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City University of Hong Kong

香港城市大學

Time Domain Integral Equations for Scattering and

Radiation by Three-Dimensional Homogeneous

Bi-Isotropic Objects with Arbitrary Shape

用於解決任意形狀雙各向同性物體散射和輻射

問題的時域積分方程

Submitted to the Department of Electronic Engineering

電子工程系

in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

哲學博士學位

by

Wu Zehai 吳澤海

July 2010

二零一零年七月

i

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Bi-isotropic and bi-anisotropic media have received considerable attention from

many researchers, because these materials have been recognized as two of the miracle

substances for fueling multi-disciplinary developments in the new century. For

example, exotic material has been used to load an antenna for enhancing channel

capacity, suppressing interferences, improving sensitivity, and reducing its size.

Electromagnetic modeling of these exotic substances is much sophisticated, especially

for the case of three-dimensional arbitrary shape. Hence, this dissertation concentrates

on devising a time-domain integral equation solver for radiation and scattering

problems associated with complex media. Simply put, the contributions of this work

are summarized into two parts: (i) to generalize the surface integral equations in the

time domain for investigating the wave scattering by homogeneous bi-isotropic

objects, (ii) to extend the coupled surface integral equations in the time domain for

predicting the radiation by bi-isotropic body loaded dipole antenna.

In the first chapter, a brief introduction of the bi-isotropic media is given. At first,

the main characteristics of bi-isotropic media are presented together with the

constitutive relations. Then the research procedure about chirality property evolved

from optical frequency into microwave frequency is explained. Next the constitutive

equations about the two subclasses of bi-isotropic material, chiral media and Tellegen

media are described. Also, the 3-dimensional tensor parameters of the bi-anisotropic

media are introduced. Finally, the objective of the study is proposed.

Chapter two is the review of the marching-on in time (MOT) and marching-on in

degree (MOD) methods for solving the time domain integral equations (TDIE). The

ii

TDIE solver outperforms finite difference time domain (FDTD) method in some

aspects, and the reason is outlined in the introduction section. In the second section,

the formulation of the integral equations incorporated with the MOT method for

scattering by homogeneous dielectric body is given at first, and then the numerical

implementation is described using the method of moment (MoM). Compared to MOT,

the MOD method is more stable and can eliminate the late-time instabilities. The

formulation and the numerical procedure of the MOD method are reported in the last

section.

Subsequently, the extension of the surface integral equation from homogeneous

dielectric objects to bi-isotropic bodies is presented in chapter 3. It is rather difficult to

make a straightforward extension because the Green’s functions are very complicated

to handle numerically. In the second section, the field decomposition scheme is used

to replace the bi-isotropic medium with two respective isotropic ones, namely the

“plus” and “minus” mediums. The procedure to obtain the parameters of the two

equivalent isotropic media is described. Using the surface equivalence principle, a set

of the coupled integral equations using the renowned Poggio-Miller-Chang-

-Harrington-Wu-Tsai (PMCHWT) formulations are eventually derived.

Following the integral equations series is the numerical implementation for the

scattering of general bi-isotropic bodies. The surface integral equations are solved

using the MoM involving separate spatial and temporal testing procedures. The

famous Rao-Wilton-Gllison (RWG) functions are selected as the temporal basis and

testing function, and the weighted Laguerre functions are chosen as the temporal basis

and testing functions. To validate the accuracy of the proposed TDIE method, the

scattering of bi-isotropic objects is analyzed, and the transient currents, far scattered

fields, and bistatic radar cross-sections are presented. At the same time, the parametric

iii

study on the convergence test is conducted, and the influence of the parameters on the

accuracy of results has been summarized.

In the fifth chapter, the surface integral equation in the time domain is further

extended for the dipole antenna in the vicinity of a bi-isotropic body. A very narrow

perfectly electric conducting (PEC) strip is placed near the bi-isotropic object as the

loaded antenna. By enforcing the boundary condition separately on the strip dipole

and the surface of the bi-isotropic materials, a series of coupled integral equations are

obtained and solved numerically using MoM. The numerical results show that the

method provides accurate prediction of the radiation compared with the previous

solutions.

Finally, the conclusion of this thesis is given in chapter 6, and the further

extensions of the proposed MOD based TDIE method are discussed.

iv

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First and foremost, my greatest debts are to my ever encouraging and supportive

supervisor Professor Edward Yung Kai Ning. Because of him, I decided to come to

this university, which is an important decision in my life that that I will never regret.

During these four years, I have learnt a great deal about both technical issues and

ethics of research. I have benefited from his insightful comments and valuable

suggestions throughout the study. All that I learned from him will accompany me in

my whole life.

There is a long list of colleagues who helped me in different stages of my work.

Most of all, I would like to thank Dr. Wang Daoxiang for his valuable advices and

useful help on mathematics and computational electromagnetics. My special thanks

also go to Mr. Bao Jian, who taught me a lot about coding using FORTRAN.

Thanks are also extended to Prof. Luk Kwai-Man, Prof. Leung Kwok Wa, and

Prof. Chen Zhining for serving as the members of the oral examination committee.

My sincere thanks go to many colleagues for their help. This list includes but is

not limited to Dr. Zhang Xiuyin, Dr. Qu Shiwei, Dr. Wang Xiaohua, Mr. Zhao Peng,

Dr. Lou Yu, and Dr. Lim Eng Hock.

Finally and most importantly, I want to give my thanks to my family. My father,

my mother, and my brother in mainland China deserve all my gratitude for their

continuously support. Without their encouragement, it would be impossible for me to

complete this work.

v

Table of Contents

Abstract…………….………….…………..………………………………….……………i

Acknowledgements………………….……..…….………………….……………………iv

Table of contents…………….……………...……………………………….…………..…v

List of Figures…………….……………..…..…………………….……………….…….vii

Chapter 1 Introduction

1.1 General View………….……….……………….………..……………….…… 1 1.2 Bi-isotropic Media……..……….……………...………………………..……... 3 1.3 Study Objectives…….…………………..…………..…………………………. 7

Chapter 2 Time Domain Integral Equations

2.1 Introduction……………………….………………………………….……... 10 2.2 MOT TDIE……………………………….…………...……………..…….. 12 2.3 MOD TDIE…………………………………………….…………....……… 16

Chapter 3 TDIE Formulation for Scattering of BI Media

3.1 Introduction……………………………….…………………….……...…….. 23 3.2 Field Decomposition……………………………………………….………... 24 3.3 Two Equivalent Problems………………………………………………...….. 28 3.4 Surface Integral Equations…………………………………………….…..… 33

Chapter 4 Scattering for BI Objects

4.1 Introduction…………………………………………….….…..……………... 38 4.2 Basis Functions and Testing Scheme…………………….………...…..…….. 39 4.3 Current and Far Scattered Field……………………...…………………..…... 47 4.4 Numerical Results and Discussions…...…………………………….……….. 48 4.5 Conclusion…………………………………………….….………….……… 68

vi

Chapter 5 Radiation from Dipole Interacted with BI Object

5.1 Introduction…………………………………………….……….…..………....70 5.2 Theory and Integral Equations…………………….………….…...…..……71 5.3 Numerical Implementation……………………………………………….….75 5.4 Numerical Results……….........................................................................88 5.5 Conclusion……………………………………………………………...……...93

Chapter 6 Conclusions and Future Work

6.1 Conclusion……………………………………..………..….………………… 98 6.2 Future Work…………………………………………….………………….... 99

Appendix A Laguerre Functions.……………………………….…………………. 101

Appendix B Analytical Formulations for Self-term Double Integrals ….………..103

Bibliography…………….………………………………………….…………………. 107

Publications……………….………………………………………….……………….. 118

vii

List of Figures

2.1 Electromagnetic pulse incident on an arbitrarily shaped dielectric object……………13

3.1 The original problem of the Bi-isotropic scatterer. A plane wave is incident on an

arbitrarily shaped 3-D homogeneous BI objects……………………….……………29

3.2 The external equivalence of the original problem..……….………...……….....…....…29

3.3 The interior equivalence of the original problem….……………….…………......….…30

3.4 The “plus” sub equivalent problem of interior equivalence………......…………..……32

3.5 The “minus” sub equivalent problem of interior equivalence……………….….…...…32

4.1 Triangular pair associated with the nth edge for the definition of RWG function….….39

4.2 3-D Bi-isotropic sphere and coordinates illustrations…………...…..…………………50

4.3 Incident Gaussian pulse with ct0=0.5 lm, and T=0.1 lm…..……………………………50

4.4 Backward scattered far-field from the dielectric sphere…………………………....…..51

4.5 Forward scattered far-field from the dielectric sphere….………....……………………51

4.6 (a). Incident Gaussian pulse with ct0=0.5 lm, and T=0.1 lm. (b). Incident Gaussian pulse

spectrum with ct0=0.5 lm, and T=0.1 lm.…...…………………..……………………53

4.7 Normalized forward scattered fields of the dielectric and Tellegen spheres with radius of

0.01 m, permittivity εr=4.0, and permeability μr=1.0…..……………..…………….…54

4.8 Forward co-polarized bistatic echo widths of the dielectric and Tellegen sphere as a

function of frequency. The parameters of the sphere are chosen with εr=4.0, μr=1.0, and

radius of 0.01 m.…………………………………………………………………….…54

4.9 Normalized forward scattered fields of the chiral sphere. The sphere has a radius of 0.01

m, and other parameters are εr=4.0, μr=1.0, κr=0.15, and χr=0….…………………55

4.10 Forward co- and cross-polarized bistatic echo widths of the chiral sphere as a function

of frequency. The sphere has a radius of 0.01 m, and other parameters are chosen with

