Computational Methods for Electromagnetic and Optical Systems (Optical Engineering)

Copyright © 2000 Marcel Dekker, Inc. Copyright © 2000 Marcel Dekker, Inc. Copyright © 2000 Marcel Dekker, Inc. Co...

Author: Partha P. Banerjee

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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


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.


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.


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


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


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


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


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


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


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

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

2.2.5

State Variable Analysis of a Radar Absorbing Layer (RAM)

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


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

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

11 *12

12 *22

2:2:35

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

2:2:36

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


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

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


Figure 5

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

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

State Variable Analysis of a Source in Isotropic Layered Media

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


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

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

1 *11 exp 1

*21 exp 1

) 1 0 1


1 * exp 1

1 21

*11 0 2:2:37a

2:2:37b

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

,% 1 *11 exp 1 *21 exp 1

2:2:38a

) 1 0 1

2:2:38b

1 *11 exp 1 *21 exp 1 1

,% 2 *12 exp 2 *22 exp 2

2:2:39a

) 2 0 2

2:2:39b

2 *12 exp 2 *12 exp 2 2

,% 6 3 *13 exp 3

2:2:40a

) 3 0 3

2:2:40b


3 * exp 3

3 13

Figure 8

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

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

1 =1 1 =1 exp 2 1

1 =1 1 =1

2:2:41

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

3 =3 2 =2 exp 2 2

3 =3 2 =2

2:2:42

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


2:2:43a

2:2:43b

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

2:2:44a

2:2:44b

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

where !( 0 s

~ 0

0 s

~ 0

2:2:46

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

1 *11 *21 2 *12 *22

1 2

2:2:47

Thus 1 2 *11 *21 * *22

!s *11 *21

1 2 12


2:2:48

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

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

!.)"

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

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

2 2

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

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

2:2:44a

2:2:44b

when .)" > !.)"

3 3

" exp 3 .)"

2

2:2:45

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


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

2.3 2.3.1

STATE VARIABLE ANALYSIS OF AN ANISOTROPIC LAYER Introduction

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


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

Basic Equations

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

1 0 exp 1 $ exp 1 exp

1

* 11 exp 11 * 21 exp 21 exp

2:3:1

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


1 ( 1 exp

0 exp 1 $ exp 1 exp

1 *11 exp 11 *21 exp 21 exp

0 1 ) 1 exp

0 exp 1 $ exp 1 exp

*11 exp 11 *21 exp 21 exp

2:3:2

2:3:3

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

3 " exp 3

exp

3

* 13 exp 13 * 23 exp 23 exp

2:3:4

" exp 3

exp

3 *13 exp 13 *23 exp 23 exp

2:3:5

) 3 exp " exp 3

exp

3 ( 3 exp 0 3

*13 exp 13 *23 exp 23 exp

2:3:6

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

0

2:3:7

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


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

0 ) ^ ) ^ ) ^ exp

2:3:8

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

2:3:9

@) ( ( @

2:3:10

) ( (

2:3:11

(

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

( )

2:3:12

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


2:3:13

2:3:14

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

2:3:15

where

11

21

# $ 2 12 22

2:3:16

2:3:17

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

2:3:18

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

1; 2

2:3:19

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

11

21

12 22

$#

(

)

$

0

2:3:20

For this to have nontrivial solutions, det


11

21

12 22

11

22

12 21

0 2:3:21

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

11

22

0:5

2 11

2

11 22

4

12 21

2 1=2 22

1; 2

2:3:22

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

12

11

2:3:23

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

( )

