2 x~ 2

R …p† ˆ

P…p† ‡ 2 2x~ x~ 2

and where p ˆ 1; 2; . . . ; Np . Because the diffraction grating is periodic, we have the important ~ ˆ n…N ~ p † and boundary conditions on the variable that n…0† ~ ~ p ‡ 1† ˆ n…1†. These equations can be used to eliminate the variables n…N ~ ~ p ‡ 1† in Eq. 7.4.16 and thus give an equation that depends n…0† and n…N ~ only on the unknowns n…p†, p ˆ 1; . . . ; Np . We thus observe that Eq. 7.4.17 ~ p ˆ 1; . . . ; Np . represents a system of Np equations in Np unknowns n…p†, The system of ®nite difference equations given by Eq. 7.4.17 can be conveniently expressed in terms of a matrix equation: 2

0 0 R…1† R‡ …1† 6 6 R …2† R…2† R‡ …2† 0 6 6 6 0 R …3† R…3† R‡ …3† 6 6 6 0 0

6 6 6 0 0 0
6 Lˆ6 6 0 0 0 0 6 6 6 0 0 0 0 6 6 6 0 0 0 0 6 6 6 0 0 0 0 4 R‡ …Np † 0 0 0  T S ˆ S…1† S…2†


S…Np †

0 0

0

0

0

R …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

0







0 0

R …Np

0 0

0

0 0 0  n ˆ n…1† n…2†

L n~ ˆ S


2†

R…Np R …Np

0 2† 1†

0


n…Np †

0

R‡ …Np R…Np T

2† 1†

R …Np †

0 R‡ …Np

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 1† 7 5

R…Np †

…7:4:18†

Inverting this matrix equation gives n~ ˆ L 1 S

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…7:4:19†

402

Chapter 7

which thus determines the electron density pro®le for a given value of y in the PR medium. ~ x; ~ y; t~† is determined [n…p† ~ is The algorithm proceeds as follows. Once n… ~ ~ x; ~ y; t~†] and its derispeci®es n… determined from matrix inversion and n…p† ~ x; ~ y; t~†=@x~ is calculated, n… ~ x; ~ y; t~† and @n… ~ x; ~ y; t~†=@x~ are substituted vative @n… ~ y; t~ ‡ t~† is found from Eq. back into Eq. 7.4.13. Once a new value of
…x; ~ y; t~ ‡ t~† is calculated for all values of y (all 7.4.13, this new value of
…x; discrete layers of the PR slab). Once this step is completed, RCWA is used ~ y; t~ ‡ t~†, and thus a new to study diffraction from the new value of
…x; ~ y; t~ ‡ t† is found. The new intensity I…xy; ~ t~ ‡ t~†, optical intensity value I…x; ~ y; t~ ‡ t~† and its x derivative, is substituted into Eqs. 7.4.17 along with
…x; ~ x; ~ y; t~ ‡ t~† is found. By repeating the above and 18 and a new value of n… steps for many iterations, the time evolution of the PR material and the optical diffracted intensity can be found. 7.4.2

TM Numerical Simulation Results

We consider the photorefractive grating that occurs when two in-phase TM plane waves ( ˆ 0:633 m) of amplitudes E0 and E1 impinge on BaTiO3 crystal at an angle of incidence i 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 r42 ˆ 1640 10 12 m/V, r13 ˆ 8  10 12 m/V, r33 ˆ 28  10 12 m/V, nO ˆ 2:437, nE ˆ 2:365, and E00 ˆ O00 ˆ 2:42  10 6 (these values of O00 and E00 correspond to an absorption coef®cient p ˆ 1 cm 1 ), which are the values given for BaTiO3 by [8] at  ˆ 0:633 m. In the simulation we have assumed that the BaTiO3 c-axis makes a 45 angle with the y-axis of Fig. 1. We further assume [31,32] R ˆ 5  10 14 m3 /s,  ˆ 0:5  10 4 m2 =V s, NA ˆ 3  1022 , ND ˆ 200NA , s ˆ 1  10 5 m2 =J s, and ˆ 2 1=s. With these values we ®nd that the dark current C ˆ =s ˆ 2  105 W/m2 . For BaTiO3 and for the geometry shown in Fig. 1, the effective value of 0 0 0 cos2 c ‡ s3 sin2 c , where s1 ˆ s2 ˆ 36000 ; s3 ˆ 1350 [8, Table s ˆ s2 0 0 0 1.2, p. 28], and s ˆ ‰si i;j 0 Š, …i; i † ˆ 1; 2; 3 is the static dielectric permittivity tensor when the BaTiO3 PR crystal has its c-axis aligned along the z 0 -axis. For c ˆ 45 , s ˆ 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 L ˆ 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  ˆ 1; 2; 5; and 10 at y ˆ L=2. The power intensity has the approximate intensity pro®le of a squared sinusoidal wave. The intensity