εr=4.0, μr=1.0, κr=0.15, and χr=0……………………..……..……………….…55

viii

4.11 Normalized forward scattered fields of the BI sphere. The sphere has a radius of 0.01 m,

and other parameters are εr=4.0, μr=1.0, κr=0.3, and χr=0.5..………....…………..57

4.12 Forward co- and cross-polarized bistatic echo widths of the BI sphere as a function of

frequency. The sphere has a radius of 0.01 m, and other parameters are εr=4.0, μr=1.0,

κr=0.3, and χr=0.5……………………………….…………….……………….57

4.13 Co-polarized bistatic echo widths of the BI sphere as a function of the evaluation angle.

The BI sphere has a radius of 0.01 m, and other parameters are εr=4.0, μr=1.0, κr=0.3,

and χr=0.5..………………………………………………………..………..……….…58

4.14 Cross-polarized bistatic echo widths of the BI sphere as a function of the evaluation

angle. The BI sphere has a radius of 0.01 m, and other parameters are εr=4.0, μr=1.0,

κr=0.3, and χr=0.5………………………………………….……….……………….…58

4.15 Transient currents at the point (0.0096, 0.0022, 0.0005) on a 0.01m radius BI sphere.(a)

linear scale, (b) logarithm scale ……………………………………...……..…59

4.16 Computed forward co- and cross-polarized radar cross-sections of a 7.2 cm radius

chiral sphere with comparison with FDTD method. The parameters of the sphere are

εr=4.0, μr=1.0, and κr=0.25.………………………………………………………….…60

4.17 Forward co-polarized bistatic echo widths of the BI sphere with different numbers of

meshed triangles. The sphere parameters are chosen with εr=4.0, μr=1.0, κr =0.3, χr =0.5,

and radius of 0.01 m. The maximum temporal order is M=120.………………………62

4.18 Forward cross-polarized bistatic echo widths of the BI sphere with different numbers

of meshed triangles. The sphere parameters are chosen with εr=4.0, μr=1.0, κr =0.3, χr

=0.5, and radius of 0.01 m. The maximum temporal order is M=120……….….……63

4.19 Forward co-polarized bistatic echo widths of the BI sphere with different values of M.

The BI sphere has 0.01 m radius, and other parameters are εr=4.0, μr=1.0, κr=0.3, and

χr=0.5………………………………………………………………………….………63

4.20 Forward cross-polarized bistatic echo widths of the BI sphere with different values of

ix

M. The BI sphere has 0.01 m radius, and other parameters are εr=4.0, μr=1.0, κr=0.3,

and χr=0.5…………..………………………………………………..………….…….64

4.21 Forward co-polarized bistatic echo widths of the BI sphere with different values of M.

The BI sphere has 0.01 m radius, and other parameters are εr=4.0, μr=1.0, κr=0.5, and

χr=0.5……………………………………………………………….…………...….…64

4.22 Forward cross-polarized bistatic echo widths of the BI sphere with different values of

M. The BI sphere has 0.01 m radius, and other parameters are εr=4.0, μr=1.0, κr=0.5,

and χr=0.5……..…………………………………………………….……………..…65

4.23 Forward co-polarized bistatic echo widths of the BI sphere for different values of M.


and χr=0.8..………………………………………………………….…………….…65

4.24 Forward cross-polarized bistatic echo widths of the BI sphere for different values of M.


and χr=0.8………………………...……………………………………………….…66

4.25 3-D Bi-isotropic cylinder and coordinates illustrations..………………………....…66

4.26 Forward co- and cross-polarized bistatic echo widths of the BI cylinder as a function of

frequency. The BI cylinder has a radius of 0.02 m with height of 0.04 m, and other

parameters are εr=4.0, μr=1.0, κr=0.3, and χr=0.5…….. ……..………………………67

4.27 Co-polarized bistatic echo widths of the BI cylinder as a function of the evaluation

angle. The cylinder has a radius of 0.02 m with height of 0.04 m. Other parameters are

εr=4.0, μr=1.0, κr=0.3, and χr=0.5……………………………..………………………67

4.28 Cross-polarized bistatic echo widths of the BI cylinder as a function of the evaluation

angle. The cylinder has radius of 0.02 m with height of 0.04 m. Other parameters are

εr=4.0, μr=1.0, κr=0.3, and χr=0.5………………………………………..……...….…68

5.1 A strip dipole near a BI sphere……..………………..………..…………………..….…71

5.2 Currents on the dipole and the surface of the BI sphere……..……..……..……..…71

x

5.3 Triangular patching of a strip dipole.………..…………………………………………74

5.4 The excitation Gaussian voltage pulse of T=4 lm and ct0=6 lm……………………..…88

5.5 The spectrum of the Gaussian voltage pulse with T=4 lm and ct0=6 lm................……88

5.6 Transient current at the center of the strip dipole without BI sphere. The strip dipole has

a length of 1 m with 0.01 m width, and the Gaussian voltage pulse of T=4 lm and ct0=6

lm is used……………………….………………………..….………………………...89

5.7 Current distribution along the dipole at the frequency of 150 MHz. The scatter

represents the data obtained from the proposed TDIE method, and the line denotes the

data from reference……………...………………………………………………..….…89

5.8 Radiation far-field of the single dipole in free space at θ=900, φ=00 when the excitation

is a Gaussian pulse………………………………………………………………..……90

5.9 Real part current along the dipole loaded with dielectric sphere and BI sphere at the

frequency of 150 MHz…………...……………………………………………………..90

5.10 Imaginary part current along the dipole loaded with dielectric sphere and BI sphere at the

frequency of 150 MHz.………………………………………………..………………..92

5.11 Current amplitude of the dipole loaded with dielectric sphere and BI sphere at the

frequency of 150 MHz..…………...………...………………………………………….93

5.12 Current at the center of the dipole loaded with BI sphere.……………...………...…….93

5.13 Current distribution variations along the dipole loaded with a BI sphere with different

chirality parameters…………………….……………………………….…………...…94

5.14 Current distribution variations along the dipole loaded with a BI sphere with different

Tellegen parameters………………………………………………………………...…94

5.15 Transient far fields at θ=900, φ=900 from a dipole loaded with dielectric sphere and BI

sphere. (a) θ-component, (b) φ-component……………………….………………95

5.16 Radiation pattern of the dipole loaded with BI sphere and DR sphere at the frequency of

150 MHz. (a) XOY plane, (b) YOZ plane………………….……………….………96

B.1 Geometrical representation for double integral over two triangles Tp and Tq…………..102

Chapter 1 Introduction 1

Introduction

1.1 General View

In the last two decades or so, complex media has emerged as one of the most

challenging topics in electromagnetic research in terms of theoretical problems and

potential applications [1]. Among these new materials, chiral media and

bi-isotropic/bi-anisotropic materials have been drawn considerable attention from

many researchers, and the advance has scattered in different journals [2]-[33]. The

major characteristic of these complex medium is the bi-isotropic relationship, which

enforces an additional coupling between the electric and magnetic fields. The

auxiliary coupling provides us an extra room of maneuver in our design of antennas

and components at microwave and millimeter wave frequencies.

Nowadays, as a result of the proliferation of wireless system for mobile

communications, wireless local area network, wireless access, radio and television

broadcast, satellite communications, radio frequency identification, remote control,

and other wireless applications, the spectrum assigned for civilian uses will be fully

occupied in no time. To meet the ever stringent demand on electromagnetic

interference and electromagnetic compatibility, many attempts to use chiral materials

in the design of antennas and microwave/millimeter wave devices were found in

literature, including radome [16], microstrip antennas [17], waveguide mode

transformers [18], polarization rotators [19], microstrip lines [20], chiroshield [21],

Chapter 1


and phase shifters [22].

Bi-isotropic (BI) medium is a special class of complex materials that can produce

both electric and magnetic polarizations when excited by either an electric or

magnetic source [2]. Among all, Chiral and Tellegen materials represent two

subclasses of BI medium. Chiral medium is optically active, which means that the

polarization plane of an electromagnetic (EM) wave is rotated when propagating

through it. Investigations show that chiral medium possesses the property of

reciprocity, while Tellegen medium is nonreciprocal. The optical activity didn’t been

introduced to microwave range until Lindman conducted his experiments in the

1910th [23]. After that, the chiral materials were extensively studied in EM

community. Bohren carried out several investigations on the scattering from chiral

cylinders [24] and spheres [25], [26]. The theorems for radiation, scattering,

equivalence and duality have been studied by Lakhtakia et al. [27]-[29], while the

dyadic Green’s function was obtained by Bassiri et al. [30]. In contrast to the

abundant works on chiral materials, fewer works have been conducted on Tellegen

materials since Tellegen proposed this kind of material in 1948 [31].

