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Nondestructive Measurements of Electromagnetic
Parameters of Anisotropic Materials
Using An Open-Ended Waveguide Probe System

presented by

Chih-Wei Chang

has been accepted towards fulfillment
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NONDESTRUCTIVE MEASUREMENTS OF ELECTROMAGNETIC
PARAMETERS OF ANISOTROPIC MATERIALS
USING AN OPEN-ENDED WAVEGUIDE PROBE SYSTEM

By

Chih-Wei Chang

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1995

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ABSTRACT

NONDESTRUCTIVE MEASUREMENTS OF
ELECTROMAGNETIC PARAMETERS OF
ANISOTROPIC MATERIALS USING A WAVEGUIDE
PROBE SYSTEM

By

Chih- Wei Chang

A non-destructive measurement of electromagnetic (EM) properties of anisotropic
materials using an open—ended waveguide probe has been conducted. Two different
techniques, based on the Hertzian potential method and the transverse field method, are
developed to facilitate the investigation of the subject. The technique employing Hertzian
potentials is applied to isotropic materials only, however, the technique employing the

transverse field method is suitable for both isotropic and anisotropic materials.

When a waveguide probe, which consists of an open-ended waveguide terminated on a
flange, is placed against a material layer, two coupled electric field integral equations
(EFIE’s) for the aperture electric field can be derived by matching the boundary conditions
at discontinuity interfaces. These EFIE’s can be solved numerically with the Method of
Moments (MoM) when the electric field on the waveguide aperture is expressed as a sum
of waveguide modes. The reflection coefficient of the incident wave or other relevant

quantities of the waveguide probe can be expressed as functions of the EM properties,

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such as the permittivity, permeability, and conductivity, and the thickness of the material
layer. Therefore, the EM parameters of the material layer can be inversely determined if

the reflection coefficient of the incident wave is experimentally measured.

A series of experiments have been conducted using the waveguide probe system
constructed at MSU electromagnetics laboratory. The experimental results of the probe
input admittances, with the probe attached to various isotropic and anisotropic material
layers, using an HP 8720B network analyzer are presented. These probe input
admittances are then used to determine the complex permittivity inversely by a numerical
inverse procedure based on the Newton’s iterative method. The inverse results on the EM
properties of some known materials are found to be quite satisfactory. The results on the
tensor permittivities of some anisotropic materials are found to be reasonable even though
their exact values have not been determined before. Finally, an analysis of the effects of

material parameters on the input admittance is presented.

To my wife
and

our lovely sons

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ACKNOWLEDGEMENTS

I would like to express my heartfelt appreciation to my major professor, Dr. Kun-Mu
Chen. His knowledgeable advice, technical competency as well as his patience and timely
input made it possible for me to complete this work. I am also indebted to Dr. Dennis P.
Nyquist, Dr. Edward J. Rothwell and Dr. Byron C. Drachman for their generous support
and constructive suggestions during the course of this study. A special note of thanks is

due Professor Jian Qian for his generous assistance and kind concern.

I owe a great deal to my parents for their love and continuous support. I must also
acknowledge my wonderful wife, Uan-Li. Her love, sacrifice, constant encouragement
and taking care of our children, Timmy (Ching-Chen) and Peter (Yao-Chen), have been

invaluable in the successful completion of my graduate study.

Finally, the fundings of this research by the State of Michigan Research Excellence
Fund through Michigan State University and Boeing Airplane Company are most

gratefully acknowledged.

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Chap

TABLE OF CONTENTS

LIST OF FIGURES ..................................................................................................... viii
LIST OF TABLES ....................................................................................................... xiv
Chapter 1. Introduction ............................................................................................... 1
Chapter 2. Basic Theories ............................................................................................ 5
2.1 Introduction ........................................................................................................ 5
2.2 Derivation of EM Fields via Hertzian Potentials ............................................... 7
2.2.1 Introduction of Hertzian Potentials ........................................................... 7

2.2.2 EM Fields inside A Rectangular Waveguide ........................................... 11

2.2.3 EM Fields inside A Homogeneous, IsotrOpic Medium ........................... 14

2.3 Derivation of EM Fields via Transverse Field Method ................................... 18
2.3.1 Dielectric Tensor Properties of Anisotropic Media ................................. 18

2.3.2 General Eigenvalue Problem ................................................................... 21

2.3.3 Degenerate Eigenvalue Problem ............................................................. 30

Chapter 3. Open-Ended Rectangular Waveguide Probe to Measure

EM Properties of Isotropic Materials ................................................... 39

3.1 Introduction ...................................................................................................... 39
3.2 Coupled Integral Equations for Aperture Electric Field ................................. 40
3.2.1 Reflection Coefficients. at a Discontinuity Interface ............................... 40
3.2.2 Matching Boundary Conditions at Waveguide Aperture ........................ 43

3.3 Numerical Simulation ...................................................................................... 54
3.3.1 Application of Method of Moments ........................................................ 55
3.3.2 Evaluation of Matrix Elements ............................................................... 58

vi

Ch

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3.3.3 Numerical Results and Comparison with Existing Results .................... 72

Chapter 4. Open-Ended Rectangular Waveguide Probe to Measure

EM Properties of Anisotropic Materials ............................................... 81

4.1 Introduction ....................................................................................................... 81
4.2 Coupled Integral Equations for Aperture Electric Field .................................. 82
4.2.1 Matching Boundary Conditions at z=t ..................................................... 82

4.2.2 Matching Boundary Conditions at Waveguide Aperture ......................... 87

4.3 Numerical Simulation ....................................................................................... 93
4.3.1 Application of Method of Moments ........................................................ 94

4.3.2 Evaluation of Matrix Elements ................................................................ 97

4.3.3 Numerical Results and Comparison with Existing Results .................... 110
Chapter 5. Experiments ........................................................................................... 122
5.1 Introduction ..................................................................................................... 122
5.2 Experimental Setups and Calibration .............................................................. 123
5.2.1 Fixed-Stub Calibration ........................................................................... 127

5.2.2 Adjustable Shorter Calibration .............................................................. 131

5.3 Experimental Results for Materials ................................................................ 138
5.4 Inverse Technique and Results ........................................................................ 150

5.5 Analysis on the Effects of Material Parameters .............................................. 164
Chapter 6. Conclusions ............................................................................................. 179

Appendix : 2-D Fourier Transform of Sinusoidal Functions
over Waveguide Aperture ..................................................................... 183

Bibliography ............................................................................................................... 193

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3.1(a).

3.1(b).

3.2(a).

3.2(b).

3.3(a).

3.3(b).

LIST OF FIGURES

The geometric structure of nondestructive measurement using a waveguide
probe system. ..................................................................................................

Input conductances of a waveguide probe (a = 0.4 in, b = 0.9 in) as a

function of frequency when the probe is placed against a material layer with
a dielectric constant of er = 3.76 and a thickness of 0.1299 in. The com-
parisons are made between our numerical results and theoretical and experi-
mental results of Croswell et al. [22]. .............................................................

Input susceptances of a waveguide probe (a = 0.4 in, b = 0.9 in) as a

function of frequency when the probe is placed against a material layer with
a dielectric constant of 8r = 3.76 and a thickness of 0.1299 in. The com-
parisons are made between our numerical results and theoretical and experi-
mental results of Croswell et al. [22]. .............................................................

Input conductances of a waveguide probe (a = 0.4 in, b = 0.9 in) as a

function of frequency when the probe is placed against a material layer with
a dielectric constant of er = 2.25 and a thickness of 0.3201 cm. The com-
parisons are made between our numerical results and theoretical and experi-
mental results of Bodnar et al. [24]. ................................................................

Input susceptances of a waveguide probe (a = 0.4 in, b = 0.9 in) as a

function of frequency when the probe is placed against a material layer with
a dielectric constant of er = 2.25 and a thickness of 0.3201 cm. The com-
parisons are made between our numerical results and theoretical and experi-
mental results of Bodnar et al. [24]. ................................................................

Input conductances of a waveguide probe (a = 0.4 in, b = 0.9 in) as a
function of frequency when the probe is opening onto free space. The com-
parisons are made between our numerical results and theoretical of
Baudrand et al. [30] and experimental results of Bodnar et al. [24]. ...............

Input susceptances of a waveguide probe (a = 0.4 in, b = 0.9 in) as a
function of frequency when the probe is opening onto free space. The com-

viii '

6

74

75

76

77

79

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,1:

4.4.

4.5.

4.1(a).

4.1(b).

4.2(a).

4.2(b).

4.3(a).

4.3(b).

4.4.

4.5.

parisons are made between our numerical results and theoretical of
Baudrand et al. [30] and experimental results of Bodnar et al. [24]. ...............

Input conductances of a waveguide probe (a = 0.4 in, b = 0.9 in) as a

function of frequency when the probe is placed against a material layer with
a dielectric constant of er = 3.76 and a thickness of 0.1299 in. The com-
parisons are made between our numerical results and theoretical and experi-
mental results of Croswell et al. [22]. .............................................................

Input susceptances of a waveguide probe (a = 0.4 in, b = 0.9 in) as a

function of frequency when the probe is placed against a material layer with
a dielectric constant of er = 3.76 and a thickness of 0.1299 in. The com-
parisons are made between our numerical results and theoretical and experi-
mental results of Croswell et al. [22]. .............................................................

Input conductances of a waveguide probe (0 = 0.4 in, b = 0.9 in) as a

function of frequency when the probe is placed against a material layer with
a dielectric constant of er = 2.25 and a thickness of 0.3201 cm. The com-
parisons are made between our numerical results and theoretical and experi-
mental results of Bodnar et al. [24]. ................................................................

Input susceptances of a waveguide probe (a = 0.4 in, b = 0.9 in) as a

function of frequency when the probe is placed against a material layer with
a dielectric constant of Sr = 2.25 and a thickness of 0.3201 cm. The com-
parisons are made between our numerical results and theoretical and experi-
mental results of Bodnar et a]. [24]. ................................................................

Input conductances of a waveguide probe ( a = 0.4 in, b = 0.9 in) as a
function of frequency when the probe is open into free space. The compari-
sons are made between our numerical results and theoretical results of
Baudrand et al. [30] and experimental results of Bodnar et al. [24]. ...............

Input susceptances of a waveguide probe (0 = 0.4 in, b = 0.9 in) as a
function of frequency when the probe is open into free space. The compari-
sons are made between our numerical results and theoretical results of
Baudrand et al. [30] and experimental results of Bodnar et al. [24]. ...............

Input admittance of a waveguide probe (a = 0.4 in, b = 0.9 in) as a
function of frequency when the probe is placed against an assumed known
anisotropic material layer with a thickness of 0.11 inch and three principal
permittivities of t»:l = 5.4 —j0.3, £2 = 5.8 —j0.4 and 83 = 3.8 —j1.7.

Input admittance of a waveguide probe (a = 0.4 in, b = 0.9 in) as a
function of frequency when the probe is placed against an assumed known
anisotropic material layer with a thickness of 0.053 inch and three principal
permittivities of 8] = 5.4 —j5.3 , 82 = 5.8 —j6.4 and £3 = 30.8 -j100.5 ..

ix

80

111

112

114

115

116

117

118

120

5.4.

5.5.

5.6.

5.8.

5.9.

5.10.

5,12.

5.1.

5.2.

5.3.

5.4.

5.5.

5.6.

5.7.

5.8.

5.9.

5.10.

5.11.

5.12.

Experimental setup of a waveguide probe system to measure the EM param-
eters of materials. ............................................................................................ 124

Representation of the equivalent two-port network between the waveguide
aperture and the measurement reference plane of a network analyzer. ........... 126

Signal low diagram of the calibration scheme for the measurement system. .. 128
The block diagram of the slotted-line measurement. ...................................... 132

The equivalent shorting locations of the adjustable shorter obtained by the
measurement of the slotted line over frequencies between 8.5 GHz to 12

GHz. The ’+’ marks represent the measured equivalent shorting locations of

the adjustable shorter at discrete frequencies. The solid line represents the

fitting curve of the equivalent shorting locations. ........................................... 134

The reactive characteristics of the adjustable shorter over frequencies be-
tween 8.5 GHz to 12 GHz. The discrepancy of these two curves shows that
this adjustable shorter is not a simple waveguide shorting terminator. ........... 135

Calibration procedure of the waveguide probe system using an adjustable
shorter. The 8 parameters of the transmission network is determined to con-

vert the reflection coefficient or the input admittance measured at the refer-

ence plane of the network analyzer to that at the aperture plane of the

waveguide probe. ............................................................................................ 137

Experimental input conductances and susceptances of a waveguide probe
(a = 0.4 in, b = 0.9 in) as functions of frequency when the probe is
open to free space. .......................................................................................... 140

Comparison of input conductances and susceptances of a waveguide probe
(a = 0.4 in, b = 0.9 in) as functions of frequency when the probe is
open to free space with the existing experimental results of Bodnar et al.[24]. 141

Experimental input conductances and susceptances of a waveguide probe
(a = 0.4 in, b = 0.9 in) as functions of frequency when the probe is
placed against a material layer (acrylic) with a thickness of 0.06 inch. ....... 142

Experimental input conductances and susceptances of a waveguide probe
(a = 0.4 in, b = 0.9 in) as functions of frequency when the probe is
placed against a material layer (plexiglass) with a thickness of 0.06 inch. .. 143

Experimental input conductances and susceptances of a waveguide probe
(0 = 0.4 in, b = 0.9 in) as functions of frequency when the probe is
placed against a material layer (teflon) with a thickness of 0.51625 inch. .. 144

5.11

5.17.

5.30,

5.13.

5.14.

5.15.

5.16.

5.17.

5.18.

5.19.

5.20.

Experimental input conductances and susceptances of a waveguide probe

(a = 0.4 in, b = 0.9 in) as functions of frequency when the probe is

placed against two liquid material layers (distilled water) with thickness’s of
0.2344 inch (dash line) and 0.3125 inch (solid line). ............................... 146

Experimental input conductances and susceptances of a waveguide probe

(a = 0.4 in, b = 0.9 in) as functions of frequency when the probe is

placed against a liquid material layer (acetone) with a thickness of

0.345 inch. ................................................................................................... 147

Experimental input conductances and susceptances of a waveguide probe

(a = 0.4 in, b = 0.9 in) when the probe is placed against a layer of
anisotropic material (Epoxy/Glass-Fiber) with a thickness of 0.1105 inch
measured at four different orientations with respect to a reference axis of the
material layer. ................................................................................................. 148

Smoothed curves of the experimental input admittances of a waveguide

probe (a = 0.4 in, b = 0.9 in) when the probe is placed against an
anisotropic material layer (Epoxy/Glass-Fiber) with a thickness of

0.1105 inch measured at four different orientations with respect to a refer-
ence axis of the material layer. ........................................................................ 149

Experimental input conductances and susceptances of a waveguide probe

(a = 0.4 in, b = 0.9 in) when the probe is placed against a layer of
anisotropic material (dielectric-fiber from Boeing) with a thickness of

0.053 inch measured at four different orientations with respect to a refer-

ence axis of the material layer. ........................................................................ 151

Smoothed curves of the experimental input admittances of a waveguide

probe (a = 0.4 in, b = 0.9 in) when the probe is placed against an
anisotropic material layer (dielectric-fiber from Boeing) with a thickness of
0.053 inch measured at four different orientations with respect to a refer-

ence axis of the material layer. ........................................................................ 152

The inverse results of the complex relative permittivities of free space deter-
mined by employing two different methods. The ’*’ marks represent the re-
sults based on the Hertzian potential method, while the ’o’marks represent

that based on transverse field method. The upper curves show the real part

of the complex relative permittivities, while the lower curves show the imag-
inary part. ........................................................................................................ 156

The inverse results of complex relative permittivity of a material layer

(acrylic) which has a dielectric constant of er = 2.5~2.7 and a thickness of

0.06 inch. The upper curve shows the real part of complex relative permit-
tivity, while the lower curve shows the imaginary part. .................................. 157

5.21.

5.22.

5.23.

5.24.

5.25.

5.26.

5.27.

5.28.

The inverse results of complex relative permittivity of a material layer
(plexiglass) which has a dielectric constant of er = 2.59 and a thickness of

0.06 inch. The upper curve shows the real part of complex relative permit-
tivity, while the lower curve shows the imaginary part. .................................. 159

The inverse results of complex relative permittivity of a material layer (te-

flon) which has a dielectric constant of er = 2.046 and a thickness of

0.51625 in ch. The upper curve shows the real part of complex relative
permittivity, while the lower curve shows the imaginary part. ....................... 160

The inverse results of complex relative permittivities of two material layers
(distilled water) with two different thickness’s of 0.2344 inch and

0.3125 inch. The upper curves of the graph show the real part of complex
relative permittivity, while the lower curves show the imaginary part. .......... 161

The inverse results of complex relative permittivity of a material layer (ace-
tone) which has a thickness of 0.345 inch. The upper curve shows the real
part of complex relative permittivity, while the lower curve shows the imag-
inary part. ........................................................................................................ 162

Measured complex tensor permittivities of an anisotropic material layer (ep-
oxy/glass-fiber) with a thickness of 0.1105 inch. The solid lines represent

the measured results inversely determined by employing three measure-

ments of the probe input admittances with the probe orientations of 0, 45 and

90 degrees with respect to a principal axis of the material, while the dash

lines represent the corresponding results with the probe orientation of 0, 30

and 90 degrees. ............................................................................................... 163

Measured complex tensor permittivities of an anisotropic material layer (di-
electric-fiber) with a thickness of 0.053 inch. The results are determined
inversely by employing three measurements of the probe input admittances

with the probe orientations of 0, 30 and 90 degrees with respect to a princi-

pal axis of the material. The complex permittivity in the direction perpen-
dicular to the waveguide aperture is determined to be unstable and is

omitted. ........................................................................................................... 165

Input admittance of a waveguide probe as a function of the material thick-

ness at 10 GHz when the probe is placed against an anisotropic material lay-

er supplied by Boeing Airplane Company with various conductivities in the

x direction. ...................................................................................................... 167

Input admittance of a waveguide probe as a function of the material thick-

ness at 10 GHz when the probe is placed against an anisotropic material lay-

er supplied by Boeing Airplane Company with various conductivities in the

y direction. ...................................................................................................... 169

xii

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5.30.

5.31.

5.32.

5.33.

5.34.

Input admittance of a waveguide probe as a function of the material thick-

ness at 10 GHz when the probe is placed against an anisotropic material lay-

er supplied by Boeing Airplane Company with various conductivities in the

z direction. ...................................................................................................... 170

Input admittance of a waveguide probe as a function of the material thick-

ness at 10 GHz when the probe is placed against an anisotropic material lay-

er supplied by MSU Composite Material Laboratory with various

conductivities in the x direction. ..................................................................... 172

Input admittance of a waveguide probe as a function of the material thick-

ness at 10 GHz when the probe is placed against an anisotropic material lay-

er supplied by MSU Composite Material Laboratory with various

conductivities in the y direction. ..................................................................... 173

Input admittance of a waveguide probe as a function of the material thick-

ness at 10 GHz when the probe is placed against an anisotropic material lay-

er supplied by MSU Composite Material Laboratory with various

conductivities in the z direction. ..................................................................... 175

Input admittance of a waveguide probe as a function of the material thick-
ness at 10 GHz when the probe is placed against an isotropic material layer
with various conductivities. ............................................................................ 176

Input admittance of a waveguide probe as a function of the material thick-

ness at 10 GHz when the probe is placed against an isotropic material layer
with various permittivities. ............................................................................. 177

xiii

Al

LIST OF TABLES

A. 1. Summary for simplified expressions of notations used in Chapter 4. ............. 190

A2. Summary for simplified expressions of notations used in Chapter 3. ............. 191

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CHAPTER 1

INTRODUCTION

The objective of this research is to conduct nondestructive measurement of the
electromagnetic (EM) properties of material layers using a waveguide probe system. Two
different techniques are established to facilitate the investigation of the subject. The
technique employing Hertzian potentials is applied to isotropic materials only, however,
the technique employing the transverse field method is suitable to both isotropic and

anisotropic materials.

Techniques for the measurement of the material EM properties have been developed
by numerous investigators using various structures such as open-ended coaxial line [1]-
[12], stripline [l3], microstrip line [14], cavity resonator [15]-[17] and waveguide [18]-
[20] during the last two decades. In recent years, due to the rapid advance in material
manufacturing, an increasing demand for accurate nondestructive measurements of the

material parameters has emerged.

Several methods are available for measuring the EM properties of dielectric materials
nondestructively. Some researchers employed an open-ended coaxial probe [1]-[12],

while others used an open-ended waveguide probe [19]-[20]. However, all of them were

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limited to the measurement of the EM parameters of isotropic materials only. In this
research, we present an technique for accurate nondestructive measurement of the EM
parameters of isotropic materials, and another technique for anisotropic materials, where

both of them employ an open-ended waveguide probe system.

In the study of the nondestructive measurement of the EM parameters of materials
with a waveguide probe system, we divide the study into steps: the forward and the
inverse procedures. The forward procedure refers to the theoretical and numerical
analysis of the waveguide probe system, which consists of an open-ended waveguide
terminated on a flange, placed against a layer of material [21]-[32]. In this procedure, the
reflection coefficient or the input admittance of the waveguide probe can be expressed as a
function of the assumed EM parameters of the material layer. On the other hand, the
inverse procedure deals with the problem of inversely determining the EM parameters of

the material layer from the measured reflection coefficient or input admittance of the

waveguide probe placed against the material layer [12].

This study is presented in three parts: the first part, Chapter 2, presents the basic
theories for this research; the second part, Chapters 3 and 4, develops the forward
procedure of the measurement for determining the EM properties of isotropic and
anisotropic materials; and the last part, Chapter 5, describes the experiments and the

inverse procedure of the measurement.

Chapter 2 presents the basic theories which are used to establish two different
techniques for determining the EM parameters of materials using a waveguide probe
system. Hertzian potentials are first reviewed in the chapter. The EM fields in the
waveguide, the material layer and the open free space are then determined in terms of
Hertzian potentials. These fields are used in Chapter 3 to match the boundary conditions
at the interfaces between different regions. The transverse field method is also reviewed

in this chapter. It generates a general case dealing with the EM fields inside an anisotropic

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medium and a degenerate case with the EM fields inside an isotropic medium. The EM
fields derived by this method are then used to match the boundary conditions at the

discontinuity interfaces in Chapter 4.

In Chapter 3, the forward procedure of the measurement for determining the EM
parameters of isotropic materials is studied. The EM fields expressed in terms of Hertzian
potentials within various regions are first recalled from Chapter 2. Two coupled integral
equations for the electric field are then derived by matching the tangential EM fields
across the waveguide aperture. The method of moments is implemented to solve these
coupled electric field integral equations (EFIE’s) numerically. Finally, the numerical
results on the input admittances of the waveguide probe placed against material layers are

compared with the existing results published by other workers.

Chapter 4 deals with the forward procedure of the measurement for determining the
EM parameters of anisotropic materials. The EM fields derived with the transverse field
method are recalled from Chapter 2. The matching of the tangential EM fields at the
probe aperture results in two coupled integral equations for the unknown aperture electric
field. The method of moments is implemented to solve these EFIE’s numerically. Also
the numerical examples illustrated in Chapter 3 for isotropic materials are recalculated and
compared with the existing results to validate the technique. In addition, the numerical
results on the input admittances of the waveguide probe in contact with a layer of assumed

known anisotropic material are presented.

Finally, the experiments and the inverse procedure of the measurement are described
in Chapter 5. The experimental setups and the calibration procedures of the waveguide
probe system are first described. The experimental results of the probe input admittances,
with the probe attached to various isotropic and anisotropic material layers, using an HP
8720B network analyzer are then presented. In addition, the inverse procedure of the

measurement to determine the EM parameters of the measured materials is developed.

