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Electromagnetic Waves in Chiral and
Bi-Isotropic Media
I. V. Lindell, A. H. Sihvola,
S. A. Tretyakov, A. J. Viitanen /
Artech House
Boston • London
Library of Congress Cataloging-in-Publicalion Data
Lindell, Ismo V.
Electromagnetic waves in chiral and Bi-isotropic niedia/I. V. Lindell,
A. 11. Sihvola
Includes bibliographical references and index.
1SBN0 89(X)6-684-l
1. Electromagnetic waves-Mathcmatics. 2. Chirality.
I. Sihvola, A. H. II. Title
QC661.1.52 1994	94-7656
620. Г1297-dc20	C1P
A catalogue record for this book is available from the British Library
© 1994 ARTECH HOOSE, INC.
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All rights reserved, hinted and bound in lite United Stales of America. No pan of litis book may be
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ing. recording, or by any information storage and retrieval system, without permission in writing
front lhe publisher.
International Standard Book Number: 0 891X16 684 1
Library of Congress Catalog Card Number: 94 7656
10 98765432 1
To the memory of Professor Karl F. Lindman (1874-1952)
who first demonstrated the effect of man-made chirality
on electromagnetic waves
Karl Lindman, the good old admiral1
taught us that the whole world is chiral:
when sailing the seas
in the night, one secs
that the whole universe is a spiral
'In nonort hod ox ford pronouncintion

Contents
1	Introduction	1
1.1	Early	Studies on Chiral and Bianisotropic Media	1
1.1.1	Optical Activity	2
1.1.2	Electromagnetic Activity	4
1.1.3	Magnetoelectric Activity	7
1.1.4	The Fedorov Schoo)	8
1.1.5	Bianisotropic Media	12
1.2	Characterization	of Bi-Isotropic	(BI)	Media	13
1.2.1	Constitutive Relatione	13
1.2.2	Material Structure and Characteristic Figures	14
References	18
2	Fields in Homogeneous	BI	Media	23
2.1	Wavefield Decomposition	23
2.1.1	Wavefield Postulates	24
2.1.2	Wavefield Vectors	28
2.1.3	Sources of the Wavefields	29
2.1.4	Wavefields as Self-Dual Fields	31
2.2	Plane	Waves in Homogeneous BI Media	32
2.2.1	Plane-Wave Relations	33
2.2.2	Polarization Rotation	35
2.2.3	Angle Between Field Vectors	36
2.2.4	Wavelengths of the Plane Wave	38
2.3	Green	Functions	39
2.3.1	Dyadic Sources and Fields	39
2.3.2	Solving for the Green Dyadics	40
2.3.3	The Far Field from a Dipole	42
2.4	Reciprocity and Nonreciprocity	43
2.4.1 The Reciprocity Theorem	43
2.4.2 Demonstration of Nonreciprocity	46
vii
viil	Contents
2.5	Huygens’ Principle	47
2.6	Power	and Energy in BI Media	49
2.6.1	Conditions for Medium Parameters	49
2.6.2	Wavefield Decomposition of Power	51
2.7	Electromagnetostatics	in	BI	Media	53
2.7.1	Basic Equations	53
2.7.2	Fields Due to Charges	53
2.7.3	Bi-Isotropic Transmission Line	55
References	57
3	Plane Waves in Layered	Media	59
3.1	Normal Incidence, Single Interface	59
3.1.1 Circular Polarization	60
3.1.2 General Polarization	66
3.2	Nonsymmetric Transmission-Line Theory	71
3.2.1 Transmission-Line Equations	71
3.2.2	Input Impedance of a Terminated Line	73
3.2.3 Transmission Through a Line Section	76
3.3	Normal Incidence on BI Slab	77
3.3.1	Reflection and Transmission	77
3.3.2	BI Slab Between Simple Isotropic Half Spaces	79
3.3.3	Polarization Rotation	83
3.4	Vector Transmission-Line Theory	91
3.4.1	Systems of Plane Waves	92
3.4.2	Vector Voltages and Currents	93
3.4.3	Propagation Dyadics	95
3.4.4	Dyadic Characteristic Admittances	96
3.4.5	Reflection Dyadic	98
3.4.6	Transmission Dyadic	100
3.4.7	Input Admittance Dyadic	101
3.5	Oblique Incidence, Single Interface	103
3.5.1	The Characteristic Admittances	104
3.5.2	Reflection and Transmission	104
3.5.3	Eigenproblem of the Interface	106
3.5.4	Brewster Angles	109
3.6	Oblique Incidence on BI Slab	114
3.6.1	Dyadic Admittance and Reflection	114
3.6.2	Special Case	115
References	116
vii
Contents	ix
4	Waveguides	119
4.1	Guided-Wave Solutions	,	120
4.1.1 Field Decomposition	and Boundary Conditions	120
4.1.2 Vector Circuit Approach	124
4.2	Slab Waveguides	127
4.2.1	Closed Planar Bi-Isotropic Waveguides	127
4.2.2	Open Bi-lsotropic Plane Guides	133
4.3	Circular Waveguide	137
4.3.1 Isotropic Boundary Impedance	138
4.3.2 Anisotropic Boundary Impedance	143
4.3.3 Open Circular Bi Isotropic Waveguide	145
4.4	Rectangular Waveguide	147
References	149
5	Propagation in Inhomogeneous Media	153
5.1	Geometrical Optics for BI Media	154
5.2	Polarization-Rotating Lens Antennas	155
5.2.1	Maxwell Fish-Eye Lens	158
5.2.2	Brown Lens	161
5.2.3	Luneburg Lens	163
5.2.4	Gutman Lens	167
5.3	WKB	Approximation for Normal Incidence	171
5.3.1	The Coupling Equation	172
5.3.2	Reflection Dyadic	176
5.3.3	Special Cases	177
5.4	WKB	Approximation for Oblique Incidence	180
5.4.1	WKB Approximation for Wave Propagation	182
5.4.2	Reflected Fields and Corrected Propagating Fields	184
5.4.3	Special Cases	187
References	191
C	Scattering and Mixing Theories	193
6.1	Polarizabilities of Small BI Scatterers	193
6.1.1	Bi-Isotropic Sphere	194
6.1.2	Bi-Isotropic Ellipsoid	197
6.1.3 Layered Chiral Sphere	198
6.1.4	Chiral Sphere in Chiral Background Material	200
6.2	Interpretation of Polarizability Expressions	201
X
Contents
6.2.1	Effect of Inclusion Parameters	202
6.2.2	Paramagnetic Parte Can Generate Diamagnetic Whole	203
6.2.3	Similarity of Bi-Isotropic and Anisotropic Polarizabilities	205
6.3	Modeling of Bi-Isotropic Mixtures	208
6.3.1	Effective Permittivity of a Mixture	208
6.3.2	Effective Parameters of BI Mixtures	209
6.3.3	Perturbation Expansions of Effective Parameters	215
6.3.4	Remarks on and Illustrations of Effective Material Parameters 218
6.3.5 Alternative Mixing Laws	223
6.3.6 Mixing Rules in Other Constitutive Relations	225
6.4	Dispersive Behavior of Chiral Materials	227
6.4.1 General Features	227
6.4.2 One-Resonance Condon Model	230
6.4.3 Effect of Mixing on Dispersion	233
6.5	Scattering by Large Objects	235
6.6	Scattering by Helices	239
References	243
7	Measurement Techniques	251
7.1	Ellipsometric Measurements	252
7.1.1 Linear Polarization Measurements	253
7.1.2 Use of Eigenpolarizations in Reflection	256
7.1.3 Differential Circular Reflection	256
7.2	Reflection and Transmission from Slab	258
7.2.1	Pasteur Media	258
7.2.2	Small Tellegen Parameter	260
7.2.3	Large Tellegen Parameter	261
7.3	Waveguide and Resonator Techniques	262
7.3.1	Waveguide Perturbation	263
7.3.2	Resonator Perturbation	265
7.3.3	Chirality and Nonreciprocity Parameters	267
7.4	Practical Implementation Aspects	270
References	272
8	Uniaxial Bianisotropic Media	275
8.1	Plane Waves in a Uniaxial Medium	275
8.1.1	Medium Parameters	275
8.1.2 Basic Equations	277
Contents	XI
8.1.3	Axial Impedance and Admittance	278
8.1.4	Effective Anisotropic Media	281
8.1.5	Eigenpolarizations	284
8.2	Polarization Transformer	286
8.2.1	Transverse Eigenwaves	287
8.2.2	Polarizations of the Eigenwaves	288
8.2.3	Propagating Plane Wave	289
8.2.4	Quarter-Wave Transformer	290
8.3	Green Dyadic	298
8.3.1	The Operator Equation	298
8.3.2	Solving the Green Dyadic	300
8.3.3	The F Dyadic	301
8.3.4	Field from a Dipole	303
References	304
Appendix A	Notation	307
A.l The Present Notation	307
A.2 Other Notations	308
Л.З Bianisotropic Media	311
References	312
Appendix В	Complex Vectors	313
B.l	Ellipse of a Complex Vector	313
B.2	Polarization Vector	314
B.3	Two-Dimensional CP Vectors	315
B.4	Vector Bases	316
References	316
Appendix C	Dyadics	317
C.l Basic Properties of Dyadics	317
C.2 Two-Dimensional Dyadics	318
References	320
Appendix D	Collection of Basic Formulas	321
D.l Constitutive Equations	321
D.2 Wavefields in Homogeneous BI Media	322
D.3 Plane-Wave Relations	322
xii
Contents
D.4 Green Dyadics	323
D.5 Guided Waves	323
D.6 Inhomogeneous DI Media	323
D.7 Polarizabilities for Small BI Sphere	324
Preface
In a decade or so, chiral and bi-isotropic media have emerged as one of the most
challenging topics in electromagnetics research in terms of theoretical problems and
potential applications. So far, the basics needed for understanding and developing
new applications have not been thoroughly covered in textbooks, but have rather
been scattered in different journals. The purpose of the present book is to give the
reader a working knowledge of how chiral and bi-isotropic media affect electromag-
netic fields and waves, to apply the theory to basic problems in waveguide, antenna,
and scattering analysis, and to discuss methods of measurement. Although the ab-
stract analysis is valid for lossy media, many examples deal with lossless media
which is a valid approximation outside the resonance frequencies.
The contents of the present book reflects the activities of the Electromagnetics
Laboratory at the Helsinki University of Technology, Finland, with one of the au-
thors (S.T.) visiting from St. Petersburg, Russia. Writing the text has been a joint
effort of all four authors. However, the final responsibility of different chapters lies
on the authors as follows: Chapters 2, 3 and 8: I.L., Chapter 4: S.T., Chapter 5:
A.V., and Chapter 6: A.S. Chapters 1 and 7 have been produced through collective
work: Chapter 7 by A.S., S.T. and A.V, and Chapter 1 by A.S. and S.T., with a
small contribution by I.L. While the style may differ from one chapter to another,
the mathematical notation has been kept coherent.
The present book can be effectively used as a source for scientists and engineers
working in various fields of electromagnetic engineering. The basic formulas needed
in the analysis have been compiled in Appendix D at the end of the book for
easy reference. Due to the different systems of notation existing in the literature,
formulas have been given in Appendix A for transforming equations and results
from the present notation to other notations. The book can also be used in graduate
courses in universities. The omitted steps in the analysis serve as suitable homework
exercises.
Information on any misprints and errors found by the readers of the book is
welcomed by the authors at the Electromagnetics Laboratory, Helsinki University
of Technology, Otakaari 5A, Espoo, Finland, 02150. The updated errata will be
sent to all informants in return mail.
xiii
xiv
Contents
The Authors Acknowledge comments And technical help from many individuals.
Even with the risk of being incomplete, the following list must he mentioned: M. Er-
mutlu, M. Flykt, II. Frestadius, A. Hujanen, T. Huttunen, T. Kharina, P. Koivisto,
A. Lakhtakia, A. Lindell, K. Lumme, F. Mariotte, M. Oksanen, I. Semclienko,
M. Silverman, K. Simovski, J. Sten and W. Weiglhofer. Finally, the members of
the families of all the authors deserve a bear hug for their support.
Helsinki, December 17, 1993
I.V. Lindell, A.H. Sihvola, S.A. Tretyakov, A.J. Viitancn
Chapter 1
Introduction
What are the directions of microwave technology in the 21st century? As always,
future predictions tend to be uncertain, but there are certain areas within the elec-
tromagnetics research of today that contain potential for diverse new applications
in engineering. One of these fields is that of novel material effects. The progress
of theoretical understanding of the wave-material interaction has been swift in the
1980’s and 1990’s, and the electromagnetic phenomena that have been discovered do
promise new solutions to problems that microwave engineers have been struggling
with during the last fifty years.
Important classes of the novel, complex materials are chiral and bi-isotropic
media, the electromagnetic treatment of which forms the topic of this book. In
the chapters to follow, the analysis is self-contained and starts from the basic laws.
The text aims at presenting the principles of how electromagnetic waves interact
with bi-isotropic media. Once the dependencies are known, they can be used for
measurements of novel materials. Practical aspects like this will be discussed in the
text, as will the ranges of values for material parameters that can be reached for
actual samples.
Where are the limits now? Where do we stand? This type of question may
puzzle the engineer or scientist who is starting to work in this field. A look at past
and present research may help in finding an answer. Let us therefore start with a
look at the earlier research on chiral and bi-isotropic media.
1.1	Early Studies on Chiral and Bianisotropic
Media
It is difficult to outline the history of the research on electromagnetics of bi-isotropic
media for at least two reasons. Firstly, during the two hundred years when the sci-
entific community has been aware of the strange and exciting world of these media,
2
Chapter 1. Introduction
so many electromagnetics specialists have contributed to the subject that the task is
beyond reach. Secondly, due to the interdisciplinary nature of the field, researchers
from other disciplines like optics, chemistry, and biology, have published results that
bear strong relevance also to electromagnetics and microwave applications.
The aim of this review is therefore to give the main milestones of bi-isotropic
electromagnetics during this and the last century. In addition to the long-term
development, the present wave of interest that the microwave community directs to
chiral media and applications has produced scores of publications. This movement
takes place not only in Europe, the Unites States, and Russia, but also in China,
Israel, and South Africa. The results of the present flow of research will not be
discussed here in the introduction since the main text of this book tries to convey a
complete presentation of the electromagnetics of bi-isolropic media, and the latest
results are interwoven with the text of the chapters to follow.
1.1.1 Optical Activity
Electromagnetism in chiral and bi-isotropic media differs from the behavior of sim-
pler, isotropic materials in several ways. One of the aspects characterizing chiral
media is the phenomenon of optical activity. Optical activity is connected with the
behavior of the polarization of light illuminating the material. In particular, the
special property of optically active media is that the polarization plane of linearly
polarized light will rotate as light passes through the medium.
The early 19th century saw the origins of the scientific study of optically ac-
tive materials. This required, however, a means to produce polarized light out
of the ubiquitous natural and man-made sources of unpolarized optical radiation.
Although Huygens and Newton had noted the special properties of light refracted
from Iceland Crystal, it remained for French scientists, more than a hundred years
later, to formulate the laws and start a systematic analysis of polarization. Malus
recorded in 1809 a method of imparling to the waves of light “a certain form or dis-
position” using — to express the fact anachronistically in present-day terminology
— the ordinary and extraordinary waves of doubly-refracting anisotropic crystals.
These rays would no longer separate into two as they were transmitted another time
through a birefringent crystal; they were “modified by polarisation.”
And once sunlight and a Malus-type polarizer were available, the scene was set
for investigations on the behavior of polarized light in materials. The first effects
of optical activity were discovered in anisotropic gypsum and quartz crystals by
Arago (1811) and Biot (1812). In their experiments, the rotation of the plane of
polarization was not directly observable; rather, the rotatory dispersion could be
seen. The different components of polychromatic light suffer difierent amounts of
rotation, which, having passed through the material, lead to colored light rays with
1.1. Early Studies on Chiral and Bianisolropic Media
3
"tints of Newton’s rings.” A little later, Biot’s interest was attracted to the optical
properties of liquids and gases: he suspected that optical activity was a property
of molecules, not of their aggregates in solid state. In 1817, he observed the optical
rotation caused by the exhausts of a furnace boiling turpentine.
Free turpentine gas is certainly isotropic, and in this later experiment, Biot did
not need to care about the symmetry directions in which the crystals had to be
cut in order to achieve the greatest rotatory effect. Now it was proven that there
are materials with strange, powerful abilities. They could change the properties
of light more radically than what was the result of the earlier known plain effects:
attenuation, reflection, and refraction. These new media exhibit optical rotatory
power, and the phenomenon came to be called natural optical activity.
This label may be compared with magnetic optical activity, where the polariza-
tion rotation is caused by an external magnetic field. In microwave technology, this
phenomenon is known by the name Faraday rotation in ferrites. Radio engineers
are well acquainted with the similar effect of polarization rotation in the ionosphere
which is caused by the Earth’s static magnetic field. There it is a gyroelectric
phenomenon. But an external field breaks the spatial symmetry, and the effect
of magnetic activity is therefore not isotropic. It is different from natural optical
activity, although some optically active media may be anisotropic as well.
A formal treatment of the concept of polarization was given by Fresnel, with the
fundamental discoveries of the transverse nature of the field vectors of light (1821)
and that of circular polarization (1822). With the conjecture of right-handed and
left-handed circular polarization states, it was easy to explain the rotation of linear
polarization. Fresnel also constructed a prism from three pieces of dextro- and levo-
rotatory quartz to separate the two circularly polarized components of a linearly
polarized light ray, hence providing an experimental proof of the existence of circular
polarization.
But what is the strange properly of media that makes them “powerful”, or “ac-
tive”? This enigma was solved by Pasteur, as he began, in the 1840’s, to study the
crystal structure of materials and their relation to optical activity. In the rigorous
classifications of chemical substances of the time, it was the nature, arrangement,
and distance of the atoms in the compound that were essential in the distinction
between different elements. But two substances — chemically identical in this
classification scheme — could still be mirror images of each other. Spatially non-
symnictric forms of crystals were indeed known to exist in natural media although
these stereoisomers were not considered to belong to different classes.
Pasteur was interested in the properties of tartrate solutions, known to be opti-
cally active. Studying the crystallized forms of a certain optically inactive paratar-
trate he found both left- and right handed crystals, roughly in the same proportion,
whereas in the crystal composition of the Optically active sample, one type of hand-
4
Chapter 1. Introduction
edness dominated over the other. The solution with equal amounts of left- and
right-handed elements has been called a racemic mixture, and now it was shown
that racemic solutions did not display optical activity. Pasteur found, in other
words, the link between the optical rotatory power and the left-right imbalance.
The often retold history of Pasteur’s great discovery culminates in the experiment
that he demonstrated to the grand old man of French science, Biot, who, deeply
affected, realized the connection between handedness and natural optical activity.
Handedness, in other words chirality, is a property that is often encountered
in the organic and biological nature. At different levels, there exists nonsymmetry
with respect to spatial inversion. Seashells, bacteria and amino acids, and even
the smallest-scale elementary particle interactions display different proportions of
left- and right-handed occurrences. Sometimes the preferred one is a left-handed
enantiomer, sometimes a right-handed one. On the other hand, from building blocks
of single handedness, Nature is able to grow an equal amount of left and right
hands for human beings. Racemication processes oppose the mysterious forces that
shamelessly discriminate against one or the other of the enantiomers.
In the latter part of the 19th cenfury, after Maxwell had unified optics with elec-
tricity and magnetism, the connection could be established between optical activity
and the electromagnetic parameters of materials. Drude was the first to suggest
constitutive relations for optically active media as well as a model for the wave-
length dependence of optical rotatory power. The measure of optical activity in
these relations can be called chirality parameter.1 The same parameter also con-
tains information on circular dichroism, which means that handed media not only
rotate the polarization but they also distort it. Circular dichroism is a consequence
of the unequal absorption of the. left- and right-circularly polarized waves traveling
through the medium.
The later history of optical chirality has ramifications to diverse fields like stereo-
chemistry, quantum physics, and the drug industry. The present exposition does
not attempt to cover these directions; general literature is abundant [2]. Let us
rather continue towards the direction of chiroelectromagnetics.
1.1.2 Electromagnetic Activity
As it became possible for scientists to generate radio waves after Hertz’s discovery
of 1888, it was natural to look for the rotatory power in materials that would be
effective at these wavelengths. Electromagnetic activity was supposedly inherent
in materials with handedness, just as in the case of optically active media, but the
crucial question was how the enormous difference in wavelengths affected the ob-
’The tcim "chirality” in this context was first used by Lord Kelvin (William Thomson) in his
Baltimore lectures [1].
1.1. Early Studies on Chiral and Bianisotropic Media
5
servability of this strange phenomenon. There was a gap of six orders of magnitude
in the wavelength scale between optics and Hertzian radio waves. Although the
students of today arc imbued with the notion that the Maxwell equations govern
all electromagnetic phenomena from statics to X rays, the extrapolation requires a
lot of courage, at least in those days of a hundred years ago.
Karl F. Lindman was the first to look for radio wave activity, and his mission
was successful. In 1914 he made his first experiment in Helsinki [3]. Lindman’s
publication of this work in 1920 [4] has become an often cited and followed report
in the microwave community of experimental chirality.
Lindman was aware of Drude’s model for the refractive properties of optically
active media. According to Drude, the wavelength dependence of the amount of
polarization rotation ф followed the law:
* = Lj—<”)
Here A is the wavelength of light, and the summation i takes into account all the
“characteristic vibrations of the electron groups” in the parlance of late-nineteenth-
ccnttiry physics, А,- is the corresponding resonant wavelength and k; the strength of
this resonance. Lindman was interested in testing the validity of Drude’s model at
lower frequencies, on radio waves, or, as he himself calls it “the Hertzian waves.”
He notes that this means a similar situation as with “ultraviolet electrons,” i.e., the
case that in Drude’s formula the wavelength is much longer than the characteristic
wavelengths of the resonant modes, in which case Drude’s model gives
(12)
where k' is now the strength of the lowest order mode. And because of the strong in-
verse wavelength dependence of ф, at Hertzian waves no measurable optical rotation
can be expected. This formula (1.2) was known as Biot’s first law’.
Lindman simply scaled the problem. If molecular dissymmetry produced optical
activity, radio-wave activity should result from dissymmetry that manifests itself in
spatial scales of the order of centimeters. The artificial creation of this kind of a
material was the next step.
Lindman synthesized a chiral medium by twisting small helices from copper
wire, immersing these in cotton balls, and then positioning the balls with random
orientation in a cardboard box. The length of the straight wire of the helices was
9cm and the thickness 1.2mm. The diameter of the spirals was 10mm and there,
were 2.5 circles in one spiral. The cardboard box had length of 26cm and the
total number of spirals in the game was 700. Lindman made both left handed and
right-handed helices.
6
Chapter I. Introduction
Lindman put his chiral box into his measurement system, shown in Figure 1.1.
There he directs linearly polarized radio waves through the metal guide where the
sample is located, and measures the linearly polarized component of the intensity of
the received signal as a function of the rotation angle of the receiving linear antenna.
For a linearly polarized wave this function should be a squared cosine, and from the
maximum point of this curve, the polarization rotation can be inferred.
Figure 1.1 Lindman's measuring equipment from his original article in 1914. The trans
matting oscillator О stands in front of a reflector, which is bounded by the “standard
indicator” I on one side, and a metal screen S on the other. В and U are hollow metal
tubes with a circular opening A. The sensor dipole И can be turned with the slick T,
and the rotation angle is read in the display formed by a pointer V and a protractor </.
At is the box where the chiral sample is put in. Reprinted with permission from Societas
Scientiarum Fennica (The Finnish Society of Sciences and Letters).
The results proved the existence of polarization rotation and the dispersion of
the activity around 1 to 3 GHz. Lindman also proved that the racemic cotton-ball
mixture (equal amounts of left- and right-handed helices) was electrornagnetically
inactive.
In 1956, Winkler was able to duplicate Lindman’s results over a wider frequency
band |5). He also observed that a chiral arrangement of a set of irregular tetrahedra
did nol rotate the plane of polarization, which is a counterintuitive result. The
next year, Tinoco and Freeman [6] performed a careful experiment using arrays
of oriented helices. They worked at X band, and sought especially to eliminate
the effects of anisotropic scattering. Tinoco and Freeman confirmed the chirality
hypothesis and gave further results on the frequency dependence of the rotation.
The existence of microwave activity was now given a solid experimental proof,
but it was not until the 1990’8 that new results on the electromagnetic activity of
artificial chiral materials started to appear. The flow of publications has expanded,
1.1. Early Studies on Chiral and Bianisotropic Media
7
witnessing a new wave of chiral research at the end of this century. This activity
embraces also more general bi-isotropic and bianisotropic media electromagnetics,
and the present book is part of this movement. For reviews and literature references
of the latest results, see [7, 8].
1.1.3	Mag net о electric Activity
Already in 1894, Pierre Curie speculated about the symmetry properties of the po-
larization mechanisms of matter, and suggested that an electric field could possibly
generate a magnetization [9]. In a more systematic study, Landau and Lifshitz
(1956) predicted a coupling phenomenon between the electric and magnetic quanti-
ties in antiferromagnetic materials that they called magnetoelectric effect [10]. Their
reasoning was based on the thermodynamic potential in crystals, in particular on
the tensor coefficient of the linear term both in the magnetic and electric fields. If
this tensor is nonsymmetric, a magnetization proportional to the electric field exists
in the matter, and vice versa.
The magnetoelectric effect depends on the crystal structure of the magnetic
material classes [11]. Dzyaloshinskii was able to show that among the well-known
antiferromagnetic substances there is one, namely Cr2O3, where the magnetoelectric
effect should occur from symmetry considerations [12]. In 1961, Astrov experimen-
tally confirmed this prediction [13], as he measured the finite magnetic moment of
a sample of single-crystal chromium oxide, placed in an electric field. The measure-
ments were carried out at the frequency of 10 kHz.
Later, the magnetoelectric effect was found for titanium oxide [14], although in a
weaker form than in Cr2O3. Their sample was, however, polycrystalline. Shtrikman
and Treves [15] annealed Cr2O3 powders with heat-treatment, and showed that the
resulting ceramic also displayed magnetoelectric effect. Also several other magnetic
crystal classes were shown to exhibit the magnetoelcctric effect, and it was discov-
ered that the effect was not a unique property of antiferromagnets but it could be
found in ferromagnetic materials, too.
As the similitude of the terms “electromagnetic activity” and “magnetoelectric
activity” hints, there is much common of this magnetoelectric effect with chirality.
For both phenomena, the ability of electric field to create magnetic polarization is
the essence. However, while the effect in optically active media emerged simply
through geometry, the explanation for magnetoelectric coupling is not that intu-
itive. It was Tellegen who started to ponder on the ways to create macroscopical
magnetoelectric effect in materials [16].
Tellegen suggested various ways to generate a nonreciprocal four-pole, or two-
port. A circuit element of this type he called a gyrator. A device like this was
revolutionary in circuit theory, where one was used to playing with the classical
8
Chapter 1. Introduction
reciprocal passive elements like resistors, inductors, capacitors, and transformers.
Л general theory for circuits requires the existence of gyrators, just like in the most
general exposition of field theory, nonreciprocal media effects have to he faced.
1.1.4 The Fedorov School
In the early fifties, Fedor I. Fedorov of Byelorussia founded his famous and very
active school in theoretical physics. Л very important direction of the activity
has been in the electromagnetics of bianisotropic media. Fedorov developed the
so called covariant methods which allowed one to study electromagnetic fields in
anisotropic crystals independently of any coordinate system. In fact, he introduced
and developed dyadic formalism in the optics of crystals [17].’ In the following,
a review of the Fedorov school research will be presented. The detailed extent of
the exposition to follow can be motivated by the lack of information in the western
scientific community of the Former Soviet Union microwave and electromagnetics
research.
In the Byelorussian school, the emphasis was, until recently, on optical activity
in crystals. In the optics of chiral trfedia, the theoretical studies started from plane
electromagnetic waves propagation in uniform media and reflection and transmission
at interfaces. For plane waves, the optical activity can be considered, as an effect of
the first-order spatial dispersion, i.e., it can be described in terms of the dielectric
permittivity dyadic which is a function of the propagation vector. The early Soviet
studies were based on the phenomenological constitutive equations,3 expressed in
tensor notations as
DB ~ e«sFs + ааксХ7ьЕс, В = II.	(1.3)
Here E,H are the electric and magnetic field strengths, D,B the electric and mag-
netic displacements, e refers to the permittivity, and the tensor a measures the
components of the spatial dispersion. The Onsager principle (symmetry of kinetic
coefficients in the linear theory) was applied in [19] to determine physical limitations
on possible forms of the tensor a in (1.3). Because of certain symmetry properties,
valid in reciprocal media, the third-rank tensor a can be reduced to a second-rank
tensor, and (1.3) can be rewritten in dyadic form:
D ~ • E-f-5 • V x E, В = H.	(1.4)
’In referring to Russian language journals we give page numbers of the original Russian versions.
In reprints translated into English the page numbering may be different.
’These relations were introduced already in the last century by Gibbs [18] as a generalisation
of the usual linear relation between the field vectors.
1.1. Early Studies on Chiral and Bianisotropic Media
9
In isotropic media, the material parameters t and a are scalar quantities. The
theory based on (1.4) correctly describes the optical activity and other properties
of plane electromagnetic waves in uniform media. However, as was revealed in
the early papers by Fedorov, the relations (1.4) are noninvariant with respect to
the Lorentz transformation, and are not fully sufficient for nonuniform media. To
correct the phenomenological theory [19], he suggested the relations [20]
D = f(7+a-Vx7)-E, B = ^(7-|J-Vx7)H	(1.5)
with two optical activity dyadics a and ft. 1 is the unit dyadic, and now the perme-
ability dyadic Ji makes the relations symmetric with respect to electric and magnetic
quantities. Using certain limitations following from the energy conservation con-
cept, it was shown that for reciprocal bianisotropic media, simpler relations with
only one coupling tensor of the second rank describe the phenomena correctly. In
the modern dyadic notations, the relations introduced in [21] read:
D = ?(7 + a-VxI)E, B = ?(7 + aTVx7)H, (1.6)
where the index T denotes the transpose of the optical activity dyadic a.
In more recent papers, the constitutive relations with redefined parameters were
introduced ([22] to [25]) for reciprocal media:
D = ё • E 4-ja • H, В = Ji • H — jaT • E.	(1.7)
These relations are Lorentz invariant and probably the most simple and convenient
ones. Accepting the form (1.7) or (1.6), the boundary conditions on interfaces have
the usual form, while they have to be modified if one uses (1.4).
Possible alternative constitutive relations were analyzed. In the 1970’s, there
was an intensive discussion in the Soviet literature [24, 26] about this issue. As a
result, it was established that the different constitutive equations (1.4), (1.6), (1.7)
are in fact equivalent for uniform media, after appropriate redefinitions of the field
vectors and the material parameters. These studies were summarized in a book by
Fedorov [22]. At present, the form (1.7) is widely adopted in the Russian literature,
although with some fluctuations in notations.
The symmetry of crystals imposes certain restrictions on possible forms of the
constitutive dyadics in (1.7). The list of the 11 possible classes of constitutive
relations can be found in [22].
Plane waves in bianisotropic crystals were analyzed using the covariant approach
developed by Fedorov in [17]. General eigenvalue equations for nonmagnetic crystals
with a discussion of eigenpolarizations in crystals of different symmetry can be found
in [22]. In the same reference, the problem of reflection from an interface between
isotropic and general bianisotropic media was covered.
10
Chapter 1. Introduction
To the best of our knowledge, the time domain Green function for isotropic
chiral media was first presented in [27], where also some scattering problems were
considered. The lime-domain dyadic Green function was found for the vector and
scalar potentials, and the result is valid for dispersionlese media. It was also es-
tablished that two transverse circularly polarized electromagnetic waves propagate
from the source point with the phase velocities = c/n±, whereas the longitudinal
wave components have the velocity t'o — c/v/npn_. Here, c is the velocity of light,
and the two refraction indices read
where po and c„ arc the free-space permeability and permittivity, and a is the scalar
optical activity coefficient.
In [27], retarded potentials and the dipole radiation in gyrotropic media were
considered. Note that in the Russian electromagnetics literature, the term “gy-
rotropy” has been preferred over “chirality.”
The Green function, which takes into account frequency dispersion in chiral
media, was obtained in [28], where the Green function and dipole radiation were
studied for sources with arbitrary time variation.
The Green function [27] gives a possibility to consider electromagnetic wave
scattering by fluctuations of the dielectric permittivity and the chirality parameter
in isotropic optically active media. Consequently, the extinction coefficients for
the left- and right-hand circularly polarized waves were found [29]. The difference
in two extinction coefficients leads to circular dichroism of scattering phenomena
in lossless media. Energy dissipation due to the scattering as well as the circular
dichroism effect can be described by introducing effective material parameters [27].
The fluctuation dissipation theorem and the Kramers-Kronig relations for chiral
media were considered in [30]. It was shown that the fluctuations of the electric
and magnetic flux densities are correlated, in contrast to nonchiral media. As
follows from the theorem established in [30], in isotropic chiral media with nonzero
imaginary part of the chirality parameter a, the dielectric permittivity e and the
magnetic permeability p must be complex, hence, in lossy isotropic chiral media,
all three material parameters are always complex.
Some general theorems for material parameters in chiral media, based on the
Kramers Kronig relations in chiral media and the Onsagcr Kasimir symmetry re-
lations were established in [31]. It was shown that there cannot exist absolutely
— i.e., over the whole frequency spectrum — right- or left-hand rotating media.
Consequently, terms like right-hand rotating or left-hand rotating are not absolute,
and one has to define the frequency range where the sense of rotation holds.
Spherical waves in source-free isotropic chiral media were studied in [32]. There
1.1. Early Studies on Chiral and Bianisotropic Media
11
exist two spherical wave solutions with different wave numbers, which are both
hybrid with nonzero longitudinal electric field components. In [32), the resonant
frequencies and eigenmodes of spherical resonators with ideally conducting walls
and filled with isotropic chiral media were found.
The constitutive equations (1.7) and other equivalent relations discussed above
are adequate for small spatial dispersion only, since only the first-order spatial
derivatives of the fields are taken into account. For larger optical or microwave
activity, more complex material equations with higher-order derivatives should be
used. Spherical waves in isotropic media modeled by the second-order material
equations
D = eE + jaH + (aV27 + bVV) • E,
В = fill — jaE	(1.9)
were studied in [33). Here, a and b are two extra complex material parameters
that characterize the higher-order magneloelectric behavior. Solutions to the corre-
sponding wave equation for the electric field were expressed as series of transversal
and longitudinal waves. In the Western literature, Monzon has later considered
media with second-order terms in both electric and magnetic polarization [34].
Cylindrical wave solutions in chiral media were considered in [35], based on
the second-order material equations (1.9). The solutions give cylindrical waves
and potential fields. In the far-field zone, the solutions reduce to two transversal
cylindrical waves. In far field, the waves are locally circularly polarized.
Spherical waves in chiral bianisotropic media with scalar dielectric permittivity
and dyadic chirality parameter were studied in [36]. The phase velocities of quasi-
spherical electromagnetic waves in the far-field zone were found. Some special
examples of real crystals were analyzed and local phase velocities were calculated
as functions of crystal axis orientation.
The analysis of chiral properties becomes much more involved for biaxial crys-
tals. It was shown in [37) that in biaxial chiral crystals of the symmetry class 222,
eigenwaves in the directions of the optical axes are elliptically polarized, and the
ellipticity depends on the anisotropy of the dielectric permittivity and on the angle
between the optical axes. Circular eigenwaves may exist only at the frequencies
where the crystal becomes uniaxial or bi-isotropic.
Chiral media with intrinsic magnetic structures or induced spiral structures [38]
are of special interest. In such magnetic materials, both the optical activity and
the Faraday effect cause polarization rotation. Eigenwaves in these optically active
magnetic crystals, for which the name “chiroferrites” was adopted in the Western
literature, were studied in [39].
12
Chapter 1. Introduction
1.1.5 Bianisotropic Media
The concept of bianisotropic medium, which actually is synonymous to “linear
medium,” was coined in 1968 by Cheng and Kong [40, 41), when a suddenly started
research boom on electromagnetics of moving media had reached a level of cer-
tain maturity. In fact, in the 1960's, problems involving moving media look quite
a central role in the world-wide theoretical research on electromagnetics, starting
with the problem of moving plasmas such as the ionosphere. Unz [42] extended
in 1962 the well-known Appleton-Hartree formulas from stationary to uniformly
moving magnetoplasmas and, in 1963, considered the problem of a moving Tellegen
medium [43]. In a short time, there followed a flow of papers on electromagnetics
of different media, dispersive or nondispersive, in relativistic or nonrelativistic mo-
tion, in free or bounded space, in frequency and time domain, describing solutions
for radiation fields, wave numbers and Green functions, and antenna impedances;
see, e.g., [44]. It was shown that a medium in uniform motion could be described
through constitutive equations of a stationary medium with dyadic medium param-
eters [45], which made working with moving frames and Lorentz transformations
obsolete.
The paper [40] was not the first one to define a medium with the most general
linear constitutive equations. In fact, Tellegen in his 1948 paper [16] already pre-
sented a set of linear relations generalizing the isotropic (actually bi-isotropic) ones,
which he applied for the realization of his gyrator. Also, Fano, Chu, and Adler in
their book [46] had the idea of replacing a moving medium by a stationary medium
with more general medium equations. However, since the emergence of the concept
of bianisotropic medium in [40] and [41], with the constitutive equations of the form
with general dyadics P, L, M, and Q, the two separate branches of research on
moving media and magnetoelectric crystals were united under the same topic.
The main theorems and principles governing the bianisotropic medium were
subsequently worked out in the paper by Kong in 1972 [47], which more or less fin-
ished the grand era of moving medium research. The reference covered in textbook
fashion basic electromagnetic properties of bianisotropic media, like the conditions
for the medium parameter dyadics of lossless or reciprocal media, image theory,
duality transformation, and the dyadic Green function. In due time, the materia]
was transferred to the widely cited monograph by Kong [48].
1.2. Characterization of Bi-Isotropic (BI) Media
13
1.2	Characterization of Bi-Isotropic (BI) Media
1.2.1 Constitutive Relations
On the macroscopic level, the description of bi-isotropic media (in short, Bl media)
is contained in the material parameters of the constitutive relations, written in this
book in the following form:
D = eE 4 (x - П, В =/‘H l (x l	E- (1П)
The dielectric response of the material is contained in the permittivity c, and the
permeability p is the corresponding magnetic parameter. But the essence of bi-
isotropic media are the magnetoelectric material parameters x and к, which are
dimensionless in the representation (1-11) as the free-spacc constant	is sep-
arated. The imaginary unit j emphasizes the frequency domain character of the
equations, and comes from the time-harmonic convention exp(j’wt).
The chirality parameter к measures the degree of the handedness of the mate-
rial, and for racemic media4 this parameter vanishes. A change in the sign of к
means taking the mirror image of the material. The other parameter x describes
the magnetoelectric effect, discussed in Section 1.1.3. Materials with x 0 are non-
reciprocal. Depending on the values of these parameters, Table 1.1 can be written
to classify BI media. In this table, and the analysis of the present book, pioneers of
magnetoclectric studies are honored with labels for different types of these complex
media.
	non chiral (a = 0)	chiral (^0)
reciprocal (x-o)	simple isotropic medium	Pasteur medium
nonreciprocal (x/o)	Tellegen medium	general bi-isotropic medium
Table 1.1 Classification of bi-isotropic media.
In the modern electromagnetics literature, notations other than these consti-
tutive relations are also used in the characterization of chiral media. In relations
4Media made of chiral elements of both handednesses in equal number so that they do not
show any macroscopic chiral properties.
14
Chapter 1. Introduction
(1.11),	the electric and magnetic displacements are represented as functions of the
field strengths. As earlier variations, this form of characterization has been chosen
by Condon [49], Tellegen [16], and Kong [47].
One alternative set of the constitutive relations are those named after Post [50],
which measure chirality by an admittance
D = eE - jfcB + V’„B, H =	(1.12)
Here, the fourth bi-isotropic parameter, the nonreciprocity susceptance V'n, has been
included [51].
Another choice, much used in the analysis of reciprocal chiral media, are the
Drude-Born- Fedorov [52] relations
D = t(E+/iVxE), B=p(H+/3VxH).	(1.13)
where fl gives the amount of (he chirality of material, in terms of length. Note
that due to the different way of representing the connections of the field and flux
quantities, the permittivity e is not the same in (1.11), (1.12), (1.13) for a given ma-
terial, nor the permeability /». Transformations between the parameters of different
systems are given in Appendix A.
1.2.2 Material Structure and Characteristic Figures
The connection between the bi-isotropic material parameters and the structure in
the media can "be illustrated with practical examples.
Pasteur Merlin
Figure 1.2 shows a sample of Pasteur medium at microwave frequencies.
The sample in Figure 1.2 is made by mixing metal helices of one handedness
into a resin matrix. Care has been exercized to secure isotropy: the helices must
be randomly oriented so that there is no special direction. A closer view of the
individual helices and their orientation is shown in Figure 1.3. From this figure, the
validity of the randomness and isotropy assumption can be judged. If the helices are
set in arrays, in aligned configuration, the result is a macroscopically bianisotropic
material, leading to dyadic (or matrix) coefficients in relations (1.11). This type of
media are discussed in Chapter 8.
The magnetoelectric coupling can be intuitively grasped from the behavior of
a helix as it is exposed to the electromagnetic field. If an electric field excites the
helix, it separates charges, creating an electric dipole moment. This contributes to
the permittivity of the medium, but the shape of the helix forces the charge to move
1.2. Characterization of Bi-Isotropic (BI) Media
15
Figure 1.2 A sample of chiral material manufactured by the Finnish company Finnyards
Ltd. Materials Technology. The sample measures 15 cm in diameter.
along a circular route, in addition to the linear path. This electric current loop is
equivalent to a magnetic dipole, and if all helices of the mixture have the same
handedness, the magnetic polarization effect will be enhanced. The corresponding
appearance of both types of polarization results also for magnetic field excitation.
Metal helices are often used in the present prototype samples of synthetic op
tically active media and chiral composites for microwave and millimeter wave ap-
plications. But since chirality is a geometric effect, this is not the only possibility.
Handedness can also be generated by structures with dielectric contrasts. Experi-
mental chiralists have indeed sintered ferroelectric ceramics in spiral forms and used
these as components for sample preparation. Nature is especially proficient in mass-
producing chiral elements, and engineers have learned to use her yield: polymers
have been exploited in the plastics industry for decades, and technologies are there
to play with handed polymers for chiral composites.
Figures 1.2 and 1.3 display clearly a property that is characteristic of today's
synthetic chiral structures: there is a distinct scale, the dimension of the helices. As
the novel electromagnetic properties hinge upon the scattering due to these handed
structures, it can be anticipated that this scale also determines the wavelengths
at which the structure is electromagnetically active. This is indeed true, and the
dispersion is strong around the resonant frequencies of the elements. To achieve
broader optically active frequency ranges, different sizes of helices are needed in the
sample. The problem, however, with polydisperse size distributions is congestion,
16
Chapter 1. Introduction
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o*tfSoo4d
% <0° ^'r
oV W
VO Л /
4	$
.	0°
j. V *
,3
£ 4? 0 £ ,
* (Ж $
(HU .
^4<>*
o* 1 Л "*
? \$- Ч ъ *
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b £ 0
£ 0 J*
6V
*
О
Figure 1.3 Л blow-up of the chiral sample in Figure 1.2. The helix dimension is 1.6 tnm.
as more helices are needed. Л helix becomes ineffective if the wavelength differs
greatly from the length of the helix. The issue of scattering by helices is discussed
in Section 6.6.
Tellegen Media
The interpretation of the nonreciprocity parameter % in bi-isotropic and Tellegen
media is not as straightforward as the concept of handedness. As can he seen from
the constitutive relations (1.11), there is a phase difference between these two effects.
If the magnetoeleclric effect is cophasal with the exciting field, the Tellegen effect
appears. Therefore, a phenomenologically acceptable scmimicroscopic picture of a
magnetoeleclric material with x 0 consists of such units where a permanent elec-
tric. dipole is lied — by a non-electromagnetic force — with a permanent magnetic
dipole. If these units are randomly dispersed in the material, there is no special
direction, and the material is bi-isotropic. This model is illustrated in Figure 1.4.
If an electric field is exciting the medium, the electric dipole feels a torque and it
turns. Hence also the juxtaposed magnetic dipole turns, and magnetic polarization
is created. Л similar effect happens for the incident magnetic field.
There are two samples of Tellegen media in Figure 1.4 that have the same
permittivity t and permeability )i but the nonreciprocity parameters XnXr °f the
samples are different: Xi = ~Хг- The adjacent samples of Tellegen media in the
figure display clearly the analogy between the two different sign changes: к —i —a
takes the mirror image, and x — X changes the dipole constellation.
In the early studies of chromium oxide where the magnetoeleclric effect was dis-
covered, the solid state crystals were anisotropic. Hence also the Tellegen effect was
of different magnitude along the crystal axis and on the basal plane. The anisotropy
1.2. Characterization of Bi-Isotropic (BI) Media
17
Figure 1.4 Л phenomenological model for Tellegen material. The two samples shown
here are isotropic, and have the same magnitude of the nonreciprority parameter \ but
of opposite sign.
can be of the order of one magnitude [13], and the nonreciprocity coefficient has
even different signs for these two directions [11]. The anisotropic Tellegen effect
in antiferromagnetic crystals is a consequence of the special spin distribution in
the unit cell. The spins are antiparallel, and an applied electric field perturbs the
magnetic moments slightly. Normally the average magnetization is zero but the
particular symmetry of chromium oxide, for example, is such that the oppositely
directed moments change their amplitudes in different amounts.
The magnetoelectric effect is sensitive to the ambient temperature. The thermal
vibrations become al some point strong enough to break the antiparallel coupling of
the magnetic moments. For chromium oxide, 34° C is this critical Neel temperature,
corresponding to the Curie temperature for ferromagnetic materials.
The isotropic material parameters can be classified as true scalars and pseu
doscalars, depending on bow they transform as coordinates arc inverted [r’3|. It
is easy to understand that the chirality parameter к changes sign as the spatial
coordinates arc inverted (r —» —r). The duality with the other magnetoelectric
parameter shows in the fact that the nonreciprocity parameter у changes sign for
time inversion (t —♦ —t).
Table 1.2 gives some practical values for the magnetoelectric parameters cither
observed in the nature or achieved with man made materials. It shows clearly
18
Referen ccs
the difference between the magnitudes of magnctoelectric parameters of different
samples. Natural media display к and x values with orders of magnitude lower
than man-made media. In spite of the quantitatively low values for the chirality
parameter, optical activity was visible to scientists already nearly two hundred years
ago.
Table 1.2 also tells the frequency at which the magnetoelectric effect has been
observed. This piece of information is essentia! in evaluating the magnetoelectric
performance of materials, due to the strong wavelength dependence of the manner
in which electromagnetic waves interact with chiral and bi-isotropic media. The
dispersion effects are discussed in more detail in Chapter 6.
к	n/n	X	Frequency	Reference	Note
3.910B		0	549 THz	[54|	optical wave in Quartz
0.05		0	1.2 GHz	[4|	Lindman in 1914
0.44	0.27	0	10 GHz	[55|	sample of Fig. 1.2
1.78	0.34	0	15 GHz	[56]	metal helices in epoxy
0.16		0	15 GHz	[57]	Copper strings in diclcxlric
0.30	0.15	0	8 GHz	[58]	Ferroelectric ceramic strings
		610-®	10 kHz	[13]	anisotropic Сг20з
'table 1.2. Magiictoelectric parameter values that have been observed in natural and man-made
samples. For Pasteur media, often the relative chirality value ar = к/п is a better measure for
chirality than the absolute value a. The refractive index is n - По*о- The value of the
chirality parameter has been calculated in some cases from the amount of polarization rotation.
References
[1]	Barren, L.D., Molecular Light Scattering and Optical Activity, Cambridge, England, Cam-
bridge University Press, 1982, p. 24.
[2]	Lowry, T.M., Optical Rotatory Power, New York, Dover, 1964. Appleqnist, J., “Optical
activity: Biot’s bequest,” American Scientist, Vol. 75, 1987, pp. 59-67. Hegstrorn, R.A., and
D.K. Kondepundi, “The handedness of the universe,” Scientific American, Vol. 262, No. 1,
January 1990, pp. 108 115. Mason, S.F., “General principles,” Proc. Royal Society of London,
Series Л, Vol. 297, No. 1448, 1967, pp. ,3-15. For a reproduction of historical and recent
articles on optical activity, sec Lakhtakia, Л. (editor), Selected Papers on Natural Optical
Activity, SPIE Milestone Series, Vol. MS 15, SPIE Optical Engineering Press, Bellingham,
Washington, 1990.
[3]	Lindman, K.F., “Om en genom ett isotropt system av spiralformiga resonatorer alstrad
rotationspularisation av de elektromugnetiska vagorna,” Ofversigt af Finska Vetenskaps-
Socieletens forhandlingar, A. Matematik och naturvetenskaper. Vol. LVII, No. 3, 1914 1915,
pp. 1-32. For a historical look on Professor Lindman’s career, see Lindell, I.V., Л.И. Sihvola,
i
г
References	ig
and J. Kurk(jarvi, "Karl F. Lindman — the last Hertsian and a harbinger of electromagnetic
chirality,” IEEE Antenna» and Propagation Magazine, Vol. 34, No. 3, 1992, pp. 24-30.
[4]	Lindman, K.F., “Uber eine dutch ein isotropes System von spiral for migen Resonatoren
erseugtc Rotationspolarisation der elektromagnetischen Wellen,” Annalen der Physik, Vol.
63, No. 4, 1920, pp. 621-644. See also the successive paper where Lindman compared his
measurements against a refined model for optical activity (including a molecular damping
term): “Uber eine durch ein aktives Raumgitter eneugte Rotationspolarisation der elektro-
magnetischen Wellen," Anna/en der Physik, Vol. 69, No. 4, 1922, pp. 270-284.
[6]	Winkler, M.H., “An experimental investigation of some models for optical activity,” Journal
of Physical Chemistry, Vol. 60, 1966, pp. 1656-1669.
[6]	Tinoco, L, and M.P. Freeman, “The optical activity of oriented copper helices; I. Experi-
mental,” Journal of Physical Chemistry, Vol. 61, 1967, pp. 1196-1200.
[7]	Lakhtakia, A., “Recent contributions to classical electromagnetic theory of chiral media:
what next?” Speculations in Science and Technology, Vol. 14, No. 1, 1991, pp. 2-17. Sec
also, Lakhtakia, A., V.K. Varadan, and V.V. Varadan, Time-Harmonic Electromagnetic
Fields in Chiral Media, Lecture Notes in Physics, 336, Springer-Verlag, Berlin, 1989; Special
Issue on Wave interactions with chiral and complex media, Vol. 6, No. 6/6 of Journal of
Electromagnetic Waves and Applications, (Engheta, N., Guest Editor), 1992; Lakhtakia, A.,
“Electromagnetic theory for chiral media,” in Spiral Symmetry, (Hargittai, I. and C.A. Pick-
over, Editors), pp. 281-294, Singapore, World-Scientific, 1992; Lakhtakia, A., Beltrami Fields
in Chiral Media, Singapore, World-Scientific, 1994. Bi-isotropic media and applications, (A.
Priou, Editor), in Progress in Electromagnetics Research, to appear in 1994, Elsevier.
[8]	Proceedings of Bi-isotropics’93, Workshop on Novel Microwave Materials, Helsinki Univer-
sity of Technology, Electromagnetics Laboratory Report Series, No. 137, February 1993; Pro-
ceedings of Bianisotropics’93, International Seminar on the Electrodynamics of Chiral and
Bianisotropic Media, Helsinki University of Technology, Electromagnetics Laboratory Report
Series, No. 169, December 1993. For a comprehensive list of references, see Unrau, U.B.,
“A bibliography on research in the field of bi-anisotropic, bi-isotropic, and chiral media and
their microwave applications,” published electronically by llstservCllstaerv.earn.net in
the list “CHIRAL-L” on 1 February 1994.
[9]	O’Dell, Т.Н., The Electrodynamics of Magneto-Electric Media, Vol. XI, Selected Topics in
Solid State Physics, Amsterdam, North-Holland Publishing Co., 1970.
[10]	Landau, L.D. and E.M. Lifshits, Electrodynamics of Continuous Media, Second Edition,
Oxford, Pergamon Press, 1984.
[11]	Freeman, A.J. and H. Schmid, Magnetoelectric Interaction Phenomena in Crystals, London,
Gordon and Breach, 1976. This book contains the lectures of the Symposium on Magneto-
electric Interaction Phenomena, held at the Battelle Seattle Research Center, Washington,
USA, 21-24 May 1973.
[12]	Dsyaloshinskii, I.E., “On the magneto-electrical effects in antiferromagnets,” Soviet Physics
JETP, Vol. 10, 1960, pp. 628-629.
[13]	Astrov, D.N., “Magnetoelectric effect in chromium oxide,” Soviet Physics JETP, Vol. 13,
No. 4, 1961, pp. 729-733.
[14]	Alehin, B.I. and D.N. Astrov, “Magnetoelectric effect in titanium oxide TijOj,” Soviet
Physics JETP, Vol. 17, No. 4, 1963, pp. 809-811.
[16]	Shtrikman, S. and D. Treves, “Observation of the magnetoelectric effect in Сг^Оз powders,”
Physical Review, Vol. 130, No. 1, 1963, pp. 986-988.
[16]	Tellegen, B.D.F., “The gyrator, a new electric network element," Philips Research Reports,
Vol. 3, No. 2, 1948, pp. 81-101.
20
References
[17]	Fedorov, F.I., Optic» of Anisotropic Media, Minsk, Byelorussian Academy of Sciences,
380 pp., 1958 (in Russian).
[18]	Gibbs, J.W., American J. Science, 111, Vol. 23, 1882, pp. 460.
[19]	Landau, L.D. and E.M. Lifshits, Electrodynamic» of Continuous Media, Pergamon Press,
1984, (the first Russian edition, Moscow, Goslechisdat, 1957).
[20]	Fedorov, F.I., “On the theory of optical activity of crystals. I. Energy conservation law and
optica] activity tensors," Optic» and Spectroscopy, Vol. 6, No. 1, 1959, pp. 85-93.
[21]	Bokut, B.V., A.N. Serdyukov, and F.I. Fedorov, “On the phenomenological theory of opti-
cally active crystals,” Soviet Physic» Crystallography, Vol. 15, No. 5, 1970, pp. 1002-1006.
[22]	Fedorov, F.I., Theory of Gyrotropy, Minsk, Nauka i Tekhnika, 1976 (in Russian).
[23]	Bokut, B.V. and A.N. Serdyukov, “On the phenomenological theory of natural optical activ-
ity,” Soviet Physics JETP, Vol. 61, No. 5, 1971, pp. 1808 1813; Bokut, B.V., A.N. Serdyukov,
and F.I. Fedorov, “On the form of constitutive equations in optically active crystals,” Optic»
and Spectroscopy, Vol. 37, No. 2, 1974, pp. 288-293; Bokut, B.V., A.N. Serdyukov, and V.V.
Shepelevich, “On the phenomenological theory of absorbing optically active media,” Optics
and Spectroscopy, Vol. 37, No. 1, 1974, pp. 120-124; Bokut, B.V., A.N. Serduykov, F.I. Fe-
dorov, and N.A. Hilo, “On the boundary conditions in electrodynamics of optically active
media," Soviet Physics Crystallography, Vol. 18, No. 2, 1973, pp. 227-233.
[24]	Bokut, B.V., A.N. Serdyukov, and F.I. Fedorov, “Comment on the paper by V.M. Agranovich
and V.L. Ginsburg, ‘On the phenomenological electrodynamics of gyrotropic media,’” Zhur-
nal Prikladnoi Spektroskopii, Vol. 19, No. 2, 1973, pp. 377-380, (in Russian).
[25]	Bokut, B.V. and A.N. Serdyukov, “On the theory of optical activity of non-uniform media,”
Zhurnal Prikladnoi Spektroskopii, Vol. 20, No. 4, 1974, pp. 677-682, (in Russian).
[26]	Agranovich, V.M. and V.L. Ginsburg, Kristallooptics with Spatial Dispersion and the Theory
of Excitons, Nauka, Moscow, 374 pp., 1965 (in Russian).
[27]	Gvosdev, V.V. and A.N. Serdyukov, “The Green function and fields radiated by moving
charges in gyrotropic medium," Optics and Spectroscopy, Vol. 47, No. 3, 1979, pp. 544-548.
[28]	Gvosdev, V.V. and A.N. Serdyukov, “Electromagnetic waves in dispersive gyrotropic media,”
Optics and Spectroscopy, Vol. 50, No. 2, 1981, pp. 349-353.
[29]	Gvosdev, V.V., V.A. Penyas, and A.N. Serdyukov, “On the scattering of electromagnetic
waves in gyrotropic crystals," Optics and Spectroscopy, Vol. 49, No. 6, 1980, pp. 1168-1171.
[30]	Serdyukov, A.N., Wave Phenomena tn Gyrotropic Media, Doctoral Thesis, Minsk, 343 pp.,
1985 (in Russian).
[31]	Bokut, B.V., V.A. Penyas, and A.N. Serdyukov, “Dispersion summation rules in the optics
of naturally gyrotropic media,” Optics and Spectroscopy, Vol. 50, No. 5, 1981, pp. 929-933.
[32]	Godlevskaya, A.N., V.A. Karpenko, and A.N. Serdyukov, “Spherical electromagnetic waves
in naturally gyrotropic crystals,” Optics and Spectroscopy, Vol. 59, No. 6, 1985, pp. 1262-
1265.
[33]	Godlevskaya, A.N., “Some peculiarities of spherical electromagnetic wave propagation in
naturally gyrotropic media,” Doklady Akademii Nauk BSS, Vol. 31, No. 7, 1987, pp. 616-
618, (in Russian).
[34]	Monson, J.C., “Radiation and scattering in homogeneous general biisotropic regions,” IEEE
Transactions on Antennas and Propagation, Vol. 38, No. 2, 1990, pp. 227-235.
[35]	Kondratova, N.L. and A.N. Serdyukov, “Cylindrical electromagnetic waves in media with
spatial dispersion," Zhurnal Prikladnoi Spektroskopii, Vol. 47, No. 2, 1987, pp. 326-329, (in
Russian).
References
21
[36]	Godlevskaya, A.N. and A.N. Serdyukov, “Quasi-spherical electromagnetic waves in gy-
rotropic crystals with scalar dielectric permittivity,” Optica and Spectroscopy, Vol. 69, No.
4, 1990, pp. 8Б1-8ББ.
[37]	Bokut, B.V., “On the polarisation plane rotation in gyrotropic crystals,” Doklady Akademii
Nauk BSSR, Vol. 34, No. 9, 1990, pp. 790-793, (in Russian).
[38*	] Akhramenko, I.N., I.V. Semchenko, and A.N. Serdyukov, “Selective interaction of electro-
magnetic waves in crystals with spiral-modulated non-stationary dielectric permittivity,”
Optica and Spectroscopy, Vol. 66, No. 3, 1989, pp. 618-622.
[39]	Bokut, B.V. and S.S. Girgel, “Electromagnetic waves in magnetic crystals with optical ac-
tivity,” Optica and Spectroscopy, Vol. 49, No. 4, 1980, pp. 738-741; Girgel, S S., “Refractive
indices and refraction vectors of eigenwaves in optically active magnetic crystals,” Optics
and Spectroscopy, Vol. 60, No. 4, 1986, pp. 777-780.
[40]	Cheng, D.K. and J.A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE,
Vol. Б6, No. 3, March 1986, pp. 248-261.
[41]	Cheng, D.K. and J.A. Kong, “Time-harmonic fields in source-free bianisotropic media,” J. of
Applied Physics, Vol. 39, No. 12, November 1968, pp. 5792 Б796.
[42]	Uns, H. “The magneto-ionic theory for drifting plasmas,” IRE 7Yans. on Antennas and
Propagation, Vol. 10, No. 4, July 1962, pp. 4Б9-464. See also comments by Epstein, M., T.
Bell, R. Smith and N. Brice and Ulis’s replies in IEEE Trans. on Antennas and Propagation,
Vol. 11, No. 2, March 1963, pp. 193-195.
[43]	Uns, H., “Electromagnetic radiation in drifting Tellegen anisotropic medium,” IEEE Trans,
on Antennas and Propagation, Vol. 11, No. Б, September 1963, pp. Б73-Б78.
[44]	Tai, C.T., “Electrodynamics of moving anisotropic media: the first-order theory,” Radio
Science, Vol. 69D, No. 3, March 196Б, pp. 401-405; Collier, J.R. and C.T. Tai, “Guided
waves in moving media,” IEEE Trans. on Microwave Theory and Techniques, Vol. 13, No.
3, July 196Б, pp. 441-44Б.
[45]	Chen, H.C. and D.K. Cheng, “Constitutive relations for a moving anisotropic medium,”
Proc. IEEE, Vol. Б4, No. 1, January 1966, pp. 62-63; Lee, S.W. and Y.T. Lo, “Radiation in
a moving anisotropic medium,” Radio Science, Vol. 1, No. 3, March 1966, pp. 313-324.
[46]	Fano, R.M., L.J. Chu, and R.B. Adler, Electromagnetic Fields, Energy and Forces, New
York, Wiley, 1960.
[47]	Kong, J.A., “Theorems of bianisotropic media,” Proc. IEEE, Vol. 60, No. 9, September 1972,
pp. 1036-1046.
[48]	Kong, J. A., Theory of Electromagnetic Waves, New York, Wiley, 1975. Chapter 7.
[49]	Condon, E.U., “Theories of optical rotatory power,” Reviews of modern physics, Vol. 9,
October 1937, pp. 432-4Б7.
[Б0]	Post, E.J., Formal Structure of Electromagnetics, Amsterdam, North-Holland, 1962. See
also, laggard, D.L., A.R. Michelson, and С.И. Papas, “On electromagnetic waves in chiral
media,” Applied Physics (Springer-Verlag), Vol. 18, 1979, pp. 211-216.
[51]	Sihvola, A. II. and I.V. Lindell, “Bi-isotropic constitutive relations,” Microwave and Optical
Technology Letters, Vol. 4, No. 8, 1991, pp. 295-297.
[Б2] Drude, P., Lehrbuch der Optik, Leipzig, S. Hirsel, 1900, p. 371. Born, M., “Uber die naturliche
optische Aktivitat von Flussigkciten und Gasen,” Physikalische Zeitschrift, Vol. 16, 1915, pp.
2Б1-258. Fedorov, F.I., Teoria girotropii, Minsk, Nauka i Tekhnika, 1976, p. 206. See also.
“The editor’s apology,” in Lakhtakia (1990) of Reference [2].
[БЗ] See the book by Post in Reference [50].
[54]	Lowry, T.M., Optical Rotatory Power, New York, Dover, 1964, Chapter 1, Table I. Measure-
ments made already by Biot in 1817, repeated by Lowry and Coode-Adams in 1927.
22	References
[55]	Hiyanen, A., private communication.
[56]	Ougier, S., I. Chenerie, and S. Bolioli, “Measurement method for chiral media,*1 Proceedings
of the 22nd European Microwave Conference, Espoo, Finland, August 24-27, 1992, pp. 682-
687.
[57]	Umari, M.H., V.V. Varadan, and V.K. Varadan, “Rotation and dichroism associated with
microwave propagation in chiral composite samples,*’ Radio Science, Vol. 26, No. 5, 1991,
pp. 1327 1334.
[58]	Guerin, F., “Microwave chiral materials: a review of experimental studies and some results on
composites with ferroelectric ceramic inclusions,*1 to appear in the Special Issue of Progress
in Electromagnetics Research on bi-isotropic media and applications, Priou, A. (editor),
Elsevier, 1994.
Chapter 2
Fields in Homogeneous BI Media
In this chapter, basic relations between the BI medium and the electromagnetic
field are considered. It is seen that the field in a homogeneous BI medium can be
split into two partial fields, the wavefields, each of which sees the BI medium as an
isotropic medium. This makes it easy to solve electromagnetic problems; in fact,
computer programs made for isotropic media can be readily applied to each of the
wave fields separately.
The notation applied in the theory is summarized in Appendix A. It is easy to
transfer from the present one to other existing notations through the transformation
formulas given in Appendix A [1].
2.1 Wavefield Decomposition
The Maxwell equations in frequency domain can be written as
V x E = —- M,	(2.1)
V x H = ju?D + J,	(2.2)
where E, B, D, H are the electromagnetic field vectors and J,M the electric and
magnetic current source vectors. In a BI medium, there exists a linear relation
between the four field vectors, which in the notation adopted in the present book
(see Appendix I)) can be written as
D = cE I (H,	(2.3)
B = <E + pH.	(2.4)
Inserting these in (2.1), (2.2), we obtain
V x E = —juj/ill — ju^E — M,	(2-5)
23
24
Chapter 2. Fields in Homogeneous BI Medin
V x H = jweEl jw(H + J.	(2.6)
For writing these and other equations in other notations existing in the literature,
the reader is advised to apply the transformation tables of Appendix A.
Later we shall replace the parameters ( and ( by
( = (X -	(2.7)
C = (x + >«)Vi^»	(2-8)
in terms of the Tellegen parameter x and the chirality parameter к, which can be
interpreted physically in a simple way. It will turn out that, for lossless media, the
parameters x and K> as we" as /' an,l £> have real values.
2.1.1 Wavefield Postulates
It is most convenient not to work with the electric and magnetic field vectors E, H
but with two other field quantities, the wavefields, E+1H+ and E_,II_, which make
up the total field as
E = E++E_,	(2.9)
H = H++H_.	(2.10)
It will be shown later that, in a homogeneous lossless HI medium, the power in
an electromagnetic field can be pictured as consisting of two parts: the “plus”
wavefield carries part of the power in right-hand elliptical polarization and the
“minus” wavefield carries the rest of the power in left-hand elliptical polarization.
The wavefield decomposition of an electromagnetic field in a BI medium1 can
be uniquely defined through two postulates:
1.	Each of the two wavefields E.|.,H+ and E_,H_ sees the BI medium as an
equivalent isotropic medium with respective medium parameters e+,/i+ and
2.	The two wavefields are independent: they do not couple in a homogeneous BI
medium.
Let us consider the implications of these postulates separately.
'The definitions can be generalised to the more general bianisotropic medium where the chi-
rality parameter can be a dyadic and other medium parameters scalars.
2.1. Waveficld Decomposition
25
Figure 2.1 The field in a BI medium can be split into two wavefield components.
Equivalent Isotropic Media
If the two wavefields see the BI medium as equivalent isotropic media with respective
parameters e+,/t+ and the medium parameters and the electric and magnetic
field vectors must satisfy special conditions. Writing
D+=<E++(H+ = «+E+,	(2.11)
B+ — (E+ + pH+ = p+H+l	(2.12)
D_ = eE_ + (II_ = e_E_,	(2.13)
B_ = (E_ + pH_ = p_H_,	(2.14)
we see after eliminating the field vectors that the equivalent parameters c±,/»± must
satisfy the two conditions
(« —«4 К/* — /*-»)—4C = 0.	(215)
(s-e_)(/i -/i_)-(( = 0.	(2.16)
Also, the wavefield vectors must satisfy relations which can be written in the form
E+ = -Л7+Н.М
E_ =
with the wave impedance parameters defined as
(2.17)
(2-18)
(2.19)
(2.20)
Simpler definitions will be obtained below.
The relations (2.17), (2.18) are similar to those of circularly polarized plane wave
vectors, which gives us a reason for calling this kind of fields wavefields
26
Chapter 2. Fields in Homogeneous HI Media
No Coupling
Let us now consider the second postulate that there be no coupling between the two
wavefields in a homogeneus BI medium, which means that the Maxwell equations
split into two independent sets. This happens if the wavefields individually satisfy
the Maxwell equations, which in the sourceless case are
Vx E| +	= 0, V x E_ d-= 0,	(2.21) (2.22)
V X II+ —	= 0, V x II_ — ya»e_E_ = 0.	(2.23) (2-24)
Because the electric and magnetic fields are interrelated through the conditions
(2.17), (2.18), there arise certain restrictions for the parameters. Actually, (2.23)
must be the same equation as (2.21) and (2.24) the same as (2.22). Inserting (2.17)
in (2.23) gives us
V x II f — jo»e+E+ = —(V x E+ +o>e+qjH+) = 0, which should coincide with (2.21). This leads to the relation	(2.25)
Mt v+ = J— V £t Another condition is obtained similarly:	(2.26)
	(2.27)
Equivalent Isotropic Parameters
Now it is possible to express the as yet unknown equivalent parameters c± and /i±
in terms of the original medium parameters. In fact, substituting q, from (2.19)
and if from (2.15) in terms of p, in (2.26), we have an algebraic equation for /г (:
г - о*»	~ 1+ e £+ £+-*<-'	(2.28)
Not surprisingly, the same equation is obtained also for ;r_. They are of the second
order and can be written as a single equation
M± - /*)’ + 2q<Xr(/'± - fi) + CV = 0,
(2.29)
2.1. Wavefield Decomposition
27
when writing ( + ( =	With the notation Xr = sind, the solutions for
and /r_ turn out to be of the form /ii j(qe±jtf, *7 — \Г^Ге- There is the question of
labeling the two solutions by and . We choose the following convention:
Д+ = /1 - j(qe"J* = pfcos + Kr)e~,d.	(2.30)
Ц- = ц + jCyei'> = /r(cosd - K1.)eJI’.	(2.31)
The corresponding equivalent isotropic permittivities are
	e( = r(cosd -f- кт}е’л, e_ = e(cosd — K,)e-jd.	(2.32) (2.33)
Thus, the two impedances	are of the form	
		(2-34)
and the wave numbers		(2.35)
«7 = A:(cos d 4 Kr),
(2.36)
(2.37)
к- =	= fc(cosd — Kr).
From (2.34) and (2.35) we can write useful relations for the two impedance quanti-
ties,
q(q_ = q1, q+ + q_ = 2qcosd.	(2.38)
The following consequences are seen to follow from the previous analysis:
•	It is interesting to note that while the wave numbers are real for lossless
media, the wave impedances as well as the parameters t±, /i±, arc complex
for nonzero values of the Tellegen parameter x- This will be discussed in a
further section.
• The impedance parameters q± do not depend on the chirality parameter к at
all. Change of sign of the Tellegen parameter x or transforms q+ to q_ and
conversely.
28
Chapter 2. Fields in Homogeneous BI Media
•	The wave numbers k± depend on both chirality and Tellegen parameters.
However, change of sign of % does not affect them but that of к transforms
k± to k_ and conversely. Because we can write
— fe_ = 2ккг, kt +	= 2fccosi?,	(2.39)
it is seen that the difference in wave numbers depends on the chirality of the
medium while the arithmetic mean of the wave numbers depends solely on the
Tellegen parameter. This property will be further discussed in conjunction
with plane wave propagation.
•	For lossless media, the wave numbers k± are both positive if the parameters
satisfy the inequality cosi? > |кг|, or
x’ + Kr < 1.	(2.40)
This can be seen as a certain limiting condition for the relative chiral and Tel-
legen parameters beyond which the nature of the medium is radically changed.
This inequality is tacitly assumed to be valid in the present book.
2.1.2 Wavefield Vectors
Now we are in a position to obtain expressions for the wavefield vectors correspond-
ing to a given electromagnetic field E, H. In fact, from
E = E+ + E_, H = H++H_,	(2.41)
with the relations (2.17), (2.18) between E± and H± inserted, we can also write
E = -jq+H+ + jn_H_, H = —E+— —E_.	(2.42)
•7+	4-
From these we can solve for the wavefield vectors. For example, eliminating E_
gives us
E+ = —~~ (E-jq-H).	(2.43)
’/+ +
This and other wavefield vectors can be written in simpler form as
E+ = ^(C^E->”H)’	(2-44)
2 cos v
Ё_ = -Ц(^Е + >ЧИ),	(2.45)
2 cos v
2.1. Wavcfield Decomposition
29
H, = —4 -3-E),	(2.46)
'	* 2cosv	q	V '
H_ = —1—(e- ^E).	(2.47)
2cosv	q '	'	’
Expressing electric and magnetic fields in terms of their complex linear combinations
in electromagnetic theory was probably first applied by Silberstein [2, 3] and, more
recently in the context of wave propagation in chiral media, by Dohren [4].
It is quite easy to check that the two wavefields satisfy the Maxwell equations
in their respective equivalent isotropic media whenever the field E,H satisfies the
Maxwell equations in the original Bl medium. As an example, we can write after
some algebra the identity
V x E+ + jo)/i+II+
=-------(V x E + joj/iH + jui(E) — ----j(V x И — yuieE —	— 0. (2.48)
2 cos v	2 cos v
This shows that, if E and II satisfy the sourceless Maxwell equations in the BI
medium, E+,II+ satisfy the Maxwell equation in the isotropic “ |-” medium. The
other equations can be handled likewise.
For a reciprocal isotropic chiral medium (Pasteur medium) we have i? = 0 and
simpler expressions for the wavefields can be written:
E+ = J(E - jqll),	(2.49)
E_ = 1(E bjqll),	(2.50)
H+ = |(H + -E),	(2.51)
2 q
H_ = 1(11 - |E).	(2.52)
It must be noted that the wavefield decomposition is uncoupled only in homo-
geneous media. If the medium parameters are functions of position, coupling takes
place. This point is elaborated in Chapter 5.
2.1.3 Sources of the Wavefiehls
The previous analysis was valid for regions without sources. However, it is easy to
extend the wavefield decomposition to include sources. This is essential if we want
to know what kind of sources actually give rise to the two wavefields.
Let us assume that the sources can be split into two parts
30
Chapter 2. Fields in Homogeneous BI Media
J = J++J_,	(2.53)
M = M++M_,	(2.54)
where M denotes magnetic currents. Writing the Maxwell equations for the wave-
fields together with sources,
VxEt = —jo>/r+H+ — M+,	(2.55)
V x E_ = ->wp_H_ - M_,	(2.56)
V	x Ht = ju>«+E+ + J+,	(2.57)
V	x II_ = jwe_E_ I J_,	(2.58)
and applying the relations (2.17), (2.18) for the wavefields, we can write another
set,
V	x H+ = M+E+ - —M+,	(2.59)
9+
V	x H_ = ja>t_E_ + — M_,	(2.60)
V-
V	x E+= —ju>/r4 H+— ;q+J+1	(2.61)
V	x E_ = —yaip_H_ -f	(2.62)
These equations must be the same as those above, which implies the following
relations
M+=j>J+J + ,	=	(263)
giving us the possibility to find the sources of the wavefields from (2.53) and (2.54):
J+ = 9~Ц(Л - -M),	(2.64)
2 cos v	T]
J- = ^-Ц(е ЯЛ b ' M),	(2.65)
2 cos v	tj
j4J),	(2.66)
2 cos v
M_ = —L-(e"M-j4J).	(2.67)
2 cos v
For example, an electric current J gives rise to the following four components:
2.1. Wavefield Decomposition
31
e	7 n
J_ = ------=-M+.	(2.69)
2 cos v	2 cos v
The sum of these is clearly seen to give the original current J.
The Maxwell equations in a homogeneous medium can be written for electric
wavefields and sources alone, by eliminating the magnetic wavefields and sources,
in the form
V x E+ — A;+E+ =	(2.70)
V x E_ + A_E_ =	(2.71)
This is not, however, the case for inhomogeneous media. As will be shown in
Chapter 5, when the two wave impedances and depend on the position vector
r, the equations are of the more complicated form
V x Et - E+
eif	e~jt
---Vln^/q^ x E+ =	-Vln^qT x E_ - J4+J+, (2-72)
COS V	COS V
Vx E_ 1 A:_E_------------jVln x E_ =----------------jVln JtT x E+ +	(2.73)
COS V	cos V
and, thus, there is coupling between the two wavefields. In a slowly varying medium
the coupling is small and can be taken into account approximately, a topic of Chap-
ter 5. It should be noted, however, that inhomogeneity of the medium does not
imply coupling of the wavefields if the wave impedances are constant. In particu-
lar, because the impedances do not depend on the chirality parameter, к may be a
function of position and yet the wavefields can be uncoupled.
2.1.4 Wavefields as Self-Dual Fields
The wavefield decomposition defined above can also be introduced through the
duality transformation. It can be shown that there exist two duality transformations
which change fields, sources and media but do not change a chosen BI medium with
parameters tj, /q, (p The two transformations are defined as 7),, T>t through
|5]
( E 'l = x- 7	(
r,( \ H } cos i?, \ —l/»7i
— sin
(2.74)
32
Chapter 2. Fields in Homogeneous BI Media
Here, the subscripts , and । refer to the two duality transformations labeled ast
“right-hand” and “left-hand”, corresponding to the respective signs ” and
on the right-hand side. For xi = 0, i-e., ’’1 = 0> this reduces to the ordinary duality
transformations for isotropic media [7, 8, 6].
The two duality transformations are seen to differ only by the sign. It is easy
to see by forming the square of the matrix operator of (2.74) that we have