Many works have been contributed in an effort to develop an efficient numerical

technique to predict accurately the field’s interaction associated with such materials.

Over recent years, several numerical methods have been presented for solving the EM

problems related to three-dimensional (3-D) bi-isotropic media. Monzon developed

dyadic Green’s function for bi-isotropic materials [32]. In [33], the transition matrix

(T-matrix) method has been modified for chirally coated spheroids. The problem of

convergence restricts its application to nearly spheroid bodies as an infinite number of

expansion modes are required to truncate. More recently, surface integral equation

(SIE) based analysis of 3-D chiral bodies has been proposed in [11], and the method

was further extended for general bi-isotropic bodies by Wang et al. [7], [8], where the

integral equations were derived through the combination of the surface equivalence

principle and the field splitting scheme.


1.2 Bi-isotropic Media

BI medium is a subset of the general linear media, namely bi-anisotropic (BIA)

media. BIA media has two mechanisms, which are represented as anisotropy and

bi-isotropic [34], [35]. The former means the material parameters are dependent on

the direction, and the latter refers to the magnetoelectic between electric and magnetic

fields. This section will provide the background on BI media and its two important

subclasses including Chiral and Tellegen media.

The expression of electric and magnetic fields inside the BI region are relatively

complex because of the introduction of bi-isotropic constitutive relations, namely

1 1 1( )r rD E j Hε χ κ ε μ= + − (1.1)

1 1 1( )r rB j E Hχ κ ε μ μ= + + (1.2)

where rχ and rκ are Tellegen and Pasteur parameters, respectively, and ε1 and μ1

are the permittivity and permeability of the BI medium. For a lossy material, these

parameters are complex. It is noticed that the above relations reduce to a conventional

isotropic medium when both rχ and rκ are equal to zero. A bi-isotropic medium

with rχ 0= and rκ 0≠ is named the Pasteur medium or chiral medium, and it is

named Tellegen medium conversely.

The constitutive relations are frequency-domain expressions, which implicitly

assume time-harmonic excitation in the ejωt convention. The time dependence comes

through the inverse Fourier transform, and in order to render real electromagnetic

fields, the parameters have to be real functions of ω. On the other hand, the four

material parameters of a lossless medium have to be real. The time-domain

expressions of the constitutive relations are given as follows [36]:

1 CT CTHD E Ht

ε γ χ= + − ∂∂ (1.3)

1 CT CTEB H Et

μ γ χ= + + ∂∂ (1.4)

For time-harmonic field dependence, the connection of these material parameters to

those parameters in (1.1) and (1.2) is


1 1CT rγ χ ε μ= (1.5)

1 1 /CT rχ κ ε μ ω= (1.6)

1.2.1 Optical and Electromagnetic Activity

The BI medium has been known for nearly one hundred years in EM community. As

one subclass of BI medium, chiral medium is optically active, and this is a geometric

notion that refers to the asymmetry between an object and its image. In fact, this

chirality property can be described by another word handedness. There are two kinds

of handedness, left-handedness and right-handedness. Many objects in nature,

especially those of organic biologic nature, have handedness, such as amino acid and

DNA.

The chirality was originally found in optical frequency. In 1811, Argo discovered

that the polarization plane of a linearly polarized light ray will rotate as it passes

through a quartz crystal [37]. This gives rise to a new phenomenon called optical

activity. A formal treatment of the concept of handedness was given by Fresnel in 1822

[38]. He observed that when a linearly polarized light is travelling along the axis of a

quartz crystal, it is divided into two rays of circularly polarized light, one is

right-handed, and the other is left-handed.

In the late part of nineteenth century, Maxwell presented the famous Maxwell’s

equations and unified optics with electricity and magnetism [39]. Since the Maxwell’s

equations govern all electromagnetic phenomena from statics to X rays, theories of

optics and electromagnetics are essentially the same. Therefore, many researchers

have done a lot of work to extend the phenomenon of the polarization plane rotation to

lower frequency ranges, which can be called electromagnetic activity.

Lindman made an experiment in 1914 and demonstrated chirality at microwave

frequencies [23]. He wrapped a tiny copper helix with cotton at first, and then put

many cotton balls with random orientation in a cardboard box which was placed

between a transmitter and a receiver. The experiment demonstrated that a collection of

helices with a given handedness would rotate the plane of polarization of a linearly

polarized microwave in one direction while a collection of the helices with the


opposite handedness would rotate the plane of polarization in the other direction. The

experiment was carried out in the frequency band ranging from 1 to 3 GHz. This is a

remarkable breakthrough extended from optical frequency.

1.2.2 Chiral Materials

In the pioneering stage, most studies on EM activity were carried out

experimentally until the ground-breaking work done by Jaggard, who formulated a

relatively rigorous theoretical base for this phenomenon in 1979 [40]. For chiral

medium, the constitutive relationship between the field intensity and the flux density

must be modified accordingly; that is

1 1 1( )rD E j Hε κ ε μ= + − (1.7)

1 1 1( )rB j E Hκ ε μ μ= + (1.8)

The above equations describe the relationship of chiral media in the frequency

domain, and the time dependence relations are given as follows

1 CTD tHEε χ∂

−∂

= (1.9)

1 CTB tEHμ χ∂

+∂

= (1.10)

where the parameter CTχ is given in (1.6).

Based on the common reference between electricity and magnetism, the

electro-magnetic force should be expressed in terms of the electric field intensity and

the magnetic flux density. Nevertheless, these equations can be rewritten in a

symmetric form as

1 ( )rD E j Bε ξ= + − (1.11)

1

1 ( )rH B j Eξμ= + − (1.12)


where 11

r rεξ κμ

= .

Another choice that is used commonly in the analysis of reciprocal chiral media is

the Drude-Born-Fedorov (DBF) relations [2]. The DBF relations are used to describe a

simple relationship between the field intensity and the flux density, and the

constitutive equations can be expressed as

( )DBFD E Eε β= + ∇× (1.13) ( )DBFB H Hμ β= + ∇× (1.14)

where β denotes the chirality of the material in terms of length. An advantage in these

relations is that they are not restricted to Fourier space, and it is noticed obviously that

the chirality vanishes for electro- and magnetostatics. The conversion of the DBF

material parameters to the Pasteur medium parameters εr, μr, and κr is referred to [2].

1.2.3 Tellegen Media

As another subclass of BI media, Tellegen medium was firstly proposed by B. D. F.

Tellegen in 1948 [31]. This nonreciprocal medium came out when Tellegen defined a

new concept in circuit theory, which is called the gyrator. A gyrator is a device which

gyrates currents into voltage and vice versa. The constitutive relationship of the

Tellegen medium can be given as

1 1 1rD E Hε χ ε μ= + (1.15)

1 1 1rB E Hχ ε μ μ= + (1.16)

The Tellegen media exhibits a distinguishing magnetoelectric phenomenon; that is,

the angle between a linearly polarized electric and magnetic field is no longer 90

degree as it travels through the Tellegen material. Another word to describe this

nonorthogonality is nonreciprocity, and it is similar to microwave ferrite devices.

1.2.4 BIA Media

Generally speaking, Chiral media and Tellegen media both are sub-classes of BI


material, which is characterized by four parameters: permittivity εr, permeability μr,

chirality parameter κr, and Tellegen parameter χr, as shown in equations (1.1) and (1.2).

In the two equations, it is noticed that the electric field is coupled with the magnetic

field. With this extraordinary constitutive relationship, bi-isotropic materials have

several important properties. The first one is electromagnetic activity, which can rotate

the polarization plane of a linear polarized wave as it passes through a BI material.

The second property is circular dishroism, and it is attributed to the different

absorption coefficients of a right-handed and left-handed polarized wave in

bi-isotropic medium. The final one is the nonreciprocity. The angle between the

electric field and magnetic field of a linearly polarized plane wave is no longer

orthogonal as it travels through bi-isotropic materials.

The four parameters shown in equations (1.1) and (1.2) are all scale. When all these

four parameters become vectors, the resulted new constitutive relations represent a

new medium called BIA media. The constitutive equations would be expressed as

( )D E j Hε χ κ= ⋅ + − ⋅ (1.17) ( )TB H j Eμ χ κ= ⋅ + − ⋅ (1.18)

where ε , μ , χ , and κ are all 3-dimensional tensors, and the superscript “T”

denotes the transpose operation. In fact, BI media can be seen as a special case of BIA

media. The electromagnetic properties of BIA media have been extensively studied by

Kong. In his works, Kong has derived various theorems which characterize

electromagnetic fields in general linear, non-conducting BIA medium.