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The probe input admittances are then used to inversely determine the complex permittivity
by this numerical inverse procedure. The inverse results on the EM properties of some
known materials are found to be quite satisfactory. The results on the tensor permittivities
of some anisotropic materials are found to be reasonable even though their exact values

have not been determined before.

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CHAPTER 2

BASIC THEORIES

2.1. Introduction

In this chapter we review the basic theories which will be used to establish two
different techniques for determining the electromagnetic (EM) parameters of materials
using a waveguide probe system. As mentioned in the preceding chapter, the study of the
first technique is limited to isotropic media. The second technique, however, is suitable

for the application to both isotropic and anisotropic media.

The geometry of the problem is shown in Fig. 2.1 which depicts the cross-sectional
view of the system. A layer of unknown dielectric material of thickness d which is backed
by free space, is placed against a waveguide probe consisting of an open-ended
rectangular waveguide terminated by an infinite metallic flange. Inside the waveguide
region, a dominant TE10 mode of field is excited and it propagates toward the probe
aperture. In addition to the reflected TE10 mode, higher order modes of fields are excited
near this aperture due to the discontinuity between the waveguide and the slab material.
The electromagnetic wave carried by the waveguide radiates into the material layer and

through to the open space backing it. If the electromagnetic fields in the material layer

PLANE PLANE
INTERFACE INTERFACE
V ‘L
infinite dielectric free
flange —>. material space

 

 

J

TE”) ———> ——-> ——>
<——— <——

rectangular ‘

waveguide

 

higher
order
modes

 

z: =d

Figure 2.1 The geometric structure of nondestructive measurement using a
waveguide probe system.

and

poiei

and the open space are established based on Maxwell’s equations, the problem can be

solved by matching the fields at the waveguide-material and the material-air interfaces.

In this chapter, the EM fields in the waveguide, the material layer and the open free
space will be determined. The application of the boundary conditions at the interfaces

will be discussed in the following chapters.

Section 2.2. presents the EM waves in an isotropic medium in terms of Hertzian
potentials. In this section, the EM fields in both the waveguide region and the material
layer are examined. In section 2.3. a partial differential equation is derived directly from
the Maxwell’s equations for the spectrum-domain transverse EM fields. This partial
differential equation then leads to an eigenvalue problem for the EM fields in an
anisotropic medium and a degenerate eigenvalue problem for the EM fields in an isotropic

medium.

2.2. Derivation of EM Fields via Hertzian Potentials

For an isotropic medium, it is well known that the total EM fields inside the region can
be regarded as a superposition of two types of waves: the transverse electric (TE) modes
and the transverse magnetic (TM) modes which can be derived in terms of Hertzian

potentials [33].

The definition of the TB and TM modes of waves will be addressed in the following
subsection [34]. Also the definition of Hertzian potentials is introduced, and the EM fields

based on Hertzian potentials are defined.

2.2.1 Introduction of Hertzian Potentials

In a homogeneous and isotropic medium, we can classify waves into two types; TE
modes and TM modes. TE modes are field configurations which electric field components

lie in a plane that is transverse to the direction of propagation; while for TM modes the

30"
U-
y

[0 SC

prob}

peter.

EM ii

In

satisfi:

r:

and tilt

on

2:4.

“here 1;

Sim

did [he 1

”7+
II

2:4.
II

magnetic field components are transverse to the direction of propagation. Generally, the

TB and TM modes that satisfy Maxwell’s equations and the boundary conditions are used

to solve electromagnetic boundary—value problems, especially for waveguide-related

problems.

In general, there are two types of Hertzian potentials, the electric type Hertzian

-> ->
potential 1'1, and the magnetic type Hertzian potential 11;. , commonly used to describe the

EM fields in a homogeneous and isotropic medium.

. . . . . . —>
In a time harmonic and source free region, the electric type Hertzran potential, I'le,

satisfies
Vzltie + kzfi. == 0
and the related electric and magnetic fields are derived from
i: = W - ii. + 131—i. = Vxini.
F1 = jmerfie
where k = mJt—t—e is the wavenumber in the medium.
Similarly, the magnetic type Hertzian potential, F1}, , satisfies
VZI-Ih + kzfih = 0
and the related electric and magnetic fields are derived from
E = —jmqul_'>Ih

ii vv.fi,,+k2ii,. = VxVxfih

(2.2.1)

(2.2.2)

(2.2.3)

(2.2.4)

(2.2.5)

(2.2.6)

Ct]

75’

:J»
.3
H

Combination of eqs. from (2.2.1) to (2.2.6) shows that a general solution to Maxwell’s

equations can be written as
-> —> —>
E = VxVxIIe -j(oquHh

-) -> —>
H = jwerI‘le + VxVxHh

(2.2.7)

(2.2.8)

. . . . '9 —) . .
where the electric and magnetic Hertzran potentials, II). and He , are pornted in a constant

direction it [33]:

He = fine
a
II}, = an,

A

(2.2.9)

(2.2.10)

If the unit constant vector (1 = z is chosen where z is in the direction of wave

_)
propagation, F1}, can produce TE modes while TM modes can be generated from 11..

Using eqs. (2.2.9) and (2.2.10) and the choice of £1 = 2, these two kinds of modes can be

modified as follows:

Transverse Electric Modes (E z = 0, Hz ¢ 0)

The TE modes can be derived from a magnetic Hertzian potential by letting

.9
II}, = 211,! and be represented as

V211,, +k21'lh = 0

9
E:

-> 82 2 a
H=2—2+k Hh+V—II

82 '32 h

where

(2.2.11)

(2.2.12)

(2.2.13)

10
V = v_2_ (2.2.14)

Transverse Magnetic Modes (Hz = 0, E2 ¢ 0)

The TM modes can be derived from an electric Hertzian potential by letting

a
II. = 2n, and be represented as

V2ne+k2ne = 0 (2.2.15)
E-E az+kZII+VaII (2216)
_ 822 e fa—E 8 . .

a o A
H, = —j(r)|.t(z>< Vt) He (2.2.17)

Total Fields (TE modes + TM modes)

The total EM fields within a homogeneous and isotropic medium can therefore be
expressed as the combination of eqs. (2.2.12), (2.2.13), (2.2.16) and (2.2.17).

Components of the EM fields are shown as follows:

E a a2
x = —jmua—y —H h+-a——xazl‘l (2.2.18)
2
Ey = jmuaa—x II h+-a—Z-fafl e (2.2.19)
2 2
EZ = -[-a—2+—a—]He (2.2.20)
8x2 ayz
H _ a_2n a
x _ ax 82 h-i-jmeay —11 (2.2.21)
32 . a

= __ .. _ 2.2.22
H), ayazl'lh )weaxfle ( )

.‘J
o

53

r 2
£0

“ii:

for T

ll

32 32]
H = __ _+_ 1] (2.2.23)
z [8x2 avz h

a

2.2.2 EM Fields inside A Rectangular Waveguide

As mentioned in the previous section, the scalar Hertzian potentials, I'Ih and He,
satisfy two scalar Helmholtz equations as given in eqs. (2.2.11) and (2.2.15). Since both
TE and TM modes in a rectangular waveguide are sought, the solutions for [Th and He

can be assumed to have a form of

Z

2r
Uh wk (x, y) e (2.2.24)

trz

He — we (x, y) e (2.2.25)

where the scalar wave functions Wh and we satisfy the two-dimensional Helmholtz

equations;
V2 2 _
iWh+chh - 0 (2.2.26)
V2 k2 — o
:‘Ve+ c‘Ve - (2.2.27)

. 2 2 2 2 2
With k c = k0 + 1" and V I as the transverse part of V operator.

Thus the EM fields represented in eqs. (2.2.12), (2.2.13), (2.2.16) and (2.2.17) can be

written as
2 411‘:
Hz = k the
—> $1“:
H, = ¥FV,\V;,€ (2.2.23)

-> . —)
E, = :pzhiz th)

for TE modes, and

for T

admi‘

F.
W, n

electr

TM n

(72‘)

12

2 $1“:
EZ — kcwee
-’ ““2
E, = $I‘Vtwee . (2.2.29)

—> .. a)
H, ZFYeLZXEI

for TM modes, where Zh is the TB mode wave impedance and Y e is the TM mode wave

admittance.

For a waveguide with perfectly conducting walls, we observe from eq. (2.2.28) that
wh must satisfy the boundary condition if awh/an = 0, in order for the tangential
electric field to vanish in the case of TE modes. Also from eq. (2.2.29) we see that
w e = 0 is the condition to make the tangential electric field vanish on the boundary for
TM modes. Thus, the scalar wave functions wk and w e can be determined from eqs.

(2.2.26) and (2.2.27), using the separation of variables method, to be

 

' ' n' = O l 2

Whmm = cos(w)cos(m—:z),{m' = 0, 1,2, (2.2.30)
a m'¢n' = 0
" " " = 1,2,

we,” = sin('—'—an—x)sin(mbny),{;n _ 1 2 (2.2.31)

The scalar Hertzian potentials can then be derived from wh and w e by means of eqs.

(2.2.24) and (2.2.25) and are expressed as

-r r. .
l'Ih = alocosC-EXe loz+ Rerwz)+ ZZAnH'LZ—mcos( x)cos("1512)e "'"Z(2.2.32)

 

_ . n"1tx . m"1ty rn"m"z
He—gan..m.sm(—a—)sm( b )e (2.2.33)

where

l3

. I 2 1t 2
. 2 n 2 1t 2
win-(e) —('—'—:—i

k0 = a) 11on (2.2.36)

 

Also alo and A ".m. represent the amplitude coefficients of the dominant TE 10 mode and
the TB higher order modes, respectively; and B n. m. represents the amplitude coefficients
of the TM higher order modes. R denotes the reflection coefficient of the incident
dominant mode and it is required that Re (I‘m) 2 0 and Im (I‘m) 2 0 , for all values of n

and m.

The EM fields inside the waveguide can then be derived via the substitution of

Hertzian potentials into eqs (2.2.18) to (2.2.23):

The EM Fields inside the waveguide

 

 

n m
"1: "n . "n rn~m--Z
+ 22" Futomr( ’17)Bn"m" COS(’z—a-x)51n(mb y)e (2.2.37)
n m

Ey(x’y’z) = “mom-E )“iosiniia'lie “02+ Rer'ozl
"n mtty
.‘JGHOZZ(’%)A,,. "{sinCig- x)cos(— b )el‘,,,,.z
m 1C _ "flux many rum-Z
+22; r""""(T)B""'""Sm(—a )cos( b )e (2.2.38)

 

E, (x. y. z)

 

ll
=M
a
Ii:

2 "My - ("ml ' (mum) T”: (2.2.39)
a l ”'17 We“ 7 5‘“ b e

l4

 

n'tt . n'nx m'ny Fn.,,,.:
2.2.r...(-.-)A-sm(—.—iws( .ie
. H H I. r u "z
+ 1(08022( flbi‘ )Bn"m" sin( "7151‘ ) COS( m b” )8 " m (2.2.40)

 

 

 

= __ m1: ' x , m'ny rmz
H). (I, y, Z) "Z; Fn.m(—b-)An.m.cos(———)srn( b )e
. " " " r . ..
’1w8022(E—E)Bnnmucos(w)sin(m nyje ""‘Z (2241)
nu mu a a b
_ 1C 2 TEX —r10Z . r103)
HZ (x, y, z) — (2)“10C°S(7)(e +Re

 

. 2 r 2 ' ' r .
+2211"; +1“)1A....cosin—zxicosiwie
n m

T b

2.2.3 EM Fields inside A Homogeneous, Isotropic Medium

The scalar Hertzian potentials inside a homogeneous and isotropic medium satisfy the
scalar Helmholtz equations of eqs. (2.2.11) and (2.2.15). Since the isotropic medium is
uniform and unbounded in the transverse directions, Fourier transform will be applied to
the x and y directions. Let’s define f1 (er k), z) to be the two-dimensional Fourier

transform of II (x, y, 2) along the transverse directions, i.e.,

oo

H(kx,k),,z) = [[n(x,y,z)e

-oo

—'k -'k\.)'
I ‘Xe j ' dxdy (2.2.43)

Then

2 °° .. 'k .kJ
II (x, y, z) = (it) I I 1'] (er k), z) e] ’er "‘dkxdky (2.2.44)

15

If we take the two-dimensional Fourier transform of the two scalar Helmholtz

equations, we have

II
CD

1 2‘” 82 2 2 2 ~ jkarjko'
(ET—[)U[S?+kl—kx—ky]nhe e dkxdk),

{-21— “ff“; +1.? -k f—kfjfi, ejk'exJk"dkHdk 0

or
32 2 k2 " k -
——2—+k2 —kx- y II),( x,ky,z) — 0
82
82 2 2 ~ _
_7-i-k2 —kx-k), I'Ie(kx,ky,z) — 0
82
The solutions of 1:1,, and He of eqs. (2.2.47) and (2.2.48) can be written as
.. irlz
11;. (k. k,. z) = F (k,. k,,> e
and
.. :r,z
119(16):, kyr Z) = G (er ky) e
where
. 2 2 2
1"](kx, ky) = -j,/kl -kx-ky
k1 = 0° “131

(2.2.45)

(2.2.46)

(2.2.47)

(2.2.48)

(2.2.49)

(2.2.50)

(2.2.51)

(2.2.52)

Notice that F1 should provide positive real and positive imaginary parts to satisfy the

radiation condition of fields derived from eqs. (2.2.49) and (2.2.50).

16

The substitution of the two-dimensional Fourier transformed Hertzian potentials, fl),

and Fl, , into eq. (2.2.44) gives the expressions of space domain Hertzian potentials inside

a slab, isotropic medium as

II, (x, y, z) = i2]: [ F(kx,kye)k e +Rae dkxdky (2.2.53)
2 °° 1k .x k —l‘l I":
r18 (x, y, z) = (217:) J’ [ G(kx,k )e e’ ‘(e Z+Rbe )dkxdky (2.2.54)

Here the quantities R a (er ky) and R b (er ky) represent the ratios of the backward-to-

forward wave at the plane interface 2 = d as shown in Fig. 2.1.

Thus, the total EM fields within an isotropic material slab can be obtained by

substituting eqs. (2.2.53) and (2.2.54) into eqs. (2.2.18) to (2.2.23) as follows:

Fields inside the isotropic material slab

I 2 kxx ejkyy -r,z F,Z
Ex(x,y,z) = (27:) [(61101]. ,F(kx,k,ej) (e +Rae )dkxdky

1k 1" jk) —Fz Fz
—jJ:Jk F lyeG(kx.k) e (e l —Rbe ' )dkxdky] (2.2.55)

] 2 °° 1" xx ejky -I‘|z I‘lz)
E), (x, y, z) = (it) [4011ij kxF(k, k y)e (e +Rae dkxdk),

e e

1‘12) -
-J'] my] k F G (k,, k ,e) —Rbe dkxdky (2.2.56)

1 2 °° 2 2 2"” jk,> -I‘.z 1“,:
Ez(x,y,z) =(2—1-r)J-J(kx+ky)6(k,kye) e (e +Rbe )dkxdky(2.2.57)

17

1 ngk \‘\'( —l‘lz rig)
Hx(x,y,z) = (2—2 —jj:jk r F(kr,k )ejk‘ e -Rae dkxdky

°° 1'ka jk _\ 4“,: Fl:
“(DELI kyG(k, k )e e (e +Rbe )dkxdk),] (2.2.58)

1 2 ,°° Ikefi‘jk3 4“,: m)
Hy(x,y,z) = (in) [1U kyl‘1F(kx,k )e (e —Rae dkxdky

0° kxx 11") -rlZ rlz
+coeJ' Ika(k, k)e e e +Rbe dkxdk), (2.2.59)

l 2” ( 2 2) jkx" ( -r,z m)
Hz(x,y,z) = 21?) U kx+ky F(kx,k),)e e e +Rae dkxdk),(2.2.60)

The symmetry condition in the waveguide excitation can be used to simplify the
problem. Let‘s observe the field distributions at the waveguide’s aperture, (2 = O) . The
transverse electric fields derived from eqs. (2.2.28) to (2.2.31) for both TE and TM modes

are proportional to the following sinusoidal functions.

.. (nttx) . (mm)
xcos — sm —'—
a b

_)
Er °¢ (2.2.61)
. Sin(nnx)cos(m__1ty)
y a b
Since the dominant T E 1 0 mode,
_)
Em cc 9mg?) (2.2.62)

which is evenly symmetrical to the center of the aperture located at (a/ 2,b/ 2) , is used to

excite the waveguide, only the higher order modes with add it and even m can be excited.

18

Thus, the x-component of the electric field of higher order modes appeared at the
aperture is antisymmetric with respect to the center of the aperture, and the y-component
of the electric field of higher order modes is symmetrical with respect to the aperture
center. The excited modes are determined to be (1, O) , (3, O) , (1, 2) , ,
(2k + 1, 2k) , , for k 2 0 , in an ascending order of cut-off frequencies. Expressing the
aperture field in terms of these excited modes will result in an accurate and rapidly

convergent solution.

2.3. Derivation of EM Fields via Transverse Field Method

Due to rapid technological advance in material manufacturing in recent years, the
study on the behavior of electromagnetic fields in anisotropic materials has become
increasingly important. Numerical methods are commonly used to solve the problems

dealing with materials with complex structures.

In this section the electromagnetic fields inside an electric anisotropic medium, which
electric properties vary in different directions, is investigated. The method used in this
section is based on the concept of eigenmodes of the spectrum-domain transverse EM
fields. The EM fields derived in this section are used in chapter 4 to match the tangential

electric and magnetic fields at the interfaces between regions as illustrated in Fig. 2.1.

2.3.1 Dielectric Tensor Properties of Anisotropic Media

The constitutive relations for a homogeneous, nonmagnetic anisotropic medium are

derived as

('0 ll
[11+

.9
D = (2.3.1)

-> a
B = ”OH (2.3.2)

19

where {-5 is called the tensor permittivity and is a complex constant dyadic or tensor if the
medium is a lossy material. The basic form of the complex tensor permittivity containing

only the diagonal elements is of the form

 

 

£1 0 O
E = o 52 0 (2.3.3)
0 0 83
where
8}. = er—joj/m (2.3.4)

and 8,, is the real-valued dielectric constant and 0']. is the real-valued conductivity. The
orthogonal coordinate axes in which the diagonal tensor E takes are referred to as the

principal axes.

Let us now consider that coordinates of the medium are lines up with the principal
axes of the anisotropy and the z-axis is considered to be the direction of wave propagation.
To rotate the transverse plane of the medium with a rotated angle 6, it will give us an

alternative form of the tensor permittivity:

_ an 33.3 0
= 2.3.
8 ny £33, 0 ( 5)
O 0 82
where
2 _ 26
em = alcose +8281!) (2.3.6)
_ . 2 29
a”, - alsme +£2cos (2.3.7)

8 = 23 (2.3.8)

20

8x3 = a” = (81-82) sinecose (2.3.9)

The relationship of matrix components between eqs. (2.3.3) and (2.3.5) is based on the

change basis of the transverse plane. It can be easily found in the following derivation.

Let W be a square matrix that transforms vectors in accordance with the equation
->
u

:10:
3* , and there is a matrix W that can transform vectors 3* and ii" in accordance with the
2*

equation 3* = W - 3‘. Now if we know that the coordinates in the new basis relates to
the coordinates in the old basis by a transform matrix T’, we can then write relations for

the coordinate transformation as

Y 3* (2.3.10)

xv
ll

CW
II

33-13“ (2.3.11)

Simply applying the vector transformation of 3 and 3* for the new and old basis
respectively into eq. (2.3. 10) and taking the inverse coordinate transformation leads to the

:1
solution to the matrix W , i.e.,

:-1
9* = Y 9
:-l =
= .w-%
:-l = : 9
= . W. Y v“ (2.3.12)
01'
:3: :-—l = :
W = Y W-Y (2.3.13)

If we rotate an angle 6 of the transverse plane, the coordinate transform matrix T’ can

be represented as

21

3, = cosO sine (2.3.14)
—sin0 cost)

:—1
and the inverse matrix Y is

1f] ____ cos6 —sin9 (2315)
sin9 cosB

: :—l 2*
As a result, with the aid of Y and Y , the vector transform matrix W in the new basis is

derived as

W“ = Til-1:17;: cosO —sin6 31 0 cost) sine
sine cost) 0 £2 —sin6 cosG

8100829 + 82811129 (El — £2) sinBcosG

(2.3.16)
(81 — 82) sin9cosG e] sin29 + ezcoszfl
2.3.2 General Eigenvalue Problem

Assuming a suppressed elm: harmonic time dependence of the fields, the electric and

magnetic field vectors satisfy the Maxwell’s equations

-> _ ->

VxE = —1mu0H (2.3.17)
-> . = a

VxH = 1m - E (2.3.18)

Expanding above equations gives

a1:Z 313), . H
__.___ = _ 2. .1
3E), a1;z

az — ax = —J(1)uOHy (2.3.20)

22

(DE“ 815:3r . H

X ._ .5; _ _J(ou0 2 (2.3.21)
8Hz aH‘. . .

7.97 — —az— = JanszJ‘C +Jmex),Ey (2.3.22)
BHx 8Hz . _

—a—Z— — 3x— = wa).xEx +1wewa (2323)
8H), 6H)[ . E

__ _ — = (1)8 . 2.3.24
ax ay 1 z z ( )

As mentioned before, a two-dimensional Fourier transform can be used to facilitate the
solution of the three-dimensional Maxwell’s equations derived above if an uniform and
unbounded medium along the transverse directions is involved. Let us now define a two-

dimensional Fourier transform pair as

.. °° —'k —'k.'

f(kx, k), z) = Jlflx, y, z) e j Rte 1 “"dxdy (2.3.25)
2 00 ~ .k X .k .

f(x, y, z) = (21ft) J Jf(kx, ky, z) e] I e} ”‘dkxdky (2.3.26)

where f (x, y, 2) denotes a three-dimensional spatial function and }(kx, ky, z) is defined

to be the two-dimensional Fourier transform of f (x, y, z) along the transverse directions.

If we take the two-dimensional Fourier transform of eqs. (2.3.19) to (2.3.24), we then

have
, - at, , -
jkyEZ—E- = -}cou0Hx (2.3.27)
ais,r , .. , -
E-kaEZ = —j(D}.l0H). (2.3.28)

23

 

jkxE).—jkyi~3x = -jwll0i13 (2.3.29)
. " 3i!) . " . "'
1k),Hz—-a—z- = waxxEx +Jmex),E). (2.3.30)
air, - , .. -
dz -1ka, = finemEx+}(0enyy (2.3.31)
jkay—jk),Hx = jooszEZ (2.3.32)

Among these expressions, the partial derivatives of the spatial domain electromagnetic
fields with respect to x and y are replaced by jkx and jky and the derivative with respect to

z is left as the only variable in the spectrum-domain. Rewriting eqs. (2.3.29) and (2.3.32)

 

gives
i: "y i1 "x i1
= —— + , 2.3.33
2 mez ‘ wez ’ ( )
- k, - k -
= 4 E _ X E (2.3.34)

The substitution of eqs. (2.3.33) and (2.3.34) back into eqs. (2.3.27), (2.3.28), (2.3.30) and
(2.3.31) leads to

 

5%? = ’5: [kxkyfix + ((02.31er — kx2)i1y] (2.3.35)
83;? = .31: (40231er + kyzjilx — kxkyfiy] (2.3.36)
85:, = —;{Tl [(—m2uoeyx - kxk),)f£Jr + {—002 1108),), + kx2)f£y] (2.3.37)
88—? = 701: [( (021108” - ky2)f£x + (wzuoexy + kxk),)l§).] (2.3.38)

24

These equations constitute a set of one-dimensional linear partial differential equation
for the spectrum-domain transverse electromagnetic fields, and they can be expressed by a

matrix equation as

if = 3- 2 (2.3.39)
dz
where
~ ~ ~ - ~ T
T 5 (15,, E), nOHx, “0113.) (2.3.40)

with no = 120n,and

 

 

 

pO 0 a bT
32- O 0 c d (2.3.41)
0: B O 0
3509
k k (0 e —k
[a b] Ewe} x 3 “0 Z X (2.3.42)
6 d [”0 -c0 uoez + k), —kxk‘
. 2 2 2
[a ] E J_%_ -0) 1108”- k,k). —(0 “0833+ kx (2343)
5 (Duo 2 2 2 ' °
7 0) “08n— k), (0 ”08x3“ kxk),
The longitudinal components, E z and Hz , can be obtained from
.. k .. kx ..
E2 = — 4H, + —H). (2.3.44)
(10:5z (er
.. k, .. k ..
= _>_ _ x E (2.3.45)

25

To find the solutions for the spectrum-domain transverse EM fields we need to solve
the eigenvalue problem of the one-dimensional partial differential equation. Its general

solution can be represented as
... 4 .) A 7
T __. Z ,4me e '" (2.3.46)
m = l

where Am denotes the unknown coefficient of the mm eigenvector, 1,2,", which with a
corresponding eigenvalue 7""; satisfies the relation S ~ i2", = lmizm. Since S is a 4x4
matrix in our case, there exists four eigenmodes in this problem and the complete solution
of the spectrum-domain transverse EM fields is a linear combination of these four

eigenmodes.