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Figure 11 (a) The optical power intensity (normalized to the dark current C) in the grating when the grating period is  ˆ 1, 2, 5, and 10 as a function of the normalized grating distance xN ˆ x=. The incident power (evaluated at y ˆ L=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 y ˆ L=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].

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Chapter 7

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  ˆ 1, the curve marked ``2'' corresponds to a grating period  ˆ 2, and so on. The
 of each of the curves has been plotted as a function of the normalized variable xN ˆ x= 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  ˆ 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 n and
, namely Eq. 7.4.10, contains zero-, ®rst-, and second-order x derivatives. Thus when  is small …K ˆ 2= is large) the second-order x 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 E0 and E1 ˆ 0:1E0 are incident on an index matched PR crystal of length L ˆ 1530,  ˆ 0:633 m. The angle of incidence is such as to make the grating period  ˆ 5. Figure 12c shows the normalized power transmitted in the T0 and T1 directions as a function of time step, and Fig. 12d shows the normalized power re¯ected in the R0 and R1 directions as a function of time step (t ˆ 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 R0 and R1 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 T0 to the T1 in a period of about 35 ms, at which time the grating dynamics rapidly approaches the steady state. The power in the

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Figure 12 The dielectric perturbation function
 (b) and the power transmitted in the T1 , T0 , (c) R1 , and (d) R0 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].

T0 and T1 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 t ˆ 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 y ˆ 0 to a peak-to-peak value of
max
min ˆ 0:00642, which occurs at about y  1000. It may be noticed that the grating pro®le is slightly skewed at y  1000. This may be a slight nonlinearity effect. The simulation grid used 34 x 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 NL ˆ 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

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406

Chapter 7

over 10 points in each y ˆ L=NL to ensure that a smooth and physically realistic optical intensity pro®le I…x; y; t† 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 t ˆ 1 ms. Very stable numerical results were obtained for the grid parameters and time step used. The RCWA analysis was carried out for MT ˆ 2 (i ˆ 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 (i ˆ 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 ED 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 E0 and E1 ˆ 0:1E0 are incident on a PR crystal,  ˆ 0:633 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 ˆ 3 ˆ 1). In this case optical energy is multiply re¯ected and subsequently diffracted from the y ˆ 0 and y ˆ L 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 L in order that the effect of crystal length on grating formation could be fully studied. For the values of Lp ˆ 1482:1875 ‡ p
L,
L ˆ 9:5625, p ˆ 1; 2; 3; 4; 5; 6, Figs. 13a and 13b show the power transmitted in the T0 and T1 directions, respectively, of Fig. 12a, and Figs. 13c and 13d show the power re¯ected in the R0 and R1 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 L of the PR crystal can

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Figure 13 The power transmitted in the T0 , T1 , R0 , and R1 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 ˆ 3 ˆ 1) in (a), (b), (c), and (d), respectively, for six slightly different PR crystal lengths L. All grating parameters used in the simulation not listed on the ®gure are given in Section 7.3. The starred line on the L5 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 T1 direction for the L4 length is small (about 12% in steady state), whereas when the length is increased to L6 the transmitted power jumps to a large value (about 50%). A length change of only about 2
L  19 has occurred. A second interesting feature of the plots is that depending on the length L, the power transmitted (T0 and T1 directions) or re¯ected (R0 and R1 directions) may go into an oscillatory steady state or a nonoscillatory steady state. In Fig. 13b, it is observed that the L1 and L5 lengths form oscillatory steady states, whereas the lengths L2 , L3 , L4 and L6