(2.75)
E \
И )
E
H
E
H

= ВД
(2.76)
Field quantities Q which are invariant in one of these duality transformations
are called self-dual with respect to the transformation in question. It is easy to see
that the quantities Q t and Q_ defined by
Q+ = l(Q + vrQ), Q- = ’(Q + ©<Q)	(2.77)
are self-dual with respect to the right-hand and left-hand transformations, respec-
tively. Also, any field quantity can be written as a sum of its two self-dual parts,
Q = Q++ Q-	(2.78)
By forming the self-dual parts of the electric and magnetic field vectors in terms
of the transformations (2.74) we have
____1
2 cos
/ e*’’1
\ ±j7n
TJVi \ ( E \
e1’’’ J \ II )
(2.79)
These expressions can be seen to coincide with the definition of the wave fields in
a medium denoted by the subscript Thus, instead of “wavefields” we could also
call them “self-dual fields”. Because of convenience, the former name is adopted
here.
2.2 Plane Waves in Homogeneous BI Media
In terms of the wavefield decomposition discussed in the previous Section, it is easy
to find the effect of the BI medium on any electromagnetic field. Let us consider
simple plane-wave solutions, i.e., fields of the form
2.2. Plane Waves in Homogeneous BI Media
33
E(r) = Ee’kr, H(r) = He'kr,	(2.80)
with constant amplitude vectors E and II. It can be shown that, unlike the case
in simple isotropic media, such solutions in a BI medium are only possible for
certain polarizations, circular in fact, which coincide with those of the waveficlds.
Thus, in a strict sense, there only exist plane waves of wavefield polarization. In a
broader sense, let ms call a combination of two plane wavefields possessing the same
propagation direction a plane wave. This means that the polarization of the plane
wave need not be constant but may change during the propagation.
2.2.1 Plane-Wave Relations
Since the wavefield components of a plane wave do not couple in a homogeneous
medium, we can treat them as independent plane waves, i.e., write
E+(r) = E+e~** r,
E_(r) = E_e-,k-r,
H+(r) = H+e->k’r,
H_(r) = H_e’k’.
(2.81)
(2.82)
Assuming that both wavefield components propagate in the same direction defined
by the real unit vector u, the wave vectors are
k+ = uky,
= и’\/Я+«+ = k„n+,
(2.83)
k_ — ufe_, fc_ =	= k„n_,	(2.84)
with k„ =	and the two refraction factors defined as
ny = cos i? ± к — n(cos г? ± кг),
(2.85)
n = y//ircT. We assume that the condition (2.40) is valid for lossless media so that
the refraction factors satisfy 0 < ny < 1.
The relation between the electric and magnetic field vectors of the wavefield
components are obtained from the Maxwell equations
kj.	1
II± = ------ xE± = —u x E±,	(2.86)
u>p±	q±
k
E± =--------- x H± =x H±.	(2.87)
It is seen that the field vectors Ei and Hi are orthogonal to the direction of
propagation,
ii • E+ — 0, ii • Ну = 0.
(2.88)
34
Chapter 2. Fields in Homogeneous BI Media
This means that the plane-wave is а ТЕМ wave since both of its components are
transverse to the direction of propagation. Also, from(2.86) and (2.87) we see that
the fields are orthogonal in the sense
E+ • H+ = 0, E_ • H_ = 0.	(2.89)
Polarization
Because the wavefields obey the relations
H+ = —E+, H_ =	E_,	(2.90)
0+	4-
from the previous equation (2.86) we see that the electric fields satisfy
uxE(.=jE(, и x E_ = —j'E_.	(2.91)
From this equation, the eigenpolarizations of the two plane-wave components can
be found. Because the fields satisfy
E+  E+ = 0, E_ • E_ = 0,	(2.92)
they are circularly polarized. The direction of rotation can be obtained from the
polarization vector (see Appendix A), which is a simple real-valued vector function
which gives information of the polarization corresponding to a complex vector a:
p(a) = JT^-	(293)
p(a) points into the right-hand normal direction of the ellipse of a, Figure 2.2, and
its length is simply related to the axial ratio of the ellipse, e [9].
Figure 2.2 The polarization vector p(a) changes direction if the sense of rotation on the
ellipse is changed.
Inserting E± from (2.91) in a of (2.93) gives us the simple result
2.2. Plane Waves in Homogeneous BI Media
35
._ . E± x E±* E± x (u x E±*)	. „ .
p( ~ }E± -E±* ~ E±  E±*	“ ±U’	^2’94^
This means that the wavefield E+ is a right-hand circularly polarized vector with
respect to the direction of propagation u. On the other hand, E_ is a left-hand
circularly polarized vector since it is right handed when looking in the direction —u.
2.2.2 Polarization Rotation
Il is now easy to see how a plane wave of genera] polarization propagates in a BI
medium. Let us assume that the direction of propagation is along the positive z
axis, i.e., u = u,. Taking a linearly polarized electric field with an amplitude vector
E satisfying E • ux = 0, we can decompose it into two circularly polarized fields.
We can define the two complex CP unit vectors (see Appendix A)
u+ = ~Juv) = U- = ^(u«= u*.	(2.95)
which satisfy u+ • u+ = u_ • u_ = 0, u+ • u_ = 1 and p(u±) = ±u,, whence u+ is
a right-hand circularly polarized and u_, left-hand circularly polarized unit vector
when looking in the direction n,.
A plane wave polarized along u, at z = 0
E = u.E = (u(|u_)^,	(2.96)
can be written for subsequent values of z as the sum of two plane waves:
E(Z) = U+^e ’*+* 4 U-^e
=	cos(x,A:z) - uvsin(xrJlz)].	(2.97)
The vector in square brackets is a unit vector which has been rotated from the posi-
tion u, by the angle ф = —KTkz = ~nkoz in the right-hand direction when looking
in the direction of propagation. Denoting the rotation dyadic by (see Appendix B)
77(a) = It cos a -J- J sin a = eJa,	(2.98)
where J — u, x I is the 90” rotator, we can write in general
E(z) = е~й*с”,<’77(-к,Ь)  E(0).	(2.99)
The resulting expression (2.99) shows us that
36
Chapter 2. Fields in Homogeneous BI Media
•	The Tellegen parameter x affects the phase of the propagating electric field
and not its polarization.
•	The chirality parameter к affects the polarization of the propagating field and
not its phase.
•	There is a simple interpretation to the chirality parameter к in terms of the
polarization rotation of the propagating wave: it equals the angle of rotation
relative to the phase angle in free space |k| = 4>lkcz. The sign of к is positive
if the rotation is left-handed in the direction of propagation and negative if it
is right-handed.
Figure 2.3 Polarization of a propagating plane wave defines a helical curve in a lossless
BI medium.
Since the expression (2.99) is valid for any linearly polarized vector E(0), it is
also valid for any linear combination and, actually, for any polarization of the vector
E(0). Thus, the effect of a lossless BI medium for a propagating plane wave is to
rotate its polarization ellipse. In a lossy medium the field is attenuated. Usually
the two eigenwaves are attenuated differently, which makes the polarization ellipse
change while attenuating.
2.2.3 Angle Between Field Vectors
The relation between the electric and magnetic fields of a linearly polarized plane
wave can be written in terms of the wavefield components as
H(z) -= H+(z) + H_(z) = ±ЕДх) - —E_(x)
9+	4-
= -[eJ<’u.Fu_ — e~-’<,u_u+] • E(z).	(2.100)
The dyadic in square brackets can be expressed in terms of a rotation dyadic,
because we can write
2.2. Plane Waves in Homogeneous Bl Media
37
u+u_ — m_u+ = j(uriiF - UyU,) = —ju, x 1 — —jJ, (2.101)
«1,11. 4 « «1, =11.11.-1	— I,,	(2.102)
Л, ii_ — e^ii.n i = j sin df, — j cos d.7
= -j(cos(i?| ^)7, I sin(d I *)J].	(2.103)
Thus, the magnetic field of a linearly polarized plane wave in a Bl medium can
be written as
вдЛ(п£)-Е(г),	(2.104)
4	2
which means that
•	In a propagating plane wave with a linearly polarized electric field, the mag-
netic field is also linearly polarized.
•	Its amplitude equals that of the electric field divided by t).
•	The direction of the magnetic field makes the angle d + rr/2 with that of the
electric field, Figure 2.4. For a reciprocal medium with d = 0, the electric
and magnetic fields are orthogonal. For y> 0 the angle is greater and for
X < 0 smaller than the right angle. The angle between the field vectors is not
affected by the chirality parameter к.
•	The magnetic field vector rotates frozen to the electric field vector when the
wave is propagating in the BI medium.
Figure 2.4 The angle between linearly polarized electric anil magnetic fields differs from
the right angle by the Tellegen angle d in the general BI medium.
Because (2.104) is valid for any linearly polarized electric field, it is also valid for
a linear combination of such fields and, hence, for a plane wave of any polarization.
If E has elliptic polarization, II defines a similar ellipse whose axes make the angle
d | tt/2 with those of the E ellipse.
38
Chapter 2. Fields in Homogeneous BI Media
2.2.4 Wavelengths of the Plane Wave
A plane wave propagating in an ordinary lossless isotropic medium has a wavelength,
which is defined as the distance of periodicity in the direction of wave propagation.
If the wave number of the wave is k, the field is at the same state at multiples of
the distance A = 7.r/k in the direction of propagation.
In a lossless BI medium the situation is not so simple. Considering the expression
of the propagating electric field (2.99), it can be seen that there is no distance in
general where the field is exactly returned to the original state. This is because the
phase and the polarization of the wave are both changed in the propagation and
with a different rate. Thus, there is no universal wavelength, in general.
However, we can define two distinct wavelength quantites, one for the polariza-
tion and the other one for the phase. The polarization wavelength Apo; is defined
as twice the shortest distance at which the polarization is returned to its original
state,2 Figure 2.5:
„	2rr
Щ-^кХ^) = It => A^ = j——.	(2.105)
I'M*
The phase wavelength Apa is defined in the same way as for the simple isotropic
medium when the rotation dyadic is omitted. Thus, we have
2тг
V = -.-~V	(Z.JDG)
к cos v
There exists a universal wavelength only when the polarization wavelength and
the phase wavelength are commensurate, i.e., when their ratio is an integer. For
|ar| = cost? the two wavelengths are the same.
Ары/2
Figure 2.5 At the distance of a multiple of half polarization wavelengths the polarization
of the field is returned to the original state in a lossless BI medium. 3
3VVe could define half of this distance as the polarisation wavelength but, then, the definition
of the phase wavelength would not coincide with that in the simple isotropic medium.
2.3. Green Functions
39
As is the case in the simple isotropic medium, these wavelength concepts can be
approximately used also when the medium is slightly lossy. In this case, the real
quantities |кг |A- and A: cost? in (2.105) and (2.106) should be replaced by their real
parts |5?{кгА}| and S?{/ccost?}, respectively.
2.3 Green Functions
One of the basic electromagnetic problems in a linear medium is to find the Green
functions of the medium, because the solution of any source problem can be written
in terms of integrals with the Green functions corresponding to the medium in
question. Basically, the Green function is the field corresponding to a point source
of unit amplitude in an arbitrary location.
2.3.1 Dyadic Sources and Fields
For tiine-harmonic electromagnetic fields Green functions are dyadic functions, be-
cause they must give a vector field due to a vector source, i.e., map one vector
function to another. Let us write the vector electric and magnetic sources in terms
of source dyadics J and M as
J(r) = 7(r) • a, M(r) = J7(r) • b,	(2.107)
where a and b are suitably chosen constant vectors. Because of linearity of the
Maxwell equations, the electric and magnetic field vectors can be written in terms
of four Green dyadics in integral form
/ E(r) \ = f ( 3..(r-r')
V H(r) ) J Gme(r - r')
G.m(r - Г') \
Gmm(r r) J
\ Af(r')  b
dV.
(2.108)
The Green dyadics are functions of one vector variable r — r' when the medium is
homogeneous, because a translation of the source gives rise to the same translation
of the field.
Keplacing J by the delta function dyadic current j — 7A(r') and taking X? = 0,
shows us that, in this case, the electric field E(r) equals G„(r) • a and the magnetic
field 11( r) equals Gme(r) • a. Because a is arbitrary, these two Green dyadics satisfy
the Maxwell equations, with the electric dyadic delta source:
(V X I I jw(f) • (7„(r) = ;<x>/zG,nc(r),
(V X 7 - jo-(7) . Sme(r) = ja-cG„(r) + 76(r).
(2.109)
(2.110)
40
Chapter 2. Fields in Homogeneous Bl Media
2.3. Green Functions
Similarly, defining the magnetic source as M = 75(r) and taking J = 0, we have
(V X 7 + Ml)  CL(r) = -jivp(7mm(r) - 7«(r),	(2.111)
(V x 7 - M7) • Smm(r) = ja>£S.m(r).	(2.112)
It is seen that the off-diagonal Green dyadics can be expressed in terms of the
diagonal ones as
Sm« = ^-(Vx7 + JX7)S„,	(2.113)
= J-(V x 7 - ya>f7) - U-----	(2.114)
Also, because the diagonal Green dyadics satisfy the differential equations
M = M + 4 Af_ = 0, Af± =	(2.121)
2 cos v
Mid the dyadic electric and magnetic fields and, hence, the Green dyadics G„ and
Gme, ме split correspondingly into plus and minus components. Each pair sees
the medium as an effective isotropic medium with parameters e±, Thus, the
two Green dyadic components satisfy isotropic equations with both electric and
magnetic sources:
[-V X (V x 1) 4 ф] • G„± =	4 V x M±
= (j^±7 4 j^V x7) J4 =T^7±(^H(r),	(2.122)
where T denotes the transpose operation on a dyadic.
Now the solution is obvious. Because the isotropic Green dyadics
[(V x I - Mf) • (V x 7 4 jw<f) - k1!] - G„(r) = -jw^(r),	(2.115)
[(V x 7 4 >0,(7) • (V x 7 - M7) - fc’7] • Smm(r) = -><ve7«(r),	(2.116)
they are simply related through
-lm(r) = ~G„(r) = G(r).	(2.117)
jive	j iv ft
Thus, it is seen that all four Green dyadic can be solved if only we can solve for
the Green dyadic G(r). The dyadic G satisfies the Helmholtz equation
Z+(V) • T_(V) • S(r) = -7«(r),	(2.118)
T+(V) = (V x 7 - fc+7), Z_(V) = (V x 74 *-7).	(2.119)
These two dyadic operators commute: £4(V) • £_(V) = L_(V) • L+(V).
2.3.2 Solving for the Green Dyadics
The electric-electric Green dyadic Ge<(r) can be solved easily by decomposing it
into two wavefield problems corresponding to two equivalent isotropic media. First,
the dyadic electric source J — I6(r) is split into two pairs of electric and magnetic
sources (2.68), (2.69) as
=	=	=	= e±’e =
J = J+4J-, J± = ------------.J,	(2-120)
2 cos v
G±(r) = (I + ^-W)G±(r), G±(r) = 4xr satisfy	(2.123)
[-V x (V X 7) + kl7] • G4(r) = -7«(r), we can write	(2.124)
The final expression for the Green dyadic has the form	(2.125)
= Gee+ 4- G„_ = - —z[(fc4G+ 4 k_G~)I 2. cos v I 4 V(G+ - G_) x 74VV 4	1. у K^.	K_ j J This expression has been derived by several authors applying proaches [10, 11,9, 12, 13]. The other three Green dyadics are	(2.126) different ap-
3mm = —=Ц[(*,С+ 4*_GJ7 21] cos v I 1 V(G+ - G_) x 7 4 VV 4	], у fC.|.	ifc_ у J	(2.127)
. Reciprocity Mid Nonreciprocity	43
42
Chapter 2. Fields in Homogeneous BI Media
4- V(e~J*G+ 4 ?*G_) x I 4 VV (],	(2.128)
у Л|.	л_ J J
On. = ^-4[(*te"G+ - fc_e-^G_)7
2 cos v L
_	/e>*G	e~**G \ 1
4 V(eJ*G+ + e-^G_) x I + VV I -z—------—- ) L (2.129)
\ k— J *
The Green dyadics corresponding to the Bl medium satisfy the following condi-
tions:
G„(r — r') = Gj«(r'- r),	(2.130)
Smm(r-r') = ^m(r'-r),	(2.131)
^.m(r - r') 4 C7„.(r' - r) = --sinrf GX(r - r').	(2.132)
In verifying the last one, the following identity is needed:
(V/(r - r') x 7]T = —V/(r - r') x 7 = V'/(r' - r) x 7,	(2.133)
valid for any function /(r) satisfying /(r) = /( — r). Here, V' differentiates functions
of the variable r'.
2.3.3 The Far Field from a Dipole
Let us consider the field far from a point source at the origin. To obtain the
asymptotic expressions for the Green dyadic, we can approximate, by assuming
(k^r > 1),
VG±(r) = -(jkt 4 -)G±(rb « -j*±G±(r)u„	(2.134)
T
VVG^t) = [(jfc± 4 -)’ + 41Gi(r)urur - (jfc± + -)-G±(r)(7 - u,u,)
T V	T T
« -*iG±(r)urur,	(2.135)
and the Green dyadic can be written as
S„(r)
I * G-)7 (jfc.G,-й_С_)..гх7- (*,G, 4 *_G_)ur.i,|
Z COS V
=	- >’*' x	1 JU, X 7)1,	(2.136)
г
2.4.	Reciprocity and Nonreciprocity
43
with the two-dimensional unit dyadic defined by Ir = I — u,ur. The asymptotic
Green dyadic is seen to be a two-dimensional dyadic transverse to the radial unit
vector u,.
Thus, far from the source, the field is split into two spherical waves, which locally
act as plane waves. The electric field far from a dipole at the origin J(r) = u,/£J(r)
in the xy plane perpendicular to the dipole is
E(r) = G„(r) - u JL
~	“Л,г(”> + >”v) I	- ju„)].	(2.137)
OTii LOS I/
The T wave can be recognized as having the right-hand circular polarization and the
— wave, the left-hand circular polarization, in accord with the plane-wave analysis,
Figure 2.6. These components have different amplitudes whence the total polariza-
tion is elliptic. In the isotropic limiting case with k+ —» k_ —» k, the field is seen to
become linearly polarized.
A
Figure 2.6 The far field from a dipole is naturally split into two wavefield components.
2.4 Reciprocity and Nonreciprocity
All simple isotropic media are reciprocal in the sense that changing the positions of
a source and a measuring instrument does not change the reading in the instrument.
Unlike the simple isotropic medium, the most general BI medium is not reciprocal;
the fields and sources do not satisfy the reciprocity theorem. In this sense the BI
medium resembles certain anisotropic media which have the same property. Also,
as is the case for anisotropic and even bianisotropic media, we are able to form a
generalized reciprocity theorem by making the two measurements in two different
but related BI media. (7, 14]
2.4.1 The Reciprocity Theorem
The reciprocity theorem, originally formulated in simpler form by Lorentz in 1896,
states that if for any sources Ji, M,, which produce the fields Eh II, and any
Chapter 2. Fields in Homogeneous HI Media
44
sources Jj, M;, which produce the fields Ej, Ha, the following integral expression
vanishes [15, 7, 14, 16]:
/к"н')(-м,)-'Е"'н‘>(-м, )1л'
(2.138)
the medium is reciprocal. V here denotes the whole space. The latter expression of
(2.138) is obtained from the former one by replacing the source vectors by the field
expressions through the Maxwell equations and assuming that the ensuing surface
integral in infinity vanishes.
Figure 2.7 Testing the reciprocity of the medium.
Thus, for the medium to be reciprocal, the expression (2.138) must vanish for all
possible sources and their fields. It is seen that the right-hand side vanishes for all
field vectors only if k„x = 0, which in the nonstatic case requires that the Tellegen
parameter be zero: % = 0. Thus, the Tellegen parameter can also be called the
nonreciprocity parameter of the BI medium.
Defining the reaction of fields created by sources i on sources j by [15]
< i,j >= I(E„H.) • ( jJjj . ) dV,	(2.139)
v	\ i /
we can express the reciprocity condition simply as the symmetry of reaction between
two sources:
<1,2 >=< 2,1 > .
(2.140)
The reciprocity condition above can be generalized by assuming that the sources
and fields 1 and 2 are in different BI media [7, 14], different only in their Tellegen
parameters Xi and Xr- ln this case, in place of (2.138) we have

2.4. Reciprocity and Nonreciprocity
45
V
о -j'Mxi + Xi) \ ( E, \ 1
j’Mxi + Xi) 0	) \ H, ) J
(2-141)
The right-hand side is seen to vanish for Xi = —Xi> *«., when the two media
are otherwise the same except that their Tellegen parameters have opposite values.
Thus, the reciprocity condition (2.140) could be considered valid in the more general
sense. However, in the sequel, we assume that the sources 1 and 2 are in the same
medium.
Nonreciprocity of BI media can also be defined through their Green dyadics. It is
not, however, easy to see from the appearance of the four Green dyadics whether the
medium is reciprocal or not. Eliminating the field vectors, the reciprocity condition
can be written in terms of sources and Green dyadics as vanishing of the following
integral expression for all possible sources Jt, Mj, Jr,M2:
(h2 ) -(Ji.-Mr).
Er
Hr
dV
G„(r-r') Gem(r-r')\
Gm,(r - r') Gmm(r - r') /
( Ji(r') \
M,(r') )
The condition for reciprocity is vanishing of the difference of the two dyadic matrices,
which implies the following conditions for the Green dyadics:
(2.143)
(2.144)
(2.145)
It was seen in (2.130) and (2.131) that properties (2.143) and (2.144) are actually
always satisfied by the Green dyadics of BI media. Thus, the reciprocity condition
for the four Green dyadics is reduced to (2.145). To see the connection to the
previous condition, it is noted that (2.145) makes the left-hand side of (2.132) zero,
whence in the reciprocal case we must have sin d — 0, i.e., vanishing of the Tellegen
parameter.
ll.iweana’ Prlnrinle
47
46
Chapter 2. Fields in Homogeneous BI Media
2.4.2 Demonstration of Nonreciprocity
It has been sometimes under doubt whether a 131 medium can really be nonrecip-
rocal. Actually, such a medium is always electrically and magnetically reciprocal.
This follows from the fact that the Green dyadics Gce and satisfy the reci-
procity conditions for values of the medium parameters. Thus, in terms of two
electric or magnetic sources alone, the reciprocity condition is always valid. How-
ever, if we have one electric source and one magnetic source, the reciprocity is not
valid unless the Tellegen parameter vanishes. This is seen through the following
simple example.
- d2 0	dt
Figure 2.8 Nonreciprocity of a Bl medium can only be demonstrated with an electric
and a magnetic source.
Let us define electric and magnetic dipoles parallel to the z axis as follows:
ЛДгНМОД^-Л,), M2(r) = u.ImL6(p)8(z | d3),	(2.146)
with the distance between the dipoles D = d2 d2 large enough so that the dipoles
are in the far fields of one another. The reactions are of the form
<	1,2 >= -(u.IL) - [Sm.T(u,(d1 + d2))  u„/m£|,	(2.147)
<	2,1 >= (u.ImL)  [am(u,(d2 + d,)) • uj£].	(2.148)
After elaboration on the far-field expressions of the Green functions (2.128), (2.129),
we obtain
S.,„(r)	« 5-Ц|е ^+G+(r)(A - jj) - е*к_а_(г)(12 + p)l, 2 CO8 V	(2.149)
Sm.(r)	« --Ц[Лчг>,(г)(7. - p) - e**_G_(r)(7, b j7)l, 2 сое 17	(2.150)
with It = I ~	u,u,. Inserting these in (2.147) and (2.148) gives us	
2.5. Huygens* Principle
47
<	1,2 >« -^=^[e^+G+(u.C)-e-,d*:_G_(u,JD)),
2 cob v
<	2,1 >~ -^=^Je-’%G+(u,D) - ei’’fc_G_(u,P)),
2 COB V
(2.151)
(2.152)
which arc not identical in general, since their difference reads
< 1,2 > - < 2,1 >ss -jtainW^T’l^GJu.D)! fc-G-(u.P)].	(2.153)
This expression vanishes for nonvanishing sources and all values of D only if tan «1 =
0, i.e., x = 0. It is easy to find parameter values such that, for example, <1,2 >= 0
and < 2,1	0.
2.5 Huygens’ Principle
Huygens’ principle can be applied to problems involving a region containing electro-
magnetic sources and boundary structures separated from the BI space by a surface
S. According to the principle, the region is replaced by certain surface sources on
S, called the Huygens sources [17], in the homogeneous BI medium. Derivation of
the well known principle is done here in differential form following that given in
reference [9].
Figure 2.9 Geometry of the problem.
Let us define a pulse function, Pv(r), corresponding to a region V in the homo-
geneous BI medium, by
Py(r) = 1 for r G V,
Py(r) = 0 for r V.
(2.154)
(2.155)
Multiplying the Maxwell equations by the pulse function gives us the set of trun-
cated equations
V x (PVE) = —juiPyli - /VM -| (V/V) X E,
(2.15G)
48
Chapter 2. Fields in Homogeneous 131 Media
V x (PyH) = jwPvD + PyJ + (VPy) x H,	(2.157)
V • (PyD) = Pyg + (VPy) • D,	(2.158)
V • (PyB) = PvQm + (VPy) • B.	(2.159)
This set of equations defines certain electromagnetic fields due to certain electro-
magnetic sources which bear some relation to the original electromagnetic fields and
sources.
•	The fields PyE, Pyll etc. vanish outside the region V. Inside V they coincide
with the original fields E, II etc. Thus, they represent the original fields
truncated to the region V.
•	The volume sources PyJ, PyM etc. vanish outside V. Inside V they coincide
with the original volume sources J, M etc. Thus, they represent the original
sources truncated to the region V.
•	The terms (VPy)xE, (VPy)xII etc. can beinterpreted as additional sources
which arise due to the truncation of the original problem. They turn out to
be surface sources and arc called the Huygens sources.
The gradient of the pulse function Py(r) is zero everywhere except at the surface
S of the region V and can be written as
VPy(r) = nf$(r).	(2.160)
Here, 6s is the surface delta function
fs(r) = «(/(u,))	(2.161)
when the	surface	S	is	defined by the equation r —	=	0.	The	unit	vector n
is normal	to	the surface S and points into V, the region	where the	truncated fields
are nonzero. The additional surface sources arc
J, = n x H, M, = -ii	x E,	(2.162)
g, = n • D, gm, = n •	B.	(2.163)
They are called the Huygens sources and the principle of replacing volume sources
outside V by these surface sources without affecting the fields inside V is called
Huygens’ principle because the basic idea was first given in 1690 by Christiaan
Huygens [18|.
Taking the divergence of (2.156) and (2.157) we obtain relations for the four
Huygens sources on the surface S in the form of continuity conditions
2.6. Power and Energy in BI Media
49
VJ, = -jug, -nJ,	(2.164)
V  M, = —	— n  M,	(2.165)
which shows us that the surface currents arise both from the movement of surface
charges and the injection of volume currents.
2.6 Power and Energy in BI Media
In the general linear, i.e., bianisotropic, media the real part of the Poynting vector
S = ^Ex II*	(2.166)
is known to give the average power density (Watts over square meter) carried by
an electromagnetic field together with its direction of propagation [7]. At a point
in a HI medium containing no sources, we can write, by substituting the curl terms
from the Maxwell equations, the complex power balance equation
V • S =	- (E x II*) = ^(V x E) • H* - 1(V x II)* • E
= ~k*|E|’ - /t|H|’ + (f* - <)E • II*].	(2.167)
The real part of the divergence of the Poynting vector represents the power per
unit volume created at a point. If it is a positive number, the medium at that point
is active and gives electromagnetic energy to the field. If it is a negative number,
the medium is passive (lossy) and absorbs power from the field. Л medium which
is neither active nor lossy is called lossless and for such a medium the real part
of V - S = 0 is zero at every point for all possible electromagnetic fields. These
conditions induce conditions for the medium parameters.
2.6.1 Conditions for Medium Parameters
Lossless BI Media
The condition for the lossless BI medium is obtained from (2.167) by requiring its
real part to be zero:
- p|H|2 t (C - ()E  H*]} = 0	(2.168)
at all points of the medium for all complex vectors E and II. So, for example,
taking II = 0, we have the following condition for all vectors E:
50
Chapter 2. Fields in Homogeneous BI Media
31{>£*}|Е|2 = 0,	(2.169)
whence e must be real. Setting E = 0, we have a similar condition for fi. What
remains is
- 0E • H*} = 0 for all E, H.	(2.170)
Choosing first И — rE and then II = jrE with real r, we obtain both Sl{j(£* —
£)} = 0 and ${j(4* — C)} = 0, which together lead to the condition 4* = (
Writing
( = (x~	< = (x b Ук)а//'о«о,	(2.171)
the conditions for the parameters of a lossless HI medium become
£* = £,	/?=/!,	x*=x, n* = n,	(2.172)
i.e., all the parameters e, fl, x> K must be real for a lossless medium [7, 9|.
The divergence of the Poynting vector (2.167) has a simple form for the lossless
medium
V-S = ^[£|E|2-/i|H|2).	(2.173)
This is an imaginary quantity and coincides with the expression of a lossless isotropic
medium.
Lossy Media
Conditions for the parameters of a lossy medium can be obtained by requiring that
J?{V • S} < 0 for all non-null fields. From considerations similar to those above we
have
/«.,„< 0,	(2.174)
if denotes the imaginary part. A third condition is obtained, after some effort,
in the form [9]
(2-175)
which sets an upper limit to the imaginary parts of the chirality and Tellegen
parameters. If t and /1 are real, the medium cannot be lossy.
In the lossless case, there is a similar inequality (2.40) for the real x an(l K
parameters, when requiring the wave numbers k± of a propagating plane wave to
2.6. Power and Energy in BI Media
51
be positive. To see the connection to (2.175), the inequality (2.40) can be written
as
2 + к2 <
(2.176)
2.6.2 Wavefield Decomposition of Power
Let us study the Poynting vector of an electromagnetic field in terms of its wavefield
components. Substituting E = E+ 1 E_ and H — H+ -f H_ with H± = TyE±/»?±
in the expression of the Poynting vector above, we have
S = E‘ X +	* E: " 2ЙЕ- * E' + 2?E- X E’-	<2””
If the medium is lossless (real i? and r;* = r/T), we can further write
e-ia	e’6	i
S = —-|E+|2p(E4 ) - —|E_|2p(E_) + ^{e'% x E!},	(2.178)
where p(a) denotes the real polarization vector of a complex vector a, as explained
in Appendix A.
The real part corresponding to the propagating power density of the field is now
«{S} = ^-|E+|’P(E+) - ^|E_|2P(E_).	(2.179)
2г;	2т)
This expression gives rise to following conclusions:
Figure 2.10 Splitting of the electric field into two wavefielda.
52
Chapter 2. Fields in Homogeneous BI Media
•	The wavcficld components do not couple in a homogeneous BI medium and
carry power independently. This, of course, is the postulate made in the
beginning.
•	The power associated with the E., field propagates in the direction of the
p(E+) vector, which is normal to the ellipse of E( and points in the right-
hand screw direction. The power associated with the E_ field propagates in
the direction of —p(E_), which is normal to the ellipse of E„ and points in
the left-hand screw direction, Figure 2.10.
•	Thus, we can say that the “plus” and “minus” wavefields carry power with
respective right-hand and left-hand polarizations.
•	If a wavcficld component of the electromagnetic field is linearly polarized, the
corresponding polarization vector vanishes: p = 0, and no power is prop-
agated. Thus, a linearly polarized wavcficld component is a standing wave
which does not carry energy.
•	The case cost? —> 0, or |xr| ~+ 1, implies J?{S} —> 0. Thus, no power propa-
gates in such a medium, even if the field has a finite amplitude.
Since the wavefields see the BI medium as equivalent isotropic media whose
permittivity and permeability have complex values in general, it may appear con-
tradictory that this kind of a medium is lossless because, as concluded above, a
lossless isotropic medium must have real permittivity and permeability. However,
the equivalent media do not apply for arbitrary fields, only for the waveficlds which
have certain relations between the electric and magnetic fields. Because the prod-
ucts р±е± = (cost? ± KrYfie are real, the wave numbers arc real and the waves
propagate when pc > 0 and evanesce when pc < 0. The power balance equation
(2.167) for the waveficlds in their equivalent media
V • S± =	 (E± x l£) = ^(cosi? ± кг)еТ*(с|Е±|’ - 4|Е±|’] = 0 (2.180)
L	L	7]
shows us that the equivalent media are lossless for the wavefields in spite of their
complex parameters.
The phase velocities of the two waveficlds can be defined as
v„± = —1== = ;;--------j--	(2.181)
^/р±е± (cos v ± кт)у/це
It is seen that when cost? — |кг| > 0, or № + x2 < pe/poco, both phase velocities
are positive and finite. For к2 + x2 > Pf one °f the phase velocities has negative
sign corresponding to a backward wave, whose energy and phase travel in opposite
directions.
2.7. Electroniagnetostatics in BI Media
53
2.7 .Electromagnetostatics in BI Media
Electrostatics and magnetostatics are two independent branches of science in an
isotropic medium. In а Ш medium, the electric and magnetic fields are coupled
together to what can be called electroniagnetostatics [19]. In pure statics there is no
chiral effect because, as can be shown, the chirality parameter к is an odd function
of frequency and vanishes for zero frequency [9]. However, considering statics as
the limiting case of decreasing frequency through a quasi-static asymptotic method
introduced by Stevenson [20], chirality can be taken into account as a perturbation.
In the following, the BI medium parameters are taken in the form ( and ( without
writing them in terms of the Tellegen parameter x and chirality parameter «.
2.7.1 Basic Equations
In the static limit, the Maxwell equations become
VxE=-M, VxH = J,	(2.182)
V-D = e, V • D = pm.	(2.183)
Poisson equations for the electric and magnetic fields are obtained by substitut-
ing D and В from the medium equations and taking the curl of the curl equations
above:	V2E = VxM + ^V₽-(Vfm], V’H = -V X J + ^[cVe„ - CV₽],	(2.184) (2.185)
with	д = /<« - (.(,	(2.186)
2.7.2 Fields Due to Charges
In the case of no currents, the fields E and II from electric and magnetic charges
are irrotational and can be expressed in terms of scalar potentials,
E=-V& H = -V^m,	(2.187)
which satisfy the Poisson equations
v2<A -
(2.188)
54
Chapter 2. Fields in Homogeneous BI Media
V4„ =-^[egm - <d-	(2.189)
For example, an electric point charge at the origin, g = Qt>(r), gives rise to the
following potentials which vanish at infinity:
Ф=~~	(2.190)
4х€,Г	/I
The potential function ф is similar to that in a simple isotropic dielectric medium
with effective permittivity e,
e, = — = e — —.	(2.191)
Thus, surfaces of constant electric and magnetic potential are concentric spheres.
The fields and flux densities are
E = — V^> = -и,— 4irt,r2 II - -V^n = u.	, 47T/16erX D = t.E = -(r - 4)V^ = -u,-%, /1	47ГГ2 B = 0.	(2.192) (2.193) (2.194) (2.195)
It is to be noted that D and 13 are not affected by the nature of the medium.
Thus, metallizing two spherical surfaces r — rt and r = r2 > r2 gives us a spherical
capacitor with surface charge densities in, • D on each surface. The capacitance of
the capacitor is
Q ______ _ 4лсеГ|Г2
^(п) ф(г2) r2 -r,
(2.196)
This can be generalized to a capacitor of any geometry. If the capacitance of a
capacitor with air filling is C„, one filled with a BI medium is Coerlea. All the
parameters £, e and p are seen to affect the effective permittivity of the medium
and the capacitance of the capacitor. For the Tellegen medium with ( = f
we can write
e. = t(l-x,2) = «cos2!?.	(2.197)
It is interesting to see that in the limiting case x = n or cos = 0, we have t, = 0,
which means that the medium becomes a perfect magnetic conductor.
2.7. Electrornagnetostetks in BI Media
55
2.7.3 Bi-Isotropic Transmission Line
Let us consider a two-dimensional transmission-line type of problem with conducting
cylinders of arbitrary cross section parallel to the z axis in a BI medium with no
sources in the medium between conductors. The medium is assumed homogeneous.
Since there are no magnetic charges, V • В = 0, this motivates use of the vector
potential. Assuming fields transverse to U, and with no dependence on z, the vector
potential must be parallel to z:
В = Vx A = VAxu,.	(2.198)
Because there are no magnetic charges on the conductor surfaces, the boundary
condition ii • В = 0 implies
1 - В = -t  VA = 0,	(2.199)
with t the tangential unit vector on the boundary surface. Thus, the potential A
must be constant on each conductor surface.
The electric field satisfying V x E = 0 can again be derived from the scalar
potential
E =	(2.200)
and the boundary condition following from n X E — 0 is ф constant on each con-
ductor.
From the medium equations we can solve for the two other vector quantities
II =-~u, x VA +(2.201)
M	/1
D = —erV$ - -u, x VA.	(2.202)
Equations for the potentials in the medium between the conductors are obtained
from V • D — 0 and V x II = 0 and they turn out to be simple Laplace equations.
V2^(p) = 0,	(2.203)
V2 A = 0.	(2.20-1)
Circuit Quantities
Let us consider circuit quantities of the transmission line, i.e., certain integrated
field quantities. The voltage between two conductors a and b is defined as the
potential difference
56
Chapter 2. Fields in Homogeneous 1)1 Medin
ь	ь
Uab = У E  de = - У V^-dc = фа-фь,	(2.205)
a	a
when we denote the potential on conductor i by ф{. The path of integration from
conductor a to conductor b is arbitrary because conductors have constant potentials
and a closed-loop integral of the electric field E gives zero due to the condition
V x E = 0.
The magnetic flux per unit length between the conductors can be written as
s	ь
= У В • (u, х de) = — у(u, x \7Л) • (u, x de) = A„ - Ль. (2.206)
a	a
The total charge per unit length on conductor a is
qa = У и • Ddc = —ee У и • Xtydc — — f n  u, X VAdc
= —ee У n • \7<f>dc + — У VA  de = —ee £ n - \7<fidc. (2.207)
a	H a	a
The gradient of the magnetic potential vanishes along the surface.
Finally, the total current on conductor a is obtained from the vector potential
as
= У H  de = — — У u, x Х7Л • de + — У • c =--n  VAde.
a	! a	I a	a
This time the contribution of the scalar potential vanishes.
The quantities capacitance and inductance per unit length are now
=	^-Zn-V^dc,
I _	_	A°~ Ab
/„	f‘fn-VAdc'
They can computed as soon as the potentials are known.
Because the two potential functions ф and A both satisfy the Laplace equation
and similar boundary conditions (constant potentials on the conductors), there is
a simple relation between their solutions. For example, if both ф>, = 0 and Ль — 0,
we can obviously write
(2.208)
(2.209)
(2.210)
References
57
Л(р) = ^(р).	(2.211)
Фа
The capacitance and inductance arc independent on the boundary values. It is
easy to see that, from (2.211) inserted in (2.209), (2.210), we can write the relation
Cat,Lab - (if, - це - £<	(2.212)
This condition is a generalization from that valid in isotropic media.
The theory of quasi-TEM modes in inhomogeneous transmission lines [21, 22,
can be generalized to lines containing Bl media [23] by applying the static capaci
tance and inductance introduced in the present Section.
References
[1]	Sihvola, Л.П. and I.V. Lindell, “Bi-isotropic constitutive relations”, Microwave Optics and
Technology Letters, Vol. 4, No. 8, pp. 295-297, July 1991.
[2]	Silberstein, L., “Elektromagnetische Grundgleichungen in bivektorieller Behandlung,” An-
nalen der Physik, Vol. 22, No. 3, 1907, pp. 579-587; Vol. 24, No. 14, 1907, pp. 783-784; Phil.
Mag., Vol. 23, No. 137, pp. 790-809, 1912.
[3]	Stratton, J.A., Electromagnetic Theory, New York, McGraw-Hill, 1941.
[4]	Bohren, G.F., “Light scattering by an optically active sphere,” Chemical Physics Letters,
Vol. 29, No. 3, pp. 458 452, 1974.
[5]	Lindell, I.V. and A.J. Viitanen, “Duality transformations for general bi-isotropic (nonrecip-
rocal chiral) media,” IEEE Trans. Antennas and Propagation, Vol. 40, No. 1, January 1992,
pp. 91-95.
[6]	Jaggard, D.L., X. Sun and N. Engheta, “Canonical sources and duality in chiral media,”
IEEE Trans. Antennas and Propagation, Vol. 36, pp. 1007-1013, 1988.
[7]	Kong, J.A., Electromagnetic Wave Theory, New York, Wiley, 1986.
[8]	Lindell, I.V., “Asymptotic high-frequency modes of homogeneous waveguide structures with
impedance boundaries,” IEEE Trans. Microwave Theory and Techniques, Vol. 30, No. 10,
pp. 1087-1093, October 1981.
[9]	Lindell, I.V., Methods for Electromagnetic Field Analysis, Oxford, Clarendon Press, 1992,
pp. 10-11.
[10]	Bassiri, S., N. Engheta and C.H. Papas, “Dyadic Green’s function and dipole radiation in
chiral media,” Alfa Frequenza, Vol. LV, No. 2, pp. 83-88, March-April 1986.
[11]	Monson, J.C., “Radiation and scattering in homogeneous general biisotropic regions,” IEEE
Trans. Antennas and Propagation, Vol. 38, No. 2, pp. 227-235, February 1990.
[12]	Weiglhofer, W.S., “A simple and straightforward derivation of the dyadic Green’s function
of an isotropic chiral medium.” ЛгсЛгг far Elektrische Ubertragung and Elektronik, Vol. 43,
No. 1, pp. 51-52, Jan.1989.
[13]	Lindell, I.V. and A.J. Viitanen, “Green dyadic for the general bi-isotropic (non-reciprocal
rhiral) medium,” Helsinki University of Technology, Electromagnetics Laboratory Report 72,
October 1990.
58
References
[14]	Altman, C. and K. Suchy, Reciprocity, Spatial Mapping and Time Reversal in Electromag
netics, Dordrecht, Kluwer Academic Publishers, 1991.
[15]	Rumsey, V.H., “Reaction concept in electromagnetic theory,” Physical Review, Vol. 94, No.
6, pp. 1483 1491, 1954.
[16]	Lindell, I.V., “On the reciprocity of bi-isotropic media,” Microwave and Optics Technology
Letters, Vol. 5, No. 7, pp. 343-346, June 1992.
[17]	Lakhtakia, A., V.V. Varadan and V.K. Varadan, “Field equations, Huygens’s principle,
integral equations and theorems for radiation and scattering of electromagnetic waves in
isotropic chiral media,” J. Optical Society of America A, Vol. 5, pp. 175-184, February 1988.
[18]	Huygens, C., Tratte de la Lumiere, Leiden, 1690. English translation Treatise on Light, New
York, Dover.
[19]	Kong, J.A., “Charged particles in bianisotropic media,” Radio Science, Vol. 6, No. 4, pp.
1015-1019, November 1971.
[20]	Stevenson, A.F., “Solution of electromagnetic scattering problems as power series in the ratio
(dimension of scatterer/wavelength)," J. Applied Physics, Vol. 24, pp. 1134-1142, 1953.
[21]	dos Santos, A.F. and J.P. Figanier, “The method of series expansion in the frequency do-
main applied to multidielectric transmission lines,” IEEE Trans. Microwave Theory and
Techniques, Vol. 23, No. 7, pp. 753 -756, September 1975.
[22]	Lindell, I.V., “On the quasi ТЕМ modes in inhomogeneous multiconductor transmission
lines,” IEEE Trans. Microwave Theory and Techniques, Vol. 30, No. 8, pp. 812-817, August
1981.
[23]	Koivisto, P.K. and J.C-E. Sten, “Quasi-static image method applied to bi-isotropic mi-
crostrip geometry,” Helsinki University of Technology, Electromagnetics Laboratory Report
136, February 1993.
Chapter 3
Plane Waves in Layered Media
In this Chapter, plane wave propagation through and reflection from a structure con-
sisting of homogeneous BI slabs with plane-parallel interfaces is considered. Waves
with normal and oblique incidence are studied separately because of the great dif-
ference in complexity. Problems with normal incidence are simpler because the
fields are tranverse electromagnetic (ТЕМ) and have the same circular eigenpo-
larizations in each slab. In this case, the problems can be split in two uncoupled
scalar problems. For oblique incidence, the cigenpolarizations are different in each
medium and, because of their coupling at each interface, the problems are much
more complicated and inherently of a vector nature.
3.1 Normal Incidence, Single Interface
Let us consider a plane wave incident from a half space of BI medium 1 (z < 0)
with parameters ti, /ц, Xi> to the interface of another BI medium 2 (z > 0)
with parameters r21 Kz> Хг- Part of the wave is reflected back in the medium 1
and the rest is transmitted into the medium 2, Figure 3.1. For a normally incident
wave, the amount of reflection and transmission depends on the polarization of the
wave as well as the parameters of the two media.
The polarizations of the two plane wavefields are circular before and after re-
flection and transmission. Because of the conditions of continuity at the interface,
the directions of rotation of the field vectors must be the same for the propagating
incident, reflected and transmitted waves. Thus, the handedness of the transmitted
wave must be similar, and that of the reflected wave opposite, to that of the incident
wave. For example, in normal incidence, a right-handed wave is transmitted as a
right-handed wave and reflected as a left-handed wave, without any coupling to the
left-handed transmitted wave or the right-handed reflected wave. Thus, for normal
incidence, any polarization can be split into two components which do not couple
59
60
Chapter 3. Plane Waves in Layered Media
Figure 3.1 Basic problem of plane wave with normal incidence on an interface of two BI
media.
at the interfaces of a layered structure.
3.1.1 Circular Polarization
Let us consider the two circularly polarized ТЕМ wavefield cases at the same time
in terms of a double subscript. The incident wavefields propagating in the medium
1 positive z direction arc defined as
E‘. (z) = E'.e-^11*,	H’±(z) =	(3.1)
and they satisfy u,-E± = 0, u,-H± = 0 with kt± — u>R,'d >;i± =
Reflected and Transmitted Fields
The reflected wavefields are denoted by
El(z) = E;^11*, IE (z) = ±—Е^е'*'**,	(3.2)
»?i±
and the transmitted wavefields by
E^(z) = Е^е-^’*',	H‘±(z) = ±-^-Е±е_,*,±ж.	(3.3)
»7г±
The reflection and transmission coefficients denoted by fl+_, f?_ + ,	T are
defined through the linear relations
E; = RT±E’±,	E‘± = T±±E’±.	(3.4)
Thus, gives the transmitted + wave and the reflected — wave, for the
incident 1 wave.
3.1. Normal Incidence, Single Interface
61
Relations between the magnetic field components arc obtained by substituting
in (3.4) the electric fields in terms of the magnetic fields:
Щ	Hl =	(3.5)
71т	7i±
Reflection nn<i Transmission Coefficients
Expressions for the coefficients Лт±, T±± can be derived from the continuity of the
ТЕМ fields at the interface:
E*± + E^ = E^,	+ H; = H'±.	(3.6)
Substituting (3.4) and (3.5), valid for any E’± and respectively, gives
1 —	— 21iT
1	„ У1т± —------7±±,
4it 7>±
1 4 Лт± — 7±±,
(3-7)
which can be solved in the form

=	-4ucT>.».
4j± + 4it
—cosrJ^^1.
7>± + 4it
(3.8)
(3.9)
_ 41t(41± ~ 4h)
4i±(4j± + 4it)
г)и(Уч + 41т) _
4i±(4i± + 4it)
Note that in terms of wave admittances 1,“’ the expressions have a somewhat simpler
form:
(3.10)
r? _ 7i± 7зт	_ 4i ± 4- 4it
тi — -1 , -1 >	----—,
41т + 4i±	q2± + 41т
The difference is due to the fact that the coefficients were defined in terms of the
electric fields. The formulas would have been simpler in terms of wave impedances
if defined through the magnetic fields.
As special cases we see at once that if the medium 2 approaches PEC with
724 ~~* 0, we have R^± —» —1 and Tj-j. —» 0, while for a PMC medium with
7i± -» oo the corresponding limits are Лт4 —> e*2’*». Also, the expressions for
the reflection and transmission coefficients become similar to the ones in simple
isotropic media if the medium 1 is reciprocal, i.e., a Pasteur medium, with y, — 0
implying r/1+ =	= гц:
(3.U)
7>± + 7i	7j± + >7i
For two general BI media, the expressions (3.8), (3.9) can be further elaborated
in the more explicit form
62
Chapter 3, Plane Waves in Layered Media
41 - 4i cos(i92 -19,) T p]i sin(t92 -
>/2 I 7,cos(i92 f i9,) 1 jilt ship] I
7±± =
272 cos t9,
_______________________________
7i f 7, cos(i92 + 19,) ± jrji sin(i92 + i9,)
(3.13)
Lossless BI Media
Гог lossless 131 media with real ijit i/2> i9b i?2, the reflection and transmission
coefficients are complex conjugates of one another:
R*+ = R^,	Г;+=Г__,	(3.14)
which means that they have the same absolute values:
|Л_+| = |Я+_| = R,	|T++| = |T__| = T,	(3.15)
and the phase angles are real with opposite signs:
Я_ + = fie’*’", Rt_ = Re~ivR,	T+ + = Те’*’1,	T__ = Te~ivT. (3.16)
The absolute values of the reflection and transmission coefficients (3.12), (3.13)
can be written as

71 =
41 + ’ll - 2’11'12 COSpi - 19,)
7i + 4i I 2’11’12 cos(i9, + i92) ’
21/, cos 19,
/7? 1 4i I 2r/li/2cos(t91 I i92)
and the phase angles satisfy
,	41s,n “ ’ll »in $i , ,	1
¥>я - tan ----------------------- I tan
r)2 cos v, — 7, cos v2
t]i sin i9, — 7, sin i92
t/2 cos 19, -I 7i cos i92
(3.19)
„	, n2 sin 19, — и, sin t92	._______
VT = -i92 + tan-1 -5--------1/1	2	(3.20)
72 cos v, + 71 cos i92
Examples of the reflection and transmission coefficients R, T, as functions of
the parameters of a lossless Pasteur medium (1) and a lossless HI medium (2), are
given in Figures 3.2 and 3.3. It is seen that, for the limiting case t92 = тг/2, i.c.,
^2r = 1, we have H = 1 and T = 0 so that waves of all polarizations are totally
reflected back.
3.1. Normal Incidence, Single Interface
63
Figure 3.2 Absolute values of the two reflection coefficients + and R±_ are the same
= R for lossless BI media. The figure depicts R for a Pasteur-BI medium interface
(i?l = 0,	/ 0), corresponding to the Xlr = sintfj values given in the figure, ns a
function of eta = rjj/rjr-
From (3.8) it can be seen that Л_+ — 0 (or = 0) is satisfied only if the
respective wave impedances are continuous: >;I+ = r?2+ (or »/!_ = r/2_). For lossless
media one of these conditions implies the other and there is no reflection at all.
Actually, in this case, щ = rj2 and Xi — Хг- И is interesting to note that there is no
condition for the chirality parameter к and it may actually be discontinuous. For
example, if we have an interface of two Pasteur media realized with similar helices
of opposite handedness inserted in a similar base medium, the interface does not
reflect electromagnetic waves in normal incidence.
Total Fields
The total field in the region z < 0 is the sum of the incident and reflected waves.
Since both field components rotate in the same direction, the total field also has the
same circular polarization and can be interpreted as a propagating wave rotating
within a standing-wave envelope. Writing for both incident polarizations in double
subscripts, the total electric field has the form
E±(z) = e-><,,i'E,±(0) + eJ*‘T*E;(0)
= e^’[e->A’ I e'A*flT±]E’±(0).	(3.21)
Here we define kli: = fcj(cos i?i ± afr) =	± by
64
Chapter 3. Plane Waves in Layered Media
Figure 3.3 Absolute values of the two transmission coefficients T++ and T___are the
same = T for lossless BI media. The figure depicts values corresponding to d, = 0
(Pasteur medium 1) for the same values of Xzr as in the previous Figure, as a fimction of
eta =
= |(*i++ *i-) = *iCosi?,, 7>=K,fc1.	(3.22)
The term in the square brackets in (3.21) resembles that in the simple isotropic
case and is responsible for the standing wave envelope when the medium 1 is lossless,
which is assumed here. Extremal values of the field are obtained at the distances z
for which the following condition is valid:
arg{^tт±eI’*’,*} — nw.	(3.23)
Maxima are obtained when n is an even integer, minima when n is an odd integer.
For example, if the medium 2 is PEC (perfect electric conductor) with JlT± —
— 1, the total fields are
E±(z) = -2jeT™* sin(/?iz)E’±(0),	(3.24)
H±(z) = 2eTj(1"'H’’)cos(/?12 T^JH^O).	(3.25)
These expressions show us that the zeroes of the electric field and the maxima of
the magnetic field do not coincide unless d, = 0, i.e., the medium 1 is reciprocal.
The zeroes of the electric field appear at the distances
3.1. Normal Incidence, Single Interface
65
	П7Г n = 0,1,2,...,	(3.26) «1 COB Vj
and maxima at	, m = 1,3,5,...,	(3.27) ZAriCosd/
for both polarization*. The zeroes of the magnetic field are at the distances
mr ± 2d,
=	----Г’ rn = 1,3,5,...,	(3.28)
2fci cos d]
and maxima at
nir ± di
Z"± = “T—Г.	n = 0,1,2,....	(3.29)
kt cos
These distances are different for the right-hand (+) and left-hand (-) incident
polarizations. The standing wave pattern is depicted in Figure 3.4.
Figure 3.4 Standing-wave amplitude pattern of electric and magnetic wavefields, incident
normally on a PEC wall on the right. The locations of the magnetic field maxima are
different for the two circular polarizations.
If the medium 2 is PMC (perfect magnetic conductor) with = co, we have
= ’Ziy/’7i± — е±2-’Л| and the total fields are now
E±(z) = 2e^'‘^cOs(ptz ± tMEi(O),
(3.30)
66
Chapter 3. Plane Waves in Layered Media
IIjU) = -2je»^*Sin(/?12)irt(0),	(3.31)
or dual to the PEC case. Thus, the zeroes and maxima of the electric field occur
at different locations for the two polarizations while for the magnetic field they are
independent of the handedness of the incident polarization.
3.1.2 General Polarization
The incident field with the general ТЕМ polarization can be handled by decompos-
ing it in two CP fields with the unit vectors
»1 = ^(u.1Jui,)	(3-32)
satisfying (see Appendix 11)
7, = tipi I u_ufl J = u, x I = j(n+u_ — u_u+),	(3.33)
in terms of which the incident wave can be decomposed as
E‘ = 7, • E‘ III Л" | II	= III  E*.	(3.34)
Reflected Fields
The reflected field can be expressed in terms of a two dimensional reflection dyadic
II as
Er = f7-E‘,	(3.35)
Il = 1Ц11-А I ti ii, Ilt _ = RcoIi 3 IicrJ,	(3.36)
with the co- and crosspolarized reflection coefficients defined as1
Rco . *(7L, I llt), llc, =	- ILi ).	(3.37)
For lossless media we can also write
Hco = '(ReJVH I /fe JVH) = Kcos^h, Il„ = ~(He)v - He 3V") = Rsin^H-
2	2j
(3.38)
Substituting from (3.12), these can be expressed as
’Note that the labels “co” and “cross” are literally valid only for linearly polarized fields.
3.1. Normal Incidence, Single Interface
67
= cos 2i?, - rf | 2>Ma ein ein i?2
’?? + %’ + 2^Icos(1>; + >>J)	1	1
_ 2q2 cos 1^(42 sini?i -r)i sinil2)
"	4? f ^ + 2»?lI?2cos(^+i?2) •	1	'
It is easy to see that, for reciprocal chiral media with = i92 = 0, we have R„ =
0, and Rco reduces to the reflection coefficient expression of the simple isotropic
medium. These coefficients are depicted as functions of 772/^1 for some values of 1?2
with i?i — 0 (Pasteur medium 1) in Figure 3.5.
Figure 3.5 Co- and crosspolarized reflection coefficients Rco, Rcr, for an interface of
lossless reciprocal and nonreciprocal BI media, as functions of the ratio of wave impedances
eta = 92/91, with sind2 as the parameter.
Actually, Rcr 0 is a nonreciprocal effect. This is seen from the reflection
dyadic by writing it as a multiple of the rotation dyadic, in the form
R = 7?(совуд7( | sin nJ) = ЛК(уд) = ReJvR
(3-41)
with R and уд obtainable from (3.17), (3.19). For the phase angle <pn we can also
write
tan уд = — =	29z cos ^1(92 sin i?, -91 sinr?2)
Reo 9I cos 2т?! —	+ 2i)]>72 sini^i sini?2
For lossless media with real R and уд, the polarization of the reflected field is
that of the incident field rotated by the angle уд, without any change in its axial
ratio or direction of rotation (handedness is reversed). This is a nonreciprocal effect:
68
Chapter 3. Plane Waves in Layered Media
Figure 3.6 The angle of rotation of the reflected polarization, as a function of the
relative wave impedance eta = Tjj/r/j, with = 0 and sindj as the parameter.
if an incident LP wave is polarized at an angle <p = 0 and reflected at the angle
<p = Vn, the wave incident at the angle tpn is not reflected at the angle tp = 0, as it
should if the medium were reciprocal, but at <p = 2^n.
The coefficient R and the angle <pn can also be defined for lossy media with
complex values. In fact, (3.41) with (3.17) and (3.42) continue to be valid in the
lossy case.
Total Fields
The total field in the region z < 0 can be expressed as the sum of the incident and
reflected fields. Because from (2.99) we can write for the propagating incident field
E'(z) = e-J**’e“<’1K(-rcrJt1z) • E'(0),	(3.43)
and, similarly, for the reflected field
Er(z) = eJ*,''o,dIft(M1z) • Er(0),
combining these, we have for the total field when z < 0
(3.44)
E(z) = [e--’*1''M(’17Z(-«rMz) + е^*то,Л17г(«гМ^) • Я] - Е;(0).	(3.45)
Inserting (3.41), and applying the multiplication rule of the rotation dyadics (Ap-
pendix C), we have
3.1. Normal Incidence, Single Interface
69
E(z) = (е-л,с“*Я(-кЛ4 + Reik'‘^'H(KrklZ + y>n)] • Ef(0).	(3.46)
The dyadic in square brackets can also be written formally as a multiple of a
rotation dyadic with the rotation angle 4r-
E(z) = SnH(4>)  E*(0)	(3.47)
with
SrR.(4>) = v/fl7i(^)(vZneA'c,>,,’,7i(Krfc,zcosi?1 +
z	z
+ -^e--ilk*’co**1K(-K,fc1zcoei?l - ££)].	(3.48)
Writing \rh = expjln ч/Л], we arrive at the simple form
SrR\4>) = 2v<R77(y )[cosh(ln y/H -f- jktz cos t?j) cos(«,A:)z)/(
+ sinh(ln \//? +• jktz cos i?i)sin(ar/;iz)J|,	(3.49)
from which we can identify after some steps
Sr = 2ч//ЛусовЬ1(1п у/R + jktz cos i?i) — Bin,(Kr/c1z 4- —),	(3.50)
tan 4' — tanh(ln VH -4 jk\Z cos i?i) tan(ar/ciz +	(3.51)
It must be noted that, in general, the factor Sr as well as the angle 4' arc complex
quantities.
Transmitted Fields
The transmitted field can be expressed in terms of a transmission dyadic defined by
E* = T • E’,	(3.52)
T — u+ 4- u u+T— — TeeIt + T„J,	(3.53)
with the co- and crosspolarized transmission coefficients1
T. = |(T++ 17_. ) = 1 4 Л„,	(3.54)
2As for the reflection coefficients, the labels “co” and “cross” can be taken literally only for I,P
fields.
70
Chapter 3. Plane Waves in Layered Media
	Q ' II 1 + 1 Г* II 4. г	(3.55)
and, in the lossless case,	1	(3.56)
T	= -(TeWT + Te~1VT) = Tcos у>г,	
T л cr	= ~(TejVT - ТеivT) = Tsin^r.	(3.57)
Inserting the expressions (3.9) in (3.54), (3.55), they can be written as
	_ arfcCOSlM’lzCOSr?! + JJjCOSlJj) °	’?? + %, + 2J?1J?2cos(i9,+i?2) ’ _ 2q2 cos Л(>?2 sin 1?! - rj! sin i?2) ’ll + 4i + 2»?1 T)1 cos(i?r 4 1?2)
Again, in the reciprocal case with X\ — Xi ~ 0, Tcc reduces to an expression cor-
responding to simple isotropic media while vanishes. In Figure 3.7, co- and
crosspolarized transmission coefficients are depicted for an interface between recip-
rocal and nonreciprocal media. It can be noted that, for rjj — t]i, we have Tco — 1
for any value, while T„ = — tan(i?j/2).
Figure 3.7 Co- and crosspolarized transmission coefficients Tco, Ter, for an interface
of reciprocal and nonreciprocal BI media as functions of eta = Ч2/Ч1 and sin 1^2 as a
parameter.
The transmission dyadic can be written as a multiple of the rotation dyadic in
the form
3.2. Nonsymmetric Transmission-Line Theory
71
T = T(cosVTIt 1 sin y>T J) = ТЩу>т) = TevrJ,	(3.60)
with
2»Jj COS1?j
+ 4» + fyl»?» COs(l?! + d,)
T„ TJ2 sini?i — i]i sin
tanyjy = — = ---------------------------—.
lco cos + ’ll cos ®1
(3.61)
(3.62)
The coefficient T and the angle tp? can also be defined for lossy media with complex
parameter values. In fact, (3.60) together with (3.61) and (3.62) continue to be valid
in the lossy case.
3.2 Nonsymmetric Transmission-Line Theory
Transmission-line analogy appears appealing for plane-wave propagation problems
in layered media, because three-dimensional problems can be analyzed in terms of
one-dimensional problems. When dealing with simple isotropic media, the problem
can be reduced to two individual scalar transmission lines corresponding to the ТЕ
and TM components of the plane wave, known not to couple at the interfaces.
For BI media this is no longer true, except for normal incidence, in which case
the two circular polarizations do not couple. Thus, a plane-parallel structure with
plane waves propagating in the direction normal to the interfaces can be analyzed
in terms of two non-interacting transmission lines corresponding to the two eigen-
waves, much in the same way as for simple isotropic layered structures. There is a
marked difference, however: because the eigenwaves with circular polarization ro-
tating in the same spatial direction have different handedness when propagating in
the opposite directions, they see different effective media. Thus, the corresponding
transmission line becomes nonsymmetric with different parameters for waves prop-
agating in the opposite directions. It is necessary to find the basic properties for
such transmission lines to be able to express reflection and transmission of waves
at junctions of different lines. This theory was introduced in Reference [1]. For
oblique incidence, the two eigenwaves couple at each interface and a more compli-
cated vector transmission line concept [2] must be introduced, as will be seen in
Section 3.4.
3.2.1 Transmission-Line Equations
For the two circularly polarized ТЕМ eigenwaves depending only on the z coordi-
nate, the source-free Maxwell equations can be written as
72
Chapter 3. Plane Waves in Layered Media
u, x E'±(z) = -ju>/t±H±(z), u, x H^(z) = ju>e±E±(z),	(3.63)
where the prime denotes differentiation with respect to z. In terms of the CP unit
vectors u± satisfying u, x tij. = J  n± = iju± (Appendix B) we have, writing
E+ = UfA'l and IIt = и±Я±, the scalar equations
[Т>Я±(2)] = -j^±E±(z).	(3.64)
They resemble the transmission-line equations
V'(z) = —jwLI(z), I'(z) = -jwCU(z),	(3.65)
where U and I denote voltage and current on the transmission line and, L and C,
the distributed inductance and capacitance per unit length of the line.
If we identify the electric field with voltage, U = E.t or E., the current must
be defined as I = or jlE, respectively. In these cases, the distributed
transmission-line parameters coincide with the effective permittivity and perme-
ability, and they are different for waves traveling in the opposite directions. To
distinguish between the cigcnpolarization sign ± and the direction of propagation
sign ±, the latter is written as a superscript. Thus, U+ is traveling in the positive
z direction and U~ in the negative z direction.
Because the normally propagating fields can be split in two eigensets each of
which has an individual transmission line, we can treat both with the same notation.
For example, considering the set with RII waves propagating in -fz direction and
LII waves in — z direction, the voltage wave amplitudes are given by f/+ = E+ and
U~ — E_ and the current wave amplitudes by 7+ — —and the
line parameters are L± =	(7* = e±. The propagation factors are fc* = fcj. and
Z± =	= qexp(Tj^). The total voltage and current for this set can be written
as
lf(z) =	+ 1/-^*-', f(z) =/+е-'*+'-	(3.66)
Similarly, for the case of LH waves propagating in the +z direction and RH
waves in the — z direction, the wave amplitudes are I/1 — Er, J* — and the
parameters = pT, C4 = eT, k* — Z± =	= qexp(±ji?). The total voltage
and current are in this case
l/(z) =	I(z) = l+e-i*-’-Г^’.	(3.67)
Thus, both cases are included in the nonsymmetric transmission-line notation.
The well-known formulas corresponding to symmetric lines are slightly changed for
the nonsymmetric lines. Reflection and transmission coefficients at the junction of
3.2. Nonsymmetric Transmission-Line Theory
73
two transmission lines 1 and 2 when the incident wave conics from line 1 can be
written analogously to (3.8), (3.9):
и	и c;	w	-	~	- P-*?	C, CRl ~ Zf(Z} 4 Zt) УГ + У/’ Z^Zf |Z.) УГ 4 У.' = W*N^r^4>r	(3-69>
Again, the admittance expressions appear slightly simpler.
3.2.2 Input Impedance of a Terminated Line
To solve problems involving a layer of BI medium, a formula giving the input
impedance of a nonsymmetric transmission line terminated by a load impedance 7,^
is needed. Let us consider the nonsymmetric transmission line between — L < z < 0
of wave impedances and propagation coefficients 0i, loaded at z — 0 by the
impedance Zl, Figure 3.8. The reflection coefficient at the load is according to
(3.68)
fl1
fl(-L)
Л(0) =
z = 0
Figure 3.8 The loaded nonsymmetric transmission line.
L7(0) = Z~(ZL-Z+)
tM(0) Z+(Zb + Z-)’
(3.70)
Transferred to the beginning of the transmission line, the reflection coefficient is
R{ L> - U^Lj - ТЩор^’ ~ ( }
(3.71)
with
0=^ +/?”)
(3.72)
74
Chapter 3. Plane Waves in Layered Media
(3.71) corresponds to a certain input impedance of the terminated transmission
line and its expression can be developed through the following steps:
7 ,	_ U*(—L) + U~(—L)
,nl 1	I(-L)
= U't-V + U-l-I’) = z+z- 1 + Я(-Ь)
Y4J+(-L)-Y-U~(~L) Z~ -Z>R(-L)
Z+(ZL + Z~)e>pL + Z~(ZL - Z^)e~ipL
(ZL + ZZ)eiPL -(ZL-Z*)e^
_ ZbiZ* + Z ) cos flL-f-j[2Z+Z -j Zl(Z+— Z )]sin/3£
~	(Z1 t Z) cos/П, 1 j[2Zt - (Z+ - Z-)]sin/3Z ‘	1 ‘ '
This result is applicable in the analysis of layered structures. From the expression
it is seen that, for ftL = nit with integer n, we have Z,„ = ZL and the transmission
line has no effect on the input impedance. Also, the conventional transmission-line
formula can be obtained as the limit Z+ —» Z~ —» Z:
Z[, cos fl I, | jZ sin fl L
*" Z cosflL f jZ^ sin flL
(3 74)
Z* Z, R„
fli zi
fli Z2
Figure 3.0 Reflection at a loaded transmission line.
If the loaded transmission line with characteristic impedances Zj , Figure 3.9, is
connected to the end of another transmission line with the characteristic impedances
Z*, we can apply (3.70) with Z1 replaced by Z* and Z/, replaced by Z,-„ from (3.73),
where Zl must be replaced by Z2 . After some algebra, the following expression for
the reflection coefficient at the junction can be written
Zf (ZL - Zj )(Zj -I Za~) + j[(ZL + Z^)(Zj - Z{) - (ZL - Zj)(Zj + Z,4 )] tan PL
Z'f (ZL Г Z'- )(Z4 + Zj) + j[(Zt + ZfKZ4 I Z,) - (ZL - Zi)(Z; - Z-)j tan/U,’
(3.75)
3.2. Nonsymmetric Transmission-Line Theory
75
Again, the trivia! check of (3.75) shows that, for tan ftL = 0, the reflection coef-
ficient reduces to the simple reflection coefficient of the form (3.68) with apparent
changes in the notation. Thus, the transmission line 2 in between has no effect to
the reflection. A second check by setting Z2 = Z± reduces (3.75) after some steps
to the simple form (3.71), which means that there is no junction at z = —L.
Impedance Transformer
Let us study the possibility of generalizing the quarter-wavelength impedance trans-
former to nonsymmetric transmission lines. To have zero reflection for a given load
impedance Zi and impedance Z*, from (3.75) the condition
(ZL - Z?)(Z< + Z-) + j[(ZL + Z~)(Z2+ - Zf)
- (ZL - Z+)(Z2 + z+)] tan/?L = 0	(3.76)
is seen to follow. Denoting Z2 = Z2ei’° with real Zj and a, and assuming Z2 —
Z2 = Z„ ZL and ftL also real, (3.76) reduces to the real and imaginary parts
tan a tan ftL = —--—,
Zl 4 Zj
(Z, - ZtZi)tan ftL - 0.
(3.77)
(3.78)
Figure 3.10 Parameter a of the lossless impedance transformer as a function of the
electric length flL of the line, for different real values of the relative loading impedance
Hl/Zi.
76
Chapter 3. Plane Waves in Layered Media
Solutions are
a = tan-1 (?L	cotpL\ .
(3.79)