1.3 Study Objectives

This thesis is concerned with the EM scattering by general BI objects and EM

radiation by BI body loaded antenna. Although an analytical method has been

previously proposed for BI cylinders scatterer by Monzon where a contour integral

technique is combined with the dyadic Green’s function [32], [41], it can only

calculate two-dimensional BI bodies at normal incidence. Kluskens considered the

scattering of a chiral cylinder with arbitrary cross section [42]. Integral equations are


formulated for it, along with its solution in the method of moment (MoM). Later this

method was extended to three-dimensional chiral scatterer by Worasawate [43] and

chiral revolution by Yuceer [44]. Wang applied MoM to solve the surface integral

equations which were incorporated with the renowned Poggio-Miller-Chang-

-Harrington-Wu-Tsai (PMCHWT) formulations for the scattering by general BI

objects [6] and BI coated conductors [7]. Although there have been many

frequency-domain techniques reported for the scattering of BI media, very little work

has been done in the time domain. Most of the time-domain schemes available for BI

media focus on the finite difference methods, such as finite-different time-domain

(FDTD) [45], conformal FDTD [46], BI-FDTD [47], and so on. The available

examples are restricted to chiral spheres whose solutions can be analytically

calculated using the modal expansion theory. Therefore, the applicability of the FDTD

method still needs verification for general BI objects.

The time-domain integral equation (TDIE) solver is commonly used for analyzing

complex EM scattering and radiation phenomenon [48]-[69]. Although the FDTD

method has been the dominant tool for time-domain simulations, the TDIE approach

is preferable in some applications especially for the analysis of transient scattering by

large-size bodies. The reason is that the TDIE method solves fewer unknowns using

surfaces discretization and requires no artificial absorbing boundary condition (ABC).

The most popular method to solve a TDIE is the marching-on in time (MOT) scheme

[48]-[61]. However, many researchers have pointed out that the MOT method may

suffer from late-time instabilities in the form of high frequency oscillation. Recently,

the marching-on in degree (MOD) method [62]-[69] using a set of scaled Laguerre

polynomials as the temporal expansion and testing functions is proposed for the TDIE,

and stable results can be obtained even for late time. To the best of our knowledge,

this TDIE solver has not been used to deal with the scattering and radiation problems

associated with BI media. For the first time, this work presents the application of the

MOD-based TDIE method for three-dimensional homogeneous BI objects with

arbitrary shape.

In this thesis, a rigorous analysis based on integral equation method is developed


for modeling three-dimensional bi-isotropic media with arbitrary shape. A pair of new

sources and two integro-differential operators is first defined and, later, they are

introduced to formulate the far scattered fields by homogeneous dielectric objects in

time domain. Then the method is extended for constructing scattered fields inside and

outside the BI medium. A field splitting scheme [2], also known as Bohren

Decomposition [25], is employed to simplify the expression of the EM fields inside

the bodies. In order to achieve stable solutions, the renowned PMCHWT formulations

[70]-[72] are used to construct the surface integral equations. After enforcing

boundary conditions, a series of coupled integral equations are established, and they

are solved numerically by the MoM involving separate spatial and temporal testing

procedures. To verify the correctness of the proposed method, the numerical results

are compared with the exact solutions or the solutions available in the existing

literatures. For the sake of safety, the developed method is also used to analyze the

structures with isotropic media. In consequence, good accuracy is obtained in

comparison with other methods.

Chapter 2 Time Domain Integral Equations 10

Time Domain Integral Equations

2.1 Introduction

In recent times, the transient analysis of scattering has attracted considerable

interest in the electromagnetic fields of millimeter and microwave frequency range.

Calculating and predicting transient electromagnetic behavior are very useful. For

example, many radar systems use shorter pulses for target identification, high-range

resolution, and wide-band digital communications.

Although the transient data can be obtained by solving the problem in the

frequency domain and then using an inverse Fourier transform, direct time domain

methods may be suited better and applied more straightforwardly. First of all, direct

time-domain techniques are more convenient for dealing with nonlinearities. Direct

transient analysis provides an opportunity for us to observe and interpret

electromagnetic scattering behavior. Furthermore, direct time-domain techniques are

more efficient in broadband cases. Finally, the direct time-domain technique may be

viewed as another way to understand the nature of EM scattering behavior.

Among all the available solution techniques for obtaining transient responses, the

most popular are the finite-difference time-domain (FDTD) approach, the

transmission-line matrix (TLM) method, and the time-domain integral equation

Chapter 2


(TDIE) technique.

In the FDTD method, which was firstly proposed by Yee [73], the scatterer and the

surrounding infinite space are discretized into rectangular cells. The spatial and the

temporal derivatives in Maxwell’s equations can be achieved by leapfroging the

electric and magnetic fields in space and time. The FDTD approach can be used to

easily model complex bodies, such as three-dimensional inhomgeneous materials.

However, to yield a numerical solution the discretization grid must be truncated at a

suitable distance from the scatterer by using absorbing boundary conditions. Also,

modeling of curved sturctures with stair-step approximations usually brings about

problems. What is worse, the memory requirement is large in three-dimensional cases.

The TLM method, in which the space is modeled as an interconnection of

transmission lines, was firstly developed by Johns and Beurle [74]. Like the FDTD

approach, the TLM method is a time-stepping procedure in which the currents and the

voltages at one time instant are represented in the form of the currents and voltages of

previous time steps. Unfortuneletly, the solution must be performed again if the

polarization of the incident wave is changed. Also, the TLM method has the same

storage problem like the FDTD method.

Bennett formed an integrodifferential equation by enforceing the boundary

conditions on the surface of the scatterer and solved it directly in the time domain [75].

Like the MoM based frequency-domain analysis, this time-domain integral equation

(TDIE) solver discretizes the scatterer into segments or patches. The time axis is

generally divided into equal time instant. In this method, the currents on the scatterer

at a certain time t=t1 are related to the currents at t


2.2 MOT TDIE

In this section, the time domain integral equation solver for the scattering of

dielectric bodies is presented. The formulation of the integral equations is given at first

using the equivalence principle [76], and then the numerical implementation is

described using MoM [77]. The scattering involving dielectric bodies should interest

us for serveral reasons. First, most of the antenna structures nowdays comprise some

kind of dielectric material for the purpose of either feed support or ground plane

coverage. Second, many conducting scatterers are either partly or entirely coated with

radar-absorbing materials to reduce the radar cross-section. Third, almost all the

biological applications involve dielectric objects.

2.2.1 Integral Equation Formulation

Consider a homogenous dielectric body with a permittivity of ε2 and a permeability

of μ2 in an infinite homogenous medium with a permittivity of ε1 and a permeability of

μ1. We assume that the dielectric body is a closed body so that a unique outward

normal vector can be defined unambiguously. Employing the equivalent principle, the

body may be replaced with two sets of electric and magnetic currents. It can be easily

proved that the continuity of the tangential fields requires that

e dJ J J= = − (2.1)

e dM M M= = − (2.2)

Using the potential theory, the scattered fields are given by

( ) 1= ( , ) ( )s vv v v

v

A r,tE r t F r,tt ε

∂∇Φ ∇×

∂∓ ∓ ∓ (2.3)

( ) 1= Ψ ( , ) ( )s vv v v

v

F r,tH r t A r,tt μ

∂∇ ∇×

∂∓ ∓ ∓ (2.4)

where vA and vF are the magnetic and electric vector potentials, respectively, and

vΦ and vΨ are the electric and magnetic scalar potentials, respectively, given by

s

( , / )( , )=4

v vv

J r' t R cA r t dS'R

μπ

−∫ (2.5)


s

( , / )( , )=4

v vv

M r' t R cF r t dS'R

επ

−∫ (2.6)

( / )1( )

4e v

vv s

q r',t R cΦ r ,t = dS'Rπε−

∫ (2.7)

( / )1( )

4m v

vv s

q r',t R cr ,t = dS'Rπμ−

Ψ ∫ (2.8)

where for v=e or v=d; the “–” sign is for v=e, and the “+” is for v=d, R=|r - r' |

represents the distance between the observation point r and the source point r' .

Figure 2.1 Electromagnetic pulse incident on an arbitrarily shaped dielectric object


The electric surface charge density qe and magnetic surface charge density qm are

related to the electric current density J and magnetic current density M ,

respectively, by the equation of continuity

( , )= ( , )eJ r t q r tt∂

∇⋅ −∂

(2.9)

( , )= ( , )mM r t q r tt∂

∇⋅ −∂

(2.10)

We may eliminate qe and magnetic surface charge density qm from equations (2.7) and

(2.8), respectively, by defining

( )1 1,

4( / )v

vv

SvJ r' ,t R cΦ r t dS'Rπε

= − ∇⋅ −∫ (2.11)

1 1( ) ( / )

4v v vv Sr ,t = M r' ,t R c dS'

RπμΨ − ∇⋅ −∫ (2.12)

By enforcing the continunity of the tangential electric and magnetic fields at the

dielectric interface and taking an extra derivative with respect to time, the following

integral equations are derived as

tantan

e d incE E Et t∂ ∂⎡ ⎤− = −⎣ ⎦∂ ∂

(2.13)

tantan

e d incH H Ht t∂ ∂⎡ ⎤− = −⎣ ⎦∂ ∂

(2.14)

where incE and incH are the incident electric and magnetic fields, respectively, and

the subscript “tan” defines tangential components.