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Chapter 7

Figure 14 The dielectric perturbation function
 that results in the index mismatched case of Fig. 13 when the PR crystal length is L5 ˆ 1530 and (a) t ˆ 56 ms, (b) t ˆ 90 ms (oscillatory steady state, see Fig. 13, L5 ˆ 1530). (c) The dielectric perturbation function
 that results in the index mismatched case of Fig. 14 when the PR crystal length is L3 ˆ 1510:875 and t ˆ 90 ms (nonoscillatory steady state, see Fig. 14, L3 ˆ 1510:875). 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 L3 ˆ 1510:875 and when the crystal length is L5 ˆ 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 L. For example the period of the length L1 is about 50 ms, whereas the length L5 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 L5 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 L ˆ L5 ˆ 1530 (the grating system for this

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409

length is an oscillatory steady state, see Figs. 13a±d) at times t ˆ 56 ms and t ˆ 90 ms. Figure 14c shows the
 dielectric modulation function that results when L ˆ L3 ˆ 1510:875 at a time t ˆ 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 T0 , T1 , R0 , R1 . In Fig. 14a at time t ˆ 56 ms (L ˆ L5 ˆ 1530) we notice
max
min ˆ 1:57  10 3 , whereas in Fig. 14b at t ˆ 90 ms (L ˆ L5 ˆ 1530†
max
min ˆ 5:31  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 (t ˆ 1 ms) when the PR crystal length is L ˆ L3 ˆ 1510:875 and when the PR crystal length is L ˆ L5 ˆ 1530. As can be seen from the plots of Fig. 14d, the L ˆ L5 ˆ 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 L ˆ L3 ˆ 1510:875 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 L 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 E0 and E1 ˆ 0:4E0 are incident on a PR crystal ( ˆ 0:633 m, L ˆ 1530 ˆ L5 ) that is not index matched to the surrounding medium (free space is assumed to surround the PR crystal, 1 ˆ 3 ˆ 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 (L ˆ L5 ˆ 1530 curves) only in that the signal amplitude

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410

Chapter 7

Figure 15 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 E0 and E1 ˆ 0:4E0 are incident on a PR crystal ( ˆ 0:633 m, L ˆ 1530 ˆ L5 ) that is not index matched to the surrounding medium (free space is assumed to surround the PR crystal, 1 ˆ 3 ˆ 1). (a) The dielectric modulation function
 that results when NL ˆ 160 layers is used. (b) The results when NL ˆ 640 layers is used. (c) The transmitted and re¯ected power diffracted in the zero and ®rst orders when NL ˆ 640. (d) The power transmitted in the second order when NL ˆ 160, 320, and 640 layers. The dashed line shown in (d) shows the transmitted power that is diffracted in the second order when NL ˆ 160 layers and MT ˆ 3. Used with permission of JOSA-A, 1996 [34].

E1 is E1 ˆ 0:4E0 rather E1 ˆ 0:1E0 . In this case, as can be seen from Figs. 15c and 15d, power is initially diffracted into the ®rst-order diffraction mode directions T1 and R1 and then later at about t ˆ 100 ms diffracted into the T2 and R2 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 T0 attains a quasi±steady state while there is power exchange

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between T1 and T2 . 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] I2 ˆ