The impedance Z2 obeys the same law as for the symmetric transmission line.
The electric length ftL of the transformer can be chosen to find the angle a satisfy-
ing the condition (3.77), as shown in Figure 3.10. Thus, it is possible to make the
impedance transformer shorter than quarter wavelength by introducing nonsymme-
try'through the parameter a. However, as is seen from the figure, the value of <i
tends to ir/2 as the line becomes very short.
3.2.3 Transmission Through a Line Section
As a second problem, let us consider a section of nonsymmetric transmission line 2
of length L between two transmission lines 1 (z < 0) and 3 (z > L), Figure 3.11. A
voltage wave £// is incident on line 1. The transmitted voltage wave U2 at z = L
on line 3 can be written in terms of a transmission coefficient as
Figure 3.11 Section of nonsymmetric transmission line 2 between lines 1 and 3.
= T31l/+.	(3.80)
The transmission coefficient is obtained from the equations of continuity at z = — L
and z — 0 for the total voltage and current:
£/+ + U,- = £/+ + t/f, _ t/f _ tv tv	(3.81)
	
zi z^~zi zi'	^«5 .OZ у
U^e~i0iL + f/2-e-jZ’>“t = f/+,	(3.83)
Zf	Z2	z+ 	(3.84)
3.3. Normal Incidence on BI Slab
77
In fact, eliminating all the other voltage quantities, the transmission coefficient
can be solved in the form
т zj 	+ zt~)(zj + z,~)__________________
31 ^(Zr + Z+)(^"4 23+)е^ + (Я1--7Г)(71+-г,’)е-№1’
To check this expression we can set the transmission lines 2 and 3 equal and
obtain
7’sr =
z+ + Zi
(3.86)
which coincides with (3.69) at z = L. As another check we set the transmission
lines 1 and 2 equal and obtain a similar result:
T31 =
zj Zj -I zt
zi zi + zi
(3.87)
3.3 Normal Incidence on BI Slab
With the aid of the previous impedance formula (3.73), problems involving plane
waves in layered planar structures can now be analyzed analogously to problems
with multiple transmission-line sections. Use of the formula is limited, however, to
directions of propagation normal to the interfaces.
3.3.1 Reflection and Transmission
Let us consider the layered BI structure of Figure 3.12, consisting of medium 1
(z < — L), a slab of medium 2 ( — L < z < 0) and medium 3 (0 < z). We wish to
find the reflection and transmission of a plane wave, ТЕМ to u, and incident from
—oo.
Applying the input impedance formula (3.73) and the following notations
Zl = 1)3±, Z = q2± - q2eT,<’’, Z* = q2T = rj2eb*’, (3.88)
Z+ T Z~ = 2>jjcosi9j, — Z~ = f2jri2 sin t?2,	(3.89)
fl¥ = k1± = fc2(cosi?2 ± к2г), ft~ = fc1T = fc2(cosr?2 f a2r),	(3.90)
fl = k2 cosi?2,	(3.91)
to the two eigenfield components separately, gives us the input impedance in the
two respective cases (upper signs refer to the RII incident field and lower signs to
the LII incident fields)
Incidence on BI Slab
78
Chapter 3. Plane Waves in Layered Media
j/31 cos i?2 cos(fc2Z> cosiJ2) 4 j(i]2 T jt]3± sini>2)sin(fc2Zcosi>2)
r/2 cos d2 cos(£2£ cos i?2) 4 4(431 i jt]i sin i?2) sin(fc2L cos i?2)
4з1 cos(i?2 J k2 L cos i?2) I jt]j sin(fc2 L cos t?2)
cos(iJ2 ± kjL cosd2) 4 J431 sin(fc2£ cos i?2)
(3.92)
Figure 3.12 Bi-isotropic slab between two bi-isotropic media with normally incident
plane wave.
The corresponding reflection coefficients at the interface are obtained from (3.8)
as
до ± _ 911(^ml " 411) _ Zini ~ 411 clijJi
4i±(^inl + 4ii) ^«ni 4 4ii
= 92(431 cos Ф^ - д^совф^.) 4 j(42 - 411431) sin(A:2£ cos ^2)с±а;Д1
42(431 “s*? I 4it собФ±) 4 j(4t I 411431)sin(^2£cosi?2)
where we denote for short
Ф± = i?2 ± £2f.cosd2.
(3.94)
The two reflection coefficients can be combined into a reflection dyadic, al the in-
terface of the media 1 and 2, /1ц — RcoIt 4 RCrJ, with
И ~ 1	9n	“ 9ь- -ijj,
c° 2\ZinV14i- ^1411
Z;o_ cos 2i9) - rtf 4 У41(^1П1 - z,„ ) sin i?,
Z;„+Zi„- + 41 1 4il^>ni + 41 -Zin-
Zin^ ~ 411 c32'tf 1 _ Zin~ ~ 41-^-2111,
T 41 -	+ 411
3.3. Normal Incidence on BI Slab
79
jZi„^Zin sin2i9, + 4t(Z{n+ - ^п)соб02
^in+^.'n- + 4l + 9l+^in+ + TJ1_Z,n_
After inserting the input impedances (3.92), the expressions will grow quite exten-
sively.
In a similar fashion, the transmission coefficients Г-ц. can be written from (3.85)
as
у = ,,3t_________________________4r>lT>2 cos 1,1 cos ^2gT''K't,t_________________
11	91± (91т + 921 )(2?2T 4 93±)e>*>tcM,’« + (9^ - 92T)(921 - 9з±)е--’‘’£с<>,л’
__	Zr/jqacosi?! coei?3e±-’^l-d,*e:F-’'''*1£'
92(91T cos Ф± -| q3±cos*T)-| j(>?’4 9iT93i)sin(V'Cosi?2)’
These transmission coefficients can be combined with the transmission dyadic
7’з1 = 7++U+U_ 4 T—u_u+ = TcoIt 4 T„J with the coefficients TCo,T„ defined in
analogy to (3.54), (3.55) as
7„=l(Tt++r__), Tcr = -|(T++-T„).	(3.98)
3.3.2 BI Slab Between Simple Isotropic Half Spaces
Let us consider the interesting and important special case: a slab of thickness L
between two simple isotropic half spaces. In fact, if the medium 3 is isotropic with
Чз = Уз- = 9з, we can write from (3.92), applying the notation (3.94)
_ 9з cos(i?2 ;p kjLcosOi) + J92sin(A2£cosi?2) _
ml ^2^2cos(i?2 ± kjLcosOj) 4- ЗУз sin(fc2L cos i?2j
__________[j92(92 4 9з) + УзУз cos ^2] cos(i?2 T ferb cos fl2)___
t)l cos2i?2 — (q2 4- q3) sin2(fc2L cosi?2) 4- j 1)143 cosi?2 sin(2fc2L cosi?2)
When the medium 2 is reciprocal with i?2 = 0, we have Z,n+ = Zi„_-
The reflection coefficients (3.95), (3.96) with гд + = 91- = yt have the form
R — J 1	j ^«n- ~~ 9i |	Zjn~ ~ 9i	z„ innt
2 ^.n||rn *Zin. I 4J~ (Z.„, I »/,)(^„_ I	’
n J I ~ 91 Z{n— “ 91 I •	~~ %in-	ln.\
Rcr = I 7----------Z----7------T--J = ~JT>lT7-----1----W7----1-----V (3101)
2 \^in+ T 9t Zi„- + 91/	(Zjn+ + 9i)(Z'm~ 4- 91)
Combining these, we can write the reflection dyadic Rlt in the product form
Ж1 = IiJt 4 ItcJ = (.	(З.Ю2)
\	91 + Z^-	)	\ 9i + Zin3 /
80
Chapter 3. Plane Waves in Layered Media
Inserting the impedance expressions (3.99), a more explicit but complicated form
for the reflection dyadic can be obtained.
The transmission dyadic parameters are obtained similarly from (3.97) as
2тмз cos d2eTj''’*lt
Ту у ~ —7-----\--------“r—7---------------7—•—77—r--<i"”7’	(3.103)
Oil’ll совФу + 1?3 cos Фт) +7(175 + i^aJsin^jLcost?,)
with Фу = i?2 ± fc2Lcosi?2, and can be combined into a transmission dyadic T3, =
TCBIt + T„ J. After some algebra, the result can be written in the form
T31 — Тз1_77(—arfcj£) + Тз1у77(лгЛ2Т),	(3.104)
with
T31_ = -рЧгЧз «os 1?2(173(171 + Оз) cosi?2 cos(fc2L cos t?2)
+ A’/’ + ’Zi’Zs) sin(fc3£ cos i?2)],
З31+ = -^0i03cosi?2[yi72(o1 - i/3) sin i?2 sin(J:2£ cosi?2)],
D = [0i(0i cos Ф+ + т)з cos Ф_) + j(t), + 01 Оз) sin(Js2T cos t?2)(
’ [»?i(0i «os Ф_ + Оз cos Фу ) + j(oj + 01 Чз) sin(k2T cos i?2)].
(3.105)
(3.106)
(3.107)
The most interesting special cases of reflection from and transmission through a
BI slab are encountered when the media 1 and 3 are both air or when the medium
3 is perfectly conducting.
BI Slab with PEC Backing
The previous formulas become simpler if the medium 3 is PEC with оз = 0, Figure
3.13. In this special case we have from (3.73),
_ . sin(fc2Lcost>2) _ . sin(fc2Lcosi?2)
cos Фу ^^3cos(i?2 ± k2£costl2)’
(3.108)
which for a reciprocal slab reduces to the familiar form 7’172 tan(fc2L). Substituting
(3.108) in the reflection coefficient expressions (3.100), (3.101), gives us
17 J cos Ф+ совФ_ + у* sin2(fc2Lcos t?2)
[r;2 cos Ф+ + jijj sin(k2T cos *9а)Ц»71 cos Ф_ f ji)2 sin(L2L cos tf2)|
_______________(Vi ~ Чз) sin2(fe2L cos tl2) - rf cos2 t?2__________
17? cos2 tf2 - (17? + ^)sin2(A:2Lcosi?2) + 7’17,173 cos i?2 sin(2l2Lcos t?2)’
3.3. Normal Incidence on BI Slab
81
2qiq2 sin i?2 sins(fc2Z/cos t?2)
[772 cos Ф+4-У»?181п(Аг2Ьсо8 192)][т;2 cos Ф_ + jifa sin(fc2L cos i?2)J
_____________________27!>Mini?, sin’(fe2Z,cosr?2)__________________
71 cos’tf2 - (7? + ^)sinl(/:J£coBi?2) + cos d2 sin(2fc2Lcos	'
Figure 3.13 Conductor-backed BI slab in an isotropic medium.
The reflection dyadic (3.102) can be given quite a compact form:
== _ I 71 cos 9_It + г)г sin(k2L cos i?2)J |
\ —171 совф_ + q1sin(l:17,co8i?1) I
(71 сов Ф Д - ft sin(fe2£ cos i?2) J\
—j’71 cos Ф+ + q2 sin(A2£ cosi%) I ’
In the lossless case, the reflection dyadic can be written in terms of two rotation
dyadics:
Il =	 <7°+тг(-^+) = e*"*+a-’7?(^ - ф+),	(3.112)
where the real rotation angles are defined through
.	, q2 ein(k2L cos t?2)	7, sin(l2£ cos d,)
tan <4+ =  ---— --------ii = ----------------U—.	(3 1131
71 cos Ф.(	7! cos(i?2 ± k2Lcos i?2)	'	'
and the phase angles through
e^»i =	+ 7*«n*(*2£^7J	}
71Со.вФ± 1 J72 sin(fc2T, cos i?2)	' ’	*
or
71 cos Ф± 7j cos(tl2 ± k} L cos i?2)
tana± = ---------------------------\2H~T-----TV22’	(3115)
72 sm(*:2£ cos v2) q2 sin(fc2/ycos i?2)
82
Chapter 3. Plane Waves in Layered Media
The reflection dyadic of the form (3.112) with real angles a±, ф± is a unitary
dyadic, i.e., it satisfies
RT - R* = (Л„/, - R„J) - (СЛ + RtJ) = (l«~l’ + 1«~1’)Л = Л, (З.П6)
as can be most straightforwardly verified. The reflection coefficients thus satisfy
|«co|’ I Rier I’ = 1,	- HcrlZ = 0,	(3.117)
and the quantity Rco 0, is seen to determine Iicrexcept for the sign:
= 4|ficJ’ = 4(1 - Ifieol’)-
ilCT	llco
(3.118)
In the case Rco — 0, the quantity R„ is a phase factor e,a.
Unitarity of the reflection dyadic means that the power propagating in the
isotropic medium, which is proportional to |E|2, is the same in the incident and
reflected fields:
|Er|2 = 17? • E‘|2 = E*  RT • R* • E* = |Е;|2.
(3.119)
BI Slab in Air
In this special case we have 7i = q3 = t)o
have the appearance
7<o/e<>. The input impedances (3.99)
z ________________> Vo) । >M><-<»si>2|cos(lVr feibcosJ,)_________________
,ni q2 cos3 i?2 — (q2 4 q2)sin2(fc2£cosi92) 4 cos ^2 sin(2fc2 L cos i?2)
(3.120)
and the reflection dyadic (3.102) becomes
fi.. = /P}Jl I	/ jqo7t - Zin t J\
\ i)o + 4>- у У О» 1 Zin г )
in which (3.120) can be substituted. The final result is obtained, after some algebra,
in the simple form
-jsin(fc2/,cosi>j)((q2 - qj)f( 4 2q2qosm i?2./|
Jjj_________-J »“Ц’Ч*' co» Vinvio ~ 'lipt T 2.7270 o'" v2.'j	_	j
11 2q2qo cos i?2 cos(I-2T cos i?2) f j(q2 4 q3) sin(fc2Z, cos i?2)
(3.122)
3.3. Normal Incidence on BI Slab
83
To convince oneself of this expression, we see that for k2L = 0, or vanishing slab,
we have Йц = 0. For a Pasteur slab with i?2 = 0, the expression reduces to
W - Г	=
сЦ| J о	 r  'i 3 ax * l	(o.lzoj
2>?ai7<,coe*iL ++ >?2)sinA:2T
which coincides with the result readily obtained from the ordinary scalar
transmission-line theory.
The transmission coefficients for the BI slab in air, (3.103), can be written in
the form
(3.124)
(3.125)
(3.126)
T _ ___________ 2q2r;,,cost92eT1"-t>b______________________
2i?jVo cos i?2 cos(A:2 L cos t?2) 4-	-f- q’) sin(A.j£ cos r?2) ’
whence the transmission dyadic is
Тэг = 7з17?(—кг fc2L),
T _ __________________________2r?oy2 cos i?2_______________
2i7o>72 cos(fc2Z, cos i?2) cosi?2 + j(ij’ + q2)sin(k2I,cosi?2)
Thus, the transmitted polarization equals the incident polarization rotated by the
real angle -кгкгЬ. The amplitude is multiplied by |7’31|, which is equal or less than
unity, as can be shown.
3.3.3 Polarization Rotation
The previous formulas will now be applied to study polarization rotation of the
incident field in reflection from a BI slab with a PEC back plane and transmission
through a BI slab free space.
BI Slab with PEC Backing
The unitary reflection dyadic corresponding to the lossless BI slab can be written
in the form
Й = е*П(ф),	(3.127)
and the rotation angle ф is obtained by substituting (3.113) in
tan ф - tan(^_ - ф+) = т , p т 	(3.128)
1 T IRH 0— фц.
84
Chapter 3. Plane Waves in Layered Media
The polarization is not changed in reflection if sin t?2 sin(k-2L cos d2) — 0, which
means that the slab is cither reciprocal (d3 = 0) or a multiple of half phase wave-
length (L — mr/k2 cost?2). More interesting is the case ф = ±rr/2 so that a linearly
polarized incident wave is reflected totally crosspolatized. Thus, a linearly polar-
ized antenna does not see its reflection from the slab. Such a device is called twist
polarizer, Figure 3.14, and it has applications in antenna engineering [2, 3, 4].
Figure 3.14 Twist polarizer reflects a normally incident plane wave with polarization
rotated by 90°. It can be realized by a slab of HI material with a perfectly conducting
back plate.
Twist Polarizer
The condition /?<„ = 0 can be written from (3.100) as
= rf,	(3.129)
which is tantamount to
(qj — >7j) sin2(fcj/, cos t?j) — rtf cos2t?2,	(3.130)
or
sin(A:2£cost?j) = —.	2 —	(3.131)
/i-ЬА.)2
This equation has real solutions for L only when the right-hand side is real and
smaller than unity in magnitude, which leads to the following condition for the
medium parameters:
^<|sintf2| = |»l-	(3.132)
Vi
Thus, first of all, we must have q2 < ту, and the Tellegen parameter |xir| must be
large enough.
3.3. Normal Incidence on BI Slab
85
If the twist-polarizer condition (3.131) is valid, the reflection dyadic (3.112)
reduces to
l=eJ“J=e>(“t4“-)J,
with oj satisfying (3.115). After inserting (3.131), the phase angle a
can be computed from the simple expression
Vi
cos a =------
t/1 Stn Vi
(3.133)
= o+ I a.
(3.134)
Figure 3.15 Copolarized reflection coefficient Rco for normal incidence, corresponding
to a HI slab backed by a PEC plane, as a function of the normalized thickness kjL, for
Xjr = 1, and rjz/rji values from 0.5 to 2.0. Zero copolarized reflection is obtained only
when rjz/’Ji < Xir — 1 is satisfied.
Numerical Examples
The twist polarizer effect is clearly seen from the computed figures corresponding to
a BI slab of thickness L backed by a PEC plane. Since the chirality parameter has no
effect, we assume кг — 0. Co- and crosspolarized reflection coefficients	for
normal incidence are first shown in Figures 3.15 and 3.16, for the extreme Tellegen
parameter value Xir ~ sintfj ~ 1 and various values for the normalized impedance
parameter qi/rfo. It is seen that, in this case, the ideal twist-polarizer effect Rro = 0
is obtained for values of the normalized thickness kjL = r/i/y/r/? — »?j, as predicted
Chapter 3. Plane Waves in Layered Media
by (3.131). For >7j/>7i values close to unity, the effect is broadbanded, but requires
a thick slab.
Figure 3.10 Crosspolarized reflection coefficient for the same parameter values as in
Figure 3.15. For lossless Bl media the condition (Ясо!1 + |Я„|2 = 1 is valid.
Figure 3.17 Same as in Figure 3.15 but for normalized impedance parameter 1J2/>J1 = 1
and varying relative Tellegen parameter x?r values.
3.3. Normal Incidence on BI Slab
87
Figure 3.18 Same as in Figure 3.17 but for 173/171 = 0.8. RCQ is small around kjL =
n>r/2cosi?2 when Xir is slightly larger than rft/i/i. The polarization rotation effect in
reflection is more broadbanded than for the parameter values in Figure 3.15.
In Figure 3.17, the magnitude of the copolarized reflection coefficient corre-
sponding to a PEC-backed BI slab is given for a fixed ratio of the normalized wave
impedances Va/7?! = 1 and varying normalized Tellegen parameter Xir — 0.2...1.0.
It is seen that the copolarized reflection coefficient does not have a zero for
Ix'irl <172/1/1 in accord with (3.132). In Figure 3.18, the same is given for 172/1?! =0.8
and |хз*| > 0.8, whence the zero is clearly seen around kjL = ?r/2cos i?3. The effect
is broadbanded if хз* is chosen slightly larger than 173/171.
The twist-polarizer effect is obtained even for small values of the relative Tellegen
parameter, if 173/171 equals or is slightly below the value of Xa*- The effect occurs
at k2d ~ nir/2, where n is an odd number. This is demonstrated by Figure 3.19
showing both the co- and crosspolarized reflection coefficients for xz* — 0.1 and
1/2/171 either 0.1 or 0.09. The effect is seen to be basically narrowbanded: it can be
broadened only slightly by choosing 172/1/1 smaller than Xzr-
The twist polarizer can also replace an absorber if its only function is not to
return the copolarized wave. Absorbers based on lossy chiral materials have been
studied in the literature [5]. .
BI Slab in Air
The BI slab may act as a polarization rotator for the transmitted wave incident in
the normal direction. By dimensioning the parameters in a suitable way, there will
88
Chapter 3. Plane Waves in Layered Media
Figure 3.19 Co- and crosspolarized reflection coefficients Re„, R„ for low parameter
values Xir — 0-1 and tja/zyi = 0.11, 0.1, 0.09. The twist-polarizer effect exists but it is
narrowbanded.
arise no reflection of the plane wave. Since the Tellegen parameter has no essential
effect on the polarization rotator, we can assume = 0.
To have no reflection, the factor Tai of the transmission dyadic Тл in (3.126),
Y’ _____ ________
31	2^, сов(Л2£) + >(ij’ + j/j) sin(fcjb)’
must have unit magnitude,
|73‘ *’ = l + K^-^/ZtZ.ihl’BinW) = L	(3136)
If	this requires sin(fc3L) = 0, or the thickness of the slab must be a multiple
of half phase wavelengths:
T = ^, n = 1,2,3,...	(3.137)
If the required rotation angle 4> — —KrktL in (3.125) is given, the chirality can
be chosen as
Ф   ф_
k2L	П7Г
(3.138)
3.3. Normal Incidence on BI Slab
89
Figure 3.20 Transmission coefficient |7’з1| for a chiral slab 90" polarization rotator, as
a function of electric thickness k2L (log scale), with relative impedance eta = q2/qo as
a parameter. Integers n correspond to the required relative chirality parameter values
Kr — -l/2n.
Recalling that |кг| < 1 seems to be a practical limitation for the realization of
chirality, for the smallest L corresponding to n = 1, the largest rotation angle
realizable is |</>[ = rr.
On the other hand, by making n large, the chirality needed to realize the required
rotation angle can be made as small as we wish. However, the relative bandwidth
becomes small for large n. This is demonstrated by the example in Figure 3.20
depicting transmission through a chiral slab. Transmission is close to unity at
certain frequency bands which become narrower as the integer n grows.
Let us also study reflection from a BI slab with i?2 / 0 in air, through the reflec-
tion dyadic Нц, from (3.122), it is seen that the copolarizcd reflection coefficient
Rc„ vanishes exactly at q2 = t]„. In this case, the reflection dyadic reduces to
—j sin(Aj L cos r?2) sin i?2
Jji _ /J J —	-J МЩ1Чо cm v2) » vi_____j
cos t?2 cos(A2jLcos i?2) + j sin(/c2£ cos i?2)
(3.139)
The magnitude of the crosspolarized reflection coefficient can be expressed as
KJ =
sin(fc27j cos d2)| sin i?2|
^/cos2 i?j cos2(Ar2£cos i?2) + sin2(fc2L cos i?2) A
(3.140)
__ cot3 dj
•;n’( Ar j L сот t? j)
Chapter 3. Plane Waves in Layered Media
This shows us that total reflection in crosspolarization with |/<’сг| = 1 for a lossless
slab is possible in the limit cos —» 0, in which case we have
1
\Л+(1/мУ
(3.141)
Total reflection in crosspolarization from a BI slab in air is thus only obtained in
the limit of an infinitely thick slab L —> oo, i.e., for a reflection from an interface of
a BI half space with Т)г = and X'Jr — 1-
The maxima for both |/?c„| and |/?cr| of (3.122) occur for k3L satisfying
cos(kj£ cos dj) = 0 and the nulls for sin^-ji cos dj) = 0, which correspond to
the respective odd and even values of the integer n in
2 cos dj
(3.142)
The values of the maxima can be solved quite easily:
I'M
inax —
| ' 'cr |rnaz
>/Z + 42
(3.143)
>?o +
and they are nicely reproduced in the subsequent computed diagrams.
Figure 3.21 Crosspolarized reflection coefficient |f?„| for a BI slab in air as a function
of the normalized thickness kjL, for ifo = r)„ with %Zr as a parameter. Ilco = 0 for >/z = q„
regardless of values.
3.4. Vector Transmission-Line Theory
91
Numerical Examples
The magnitude of the crosspolarized reflection coefficient R„ for a BI slab in air is
depicted in Figure 3.21 for q2 — r]a for some values of \2r = sind}. For >/j = J?o, Rco
vanishes regardless of the x2, value. It is seen that, as predicted by the theory, the
maxima are at odd multiples of k2L = x/2cost?2 while the nulls occur at integer
multiples of rr/cosr?j. The maximum values of |Л„| equal those of Xir-
Figure 3.22 |f?co| (solid) and |/?cr| (dashed) as functions of the normalized thickness
frjbcostfz of a BI slab in air, for X2r = 0.9 and qj/qo as the parameter. For k2Lcosti2 =
nx with integer n, the slab is totally transparent.
In Figure 3.22, both Rco and R„ magnitudes are given when q2/r/o is varied and
X2r — 0.9 is fixed. Л similar set of curves is obtained for »;o/q2 = 0.9, as can be
seen from the reflection dyadic expression (3.122). For a practical polarizing shield,
which prevents radiation transmitting through the slab and kills the copolarized
reflection, both y2r and q2/qo must be as close to unity as possible, as is evident
from the figure.
3.4 Vector Transmission-Line Theory
When considering obliquely propagating plane waves in a layered medium, the scalar
transmission-line theory described above becomes cumbersome because at the in-
terfaces energy is coupled between the scalar voltages and currents corresponding to
the eigenwaves. Thus, each interface must be represented by a four-port with two
transmission lines entering on each side. However, the four-ports can be avoided by
92
Chapter 3. Plane Waves in Layered Media
combining the two coupled transmission lines to a concept involving vector-valued
voltages and currents: a vector transmission line [2]. It can be visualized as having
two scalar transmission-line components and the vector voltage and current have
scalar components on these two lines. Plane-wave systems with different transverse
wave-vector components correspond to different noninteracting vector transmission
lines and can be analyzed separately. What is gained with this concept is a more
compact notation and a better insight to the problems. The price is that one
has to learn to work with two-dimensional dyadics [6]. Л companion to vector
transmission-line theory is vector circuit theory which basically treats circuits with
dyadic elements [7, 8]. It is applied in Chapter 4.
3.4.1 Systems of Plane Waves
Let us assume that the interfaces are again orthogonal to the z axis and the plane
waves are propagating normal to the x axis, which imposes no restriction to gener-
ality.
Figure 3.23 All plane wave components belonging to the same system of plane waves in
a layered BI medium have the same transverse wave number.
Considering a single incident plane wavefield with the wave vector к =
k-lju.cos# + «„sin#), where fcy stands for either or k_, it is reasonable to
assume that it creates a system of reflected and transmitted plane waves in all
regions of the layered structure with the same transverse wave vector component
= fey. sin#, Figure 3.23. In fact, were this not the case, the wave would propagate
in different sections at different transverse velocities and the continuity conditions
at the interfaces could not be satisfied. Thus, a single eigenwave creates a system
of plane waves with the same kv.
Л plane wave with the general polarization, incident at an angle #', creates two
systems of plane waves because the two eigenwaves in general have different
values: k+ sin 6' and sin #'. The two systems coincide only if (k+ —	) sin #' = 0.
This can happen if either #' — 0, which means normal incidence, or = k_
implying к = 0, which means that the medium of incident wave is nonchiral. Thus,
3.4. Vector Transmission-Line Theory
93
the problem where a plane wave is incident from a simple isotropic medium on a
layered Bl medium can be treated with a single system of plane waves.
In the following we consider only one system of plane waves with a constant
component. Let us denote any vector q associated with waves propagating in the
positive or negative z direction by <1 or <1, respectively. In the same manner, the
dyadics I) are denoted either I) or D.
In any one single layer with the parameters e, /i, к and y, there may exist
four plane waves belonging to the same system corresponding to a particular
component. The total fields are
E(r) =E+ e'ik* ’ + E- e“>k- r+ E+ e-'»k+ r+ Ё- e_,k- r,	(3.144)
H(r) =Й+ e~ik* r+ H- e->k- ’+ H+ e’ik+ r+ H- e->k-r,	(3.145)
when the propagation vectors are denoted by
k± = u,Pt +	к±= -и,Р± + Uvkv,	(3.146)
fl+ = \/k+ ~ k}, fl- ~ \/k- ~ k}> ki = *(c°st9 ± к,).	(3.147)
It is assumed that the propagation-vector components along the z direction, /3+ and
/?_, have a positive real part.
3.4.2 Vector Voltages and Currents
In the vector transmission-line theory the scalar voltage and current concepts are
replaced by the two-dimensional quantities “vector voltage” e and “vector current”
j, Figure 3.24. They are related to the transversal components of the electric and
magnetic fields on the z axis (z = у — 0) as
c(z) = 7( • E(0,0, z), j(z) = —u, x 11(0,0,z) — -J • 11(0,0,z).	(3.148)
The definition for the vector current has the analogy of currents flowing into the
positive z direction, which must be remembered when considering waves propagat-
ing in the negative z direction. Note that, actually, j is a two-dimensional vector in
the transversal zy plane.
Vector voltage and current waves propagating in the positive and negative u,
directions are denoted by
e± (z) = 7, E±	e± (z) = 7, E± e’0*’,	(3.149)
j ± (z) = -J- Й± e ji(z) = J-Н±е^'-	(3.150)
- -> j(z)
Figure 3.24 Vector transmission line.
Because the electric field vector of a plane wave is perpendicular to the respective
к vector, knowing k± and e± gives us enough knowledge to determine the total
electric fields E± in the respective effective isotropic media:
Ei =
E±=
e±.
e±.
(3.151)
(3.152)
as is easily seen by solving for the z components of the fields from k± • E± = 0.
The vector voltages e,- are elliptically polarized, corresponding to the projections
of the circularly polarized E±, E± vectors. Their polarizations can be represented
in terms of two two dimensional vector bases {a + ,a_}, {a( , a ), together with
their reciprocal bases (a + , a }, {a^, a }, satisfying
a+a+ J a_a_=a^a+ ( a_a_=	(3.153)
so that any two-dimensional vector a can be expanded as
a =a+ (a+ -a)-f a_ (a_ -a),	(3.154)
a =a+ (aF -a)+ a_ ("a -a).	(3.155)
The vector voltages have the same polarization as the base vectors and can be
written as
e± = a± (ai • e±),	е±=а± (a'± • e±).
The definitions of the different base vectors are
(3.156)
»i = T >c±uv,
utx aT
a±= [--------±5-
u, a । x a
Tu„ h>cTu„
i(cl Iе-)
(3.157)
rrtiur iransimseion-Line Theory
95
*-	. .	—*	. uxx aT Tuv — jcTu_
»i= «. ± Jc±uv> a±= ±------------z;---Eqr- =----——^y-.	(3.158)
u,- a+ x a_ j(c+ + c_)
Here we denote for short by c± the cosines of the axial angles 6± of propagation:
c+ = cos0+ = —,	c_ = cos0_ = —.	(3.159)
K^.	K_
The two sets of base vectors a^ and a± are obviously related. In fact, by
changing the signs of c± = cos$±, i.e., changing from to ir — the two bases
are interchanged, which can be interpreted as change of propagation direction along
the transmission line. The relation can also be expressed by
a±= К- a*,	a±= К- a’±, К — urii, — uvuv (3.160)
with K, a reflection dyadic, satisfying К = It. For real c± values this is tantamount
♦“	—•*
to the simple relation a± = a±. 7'he base vectors also satisfy the orthogonality
relations
a+ • a_ = a_ • a + = 0, a+ • a + = a_ • a_= 1,
(3.161)
with similar relations for the other base vectors {a + , a_}.
In the special case of normal incidence we have = 1 and the base vectors are
associated with the CP unit vectors rij. defined in (3.32) and Appendix В through
a±= y/2u± =a
1 1
a. = T —7=u, x uT = -7=111 =ax .
* jy/2 T Л T
(3.162)
(3.163)
3.4.3 Propagation Dyadics
By defining two propagation dyadics for the vector voltage waves as
0- a.t a+ -(-/?_ a_a_,	0- 0+ a+ a+ +/L a_ a_,	(3.164)
we can compactly express the total vector voltage in the line as
e(z) = e e (0) + e’/u- e (0).
(3.165)
The exponential function of a two-dimensional dyadic is defined here as a two-
dimensional dyadic and can be understood in terms of its Taylor series expansion
(see Appendix C). Because of the orthogonality properties of the base vectors, the
dyads a±n± and Sj a±, are projection dyadics allowing us to write
96
Chapter 3. Plane Waves in Layered Media
e-^« = e ^+i+:'+« .	=a + a^ е-Я»+«+ a_a'_	(3.166)
This implies
e"^* • (e+ + e_) = e~iP^ e+ ye"#-* e_,	(3.167)
and similarly for the waves propagating in the reverse direction. Thus, the dyadic
propagation factor gives the correct scalar propagation factor for each of the wave
components. The two propagation dyadics are related by
/?= T- fl K, fl= It- fl -T.	(3.168)
If fl± and k± are real, these reduce to the simpler relation fl=fl *.
In the special case of normal incidence, the propagation dyadics arc of the simple
form
=> — _ ;=
fl= fc+u+u_ f k_u_u+ — fc(cosr?fe — jxrJ),
<=	=z
fl— fc+u_u+ 4- fc_u+u_ = l:(cosil/t + JK,J),
whence the exponential terms can be written simply as
£~i0‘ = е-Л«со.Ле-КД.7 =
=	= e->*'*'“<’7J(Krb))
in terms of a phase factor and a polarization rotator dyadic. These expressions are
in accord with the expression (2.99) for the propagating field in a BI medium.
(3.169)
(3.170)
(3.171)
(3.172)
3.4.4 Dyadic Characteristic Admittances
The dyadic characteristic admittances for the vector waves propagating in the two
directions are defined through
j(z)=y.e(z), j (z) = -y .e(z).	(3.173)
The minus sign is due to the fact that the direction of the returning current wave
is defined in the direction. The expressions of the admittance dyadics can be
identified from
j (z) = ~J- h (z) = -u, X
3.4. Vector Transmission-Line Theory
97
(a.n,	a_a \	=> _
----t------— • e (z) =У • e (z),
7+ V- I
(3.174)
whence the characteristic admittance dyadic Y is
(3.175)
Similarly, we have
(3.176)

The relation between the two dyadic admittances can be written as
Y=K-Y-K, Y—K-Y-K.
(3.177)
For real medium parameters and c± values, this relation is more simply Y=Y T*
In x,y coordinates the dyadic characteristic admittances have the form
-----------------------|tirnI2c+c_ cost? — uruv(j(c_ — c+)cost? I- (c+ 4 c_)sind)
7(ct + c~)
- Uj,ur(j(c_ - c+)cost? — (c+ 4 c_)sintl) 4 tivuv2cost?|,
У= -----------r|utuI2c+c_ cost? и^иДДс- — c+)costl 4 (c+ 4 c_)sint?)
t/(c+4 c_)
4 и„иДДс_ — c+) cos t? — (c+ 4 c_) sin t?) 4 uvuv2 cos t?].	(3.179)
In particular, it is seen that, for a reciprocal DI medium (Pasteur medium) with
i? — 0, the У and Y dyadics are symmetric satisfying the simpler relation У=У *,
while for a Tellegen medium with Cy = c_ = c, the characteristic admittance dyadics
satisfy
(3.178)
I rz	1 .	„ —у	, т
Y= -|(ueiirc 4 uvtiv-)cos v 4 Jsinil] =y .
V	c
For normal incidence, the characteristic admittance dyadics (3.175), (3.176) sim-
plify to
(3.180)
У= —4f.u_ 4-----------u Uy = - (cos А/, I sin OJ) =	(3.181)
’ll	4-	V	4
Y— —U|U_ |--------u_ii( = -(cosr?f( — sintM) = -e <’J.	(3.182)
4- Vr '/	4
Thus, the admittance dyadics are just polarization rotators multiplied by l/т). To
see the connection to the expression (2.104) presenting the relation of the magnetic
field to the electric field in a propagating wave, in the normal incidence case we can
write from (3.148)
H = J j = 1 eJ’/2  e7’’ • E = -ftp + -) • E,	(3.183)
4	»?	2
which equals (2.104).
3.4.5 Reflection Dyadic
Scalar parameters of the conventional scalar transmission line, like the reflection
and transmission coefficients and impedances, can now be generalized to the vector
transmission line. In this process, scalar parameters become dyadic quantities.
Let a vector transmission line, defined for z < 0, be terminated at z = 0 in a
dyadic admittance Yt, or impedance Zl = Yl -I (the inverse must be understood in
two- dimensional sense, Appendix C), Figure 3.25. The load admittance is defined
through the relation
j(0)	e(0).	(3.184)
Figure 3.25 Loaded vector transmission line.
The total vector voltage and current on the line z < 0 can be written in terms
of the incident and reflected vector voltage and current waves:
e(z) = e e (0) + e (0),
j(z) =y -e-’^- e (0)- Y e"'- e (0),
(3.185)
(3.186)
whence both quantities are known for any z, provided the vector voltage is known
at z — 0:
e(0) =e (0)1 e (0).	(3.187)
At this point it is proper to point out that while the transverse magnetic field shares
the propagation dyadics of the vector voltage:
h(z) = h (0) 4 ei0‘- h (0),	(3.188)
the vector current function does not. In fact, from (3.186) we can write
j(z) =Y e~i0‘ Y -1- j (0)- Y Y “* j (0)
= j (0) - j (0),	(3.189)
=> <= ==»= = <= =
which amounts to replacing fl and fl by -J- fl -J and -J- fl -J, respectively. The
property of obtaining different propagation coefficient matrices for voltage and cur-
rent distributions in multiconductor transmission-line theory is well known [9]. It is,
however, sufficient to work only with the vector voltage waves and the corresponding
propagation dyadics fl, fl.
Defining the reflection dyadic R (z) for waves reflecting into —u, direction by
e (z) =R (z)- e (z),	(3.190)
we can find its expression in terms of admittance dyadics when the transmission
line is terminated at z = 0, Figure 3.26. From continuity of the voltage and current
at the load we have
j(0) =Y  e (0)- Y - e (0) =Ft -c(0) =yL -[e (0)4 e (0)],	(3.191)
whence
(У - Гг.)-e (0) = (v 4-rt)-c (0),	(3.192)
which gives us finally
R(0) = (Y 4 Yl)-' (Y -Yl).	(3.193)
Knowing the reflection dyadic at z = 0, (3.185) allows us to determine its
expression for any z < 0:
e (z) = eip‘- R (0). e (0) =/l (z)- e (z) = R (z)  е~’р‘- e (0),	(3.194)
100
Chapter 3. Plane Waves in Layered Media
z = — L	z — 0
Figure 3.26 Reflection from a terminated vector transmission line.
whence for z = — L,
R (-L) = e~^L- R (0) • e~^L.	(3.195)
As a simple check, (3.193) and (3.195) are seen to reduce to the conventional
scalar transmission-line expressions for normal incidence, (3.70), (3.71)
Л(°) =	R(z> = e-Wtfi(0).	(3.196)
r + il
The reflection dyadic for vector current waves would be different, of the form
Y  R (z)- У . Also, in terms of impedance dyadics, the reflection dyadic (3.193)
has a slightly more complicated form:
R (0) =Z (Zl + Z)"*  (Zl - Z)- Z ' -	(3.197)
3.4.6 Transmission Dyadic
If in the previous example we denote the vector transmission line with the subscript
1 and the load impedance is another vector transmission line 2, Figure 3.27, the
reflection dyadic (3.193) at the junction can be written as
Лп= (Ki + f»)’1 • (Vi - ?i),	(3.198)
which coincides in the scalar case with (3.68). The field transmitted into the line 2
can be written as
e2=T2i  e„	(3.199)
which defines the transmission dyadic 7’2I. Its expression in terms of the reflection
dyadic is obtained from the continuity of the vector voltage at the junction:
ег ~(ei + ei) = (Tzt —Ii~ Rit)- e2= 0.
(3.200)
3.4. Vector Transmission-Line Theory
101
— > 7'21
7i Zi
z = 0
Figure 3.27 Reflection from and transmission through a junction of two vector trans-
mission lines.
This is valid for all et, whence we must have
T«= Л+ Лп= (Vi + Г»Г*  (Vi + Kj).
(3.201)
This expression reduces to that of the scalar transmission line, (3.69).
3.4.7 Input Admittance Dyadic
The reflection dyadic R (0) has the same relation to the terminating admittance
Yl as the dyadic R (z) to the input admittance y;„ at z = — L, Figure 3.28. An
expression can be obtained from the vector transmission-line equations:
Yl
z = -L
z = 0
Figure 3.28 Input admittance of a loaded vector transmission line.
j(-i) =r.n -e(-I) =y • c (-£)- у - R (-£)• e (-£)
= [Y-YR (-£)]  [lb R (-£)]-’ • e(-L),
(3.202)
which gives the result
F.„= [K -Y R (- A)| • (/, + R (-L)]'.
(3.203)
Г"	.......................~
This resembles the corresponding expression for simple isotropic media with у=У=
У, in which case it reduces to
r.„= F- [7. - Я(-£)]  [7, +	(3.204)
Substituting (3.195) in (3.203), we arrive at the representation
I	Yi„= [У jpL- Y e~ipL- R (0)] • [<A + e~’pL- R (О))"1.	(3.205)
Finally, inserting (3.193), we can write the input admittance in terms of the char-
acteristic admittances and propagation dyadics in the form
K.n= [У e,0L  (Y ~ El)’1- Y e’pL • (Y 4 Уь)'']
[e^L  (У - Yl)' 4 e~ipL  (У + Уь)-']-*.	(3.206)
With too many inverse dyadics, this expression, though symmetric, may he difficult
to apply. With some algebra, the following equivalent result can be derived:
У.п= (У 4 У) • [ (у е-^ь4- У eipL- Yl (e~ipL - ejpL)j • (У + У ) - e"^b]
•(e-^t-eJ^b) *4-У -	(3.207)
'Го check the developed expressions for the input admittance, (3.206) and (3.207),
we study the following special cases with known results:
•	For matched termination with Yl—Y, we obtain Yin~Y- In this case there
is no reflection.
•	For PEC plane at z = 0 with Y l— oo and simple isotropic medium, the input
admittance becomes — {j/rj) cot(ftL), or the impedance is jr/ tan(/?£), which
is a well-known result.
I	<=
•	Neglecting the terms e~’pL and e~’pL in the case of a long lossy line (large L
> < .=>=>. . . >
and complex /?, ft), we end up in Yin=Y, i.e., the loading admittance Yl is
not seen through a long lossy line.
The expression (3.207) gives a mapping between two admittance dyadics,
Yl—>У.„ through a section of vector transmission line. It can be applied consecu-
tively to find input admittance of a series of BI slabs by mapping the admittance
over each slab. Л similar expression could be written for the input admittance when
looking into the negative z direction, У,„.
—~ouirque “Incidence, Single Interface	ЮЗ
3.5 Oblique Incidence, Single Interface
The vector transmission-line notation can be applied to the problem of an obliquely
incident plane wave propagating a BI medium 1 z < 0 towards the interface of
another BI medium 2, z > 0. However, if the medium 1 is chiral with «i -/ 0, the
problem must be treated in terms of two vector transmission lines corresponding
to the two eigenpolarizations of the incident wave, because they give rise to two
different systems of plane waves, in general. Let us, however, limit the problem to
the case when the medium 1 is simply isotropic with zq = 0 and Xi = 0, Figure
3.29. The incident polarization may now be arbitrary and yet the fields belong to
the same system. Problems involving reflection and transmission of an obliquely
incident plane wave upon an interface of isotropic and bi-isotropic media have been
studied with scalar equations by many authors, [10, 11, 12, 13, 14].
The direction of propagation of the incident plane wave is defined by the unit
vector
Figure 3.20 Plane wave incident from a simple isotropic medium 1 to the interface of a
BI medium 2.
In the simple isotropic medium 1 we have c1+ = Cj_ = c, = cos#, in (3.157),
(3.158), and the base vectors are related by
a»±= T jcos^jUy = a1T .	(3.209)
—♦ —»/	—♦ —tf	*	e—
Because of a,+	/, and similarly for a1±, we have from (3.164)
Pit = Pi- - Pi = y/tf - tf = kt cos#,,	(3.210)
P=P~ Д1(»1+«'1+ + ai-a'i-) = Р1Л-	(3.211)
104
Chapter 3. Plane Waves in Layered Media
3.5.1 The Characteristic Admittances
The dyadic characteristic admittances of the vector transmission line corresponding
to the medium 1 (z < 0) are the same and, after substituting the base vectors in
(3.175), they can be written as
Pi=yt= Fi = — (cos6,u.ur + -^-t-uvu )
7Д	COS t/j
=	(3.212)
Mi Pi’ll
This dyadic has the two-dimensional determinant (Appendix C)
spmFI = lF1;FI:7=l	(3.213)
Z
In the medium 2 there are two CP plane waves refracted at the angles 0ц defined
by the Snellius law
sin62± = sin 6i,	(3.214)
«2±
or
/32± = кц. cos 0ц — \Ai± — fcjsin26i-	(3.215)
The forward admittance dyadic of the BI medium 2 is, from (3.175),
У2= -u. X J f - a*-"2-
\ 11+	11-
and its two-dimensional determinant function can be evaluated to have the simple
result
(3.216)
spm y2=---------((u,x a2+) x (u.x a’2_)) • [a‘2+ x a’2_]  ----------- —. (3.217)
lirli-	11+11- 1i
3.5.2 Reflection and Transmission
The reflected and transmitted waves are defined through the reflection and trans-
mission dyadics
«1 = 7/11
ei>
ei—7'ii  ei>
(3.218)
and their expressions can be simplified from (3.198), (3.201) in the form
3.5. Oblique Incidence, Single Interface
105
Ki>=(FI+yI) ’ (rI- П),
72i=2(V, I Yi)'* V,.
(3.219)
(3.220)
Substituting the admittance expressions from (3.212), (3.216), we can write the
reflection and transmission dyadics in explicit form. It is not easy to find simple
expressions because the eigenvectors of the У, and Yi dyadics are different. In
terms of the eigenvectors of the dyadic, U, and uv, we can write (15]
Hii = ’М’«Л„ I и,чуЛги I ’4’1,4 4vuvnvv,	(3.221)
with
fi = № ~ *71)C«(C2+ + c»~) + 29;91(C1 - c2tc2_)cost9
(’/i I ’ll )ci(c»r + c,_) | 27,7, (c| c2+c2_ j cos 19 ’
Л = _ 29г’?1Ь'(сг+ ~ c;. )cosi? - (c2+ 4 c2_ )sini?)
*V (’la + ’li)ci(c2+ 4 CJ-) + ^V1V1(C1 + c2+Ci_)cosi?’
„ _	2’12’11 c?b'(c2+ - c2 _) cos 19 4 (c24, 4 c2_) sin i9]
(’ll 4 7i)ci(c2+ + C2-) 4 21727,(c? 4c21c2_)cosi9’
/? =	~ l?l)C’(C2t + C1~) ~ 212’1i(c? ~ c2tc2_)cosi9
v (’12 + ’ll )ci(c24 4 c2_) I 2^,7,(c? 4 c2+c2_) cos r9 ’
and
(3.222)
(3.223)
(3.224)
(3.225)
721= hl Ли,	(3.226)
T =	272[27icfcost9 4 72Ci(c2-| 4 er-)]	(3 227)
(’12 4 ’li)ci(c2+ 4 c2_) 4 2717,(c? 4 c2+c2_) cos i9 ’
T =	-- C2_)cosi9 - (c24 4- c2_) sin 2?)	2
(’12 4 ’Ji)ci(c2+ 4 С2_) 427г7,(с? 4 c2+c2_)cos t? ’
T = 21?2’?’cib'(c2+ - c2 Jcosi9 4 (c24 4 C2, )sin 19]
*" (’ll + ’ll)C1(C2+ 4 C2_) I- 21/27,(c? 4 c2+c2_)cos 19’
_	272[27ic2+c2_ cqsi9 4 72c,(c,, 4 c2.)]_____
('ll + ’1?)C1(C21 4 c2_ ) 4 27271 (c? 4 Cj+c2_)cosi9'
It can be noted that, while for a reciprocal BI (Pasteur) medium 2 the wave
admittance dyadic is symmetric, neither the reflection nor the transmission dyadic
are, in general. They are symmetric only if the medium is also nonchiral, i.c., if it
is a simple isotropic medium.
3.5.3 Eigenproblem of the Interface
The two-dimensional eigenvectors e — e„ and e = es of the reflection dyadic /?ц
corresponding to the eigenvalues R — Ra and R = Rt and satisfying
R -e = Re,	(3.231)
give the transverse components of the polarizations of the incident electric field
which do not change in the reflection. The eigenpolarizalions of the three-
dimensional electric fields are obtained from the eigenvectors ea, es through (3.151).
The eigenvalues Ra and fij are the corresponding scalar reflection coefficients of the
eigenpolarized fields.
Another form for the eigenproblem (3.231) is obtained by expressing the reflec-
tion dyadic in terms of the two admittance dyadics from (3.219):
У2 e = aFr - e, a =	(3.232)
This can be interpreted so that we arc looking for such vector voltages whose vector
currents in both media arc scalar multiples of each other.
Eigenvalues
The two-dimensional eigenvalue equation (3.232) can be evaluated by applying
dyadic algebra [6]
spm(y2 —aKj) = spm У2 —a Yt JEj : u,u, f- a’spmY] = 0,	(3.233)
which, invoking the properties of the admittance dyadics (3.213), (3.217) and
Y2 : u.u. = — Qcosi?2, Q = —(3.234)
9192	C1(C2++C2~)
can be written as
a1 - 2a~Q cos i?2 4	= 0.	(3.235)
92	9r
The two solutions aa and сц, satisfy ао«ь = (»;i/»;2)2 and thus can be written as
ao = —eT,	<»s = — e~T,	(3.23C)
92	92
with
cosh т = Qcosi92 = —~—Ц-совгУ2.	(3.237)
cl(c2 । + c2_)
vuuipie incidence, Single Interface .
107
The eigenvalues have the following relations:
1 - a„ - »дег	1 at	г)г - i)2e~r
lia = --------- = ---“-----, rts — Z------------“ -----------
1 I a„	v3 + »;ier	1 + as	i)2 + где T
They also have interesting analytic properties (16].
(3.238)
Eigenpolarizations
The eigenvectors ea, es are obtained from (3.231) or (3.232). In fact, when a satisfies
(3.233), (3.232) can be written in the form
[uulu,, +	— «У1И,)][и1(Уг„ — <il'i„) Uyl^xJ • e = 0,	(3.239)
whence the two eigenvectors еаЬ must be orthogonal to the vector in the second
square brackets with a, the respective eigenvalues ao s- Without any normalization,
the eigenvectors are thus proportional to expressions which can be written in a
symmetric form
®-o,S Г1Х X (uz(12xx ' aO|sllxx) T riyljxy]	n^(ljxx	ао^1 Ixx) Г1Х1 2xy
(сг+-с2_	Г)гл-Т)г\	/	2cI+c2- 2гд(>д„е±т \
I	11^ UyC) I .	.	|
C2+ t Cj_ T/2+ 4	/	\C1(C2+ "Ь C2-)	’/2V/2 + + ’/2-)/
f -c2 1 — Ф-	л 1	/ ZCjaCj- Ci 4 \	.	.
— «X I J------------tan 1?2 ) + U„ I-------------—I. (3.240)
\ C2+ + c2_	)	\C2+ + c2_ cos v, )
The eigenvectors are elliptic, in general. If the medium 2 is a lossless Tellegen
medium with real c2) = c3_, they are, however, linear.
It is interesting to note that the eigenvectors do not depend on the impedance
ratio so that, regardless of the impedance level of the BI medium 2, the
eigenvectors are the same.
For the Pasteur medium with »?2 = 0, the eigenvectors e„ and es possess an
orthogonality property. In fact, for 11 = 0 the admittance dyadic E was seen in
(3.178) to be symmetric, whence we can write
 (i'l - Yi 2) • <‘b = e„- у2 -et - eb- Yi eo = (as - a„)(ea • У, • eb) = 0. (3.241)
Thus, unless a„ = ak, there arises the orthogonality condition
e„  1'j  et = 0.
(3.242)
108
Chapter 3. Plane Waves in Layered Media
The case cro = ab turns out to he possible only if Yj is a scalar multiple of the 1 ।
dyadic. From the expression of the characteristic admittance dyadic (3.178) this
is seen to imply c2+ — c2_ = c2, or there is no refraction at the interface. This
restricts the medium 2 so that it must be simple isotropic and have /it continuous
at the interface.
Simple Isotropic Media
Let us apply the previous theory to the interface between two simple isotropic
media to check that the well-known results are recovered. Thus, if the medium 2
is isotropic with c2|. = c2_ = c2, r/2+ = r/2_ — T]2 and i?2 = 0, we can write from
(3.237)
Q = 5t±51 = 1(51 + 51) = cosh r => eT = -,	(3.243)
2cicj 2 с2 ci	Ci
whence the eigenvalues are
(3.244)
(3.245)
(3.246)
— C —	у	— С —	у
’ll ’ll0!	’ll ’ll0!
„	- ’ll ’?iC> — ’ll0! n ’/b-’/l	’ll0! ~ ’ll0!
lla = -------- = -———  , lib — ——— — ——-------------------------.
Va + Vl ’ll0! + ’ll0!	% + ’ll ’ll0! + ’ll0!
The expressions (3.245), written more explicitly as
r)i cos P2 — cos Pi	sec P2 — i?i sec P2
°	Г]2 cos P2 + rj2 cos Pi ’	1	t]2 sec Pj + r)i sec Pt ’
can be recognized as the well-known reflection coefficients of the respective TM and
ТЕ polarized plane waves at the interface.
The eigenvector formula (3.240) applied to a simple isotropic medium gives
ea ~ uv and ~ 0, of which the latter is useless. However, taking the limit as the
parameters tend to those of the isotropic medium gives us the limit es ~ li,. The
corresponding cigcnpolarizations of the incident field can be seen from (3.151) to
be
sinP,
u,uv-------
»i cos Pj
u„ ~ (”v cos &l ~ u« 6’n ^l)>
-•	/=	k2 sin Pi \
Es~ I - u,u„----------— I • u, = u„
\	K] COS Pl J
(3.247)
(3.248)
from which we immediately see .that b corresponds to the ТЕ polarization. Since Eo
and ki are orthogonal to u„ Ua is parallel to 11, and the a case thus corresponds
to the TM polarized incident field.
3.5. Oblique Incidence, Single Interface	109
3.5.4 Brewster Angles
For an interface between two simple isotropic media, impedance matching of one
of the eigenpolarizalions, ТЕ or TM, at the interface results in zero reflection for
that polarization. This happens at a certain angle of incidence = flp, called the
Brewster angle. The concept can be generalized for the isotropic-Bl interface by
requiring that one of the eigenvalues of the reflection dyadic, R„ or 7?s, be zero, which
gives us a condition between the medium parameters and the angle of incidence.
The same condition is obtained by requiring oa = lorOk = l, both of which are
seen from (3.234) and (3.237) to lead to the symmetric condition
(3.249)
cl(c24 +C2-)	^1(^2+ + *12-)
This can also be written in the form
=fot-*;;*--»;.	(32M)
(cJ+ -4 cj)(cj_ 4 Ci) (i?i+ 4- 7i)(q2_ +7i)
Corresponding equations for the Brewster angle have been derived by various au-
thors [17, 18, 19]. The equation can, however, be solved analytically, as was first
shown in [20].
Analytic Solutions
The left-hand sides of the equations (3.249), (3.250) depend on the angle of incidence
Pi, while the right-hand sides do not and can be considered as constants when trying
to find the angle Pi. Defining Si = sin fl], we can write
CJ+ -
c2- =
(3.251)
and substitute in the left-hand side of (3.250), which becomes an equation for the
unknown quantity Si- After some algebra, two analytic solutions can be found and
written in the form [20, 21]
---------------------------WM- - s,)-----------	,3 2S2)
or, equivalently, as
tan’ fl - I R /G) 1 fi»H(G-/C2+) 1 Яр]
tan fln _ 1/?B_---_______-------------.	(3.2j3)
The parameters in these expressions are defined as
n2± =
(l»t -- 4i)(4»~ - yi)
(42+ + 4i)(%- t 41)
4? + 4r ~ 24i4a cos t? 2
4? 1 4a + cos
(3.254)
(3.255)
(3.256)
The double sign in front of Rg in (3.253) can be understood as corresponding to
the two branches of the square root if (3.256) is defined to have a positive real part.
To obtain real or purely imaginary3 Brewster angles, a lossless Bl medium is
assumed, which makes fig and C± real. From (3.252) it is seen that, to have a real
right-hand side, C2+C2_ should also be real, which requires the following inequality
to be satisfied:
(n’+ - "1)(«L - ”?) > 0.	(3.257)
Inserting n2± = n2(cosi?2 dr л2г), after some algebra we obtain the equivalent con-
dition
|«2rl < к»,	= |cosi?2 — —|.	(3.258)
n2
Thus, real or imaginary Brewster angles are only possible if the relative chirality
parameter of the BI medium is within the interval
 - ns < к2г < Kg.	(3.259)
At the limit points к2г = ±кв we have either n2+ = n2 or n2_ = n2 and, respectively,
C'2(. = 0 or C2- = 0, whence both solutions of (3.253) satisfy tan fig — oo and, hence,
йд — тг/2, i.e., the Brewster angle equals the grazing angle 90°.
It turns out that, in the lossless case, one of the solutions is real in the whole
interval — кд...кв while the other one is partly imaginary. The former solution
is labeled as the ordinary Brewster angle 0bo, and the latter, the extraordinary
Brewster angle 0цх- И is easily seen that for к2г = 0, whence C2). — C2_ = C2, the
extraordinary Brewster angle is always imaginary, because the product
16Я’В(Л’В - 3£)(Л’е - J)
tan eBotan 0VX ------------________ —
aAn imaginary Brewster angle corresponds to an inhomogeneous plane wave in the region 1.
The surface waves, to be discussed subsequently, correspond to a part of such solutions.
in iruece
111
has the value — Xfillg/C*, which is negative. Thus, for simple isotropic and Tellegen
media, there exists only one real solution, the ordinary Brewster angle.
It can also be seen that when ic2r increases from zero, <7’+ and have the
same sign until one of them becomes zero at |к2г) = Kf. Thus, within this interval,
one of	and С£_/С2+ approaches monotonically 0 as |k2,| increases, and
the other one oo, so that at some value |к2гI — к0 < Kg one of the factors in the
numerator of (3.260) becomes zero. After this, in the interval «0 < |к2г| < *g, its
value is positive and both Brewster angles arc real.
Thus, the extraordinary Brewster angle has a null at the values «2r = ±«0 of
the chirality parameter satisfying one of the equations
= й’В = ^-	(3.261)
C2+
These lead to two fourth-order equations for the nonnegative number k0:
( - ± Ko)(— ± Ko) = —|(cos2i?2 - к2)2 + sin’rJj,	(3.262)
4i 4i	ni
whose solutions in the range 0 < Ko < K„ can be found numerically. This effect is
demonstrated in subsequent examples.
Simple Isotropic Media
The double expression (3.252) or (3.253) can be tested for the simple isotropic
interface, in which case, by setting
C2+ = C2_ = C2 =	RB = 21—(3.263)
V lllellllel	'12 + 41
we have
lan2/’B = ±ё2(к~л т'п2 = и	-1].	(3-264)
leading to the two solutions
tan 6Bl =	/ц(/<гС1 ~<2/ti) ,	(3.265) /'l(/Oe2 ~ /*lel)
tan 0B2 =	CzfcrPl ~/<2Cl)	,,	.. ,	(3.266) C1(/*2C2 - Hl<l)
which coincide with the well-known isotropic results [22]. Because the product	
tan#B! tan6/j2 is obviously imaginary for lossless media, only one of the solutions
is real: the ordinary Brewster angle 6 во, as observed above.
112
Chapter 3. Plane Waves in Layered Media
Examples
Figure 3.30 gives the ordinary and extraordinary Brewster angles for the interface
between air (ni = 1) and Pasteur medium as functions of the relative chirality
parameter к2г [20, 23].
Figure 3.30 Ordinary (“O") and extraordinary (“X”) Brewster angles as functions of
the relative chirality Kjr, for an interface between air and a BI medium with parameters
/z21. = 1> X2 = 0 and ci, = 2, 4, 10. The extraordinary Brewster angle exists only in the
intervals «0 > |к2г| < к₽.
As predicted by the theory, it is seen that both Brewster angles increase with
increasing |x2r|, and attain the grazing angle 90° when the chirality parameter
satisfies «2. =	= ±(1 — l/n2) according to (3.258). For larger |к2г| values both
Brewster angles are complex.
The extraordinary Brewster angle Opx is real only within the two intervals k0 <
|«2r| < kb and passes through all values 0 < дцх < rr/2 while the ordinary Brewster
angle is limited to a range of values 0mtn < впо < x/2. The interval к0 • •  ке may
be very small, as can be seen from the c2r = 2 curve in Figure 3.30.
Increasing the Tellegen parameter Xi reduces the grazing chirality кд, which is
demonstrated by Figure 3.31. For cosdj = l/n2 we have actually ке = 0 and no
real Brewster angles appear, as is seen from (3.258).
Polarization
The eigenvectors corresponding.to the Brewster angles can be obtained from (3.240)
by inserting the solution (3.252). To find the eigenvector corresponding to the zero
3.5. Oblique Incidence, Single Interface
113
Figure 3.31 Ordinary (“0”) and extraordinary (“X”) Brewster angles as functions of
the relative chirality K2,, for an interface between air and a BI medium with parameters
/11, — 1, tir — 10 and xi, = 0, 0.4, 0.7.
in the reflection coefficient, say a, = 1, whence from (3.244) c»s = (q,/q2)2, we see
from (3.237) that er — 71/7,. This inserted in the eigenvector expression (3.240)
and invoking (3.249) leads to the corresponding eigenvector
o„
. f c2+ - c2_
«.7 I-----:---
\Clf + c2_
7г+ — 7г-
72+ + 72-
+ U„C1
71
71 COS 1?2
2c,
C2+ + C2_
(3.267)
in which we must substitute c± — ^/к2± — fc, sin2 fig.
Surface Waves
If the solution of (3.252) for lossless media gives a negative value for tan2 вд, the
Brewster angle becomes imaginary. This happens for the extraordinary Brewster
angle in the interval of the relative chirality parameter — к0 < к2, < k0. Writing
the propagation vector of the incident plane wave in medium 1 as
k,=u, cos^bjc tuvsin$Bjr
u, 4 ut tan fil)x
c/1 + tan2 Ggj
= u,k,, 4 и,,*,,,,
(3.268)
we can distinguish two possible cases of negative tan2 6g. In the first case,
— 1 < tan2 6gx < 0, we have real and kiv imaginary. This corresponds to
an inhomogeneous plane wave which has exponential growth along the surface and
propagates normal to the interface. The other case, —oo < tan2 Obx < — 1, cor-
responds to a solution which decays normal to the interface and propagates along
the surface. This kind of an inhomogeneous plane wave is called a surface wave or
Zenneck wave.
Because tan20Bj >8 complex for |к2г| > and positive real for кд > |«2r| > Ko,
possible k2t values for surface waves to exist must occur in the interval — к0 < к2г <
к0. The condition (3.253), can be applied to find more explicit condition.
3.6 Oblique Incidence on BI Slab
The vector transmission-line theory can be applied to problems involving obliquely
propagating plane waves in layered BI media with plane parallel interfaces so that
each plane-wave system with the same transverse propagation factor is treated sep-
arately. Let us consider as an example of such a system a BI slab terminated by a
PEC plane at z=0, Figure 3.32. The incident field is a plane wave in front of the
slab in the simple isotropic half space z < — L.
- L 0
Figure 3.32 Obliquely incident plane wave from a simple isotropic medium onto a
conductor-backed 111 slab.
3.6.1 Dyadic Admittance and Reflection
To see the effect of the slab to an incident plane wave, we have to find the reflection
dyadic. It can be found after forming first the expression of the input admittance
dyadic through the vector transmission-line theory.
Input Dyadic Admittance
The input dyadic admittance of a Bl slab with the parameters /t2,	*2, Xz> termi-
nated by a PEC slab with the dyadic admittance can be obtained as a special case
from (3.206), by letting Y t,—* oo, as
115
У.„= (У2 -<АЬ+ У,  (e^’r - e~^L)-'
=	- У2) 4 |(Уа 4 У2) •	4 e~’P'L) • (ej/,,t - e iP^L)~'. (3.269)
We can check this expression by first noticing that when L —> 0, the difference of
exponentials becomes zero and its inverse becomes infinite making Ym infinite. This
is just the admittance of the PEC plane. As a second test, let us assume L —» oo
and losses in the slab which make the negative exponentials vanish. After this, the
positive exponentials are seen to cancel leaving the anticipated input admittance
У2. Thus, there is no reflection from a PEC plane through a thick lossy slab, which
is a known result.
Reflection Dyadic
The reflection dyadic at the interface of the simple isotropic medium 1 and the HI
slab 2 is
Rn=(r1+Fi„)-1(rl-r.„).
(3.270)
Inserting (3.269) gives a possibility to different expressions for the reflection dyadic:
/111— — It + 2(Ki+ Уin) 1 • 1 i
= -Z, 4- 2(e»‘*b - e->^L) • |(у2 4-Г1) •	+ (У2 -У,) -	‘  У,
= -(Л’ь - e~’p,L)  |(У2 4-F.) • e^’L + (У, -F,) •	’
 1(У2 -F,) 	4 (y2 4 F,) . e-^£] • (e^L -	(3.271)
Writing the corresponding expression in component form would take a lot of space,
[24]. However, to get the final numerical expression, the dyadics must be expressed
in component form. To get full use of the present vector transmission-line method,
handling of formulas can be done by applying one of the existing programs doing
both symbolic and numerical analysis.
3.6.2 Special Case
Vector transmission-line formulation gives direct access to special cases. To see the
relation between (3.269) and a previously analyzed result for normal incidence, we
invoke the dyadic wave admittance and exponential propagator expressions (3.171),
(3.172), (3.181) and (3.182):
116
References

। fe^iJ |	• fef*J^c<”^aeKr*’^ । е-л*»Ь<о.^>е«.ЬГ'\
2>/j \	/ '	'
. ^jkjLco.^j екг1<зЬ _ ^-jkiLcoe^t^nrkiL^ 1
= — [sintPjJ - jcosi52cot(fej£cosi9j)I(].	(3.272)
41
If the reflection dyadic at the interface of the BI slab and a simple isotropic
medium 1 is written according to
7lu= (F.+	 (Г,- ?.„) = HrJt d R„J, (3.273)
the same expressions obtained through the scalar transmission-line theory, (3.109),
(3.110), are recovered.
References
[1]	Lindell, I.V., M.E. Valtonen and Л.Н. Sihvola, “Theory of nonreciprocal and nonsymmetric
uniform transmission lines," IEEE Trans. Microwave Theory and Techniques, Vol. 42, No.
2, February 1994.
[2]	Lindell, I.V., S.A. TYetyakov and M.I. Oksanen, “Vector transmission-line and circuit theory
for bi-isotropic layered structures," J. Electromagnetic Waves and Applications, Vol. 7, No.
1, 1993, pp. 147-173.
[3]	Lindell, I.V., S.A. lYetyakov and M.I. Oksanen, “Conductor-backed Tellegen slab ns twist
polarixer,” Electronics Letters, Vol. 28, No. 3, January 1992, pp. 281-282.
[4]	Tretyakov, S.A. and M.I. Oksanen, “A bi-isotropic layer as a polarisation transformer,”
Smart Materials and Structures, Vol. 1, 1992, pp. 76-79.
[5]	Jaggard, D.L., N. Engheta and J. Liu, “Chiroshield: a Salisbury / Dallenbach shield al-
ternative," Electronics Letters, Vol. 26, 1990, pp. 1332-1334, Errata, ibid., Vol. 27, 1991,
p.547.
[6]	Lindell, I.V., Methods for Electromagnetic Field Analysis, Oxford, Clarendon Press, 1992.
[7]	Oksanen, M.I., S.A. Tretyakov and I.V. Lindell, “Vector circuit theory for isotropic and
chiral slabs,” J. Electromagnetic Waves and Applications, Vol. 4, No. 7, 1990, pp. 613-643.
[8]	Tretyakov, S.A. and M.I. Oksanen, “Electromagnetic waves in layered general bi-isotropic
structures,” J. Electromagnetic Waves and Applications, Vol. 6, No. 10, 1992, pp. 1393-1411.
[9]	Lindell, I.V., “On the quasi-TEM modes in inhomogeneous multiconduc.tor transmission
lines," IEEE Trans. Microwave Theory and Techniques, Vol. 29, No. 8, August 1981, pp.
812-817.
[10]	Bokut, B.V. and F.I. Fedorov, “Reflection and refraction of light in optically isotropic active
media,” Optics and Spectroscopy, Vol. 9, 1960, pp. 334-336.
References
117
[11]	Lakhtakia, A., V.V. Varadan and V.K. Varadan, "A parametric study of microwave reflection
characteristics of a planar achiral-chiral interface,”IEEE Tran». Electromagnetic Compatibil-
ity, Vol. 28, No. 2, May 1986, pp. fl0-95.
(12)	Silverman, M.P., “Reflection and refraction at the surface of a chiral medium: comparison of
gyrotropic constitutive relatione invariant or noninvnriant under a duality transformation,"
J. Optical Society of America A, Vol. 3, No. 6, 1986, pp. 830-837.
[13]	Bassiri, S., C.H. Papas and N. Engheta, “Electromagnetic wave propagation through a
dielectric-chiral interface and through a chiral slab,” J. Optical Society of America A, Vol.
5, No. fl, September 1986, pp. 1450-1459.
(14)	Viitancn, A.J., I.V. Lindell and Л.II. Sihvola, “Eigensolutions for the reflection problem
involving the interface of two chiral half spaces,” J. Optical Society of America A, Vol. 7,
No. 4, April 1990, pp. 683-692.
[15]	Lindell, I.V., A.H. Sihvola and A.J. Viitanen, “Plane-wave reflection from a bi-isotropic
(nonreciprocal chiral) interface,” Microwave and Optics Technology Letters, Vol. 5, No. 2,
pp. 79-81, February 1992.
[16]	Sihvola, A.H. and l.V. Lindell, “Properties of bi-isotropic Fresnel reflection coefficients,”
Optics Communications, Vol. 89, No. 1, 1992, pp. 1-4.
[17]	Bassiri, S.f “Electromagnetic waves in chiral media,” in Recent Advances tn Electromagnetic
Theory, eds. H.N. Kritikos and D.L. Jaggard, New York, Springer, 1990, pp. 1-30.
[18]	Lakhtakia, A., V.V. Varadan and V.K. Varadan, “Reflection of plane waves at planar achiral-
chiral interfaces: independence of the reflected polarisation state from the incident polarisa-
tion state,” J. Optical Society of America A, Vol. 7, No. 9, 1990, pp. 1654-1656.
[19]	Lakhtakia, A., “On extending the Brewster Jaw at planar interfaces,” Optik, Vol. 84, 1990,
pp. 160-162.
[20]	Lindell, I.V., Л.II. Sihvola and A.J. Viitanen, “Reflection and transmission of plane waves at a
planar interface between isotropic and biisolropic media,” Helsinki University of Technology,
Electromagnetics Laboratory Report 100, August 1991.
[21]	Lindell, I.V., A.H. Sihvola and A.J. Viitanen, “Explicit expression for Brewster angles of
isotropic-bi-isotropic interface,” Electronics Letters, Vol. 27, No. 23, November 1991, pp.
2163 2165.
[22]	Kong, J.A., Electromagnetic IVntir Theory, New York, Wiley, 1986.
[23]	Sihvola, A.H. and I.V. Lindell, “Novel effects in wave reflection from bi-isotropic media,”
Microivave and Optics Technology Leiters, Vol. 6, No. 10, August 1993, pp. 581-585.
[24]	Lakhtakia, A., "Time-harmonic dyadic Green’s functions for reflection and transmission by
a chiral slab,” Arkhiv der Elektrischen Ubertragung und Elektronik, Vol. 47, No. 1, 1993, pp.
1-5.