1; 2

2:3:24

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

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

* 12 exp 12 * 22 exp 22 exp

2 ( 2 exp

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

*12 exp 12 *22 exp 22 exp

2:3:25

2:3:26

0 2 ) 2 exp

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

*12 exp 12 *22 exp 22 exp

2:3:27

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


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

2:3:28

0 $ *1 )1 *2 )2

2:3:29

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

3

2:3:30

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

2:3:31

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

2:3:32

3 0 exp 1 ) ( 1 *1 exp 2

3 1 3 ) ( 2 *2 3 2

2:3:33

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

Numerical Results

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


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

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


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

%

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


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

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


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

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


2.4 2.4.1

STATE VARIABLE ANALYSIS OF A BI-ANISOTROPIC LAYER Introduction

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

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


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

2.4.2

General Bi-Anisotropic State Variable Formulation

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

2:4:1

~ l~

2:4:2

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

~ =0

~ 0 l l l l=

~ 0 0 0 !

~ 0 ~ 0 0 m m !

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

2:4:3

2:4:4

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


exp ( exp

l ) (

2:4:5

exp ) exp

( )

2:4:6

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

2:4:7

0 ) exp

2:4:8

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

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

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

)

(

(

) ) 2:4:10

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


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

/22

#

( )

$

(

( ,24 )

2:4:12

)

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

#

( )

$

(

# ( &11 / 1 , ) &21

&12

&13

&22

&23

)

(

$ &14 ( &24 ) )

2:4:13

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

'11

'12

'13

'14

@5 '21 @ '31 '41

'22

'23

'32

'33

'42

'43

'24 5 +5 '34 '44

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


2:4:14

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

2:4:15

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

5

4

* 5 exp

1

(

( 5 ) )

The EM ®elds in Region 2 are given by


2:4:16

2:4:17

4 *

2:4:18

1

4 *

2:4:19

1

where ( ^ ( ^ ( ^ exp

1 ) ^ ) ^ ) ^ exp

0

2:4:20

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

and

(

)

2:4:22

724 5

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

2.4.3

Incident, Reflected, and Transmitted Plane Wave Solutions

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

0 I ) exp I

2:4:23

I 1

2:4:24

where I ^ 1 ^ ^


^ ^ ^

2:4:25

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

2:4:26

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

(

( ( ( ( 1 1

2:4:27

)

1 1 ( ( 1

2:4:28

1 ( ( 1 1 ) ( 1 ( 1

)

2:4:29

2:4:30

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

$ ;

0 $ ) $ exp

$ 1

$

2:4:31

2:4:32

where $ ^ 1 ^ ^

2:4:33

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


($ ) $

( $ ($ ( $ ($ 1 1

2:4:34

1 1 ($ ($ 1

2:4:35

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

)$

2:4:36

)$

2:4:37

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

T

0 T ) T exp

T

T ^ 3

2:4:38

2:4:39

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

2:4:40

2:4:41

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

( " (" ( " (" 3 3

2:4:42

) "

1 3 (" (" 3

2:4:43

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

)"

2:4:44

)"

2:4:45

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


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

0

0

$

0

$

0

4

4

4

4

*

1

0

*

1

0

*

1

1

2:4:46

0

*

0

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

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

2:4:47

After a small amount of algebra, it follows that

where

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

2:4:48

2:4:49

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


, ,

4

1

4

1

& ' * ( ( 1 1 )

& ' * ( 1 ( 1 )

2:4:50

The boundary conditions at the Region 2±3 interface are

" " " "

4

4

4

4

*

1

*

1

*

1

1

2:4:51

*

Substituting

( " ("

4

* exp (

1

4

2:4:52

* exp (

1

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

0 0

4

1

4

1


& ' * exp ( ( 3 3 )

& ' * exp ( 3 ( 3 )

2:4:53

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

Numerical Example

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

0:3 0:2

0:1 0:05

0:3 0:1 0:1

0:2 0:2

0:6 0:65

0:25

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

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

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

0:05

0:15

2:4:54

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


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

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


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

2.5 2.5.1

ONE-DIMENSIONAL k-SPACE STATE VARIABLE SOLUTION Introduction

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


2.5.2

k-Space Formulation

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

( ; exp

2:5:1

) ; exp

2:5:2

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

0

0

)