The nontrivial solutions to eigenvalues and eigenvectors can be determined by solving

detUi—i) = 0. This leads to

 

 

32:: ch Old 01c
detaBAO :2. [33,0 +a 01130 -b 01(3)»
_780% 801 761 780

7&4—K2(a0t+by+c[3+d8) + (ad—bc) (afi—BY)

a x4 — 2.22 + AS = 0 (2.3.47)

With the aid of the quadratic formula, four eigenvalues are obtained from the solutions of

 

 

eq. (2.3.47) as
2
+,/ —4AS
A'l (kx’ ky) 5 1“1+ = J2 2 (23.48)
12(er ky) -=- I"- = —I‘1+ (2.3.49)

26

2 —,/22—4AS

733 (kx, ky) 213+ = J 2 (2.3.50)

 

 

 

(2.3.51)

+

714(kx,ky) .=.r2_ = —r2

These eigenvalues represent right and left traveling waves which are denoted by the

propagatlon constants 1"” and 1“,; for the subscript t = 1 or 2.

. . T . .
The corresponding eigenvector, denoted by i2", = [e, f, g, h] , satisfies a matrix

equation as

 

 

 

 

'xmo a If:
02. d
”'6 f=0 (2.3.52)
01 BXmO g
y 8 Okm .h.

For a specnfic eigenvalue Am , expandmg this matrix equation gives four homogeneous
linear equations for four unknown components of the corresponding eigenvector. It can

be written out as

Homogeneous system of four linear equations:

ekm+0+ag+bh = O

O+fkm+cg+dh = 0

ea+fl3+gkm+0 = O

ey+f6+0+hlm = O

The algorithm used to solve this homogeneous system is that one of the equations is
selected and the unknown term containing the eigenvalue is transferred to the right-hand

side and treated as a known value. Next by transferring the selected unknown terms on

the left-hand sides to the right—hand sides of the remaining equations, we can solve the

27

remaining three unknowns in terms of the unknown treated as a known value. Thus, four
different sets of nontrivial solutions, or four different eigenvectors, can be constructed.
Since the four eigenvalues represent two pairs of forward and backward waves of two
different waves, two different eigenvectors are used to construct four possible eigenmodes
in the medium. The selection of appropriate eigenvectors to create eigenmodes is dictated

by the numerical convergence.

Case]
ag+bh = —ele (2.3.53)
Am C d f 0
B m 0 ' g = —e0t (2.3.54)
8 0 2. h -eY

Solving these equations with the Gauss elimination method leads to,

p 2 —
(Am-CB-da) 0 O f e(ca+dy)

[3 Km 0 ' g = —ea (2.3.55)
8 0 71 h l. —eY

L ”L

 

 

The value of unknown f is first determined from the top of above expression, and

unknowns g and h are then solved by substituting f back into the subsequent equations.

 

 

 

Thisgives
f= 2coz+aVY 8 (2.3.56)
lm-cB—dfi
g = 7— a+ 2 Y = - "’ , 2 5 e (2.3.57)
m xm—cs—da MUM—C1345)

28

2.2 — —d5 5 +d
h = _}% 7+ 52(Ca+dY) = __ K m Cf: 2 J+ (CE Y) 8 (2.3.58)
m Am—cB—ds AMLAm—cs—dsj

 

If we multiply MILK: — cB — d5) to and divide e from above equations, the eigenvector

 

 

 

 

becomes

_ 2 -
r1 Xm(}»m—CB-d5)
e

71m(coc+dy)
1;", = f = 2 (2.3.59)

8 -[a(xm-cB-d8)+B(ca+dy)]
h
" -[y(k,2n—cB—d8)+8(ca+dy)]

Notice that e should be a nonzero value.

 

 

Case 11
ea +113 = —g}.n (2.3.60)
kn O b e —a g
0 A" d ' f = —cg (2.3.61)
7 8 A" h 0
Using the Gauss elimination method, eq. (2.3.61) becomes
3L" 0 b H
e —ag
0 A" d . f = _cg (2.3.62)
0 0 (xi—by—da) h (“+650

 

 

From eq. (2.3.62) the components of the eigenvector are then solved in terms of the

selected unknown g as

29

 

 

 

 

 

h = [1201+ 6:8]8 (2.3.63)
n- Y-
2
}. —b —d8 +b + 5
e = -513- a+b2(ay+c5) = - at n Y 2 ) (d7 c ) g (2.3.64)
n An—by—dfi ln(An—by—d8)
_ g d(a'y+c5) _ c(li—by—d§)+d(ay+c5)
f - -— 6+ 2 — - 2 g (2.3.65)
n An-by-db xn(xn-by—d5)

Finally, multiplying 23,1(ki—b'y—d5) to and dividing nonzero value g from each

component of the eigenvector, we have

 

 

r- —[a l:-by—d5 +b(a'y+c5):fl
e
2
3,” = f = —[c kn—by-dé +d(ay+c5)] (2.3.66)
: xiii-3345
H kn(ay+c5)

 

 

th . .
Let us now denote the n e1genvector of the four e1genmodes of the system as

vn2
, n=1, 2, 3, 4 (2.3.67)

CW
3
ll

 

 

We will generate the first and second eigenvectors based on eq. (2.3.59) and the third and
fourth eigenvectors on eq. (2.3.66). Finally, the spectrum-domain transverse

electromagnetic fields represented in eq. (2.3.46) for an anisotropic medium can be

rewritten as

3O

 

F ~ 7

Ex

E. A; la Az A:

i = A1121 6 1+A21/28 ‘ +A3$3€ 3 +A41/4e 4 (2.3.68)
"OHx
"OI-I),

 

2.3.3 Degenerate Eigenvalue Problem

In this section the electromagnetic fields within an isotropic medium will be derived in
the same way as we did in the preceding section using the transverse Fourier transform
and eigenmode representation. As shown in Fig. 2.1, the free space backing the material

layer will be recognized as an isotropic medium and used as an example.

In free space the permittivity tensor is reduced to a complex scalar value by setting
£1 = 82 = 83 = so and the permeability remains as u = 110. Substituting back into

eqs. (2.3.6) to (2.3.9), we have

em = a”, = 82 = 80, e = e, = 0 (2.3.69)

The two-dimensional Fourier transform of Maxwell’s equations expressed in the

preceding section gives the following simplified relations.

 

did 2 ~
32 =,gr7; osjm
.. k), .. kx ..
Eza = ‘m—EBqu' (0801713... (2.3.71)
.. k‘, - kx ..

w - (OMEN—MICE)“ (2.3.72)

where

31

~ ~ ~ ~ ~ T
T a 2 (EM, Em, nOHm, 11011330)

 

 

 

and
0 0 a’ b’
505— O 0 ed
01' [3' 0 O
-7’ 5' 0 0.

, , _ 2 2
a b E J kxky (D l'I’OEOukx
c d —0) ”080+k). —kxky

. . 3 2 2
a 13 = £2 —kxky —co u080+ kx

. . “(0
y 8 “0 (1)211060—k:z kxkv

(2.3.73)

(2.3.74)

(2.3.75)

(2.3.76)

The nontrivial solution to the eigenvalues of matrix 50 can be proven to have

degenerate A. This means that multiple eigenvectors of Sa correspond to the same

eigenvalue. Let’s calculate the coefficients of the quadratic equation (2.3.47) and find the

solutions for eigenvalues via eqs. (2.3.48) to (2.3.51).

-~ 1 2 2
act = 2 kxk,
(0)1080

 

 

. . —1 2 2 2 2
by = 2 (to “050-181 X 0) pogo—k), )

(0 11on

 

-1 2 2 2 2 2 2
= 2 [(0011080) —(kx+k),)00 [1080+]:ka

0) 1.1080

—1

 

. . 2 2 2 2
c3 = 2 (40 uoeo+ky X —(0 [1080+ kx)

(0 11on

32

—1 2 2 2 2 2 2
= 2 [(00 [1080) —(kx+k),)m 1108044ku
0) “080

 

.. 1 2 2
d8 = kxk),

2
a) “080

 

=> 2 = (a'a'+b'y'+c’[3'+d'6') = —2[(02p.0€0—k: —k:]
a’d' = ——k k,

. . —1 2 2 2 2
b c = ;)2—((0 ttoeo-kJIC X —m u0£0+kyj

1 2 2 2 2 2 2
= —2 2[(o) “050) —(kx+ky)m [1080+kx kg
to 80

068' = —k k

II
I H

. 9 — 2 2 2 2
By 2 2(-0‘) “080+kx )( (I) “Geo-Icy)

1 2 2 2 2 2 2
= —2—5[(w “030) -(kx+ky)w “080+kx kfl
o

, ,, ., ,, .. 2
=>AS = (ad —bc)(a5 —[37) = [wZHOSO-k: -k:]

From eqs. (2.3.77) and (2.3.78) we obtain

( 2’)2—4As’ = 0

(2.3.77)

(2.3.78)

(2.3.79)

If we substitute eq. (2.3.79) back into the formulas for the roots of a quadratic equation, it

will lead to a condition of eigenvalue degeneracy meaning the multiple roots of the

eigenvalues. The eigenvalues are

33

/ . 2 2 2
A] = 132?.“ = 22; = —1Jm poeo—kx -k), (2.3.80)

. 2 2 2
2.2 = 142-Ad =1Jw quO—kx —k), (2.3.81)

 

 

The corresponding eigenvectors associated with these two degenerate eigenvalues can
also be determined by using a similar algorithm as shown in the preceding section. Since
eigenvalues are degenerate within an isotropic medium, this algorithm doesn’t need to
yield linear independent eigenvectors. Let’s write one eigenvector by setting zero in one
component and normalizing the other two nonzero components by the other remaining
nonzero component in a form like [0 f 1 [1:11 , or e 0 g 1] T, or [1 f 0 hJT, or
[e 1 g 0] T. The other eigenvector that has the same eigenvalue can be constructed with a
0 where the first eigenvector has a 1 and with a l where the other has a O. The derivation

of these eigenvectors is given below.

Qase I

Let the first eigenvector which corresponds to the eigenvalue la be chosen as

T
i210 = [0 f} h] . Substituting this into eq. (2.3.52) gives

 

 

plaOab-b‘
02» d
ac f=0 (2.3.82)
a [3100]
y 6 Okafi.

 

 

This will give us four linear equations to determine unknowns f and h. Expanding the

matrix equation yields a system of four homogeneous linear equations:

a+bh = O
fita+c+dh = O

34
fB+ka = O
fB-I-hka = 0

It is obviously that the solution to f and h are
f = __a (2.3.83)
B

5
- (2.3.84)
B

The other eigenvector which corresponds to the same eigenvalue is then expressed as
3

v30 = [1 f 0 h]T. Substituting this back into eq. (2.3.52) and expanding this matrix
equation gives

la + bh = 0

fit“ + dh = O

a+fl3 = 0

y+f6+h7ta = 0

Similarly the SOlUthDS t0 unknownsfand h are obtained as
h = ___a 2. .
l ( 3 85)
— -E - d. 2.3.
f l ( 86)

Finally, the eigenvectors in this case are summarized as

T
t»... = [ 3:2 1 8] (2.3.87)
[3 l3

35

320 = [0 .1“ 1 §]T I (2.3.88)
B l3
9 d 7» T
V30 = [I I; O _f] (2.3.89)
9 d 2» T
V40 — [I I; 0 3“] (2.3.90)
Qase II

Now let the first eigenvector which corresponds to the eigenvalue la be chosen as

T
310 = e 0 g 1] . Substituting this into eq. (2.3.52) leads to

1
I

FKaOab-
Olacd
aBAaO
“5010-..-

= 0 (2.3.91)

 

 

~OQON

 

 

Expanding the matrix expression gives a system of four homogeneous linear equations:

ela+ag+b = O

cg+d = O
ea-l-gla = O
e'y-I-la = O

The solutions to unknowns e and g are obtained as

e = --3 (2.3.92)
7

g ___ _6_1 = 9 (2.3.93)
C 7

36

The other eigenvector corresponding to the same eigenvalue is then written as
i230 = e 1 g OJT. With this eigenvector, four linear equations are obtained when the
matrix equation is expanded:

eka+ag = O

la+cg = 0

ea + B + gka = O

ey+5 = O

The solutions to unknowns e and g are obtained as

la
= __ 2.3.94
8 c ( )
e = _§ = 9 (2.3.95)
'Y C

T
31a = [.92 o 9 1] (2.3.96)
7 'Y
A. T
320 = [J 0 ‘1 1] (2.3.97)
7 'Y
A, T
33a = ‘3 1 __£ 0] (2.3.98)
C C
A T
340 = E 1 .8 0] (2.3.99)
C C

. . th . . . . . .
We Will use the notation of the n eigenvector of the eigenmodes Within an isotropic

medium as

37

nla

n20 n=1, 2, 3. 4 (2.3.100)

na ’

CV
II

n30

 

 

_ "40.

Let’s use the eigenvector of case I to create the first two eigenmodes with two different
eigenvalues and the eigenvector of case 11 to generate the last two eigenvectors with two
different eigenvalues. The spectrum-domain transverse electromagnetic fields can then be

expressed as

Exa
E» l (z—t) -7\ (z—l) Ada—1) -ka(z-z)
in = 813108 0 323208 a +B3a3a€ +3494ae (2.1101)
“0 xa
.nOqu

 

 

If we let la satisfy Re {la} > O and 1m {10} > 0, the terms with complex-valued
coefficients B1 and B3 will represent the backward propagating waves, while the terms

with 32 and B 4 will represent the forward prepagating waves.

By considering only the forward plane waves in an infinite half space, the spectrum-

domain transverse fields inside free space can be expressed as

 

 

xa
E —la(z—t) -l (z-t)
{“ = 321320 e + B4340 e " (2.3.102)
nOI-Ixa
moi-lye

In summary of this chapter, the EM fields inside a homogeneous medium have been

derived based on two different techniques. Those derived in section 2.2. will be further

used ii
an 1501
chap-1:

3111301

38

used in matching the boundary conditions in chapter 3 for measuring the EM properties of
an isotropic slab material. On the other hand, those derived in section 2.3. will be used in
chapter 4 to match the boundary conditions for measuring the EM properties of an

anisotropic slab material.

CHAPTER 3

OPEN -ENDED RECTANGULAR WAVEGUIDE
PROBE TO MEASURE EM PROPERTIES OF
ISOTROPIC MATERIALS

3.1. Introduction

There are demands for measuring the EM parameters of materials nondestructively. A
waveguide probe which consists of an open-ended waveguide terminated with a flange
can be used for this purpose. When a waveguide probe is placed against a material layer
and the reflection coefficient of the incident wave at the waveguide aperture is determined,

the EM properties of the material layer can be inversely determined.

The goal of this chapter is to derive two coupled integral equations for the electric field
on the waveguide aperture. After the determination of the aperture electric field by
solving the electric field integral equations (EFIE’s) numerically, the reflection coefficient
of the incident wave can be expressed as a function of the EM parameters of the material
layer. Thus, the EM parameters of the material layer can be inversely determined if the

reflection coefficient of the incident wave is experimentally measured.

In section 3.2., the electromagnetic fields in the waveguide, the material layer and the
free space derived using Hertzian potentials in section 2.2. are employed to match the

boundary conditions at the material-air interface and the material-waveguide aperture

39

4O

interface. This will lead to two coupled electric field infegral equations. In section 3.3.,
the moment method is implemented to solve the EFIEs numerically. Numerical results on
the input admittance of the waveguide probe are then compared with the existing results

published by other workers.

3.2. Coupled Integral Equations for Aperture Electric Fields

The geometry of the problem is shown in Fig. 2.1, where there are two discontinuity
interfaces; the material-free space interface at z = d where the material to be tested is
backed by free space, and the waveguide-material interface at z = 0 where the material

slab is placed against the waveguide flange.

Using the EM fields in the waveguide, the material layer and the free space derived in
section 2.2., the tangential components of the electric and magnetic fields can be matched
at the two interfaces. This will lead to two coupled integral equations for aperture electric
field. After the aperture electric field is determined, the reflection coefficient or other
quantities of interest such as the input impedance and admittance of the waveguide probe

can be determined.

3.2.1 Reflection Coefficients at a Discontinuity Interface

When a TE or a TM wave is incident upon an interface plane which separates two
different media, a part of the wave is reflected back from the discontinuity. The ratio of
the complex amplitudes of the reflected wave to the incident wave is defined as the

reflection coefficient.

In section 2.2.3 we have derived the EM fields in a layer of material which is backed
by free space. In that section two unknown quantities, R a (kx, ky) and R b (kx, Icy) ,
appeared in the field components given in eqs. (2.2.55) to (2.2.60) represent the reflection

coefficients of propagating TE and TM modes, respectively, at the plane interface at

41

= (I. These quantities can be determined when the transverse EM fields within the
material layer and that in free space are matched at the interface at z = d . Since the free
space is of infinite extent, only the forward propagating wave is assumed in the free space

region.

The forward propagating transverse fields inside the free space can be derived as

follows:
1 2°° , —r,(z— d) ejkx ejk)
E: = (it) LI [wuokyI-I (kx, ky) —jer21(kx, ky) ] e dk dk (3.2.1)
1 2 w , (z— d) ejkx ejk)
E: = (27:) LJ [—wu0ka(kx,ky) —jkyl“21(kx,k),)]e r, dk Hdk, (3.2.2)
1 2°° . —F(z- d) jkxjk1
H: = (27:) L] {—jer2H(kX, ky) —0080ky1(kx, ky) ] e 2 dk dk (3.2.3)
’12” . ~21“(-z d)ejkxxejk\.’,1
H: = ('27:) L] [—jk),l‘2H(kx,ky) +0080kx1(kx,ky)]e dk xdky (3.2.4)
where
I‘z(k, k) = — j,/k0 -k: —k: (3.2.5)
k2 = 00 11080 (3.2.6)

The quantities of H (kx, ky) and I (k)e ky) represent the corresponding coefficients of
fields of TE and TM modes at the spectrum location (kx, ky) . The superscript, a,

appeared in the field components signify that existing in the free space.

Using the transverse EM fields in the material layer given in eqs. (2.2.55) to (2.2.60)

and the transverse EM fields in the free space given in eqs. (3.2. 1) to (3.2.4), the matching

42

of these transverse EM fields at the interface at z = d will generate the following

 

 

 

relations:
for E)I = E:
z=J z=d+
21‘1d , 21‘,d -r,d ,
[mp/(yr? 1+Rae —)er10 l—Rbe ]e = [muckyH— 111x131] (3.2.7)
for E, = E0
42=d‘ )z=d+
2nd , 21‘,d -r,d ,
[-0)ukxF 1+Rae —jk)_I‘]G l-Rbe ]e = [—01uoka—jk),l“21] (3.2.8)

 

for Hxl = H:
z=J

z=d+

, 21‘,d 2r,d —r,d ,
[‘karlF 1 — Rae — (flakyG 1 + Rbe ]e = [—jkxF2H — (080k),l] (3.2.9)

forHvl =Hj',
.2“, .

 

z=d+
—r,d

[ , 2nd 2nd , ,
—jk),I“1F l—Rae +018ka 1 +Rbe Je [—jkyI‘2H+0)ekaI] (3.2.10)

The unknown quantity I (kx, ky) can be eliminated from eqs. (3.2.7) and (3.2.8) by

subtraction to yield
2r,d -r,d
11F l +Rae e = 00H (3.2.11)
Similarly, I (kx, k) can be eliminated from eqs. (3.2.9) and (3.2.10) to yield

2nd -r,d
FIFLI —Rae )e = FZH (3.2.12)

Now, the reflection coefficient of propagating TE modes, R a , at the interface at z = d can
be obtained from eqs. (3.2.11) and (3.2. 12) after eliminating H (k x, ky) . The result for R a

is

43

11 F —uF -2r,d
L3] (3.2.13)

R (kx,k,) = e
a ) [Flori + “F2

On the other hand, after eliminating the unknown quantity H (kx, ky) from eqs. (3.2.7)

and (3.2.8), we have

21“,d —r,d
F1G(l—Rbe )e = 1‘21 (3.2.14)

Similarly, eliminating H (k x, k\,) from eqs. (3.2.9) and (3.2.10), we have

2nd —r,d
6

80(1 + Rbe = 80] (3.2.15)

The reflection coefficient of propagating TM modes, Rb, at the interface at z = d can
now be obtained from eqs. (3.2.14) and (3.2.15) after eliminating l (er ky) . The result

for Rb is

8 1“ —EF —21‘d
0‘ 2Je ‘ (3.2.16)

Rb (kx’ ky) = [801‘l +8F2

3.2.2 Matching Boundary Conditions at Waveguide Aperture

After deriving the reflection coefficients of the TE and TM modes at the interface at
z = d by matching the tangential EM fields across this interface, we can now proceed to

match the boundary conditions at the waveguide aperture at z = O.

The field components derived in section 2.2. for both the waveguide and the material
layer are recalled in this section. Since a discontinuity is presented at the waveguide
aperture, higher order modes are excited near the aperture. Therefore the aperture fields
should contain not only a dominant mode but also an infinite number of higher order
modes. Due to the orthogonality property of waveguide eigenmodes, the unknown

amplitudes of these higher order modes can be derived in terms of the aperture fields. As

44

a result, two coupled integral equations for the aperture fields can be obtained after

matching the boundary condition at the aperture.