 2  sin2  Q

…7:4:20†

where Q ˆ K 2 L=k0 and  ˆ k0 …
n†pp Leff =4. In the above expression K is the hologram wave number, Leff is the effective grating thickness, and …
n†pp denotes the peak-to-peak change in induced refractive index. In our case, with the hologram spacing  ˆ 5 and Leff  L=2 ˆ 1530=2,  ˆ 0:633 m and Q ˆ 200. Also from the plots (Fig. 16b) the peak value of …
n†pp  0:00134 (since (
n†pp  …
†pp =…2n0 †,
pp 
max
min ˆ 0:00654 (see Fig. 16b), n0 ˆ 2:437) at t ˆ 200 ms implying   1:6. Note that this approximately corresponds to the condition for maximum diffracted power in the second order. Equation 7.4.20 shows that as Q 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 ˆ 3 ˆ 1) when the incident amplitudes were E0 and E1 ˆ E0 for the cases when L ˆ 1500, In these cases very interesting results occurred. In the case when L ˆ 1500; the power in the T0 order was diffracted and the mode converted into the T1 order, and no other appreciable diffraction occurred. The diffraction in the T0 and T1 directions was observed to be in a nonoscillatory steady state as time increased. For L ˆ 1500, the peak-to-peak …
†pp 
max
min ˆ 3:86  10 3 which made (
n†pp  0:79  10 3 . Further, it was observed that the peak-to-peak dielectric modulation function (
†pp 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 Leff  L=2 ˆ 1500=2. In the case when L ˆ 1530, power was initially diffracted from the T0 order to the T1 order and then subsequently at

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412

Chapter 7

about t ˆ 75mg was diffracted from the T1 order to the T2 second order. For this case, the peak-to-peak dielectric modulation function (
†pp 
max
min ˆ 5:33  10 3 , which made …
n†pp  1:09  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 L ˆ 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 n~ ˆ n=NA that result from the simulation shown in the matched case of Fig. 13 (t ˆ 113 ms, y ˆ L=2, L ˆ 1530,  ˆ 0:633 m,  ˆ 5) and the mismatched case of Figs. 14 and 15 (t ˆ 200 ms, y ˆ L2, L ˆ L5 ˆ 1530,  ˆ 0:633 m,  ˆ 5). As can be seen from these plots, we ®rst notice that the results of the simulation show that the normalized electron density n~ ˆ n=NA assumes a very small value of n~  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 n~ and
 are related by a ®rst space and time derivative equation. 7.5

REFLECTION GRATINGS [37,38]

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. 7.5.1

RCWA Optical Field Analysis

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

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Figure 16 (a) Plot of the normalized electron density n~ ˆ n=NA that results from the simulation shown in the matched case (t ˆ 113 ms, y ˆ L=2, L ˆ 1530,  ˆ 0:633 m,  ˆ 5) and the mismatched case (t ˆ 200 ms, y ˆ L=2, L ˆ L5 ˆ 1530,  ˆ 0:633 m,  ˆ 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].

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Chapter 7

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. ~ y ˆ k0 y, ~ etc., Normalizing all space coordinates according to x ˆ k0 x, where k0 ˆ 2= and  in meters is the free space wavelength, the EM ®elds E and H in the `th layer (the ` index is suppressed) of the RG are given by S~ ˆ

X ‰Sxi a^ x ‡ Syi a^ y Š exp… jkxi x ‡ ji y† i

X ‰Uzi a^z Š exp… jkxi x ‡ ji y† U~ ˆ

…7:5:1†

i

  0 ˆ 377 , kxi ˆ p ~ sin…'†, where S ˆ E, U ˆ 0 H; 1 sin I i…=† ~ cos…'†, and  ˆ 0 . The angle ' [39±41] is the tilt angle of i ˆ i…=† 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 S and U, 2

r  S~ ˆ

j U~

r  U~ ˆ jS~

xx where  ˆ 4 ys 0

yy yy 0

3 0 0 5 zz …7:5:2†

In this chapter the anisotropic permittivity tensor is assumed to have its caxis at C ˆ 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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Figure 17

415

Problem geometry.