Chapter 4
Waveguides
In the literature, there has been an extensive discussion on wave propagation in
waveguides filled with reciprocal chiral materials, so-called chirowaveguides. First,
the simplest cases of closed planar reciprocal structures were analyzed [1, 2]. Cir-
cular chiral waveguides were studied in [3] to [6]. Open waveguiding structures
was the topic of papers [7] to [11], see also [5]. Chiral waveguides with lossy walls
and anisotropic impedance surfaces were analyzed in [12] to [14], and the effect of
losses in chiral filling was analyzed in [15]. Coupled-mode theory was developed
and mode conversion analyzed in [16] to [19]. In [20, 21] some special issues such as
mode orthogonality and bifurcation of modes were addressed. The cut-off frequen-
cies and the field distribution in parallel-plate waveguides partially filled with chiral
materials were found in [22]. Nonreciprocal phenomena can be seen not only in
waveguides with nonreciprocal bi-isotropic filling but also in so- called chiroferrite
waveguides [23, 24]. In these waveguides the filling medium is bianisotropic, and its
parameters can be in principle electrically controlled by biasing the ferrite fraction
of the composite filling material.
The interest of researchers, besides the academic one, lies in some prospective
possible applications in mode transformers and directional couplers in millimeter
waves and the suboptical band.
In this Chapter the fundamental eigenvalue equations and field patterns of the
eigenwaves for general bi-isotropic waveguides are derived. The waveguide surface
impedance is assumed to be either isotropic or anisotropic or the waveguide is open.
The general field analysis for arbitrary cross sections is exemplified for plane and
circular waveguides, either open or closed. For the rectangular cross section, an
approximate analytical solution is given.
119
120
Chapter 4. Waveguides
4.1	Guided-Wave Solutions
Waveguide solutions for different types of closed and open waveguiding structures
with general bi-isotropic filling can be found using the wavefield decomposition
approach, discussed in Chapter 2. The decomposition into wavefield components
reduces bi-isotropic waveguide problems to two equivalent isotropic waveguide for-
mulations. Instead of working with longitudinal components of the electric and
magnetic fields we deal with longitudinal wavefield components now. The corre-
sponding system of equations is coupled due to the boundary conditions on the
waveguide wall.
An alternative approach, designed for plane layered structures, which is called
vector circuit theory, will be formulated and the results obtained for plane waveg-
uides with different types of impedance boundaries or for open waveguides will be
discussed.
4.1.1 Field Decomposition and Boundary Conditions
The conventional method of solving waveguide problems is based on division of
the fields into longitudinal and transversal components. In the case of bi-isotropic
filling, the corresponding equations for the longitudinal field components turn out
to be more complex than for isotropic guides. The analysis can be simplified if
we first decompose the fields into wavefield components and only then split these
wavefields into transverse and longitudinal parts. This approach leads to uncoupled
Helmholtz equations for two longitudinal wavefield components. Eigensolutions
satisfy appropriate boundary conditions, which in general involve both longitudinal
wavefield components.
Let the z-axis (the corresponding unit vector is u,) of a rectangular or of a
cylindrical co-ordinate system be directed along the waveguide axis, see Figure 4.1.
Figure 4.1 The waveguide geometry.
4.1. Guided-Wave Solution»
121
We express the electromagnetic field inside a uniform bi-isotropic material filling
the guide in terms of wavefield components Ej., subject to the sourceless uncoupled
first-order wave equations (2.70), (2.71)
VxEtT^Et = 0	(4.1)
as discussed in Chapter 2. Looking for propagating modes with the propagation
factor P, the wavefields can be divided into longitudinal and transverse parts
Ee.|. like
E±(r) = (E,±(p) 1 u,E,±(p)]e-^*	(4.2)
where p stands for the transverse coordinate. Inserting (4.2) into (4.1), each of the
equations in the pair (4.1) separates into two parts
V, x E(-£	= 0	(4-3)
and
+ (jPIi T fc±u, x /() • E(± = 0.	(4-4)
Jt stands for the transverse unit dyadic and Vt is the transverse gradient operator.
Taking the inverse of the two-dimensional dyadic in (4.4) expresses the transverse
wavefield components in terms of the longitudinal parts:
E.± - (-M ± x 7,)-1 - V,E,± = ~-(-jpl T *±Ul x 7,)  V,EI±. (4.5)
*c±
Here, kci =	— /?’ are the transverse wave numbers.
Eliminating the transverse field, we are left with two Helmholtz equations for
the longitudinal wavefield components
(V,’ -b = 0.	(4.6)
If the guide is open, i.e., the fields extend to infinity, we will have similar equa-
tions for the outer wavefields as well (it is convenient to use a similar wavefield
decomposition for isotropic cladding when the core is bi- isotropic). Boundary con-
ditions relate the physical fields E, II, and they should be expressed in terms of
their wavefield components. In the following two subsections we specify the bound-
ary conditions for closed and open BI waveguides. In general, the two Helmholtz
equations (4.6) are coupled because of the boundary conditions.
After solving for the longitudinal components from (4.6), all the other wavefield
components can be found using (4.5). The actual physical fields can finally be
recovered from (2.9) and (2.10) as linear combinations of the wavefields Ej..
Impedance Boundary Condition
The boundary condition on the waveguide walls for closed guides can be written as
an impedance condition, which states the relation between the tangential electric
and magnetic fields on the boundary surface
u„ x E = Z, • II,	(4.7)
where un is a unit normal vector pointing outwards from the waveguide bound-
ary, see Figure 4.1. Assuming the impedance dyadic to be two-dimensional and
containing diagonal terms only, we have
Z, = ZrtlTUT | Z, 11,11,.	(4.8)
The tangential unit vector is denoted by ur. The unit vectors are related according
to ti, = u, X u„. Equation (4.8) can be used for modeling lossy and (or) corrugated
surfaces, for example.
In terms of the waveficld components, the boundary impedance relation (4.7)
splits into two equations
E’r+ + Er_ = -jZ,(e>aE,+ - e~iaE,_),	(4.9)
E,+ + E,_ = jZT(e>aEr^ - e~iaEr.),	(4.10)
where ETi stand for the wavefield components along the u, vector (see Figure
4.1) and the normalized surface impedance components are introduced as Zr =
Z,l-q, Zt = Z./ti, q = \Jnlt- Using (4.5), the following relations valid for the
fields inside the waveguide and on the boundary can be obtained:
 P dE.j	k± dE,±
T±	dr	ket dn ’
. fl dE,± k± dE,±
"*	3 dn	dr
(4.11)
(4.12)
Next, eliminating the tangential components from (4.9) and (4.10) and writing the
boundary condition in terms of the longitudinal wavefields only, results in a set of
two equations, valid on the boundary
1 - e>aZr
k*t dr 3 k\ dn)

1 T e’^Z,
(fl <9 _ . JL
dr ~3^. dn/]
E,_ = 0,
(4.13)
123
AA-AlA
к2, дт ^ 1 к2, дп Z‘
\ fe’_ дт к2_ дп
+ e~ieZ, Е,.
= 0.
(4.14)
Il is easy to sec from expressions (4.13) and (4.14) that for ideally conducting
boundaries (Z, — 0) the explicit ^-dependence disappears. Therefore both the non-
reciprocal and reciprocal ideally conducting waveguides have eigenvalue equations
of the same form.
Open Waveguides
Consider an open bi isotropic waveguide of an arbitrary cross section and sur-
rounded by an isotropic nonchiral homogeneous medium with the parameters £j, pi.
For the fields inside the bi-isotropic core (marked with the superscript ,n) we ap-
ply the field decomposition technique described above. The longitudinal wavefield
components satisfy (4.6):
(V.2 + fec2±)£i"± = 0.	(4.15)
Although the surrounding dielectric is assumed to be nonchiral and reciprocal, it
is handy to use the same type of field decomposition as for the fields inside the
bi-isotropic core. The longitudinal components of the wavefields outside the bi-
isotropic core region (superscript°u') are subject to the equation analogous to (4.15):
(V,2 4 fec22)^‘= 0.	(4.16)
Here, feci = jq = ^/fe2 — fl1 is the transverse wavenumber in the dielectric (fej =
u'y/eipi). I*1 isotropic dielectric, the two longitudinal wavefield components E^
will obviously have the same coordinate dependence, and they may differ only by
constant amplitude coefficients. The other tangential components of the fields inside
the bi-isotropic rod come from (4.11). For the fields in simple dielectric the equation
(4.11) reads
= , A T k' dE^
T± q2 (jr q2 dn
(4-17)
Finally, the longitudinal wavefield components are subject to the boundary condi-
tions at the interface:
4- Ein = E™‘ 4- Е°'л,
(4.18)
124
Chapter 4. Waveguides
E'n + Ein =	+ E"”*	0.19)
q, (e*E" -	= q (E£‘ - E?) ,	(4-20)
q, (e*E" - e-jdE“) = q (E™‘ - E™‘) ,	(4-21)
where qi = ypi/«i and the т-components are related to the longitudinal wavefields
as in (4.11) and (4.17). These conditions equate the tangential field components on
the sides of the interface.
4.1.2 Vector Circuit Approach
The vector circuit method [25, 26, 27j can be used for modeling planar chiral and
BI structures.
E,‘ H,'
e/r к x
Figure 4.2 Plane bi-isotropic layer.
In this method, a BI slab is modeled as a two-port circuit with dyadic equivalent
impedance, admittance, or transmission parameters. Fourier-transformed compo-
nents of the electric and magnetic fields tangential to the interfaces arc interpreted as
vector voltages and currents, like in the vector transmission-line theory, see Section
3.4.2. The approach is effective in calculating reflection and transmission coeffi-
cients of layered structures [26, 27], and can be used in studying closed and open
plane bi-isotropic waveguides [11, 13].
The half spaces or boundaries on different sides of the structure correspond to
equivalent dyadic impedances or admittances. The relation between the tangential
fields on opposite surfaces of the structure is expressed in terms of the dyadic chain
matrix
4.1. Guided-Wave Solutions
125
/	E,+
u„ x H,+
( «II
\ 511
«11
5ц
( ЕГ
\ U„ X Н,-
(4.22)
Figure 4.3 Vector-circuit model of a BI «lab.
Subscript t denotes Fourier-transformed tangential field, superscripts 4 and mark
the fields on the upper and lower side» of the slab, respectively, and u„ is the normal
unit vector pointing upwards from the upper interface.1 Inside the slab, the vector
voltage and currents can be expressed as two vector waves, which travel in opposite
directions as expressed in Equations (3.185), (3.186). Solving the system of (3.185)
and (3.186) for the fields on two interfaces, gives the dyadic components of the
transmission matrix modeling the bi-isotropic slab of the thickness d [27):
«11 =	+ •£>-)- | tan t?(P+ - D_),
5., = -'’ (Ё+-Ё_)хип,
2 CO8 V
1	j	=====
«11 =---x a12 X Un = -------—u„ X p+ - /?_),
if	2т/ сов v
«11 = -Un X [|(E’+ + D-) + tan d(£>+ - Z)_)j x un. (4.23)
The dyadics D± define the relation between the transverse components of the wave-
fields on the interfaces, and they read (25, 27]
D± = cos(kc±d)I t T sin(fcc±d) (^i-uTu, — ^^uxur
\
’The superscripts + and must not be mixed with the notation of the wavefields.
(4.24)
__     ___________ __*  .» ^«vyujpageijuirшстог Bjong tne plane waveguicfe,
and
kc± = /4-Д1	(4.25)
are the transversal components of the propagation factors of the two wavefields.
The unit vector 11, points in the propagation direction along the guide, and the
vector u, satisfies u, = u„ X u, (Figure 4.2).
To derive the general eigenvalue equation for a bi-isotropic slab waveguide with
boundary impedance conditions, we assume that at the slab interfaces the tangen-
tial electric and magnetic fields are related to each other through the boundary
conditions
E+ = -fcu-unxH,+ , Er=fd u„xH,“	(4.26)
with diagonal dyadic boundary impedances
X. = I	• = u,l.	(4.27)
Here 7’£’ and TM refer to TE- and TAf polarized field components.
Using (4.22) and the boundary conditions (4.26) we find the general eigenvalue
equation for bi isotropic slab waveguides
( 7TM 7TM TM . i7TM . 7TM\TM . „TM\
(Au Al Al + (Au + A| )all + “12 )
I7TE7TETE , i 7TE . 7TE\ TE , TE\
(Au zd At + (z™ 1 Ai )au +an )
I7TM7TE ME , 7TM ME . 7TE ME . „ME\
- (Au Ai At + Au an + Ai “и 1 au )
(7tm7tejem , 7TE EM , 7TMEM , EM\ _ n	r. 9Rr
•(Al Au Al + Au Al + Al Al + °12 J = °’	(4.28)
Here, the a-dyatlics (4.23) have been divided into parts as
«v ~	1-о,УБч,Чг 4	*,J — 1>2,	(4-29)
where the subscripts ТЕ and TM correspond to the ТЕ- and TM-polarized field
components, as in the definition of the impedance dyadic (4.27), and ME and EM
stand for the coupling terms from ТЕ to TM and from TM to ТЕ, respectively.
This model covers symmetric and non symmetric bi-isotropic slab waveguides,
waveguides with ideally conducting or lossy boundaries and corrugated boundaries.2
2Provided that the period of corrugations is much smaller than the wavelength.
1
4.2 Slab Waveguides
The simplest waveguide solutions which demonstrate most features typical for chiral
and bi-isotropic waveguides can be obtained for planar slab waveguides. We start
from analysing closed bi-isotropic waveguides.
4.2.1 Closed Planar Bi-Isotropic Waveguides
For a plane waveguide (see Figure 4.2), assuming no у-dependence, we express the
general solution for the longitudinal wavefields (4.6) as
sin(fcc±z) + Il± cos(fcc±z)	(4.30)
with the amplitude coefficients A± and B±. Assuming that on the boundaries the
impedance condition (4.7) holds, we substitute (4.30) into the boundary conditions
written in terms of the longitudinal wavefield components (4.13), (4.14). Equating
the determinant to zero, we see that the determinant can be expressed as a product
of two smaller, 2x2 determinants and we are left with two eigenvalue equations for
two sets of eigenwaves
\
-— sin(Ac_d) cos(fcc+d) I
*c-	)
_ _	/	f.
(1 + Z.ZT) -!-sin(V<i) cos(fcc+ d) + -— sin(fc<.+cos(fcc_ J)
+ 2jcos«? ( Z, sin(fcC| d) sin(lc_d) — Z.  *—cos(fcc+d) cos(fcc_d) j =0, (4.31)
\	*c+ kc_	)
(1 + Z,ZT) ( : — sin(fcc(d) cos(kc_d) I
\«c+
(-	k k_	\
Z, cos(A:c| d) cos(fcc_d) -	—ein(fccyd)sin(fcc_d) j =0. (4.32)
кс+кс-	/
Equation (4.31) corresponds to odd longitudinal wavefields E,± distribution (Z?± =
0), and (4.32) corresponds to even wavefield function or А± = 0. Of course, the
actual electric and magnetic fields are neither odd nor even for both sets.
The eigenvalue equations for slab waveguides with ideally conducting walls follow
from (4.31) and (4.32) after letting Z, = Z, = 0, which simplifies (4.31) and (4.32)
to
к+кс_ tan(fcc_d) + к_кс* tan(fce4d) = 0,	(4.33)
к)кс_ cot(kc_d) 4- к_кс} cot(fcc+d) = 0.	(4-34)
Multiplying (4.33) by (4.34), the product reads
128
Chapter 4. Waveguide»
(k\ кг\
sin(2fcc+ d) sin(2fcc_ d)
L L
2, + ~ (1 — coe(2A^+d) coe(2fcc_d)) = 0.	(4.35)
«<._|Ле-
The form of the last equation coincides with the dispersion relation for plane re-
ciprocal chiral waveguides with ideally conducting surfaces [1, 2, 4], although the
wavenumbers in (4.35) depend on the Tellegen parameter. Assuming sin d =0 or
cosd = 1, the equation (4.35) and the corresponding result for reciprocal chiral
guides coincide.
Field Patterns
In the case of general boundary impedance, for the modes given by (4.31) the
longitudinal wave fields can be written as
E,+ = Asin(fce+z),	= Aasin(fce_i),	(4.36)
where A is the amplitude coefficient, and
sin(fc<4.d) —	cos(fee+d)
sin(fcc_d) — je~’eZT^ cos(ke_d)
cos(fce+d) — e^Z, sin(fcc+d)
j£=- cos(fcc+d) — e~i*Z, sin(fcc_d)
The longitudinal electric field can then be found from (2.9):
E. = A (sin(fcc+z) + aein(fct_a:)).	(4.38)
Transverse wavefield components are given by (4.5) and, using (2.9), we find the
transverse electric field
—j/M {у— сов(к')-х) + a-— c
\fcef	*c-
—A |	cos(fce^r) — ci-—~ cos
\ kc^	kc_
(4.39)
(4.40)
Magnetic field components are related to the wavefields as in (2.100) and they
read
4.2. Slab Waveguides
129
H, — A- (e’* sin(ke+x) — ae sin(fcc_x)) ,
Hx = Л—| e’^r—cos(feexx) — ae'^y—cosfA^x) |
r] \ kt	)
i f  к	к
Hv = — Л-[е,<’-—cos(fce4 x) 4 ae~’e-—cos(fcc_x)
*7 \ kc±	kc~
(4.41)
(4.42)
(4.43)
Similarly, the field patterns for the modes subject to (4.32) can be expressed as
-—sin(Ac_x)|,	(4.46)
ke-	/
E, = A(cos(fcc+x) I a cos(fc<._x)),	(4.44)
£’» = jflA [ 7—sin(fcc+x) I «’7— sin(At_x)j ,	(4.45)
\ ^e+	“e-	/
—-sin(fcc^x) - a'
k't
H, = A-(e’f cos(Actx) — a'e"** cos(fcc_x)) ,	(4.47)
4
II, — — Л— (T~~ sin(Ac|.x) — a'e-’*-—ein(fec_x) J ,	(4.48)
V \	"c+	*c-	/
Hy = A- 1	7— sin(fcc4 x) 4	sin(fcc_x) ] ,	(4.49)
17 \ fcc+	kc_	)
where
cos(kc+d) 4- je’fZT^ sin(Ac+d)
cos(fcc_d) +	sin(fcc_d)
sin(fcc4 d) 4- e’^Z, eos(fcc+d)
J17=-sin(fee+<7) 4- e~^eZt cos(fct_d)
(4.50)
All the modes are hybrid. The cut-off frequencies can be determined from (4.31),
(4.32) by setting the propagation factor fl — 0. In the case of ideally conducting
walls (Z, — ZT — 0) the corresponding equation is simple:
sin(2fcodn cos d) = 0	(4.51)
and the cut-off frequencies are same for both sets of eigenwaves:
mr
2dv/c^/ion cos d
m = 0,1,2,...,
(4.52)
where the refraction index is n = y//ic//ioco.
The fundamental mode has zero cut-off and an odd longitudinal wavefield distri-
bution. This mode can be named as guasi-TEM mode, since the longitudinal fields
are small compared with the transverse components for small waveguide height, as
is seen from Equations (4.38) to (4.43).
Numerical Examples
Typical dispersion curves3 for parallel-plate bi-isotropic waveguides with ideally
conducting boundaries are shown in Figure 4.4.
Figure 4.4 Dispersion curves for parallel-plate bi-isotropic waveguide with ideally con-
ducting boundaries. Relative chirality parameter = 0.1. Dotted lines show dispersion
of two eigenwaves in the uniform medium with the same parameters.
Except for the fundamental mode, all other modes in chiral waveguides are
bifurcated. This means that there exist two modes which both correspond to a
single mode of a nonchira) waveguide (in the limit к —» 0). In guides with ideally
conducting boundaries, according to (4.52), the cut-off frequencies of the bifurcated
pairs are the same. There is an analogy with eigenwaves in uniform media, since we
also have in chiral media two eigenwaves with different propagation factors k± —
ho(n ±«), in contrast to simple isotropic dielectrics where the eigenwaves have equal
propagation factors. If the chirality parameter tends to zero, both waves have the
same propagation constant. The mode bifurcation phenomenon was discussed in
[4], and analyzed in more detail in [21].
Numerical examples in Sections 4.2 and 4.3 have been computed by Paivi K. Koivislo.
Notice that, because of the lack of explicit ^-dependence in the eigenvalue equa-
tion, the term cos d can be included in the г-axis argument values. Therefore Fig-
ure 4.4 represents the dispersion curves for nonreciprocal ideally conducting planar
waveguides with every possible real d value.
In Figures 4.5 and 4.6, phase constant and attenuation curves of a reciprocal
chiral parallel-plate waveguide with к, = 0-1 and lossy boundaries are depicted [13].
Figure 4.Б Phase constant curves of a parallel-plate Pasteur waveguide with lossy bound-
ary impedances, к, = 0.1 and M = 10-4.
The surface impedance was adopted to be isotropic: Z. = ZT = (l+j)^kdM/2
with the normalized parameter M = 0.0001. The dashed curves in these figures
stand for the wavenumbers k± in the corresponding unbounded cliiral medium.
The modes are marked according to increasing cut-off values of the corresponding
ideally conducting walls waveguide as f/°, H1, №... The indices a, b are used to
distinguish bifurcated modes with the same cut-off frequencies. Due to losses on
the waveguide walls, the sharp cut-offs disappear. Comparison of Figures 4.4 and
4.5 helps to see the effects of losses, since the other parameters are kept the same
(the Tellegen parameter in lossy guide was assumed to be zero).
The attenuation curves shown in Figure 4.6 reveal noticeable differences between
the two (marked by a and 5) sets of modes. The losses for the 6-modes have minima
at certain frequencies and then the losses grow in the high frequency region, while
the losses tend to zero for the а-modes. Thus, these modes (although hybrid)
are in this sense analogous to the TM and ТЕ modes of parallel-plane isotropic
waveguides, respectively.
132
Chapter 4. Waveguides
Figure 4.6 Attenuation curves of a parallel-plate Pasteur waveguide with lossy boundary
impedances. кг = 0.1 and M = 10“4.
Plane chiral waveguides with ideally conducting boundaries but lossy filling ma-
terial were considered in [15). In fact, the above theory holds for arbitrary media
parameters, and for complex values appropriate for lossy materials as well. The
numerical solutions, however, are naturaly more involved than for lossless media.
In [13], plane chiral waveguides possessing corrugated surfaces on both sides
of the guide were analyzed. In that case, the cut-off frequencies were moved, the
mutual position of the dispersion curves was changed due to chirality. In addition,
the dispersion curves could even cross each other at certain frequencies.
4.2. Slab Waveguides	133
4.2.2 Open Bi-Isotropic Plane Guides
Open plane Pasteur and bi-isotropic waveguides can be studied using the wavefield
decomposition or the vector circuit method, as described in Section 4.1.
Waves Along a Chiral Slab
Here we use the vector circuit approach (Section 4.1.2). In that method, the two
half spaces on different sides of the slab are modeled by two dyadic impedances
with i — и, I, or admittances K^, which are the two-dimensional inverses of the
characteristic impedances [25, 26]. The indices u and I refer to the upper and lower
half spaces, respectively. The impedances read
= — uTur + u,u, = Z™uTuT + ZjJMu,u„ i=u,f, (4.5.3)
Ii
— y/^i ~ 01) i — u,l
with the standard notation kt = and = фц/ц, i — и,I. fl is the prop-
agation factor along the z-axis and ТЕ and TM refer to ТЕ- and TM-polarized
fields.
The eigenvalue equation is given by (4.28) after substituting the impedances
and the transmission matrix a,j components. If the upper and lower half spaces
are identical, Z^ = Z^ = Z\, the eigenvalue equation (4.28) takes the form
(/ 7TM\1 TM . r)7TM TM , „TM\ h7TE\t„TE , n7TETE , TF'l
((Zl ) a2I 4 an 4 an j • ((Z, ) a21 4 2Zt an 4 a12 J
J	u. 7TM EM . 7TEME , _ЛГЕ1’ n	/, r.\
+ Vh°u + zi an + z\ an +“u ) =0-	(4.54)
Explicit form of the eigenvalue equation is obtained after substitution of the a-
dyadics (4.23). In terms of normalized parameters from the optical waveguide lit
erature [28], the eigenvalue equation (4.54) reads [9, ll]4
2w’u4u_(l — a’){(n’ — nJ)’ - |(n’ + nJ)’ 4 4n’nJ]cosuF cosu_)
+ {w’(n2 1 nj)’[uj(l - к,)2 4 ul(l 4 к,)2]
-4[uju2 nJ 4 w4n4(l — «J)’| J sin u+ sin u_
4In [9] the equation (4.55) was obtained using the field decomposition approach. An alternative
derivation waa given in [8], where no explicit forms for the eigenvalue equation were presented.
4 4(n2 У nj)wu_(l — кг)[и2 nJ — w2n2(l 4- *»)’) cobu_ sin u+
4- 4(n2 4 nj)wu+(l + Kr)[uJ_nJ — w2n2(l — Kr)2|cosu+ sinu_ = 0.	(4.55)
The normalized parameters have been defined as follows: the normalized frequency
V and the parameters u,w,U£
v = djk1 - ki = v/uj+w2,
u = d\Jk* — fl1, w — dyjfl1 — k1,
U£ = dy/fc2 — fl1.	(4.56)
The normalized propagation constant is w/ V, and the index 1 marks the parameters
of the isotropic dielectric.
Division of the Modes, Dispersion Curves
It was shown in [9] that (4.55) can be divided into two parts: the one corresponding
to symmetric, or even, TE and TM inodes, and the other corresponding to antisym-
metric, or odd, ТЕ and TM modes of the nonchiral slab counterpart. “Symmetric”
modes are given by solutions to
2 Ju, u.nj sin(u, /2) sin(u_/2) | u>2n2(l — к2) cos(ut /2) cos(u_/2)]
—w(n2 + nj)(u_(l «,) cos(u+/2) sin(u_/2)
f ut(l — Kr)sin(u|/2)coe(u_/2)] = О	(C57)
and “antisymmetric" ones come from
2 Ju ,u nJ cos(u,/2) cos(u_/2) | w2n2(l — Kj)sin(u+/2)sin(u„/2)j
+w(n2 + nj)|u_(l 4 ar)sin(u+/2)cos(u_/2)
4-u+(l — Kr)cos(u+/2)sin(u_/2)] = 0.	(4.58)
The formula for the cut-off frequencies can be obtained by setting w — 0 in
the eigenvalue equation (4.55), or in (4.57) and (4.58). The cut-off values are
Vr»n
ci = ™
N
n2 — nJ
п2(1Тк,)2~ nJ’
ni = 0,1,2,....
(4.59)
The cut offs for the modes (4.57) are given by even values of m, and odd indices
m correspond to the modes (4.58). These normalized cut-off values are exactly the
same as given in (8|, where different constitutive relations for the chiral material
135
Figure 4.7 Dispersion curves for a chiral slab waveguide. The relative indices are n = 2.1
and nj = 2. The dashed lines stand for the corresponding nonchira] slab. The relative
chirality parameter = 0.01.
were used, and the cut-off values were derived for the angular frequencies, and not
for the normalized frequencies as here.
Figures 4.7 to 4.9 show how the chirality effects on the dispersion characteristics
of an open chiral slab.
As can be seen, the cut-off numbers are removed, except for the fundamental
mode, and the original ТЕ and TM curves being almost degenerate due to the
very small index contrast are now shifted from each other. The hybrid modes are
classified according to the cut-off index so that the fundamental bifurcated mode is
marked by /7°, the next two by and so on. The superscript refers to the index m
and the .1 sign to the ;|- sign in the cut-off number formula, respectively. In [8], these
modes were termed, accordingly, Rm and Ln modes. With a high enough degree of
chirality, such as кг = 0.5 in the given example (Figure 4.9), single-mode regime is
realized, since one of the bifurcated modes in each pair becomes evanescent. For
small dielectric contrast between the core and cladding materials, already very small
chirality dramatically affects the dispersion curves.
In [8] and [9] more dispersion curves for open chiral slabs are shown at various
chirality parameter values. The field pattern structure was analyzed in more detail
in (9|.
136
Chapter 4. Waveguides
Figure 4.8 Dispersion curves for a chiral slab waveguide. The relative indices are n = 2.1
and nj = 2. The dashed lines stand for the corresponding nonchiral slab. The relative
chirality parameter кг = 0.042.
Figure 4.9 Dispersion curves for a chiral slab waveguide. The relative indices arc n ~ 2.1
and nj — 2. The dashed lines stand for the corresponding nonchiral slab. The relative
chirality parameter kt = 0.05.
4.3. Circular Waveguide
137
Chiral Slab on a Conducting Surface
Next we obtain the eigenvalue equation of the guided waves in a chiral slab which
is placed on an ideally conducting surface. The propagating modes in the guide
will refer to antisymmetric modes in a corresponding nonchiral dielectric slab. The
eigenvalue equation can be found from (4.28) by setting — Z™ = 0 and
renaming to Zlt which gives us
(„ТМ J 7™ „TM\(„TE , 7TETE\
(°ii + zi “m )(au + “u )
4. (nME 4 7™ nME\( „EM 4 7TE„EM\ - П
+ Wil + Z1 °n Aan + zi °11 ) = f-
(4.60)
The eigenvalue equation for a chiral slab of thickness d/2 on an ideally conduct-
ing surface reads now, in terms of the normalized optical parameters
2 [n1 — nJ 4- (n2 4- nJ) cos — cos —]
I 1 '	17	2	2 J
1 — кг u. . u+sin yt 1 кг u_ u_ sin
FTZTe,n	nn 7 V
+wn' (l+«r)cos
u_ sin
---H1 -kJcos 2
u+ sin y-
nJ cos'‘;'t<_ . u_	ns“-u,	u.
-4------	--l — sin — 4- ------’‘-—-sin--
1 -к, 2	2	1 + s, 2	2
= 0.
(4.61)
w
To see that this equation refers to the antisymmetric modes in a nonchiral di-
electric slab, we set кг = 0 in (4.61). Then we are left with two dispersion equations
corresponding to antisymmetric TM or ТЕ modes, respectively, in a dielectric slab
guide of n with identical upper and lower half spaces of the refractive index n, [29]
titan = w-j (TM mode),	(4.62)
ucot = — w (ТЕ mode).	(4.63)
Examples of calculated dispersion curves of a chiral slab on an ideally conducting
tnetal surface can be seen in (7, 11].
4.3 Circular Waveguide
Here we apply the general theory of BI waveguides (Section 4.1) to waveguides with
circular cross-section. This case is probably most important for applications For
bi-isotropic fdling, where the circularly polarized waves represent eigensolutions in
free space, the circular geometry of waveguide elements appears most promising
for engineering practice. In the following, we discuss closed and open waveguides,
uniformly filled with general bi-isotropic media.
4.3.1 Isotropic Boundary Impedance
Consider a closed circular bi-isotropic waveguide with impedance conditions (4.7)
held on the wall. In this section we assume that the impedance dyadic is isotropic:
Z, = Z,(urtir 4- u,u,) and ZT = Z, = Z, in (4.8).
Figure 4.10 Geometry of circular waveguide. Coordinate system.
For a circular waveguide of the radius a, we have in cylindrical coordinates (p, ф),
see Figure 4.10,
did d d	,
o" = —Й7,	7Г=л~>	4M)
От а дф	On dp
and particular solutions of the Helmholtz equations (4.6) read
£’J± = AtJ^kctP)e^,	(4.65)
where J„ stands for the Hessel function of the first kind and order i>.
Dispersion Equation
The dispersion equation can be obtained by substitution (4.65) in the boundary
conditions (4.13), (4.14) and equating the determinant to zero. This leads to lire
relation
473. (Circular Waveguide
139
Л(*:с+а) + je,aZ,-^-Jp(fcc+a) + je>*Z,£±-J^(kc+a)
a*c+	*c+
• |^р~Л(М:-а) - fc~Jv(fcc-a) + je~J"Z,J^(kc^a)
- Mkc-a) - je-^Z.^JJk'.a) t- je~’*Z^f^a)
Clk*_	kc_
 bpTJ,-(k+a) + ^_JL(l'c+a) - Уе,*2,Л.(ке+а)1 = 0,
(4.66)
where Jv is the derivative of the Bessel function. Introducing the normalized
wavenumbers kc± = A:c±a> fc± = k±a, /3 = fla and adopting the following no-
tation for the normalized function5
=$,(&.*) =	(4.67)
*С±«Л/(Кс±)
the eigenvalue equation can be written as
(1 + j^Z.^- + j^Z,ktSvA (^--k_Sv_+ je *z.}
-II- je~iazj£- + je->*£,£_&-) f	+ k+S„+ - je^Z.} = 0	(4.68)
\	’‘'C —	/ \ I	/
or
+& Li I z,2) Ц- - 4-V zjeosoz.
Г	\fc2- fc2J к k2_	л2+ л
- (! + Z2,)(k_S^ + k+S^) + 2jcosi?z.(l - k+S^k_S„_) = 0.	(4.69)
The term proportional to ftv is typical for eigenvalue equations of circular chiral
waveguides. When changing the propagation direction, we have the same propaga-
tion factors and field patterns but for the opposite sign of r*.
БТЬе impedance is normalised ns Zt =
140
Chapter 4. Waveguides
Field Putterns
The field patterns can be derived from (4.11), (4.12) and (2.9), (2.10):
E. = A\Jv(kc+p) +
Er = А р-Л(*е+р) + ap“J-
^»(^e p)j
k*
	+	тт-ХХЬг+р) - “r-Xtf'-P) Ke+	Kc+	► e’^*,
£„ =		-Jz4 J J-j'(fce+p) + a^-J^(fcc_p)j	
	iz h- p	«7р(^с+р) aЛ/(^с-Р)|	> e-Jru^
Я.		A1- [e>U( W - ae-^J„(k^ 'I 1	p)] в--’1'*,
Ur	—	.1 (Р” Ге5* j	л e-’* 4n“ IT к’ л{ '+p) “ a~P~ 4 I P [лс+	Jv^e— P)j
+	,’zi 1	6	,'zi fc+ 4(k+p) + « k JAk'-p		|е-л#
(4.70)
(4.71)
(4.72)
(4.73)
(4.74)
1
Л ( Г ,	e~’e ,	]
B„=	—JJk'+p) ~ “7—Jv(Vp)
1 ( IAi	k'~ J
I - [С^Л(^+р) + a^^JJVp)] ] e"**,	(4.75)
P *e+	"e+	J J
where
Л(*е+ a) -	a)
Л(к£_а) + ;е-^г,^Л(^_а) - je-^Z,^j;(fcc_a)
and A is the amplitude coefficient.
As a test, we let к —» 0 and % —♦ 0 whence the chirality and nonreciprocity
disappear and kc_ = fcc+ = kc as well as k_ = F+ = k. The dispersion equation
simplifies now to the classical form of the circular waveguide with isotropic surface
impedance [30]
4.3. Circular Waveguide
141
' ^ЙЛ,(М) ~ Z,fce J,.( fcf<l)j
/Й1/	\2
+ ~J„(kca) = 0.	(4.77)
\ a	)
If the boundary is ideally conducting, that is, Z, — 0, we have the eigenvalue
equation for a general bi-isotropic circular waveguide in the form
«j	—й)
fce_
— Л(А-е. а)	—/„(*,., а) | - 4 -7,’(А>, <i)l = 0
1«е+П	">+	J
or, using the normalized parameters:
(4.78)
& I 4- -	+ £+SM+) = 0.	(4.79)
\ А? к. f '	'
As was already mentioned in Section 4.1. for an ideally conducting boundary
the eigenvalue equation has the same form as for nonreciprocal chiral filling, since
the parameter i? cancels out. Of course, the wavenumbers and depend
on both coupling parameters к and x, but the Tellegen parameter x modifies the
refraction index only. This means that the bi-isotropic circular waveguide with
ideally conducting wall is reciprocal in terms of the propagation factor and acts as
a chiral one.
Calculated Examples
In Figure 4.11 calculated dispersion curves for circular bi-isotropic waveguide with
ideally conducting boundary (Zr =	= 0 in Equation (4.78)) are depicted for the
normalized chiralit,' parameter Kr — к/n = 0.35.
Modes with v = 0, ±1,±2 are included. Because the wall is ideally conducting,
Figure 4.11 represents the dispersion curves for all possible real I? values. The effect
of nonreciprocity can be seen from renormalizing the frequency. More precisely, one
can get the solution for a nonreciprocal waveguide by changing na — + na cos t?, if
the corresponding solution for the reciprocal guide is known. In Figure 4.11, this
corresponds to the change in the scale of the z-axis. Consequently, with an increase
of the nonreciprocity parameter x/n — sin if, the propagation factor decreases,
broadening the single mode frequency band. This feature is the same as in plane HI
waveguides, see the previous Section. Also, the modes with | r |> 0 are bifurcated,
as in planar waveguides.
Figure 4.11 Dispersion curves for bi-isotropic circular waveguide with ideally conducting
boundary. The normalized chirality parameter Kr = 0.35. Solid curves are for the modes
v - 1 1, dashed for v — 1 2 and dashed dotted for n = 0. For curves with the same cut-off
values, the upper refers to the positive v, and the lower refers to negative v values.
Figures 4.12 and 4.13 present calculated complex propagation constants for a cir-
cular bi-isotropic waveguide with lossy isotropic boundary impedance as functions of
the normalized chirality parameter nr for different nonreciprocity parameter values
[31]. ЛИ these curves are calculated with the values ka = 3.5 and Z, = 10"4-(l 4 j).
Examination of Figures 4.12 and 4.13 shows that the real part of the propagation
constant is increasing with increasing к for modes with zero and positive v values,
while for negative о it has a flat minimum. The value of кг of the minimum point
varies between 0.2 •• •0.4, depending on the mode number i? and the value of x
parameter. Modes with v — 0 and 1 have quite low attenuation, which is still
decreasing with increasing a, while the attenuation for the inode with v - -1
is increasing for small к and again decreasing for higher degree of chirality. The
attenuation for modes v = i2 is rather high al small к but it is decreasing rapidly
at higher re values.
For ideally conducting circular waveguide, the attenuation naturally equals zero,
and the real parts of the propagation constants change only very slightly as func-
tions of the surface impedance. The change in the real parts is approximately
8  10 6 •  -6.2 • 10 4, when the impedance varies from zero to Z, = 10 4 • (1 4 })
Numerical calculations [31] show that the real parts of the propagation factors
143
Figure 4.12 Real and imaginary parts of propagation constant in circular BI waveguides
as functions of at different Tellegen parameter values. Solid curves correspond to
cost? = 1, dashed curves refer to cost? = 0.9 and dash-dotted for cosi? = 0.8. The two
lowest modes у = ±1 are shown. Curves for the mode v = 1 are indicated by crosses, and
curves for the mode v — - 1 are indicated by circles.
decrease and the attenuation increases with increasing % values. Л very sharp
increase of attenuation occurs near the points where ka cos 1? = k*,n. Here k*,n =
are the corresponding transverse propagation factors for the nonchiral
and reciprocal circular waveguide with the ideally conducting boundary (i.c., the
solutions of the equations	= 0 or	= 0).
4.3.2 Anisotropic Boundary Impedance
Inserting the longitudinal wavefield solutions (4.65) into the anisotropic bound-
ary conditions (4.13), (4.14), performing the differentiations, and writing out the
determinant leads to the following eigenvalue equation for a bi-isotropic circular
waveguide with an anisotropic surface impedance (4.8):
1-14
Chapter 4. Waveguides
Figure 4.13 The same as in Figure 4.12 for the inodes v — 0, ±2. Curves for the mode
i- = 2 are marked with crosses, for i> = -2 with circles, and for v = 0 with asterisks.
k(*e+a) + je^Zr-^-J^a) +	—-ЛК: «)|
L	akcV	J
- [•№-<) - je^Zr-^J^a) + je-» ZT^-ji(kc_a)
 |^2-Jl'(fcr+a) +	«) - je,tfZtJv(fc€+a)j = 0
or, using the normalized parameters, in more compact form:
(4.80)
4.3. Circular Waveguide
145

COS 1?
О.
(/ 1	1 \	/ L С
(1 I Z,ZT) — + 2j cos0ZT -4-it
\fce_ K*.J	\ k*
— (1 + Z,2T)(k_S„_ + kt S„i ) -I- 2j cos ^(Z, — ZTkj k_ S„_) = 0.
(4.81)
To check the result, we can substitute = kci — ke end к_ — k^ = к for the
nonchiral guide. The dispersion equation simplifies now to the known equation of
the circular waveguide with anisotropic surface impedance [30]:
(0v)’Z 1 + j(l + Z.ZT)kSv 4 (Z. - Z^Sl) = 0.	(4.82)
As special cases, the above equations cover bi-isotropic circular guides having
an ideally conducting surface with transverse corrugations, which corresponds to
Z, — 0 in (4.80) and (4.81), or with longitudinal corrugations when Z, — 0 .
4.3.3 Open Circular Bi-isotropic Waveguide
Particular solutions of the Helmholtz equation for the longitudinal waveficlds inside
a bi-isotropic waveguide (4.15) read, as in (4.65),
= Л±Л(кс±Р)е->-'*
(4.83)
and for the surrounding isotropic dielectric solutions to (4.16) can be written as
E% = D±K^qp)e^.
(4.84)
Here K„ is the modified Bessel function of the order v and kel = jq = ./kJ — /32 is
the transverse wavenumber in the surrounding dielectric (kt —	).
Substitution of the solutions (4.83) and (4.84) in the boundary conditions (4.18)
to (4.21) leads to the system of equations for the constants Af and Pj.:
f /or	/о-	—go
Vi^fot -qie-’af0- -ggo
Eo r	Fo-	— Go+
< Vie,fE0i -t)le~}eF„^ -i)Go+
-go
ggo
-Go_
gGo-
(4.85)
\V- /
where
/о± = Л(^屫)>	Foi = -“у~Л(&е*«) i
kri
9о = Л'^(да),	Go± = ^—-K„(qa) _fc — K'(ga).
n4	4
The eigenvalue equation is then obtained by enforcing the determinant of the matrix
(4.85) equal to zero, which gives in the normalized form:
- fc	- ktf I	(i^Sv+ + L5„_) = 0.	(4.86)
ZJ/T/l COS V'	'
Here, q and k^ stand for the normalized wavenumbers in dielectric (q = qaf kx = k^a)
and the normalized function has been introduced as
. __K(9).
(4.87)
The normalized transverse wavenumbers satisfy two additional equations
£’± I 9’ = *1 -	(4.88)
so that one of the three transverse wave numbers kctl kc. and q can be eliminated
and we actually are left with the system of two equations ((4.86) and one of the
(4.88)), just like in the case of a nonchiral guide.
The eigenvalue equation for the reciprocal chiral waveguide can be obtained
from (4.86) by setting sin d = 0 and cos 1? = 1. This special case of the chiral
open reciprocal waveguide was studied in [5], but the eigenvalue equation in [5] was
obtained only in a form of a determinant of a 4 x 4 matrix.
As is seen from (4.86), the bi-isotropic circular waveguide possesses the same
propagation factors for both the propagation directions. The situation is exactly the
same as for the chiral reciprocal guide: when one changes the sign of the propagation
factor p and the sign of 1/, the equation is preserved. The modified reciprocity
theorem (see, for example, [32)) slates that the reaction of source a caused by source
i in a bianisotropic medium is equal to the reaction of source b caused by source
<1 in the complementary medium. For a bi-isotropic medium the complementary
medium differs from the original one by the sign of the nonreciprocity parameter
X". As is seen from (2.36) and (2.37), the equation (4.86) does not depend on the
sign of the non reci procity parameter sini? — x/n, and the propagation factors are
not changing when one transforms the original medium into the complementary
medium.
Го check the result we can let £ = £ — 0, whence the Equation (4.86) simplifies:
+ X)
\? *’/ V?1 k*J
— (fiS^ —	— t„) = 0,
(4.89)
where e = e/f|, fi = /«//<>. Equation (4.89) coinrides with the well known dispersion
relation for the isotropic dielectric circular waveguide (see, for example, [28]).
4.4 Rectangular Waveguide
The genera] procedure, introduced in Section 4.1 and based on the decomposition of
the wavefields into the longitudinal and transverse components, can he in principle
applied to bi-isotropic waveguides of any cross-section shape. However, to satisfy
appropriate boundary conditions for the rectangular cross section, the solutions of
the Helmholtz equations for the longitudinal wavefield components (4.6) have to be
built as infinite Fourier series [33] of the corresponding trigonometric or exponential
eigenfunctions.
For simple dielectric filling, plane and rectangular waveguide solutions are very
simple, which is because the plane waves are eigensolutions in that media. As
a consequence, superpositions of just a few (two or four) plane waves satisfy the
boundary conditions in isotropic guides. In chiral and general bi isotropic rectan-
gular waveguides, there is no such combination of any finite number of circularly
polarized eigenwaves in the filling media. Since circular polarizations are the eigen-
polarizations in bi-isotropic media, it is easier to solve circular chiral waveguides,
compared to the rectangular guides.
There is an analogy between chiral waveguides and waveguides filled with mag-
netized ferrites in a sense that no exact solution exists for longitudinally magnetized
waveguides, although the problem can be solved in a closed form for transversally
magnetized waveguides. Involved numerical procedures are required to find the
dispersion relation for chiral rectangular waveguides [34] (see also [33], where a so-
lution was given for a special case of reciprocal chiral waveguide with square cross
section).
Here we give and discuss an approximate analytical solution for bi-isotropic
waveguide of low profile, Figure 4.14. The following theory [35] applies to the cases
when one of the transverse dimensions (the height h) may be assumed to be small,
so that the fields (more precisely, their z and у components) change little over the
height of the waveguide. The approach is an extension of the averaging method,
used for waveguides with anisotropic filling, such as ferrite guides [Зв].
For the analysis, we split the waveguide fields into two orthogonal components
E = E„n„ 1 Er, Il = f/„u„ 4 Hr,	(4.90)
148
Chapter 4. Waveguides
Figure 4.14 Cross section of the waveguide.
where the tangential parts lie in the xy plane. The method is based on averaging
the fields over the waveguide height:
„	1 rb	„	1 rb
E = 7 / Edz,	H = 7 / Hdz	(4.91)
о Jo	b Jo
and on the use of approximate relations between the averaged tangential components
and the values of the same components on the boundaries of the region of averaging
(marked by the superscripts + and respectively):
Ёт = -(Б; +E;), Ht = i(lV+H;).	(4.92)
2	2
(4.92) means that the distribution of the tangential components is assumed to be
locally quasi-static— the average field at a point (ж, у) is determined by the values
of the field on the boundaries at the same point (ж, у). We note that this assumption
is the physical basis for the conventional transmission-line equations.
Integrating the Maxwell equations and using (4.92), we have, for ideally con-
ducting walls,
Vr x u„E„ = -joi(/iH d («Л),
VT x H + | x H+ - u„ x H;) = ju>((unE„ + (II),
O'	'
(4.93)
(4.94)
where VT is the two-dimensional gradient operator in the ту plane. Separating
the normal and transverse components in (4.93) leads to the relations between the
averaged electric and magnetic fields:
Hr = — vr x u„E,
‘ (Vfl
(4.95)
nn
4.4. Rectangular Waveguide
149
Eliminating the normal field components from (4.94) we obtain the equations for
the transverse magnetic field:
VT X (VT x Hr) =	Hr,	(4.96)
un x H* - un x H; = ju>b(( - C)Hr.	(4.97)
The first relation (4.96) is the vector wave equation, whose solution determines
the propagation factors of the modes
/?’ = fcj(n* - x1 - к*) - (™) , m = 1,2,3,....	(4.98)
It is obvious that for the case under study the effect of the nonreciprocity and
the chirality parameters on the propagation factor is merely to change the effective
refractive index of the medium. The relation (4.97) determines the change in the
tangential component of the magnetic field over the height of the waveguide. The
change is proportional to the chirality parameter and to the average magnetic field:
H) — Hr = —2кк0Ьи„ x HT.
(4.99)
Finally, the field structure can be seen from the last equation and (4.95). Setting
E„ — cos
e~^
(4.100)
we find the other averaged components:
Wn =	C	/ттг \ 	cos 1	у ) fi	\ a /	e-^,	(4.101)
nv =	fl	(in* 	 COB I	у u)fi	\ a	)	(4.102)
/7, =	j , /Ш7Г ' 	sin I	у ыц \ a i	)	(4.103)
The field structure in the	bi-isotropic waveguide differs from		that in the
waveguide with simple dielectric filling by the fact that the normal component
of the magnetic field H„ is non-zero. For lossless Pasteur filling it is 90° out of
phase from the normal electric field component. The field pattern (4.100), (4.103)
can be termed "twisted,” since the tangential component of the magnetic field Hr
is non-uniform over the height and the difference (4.99) changes sign when the sign
of the average magnetic field II. changes.
More discussion on the averaging method, which can be also used in simuiat
ing thin bi-isotropic and bianisotropic layers by approximate impedance boundary
conditions, can be found in (25), (37) - |39).
iLCierences
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[2) Engheta, N., and P. Pelet, “Modes in chirowaveguides,” Optic» Letter», Vol. 14, 1989, No.
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Propagation, Vol. 38, 1990, No. 1, pp. 90-98.
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[6]	Svedin, J.A.M., “Finite-element analysis of chirowaveguides,” Electronics Leiters, Vol. 26,
1990, pp. 928-929.
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— application to a grounded chiroslabguide,” Electromagnetics, Vol. 11, 1991, No. 11, pp.
209-221.
(8]	Cory, II., and I. Rosenhouse, “Electromagnetic wave propagation along a chiral slab,” IEE
Proc., Part II, Vol. 138, 1991, No. 1, pp. 51-54.
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waveguide,” IEE Proc., Part II, Vol. 138, 1991, No, 4, pp. 327-334.
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No. 10, pp. 723-725.
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68-72.
[14]	Mahmoud, S.F., “Mode characteristics in chirowaveguides with constant impedance walls,”
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[15]	Mariotte, F., and N. Engheta, “Effect of chiral material loss on guided electromagnetic modes
in a parallel-plate chirowaveguide,” J. of Electromagnetic Waves and Applications, Vol. 7,
1993, No. 10, pp. 1307-1321.
[16]	Chien, M., Y. Kim, and II. Grebel, “Mode conversion in optically active and isotropic waveg-
uides,” Optics Letters, Vol. 14, 1989, No. 15, pp. 826-828.
[17]	Pelet, P., and N. Engheta, “Coupled-mode theory for chirowaveguides,” J. of Applied Physics,
Vol. 67, 1990, pp. 2742-2745.
[18]	Cory, II., and T. Tkinir, “Coupling processess in circular open chirowaveguides,” IEE Proc.,
Part II, Vol. 139, 1992, No. 2, pp. 165-170.
[19]	Cory, H., and S. Gov, "Mode energy transfer along a circular open chirowaveguide,” Mi-
crowave and Optical Technology Letter», Vol. 6, 1993, No. 9, pp. 536-641.
[20]	Engheta, N., and P. Pelet, “Orthogonality relations in chirowaveguide,” IEEE Trans. on
Microwave Theory and Techniques, Vol. 38, 1990, pp. 1631-1634.
[21]	Mahmoud, S.F., “On mode bifurcation in chirowaveguides with perfect electric walls,” J. of
Electromagnetic IVaues and Applications, Vol. 6, 1992, No. 10, pp. 1381-1392.
[22]	Toscano, A., and L. Vegni, “Effect of chirality admittance on the propagating mode* in a
parallel-plate waveguide partially filled with a chiral slab,” Microwave and Optical Technology
Leiters, Vol. 6, 1993, No. 14, pp. 806-809.
[23]	Engheta, N., D.L. Jaggard, and M. Kowars, “Electromagnetic waves in Faraday chiral me-
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[24]	Maiur, J., “Nonreciprocal phenomena in coupled guides filled with chiroferrite media,” J. of
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[26]	Oksanen, Ml, J. Hanninen, and S.A. Tretyakov, “Vector circuit method for calculating
reflection and transmission of electromagnetic waves in multilayer chiral structures,” IEE
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[27]	Tretyakov, S.A., and M.I. Oksanen, “Electromagnetic waves in layered general biisotropic
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1411.
[28]	Adams, M.J., An introduction to optical waveguides, New York, John Wiley к Sons, 1981.
[29]	Marcuse, D., Light transmission optics, New York, Van Nostrand Reinhold, 1982.
[30]	Elsherbeni, A.Z., J. Stanicr, and M. Hamid, “Eigenvalues of propagating waves in a circular
waveguide with an impedance wall,” IEE Proc., Part II, Vol. 135, 1988, No. 1, pp. 23-26.
[31]	Koivisto, P.K., S.A. Tretyakov, and M.I. Oksanen, “Waveguides filled with general biisotropic
media,” Radio Science, Vol. 28, 1993, No. 5, pp. 675-686.
[32]	Kong, J.A., Electromagnetic wave theory, New York, John Wiley к Sons, 1986, pp. 402-405.
[33]	Cory, H., “Wave propagation along a closed rectangular chirowaveguide,” Microwave and
Optical Technology Letters, Vol. 6, 1993, No. 14, pp. 797-800.
[34]	Pelet, P., N. Engheta, “Modal analysis for rectangular chirowaveguides with metallic walls
using the finite-difference method," J. Electromagnetic Waves Applic., Vol. 9, 1992, pp.
1277-1285.
[35]	Tretyakov, S.A., “Electromagnetic waves in a low rectangular waveguide filled with a bi-
isotropic (nonreciprocal chiral) medium ” Soviet J. of Communication Technology and Elec-
tronics, Vol. 37, 1992, No. 6, pp. 25-29.
[36]	Kontorovich, M.I., and A.S. Cherepanov, “An averaging method for analysing the processes
occuring in a waveguide with a gyromagnetic filling,” Soviet J. of Communication Technology
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[37]	Tretyakov, S.A., A.S. Cherepanov, and M.I. Oksanen, “Averaging method for analysing
waveguides with anisotropic filling,” Radio Science, Vol. 26, 1991, No. 2, pp. 523-528.
[38]	Tretyakov, S.A., M.I. Oksanen, and A.S. Cherepanov, “New ferrite-filled waveguiding struc-
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applications,” Microwave and Optical Technology Letters, Vol. 6, 1993, No. 2, pp. 112-115.