First, we will give the aperture fields at the side of the waveguide, i.e. at z = 0'.
According to eqs. (2.2.37) and (2.2.38), the aperture electric fields E and E110 can be

X0

expressed as:

m1(x y) =jwlLOZZ(I—n-'— b RAH.) ”€04 —:—x)sin("—1;t—y)
+nZZFn" m(—1-t)8nn “cos(£g£)sin(mnny) (3.2.17)

a b

 

Eya(x, y) = —jmu0(:)alosin(% x)“ +R)

24—11 101w)
7W

b
m"

+ZZrn..m(T-E’—‘)Bn.m.sin(”—Z‘— )cos( b ) (3.2.18)
n m

 

Since the TM modes do not exist if one of the mode indices is zero, we can write the

above equations in more compact forms as

- '2'! L“ ,, 1'55 - '71—'19?
8811111 11111110136891 . )...( . 1

(3.2.19)

Ex, ()6 y)

Eyo(x,y) = —jco|.10(: )amsin(n7 x)(1+R)
_- "'1 "311283 m_'ny
+§§[1muo( a )An.m.+l“n.m( b )Bnm]sm( a )cos( b )
(3.2.20)

By using the orthogonality properties of the sinusoidal functions, we can determine the

unknown amplitudes a10 , Am". and Bn.m. from eqs. (3.2.19) and (3.2.20). We have

45

a . 1tx . 11 b
gE),osm(-EI-)dxdy = —jwpo(5)%alo(l +R) (3.2.21)

d—.°‘

ab . m'n nn
:1" [181111 -,,—)e,.A,... + 34171314]

(3.2.22)

ba in o i
m ab . n 1: m It
£&E)081n(’ig- —)COS( T)dXdy — I [-](Dll0( 71—)Em'An'm' + 1“".m( T) Bn'm']

(3.2.23)

b0 I i
MExocos('-E7nJ—C)sin( Eggdxdy

or these unknown amplitudes can be expressed in terms of the aperture electric fields as

2/—j(i)110 ba , (m)
010 — mglEyOSfll -a— dXdy (3.2.24)

41111118111511 1'— .1151]

 

 

 

An'm' = ab [8m (2%— 112+) 8,1("11‘ )2 ] (3.2.25)

b

n'n

1 _ WW)” (81115111
"""' .1, 1 1”): ("”121 >

e . — +8. —

m a " b

where for the sake of breVity,

E _baE n'nx . m'ny dd

H m...“ xocos —a- Sln b x y (3.2.27)

b0 i I
NEW 2 uEyosin('—I§5)cos(m7m’)dxdy (3.2.28)

46

2 n'=0 2 m'=0
n'¢0’ m. l m'¢0

(0
III
PM
01
III
A

(3.2.29)

In eq. (3.2.24) if we let the amplitude of the incident wave, “10' to be 1, then we can
obtain an expression for the reflection coefficient of this incident TE10 mode in terms of

the aperture electric field. That is

2/—'00 ba
R = {—L—BBME, sin(1t—x)dxdy]—l (3.2.30)
b1t >0 (1

Substituting A n. m. , B . . and R back into the aperture magnetic field in eq. (2.2.40) leads

nm

10

Hx(x, y) L0- = (:)Flosin(1-:1—)(l—R)

2211—11—111—11—1

 

 

 

 

 

 

2r b0 I
-15 -12__10_-12 -n_x11
_ 2(a)I’losm( a )+jcou0absm( a )MEyosmE’t a )dxdy

 

-1191r:1-1e11:111311

47

Employing the following identities,

anm.+k(2) = (’12—th2 +("—'b—")Z (3.2.32)

1911210114 11111 "11 121-11-1- 11
111 (m1;— 111 1 ~11

and substituting them into eq. (3.2.31) yields

1
7

H

 

 

- ‘ {Ll-1’2)” 1’5.- *1
2:0. — muoab[2bnmuorl 0sin( —j2F0sin uEwsin dxfldy

+ 81:11::3111—1111—11111111101151
+ [hi—(5211‘)z JHEW ] ) (3.2.34)

 

where

 

(3.2.35)

Similarly, substituting An.m., B . . and R back into the aperture magnetic field in eq.

(2.2.41) leads to

_ m'rc . m‘it n'nx . m'n
”10"“ -' 2.2. .1r,.,,.(_b_)A,.m.-161(T)B,.m.]111(T)sm(__by)
n m
( y

 

 

 

 

48

w1.i1sm1a.—1 _. _. .2 .
mm W W ) 1:111M 1.1111-

_("__)( “Mr” +kOJHEw) (3.2.36)

 

Using the identities of
2 2 n'1t m'1t
rn'm' +k0 = (7} + (7)2 (3.2.37)

1%“)213m~1"—.’-‘7:= 1—111’? +1¥1-kél-1'%‘)k:
“11%“? 4‘1] @123]

and substituting them into (3.2.36) yields

 

#48 - 1 (n'nx) . (m'ny)([ 2 (Mn)?
H = ——"—’1'——cos — sm — — E
0
ylz = 0' ggabwuornw. a b b II x0
n'n m'1t
+ (7)177me ) (32-39)
As a consequence, eqs. (3.2.34) and (3.2.39) show that the aperture magnetic field at

the waveguide side are related to the integrals of the aperture electric field as defined in

eqs. (3.2.27) and (3.2.28).

Next, we will give the aperture fields at the side of the material layer, i.e. at z = 0+.

From eqs. (2.2.55) and (2.2.56), the aperture electric fields Em and Eyo are expressed as:

Exo(x,y) =(2i J13 [toukyaF(1+R) jk FG(1— Rb)]ejkexx1k’yxdk dk

(3.2.40)

49

1 2°° _ flax ejk)
E)_0(x,y) = E U [—(oukxF(1+Ra) —jk),I]G(1—Rb)]e dk dk

(3.2.41)

Using the orthogonality properties of the exponential functions, the above two equations

can yield the following relations:

five-1*.) .
[WE dx dy mpkyF(1+Ra) —-jer1G(l-—Rb) (3.2.42)

[We -jk 0d .
1:105 dxdy —(oukxF(1+Ra) —jk\,I}G(l—Rb) (3.2.43)

Since the aperture electric field components E x 0 and Eyo only ex1st over the aperture
region, 0<x<a, 0<y<b, and vanish on the flange, the double infinite integrals can be

replaced by the two double finite integrals as

b“ -1: k

a W I[15 e’ ”e " "‘dx d): 13.2.44)
b0 . .

_ -Jk.x -1’<,..v

E):2=‘[W[Eyoe e dxdy (3.2.45)

The unknown amplitudes F and G can then be solved from eqs. (3.2.42) and (3.2.43) as

 

F(kx,ky) = (122+ 1:51qu [kyflEfo- k M] (3.2.46)

 

C(kx,ky) = (W +k jam-R Rb) [k ”E; +kWHEfo] (3.2.47)

50

Substituting F and G back into the aperture magnetic field given in eqs. (2.2.58) and

(2.2.59) leads to

+

      

1 2°° . jkxx
=0 = (5;) LL! {—karl F(1_Ra) ’wEk),G(l+Rb)]e

=13.) 1:1 [:12 1.12:; 5:14 4142- (3115:.)

Rb 81k} jk .1

1m” e 2;“ k)”Efo+k:j:Ef:)]dkHdk

Ii 1’Rbe:k +k:

 

 

 

'k 'k.‘
-1—)WH :51:
_ 2113 - (OI-l F(k2+y2k) 11+ kll_——R x0
,

1 221—R0221+::J
+ Fk—— kzk E dkxydk
1
Riki-+14% xl+Ra 1‘1- 1“ yo

 

01'
H -1.” 12.1,)..[M k k E"+N (k k) E‘]dk dk
M“ . 1. .111- Mu... . .1
where
M(k k): k/(21t) [Wt—41W [(21443)]
x )’ nx(k:+k,) 11+R ll—Rb
1
5 2 l-R 1+R
N)'(kx’k>') E 1/(221t)2 [n2k:1+Ra- f:l-ij
Fl kx+ky a b
and

1 2 °° . 178* J
Hy(x,y)l 0. =1fi)” {—1181} F(1—Ra) +weka(l+Rb)]e e
Z: -°°

flay
e ’ dkxdky

(3.2.48)

(3.2.49)

(3.2.50)

(3.2.51)

'kWy
dkxdky

51

 

—]r 1_ejkxejk_1
=2(1112).[:.’I [qu1+R: k2 «1k: 2“" ”Em kxk1HEyo)
jk k
+j1“-£i)—§::R:ek2+ "e’ 11:29 j 1%”ka JE;0):ldkxdky

 

 

‘k 'k.'
_ (1)2101"; 1x8} 0 , ‘ [ft—1:112:14 +k2k2—1+Rb]”Ee
21! -00 0.)“. ELki‘Fki) l y1+Ra lxl—Rb x0
kk l-R 1+Rb
+ :1 2 [fir—Rf:+kf-l—-R :JHE;0 dk dk (3.2.52)
[‘1 kx+kv + -
0r
_ -f °° ””1".” e k k ‘ d dk
Hyz=0+ — $11.. I e e [Nx(ky,kx)”Em+M( y, QHEW] 1‘. y (3.2.53)
where
2
= 1/(21t) 2 zl'Ra 2 21+Rb
Nx(k),,kx)_rl k2+k2 [Fl k>‘1+Ra IkXI—Rb (3.2.54)
x )1

Equations (3.2.49) and (3.2.53) indicate that the aperture magnetic field at the material
side can also be expressed in terms of the aperture electric field as defined in eqs. (3.2.44)

and (3.2.45).

It is also important to note that the roots of 1 + R a = O and l — Rb = O , the factors
appeared in eqs. (3.2.50), (3.2.51) and (3.2.54), lead to the surface wave pole singularities
in an isotropic dielectric slab. Let us substitute eqs. (3.2.13) and (3.2.16) into 1 + R a = O

and l—Rb = 0. We have

F - F -2rd
1+R = 1+ M}. ‘ = 0 (3.2.55)
“Orl+”r2

 

I”: 7 J

 

52

GOP] —€r2 -2rld
l-Rb = 1— —— e = 0 (3.2.56)
£0F1+8F2

These relations can be rearranged to become

-2r,d

1— ll F
__e___2_r_2 = "OT—l (3.2.57)
1 + e ‘ l1 2

-2r d
1 — e I erg

—2r,d ‘ ”a F (3.2.58)
1 + e 1

If surface wave modes are excited in the dielectric slab, we can let Fl = jK1 in above

two equations. This leads to

K
tanKld = 31-1 (3.2.59)
M‘z
eI‘
tanKld = —2— (3.2.60)
30K]

It is noted that eq. (3.2.59) is the well known TE-odd mode eigenvalue equation and eq.
(3.2.60) is the well known TM-even mode eigenvalue equation for the guided surface

waves.

Next, we will derive two coupled integral equations for the aperture electric field.
Since the tangential magnetic field components, Hx and Hy, are continuous at the
aperture plane, 2 = O , the equality of eqs. (3.2.34) and (3.2.49), and that of eqs. (3.2.39)
and (3.2.53) will yield two coupled integral equations for the aperture electric field. We
have

For Hyl = H‘,
2:0.

. +
z=0

 

53

22—(—)(—X[ 4% Min +':;( lamb-w)

.—_ 0%.]: ejk“xejk-“”[Nx (k), kx) ”1550 + M (k), kx) j j 15:0] dkxdk), (3.2.61)

01'

22.1——Z§:,,;..(“)wos(".fl)sm('-"gfl)(l3%MM (—.—X%.—)na..)

+ fleik xejk y [N (k k) “E:0+M(ky, kx) ”1350] dkxdk), = 0 (3.2.62)

         

 

 

 

_ 5151:!$1}kaJkWJEfoJrNyMX’ ky) HEdekxd/cy (3.2.63)

— i)m(’%)fiw(¥) ' ,,, f:;,:.(.—t:)x
sin(tix)cos('2:—ylt(¥)(¥)m(ks-(”muted

+178” xejkwmk 19,)qu +N (k k)”E;o]dkxdky= Csin(1-:—:E) (3.2.64)

           

where

54

C .=_ — 'gCZ—tcour‘m (3.2.65)
Equations (3.2.62) and (3.2.64) are the key coupled electric field integral equations to be

used in the further development.

3.3. Numerical Simulation

In our study, the numerical simulation includes two procedures, the forward procedure
and the inverse procedure. In the forward procedure, the reflection coefficient or the input
admittance of the waveguide probe is determined with assumed EM properties of the
material layer. In this procedure, two coupled electric field integral equations (EFIE’s) for
the aperture electric field are numerically determined. The unknown aperture electric field
is expressed as a sum of incident and reflected dominant mode and a number of higher
order modes due to the discontinuity presented at the aperture. In the inverse procedure,
the reflection coefficient or the input admittance of the waveguide aperture is measured,
and the EM parameters of the material layer are inversely determined through an inverse
technique. In this procedure, a number of higher order modes are included in the
unknown aperture electric field to improve the accuracy of the solutions for the EM
parameters of the material layer. The forward procedure will be discussed in the present

and subsequent chapters, while, the inverse procedure will be discussed in Chapter 5.

In section 3.3.11, the method of moments is employed to convert the EFIE’s to a matrix
equation. Here the EFIE’s are solved by the Galerkin’s method using the waveguide
eigenmodes as the basis and testing functions. Section 3.3.2 develops formulas which are
needed in carrying out the numerical calculation of the matrix equation transforming from
rectangular coordinates into cylindrical coordinates. Finally, section 3.3.3 presents
numerical results of this study, and compares them with the published results to verify the

accuracy of this technique.

55

3.3.1 Application of Method of Moments

The coupled integral equations given in eqs. (3.2.62) and (3.2.64) for the unknown
aperture electric field are solved by using the moment method technique. The unknown
aperture electric field is first expanded into a set of appropriately chosen basis functions
{e5 (x, y) } . Since the aperture fields of a waveguide can be expressed as a sum of a
dominant mode and a number of higher order modes, the appropriate basis functions are
the eigenmodes of the waveguide. Let us expand the unknown aperture electric field

components into two finite sums of eigenmodes of the waveguide as follows:

Exo(x’~v) = ZaBeEUJ) (3.3.1)
B
E,.o(x,y) = ébgegb’cfl) (3.3.2)

where 2 represents 22 for summation of all possible higher order modes which
i3 P q

have mode indices {(2p-1), (2q—2)] with p,q = 1,2,3, The set of basis

functions are expressed as

x _ (2p—l)1tx . (2q—2)Tc_v
2,6, y) z co.[__a__] [_b-] (3.3...

cg (x, y) 5 sin [LEE—71L”? cos [Ea-7332] (3.3.4)

Substituting eqs. (3.3.1) and (3.3.2) into the coupled EFIE’s yields

261%- [W42 + W;] + gbfl- [W33 + Wei] = 0 (3.3.5)
ga, [de. + wggj + g3, [wg + w; = aging) (3.3.6)

where

56

x _ -48,,om- p. (n'nx) . (m'ny)[ 2 (mm)?
de—ggabrn,m,(flojcos 7 sm —b_ [(0— —b— IIBX (3.3.7)

 

WC: sf] e’k‘xe "“ "N (k k )(He :)dk dk . (3.3.8)

111-2233: 6—101 '—:x)sm(e)(%)(e)ne.

 

E f] ejkfiejky)’M (ky, kx) (J'J‘e;)dkxdky (3.3.10)

 

W53 £2.12?" (1%)“? 4lecos(’%9l(?l(’%lflex mm
wcjgfle “‘1 ejm‘)M(kx,k)(JI jdk dk (3.3.12)

21“] ba
y H ) 1tx
Wdy= ab “0 —)sin(%— —)1[([efl (x', y') sin(—x'a )dxfldy

+ZZab::.m m(ll— 7))Sin(L:x)cos(’-Zl%l)[k3_(¥7].Ue)' (3'3-13)

 

 

°° 1k} jkyy
EU e e - Ny(kx,ky)(Hef)dkxdky (3.3.14)
and
=ba "(1 .) (GE) ' ("”de 'd' 3315
”ex-“65 x,y cos a srn b x y (.. )

 

b0 1 I 1 1
Hey E- gle; (x', y') sin(E—;E-)cos(mgy )dx'dy' (3.3.16)

57

b0 _-k 1 —.k. '1
“6:2 ‘MeEU', y') e j "x e j “) dx'dy' (3.3.17)

’9“ . —'k ' :1."
”eyes He; (x', y') e j "Xe I Q dx'dy' (3.3.18)

The EFIE’s for the aperture electric field are now expanded into the waveguide
eigenmodes with unknown expansion coefficients a‘3 and bB which represent the
amplitudes of the eigenmodes for the x and y components of the aperture electric field,
respectively. Next, we will use the Galerkin’s method to determine al3 and b3. Since the
Galerkin’s method uses the same set of basis functions as the testing functions, we write

the set of testing functions as

I; (x, y) 5 cos [La-T671225] sin [fig—)9] (3.3.19)
1; (x, y) a sin [$21123] cos [gig—12] (3.3.20)

After taking the inner products of eqs. (3.3.5) and (3.3.6) with the set of testing functions,

we have the following results.
xx xy

gas ‘ DGB + ébfl ’ Dab = 0 (3.3.21)
N )2)‘

gm” - DaB + @630,1B = Fa (3.3.22)

which can be represented in a matrix form of

)0: xy

D0113 Dali “B = [0] (3.3.23)
N N b
Da 0015 B

58

where

ba
0;; a Mr; (x. y) [Wj‘y <x.y) + W; (x. y)]dxdy

ba
0;; a (it; (x. 3) [W51- (x, y) + W3, (x. y) ] dxdy
and

ba ‘
Fa a C Mt; (x, y) sin( 1%)dxdy

3.3.2 Evaluation of Matrix Elements

(3.3.24)

(3.3.25)

(3.3.26)

(3.3.27)

(3.3.28)

In this section, elements of the matrix will be evaluated into compact forms to

facilitate the computer programing. Among the evaluations in this section, two-

dimensional Fourier transforms of the sinusoidal functions over the aperture are evaluated

analytically in Appendix.

‘1

In the matrix, the component 0:113 which represents the self-interaction of the y-

component of the aperture electric field is first discussed. Let us divide Dz; into three

parts:

ba
D33 5 ”I: (x. y) [ij (x, y) + W:). (x. y) ] dxdy

59

= (1)31" + <1)"; + (1);" (3.3.29)

where <1)";y represents the contribution coming from the material layer which has
continuous spectrum property, (I)? represents the contribution from discrete higher order
modes of the waveguide and (1%y represents the contribution from the incident dominant

mode. They are expressed as

(DigiI‘ (x ”We: ()6 y)dxdy
fit: ()6, 3’) [J3 81k,xejk,)'1v (kx, ky) (Hejjdkxdkyjdxdy
=J:IN)- (kx’ k ,-_,)(”t )(”e jdk dk (3.3.30)

 

 

waiiw(22;:i~':.(.3.)sm(";—:t—)cos(mr)1k:-(a)1113.)”)
=§§:§::(rz)lké—(¥1161.1613)

 

= :a—El- ffliitz; (x, y) sin(1%)dxdy][ileg(x', y') sin(1%)dx'dy']
[(2144011) cos [_____(2r -b2)1ty] sin(1-:1—x)dxdy] x
[IIE—gsinu [__(2p ’01) nx'] cos s[——(2q —b )ny] sin(1-%- f‘)dx dy]

= 2712' 611351.153 1] [Cl—b 2 5101511]

II
“I
3‘l,_
tr:
0
i
h
d—afi"
d—.::
m

60

ab
= "(5'02)r1051, 15r15p15q1

where some shorthand notations are expressed as
ba
1 1k. ejk. 1
m=lgta (X, y)e dXdy

= [15in [Ly—712B] [cos (kxx) +jsin (kxx) ] dx] x

[Icos [W] [COS (kyy) +jsin (kg) 1611’]
E [511081) +s,2(k,)1 151303.) +s14(k,.)]
ysfité (x y) sin(n :x )cos( thyjdxdy
= Elsi“ [32—12235] cos [fig-13],, in( "'Zx)cos( m'bny)dxdy

8. 8

n, (21- l) m', (2r- 2)8 m'

 

_ab
’4

311 (k) =Isin [Ly—711m]cos (kxx)dx

512(kx) stsin [ail—m] sin (kxx) dx

313 (ky )— = icos [filth—2m] cos (kyy) dy

314 (ky ) =jicos [Qlefl] sin (kyy) dy

(3.3.32)

(3.3.33)

(3.3.34)

(3.3.35)

(3.3.36)

(3.3.37)

(3.3.38)

6]

Similar forms for He; and He), are expressed as

Hefafieév' .1.-

dx dy

=1“ sin (2” ——1—:xe)n::|[cos(kxx')—jsin(kxx')]dx']x

(261 211W

61 -.

] [cos (k,.y') —j sin (kyy') 1 dx']

'=- [321(k) +322 (1%)] [523081) +524(k)”

8

 

= T 11', (210—1) m',(2q—2)8m'

321 (kx) E Isin [fly—m] cos (kxx') dx'

s22 (kx) a {1 sin [LZL—a—lfli] sin (kxx') dx'

b ]cos (kyy') dy'

323 (ky ) =icos [M

324 (ky) a -'icos[

(2(1- 2) 1t)"

b

] sin (kyy') dy'

 

(3.3.39)

(3.3.40)

(3.3.41)

(3.3.42)

(3.3.43)

(3.3.44)

62

If we further define
¢12u(kx) 215,109.) +s12(kx)] [s21(kx) +s22(kx)l (3.3.45)
¢12v (ky) E [313 (k),) +sl4(k),)] [323(ky) +524 (k),)] , (3.3.46)

the multiplication of the shorthand notations given in eqs. (3.3.30) and (3.3.31) become

(1.153111%?) Eq’12u (kx) $121418) ' (3.3.47)

and

ab 2 2
(JJGXII‘L’N) = (71—) 8n'. (21— l)5m‘, (2r—2)5n', (2p—1)8m', (24-2)5m' (33.48)

Substituting them into eqs. (3.3.30) and (3.3.31) gives

ch? = j j Ny(kx, ky)¢>12u(kx)¢12v(ky)dkxdky (3.3.49)

—00

-4en

“"3"ééabr.'.:'.(i)[ké—(¥l101511114

2
_ ab [1 8n'm'Em' n.“ 2
" 7([)22 p. . [(7) -k0:|6n',(21—1)5m',(2r—2)8n',(2p-1)6m', (2q-2)
0 n' 01' nm

(3.3.50)

 

 

Moreover, since

N,(k,k,)-=-: 1 —— 1k—
) x ) 1"l ki+k2 x1+Ra yl-Rb

1/(21t)2 [rzkzl’Ra 2 21+Rb]
,

is an even function of kx and ky, (I)? can be further simplified to

63
(b? = 4flNy(kx, ky)<i>;92u(kx)(1)182v(ky)dkxdky (3.3.51)
where (bqu (kx) and $162,, (ky) denote the even components of ¢12u(kx) and ¢12v (ky) ,

respectively, and are derived in detail in Appendix.

Since it becomes highly oscillatory if the two-dimensional semi-infinite spectral
integration given in eq. (3.3.51) is calculated in rectangular coordinates, this spectral
integration is computed in cylindrical coordinates to sidestep this difficulty. Let’s change

the coordinates from rectangular to cylindrical with the relations of

(3.3.52)

{kx = kcosrp

k ksintp

where (p is a real variable with range of [0, 21:] and k is a real variable of [0, oo] . The

substitution of eq. (3.3.52) in eq. (3.3.51) yields

eon/2
ch? = 4 (i i N,.<k.(p>¢f2,(k.(p)¢f2,.<k.<p)kd<pdk

= 4 {0”, (k) kdk (3.3.53)
where
rt/2 e e
0,, (k) a l N, (k, «m ¢12u (k, <9) $12.. (k. (9) dm (3.3.54)

Next we will discuss the component D3; which represents the self-interaction of the
x-component of the aperture electric field in the matrix. Divide D33 into two parts as

follows:

64

ba
BE Dauta x (x, y) 1:ij (x, y) + W; (x, y) ] dxdy

XX XI

= (bl +<l>2 (3.3.55)

where

(blxfit; 0" Y) We: (x, Y) dxdy
III; (x, y) [1:] 311.381ng (k), kx) ( Hefjdkxdkyjdxdy
= J J N"(k-V'k‘)(J-1t:)(116:)dkxdky (3.3.56)

 

 

=11W122.;Z§:':1(i)cos("'—::—’f)sm('"2”)[ e m )M
= gg'j—Ii—Z: (lit—()Hkg-(mbltFMH’xXHex) (3.3.57)

and the shorthand notations are expressed as

6 ba jkx ejk )
5:5 t (x, y)e dxdy
lia
= [Icos [Ly—#1 [cos (kxx) +jsin (kxx) ] dx] x
[Isin [Q—b-M’] [cos (kyy) +jsin (kyy) ] dy]

5 [531 (k) + 332 (k)] [333 (ky) + 334 (k),)] (3.3.58)

65

(1! [Him] 8‘“ [m] cos('g’3)sin(r2%rz
ab

= 7%, (21— 1)5m', (2r—2)

531(kx) a Icos [Lg-1:703] cos (kxx) dx

332 (k) ajIcos [flip—E] sin (kxx) dx

1?
s33 (ky) a {sin [W] cos (kyy) dy

b
. . 2 -—2 1t .
s34 (ky) 51‘! sm [(—rE)—-Y:l srn (kyy) dy

Similar forms for I I e If and ”ex are expressed as

a —'k . -'k."
1.1425 legU', y') e 1 ‘xe J 9 dx'dy'

 

icos [ (2p -al)7tx'] [cos (kxx') —jsin (kxx') ] dx'] x

 

b I
(ism [ (2q —b2)1ty] [cos (kyy') —jsin (kyy') ] dy']

s [341(kx) + 342 (kx)] [s43 (ky) + 344 (k),)]

”ex 5:185:05, y') COS('—I¥)sin(flifli)dxvdy.