* ( ; exp l ) ( exp

)

* ) ; exp ) ( exp

2:5:3

2:5:4

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

l ) (

exp ) exp

) (

2:5:5

2:5:6

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


2.5.3

Ground-Plane Slot-Waveguide System

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

0

0 exp 3

3

$ $0 exp 3

Figure 18

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


2:5:7

2:5:8

2:5:9

0 $

$0 exp 3

3

2:5:10

3 $

2:5:11

3 $

2:5:12

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

+

11

12

21

22

2:5:13

where

11

21

#

2 12 22

$

2:5:14

2:5:15

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

2

2 0 2


# 2

1

# 2

1

# 2

1

$

* ( exp exp

2:5:16

$

* ( exp exp

2:5:17

$

2:5:18

* ) exp exp

where ( 1 11

)

(

2:5:19

2:5:20

12

( )

1; 2

2:5:21

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

11

22

11

2 22

4

12 21

11

2 22

4

12 21

1=2 1=2

2:5:22

2:5:23

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

1 1

) exp 1 1 1

) exp 1 1 ) 1 exp 1

1

1

0 1

2:5:24

2:5:25

2:5:26

where 1

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

1 1 2 0 1 1 2 < 0

2:5:27

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


2

^

3

s

2:5:28

where 3

0

2:5:29

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

' rect

^ 2

2:5:30

where rect 2

1

<

0

>

2:5:31

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

exp

2:5:32

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

' sin

2:5:33

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


2 1 1

) * (

1

1

) 1 2

1

2

2:5:34

2:5:35

* )

1

* ( exp

2:5:36

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

"2

"1 exp 2 "2 exp 1

2:5:37

"1

"1 exp 2 "2 exp 1

2:5:38

where "

12

1 1

11

1; 2

2:5:39

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

0 3

2:5:40

2

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

0 2

1 3

2 0

2:5:41

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

0 3


1 0 $0 3

2:5:42

Thus

1 1 $0 3 0 2

0 2

2:5:43

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

~ 3

3

2:5:44

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

$0 exp 3

~ 3

0 3 0 exp 3

$0 exp 3

2:5:45

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

3~ '`

3

3

3

'

" 1 1 1 2

0 ' 2 0


1 0 $0 0 3 '

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

we ®nd that 3~ '`

%

2:5:46

0 2 using Eq. 2.5.43,

+

2:5:47

De®ning a normalized aperture load admittance we have

3'`

0 3~ '`

1=2

%

0 2

'

2:5:48

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

3

2:5:49

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

2:5:50

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

1 0 $0 3 0 $0

2:5:51

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

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

2:5:52

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


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

2:5:53

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

2

~ ~

2

~

1 ~ 2

~ ~ ~

2:5:54

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

2:5:55

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

1 sin 2 1

2:5:56

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


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

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

2:5:57

where ) 1

' exp 1

2:5:58

and where ' "2 "1 sin

' 1 1 "1 exp 2 "2 exp 1

2:5:59

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

1=2 2 1 sin 'c

exp 1 1 4

2:5:60

where 1 sin c

1 cos c ' 1 sin c

' "2 "1 sin

1 "1 exp 2 "2 exp 1


2:5:61

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

1 sin c

2

2:5:62

2 =2 1 sin c

c

2.5.4

Ground-Plane Slot-Waveguide System, Numerical Results

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

2 0

0

0 0

2:5:63

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


Figure 19

A plot of the 3 aperture admittance integrand.

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

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


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

Figure 21

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


2.6 2.6.1

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

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

Figure 22

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


2.6.2

The Field Inside the Anisotropic Layer

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

2:6:1

~ ~ !l

2:6:2

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

~ ~ ~ ~ exp ~ ~ ~

;

2:6:3

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

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

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

;

2:6:4

~ !l~ ;

~ ;

2:6:5

where

0 @=@~ ~

@=@~ ~ 0 ~ ~ 0

2:6:6

De®ning the auxiliary ®eld column matrices

~ $


#

$ ~ ;

^ ~

;

3

#

$ ~ ;

^ ;

~

2:6:7

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

~ , ~ 5 ~ $

0 !