Chapter 5
Propagation in Inhomogeneous
Media
Wave propagation in inhomogeneous bi-isotropic media is the topic of this chapter.
ЛИ the medium parameters may be functions of the position, they are no longer
constant as in previous chapters. This property leads to more complicated equations
in analyzing plane wave propagation and gives us new effects and the possibility
for many new applications. The equations in an inhomogeneous medium can be
written for wavefields by substituting E = E4 4 E_ and H = j(E4/q4 — E_/q_)
into the Maxwell equations, which leads to the coupled equations
e’*’	e’e
V x E+— fc4E4 =-------Vln«/q4"xE4----------VlnJ^xE., (5.1)
cos v	cos v
ei»
V x E_ + fc_E_ =------Vln v/qT x E_--------Vln ./5)7 x E+.	(5.2)
cos v	cos V
The analysis in this chapter is based on these two equations. Also plane wave prop-
agation is considered through so-called normalized wavefields, which means that the
wavefields introduced in the beginning are normalized with respect to the square
root of the wave impedances. In inhomogeneous media these normalized wavefields
are coupled. The normalization is seen to lead to simple geometrical optics equa-
tions for slowly varying media. Assuming small inhomogeneity it is seen that the
paths of the two geometrical optics eigenrays actually coincide and the main effect
is seen as a change in polarization of the geometrical optics field. For greater inho-
mogeneity, the geometrical optics eigenrays are separated and the total propagated
field is a combination of the two eigenfields propagating in inhomogeneous media.
_____________________—-------Uliapter 5- —Propagation in Inhomogeneous Media
5.1	Geometrical Optics for 131 IVIeclia
Let us consider the solution of Maxwell equations in an inhomogeneous bi-isotropic
medium in terms of the wavefield vectors. Assuming the medium to be slowly
varying, the normal Anru/z for the geometrical optics electric, field is
E±(r) = E^(r)	(5.3)
The vectors Е^(г) are assumed to be slowly varying and <^±(r) is the phase function
for each geometrical optics ray. Inserting this formula into the Maxwell equations
in inhomogeneous media and proceeding as in [1] the following geometrical optics
equation is obtained
HW±(r) x 7 T n±(r)7|. El е-Л'Мг) = 0.	(5.4)
These equations have a nonzero solution if the determinant of the square bracket is
zero, which leads to the characteristic equation, the eikonal equation
V<J±(r) - V^>£(r) = n^(r).	(5.5)
The solution for the phase factor is
•Ыг) = jn^ds.	(5.6)
о
It is seen that the two geometrical optics wavefields have different propagation
factors which coincide for к = 0.
Assuming small inhomogeneity the two geometrical optics eigenrays coincide.
By denoting the unit vector parallel to the direction of the propagation direction
of both eigenrays u =	the polarizations of the two wavefields are right
handed and left handed circularly polarized as seen from the equation
u x El = ±jEl-	(5.7)
In terms of the wave propagation direction u, the circular polarization of the wave
vector E° is right-hand and that of E°. left-hand. The phase velocities of the
two wave fields differ from each other, which makes the polarization of the elec-
tromagnetic field turn during the propagation. This is a Faraday-like rotation but
independent of the direction of the propagation (2|. The polarization change of the
field is seen by forming the expression for the electromagnetic field on the ray with
the path length parameter s
E(s) = E+ -I E.
5.2. Polarization-Rotating Lens Antennas
155
= E“J exp(-у a(r)ds) E" expfjfe,, у к(г)Йл) exp(—J n(r) cos i?(r)ds).
о	о	о
(5.8)
Along the ray the wavefield vectors can be written with the right hand and left-hand
circularly polarized unit vectors as
EX = E°u+ , El = E°u_	(5.9)
and the electromagnetic field is
E(s) = exp(—jko j n cos i?ds)R(—t) • E(0).	(5.10)
0
This is a local plane wave. The electric field vector at the point s is rotated by the
angle т with respect to the field vector at origin. The angle r is defined as
т = k„ У a(r)ds.	(511)
0
The chirality rotates the original linear polarization counterclockwise if к is positive
and clockwise if a is negative [2]. Note that the Tellegen parameter does not affect
the rotation.
5.2	Polarization-Rotating Lens Antennas
Let us consider wave propagation in an inhomogeneous lens medium, with all pa-
rameters functions of the radial position vector r. Since the Tellegen parameter has
no effect to the rotation of the propagating field we consider only the effects due
to the chirality parameter a(r). The chiral media can be applied to eliminate the
inherent crosspolarization in inhomogeneous lens antennas. The idea of the anal
ysis is to consider rays in a nonchiral dielectric lens and add chirality to adjust
the polarization at the aperture with the assumption that the chirality does not
affect the rays. This assumption is valid if the lens is large enough in terms of the
wavelength so that the chirality needed is very small. The polarization change due
to the chirality is an integrated effect along the ray path.
At the source point of an inhomogeneous lens the starting direction of each ray
is defined by the value of its tangent unit vector
u = up sin V’o + «. cos V'<
(5-12)
156
Chapter 5. Propagation in Inhomogeneous Media
Figure Б.1 Ray geometry in an inhomogeneous medium with a dipole source.
as seen in Figure 5.1, where the angle V1» is the elevation angle of the ray.
For a dipole source situated at origin with the moment p = puv the propagating
field at the beginning of the ray is proportional to the transverse component of the
vector —uv on the ray:
eo = -(I - uu)  Uv = -(nn + bb) • uv
= n cos sin b cos y>,	(5.13)
n and b are the normal and binormal (b = u X n) unit vectors to a ray, respectively
[2]. The path of the ray is uniquely determined by the starting direction and the
angle with respect to the direction ux. In the aperture all the rays are propagating
in z-direction and the field is now proportional to the vector (5.13) with n and b
replaced by —up and —uv, respectively, thus,
e = — Up cos sin — uv cos
= u, sin 2y>sin2 (V’o/Z) — uv(cos’+ sin’y>cos V’o).	(5-14)
This leads after trigonometrical considerations to the crosspolarization ratio X of
the field in the aperture which is the x-component of the field divided by the y-
component of the field:
sin 2y?
cos2y> + cot2 (^„/2)
(5.15)
5.2. Polarization-Rotating Lens Antennas
157
Figure 5.2 Field pattern in the aperture plane for a dipole source and the corrected field
pattern.
The crosspolarization for a dipole source in the aperture due to the curvature of
the path of the ray inside the lens is shown in Figure 5.2. This inherent cross-
polarization can be eliminated by using chiral lens material. Consider the polariza-
tion rotation angle
r = arctan X = arclan
sin 2y>
cos 2<p -J- cot’ (V’o/2)
(5.16)
defined in counterclockwise sense from the direction —uv [2]. By requiring the
polarization rotation caused by the chirality to be equal, but of opposite sign, as
the polarization rotation due to the curvature, the inherent crosspolarization can
be eliminated. Because it is the integral of the chirality parameter that affects the
polarization rotation, the chirality can be distributed in infinitely many ways along
the rays. In this presentation, the crosspolarization is eliminated through use of
an inhomogeneous chirality distribution, but the chirality parameter could also be
chosen constant on each ray leading to different distributions. Then, the angle of
the polarization depends on the length of the path of the ray inside the lens. In
the following subsections this theory is applied to some spherically and rotationally
symmetric lens antennas.
5.2.1 Maxwell Fish-Eye Lens
As a first example, let us consider the Maxwell fish eye lens. It is a half-spherical
lens which is characterized by the refractive index
"w=l^r	<*’>
and the raypaths inside this lens are segments of circular arcs [3]. The refraction
index parameter n„ is constant. The equation for the crosspolarization angle (5.16)
can be written in an integral form
x
/ax
ГГР-	<518>
0
By changing the integral variable to y> through
sin2y>
cos 2y> + cot1 (^/2)
(5.18) can be written
,.?****# (5.m|
J 2(1 — sin2 у>Б1п2
Figure 5.3 Hay geometry of the Maxwell fish-eye lens.
ГХИЮШвУ
159
For the angle ip one can give the geometrical interpretation as the central angle
of the circular ray as seen in Figure 5.3, where p = i}io corresponds to the starting
point and ip = 0 the ending point of the ray. This gives us the possibility of
interpreting the integral (5.20) as an integral along the ray if we define the path
length along the ray as
V’» - V’
« = “v—Г-
sin^o
(5.21)
with
(5.22)
a
d, = —— difi.
sin ip„
Substituting this in (5.11) and comparing the resulting integrand with that in the
expression (5.20) for r leads to an expression for the chirality parameter к required
for the polarization correction of the Maxwell lens:
1 sinV’oSin^ sin2y>
2k„a 1 - sin’ V’sin’yi
This can also be written as a function of the spherical coordinates (r, 6, by
applying the following relations from the geometry in Figure 5.3,
. cos ill	cos V’ — cos ip,
r sin 6 — a-—----a cot ip„ = a---;—------
siny>„	SHlV'o
(5.23)
(5-24)
sin^
r COS U = a—--— .
sin V’o
IrasinO
tanV’» =
From these formulas we finally obtain
2rasin6
sin V'o =	~ .........-	==
J(a2 — r2)2 + (2ra sin 0)
(5.25)
(5.26)
(5-27)
г2 sin 26
(5.28)
Sin^/l = - -__
^/(а2 — r2)2 + (2ra sin 6)3
Inserting these expressions into (5.23) we have the distribution of the chirality inside
the Maxwell lens for compensating the crosspolarization due to a dipole source
1	ar3 sin 0sin 26 sin 2y?
’ ’V koa (a2 — r2 cos 26)2 + (r2 cos y>sin 26)2
The distribution of the chirality is shown in Figures 5.4 and 5.5. Near the special
points (a,0, JLir/2) the chirality distribution л(6) acts like a delta function. This is
(5.29)
160
Chapter 5. Propagation in Inhomogeneous Media
because at these points in the aperture the crosspolarization angle is 90°, although
the field is zero. Practically it is not important to realize this singularity very
accurately, and in many cases the chirality function can be approximated through
a piecewise constant distribution.
0	30 0 (degrees) 60	90
Figure 5.4 Normalized chirality parameter of the Maxwell lens as a function of the
angular coordinate 0 for different radii r, on the constant = 45” plane.
0.40
0.30
J* 0.20
0.10
0.00
30 0 (degrees) 60	90
Figure 5.5 Normalized chirality parameter of the Maxwell lens as a function of the
angular coordinate в for different angle <p, with constant radius r = 0.8 a.
5.2. Polarization-Rotating Lens Antennas
161
5.2.2 Brown Lens
Figure Б.В Ray geometry of the Drown Jens.
The short focus horn antenna, or Brown lens, is a rotationally symmetric lens in
which the index of refraction obeys the law (3)
n(p) =-------"°	(5.30)
cosh(rrp/2/)
and which has a plane surface distance / from the focus as an exit aperture as seen
in the Figure 5.6. The ray equation of the lens is
sinh(rrp/2/) = tan V'esin(xz/2/), 0 <	(5.31)
The endpoint coordinates (p»,<p) of each ray in the aperture are determined by the
angle of incidence V'o and the coordinate ip in the xy plane. From the ray equation
(5.31) the endpoint coordinate p in the aperture is
Po = — In (tan V’o +	+ tan2 V'o )	(5.32)
Assuming the chirality distribution constant on each ray is, the polarization
rotation angle may be written as
•	a
r ~ ko J = koK У ds ~ кокя,	(5.33)
о	0
where л it the length of the path of the ray inside the lent defined by the angle of
incidence
л(^о) = / \A + (~r) dz = [ ' i ,	~	—•	(5.34)
/ V <fz /	_ sin’ ^,o Cos3 (rrz/2/)
After making the change of variable t = cos(xz/2/) (5.34) becomes
, - -- - v ~ ~ A'(sin’ V’»).	(5.35)
’ I У(1-/’)(!-sin1 ^f’)	’
where A'(m) is the complete elliptic integral of the first kind [4]. For normal inci-
dence, e.g., for V’o = 0, the length s = f and as the angle of incidence approaches
Tc/2, s —» oo. From (5.33) the chirality distribution for a dipole source may be
written in terms of the angle of incidence and the angle y> as
rr 1	Г sin 2^	1
«(V’o,V>) =	arctan I-------------5-7-—т-г|.	(5.36)
2k„f A (sin1 V'o) [cos2y>4 cot3 (^>O/2)J
The chirality function may also be written in terms of the cylindrical coordinates.
From the ray equation (5.31) we may introduce the parameter m,
1	1	 1 1	sinh1 (>rp/2/)
TFl(p. Zf — Sin Wo = -------Z-------r------------------
sinh1 (*p/2 f) T sin1 (irz/2/)
(5.37)
satisfying 0 < m < 1. By using this parameter m which depends only on the
coordinates p and z the angle of incidence is eliminated from (5.36). The required
chirality distribution is:
к(р’^)=^^“с1аП
(1 — v^l — tn) sin 2y?
1 + cos 2<p + \/\ — m(l — cos 2y>)
(5.38)
The crosspolarization in the aperture is zero on the diameters <p = 0, ?r/2, it, Зя/2
and of alternating sign in the adjacent sectors. Also the chirality distribution has
alternating sign in adjacent sectors. In Figure 5.7 the chirality distribution is shown
as a function of the coordinate z with different values of the coordinate p, the angle
ip = 60°. The maximum value of the chirality is found to be limited to \nkof | < ir/2
everywhere inside the lens.
<р = 60‘
Figure 5.7 The normalized chirality distribution of the Brown lens as a function of p
and z in the plane of constant <p.
5.2.3 Luneburg Lens
The geometrical optics method for the wave propagation in inhomogeneous chi-
ral media introduced in the previous section will now be applied to the Luneburg
lens, which is the most generally used inhomogeneous lens antenna. The cross-
polarization can be eliminated through use of an inhomogeneous chirality distribu-
tion.
The dielectric Luneburg lens of radius a is spherically symmetric and obeys the
refractive index law [3]:
(5.39)
The ray equation for the Luneburg lens is in terms of cylindrical coordinates (p, ip, z)
z2 -f- p2(l + 2cot’^„) — 2zpcot^„ = a2.	(5.40)
The rays in the Luneburg lens are sections of ellipses determined by the starting
direction of the ray at the angle {>„. The plane of the ray is determined by the
coordinate angle <p. The endpoint coordinate p = p„ of the ray on the aperture
plane is
p„ = a sin
(5.41)
164
Chapter 5. Propagation in Inhomogeneous Media
Figure Б.8 Ray geometry of the Luneburg lens.
It is also found that for the Luneburg lens the crosspolarization ratio X for a dipole
source can be written in the form (5.15), which is the same expression as for the
Maxwell lens [2],[5]. The required polarization rotation angle is r — arctanX.
Let us assume that the chirality parameter к is constant on each ray. Thus, the
polarization rotation angle is written
г = k„ j Kds = k„Ks,	(5.42)
о
where s is the arc length of the ellipse defined by i]>o. It can be shown that the ray
equation (5.40) defines an ellipse which is tilted by an angle V’o/2 and centered at
origin [3], [5]. The length of the main axes of the ellipse are
A — Viacos^-,	(5.43)
В = \/2asin	(5.44)
By using the coordinates (p, q) inside the lens, the arc length of the ellipse
5.2. Polarization-Rotating Lens Antennas
can be written with the parameter as
a coi Л®
(5.16)
After making the change of variables [6]
p — Asint, q = Bcost
(5-47)
(5.46) is of the form
/« _____________
s = 2A j \/l - msin2 t<H — 2A Л’(гг/4,m)	(5.48)
о
with
m = m(V>e) = AiB = 1 - tan2 у	(5.49)
and E(t,m) is the elliptic integral of the second kind. Some special values for m(V'<>)
are m(0) = 1, m(rr/4) = 2(y/2 — 1) and m(ir/2) = 0. (5.48) gives the length of
the path of the ray inside the lens and may be checked in two special cases. When
ij’Q = 0, which means normal incidence, the ray travels through the lens and the
length is s = 2a, and, when — тг/2, which means tangential incidence, the length
is s — rra/2.
From (5.42) the required chirality distribution function can be written as
sin 2tp
cos 2лр 4- cot2 (V’o/2)
= —=------arctan
2\J2k„a
«(V’o.V’) =
—-----------------------V-	(5.50)
cos& £(w/4,l-tan2^)
Written in terms of spherical coordinates the expression of the chirality becomes
quite complicated
к(г, 6, <p)
1	(1 — д/l ~ <?)sin2y>	1	- v/1 - <3
2fcoa аГС аП [(1 - v/1- <5)со8 2у>+ 1 4- Q VQ
The parameter Q is here defined as Q — sin2 i/>0 to obtain the expression for к.
There may be other choices for Q as well. The parameter Q is expressed with the
basic coordinates (г,в) as follows:
2r2 sin2 fl[r2 sin2 0 4 a3 4 r cos 2a2 — r2(l 4-sin2 0)]
Q(r, в) =	r4 — 2r’a2 cos 26 4 a*	'
Figure 5.fl Normalized chirality parameter of the Luneburg lens as a function of the
angular coordinate в for different radii r, on the constant ip plane.
The value of the elliptic integral of the second kind in (5.50) and (5.51) can easily
be calculated by using binomial expansion. Inside the integral (5.48) nisin’f < |
is valid during the integration. Thus,
<4____________
E(rr/4,m) = У yfl — msin2 tdt
о
» /	\	»/4	°°	/	\
= Ё (	T	/ «»’*** =	E	(	1L2	)(-l)*/z*mk,	(5.53)
k=o \	*	/	'	*=o	\	K	/
(1/2 \
I are binomial coefficients [4] and the coefficients can be obtained
from the recursive equation
'-V'-aG1)'
(5.54)
г.ж. x'oiarixai»on;Rotating Lens Antennas
167
with the initial value IB = rr/4.
Distribution of the chirality is shown in Figures 5.9 and 5.10 as a function of 0
and tf>. The distribution function is seen to be much like that for the Maxwell lens
[2].
0.100
0.075
•f 0.050
0.025
0.000
0 (degrees)
Figure 5.10 Normalized chirality parameter of the Luneburg lens as a function of the
angular coordinate G for different angle y?, with constant radius r.
5.2.4 Gutman Lens
The Gutman lens is a spherically symmetric lens with radius a and the focus inside
the lens. The focus length satisfies f < a for the Gutman lens, the special case
f = a gives the Luneburg lens. The rays inside the lens are sections of ellipses as
for the Luneburg lens, but in this case the situation is no longer symmetric and the
expressions for the chirality become more complicated. The refractive-index law for
the Gutman lens is
0 < r < a , 0 < f < a
(5.55)
and the ray equation [3]
z’+p’
/’ + a’ cos2 V’»
a2 sin2 J’o
2-zpcot^, =
(5.56)
168
Chapter 6. Propagation in Inhomogeneous Media
The ellipse is determined by two parameters: the angle of incidence and the
focus length f. The endpoint coordinates of the ray on the aperture plane
and the cross-polarization ratio X for a dipole source are found to be the same as
for the Luneburg lens.
Let us again assume constant chirality distribution on each ray. The polarization
rotation angle may be written as
(5.57)
where a is the arc length of the ellipse defined by and f. The ray equation for
the Gutman lens defines an ellipse shown in Figure 5.11, which is tilted by an angle
7 and centered at origin [3], [7]. The angle 7 and the axes Л and В of the ellipse
obey the expressions:
j Kda = k„M.
0
7 =
- arctan
2
a2 sin 2^’о
a2cos2^0 -f J*
(5.58)
Л = -/=^/a1 + /’ -f- \/(а2 T /’)5 — 4a2/2 sin2 V’o,	(5.59)
В = аг + /’ — y/(a2 + /2)2 — Aa,f1 sin2 V’o-	(5.60)
As a check note that for f — a the above expressions reduce to those of the Luneburg
lens (5.43) - (5.44).
Б.2. PolnrizationBotating Lens Antennas
169
The arc length of the ellipse in coordinates (p, q)
2L + ±.
Л2 B1
(5.61)
inside the lens can be written as
(5.62)
which is, after making the change of variables (5.47), of the form
a
0
msin’ tdt
0 ____________ '
+ У — msin’tdt
0
= A (E(a,m) -f E(/?,m)]. (5.63)
The eccentricity parameter is defined as
, В1 a1 + /’ - \/(a2 Ь /2)2 - 4а1/2 sin’ V'o ,	.
m = 1 - — = 1 - ------------ 	. - -  -	(5.64)
a’ + /’ + ^(a2 I- /2)2 - 4a2/2 sin’ i]io
and the integration limits are
, . / cob 7 4
a = arcsin (———),	(5.65)
/3^arcsin(aC°S(^2j).	(5.66)
The chirality required for compensation of the crosspolarizalion in the aperture
is now of the form
. ,	, т 1 f sin2u>
KlV’o.V’) =	= т-arctan I--------------rr-—ft
[cos2y> + cot2(V’„/2)
1
Л(Е(о,т) + Е(Д,т))'
(5.67)
Values for the elliptic integral of the second kind E(-,m) can easily be calculated
by using the binomial expansion similarly as in the Luneburg lens analysis.
The chirality function can also be written in terms of spherical coordinates. From
(5.56) the path of the ray is expressed with the spherical coordinates by choosing
the parameter Q = sin2^'o which depends only on the coordinates (r, 6) and the
focus length f. By using the parameter Q, the angle V’o may be eliminated to obtain
the chirality distribution for the Gutman lens in the form
к(г,в,<р) = —arctan
*^о
(1 -	- <?)sin2y>
(1 - т/1 — Q) cos 2y> + 1 + \/i — Q
1
Л[£(а,т) + £(0,т)]’
(5.68)
The parameter Q expressed with the basic, coordinates (r, 6) is
)	(r2 cos20 —/2)2 T (r2 sin20)2
[’’ +	- (//a)2(r2 cos20 -/2)
t 2r^//’ cos’ 0(1 | (f/а)1) — r2cos2 0(//a)2(l I (//a)2 sin2 0)1 .
(5.69)
The largest value of the chirality is obtained at tangential incidence in the case
= тг/2. Because the maximum value for the required rotation of polarization to
eliminate the crosspolarization at the aperture is ±x/2 it is seen that the maximum
value of к for the Gutman lens is limited to
I =
max .	,
ko f + a
(5.70)
Distribution of the chirality is shown in Figures 5.12 and 5.13 as a function of
i/i„ and <p with two different values of focus / = 0.8a and / = 0.4a. The shape of
the distribution functions is seen to be similar to those of the Luneburg lens.
0 10 20 30 40 50 60 70 80 90
(degrees)
Figure 5.12 Normalized chirality parameter of the Gutman lens as a function of the
angle of the plane of the ray, with different values of the ray parameter i/’o and the
focus /.
.. >.1>approximation for Normal Incidence
Figure 5.13 Normalized chirality parameter of the Gutman lens as a function of the ray
angle V’o! with different values of the angle and the focus /.
It is seen that for large lenses with koa >> 1, the required chirality is very
small since the values for the chirality are inversely proportional to koa, which is
in accordance with the asumption of the geometrical optics. In these examples of
inhomogeneous lens antennas we have used the dipole source and evaluated the
chirality distribution to correct the crosspolarization. For other types of sources
this kind of procedure can also be used. The chirality functions to eliminate the
crosspolarization obviously differ from those given in this section.
5.3 WKB Approximation for Normal Incidence
The WKB method is applied to wave propagation in stratified bi-isotropic media, or
nonreciprocal chiral media, for waves with normal incidence. The method is based
on the wavefield expansion. The propagating fields are derived with the WKB
method. For sufficiently slowly varying media, the reflected fields are obtained as
a first-order correction to the WKB solution. For normal incidence, the reflection
effect is independent of the chirality parameter к but depends on the Tellegen
parameter x as was seen in Chapter 3.
Instead of the wavefield vectors E±, Jet us consider the solution of the Maxwell
172
Chapter 5. Propagation in Inhomogeneous Media
equations in terms of the normalized wavcfield vectors defined by
E± .
y/Vt
(5-71)
Conversely, the electric and magnetic fields can be obtained from the normalized
wavefields as
E = ^F+ + v^F_,	(5.72)
F F
H = j(-±z + --^).	(5.73)
Substituted in the Maxwell equations leads us to the following equations for the
field vectors Fy.:
V x F± T A:±F± —------—-Vln	x FT ± j tan 19 V In x F±.
(5-74)
This is a pair of coupled equations where the coupling is proportional to the deriva-
tive of the functions In In Pasteur medium with i? = 0, the coupled equations
reduce to the form
V x F± T fc±F± = — Vln y/ij x FT,
(5.75)
with k± = k„(n zt «) = k„n±. Here we again note that for constant q there is no
coupling between the two normalized wavefields. This means that the wavenumbers
k± can be functions of position without any coupling effect between the wavefields.
5.3.1 The Coupling Equation
The WKB method 1 is suitable for obtaining the approximate solution for the wave
equation in inhomogeneous media [8] - [11]. Also, the reflection of the electro-
magnetic waves from inhomogeneous layered media can be done by using a matrix
Riccati equation [12]. It is assumed that all medium parameters e, /i, x and к
of the bi-isotropic media are real (lossless medium) and can be dependent on the
coordinate z , whence,
fc±(z) =	[n(z) cos d(z) ± k(z)| ,	q±(z) = rj(z)eri^‘\ (5.76)
In this case waves are propagating along the z direction. The equation (5.74) can
then be written in the form
lAIso called the WKBJ method; the name is derived from the initials of the authors Wentrel,
Kramers, Brillouin and Jeffreys.
5.3. VVKB Approximation for Normal Incidence
173
Figure 5.14 Reflection from inhomogeneous bi-isotropic half space.
VF±± k±(z)J - j tan d ~ (in y/qZ(z)) h -F± =----------------V’iZF (,п\А»=г(2)}
dz	dz \	*	/	cosv(z) dz X v /
(5.77)
with J = u, x I. The normalized wavefields F^ are transversal to the z direction,
which is easily seen by decomposing the fields F± in (5.74) into two components:
transversal and parallel to u,.
Since the dyadics 7( and J commute, the following rule of differentiation is valid:
A AA = (7, J + 7^) • eT”*J* = e1'^  (?Л -| A.
dz \	) dz dz	dz dz
Now it can be verified that the equations (5.77) are equivalent to the following
equations:

---Цу- (1п\/Ы2))
cos d dz \	’	/
е±(7.-»г+Jfi)
(5.79)
with the phase functions defined as
7±(2) = -j /
J	2n
d'
т 7y)rf2 = ±in
cos d(z)
COSI^Zp)
J7(z)>
(5.80)
(5.81)
(5.82)
and
4>AZ) = f k±(z')dz‘.
The reference point zo depends on the boundary conditions. Because the dyadic
exponential function is
e±b-r±
= cosJ(z) CT»(.)7
\ cosd(zo)
(5.83)
Equation (5.79) can be written in compact form as
^-G±(z) = -l±(z).GT(*)
az
(5.84)
with
G±(z) =
COSt,(*) етъ(.) e±J#r(.)
\ cosr?(z„)
F±,
(5.85)
j4±(z) =-------flny/^fz)^ eT^*> expl±2Jfco [ ncosddz'l. (5.86
cosv az \ *	)	J
(5.84) is the coupling equation equation which forms the basis for the followirq
analysis. It is seen that the dyadic ЛДг) does not depend on the chirality parametei
к. Since the quantity £ (in y/qi) is assumed to be small, the norm of the dyadic
A±(z) is small. For a step discontinuity, in the parameter values at the interface
problem of two homogeneous media the quantity £ (in^/ijj) is zero everywhere
but at the interface. The propagating fields can be calculated in the two regions
from the boundary conditions for the electric and magnetic fields at the interface.
The WKB expression for the reflection dyadic is obtained also in this case because
the reflection coefficients are functions of the integral of £ (in which is a
well behaved function.
Zeroth-Order Solution
As a zerolh-order approximation the dyadic A±(z) can then be replaced by zero.
This means that G^ is a pair of constant transversal vectors C±, which leads to
the solution for the normalized wavefields
(5.87)
F°(z) =	. C. .
^cos^(z)
The constant transversal vectors C± are expanded with right-hand and left-hand
circularly polarized unit vectors u+ and u_ in Appendix B. The u4 (u_) corresponds
to a right-hand (left-hand) circularly polarized unit vector when looking in the
positive z direction. This is reversed when looking in the negative z direction. By
using the relation (13]
= еЪ#± u+u + u u+i	(5,88)
the F“j field vectors can be written in the form
F±(*) = -7——7	+ e±J*±U-l-	<5-89)
у/совф)
The constants CJ. and Cj. depend on the boundary conditions of the problem. The
zeroth-order expressions for the propagating electric and magnetic fields can be
written as
E°(z) =	e->*+u+ + O' e*u.|
V cos v v
+сЯГт) [(7 e>#-u4 + c'_ e->*-u_l) ,	(5.90)
ir(z) =	(еЛ?+^[С; -I c1, e>*‘ll_]
V’/CO8V '
_e-X?-r) [Cl ?*-u+ + C'_ e^-11-]) .	(5.91)
It is seen that the total fields consist of four circularly polarized components, two
propagating in the positive z direction and two in the negative z direction.
Let us consider the electric and magnetic fields propagating in the positive z
direction and assume linear polarization at the point z — zo, which implies to
C; = Cl_ = E°. Since
= k„ j n(z') cos d(z')dz' ±r(z)
the angle r is defined as
Я
r(z) = k„ j K.(z')dz'.
(5.92)
(5.93)
176
Chapter 5. Propagation in Inhomogeneous Media
The zeroth-order fields at the point z are
E°(z) = £”(z) pos(r + -)u. - sin(r + -)uvj
Ft?
He(z) = n _ 7) u« + co® (r “ 9) uv
*7(z/ I z	z
where
E‘,(z) = —j== exp(—jk„ I ncosiJdz').
»/cosd(z)
(5.94)
(5.95)
(5.96)
For Pasteur media, 15 = 0, the electric and the magnetic fields are seen to rotate in
the same direction, counterclockwise if к > 0 and clockwise if к < 0 with respect to
the direction of propagation in analogy with waves propagating in a homogeneous
medium as was seen in Chapter 2. This rotation of the plane of the polarization
is frequency dependent. We have omitted the term J tan (in ^/fj^ dz' which is
assumed to be very small. This small term behaves in the same wny as the chirality
parameter, making a small contribution to the effect caused by the chirality. On
the other hand, if the Tellegen medium is considered, then r = 0, and the Tellegen
parameter is seen to change the angle between the electric field and the magnetic
field during propagation. The angle between the electric field and the magnetic field
is rr/2 + d(z) at the point z > 0.
5.3.2 Reflection Dyadic
The zeroth-order solution does not give the reflection from the stratified medium
because the two fields do not couple in this approximation. Since the zeroth-order
solution for the field Gj(z) is the constant transversal vector G^ = C±, from (5.84)
the first-order correction for the fields is obtained in integral form
	G* (z) = -1 A±(z')dz'  G®	(5.97)
with zi dependent on the boundary conditions. Now, the corrections for the nor-
malized wavefields are
J cosi? eXP z	ПМ = ±2 У[j tan t? — (In уЛ/) It — k„n cos d Jjdz" dz'  F" (z). (5.98)
5.3. WKB Approximation for Normal Incidence
177
The correction for the electric field ie
E* =
and knowing that the propagating electric field is
E° =	+ ^f°_,
(5.99)
(5.100)
we can calculate the relation between the corrected field and the propagating field
in the form
Е‘ = ЛЕ°.	(5.101)
The expression for fi is the reflection dyadic for the electric field in WKB method
written in complete form as
Л(х) =
—j'2 [ |k„n cos 19 + tan 19 — (In \A/)]dz"
J	dz
dz' ti_n4
cosi9
,n cos 19 — tan 19— (In y/y)]d.
dz
dz' U+U_.
(5.102)
The reflection dyadics arc off diagonal in the basis of right-hand and left hand
circularly polarized unit vectors, which means that there are no crosspolarizcd terms
in reflection for normal wave incidence.
Similarly, the reflection dyadic for the magnetic field is obtained. The expression
for the reflection dyadic for the magnetic field is very much similar to that of the
electric field. The reflection dyadics for the electric field and for the magnetic field
differ from each other by sign and the phase factor exp(ji9).
5.3.3 Special Cases
To check the result, as a first example, the reciprocal special case i9 = 0 is considered.
This gives the reflection dyadic for the interface of an isotropic stratified medium.
Then i/jt = r), and the reflection dyadic is
kon(z")dz"]dz' lt
(5.103)
which is the same result as the WKB expression for the reflection dyadic for the
Pasteur medium because for normal incidence the coupling dyadic Л±(х) is inde-
pendent of the chirality parameter k(z) (13).
As a second example, the case where rf = 0 but r?(z)	0 is considered. This
gives the reflection effect due to the Tellegen parameter only. The reflection dyadic
(5.102) reduces to the form
=. , t d I, I rr,
R(z) = ~JTz Wtan(- + -)
exp[—j2k„ j ncosddz"]dz''J. (5.104)
The reflection dyadic rotates the incident field by the angle тг/2 | t?. For a step
discontinuity of the Tellegen parameter from d = 0 to i?(z) = at z = 0, the
following expression for the reflection dyadic is obtained:
R(z) =	ln^tan(^ +J) j
(5.105)
In this case the reflection dyadic rotates the incident field by 90°. This WKB
expression for a step discontinuity of the Tellegen parameter can be compared to
the exact result given in [14] :
Л _
f?(0) = — tan J.
(5.106)
In Figure 5.15 the exact and WKB solutions for the reflection dyadic are presented
as a function of The results are shown to coincide quite well for values up to
Л, = 30°.
Figure 5.15 Comparison of the exact and WKB solutions for a step discontinuity in the
Tellegen angle i?o.
As a third example, the expression of the reflection dyadic for a linear distribu-
tion of d(z) is calculated. The Tellegen angle is defined by
10	z < 0
az	0 < z < Zp	(5.107)
azj = i?i z > zt
This corresponds to a sinusoidal variation of the Tellegen parameter \(z) in the
interval 0 < z < zP The medium parameters c and p may have a step discontinuity
at z = 0 . After some algebra, the reflection dyadic (5.102) reduces to the form
Re(R)
Figure 5.16 The value of the rotating part in the reflection dyadic in the case of linear
stratification of the Tellegen angle d.
The value of the last term of (5.108) is presented in complex plane as a function
of in Figure 5.16; the angle ilj is assumed to be less than rr/2. The spiral curve
in complex plane is modified by the real parameters a and i?p These parameters
may obviously be chosen so that the reflected field vanishes for either right-hand
circularly polarized or left-hand circularly polarized incident field. The condition
for the parameters a and d) in this case is then
2 J cos d	у 7/i
(5.109)
180
Chapter 5. Propagation in Inhomogeneous Media
where the + sign refers to the right-hand circularly polarized and — sign to the left-
hand circularly polarized field. This kind of effect of separating the right-hand and
left-hand circularly polarized waves may have use in some applications for antenna
engineering.
5.4 WKB Approximation for Oblique Incidence
Let us now consider the more general case of obliquely incident plane wave in an
inhomogeneous layered structure with the same method as in the previous section.
Again, it is assumed that all medium parameters c, /t, x and к of the bi-isotropic
media are real and are dependent on the coordinate z. For sufficiently slowly varying
media, the reflected fields and the corrections to the propagating fields are obtained
as a first-order correction to the WKB solution. It will turn out that, while for
normal incidence, the reflection effect is independent of the chirality parameter к
but depends on the Tellegen parameter y, for oblique incidence all four medium
parameters have an effect on the reflection dyadic.
Figure 5.17 Reflection from the stratified bi-isotropic half space with oblique incidence.
For obliquely incident waves, the field is no longer a function of just the coordi-
nate z but also of the transverse position vector p. In the oblique case the transverse
field components F±(z) are not sufficient in the analysis of the problem. Instead
let us consider the fields [Г,±(г)и, + F±(z)] exp(—j’K  p) and try to find the solu-
tion of the coupled wave equation. Vector К can be interpreted as the transverse
component of the wave vector of a local plane wave. The field component FI±(z) is
5.4. WKB Approximation for Oblique Incidence
181
parallel and F.j.(z) perpendicular to z-direction. Substituting this expression to the
equation of the coupled wave equation (5.74) the following equation is obtained:
V dz	tai”? (ln J ~ 7K X Jtj  [F,±u, -I Fj
Writing the equation (5.110) in parts parallel and perpendicular to u, we obtain an
equation for the parallel component
(u, x K) • F±(z)
M*)
(5.111)
and an equation for the perpendicular component
F±(z)
S 0" /??(*))
~~ co7d----------F*(4
Here i*±(z) is a diagonal two-dimensional dyadic defined as
75 ГИ - M*) KK . M*) (“» x K)(u« x K)
(5.112)
(5.113)
and fc,±(z) = yfc±(x) — №, where К is the absolute value of the vector К defined
as К = у/К • К. As is seen, the parallel field component can be obtained through
the transverse field component. In the following only the transverse part of the field
is considered. Because the equation (5.112) is very similar to that in the normal
incidence case (5.77), let us try to find a solution to this equation in the form
Fi(z) = _l_/j±(2) • e^W+^rOl. G±(z),
vm*)
(5.114)
with the phase function 7±(*) defined as in (5.80) and
Mz) = /\±(*')rfz'-
(5.115)
With these definitions the equation (5.112) can be written in a. corn pact way~
182
Chapter 6. Propagation in Inhomogeneous Media
4<М*) = -4±(z) • GT(z) - Bt(z)  G±(z)
az
with
(5.116)
4±(z) =
“ V m 'L eTz*»(.)
COB 1?(z)
e±J#±(’).£);t,(z) 2)T(z)-e±J*Tl*), (5.117)
B±(z) = 2^
(5.118)
. KK
№

and the transverse fields are
F±(z) =
cosj^zJе±ь(х) £>±(z)
\ COS1?(z) y/k^z)
(5.119)
етУФН«) . G±(z).
Since the scalar quantities (in and £ (in ^/^±/fc±) inside the dyadics
A±(z) and Bf(r), respectively, are assumed to be small, the norms of these two
dyadics are small.
5.4.1 WKD Approximation for Wave Propagation
For slowly varying media, as a zeroth-order approximation the dyadics A±(z) and
B±(z) can be replaced by zero. This means that G± is a pair of constant transversal
vectors Cjt, which leads to the zeroth-order solution for the normalized wavefields
F^z) = -==.l=. =:	. C±.	(5.120)
y/cosi?(z) yjk^z)
As in the normal incidence case, the constant transverse vectors C± can be expanded
in terms of right hand and left-hand circularly polarized unit vectors which are
defined in this case as
К u, X К
~K ~J К
К ,u,xK
К + }~К~
Ну using the relation [15] (see also Appendix В and C)
(5.121)
(5.122)
e±J#± = СЪФ± U(,,	„ U)>
and denoting the elliptically polarized eigenvectors of the dyadic • J by
5.4. WKU Approximation for Oblique Incidence
183
e±(2) =-7ГТтП±(г)и±>
Vfc±(2)
(5.123)
the field vectors can be written in the form
Р±Ут(«)
П(2) = ' 7—+ C\ e^(*)e±(z)].	(5.124)
«/cos v(z)
The constants Cj and Cj. depend on the boundary conditions of the problem.
The transverse field is a combination of two elliptically polarized waves which arc
propagating in both positive and negative z-direction; the 4- wave is right-hand
elliptically polarized and the — wave is left-hand elliptically polarized with respect
to the propagation direction. Equation (5.124) is the zeroth-order result from which
the propagating fields arc obtained.
The expressions for the propagating transverse electric and magnetic fields can
be written by using the normalized wavefields as
E‘”(r) =	p	+ C'+
V COS v
4	((7 ?*-e_ 4 C'_ e-J*-e_]) ,	(5.125)
|c: ew-e. + Ct e_rf-e_]) .	(S.126)
The parallel components of the fields E°(z) and ff“(z) are obtained straightfor-
wardly from the transverse fields by using (5.111). It is seen that the total fields
consist of four elliptically polarized components, two propagating in the positive z
direction and two in the negative z direction. Also from (5.125) and (5.126) we see
that
E1(P, *) = Tjtfz)^') Щ(р, z)	(5.127)
is valid. This shows us that the wavefield relations (2.17) and (2.18) also hold
approximately in an inhomogeneous medium.
Considering the polarization properties during propagation in bi-isotropic
medium, let us assume, for example, linear polarization at the interface. Then
the constants CJ* arc related to each other, in particular = Cl_ for TM, and
CJ. = —C\_ for ТЕ polarization. For Pasteur medium, d = 0, the electric and the
magnetic fields are seen to rotate in the same direction, counterclockwise if к > 0
184
Chapter 5. Propagation in Inhomogeneous Media
and clockwise if « < 0 with respect to the direction of propagation. The Tellegen
parameter is seen to change the angle between the electric field and the magnetic
field during propagation.
5.4.2 Reflected Fields and Corrected Propagating Fields
The zeroth-order solution does not give the reflection from the stratified medium
because the two fields do not couple in this approximation. The reflected field
and the correction for the transmitted field are now obtained as the first-order
correction to the WKB solution. Generally, the higher-order corrections can also
be used, which leads to an iterative procedure based on the equation (5.116) for
calculating the corrections to the reflected and the transmitted fields [13]. Even
for simple stratification the higher-order terms become very complicated and we
are restricted here only to the first iteration step. Since the zeroth-order solution
for the field Gj,(z) is the constant transversal vector G± = Cj., from (5.116) the
first-order correction for the fields is obtained in integral form
G±(2) = ~ /X=(2')<*2' • CT - I Ti^z'jdz'  C± (5.128)
with dependent on the boundary conditions. The corrections for the normalized
wavefields can be written in terms of the basis vectors е± and the reciprocal basis
vectors Ry corresponding to e±, introduced as
s±(z) = yjk±(z) D±\z)  u±,	(5.129)
and satisfying the orthogonality conditions
е±-в^ = 1,	e±s±=0.	(5.130)
Let us assume that we have electric and magnetic fields propagating in the pos-
itive z-direction. The transverse fields are obtained from the zeroth-order solution.
These fields arc expressed as a combination of right-hand (-1 wave) and left-hand
(— wave) elliptically polarized fields
	E‘" = Ej"e“’*+(,)c+ +	(5.131)
and H*’ = Я£е-'*+<*’е+ + where the coefficients arc		(5.132)
	e->K P, H%(z) =	E± (z)- costf(z)	J q(z)	1	(5.133)
5.4. WKB Approximation for Oblique Incidence
185
From the coupled equation for the first-order correction (5.128), using the normal-
ized wavefields, we finally obtain after some algebra the first-order corrections for
the transverse electric and magnetic fields. These can be written in dyadic matrix
form
(5.134)
(5.135)
(5.136)
(5.137)
(5.138)
(5.139)
These two matrix equations show us how the corrected 4- and — fields are related to
the zeroth-order field components. The total correction for the transverse electric
field is E" 4- E'1 and, for the transverse magnetic field, И" -| II'1 , from which
the reflected field and the correction to the transmitted field can be identified.
The reflected field is propagating in the negative z-direction, -f wave right-hand
elliptically polarized and — wave left-hand elliptically polarized with respect to the
propagation direction:
Eref = (eWa(z)E\° + b(z)E!°) е*~(,)е_ 4- (c(z)E\a + e-^^dfzjE1’) e’*’(,,e+
(5.140)
and
Href = -^j^a(z)H‘° + b(z)Hu) e*We_- (c(z)H{° 4 eJ<’<*>d(2)/7,°) e>*+t*’e+1
(5.141)
where the reflection coefficients are
“ 186
Chapter 5. Propagation in Inhomogeneous Media
a(z) = I—------------exp(~j f[kt+ + &*- — 2tant9 — (inv/5)]tfz")(e+ 8-)dz\
J cos v	J	dz
i	*
w ~ 7 j! (in/r^) exp(~j2 / k‘ dz"'>dz'<
4 . i I.--------\	•'
c(z) ~ / У Un МТУ ) exP(~.?2 / kt^dz")dz',
J dz у у K.|_ J	J
‘r T (in л/у) /	d
d(z) = I—-------------- exp(—j f[k.+ + *,_ + 2tani5 — (In ^))</г")(е_  s+)dz'.
J cos v	J	dz
For normal incidence the functions b(z) and c(z) vanish, and the results for reflected
fields reduce to those in the previous section. Also the contribution of the chirality
parameter k(z) vanishes in normal incidence (16].
The corrections to the transmitted fields can be identified similarly. The first-
order correction to the total electric field can be written as
(ГТ	\	IZ
C(z) + ~ P(z) E‘;e-^l*)(e+ - — u,|
C4.	/	y2«+
C	\
B(z)+-±A(z)
C—	/
к
---r=-u«i
(5.142)
similarly the correction to the total magnetic field is obtained from (5.135) or by
using the duality condition for the correction for the electric field
(л* \	к
C(z) + ~ D(z)j	__Uj]
(c \
B(z) + -±A(z)}
-----7=--u« h
v/2fc_ *’
(5.143)

where the coefficients
A(z) = / —------- exp (~j f (fe,+ - fcx_ - 2 tan t?—- (In v/^))dz")(e+ • s'_)dz',
J COS V	J	dz
u(z} = /zz(!n^)dz'’
5.4. WKB Approximation for Oblique Incidence
187
c(z| =
;(in./nz) <	d
£>(z) — I —------.—- exp (j I [k,y —	- 2 tan д-j- (In v/q)]dz")(e_ • s* )dz‘.
J cos v	J	dz
For normal incidence all the functions A(z), E(z), C(z) and D(z) vanish, which
means that the first-order corrections to the WKB solution do not give any contri-
bution in the propagating direction.
5.4.3 Special Cases
In the following the reflection from the stratified bi-isotropic half space with r(z),
/t(z), k(z) and x(z) *Б considered in some special cases in detail. The plane wave is
coining obliquely from the free space to the interface at z = 0. In the isotropic free
space (z < 0) where the reflected fields are considered, the vectors e± are constant
right-hand (-4) and left-hand ( —) elliptically polarized vectors and the reciprocal
vectors satisfy the condition s‘, = sT. The orthogonality conditions reduce to the
form
c± • sT = 1, e±  sj. = 0.	(5.144)
The transverse reflected electric field at the interface z — 0 is
Ercf(0) = («(zJEf | HzOtf’le, I HzOE'” I-	(5.145)
since the incoming wave is
E‘"(0) = Е(°е¥ + E%_,	(5.146)
where E‘° = s_  E'“ and E‘° = S+ • E‘“. After eliminating the coefficients E± and
calculating the parallel components of the electric field the WKB expression for the
reflection dyadic for the total electric can be written in the form E’(0) = jR(0)-E<’(0)
in terms of the right-hand and left-hand elliptically polarized basis vectors. The
reflection dyadic for the magnetic field differs in this case only by sign according to
(5.141):
ft(0) — a e+s_ + fce+Sr 4 ce.s. -1 de_s+ — ~(a 4 b 4- c |- djti.u,. (5.147)
In the basis of unit vectors (K/A\(ux X K)/A',uI), the reflection dyadic can be
written in the form
188
Chapter 5. Propagation in Inhomogeneous Media
==, 4	1,	, „КК j ,	,	,4(u, xK)K
+1^(O -11. - 4	K-> +1(« d)
- ifa + b + c + dju.u,.	(5.148)
The reflection coefficients contain integral functions of the medium parameters.
As a first example, the special case «(z) —» 0 and il(z) —> 0 is considered.
This limit case gives the reflection dyadic for the interface of an isotropic stratified
medium. Then k,±(z) —> fc,(z) and ^(z) —> ’;(z), and we have the coefficients
a = d
(5.149)
exp (—j2 У k,(z")dz"]dz'.
о
(5.150)
The reflection dyadic reduces to
= 1 Tz (,n\	ex₽H2 / k,(z")dz"]dz' I— - u,uj
OX»''/	0	L	*
0	\	’	*' ’	)	0
The scalar functions in this reflection dyadic are the well-known WKB expression
for the reflection coefficients for TM and ТЕ waves in stratified medium [11].
As a second example lei us consider the reflection from the stratified Pasteur
medium. The Tellegen parameter x(z) ~* 0» but the chirality parameter «(z)	0,
which means that i?±(z) -» r/(z), and fc,±(z) = fe„y[n(z) ± a(z)]’ - sin2 0. The
reflection coefficients reduce to

5.4. WKB Approximation for Oblique Incidence
189
exp(— j2 J kx±(z")dz")dz\
о
0	' ’ ,+v '	0
Figure 5.18 Exact result and WKB approximation for the reflection coefficient b for a
step discontinuity in the medium parameters of a Pasteur medium, P = 30° and n =2.24.
For normal incidence the coefficients b — c = 0, and, a = d is independent of the
chirality parameter. For oblique incidence the chirality parameter also has an effect
to reflection. In Figure 5.18 the reflection coefficient b is compared to the exact
result for a step discontinuity in the chirality parameter к available in the literature
[15].
As a third example, Tellegen medium is considered. Then k(z) = 0, but х(г) /
0, and
k,±(z) —> k,(z) — k^n^z) cotd i?(z) - sin’ 0.	(5.152)
In this case the splitting into two separate waves vanishes, and we have the reflection
coefficients
190
Chapter 5. Propagation in Inhomogeneous Media
/ '*	~ cxp(~J2 jlW) - tan Ф") 4- (in \fil(z^}]dz")dz',
J cosvlz )	J	az \	’	/
о	'	о
(5.153)
b = c
exp] j2 У fcI(z")rfz"|dz/,
о
(5.154)
7 (in \/'/(z'))	‘r	d /	/----\
d = I —--------------cxp(-J2 f[k,(z") + tani?(z")y- (in ^(z"))]dz")dz'.
J cos viz'I	J	dz \	'	/
0	'	'	0
(5.155)
As a special case where drj/dz = 0, but dy/dz / 0, the effect of the Tellegen
parameter x(-i) is clearly seen. Then
exp[j2 У k,(z")dz"]dz'.
о
(5.156)
The reflection dyadic for this special case is
7?(0) /
J dz
о
/	.d(z') n.
In \ tan (----- |—1
V ‘ 2	4*
exp[—j’2 у k,(z")dz"\dz'
о
[ K(u, x K)
[cosfl——---
1 (u, x K)K
cos в К2
7 d [	kt(z')\	7	,	,, [KK (u, x K)(ux x K)
Vsh гм	кг-*-=—
0	\	'	' /	0
(5.157)
The first part of the reflection dyadic gives us the rotation of the polarization in
reflection, the effect known to the reflection from a nonrcciprocal chiral half space
for normal incidence caused by the у factor (14]. In this case the crosspolarization
is dominant. For normal incidence the copolarizcd terms vanish and the crosspolar-
ized terms arc left, which are responsible the effect of rotation of polarization. In
addition to the rotation effect in reflection there is also a phase shift caused by the
stratification. This gives us in the general case elliptically polarized reflected field.
6.4. WKU Approximation for Oblique Incidence
191
In Figures 5.19 and 5.20 the values of the integral functions in (5.157) are compared
to the exact results for a step discontinuity in the Tellegcn parameter x given in [17),
[18]. The WKB method is seen to fail near the turning point, but gives reasonably
good results elsewhere.
Re(Rco)
Im(Rco)
8 (degrees)
8 (degrees)
Figure Б.19 Exact result and WKB approximation for the copolarized reflection coeffi-
cient for a step discontinuity in the Tellegen angle, 6 = 30° and n =2.24.
Figure Б.20 Exact result and WKB approximation for the crosspolarizcd reflection co-
efficient for a step discontinuity in the Tellegen angle, 0 = 30° and n = 2.24.
192
References
References
[1]	Kong, J.A., Electromagnetic Wave Theory, New York, Wiley, 1986.
[2]	Lindell, I.V., A.H. Sihvola, A.J. Viitanen, and S.A. Tretyakov, “Geometrical optics in in-
homogeneous chiral media with application to polarisation correction in inhomogeneous mi-
crowave lens antennas,” J. of Electromagnetic Waves and Applications, Vol. 4, No. 6, 1990,
pp. 633-648.
[3]	Cornblect, S., Microwave Optics, London, Academic Press, 1976.
[4]	Abramowits, M.( and LA. Stegun, Handbook of Mathematical Functions, New York, Dover,
I960, pp. 14-15.
[5]	Viitanen, A.J., I.V. Lindell, and A.H. Sihvola, “Polarisation correction of Luneburg lens
with chiral medium,” Microwave and Optical Technology Letters, Vol.' 3, No. 2, Feb. 1990,
pp. 62-66.
[6]	Arfken, G,, Mathematical Methods for Physicists, Orlando, Florida, Academic Press Inc.,
1985, pp. 321-325.
[7]	Viitanen, A.J., “Polarisation correction of Gutman lens with chiral medium,” Microwave
and Optical Technology Letters, Vol. 3, No. 4, April 1990, pp. 136-140.
[8]	Budden, K.G., The Propagation of Radio Waves, Cambridge, England, Cambridge Univ.
Press, 1985, pp. 165-174.
[9]	Bremmer, H., “The W.K.B. approximation as the first term of a geometric-optical series,”
The Theory of Electromagnetic Waves, Editor M. Kline, New York, Dover, 1965, pp. 105-115.
[10]	Bahar, E., “Generalised WKB method with applications to problems of propagation in
nonhomogcncous media," J. of Math. Phys., Vol. 8, No. 9, Sept. 1967, pp. 1735-1746.
[11]	Wait, J.R., Electromagnetic Waves in Stratified Media, London, Pergamon Press, 1962.
[12]	Jaggard, D.L., X. Sun, and J.C. Liu, “On the Chiral Riccati Equation,” Microwave and
Optical Technology Leiters, Vol. 5, No. 3, March 1992, pp. 107-112.
[13]	Lindell, I.V., and A.H. Sihvola, “Generalised WKB approximation for stratified isotropic
chiral structures,” J. of Electromagnetic Waves and Applications, Vol. 5, No. 8, 1991, pp.
857-872.
[14]	Lindell, I.V., and A.J. Viitanen, “Duality transformations for general bi-isotropic (non-
reciprocal chiral) media,” IEEE Trans, on Antennas and Propagation, Vol. 40, No. 1, Jan.
1992, pp. 91-95.
[15]	Viitanen, A.J., I.V. Lindell, and A.H. Sihvola, “Generalised WKB Approximation for Strat-
ified Isotropic Chiral Media with Obliquely Incident Plane Wave,” J. of Electromagnetic
Waves and Applications, Vol. 5, No. 10, 1991, pp. 1105-1121.
[16]	Viitanen, A.J., “Generalised WKB Approximation for Stratified Bi-isotropic Media,” J. of
Electromagnetic Waves and Applications, Vol..6, No. I, 1992, pp. 71-83.
[17]	Lindell, I.V., A.H. Sihvola, and A.J. Viitanen, “Explicit expression for Brewster angles of
isotropic-bi-isotropic interface,” Electronics Letters, Vol. 27, No. 23, 7 Nov. 1991, pp. 2163-
2165.
[18]	Lindell, I.V., A.H. Sihvola, and A.J. Viitanen, “Plane-wave Reflection from a Bi-isotropic
(Nonreciproca! Chiral) Interface,” Microwave and Optical Technology Ijctters, Vol. 5, No. 2,
Feb. 1992, pp. 79-81.
Chapter 6
Scattering and Mixing Theories
This chapter focuses on bi-isotropic inhomogeneities that are more random in char
actcr than the deterministic layered structures discussed in the earlier chapters.
Small-scale inhomogeneity is a natural property of chiral and bi isotropic materials,
due to the origin of the magnctoelectric coupling which is induced by geometry and
asymmetry. Macroscopic modeling of bi-isotropic media requires the analysis of the
response of bi-isotropic samples in an electromagnetic field. The response can be
described with polarizability coefficients of material objects. Once the polarizability
behavior of small inclusions is known, the tools arc there to design materials with
desired magnctoelectric material parameters.
The chapter starts with the analysis of the polarizability of bi-isotropic spheres
and ellipsoids. The results arc used to derive Maxwell-Garnett mixing rules that
predict the effective magnetoelcctric parameters of a mixture, given its composition.
In addition, the chapter discusses the physical interpretation of the polarizability
and macroscopic parameter expressions, and it is shown that these results display-
profound symmetries and dualities in the electric and magnetic responses of the ma-
terial. Furthermore, dispersion effects in mixtures arc discussed as well as analogies
of bi-isotropic materials with anisotropic media, like ferrites.
The scattering behavior is strongly dependent on the size of the obstacles. Ob-
jects with dimensions that arc comparable to the wavelength require more com
plicated treatment than small inclusions. Towards the end of the chapter, these
problems are addressed. Finally, one of the basic problems in the study of chiral
media, the scattering from a helix, will be discussed.
6.1 Polarizabilities of Small BI Scatterers
No real media arc homogeneous. If one takes a look at a sample of any material
medium on the small scale level, the structure always displays irregularities. Л
193
I94	Chapter 6. Scattering and Miring Theories
classical approach in modeling is to treat the molecular structure by a collection of
polarizable entities, called inclusions, or more complicated clusters of inclusions in
the case of dense media. On the other hand, the essence of the macroscopic electro-
magnetic characterization of media are the four scalars c,/r,X>K in the constitutive
relations, which is in accord with the message of this book so far. ЛИ polariza-
tion phenomena are “hidden” beyond the values of these quantities. Therefore, one
needs a macroscopic electromagnetic description of the materia). In other words,
the average electric and magnetic polarizations that are induced in the medium by
the presence of an electromagnetic field have to be calculated. These polarizations
are the result of the dipole moment densities. Therefore the polarizabilities of the
inclusions composing the mixture have to be known. The electric, and magnetic
polarizabilities, on the other hand, depend on the properties of the inclusions: on
their volume, shape, and refractive index. These dependencies are governed by laws
that will be discussed next.
6.1.1 Bi-isotropic Sphere
Calculating the polarizability of a homogeneous bi-isotropic sphere (described by
parameters e,/r,K,x), requires solving the quasistatic problem of this sphere in vac-
uum with the presence of an electromagnetic field. Although the analysis in this
report focuses on time-dependent fields, static field solutions can be used in calcu-
lating the polarization densities for small particles. Л static field solution for the
electric field inside a spherical inclusion in a constant external field is also constant.
The quasistatic assumption means that the particle will be approximated electro-
magnetically by electric and magnetic dipoles; higher multipoles are ignored. The
quasistatic approach is equally valid for magnetoelectric problems as in purely di-
electric cases. This quasistatic assumption, which has sometimes been challenged
with the argument that in statics there is no chirality, has been thoroughly elabo-
rated in (1 ], where solutions for electric and magnetic potentials inside and outside
the sphere are solved. That analysis is not repeated here; rather, an analogy is
made to purely dielectric mixtures.
In the case of a plain dielectric sphere with permittivity c, the classical way
to enumerate its (electric) polarizability is closely connected to the electric field
inside the sphere E, [2] to |4|. If the sphere with radius a is located in free space
(permittivity t0), the relation of the interna! field E, to the external field E is
The internal field is smaller than the external one if € > co> which is normally
the case. Because the electric polarization density Pe inside the sphere is
6.1. Polarizabilities of Small BI Scatterers
195
P. = (« - €„)E„	(6.2)
which is a constant throughout the sphere, as also E, is constant, it follows that
the internal field relation can also be written as
e-=e-£- (6-3>
The electric dipole moment p, of the sphere, on the other hand, is the volume
integral of the dipole moment density: pe = VPe, and the polarizability a is the
relation between the outside field and the dipole moment, p, = <«E. This leaves us
with the classical result for the dielectric polarizability of a sphere
° = 3^S£’	(64)
where V = 4rra3/3 is the volume of the sphere.
As a straightforward generalization from this dielectric case, the bi-isotropic
sphere (with parameters e,p,X>K> 5ce Figure 6.1) can be solved. The incident
electric and magnetic fields E,H, create electric and magnetic polarization Pe,Pm
in the bi-isotropic sphere. Let the sphere be located in background medium which is
isotropic with parameters of free space (e„, po). The internal fields can be calculated
from the external fields and the polarization inside the sphere in the following way
(cf. Equation (6.3)):
E, \ ( E A	1 / PJeo \
и J \ н / з p„/p. ) 
(6-5)
Mo
Figure 6.1 Л bi-isotropic sphere exposed to a quasistatic electromagnetic field.
On the other hand, as was the case in Equation (6.2), the internal polarization
densities are related to the internal fields as
196
Chapter 6. Scattering and Mixing Theories
Pt \	/ t-«O (X - J*)y//V<> W EiI \
I’m /	\ (X + jK)#«« I1 ~ llo ) \ И. ) '
From these two coupled equations, the internal fields can be solved:
( E'1 - A (	£»(P 4 2p„)	-/»»(x ~ J«)y//‘»£o W E
\ H, / Д \ ~£o(x +.1«)у/Ро£о llo(e + 2fo) ) \ И
(6.6)
(6.7)
with
A = (p + 2p„)(s + 2e„) - (x1 4 «2)po£o-	(6.8)
Outside the scatterer, the polarization densities have an effect that is similar in
spatial dependence to the scattered fields of electric and magnetic dipole moments
p,,pm. These dipole moments are proportional to the incident field:
P.
Pm
«re
^em
mm
(6-9)
Note that the polarizability — which was a single scalar с» in the dielectric case
— has now expanded to a 2 X 2 matrix as the magnetoelectric coupling emerges.
In the polarizability symbols ay, there are two indices: the first (i) denotes the
polarization type, and the second (j) is for the origin of the polarization. The
dipole moments are integrals over the sphere volume V of the polarization, which
is constant:
(6.10)
The co- and crosspolarizabilities can be solved from the previous equations by
relating the dipole moments and the incident fields:
Q = Зе у (£ ~	1 2I'«) - (X7 I «2)/'o£Q (p + 2p»)(e + 2eo) - (x2 + x’)/ioro ’	(6.И)
_,	,,	з(х - ^em —	/	. о \/	n \	/ 9 ,	’ (p + 2p„)(e 4 2e„) - (x2 4 к2)рое„	(6.12)
о v	3(x + J^y/РЛ (p 4 2p„)(e + 2c.) - (x2 -I- «2)po£o ’ - -i у ~	। 2t„) - (x2 4 «2)pof<, (ц 4 2p„)(c 4 2c„) - (x2 4 x2)/t„e„ ’	(6.13) (6-14)
6.1. Polarizabilities of Small BI Scatterers
197
These polarizabilities are very essential in the modeling of bi-isotropic media, as
the mixing relations arc derived. For the noncliiral reciprocal limit к —» 0, у- > 0,
the polarizabilities simplify to the expression (6.4) and its counterpart in magneto-
statics:
с I 2c.	(6.15)
_ •> V I1 ~ I'o + 2//o	(6.16)
Ctme ~ ^rrn = 0.	(6.17)
Note here the decoupling of the electric and magnetic quantities, compared with
the bi-isotropic case.
6.1.2 Bi-isotropic Ellipsoid
In the case of ellipsoidal bi-isotropic scatterers, the internal fields are also constant
for uniform external fields. Therefore the previous analysis can be fairlv easily
extended from spherical to ellipsoidal geometry. The difference with respect to
spheres is that the polarizability matrix, relating exciting fields and dipole moments,
does not consist of four scalars <•»,, but rather of four dyadics S,,-. This is because
the internal field components of an ellipsoid are depolarized differently along the
three axes. The way this happens is determined by the depolarization dyadic which
contains the depolarization factors of the ellipsoid. These depend on the axis ratios:
If the semiaxes of the ellipsoid are o,b,c, the depolarization factor in the direction
of the a axis is [5] and [6]
°’ 2 i (’ 1 «’)/(•’ I «’)(•’ I b’)(.’ l r’)'
For depolarization factors Nt, and Nrj interchange b and a, and c and a in
Equation (6.18), respectively. The depolarization factors satisfy N„ I 4- Nc = 1
for any ellipsoid, and for a sphere, these are equal: Na = Nt, = Nc = 1/3. The other
two special cases are a disk (depolarization factors 1,0,0) and a needle (0,|,|).
Closed-form expressions can be written for ellipsoids of revolution [7]. Osborn and
Stoner have given tabulated values for the depolarization factors of general ellipsoids
(8| and [9].
The depolarization dyadic is
A = 7V„u„ua -I TVsUjiu 1 Ncueuc	(6.19)
_________________________ Chapter 6. Scattering and Mixing Theories
with ualiib)u<; being the unit vectors along the axes.
Using the same reasoning as for bi-isotropic spheres, the polarizability compo-
nents of the bi-isotropic ellipsoid can be calculated. The internal fields are constant,
and Equation (6*5) reads in the ellipsoidal case as
Therefore the dipole moments can be written as functions of the incident fields,
and the polarizability matrix components are now dyadics [lOj:
( P' ) = (	® ) .	(6.21)
\ Pm /	\ ^rnc &tnvn J у ff у
Written explicitely, the components of the polarizability dyadics
з
= J2«rMu»n«	(6.22)
•-1
are
c V f	1
~ {(c - £о)1^/1 1 (1 -	I «2)/«o«o} ,	(6.23)
amm,i =	{(/* - /‘«)[^е T (1 - W.)e„] - N,(x2 । «2)м«,‘о} ,	(6.24)
a.ro,; = ^(X - J>)^, «me, = ~\x +	(6-25)
with
A, =	1(1-	4- (1 - M)t»J - Л?(х’ ।	(6-26)
and V = АтгаЬс/З being the volume of the ellipsoid. These expressions reduce to
Equations (6.11) to (6.14) as the ellipsoids degenerate into spheres:	1/3.
6.1.3 Layered Chiral Sphere
If a polarizable inclusion has an inhomogeneous internal structure, the quasistatic
problem is more complicated, and the polarizability expressions become lengthier.
The problem of a partially inhomogeneous sphere, consisting of a spherical shell
and a spherical layer, both reciprocally chiral, has been solved [11]. Consider this
6.1. Polarizabilities of Small BI Scatterers
199
Figure 6.2 Structure of layered chiral sphere.
type of sphere — depicted in Figure 6.2 — with radius «ц located in isotropic free
space (parameters co,/«„). Let the three material parameters (relative permittivity,
relative permeability, and chirality) of the shell he	and those of the core
cr2l/trj>KJ- Let the radius of the core be a2 and let 6 = aj/aj. The polarizability
components of this sphere, defined by Equation (6.9) are given by

(6.27)
x {([(/M + 2)(G, - 1) - kJ] 6 F [(/.„ - l)(2crl + 1) - 2kJ])
x ( (/‘r2 4- 2pri)(cr2 T 2e,i) — (k2 4- 2ki)2] 6
t2 (pr2 - p,i)(e,2 - c,i) - (k2 - к,)2])
— 96 [2( eri/iri — kJ)2 -|- (eri/i,i — kJ)(/ip2 — 2er2) I- kJ — £.2/1.2]} ,
^em ^me
= --V^^{(k,6 6 2k,)	(6.28)
x ([(/'’J f 2prl)(er2 I 2e,i)	(k2 -| 2kj)2] 6
- [(/‘.2 - /'ri)(cP2 - tn) ~ (k2 - Kt)2])
4 96 [kJ(k, 4- k2) + e,iprI(K2 - kJ - к1(ег1/<г2 4 tr2/'rl )] } 1
(6.29)
X {([(p„ - l)(crl 4- 2) - kJ] 6 4- [(2р.. 4 l)(e., - 1) - 2kJ])
x ([(/i.2 4 2prl)(er2 4- 2eri) — (k2 4- 2k()2] 6
200
Chapter 6. Scattering and Mixing Theories
+2 [(Prj — /*ri)(«ri — «ri) — (k2 — Ki)2])
—96 [2(erl/rrl — kJ)2 + (c,ipri — kJ)(c,j ~ 2/<r2) + kJ — er2/(r2]} ,
where
& = ([(/*г1 d 2)(сн + 2) — kJJ 6 + 2	— l)(c,i - 1) - kJ])
x ([(/'ггЧ- 2//ri)(erj + 2erl) — (k2 4 2k2) j 8
+2 [(Pri — Дг1)(£г» — fri) “ (K2 ~ л1)2])
-186 [(e,,prl - kJ - er2)(eri/«,i - kJ - pr2) - kJ] .	(6.30)
This general result satisfies the check of reducing to all known simpler polariz-
abilities. The case of homogeneous chiral sphere can be achieved in four ways from
the layered case:
•	Ki —» 0,c,i —» 1,/Vi —» 1, whence the shell vanishes and only the chiral core
(parameters 2) exists
•	6 -» 1, in which case the layer is squeezed to zero and again the inner sphere
(parameters 2) only exists
•	8 —» oo, and the inner sphere vanishes, the sphere being of medium 1
•	Kt —» K2,crl —» cr2,/iri —* pr2, meaning that the shell and the core are of the
same chiral material (parameters 1=2)
It is straightforward, albeit tedious to show that indeed in all these four cases
the polarizability matrix reduces to the result of Section 6.1.1.
The other special case is the layered dielectric (or magnetic) sphere, whence the
chirality parameters vanish: K! = 0,k2 = 0. In this case the polarizability matrix
becomes diagonal: the magneloelectric terms (6.28) vanish, and the electric and
magnetic copolarizabilitics decouple, electric polarizability (6.27) depending only
on the permittivities, and the magnetic polarizability (6.29) depending only on the
permeabilities, yielding the correct result [12].
6.1.4 Chiral Sphere in Chiral Background Material
In the previous analysis, it was assumed that the background material in which the
inclusion was embedded was isotropic. If the background is also bi-isotropic, the
steps in the analysis and the complexity of matrix algebra remain exactly the same
6.2.1	КЧГ,,..4 _ГТ
Copter 6. Scattering and Mixing Theories /
6.2. Jnlrrprrtntion of Pnlariznbility Expressions
201
as in the previous subsections. But now the electric and magnetic polarizations have
to include also the magnetoeleclric coupling in the background medium. The prob-
lem of a bi-isotropic-in bi-isotropic sphere has been solved in [13]. The following
is the polarizability result for the special case of Pasteur sphere in Pasteur medium
(Xi - 0)- Let 4|C material parameters of the background be гг,,/>г|,к, and those
of the spherical inclusions er2,/>r2,K2. Then the polarizabilities (generalizations of
(6.11) to (6.14)) read:
- МХ/М + 2/»rl) - t,l(«2 - *i)* - 3(frI - crl)«?
O'r2 I 2/irl)(erJ + 2erI) - (k2 ! 2k,)2	'
CTem	— 3V^//f0f0
x -3j/lrlCrl(K2 - «1)	j«l[(/‘r2 ~/2г|)(М - t,l) - («2 - «|)(«2 I 2k,)|
(/'r2 + 2/<rl)(frj p 2cr,) — (k2 b 2k,)2
_qI/ /*н(/£г2 ~ /'н)(м I 2ff,) - /м(к2 - «i)? - 3(рг2	/<г,)к2
"""	'°	(/Ml 2pr,Хм-I 2си)-Тк2 | 2k,)2	• (G
These expressions give the bi-isotropic in isotropic special cases (Equations
(6.11) to (6.14)) for к, —» 0.
G.2 Interpretation of Polarizability Expressions
After the analysis of all the different inclusions that have been treated above, the
components of the polarizability matrices may give the impression of being only
complicated quotient expressions with interminable divisional lines. Even in the
simplest, homogeneous spherical case, the electric and magnetic parameters be-
come coupled. Nonetheless, a closer look at the polarizability expressions reveals
connections to physically interpretable phenomena.
The polarizabilities of bi-isotropic inclusions arc dependent on the four material
parameters of the inclusion material, and trivially — linear proportionality — on
the volume of the sphere. In addition, the ellipsoidal inclusions display the shape
effect as in the isotropic dielectric case: the smaller the depolarization factor, the
larger the polarizabilities. Of course, in the more complicated structures of layered
chiral sphere and chiral background material, the extra parameters of the geometry
have an effect.
'•2. Interpretation nC ......
202
Chapter 6. Scattering and Mixing Theories
6.2.1 Effect of Inclusion Parameters
First of all, it is easy to note the similarities in the appearance of the components
of the polarizability matrix: ate and arnrn have the same mathematical form, with
different coefficients. This is a consequence of the duality that the electric response
of a bi-isotropic sphere due to an electric excitation possesses with its magnetic
response due to a magnetic excitation. This duality will be discussed more deeply
in the next section where the effective parameters of mixtures are analysed.
The main effect of the material parameters on the polarizability components is
the increase of the electric copolarizability ctee with increasing permittivity c, and
the corresponding monotonous effect of [L on In other words, a “dielectrically
heavy” sphere is able to generate a larger dipole moment than a more diaphanous,
transparent inclusion.
0.00 “ ~Г ГГТ| i ГГ"Г | I I I I I I I I I pTI 'I' j I FI T I I TT_r I
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Relative permeability
Figure 6.3 Normalized crosspolarizability of a chiral sphere as a function of the relative
permeability of the sphere р/Цо for two permittivity values € = 1.5co,c = 5co- Solid line:
к ~ 1; dashed line: к - 0.5; for the case к — 0, the crosspolarizability vanishes.
Complementing this copolarization effect, the magiictoclec.tric parameters \ and
к mostly affect the crosspolarizability terms aem and ame. In most practical cases,
these chirality and nonreciprocity parameters are small compared with the relative
permittivity and permeability parameters. And since these affect the copolarizabil-
ities Q«,Q,nm through squares, д2 and к2, the effect is rather small. 4’he overall
effect of increasing к or is that both copolarizabilities decrease. In addition, the
6.2. Interpretation of Polarizability Expressions
203
Relative permeability
Figure 8.4 Normalized electric copolarizability of a chiral sphere a„/(3coIz) as a function
of the relative permeability of the sphere p/po for two permittivity values t = 1.5<o,c =
5c„. Solid line: к = 1; dashed line: к = 0.5; dotted line: к = 0.
introduction of a nonzero к or у leads to the crossdependence of a„ on /i and also
amm on e.
The increase of the crosspolarizabiiities crem and <>,„e is linear on increasing
both X and к, for small values of x and к. The crosspolarizabilities decrease for
increasing e and /i, which is visible in (6.12) and (6.13), and clearly seen in Figure
6.3. There, for simplicity, the sphere is reciprocal (\- = 0). The crosspolarizability
is normalized, and the value <*n,t/(3j,yji^c2V) is shown.
Figure 6.4 shows the electric copolarizability. The dominant parameter affecting
a„ is indeed permittivity c, and for large values of c, there is negligible effect of p
and к. Although the example in Figure 6.4 is a Pasteur sphere, the effect of к can
be identified with the x effect for a corresponding Tellegen sphere because these
quantities appear in the expression for a„ in the form x1 -| №.
6.2.2 Paramagnetic Parts Can Generate Diamagnetic
Whole
Perhaps the most interesting effect in the polarizability expressions is the way the
electric material parameter € changes the magnetic copolarizability and vice
versa, how ft affects a<e. This is illustrated in Figure 6.5 where the magnetic copo
204
Chapter 6. Scattering and Mixing Theories
Relative permeability'
Figure 6.5 Normalized magnetic copolarizability of a chiral sphere amm/(3p„V) as a
function of the relative permeability of the sphere /i/p„ for two permittivity values c =
1.5ro,c = 5eo. Solid line: к ~ 1; dashed line: к ~ 0.5; dotted line: к = 0. Note the
diamagnetic behavior at sufficiently small values of fi.
larizability a,nnl is plotted as a function of the inclusion permeability, with different
permittivities. The monotonic increase of amm with /i is evident, but for sufficiently
small values of p, the magnetic copolarizability can be negative. In other words,
for a bi-isotropic sphere, the magnetic polarization response can be opposite to the
incident magnetic field, although the material permeability is larger than that of
the vacuum. To paraphrase, the depolarization field — which is opposite to the in-
ternal field — has an amplitude larger than the incident field. Figure 6.5 shows also
that only in the case of nonchiral and reciprocal sphere, is always nonnegative.
Another way of stating this phenomenon, as has been noted elsewhere [14], is that
a mixture with bi-isotropic inclusions will have magnetic properties, even though
the materials composing it are nonmagnetic.
Thus a paramagnetic sphere can display a diamagnetic behavior if it is chiral
or nonreciprocal or both. The effect is more pronounced if the chirality or non-
reciprocity is increased. The figure also shows the manner in which t affects this
phenomenon: making the permittivity larger decreases the effect.
Also, due to the duality, or the fact that Equations (6.11) and (6.14) switch
places if the electric and magnetic quantities arc interchanged, a similar phenomenon
exists for the electric copolarizability: a “diadielectric” material — “diadiclectric”
6.2. Interpretation of Polarizability Expressions
205
behavior meaning a negative electric susceptibility — can be manufactured from
“paraelectric” bi-isotropic spheres that have a permittivity sufficiently close to that
of vacuum, in a way which corresponds to a state below the zero crossing point of
Figure 6.5.
Л further property of the polarizability matrices for the Pasteur case is reci-
procity. The reciprocity manifests itself in the fact that the crosspolarizabilities
a,,n and are imaginary, and opposite numbers of each other. This is visible in
the expressions (6.28) and (6.32).
6.2.3 Similarity of Bi-Isotropic and Anisotropic Polariz-
abilities
Bi-isotropy is not the only direction away from isotropic materials. Although non-
isotropy is the topic of this book, bi-isotropic media only form one class of novel,
complex media. As has been thoroughly discussed, this class of media retains
isotropy but allows magnetoelectric interactions. The second path toward more
general media takes another direction: there is no natural chirality or isotropic non-
reciprocity but the polarization.may depend on the electric or magnetic field vector
direction. These materials are anisotropic.
Dielectric anisotropy means that the polarization caused by an electric field is
generally not in the same direction as the field itself. Correspondingly, in anisotropic
magnetic materials, the average magnetic dipole moment density is only in principal
axes directions parallel to the magnetic field, fn bi-isotropic media, on the other
hand, there are no special axes, or directions. Therefore, it may seem strange
that for such different polarization mechanisms as in these two different classes of
materials, there exist similar laws in the polarizability descriptions. But this is
indeed the case, as will be shown here.
The constitutive relations of anisotropic media are formally simpler than bi-
isotropic ones. For dielectrically anisotropic media (for example magncloplasma),
the permittivity is dyadic:
D —?E	(6.34)
and for magnetically anisotropic media (for example, ferrites), the permeability is
dyadic:
B = II.	(6.35)
Let us consider an anisotropic sphere, with permittivity dyadic
ё - -I	t r.nji, | ecirz x 7.
(6.36)
Chapter 6. Scattering and Mixing Theories
206
This consists of a symmetric (biaxial) part and an antisymmetric (gyrolropic)
part. The gyrotropy axis is here assumed to he aligned with one of the symmetry
axes u, and the measure of gyrotropy is eg. The gyrotropic1 character of the per
mcability in ferrites can be exploited in nonreciprocal microwave applications, like
circulators and isolators, but also nonreciprocal permittivities are being studied for
the use of gyroelectric waveguides [16}-
The polarizability dyadic of this sphere can be shown to be [17]
(6.37)
with components
ct„
am
„ ,z К ~ to)(fv I 2co) + t*
° (e. | 2ro)(4 T2c„)
4 v	1 2c°) 1
° (£, |2ам2«.)кГ
(6.38)