 

 

)dxdy

(3.3.59)

(3.3.60)
(3.3.61)
(3.3.62)

(3.3.63)

(3.3.64)

 

= :Icos [ (2P -a1)1tx'] sin [(Zq -b2) 10"] COS('L:£)81D(mZCy')dx'dy'

66

 

 

 

 

ab
= 75"., (2p—1)5m'. (2‘14) (3.3.65)
341 (k) 2 {cos I: (2p :11) nx] cos (kxx') dx' (3.3.66)
0 0
.542 (k) a —jt[cos [ (2p — DEX] sin (kxx') dx' (3.3.67)
a
b . (2q—2)7tv'
343 (k),) 5 ‘[sm[ b ' :lcos (kyy') dy' (3.3.68)
.” . (2q—2)1tv' .
s44 (ky) E 1]! srn[ b ' :l Sln (kyy') dy' (3.3.69)

If we define the following notations,

¢34u(kx) E [331(kx) +532(kx)] [541(kx) +S4Z(kx)] (3.3.70)

¢34,(k,,) 2 [333(5) (3.934(5)) [s43 (ky) +544(k),)] (3.3.71)

the multiplication of the shorthand notations which given in eqs. (3.3.56) and (3.3.57)
become

UJ’QUPU E ¢34u(kx) $341418) 1 (3.3.72)

and

ab 2
(”’x)(”"x) = (T) 5'13 (21-08%. (2r-2)8n'. (2p-1)6m', (2q-2) (3.3-73)

Substituting them in eqs. (3.3.56) and (3.3.57) gives

67

of = j j Nx(ky, kx)<1)3‘m(kx)(1>_.54V(ky)dkxdky (3.3.74)

 

 

n m' 0
ab [1 8n'm' m.“ 2
= 7(f)22r [(7)2 ’k015n'. (21-1)5m'. (2r—2)5n: (2,,_1)5m', (2.1-2)
0 n' m' n'm'
(3.3.75)
Since
2 1 —R l +R
[773—- 73—)
' I] kx+ky ' a b

is an even functions of k x and ky, (1)):Jr can be modified to
<61" = 4 H Nx (k), k) (1);,“ (k) (1);“ (ky) dkxdk), (3.3.76)

where $3,473“) and ¢;4v(ky) represent the even components of ¢34u(kx) and
¢34v (ky) , respectively. As done earlier, changing coordinates from rectangular to

cylindrical, the two-dimensional semi-infinite spectral integration becomes

ecu/2
cf,“ = 4i ,1 N,(k.<p)¢§.,(k.<p>¢;4,(k.(pikdwdk

= 4‘10”“) kdk (3.3.77)

where

1t/2
0,, (k) a l N, (k, <9) (93‘4“ (k, (p) 4’34. (k, <9) dcp (3.3.73)

68

Next, let’s evaluate the component D23 in the matrix which represents the mutual-
interaction of the x—component of the aperture electric field as the testing function and the
y-component of the aperture electric field as the basis function. Divide Déé into two parts

as follows:

333:1: (x, y) [ng (x. y) + W; (x. y) ] dxdy

HIV.
Cit—5°-

ll
'6-

?" + (I)? (3.3.79)

where

WE}; (x, Y) W510, y) dxdy
Kit; (X, y) [J3 ejkgejkyy M (ky, kx) (Jje:)dkxdky]dxdy
D M US" kx) (11t:)(”e;)dkxdky (3.3.80)

63231: 730:, y) Wd:(x, y) dxdy
= 31,363) [223:n.:(r110)(i)(m—bfl)(_n)(m7n)”ey]dmy
22.32:?" (:.)("“)('”)mr)(n)

 

 

According to eqs. (3.3.39) and (3.3.58), we obtain

UPUUIC’U E ¢32u (kx) ¢32v (ky) (3.3.82)

with the new shorthand notations defined as

69

¢32u(kx) E [331(kx) +532(kx)] [521 (k) +522(kx)] (3.3.83)
¢32v(k ,) s [.933 (ky) +534(k),)] [523 (k),) +sz4(ky)] (3.3.84)
and according to eqs. (3.3.40) and (3.3.59), we get
— “b 25 5 5 5
”’3: Hey ‘ ‘4- 12', (21—1) m', (Zr-2) n', (2p— 1) m', (2q—2)€m' (3'3-85)
Substituting them in eqs. (3.3.80) and (3.3.81) gives,
(3.3.86)

o3" = j j MUS,kx)(5mm)(53241595115311:y

 

= 2;X§:’.:.(fi:)(£-‘)(i"£~‘ (11001.3)
8 m 5.

’ 'T(—)ZZF.M (la—X b )572 (21— li5m'. (2r— 2)5n' (2;)— I) "1' (24 2)
n m nm
(3.3.57)

 

Since

 

k/(27t) 21-R 213mb]
M(k.,k)-=- 2(1‘ ——"+ —
> It 1:02”); 113 R ll-Rb

X

is an odd functions of kx and ky, (I)? can be written as

chi" = 4 [J M (k), kx) 53",“ (kg) 4’32) (ky) dkxdk), (3.3.88)

where (1)302“ (k) and ¢§2v(k) represented the odd components of ¢32u(kx) and

(1)3” (k ) respectively. As before, changing coordinates from rectangular to cylindrical

(I)? in eq. (3.3.88) becomes

70

003/2
xy

<I>l = 4i [ Mac,cp)¢§’2,<k,<1)¢;’2.,(k.wicdmdk

00

= 4 {on (k) kdk (3.3.89)
where
n/Z 0 0
6,, (k) a 1 M (k, (p) «132,, (k, (p) «132., (1., (p) .75 (3.3.90)

The last step is to evaluate the component Dig in the matrix which represents the
mutual-interaction of the y-component of the aperture electric field as the testing function
and the x-component of the aperture electric field as the basis function. Let’s divide D33
into two parts as follows:

a
Dig {1606’ Y) [Wd: (3‘, .V) + We: (X, y) J dxdy

Ill
Cit—w-

x \‘X

+ (Dz (3.3.91)

V
(Di

where
yx ha y x
b0 y °° jkxx jky)» e
l ([10! (x, y) j j e e - M(kx,k),) ”ex dkxdky dxdy

D M “I" k)') (.11 t;)(j.1e:)dkxdky (3.3.92)

ba
<19; 2 1111:“ y) W); (x. y) dxdy

71

 

 

Based on eqs. (3.3.33) and (3.3.64), we have

thflkflea 5 $14“ (kx) 4’14); (1‘)») (3.3.94)

With the new shorthand notations defined as
¢14u(kx) E [511(k) +512(kx)] [541(k) +542(kx)] (3.3.95)
¢14v(ky) a [$13 (ky) +sl4(ky)l [s43 (ky) +544(ky)] (3.3.96)

and with eqs. (3.3.34) and (3.3.65), we obtain

ab 2
(JIt3')(JJIex) = (I) 8'1', (21— l)8m'. (2r—2)5n', (2p-1)5m', (2q-2)Em' (3.3.97)

Substituting them in eqs. (3.3.92) and (3.3.93) gives

0}" = J I M(kx,ky)¢l4u(kx)¢14v(ky)dkxdky (3.3.93)

237;: (—)<-2)(9)(n)(n)

ab p 8" n'n m'n
2—T(E)22an=(7)(7)5n',(21-1)5m3(Zr-2)5n'.(2p—1)8m1(29-2)
0 n' m' n'm'

(3.3.99)

 

Since

M (kx, ky) E

 

2
-k k,/(21t) l-R 1+R
1,)2 2.[1‘12—_1 Ra+k12—1 Rb]
1‘1(kx+ky) + a - b

72

is an odd functions of kx and k)" (I)? can be written as
dy‘," = 4 j 0] M (k1, ky) 0;“ (kx) 01"“,(10‘) dkxdky (3.3.100)

where (1)104u(kx) and of 4v(ky) are the odd components of ¢14u(kx) and ¢14v(ky) ,
respectively. If we change coordinates from rectangular to cylindrical as before, (I)? in

eq. (3.3.100) becomes

con/2
qr? = 4i (i Mac, 0)¢f4,(k. cp)¢104v(k, <0) kdcpdk

oo

= 4([Gw(k)kdk (3.3.101)
where
n/Z 0 0
ny(k) a 1i M(k, (0)014u(k, (0)014v(k, (0) dcp (3.3.102)

3.3.3 Numerical Results and Comparison with Existing Results

For a specific material layer, the numerical evaluation of the components of the matrix
equation given in eq. (3.3.23) for solving the aperture electric field and other relevant
quantities of the waveguide probe is performed in a FORTRAN computer program. Once
the matrix equation is solved, the aperture electric field is first obtained by summing up a
finite number of modes that we have taken into account in the program. Then the
reflection coefficient or the input impedance of the waveguide probe is determined via

these aperture electric field.

To verify the accuracy of this technique, as well as the validity of the computer

program, data of the input admittance of the waveguide probe placed against various

73

material layers which have been published using a waveguide (RG 52U rectangular

waveguide, X band) are recalculated and compared with the existing results.

Figures 3.1(a) and (b) show the real and imaginary components of the input
admittances of a waveguide probe when the probe is placed against a layer of quartz with
a dielectric constant of er = 3.76 and a thickness of 0.1299 inch. Our results are
compared to the theoretical and experimental results of Croswell et al. [22] where only a
dominant TE10 mode was considered in their theoretical calculation. These figures show
that our numerical results compare quite well with the theoretical value of Croswell et a].
when we considered only a dominant mode in our calculation. However, when higher
order TE and TM modes are taken into account in our calculation, our numerical results

match very well with their experimental results, much better than their theoretical results

do.

We considered two multi-mode cases; the one with a dominant TE 10 mode plus three
higher order modes (T5530, T1512 and TM 12) and the other one with a dominant TE 10
mode plus eight higher order modes (T1330, TE”, TMIZ, TESO, TE32, TM32, TE52 and
TM52 ). It is noted in Figs. 3.1(a) and (b) that the 4 modes case and the 9 modes case
yielded almost identical results. This indicates that a good convergence can be obtained

when only the first four modes are used in the numerical calculation.

Figures 3.2(a) and (b) show the real and imaginary components of the input
admittances of a waveguide probe which is placed against a material layer with a
dielectric constant er = 2.25 and a thickness of 0.3201 cm. For this case our numerical
results obtained by using one, four and nine modes are compared to the theoretical and
experimental results of Bodnar et al. [24]. These figures show a good agreement between
our numerical results with multi-mode consideration and their theoretical and

experimental results.

74

 

 

 

 

 

4.5 . , , P T
;XX§xXXXXXXXXXX
4,_ ,
3 3.5 ~
C
3
Q
3
s 3'
U
1:
Q)
.9}.
E 2.5 —
0
Z
2..
Present theory. : g
. _ dominant mode only : . ___
1.53.... ,, .. . dominant+3 h1gher-ordermodes. I —.—-—._
3 dominant 47 8 higher-order modes ; ++++
1 3‘ 1 1 1 4
7 8 9 1o 11 12 13
Frequency ( GHz)

Figure 3.1(a) Input conductances of a waveguide probe (a = 0.4 in, b = 0.9 in)
as a function of frequency when the probe is placed against a material layer with a
dielectric constant of er = 3.76 and a thickness of 0.1299 in. The comparisons are
made between our numerical results and theoretical and experimental results of
Croswell et al. [22].

75

 

T l l T l
4.5 ._ ...... , , . . . . . . _ . . . . . ExperimentbyCI-Oswefluet. a] ........... ..... 0000 . . . _
Theory by Groswell et al. XXXX
4 ,_ ....... . , . . ......... . .(dominantmode onlfly) . ............... ............... _
Present theory : , g i
3.5.. ....... XXI-'7. ... dominantmodconly ___._.....
\ §xx dominant j+ 3 hi gher-ojrder modes —

(ID
1

Normalized susceptance
N
N 01
l l

—L
01
I

 

 

 

l l l

)—

 

10 11 12 13
Frequency ( GHz)

Figure 3.1(b) Input susceptances of a waveguide probe (0 = 0.4 in, b = 0.9 in)

as a function of frequency when the probe is placed against a material layer with a
dielectric constant of er = 3.76 and a thickness of 0.1299 in. The comparisons are
made between our numerical results and theoretical and experimental results of
Croswell et al. [22].

76

 

 

 

 

 

I l I I I l l f l l
2.25 '
2r _.
1.85 _
d.)
U
€51.5— ,.
2'3
:3
-o
81.40- ~
0
1::
o 3 - _ _
S12- ..... ._.
a ' Experiment by Bodnar et al. 3 , 0000;
2 1_ ..TheorybyBodnaretal. .3 ...... ..fxxxxf. ...-
w : (1.5m0des) . . z 1 i
0.8-..................gPresent.theory.: ..... i ......... i ...._
’ ~ dominantgmode only ; — — —§
dominant:+ 3 higher-order modes ' —: j
0., . . . . ; ; . ; . 3
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)

Figure 3.2(a) Input conductances of a waveguide probe (a = 0.4 in, b = 0.9 in)
as a function of frequency when the probe is placed against a material layer with a
dielectric constant of er = 2.25 and a thickness of 0.3201 cm. The comparisons are
made between our numerical results and theoretical and experimental results of Bodnar
et al. [24].

77

 

1.6 I I I I I I j I I I

14-

...L
N
I

 

Normalized susceptance
._L
I

Experiment by Bodnar etal.i '- ioooo
0.8— ....... TheorybyBodnaI-eta] ......... xxxx..i... _
(15modes) 3 ~ ‘ , -

Present _i:theory . . _
dominant mode only i _ _ _

 

 

 

0°65" dominant-1+3 h1gher-ordermodes" — ' i " ” ””7
dominant + 8 higher-order modes i-H-++ E ’
Q4 . . . ; ; ; . ; ; 1
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)

Figure 3.2(b) Input susceptances of a waveguide probe (a = 0.4 in, b = 0.9 in)
as a function of frequency when the probe is placed against a material layer with a
dielectric constant of Sr = 2.25 and a thickness of 0.3201 cm. The comparisons are
made between our numerical results and theoretical and experimental results of Bodnar
et al. [24].

78

The last comparisons are shown in Figs. 3.3(a) and (b) which show the real and
imaginary components of the input admittances of a waveguide probe that is open to free
space. Our numerical results are compared to the theoretical results of Baudrand et al.
[30] and the experimental results of Bondar et al. [24]. Our results using four modes and

nine modes agree well with above published results.

In this chapter, we established a forward procedure for the theoretical study of
measuring EM parameters of isotropic materials using an open-ended waveguide probe
system. The accuracy of the technique and the validity of the computer program have

been verified by comparing our numerical results with the published results.

79

 

 

 

 

1 I I I I I I I I I
09_ d
0.8" -

O)

U

C

:3 . . , . .

g I T . I i I I i .

‘0: ' : t : : :

o : Experiment by Bodnar etal. oooo

’7; ' Theory by Baudrand et al. : XXXX

E (15modes) . : : :

2’ 0 5 .. .................... Prsssntthsorx ................... ; .......... ; .......... ; .................. _

' dominant mode only : - - ;_ _

dominant + 3 higher-order modes j—
dominant + 8 higher—order modes: ++++ i
03 1 1 1 1 1 1 1 1

 

I
7.5 8 8.5 11 11.5 12 12.5

Fgresquency0( GH129'5

Figure 3.3(a) Input conductances of a waveguide probe (a = 0.4 in, b = 0.9 in)
as a function of frequency when the probe is opening onto free space. The comparisons
are made between our numerical results and theoretical of Baudrand et al. [30] and
experimental results of Bodnar et al. [24].

80

 

0-5 I I I I I I I I T

0.45- ....... . ......... ; ......... ; ........ ..... ......... .......... ......... .......... ; ........ _.

 

 

 

 

8
g 0.35 .. O O ........................... ..
‘5.
0)
U
S
‘3 03.. ................ Exper1mentbyBodnaretaloooo ............... _1
O
N .' i I ,
‘28 025? iTheorybyBaudrandetal xxxx ....J
E ' (15 modes) . : : -
o .
Z Present theory . ; . _ _
0.2—. dominantmodeonlyn --_ ....... ...-
dominant + 3 higher-order modes 3 ——: -
015 7 - dominant + 8 higher-order modes ++++ 7
0.1 l l l l l l J i l
7.5 8 8.5 9 10 10. 5 11 11.5 12 12.5

Frequency ( GHz)

Figure 3.3(b) Input susceptances of a waveguide probe (a = 0.4 in, b = 0.9 in)
as a function of frequency when the probe is opening onto free space. The comparisons
are made between our numerical results and theoretical of Baudrand et al. [30] and
experimental results of Bodnar et al. [24].

CHAPTER 4

OPEN -ENDED RECTANGULAR WAVEGUIDE
PROBE TO MEASURE EM PROPERTIES OF
ANISOTROPIC MATERIALS

4.1. Introduction

In Chapter 3 we have studied the forward procedure of the nondestructive
measurement of material parameters using a waveguide probe system. However, it is
limited to isotropic materials because the EM fields inside the material layer are expressed

in terms of Hertzian potentials which are only suitable in an isotropic medium.

In this chapter we will employ the similar procedure to conduct the forward procedure
for measuring the parameters of anisotropic materials using the transverse field method to
express the EM fields inside the material layer. The EM fields derived with the transverse
field method were presented in section 2.3. where a general case dealt with the fields
inside an anisotropic medium and a degenerate case with the fields inside an isotropic
medium. As presented in Chapter 3, after matching the boundary conditions at the
discontinuity interfaces, two coupled electric field integral equations (EFIE’s) for the
aperture electric field can be obtained. Similarly, this chapter will include the derivation

of the EFIE’s, as well as the numerical solutions to these EFIE’s. Also the numerical

81

82

examples illustrated in the preceding chapter will be recalculated and compared with the

existing results to validate this technique.

The derivation of the coupled EFIE’s for the aperture electric field is first presented in
section 4.2.. After the EFIE’s are derived, the reflection coefficient or other quantities of
interest such as the input impedance or admittance of the waveguide probe can then be
determined to be a function of the aperture electric field. In section 4.3. the method of
moments is implemented to convert the EFIE’s to a matrix equation. In that section, the
elements of the matrix are evaluated in detail. Finally, some numerical results are
compared to the published results by other workers to verify the accuracy of the
technique. The numerical results on the input admittances of the waveguide probe

attached to a layer of assumed known anisotropic material are also presented.

4.2. Coupled Integral Equations for Aperture Electric Field

Two coupled EFIE’s via matching the tangential electric and magnetic fields at
discontinuity interfaces are derived in this section. The EM fields inside the waveguide
are given in the previous chapter, while the EM fields inside the material layer and the
open free space are established based on the transverse field method as described in

section 2.3..

4.2.1 Matching Boundary Conditions at z=t

Let’s assume that the waveguide probe is placed against a layer of anisotrOpic material
which is backed by open free space and has a tensor permittivity 1:: and the free-space

permeability 110.

Recalling from section 2.3., the spectrum-domain transverse EM fields inside an

anisotropic medium can be expressed as

83

I. 714:

-A) A9 123A) 32A)
— lvle + 2v2e + 31238 + 4V4€ (4.2.1)

 

 

+ 341/40 8 (4..22)

 

 

. 9 9 .
where the e1genvectors, vn and v are g1ven as

ma,

V V

nl mla

1,2, 3,4
2,4

vn2 9 = vm2a n
m
vm3a

cw
ll
v:

(4.2.3)

vn3

 

 

 

 

vn4 Lvm4a_

b .1

and A1 , A2 , A 3 , A 4, 32 and B4 represent the unknown amplitudes of eigenmodes for

both regions. Notice that these unknown quantities are all functions of kx and ky.

At 2 = t, the spectrum-domain transverse fields satisfy

13:,(kx, ky, t) = i5m(kx, k), 1) (4.2.4)
Ey(kx, k), t) = Eya(kx,ky,t) (4.2.5)
nofixwx, k), 1) = noilxa(kx,ky,t) (4.2.6)

110171), (kx, k), t) = no”)... (kx, k), 1) (4.2.7)

84

Substitute eq. (4.2.3) into eqs. (4.2. 1) and (4.2.2) and define the following notations for the

field components as

A'1’ ll!
’11="119 ’ ’21="12"
2.21 2.21
’12="21€ ' ’22=V228
2.31 2.31
'13="31" ' ’23="32"
2.41 2.41
r14=v4le ' ’24="4ze
’155V21a' ’255V22a
’10 E V4101: ’26 E V420 (4'2'8)
Alt 1.11
’31="13e ' ’41="14"
2.21 1.21
’32 = ”233 ' ’42 = ”24"
2.31 A31
’33 = V33e ' ’43 = ”348
A41 7141
r 35 E ”2322’ r 45 E V240
’305V43a’ ’405V440

Then eqs. (4.2.4) to (4.2.7) can be rewritten as

Air11 +A2r12+A3r13 +A4r14 = Bzr15 +B4r16 (4.2.9)
A1rz1+A2"22 -1-A3r23 +A4r24 = 32'25 + 34’26 (4.2.10)
Alr31+AZr32-1-A3r33+A4r34 = Bzr35-1-B4r36 (4.2.11)

Alr“ +A2r42 ~1-A3r43 + A4r44 = Bzr45 + B4r46 (4.2.12)

85

If we transfer all the terms with unknown amplitudes Al and A2 on the left-hand side of
the equations to the right-hand side and transfer the terms with B2 and B 4 on the right-

hand side to the left-hand side of the equations, we have

A3rl3 ~1-A4rI4 — Bzr15 - B4r16 = —A]rH—A2r12 (4.2.13)
A3r23 +1114r24 —Bzr25 —B4r26 = —A]r21—A2r22 (4.2.14)
A3r33 -+-A4r34--Bzr35 —B4r36 = —A1r3l—A2r32 (4.2.15)
A3r43 +A4r44 — Bzr45 —B4r46 = —A]r4l —A?_r42 (4.2.16)

To eliminate the unknown quantity 32 (kx, ky) first, subtract eq. (4.2.13) multiplied

by r45 from eq. (4.2.16) multiplied by r15:
A3(’15’43"13’45) +A4(’15r44"14r45) +B4(’16’45‘r15’46)
= A1(’11’45"15’41)+A2(’12’45"15’42) (42-17)
Subtracting eq. (4.2.14) multiplied by r45 from eq. (4.2.16) multiplied by r25 yields
A3(’25’43"23’45) +A4(’25’44“’24’45) +B4(’2csr45"’25r40)
= A1(’21’45“r25’41) +"122(r22’45‘r25’42) (42-18)
Subtracting eq. (4.2.15) multiplied by r45 from eq. (4.2.16) multiplied by r35 yields
A3('35r43"33’45) +A4(’3sr44"r34"45) +194(r36’45"35'40)
= A1(’31"45‘r35’41)+A2(’32’45-’35r42) (42-19)

By defining the following notations

'05’15’43"13’453 ’15r21’45"25r41
’05’15’44‘fi4’45’ ’j5'22’45"’25’42

’c5’10’45"15’40' 'k5’35’43"33'45

86

’d5’11r45"15’41’ ’15’35’44"34’45 (42:20)
re5’12’45‘r15’4r 'mE’30r45‘r35’40
75’25’43‘53'45' rnar31’45"35’41
’g5’25’44"24’45' '05’32’45"35r42
'h5’20’45"25’40'
eqs. (4.2.17) to (4.2.19) are simplified to
A3ra+A4rb+B4rc = Alrd-t-Azre (4.2.21)
A3rf+A4rg+B4rh = Alri-t-Azrj (4.2.22)
A3rk+A4rl+B4rm = Alrn +A2r0 (4.2.23)
Similarly, B 4 (kx, ky) can be eliminated from eqs. (4.2.21) to (4.2.23) to give
A3 (rarh—rcrf) +A4 (rbrh—rcrg) = A1 (rdrh—rcri) +A2 (rerh—rcrj) (4.2.24)
A3 (rarm—rcrk) + A4 (rbrm—rcrl) = A1 (rdrm—rcrn) + A2 (rerm—rcro) (4.2.25)
Let us define
rp E rarh—rcrf, rv a rarm—rcrk
rq E rbrh—rcrg, rw E rbrm—rcrI (4.2.26)
rr E rdrh_rcri ’ rx 5 rdrm_rcrn
ru 5 rerh—rcrj, r), E rerm—rcro
and rewrite eqs. (4.2.24) and (4.2.25) as a form of
A3rp+A4rq = Alrr+A2ru (4.2.27)

A3rv + A4rw = A 1 rJr + Azr), (4.2.28)

87

Now, using eqs. (4.2.27) and (4.2.28), two unknown quantities A3(kx,k\,) and
A4(kx,ky) canbeexpressed in terms ofA1 (kx, ky) and A2(kx,k),) as follows:

rr —rr rr —rr,

A3(kx,kv) = A][J—w——q5J+A2[—“—W—q—’]EAlrl+Azr2 (4.2.29)
- rr —rr rr —rr
P W 4 V P W q V
rr —rr rr,—rr

214(k), k\,) = A,[l’—¥J+42[L‘——L¥)541r3+42r4 (4.2.30)
. rr —rr rr —rr
p w q v p w q v

Substituting eqs. (4.2.29) and (4.2.30) back into eq. (4.2.1) yields

21,2 2» A

9 9 z 9 l: 9 NZ 9 _z 9 712
=Al(v]e +r1v3e3+r3v4ei)+A2(vze +r2v3ei+r4v4e4) (4.2.31)

 

 

It is important to note that to derive these formulas for numerical calculations, it needs
to avoid creating singularities during the process of elimination of a term from a set of
equations. Once a singularity occurs in the elimination process, an alternative process
needs to be found. Since the process of choosing an alternative elimination is

straightforward and explicit, it will be considered directly in the computer program.