2.2.2 Numerical Implementation

MoM is used to solve the combined field equations (2.13) and (2.14). For this

purpose, the body is approximated by planar triangular patches. As noted earlier,

triangular patch modeling is capble of approximating any arbitrarily body accurately

and efficiently. Furthermore, for the application of the numerical procedures, we use

the well-known Rao-Wilton-Gllison (RWG) functions [78], [79] as the expansion and

testing functions.


As a first step, we define the electric and magnetic currents on the stucture as

( ) ( )1

, ( )N

k kk

r t I t fJ r=

= ∑ (2.15)

( ) ( )1

, ( )N

k kk

r t M t fM r=

=∑ (2.16)

where N is the number of inner edges, and ( )kf r is the RWG function. For the

implementation of the MoM, the surface of an arbitrarily shaped object is required to

mesh using a number of triangulated surface patches, in each of which the electric and

magnetic sources are expressed in terms of known triangular basis functions, namely,

the well known RWG functions.

To continue the MoM procedure, we test equations (2.13) and (2.14) with ( )mf r

by multiplying it and integrating in the triangle pairs mT± , and appromate the time

derivative of the potential functions by use of finite differences,

, ( , ) ( , ) , ( )E E

m e d m inc nf L J M L J M f E tt∂

+∂

= (2.17)

, ( , ) ( , ) , ( )H H

m e d m inc nf L J M L J M f H tt∂

+∂

= (2.18)

where

1 12

1

1( , ) ( ( , ) ( , ) ( , )) ( , )

1 1 ( ( , ) ( , ))

Ee v n v n v n v n

v n v nv

L J M A r t A r t A r t r tt

F r t F r ttε

+ −

+

= + − +∇ΦΔ

⎡ ⎤+ ∇× −⎢ ⎥Δ⎣ ⎦

(2.19)

1 12

1

1( , ) ( ( , ) ( , ) ( , )) ( , )

1 1 ( ( , ) ( , ))

He v n v n v n v n

v n v nv

L J M F r t F r t F r t r tt

A r t A r ttμ

+ −

+

= + − +∇ΨΔ

⎡ ⎤− ∇× −⎢ ⎥Δ⎣ ⎦

(2.20)

After some mathmatical manipulation, the previously equations (2.17) and (2.18) can

be rewritten as

2/ (4 ) , ( , ) (( , )) ( ) E Em inc e dm e d m n f E L J M L J Mtk t I tπ μ μ ∂ − −

∂Δ + = (2.21)


2/ (4 ) , ( , ) (( , )) ( ) H Hm inc e dm e d m n f H L J M L J Mtk t M tπ ε ε ∂ − −

∂Δ + = (2.22)

where

0.5c

m m mm mk l K ρ±

±

= ∑ (2.23)

2 1 12

1

1( , ) ( ( , ) ( , ) 2 ( , )) ( , )

1 1 ( ( , ) ( , ))

Ee v m n v m n v m n v m n

v m n v m nv

L J M A r t A r t A r t r tt

F r t F r ttε

− − −

−

= + − +∇ΦΔ

⎡ ⎤+ ∇× −⎢ ⎥Δ⎣ ⎦

(2.24)

2 1 12

1

1( , ) ( ( , ) ( , ) ( , )) ( , )

1 1 ( ( , ) ( , ))

He v m n v m n v m n v m n

v m n v m nv

L J M F r t F r t F r t r tt

A r t A r ttμ

− − −

−

= + − +∇ΨΔ

⎡ ⎤− ∇× −⎢ ⎥Δ⎣ ⎦

(2.25)

/kmm k mTK f r r' dS'

±

±

= −∑∫ (2.26)

It is obviously observed that equation (2.21) only has Im(tn) as the unknown, and

equation (2.22) only has Mm(tn) as the unknown. Hence, the electric and magnetic

currents have been decoupled in the present-time sense. However, the currents are still

coupled with previously occurring currents. The decoupling between the electric and

magnetic currents results from the self-term cancelation of the curl terms. This is an

advantage of the PMCHWT formulations, since the self-term of the curl operators is

not required to calculate.

The currents are obtained using the MOT technique. Once the currents at tn for all

triangle edges are calculated, the time step is increased, and the currents at tn+1 can be

obtained in the same manner. It is important to note that the time step should be

Δt≤Rmin/max(ce,cd) to generate a stable numerical results. In addition, an implcit

formulation can also be developed, which would need a matrix inversion.

2.3 MOD TDIE

Although the most popular method to solve a TDIE is the MOT method using the

RWG functions as vector basis functions, many researchers have pointed out that this


MOT method may suffer from late-time instabilities in the form of high frequency

oscillation. The cause of this instability is not exactly known, and most of the

researchers think that the MOT methods that diverge place some eigenvalues of the

system matrix outside the unit circle, which causes the instability according to the

Neumann analysis.

Many efforts have been paid to eliminate the instability of the MOT methods. An

explicit solution of the time domain PMCHWT formulation has been presented by

differentiating the coupled integral equations and using second order finite difference

[49]-[61]. The late-time oscillations could be eliminated by approximating the average

value of the current. In addition, a backward finite difference approximation for the

magnetic vector potential term in the time domain electric field integral equation has

been used for the implicit technique to minimize these late-time oscillations [81], [82].

Even though employing the implicit technique, the solution obtained by using MOT

still has a late-time oscillation that is dependent on the choice of the time step.

Recently, the marching-on in degree (MOD) method using a set of scaled Laguerre

polynomials as the temporal basis functions is proposed for arbitrarily shaped 3-D

dielectric bodies [62]-[69]. In this MOD method, the transient electric and magnetic

currents can be spanned by the orthogonal basis functions derived from the Laguerre

polynomials. Then a temporal testing procedure is introduced and the testing

functions are the same as the temporal basis functions. The numerical instabilities can

be eliminated by applying the temporal testing to the integral equation series. Instead

of marching on in time, a procedure of marching on in degree is employed by

increasing the order of the temporal testing functions.

2.3.1 Integral Equations

We define a pair of new sources ( , )e r t and ( , )h r t on the surface of the

dielectric body by

( ), ( , )J r t e r tt∂

=∂

(2.27)


( ), ( , )M r t h r tt∂

=∂

(2.28)

where ( ),J r t and ( ),M r t are the equivalent electric and magnetic surface

currents. According to the equation of continuity, the electric charge density ( ),eq r t

and magnetic charge density ( ),mq r t will be

( ), ( , )eq r t e r t= −∇ ⋅ (2.29)

( ), ( , )mq r t h r t= −∇⋅ (2.30)

By using the equivalent principle [76], the scattered fields will be formulated in terms

of the equivalent sources ( , )e r t and ( , )h r t on the surface S of the dielectric body

by

1( , ) ( ) ( )sE e h L e K h= − −

(2.31)

( ) ( ) ( )1 21

1,sH e h K e L hη

= − (2.32)

where η1 is the wave impedance in the medium surrounding the scatterer, and 1sE

and 1sH are the scattered fields outside the dielectric body. The two

integro-differential operators L and K in equations (2.31) and (2.32) are defined as

( )( ) ( )2

1 21

, ,14 4S S

X r' X r'L X dS dS'

t R'

Rτ τ

μπ ε π

∂ ∇ ⋅∇= −

∂∫ ∫ (2.33)

( ) ( ) 01 ( , ) 1ˆ , [ ]2 4πS

X r'K X n X r t dS't t R

τ∂ ∂= × + ∇×

∂ ∂∫ (2.34)

where R =| r - r' | represents the distance between the observation point r and the

source point r' , 1t - R cτ = is the retarded time, 1 1 11/c ε μ= is the velocity of

the propagation of the electromagnetic wave in the space, and S0 denotes the surface

with the singularity at r r'= removed from the surface.

By enforcing the continuity of the tangential electric and magnetic fields at S, the

following PMCHWT integral equations are obtained,

2 1 tantans s

incE E E⎡ ⎤− =⎣ ⎦ (2.35)


2 1 tantans s

incH H H⎡ ⎤− =⎣ ⎦ (2.36)

where 2sE and 2

sH are the scattered fields inside the dielectric body.

2.3.2 Numerical Implementation

For the implementation of the MoM, the equivalent electric and magnetic sources

( , )e r t and ( , )h r t are represented in terms of RWG functions by

,1 0

( , ) ( ) ( )N

n j j nn j

e r t e st f r∞

φ= =

=∑∑ (2.37)

,1 0

( , ) ( ) ( )N

n j j nn j

h r t h st f r∞

φ= =

=∑∑ (2.38) where N is the number of the inner edges, en,j and hn,j are the unknown coefficients,

and ( ) /2 ( )stj jst e L stφ −= is the causal temporal basis function, and ( )nf r represents

the RWG function. Lj(st) is the Laguerre function [83] of order j with a scaling factor

s.