<r,v = -aVI = 3tJz;---------—-------------
(G I 2со)(г„ 1 2c„) I e’’
«XX = «zx — Oyz = «zy = 0-
There are striking similarities as one compares the gyrotropic polarizability com-
ponents of (6.38) to the polarizability matrix of a bi-isotropic sphere (6.11) to (6.14).
In (6.38) the gyrotropy parameter cg affects the polarizability components. If it
vanishes, the matrix becomes diagonal and the components become simple functions
of the permittivities like in the perfect isotropic case. However, in the gyrotopic
case, there is one component that is not affected by eg. This is the z-dirccted
copolarizability which is the same as the isotropic polarizability. It means
that, for example in the case of a ferrite sphere, the gyrotropy has no effect on the
copolarizability in the external magnetic field direction.
On the other hand, gyrotropy affects the transversal components and ayy.
It also has an effect on the off-diagonal components axl/ and ayx. Here tg plays a
similar role as the chirality parameter л or nonreciprocity parameter x the case
of bi-isotropic sphere. However, there is a change of sign: the denominator of the
gyrotropic case is
lIn a more broad sense, the term gyrotropy can be affiliated with media that are capable of
rotating the polarization plane of the propagating wave. Therefore both chiral media and ferrites
or magnetoplasma can be characterized as being gyrotopic. Il has been typical of the terminology
of the former-Soviet Union physics to call chiral media “gyrotropic.*' In the West, often only
nonreciprocal anisotropic materials have deserved the gyrotropic label {15].
6.2. Interpretation of Polarizability Expressions
207
(cr 4 2€o)(t1/ 4- 2eo) I e’
whereas the corresponding expression for the chiral (Pasteur) case is
(t I- 2to)(/t + 2/4o) - «2p„c„
anti in the nonreciprocal (Tellegen) case
(e + 2c„)(p + 2/t„) - x2/r„e„
and in the general bi-isotropic case
(c + 2e„)(/i + 2/io) - (x2 + k’)/Io€„.
These quantities, and also the polarizability expressions, have full correspon-
dence if the gyrotropy is imaginary: te = jg where g is real. This is in fact the case
in magnetoplasma [6] and [18] or in the case of the permeability of ferrites [19]:
Ji = /roil,u, + fit(I - u,u,) + jgu, x I.	(6.39)
The imaginary nature of the gyrotropy in the permittivity/permeability expres-
sion completes the analogy with respect to bi-isotropic media. We can write Table
6.1 for the correspondence between the quantities.
Pasteur Tellegen Dielectrically Magnetically
medium medium gyrotropic gyrotropic
.E = “ j « « a. o’ oE c“ o“	X € «ее ®rnm ^ern ^rne	I [	a 8 2 ? S. J* 4? 0 0 0 0	s 5 о o = t i-' к с н с к *5 : н с *г н	о j
			
Table 0.1. Correspondence between polarization characteristics of bi-isotropic and
anisotropic media.
The message of Table 6.1 is that the results and conclusions about BI media
that have been presented in the analysis of the present chapter can be used mu-
tatis mutandis for spheres made of gyrolropically anisotropic material, equally for
electrically and magnetically gyrotropic media.
208
Chapter 6. Scattering and Mixing Theories
6.3 Modeling of Bi-Isotropic Mixtures
The determination of the polarizability matrices of different types of bi-isotropic
inclusions is the first step on the way to the modeling of heterogeneous complex
materials. The polarizability components express the dielectric, magnetic, and
niagneloelcctric responses of the inclusion element, and once these are known, it is
possible to characterize mixtures that are composed of this type of inclusions em-
bedded in a host medium. The macroscopic parameters depend, in addition to the
polarizabilities, on the fractional volumes of the components making the mixture.
Furthermore, in the case of ellipsoidal inclusions, their orientation distribution may
have a strong effect on the macroscopic effective parameters of the mixture. Since
the analysis has been based on the quasistatic assumptions, the mixing rules are
also restricted by it. Therefore the limitation of the formulas to follow is that the
size of the inclusions — or the correlation length of any of the global electromag-
netic parameter functions с(г),/г(г),х(г),к(г) of the random mixture — has to be
smaller than the wavelength of the operating electromagnetic field. Л safe limit is
that the size parameter is less than unity: krnd < 1, where d is the average diameter
of the inclusion, and kru is the wave number within the mixture.
Before delving into the full bi-isotropic mixing laws, it is instructive to give a
brief reminder of how plain dielectric mixtures have been modeled in materials sci-
ence. This project has always been approximate: due to the fact that mixtures can
be deterministically inhomogeneous or random, their macroscopic behavior cannot
be exhaustively predicted with few parameters, like the volume fraction of the com-
ponents. Therefore there arc several mixing rules in use in different application
areas, for example, remote sensing and composite material design [20]. In the fol-
lowing, the focus is only on those that can be derived through theoretical analysis
and have been extended into the regime of bi-isotropic materials.
6.3.1 Effective Permittivity of a Mixture
The simplest classical mixing rule is the Maxwell-Garnett formula [21] which gives
the effective permittivity ecn of a dielectric mixture where spheres of permittivity e
occupy a volume fraction f in a host medium of permittivity e„:
_	E - fO
Ceff — Cp + 3f<O „	.
e f- 2c„ - /(e - c„)
or, in the form labeled as Rayleigh formula
^eff	- € €o
. c.a 4 2c„ e | 2c„
(6.40)
(6.41)
6.3. Modeling of Ui-lsotropic Mixtures
209
The Maxwell-Garnett formula can be easily derived from the following collection
of interdependencies: the effective permittivity is the relation between the electric
field and the average (electric) flux density:
< D > = ecfrE r„E I < Pr > .	(6.42)
The average polarization is the dipole moment density,
< P, > = npe,	(6.43)
where n is the number density of electric dipoles. The electric dipoles are determined
by the electric polarizability a, given in Equation (6.4), and the field exciting the
inclusions: pt = oEt. This exciting field Et, also called as the Lorentzian field,
is not the same as the average field E but larger than that due to the fact that it
includes the contribution from the surrounding polarization [2], [22], and [23]:
P,
Et = E|-'.	(6.11)
3e„
The combination of these conditions yields the dielectric mixing rules (6 40) and
(6.41).
Note also that if the effective permittivity is given in terms of the polarizability,
we are left with the Lorenz-Lorentz formula [24] and [25]
na
W = e<,4---------- (6.45)
na
1	3r„
or
6.3.2 Effective Parameters of 131 Mixtures
The electromagnetics community has put considerable effort on bi-isotropic mix-
tures in the 1990’s. The dielectric Maxwell-Garnett formula (6.40) has been gen-
eralized to chiral, nonreciprocal, and general bi-isotropic mixtures |10], [26], and
[27]. Even bianisotropic mixtures have been treated [28]. In the following, a review
of the derivation of bi-isotropic mixing rules is given first, followed by conclusions
about the predictions of these equations. Also the mixing rules arc presented using
other constitutive relations, and alternative mixing rules are shortly touched.
212
Chapter 6. Scattering and Mixing Theories
The generalized Maxwell-Garnett formulas express the effective parameters as
functions of the coinpohent material parameters. These can be written from the
Lorenz-Lorentz results by substituting expressions (6.11) to (6.14) for the homoge-
neous sphere polarizabilities. The result is, using the relative values for the permit-
tivities (cr = c/c„, e,n.r —	and f°r the permeabilities (fi, = p/p„, p,n> =
(e, - l)[p, + 2 - /(p, - I)] - (Xa + «»)(! - f)
[p, + 2 - /(pr - l)](fr + 2 - /(£, - 1)] - (X2 + «2)(1 - /У ’
(6.55)
Pcff.r = 1+3/
(/tr - l)[er + 2 - /(er - 1)] - (x* + «’)(! - /)_________________
[p, + 2 - /(p, - l)][c, + 2 - /(c, - 1)] - (x1 -I «2)(1 - /У ’
(6.56)
(pr + 2 - /(p, - l)J(e, + 2 - /(c, - 1)] - (x1 + K’)(l - fy
[pr + 2 - /(p, - 1)][£, + 2 - /(£r - 1)] - (x2 + к’)(1 - /У (6'58)
with / = nV being the fractional volume of the bi-isotropic inclusion phase in the
mixture.
Rnyleigh Formulas
As in the purely dielectric case, where the Maxwell-Garnett formula (6.40) could be
written in the symmetrical form (6.41), the bi-isotropic results (6.55) to (6.58) can
also be rewritten. It requires substantial bookkeeping to prove that the equations
are algebraically equal to the following ones:
- l)(/>eir,r + 2) - (x2ff + >c2ff) _ (er - l)(p, + 2) - (x2 1 *2)
(c«g.r + 2)(peg,r + 2) - (x2ff + к2я)	} (er + 2)(pr + 2) - (x2 I k2 j’
(p.ff,r - 1)(<«и> + 2) - (x?rr +	_ Al'r -!)(<> I 2)	(x2 + K2)
(/‘e<r,r I 2)(сгГГ|Г + 2) - (х2(г + К2(т)	1 (p, I 2)(cr I 2) - (x2 I «’)'
____________________Xcff .	_______________ f__________________X__________________
(PrfT.r + 2)(£.fr,r + 2) - (х2я + К2Я) (fir + 2)(cr + 2) - (x2 + K2)’
(6.61)
6.3. Modeling of Bi-Isotropic Mixtures
213
(/«.IT,. + 2)(eelr.r + 2) - (x’ff + k’(t)	I 2)(er I 2) - (*’ t- «’)'
This set is the generalization of the Rayleigh mixing formula (6.41). The de-
coupling into separate electric and magnetic Rayleigh formulas is easily seen for
к -» 0,x -• 0.
These Rayleigh mixing formulas also have a physical interpretation. The quan-
tities that are treated on both sides of the four mixing expressions are proportional
to the components of the polarizability matrix. The left-hand sides are written
about the mixture, and the corresponding expression for the inclusion multiplied
by the inclusion volume fraction builds up the right-hand side. Interpreted in words,
the Rayleigh mixing rules say that the mixture parameters are such that a sphere
made of homogeneous material with these parameters would exhibit polarizability
amplitudes a certain number of limes smaller than an equally large sphere made of
the inclusion material. This proportionality number is the volume fraction of the
inclusion phase.
One of the most intellectually satisfying features of the Rayleigh formulas -- in
addition to their charming symmetry — is that they meet beautifully the limiting
cases of the mixture: “no inclusions” means f = 0, which leads to the effective
parameters being those of the background; and full occupation of guest material
(/ = 1) gives the result that the macroscopic parameters are exactly the same
as the inclusion parameters. This is evident from (6.59) to (6.62), and visually
illustrated in the figures to follow.
Mixtures with Ellipsoidal Inclusions: Aligned Orientation
If the bi-isotropic inclusions in the mixture have ellipsoidal form, the polarizability
dyadics (6.23) to (6.25) have to be used. The orientation distribution of the inclu-
sions affects the macroscopic parameters: depending on the manner in which the
inclusions are embedded, the macroscopic behavior is changed because the polariz-
abilities of the ellipsoids depend on the depolarization factor in the given direction.
Consider first the case in which all ellipsoids have the same alignment of their
axes within the mixture. The consequence is that the medium is also anisotropic, or
rather, bianisotropic. The average polarizations P,,Pm can be calculated along the
lines of the case of spheres. There is a difference with the Lorentzian fields Er,, lit
due to the shape effect. The depolarization dyadic (6.19) needs to be taken into
account:
< P. > /<»
< Pm > /Цо
(6.63)
214
Chapter 6. Scattering and Mixing Theories
The effective parameters are now dyadic with the principal coordinate system
spanned by the axes ua,ils,uc of the single inclusion ellipsoid:
>Lr = У? ’’«rr.iU.’b.	(6-64)
i=a,b,c
where d stands for	The results for the components are
= U ~{(er - 1)[1 4 (pr - 1)^(1 - /)] - (x’ 4 «’)№(] - /)},	(6.65)
= ! Z{(/lr _ i)(1 |. (fr _ i)M(i _ /)] _ (/ 4 ^)/V<(1 _ /)},	(6.66)
/'<. di
/x	fK	tr,
Xeff.i =	= -j7>	(CC7)
where
d. = [1 + (/*, - l)/V.(l - /)][1 т К - 1)M(1 - /)] - (x2 F «2)/V2(l - f)2 (6.68)
with f = nV being the fractional volume of the bi-isotropic inclusion phase in the
mixture.
As the ellipsoids degenerate into spheres (N, —» 1/3), it can be seen that all
three axial components become equal, and the spherical Maxwell Garnett formulas
(6.55) to (6.58) are recovered.
For ellipsoids with an orientation distribution within the mixture, the dipole
moments have to be averaged with this distribution function as the polarization is
integrated. In this case the final formulas cannot be represented in such compact
form as the spheres or aligned ellipsoids. However, a closed-form solution still exists
|10|.
Randomly Oriented Ellipsoids
In the case that all ellipsoids are randomly oriented, there is no preferred global
direction in the macroscopic picture, and the effective parameters are multiples of
unit dyadic, equivalent to scalars. The mixture is bi-isotropic in this case. Л full
expression for the effective parameters is rather complicated (26| but if the second-
order niagnetoelectric terms in the denominator (which are small, in general, even
to the first order) are neglected, the following expressions can be written:
6.3. Modeling of Bi-Isotropic Mixtures
215
/ Л________(ее - 1)PU 4 (1 - M)J - M(x2 4 к2)
3d [Niti, -| (1 - TV.)][TV.c, 4 (1 - TV,)] - N?(x2 + «’)’
.. =. . / у___(^-IW + O-WB-W + K8)
3d [Niltr | (1- Nt)][TVltr 4(1- /V;)] - ДГ?(Х« + K1)’
x
= /Л______________________________________
X'ff 3d [Niflr 4(1- W,)|(/Va 4(1- TV4)] - N?(X2 4 к’) ’
к _ / y'__________________________K_____
“ 3<f \Nd‘r 4(1-	4 (1 - /V.)] - Л?(А’ i
(6.69)
(6.70)
(6.71)
(G.72)
with
d = 1 - L v TV - 1W, + (! ~ ^)] - N\x2 + n2)
з ‘ W/*, 4 (1 - M)]|TVi£r 4(1- TV.)] - TV,’(A2 + «’) .
lyN. (Be ~ 1 Wr 4 (1 - M)| - M(x2 4 *’)_______________
3.tt ’j^B, 4 (1 -Л)>Д + (1 -Л,.)]-Л?(х24*2)'	1	'
The treatments and results above only allowed one type of inclusions, such that
the spheres (or ellipsoids) had to have the same bi-isotropic material parameters.
This restriction can be relaxed. If there are different guest phases in the host
medium, their average polarization contributions can be added together, the other
parts of the analysis remaining as before.
6.3.3 Perturbation Expansions of Effective Parameters
The attractiveness in the Maxwell-Garnett relations was already emphasized: the
fact that they satisfy the limiting cases of low and high volume fractions; and this
piece of beauty is shining even more in the full bi-isotropic formulas. In these
extreme cases, the effective parameters reach the values of the background, and the
guest, respectively. However, in actual practice, in the work on novel microwave
materials, for example, the global behavior prediction is too general. The engineer
is often working with component materials having a limited range of parameters,
or, due to nonelectrical design objectives, she is only interested in a small range of
volume fractions of a component. In these cases, it is more informative to see what
the dominant quantities determining the macroscopic behavior are.
The most common cases in working with real chiral and bi-isotropic materials
is the case that either the magnctoelectric parameters (к and %z), or the volume
fraction of the guest phase (/), are small. Let us treat these cases in the following.
216
Chapter 6. Scattering and Mixing Theories
Weak Magnetoelectric Coupling
In the previous discussion about the polarizability expressions, the effects of small
chirality and nonreciprocity values were analyzed. Those conclusions also affect ef-
fective medium modeling. There are important consequences of the fact that many
of the functional dependences of the macroscopic parameters arc even functions of
inclusion parameter values. If the chirality and/or the nonrcciprocity of the inclu-
sions are small, their effect is weak. On the level of effective parameter expressions,
this leads to the fact that perturbation expansions start with the second-power term
of к and x-
The full expressions converge uniformly to the classical dielectric (and magnetic)
mixing relations as the chirality and nonreciprocily vanish. Hence, for values of x
and к much less than 1, the dominant effect is the dielectric polarization in £,e
and magnetic polarization in This fact and also the decoupling of electric and
magnetic polarizations is evident in the perturbation expansions of (6.55) to (6.58):
" 1 + 3Л~^7(Ь1)	(6 74)
_(x’ + K’)—MbZ)__________________,
W Ъ + 2 - f(er - l)]’[p, + 2 - /(pr - 1)]’
~ 1 + 3/ -	----	(6.75)
/i, + 2 - /(pr - 1)
-(x1 + re»)	9/(1
U V + 2 - /(Mr - 1)]’(C, + 2 - /(£r - 1)1 ’
9/«
K'* ~ k + 2-/(/tr-l)][tr + 2-/(£r-l)]’	(6'76)
[/<r + 2-/(pr-l)][£, + 2-/(£r-l)]-
In other words, the chirality of the inclusion phase has little effect on the macro-
scopic permittivity and permeability, at least for high permittivity and permeability
contrasts between the inclusion and background phases. On the other hand, nat-
urally the chirality of the inclusion is the dominant factor defining the effective
chirality parameter of the mixture. Thirdly, the effective chirality parameter of
a mixture is decreased by high permittivity and/or permeability of the inclusion
phase. These phenomena are illustrated in the figures to follow.
6.3. Modeling of Bi-Isotropic Mixtures
217
Dilute Mixtures
The other case is that of dilute mixtures: the volume fraction of the guest phase is
small ( f <4 1). This question has been treated in [14] and [26] for Pasteur (reciprocal
chiral) mixtures; the following is an extension to the full bi-isotropic domain. On
the other hand, the values for к and x are not assumed to be small. The diluteness
of the mixture means that the effective material parameters arc mostly those of the
background, and there arc only perlurbational corrections to these:
«eir.r — 1 +
(cr - !)(/., 4 2) - (x2 I x2)
J(cr4 2)(/tr 4 2)-(x2 4x2)
	!)(/>, ! 2) "(x2 + *2)]2 + 9(x’ + «’)
[(cr + 2)(/.r + 2) - (x2 4 x2)]2
(6.78)
PelT.r — 14
(/ir - l)(cr 4 2) - (x2 4 к2)
(cr 4 2)(/i, 4 2) - (x2 4 к2)
|f2K/*r ~ !)(* + 2) ~ (X2 1 x2)]2 4 9(X2 I x2)
1	[(e, 4 2)(,tr 4 2) — (x2 4 x2)]2
(6.79)
~ f__________________9X_______________
Xtn ~ 7 (cr 4 2)(/tr 4 2) — (x2 4 x2)
4-0	~ })(е-	2) + (fr ~	+ 2) ~ 2(X2 I X2)
7 X	[(cr 4 2)(/rr 4 2)- (x2 4 a2))2
(er 4 2)(,/r 4 2) - (x2 4 x2)
ozU/'r -t)K4 2) 4 (e,-!)(/», 4 2) - 2(x2 4 x2)
*	|(er 4 2)(Mr 4 2) - (x2 4 x2)]2
(6.81)
To the first order, these can be written as
feir = f„ 4 na„, ii,n - /<„ 4 namm,	(6.82)
Хиг =
n3?{a,n,}
«иг =
nS{o,.,}

(6.83)
218	Chapter 6. Scattering and Mixing Theories
6.3.4 Remarks on and Illustrations of Effective Material
Parameters
The Maxwell-Garnett mixing rules have gained length and complexity in the gener-
alization from isotropic mixtures to the bi-isotropic regime. Although they satisfy
the tests of simpler limiting cases, in the full form they are not really transparent.
It may be difficult to sec the effect of a given mixture parameter in the mixing
rules (6.55) to (6.58), or (6.65) to (6.67). This is because of the coupled nature of
the macroscopic equations for the effective parameters: for example, the effective
permittivity depends not only on the permittivities of the constituent phases but
also on their permeabilities, and the guest chirality and nonreciprocity.
However, a closer look, again, reveals order and regularities in these mixing
rules. As a matter of fact, after some examination it may even seem that intuition
can produce reasons for all details in the effective material parameter expressions.
Even and Odd Effects of Handedness
Perhaps the most impressing property of Maxwell-Garnett rules is the set of multiple
dualities in the macroscopic material formula expressions. For example, the way the
permeability of the inclusion ft affects the effective permeability /i,n, chirality Kelr,
uonrcciprocity and permittivity cea, is the same as the way the permittivity
of the inclusion e affects the effective permittivity, chirality, norireciprocity, and
permeability. Also, the functional dependence of the macroscopic permittivity on
the inclusion chirality is exactly the same as on the inclusion nonreciprocity. This
property can be expressed clearly on the level of symbols:
= «<rr(c, A, /»<.)	(6.84)
Of course, the same dependence on the inclusion parameters applies for the
macroscopic permeability Furthermore, the macroscopic chirality depends on
inclusion nonreciprocity identically with macroscopic nonreciprocity dependence on
inclusion chirality.
It is important to observe that the effective ( — macroscopic) permittivity and
permeability of a mixture are even functions of both the chirality and nonreciprocity
of the component material. Hence, firstly, the sign of handedness, i.e., whether left
or right handed, should not have effect on these parameters, and they are true
scalars, invariant of spatial inversion. This is obvious: samples of media that are
mirror images of one another should have the same permittivity and the same
permeability; Nature should not prefer left to right. Although in the subatomic
level, in the weak interaction process, the asymmetry between left and right has been
predicted by Lee and Yang in 1956 [30], and experimentally observed in 1957 [31],
6.3. Modeling of Bi-Isotropic Mixtures
219
one would expect this not to happen at the macroscopic level where racemization
processes tend to wash out handed effects.* 2
But the equations above clearly show that the dependence of ecff and /reff on
the inclusion nonreciprocity parameter is identical to the dependence on inclusion
chirality: changing the sign of x does not affect the macroscopic permittivity or
permeability:
/'m(x) = /'.rr(-x)-	(6.85)
This phenomenon is not as intuitive as in the case of handedness and chirality.
One can, however, refer to the phenomenological model of nonreciprocal media
depicted earlier in Figure 1.4. There the microstructure is represented by rigid
units that consist of bound electric and magnetic dipoles. The polar mechanism
is as follows: As the electric field exerts a torque on the electric dipole, it also
produces cophasal magnetic polarization. If all these hybrid units have the same
“state,” i.e., in all elements, the permanent electric and magnetic dipole moment
vectors are of the same direction (or opposite), but not mixed, the nonreciprocity
effect is nonzero. Changing the sign of x corresponds to reversing the electric (or
magnetic) dipole moment vector direction in all units. In this view it is a vaguely
similar operation as taking the mirror image of a material, and means a change of
the sample from Figure 1.4.
The evenness of ecfr and /re(T on x can be understood from Figure 1.4. The change
in the sign of x does not change the overall copolarizability characteristics of the
medium, and hence also the macroscopic permittivity and permeability should keep
their amplitudes unchanged.
The effective chirality parameter is an odd function of the chirality of the
inclusion material.3 The intuitive support for this is that a change in the handedness
of the component changes the handedness of the mixture, к.п is an even function of
the inclusion nonreciprocity. Therefore handedness and nonreciprocity are separate
geometrical properties in material structure. And again, a similar conclusion holds
for the effective lionreciprocity: Xcff is an odd function of x and even of к.
Illustrations on Effective Parameter Behavior
Figures 6.7 to 6.9 illustrate the effect of mixture structure on the effective macro-
scopic material parameters e.g, and Ktfr. The Maxwell-Garnett mixing rules
aThis issue is, however, more subtle Ilian it may seem from niacroscopica! observation. The
early 1980’s saw the experimental confirmation of the unification of the electromagnetic force
and the weak interaction. Therefore electromagnetism is only one aspect of a more fundamental,
electroweak force, and this force is chiral.
3This means that ксд is a pseudoscalar.
220
Chapter 6. Scattering anil Mixing Theories
Volume fraction
Figure B.7 Relative effective permittivity ct(f/re of a mixture with spherical reciprocal
chiral inclusions as a function of the volume fraction of the inclusions. Both components
have the same permeability p„. Two different inclusion permittivity and chirality values
arc shown. (Note that the figure indeed displays four curves! Two pairs are almost
indistinguishable.)
(6.55) to (6.58) are used to predict the parameters for a reciprocal chiral mixture
(x — 0)> '"here Pasteur spheres are occupying a fractional volume f in simple
isotropic background medium with parameters co,/<o. Because of the duality be-
tween chirality and nonreciprocity, the illustrations also convey information about
Tellegen mixtures.
In the figures, the inclusions ate assumed nonmagnetic (p = /<„) for simplicity.
Two dielectric contrasts and two chirality values are treated (e = 2c„, 5c„) and
(к = 0.1, 0.5). Figure 6.7 displays the effective permittivity behavior, showing the
continuous transition between the background and inclusion values as the fractional
volume increases from 0 to 1.
Figure 6.8 shows the effective permeability of the mixture. The diamagnetic
nature of the composite is indeed visible although the magnitude of the magnetic
susceptibility is small. Note that to begin with, both components of the mixture
are nonmagnetic. The figure also tells that the magnetic effect is larger for larger
inclusion chirality and for smaller inclusion permittivity.
In Figure 6.9, the curves for the effective chirality parameter are shown. The
limiting cases f 0 and f = 1 are the constituent chirality values as they should,
6.3. Modeling of Bi-Isotropic Mixtures
221
£4.02
1 1-01
g
£ 1.00
(D
£ 0.99
----- e=2eo /6=0.1
— — £=2eo /6=0.5
-----e=5eo k=0.1
----- e = 5eo /c=0.5
о	\
Ф	“	_____ __
Ы 0.98	। । । । । । । । I | । । । । । ।
0.00	0.20	0.40	0.60
T-1
1.00
Volume fraction
Figure 6.8 The same as Figure 6.7, for the relative effective permeability p,ir//r„ of the
mixture.
and the behavior in between is fairly linear, at least for small dielectric contrast.
The effect of increasing the inclusion permittivity* is the decrease of the effective
chirality parameter, and the consequent increasing nonlinearity of the к,л(/) curve.
Perturbationnl Effects
How can the perturbation rules (6.74) to (6.83) be illustrated? Figures 6.7 to 6.9
also give information about the small chirality and small inclusion volume fraction
regimes. The conclusions made in Section 6.3.3 about small-chirality effects are
supported by the figures. For small values of к the effect of chirality on the macro-
scopic permittivity is extremely small: the four curves in Figure 6.7 appear almost
as two indistinguishable pairs. The minute effect of к is to decrease the effective
permittivity.
The effective permeability curves show also similar behavior. Very little effect
can be seen in curves of Figure 6.8 for chirality к = 0.1. And finally, Figure 6.9
shows the linear relation between the effective chirality parameter and the volume
fraction for dilute mixtures (/	1).
This linear relation can also be illustrated by combining the perturbation for
mulas (6.74) to (6.76) for low chirality values. To the first order, one can write
«The same effect arises from increasing the inclusion permeability.
222
Chapter 6. Scattering and Mixing Theories
^eff ‘	/^cfT /^o
/ C - fl - /I.
(6.86)
This relation can be useful in the design of chiral composites: the effective
chirality parameter can be determined from the measurement of the composite
permittivity and permeability values, once the inclusion chirality is known. The
other possibility to exploit Equation (6.86) is to determine the chirality parameter
of the inclusion phase, using the measurement data of the composite (ecfr,pcn>'srfr).
Volume fraction
Figure 0.0 The same as Figure 6.7, for the effective chirality parameter of the
mixture.
Mixtures with Nonsphcrical Inclusions
How docs the material behavior change if the inclusions are nonsphcrical? The
only other forms of homogeneous inclusions whose polarizabilities can be solved in
closed form are ellipsoids. By using chiral inclusions of ellipsoidal shapes, a further
range of mixture parameters can be tailored. The effects vary: by using needle-
shaped or disk-shaped inclusions, larger effective parameters than with spheres can
be achieved, although the shape effect depends on the dielectric and magnetic con-
trast between the inclusion and host phases. One observation is, however, always
valid: spherical inclusions produce minimum effects in the macroscopic properties,
and each deviation from this extremum shape increases the value achieved by the
spherical geometry.
6.3. Modding of Ш-Isotropic Mixtures
223
As an example of how the shape effect can dominate the mixture parameters,
Figure 6.10 shows the effective chirality parameter of a mixture as a function of the
volume fraction of the chiral inclusions. The dependence of л on «сц is clear, again
as in Figure 6.9, but the large shape effect (spheres, needles, disks) of the inclusions
can only be compensated by extremely large variation in the inclusion permittivity
c.
Figure 0.10 The effective chirality parameter of a mixture composed of reciprocal chiral
bi-isotropic (x = 0) inclusions immersed in isotropic background medium. The volume
fraction of the inclusion phase ranges from 0 to 10%. The inclusions are randomly oriented
in the mixture, and the line type of the curves denotes the shape of the inclusions: solid -
spheres; dashed - needles; dotted - discs. Inclusion permeability relative to background
is 1.1 and inclusion chirality parameter is к = 0.1. The numbers refer to relative inclusion
permittivity; c = 2c„ (1); c = 10co (2); с = 100с» (3).
It is also possible to treat mixtures with layered chiral spheres and enumerate the
effective parameters that emerge. To do this, one needs to combine the bi-isotropic
Lorenz-Lorentz formulas (6.51) and the polarizability expressions (6.27) to (6.29).
6.3.5 Alternative Mixing Laws
As has already been noted, Maxwell-Garnett (MG) is not the only mixing rule in use.
In fad, there have been attacks against the careless use of MG in dielectric mixtures,
and many advocates of this venerable formula also admit that there arc mixtures
224
Chapter 6. Scattering and Mixing Theories
which MG is not able to handle. It may be correct to say that the safe domain
for MG are mixtures where the volume fraction of the guest phase is small, and
where the dielectric and magnetic contrasts between the inclusion and background
are reasonable, in other words, not too large.
The extreme case of mixtures with contrasting components arc the conductor-
insulator composite systems that are being studied presently. Applications for these
mixing theories arc found in the study of, for example, conducting polymer mate-
rials. For large-contrast mixtures, the phenomenon of percolation becomes visible,
which means that the electrical behavior of the composite changes dramatically
with a small variation of volume fractions of the component materials [32]. The
state of the system where this happens is called percolation threshold. To explain
percolation has always caused problems for MG users [33].
For this and also other reasons, several rivaling mixing laws other than MG are
used in the studies of dielectric mixtures. Many of these rules are based on exper-
imental studies, but several laws also have theoretical support. Labels and names
arc attached to mixing rules, such as effective medium theory, coherent potential,
quasicrystalline or average T-matrix approximation, etc.; see, e.g., [34] and [35] for
the relations between different rules. It seems, however, that the literature only
contains one attempt beyond MG to generalize a dielectric mixing theory to cover
bi-isotropic mixtures.
In [36], the macroscopic properties of chiral composites are studied according to
the so-called IJruggcinan model. There the mixture under study consists of chiral
spheres embedded in chiral background material, and the result is a set of coupled
implicit equations for the effective parameters, which have to be solved numerically.
In the dielectric mixing case, the Bruggeman model is also known as Polder -
van Santen mixing formula (or Bottcher mixing rule, or effective-medium theory
rule) [37], and in this case the effective permittivity can be solved from the
second-order equation
= (c-87)
3c«rr e + 2c<.ff
However, the generalization into the reciprocal chiral regime does not yield ex-
plicit formulas. Aside this obstacle, the computer-aided numerical illustrations in
[36] show that the Bruggeman model predicts slightly larger results for the effec-
tive parameter values than Maxwell-Garnett estimates, and the differences become
visible as the chirality of the inclusion phase increases.
In spite of the fact there are several acceptable mixing rules in scientific and
engineering use, Nature has dictated upper and lower bounds for the predictions
of the different mixing formulas. Using variational principles, the limits for the
effective material parameters have been derived in the literature [38].
6.3. Modeling of Bi-Isotropic Mixtures
225
6.3.6 Mixing Rules in Other Constitutive Relations
The mixing-related formulas can be written as well by employing the two other sets
of constitutive relations that were introduced in Section 1.2. Use of the translation
formulas of Appendix Л, the polarizabilities and Maxwell Garnett mixing rules
change their appearance in the following way.
Post Rclntions
The generalized Post relations
D = ePE + V'„B-y(cB,	(6.88)
II = —В V’nE - y(.E	(6.89)
f'r
use (the chirality admittance) and V’n (the nonreciprocity susceptance, [39]) to
characterize the magnetoelectric coupling in the medium.
The polarizabilities of spherical scatterers (6.11) to (6.14) change their appear-
ance to
3 v 1 2**d.
(/<F 4 2/i„j(cP 4 2f„) 4 2(V’’ I
- 3 v (/'’ ~ P°)(fr 1 2f°) ~	4 Cc)l‘rl>o
'‘° (/IP 4- 2/i„)(cP + 2e„) 4 2(V>’ 4 (’)/<₽/<.’
(6.90)
(6.91)
(6.92)
— 3iz_____________________________________________
(/'r 4 2p„)(cP I 2c„) I 2(У>’ 4
3(tf’n I	_________
Q — зу-__________________________________
(/'г + 2p»)(cp 4 2c„) 4 2(V’’ 4 C2)/'r/'o
The Maxwell-Garnett formulas for spherical scatterer geometry (6.55) to (6.58)
look like, using the relative permittivities and permeabilities:
(6.93)
Qn.r
(cp„ - 1)[/,P, 4 2 I 2/(pP„ - 1)) 1 2(# 4 Сг)/>гЛ2(1 ~ /) ____________
[/!„.. 4 2 4 2/(/zP„ - l)](fp,. 4 2 - /(c,. - 1 )J + 2(V’2 4 (’Ьл’(1 - /)’ ’
(6.94)

liapter 6. Scattering and Mixing Theories
. 3,________(/*>.-+ 2-/(cF„ —I)) — (^ I	-/)
[/<₽.. I 2 -	- lj|k, I 2 - 7(tr_. - 1)] | (ft I	- /)(2 1 /)’
(6.95)
I __________________________________________9/У’пДр,»_____~_____________________________
I- 2 t 2/(/<P.. - l)lf(,„ + 2 - /(^ - 1)] I 2(ft -I f’)M2(l - /)”
(6.96)
=_________________________________9/6яг,-________________________________
[я,,. 4 2 + 2/(/rP.. - l)](£p„ + 2 - /(.„,. - 1)) F 2(ft f (’)/«Р.Л2(1 - /)’’
________________	(6.97)
Note that i)„ =
DrudeBornFedorov Relations
The Drude-Born-Fedorov constitutive relations take into account the chirality
through the parameter fl in the following equations
D = eD8P|E + /3V x E),	(6.98)
В = Яовр[П p/JVxII],	(6.99)
These relations are for Pasteur media. For spherical scatterers, the polarizabil-
ities are
	— Зс у (fDBF fo)(/fDBF I' 2/Jo) -|" 2/c2/32/ioeo (Fdbf Ь 2/io)(cdbf 4- 2fo) — 4fc2/326o/to	(6.100)
	□ .z (/'dbp - g«)(<D!f + 2to) -1 2k2fl2/i„eo	(6.101)
	* ° (/*DBF + 2/io)(cdbf 4 2co) - 4A2/326O/£O ’ — 3V	-j3^U'/inBFCDBF	
		(6.102)
	(/*dbf b 2/1о)(б1)И1» | 2eo) - 4Аг2/32€<,/4о	
	„ ^У	^^^/^DBF^DBF	 (/^dbf T 2/2o)(cdbf 4- 2eo) — 4fc2/32eo/io	(6.103)
with	A: = u;	(6.104)
and o> is the angular frequency of the field variation. These translated forms of the
polarizabilities (6.11) to (6.14) agree with those of (40] where the polarizabilities
were studied using DBF relations in the first place.
6.4. Dispersive Behavior of Chiral Materials	227
The Maxwell-Garnett formulas arc
W£ = 1+3/
___________(tp-e,. -	+ 2 + 2f(/<nBP, - 1)] +	- f) _
(/«dbf,. + 2 4- 2/(pDBPi, — l)][cDBr,. + 2 — /(eDBPi, — 1)] — 2k2fl2(2 4 /)(! — /)’
(6.105)
^*=14 3/
Po
__________(/«dbf,. ~ 1)[грВР,. 4 2 2/(cDBP, — 1)] 4 212/?z(l ~ f )
' (/«dbf,. + 2 - /(//„„.. - 1)1[£dbp,+T4^(^F~;nj]^^2T2T/Xl^:7j’
(6.106)
_______________________0//iDBP f6DBP,/?
~ (/«dbf,. 4 2 4 2/(/«DBPi, - 1)1[£dbp„ 4 2 + 2/(eDBP., - 1)] - 4*’/J’(l - /)’’
(6.107)
The Rayleigh forms (Equations (6.59) to (6.62)) of these mixing rules are
straightforward extensions from the polarizability expressions.
6.4 Dispersive Behavior of Chiral Materials
In this section, we concentrate on the question of how the electromagnetic descrip-
tion of complex materials depends on the frequency of the operating field. The
magnitude of the macroscopic parameters is determined by the temporal change of
the incident field, the reason being that the polarization mechanisms within me-
dia take their time in growth and decay. It may even be sensible to conduct the
electromagnetic analysis in time domain from the beginnings (41).
Chiral dispersion must exist: it is intuitively clear that helices of centimeter-
range sizes cannot be optically active. Hence к is dependent on frequency, and may
even be a strong function of it. In this section, the focus is on how the mixing
process affects temporal dispersion. How does the dispersion curves of composite
chiral media differ from those of the bi-isotropic bulk material? The dominant
characteristics of the dispersion behavior will be reviewed here; for a more complete
exposition, cf. (42).
6.4.1 General Features
It is not only of academic interest to bother about changes in dispersion curves.
Already in classical mixing studies on dielectrically heterogeneous media, it is well
228
Chapter 6. Scattering and Mixing Theories
known that the relaxation-type phenomena occur al drastically different frequency
ranges for bulk and composite media. Relaxation is a frequency-dependent polar-
ization phenomenon that occurs in liquids that consist of polar molecules. Water
is one example, and the model used for the permittivity is called Debye model [43]
and [44]. Mixtures with spherical inclusions change their relaxation behavior most
strongly. It is worth noting that the shape of the inclusion particles is extremely
crucial in the magnitude of the shift in relaxation frequency: in [45] the effect of the
nonsphericity has been studied, and there the conclusion was that for ellipsoids de-
viating from the spherical shape, the relaxation frequency always was lower than for
sphere-mixtures. For the case of needle- and disc-shaped inclusions, the relaxation
frequency would come very close to the relaxation frequency of bulk water.
But chiral dispersion is different from the dispersion in permittivity and perme-
ability. This is because chirality is embedded in nonchiral background, and if the
guest vanishes, there is no handedness anymore. On the other hand, permittivity
is floating in the background medium, which possesses certain host permittivity, at
least that of vacuum. Therefore the dispersion in permittivity is modified in in-
teraction with the host permittivity which does not vanish in any case. The chiral
dispersion transformation is hence radically different from permittivity dispersion.
Historically, dispersion was a phenomenon that was discovered early in the op-
tical activity research. This was due to the broad spectrum of visible light, in
constrast to the narrowband character of early-times applications of microwave
technology. The terms used to characterize the optical activity of Pasteur media,
ORD (optical rotatory dispersion) and CD (circular dichroism) contain the assump-
tion that there is spectral structure in the chiral behavior of these media. The early
19th century observed the spectrum of polarization rotation. Later, the absorption
counterpart of optical rotation, which is manifest in the ellipticity of transmitted
radiation, was first observed [46] in crystals by Haidinger (1847) and in liquids by
Cotton (1895).
ORD is the measure for the rotation of the polarization plane of linearly po-
larized light, and therefore proportional to the real part of chirality factor 9?{k).
Correspondingly, CD denotes the difference between the absorption coefficients of
the two circularly polarized eigenwaves. It is proportional to 9{«}.
The strong frequency dependence of the real and imaginary parts of the chirality
factor in optically active substances is illustrated with the example in Figure 6.11.
There, the dispersion curves are shown for a certain organic alcohol liquid solution
[47]. Л qualitative look at the curves shows a very strong spectral chiral behavior
within a band less than an octave.
As can be seen from the units used in Figure 6.11, different terms are used
by the microwave engineering and physical chemists communities for describing
chiral behavior of materials. Rather than the chirality parameter к, the amount of
6.4. Dispersive Behavior of Chiral Materials
229
Figure 6.11 Ultraviolet absorption u.v., optical rotatory dispersion o.r.d., and cir-
cular dichroism c.d. curves for the lignan, dimethyl-o-conidcndryl alcohol (XXIV A) in
methanol. Reprinted with permission from the Royal Society (London).
polarization rotation ф appears in reports of optical activity studies. 'Phis quantity
is expressed in degrees, and is usually displayed as the molecular rotation, in which
case it is given in square brackets:
io д/
M = Ф ~	(6.108)
7Г C
where the first ratio translates radians into degrees, Л/ is the molecular weight of
the optically active material, and C is its concentration in the liquid. Likewise, the
circular dichroism is often expressed in specific terms. The quantity c appearing in
Figure 6.11 describes the difference in the absorption constants of the two circularly
polarized eigenwaves (481.
The optical rotation can change sign with frequency, which even happens two
times in Figure 6.11. Hence, the chirality parameter к also has positive and negative
bands in the frequency scale. In other words, a material sample which is right
handed from the microscopical geometric picture, is able to rotate (he polarization
of the electromagnetic wave into right or into left, depending on the wavelength.
230
Chapter 6. Scattering and Mixing Theories
Hence one should be careful in terminology and not associate any geometric meaning
for к if one wishes to call it “chirality.” A more appropriate term would be “chirality
parameter."
On the other hand, Figure 6.11 shows that the circular dichroism remains of the
same sign throughout the whole frequency range. The imaginary part of к does
not change sign. One interpretation for this fact is the following: as a linearly
polarized electromagnetic wave traverses through optically active medium and the
rotated polarization also suffers from losses and becomes an ellipse, the handedness
of this ellipse does not change with wavelength. The analysis further in this section
provides more examples of the nonnegativeness of the CD curves.
6.4.2 One-Resonance Condon Model
To analyze the dispersive behavior of the bi-isotropic material parameters, one needs
to study the physical mechanisms responsible for the electric, magnetic, and mag-
netoelectric polarizations. It has been observed, and can be also seen from Figure
6.11, that there are similarities between the ordinary dispersion and absorption on
one hand, and rotatory dispersion and circular dichroism on the other. The models
for optical activity arc intimately related to the electronic transitions that determine
at the same time the refractive index of the material. Also the Kramers Kronig
relations can be written for medium parameters responsible for the optical activity
[49].
Therefore resonance models have been used to describe chiral dispersion of ma-
terials by many scholars since Drude’s work in the last century. This can be easily
accepted phenomenologically, since the element that is used to achieve macroscopic
chirality, the helix, displays also resonances very effectively. This will be discussed
more closely in Section 6.6.
Homogeneous Chiral Medium
One of the basic models for the dispersive character of optical activity was pre-
sented by Condon in 1937. His quantum-mechanical theory gives a result where
the transitions between the states of the optically active molecules and rotational
strengths of the absorption lines combine in a miiltircsonant expression [f>0|.
In the following, assume that the dispersion of the chiral medium follows the
Coudon model with one dominant resonance that lies far away from other molecular
transitions. Then the following frequency behavior for the chirality parameter «(o')
can be written
k(u>) oc
u>Il
(6.109)
u/g — U)1 I JU>r ’
6.4. Dispersive Behavior of Chiral Materials
231
where R is the rotational strength of the molecular transition, cv0 is the resonant
frequency, and Г measures the damping associated with the transition. For a more
complete description of the rotatory dispersion, one has to take the sum over the
molecular transitions of this type, weighted with the associated amplitudes fi,-.
Using dimensionless parameters in expression (6.109), we get
(6.111)
(6.112)
(6.113)
where x = w/w0 is the relative frequency and d = V/ui0 is again a measure for
the damping, r is a characteristic time constant describing the magnitude of the
chirality. This model clearly shows the absence of handed effects in statics: a
disappears for DC (x = 0).
Because of the absorption term, the chirality is now (in the frequency domain)
a complex number, indicating real and imaginary parts, a' is responsible for optical
rotatory power and a" produces circular dichroism. The real and imaginary parts
can be written explicitly:
a(u/) = k'(w) — ja"(a>)
with
a'(u/)	x(l — x2)
tw0 ~ 1- (2 d2)x2 I x4’
a"(u>)	dx2
two 1 — (2 - d2)x2 + x* ’
It is clear that the “resonant frequency” <v0 is the center for dispersion. At this
frequency the imaginary part attains its maximum value (a" always is positive).
At the same frequency, the real part vanishes, and it behaves symmetrically above
and below this frequency if it is drawn with the logarithmic scale. The effect of the
increasing damping term d is to broaden the resonance peak and to make it lower
at the same time. The imaginary part of the chirality parameter in this Condon
model (6.110) has the maximum value
«"(wo) = ~7°	(6.114)
a
at the resonant frequency u>0.
Heterogeneous Chiral Medium
Now, given the bulk medium behavior contained in Equations (6.112) and (6.113),
let us study the mixture dispersion. To appreciate the most important effect in this
232
Chapter 6. Scattering and Mixing Theories
dispersion transformation, keep all other variables constant in the mixing process.
Therefore, consider a mixture with a small amount of chiral guest material in a
nonchira! background medium, and let all the frequency behavior be contained in
the chirality of the inclusion phase. The other material parameters will be kept
nondispersive in the following analysis, i.e., the dielectric and magnetic dispersions
are assumed to take place at other frequency bands.
To be sure, this approach suffers from the following weakness: The wave numbers
of the two eigenwaves in a bulk chiral medium, are kj = k0(n £ «), (cf. Equations
(2.83) and (2.84)). These wavenumbers will become complex numbers for dispersive
media, due to the imaginary part of the chirality к. If the dispersion in permittivity
and permeability are neglected, e and p are real constant quantities. Consequently,
n will be real, and hence one of the eigenwaves will have positive, and the other neg-
ative imaginary part of the wave number. This means gain in one of the propagating
modes, which contradicts with an assumption of a nonactive material. Therefore,
in a real physical situation where optical rotatory dispersion and circular dichroism
are present, dispersion has to appear also in e and /», not only in к. This reasoning
sets a limit for the imaginary part of the chirality factor which can be seen to be
equal to that given in relation (2.175), by setting there у = 0.
Let us accept this shortcoming and add the simplification to the list of assump-
tions made so far in the dispersion model. Due to the dual character of the chirality
and nonreciprocity in material effects, it is sufficient to study the dispersion in
chirality. From this, one is also able to glean information about the effect of non-
reciprocity, which has been the philosophy in the earlier numerical analyses of this
chapter as well.
Consider a two-phase mixture where the background is nonchiral of permittivity
to and permeability po, and spherical chiral (y- = 0) inclusion particles of material
parameters occupy a fraction f of the total volume. Taking the assumption
/«Si, and using (C.81), the macroscopic chirality of the mixture follows the formula
9/k
~ (e, + 2)(pr | 2) -	(C115)
Here er =	= p/po are the permittivity and permeability of the inclusion
relative to the background. Assuming the model (6.110) for the chirality dispersion,
the macroscopic chirality can be calculated. The following frequency dependence is
the result:
«sir = <IT - X'ff	(6.116)
with
6.4. Dispersive Behavior of Chiral Materials
233
,	x(l r’)[l-(2 d1 I Л)х2 I x4]
~ 7 I' “ (2 । d2 > Л)г’ < r4]M 4rf2x’(T- г2)2’
</x2[l — (2 - J2 - Л)х2 + x4]
к" n___________________________________________________
гП [1 -(2 + </2 + Л)х2 + x4]2 + 4<72х2(1 -х2)2'
(6.117)
(6.118)
Here, the mixture parameters that have effect on the frequency behavior are
Гь, л _	(^o)2_____VJtWq
<^o ’	(/«,. +2)(e, + 2)’	(/i, 4 2)(e, 4 2)'
(6.119)
These are the formal results of the simple chiral mixing model.
6.4.3 Effect of Mixing on Dispersion
Interpretation of the formulas (6.117) and (6.118) shows that in the dispersion of
the real part of composite chirality, which is responsible for the optical (or electro-
magnetic) activity, the main effect in the composite dispersion is that the frequency
range is broader than the bulk chirality dispersion. The dispersion surrounds the
center frequency ufo. Also it has more structure in its shape. For low damping fac-
tors d — 0.01 •••0.1, four extrema in the Krn(u') curve appear (instead of only two
in the case of bulk chiral material): there appears one maximum and one minimum
below cj0, and also one maximum and one minimum above. The value for at the
resonant frequency ur0> however, always remains at 0, and this point in the curves
seems like a pivot around which the curves twist as the parameters change. Also
one could say that at the resonant frequency there is no chiral effect, although the
medium itself is surely chiral! This, again, underlines the warning against assigning
any geometrical meaning to the chirality parameter.
Also, conclusions can be drawn about the effect of the mixing process on the
dispersion of the imaginary part of the composite chirality First of all, the
imaginary part does not change sign: it is negative all the way throughout the
frequency range. The positive values in the figures arc due to the convention in
Equation (6.116).R As in the case for the real part of the structure of the
composite chirality frequency function is more detailed: there appear new maxima
and minima in the к"п curves.
The conclusions are illustrated in Figures 6.12 and 6.13. These show an ex-
ample of the composite chirality behavior as functions of frequency. Although all
possible quantities have been normalized, there remain two parameters {cf. defini-
tion (6.119)) affecting the behavior of the curves: d is the relative damping of the
RThis is a similar convention to that for dielectric properties of materials in th*- frequency
domain, as the time-harmonic notation exp(yo’t) is adopted. The complex permittivity is often
written as c = t' — jf" because c" stays positive for passive (dissipative) materials.
234
Chapter 6. Scattering and Mixing Theories
Relative frequency
Figure 6.12 The frequency dependence of the real part of the chirality parameter of a
mixture where chiral inclusions obeying the Condon model are embedded in a nonchiral
background medium. The frequency axis is shown relative to the resonant frequency u?q.
'ГЬе mixture is dilute, i.e., the volume fraction of the chiral phase is small. The damping
factor of the resonant model is d — 0.8 for the curves in the figure, and the normalized
chirality amplitude A is varied. The effective chirality parameter is shown relative to
the value В defined in the text.
resonance transition (being 0.8 in the figure), and A is the measure of the chirality
in the inclusion phase. The illustrated effective chirality parameter is normalized
with the value of If in Equation (6.119).
Figures 6.12 and 6.13 also show the dispersion of bulk chiral medium. This turns
out to be the case A = 0. As A increases, dispersive variation increases, although
it docs not yet reach the details of real-life characteristics of Figure 6.11.
The Condon model also allows us to analyze more complicated geometries than
mixtures with spherical inclusions. Using the mixing formula (6.67), the Condon
model for chiral dispersion can be used to see the effect of ellipsoidal mixing on
the chirality dispersion, as was done above for spherical mixing. The qualitative
changes in the dispersive behavior become more detailed, which is a reflection of
increasing structure of the mixture: the shape of the inclusions have lost part of
their symmetry. The curve features become functions of the shape parameters. fI wo
more parameters arc there now, compared with the spherical case.
6.5. Scattering by Large Objects
235
Figure 6.13 The same as Figure 6.12, for the imaginary part of chirality parameter of
the mixture, K"ff.
G.5 Scattering by Large Objects
The inhomogeneities encountered in this chapter so far have been of small scale, as
has been emphasized already several times. The wavelength has been assumed to be
large compared with the dimensions of the inclusions, whose polarization responses
were determined. The present section reviews briefly what type of procedures are
needed in the analysis if quasistatic conditions are violated. Once this assumption
does not hold, the inclusions take a more active role in the electromagnetics game;
they become scatterers.
As a matter of fact, it is not incorrect to term small, low-frequency inclusions
also as “scatterers,” and sometimes very minute inhomogeneities are indeed called
“Rayleigh scatterers” although they only measure a fraction of the wavelength.
And there exists a justification for treating inclusions — like those of the previous
sections in this chapter — as scattering elements. Il is based on the fact that the low-
frequency dipole moment expressions can be exploited in calculating the equivalent
radiating electric and magnetic current elements. These scattered fields are weak,
with a very strong frequency dependence; they appear as Rayleigh scattering, which
is the very source for the blueness of sky, and the radio wave attenuation by rain.
In this section, however, the higher order scattering effects are of interest. It
is strange that from the historical point of view, the scattering analysis of chiral
236
Chapter 6. Scattering and Mixing Theories
and bi-isotropic materials has evolved in opposite directions compared with that
of classical media. In dielectric analyses, the great studies by Lord Rayleigh and
others from the last century [51] came first. Afterwards, a full treatment of large
spherical scatterers was given by Mie [52]. In comparison, for bi-isotropic media,
the polarizability analysis d In Rayleigh has been a recent topic of 1990’s, but the
large-object chiral scattering was analysed already in the early 1970’s by Dohren.
Mie Theory and Bohren Decomposition
Mie theory is a mathematically straightforward approach to solving the electromag-
netic fields in the presence of a material sphere [53] and [54]. On the other hand,
the resulting bookkeeping is cumbersome, and includes infinite sums, which have to
be truncated and enumerated with a computer. The unknown fields arc expanded
in vector spherical harmonics, as these arc solutions of the vector wave equation in
spherical polar coordinates. The amplitudes of the scattered field components arc
determined through the use of electric and magnetic boundary conditions at the
sphere boundary.
Dohren [55] was bold enough to go further and extend the Mie solution to large
spheres that were optically active (or reciprocal chiral, Pasteur spheres, in today’s
terms). The mathematical obstacles he was facing were similar to the classical
case; now it is only the wavcficlds that have to be expanded in vector spherical
harmonics rather than plane wave electric fields.’ The resulting scattered field is
expressed with the expansion coefficients, but the number of the coefficient types is
eight, compared with four in the dielectric-magnetic (Mie) case.
Later, Dohren also solved the problems of electromagnetic scattering from an
optically active spherical shell [58], and also the scattering from an arbitrary Pasteur
cylinder [59|. In the 1990’s, even the more general problem of electromagnetic
scattering by arbitrary chiral spheroids was solved [60].
Application to Radio Waves and RCS Reduction
Dohren was mainly talking in terms of classical optics; however, his results are
general and apply to radio waves as well. The results have been later found useful
in calculating the radar cross sections (RCS) of chiral spherical scatterers that arc
not electrically small. Can the sphere be made invisible using chirality? Two studies
try to find an answer to this question.
eSometimes the wavefield decoinposition is called "Bohren transformation” [Sfij. The decom-
position (Equations (2.70) and (2.71)) in source-free homogeneous regions shows that a wavefield
is proportional to its own curl. This type of vector field is sometimes known as “Beltrnmi field”
6.5. Scattering by Large Objects
237
Bhattacharyya [61J used the chiral Mir coefficients to deter mine the bistatic
RCS of a homogeneous chiral sphere with radius equaling the wavelength; a case
clearly beyond the Rayleigh scattering limits. His numerical results show that
the scattering properties are aspect-dependent, and there exist large crosspolarized
scattering (the scattering coefficient 5WV can be even larger — for some scattering
directions — than the copolarized scattering coefficients Shh and Syy). Also, there
can be a significant difference in the vertical and horizontal copolarized scattering
coefficients for certain aspect angle.s.
Spherical Chiroshicld by .laggard ct al. is a trademark, which consists of a re
ciprocal chiral layer on a metal sphere (62]. The scattering due to this kind of
structure can be determined with the generalized Mie results, and the calculations
of show that considerable reduction of the monostatic RCS can be achieved by in-
cluding chirality on the covering layer. The backscattering can be reduced by 15 dB
25 dB. In these calculations, the radius of the core is one wavelength and the
Pasteur layer is one-fifth of a wavelength thick. In backscattering, lossless recip-
rocal chirality docs not help in reducing RCS, as has been noted in Chapter 3, in
the analysis of planar structures. The reason for this is that the effect of chirality
is only to rotate the vector direction of the linear polarization of the propagating
wave, which rotation unwinds for the backscattering case on the return path. For
oblique incidence from planar structures, on the other hand, the reflected wave may
be strongly affected by reciprocal chirality [63].
Therefore, to produce the absorbing effect, the layer in [62] possesses mag-
netic loss. What may make Chiroshicld impractical, however, is that the chirality
admittance7 needed in calculations was
1/rj.	(6.120)
This figure is quite high according to the standards of today’s technology.
Outside the Western line of chiral research, the Fedorov school in the former
Soviet Union were also investigating problems of scattering of large chiral structures.
Solutions to spherical wave problems enabled researchers to consider scattering of
plane circularly polarized electromagnetic waves by spherical scatterers in chiral
media [64]. In particular, such scatterers as ideally conducting spheres, isotropic
chiral spheres, dual-layer particles with metal cores and chiral coatings, and dual-
layer isotropic chiral spherical particles were considered. The exact solutions were
expressed in terms of series of vector spherical eigenfunctions. Scattered fields in the
far-field zone were examined in detail for limiting cases of the particles which wer**
small or large compared to the wavelength. Scattering cross sections, extinction
and absorption coefficients wore calculated. The results can be used for studying
scattering of ensembles of spherical particles in uniform chiral media.
7See the material parameters of Post relations in Appendix Л.
238
Chapter 6. Scattering and Mixing Theories
In light of the promising effects on RCS, it is understandable that ehiral ma-
terials have been heavily advertised for radar cross section reduction and stealth
applications. This strategy for motivating electromagnetics research projects was
conspicions in the 1980's. However, objecting comments have also been raised. As
was already mentioned, it is mostly the transmission characteristics, rather than
reflection, that the chirality parameter of material layers affects. Therefore it is
more advantageous to focus on the Tellegen parameter, and not the Pasteur param-
eter when optimizing RCS-minimizing structures. Reference |65| is one example of
the criticizing comments. There the planar structure is studied where a chiral slab
covers the conducting surface. The conclusion is that the chirality of spring-loaded
composites provides no design advantages for absorbing materials over nonchiral
ones. On the other hand, the authors qualify this by stating that inclusions in
the form of conducting helices do seem to have desirable properties in absorber
design through the inagnctoclcctric coupling of helices. This would result in in-
creased permeability without the weight penalties of the usual magnetic materials.
However, this claim of [f>5| has not been confirmed due to the lack of permeability
measurements on chiral media.
Scattering by Irregular BI Objects
Chiral structures with canonical spherical geometries described above can be treated
analytically: the methods based on field expansions to solve scattering are exact, but
require numerical approximation of infinite sums. However, these methods cannot
handle the tougher problems which are encountered in the presence on nonspherical
scatterers. In 1980’s and later, there has been work in this direction, towards
arbitrary irregular chiral and bi isotropic scatterers.
The T-iuatrix method of Waterman (66] has been extended to chiral scatter-
ers [67]. The T matrix method, sometimes also known as the Extended Boundary
Condition method, or Null-field method, is based on the equivalence principle. Ma-
nipulating the surface integral equation for the scattered fields and using vector
spherical functions, the transition matrix for a given scatterer can be determined.
With this matrix, it is possible to calculate the scattering for any incident field
combination.
The T matrix method can also be used for permeable objects and numerically
effectively for axisymmetric scatterers [С8]. The Method of Moments [69], often
used with success to scattering problems involving conducting regions, means a
more number-crunching-oriented attack to the problem than the T-matrix method.
In the Method of Moments (MoM), instead of expanding the unknown fields in
harmonical functions, the integrals are discretized with more local basis functions.
Hence, with MoM, the problem is reformulated into matrix form at an “earlier"
stage than with the T-matrix method.
6.6. Scattering by Helices	239
It is also possible to analyze large scatterers by replacing them by a collection
of dipoles that are interacting with each other. The coupled-dipole method, or
discrete-dipole approximation, also carries the пашё Purcell Pennypacker technique
[70| and [71]. PPT has been used in trying to understand the optical behavior of
interstellar dust [72]. For optically active spheres, Singham showed that the coupled-
dipole method was in good agreement with the exact theory [73]. Lakhtakia has
extended PPT to cover inhomogeneous and even bianisotropic scatterers [74].
The number of discrete dipoles needed in the implementation of PPT can grow
large, up to thousands, or even tens of thousands |72], The method requires effi-
cient computer capacity. From another point of view, there are also various ways
to include the size-dependent terms in the dipole polarizability, and in fact, the
appreciation of dipoles as individual elements brings forth conceptual connections
between PPT and quasistatic mixing rules. There is no sharp line between Rayleigh-
type statics and scattering problems: the low-frequency theory of Maxwell-Garnett
has been, in a natural way, extended [75] up to scattering frequencies. No wonder
then that there exists a close connection from MoM to PPT, as has been shown [74]
for the full bianisotropic case.
6.6 Scattering by Helices
The discussion in the present chapter concentrated so far on the scattering and
electromagnetic response of bi-isotropic objects with relatively simple, or canonical
shapes. The chirality or nonreciprocity of the scatterers was inherently included in
the parameters that describe the material. For example, in the analysis of mixing
rules, which aim at predicting macroscopic chirality parameters, the inclusions were
assumed to be chiral, a priori. The chirality parameter was taken to be known in
the first place.
Hut the ultimate cause for chirality is mirror-asymmetrical geometry. Therefore
it is fair to ask what is the electromagnetic effect of a given left-handed or right-
handed structure. How differently does a specific helix scatter incident left-handed
electromagnetic radiation compared to right-handed waves? How can one predict
the effective permittivity, permeability, and chirality parameters of a medium from
the knowledge of the amount and dimensions of helices embedded in the sample?
To attack the electromagnetic problem involving chiral geometries and struc-
tures, one needs to write boundary conditions on handed surfaces. It is obvious
that the formulation of the. conditions on a helix surface leads to a problem with
far greater analytical difficulties than in the corresponding nonchiral geometries.
Therefore no analytical solutions for the helix scattering problem exist. However,
numerical and approximate efforts in the literature are numerous.
240
Chapter 6. Scattering and Mixing Theories
Modeling of n Single Helix
Starting front the study by Jaggnrd et al. [76], there has been work on a qua-
sistatic approach to the modeling of helix scattering. In these investigations, the
polarizabilities” of the helix are calculated as functions of the static capacitance and
inductance of the straight portion and loop of the helix, and the crosspolarizabilitics
arc connected with the copolarizabilitics. As the polarizability matrix elements are
known, the effective parameters can also be calculated, as has been shown earlier
in this chapter, and discussed in [77] for more complicated scatterers. Qiiasistatic
helix models that arc more detailed than the one in [76] can be found in [78] and
[79].
The problem with the early models has been the narrowband character of their
validity around the resonant frequency of the helix. The helix ensembles are recip-
rocal, and therefore the effective medium must be classified as a sample of Pasteur
medium. This means that the crosspolarizabilitics arc related to each other as
aem = -amf) as can be seen from Equations (G.12) and (6.13). However, this con-
dition has only been satisfied for the resonant frequency of the helix. More work
therefore is needed on the helix polarizabilities, which is indeed being done at the
moment in several places in France and Russia [80]. In these works, solutions for
the current distribution for one-turn helix have been found and the broadband scat-
tering cross-section of the helix has been evaluated, which has good agreement with
the measured values. The limiting equations of the low-frequency polarizabilities
follow the correct frequency behavior, and also the reciprocity condition is satisfied
across the total spectrum.
Consider a helix that consists of two straight sections of a wire and a loop. The
loop has an area S and the straight parts are both of length 1/2. The straight
portions arc connected, parallel, to the ends of the open loop, sec Figure 6.14. If
the unit vector along the straight part direction is u, the dipole moments of the
helix satisfy
Pe = aeeuu  E 4 ererntlii • II,	(6.121)
Pm = «mmHIl • II 4 ttmeUll • E.	(6.122)
The polarizabilities can be expressed in terms of the inductance of the loop L,
the capacitance of the wires (7, and the radiation resistance 7?, which is equal to
the sum of the radiation resistances of two equivalent antennas. The expressions
for ideally conducting wire elements read
*The terms “self-susceptibilities” and “cross-susceptibilities" are used in [76] and the subsequent
publications.
6.6. Scattering by Helices
241
Cl2
1 ~w4C I juCll'
~ 1 - шЧ.С I jwRC’
j/iuiCSl
= 1 - w*LC 4 jjnc'
(6.123)
(6.124)
(6.125)
where /I in the bnrkground permeability. Il in very interesting to note Hint in the
present case the co- and crosspolarizabilitics are not independent but related to
each other by
— UemUfiw"	(6.126)
This reflects the symmetry of the scattering elements: the loop is connected to the
wire in its center, and therefore the input current is the same, both for the straight
parts and the loop. The condition (6.126) is valid for any' current distribution. If
the current distribution can be assumed uniform, condition (6.126) holds even if
the loop is connected at any arbitrary point to the straight part of the wire.
Since the polarizability components are now known, the effective parameters of a
mixture with helices can also be determined using the the Lorenz Lorentz formulas
from S’cclion 6.3. Note that due to (6.126), the effective parameter formulas (6.51)
become simpler as the terms disappear which are quadratic in the number density
of the helices n. The mixture with randomly oriented helices has the effective
parameters®
e'ff - e° + 3D ’
/<elT = /'o +
"«mm
3D ’
(6.127)
к'*т- 37F^d’
where
jrj ________________________________ ।	^^rnrn
9ert 9/r „
Using the polarizability expressions for the helices, it can be seen that the effec-
tive parameters of a mixture obey a Condon-type dispersion, discussed in Section
6.4. In particular, the macroscopic permittivity and permeability will also become
dispersive with the same resonant frequencies as the chirality parameter.
The quasistatic model with the wire capacitance and the loop inductance pre-
dicts well the polarization behavior of the helix up to the first resonance. This reso-
nance corresponds approximately to the condition that the total length of the wire
°Due to the fact that the helix polarizabilities are uniaxial, in the Lorenz Lorentz formulas the
averaging brings forth an extra factor of 1/3, compared to the isotropic case of Section C.3.2.
242
Chapter 6. Scattering and Mixing Theories
composing the helix is one half of the wavelength. To construct a more broadband
helix model, one needs to replace the capacitance and inductance in the polarizabil-
ity expressions by the input impedances of the corresponding wire antennas. These
radiating elements are no longer electrically small.
Figure 6.14 shows the comparison of the calculated scattering properties of a
single helix with the anechoic-chamber measurement result, both performed at the
French Atomic Energy Commission, C.E.A.-CESTA, Le Barp [81]. The one-turn
helix under study consists of two straight parts of a metal wire, and between these
there is a loop. The figure proves clearly the strong and broadband resonance struc-
ture in the RCS of the helix scatterer. Here the calculated curve is a result of a
numerical solution of the surface integral equation, using the thin-wire approxima-
tion for the helix current.
о
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
— — ~ Code Ficelle
------- Measurements
10	15	20
Frequency (GHz)
Figure 6.14 Comparison of the calculated and measured backscattering of a helix from
C.E.A.-CESTA, Le Barp, France. The helix consists of two straight wire parts (each of
length 12 mm) and a loop with diameter 6 mm. The thickness of the wire is 0.2 mm.
(Reproduced by courtesy of F. Mariotte.)
Ensemble of Helices
How, then, does a collection of helices respond to electromagnetic excitation? The
helix polarizabilities can be integrated to get the macroscopic parameters as was
6.6. Scattering by Helices
243
done above. But brute-force numerical approaches for helix aggregates have also
been presented in the literature.
An ensemble of helices scatters differently if illuminated with left- or right-
i handed circularly polarized wave. Physical chemists have directed interest in the
| numerical determination of this difference. Circular Intensity Differential Scattering
i (CIDS) is the term for preferential scattering of light of circular polarization by
chiral structures |82j. The helical structures are not restricted to be electrically
small, but they are modeled with helical arrays of dielectric spheres or ellipsoids
I |83], treated as point dipoles. Using Born approximation and rotation matrices,
।	CIDS from the ensemble can be enumerated. |84] goes beyond Born approximation
•	in calculating the internal fields of the elements, which are disks in this reference.
. The numerical analyses have covered even hierarchical chiral structures, e.g., a helix
made of helical wire [85].
Numerical leatments of helical ensembles have also been performed from the
electromagnetic point of view. In addition to the low-frequency approach to helix
polarizabilities discussed above, the efiective electric and magnetic dipole moments
of the helix can be calculated from the far fields radiated by the helix. In |86], this
has been done using multiple scattering theory for the element scatterers.
The electromagnetic scattering from a system of helices is difficult and for nu-
merical analysis, large computer capacity is required. On the other hand, the re-
ward is great: effective material parameters for a microscopically well defined chiral
sample. There exists the result by Whites [87], which assigns magnitudes of effec-
tive chirality quantities for a given helix configuration. Here, a classical Method
of Moments solution with triangular basis functions for the helix wires involves
Monte-Carlo simulations and tries to replace the helix ensemble with similarly be-
having homogeneous chiral sphere. The objective is to relate the helix dimensions
and concentrations with macroscopic material parameters. The preliminary results
predict values of the order of tenths of millimeters for the chirality parameter fl
at microwave frequencies. Using the formulas of Appendix A, the figure translates
approximately to к < 0.1.
In the simulations of [87], the metal volume fraction was 0.6% of the total
volume of the medium. Il is interesting to compare this value with manufactured
chiral samples [88], where the volume fraction of 3.2% seemed to result in the best
performance in terms of RCS reduction. Confirmation for the calculations of [87]
arises from the comparison of the chirality values with those of Table 1.2. There,
slightly higher chirality magnitudes than 0.1 are seen, which probably is due to the
denser helix concentrations than in the numerical study by Whites.
I
244
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of lossy dielectric, chiral, nonsphcrical objects,” Applied Optics, Vol. 24, No. 23, 1985, pp.
4146-4154.
[68]	Barber, P.W. and S.C. Hill, Light Scattering by Particles: Computational Methods, World
Scientific, Singapore, 1990.
[69]	Harrington, R.F., Field Computation by Moment Methods, McGraw-Hill, New York, 1968;
Reprint Edition, Robert E. Krieger Publ. Co., Malabar, Florida, 1982.
[70]	Purcell, E.M. and C.R. Pennypacker, “Scattering and absorption of light by nonsphcrical
dielectric grains,” The Astrophysical J., Vol. 186, 1973, pp. 705-714.
[71]	Draine, B.T., “The discrete-dipole approximation and its application to interstellar graphite
grains,” The Astrophysical J,, Vol. 333, 1988, pp. 848-872.
[72]	Lumme, K. and J. Rahola, “Light scattering by porous dust particles in the discrete-dipole
approximation,” CSC Reprint Series, 2/93, Center for Scientific Computing, Espoo, Finland,
1993. To appear in The Astrophysical J.
[73]	Singham, S.B., “Intrinsic optical activity in light scattering from an arbitrary particle,”
Chemical Physics Letters, Vol. 130, No. 1-2, 1986, pp. 139-144.
[74]	Lakhtakia, A., “General theory of the Purcell-Pennypacker scattering approach and its ex-
tension to bianisotropic scatterers,” The Astrophysical J., Vol. 394, 1992, pp. 494-499. For
dielectric scatterers, the connection between PPT and the electric field integral equation was
provided already in Lakhtakia, A., “Macroscopic theory of the coupled dipole approximation
method,” Optics Communications, Vol. 79, No. 1-2, 1 October 1990, pp. 1-5.
[75]	Lakhtakia, A., “Site-dependent Maxwell-Garnett formula from an integral equation formal-
ism,” Optik, Vol. 91, No. 3, 1992, pp. 134Л37; Shanker, B. and A. Lakhtakia, “Extended
Maxwell Garnett formalism for composite adhesives for microwave-assisted adhesion of poly-
mer surfaces,” J. of Composite Materials, Vol. 27, No. 12, 1993, pp. 1203-1213. This theory
for dielectric-in-dielectric composites has been generalised to chiral mixtures in Shanker, B.
and A. Lakhtakia, “Extended Maxwell Garnett model for chiral-in-chiral composites,” J. of
Physics, D: Applied Physics, Vol. 26, 1993, pp. 1746-1758.
[76]	Jaggard, D.L., A.R. Michelson, and C.H. Papas, “On electromagnetic waves in chiral media,”
Applied Physics (Springer-Verlag), Vol. 18, 1979, pp. 211-216.
[77]	Weiglhofer, W.S., A. Lakhtakia, and J .C. Monton, “Maxwell-Garnett model for composites of
electrically small uniaxial objects,” Microwave and Optical Technology Letters, Vol. 6, No. 12,
1993, pp. 681-684; Weiglhofer, W.S. and A. Lakhtakia, “Electromagnetic wave propagation
in super-cholesteric materials parallel to the helical axis,” J. Physics, D: Applied Physics,
Vol. 26, 1993, pp. 2117-2122.
References
249
[78]	Zouhdi, S., Л. Fourrier-Lamer, and F. Mariotte, “On the relationships between constitutive
parameters of chiral materials and dimensions of chiral objects (helices),” J. of Physics, III
(France), Vol. 2, 1992, pp. 337-342.
[79]	Arnaut, L.R. and L.E. Davis, “Chiral properties oflossy n-lurn helices in the quasi stationary
approximation using a transmission-line model,” Proc. 23rd European Microwave Conf.,
Madrid, Spain, Sept. 6-9, 1993, pp. 176-178.
[80]	See, e.g., the papers by F. Mariotte and Л. Vinogradov in the Bianisotropics '93 Workshop,
held at the Gomel State University, Belarus, Oct. 12 14, 1993. Proceedings, Sihvola. A.,
S. Tretyakov, I. Seinchenko (editors), Helsinki University of Technology, Electromagnetics
Laboratory Report Series, No. 159, December 1993.
[81]	Mariotte, F., private communication.
[82]	Bustamante, C., M.F. Maestre, and I. Tinoco, Jr., “Circular intensity differential scattering
of light by helical structures. I. Theory,” J. of Chemical Physics, Vol. 73, 1980, p. 4273;
II. Applications, Vol. 73, 1980, pp. 6046; III. A general polarisability tensor and anomalous
scattering, Vol. 74, 1981, p. 4839; IV. Randomly oriented species, Vol. 76, 1982, p. 3440.
[83]	Belmont, A.S., S. Ziets, and C. Nicolini, “Differential scattering of circularly polnrired light
by chromatin modeled as a helical array of dielectric ellipsoids within the Born approxima-
tion,” Biopolymers, Vol. 24, 1985, pp. 1301-1321.
[84]	Ilaracr, R.D., L.D. Cohen, A. Cohen, and R.T. Wang, “Scattering of linearly polarited
microwave radiation from a dielectric helix,” Applied Optics, Vol. 26, No. 21, 1987, pp.
4632-4638.
[85]	Patterson, C.W., S.B. Singham, G.C. Saliman, and C. Bustamante, “Circular intensity
differential scattering of light by hierarchical molecular structures,” J. of Chemical Physics,
Vol. 84, No. 3, 1986, pp. 1916-1921.
[86]	Varadan, V.V., A. Lakhtakia, and V.K. Varadan, “Equivalent dipole moments of helical
arrangements of small, isotropic, point-polarixable scatterers: Application to chiral polymer
design,” J. of Applied Physics, Vol. 63, No. 2, 1988, pp. 280-284.
[87]	Whites, K., “Surmounting numerical limitations in the full-wave simulation of chiral mate
rials,” Proc. 1993 URSI Radio Science Meeting, Ann Arbor, Michigan, June 28 - July 2,
1993, pp. 243.
[88]	Guire, T., V.V. Varadan, and V.K. Varadan, “Influence of chirality on the reflection of EM
waves by planar dielectric slabs,” IEEE Trans, on Electromagnetic Compatibility, Vol. 32,
No. 4, 1990, pp. 300-303.