 @V a ˆ 11 a21 @y

 a12 V a22

…7:5:3†

where V ˆ ‰fStx Utz Št , Sx ˆ ‰Sxi Š, Uz ˆ ‰Uzi Š, i ˆ h i a11 ˆ j K yy1 yx m h i a21 ˆ j xx xy yy1 yx   K ˆ kxi i;i 0   m ˆ i i;i 0

MT ; . . . ; MT , and

i K yy1 K ‡ I h i ˆ j m ‡ xy yy1 K

a12 ˆ j

h

a22 h i  ˆ  i i 0 …; † ˆ …x; y† ( 1 for i ˆ i 0 i;i 0 ˆ 0 for i 6ˆ i 0

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. 7.5.2

Material Analysis

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 x-direction and that a PR re¯ection grating is formed in the y-longitudinal direction. Following the analysis of transmission gratings above, but taking care to use the longitudinal spatial variable y (the direction of the re¯ection grating vector), we

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Chapter 7

®nd that Kukhtarev's material equations can be reduced to differential equations     @n I…y† @
 ‡ ‡‰ 1 1
 3 1‡ 4Š 3 4 ‰1 ‡ Š n @y C @y       @n I…y† ND I…y† @
 1‡ …7:5:4† ˆ 3 1 ‡ 1‡ @t C C @y NA

@2 n ‡ 2 @y2

@
 ˆ @t

1 n


2

@n @y

…7:5:5†

In these equations I…y† is the optical intensity W/m2 and
 ˆ n20 n2e r42 Es …y† is a normalized dielectric modulation function linearly related to the longitudinal electrostatic ®eld Es …y† (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 I…y† and the material variables n…y† and
…y† in a spatial Fourier series (the period of the Fourier series is the grating wavelength ~ where the Fourier amplitudes are all spatially varying functions  ˆ k0 ) of the longitudinal coordinate y. We have I…y† ˆ

iˆM Xc iˆ Mc

n…y† ˆ

iˆM Xc iˆ Mc


…y† ˆ

iˆM Xc iˆ Mc

Ii …y† exp…jiy†

…7:5:6†

ni …y† exp…jiy†

…7:5:7†


i …y† exp…jiy†

…7:5:8†

where  ˆ 2= and Mc 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…jiy† are equated, we ®nally obtain @2 ni …y† @n …y† ‡ 2ji i 2 @y @y



 2 ‡ …i†2 ni …y† ˆ Hi …y†

iˆ

M C ; . . . ; MC …7:5:9†

where

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 ˆ

3 2

…1 ‡

4†

417

1=2

and Hi ˆ

X i0

Ti

i 0 ni 0

‡ Fi

…7:5:10†

and where Ti ˆ

ji

1 2


i ‡

  Ii 2 C 3

1

4 2



 @
i ‡ ji
i ‡ @y

   X @n 0 @ni I ND 1 i;0 ‡ i Fi ˆ
i i 0 i ‡ 3 @y @t C NA 2 i0 2     1 X @
i i 0 Ii 0 0 ‡ j…i i †
i i 0 i 0 ;0 ‡ ‡ @y C 2 i0

3 4



2

ni …7:5:11†

1 …7:5:12†

~ L~ (in meters being Equation 7.5.12 for the interval L  y  0 …L ˆ k0 L, the crystal layer thickness) represents a set of 2MC ‡ 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 n…y† vanishes at the crystal interfaces y ˆ 0 and y ˆ L. This boundary condition further imposes the boundary conditions on Eq. 7.5.12 that ni …0† ˆ ni … L† ˆ 0

iˆ

M C ; . . . ; MC

…7:5:13†

To proceed further we now for the moment regard the RHS of Eq. 7.5.12 as a known function y. Equation 7.5.12 along with its boundary conditions, for each i, 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 …y y 0 †; (2) solving the resulting differential equation @2 gi …yjy 0 † @gi …yjy 0 † ‡ 2ji @y @y2

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 2 ‡ …i†2 gi …yjy 0 † ˆ …y

y 0†

…7:5:14†

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Chapter 7

with the boundary conditions and continuity condition, respectively gi …0jy 0 † ˆ gi … Ljy 0 † ˆ 0 0

gi …yjy 0 † yˆy 0 ˆ gi …yjy 0 † yˆy 0 ‡

…7:5:15†

0

where y and y ‡ represent locations an in®nitesimal to the left and right of y 0 ; (3) superposing the Green's function solutions times the nonhomogeneous RHS Hi to ®nd the overall response of the system. Regarding Hi …y† as a known function, the solution for ni …y† is given by ni …y† ˆ