Through the Galerkin’s method, we take a spatial testing with ( )mf r (m=0, 1,

2…N) and a temporal testing with ( )i stφ (i=0, 1, 2…M) to the two

integrodifferential operators L and K, respectively. With reference to [61], M is the

maximum order of the Laguerre functions that is the time-bandwidth product of the

incident waveform. In computing the integrals, the distance between two triangles R is

assumed to be constant, hence

1 2,1 ,2,pq pq

pq pqmn mnmn mn

R Rt tc cτ τ= − = − (2.39) where p and q can be either + or –, 1(2) 1(2)1(2) 1/c μ ε= , and

pqmnR is the distance

between center points of triangles pmT and q

nT . With the assumptions, we obtain the

equations as

1 , 01

12

11 0

,,

,0

,1( ), ( ), ( ) ( )

(0.25 ( ) ) ( )

N ipq

i m ij mnn p q j

jNpq

n k ij mn

pqn j mn

ipq

nk

j mp qn j

n

st f r L e e I s b

s e j k e I s aμ

φ τε

τ

= =

−

= = =

=

+ −

∑

∑∑∑

∑∑

∑ (2.40)


( ) ( ) ( )

( )

( )

1

, ,,1 0

2

, ,11

, ,

1

,1, 0 0

1

,2, 0 01

, , 0.5 0.5

(0.25 ( )

(0.5

)

)

pqmn

jipq

mnp q j k

jipq

m

N i

i m n i n kp qn k

Npq

n j n k ij mnn

Npq

n j n k ij m np q

nn j k

st f r K e s e e c

s e j k e I s d

d

c

s e e I s

φ

τ

τ

−

= =

=

=

−

= =

−

= =

⎛ ⎞= +⎜ ⎟

⎠

+

⎝

+

+ + −

∑

∑∑∑ ∑

∑∑ ∑

∑

∑

∑

(2.41)

where the inner integral ( ) ,mf r represents the spatial testing with multiplying

( )mf r and integrating in the triangle pairs mT± , and ( ) ,i stφ represents the

temporal testing, which means multiplying ( )i stφ and integrating from zero to

infinity. According to [68], the temporal integral is simplified as

/21[ ( ) ( )]

0( )

pqmns pq pq

pq i j mn i j mnij mn

e L s L s j iI = sj i

τ τ ττ−

− − −− ≤>

⎧⎪⎨⎪⎩

(2.42)

and the spatial integrals pqmna , pq

mnb , pqmnc , ,1

pqmnd , and .2

pqmnd are given by

( ) ( ) ( )' / 4pq p qmn m nS S

a f r f r' R dS'dSπ= ⋅∫ ∫ (2.43)

( ) ( ) ( )' / 4pq p q

mn m nS Sb f r f r' R dS'dSπ= ∇ ⋅ ∇ ⋅∫ ∫ (2.44)

( ) ( )ˆpq p qmn m nS

c f r n f r dS= ⋅ ×∫ (2.45)

( ) ( ),1 'ˆ (4 )pq p qmn m nS Sd f r f r' R R dS'dSπ= ⋅ ×∫ ∫ (2.46)

( ) ( )2

,2 'ˆ (4 )ppq qmn m nS Sd f r f r' R R dS'dSπ= ⋅ ×∫ ∫ (2.47)

where R̂ is a unit vector along the direction r r'− . It is noticed that the time and

space variables are separated in the computation and the time variable is replaced by

degree orders of the Laguerre functions.

We apply both spatial and temporal testing procedures to equations (2.35) and

(2.36), and the 2N×2N matrix below is obtained after some mathematical

manipulations,

,, 11

, ,1 1

E H Emn mn m in iN N N N NNE H H

n imn mn m iNN N N N N

ZE ZE e

hZH ZH

γ

γ× × ××

×× × ×

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦⎢ ⎥ =⎢ ⎥ ⎢ ⎥⎢ ⎥⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ (2.48)

where the elements on the left hand side of the matrix are


21

1

,1

,2

2

22

( ) exp( )4

( )e

12

p )4

12

x (

pqmnE pq pq

mn mn mnpq

pqmnpq pq

mn mnpq

sZE as b

s bs

a

τμ

τμ

ε

ε

= + −

+ + −

∑

∑ (2.49)

,1,1 ,2

1

,2,1

2,2

2

2

( ) ( )4

exp2

(

2

) ( )4 2

exp2

pqmnH pq pq

mn mn mnpq

pqmnpq pq

mn mnpq

ssZE d dc

ssdsc

d

s τ

τ

= + −

+ + −

∑

∑ (2.50)

2

2

,1,1 ,2

1

,2,1 ,2

1

( ) exp( )4 2

( )exp( )4 2

2

2

pqmnE pq pq

mn mn mnpq

pqmnpq pq

mn mnpq

ssZH d dc

ssd

s

cds

τ

τ

=− + −

− + −

∑

∑ (2.51)

2

2,11

1

2

2

1

2,2

2

2

( ) ( )4

) (

1 1 exp2

1 1 )4

( exp2

pqmnH pq pq

mn mn mnpq

pqmnpq pq

mn mnpq

s b

s

sZ

b

H a

sa

τμη ε

τμη ε

= +

+

−

+ −

∑

∑ (2.52)

and the elements on the right hand are

, , , ,E E E Hm i m i m i m iV P Pγ = − − (2.53)

, , , ,H H E Hm i m i m i m iV Q Qγ = − − (2.54)

where

( ) ( ), , ,iE

m i m if r st EV φ= (2.55)

( ) ( ), , ,iH

m i m if r st HV φ= (2.56)

and the elements ,E

m iP , ,H

m iP , ,Em iQ , and ,

Hm iQ are given as,

( )

( )

( )

( )

1 2

,11

1, ,

1 0

, 1

1

12

,11 0 0

2

,22

2,

12

,0

0

, 20

1

2

( )4

(

1

( )

14

( )

)

pqij mn

jN ipq

ij mnn pq

N iE pq pq

m i mn mn n jn pq j

pqn k mn

N ipq

j k

pqij mn

ji

pqmn mn n j

n pq j

pqn k mn

pqij mn

j k

s b I s

j k s I s

s b I s

j k s

P a e

e a

a e

e a I s

τε

τ

τε

μ

τ

μ

μ

μ

−

= =

−

−

= = =

−

=

=

=

=

= +

+

+ −

+ −

+

∑∑

∑∑∑∑

∑

∑

∑∑∑

1

N

n pq=∑∑∑

(2.57)


( )

( )

( )

1

, ,1 ,2 ,1 0 1

,1 ,2 ,1

2

,1

1 2

,11

1

,1 ,2 ,1 0 2

,12

0 0

2

,2

2

( 0.5 )4

)

(

(( )

0.

((

)4

)

5

pqij mn

jipq

ij mn

N iH pq pq

m i mn mn n jn pq j

Npq pq

mn mn n kn pq

N ipq pq

mn mn n jn pq j

p

j k

pqij m

m

n

qn

s d I sP d s hc

d s hc

d s hc

sj k d I s

s d I s

k dsc

j

τ

τ

τ

−

=

−

= =

=

−

= =

=

−

= +

+ +

+ +

+ −

∑∑∑

∑∑

∑∑

∑

∑

∑

( )1

,20

,2 ,01

)ji

pqij mn

j k

Npq

mn n kn pq

s hd I sτ−

= = =

+∑∑ ∑∑

(2.58)

( )

( )

( )

1

, ,1 ,2 ,1 0 1

,1 ,2 ,1 1

1

,1 ,2 ,1 0 2

2

,1

1 2

,10

,

0

2

,2

2

12

( 0.5 )4

)

( 0

(( )

(( )

.5 )4

pqij mn

jipq

ij m

N iE pq pqm i mn mn n j

n pq j

Npq pq

mn mn n kn pq

N ipq pq

mn m

nj k

pqij mn n j

n pq j

pmn

n

Q d s ec

d s e

s d I s

sj k d I s

s d I s

c

d s e

j k

c

dcs

τ

τ

τ

−

= =

=

−

= =

−

= =

=− +

− +

+

−

−

−

−

∑∑∑

∑

∑∑

∑∑

∑

∑

( )1

,2,20

,1 0

)ji

pN

q pqmn n

qij mn

j kk

n pq

s ed I sτ−

= ==

+∑∑∑∑

(2.59)

( )

( )

( )

2

,11

1

11

, ,21 01

, 1211

12

,2

2,1

0 0

2

1 02

, 2

2

1

22

,2

1 1

( )

14

( )

( )4

1

1 ( )

1

pqij mn

jipq

ij mnj

N iH pq pqm i mn mn n j

n pq j

Npq

n k mnn pq

N ipq pqmn mn n j

n pq j

pqn k mn

k

pqij mn

ij m

s b I s

j k s I s

s b I s

j k s I s

Q a h

h a

a h

h a

τε

τ

τ

μη

μη

μη

μη

ε

τ

−

= =

=

−

=

−

= =

=

= +

+

+ +

+

−

−

∑

∑∑∑

∑∑

∑∑

∑

∑

( )1

1

,20 0

jipqn

j

N

n pq k

−

= = =∑∑∑∑

(2.60)

As we can see from equations (2.48) to (2.60) that, to obtain the coefficients en,j and

hn,j, we need to solve the matrix recursively on the order of the degree of Laguerre

function. Particularly, in the first step when i=0, ,E

m iP , ,H


Hm iQ are all

equal to zero, only system matrix elements EmnZE , HmnZE ,

EmnZH , and

HmnZH are

needed and its LU decomposition can be stored for further use. In the following i-th

step, we only have to compute ,E

m iP , ,H


Hm iQ on the right side of the

matrix, which are the sums of the previous solved coefficients en,j and hn,j.