4.2.2 Matching Boundary Conditions at Waveguide Aperture

Equation (4.2.31) indicates that the spectrum-domain transverse fields inside a
material layer are expressed in terms of two unknown quantities only. If the aperture
fields are studied (i.e. at z = 0) and the electric field is chosen as the reference, the two
unknown quantities can be represented as functions of the aperture electric field first.
Then the aperture magnetic field at the material side can be expressed in terms of the
aperture electric field in terms of these two unknown quantities. Recalling from section

3.2.2 that the aperture magnetic field at the waveguide side can also be expressed in terms

88

of the aperture electric field. As a result, two coupled EFIE’s for the aperture electric field

can be derived after matching the tangential magnetic fields at the waveguide aperture.

By eq. (4.2.8), eigenvectors 31 , 32 , 1’23 and in; can be rewritten as

 

 

 

 

’11 "12 ’13 ’14
9 r —A1 9 r A! r -At 9 r A:
v1: 218', v2: 2281' 33: 23123, v4: 2463
’31 ’32 ’33 ’34
I41. .’42. 1’43. I44.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

' ~ ' f- - - - — - \
x ’11 ’13 ’14
. f A(z-1) r A (z—t) r -A (z—I)
i =Al 21el +r123e3 +r3 24c 3
11on ’31 ’33 ’34
" r r
31011,; ( _ 41_ _ 43_ _’44_ )
(- 1 r - r - )
’12 ’13 ’14
r —A (2—1) r A (Z-I) r -A (z—t)
+A2 22 e ' +r2 23 e 3 +r4 24 e 3 (4.2.32)
’32 ’33 ’34
r r r
(_42 -43. _44 }

 

 

 

 

 

 

 

Let’s write out the spectrum-domain aperture electric field components (at z = 0) as

- -A,t -A31 A3
Ex(kx,ky,0) = A1[r”e +’1’13e +r3rl4e A]
I: All -A3t Ag]
+ A2 rue + r2r13e + r4r14e (4.2.33)
.. -A,r -A31 A3
E), (kx, ky, O) = A1 [r216 + r1r23e + r3r24e i]
A1! -A31 A3
+ A2 [’2ze + r2r23e + r4r24e {:1 (4.2.34)

89

If we define

-A 1 A31

-A,t 3
C1(kx,k),) Erne +r1r13e +r3r14e

—A 1 A31

"1' 3
C2 (kx, ky) = ’1ze + r2r13e + r4r14e

—A 1 A31

"‘1’ 3
C3 (kx, ky) 5 r216 + r1r23e + r3r24e

—A 1 A31

A1’ 3
C4 (kx, ky) :—: r2263 + r2r23e + r4r24e

(4.2.35)

(4.2.36)

(4.2.37)

(4.2.38)

and employ the definition of the inverse Fourier transform of the aperture fields, eqs.

(4.2.33) and (4.2.34) become

00

JIEx(x,y,O)e

—oo

._ij —jk\')'
e ' dxdy

—jk{t —jk\'y
e ' dxdy

I] E), (x, y, 0) e

Al-C1+A2-C2

A].C3+Az-C4

(4.2.39)

(4.2.40)

As mentioned in the preceding chapter, since the aperture electric field components,

I

E 0 = E x (x, y, 0) and Eva = By (x, y, 0) , only exist over the aperture region 0<x<a,

0<y<b, and vanish on the flange, the double infinite integrals can be replaced by the two

double finite integrals as

10

b0 —jkxx -jk‘.y
“E e e 'dxdy =41-C14142-C2

5“ —'k -'k,
I[‘[Eyoe J Ixe J "ydxdy = A1 - C3+A2 - C4

Let’s denote

(4.2.41)

(4.2.42)

90

E. e e " dxdy (4.2.43)

’7“ :1. -—'k,‘
”5):, 5 ME“; I ’re 1 ’idxdy (4,244)

and substitute back into eqs. (4.2.41) and (4.2.42) to solve for the unknown quantities

A1 (kx, ky) and A2 (kx, ky) . We have

 

A1(kx,ky) C1C41C2C3[C4HE:0— C2] [15:0] (4.2.45)

 

142(k), ky) C1C4‘1C2C3 [C3HE:0 — C1 I [12:0] (4.2.40)

Similarly, the aperture magnetic fields can be derived by defining

-Al! -A31 A31

Dl (kx, ky) .=_ r31e + rl r33e + r3 r34e (4.2.47)
All —A31 A31

D2 (kx, ky) E r32e + r2r33e + r4r34e (4.2.48)
—A,t —A3t A3:

D3 (kx, ky) E r41e + r1r43e + r3r44e (4.2.49)
Alt —A31 A31

D4 (kx, ky) ‘5 r42e + r2r43e + r4r44e (4.2.50)

and taking the two-dimensional inverse Fourier transform of the spectrum-domain

magnetic field components given in eq. (4.2.32) after substituting z = 0. We obtain

_ l )2 1 0° 'k‘x jky)’
Hx(x,y)L=O+ - (21: “0!”! [Al D1+A2 Dzle' e dkxdk), (4.2.51)
H — 1 2‘ .. A D A D e’“ jk’ydk dk
y(x,y) z=0*- 5;; figiul 1' 3+ 2' 4] e x y (4.2.52)

 

91

Substituting eqs. (4.2.45) and (4.2.46) into above two equations yields

 

 

 

 

 

sz=0, - 53ij e e [M9049 ky) j JEXO+N),(kx, ky) I may/.91.), (4.2.53)
H - "j .. 11,1 5" N k k " k k " d dk
y|z=0+ _ fijje e I: X( >” X)HEIO+MX( )" I).IIE)'0] kx .1' (4'2'54)
where
CD-CD CD—CD
M>'(k1’k>')5 (011 2C4C1 C3C2’ N>'(kx’ky)5 0011 2C1C2 CZCl
11100-1!) 1 4_ 2 3 jn0(2n) 1 4‘ 2 3
CD—CD CD—CD
11411491515 (0)1 203; C46}, M,(k,,k,1a “’11 20303 clc4
111007!) 1 4— 2 3 ' jn0(21t) 1 4’ 2 3

(4.2.55)
Recalling from section 3.2.2 that the aperture magnetic field components in the

waveguide side are derived as

H

x

 

_ 1 . 1tx . . 11x b“ .(nx') , ,
2:0 — wuoab[2bnm’lor10$m( a )—12F103m( a )“Eyosm a dx dy
4%- 410 11911111" 1

.1 8‘“ “1 '7;— ”Em

1'
+2.21
nm It

. . ("'2‘
“<31 ”(1.51511114411

 

 

 

n'm'
14114114.

where

92

 

(1'31 +(—“1°
a b
En'm E 1 2 1 '1 (4.2.58)
nn m1! “
84(7) ”4177)
E =baE c (m)sn(m—n—X)d d 4259
II 210-!!! 110 05 a 1 b x)’ (.. )
E _baE . n'1tx m'Tty dd
H yo: (I! yosm 7 cos 7 x y (4.2.60)
e-{2 "i=0 8 -{2 "1.: (4201)
"w n'¢0i mi— m'¢0 H

Since the tangential magnetic field components, H x and Hy, are continuous at the
aperture plane, 2 = 0 , the equality of eqs. (4.2.53) and (4.2.56), and that of eqs. (4.2.54)
and (4.2.57) will yield two coupled integral equations for the aperture electric fields. We

have

H

     

 

ly—

z=0

 

2an 1T0)” ”(Ea— ""HEIE>'0“"( 1")‘1x‘1’ 22m: "’11? 0X)

8412151081911911'9‘1114. 4413119111140]

+ f] 811.484.3[115 (kx, 1),) ”E; +Ny(kx, ky) “2,321,111: Csin(—ax) (42 02)

9

)9
z=0+

and for H’l . = H

 

93

22:1:.:.11,1ws1"2—?1sm1fi:—¥11 11319111113»191191114...)

+ f] ej 1,18,), [Nx(ky, kx) j j Ef0+Mx(k),, kx) j 1 5:0] dkxdk), = 0 (4.2.03)

 

where
C a —'2—(:-tcou1‘m (4.2.64)

Equations (4.2.62) and (4.2.63) are the two coupled integral equations for the aperture
electric field components, Em and Em. These EFIE’s are the key equations to be used

for further development.

4.3. Numerical Simulation

In this section the similar scheme used in section 3.3. will be employed to solve two
coupled EFIE’s for the aperture electric field. As mentioned before, the unknown aperture
electric field is expressed as a sum of incident and reflected dominant mode and a number

of higher order modes due to the discontinuity presented at the aperture.

In section 4.3.1, the method of moments is employed to convert the EFIE’s to a matrix
equation. This matrix equation is then solved by the Galerkin’s method using the
waveguide eigenmodes as the basis and testing functions . Section 4.3.2 develops
formulas which are needed in carrying out the numerical calculation of the matrix
equation transforming from rectangular coordinates into cylindrical coordinates. Finally,
section 4.3.3 presents numerical results compared to the published results to verify the

accuracy of this technique.

94

4.3.1 Application of Method of Moments

The coupled integral equations given in eqs. (4.2.62) and (4.2.63) for the unknown
aperture electric field components are solved by using the moment method technique. The
unknown aperture electric field components are first expanded into a set of appropriately
chosen basis functions {13‘3 (x, y) } . Since the aperture fields of a waveguide can be
expressed as a sum of a dominant mode and a number of higher order modes, the
appropriate basis functions are the eigenmodes of the waveguide. Let us expand the
unknown aperture electric field components into two finite sums of eigenmodes of the

waveguide as follows:

Exo (x’y) = 20581; (x’-’) (4.3.1)
13

E ,,.(x 1’) = 21936 (3(x .1) (4.3.2)

where 2 stands for 22 for summation of all possible higher order modes which
13 p 9
have mode indices [p, q] with p,q = 1, 2, 3, The set of basis functions are

expressed as

P7” 470’
eB(x, y) =cos[— a ]sin[— b :l (4.3.3)
eb(x, y) = sin [p_:1t_x ]cos [(12.1] (4.3.4)

Substituting eqs. (4.3.1) and (4.3.2) into the coupled EFIE’s yields

gafl ' [Wdrx + Weir] + ébfl . [Wdyx + We);

I
O

(4.3.5)

Csin(1—;5) (4.3.6)

ZBZaB-[W‘g,+w 3+2), [WNW]

95

where

x_ ‘48,)", u (n'nx) . (m'ny) 2 (m'n)2
de=ggabrn.m.(l—l—0)COS -—a-- sm —b— [ko— —b— :|Hex (4.3.7)

 

WC:EJ°°Je ’k" e’k’N M(k),k)(”e jdk dk (4.3.8)

 

k (A) - (Mr—(M)
de_;;abrn'm(uo)cos a Sin b a b He) (4.3.9)

exajje j k} ejk )M x(kvk “DUI yex)dk dk (4.3.10)

 

w.:- 223?: ,6 —)sin("—-':.”)cos('-"—;?)("%‘)('"Ti")m
5‘1“] ejkxxejkyyMy (kx, ky) (J‘J'exejdkxdky (4.112)

2r,

b0 1
y _ y . ). - fix . .
Wdy_ = fill iojsin(1%—x)uefi (x, ) ) s1n(—a )dx dy
-4€,,om» u . n'nx m'1ty 2 n'n
+§§abr,.m.(fo)sm(7)°°s( b )[k0‘(7fl”ey ‘43-”)

we); a f] ej 1,. ej "~""Ny (kx, ky) ( j ] e;)dkxdky (4.3.14)

 

 

and

 

b0 1 v I I
= x , , n TEX . m 1ty . .
”ex—MeB (x, y) cos(—a )sm( b )dx dy (4.3.15)

96

 

b0 v v v v
. , , . n 1tx m It . ,
‘UeyEIUeé (x,y) sm(-—a—)cos( byjdxdy (4.3.16)
b0 _‘k . -.k, .'
”(2:5ue; (x', y') e 1 ’xe } ") dx'dy' (4.3.17)
ba . —'k . -'k y.
”efsueg (x', y') e I ‘xe j "" dx'dy' (4.3.18)

The EFIE’s for the aperture electric field are now expanded into the waveguide
eigenmodes with unknown expansion coefficients aB and b‘3 which represent the
amplitudes of the eigenmodes for the x and y components of the aperture electric field,
respectively. Next, we will use the Galerkin’s method to determine aB and bB' Since the
Galerkin’s method uses the same set of basis functions as the testing functions, we write

the set of testing functions as

x _ lnx . my

ta (1:, y) = C08 [7] 8111 [7] (4.3.19)
y _ . lnx my

(a (x, y) = $111 [7] C08 [7] (4.3.20)

After taking the inner products of eqs. (4.3.5) and (4.3.6) with the set of testing functions,

we have the following results.

1 yy

which can be represented in a matrix form of

97

DIX nya
043 0:13 [:B]_ [P9] (4.3.23)
D GB HD GB [3 a
where
xx ba
DGBE Mt; (x, y) [Wj (x, y) + W" (x, y)]dxdy (4.3.24)
X) ba x v )'
DGB aura (x, y) [Wéx (x, y) + ch (x, y)]dxdy (4.3.25)
VI b0 X
Daflgut: (x, y) [W(;: (x, y) + WC),(x,y)]dxdy (4.3.26)
vy ba y
Dag silt; (x, y) [W3 x,y) + WC'),(x,y)]dxdy (4.3.27)
and
ba x
F HEC‘MI (x, y) sin(— a )dxdy (4.3.28)

4.3.2 Evaluation of Matrix Elements -

In this section, elements of the matrix will be evaluated into compact forms to
facilitate the computer programing. Among the evaluations in this section, two-
dimensional Fourier transforms of the sinusoidal functions over the aperture are evaluated

analytically in Appendix.

98

In the matrix, the component Dig which represents the self-interaction of the y-
component of the aperture electric field is first discussed. Let us divide 03% into three

parts as follow:

ba
Dégsglt; (x, y) I: (x y) + W: (x y)]dxdy
-_- ¢;'>'+¢;>'+¢;>' (43.29)

where (17:), represents the contribution coming from the material layer which has
continuous spectrum property, (1):) , represents the contribution from discrete higher order
modes of the waveguide and (1%y represents the contribution from the incident dominant

mode. They are expressed as

(1)” _ be t), ( .v
1 =‘M a x, y) ch (x, y) dxdy
1": (x, y) [f] ejkxxejk‘yNy (kx, ky) (JJe)€)dkxdk),]dxdy
:j N (kx, k )( ”z 5)( ”(3:)dkxdk), (4.3.30)

II
[._‘8 d—.Q"‘
ha

 

(DHEII (x ”(z-2M1“: (11%)“ "(T )C°S(TEQ)["0 “(gflllegdxcb
=§3§abr:: it“ 0sz vigilllW (3)) (4.3.31)

 

(I);y 5 fit: (x, y) [%Ti)sin(?)fieg(xfi y') sin(%x—')dx'dy']dxdy
= —a—b ”ll—0X3}; (x, y) sin(1%)dxdy][:’:ieg(x', y') sin(1-:l£)dx'dy']

99
{[sinIEa— [llx] cos[ [J] sin(T-E£)dxdy]x
b a
[ban pnx qny nx'
(ti [— )4— l - Hm»
a b a
21“I 11 ab
= 7? (7,)[251.15r.ol[2 541540]

ab p.
= 2 —(p—O)I‘108, 152 05,, 18,] 0 (4.3.32)

2r, 11 b
‘44.)!

where some notations are expressed as

{13in [112x] [cos (kxx) +jsin (kxx) ] dx] x

(icos[% —yy][cos(k y)+jsin(k),y)]dy]

E [311(k) +512(kx)] [513(k),) +s]4(k),)] (4.3.33)

b

y ”(1513‘ (x Y) Sin(n7n- x)cos(mzty)dxdy
b131n[]cos[r_1bty] sin(%)cos(m%)dxdy

a—bé. 5 . e . (4.3.34)

4 ml m,r m

 

a
51‘ (k) E gsin [LE-x] cos (kxx) dx (4.3.35)

0
512 (k) Ej! sin I??? sin (kxx) dx (4.3.36)

b
.913 (ky) sicos [$1 cos (kvy) dy

sly4(k) =jIcos[ 2:52 ]sin (k y)dy

100

Similar forms for I I e f and Hey are expressed as

ejk \"

JIe; s Iieé (x', y'e) — ... dx'dy'

=[H sin —']x'[cos(kx)-jsin(kx")]dx]x

[i::s] [LEV] [cos (kyy') —jsin (kyy') ] dy']

E [521(kx) +522 (kx)] [523 (ky) +524(k_)’)]

 

”eysfieg (x', y')sin( n.—::—x')cos(m1t

b y )dx'dy'

{1414232424144

$95.52

4 n',','pmqm

a v
521 (kx) a i sin [2%] cos (kxx') dx'

=_-"- m - . .
522(k) .. Jlsml: a ]sm (kxx)dx

nnx ) (
cos
a

m'n

 

b

y )dx'dy'

(4.3.37)

(4.3.38)

(4.3.39)

(4.3.40)

(4.3.41)

(4.3.42)

101

b .
523 (k) s leos [fig—v] cos (kvy') dy'

324(k)) E-jicos [Lby]sin (k y')dy'

If we further define
$1222“?E [511(kx)+312(kx)] [521082) +522(kx)]

¢12v(k),) E [513 (ky) +S]4(k).)] [52308) +324(k).)]

the multiplication of the notations given in eqs. (4.3.30) and (4.3.31) become

Ul’flUH} 5 (912.04) 412412).)

and

(1144112.) = (4)25.

Substiting them back into eqs. (4.3.30) and (4.3.31) gives

Y)’ 0°
<1) 1 = _U N),(kx, ky) 512“ (kx) ¢12v(ky) dkxdky

 

¢:Y=;§;Z§j:.(i)[3- (’% HOMO! )

=“”(1..)22"’"'"[("-kl 2.2;]5 Smanpfimq

(4.3.43)

(4.3.44)

(4.3.45)

(4.3.46)

(4.3.47)

(4.3.48)

(4.3.49)

(4.3.50)

Since it becomes highly oscillatory when the two-dimensional infinite spectral

integration given in eq. (4.3.49) is evaluated in the rectangular coordinates, this spectral

102

integration is computed in the cylindrical coordinates to sidestep this difficulty. Let’s

change the coordinates from rectangular to cylindrical with the relations of

kx kcosq)
. (4.3.51)
k), ks1n (p

where (p is a real variable with range of [0, 21:] and k is a real variable of [0, oo] . The

substitution of eq. (4.3.51) in eq. (4.3.49) yields

0027!
<19." =1 N,(k,<p>¢.2.(k,<p)¢12.(k.<p>kd<pdk

= l Gyy (k) kdk (4.3.52)

where

2n

G)... (k) a 1”)“, 4)) 412.414 (p) 4.2.414 <p)d<p (4.3.53)

Next we will discuss the component D23 which represents the self-interaction of the

x-component of the aperture electric field in the matrix. Divide Dz; into two parts as
follows:

ba
0;; a (111'; (x, y) [W51 (x, y) + W; (x. y) J dxdy
where

ba
4’,“ a 111'; (x, y) W; (x, y) dxdy

103

a

I; (x, y) [In] ejkxxejknyx (k), k x) (”ejadkxdkyjdxdy

j Nx (k), kx) ( ”’31 ”(2:)dkxdk), (4.3.55)

—-00

I
1

 

 

= z. .;:§:1:.(.%.)[4-("2‘“i141144112)

and the shorthand notations are expressed as

Icos [1%1 [cos (kxx) +jsin (kxx) ] dx] x
b . my . .
[£s1n[7] [cos (kyy) +Jsm (kyy) ] dy]

a 1.23, 4k.) + s32 4k.) 1 [s33 (k,.) + s34 (15,11 (4.3.57)

”Ix 5 fit; (x, y) cos(’%£)sin(n—1?)dxdy

.115 - :12 (wj-(m_'n_y)
cos[a]s1n[b]cos a 5m b dxdy

. 8 . 2. (4.3.58)

I II

a
53, (k) a l cos [1%] cos (kxx) dx (4.3.59)

104

532 (k) Ejlcos [£13] sin (ktx) dx
a -

 

 

(4.3.60)
1’ my
533 (ky) a {sin [T] cos (kyy) dy (4.3.61)
b m
534(k ). = j‘gsin[ by] sin (k),y)dy (4.3.62)
Similar forms for ”e: and ”ex are expressed as
b“ rte-1K)" , ,
”e: E (“6506“, ,ey') ' dxdy
[ cos ][cos(kx’) —jsin(kx’)]dx']x
[Eisin[— 1%] [cos (kyy') -jsin (kvy') ] dy']
E [541(kx) +S4Z(kx)] [543(ky) +544(k),)] (4.3.63)
b“ x , , n'nx' . m'ny' , ,
J‘J'exauefluy) cos Tjsm b )dx dy
b0 I o o 1 1
= ” ' — (—) (m )
tMCOSI: ]s1n[ b ]cos b d dy
b
= 9:522',p5m'.q5n' (4.3.64)
0
s41 (kx) :gcos [p ]cos (k x) dx (4.3.65)
342 (kx) _ —} l cos [p :lsm (k x) dx (4.3.66)