…0 L

gi …yjy 0 †Hi …y 0 † dy 0

…7:5:16†

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  ˆ 2=. Investigation of the Green's function gi …yjy 0 † 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 y ˆ y 0 in the interval. Investigation has further shown that most of the exponential terms are exponentially small. After analysis it is found that gi …yjy 0 † is well approximated by [37] 0

gi …yjy † ˆ



1 s1

s2

exp‰s1 …y exp‰s2 …y

y 0 †Š y 0 †Š

L  y < y0 y
…7:5:17†

where s12 ˆ  ji. gi …yjy 0 † is signi®cantly nonzero only when jy y 0 j  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…jiy† functions. Using this assumption one notices that in the integral of Eq. 7.5.16 the Hi …y 0 † function can be approximately taken as constant over the range where gi …yjy 0 † of Eq. 7.5.17 is nonzero. With the approximations that (1) Hi …y 0 † is taken as constant and evaluated at y ˆ y 0 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 y 0 ˆ 1 to y 0 ˆ 1, it turns out that the integral of Eq. 7.5.16 can be evaluated in closed form. The result of the integration is ni …y† ˆ where

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Gi Hi …y†

…7:5:18†

Induced Photorefractive Gratings

Gi ˆ

2

419

1 ‡ …i†2

…7:5:19†

Substituting Eq. 7.5.10 into Eq. 7.5.18 and collecting terms on ni …y†, we derive a matrix equation for the normalized electron density ni …y†: X i0

i;i 0 ‡ Gi Ti

i0



ni 0 …y† ˆ

X i0

Li;i 0 ni 0 …y† ˆ

Gi Fi

…7:5:20†

To proceed further we substitute the spatially varying approximations Eqs. 7.5.6±7.5.8 for n…y† and
…y† into Eq. 7.5.5 and ®nd after equating like coef®cients of exp…jiy) to each other that @
i ˆ @t

X ‰ ni 1 i0

i0


i 0 Š

  @ni ‡ jin 2 i @y

…7:5:21†

If Eq. 7.5.20 is substituted for ni …y†, and matrix terms common to
i are collected in Eq. 7.5.21, it can be placed in the usual state variable form:  @
i X ‡ Ai;i 0
i 0 ˆ fi @t i0

…7:5:22†

where Ai;i 0 and fi are de®ned in Ref. 37, where the complete temporal numerical solution is detailed. 7.5.3

Numerical Results

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 r42 ˆ 1640  10 12 (m/V), r13 ˆ 8  10 12 …m=V†, no ˆ 2:437 and ne ˆ 2:365, which are the given crystal values for BaTiO3 by Ref. 8 at  ˆ 0:633 m. For these values the ~ ˆ 0:132 m. In the simulation we grating wavelength had the value  have assumed that the BaTiO3 c-axis makes a 45 and with the y-axis of Fig. 18. We further assume R ˆ 5  10 14 (m3 /s),  ˆ 0:25  10 4 (m2 /V s), NA ˆ 3  1022 (1/m3 ), ND ˆ 200NA , s ˆ 1  10 5 (m2 /J s) and ˆ 2 (1/ s). With these values we ®nd that the dark current is C ˆ 2  105 (W/m2 ). For BaTiO3 and for the geometry shown in Fig. 18, the effective value of 0 0 0 0 0 s ˆ s2 cos2 …c † ‡ s3 sin2 …c †, where s1 ˆ s2 ˆ 36000 s3 ˆ 1350 [8, Table 0 0 1.2, p. 28] and where s ˆ ‰si i;i 0 Š, …i; i † ˆ 1; 2; 3, is the static dielectric permittivity tensor when BaTiO3 PR crystal has its c-axis aligned along