Chapter 3 TDIE Formulation for Scattering of BI Media 23

TDIE Formulation for

Scattering of BI Media

3.1 Introduction

The formulations which are required to solve the scattering problem of general

bi-isotropic objects are derived in this chapter. There are a lot of publications scattered

in various journals for numerical solution of bi-isotropic or chiral bodies. Most of

them are restricted to 2-D infinite bi-isotropic or chiral cylinders and spheres, of which

the solution can be calculated using the expansion of vector wave functions. The

FDTD formulation was proposed for three-dimensional chiral scatterers with arbitrary

shape. However, the available examples were limited for chiral sphere whose solutions

can be analytically calculated with mode expansion theory. The applicability of this

method still needs to be verified for general bi-isotropic bodies. The early integral

equation based method was developed for Bi-isotropic cylinders on the basis of the

dyadic Green’s function. Unfortunately, the form of the Green’s function is

comparatively complex [84], [85]. Recently, the field decomposition was introduced

combined with integral equation for general chiral and bi-isotropic objects. Although

their calculated results are achieved in a satisfactory manner, the formulations are

derived in frequency domain, and few works has been done in the time domain using

the integral equation. In addition to that, a T-matrix method was extended for full wave

Chapter 3


analysis scattering by a chiral body, but this method suffers the deficiency of

convergence stability, especially when the modeling body has a surface of complex

shape.

In this chapter, a time domain surface integral equation (SIE) is developed for the

analysis of wave scattering by a homogeneous bi-isotropic object. The Bohren

Decomposition [24], is employed to simplify the expression of the EM fields inside

the bodies. It turns out that the fields in the BI media can be decomposed into two

uncoupled wave fields, both of which satisfy Maxwell’s equations individually. In this

sense, the BI media can be replaced by two isotropic dielectrics each of which is

characterized by its own isotropic parameters, and the fields for BI media can be easily

obtained as the summation of two wave fields. In order to achieve stable solutions,

PMCHWT formulations are used to construct the surface integral equations. After

enforcing boundary conditions, a series of coupled integral equations are established.

3.2 Field Decomposition

Here, we consider a homogenous bi-isotropic body with permittivity of ε2 and

permeability of μ2 which is embedded in an infinite homogenous medium with

permittivity of ε1 and permeability of μ1. The expression of electric and magnetic

fields inside the BI region is relatively complex because of the introduction of

bi-isotropic constitutive relations

2 2 2( )r rD E j Hε χ κ ε μ= + − (3.1)

2 2 2( )r rB j E Hχ κ ε μ μ= + + (3.2)

where rχ and rκ are Tellegen and Pasteur parameters, respectively. Regardless of

the medium, the electromagnetic fields in bi-isotropic media must fulfill the Maxwell

equations in the frequency domain, given by

E j B Mω∇× = − − (3.3)

H j D Jω∇× = + (3.4)

where J and M are the equivalent electric and magnetic currents.


Substituting equations (3.1) and (3.2) into (3.3) and (3.4) yields

E E M

H H J

⎡ ⎤ ⎡ ⎤ ⎡ ⎤−∇× = +⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦Z (3.5)

where

( )

1/ ( )r r

r r

j jjk

j jχ κ η

η χ κ− − −⎡ ⎤

= ⎢ ⎥− +⎣ ⎦Z (3.6)

in which /η μ ε= is the characteristic impedance of bi-isotropic medium, and

k ω εμ= is the wavenumber of bi-isotropic medium. The coupling resulted from Z

can be decoupled by diagonalizing Z such that

1 0

0k

k+−

−

⎡ ⎤= ⎢ ⎥⎣ ⎦

A ZA (3.7)

Let 21 ( 1)r rα χ χ= − ≤ and rv jα χ± = ± , we have

1 j j

v vjη η

η − +

⎡ ⎤= ⎢ ⎥

⎣ ⎦A (3.8)

1 1

2v jv j

ηηα

+−

−

−⎡ ⎤= ⎢ ⎥−⎣ ⎦

A (3.9)

and k± are related with the right- and left-handed components defined by

( )rk k α κ± = ± (3.10)

and accordingly, the right- and left- handed fields and currents

1E E

E H+ −

−

⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥

⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦A (3.11)

1M M

M J+ −

−

⎡ ⎤ ⎡ ⎤−= −⎢ ⎥ ⎢ ⎥

⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦A (3.12)

jEHη

±± = ± (3.13)

jMJη

±± = ∓ (3.14)


Substituting equation (3.9) into (3.11) yields

1 [ ]

2E v E j Hη

α±= ∓ ∓ (3.15)


1 [ ]

2jH v H E

α η± ±= ± (3.16)


1 [ ]

2M v M j Jη

α±= ±∓ (3.17)


1 [ ]

2jJ v J M

α η± ±= ± (3.18)

From equations (3.15) to (3.18), we can obtain

E E E+ −= + (3.19)

H H H+ −= + (3.20)

J J J+ −= + (3.21)

M M M+ −= + (3.22)

Equations (3.1) and (3.2) can be written in matrix form as

( )

( )r r

r r

D E

H

j

jB

χ κ εμ

χ κ εμ

ε

μ

−

+

⎡ ⎤⎡ ⎤ ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ (3.23)

Using equations (3.13), (3.19) and (3.20) to express (3.23) involving E± instead

of E and H , we obtain

( ) ( )( )

/)

( ) ( ) /(r r r r r r

r r

ED

EB

j jj j

ε α ε αεκ χ κ κ χ κ

κ η κε ηα α+

−

⎡ ⎤⎡ ⎤ + −⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥+ −⎢ ⎥ ⎣ ⎦ ⎢ ⎥⎣ ⎦ ⎣ ⎦

+ −−

(3.24)

Therefore, the right- and left- handed electric flux density vectors D± , magnetic

flux densities B± as

D Eε± ± ±= (3.25)


B Hμ± ± ±= (3.26)

where

( )r vε ε α κ± ±= ± (3.27)

( )r vμ μ α κ± = ± ∓ (3.28)

From the previously derived equations, it is not difficult to obtain

D D D+ −= + (3.29)

B B B+ −= + (3.30)

After the field decomposition mentioned above, the wavefields associated with

right- and left-handed or “Plus- and Minus-” fields are independent of each other. In

other words, wavefields and corresponding currents are individually connected with

the Maxwell equations for equivalent homogeneous isotropic dielectric, namely,

E j H Mωμ± ± ±∇× = − − (3.31)

H j E Jωε± ± ±∇× = + (3.32)

With the field decomposition, the problem associated with bi-isotropic media is

simplified into the ones with two equivalent isotropic media, that is, Plus- and

Minus-media.

Here we define a pair of new sources ( , )e r t and ( , )h r t on the surface of the

bi-isotropic body as

( , ) ( , )J r t e r tt∂

=∂

(3.33)

( , ) ( , )M r t h r tt∂

=∂

(3.34)

It is not difficult to obtain the time domain Maxwell’s equations (3.3) and (3.4) as

( , )E B h r tt t∂ ∂

∇× = − −∂ ∂

(3.35)

( , )H D e r tt t∂ ∂

∇× = +∂ ∂

(3.36)

Also, the wave splitting equations (3.31) and (3.32) are changed into

E B ht t± ± ±∂ ∂

∇× = − −∂ ∂

(3.37)


H D et t± ± ±∂ ∂

∇× = +∂ ∂

(3.38)

where e± and h± are corresponding Plus-/Minus- sources on the surface of the

bi-isotropic object, and their relations with sources ( , )e r t and ( , )h r t are governed

by

1 [ '' ]

2e v e hη

α± ±= ± (3.39)

1 [ ' ]

2h v h eη

α±= ±∓ (3.40)

where ' jη η= and '' jη η= .

It is noted that here we introduce a pair of new sources ( , )e r t and ( , )h r t to

replace the conventional equivalent electrical current ( , )J r t and magnetic current

( , )M r t . In this way, a time-integral term will disappear in formulations of the

far-scattered fields, and we can easily handle the time derivative of the electric and

magnetic vector potentials.