105

b .
343 (k\,) 2 {sin [g] cos (kyy') dy' (4.3.67)
b V
544(k ) =—j£sin[q—'b ]sin (k y")dy (4.3.68)

If we further define the following notations,
¢34u(kx) '5 [331(kx) +332 (kx)] [841(kx) +s42(kx)] (4.3.69)

$34,,(k),) E [533 (k),) +534(ky)] [s43 (ky) +344(k),)] , (4.3.70)

the multiplication of the shorthand notations given in eqs. (4.3.55) and (4.3.56) become

”PUMPS E ‘1’3412 (kx) ¢34v (ky) (4.3.71)

and

(115)11163) = (eszan'. 15mg 25,13,223»); (,5: (4.3.72)

Substituting them back into eqs. (4.3.55) and (4.3.56) gives

ch,“ = J [ Nx(ky,kx)¢34u(kx)(1)34v(ky)dkxdky (4.3.73)

"‘ £2323" (5.)[3- ("'— 74114411 )

b
ab n"m 8'n "1
=—(:O)Zzif;[(%j"(14]5'15m'r5nm5m4 (4.3.74)

 

As mentioned before, changing the coordinates from rectangular to cylindrical, the

two-dimensional infinite spectral integration becomes

106

002“
chi" = [ N.(k, (p) 434,414 45434.44 4) kdcpdk

(4.3.75)

00

= 16”“) kdk

where

21:
Gxx (k) E le (k, (P) (1)34“ (k, (P) ¢34v (k, ‘9) 61¢ (4.3.76)

Next, let’s evaluate the component D3 which represents the mutual-interaction of the
x-component of the aperture electric field as the testing function and the y-component of

the aperture electric field as the basis function in the matrix. Divide D :g into two parts as

follows:

ba , .
0.; a b)": (x. y) [m‘, (x, y) + WC". (x, y)]dxdy
= ¢T>'+¢:Y (4.3.77)
where
xy ba x y
(1)] Elli“ (x9 Y) ch (x, y) dxdy
ba x 0° jkxx J'ky)’ e
= [E {20, (x, y) j j e e M(ky,kx) Hey dkxdky dxdy
(4.3.78)

= £1w.4415:)(114544

b
«>5 2 [1436, y) W}. (x, y) dxdy

n. m,.44::(3J...('zf_4)5m("ig_)(_)('1b£)JJ J...
‘ 5543?:1:(5X“)(’57(Hr )(n )

According to eqs. (4.3.39) and (4.3.57), we obtain

 

 

 

 

(114411454344.43.4..)
with new shorthand notations defined as
432,, (k ) -=- [53, (k.) + s32 (k.)] [s21 (k.) + s.. (k.)] (4.3.81)
¢32v(k ) 2 [533 (k),) +334(k),)] [523 (ky) +sz4(ky)] (4.3.82)
and according to eqs. (4.3.40) and (4.3.58), we have
(Mme) = (3})25,‘,5m.,5n.,p5m.‘qem.en. (4......
Substituting them back into eqs. (4.3.78) and (4.3.79) yields
51>? = D Mx (k), k) 532,, (kx) ¢32. (ky) dkxdk) (4.3.34)
33’ = 22.122'.:(5)(’5‘)(5‘ (1144114)

Like before changing the coordinates from rectangular to cylindrical, (I), g1ven 1n eq

(4.3.84) becomes

«.21:
“ = J M.(k. 4)) 432.04 (p) 4.2.04 (p) kdcpdk

108

= 1G“ (k) kdk (4.3.86)
where
21:
G... (k) a J M. (k. 4) 432.414 4) 432.414 <4) d<4 (4.3.87)

The last step is to evaluate the component D “,3 which represents the mutual-
interaction of the y-component of aperture electric field as the testing function and the x-

component of aperture electric field as the basis function 1n the matrix. Let’s divide D: (1,3

into two parts as follows:

ba
DHMIE (.., 5) [W3 (4 y) +W.§.(x.y)]dxdy

YX

<14 + (142 (4.3.88)

......
5;" a fit; (4:. 5) W5; (.., y) 4444

1.; (.. 5) m .1’5."“14'-"My(4x,4y5(j).;)..x..J....

:m 444 4114M .74 ..

ll
lc._,8 c.53-

 

d—‘Ql
d—gfi
N
Q ‘~<
A
H
‘<
v
:—\
a
G"
,‘,‘—1:
3%
{1.5
22.
5
h
|:.
aI :1
H
\_/
O
O
m
/—\
l5.
°‘ :1
‘<
v
A
3
«‘4
\_/
A
o-l5
v
%
E
R.
H
a.
‘<

109

=2.2.5:.4—)4"—;:-‘)4'"."‘)414.)414.)

n m

 

Based on eqs. (4.3.33) and (4.3.63), we have

1115711185)544.444.4444

With new shorthand notations defined as
(1’14..("‘..) 5 [511084) +512(kx)] [541(kx) +S4Z(kx)]
(p14,, (1‘)) E 1513 (kv) '1' 514 (18)] [S43 (1‘)) '1' S44 (18)]

and with eqs. (4.3.34) and (4.3.64), we obtain

(144.4445) = (5)264. .. 4.1.4.. . ..

Substituting them back into eqs. (4.3.89) and (4.3.90) yields

<14" = J J M. (k. k.) 4... 4k.) 4... 4k.) .14....

22...— 54 —.)4"—:-‘)4'"5'")414.)4n4)

_ -44 4 Max:424)
— 4 (“Ojgg rn'm' a b 8n', 16m', ran', pfim', q

 

If we change the coordinates from rectangular to cylindrical as before, <19,“

(4.3.95) becomes

0021C
<14)" = J M,.4k.<4)4...4k.<4)<4,..4k.<4)kd<44k

(4.3.90)

(4.3.91)

(4.3.92)

(4.3.93)

(4.3.94)

(4.3.95)

(4.3.96)

in eq.

110

= J G” (k) kdk (4.3.97)
where
211
G... (k) a 1‘44"“ 4) <4... 4k. <4) 4....4k, <4)d<4 (4.3.98)

4.3.3 Numerical Results and Comparison with Existing Results

For a specific material layer, the numerical evaluation of the components of the matrix
equation given in eq. (4.3.23) for solving the aperture electric field and other relevant
quantities of the waveguide probe is performed in a FORTRAN computer program. Once
the matrix equation is solved, the aperture electric field is first determined by summing up
a finite number of modes that we have taken into account in the program. Then the
reflection coefficient or the input impedance of the waveguide probe is determined via the

aperture electric field.

To verify the accuracy of this technique, as well as the validity of the computer
program, numerical examples illustrated in the previous chapter are recalculated and

compared with the existing results.

Figures 4.1(a) and (b) show the real and imaginary components of the input
admittances of a waveguide probe when the probe is placed against a layer of quartz with
a dielectric constant of er = 3.76 and a thickness of 0.1299 inch. Our results are
compared to the theoretical and experimental results of Croswell et al. [22] where they
only considered a dominant TE ,0 mode in their theoretical calculation. These figures
show that our numerical results employed by two different techniques given in Chapters 3

and 4 compare quite well to each other. Also our numerical results used a dominant TE ,0

lll

 

4-5 1 ' 5 1 7

 

 

 

 

 

a) 3.5
U
G
<6
‘5
3
'2 3
o
0
'0
O)
.2!
75 2.5
2 Theory by Croswelletal. XXXX
2.. 5,(dominantrnodcpnly)....;.__..,.._.....@-. .. . -
Present theory : Previous technique .
Present technique 3 ****
,5. ............ .,...__.(domin_ant.+..3..highcr-prder. modes.) ..... ;- ............... -
1 4 . 1 1 1
7 8 12 13

10 1 1
Frequency ( GHz)

Figure 4.1(a) Input conductances of a waveguide probe (a = 0.4 in, b = 0.9 in)

as a function of frequency when the probe is placed against a material layer with a
dielectric constant of 8. = 3.76 and a thickness of 0.1299 in. The comparisons are
made between our numerical results and theoretical and experimental results of
Croswell et al. [22].

112

 

 

4-5 I 1 4 1 1
3 Experiment by Croswell et al. 3 0000
4.. ............... ....... . ...... ..
2 Theory by Croswell et al.; 2 XXXX
? (dominant? mode only) 3 3
3.5... ............. ........... ....... _ ..................... -
XXX : - Present theory : Prev1ous:techn1que
Xxxx _ Present technique ****
3 _ ............. o. . .x.>.<.x . _ . . _ . ;. (dominant .+_3. .higherrordermodes). .: ............... q
. Xxxx: : : :
00; Xx 1 ' 1

Normalized susceptance

 

 

 

 

0 11
Frequency ( GHz)

Figure 4.1(b) Input susceptances of a waveguide probe (a = 0.4 in, b = 0.9 in)
as a function of frequency when the probe is placed against a material layer with a
dielectric constant of Sr = 3.76 and a thickness of 0.1299 in. The comparisons are
made between our numerical results and theoretical and experimental results of
Croswell et al. [22].

113

mode plus three higher order modes (TE3O, TE12 and TM ,2) match very well with their

experimental results, much better than their theoretical results do.

Figures 4.2(a) and (b) show the real and imaginary components of the input
admittances of a waveguide probe which is placed against a material layer with a
dielectric constant er = 2.25 and a thickness of 0.3201 cm. As mentioned before, our
numerical results are obtained by using four modes in our theoretical calculation; and in
these figures, they compare quite good for both techniques we used. These figures also
show a good agreement between our numerical results and the theoretical and

experimental values of Bodnar et al. [24].

The last comparisons are shown in Figs. 4.3(a)-(b) which show the real and imaginary
components of the input admittances of a waveguide probe that is open to free space. Our
numerical results are compared to the theoretical results of Baudrand et al. [30] and the
experimental results of Bondar et al. [24]. Our results using four modes for both

techniques agree well to each other, as well as to the above published results.

Now if we use the waveguide probe to measure an assumed known anisotropic
material layer with a thickness of 0.11 inch and three principal permittivities of
a, = 5.4 -j0.3 , £2 = 5.8 —jO.4 and 83 = 3.8 -j1.7 , the theoretical input admittance at
the probe aperture can be obtained. Figure 4.4 shows the real and imaginary components
of the input admittance of a waveguide probe which is placed against the assumed known
anisotropic material layer at the orientation of 0 degree. In this figure, we considered three
cases; (a) with a dominant TE10 mode only, 6)) with a dominant TE10 mode plus one
higher order mode (TEZO) and (c) with a dominant TE ,0 mode and two higher order
modes ( T1520 and TE01). Since two multi-mode cases yield almost identical results over
the frequency range of 8 to 12 GHz, it is then obvious that a good convergence can be
obtained when the first two modes (TE10 and "’20) are employed in the numerical

calculation. It is also noted that the numerical results obtained by considering the

114

 

 

 

 

 

 

I I I I I fl I V I I
2_ .
1.8-
§1.6~
CB
2'5
5
'8
81.47 0
'c O
Q) . . . .
.E : ' : : i : . :
E12_ ....... ....... ExpenmentbyBodnareta] ....... 0000 ...... _
2° . Theory by Bodnar et a1. XXXX
1...“...4 ......... ........ (15modes) ....... ..
' ‘ Present theory Previous technique . i
_ . . Present technique ; ***’."
0.8—--~-; ~ (donunant+3higher-ordermodes) —
._. . . . . ; . . 1 . 1
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)

Figure 4.2(a) Input conductances of a waveguide probe (a = 0.4 in, b = 0.9 in)
as a function of frequency when the probe is placed against a material layer with a
dielectric constant of Sr = 2.25 and a thickness of 0.3201 cm. The comparisons are
made between our numerical results and theoretical and experimental results of Bodnar
et al. [24].

115

 

1.6 I I I I 1 I I I I I

1.4

—L
N

 

Normalized susceptance
a-L

 

0 8 ._. ...... Theory. .by. .B.O.d.na.r ét. 'a'l. .......................... xxxx ............................ _4
J (15 modes) ' '
Present theory: Previous technique _ . ;
0.6~~~~ - Presenttechnique ****-

(dom1nant + 3 higher-order modes)

 

 

1 1 l 1 l J L 1 1

 

0.4 4
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)

Figure 4.2(b) Input susceptances of a waveguide probe (a = 0.4 in, 'b = 0.9 in)
as a function of frequency when the probe is placed against a material layer with a
dielectric constant of 8, = 2.25 and a thickness of 0.3201 cm. The comparisons are
made between our numerical results and theoretical and experimental results of Bodnar
et al. [24].

116

 

0.9 ~

 

 

 

 

 

o 0.8 *- _
0
t:
a
8
:3
'8
80.7.. ....................... .................. -1
1: . .
g . ‘ Experiment by Bodnar et al. [5] Eoooo ‘
'5 : i :
E05- ........ ..... TheorybyBaudrandetal [4] ....... f .......... xxxx...f ........ _
° ' (15 modes) . : . :
z . .
Present theory: Previous technique 3
p _ . Present technique : g **** ._
0.5---~~ ' (dominant+3 higher-ordermodes)”w"';""" "‘
0.4 1' i ‘1 1 1 . ‘1 1 i 1
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)

Figure 4.3(a) Input conductances of a waveguide probe (a = 0.4 in, b :09 in)
as a function of frequency when the probe is open into free space. The comparisons are
made between our numerical results and theoretical results of Baudrand et al. [30] and
experimental results of Bodnar et al. [24].

117

 

 

 

 

 

 

0-5 I T 7 T ' I I I F
0.45_ ........ g .......... g .......... f .......... g .......... _
0.4 - -
8
g o 35 L— o 0 _
‘5.
O
U
m
a 0.3 ._ .................. __ ............................. a ................. ......... .................. _
'53) 5 Experiment by Bodnar et al. ; oooo
20.25 __ Theory by Baudrand et a]. .,.A . . . _. ixxxx ..... ....... _.
‘25 q (15 modes) _ i i 3
_ ' Present theory: Previous technique ' .... . ; . q _ _ q ‘1. _ . ' ’ _
0'2 ’ Present technique ' 3' *f'*** ‘ ‘ ‘ '
(dominant + 3 higher-order modes) ‘ '
0.15 ._ ............................................................................................ _.
0.1 4 L l 1 l l l l l
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5

Frequency ( GHz)

Figure 4.3(b) Input susceptances of a waveguide probe (a = 0.4 in, b = 0.9 in)
as a function of frequency when the probe is open into free space. The comparisons are
made between our numerical results and theoretical results of Baudrand et al. [30] and
experimental results of Bodnar et a]. [24].

118

 

 

 

 

 

 

 

 

I I I I T I I I I
3
_ , , 6 4L Q ,. A
5.4 ’6’ \Q‘e
0,9
s.2~ -- v ’ ~
8 a’
5 9’
H 51- ‘ / .1 -1
8 4.8* ~ ,’d -
U o
'8 : / .
,E 4.6.. ’9' . .1
7g . a .
$— 4.4" f d, . ~ ~ » .1
2 ; , dominant mode only 1 - - -
I ' .' I I
4.2- ,9 dominant + 1 higher order 000 -
6 ' : . ; 1 . 3 ; .
_ a’ . 1 dominant + 2 higher order ***
._ /.. . 1 1 ..... ,_ . ., ..... _.
3.8 l l 1 l l l 1 l 1
7.5 a 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)
(21)
4 I I I I I I I I I
61‘ V
3.5— - ‘,‘- -
G .
3» ‘a ~
‘0\ p
8 U1
5 2.5~ 0‘ _
‘5. ‘K
0 O
8 2" \Q . _.
:1 \ 1
m 0
ID \
0 1.5_ ”3.0 .. .1
£3 ‘0‘ .
a .
E 1- . a . BN1. _
o . , ‘ : ; . ‘0‘ Z
Z dominant mode only ~ - - _- K
0.5.... ..im ..... .-.‘. ......... ... ...... -
dominant + 1 higher order 000 . j‘q
. , I I I ’ ' \ .
o _ dominant +.2.higher.order; a **f*. . . ..... 1,. ..gxé ..... _
-0.5 l 1 L l l l l L l
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)
0))

Figure 4.4

Input admittance of a waveguide probe (a = 0.4 in, b = 0.9 in) as a

function of frequency when the probe is placed against an assumed known anisotropic

material layer with a thickness of 0.11 inch and three principal permittivities of
a] = 5.4-103,82 = 5.8—j0.4 and a3 = 3.8—j1.7.

119

dominant mode only in the theoretical calculation match very close with the results using
the multi-mode cases. This indicates that the aperture electric field is dominated by the
dominant TE ,0 mode and when the waveguide probe is attached to an anisotropic
material layer, higher order modes contribute insignificantly to the electric field at the

input admittance of the waveguide probe.

To further verify this observation, we use the waveguide probe to measure another
assumed known anisotropic material layer which has higher permittivity and conductivity
in the principal direction perpendicular to the waveguide aperture. Figure 4.4 shows the
real and imaginary components of the input admittance of a waveguide probe which is
placed against a layer of anisotropic material, with a thickness of 0.053 inch and three
principal permittivities of £1 = 5.4 -j5.3, £2 = 5.8 —j6.4 and £3 = 30.8—j100.5, at
the orientation of 0 degree. As observed earlier, our numerical results obtained by using
the first two and three modes in the theoretical calculation are almost identical, while the
theoretical results obtained by using only the dominant mode match very well with those
of multi-mode cases. Therefore, using a dominant TE 10 mode plus one higher order
mode TE20 in the numerical calculation, we can expect a good convergence on the probe
input admittance; even using a dominant TE ,0 mode only, good results can be obtained
over the major portion of the frequency range. As a result, in the interest of saving
computation time, only the dominant TE 1 0 mode will be assumed for the aperture field if
the waveguide probe is attached to an anisotropic material layer. This approximation will
be used in the next chapter to inversely determine the measured EM parameters of

anisotropic material layers.

In this chapter, we established a forward procedure for the theoretical study of
measuring EM parameters of anisotropic materials using an open-ended waveguide probe
system. The accuracy of the technique and the validity of the computer program have

been verified by comparing our numerical results with the published results. The

120

 

 

 

 

 

 

 

 

5 I I I I I I I I I
4,9_ ..dommantmodcqnly,_ .  --...-, ‘ , .91. _
dominant + 1 higher order ; ooo . ’92 9
q) , ‘ ’, ; : : 6’ '
g 4.8-‘*-donnnant'+~2 higher-order“;~ 1“; ~ ' 6/ ‘- ~
. f 1 /,
U /
g 4.7,. /¢/, ........... .1
S 9
o , :
B 46' C._ .. [é ........ .1
.9! -, p’
"a t l i
E 45- ......... $.. ..... .1
, /
o a
Z _’
44.. ..-./fl .. _
9 9 '
6. . .
4'3... ..... . fl ......... .1
‘a 0'6 .
4.2 1 1 1 1 1 1 1 1 1
7.5 a 8.5 9 9.5 10 1 .5 11 11.5 12 12.5
Frequency ( GHz)
(9)
2 I I I T T I I I I
a ' j . ' ' I
‘9" _ . i
1.6- 1 ‘11 : ~ . . < : -- -
\o ,
8 »‘ 2
51'4_ & ...... _1
a :‘
191.2- ' R .1
a n, -.
a 1.. .w\, .1. -
-c 9‘ _
a :‘9 E
:0.8.. ............... \. ,
cc ‘9‘ ‘ I
. , . \I .
E06,. ....... f ........ ' .......... ..... 6‘ _
g ' dominant mode only - - - ' 3
. . . . , , .
. . ; : . : ' t : Q'
0.4 .. ..... dominant+1hlgherorder ...... 000 ..... . . .. ”\Q ............ .1
..... dfiominantt.2ihighet9rder.. *** o. . ..
0.2 5 1 ..1 ..... ._ : : ,.\: . ._1
0 1 1 1 1 1 1 1 1 1
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)
0))

Figure 4.5 Input admittance of a waveguide probe ((1 = 0.4 in, b = 0.9 in) as a
function of frequency when the probe is placed against an assumed known anisotropic
material layer with a thickness of 0.053 inch and three principal permittivities of

£1 = 5.4—1'53, £2 = 5.8 — 115.4 and £3 = 30.8 -j100.5.

121

theoretical calculation on the input admittance of the waveguide probe placed against a
layer of assumed known anisotropic material was also presented and discussed in detail.
The experiments for measuring the reflection coefficient or the input admittance of the
waveguide probe and the inverse technique of this measurement method for determining

the EM parameters of the material layer will be discussed in the following chapter.

CHAPTER 5

EXPERIMENTS

5.1. Introduction

In this chapter we discuss the experiments for determining the EM parameters of
isotropic and anisotropic samples using an open-ended rectangular waveguide probe
system. The calibration procedures of this waveguide probe system and the measurement
procedures to obtain the complex permittivity e of isotropic and anisotropic materials are
explained. Also, the experimental input admittances of the waveguide probe when placed
against various material layers and their complex permittivities determined from the

inverse procedure are presented.

Section 5.2. presents, experimental setups and calibration procedures of the waveguide
probe system. In this section, two different calibration procedures for the waveguide
probe system are discussed in detail. Also the reactive characteristics property of the
adjustable shorting device used for the calibration procedure is studied. Section 5.3.
includes the experimental results of the input admittances of the waveguide probe placed
against various isotropic and anisotropic material layers. Section 5.4. develops an
inversion technique to determine the material parameters such as permittivity,

permeability and the thickness of the material, etc., using the Newton’s iterative method.

122

123

After the theoretical analysis of the parameter retrieval and technique descriptions, the
inverted results of the EM parameters for various material layers measured in the

preceding section are presented.

5.2. Experimental Setups and Calibration

A waveguide probe system for measuring the complex permittivity e of isotropic and
anisotropic materials has been constructed at MSU electromagnetics laboratory. The

waveguide probe system and associated equipment are schematically shown in Fig. 5.1.

The waveguide probe consists of an X-band open-ended rectangular waveguide with a
cross-sectional dimensions of O.4"x0.9" terminated on a 18"x18"x0.25" metallic flange. It
was experimentally found that the metallic flange is sufficient large to act as an infinite
plate as the theory assumes. An acrylic tank is also built to measure liquid materials. In
case the EM parameters of a liquid material are measured by this waveguide probe, a thin
Scotch 3M Mailing tape is taped on the aperture to prevent the leakage of the liquid into
the probe. It was also found that the effect due to this thin tape is neglible when we
measured the input impedance of the probe open into space with and without the tape on

the aperture.

An HP 87208 Network analyzer is connected to the waveguide probe and it excites a
dominant TE 10 mode of wave into the waveguide in the frequency range of 8 GHz to 12
GHz. The waveguide used in the system is 16" in length and it is sufficient long to permit
only the dominant TE 10 mode of wave to be reflected back to the network analyzer. That
is higher-order modes excited near the waveguide aperture have attenuated greatly and

will not reach to the network analyzer.

A layer of material to be measured is placed against the waveguide aperture and it is

backed by free space. The material layer is also required to be sufficiently large to reduce

124

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

metallic unknown
flange [Egan] rectangular

<— waveguide

 

 

 

 

[_17

r A

 
  
  

 

waveguide
adapter

 

HP 87 208

 

/\11
I/VV

 

 

 

 

 

 

network analyzer network analyzer cable

Figure 5.1 Experimental setup of a waveguide probe system to measure the EM
parameters of materials.

125

the possible edge effect of a layer of finite size. Due to the discontinuity presented at the
aperture, a part of the incident T510 mode of wave penetrates into the material layer and
the rest of it is reflected back to the waveguide. Additionally, higher-order waveguide
modes are excited near the probe aperture. The theoretical analysis of the EM fields at the
probe aperture was conducted in the preceding chapters and it showed that the reflected
dominant mode of wave at the aperture is dependent on the material parameters such as
permittivity, permeability and the thickness, etc.. Therefore, by measuring the reflection
coefficient of the dominant mode at the waveguide aperture, the material parameters can

be determined inversely.