~ 3

~ $

2:6:8

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

0 !

8 9

2:6:9

where 8 and 9 are 2 2 matrices with entries

~ 11

8 ~ 21

~ 12

~ 22

~ 12

9 11 ~ 22

21

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

1

'

2:6:10

~ ~ 91 8 ~ 1 $

1

'

2:6:11

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

91 81

~ exp +

2

and where the matrix + is given by

! 5 + 0 ,


2:6:12

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

2:6:13

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

3 0

~ k *k

1; 2; 3; 4

2:6:14

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

1

3

2

2

3

1

4

0

2:6:15

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

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

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

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

Solution of the Boundary Value Problem

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

~ and 0 ;

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


~ and $0

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

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

1 ~ 2 30

8 sgn ~ ~

0 4 exp ~0 ~ ~

# 1 1 0 !0

~ 2 $0

8 0

0 !0 ~0 1

0 ~2 ~0 1

$

sgn ~ ~ ~2 !0 1 0

2:6:15

$

4 exp ~ ~0 ~ ~

2:6:16

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

^ 4 ^ ^

~ 0 ;

30 ~

^ 0 ;

^ ^

~ 0 ;

$0 ~

^ 0 ;

2:6:17

and ~ ~ ~ ^

2:6:18

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

#

~ $

#


~0

0

0

!0

$#

0

~0

!0

0

$#

$

~ exp ~0 ~

2:6:19

$

2:6:20

~ exp ~0 ~

~ and $

~ are determined from ~ and ;

~ in a where 3

;

~ and $0

~ were manner similar to the way 30

determined from ~ and 0 ;

~ or 3

~ and $

~ from ~ and ; . ~ 0 ;

;

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

~ 3

~ 3

2:6:21

~ $

~ $

~ $

0

2:6:22

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

#

#

~0

0

0

!0

0

~0

!0

0

$# $#

$

~ 30

2:6:23

$

~ $0

2:6:24

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

2.6.4

Numerical Results and Discussion

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


2:6:25

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

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


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

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


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

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


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

2.6.5

Conclusion

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

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

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


2.7

2.7.1

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

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

2.7.2

Variable Transformation in the Matrix Exponential Method

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

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


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

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

@~

2:7:1

where

~ ~ ~ ~ ~ ~ ~ ~ ~

~ w ~ ~ ~ ~ ~ ~ ~ ~

2:7:2

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

~ ~ 0

~ ~ /


2:7:3

where

~ exp 1

0

0

0

0

~ exp 2

0

0

0

0

~ exp 3

0

0

0

0

~ exp 4

~ / ~ /

~ 1 /

2:7:4

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

!0 ,

~2 20 !0 : w 0

, ~2 20

2:7:5

!0

The unknown equation

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

~ w 0 : /

: 4inc w

2:7:6

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


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

2:7:7

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

1

0

0

0

0 +1 r~ 0

0

0

0

0

0

0

0 ~ 1 r 0 0

0

2:7:8

and

0

0 ~ +2 r 0

0

0

0

0

1 0 ~ 0 exp 3 2 0

0

0 0 ~ exp 4 2

~ 1 r

2:7:9

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

~ +1 w 0

1

3 2

2:7:10

4

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

2:7:11

~ + exp 2

2:7:12


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

~ w 0 ~ /

0

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

4

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

!0

0

!0

2:7:13

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

2.7.3

An Example: Scattering from a Biased Ferrite Layer

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


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 ( ; ; ; ,


7. 8.

9.

10.

11.

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


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

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

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18. 19. 20. 21.

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


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