3.3 Two Equivalent Problems

We consider a homogeneous bi-isotropic body, which is assumed to be suspended in

free space and is illuminated by a plane wave pulse at incident angle of (θi, φi). The

problem under consideration is shown in Figure 3.1. incE and incH are incident

fields. S is the surface of the BI body, and n̂ is the unit vector normal to the surface

which points outward from S. With the presence of the scattered body, the fields inside

the body are E and H , which can be determined as

inc SE E E= + (3.41)

inc SH H H= + (3.42)

where SE and SH are the scattered fields produced by the BI objects.

Similar to the method used to solve scattering problem from dielectric medium, the

original problem can be replaced with two equivalent problems: one exterior


Figure 3.1. The original problem of the bi-isotropic scatterer. A plane

wave is incident on an arbitrarily shaped 3-D homogeneous BI objects

Figure 3.2. The external equivalence of the original problem.


equivalent problem and one interior problem, as shown in Figures 3.2 and 3.3,

respectively. In the exterior equivalent problem, the whole space is occupied by the

homogeneous medium which is characterized by ε and μ, and the scattered body is

replaced by equivalent electric and magnetic sources e and h which are residing

on the outside surface of S. It is assumed that the fields inside the space of S are null,

and the fields outside the space of S are exactly the same as the fields in the original

problem. To support the discontinuity of the fields across the surface S, the equivalent

electric and magnetic sources should be defined as

ˆ ˆ( , ) ( ( , ))inc ee r t n H n H H e h

t∂

= × = × +∂

(3.43)

ˆ ˆ( , ) ( ( , ))inc eh r t n E n E E e h

t∂

= − × = − × +∂

(3.44)

where E and H are the total electric and magnetic fields outside the surface S, and

( , )eE e h and ( , )eH e h are electric and magnetic fields produced by the equivalent

surface sources e and h .

Figure 3.3. The interior equivalence of the original problem.


According to the boundary condition, the tangential fields should be continuous

across the surface S. Thus,

ˆ ˆ ( , )inc en E n E e h× = − × (3.45)

ˆ ˆ ( , )inc en H n H e h× = − × (3.46)

In the interior equivalent problem, the whole space is characterized by ε, μ, κr, and

χr. Equivalent electric and magnetic sources e− and h− are on the inside surface of

S. Null fields are assumed in the space external to surface S, and the fields in the space

internal to surface S are the same as that in the original problem. To support the

discontinuity on surface S, the equivalent sources are required to be

ˆ( ( , )) ( ( , ))ie r t n H e h

t∂

− = × −∂ (3.47)

ˆ( ( , )) ( ( , ))ih r t n E e h

t∂

− = ×∂ (3.48)

where ( , )iE e h and ( , )iH e h are the fields radiated by the surface equivalent

currents in the space internal to surface S. Again, two equations can be obtained

according to the continuity of the tangential fields across surface S, which are

ˆ ( , ) 0in E e h× = (3.49)

ˆ ( , ) 0in H e h× = (3.50)

Return to the original problem again, the tangential fields also need to be continuous

across surface S, which means that the summation of equations (3.45) and (3.49),

equations (3.46) and (3.50) should satisfy

ˆ ˆ ( ( , ))inc ebn E n E E e h× = − × + (3.51)

ˆ ˆ ( ( , ))inc ebn H n H H e h× = × + (3.52)

The next step is to split the interior equivalent problem into two sub equivalent

problems, which can be called the “Plus” sub equivalent problem and “Minus” sub

equivalent problem. As shown in Figure 3.4, the whole space in the “plus” sub

equivalent problem of the interior equivalent problem is characterized by parameters

ε+ and μ+. e+ and h+ are the electric and magnetic equivalent sources on the inner


Figure 3.4. The “plus” sub equivalent problem of interior equivalence.

Figure 3.5. The “minus” sub equivalent problem of interior equivalence.


surface of S. Electric field bE + and magnetic field bH + are the fields radiated by

them in the space internal to surface S, and the fields in the space external to surface S

are null. Similarly, in the “minus” sub equivalent problem of interior equivalent

problem as shown in Figure 3.5, the whole space is characterized by parameters ε– and

μ–. The electric and magnetic sources on the inner surface of S are e− and h−

respectively. The electric and magnetic fields radiated into the internal space of surface

S are denoted as bE − and bH − , respectively, and also null field is radiated by them in

the external space of surface S.

Based on the fields splitting scheme discussed previously, the relations between the

interior equivalent problem and its two sub equivalent problems can be expressed by

( , ) ( , ) ( , )b b bE e h E e h E e h+ + + − − −= + (3.53)

( , ) ( , ) ( , )b b bH e h H e h H e h+ + + − − −= + (3.54)

3.4 Surface Integral Equations

The fields inside the BI media have been split into two groups of wavefields, and

each group of wavefields can be treated as an equivalent isotropic medium with

corresponding parameters. As a result, the problem of scattering and radiation from BI

medium becomes a summation of the problems of EM scattering from two equivalent

homogeneous isotropic bodies. With the aid of integral equations, scattering of

isotropic objects can be solved. For certain scattering problems, different integral

equations can be implemented based upon their geometry and material characteristics,

and the PMCHWT formulations, which is a special kind of combined field integral

equation (CFIE), is employed to construct the field equation series.

Substituting equations (3.53) and (3.54) into equations (3.51) and (3.52), we obtain

ˆ ˆ ( ( , ) ( , ) ( , ))inc e b bn E n E e h E e h E e h+ + + − − −× = × − + + (3.55)

ˆ ˆ ( ( , ) ( , ) ( , ))inc e b bn H n H e h H e h H e h+ + + − − −× = × − + + (3.56)

Inserting the relations (3.39) and (3.40) between ( )e h and ( )e h± ± into two


equations above as

ˆ ˆ ( ( , ))1 1ˆ ( [ '' ], [ ' ])

2 2

inc e

b

n E n E e h

n E v e h v h eη ηα α± ±±

× = × −

+ × − ± − ±∑ ∓ (3.57)

ˆ ˆ ( ( , ))1 1ˆ ( [ '' ], [ ' ])

2 2

inc e

b

n H n H e h

n H v e h v h eη ηα α± ±±

× = × −

+ × − ± − ±∑ ∓ (3.58)

The electric and magnetic fields E and H produced by electric and magnetic

surface currents J and M , radiating into an unbounded space characterized by ε1

and μ1 are given by

s

1

( ) 1= ( , ) ( )A r,tE r t F r,tt ε

∂− −∇Φ − ∇×

∂ (3.59)

s

1

( ) 1= Ψ( , ) ( )F r,tH r t A r,tt μ

∂− −∇ + ∇×

∂ (3.60)

where A and F are the magnetic and electric vector potentials, respectively, and

Φ and Ψ are the electric and magnetic scalar potentials given by

1 ( , )( , )=

4 S

J r'A r t dS'R

μ τπ ∫ (3.61)

1 ( , )( , )=

4 S

M r'F r t dS'R

ε τπ ∫ (3.62)

1

( )1( )4

e

S

q r',Φ r ,t = dS'Rτ

πε ∫ (3.63)

1

( )1( )4

m

S

q r',r ,t = dS'Rτ

πμΨ ∫ (3.64)

where R =| r - r' | represents the distance between the observation point r and the

source point r' , 1t - R cτ = is the retarded time, and 1 11/1c ε μ= is the velocity

of the propagation of the electromagnetic wave in the space. The electric surface

charge density qe and magnetic surface charge density qm are related to the electric

current density J and magnetic current density M , respectively, by the equation of

continuity


( , )= ( , )eJ r t q r tt∂

∇⋅ −∂

(3.65)

( , )= ( , )mM r t q r tt∂

∇⋅ −∂

(3.66)

A pair of new sources ( , )e r t and ( , )h r t are defined in (3.33) and (3.34), so the

charge density will be

( ), ( , )eq r t e r t= −∇⋅ (3.67)

( ), ( , )mq r t h r t= −∇⋅ (3.68) Equations (3.61)-(3.64) will be changed as

( )1 1, ( , )

4 SA r t e r' dS'

R tμ τπ

∂=

∂∫ (3.69)

( )1 1, ( , )

4 SF r t h r' dS'

R tε τπ

∂=

∂∫ (3.70)

( )1

1 1, ( , )4 S

Φ r t e r' dS'R

τπε

= − ∇⋅∫ (3.71)

1

1 1( ) ( )4 S

r ,t = h r', dS'R

τπμ

Ψ − ∇⋅∫ (3.72)

Substitute equations (3.69)-(3.72) to (3.59) and (3.60), respectively,

( ) ( )

( )0

2'

1 21

, ,14 4

, 1 ( , )0.54π

s

S S

S

e r e rE dS dS'

R t R

h r t h rn dS't R t

τ τμ

π ε π

τ

∂ ∇ ⋅∇= − +

∂

∂ ∂− × − ∇×

Documents

Time Domain Integral Equations for Scattering and Radiation by …lbms03.cityu.edu.hk/theses/c_ftt/phd-ee-b39478609f.pdf · 2019. 7. 9. · Bi-isotropic and bi-anisotropic media have