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420

Chapter 7

the z 0 -axis. The optical and material analysis was carried out using values of ~ ˆ 0:132 mm,  ˆ =k ~ 0 ˆ 4:795, MT ˆ 5 and MC ˆ 10. Also L~ ˆ 1000   ˆ 4:858, and I ˆ 6 . In Ref. 37, we have shown the plots of the dielectric modulation function
…y; t† [proportional to the electrostatic space charge Es …y; t†] as ~ ~ in meters,  ˆ k0 , 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 t~ ˆ 0:059, 0.477, 4.97, 30.1, and 69.0 ms. Also plotted was the re¯ected diffraction ef®ciency DER …%† 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 t~ ˆ 30:1 ms. These plots were all done only for a tuned layer and only for a thickness of 1000. Also in Ref. 37, no study was done of the convergence of the numerical solution with different spatial harmonics MC (number of harmonics in the material equations) and Mt (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 MC and Mt from MC ˆ 1 and Mt ˆ 1 to MC ˆ 14

Figure 18 Transient re¯ected diffraction ef®ciency for different optical and material harmonics. Used with permission of JOSA-B, 1998 [37].

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421

and Mt ˆ 17. As can be seen from Fig. 18, reasonable convergence is achieved for values of MC ˆ 6 and Mt ˆ 3 or higher. A nominal combination of MC ˆ 10 and Mt ˆ 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, 0:25, 0:375, and 0:5. 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.

Figure 19 Transient re¯ected diffraction ef®ciency for different tuned layer thicknesses. Used with permission of JOSA-B, 1998 [37].

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422

Chapter 7

Figure 20 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, L ˆ 0) and when a nontuned layer re¯ection grating is formed (Fig. 21b, L ˆ 0:5). 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 yloc 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, L ˆ 0) and when a nontuned layer re¯ection grating is formed (same case as Fig. 21b, L ˆ 0:5). As can be seen from Fig. 21, detuning of the layer causes a shift in the dielectric modulation function
. 7.5.4

Conclusion

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

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423

Figure 21 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 L ranging from 0 to 0:5. Numerical convergence was also analyzed. We have found that a minimum number of spatial harmonics (viz., MC ˆ 6 and Mt ˆ 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.

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424

Chapter 7

Figure 22 Typical induced permittivity distributions for a tuned and detuned layer. Used with permission JOSA-B, 1998 [37].

PROBLEMS 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 pI on the righthand side, where p is the photovoltaic tensor element and I is the optical intensity. 2. Assume that a thin photorefractive layer has incident on it two plane waves of amplitude ratio p. (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 x over a spatial period  formed by the interfering plane waves. Plot the electrostatic ®eld Es 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 Es is well approximated by means of the intensity dependent time constant …I† de®ned in Eq. (7.3.8). (d) Compare your results with the ®ndings of Ref. [42].

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425

3. (a) Decouple and linearize the Kukhtarev equations (7.2.1±7.2.4) in the small contrast approximation (small m). (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 t ! 1 and compare your answer with part (c). 4. Repeat Problem 3 but now including a photovoltaic term in the Kukhtarev equations (see Problem 1). 5. 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 m). Analyze the resulting equations and ®nd an expression for the space charge ®eld in terms of the intensity and the frequency offset [8]. 6. Assume that two optical waves which are frequency offset from one another by , as in Problem 5, are incident on a photorefractive material at t ˆ 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 ? 7. 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? 8. 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 j rand j  158 and 308, respectively. Make several independent computations for these values of rand , and average these values for each E1 =E0 ratio as shown in Figure 5. Compare your results with the coherent case.