Chapter 7
Measurement Techniques
In this chapter we discuss various methods to determine the material parameters
of bi-isotropic samples using electromagnetic measurements. The theory developed
so far in this book has brought forth formulas that express the effect of material
parameters on the electric and magnetic fields. The problem to be treated in the
present chapter is the inversion of these relations; i.e. explicit relations are sought
which give the material parameters as functions of measurable quantities. Examples
of quantities that have to be measured for this purpose are the complex values of
scattering matrix elements, resonant frequencies and quality factors in resonators,
and co- and crosspolarized reflection coefficients.
The electromagnetic methods used in the characterization of media often have
the advantage of being nondestructive. Il is only invisible fields that touch the ob-
ject, although for certain waveguide measuring systems samples have to be machined
due to fixture restrictions. Classical metrology has provided us with several algo-
rithms to determine the permittivity and permeability of isotropic and anisotropic
materials. For the case of measuring more complex media, some new principles or
supplementary measurements are required. This is a natural consequence of the
fact that the characterization of bi-isotropic materials requires a broader set of pa-
rameters than for classical media. The measurement methods will be discussed in
the following according to the classification into three groups: ellipsometric reflec-
tion measurements, normal-incidence slab measurements, and waveguide/resonator
measurements. The first two could be termed free-space techniques whereas the
third type employs closed structures. The practical choice of the most suitable
measurement technique, depends on the size and shape of the sample, on the oper-
ating frequency, and other factors.
In free-space techniques, the retrieval of parameters is based on the theory of
plane wave reflection and transmission in planar Pasteur, Tellegen, and general
bi-isotropic structures. Thick and lossy samples can be analyzed using the model
of a bi-isotropic half-space, for which case Chapter 3 has already given us the
251
252
Chapter 7. Measurement Techniques
relatively simple specular reflection coefficients. In the case of a planar slab, both
reflection and transmission measurements can be made, and one can characterize the
sample using only normal-incidence measurements. Cavity resonator and waveguide
techniques arc usable for small samples of different shapes. Here, the pcrturbational
analysis shows the way to determine the constitutive parameters.
After the theoretical analysis of the parameter retrieval and algorithm descrip-
tions, practical topics will also be addressed in the present chapter. What arc. the
main sources for error in the measurements, and what is the suitability of a par-
ticular measurement system for different types of samples? Some attention will be
reserved for one existing measurement system.
7.1 Ellipsometric Measurements
The properties of a sample of unknown bi-isotropic material can be determined by
measuring how it reflects plane electromagnetic waves. The reflection coefficient
matrix was derived in Chapter 3 for the case of a planar interface between isotropic
and bi-isotropic media, and the inversion of these relations allows us to calculate
the material parameters from measured co- and crosspolarized reflection coefficients.
Using the generalized Fresnel reflection coefficients in material measurement implies
two assumptions. First, the surface of the sample has to be smooth, a plane. Second,
the sample needs to be sufficiently thick in order that the half-space reflection
coefficients are relevant.
It is a common way to measure optical properties of unknown materials us-
ing polarized light reflection principles; often the term ellipsometry is used for the
systematic determination of electromagnetic parameters [1, 2j. The principles of el-
lipsometry can be generalized for bi-isotropic material surfaces using the reflection
coefficient relations (3.221):
R -	~ 1?°)C<>(e+ + c-) + 2W<№> ~ ctc_) costf
(q2 + q2)c„(c+ + c_) + 2qq„(c2 + c+c_ ) cos i? ’
R = 2,?|?»Ь(С^ ~ c- ) cos - (c+ + c_ ) sin i?)
(’72 + ’Zj)co(c++ c_) + 2q>7o(c2 + c+c_)cosi?’
R = 2W»cotj(c+ ~ c_)cosr? + (c+ -) c_)sinr?]
VI	(l1 +	iJHT	+ c_) + 2qq„(c2 + c+c_) cos i? ’
R	-	~	+ c~) ~ 2W°(C» - cFc_)cos r?
n	(»/2 +	»?2)c„(c++ c_) + 2qq„(c2 + c+c_)cosd’
Here co = cosfl„ and c± = cosf?±. The incidence angle is 6„ and the two refracting
angles 8± are determined by Snell’s law: n±sin0± = sinfl„. As usual, nt =ncosi?±
7.1. Ellipsometric Measurements
253
к. The parameters with subindiccs о refer to the isotropic half space (air, n„ — 1)
and those with no index are connected with the bi-isotropic one according to Figure
7.1.
Figure 7.1 Reflection from a bi-isotropic half space. Note the two refracted rays due to
the Pasteur parameter splitting the refractive index.
Of course, the classical meaning of ellipsometry is narrower. The term stands
for determining the real and imaginary part of the refractive index of the unknown
material from the measured state of the reflected light from the surface of the ma-
terial. Plane polarized light becomes elliptically polarized, and the parameters of
the ellipse contain information about the material [3]. Photometric measurements
concentrate on the intensity ratios of the two linearly polarized states, and polari-
metric methods on the phase changes. In the bi-isotropic measurement mission, one
has to look for generalizations of ellipsometry, photometry, and polarimetry.
Indeed, there are four measurable quantities for four unknowns in (7.1) to (7.4),
and at least with numerical inversion, the problem seems to be solvable even with a
single incidence-angle measurement where all parameters have effect. In addition,
further accuracy can be gained by repeating the measurement for another aspect
angle. Let us dwell on this question somewhat further.
7.1.1 Linear Polarization Measurements
For normal incidence (co = 04 = c_ = 1), an incoming linearly polarized wave
will be reflected with copolarized and crosspolarized components. The reflection
coefficients can be seen to be
„2	„2
V ~Vo
7?co —	Hyy —	-	_	Л»
V + % + 2VVo cos v
(7-5)
254
Chapter 7. Measurement Techniques
= — //v«
—2гр)„ sin i?
q2 f q’ I 2qq„ cos i?'
(7.6)
where q — yjfi/e is the wave impedance of the bi-isotropic medium, qo =	is
that of the isotropic half space (where the wave is impinging from), and sin i9 = xln
gives the nonreciprocity of the medium. Figures 7.2 and 7.3 illustrate the behavior
of these reflection coefficients as functions of the material parameter contrasts.
Figure 7.2 The copolarized reflection coefficient Jlco for a linearly polarized wave, nor-
mally incident from air onto bi-isotropic half space, as a function of the impedance ratio
q/q„ and the nonreciprocity parameter 19. Note that the Pasteur parameter к of the BI
medium has no effect on the reflection coefficient.
The two straightforward reflection measurements can be directly inverted, and
two parameters of the medium can be determined:
2, 2 = (1 4 W + Hl
11‘°	(1 - Лео)’ + II2,
(T.l)
and
sin = -=======2^=========.	(7.8)
/1(1 F /С)2 I Л2,]((1 - До)2 I Л’г]
Further measurements are needed for the retrieval of the remaining parameters.
One possibility is to look for the Brewster angle. In normal optical usage, the Brew-
7.1. Ellipsometric Measurements
255
ster angle is the angle of incidence 6b for which one of the polarizations (vertical, or
parallel, polarization) does not suffer any reflection from the surface. Another, and
equivalent, way of defining (>b is the angle of incidence for which the reflected wave
is totally polarized. In other words, for Brewster incidence, one of the elliptical
cigcnpolarizations will be totally transmitted through the interface, and hence any
(unpolarized) incident field will have a definite polarization after reflection. For the
case of simple isotropic materials, this polarization state corresponds, of course, to
linear polarization.
Now it is easy to see that for Tellegen medium (к = 0), the measurement of бд
gives the total retrieval of all three parameters. The Brewster angle can be shown
to satisfy (с/. Equation (3.252)) the equation
tan2flB ±(1 _n-z)(/?T.i)Z>
(7.9)
where
(’/i : уо)(п -’/«)
(’N t q„)(’f- 4 V»)’
which is known from the two earlier measurements as — i] exp(^=jt?). Hence,
are known, or equivalently	ant^ ^Ie Tellegen problem is solved. The
Pasteur case is a little more complicated because of the two different wave numbers
in the bi-isotropic medium. More than one scalar measurement is needed, c.g. the
256
Chapter 7. Measurement Techniques
whole reflection matrix.
7.1.2 Use of Eigenpolarizations in Reflection
The inversion algorithm described above only required measurement of linear po-
larization components. If one, however, is able to perform measurements fully
polarimetrically, the material parameters can be retrieved in an alternative man-
ner. The eigenpolarizations in reflection from bi-isotropic interface arc elliptically
polarized fields, as was shown in Chapter 3. These fields diagonalize the reflection
matrix (7.1) to (7.4). The diagonalized matrix contains two reflection coefficients
Hi, III, one for each eigenpolarizalion. ЛИ four parameters of a fully bi-isotropic
sample can be determined using these quantities.
The reflection coefficients R,,R2 have been shown [4] to satisfy the relation
Ri - Rj _ y^Uos^tTi
1-RiR, Qcost? ’	'
where
cos cos fL + cos’
cos -| cos 0_
Therefore, as d and q are known from the simple normal-incidence measurement
(7.7) and (7.8), this quantity gives us Q. Knowing the Brewster angle Og, which is
related to the material parameters according to (3.252), the information is sufficient
for total retrieval: the quadruplet (>?,1?,<2,#в) yields (c,/«,«,%).
7.1.3 Differential Circular Reflection
Often, in actual practice, a rigid and formal four-parameter inversion is not the best,
or at least, not the most robust way of determining the magnctoelectric parameters.
One has to care about algorithm sensitivity on measurement uncertainties. It is
better to look for quantities on which the effect of the unknown parameter is large.
From this approach, an efficient way of retrieving the Pasteur parameter of chi-
ral materials is to measure the Differential Circular Reflection (DOR), a technique
developed by Silverman [5]. DCR is based on the observation that the chirality
parameter has weak effect on the absolute magnitude of the reflected signals. This
is evident from the analysis so far in this book, even to the extent that for nor-
mal incidence, the chirality cannot be sensed at all (see Equations (7.5) and (7.6)).
Therefore it is better not to measure the reflection amplitudes for the various po-
larizations but rather reflection coefficient differences.
In particular, a sensitive measure for the Pasteur parameter is the difference
between the left- and right circularly polarized reflected intensities IL and 1ц- This
7.1.Ellipsometric Measurements	zo<
fact can be justified with the following reasoning. In case of reflection from ordinary 'j
isotropic-isotropic interface, the eigenpolarizations arc vertically and horizontally
polarized fields, which have different reflection magnitudes. Therefore the difference
of these reflections is nonzero even for the nonchiral case. However, as one considers
the case of the interface reflecting circularly polarizedincidcnt waves, the response
is the same in magnitude for LCP and RCP, and the difference is zero. But if
chirality is added to the problem, the reflection coefficients start to differ. The
difference is linearly proportional to к for small chirality parameter values; it is zero
for normal incidence and reaches a maximum at some angle beyond the Brewster
angle. An unambigious optical measurement of the chirality in camphorquinone
methanol solution was reported in |6| where the measured value of the chirality
parameter was of the order of к ~ 10~7. Comparing this value with the figures in
Table 1.2, the DCR method can indeed be appreciated. Experimental evidence is
also provided by Figure 7.4 where the dispersion in DCR = (It,	I In) is
shown.
Figure 7.4 Differential Circular Reflection DCR (xlOE) is a measure of the chirality
amplitude, here shown after a single chiral reflection (D) and two reflections (2D) as
a function of wave number. Measurements are made by Silverman, Badoz, and Briat.
(Reprinted with permission by the Optical Society of America and the author.)
Although DCR provides us a way to sense chirality with oblique reflection mea-
surements, it seems clear that reflection principles are actually better suited for
determining the Tellegen parameter. Pasteur parameter, on the other hand, affects
most strongly the signal transmitted through the medium. Let us focus in the se-
quel on the problem of how transmission measurements can be exploited for Pasteur
parameter retrieval.
258
Chapter 7. Measurement Terhniques
7.2	Reflection and Transmission from Slab
I'hc theory of Chapter 3 for plane wave reflection and transmission serves as the
basis for calculating the material parameters from measurements of the electromag-
netic response of a bi-isotropic slab. However, it has already turned out that for
novel bi isotropic materials, it is neither a trivial nor an easy task to extract the
material parameter values from the experimental data which arc usually in the form
of complex reflection and transmission coefficients. Employing the exact formulas
for reflection and transmission coefficients at arbitrary angles of incidence, one can
in principle solve material parameters by numerical techniques. For slab measure-
ments al the normal incidence, it appears possible to find the material parameters
by direct inversion of the reflection and transmission coefficients. The latest litera-
ture already contains expositions about free-space measurement strategics of chiral
and bi-isotropic materials |7| to 110|.
7.2.1 Pasteur Media
Reciprocal chiral materials arc characterized by three complex constitutive param-
eters if losses are taken into account. Measuring these three parameters appears
to be a task not too much more complicated than isotropic material parameter
determination, and a method for material parameters retrieval from free space re-
flection and transmission measurements has been presented in [8|. Л chiral slab is
illuminated by a linearly polarized wave at normal incidence. From the measured
co and crosspolarized transmission coefficients, the complex chirality parameter
к - Kre J^iin tan be determined, both its real and imaginary parts.
Due to the polarization plane rotation and the circular dichroism, the field trans-
mitted through the slab is in general elliptically polarized. The real part of the
chirality parameter Kre can be determined through the angle ф between the polar-
ization direction of the incident electric field and the direction of the major axis
of the transmitted field polarization ellipse. This angle can be determined if the
transmitted field has been measured at two angles by rotating the receiving an-
tenna, because these complex measurements determine the ellipse of the wave. The
following relation between the angle ф and the real part of the chirality parameter
is valid:
sin(2«rcA:od) — sin(2</»),	(7.11)
where ko is the free-space wave number and d is tlie slab thickness. This relation,
expressing the fact that the rotation angle is linearly proportional to the Pasteur
parameter is in accord with the theoretical results on wave propagation in chiral
media.
7.2. Reflection and Transmission from Slab
259
The imaginary part of the chirality parameter K;m can also be found as the
ellipticity of the transmitted field has been measured. If e is the ratio between the
major and minor axes (the ellipticity) of the polarization ellipse of the transmitted
field (1 < e < oo), we have [8]
arccoth(|e|) 1	|e| -| 1
Kim = M = 2M П je|-l ’
(7.12)
To remove uncertainties in the solution of (7.11) it lias been suggested in [9] that
one extra measurement at oblique incidence be performed.
It is quite charming to note that in this method, the chirality parameter can
be measured independently of the other two materia) parameters. On the other
hand, one cannot determine the permittivity and permeability without knowing
the chirality parameter value. Provided the chirality parameter has been measured,
the other two constants can be determined from the reflection measurement at
normal incidence, combined with the copolarized transmission data. The remaining
parameters e and )i can, in fact, be determined [8] similarly to the classical methods
used for nonchiral materials (see, e.g. |H. 12])-
The following explicit inversion algorithm for all three complex material pa-
rameters is suitable for practical measurements [13]. One needs to measure the
copolarized reflection coefficient 5ц. In addition, the transmission coefficient has
to be determined for two positions of the receiving antenna: 52i is the copolarized
transmission, and 5^ is the measured copolarized transmission as the receiving an-
tenna is rotated by the angle a. The measurement situation is depicted in Figure
7.5.
d
Figure 7.5 The required S parameter measurements for the determination of the material
parameters of a chiral slab.
With these measurements, the three material parameters can be calculated from
the following equations:
260
Chapter 7. Measurement Techniques
/I	1 | Г kc	e	1 Г Av
/<„ 1 - Г fc,,’	<0 1 I 1'Av*
arctan G + rnn
к----------- ----------,	(„1 zz 0,1,2,•••)
fc„d
(7.13)
(7.M)
with
г = к d \/F2 - i,
5?,- 52,(1-1 G,2)l 1
25,,
c=5£,------5„cosa, fcc = i(Jn T + n2a), (n = O,T1, T2, • -
52i s>n a	d
5,, + 5г,уТТ(72 - Г
1 - (5„ + S„t/lW
and again, k„ is the free space wave number and d is the slab thickness.
In these formulas, the integers n and m express the multiple-wavelength ambi-
guities that can be removed by repealing the measurement at another frequency or
with another slab thickness.
It is interesting to note that this inversion algorithm reduces to the classical
nonchiral material measurement procedures, that are iccommended in the applica-
tion notes of network analyzer manufacturers. In case of a slab of nonchiral material
(a = 0), the parameter G vanishes, and the equations of |11| are recaptured.
The measurement technique becomes more involved for nonreciprocal materials.
However, the chirality parameter can still be determined by the same procedure as
for Pasteur media, since in the measurements at normal incidence the polarization
state of the transmitted field is not affected by the Tellegen parameter. On the
other hand, the above mentioned procedure for determining permittivity and per-
meability cannot be applied to the general case, because the copolarizcd reflection
and transmission coefficients depend on the nonreciprocily parameter. Also, we
need a new procedure for measuring the non reci procity parameter itself.
'Phis fact can be compared with the analogous situation noted in Section 7.1;
there, in the reflection measurement, the Tellegen parameter was easy to extract but
and к were intertwined in the measurable quantities. Now, for the transmission
case, the Pasteur parameter emerges from the results in the most natural way, but
\ affects heavily the retrieval of < and /i.
7.2.2 Small Tellegen Parameter
If the material of a slab is non reciprocal, the field reflected from a slab at normal
incidence changes its polarization state. The ratio of the cross- and copolarized field
components reflected from a slab in air is given by the analysis in Chapter 3:
7.2. Reflection and Transmission from Slab
261
(7.15)
This is proportional to the normalized nonreciprocity parameter sin il. Here, rj —
yjli/t and Г]о is the free-space wave impedance.
Il is interesting to note that the ratio (7.15) does not depend on the slab
thickness,' so the measurement can be most reliable for thicknesses (or frequen-
cies) which correspond to reflection maxima. To determine the Tellegen parameter
X from the value of a, which can be directly measured, we have to know the wave
impedance у or the values of the dielectric permittivity c and the magnetic perme-
ability ft.
As was already mentioned, the known procedure developed for isotropic mate-
rials in (11) and generalized for reciprocal chiral slabs in [8] is not suitable for the
general case. However, it can be applied if the normalized nonreciprocity parameter
sin 1? is small, so that the second order terms O(sin’d) may be neglected. Under
this approximation, the nonreciprocity gives rise to a first-order crosspolariz.cd com-
ponent in the reflected field, whereas the transmission dyadic and the copolarized
reflection coefficient do not depend on the nonreciprocity parameter у. This means
that the procedure described in (8) can be used to measure c and fi of the material.
Finally, when c and fi are known, the complex value of the nonreciprocity parameter
can be calculated through the equation (7.15). This assumption, x’/n’ “K 1> holds
probably quite well in most practical cases.
7.2.3 Large Tellegen Parameter
In the case of large nonreciprocity parameter, the permittivity and permeability in
general cannot be determined independently of the nonreciprocily parameter. The
relatively large magnitude means that |y| cannot be considered much smaller than
the refractive index n. Therefore, an extra measurement is needed. The supplrmen
tary information can be gained by measuring the co- and crosspolarized reflection
coefficients for the slab positioned on a metal surface. This method also helps to
avoid complicated numerical procedures. The ratio of the cross- and copolariz.ed
field components, reflected from the metal backed slab, equals (see (3-109) and
(3.110))
2»/>^osin dsin2(kdcosrJ)
ЯЦ Ч2 cos2 г? 1 (rj1 - r;’)sin2(A'dcos r?)’
where k — Ыу/ejl. Substituting (7.15), we arrive at the equation
(7.16)
lTherefore this ratio is also compatible with the half-space reflection equations (7.5) and (7.6).
262
Chapter 7. Measurement Techniques
1 “ Й') si,,2(Wcos’’) = Y -C;S /	(7>7)
A /	1 - I'M
Denote by fli the measurement result for the same ratio of the reflection coeffi-
cients (7.16), but for a slab of the double thickness 2d. Then the equation
a
A
sin’ (2kd cos i?) = ---——
1 Y'U'loY
(7.18)
holds. The system of two equations (7.17) and (718) can be solved for kd cos 17:
cos2(Wcosi?) =	(7.19)
4A(A - «)
After some simple algebra, the equation for the impedance 1] follows from (7.15)
and (7.17):
where
A(A___\
4 A (A ~ «)/
Finally, when the impedance is known, the nonreciprocity parameter can be found
from (7.15).
In the present method, four material constants are determined from four mea-
surements of complex scattering parameters. From the transmission coefficients the
chirality parameter can be calculated, and the other three can be solved from the
reflection coefficients of slabs in air and on a metal screen. To remove uncertainties
due to multiple, solutions of the equations (7.19) and (7.20), still more measure-
ments for slabs of different thicknesses or at oblique incidence may be necessary.
As it appears, in measurements at normal incidence the material parameters can
be solved from rather simple equations (7.15), (7.19), (7.20). lit measurements at
oblique angles the material parameters can be extracted by numerical techniques
only.
7.3	Waveguide and Resonator Techniques
lii parameter determination of simple isotropic materials, reflection and transmis-
sion measurements of waveguide sections filled with the material are often used.
7.3. Waveguide and Resonator Techniques
263
For bi-isotropic composites, however, this way appears rather difficult to imple-
ment. The most crucial problem here is the material parameter extraction from
the experimental data. The theory of chiral and bi-isotropic waveguides is rather
involved, and numerical procedures are required even al the stage of eigenwave prob-
lem solution for regular guides (sec Chapter 4). If the sample is small enough, the
perturbational analysis can be applied, and its results allow clear physical interpre-
tation [14, 15, 16J. In this section the chirality and the nonreciprocity parameters
are determined with the perturbation method.
7.3.1 Waveguide Perturbation
In this section a waveguide with an arbitrary cross section and ideally conducting
walls is considered. Field vectors of propagating modes in an empty waveguide have
dependence on the longitudinal coordinate z. A bi-isotropic rod of a small
cross section positioned in the waveguide as shown in Figure 7.6 causes a small
change in the propagation factor of the empty waveguide which can be measured.
The unperturbed electric and magnetic fields Eo and 11„ inside the waveguide
satisfy the Maxwell equations, where the e,wl time dependency is assumed and the
subindex t denotes to transverse components:
V(xE„ I = jflau, x E„,	(7.21)
V, x H„ - jwc„Eo = x H„.	(7.22)
For the perturbed fields E and II the Maxwell equations are
V( x E = j/hr, x E,	(7.23)
V< x II - jwl) jfln, x II,	(7.24)
where /1 is the perturbed propagation factor. The constitutive equations inside the
rod are those of the bi isotropic medium while outside the perturbation inclusion
these relations are those of isotropic medium, D = c„E, 11 = poII.
In the conventional way [17], by developing the expression
n; v, x e + и v, x e; - e; v, x и - e v, x h;
and integrating it over the cross-section area S of the waveguide, the relation for
the change of the propagation factor Л/1 = /? — /?» can be obtained:
лр - p I [(< • UE • e; I (/< - /'JH • и; 1 (П • e; i <e  n;j ds. (7.25)
°6S
264
Chapter 7. Measurement Techniques
Figure 7.6 The empty waveguide and the waveguide with a thin bi-isotropic rod.
Here US is the cross-section area of the bi-isotropic rod, and the asterisk denotes the
complex conjugate operation. The factor in the denominator represents the power
propagating in the waveguide
л = / (E; x но + Eo x h;) • ux ds,	(7.26)
s
where it is assumed that with an inclusion of a small sample we can approximate E
E„, H ks when integrating over the waveguide cross-section area. The boundary
integral term around the cross section of the ideally conducting waveguide walls
vanishes. The two last terms in (7.25) give first-order change in the propagation
factor with respect to the nonreciprocity parameter x and the chirality parameter
к.
Di-Isotropic Circular Rod
To determine the fields inside the bi-isotropic inclusion of a small circular cross
section the quasi-static approximation can be used for the transverse fields inside
the bi-isotropic rod (18), [19]:
/ E( \ ______________2___________
\ П« /	(/'- + !)(«r + О - (x1 + K’)
Mr + 1 __ ~(x - J'OyT'./e» \ ( E(o \
-(х + лО/^/Д» «г + 1	/ \ II<» /
(7.27)
Since the axial components of the electric and magnetic fields are tangential
to the surface of the rod, these field components are continuous on the sample
surface. The axial electric field inside the bi-isotropic rod may then be written as
Ez == Ezo and the magnetic field as IIZ = IIZO. Inserting the quasi-static fields into
7.3. Waveguide and Resonator Techniques
265
the expression for the change of the propagation factor (7.25), we have finally the
change for the propagation factor
Д/4 = у J (<« [£[(<, -	4 1) - (x’ + к’)]|Е|„|г + (r, - 1)|E„|’]
f>S
+/le [i(('tr ~1)(t'4 ° “(x’+ 'C’NIH'’I1 + ~ OH'-I’]
Ix^ [у-«{e„ • ii;j -123?{E,„//;„}]
-«v^[£g{e-»-hU + 2 9{E„//;„}])dS,	(7.28)
where Aw = (/t. + l)(e. 4- 1) — (x2 4 к2). It is clearly seen that, by using a small
perturbation approximation to achieve the first-order change in the propagation
factor due to the nonreciprocity factor %, the unperturbed field configuration for
the propagating mode must have a nonzero value for J?{E„ I1‘}. 'Го delect the first
order change due to the chirality parameter к, the unperturbed fields must have a
nonzero value for 5{EO II*} [20]. In the next subsection it is seen that this is also
valid when using cavity resonator methods for determining material parameters.
7.3.2 Resonator Perturbation
Л perturbation formula expressed in terms of electric and magnetic dipole moments
of a bianisotropic inclusion in cavity resonators with ideally conducting walls has
been published in [21]. Perturbation theory for cavity resonators with small bi-
isotropic inclusions was developed in [14|.
The resonant frequency shift Aw = w — w„ due to a bi-isotropic inclusion shown
in Figure 7.7 is given by the following exact formula in analogy with the simple
isotropic media [17]
A" =	/ [(£ “ £o)E ' E: 4 (/‘ " /Io)H ’	4	' E: 4 <E П:| dV' (7 29)
° tv
where E and II are the fields inside the sample, and E„ and II„ stand for the
unperturbed fields. If the perturbation inclusion is so small that we can replace
the fields by their unperturbed values when calculating the integral over the cavity
volume, then the denominator is proportional to the field energy in the unperturbed
resonator Hz„:
и; = | /(^e. • e; + /.„и.,  n;)</v.	(7.30)
V
The integral in (7.29) is calculated over the inclusion volume 6V only.
ZUU
Chapter 7. Measurement Techniques

Figure 7.7 The empty cavity resonator anil a small bi-isotropic sample inside the cavity
resonator.
Spherical Bi-Isotropic Sample
For a special case of a spherical bi-isotropic sample we can use the quasistatic ap-
proximation (6.7) for the fields inside a bi-isotropic sphere, which gives the following
relations between the fields inside the sample and the external fields
/ E \ =	/	P,f2 “(x-\ / Eo \	-731.
\ H / A' \ ~<X +	er + 2	/ \ П« /
with A. = (p, -| 2)(сг -J 2) — (x2 -| «’). Substituting (7-31) into (7.29) leads to the
expression for the shift of the resonance frequency
Лш = 'нУГ I h1(€r “1)('tr + 2) -(x’+ л2),|Ео,а
° T tv
+м«[(мг - l)(«r + 2) - (x2 + «2)]|HO|2
+ бх%ЛоуЛ{Еон;}-бк^оэ{Ео-н;}]</к (7.32)
To determine the coupling parameters, we should place the sample in such a
resonator and in such a position that the dot product Eo • II* is nonzero. Due to
the obvious orthogonality properties of the fields, Eo • II* always vanishes if this is
a single-mode resonator. However, if we assume that two inodes (one is II mode
and the other is E mode) with the same resonant frequency arc excited, the dot
product Eo-H* is always real for a rectangular cavity resonator And this fact makes
it possible to measure only the nonreciprocity parameter y. Hy Using, for example,
a cylindrical cavity resonator the dot product Eo-H* may be imaginary thus making
possible to measure the chirality parameter к.
7.3. Waveguide and Resonator Techniques	267
7.3.3 Chirality and Nonreciprocity Parameters
Based on the perturbation analysis of the previous subsections, we can now study
possibilities to measure the material parameters by waveguide or resonator tech-
niques. As is seen from (7.28), (7.32), to have a first-order effect on the waveguide
propagation factor or on the resonance frequency of a cavity proportional to the
chirality parameter, we should have a mode such that 9{EO  H*} is nonzero. In
contrast, to have a first-order effect due'to the'non reciprocity parameter, we must
have nonzero the real part of the dot product Eo • H*. In this section we consider
possible ways of achieving desired properties of the modes. Obviously, the inodes
must be degenerate, since otherwise the dot product E„-H* always vanishes due to
orthogonality of the modes.
In the following for simplicity the circular cross-section waveguides and cylin-
drical resonators are considered. However, the theory can be applied to the general
case of arbitrary cross-section guides by substituting corresponding solutions for the
Hertz vectors.
Circular Waveguide
Let us consider the dot product E„-H* for eigenwaves in a waveguide with a circular
cross-section of the radius a and assume ideally conducting walls. For В modes, for
example, the Hertz vector II(p, y?)u, with the potential function
П(р.У’) = Лп(ЛсР)[А cos (гну,) + В sin(rnyr)],	(7.33)
where A, В are the amplitude coefficients, /„(•) is the Bessel function of the first
kind, the transverse wave number = p'„n/a, the propagation factor fl„ =	— k*
and the free space wave number ko =	|22|. The dot product
ЕоН:=;?^^9{|Н&~;9{АН-}	(7.34)
p Oy> Op
is always imaginary. If either A or В is zero, the dot product vanishes. While using
any waveguide with only И modes, one can measure the chirality parameter but
not the nonreciprocity parameter. In analogy, the same result can be stated for the
E modes.
Let us utilize two degenerate propagating modes of different types and consider
the case when one of the modes is an E mode and another is an H mode. For
example, let us have modes Eln and HOn in a circular waveguide. These modes are
degenerate, because kc — p\n/a — Ponla [22], (23]. The dot product of the total
field reads
E„ • h; = (E,„ • h;„ + EOn • H-„) + (Eln  h;„ + e0„ . n;„),
(7.35)
268
Chapter 7. Measurement Techniques
where the indices denote the fields of the corresponding inodes. The first term
vanishes because of the orthogonality of the fields in each modes, but the latter
term remains. Writing the dot product in terms of Hertz potential If for the Я
mode and M for the E mode it is seen that this function is in general complex-
valued and we can have both real and imaginary coupling terms. Specifically, for a
circular waveguide with and Hol modes, wc substitute the Hertz potentials
1И(р,уз) = A Ji(fccp) совуз, H(p) = П Jo(frcp),	(7.36)
where = 3.832/a and the dot product is
E„ • h; = fcj cos^ [лв-[*’л,(М).Л(М) + Р№ср)^р)]
-А'Бк^(кср)^(кер)].	(7.37)
When the Hertz potentials of the two modes are in phase, the dot product is a real
quantity. This allows one to measure the nonreciprocity parameter, since there is a
first-order perturbation of the propagation factor, proportional to the nonreciprocity
parameter. When the modes are 90 degrees out of phase we have imaginary coupling
coefficient, and the perturbation is proportional to the chirality parameter.
Circular Cylindrical Resonator
Let us next study the resonator problem by considering the cylindrical resonator
with ideally conducting walls. The length of the resonator is L and the radius
a. Again, by taking two, for example, H modes, and calculating the dot product
E„ • H*, one arrives at
(7.38)
O<p op
which is always a real function, in contrast with that for the waveguides, and,
hence, the chirality parameter is not measurable. This could have been expected,
since a resonator mode is a combination of two waveguide modes, propagating in the
opposite directions. Perturbations due to chirality cancel out because of the isotropy
and the reciprocity. However, one can measure the nonreciprocity parameter у by
exciting two degenerate H modes (or two E inodes).
Let us consider two degenerate modes of different types, in a cylindrical res-
onator, for example, the modes Elnp and HBnp. Calculating the dot product E„ -H’
shows that it is in general complex-valued. As an example, considering modes EIlt
and Hoil and substituting the Hertz potentials
Л7(р,уз) = AJ^p) cosy,, H(p) = БJo(Kp)
(7.39)
7.3. Waveguide and Resonator Techniques
269
k2 . 2?rz
Е» • H„ = у cosy?sin(-—)
with kc = 3.832/a, and the resonance condition k„ — yfk’ Г (rr/L)’, results in
AB'[k2J0(kcp)Jt(kcp) - n Jo(^p)j;(/=eP)]
(7.40)
Thus, depending on the phase shift between the inodes, the dot product can
be either real or imaginary. When the dot product is a real quantity the chirality
parameter к cancels out from the expression for the frequency shift, and allows us
to measure the Tellegen parameter y. On the other hand, if the dot product is an
imaginary quantity, the nonreciprocity parameter % cancels out from the expression
for the frequency shift, which gives the way for measuring the Pasteur parameter к.
However, in order to determine the Tellegen parameter у and the Pasteur pa
rameter к, we still have to measure first /«, e and the sum x* 4 к2. To determine
these quantities, let us excite one mode (II or E) inside the cavity resonator and
assume first that a sample has such a form that the size in the direction of the
unperturbed electric field is much larger than that in the direction of the magnetic
field, i.e., the sample is a needle or a disc [14]. The electric field inside the sample
is then approximately equal to that outside E ~ E„ and the magnetic flux density
inside is approximately same as outside, В ss Together with the constitutive
relations of the bi-isotropic media we can determine the magnetic field inside the
sample. Substituting the electric and magnetic fields into the expression for the
frequency shift we obtain
^ = -4^/ l(f “	)|E»I2 1 0‘ - rZ’IHJ’U. (7.41)
4 J I	и	ii J
tv	'
If the sample is now located in such a place where the electric field is zero we can
determine the magnetic permeability p.
In analogy when the size of a bi-isotropic sample inside the cavity resonator is
small in the direction of the electric field, we have H ~ H„ and D ss c„Ep. The
corresponding shift in the resonant frequency is now
Д" =	/ [(M ~ ~	-I (‘ ~	(7.42)
° tv
By placing the sample in an area with negligible magnetic field gives us the
possibility of measuring the electric permittivity e. When the permeability p and
the permittivity e are known we are able to measure the sum x1 + K* by placing the
sample so that the magnetic field vanishes in expression (7.41), or the electric field
vanishes in expression (7.42).
270
Chapter 7. Measurement Techniques
7.4 Practical Implementation Aspects
The different measurement principles described above each have their advantages.
First of all, the previous sections have shown that the algorithms to retrieve the
material parameters are quite different for various systems. For open free-space
systems, the inversion is generally easier than for closed systems. In addition to
these aspects, there are practical issues to he considered. These will be touched in
the present section.
Sample Preparing and Bandwidth Issues
The free-space systems are nondestructive for planar samples but they require a
certain amount of noninteracting empty space around the equipment to suppress
interferences from the environment. Complementarity, the closed systems, cavity
resonator and transmission-line waveguide setups need less space and the fields are
confined within the desired structures, being hence insensitive to the surroundings.
However, in transmission-line measurements, the samples have to be machined ac-
curately to fit into the coaxial or hollow waveguides.
Achievable frequency bands are often one criterion for choosing the measure-
ment system. In the open systems, where no cutoffs restrict the propagation of
plane waves, the frequency limitations arise from the antennas and their feeding
systems that are used, or the bandwidth of the network analyzer that drives the
measurement system. In optical measurements, the detector limitations may be
somewhat different in character. In terms of bandwidth, the closed systems are less
versatile: waveguide bands are normally less than an octave, not to mention the
cavity resonators, which give the parameters to be measured at a single frequency.
And furthermore, the resonant frequency is not known a priori, but it is determined
by the very parameters that one wishes to find out.
Homogeneity Problems
A further disadvantage of the closed systems is that for microwave frequencies,
where the sample dimensions are rather small, there appear difficulties in assuming
the sample to be homogeneous. Especially for chiral samples containing macroscopic
metal helices, it may happen that only a few of these helices find their place within
the sample. In addition, as the sample has to be in touch with the metal conductor
of the waveguide or resonator, one has to avoid the galvanic contact from a helix to
the boundaries of the measurement structure. The problem of sample homogeneity
may cause troubles especially in the perturbalional measurement system.
Л.. •!.» лчИ—	*----1:
"7.4. Practical Implementation Aspects
271
metal walls. The air gap produces a large error in the measurement of especially
high-permittivity materials, and to remedy this, one has to use conductive fillers,
lubricants, or paste to eliminate the source of error. Free-space systems do not
suffer from shortcomings of this type.
However, open setups are not totally free from the inhomogeneity problem. The
helices — be these of metal or ceramic type — in the man-made chiral samples
serve as scattering centers for the incoming radiation in the measurement situation.
Diffuse scattering decreases the energy of the coherent component which is assumed
to be the only type of radiation in the analysis, and hence the parameter retrieval
will be contaminated with scattering errors. The magnitude of this error is strongly
frequency-dependent. The helices are normally much smaller than the wavelength.
Analogously with small dielectric Rayleigh scatterers whose cross section is propor-
tional to the sixth power of the radius, one can anticipate that it is at the upper
end of the frequency band where the measurement error starts to escalate.
Sample Anisotropy and Surface Roughness
Aside from the inherent small-scale inhomogeneity of man-made chiral samples,
the unwanted anisotropy may also cause problems for accurate measurement pro-
cedures. As is well known, chirality can be generated with the random mixing of
helices in a host material, but in practice, a high purity of isotropy may be difficult
to achieve. This is due to the fact that as the helices are mixed into the material,
which can be epoxy resin, for example, the difference in the specific weights of the
components may distort the orientation distribution during the hardening process.
The helices may end up being mostly oriented within a horizontal plane, and the
chirality along the vertical normal direction decreases. This is a source for error, if
the sample is assumed to be isotropic in the measurement algorithm.
In the ellipsometric measurement method, the surface characteristics determine
the accuracy to a large extent. And even if the surface roughness is sufficiently
small, within a fraction of a wavelength, there are other limitations. The signal
that is measured is assumed to have reflected specularly from a surface of a semi-
infinite half space. This is an assumption that can be accepted in cases of thick or
lossy samples, or if the incidence angle is close to grazing.
Analyzing Equipment Accuracy
If transmission measurements are also performed, more parameters appear in the
inversion algorithm. The thickness of the slab is a crucial quantity that affects the
S-parameters, and hence errors in determining the thickness of the effective ho-
272	Chapter 7. Measurement Techniques
in automatic network analyzer measurements one can overcome multiple reflection
problems. Modern network analyzer software packages use clever calibrating pro-
cedures to eliminate unwanted systematic errors in the measurement situation.
It is particularly important to emphasize that in practice, the success of the
measurement hinges on the accuracy of the network analyzer and on the precision
of the calibration procedure.
Л further source of error in practical free-space systems is the imperfect polar-
ization purity of the antennas that sense the reflected and transmitted signals. In
transmission through chiral media, the rotation of the plane of linear polarization
is essential, and hence the accuracy in measuring this rotation angle transforms
directly to the determination of the Pasteur parameter. The unwanted polarization
in the measured signals due to antenna crosspolarization also degrade the retrieval
of the imaginary part of the chirality parameter, as the axis ratio of the transmitted
ellipse is measured.
In the measurement of chiral slabs, the determination of the Pasteur parameter
к is based on finding accurately the major axis direction of the transmitted wave
ellipse. Therefore the greatest accuracy could be achieved by rotating the receiving
antenna manually and sensing the strongest signals. However, this cannot be done
in connection with the automatic multifrcqucncy sweep. Measuring the transmitted
field at two (or more) fixed rotation angles could be a sophisticated approach to
increase accuracy of the basic procedure described in Section 7.2.1.
Figure 7.8 shows the Ku-band measurement setup of the Telecommunications
Laboratory of the Technical Research Center of Finland in Espoo. It is a free-
space system where reflection and transmission from a sample arc measured and
processed with an Hewlett-Packard automatic network analyzer. The sample is a
planar slab with minimum transverse dimensions of the order of 5 cm, depending
on frequency. The sample is placed in the focus of the ellipsoidal reflector antennas.
The advantage of the system is that no anechoic chambers arc needed, nor the
compact range setups for realizing far-ficld conditions.
On the other hand, as the inversion algorithm described in Section 7.2 is used,
where plane wave reflection and transmission is assumed, a further source of inac-
curacy arises from the Gaussian beams of this measurement system. One has to
care for waist widths and phase errors at the focus point in order not to violate
the requirements of the plane wave assumption. The extra feature of the system
of Figure 7.8 compared with the measurements for classical isotropic media is that
one has to be able to rotate the receiving antenna in the polarization plane of the
field. The focusing ellipsoidal reflector antenna system shown here is not the only
possibility to achieve free-space conditions in limited space. Other solutions include
using direct radiation with focusing horns, or dielectric lenses to produce plane
waves needed al the sample position.
References
273
Figure 7.fi The free-space material parameter measurement system of the Telecommu-
nications Laboratory of the Technical Research Center of Finland, with capabilities for
chiral measurements. It consists of focusing ellipsoidal reflector antennas and is driven by
the network analyzer HP8510B.
References
[1]	Bash ar a, N.M., A.B. Buckman, and Л.С. Hall, (editors), Surface Science, Vol. 16, Proc.
Symposium on Recent Developments in Ellipsometry, Amsterdam, North-Holland, 1969.
[2]	Aeram, R.M.A. and N. M. Bashara, Ellipsometry and Polarized Light, Amsterdam, North-
Holland, 1977.
(3]	Ward, L., The Optical Constants of Bulk Materials and Films, Bristol and Philadelphia,
Adam Hilger, 1988.
[4]	Sihvola, A. and 1. Lindell, “Properties of bi-isotropic Fresnel reflection coefficients," Optics
Communications, Vol. 89, 1992, pp. 1-4.
|5]	Silverman, M.P., “Specular light scattering from a chiral medium, unambigious test of gy-
rotropic constitutive relations," Lettere al Nuovo Cimento, Vol. 43, No. 8, 1985, pp. 378-382;
Silverman, M.P. and J. Bador, “Light reflection from a naturally optically active birefringent
medium," Journal of the Optical Society of America, A, Vol. 7, No. 7, 1990, pp. 1163-1173.
[6]	Silverman, M.P., J. Bador, and B. Briat, “Chiral reflection from a naturally optically active
medium,” Optics Letters, Vol. 17, No. 12, June 15, 1992, pp. 886 888. Figure 7.4 is reproduced
from Figure 3 of this article.
[7]	Tretyakov, S.A. and D.Ya. Haliullin, “Free-space techniques for biisotropic media parameter
measurement,” Microwave and Optical Technology Letters, Vol. 6, No. 8, 1993, pp. 512-515.
[8]	Oksnnen, M. and A. Hujanen, “How to determine chiral material parameters,” Гтос find
European Microwave Conference, Espoo, Finland, August 24-27, 1992, pp. 195-199.
274	References
[9]	Ougier, S«, I. Chenerie, and S. Bolioli, “Measurement method for chiral media," Proc. 22nd
European Microwave Conference, Espoo, Finland, August 24-27, 1992, pp. 682-687.
[10]	Sihvola, A.Ii. and I.V. Lindell, “Free-space antenna measurement principles for measur-
ing novel bi-isotropic materials," Joumees Int. de Nice лиг lee Antennes (JINA'92), Nice,
November 12-14, 1992, pp. 196-197.
[11]	Ghodgaonkar, D.K., V.V. Varadan, and V.K. Varadan, “Flee-space measurement of complex
permittivity and complex permeability of magnetic materials at microwave frequencies,"
IEEE Trans. on Instrumentation and Measurement, Vol. 39, 1990, pp. 387-394.
[12]	Guire, T., V.V. Varadan, and V.K. Varadan, “Influence of chirality on the reflection of EM
waves by planar dielectric slabs," IEEE Trans, on Electromagnetic Compatibility, Vol. 32,
1990, pp. 300-303.
[13]	Htjanen, A., private communication, 1993.
[14]	Tretyakov, S.A. and A.J. Viitanen, “Perturbation theory for a cavity resonator with a bi-
isotropic sample: applications to measurement techniques," Microwave and Optical Technol-
ogy Letters, Vol. Б, 1992, pp. 174-177.
[1Б]	Tretyakov, S.A. and A.J. Viitanen, “Waveguide and resonator perturbation techniques for
measuring chirality and non-reciprocity of biisotropic materials," Electromagnetics Labora-
tory, Helsinki University of Technology, Report 134, February 1993.
[16]	Viitanen, A.J. and I.V. Lindell, “Perturbation theory for a corrugated waveguide with a bi-
isotropic rod," Microwave and Optical Technology Letters, Vol. Б, No. 4, 1992, pp. 729-732.
[17]	Kong, J.A., Electromagnetic Wave Theory, New York, John Wiley fc Sons, 1986.
[18]	Sihvola, A.H., “Bi-isotropic mixtures,” IEEE Trans, on Antennas and Propagation, Vol. 40,
No. 2, 1992, pp. 188-196. Corrections, Vol. 41, No. 7, 1993, p. 1000.
[19]	Yaglyian, A.D., “Electric dyadic Green’s function in source region,” Proc, of the IEEE, Vol.
68, No. 2, 1980, pp. 248 263.
[20]	Saadoun, M.M.I., and N. Engheta, “A reciprocal phase shifter using novel pseudochiral or
fl medium,” Microwave and Optical Technology letters, Vol. Б, April 1992, pp. 184-188.
[21]	Lakhtakia, A., “Perturbation of a resonant cavity by a small bianisotropic sphere," Int. J.
Infrared and Millimeter IVaves, Vol. 12, No. 2, 1991, pp. 109-114.
[22]	Collin, R.E., Foundations for Microwave Engineering, McGraw-Hill, Tokyo, 1966, pp. 108-
112.
[23]	Van Bladel, J., Electromagnetics Fields, New York, McGraw-Hill, 1964.
Chapter 8
Uniaxial Bianisotropic Media
The theory given in the previous Chapters was associated with the bi-isotropic
medium with properties independent on spatial directions, a fact manifested by
scalar medium parameters. The most general linear medium, however, is the bian-
isotropic medium whose parameters are dyadics. To have a taste of electromagnetics
in more general media, in this last Chapter we consider the uniaxial bianisotropic
medium, whose parameters are uniaxial dyadics. Such a medium possesses one
special direction which in the following is taken to be parallel to the z axis with-
out loss in generality. Components transverse to z are denoted by the subscript (.
The uniaxial bianisotropic medium has been studied for potential appb'cations in
polarization-transforming devices [1, 2, 3, 4].
8.1 Plane Waves in a Uniaxial Medium
8.1.1 Medium Parameters
The constitutive equations for the most general uniaxial bianisotropic medium are
of the form
D = 7 E4 £ H, B = ?-E + pH, where the parameter dyadics are uniaxial,	(8-1) (8-2)
i = eJi + e.u,u„ Д = F =	+ 6««u,,	(8-3) (8-4) (8.5)
275
276
Chapter 8. Uniaxial Bianisotropic Media
? = (J. +	(8-6)
This kind of medium dyadics can be realized, for example, by inserting metal helices
in an isotropic host medium in such a way that, although the positions of the helices
are randomly distributed, their axes are either parallel to the z axis or transverse
to the z axis, with no preferred direction in the transverse plane. Instead of four
scalar parameters as in the BI medium, we now have eight scalar parameters.
It has been shown that [5, 6], for any reciprocal bianisotropic medium, the
medium parameter dyadics must satisfy
F = ?, Д7" = Д, F = -?>	(8-7)
where T denotes the transpose of the dyadic. Thus, the uniaxial medium defined
above is reciprocal if the conditions
6 = -<i,	{. = -<.	(8.8)
are satisfied. Similarly, conditions of lossless bianisotropic media can be expressed
as [5, 6]
=T =* =T =* 77 7*
e =e , p = p , C = .( .
(8.9)
The uniaxial medium is lossless if e and p are real dyadics, and ( and ( are complex
conjugates. In the lossless and reciprocal case, J = — ( is an imaginary dyadic and
will be denoted by jKy/p„tc, where к is the chirality dyadic.
Figure 8.1 Uniaxial chiral medium made of parallel metal helices randomly distributed
in a host medium.
Without complicating the analysis too much, let us restrict to a simpler special
case, the axially bianisotropic uniaxial medium with ( and ( of the simpler form
? = (щи,.
(8.10)
(8.U)
8.1. Plane Waves in a Uniaxial Medium
277
Such a medium can be thought of being realisable by inserting metal helices in a
host medium with their axes parallel to the z axis, at least to a certain accuracy,
Figure 8.1. More general uniaxial media have also been studied [7, 8, 9]. Properties
of an electromagnetic wave propagating in such a medium and the Green dyadic
are the subject of this Chapter.
8.1.2 Basic Equations
Starting from the plane-wave assumption with a given unit vector u,
E(r) = Ee--’*u,t	Н(г) = Не-*"г,	(8.12)
D(r) = De~'*u',	B(r) = Be->*u r,	(8.13)
and inserting these in the Maxwell equations (2.1), (2.2) without source terms, wc
obtain the following relations for the amplitude vectors and the wave vector к = uk,
where к is now an unknown quantity:1
к x E = u>B, к x II = —u>D.	(8-14)
The unit vector u defining the direction of propagation makes an angle P with the
axis direction u, of the medium, Figure 8.2. For simplicity, it can be assumed that
u lies in the'j/z plane:
tl = u„ sin 6 | u, cos (?,
(8.15)
whence uB is normal to the plane defined by the axis of the medium and the direction
of propagation.
Figure 8.2 Geometry of the plane-wave problem.
1The quantity k must not be mistaken as the bi-isotropic wave number k = Wy/fii.
278	Chapter 8. Uniaxial Uianisotropic Media
8.1.3 Axial Impedance and Admittance
All the information of the plane wave is contained in the equations (8.14). Let us
first consider the ensuing orthogonality conditions