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426

Chapter 7

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18. A. Nowak, T. R. Moore, and R. A. Fisher, Observation of internal beam production in BaTiO3 phase conjugator, J. Opt. Soc. Am. B, 5, 1864±1878 (1988). 19. P. M. Jeffrey and R. W. Eason, Lyapunov exponent analysis of irregular ¯uctuations in a self-pumped BaTiO3 phase conjugate mirror, J. Opt. Soc. Am. B, 11, 476±480 (1994). 20. D. Wang, Z. Zhang, X. Wu, and P. Yeh, Instabilities in a mutually pumped phase conjugator of BaTiO3 , J. Opt. Soc. Am. B, 7, 2289±2293 (1990). 21. R. Daisy and B. Fischer, Light waves at the interface of linear and photorefractive media, J. Opt. Soc. Am. B, 11, 1059±1063 (1994). 22. E. D. Baraban, H.-Z. Zhang, and R. W. Boyd, Optical bistability by two-wave mixing in photorefractive crystals, J. Opt. Soc. Am. B, 9, 1689±1692 (1992). 23. L.-K. Dai, C. Gu, and P. Yeh, Effective of position-dependent time-constant on photorefractive two-wave mixing, J. Opt. Soc. Am. B, 9, 1693±1697 (1992). 24. H. Kong, C. Wu, and M. Cronin-Golomb, Photorefractive two-beam coupling with reduced spatial coherence, Opt. Lett., 16, 1183±1185 (1991). 25. M. Cronin-Golomb, H. Kong, and W. Krolikowski, Photorefractive two-beam coupling with light of partial spatio-temporal coherence, J. Opt. Soc. Am. B, 9, 1698±1703 (1992). 26. M. Snowbell, M. Horowitz, and B. Fischer, Dynamics of multiple two-wave mixing and fanning in photorefractive materials, J. Opt. Soc. Am. B, 9, 1972± 1982 (1994). 27. E. Serrano, V. Lopez, M. Carrascosa, and F. Aguillo-Lopez, Recording and erasure kinetics in photorefractive materials at large modulation depths, J. Opt. Soc. Am. B, 11, 670±675 (1994). 28. W. P. Brown and G. C. Valley, Kinky beam paths inside photorefractive crystals, J. Opt. Soc. Am. B, 10, 1901±1906 (1993). 29. J. Jarem and P. P. Banerjee, A nonlinear, transient analysis of two and multiwave mixing in a photorefractive material using rigorous coupled-mode diffraction theory, Opt. Com., 123, 825±842 (1996). 30. P. P. Banerjee and J. Jarem, Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled mode approach, Opt. Engr., 34, 2254±2260 (1995). 31. P. P. Banerjee and J. J. Liu, Perturbational analysis of steady-state and transient-beam fanning in thin and thick photorefractive media, J. Opt. Soc. B., 10(8), 1417±1423 (1993). 32. P. P. Banerjee and R. M. Misra, Dependence of photorefractive beam fanning on beam parameters, Opt. Commun., 100, 166±172 (1993). 33. P. P. Banerjee, H. L. Yu, D. A. Gregory, and N. Kukhtarev, Phase conjugation, edge detection and image broadcasting using two-beam coupling con®gurations in photorefractive KNbO3 , Opt. Laser Tech., 28, 89±92 (1996). 34. J. M. Jarem and P. P. Banerjee, An exact dynamical analysis of the Kukhtarev equations in photorefractive, barium titanate using rigorous coupled wave diffraction theory, J. Opt. Soc. Am. A, 13, 819±831, (1996).

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35. R. Alferness, Analysis of propagation at the second-order Bragg angle of a thick holographic grating, J. Opt. Soc. Am., 66, 353±362 (1976). 36. A. Korpel, Acousto-Optics, Marcel Dekker, New York, 1988. 37. J. M. Jarem and P. P. Banerjee, Rigorous coupled-wave analysis of photorefractive re¯ection gratings, in press, J. Opt. Soc. Am. B. (1998). 38. J. M. Jarem and P. P. Banerjee, Time domain state variable analysis of induced re¯ection gratings in photorefractive materials, Photonics West Proc. SPIE (1998). 39. M. G. Moharam and T. K. Gaylord, Coupled-wave analysis of re¯ection gratings, Appl. Optics, 20, 240±244 (1981). 40. M. G. Moharam and T. K. Gaylord, Chain-matrix analysis of arbitrary-thickness dielectric re¯ection gratings, J. Opt. Soc. Am., 72, 187±190 (1982). 41. Z. Zylberberg and E. Marom, Rigorous coupled-wave analysis of pure re¯ection gratings, J. Opt. Soc. Am., 73, 392±401 (1983). 42. N. Singh, S. P. Nader, and P. P. Banerjee, Time-dependent nonlinear photorefractive response to sinusoidal intensity gratings, Opt. Comm., 136, 487±495 (1997).

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