E • В = 0, H-D = 0.	(8.16)
After substituting D and В from the medium equations (8.1), (8.2), we have
/xrEtHf + цяЕяН, 1(^ = 0,	(8.17)
€<Et • He + cgEKHt + (Я2 = 0.	(8.18)
In passing we note that for the uniaxial anisotropic medium with ( = ( = 0,
satisfying/te£*~6tP*	0, (8*17) and (8.18) imply Et*He = 0 and ЕЯНЯ — 0. Because
the latter is satisfied when eithy Et — 0 or Ня — 0, a plane wave propagating in
the general uniaxial anisotropic medium must be either TM or ТЕ polarized with
|	respect to z. However, for a special anisotropic medium satisfying	= 0,
„	this is not valid and solutions which are neither TM nor ТЕ are possible.
Considering again the more general uniaxial bianisotropic case, eliminating Ef •
lit from (8.17) and (8.18) leaves us with an equation for the axial components
+ n.'E.H,) = д.((Я2 4 itE,II,).	(8.19)
Defining the axial impedance and admittance’ quantifies by
„	E,	,r H.
Z. = ^,	Y. = -f,	(8.20)
Лх •
from (8.19) we can write the equations	.
K,2 +	- — = 0,	(8.21)
/‘<4	/‘<4
Z} +	_ Ed = 0,	(8.22)
fiQ	£<( •
which can readily be solved. The resulting admittance and impedance expressions
obviously have a relation through the duality transformation. Let us, however,
introduce a self-dual parameter A defined by
Л=-±^ = Е^,
£i	Pe
and satisfying the equation
(Л_Е±)(Л_^) = 1С.	(8.24)
/‘i	£t £</‘«
8.1. Plane Wave» in a Uniaxial Medium
279
Figure 8.3 Plot of the normalized parameters A^/E and A-/E, with E = as
functions of M/E, with M = /<«//<!, for different values of the parameter К = |к|/п,,
corresponding to a lossless uniaxial chira) medium.
The two solutions will be distinguished by the subscripts + and —
л± = ^(Г + 7)±\
2 tll et )
(—
\P«
(8.25)
1
4
it ) fll‘<
and they correspond to two plane eigenwave solutions E±, H±, in analogy with
those of the BI medium. Let us define the two roots A+ and A_ corresponding to
the two branches of the square root so that for the lossless case they are real valued
and satisfy the condition A+ > A_. In the lossless uniaxial anisotropic medium
with ( = ( -• 0 and /tte, > the two solutions reduce to
A+	A_ ->	(8.26)
it	pl
which correspond to the respective TM and ТЕ polarized solutions of the anisotropic
uniaxial case [10, 11]. In the converse case ftte, < fitti, the'solutions reduce to
A_ -» A+ -♦	(8.27)
it	fit
This effect is seen in Figure 8.3, where normalized values of A+ and A are plotted
as functions of pxe(/ptex with different values of the parameter К =	For
JNote the difference in definition when compared to that in the references (1, 2, 4].
280
Chapter 8. Uniaxial Bianisotropic Media
example, for a reciprocal uniaxial chiral medium, the parameter is the normalized
chirality parameter К = |«|/n, with n, = '/1,6,//toco- For К —♦ 0, the ТЕ and TM
limits are clearly seen. For the other limit К —♦ 1, the solutions obviously become
A_ -+ 0 and A+ —» (b«/b<) + (6«/£«)-
Figure 8.4 Plot of the normalized impedance parameters Z±/Z\, Z\ = 7г;(п(/к, as
functions of MIE with M — p,/pi, E = t,/for different values of the normalized
chirality parameter К = |«|/пж, corresponding to a lossless uniaxial chiral medium.
The axial impedance and admittance quantites can be expressed in terms of the
two parameters A±. The impedances, Zt± = Et±IHt±, corresponding to the two
plane eigenwaves, are
and the admittances K.j. = 1/Z,±
v е‘/л £«\	£«
ч. 4e‘ к
— - —) ± . ---------I + —
2 /*< £l \ 4 \	/‘< / Wl
(8.29)
Thus, actually without solving the fields of the two eigenwaves we know the ratio of
their axial components. For lossless chiral uniaxial media with ( = —f =
8.1. Plane Wave» in a Uniaxial Medium
281
the impedances and admittances are imaginary, which means that the axial field
components Et± and Лх± are in phase quadrature.
In Figure 8.4, values for normalized axial impedance Zt±/Zi with Zt =
Зт1*у/№<> = 34tntlK, Vt =	nt =	are plotted for a lossless
uniaxial chiral medium. The line Zz± = 0 corresponds to vanishing E, component,
i.e., to the ТЕ wave, in the limiting case к —» 0. On the other hand, the normalized
Zt± is finite on the line marked “TM”, but since Zi becomes infinite for к —> 0, so
also does Zx±, making II, = 0.
8.1.4 Effective Anisotropic Media
The central role of the parameters Л-t can be seen by writing the medium equations
for the two plane-wave solutions as
D± = e • E± +	= (e + (Ух±ихчх)  E± = e±  E±>	(8.30)
B± = Д  H± + <ихЕх± = (p + CZx±u.ux) - H± =	• H±.	(8.31)
Thus, the uniaxial bianisotropic medium is seen by the two plane eigenwaves as
equivalent uniaxial anisotropic media, with respective effective parameter dyadics
e± — e«7< + (ex + £У±)ихчх —	(8.32)
p± = Htlt + (/‘x + <Hx±)uxux = ptA±,	(8.33)
with
Л± = It + A±uxux.
(8.34)
Since the effective medium dyadics e+ and (e_ and p_) are multiples of the same
dyadic Л+ (Л_, respectively), the condition et+/xx+ —/t(+cx+ = 0 (c,_/rx_ —	=
0) is satisfied. Such an anisotropic medium has been labeled as affine-isotropic
medium [C] since it can be transformed through a suitable affine transformation to
an isotropic medium. Thus, the two equivalent anisotropic media seen by the two
eigenwaves are actually two affine-isotropic media.
It now appears easy to deduce the properties of a plane wave propagating in
a uniaxial bianisotropic medium of this kind by applying our knowledge on waves
in a uniaxial .anisotropic medium. However, as noticed above, the conclusion that
a plane wave in a uniaxial anisotopic medium is either TM or ТЕ polarized with
respect to the axis, does not hold in an affine-isotropic medium. Actually, the
relation between the axial field components,
ux-H± = Kx±ux-E±,	(8.35)
282
Chapter 8. Uniaxial Bianisotropic Media
applied in defining the effective anisotropic medium parameters, must be retained as
an additional condition since it is hot contained in the effective medium equations.
The two effective uniaxial anisotropic media defined above can be transformed
to an isotropic medium through two affine transformations defined by the dyadics
Л± = lt I• ^/A±u,u,.	(8.36)
In fact, since the equations (8.14) can be written as
k± x E± = o?B± = u’/i(A± ‘Н±,	(8.37)
kt x H± = — u>D± = -о>е,Л± • E±,	(8.38)
multiplying both equations by the dyadic
=-i/i	=	1
A± = lt 4—==u,u,	(8.39)
V A±
and applying the identity [6] (see Appendix C)
Л± / • (k± x a) = k'± x (л/ • a),	(8.40)
valid for any vector a and with
1 1
kt = —===Л± • k± = A:±(u, cos в -I u„—=== sin в),	(8-41)
у/Л±	VA±
we can simply write
k'± x E'± = и>р(Н'±,	(8-42)
k'± x H'± = -u>e(E'±,	(8.43)
with the transformed field vectors defined by
EV = X7’ ' E±> H± = И± ’ 	(8.44)
(8.42) and (8.43) are equations for two plane waves, both in the same isotropic
medium with the parameters et, p(. The wave vectors in an isotropic medium satisfy
k'± • k'± = k, = w’^/coq,	(8.45)
which means that the wave-number surfaces £±(h) coincide into a single sphere with
the radius w^fittf.
8.1. Plane Waves in a Uniaxial Medium
283
Wave-Vector Solutions
To obtain the solutions for the wave vectors corresponding to the two eigenwaves
in the original medium, we substite k'± in (8.45) in terms of k± from (8.41):
*
k\ = k± • k± = - !*•- Л*.	(8.46)
u•Д±u
or
k± = -------- (8.47)
у COB2 6 | (1/Л±)в1п2 в
Substituting the original parameters we have, finally,
t, _-------------------------— (8 48)
2 сов2 6(е,м« -	+ «n + ««/»< T ~ c,/*t)2 + 4е,р,£(]
In Figure 8.5 examples of the rotationally symmetric wave-number surfaces are
given for some parameter values.. The normalized chirality parameter in the figure
is |a|/n, with n, = kf/k0 and its values are varied from 0 (nonchiral anisotropic)
to 0.9. It is seen that, for u ~ 11, (6 — 0), the surfaces touch one another which
means that k+ equals Thue, the axis of symmetry of the medium is also its
optical axis. The wave-number surfaces	are actually spheroids. This is no
surprise because the wave-number diagrams for the equivalent uniaxial anisotropic
media are known to be spheroids.
As a check of (8.48), the well-known wave number expressions for the two limiting
uniaxial anisotropic media (( —i 0 are obtained. For /i,c( > p(e, they read
k.
--------------------(8.49)
^/сов2 в T sin2 в
JL
--------------------,	(8.50)
усов2 6 + (ee/e.) sin2 0
and for p,e( < /qc, conversely. (8.49) corresponds to a wave which docs not see the
e, parameter and, hence, is transverse electric with respect to the axis u,. (8.50)
corresponds to a TM wave for a similar reason.
For lossless media, the parameters Л± are real because the term under the
square-root sign in (8-25) is positive. For (( -> ц,е, we have Л_ —* 0, which
implies /г_ —♦ 0 and one of the eigenwaves ceases to propagate. This resembles
a resonance condition in some media or propagation in a waveguide at the cutoff
frequency. We will not study this aspect any further but assume that the product
(( is small enough to make and A:_ real numbers.
284
Chapter 8. Uniaxial Bianisotropic Media
Figure 8.5 Wave-number surfaces for the uniaxial medium with c,/q = 2, p«/pt = 1
and |a|/n, = 0, 0.5 and 0.9. k+ is given in solid, fc_ in dashed lines. The axis of the
medium is also the optical axis with = к_
8.1.5 Eigenpolarizations
Having determined the wave numbers of the two plane eigenwaves, let us find their
polarizations. For an isotropic medium we know that any polarization E'± which
satisfies
k'± • E'± = 0	(8.51)
is admissible. Written in terms of original quantities this reads
[k±-X]-E± = 0.	(8.52)
The additional condition (8.35) used in defining the effective anisotropic media is
taken into account if we write
u, • (k± x E± - uB±) = (u, x kt - ыреЛ±У1±и,)  E±
= [Л± • (ux x k± — aip^-tu,)]  E± = 0.	(8.53)
The two complex vectors in square brackets in (8.52) and (8.53) are both orthogonal
to the electric field. Since their cross product is not identically zero for the general
direction of propagation u, it must have the same polarization as the electric field
vector, which is denoted by
8.1. Plane Waves in a Uniaxial Medium
285
E± ~ (A± • k±) x (A± • (u, xk± -
~ X~‘ • [k± x (u, x k± - wp^n,)].	(8.54)
The condition D± • k± = 0, following from (8.38), serves as a simple check of
(8.54). In fact, because from (8.30) D± has the same polarization as • E±, the
vector in square brackets in (8.54) must have the same polarization as D± and so
must satisfy orthogonality to k±, which is obviously the case.
Since the vector u, x k± is parallel to u., we can also write
E± ~	• (Kt X u. + а)/1,У,±и.).	(8.55)
For the magnetic held we can write in the corresponding fashion,
H± ~ Л±~* • [k± x (u, x k± + a)r(Z,±ti,)]
~	• (k± x ue — u>eeZ,±ux).	(8.56)
The expressions (8.55), (8.56) do not hold for longitudinal propagation u = u„
because the vectors giving the polarizations become zero. In this special case, the
two eigenwaves actually have the same degenerate propagation coefficient and any
polarization E± satisfying E± • u, = 0 is possible.
Orthogonality
The polarizations of the electric and magnetic fields of a plane eigenwave, (8.55),
(8.56), are not simply orthogonal to one another but satisfy the following more
general orthogonality condition:
E±-B±=0, -> E± • X • H± = 0,	(8.57)
which can also be written as
E(± • Hf± + Л±Е1±Я,± = 0.	(8.58)
For the uniaxial anisotropic case we have Et±Ht± = 0 and this reduces to the simple
orthogonality condition E± • H± = 0.
Anisotropic Medium
Let us check the polarizations of the eigenwaves in the special case ( —» ( —» 0. For
p±e( > р,еж we have from (8.28), (8.29) the limits > 0 and Z,+ —♦ 0, together
with A_ —>	and A+ -» fijiii. Thus, the polarizations of the eigenwaves in
this uniaxial anisotropic medium can be written as
хои
Chapter 8. Uniaxial Bianisotropic Media
E+ ~ u„ H+ ~ u, sin 6 — uv — cos 6,	(8.59)
ei
E_ ~ u, sin0 — uv —cos0, H_ ~ u,.	(8.60)
Pt
Obviously, the — wave has ТЕ polarization and, the + wave, TM polarization, in
accord to what we know about plane waves in uniaxial anisotropic media. In the
converse case /i,ct < the + and — polarizations are interchanged.
Transversal Waves
Considering the special case of eigenwaves propagating in the transverse direction
6 = rr/2, or u = uv, we have from (8.48) for the wave numbers the expressions
Ww, - Ю
- k’A± = ----------------------------------------------.	(8.61)
Because iu this ease we have cos 8 = 0, the eigenpolarizations (8.55), (8.56) can
be written as
E± ~ «. -	= U, - -^(Л± - е')\Д±и«>	(8.62)
u>(	ct ’
H± ~ u, + — JT±U. = u, + -^(4± - —)Ja^ux. (8.63)
uQ	Hi '
In the lossless case, the admittance and impedance are again imaginary quantities,
whence the two eigerrpolarizations are ТЕМ and elliptic. In the anisotropic special
case ( -> ( -t (I we have, for /i,et > /iee,, the limiting polarizations E_ ~ u„
E+ ~ u, and П- ~ II+ ~ ue, and conversely for /x,c( < /i(c,, in accord with
(8.59) and (8.60).
8.2 Polarization Transformer
In general, the polarization of a plane wave is changed when traveling through a
bianisotropic medium. In an isotropic chiral medium, the polarization of a propagat-
ing wave is rotated by the angle ф = —Krkz, proportional to the chirality parameter
кг and the distance, but the ellipticity of the field vector remains unchanged if the
medium is lossless. The ellipticity and handedness of the propagating wave can,
however, be changed in a uniaxially anisotropic chiral medium, which makes it pos-
sible to design a simple polarization transformer with just a slab of uniaxial chiral
medium.
8.2. Polarization Transformer
287
8.2.1 Transverse Eigenwaves
With the described application in mind let us consider plane-wave propagation in
a lossless uniaxial chiral medium, in the direction transverse to the axis of the
medium: u = uv (в = ir/2). The parameter dyadice are assumed reciprocal for
simplicity, with £ — (*, or
( = -j»tv6joe»u,u,, < =	(8.64)
In this case, the wavenumbers of the two eigenwaves can be written from (8.61) as
__________________________________________
^TE + ^TM -F y/(^TE ~	+ 4fc’fc’№
(8.65)
with the wavenumbers of a nonchiral anisotropic medium in transverse propagation
denoted by
ктм —
kTE = ui^/p.c,,
(8.66)
and
A:t = Wy/ntet = nelo, к, =	= n,fco, кТЕкТм = ktk,. (8.67)
The expression (8.65) can also be written as
Jt±= —-------------- (к/п,)»]	(8 6g)
к ТВ + ^TM T yji^TE ~ ^ТмУ + ^^ТЕ^Тм(К/п‘У
from which it is seen that, for к —» 0, the wavenumbers k+ and fc_ reduce to kj-м
and kTE 6° that k+ > k-.
For the polarization transformation applications in mind, it is necessary to de-
compose the wavenumbers as
k±=ft±1,
with the phase, wav,enumber ft defined by
/3 =
and the polarization wavenumber 7 by
(8.69)
(8.70)
288
Chapter 8. Uniaxial Bianisotropic Media
7=^(гЩ-^/лГ).	(8-71)
these wavenumbers are depicted in Figure 8.6 for values corresponding to those of
Figure 8.3.
Figure 8.6 Normalized values for the phase wavenumber /З/^тл/ and the polarization
wavenumber ч/кты corresponding to a lossless uniaxial chiral medium, as functions of
the quantity M/E — Hit,, for different values of the normalized chirality parameter
К = |к|/п,.
8.2.2 Polarizations of the Eigenwaves
The polarizations of the two eigenwaves are transverse to y, as seen from (8.62),
(8.63) and can be written as E+ ~ e+ and E_ ~ e_, with
e+ = Q+ur + jPu„ e_ = Q_u, + jPu,,	(8.72)
Q± = ^(A± - ^), P =	(8.73)
et	«e
Because in the present lossless case the parameters A± are real, the impedance and
admittance quantities Z,±, are imaginary:
f.
ZI± = -ju>—~(/1(Л± - ft,),
K0^
(8.74)
8.2. Polarization Transformer
289
Г.±=у^(с,Л±-е,),	(8.75)
кок
and the quantities Qj. and P are real. The axes of the polarization ellipses of the
E± fields are then parallel to the x and z axes and the axial ratios are Qt/P-
8.2.3 Propagating Plane Wave
Consider a plane wave propagating in the direction of u„. Let the electric field be
linearly polarized at у = 0, making an angle a with the axis of the medium u,:
E(0) — E0(ii« cos a ur sin o).	(8.76)
To see how the electric field vector is changed in propagation along the у direction,
we have to write it in terms of the eigenvectors e±, which propagate with the
wavenumbers k±, respectively. From (8.72) we obtain
_ e+ - e_	_ 1 Q+e_ - Q_e+
U'~Q+-QJ U‘~jP Q+-Q_
(8.77)
which inserted in (8.76) gives the polarization of the field at the distance у = L as
E(Z) = pin^° n Jc+0Fgina~ Q- c°s«)e iktL
~ Ч-)
+ e_(—jPsina 4 Q+ cos a)e-J*~t).	(8.78)
Substituting (8.72) back again, we can write the field in terms of its x and z com-
ponents
E(L) =
Eoe-^b
jP(Q+-Q_)
 |ux[Q+(jPsin a — Q_cosa)e 3"rL 4 Q_(—jP sin a 4-Q+cos a)e37b]
4- j Pu, [(jPsina — Q_ cosa)e-3',£ 4 (—jPsina 4- Q+ cosaje”1]}. (8.79)
The field at у = L can be written simply in terms of a propagator dyadic I’(L) in
the form
E(L) = Г(£) - E(0),
eliminating the angle a from (8.79) by inserting
EB cos a = u, • E(0), EB sin a = u,  E(0).
(8.80)
(8.81)
7290
Chapter 8. Uniaxial Bianisotropic Media
The result can be written as
=	e~iPL {
----nJ u'I<W>Pu- -
J‘ — i/-/1
c-iPL
+ Q^q2
+jPu.[(jPux - Q_u.)e-^ + (-jPux 4 Q+u,)e”t]|
_	j 11,11,) cos 7!
p(Q+ 4 <2-)(~uxux +u,u,)+	2Pu,uxj sin 7I. (8.82)
8.2.4	Quarter-Wave Transformer
At the distance of an integer multiple of half polarization wavelengths L = nA^/2
with Apoi = 2тг/7, we have sin 7! = 0, whence the propagator becomes
Г(1) = (-1)"е-^(ихч« + ч,и,),
(8.83)
which returns the polarization to that at the origin у = 0.
Figure 8.7 Quarter-wave slab of uniaxial material as a polarization transformer.
Because the greatest change in polarization can be anticipated halfway between
these distances, i.e., at odd multiples of the quarter wavelength, to make a device for
transforming the polarization of a plane wave, a quarter-wave slab of the uniaxial
chiral medium can be attempted, Figure 8.7. The thickness of the slab is thus
r __ ^pol __	__
4	27
In this case, the propagator dyadic is seen to reduce to
(8.84)
I'(I) =
e-ipL
Q~+~Q^
[j(Qt + Q-)(-uxux + u,u,) + -^(iixii,Q4Q_ + u,iixP2
(8.85)
8.2. Polarization Transformer
291
Choice of Parameters
Let us try to choose the parameters of the medium in such a way that the change
in the ellipticity of the polarization in the quarter-wave slab is as great as possible.
Because in changing continuously from linear to circular polarization the field must
go through all the axial ratios, the change is then the greatest possible. Thus, let us
require that two orthogonal linear polarizations are transformed into two circular
polarizations of opposite handedness.
For example, if the incident field is polarized as u2 with a = 0, we require the
outcoming field to have the right-hand circular polarization, i.e., T(L)-u, ~ иж4-}и,.
This gives us one condition for the medium parameters:
P(Qi +Q_) = 2Q+Q-	(8.86)
As a second condition we require that the incident polarization u, with a = rr/2
be transformed into left-hand circular polarization ~ u, — ju,. This gives us the
other condition
<?+ + <?_ = -2P.	(8.87)
Solving these two equations leads to
Q+ = (-1 + >/2)P,	<?_ = (-1-Л)Р.	(8.88)
Let us, instead, consider the equivalent conditions
Q+Q- = -P\	(8.89)
= -3 + 2^/2 = -r.	(8.90)
Substituting the definitions (8.73) in (8.89), gives us the first condition for the
medium parameters
Q+Q- = \/л+Л_(Л+ - —)(Л_-----------) = \/л+Л_( p-)
= -P2 = -^,	(8-91)
implying
Л+Л_ = 1.	(8.92)
Substituting further (8.25) in this, the first condition (8.89) becomes one for the
chirality parameter:
292
Chapter 8. Uniaxial Bianisotropic Media
- /*«*«•	(8.93)
We see immediately that, to have real chirality values, the inequality kt > kt must
be satisfied. However, to have small chirality, к. and к, must not differ too much.
The second condition for the parameters comes from (8.90). After some algebra,
this can be shown to lead to the following relation between the four permittivity
and permeability parameters:
г = 3 - 2v/2 « 0.17157.
(8.94)
The numerator of (8.94) is negative for positive values of whence positive
values for tt/ti arc obtained only in the interval r < ihlui < l/т, i.c., 0.17 <
/*,//4 < 5.8. In this interval, the inequality p.e./pjQ > 1 is everywhere valid so
that, indeed, real values for к can be found for a wide variety of parameter values.
(2, 4].
Figure 8.8 E = t,/ti (solid line), Kt = |к|/щ (dash line) and Kz = |к|/п, (dash-dot
line) as functions of M =	satisfying the quarter-wave polarization transformer
conditions. The singularity of E and Kt at M - r = 0.17 is clearly visible.
In Figure 8.8, the interval (0...3) of M — values are studied. The corre-
sponding E — t,/tt values are given in the solid line going through the isotropic
point (1,1). The two other lines give the corresponding chirality parameter nor-
malized as Kt = |a|/nt (dash line) and Kz = |к|/п, (dash-dot line), if we denote
8.2. Polarization Transformer
293
kt = ntk„ and k, = n,fco. It is seen that at the isotropic point the chirality changes
the sign. Small chirality values are obtained close to the isotropic point, which are
bound to make the quarter-wave transformer thick.
For example, M =	= 0.8 gives E — t,/et ~ 1.3 and Kt = |к|/п, ss
|к|/пж яа 0.2, which should be realizable. On the other hand, close to the endpoints
of the interval, e,/ce becomes large and |к|/п, яа 1, which sounds impractical.
Polarization Transformation
If the expressions (8.88) are substituted in (8.85), the following simple propagator
dyadic results:
= e~i0L
1 '(L) =	- U«U«) - X /].
(8.95)
This equals the transmission dyadic for the uniaxial quarter-wave slab when no
reflections from the interfaces are taken into account.
It can be shown that, by taking the polarization angle a of the incident linearly
polarized field in a suitable way, any polarization ellipse and handedness can be
obtained at the output. In fact, for the outcoming field we can write
E( £) = Г(£) • E(0) = P(i) • (и, cos a + u, sin a)Ee
e-i0L
= —^-(-u.e'“ - ju,e,a)E„.	(8.96)
Finding the polarization vector (Appendix B) of the field E(£) gives the result
E(£)xE*(Z)	,	,
=	'84">
which can get all values between —u„ and +u„. The component u„ • p is called the
polarization number or fatness factor since |p| = 1 corresponds to the completely fat
ellipse, i.e., the circle, and p = 0 the completely thin ellipse, i.e., the line. Positive
values correspond to right-handed and negative values to left-handed rotations with
respect to the positive у direction. Because cos 2a can get all values between — 1
and +1, all elliptic ratios and both handednesses are obtained for varying the angle
a. This effect is also seen from the following table [2, 4].
294
Chapter 8. Uniaxial Bianisotropic Media
a	cos 2a	polarization	handedness
—rr/2	-1	CP	LH
—x/2•• — x/4	-1-0	EP	LH
—rr/4	0	LP	-
-X/4---0	0--1	EP	RH
0	1	CP	RII
0 • ••x/4	1-0	EP	RII
x/4	0	LP	-
x/4••x/2	0 -1	EP	LH
x/2	-1	CP	LH
For example, to obtain right-handed elliptic polarization with axial ratio 1 : 2,
i.e., ux + y2u,, out of a linearly polarized field, we must have p(E(Z)) = uv cos 2a =
u„4/5, which gives a = ±0.32 radians or a — 1:18.4°. The two solutions have
different axis directions.
Axis Directions
For a linearly polarized incoming field, directions of the axes of the outcoming field
ellipse can be obtained by writing (8.96) in the form
E(L) ~ и,е-^а+?) + иж?<“+?>
= (u, + u,) cos(a +	+ j(u. - ux)sin(a +	(8.98)
4	4
Figure 8.9 The polarization of the transformed linearly polarized field is an ellipse with
axes at 45° angle with the coordinate axes.
Since the real and imaginary parts of this complex vector are orthogonal, they
also coincide with the axial directions of the polarization ellipse (see Appendix B).
Thus, the axes of the outcoming field are always along directions making 45° angles
with the x and z coordinate axes, Figure 8.9. The major axis is along the direction
of u, 4 Uj. if the condition
8.2. Polarization Transformer
295
cos’(a + —) > sin2(a + ~)	(8.99)
is satisfied. This leads to the simpler condition
sin2a<0,	(8.100)
or x/2 < a < % and —zr/2 < a < 0. In the converse case, the major axis lies in
the direction of u, — ilr. The change of direction of the major axis occurs when the
outcoming field is circularly polarized.
The ellipticity e (axial ratio) of the outcoming polarization can be written from
(8.98) in the simple form
e = tan(a|~).	(8.101)
When a is an odd multiple of irr/4, the polarization is linear and, when a multiple
of x/2, circular.
Transformation of Elliptic Polarization
It is of interest to study if the quarter-wave transformer can be used to transform a
given elliptic polarization to another given elliptic polarization. This can, however
be quite easily seen to be impossible in the general case. In fact, because the
uniaxial chiral slab is reciprocal, it also works backwards and circular polarization
is always transformed to linear polarization. Thus, circular polarization cannot be
transformed to an arbitrary elliptic polarization with a single slab. With two such
slabs it becomes possible, as will be seen.
To see how an elliptic polarization is transformed through the slab we consider
an input polarization of the form
E(0) = E0(vcos6 + jwsinfl),	(8.102)
with the orthogonal unit vectors
v = u, cos a + u, sin a, w = — u, sin a + u, cos a (8.103)
defining the axes of the ellipse. Assuming |0| < x/4, the major axis lies at the
angle a to the z axis and the axial ratio is e = |tan$| < 1. Substituting in the
polarization vector we have
p(E(o)) = Sr™ =	p(0)= -sin 26 	(8Л(И)
296
Chapter 8. Uniaxial Bianisotropic Media
To find the polarization of the outcoming field, let us, again, form the polariza-
tion vector:
p(Ef£n = E(£) * E*(£) = E(0).F(I)xfi(Z) E*(0)
1	jE(£)E*(£)	j‘E(0) • rT(L)  Г*(£)  E*(0)
The result can be written in the simple form
p(L) = cos 2a cos 26 = cos 2ayjl — p(0)’.	(8.106)
The resulting polarization vector (8.105), (8.106) shows us that, for an arbitrary
input polarization ellipticity with p(0), only a limited range of output polarization
ellipticities are obtainable. For example, for a CP input, p(0) = ±1, i.e., 6 an
odd multiple of x/4, we have p(L) = 0, an LP field. On the other hand, for an LP
input, we have p(L) = ± cos 2a, which can create all the polarization values through
suitable values of a, as was seen above. Handedness of the input polarization does
not affect the ellipticity of the output polarization.
Figure 8.10 Polarization number p(£) in the output of the quarter-wave uniaxial chiral
transformer as a function of the inclination angle a of the input polarization ellipse, for
some values of the input polarization number p(0).
In Figure 8.10, transformation of different input polarizations is depicted as
a function of the rotation angle a of the uniaxial chiral slab. For a = ±rr/4,
8.2. Polarization Transformer
297
the outcoming polarization is always linear: p(L) = 0, as is also seen from
(8.106). For example, if the axis ratio of the input polarization is e(0) = 0.5, with
|p(0)| = 2e(0)/(l + e’(0)) = 0.8, the maximal value in the output is |p(Z)| = 0.6,
corresponding to the axial ratio e(L) = 0.25. Thus, the ellipses tend to flatten in
the transformer.
Polarization Transformation with Two Slabs
As seen above, the quarter-wavelength slab can be used to change linear polar-
ization to an elliptic polarization of any ellipticity and handedness, but the axial
directions of the outcoming ellipse cannot be controlled, because they arc fixed to
the axis of the uniaxial medium of the slab. On the other hand, an arbitrary elliptic
polarization can be transformed to polarizations with limited values of ellipticity.
Using two similar rotatable slabs in series, the general transformation from one
ellipse to another one can however be done, because the first slab can transform any
elliptic polarization to a linear polarization and the second slab can transform it
further to any other elliptic polarization. In fact, denoting the polarization number
of the incoming field by p(0), the polarization number after one slab at an angle aj
is according to (8.106)
p(L) = cos 2а2/1^ p’(0).	(8.107)
If this is the input of the second slab at the angle a2 with respect to the major axis
of the polarization, the polarization of the outcoming field can be characterized by
p(2L) = cos 2aj1 — cos’ 2«i[l — p’(0)] = cos 2aj^/sin2 2ai 1 cos’ 2oip2(0).
(8.108)
It is easy to see that by choosing a> — ±x/4, we have p(L) — cos2a2, which can
have all values between —1 and +1 by rotating the second slab.
The two slabs can also be rigidly combined for some purpose. For example, by
joining them so that their material axes are orthogonal to one another (in the first
slab along the z axis and in the second slab along the z axis), a wave with circular
polarization is seen to change the handedness. In fact, the composite propagator
dyadic in this case is then
S = F.(i)-F2(i)
e~bpL
= ---~--0(и«ч» - u,u.) - Uv X /) - (j(uxu, - u,uj - uv X I)
= -je^(u.u, + 11,11,).	(8.109)
which is easily seen to produce the transformation in question.
Chapter 8. Uniaxial Bianisotropic Media
8.3 Green Dyadic
It has been known for some time that an explicit closed-form expression can be
written for the Green dyadic corresponding to the uniaxial anisotropic medium
with either electric or magnetic anisotropy [11] or both [12, 6]. Because an explicit
expression for the Green dyadic of the Ш medium is also known [13, 14, 15, 16, 17],
as was shown in Chapter 2, it can be anticipated that these two could be combined
somehow to a more general solution. That this is really the case will be seen
in this Section. Actually, it turns out that the Green dyadic can be solved in
explicit form for the uniaxial medium considered in the previous Sections [18, 19],
which makes it possible to solve for fields arising from any sources in this kind of a
uniaxial medium. For more general uniaxially bianisotropic media, such a closed-
form expression would require solving a fourth-order differential equation which
does not seem to be feasible [6, 20].
8.3.1 The Operator Equation
The uniaxial medium considered here is defined by the parameter dyadics (8.3),
(8.4), (8.10), (8.11). The relative permittivity and permeability dyadics are defined '
as i, = e/co and fir = The Green dyadic of the electric type corresponding
to this kind of a medium is defined to satisfy the differential equation [6]
17(V)S(r) = -«(r)7,	(8.110)	;
i
with the Helmholtz operator dyadic defined by	!
//(V) = -(V x I - jw() • p,-1 • (V x I + M) + k2Jr.	(8.111)
Let us follow a dyadic procedure for finding Green dyadics corresponding to
different polynomial operators, given in [6], by writing symbolically the solution as
G(r) =-//-*( V)i(r) = -
5^).
dettf(V)
(8.112)
where we define the dyadic operator
7;(,)(v) = i//(v)j//(v),
and the determinant operator reads
det77(V) =^77(V)J77(V) : 77(V).
(8-113)	;
(8.114)
8.3. Green Dyadic
299
The double-cross and double-dot products between dyadics were originally intro-
duced by J.W. Gibbs a century ago. More recent account is given in [6] and Ap-
pendix 0.
Let us now make use of the constitutive medium relations (8.3), (8.4), (8.10)
and (8.11). Writing the two dyadics and again for convenience3
= I, 4- A±ihU,,
(8.115)
(8.116)
* 2 \/*< tt) V v‘‘	£‘/	'
after some algebra, the determinant operator (8.114) can be shown to be expressable
in the factorized form
== k2
det/f(V) =
aet/xr
with the scalar second-order operators defined as
(8.117)
tf+(V) =	: VV + fc’A+. W_(V) = A_ :VV + Jfc’A_. (8.118)
(8.117)	is a generalization of the scalar operator given in [6] for the simpler uniaxial
anisotropic medium.
To form the Green dyadic, we should evaluate the adjoint of the Helmholtz oper-
ator H<21(V). To simplify the problem let us assume that the medium is reciprocal
with
C =
(8.119)
This assumption can be relaxed, as is shown in [19].
In the present reciprocal case, after some more algebraic manipulations, the
adjoint operator can be written as a sum of three terms:
HW(V) = -Mv,’(VVl Л.’-ёЛ* - x I)
/r„det/rr I	eo

+(Vj+Jt?) ~(VV+ ^-V‘) - MV, X / -	+k2
[Pt	£»
VVxU.u, j.
(8.120)
Here we denote V, = It • V and V, = u,u, • V.
8Note the change in definition of the parameters A, and A_ when compared to the references
[18. IS]-
300
Chapter 8. Uniaxial Bianisotropic Media
8.3.2 Solving the Green Dyadic
To write an expression for the Green dyadic we have to insert the expansions (8.117)
and (8.120) in (8.112). However, before writing the different terms, let us study the
inverse determinant function. Even if the operator det/f(V) is of factorized form,
it is not possible to write its inverse as a sum of inverses of second-order operators.
However, we can proceed by applying the analogy of solving the Green dyadic for
uniaxially anisotropic media [12]. For that wc need the following partial fraction
expansions
tf+(V)tf_(V) A+ - Л_ V? \^+(V)	H_(V))’	(8121)
1	=	-1_______1	________О ,o 122)
//+(V)//_(V) л+-л_ Vj + tf VMV) //_(v)/	1	>
Now we are ready to solve for the Green dyadic. Inserting (8.120) into (8.112)
and applying the above expansions we can write
(Л, - 4.)^«-(V) = |VV +	x I] (jA_ -
_(&(vv + ЧЧ-) - t.,v, x 7+(^ - ^)
_ «Л Wx’h’b ( Л+ _ Л_ \
‘ ej V? \H+(V)
Of the three resulting terms, the first two have only second-order operators in
the denominators and the corresponding Green dyadic terms can be written in a
straightforward manner in terms of two auxiliary scalar Green functions
<8™)
These functions satisfy the differential equations
[А-Л± : VV + *’]G±(r) = [-1-V’ + V: + fc’]G±(r) = -i(r)	(8.125)
A± A±
and have the form [6]
G±(r) = A±
e-ihD±
4irD± ’
D± = уЛ±Л± : rr = \jAi-p1 -| z2.
(8.126)
8.3. Green Dyadic
301
Here we denote the distance from the z axis by p.
The Green dyadic from (8.112) can now be written as
G(r) = Р/Г"* JVV +	~ x 7NG+(') - G'-(r)]
Kt v*+ ~~ 71 }	CO
- i-iTA^r^^ 1W, X I-kyit}
kt{A+-A_)pt eo
7+
where the last term is yet to be determined from the operator equation
I Лг),
(8.127)
= _	WxU,4
д.еДЛу - Л_) V?
A+ Л_
^W W)
Л(г).
(8.128)
8.3.3 The F Dyadic
To develop the unknown dyadic term F of the Green dyadic G, it appears necessary
to solve two fourth-order differential equations. This need not, however, be done
up to the last point. In fact, let us consider the fourth-order equation
TV27/±(V)F±(r) = W—V? + VJ + fc?)F±(r) = -«(r).
(8.129)
Since we know the solution for (8.125) to be (8.126), and knowing that the solution
of (8.129) is rotationally symmetric in the z axis (no dependence on the coordinate
^>), (8.129) can be reduced to (for uniqueness of solution, a radiation condition is
tacitly assumed to be satisfied)
Vf2F±(r) =
1A
G±(r) = A±
4rrD±
(8.130)
Because of D], = A^p1 г2, we have
G = ^*-±-
*	£>± dD±
je~ik,D±
4nkt
1£
je-ib'D*
4irkt
(8.131)
Thus, we may write from comparison (assuming again the radiation condition), the
expression
n p	
F4(r) = —= 7~T”e	(8 .132)
Op	47Г KiP
Integration of F± does not seem to give a closed-form solution. However, in solving
the Green dyadic G we encounter dyadic equations of the form
Chapter 8. Uniaxial Bianisotropic Media
Vj(4-	<- V’ + *,’)F±(r) = -u.u,* VtVt«(r),	(8.133)
whose solutions can be written in terms of the solutions F'± above. In fact, writing
the dyadics in the form
F±(r) = u1u.JV1V1F±(r),	(8.134)
where the scalar functions jF±(x) are solutions of (8.129), (8.134) can be seen to
satisfy (8.133). Expanding (8.134) as
F±(r) = -V x [uji^F^],	(8.135)
we see that there is really no need to solve for the functions F± since the knowledge
of FJ. suffices. Expanding (8.135) further, we finally have the following explicit
solution for the fourth-order dyadic equation (8.133) as
T±(r) = (7, - 2uvuJ-4— e~^ + uvu„G±(r). (8.136)
‘TKKtP*
It is thus seen that the Fourier transformation approach given in [11] with its intri-
cate inverse-transformation-integral evaluations is not needed for writing this term.
The same is valid for the simpler nonchiral uniaxial medium.
After these steps the Green dyadic term F can be written as
T(r) “	* (vF) ’	(8.137)
with
u, = u,x tip, т =	- e~jk,D-.	(8.138)
A more expanded form is
ct(A+ - A_)
(Л -	- G-(r)l
(8.139)
As a simple test, for the nonchiral special case with к = 0, the F dyadic given
in (8.137) and (8.139) can be seen to coincide with previous expressions given in
[6] and [12], respectively, with slight differences in notation. Also, the rest of the
Green dyadic expression (8.127) can be seen to reduce to that given in [12].
8.3. Green Dyadic
303
8.3.4 Field from a Dipole
Knowing the Green dyadic allows us to evaluate the field radiated by an arbitrary
j current density distribution in the uniaxial chiral medium. As a simple example of
| application, let us consider the field radiated from an axial electric dipole
j.	J(r) = u,/£i(r)	(8.140)
in the medium studied in this Section. The expression for the electromagnetic field
f from such a special source can also be obtained through potential expansion without
| the use of the Green dyadic [3].
| The electric field E is the solution of
77(V)  E(r) = jwpou,/£A(r),	(8.141)
and it can be written in terms of the Green dyadic as
E(r) = —jw/ro7?(r) • u,IL.	(8.142)
| Substituting the expression (8.127) we see that, in this case, the F dyadic does not
I have any* influence on the result and the electric field becomes
E(r) =	 [7 + 1VV)[(1 - -^-)G+(r) - (1 -	)G_(r)l
A+ — A_	kt L Pt-A-j.	p«A_ J
ILkon „(G+ G\	.	.
-	------— V I -----—-) x u„	(8.143)
for which the scalar Green functions G+ and G- can be found from (8.126). The
magnetic field H(r) is obtained through the Maxwell equation
H(r) = —t——p * • (V x I — M«.u.) • E(r),
by substituting (8.143), in the form
(8.144)
H(r) =
KiF*
-^-)G+-(l--^-)G_]
р«а+ p<a_ j
+ V f-(u, • V)V, - f
\p«	p. /
G_ \ 1
A J J
(8.145)
To test the expressions for the nonchiral special case, they are seen to reduce to
the better known formulas corresponding to radiation in an anisotropic medium,
304
Reference»
-	1 e~il,,D‘
E(r) = -j^ILu,  И + uVVl-777F’	(8Л4е)
Aj	47Г JL/f
е-й.в.
H(r) = /lVx(u,—),	(8.147)
D. = /(«,/e^ + A	(8.148)
Because the fields from an axial dipole are TM to the z axis, the parameter /i, is not
involved in this solution and the medium appears only dielectrically anisotropic.
References
[1]	Lindell, I.V. and A.J. Viitanen, “Plane wave propagation in a uniaxial bianisotropic
medium,” Electronics Letters, Vol. 29, No. 2, January 1993, pp. 150-152.
[2]	Viitanen, A.J. and I.V. Lindell, “Uniaxial chiral quarter-wave polarisation transformer,”
Electronics Letters, Vol. 29, No. 12, June 1993, pp. 1074-1075.
(3]	Weiglhofer, W.S., “Dipole radiation in uniaxial bianisotropic medium,” Electronics Letters,
Vol. 29, No. 10, May 1993, pp. 844-84Б.
[4]	Viitanen, A.J. and I.V. Lindell, “Plane-wave propagation in a uniaxial bianisotropic medium
with an application to a polarisation transformer,” Int. J. Infrared and Millimeter Waves,
Vol. 14, No. 12, 1993, pp. 1993-2010.
[5]	Kong, J.A., Electromagnetic Wave Theory, New York, Wiley 1986.
[6]	Lindell, I.V., Methods for Electromagnetic Field Analysis, Oxford, Clarendon Press, 1992.
[7]	Lindell, I.V., A.J. Viitanen and P.K. Koivisto: “Plane-wave propagation in a transversely
bianisotropic uniaxial medium,” Microwave and Optics Technology Letters, Vol. 6, No. 8,
June 1993, pp. 478-481.
[8]	Lindell, I.V. and A.J. Viitanen, “Eigenwaves in the general uniaxial bianisotropic medium
with symmetric parameter dyadics,” Helsinki University of Technology, Electromagnetics
Laboratory Report 148, May 1993.
[9]	Viitanen, A.J. and I.V. Lindell, "Plane-wave propagation in the general anisotropic chiral
medium with isotropic permittivity and permeability,” Helsinki University of Technology,
Electromagnetics Laboratory Report 152, June 1993.
[10]	Fedorov, F.I., Optics of Anisotropic Media (in Russian), Minsk, Belorussian Academy of
Sciences, 1958, pp. 284-299.
[11]	Chen, H.C., Theory of Electromagnetic Waves, New York, McGraw-Hill, 1983, Chapter 6.
[12]	Weiglhofer, W.S., “Dyadic Green’s functions for general uniaxial media,” IEE Proceedings,
Vol. 137, Pt.H, No. 1, February 1990, pp. 5-10.
[13]	Bassiri, S., N. Engheta and C.H. Papas, “Dyadic Green’s function and dipole radiation in
chiral media,” Alta Frequenza, Vol. 55, No. 2, 1986, pp. 83-88.
[14]	Lakhtakia, A., V.V. Varadan and V.K. Varadan, “Field equations, Huygens’s principle,
integral equations and theorems for radiation and scattering of electromagnetic waves in
isotropic chiral media,” J. Optical Society of America A, Vol. 5, February 1988, pp. 175-184.
[15]	Weiglhofer, W.S., “Л simple and straightforward derivation of the dyadic Green’s function
of an isotropic chiral medium,” Archiv der Elektrische Ubertragung and Elektronik, Vol. 43,
No. 1, January 1989, pp. 51-52.
References
305
[16]	Monson, J.C., “Radiation and scattering in homogeneous general biisotropic regions,” IEEE
Trant. Antennae and Propagation, Vol. 38, No. 2, February 1990, pp. 227-235.
[17]	Lindell, I.V. and A.J. Viitanen, "Green dyadic for the general bi-isotropic (non-reciprocal
chiral) medium,” Helsinki University of Technology, Electromagnetics Laboratory Report 72,
October 1990.
[18]	Lindell, I.V. and W.S. Weiglhofer, "Green dyadic for a uniaxial bianisotropic medium,” IEEE
Trans. Antennas*and Propagation, to appear.
[19]	Weiglhofer W.S. and I.V. Lindell, “Analytic solution for the Green’s function of a nonrecip-
rocnl uniaxial bianisotropic medium,” Archiv der Elektrische Ubertragnng und Elektronik, to
appear.
[20]	Weiglhofer, W.S., “Analytic methods and free-space dyadic Green’s functions,” Radio Sci-
ence, Vol. 28, No. 6, 1993, pp. 847-857.

Appendix A
Notation
In this appendix we summarize the notation adopted in the present book and present
its relation to that most often used by other authors.
A.l The Present Notation
The constitutive equations for the. general linear or bianisotropic medium are gov-
erned by four medium dyadics I, p:
D = f E+? H,	(A.l)
B = f-E + p-H.	(A-2)
For the bi-isotropic (BI) medium, the dyadics are multiples of the unit dyadic,
and therefore they can be expressed in terms of four scalar coefficients,
D = cE + fH,	(A-3)
В = (E + pH.	(A.4)
In most applications, it is necessary to distinguish the effects of chirality and non-
reciprocity. For this, we define the dimensionless Pasteur parameter к and the
Tellegen parameter x by writing
( = (X-J«)v4^>	(A.5)
C = (X +	(A-6)
The BI medium with к = 0 but %	0 is called Tellegen medium while the one.
with x = 0 and к 0 is called Pasteur medium in the present book.
The following quantities are defined for.BI media just as for isotropic media:
the relative permittivity and permeability are
307
308
Appendix Л. Notation
the refractive index	II o' 1 "> t II	(k.l)
the wave number	П = v/Pr^r,	(A.8)
and the wave impedance	к = Wyffii,	(A.9)
, =	(A.10) In addition to these, it is advantageous to define relative Pasteur (chirality) and Tellegen (nonreciprocity) parameters by " = ~^=,	(А.И) Xr = - =	(A.12) n y/Urf-T These quantities have absolute values between zero and one. In addition to these, the Tellegen angle i? is defined by		
sini? = Xt- For uniqueness, we assume the condition		(A.13)
	1ВД1 <	(A.14)
whence i? and x correspond to one another uniquely.
A.2 Other Notations
The constitutive relations (Л.З), (A.4) are not the only choice of characterizing the
magnetoelectric response of matter. In modern electromagnetics literature, also
other notations are used for chiral media. Here, two other very common notations
are presented with the relation to equations (A.3), (A.4). In [1] a more complete
look at the relation between various notations can be found.
Л.2. Other Notations
309
Post Relations
The constitutive relations named1 after Post measure chirality with an admittance
D = ePE-J(cB + VnB,	(A.15)
H = —B-J6E-V>„E.	(Л.16)
Pr
In these relations, the fourth bi-isotropic parameter, the nonreciprocity susceptance
^>n, has been included [1]. Both magnetoelectric parameters have the dimension of
amperes/volt.
The connections between the material parameters are the following:
and, inversely
e = cP + цг(4>гп + (’), ц = Zz„,
PrV’n	_ _ PtC
X ~	л .--•
\Jу/P'O^O
(Л.17)
(Л.18)
(A.19)
(A.20)
Drude—Born-Fedorov Relations
Another choice, much used in the analysis of reciprocal chiral media, are the Drude-
Born Fedorov relations
D - cdbf(E 4-/3V x E),	(A .21)
В = /rDar(H + /3V x H),	(A.22)
where /3 gives the amount of the chirality of material, in terms of length. An ad-
vantage in these relations is that these are not restricted to Fourier space, and the
fact that from these, it can be directly seen that chirality vanishes for electro- and
magnetostatics. Note here the limitation to Pasteur media; the material character-
ization (being reciprocal) only requires three parameters.
The conversion of the DBF material parameters to the Pasteur medium param-
eters е,р,к used in this book is the following:
'The label for the constitutive relations is a delicate choice. Here we follow the constitutive
relation names used most commonly in the literature, see, e.g., [2].
Appendix A. Notation
with n =	and ko = w2^fio€o.
The inverse transformation is
^DBF	_ /^DBF
x = o,
with k£ar = u>’pD„eDBB.
(A.23)
(A.24)
(A.25)
(A.26)
1 - к*„Г
к =
Comparison of Notations
It can be seen from the conversion rules that the Post system is “halfway between”
the present notation and the DBF system: the magnitudes of the permittivities and
permeabilities in the three notations fall into the order
£ — fF = £ПВГ>	(A.27)
/I Ровг,	(A.28)
where the equality holds for simple isotropic (reciprocal nonchiral) media.
Confusion may arise in careless usage of the permittivity and permeability values
without explicit emphasis on the constitutive relatione. On the other hand, the
differences between these values are small because the correction terms can be seen
in the above equations to be of second order in the magnetoeleclric parameters x, K>
which are generally small.
Due to the differences, it may be more appropriate to look for more univer-
sal parameters than c,p, x>* witli which the equations are expressed in analysis.
What would these be? Pasteur medium is birefringent, and Tellegen medium is
bi-impedant. Therefore, one choice for the set of four quantities would be the wave
numbers of the two eigenwaves (wave fields), with the two impedances of these
fields. This quadruplet k¥,k~,t)+,t)_ is independent of the choice of the constitu-
tive relations. These natural parameters can of course be written in each system.
In the present system,
k± — ko(ncos^ 1 л), ijx = qe’A	(Л.29)
In the Post system,
А.З. Bianisotropic Media
311
А:± = w

y//iPey + цЩ T Jt4 V’n
fr + /‘e(C + ift)
(A.30)
Note that the these impedances are dependent on the chirality parameter (here
(c), unlike in (A.29). On the other hand, there is no effect of the nonreciprocity
parameter on the wave numbers, or refractive indices.
In the reciprocal Drude-Born-Fedorov system,1
^V^DBF^DBr d: pDartDKr(3	_ MoBF
1 “ ^1/iDBFcDBF^1	V €OBB
(A.31)
A.3 Bianisotropic Media
Bow do the parameters transform in the case of bianisotropic media? As an example,
consider the reciprocal medium with three constitutive dyadics
П = ГЕ-д/^.Н,
В = iV^o л7 • E + p • H,
(A.32)
(A.33)
where the permittivity and permeability dyadics are symmetrical.
The generalization of the Post system for reciprocally bianisotropic media calls
for the constitutive relatione
D = ?, • E —	 В,	(A.34)
H = -jV-E + ^-,.B	(A.35)
with, again, symmetrical permittivity and permeability dyadics.
The translation between the constitutive parameters is the following:
"HI II "til 1 *11 "til 1 *4 T=ll II yii	(Л.36)
C41 II tl Jtll til 1	(A.37)
and	
* = +?c£r V» P-?ri	(A.38)
list llvjf II ine	(A.39)
’Here three parameters are sufficient, as the medium is uni-impedant.
312	References
From these relations, it can be seen that the nature of the different material
parameters (whether scalar or dyadic) depends on the notation used. Take, for
example chirofcrrite, where randomly oriented helices are dispersed in ferrite ma-
terial. If the permittivity and chirality are considered scalar in one system, the
transformation rules above make these parameters dyadic in the other notation,
because these become multiplied by the dyadic permeability p or Jfp. However, for
uniaxially bianisotropic media considered in Chapter 8 of this book, the nature of
all the parameters remains the same in both systems, as can be easily seen from
the formulas.
References
(1] A.H. Sihvola and I.V. Lindell, “Bi-isotropic constitutive relations,” Microwave and Optical
Technology Letters, Vol. 4, No. 8, p. 296-297, 1991.
[2] A. Lakhtakia, V.K. Varadan, and V.V. Varadan, Time-harmonic electromagnetic fields in
chiral media, Lecture Notes in Physics, 335, Springer-Verlag, Berlin, 1989
Appendix В
Complex Vectors
Complex vector notation, originally introduced by Gibbs [1] along with the real
vector notation, is suitable for describing time-harmonic vectors, which are real
vectors rotating along an ellipse in a plane. We can define a relation between a
complex vector a = a, + jot with two real vectors a, and a^, and a real time-
harmonic vector A(t) = Ai cosivt | A2 sinwt as [2]*
A(t) = J?{ae’ul} = a, cotuii — a; sinu>l.	(B.l)
The inverse of this relation is
a = A(0)-;A(^-) = A1-jA1,	(B.2)
ZU?
as can be seen by substituting (B.2) in (B.l), or conversely, to obtain an identity.
The complex conjugate of a complex vector, a* = Лг—]вц, corresponds to the time-
harmonic vector A(—t), which means that its sense of rotation along the ellipse is
reversed from that of A(t).
B.l Ellipse of a Complex Vector
Any complex vector a can be visualized by an ellipse followed by the corresponding
time-harmonic vector A(t). The real and imaginary part vectors a,, a; are lying
on the ellipse. The direction of rotation of A(t) on the ellipse equals that of the
imaginary part a; moving the shortest way to the real part a,.
The special cases are circle and line and they are defined by
‘Time convention e‘"‘ is applied here and everywhere in the book.
313
Л1Ч
Appendix В. Complex Vectori
polarization	condition	label
linear	a x a* =0	LP
circular	a • a = 0	CP
elliptic	otherwise	EP
Any non-CP vector with a-a 0 can be written in axial form as
a = e’’b
with real в and complex vector b defined by
(H.3)
It can be shown that the vector b defines the same ellipse as a together with the
same direction of rotation but the real and imaginary parts br, b, arc shifted from
those of a so that they are parallel to the axes of the ellipse: br along the major
axis and b; along the minor axis.
B.2 Polarization Vector
The real polarization vector p(a) of a complex vector a defined by
p(a) =
a x a*
ja • a*
(B.5)
gives important information on the polarization of the complex vector. In fact, it
can be shown that [2] p(a) points in the direction normal to the plane of the ellipse
of a so that the direction of rotation is right handed when looking in the direction
of p(a). The length p(a) = |p(a)| of the polarization vector satisfies 0 < p(a) < 1
and gives information on the axial ratio (ellipticity) e(a) of the ellipse:
1 - Jl - p’(a)
(B’6)
Conversely, we can write
For LP vectors we have p(a) = 0, for CP vectors |p(a)| = 1. If we write p(a) =
up(a) with u a real unit vector, the scalar p(a) is called the polarization number
(or fatness factor, because the ellipse is fatter for larger |p(a)|).
В.З. Two-Dimensional СР Vectors
315
The axial ratio of the ellipse can also be obtained directly as the length of the
ellipticity vector
( \ - a x a*
e & j(a-a* + |a • a|)’
whose direction equals that of p(a).
(B.8)
B.3 Two-Dimensional CP Vectors
The vectors u+,u_ in xy plane, defined by
«+ = ^(u- - >uv). u- = ~^(u« + >uv).	(n-9)
where unit vectors along the cartesian Coordinate axes are denoted by u„ uv, u,,
are two circularly polarized unit vectors. From the polarization vector
p(u+) = u,,	p(u_) = -u,	(B.10)
we see that u, has right-hand polarization with respect to +z direction and u_
right-hand polarization with respect to — z direction or, what is equivalent, left-hand
polarization with respect to +z axis. The cartesian unit vectors can be written as
u. =-^=(u++ u_), uv = ^(u+ - u_).	(B.ll)
The two CP vectors possess the following simple properties:
u* = n_,	u* = u+,	(B.12)
u+ • u* = 1,	u_•u* = 1,	(B.13)
u+• u+= 0,	u_-u_ —0,	(B.14)
u+ x u_ = ju„	(B.15)
u, x u+ = ju+l	u, x u_ = -ju_,	(B.16)
u+u* + u_u* = u+u_ + u_u+=	(B.17)
u+u* — u_u* = u+u_ — u_u+ = —ju, x F„	(B.18)
where 7( is the two-dimensional unit dyadic (Appendix C).
316
Reference»
В.4	Vector Bases
Three complex vectors a, Ь, c form a base in three-dimensional space if they satisfy
ax b-c / 0. In this case, we can expand any vector <1 as [1, 2]
d = a(a' • d) + b(b' • d) | c(c • d),
when the reciprocal base is defined by
, bxc ,, cxa , axb
a = -----—,	b = ---------, c =---------—.
axb-c	axb-c	axb-c
The base and its reciprocal base satisfy the properties
a-a' = l, b-b'=l, c -c' = l,
a • b' = 0, a - c' = 0, b • c' = 0, b • a' = 0, c • a' = 0, c • b'
The original base is the reciprocal of its reciprocal base: (a')' = a, (b')' = b, (c')' =
c. Any orthonormal base satisfying a-a = b-b = c-c = 1 and a-b = b-c = c-a = 0
is self reciprocal, i.e., equals its reciprocal.
For example, the vectors u+, u_, u, defined above form a base because U+ X
u_ • u, = j. The reciprocal base is
u'+ = u_,	u'_ = u+, u' = u,.	(B.23)
The three-dimensional unit dyadic can be written in terms of any vector base as
I = a a' -f- bb' + cc' — a'a + b'b | c'c.	(B.24)
References
[1] J.W. Gibbs, E.B. Wilson, Vector Analysis, (2nd ed.), New York, Scribner, 1908. Reprint
New York, Dover, I960.
[2] I.V. Lindell, Methode for Electromagnetic Field Analgeie, Oxford, Clarendon Press, 1992.
(B.19)
(B.20)
(B.21)
Appendix С
Dyadics
Dyadics are linear mappings between vectors written in coordinate-free form. The
dyadic notation applied here is due to J.W. Gibbs, the originator of the notation in
the 1880’s [1].
C.l Basic Properties of Dyadics
The basis of dyadic notation is the dyadic product of two vectors a and b in the
form ab, which is called a dyad. A dyad is defined through its operation on a third
vector c as
(ab) • c = a(b • с), c • (ab) = (c • a)b.	(C.l)
Л linear sum of dyads Я — J3«qb; is called a dyadic polynomial, or dyadic for
short. The dyadic product does not commute. The transpose operation is denoted
by (ab)r = ba.
Different products between two dyads are defined by
(ab) • (cd) = a(b • c)d,	(C.2)
(ab) : (cd) = (a  c)(b • d),	(C.3)
(ab)JJ(cd) = (a x c)(b x d),	(C.4)
and they can be extended to dyadics. Powers of dyadics refer to the dot product:
ЯП = Я-Я"-1,	(C.5)
A° = I = U,u, + UyUy + U.U,.	(C.6)
There also exists the double-cross square of a dyadic, defined by
317
318
Appendix C. Dyadics
з<2>^з;з,	(с.?)
also called “the second of by Gibbs and “the adjoint of 3” by Chen [2].
There are a number of identities which can be applied very effectively in analysis
and a collection of them is given in [3]. Some of the most important ones are listed
as follows:
(a x 7)jT = a x I,	(C.8)
= |7;7 = 7,	(c.9)
(a x 7)J(b x7) = ab + ba,	(C.10)
3:(7hF?) ж (3: г?)Л+(3: Л)?7- БТ • ё- с Т • л, (ол о
det3 = |3<’> : 3 =	: 3,	(С.12)
3	о
spm3 = 3«:7=i(3;3):7,	(С.13)
tr3 = 3 : 7,	(С.14)
=-г 3<’>г i = = „	=
А	= = —Ц=(Л1Л)Г, detA/O,	(С.15)
det3 2detA ’	Г	'	’
3-" = (3-,]" = 13"]-,I	(0.16)
3(,) • (a x b) — (3 • a) x (3 • b) = 3-1 • (a x b)det3. (017)
C.2 Two-Dimensional Dyadics
When considering propagation of a plane-wave in a BI medium, two-dimensional
dyadics are needed. They are defined as dyadics orthogonal to any given unit vector,
taken here as u,:
u, -3 = 3 u, = 0.	(C.18)
Two-dimensional dyadics do not have an inverse because det3 = 0. A two-
dimensional inverse of a two dimensional dyadic A can be defined explicitly as (3j:
ЛТх и и	—	1= — =	1= —
A~* =—spmA=-AiA:/=-AJA:u,ul7tO, (C.19)
suniA	2	2
С.2. Two-Dimensional Dyadics
319
satisfying
Т*Л=1.Т1 = 71,	(с.20)
7, = u„u. + uvu„.	(C.21)
The function spm/I, (“sum of principal minors,” in the language of three-
dimensional matrix algebra) can also be called the two-dimensional determinant
of the dyadic A.
Analytic functions of two-dimensional dyadics /i are understood in terms of their
two-dimensional Taylor expansions. Thus, the exponential function is of the form
е1 = 7, + Л+ 11’ + ^ + -...	(C.22)
It must be noted that the rule
eX+® = e<e®	(C.23)
is valid for commuting dyadics satisfying A  В = В • A. From this property, the
relation
(e1)"1 = e~*	(C.24)
is seen to follow.
И in addition to commutation, the two-dimensional dyadics A and В are orthog-
onal in the sense A • В = В • A — 0, we can write
Л+® = J- e® - 7t.	(C.25)
If the dyadic A is a projection operator so that for all n A 0 we have A" = A,
the exponential function satisfies the property
eo^=eaA + 7t-A,	(C.26)
where It — A is another projection operator orthogonal to A.
The 90° rotator dyadic J = и, x I plays a role similar to that of the imaginary
unit in complex plane:
e“^ = It сова -f- J sin а = 71(a).	(C.27)
This exponential dyadic rotates any two-dimensional vector by the angle a in right-
hand sense with respect to the positive z direction. It satisfies
Я(-а)= [77(a)]-1,
(C.28)
320
Reference»
72(a) • 71(0) = 71(0)  72(a) = 72(a + 0),	(C.29)
spm72(a) = 1,	(C.30)
e«h+pJ = e«i. .	= ^71(0).	(C.31)
The dyadic	
Я = Al, + В 7	(C.32)
is the most general two-dimensional dyadic satisfying	
M XX S S II M	(C.33)
It can be written in rotator form	
A = p72(<J$), p = VA’ + 7?’, ф = tan-1(Z?/A),	(C.34)
and its inverse as
(C.35)
References
(1]	J.W. Gibbs, E.B. Wilson, Vector Analysis, (2nd ed.), New York, Scribner, 1909. Reprint
New York, Dover, 1960.
[2]	H.C. Chen, Theory of Electromagnetic Waves.* A Coordinate-free Approach, New York,
McGraw-Hill, 1983.
[3]	I.V. Lindell, Methods for Electromagnetic Field Analysis, Oxford, Clarendon Press, 1992.
Appendix D
Collection of Basic Formulas
In the following, a collection of basic formulas applicable in the analysis of electro-
magnetic waves in chiral and BI media is given for convenience. convention is
applied throughout.
D.l Constitutive Equations
D = cE + £H	(D.l)
В = (E 4 /ill	(D.2)
( =	(D.3)
< = (X +	(B-4)
« =	P = Pr/'o, * = M, X = ХгП	(D.5)
Xr = sini9, n = y/firt.r	(D.6)
Conditions for Lossless BI Medium Parameters
P*=P, e* = e, k*=k, x*-X	(D.7)
S’ + X’ < 1	(0.8)
Conditions for Lossy BI Medium Parameters
P«n<0, e.m<o, K?m+x?m<^=	(0.9)
321
322
Appendix D. Collection of Basic Formulas
D.2 Waveflelds in Homogeneous BI Media
V x E±	= -jo>p±H± - M±	(D.10)
V x II	± = JU'C±E± + J±	(D.ll)
E = E+ + E_,	E± = —Ц(е^Ет^Н) 2 cos v	(D.12)
H = H+ +H_,	н± = —Ц(e^H ±-E) 2 cos v	r]	(D.13)
J = J+ + J_,	J± = 7-4(e±idJT-M) 2 COS V	7/	(D.14)
M = M++M_,	M± -	(eT’ M± jr/3) 2 cos v	(D.15)
Derived Medium Parameters
e± = c(cosi? ± Kr)e±-'d, p± = /i(cosd ±
fcjt =	— kon±, n± = n(cosd ± к.г)
(1X16)
(D.17)
(D.18)
D.3 Plane-Wave Relations
кх = ufcj. = ukn-j.	(D.19)
E±(r) = E±e->k*	H±(r) =	r	(D.20)
II± = —±- x E± = — u x E±	(D.21)
O>^±	Tl±	
E± = — x H± = —»;±u x H±	(D.22)
	
H+ = ^-E+, H_ = -^-E_	(D.23)
»?+	4-	
u x E+ = jE(, u x E_ = —JE_	(D.24)
u = u, : E(z) = e->**“,dK(-Krfcz) • E(0)	(D.25)
u = u,:	II(z)=17J(i? t J) E(z)‘ 7/	2	(D.26)
2>r	
f^r^^pol)	It	-^pot	j гр |xr|fc	(D.27)
e	V-Jicos.5	(D.28)
D.4. Green Dyadics
323
D.4 Green Dyadics
( E<r)	G..(r-T')
к H(r) ) J \ Gme(r - r')
fc(/-n) (м'Йк <“»)
= 2^гН(*+<?+ + fc-G )/ 4 V(G+ - C) *1 + vv (1Г + тг) ] (D 30)
^mrn
[(I,C, + U_)7 + V(C. -<;)x 7+ VV	+ £) ] (D.31)
G.m = —-1-[(*+<’% - k_e^G_)I
L COS V L
+ V(e--',’G+ + e^G.) x 7 + VV f
\ *+
sme = —Ц|(^л+ - jt_e-^G_)7
2 COS 17 I
(D.32)
+ V(e^G+ + e“idG_) x I + VV
ejdG+ e->*G_\l
к- / I
G±(r) =
e->*±r
4лт
(D.33)
(D.34)
D.5 Guided Waves
Ei(r) = [£<±(p) + u.Eti(p)]e 1Pl	(D.35)
(V? +	= 0	(D.36)
Ett =	T kiU, x 7,) • V«Et±	(D.37)
Лс±
A=e± = \Л± -01	(D-38)
D.6 Inhomogeneous BI Media
ejd	e--’*
V x E,. - *+E+ - vln x E+ =	\A?- * E- - 34+^7 (D.39)
e-j«	ei*
V x E_ + Л_Е_ — — V In л/т? x E. = —-----------rVln./n+ x E+ +jn_J_ (D.40)
cos v	cos v
324
Appendix D. Collection of Basic Formulas
D.7 Polarizabilities for Small BI Sphere
, V (£ -	+ 2,i’^ “ (x1 + K2^oto “	° (р + 2д0)(е + 2е0)_(х’ + к’Ь£о a -3ucy	Kx-j^y/fi^	 'm ' ° (в + 2д.)(е + 2eo) - (x2 + к2)до£о _ n v	3(x +		 am.	(р + 2до)(£ + 2£о)-(х2 + «2)^<. a = 3u V ~ P°^£ + 2£°* ~ (X* + (ц + 2/j„)(£ + 2£o) - (x2 + к2)д„£«	(D.41) (D.42) (D.43) (D.44)
Maxwell-Garnett Formulas for Mixtures with Spheres
[p. + 2 - /(Mr - 1 )][er + 2 - /(e, - 1)] - (x2 + «’)(1 - /)’
(D.45)
и a =1 + 3 f " 1)k + 2 -	- 1)1 ~ (X* + *2)(1 ~ Z) m 4fi)
9/x
= [дг + 2 - f(jir - 1 )][er + 2 -	- ijj - (x1 + k’)(1 - fY (D’47)
9/k
~ к + 2 - f(b - l)lk + 2 - /(e, - 1)] - (xa + K’)(l - /У ( Л8)
Dilute-Mixture Approximations
«eff = fо + ПО,
n8?{«n.e}
у/Mt+o
Keff ~
y/l^
(D.49)
(D.50)

Xeff =
Index
absorbing materials 239
absorption coefficient 237
adjoint of dyadic 318
adjoint operator 299
affine-isotropic medium
air gaps in measurement 270
alternative mixing laws 223
alternative constitutive relations 14, 225, 309
ANA (automatic network analyzer) 272
anechoic chamber 212
angle between field vectors 37
anisotropic
boundary impedance 143
crystals 16, 18
medium, effective 281
anisotropy, BI samples 271
antiferromagnetic 7
crystals 17
antisymmetric permittivity dyadic 206
aperture 156
Appleton-Hartrec formula 12
Arago 2
artifica) chirality 5, 14, 18, 270
Astrov 7
asymmetry, spatial 218
automatic network analyzer 272
average T-matrix approximation 224
axial decomposition of complex vector 314
axial impedance 280
backscattering 237
bandwidth 15, 228, 233, 240, 270
Beltrami field 236
BI (bi-isotropic)
ellipsoid 197
media, constitutive relations 13
media, homogeneous 13, 23, 32
media, lossless 50, 321
media, plane waves 32
media, power and energy in 49
medium, small sphere 324
rod in waveguide 264
slab 77, 114
sphere 194
waveguide 120
bianisotropic media 8, 12, 311
bianisotropic mixture 213
biaxial crystals 11
bifurcated modes 130
binomial coefficients 166
binomial expansion 166
biological nature 4
Biot 2, 4
Biot’s first law 5
Bohren decomposition 236
Born approximation 243
boundary impedance, anisotropic 143
Brewster angles 109, 254
Brown lens 161
Bruggeman model 224
Byelorussian school 8
Bottchcr mixing rule 224
capacitance 54
capacitance, helix 241
cavity resonators 265
CD 228
characteristic impedance 73
characteristic admittance dyadic 104
chiral bibliography 19
chiral coating 237
chiral media, hierarchy 243
chiral terminology 230
chirality 4
admittance 237
parameter 14, 155, 171, 308
parameter, alternative 14, 225, 309
parameter, sign 229
chirofcrritc II, 312
waveguide 119
325
326
Index
< hiioshicld 237
chirowaveguide 119
chromium oxide 7,16, 18
CIDS 243
circuit quantities 55
circuit theory 8
circular cylindrical resonator 268
circular dichroism 4, 10, 228, 258
Circular Intensity Differential Scattering 243
circular polarization 3, 35, 60, 154, 213
circular waveguide 137, 267
circularly polarized unit vectors 175, 182
coherent potential 224
complete elliptic integral,
first kind 162
second kind 165
complex vectors 313
Condon model 230, 234, 241
constitutive relations 8, 13, 225, 226, 308, 321
continuity conditions 49
copolarization 66, 85, 190
correction term, propagating fields 184
Cotton 228
coupled equation 153, 172
coupled-dipole approximation 239
coupling equation 172, 174
coupling of wavefields 26, 31
covariant methods 8
Cl’ (circularly polarized) vector 314
7
crosspolarization 156, 190
crosspolarized reflection coefficient
slab with PEC 86
from interface 66
crosspolarized transmission through interface
66
cross-susceptibility 210
crystal composition 3
crystal optics 8
crystal symmetry 9
Curie temperature 17
Curie, Pierre 7
current 72
current distribution 211
current in BI medium 56
current, vector 125
cut oft frequency 129, 134
cyhndriiai resonator, cii< ular 268
damping factor 231, 234
DCR 256
Debye model 228
degenerate modes, resonator 268
depolarization dyadic 197, 213
depolarization factors 197
depolarized scattering 237
determinant of dyadic 318
diadielertric 204
diamagnetic 203
dielectric lens 272
dielectric mixture 208
dielectric sphere 194
Differential Circular Reflection 256
dilute mixtures 217, 324
dipole fields 303
dipole moment, electric and magnetic 235
dipole moments, helix 240
dipole
radiation 10
source 157
electric 194
far field 42
magnetic 194
dispersion 6, 10, 15
curves 130
equations 138
in Bl media 227
in Bl mixture 231
in bulk BI 230
and mixing effect 233
dissipative media 233
Drude 4, 5, 230
Drude- Born Fedorov relations 14, 226, 308
duality 17, 202, 218, 220
duality transformation 12, 31
dyadic admittance 96
BI slab 114
dyadic impedance 96
dyadic product 317
dyadic sources 39
dyadics 317
Dzyaloshinskii 7
effective anisotropic media 281
effective medium parameters 209, 324
effective medium theory 224, 246
effective permittivity 208
eigenfunction expansion 237
Index
327
eigenpolarizations 107
of plane wave 284, 288
eigenproblem, single interface 106
eigenvalue equations 127
open BI waveguide 146
eikonal equation 154
electromagnetic activity 4
electromagnetostatics 53
electronic list of chiral publications 19
elect roweak force 219
ellipse
of complex vector 313
main axes 164, 168
ellipsoid mixture 213
ellipsoidal reflector antennas 272
ellipsometric measurements 252
ellipsometry 253
elliptic integral, second kind 169
elliptic polarization 37
eigenvectors 182
waves 183
ellipticity 259
of complex vector 314
enantiomer 4
energy conservation 9
energy in wavefieids 52
ensemble of helices 242
epoxy 18
equivalence principle 239
equivalent isotropic media 25
equivalent parameters 26
even and odd effects 218
exciting field 109
Extended Boundary Condition method 239
extinction coefficient 10
extraordinary Brewster angle 110
factorized operator 300
far field 42, 46
Faraday effect II
Faraday rotation 3
fatness factor 314
Fedorov 8, 237
ferrite 205
ferroelectric ceramic 15, 18
ferromagnetic 7
field expansion 239
field pattern of eigenmode 128
fields from a dipole 303
first-order correction 176, 184
fluctuation dissipation theorem 10
focusing ellipsoidal reflector 272
focusing horn 272
Former Soviet Union 8
fourth-order differential equation 301
free-space measurement 252, 258, 270
Freeman 6
Fresnel 3
Gaussian beam 272
geometrical optics 153, 171
Green dyadics 323
reciprocity 45
uniaxial chiral medium 298
Green functions 39
dyadic 12
time-domain 10
guided waves 120, 323
Gut man lens 167
gyrator 7, 12
gyroclectric 3
gyrotropic media 10
gyrotropic permittivity 206
Baidinger 228
handedness 218, 229, 239
helix 5, 14, 239, 270, 276
scattering 239
Helmholtz equation 40
vector 298
Hertz 4
Hertz potential 268
Hertz vector 267
hierarchical chiral structures 243
higher-order corrections 184
homogeneous media 23
Huygens 2
Huygens* principle 47
hybrid modes 129
Iceland Crystal 2
image theory 12
impedance boundary 122
impedance of Bl slab 78
impedance transformer 75
inductance 56
of helix 211
inequalities for parameters 28, 321
328
Index
in homogeneous
BI media 153, 271, 323
chirality distribution 157, 159, 162, 169
layered media 172, 180
lens 155
medium 31
input admittance dyadic 101
input impedance of terminated line 73
internal field 195, 198
interpretation
polarizabilities 201
effective BI parameters 218
nonreciprocity parameter 16, 219
Rayleigh formula 213
interstellar dust 239
inversion algorithm 259
invisible scatterers 236
isotropic boundary impedance 138
isotropic Green dyadic 41
isotropic media, equivalent 25
Kelvin, Lord 4
Kramers-Kronig relations 10, 230; 247
Laplace equation 55
layered chiral sphere 198
layered media 59
Lee, T.D. 218
left and right 218
lens antennas 155
LH (left-handed) 72
Lindman, Karl F. 4, 18
local plane wave 155, 180
longitudinal fields 121
Lorentz invariance 9
Lorentz transformation 9, 12
Lorentzian field 209
Lorenz-Lorentz formula 241
bi-isotropic 211, 223
dielectric 209
Lorenztian field, Bl 210, 213
lossless Bl medium 62, 321
lossless media 49, 172
Luneburg lens 163
magnetic crystals 7
magnetic optical activity 3
magnetoeleclric
activity 7
crystals 12
effect 7, 13, 16
inagnetoplasma 205
Malus 2
Maxwell 4
Maxwell fish-eye lens 158
Maxwell-Garnett formulas 324
bi-isotropic 211, 225, 227
dielectric 208, 224
percolation 224
measurement
methods, advantages 270
setup 272
techniques 251
eigenpolarizations 256
ellipsometric 252
Pasteur parameter 25$, 257, 260, 266
practical aspects of 270
Tellegen parameter 252, 260, 266
uncertainties 260, 262
medium parameters 49, 307
methanol 228, 257
Method of Moments 239, 243
Mie 236
theory 236
solution, terminology 247
minimum shape 222, 224
mixing effect, dispersion 233
mixing formulas 208
mixing rules, other constitutive relations 225
mixtures 208
of spheres 324
mixture, BI ellipsoids 213, 222
BI spheres 211
bianisotropic 213
dielectric 208
spherical shape as minimum 222, 224
high-frequency scattering 239
mode excitation, resonator 269
modeling of BI mixtures 208
modes in a waveguide 268
Monte-Carlo simulation 243
moving media 12
multiple scattering 243
network analyzer 271, 273
Newton 2
nondimensionalizing 231
nonreciprocal circuit element 8
Index
329
nonreciprocal effect 67
nonreciprocity 14, 43, 46
nonreciprocity parameter 16
alternative 14, 226, 308
nonreciprocity susceptance 14, 226
normal incidence 171
normalised chirality parameter 160, 163, 167,
170
normalized wavefields 153, 172, 184
notation 307
Null-field method 239
Neel temperature 17
oblique incidence 180
Onsager principle 8
Onsager-Kasimir symmetry 10
open BI waveguide 123
circular 146
planar 133
optical activity 2
magnetic 3
optical axis 11
optical rotatory dispersion 228
optical rotatory power 4
ORD 228
ordinary Brewster angle 110
organic nature 4
orthogonality of eigenpolarizations 285
orthogonality of modes 268
paraelectric 204
paramagnetic 203
parameter inequalities 50, 52
passive media 233
Pasteur 3
Pasteur medium 13, 172, 176, 183, 188, 228,
307
PEC (perfect electric conductor) 61
percolation 224
perturbation
effects, BI parameters 221
expansion, effective parameters 216
mixture 232
resonator 265
waveguide 263
phase factor 154
phase function 173, 181
phase velocities 10, 52
phase wavelength 38
phenomenological theory 9
photometry 253
planar BI structures 124, 127, 133
plane wave 32, 322
in layered media 59
in uniaxial medium
wavelength 38
system 92
PMC (perfect magnetic conductor) 61
polar molecules 228
polarimetry 253
polarizability 193
anisotropic 205
chiral sphere in chiral medium 200
effects of material parameters 202
ellipsoid 197
gyrotropic 206
helix 241
interpretation 201
layered sphere 198
matrix 196
of Bl sphere 324
Pasteur sphere 207
reciprocity 205
sphere 193, 225, 226
Tellegen sphere 207
polarization
axes 294
correction 159
number 314
purity, antennas 272
rotation 6, 35, 258
rotation in BI slab 83
transformer 286
vector 34, 293, 295, 314
wavelength 38
circular 154
light 2
Polder - van Santen formula 224
polymers 15
Post relations 14, 225, 309
postulates for wavefields 24
potential, Hertz 268
power decomposition 51
power in BI media 49
power of dyadic 317
Poynting vector 49
practical measurement aspects 270
propagation dyadics 95, 103
330
Index
propagator dyadic 289, 297
pseudoscalar 17, 219
Purcell-Pennypacker technique 239
quantum mechanics 230
quarter-wave transformer 290
quarter-wave transformer 76
quasi-TEM modes 57
quasistatic 194
fields, resonator 266
fields, waveguide 264
helix model 240
limit 208, 239
racemic 3, 6, 13
racemization process 219
radar cross section 236
random orientation in mixtures 214
Rayleigh formula, BI 212
Rayleigh formula, dielectric 208
Rayleigh scatterer 235, 271
Rayleigh, Lord 236
RCS reduction 236
reciprocal media 8
reciprocity 43
helix 241
rectangular BI waveguide 147
reflected fields 184
reflection
at loaded transmission line 74
BI plane 252
BI slab 258
coefficient matrix 252
coefficients 61, 185
copolarized 253
crosspolarized 253
dyadic 98, 105, 176, 180, 187
dyadic, BI slab 82, 89, 115
eigenpolarizations 256
from BI slab 77, 81
from interface 61
oblique 256, 257
refraction factors 33
refraction index 158, 161, 163, 167
relativistic motion 12
relaxation frequency 228
resonance model 230
resonance, helix 241
resonant frequency 15, 265
resonator
perturbation 265
techniques 262
circular cylinder 268
degenerate modes 268
retarded potentials 10
retrieval of material parameters 254, 260,
262, 267
RH (right-handed) 72
Riccati equation 172
rotation 155
rotation angle 157
rotation dyadic 35
rotational strength of molecular transition
231
rotator dyadic 319
sample anisotropy 271
scalar potential 10, 53
scatterer 235
scattering
by helices 239
chiral spheroids
cross section 237
depolarization 237
ensemble of helices 242
inhomogeneous BI samples 271
irregular objects 239
large objects 235
low-frequency limit 239
Pasteur cylinder 236
Pasteur shell 236
Rayleigh 271
single helix 240
second-order material relations 11
self-dual fields 31
self-susceptibility 240
short focus horn antenna 161
Shtrikman 7
sign of chirality parameter 218
single interface, oblique incidence 103
slab between isotropic half spaces 79
slab waveguide 127, 135
slab waveguide on PEC plane 137
slab with PEC backing 80
slowly varying media 171, 180
small scatterers 193
Solid state crystals 16
sources of wavefields 30
Index
331
spatial dispersion 8
spatial inversion 4
spherical BI sample in resonator 266
spherical waves 10
spiral structures 11
standing waves 65
statics in BI media 53
stationary medium 12
stealth applications 238
stereochemistry 4
stratified bi-isotropic media 171
surface roughness 271
surface waves 113
symmetric permittivity dyadic 206
symmetry breaking 218
symmetry of kinetic coefficients 8
symmetry, crystals 9
synthetic chirality 5, 14, 18, 270
system of plane waves 92
T-inatrix method 239
tartrate 3
ТЕ, TM 126, 278, 283, 286
modes 137
Tellegen 7, 12
Tellegen medium 13, 176, 189, 307
moving 12
Tellegen parameter 16, 24, 44, 141, 155, 171,
184, 188, 238
ТЕМ plane waves 59
tensor notation 8
tetrahedron 6
thermodynamic potential 7
thin-wire approximation 241
Thomson, William 4
time-domain gating 271
time-harmonic vector 313
Tinoco 6
titanium oxide 7
transformation of polarisation
elliptic 295
linear 293
transmission coefficients 61
transmission dyadic 83, 100, 105
transmission line 55
loaded 74
nonsymmetric 71
transmission matrix 125
transmission
BI slab 258
through BI slab 77
through interface 61
through transmission-line section 76
transveral waves 286
transverse fields 121
Treves 7
true scalar 17
truncated electromagnetic problem 48
twist polarizer 84
two-dimensional dyadics 318
two-dimensional vectors 315
ultraviolet electrons 5
ultraviolet absorption 229
uniaxial bianisotropic media 275, 312
unification of electromagnetism and weak in-
teraction 219
variational principles 224
vector bases 316
vector circuits 124
vector current 93
vector potential 10
vector transmission line 91
vector voltage 125
vector voltage 93
voltage 72
in BI medium 56
vector 125
water relaxation 228
wave impedance 27, 73
wave propagation 153, 171, 180, 182
wave vector 33, 283
wave-number surfaces 284
wavefield 24, 153
decomposition 236, 322
polarization 34
vectors 28
wave fields
as self-dual fields 32
inhomogeneous media
power decomposition 51
reciprocal media 29
waveguide 119
perturbation 263
techniques 262
circular 267
332
Index
wavelength 38
weak interaction 218
weak magnctoelectric coupling 216
Winkler 6
WKB approximation,
normal incidence 171
oblique incidence 180
expression for TM and ТЕ waves 188
X band 6
Yang, C.N. 218
zeroth-order approximation 182
zeroth-order solution 174, 184
The Artech House Antenna Library
Helmut E. Schrank, Series Editor
AdvancedTechnology in Satellite Communication Antennas: Electrical
and Mechanical Design, Takashi Kitsuregawa
Analysis, Design, and Measurement of Small and Low-Profile Antennas,
K. Hirasawa
Analysis of Wire Antennas and Scatterers: Software and User's Manual,
A. R. Djordjevic, M. B. Bazdar, G. M. Bazdar, G. M. Vitosevic, T. K.
Sarkar, and R. F. Harrington
Analysis Methods for Electromagnetic Wave Problems, E, Yamashita,
editor
Antenna Measurement Techniques, Gary E. Evans
Antenna-Based Signal Processing Techniques for Radar Systems, Alfonso
Farina
CAD for Linear and Planar Antenna Arrays of Various Radiating
Elements: Software and User's Manual, Aleksandar Nesic and Miodrag
Mikavica
Computational Electromagnetics, Konada Umashankar
Designer Notes for Microwave Antennas, Richard C. Johnson
Electromagnetic Waves in Chiral and Bi-Isotropic Media, I.V. Lindell,
S.A. Tretyakov, A.H. Sihvola, A. J. Viitanen
Fixed and Mobile Terminal Antennas, A. Kumar
Generalized Multipole Technique for Computational Electromagnetics,
Cristian Hafner