The experimental reflection coefficient at the probe aperture is automatically measured
and recorded with an HP network analyzer. As shown in Fig. 5.2, the equivalent two-port
network between the probe aperture and the measurement reference plane of the network
analyzer can be represented as a equivalent network characterized by [S] matrix. The
scattering (S) parameters of the equivalent network can be determined via a calibration
procedure. In this section two different calibration procedures for this equivalent two-port
network are developed and will be discussed in the following subsections. With these S
parameters, the measured reflection coefficient at the reference plane of a network

analyzer can then be converted to the reflection coefficient at the probe aperture.

For a layer of anisotropic material with a diagonal form of the complex tensor
permittivity as given in eq. (2.3.3), there are three complex permittivities with respect to
the principal axes of the anisotropic material layer. Since the electric and magnetic fields
of the dominant mode inside the waveguide are orientated in specific directions, the tensor
EM parameters of an anisotropic material layer can then be determined by measuring the
reflection coefficient of the dominant mode at various orientations of the waveguide with
respect to a reference axis of the material layer. Thus, for this kind of anisotropic material

layer, it is required to make measurements of the probe input admittance at three or more

126

 

 

 

 

 

 

 

 

 

metallic
flange .
unknown
material
connector rectangular
waveguide
R 5 [S] R
r . a
{P
reference aperture
plane plane

Figure 5.2

Representation of the equivalent two-port network between the

waveguide aperture and the measurement reference plane of a network analyzer.

127

different orientations for inversely determine the three unknown quantities of the tensor
permittivity. In a serious of experiments for an anisotropic material layer, we measured
the reflection coefficient or the input admittance of the probe aperture at four different
angles, 0, 30,45 and 90 degrees, with respect to a reference axis of the material layer. If a
laminated composite material is concerned, the reference axis of the material layer is

chosen to be the axis parallel to the fibers in the first ply of the laminated composite.

5.2.1 Fixed-Stub Calibration

It is common practice to use combined calibration devices of short circuit, offset short
circuit, perfect matched load and open circuit to calibrate the equivalent network between
the actually measured plane of a material sample and the reference plane of a network
analyzer. Usually, these calibration devices are assumed to be lossless and their lengths

introduce only phase changes between the terminations[43].

For a waveguide probe system, the calibration scheme using all short circuits is most
convenient. The signal flow diagram of the calibration scheme using short circuit devices
is shown in Fig. 5.3. It is noted that R: , RE and R3 denote the measured reflection
coefficients at the terminal of the network analyzer with the offset short circuits connected
in the waveguide probe, while R11), R: and R: denote the respective theoretical reflection
coefficients at the probe base plane I = II . Since the shorting circuits are assumed to be

lossless, we can express the theoretical reflection coefficients as

l

Rb = —1 (5.2.1)
--2 z.—1

R: = —e 1 B1 ‘ ') (5.2.2)
—'2 1—1

R2 = —e I B( 3 ') (5.2.3)

128

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

reference aperture base
plane plane plane
R r [S] 1 Ra
1 i N O 1
R R
r l b
O Q—
1<——>
. 11
2 :' IO 2
Rr R b
a ’2
R3 1 1 3
r e i R b
1 ‘ i
5 i4 ‘ >
I z
1 3
l = 0
Figure 5.3 Signal flow diagram of the calibration scheme for the measurement sys-

tem.

129

2_n
).

 

. A
With kg =

1 1/1- (re/112'

According to [12], the equivalent network characterized by [S] matrix as shown in

where B =

Fig. 5.2 can convert the measured input impedance or reflection coefficient at the
reference plane of a network analyzer to the input impedanCe or reflection coefficient at
the probe aperture plane, or vice verse. The relation between the reflection coefficient R,
at the reference plane and that R a at the probe aperture plane 15 expressed as

Rr — S1 1

R = (5.2.4)
0 (Rr_ 511) S22 ‘1’ 512521

 

01'

S S R
R = 12 21 0+ 1]
’ I'SZZRa

(5.2.5)
This shows that only three scattering parameters S 11 , 522 and 512521 need to be

determined if both Rr and Ra are known.

With reference to Fig. 5.3, three offset short circuits are used to determine the S
parameters of the equivalent network. Using eqs. (5.2.4) and (5.2.5) for the three short

circuits, the S parameters of the equivalent network can determined as

[R3-R3)[R1‘.—RZ]-[Rl-R3)[R2-RZ)
522 = ' 3 RZ‘LIR] R2) 3 R1 R2XR3 szRl (5'2'6)
Rr- r b_ b Rb- r_ r b_ b b

3 2
R —R 3 2
Rb—Rb

130

3
3_ 512521Rb

511 = R 3.
l-SZZRb

(5.2.8)

Based on these S parameters, the reflection coefficient of the probe aperture Ra can be

transformed to be

'2 1
R = Rbe’ B1 (5.2.9)

0

where the reflection coefficient at the probe base plane R b is

= Rr-Sll
(Rr—Sll)S22+SIZSZI

 

Rb (5.2.10)

The easiest calibration scheme using three offset short circuits is to use three fixed
shorting stubs which can be attached tightly on the probe aperture. Since the operating
frequency covers the range of 8 GHz to 12 GHz, and in order to provide sufficient phase
differences over this frequency range, three shorting stubs are chosen to have 5 mm, 10
mm and 20 mm in length, respectively. When these shorting stubs are used, we use a short
circuit at the aperture and two offset short circuits with 5 mm and 10 mm in length to
calibrate the frequency range of 8 GHz to 10 GHz. On the other hand, a short circuit and
two offset short circuits with 10 mm and 20 mm in length are used to calibrate the
frequency range of 10 GHz to 12 GHz. That is, three shorting stubs with lengths
ll=0mm, [2:5 mm and 13:10 mm, or ll=0mm, 12: 10mm and
I3 = 20 mm are used to calibrate the equivalent network for the two frequency ranges.
It is noted that the reflection coefficient of the probe aperture in eq. (5.2.9) is now becomes
R a = R b since the probe base plane is chosen to be the same as the aperture plane in the
present arrangement. The calibration scheme discussed here is referred to the 'fixed-stub

calibration" since this calibration uses fixed shorting stubs.

131

The drawback of this fixed-stub calibration is the problem caused by poor contact
between the edges of the stubs and the inner surface of the waveguide. This may affect the
accuracy of the calibration greatly. In order to overcome this difficulty, an adjustable
shorter is used instead of fixed shorting stubs. This alternate scheme will be discussed in

the next section.

5.2.2 Adjustable Shorter Calibration

An adjustable shorter with a dumbbell-shaped loading in front of the termination
provides a reactive contact with the waveguide and ensures a very large standing-wave
ratio (SWR) in the waveguide. This allows the adjustable shorter to reduce the resistive
loss at the contact point with the waveguide and become an ideal short circuit device.
However, due to the reactive characteristics of the adjustable shorter, it is difficult to
determine the actual offset shorting length of the shorter directly by measuring the
distance from the probe aperture to the shorting termination inside the shorter. Therefore,
to use the adjustable shorter in the calibration, the offset shorting length of the shorter over

the frequency range of 8 GHz to 12 GHz needs to be found first.

The offset shorting length of the shorter for various frequencies can be found by
comparing the equivalent phase of the shorter to the phase of the shorting plate at the
aperture and they can be determined accurately by using a slotted-line measurement[44].
The simplified block diagram as shown in Fig. 5.4 illustrates the arrangement of the
instrumental setup in the slotted-line measurement. An HP 809B slotted line consists of a
section of waveguide into which a small antenna, or probe, can be introduced through a
slot. The probe extracts a small fraction of the power flowing in the waveguide, and is
connected to an HP 415E SWR meter. An HP 8620B sweep oscillator is used to generate
the dominant T1310 mode of wave inside the waveguide at the X-band frequencies. Also,

an HP X532A frequency meter is used to monitor the frequency. Between the oscillator

 

HP 8620B

Sweep Oscillator

 

 

 

I

132

 

 

Type 1203

Isolator

 

 

 

HP 415E
SWR Meter

 

 

 

I

 

HP X532A

Frequency Meter

 

1

 

 

 

 

HP 8093
Slotted Line

 

0

 

 

 

Adjustable Shorter

 

Figure 5.4 The block diagram of the slotted-line measurement.

 

133

and the slotted line, a type 1203 isolator is connected to block the reflected wave back to

the generator.

When a shorting plate is connected to the end of the slotted line, using the movable
probe in the slotted line and monitoring the SWR meter, the position of a voltage
minimum at a specific frequency can be found. Replacing the shorting plate with the
adjustable shorter, the equivalent shorting location of the shorter can then be determined.
Also the measured half-wavelength at this specific frequency can be obtained by
measuring the distance between one voltage maximum and the adjacent voltage minimum.
Repeating above procedure over discrete frequencies between 8.5 GHz to 12 GHz, the
equivalent shorting locations of the shorter are obtained and they are shown in Fig. 5.5. In
this figure, the ’+’ marks represent the measured data by the slotted-line measurement at
discrete frequencies, and the solid line represents the fitting curve interpolarized from the

measured data by using MATLAB’s LEASTSQ program.

From above measured data, we can also plot a figure to show the reactive
characteristics of the adjustable shorter. In Fig. 5.6, we compared the half-wavelength
differences of the experimental results with that of the theoretical results versus frequency.
Both of them represent the differences of the half-wavelengths measured over the
frequency range of 8.5 to 12 GHz and that at 12 GHz. If this adjustable shorter is a simple
waveguide shorting terminator, the experimental and theoretical differences should be
identical. However, as shown in the figure, there are some small discrepancies between
theoretical and experimental results over the frequency range. Therefore, a proper

calibration procedure is needed for using this adjustable shorter.

To calibrate the system with the adjustable shorter, the following procedure was

designed.

134

 

20 f f I I I I I 1

I

18

Equivalent shorted location (mm)

 

 

 

0 1 1 1 1 1 1 1 1
8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency (GHz)

 

Figure 5.5 The equivalent shorting locations of the adjustable shorter obtained by
the measurement of the slotted line over frequencies between 8.5 GHz to 12 GHz. The
’+’ marks represent the measured equivalent shorting locations of the adjustable shorter
at discrete frequencies. The solid line represents the fitting curve of the equivalent
shorting locations.

135

 

14 F I I I I I I I
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' 01
I

half-wavelength differences (mm)
#

 

 

 

 

o l l l L I
8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency (GHz)
Figure 5.6 The reactive characteristics of the adjustable shorter over frequencies

between 8.5 GHz to 12 GHz. The discrepancy of these two curves shows that this
adjustable shorter is not a simple waveguide shorting terminator.

136

Based on the scales indicated on the surface of the adjustable shorter, 40 evenly spaced
positions are chosen to be the reference offset shorting locations. Also, in order to obtain
sufficient calibration data for the calibration procedure, the distance between the first and
the last reference shorting locations should be sufficient long to provide at least one

wavelength in length over the frequency range of 8 GHz to 12 GHz.

When the adjustable shorter set at a specific reference position is attached to the
waveguide aperture, an HP network analyzer automatically measures 201 measured
reflection coefficients of the waveguide probe sweeping over the frequency range of 8 to
12 GHz. Repeating the measurements for all 40 reference shorting positions, the database
(40 x 201) of the probe reflection coefficients can be obtained. Now if we replace the
adjustable shorter with a shorting plate, 201 measured reflection coefficients of the
waveguide probe placed against the shorting plate are recorded. Therefore, for each
specific frequency, the location of the equivalent short circuit corresponding to the
aperture shorting plate can be determined by selecting the equivalent phase of the probe
reflection coefficient from the database. Two reference locations whose phases lag 120
and 240 degrees with respect to that of the equivalent short circuit are also determined to
be two offset short circuits for the calibration procedure and their corresponding offset
shorting lengths are denoted as 12 and I3 as shown in Fig. 5.7. Thus, an equivalent short
circuit with 11 = 0 and two offset short circuits with 12 and 13 in length are used in the
calibration procedure. Once the equivalent network characterized by S parameters is
calibrated, the reflection coefficient at the probe aperture placed against a layer of

unknown material can be obtained using eqs. (5.2.9) and (5.2.10).

The primary advantage of using the adjustable shorter as a short circuit device is that it
provides a low resistive loss at the contact point with the waveguide and a high SWR for
the system. It was also found that the calibration procedure designed for this adjustable

shorter is a direct and stable method to deal with the reactive characteristics of this shorter.

reference
plane

 

HP 87203

 

Network Analyzer EL

137

aperture base

plane plane
\ /
rectangular
wavegu1de

 

 

 

 

 

 

 

 

 

HP 8720B

 

Network Analyzer 3”

 

Figure 5.7

 

 

 

 

shorting-plate
adjustable shorter

 

 

 

0 7,1213 ' 1

L—Q
[ S l

A)

flange
(8. u)
rectangular free space
wavegu1de
= 0.257
,e.___)
d

Calibration procedure of the waveguide probe system using an adjust-

able shorter. The S parameters of the transmission network is determined to convert the
reflection coefficient or the input admittance measured at the reference plane of the net-
work analyzer to that at the aperture plane of the waveguide probe.

138

However, this calibration procedure turns out to be rather time consuming. Moreover, a
small oscillation (less than 5%) occtrrs to follow the curve of the calibrated data. This is
probably due to the inaccuracy of the numerical interpolation process in finding the offset
shorting lengths in the calibration procedure. This may be improved by increasing the
number of the reference offset shorting locations to 400 or 800 points in the calibration
process, if this practice is not too cumbersome. A better method may be to use an HP
waveguide calibration kits for calibrating the system. Unfortunately, such a calibration kit

system is not available to us at the present.

5.3. Experimental Results for Materials

We have conducted a serious of experiments to measure the reflection coefficients or
the input admittances of a waveguide probe system attached to various material samples
as shown in Fig. 5.1. As discussed in the preceding section and shown in Fig. 5.7, the
waveguide probe system can be represented by an equivalent two-port network
characterized by [S] matrix and calibrated by using an adjustable shorter with a proper

calibration procedure.

The reflection coefficients or the input admittances of the waveguide probe attached to
various material layers are first measured at the reference plane of the network analyzer.
After calibrating the data by the scattering parameters of the equivalent network, these
quantities of interest can be converted from that measured at the reference plane to that at
the aperture plane. In this section, we present the results of the input admittances at the
probe aperture attached to various material layers over the frequency range of 8 to 12
GHz. These material layers include isotropic materials such as air, acrylic, plexiglass,
teflon and liquid materials such as distilled water and acetone, and anisotropic materials
such as an epoxy/glass-fiber manufactured by Composite Materials and Structures Center

laboratory of MSU and a dielectric-fiber manufactured by Boeing Airplane Company.

139

Figure 5.8 shows the real and imaginary components of the experimental input
admittances of a waveguide probe when the probe is open to free space. In these figures,
the solid line represents the experimental data, while the ’*’ marks represents the
smoothed curve of the experimental data. As mentioned earlier, an oscillation of less than
5% magnitude along the measured data is probably due to the inaccuracy of the numerical
interpolation process. It seems reasonable to treat the smoothed curve of the experimental

data as the measured input admittance at the probe aperture.

As shown in Fig. 5.9, the results of Fig. 5.8 are also compared with our theoretical
calculations based on two different techniques, Hertzian potential method and transverse
field method, with four mode assumption and the published experimental results by
Bodnar[24]. Figure 5.9 shows good results for the input conductance at the probe
aperture, but a little discrepancy is observed for the results of the input susceptance at the
probe aperture. Since the imaginary component of the probe input admittance is strongly
dependent on the field distributions at the aperture, the discrepancy of this component may
indicate the imperfection of our probe aperture. This is probably caused by a slight
curving nature of the flange and four non-smooth filled holes around the aperture which

are used to connect the adjustable shorter to the flange while calibrating.

Figures 5.10 to 5.12 show the real and imaginary components of the experimental
input admittances of the waveguide probe when the probe is placed against various
material layers such as an acrylic layer of 0.06 inch, a plexiglass layer of 0.06 inch and a
tefion layer of 0.51625 inch, respectively. As before, these figures show both the
experimental data and the smoothed curves of the data. It was found that the contact
between the material layer and the waveguide probe is very important. In our
configuration the material layer to be measured is backed by free space, an unintentional

air gap may exist if the material layer is not tightly attached to the probe aperture. This

140

 

 

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Figure 5.8 Experimental input conductances and susceptances of a waveguide
probe (a = 0.4 in, b = 0.9 in) as functions of frequency when the probe is open to
free space.

141

 

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(dominant + 3 higher-order modes)
0.2» ............. _
0'155 3L 8:5 9 9 10 11' i1 His 12 125
' ' Frequency [’IEHz) ' '
(b)
Figure 5.9 Comparison of input conductances and susceptances of a waveguide

probe (a = 0.4 in, b = 0.9 in) as functions of frequency when the probe is open to
free space with the existing experimental results of Bodnar et al.[24].

142

 

 

 

 

 

 

 

 

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requency ( GHz)
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7.5 a 8.5 9 9.5 10 10.5 11 11.5 12 12.5

Frequency ( GHz)

(b)

Figure 5.10 Experimental input conductances and susceptances of a waveguide
probe (a = 0.4 in, b = 0.9 in) as functions of frequency when the probe is placed
against a material layer (acrylic) with a thickness of 0.06 inch.

143

 

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7.5 8 8.5 9 10 11.5 12 12.5

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09 1 1 1 L 1 1 1 1 1
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Frequency ( GHz)

0))

Figure 5.11 Experimental input conductances and susceptances of a waveguide
probe (a = 0.4 in, b = 0.9 in) as functions of frequency when the probe is placed
against a material layer (plexiglass) with a thickness of 0.06 inch.

144

 

 

 

 

 

 

 

 

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(b)

Figure 5.12 Experimental input conductances and susceptances of a waveguide
probe (a = 0.4 in, b = 0.9 in) as functions of frequency when the probe is placed
against a material layer (teflon) with a thickness of 0.51625 inch.

145

may cause considerable inaccuracy in the measurement as well as the inverse results on

the EM parameters of the materials.

Figure 5.13 shows the real and imaginary components of the experimental input
admittances of the waveguide probe when it is placed against two layers of liquid
material, distilled water, with thickness’s of 0.2344 inch and 0.3125 inch. These two sets
of measurements will be both used later to inversely determine the complex permittivity
of the distilled water via an inverse technique. The purpose is to verify the precision of
the reading of the thickness for a liquid material layer which is difficult to determine

accurately in a tank.

Figure 5.14 shows the real and imaginary components of the experimental input
admittances of the waveguide probe when the probe is placed against a layer of liquid
material, acetone, with a thickness of 0.345 inch. The figures show both the experimental

data and the smoothed curves of the data.

Figure 5.15 shows the real and imaginary components of the experimental input
admittances of a waveguide probe when the probe is placed against a layer of anisotropic
material, epoxy/glass—fiber, with a thickness of 0.1105 inch. These figures show the
measured results at four different orientations of the waveguide aperture with respect to a
reference axis of the material layer. As defined in section 5.2., the measurements at O, 30,
45 and 90 degrees mean that the experiments are conducted when glass fibers in the first
ply of the material layer are making an angle of 0, 30, 45 and 90 degrees, respectively,

with the direction of the electric field of the dominant mode at the probe aperture.

The smoothed curves of the above experimental results are shown in Fig. 5.18. These
curves represent the real and imaginary components of the smoothed waveguide input

admittances when the probe is placed against the epoxy/glass-fiber layer. These data will

146

 

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12*

11*

    
  

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8
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0))

Figure 5.13 Experimental input conductances and susceptances of a waveguide
probe (a = 0.4 in, b = 0.9 in) as functions of frequency when the probe is placed
against two liquid material layers (distilled water) with thickness’s of 0.2344 inch
(dash line) and 0.3125 inch (solid line).

147

 

Normalized conductance

Echrimentaltesult ——_,—
curve smoothing i 3 21:91:21:

 

 

 

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7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
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_2 1 1 1 1 1 1 1 1 1
7.5 a 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)

(b)

 

Figure 5.14 Experimental input conductances and susceptances of a waveguide
probe (a = 0.4 in, b = 0.9 in) as functions of frequency when the probe is placed
against a liquid material layer (acetone) with a thickness of 0.345 inch.

148

 

 

 

 

 

 

 

 

 

 

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7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)
(b)

Figure 5.15 Experimental input conductances and susceptances of a waveguide
probe (a = 0.4 in, b = 0.9 in) when the probe is placed against a layer of anisotro-
pic material (Epoxy/Glass-Fiber) with a thickness of 0.1105 inch measured at four dif-
ferent orientations with respect to a reference axis of the material layer.

149

 

 

 

 

 

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Normalized susceptance

 

 

 

7.5 a 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)
0))

 

I
_a
1.
1.
1.

Figure 5.16 Smoothed curves of the experimental input admittances of a waveguide
probe (a = 0.4 in, b = 0.9 in) when the probe is placed against an anisotropic
material layer (Epoxy/Glass-Fiber) with a thickness of 0.1105 inch measured at four
different orientations with respect to a reference axis of the material layer.

150

be used in the next section to inversely determine the complex permittivities of the

anisotropic material layer.

Figure 5.15 shows the real and imaginary components of the experimental input
admittances of a waveguide probe when the probe is placed against a layer of anisotrOpic
material, dielectric-fiber, with a thickness of 0.053 inch. These figures show the measured
results at four different orientations of the waveguide aperture with respect to a reference

axis of the material layer.

Figure 5.18 shows the smoothed curves of the experimental results of Fig. 5.15. These
curves represent the real and imaginary components of the smoothed waveguide input
admittances when the probe is placed against the dielectric-fiber layer. These data will be
used. in the next section to inversely determine the complex permittivities of the

anisotropic material layer.

5.4.Inverse Technique and Results

Theoretical development up to this point has been to express the effect of material
parameters on the reflection coefficient or the input admittance of a waveguide probe. In
this section, the material parameters as functions of the reflection coefficient or the input
admittance of a waveguide probe are determined numerically by using an inversion

technique based on the Newton’s iterative method [45].

The newton’s method, an extension of the Newton-Raphson method, is well known to

have a key relationship of

—1
ka = xk-(fk') ~fk (5.4.1)

151

 

3.6,.....

3.44»-

Norrnalized conductance

 

 

 

 

~ L l 1
7.5 a 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)

(3)

1.4 I 1 1 1 1 1 1 I I

 

12*

0.8“ .1 . .,
0.6—-~~~

0.4»-

Nonnalized susceptance

 

 

 

9.5 11) 16.5 1L1 111.5 112 12.5
Frequency ( GHz)
0))

l

 

l
.0
«b

1.

Figure 5.17 Experimental input conductances and susceptances of a waveguide
probe (a = 0.4 in, b = 0.9 in) when the probe is placed against a layer of anisotro-
pic material (dielectric-fiber from Boeing) with a thickness of 0.053 inch measured at
four different orientations with respect to a reference axis of the material layer.

152

 

3.8- '

3.2 ~

Normalized conductance

 

2.8” ... ... , .. ,. . .‘ . ......1

 

 

 

| l l J l
9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)

(a)

7.5 8 8.5

 

Normalized susceptance

 

 

 

—o.4 1 1 1 1 1 1 1 1 1
. 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)
(b)

 

Figure 5.18 Smoothed curves of the experimental input admittances of a waveguide
probe (a = 0.4 in, b = 0.9 in) when the probe is placed against an anisotropic
material layer (dielectric-fiber from Boeing) with a thickness of 0.053 inch measured
at four different orientations with respect to a reference axis of the material layer.

153

It also follows that, in order to extend the method to higher dimensions, an analogue of the
derivative f '(x) is developed to be an n Xn matrix Jk which can be determined by
evaluating the Jacobian matrix J (3:) whose entries are the partial derivatives

_ 8f,- (3:)

(J) :31 — ax. .19,an (5.4.2)
I

As an example, suppose that we are using the Newton’s method to solve three

equations for three unknowns simultaneously, say,

f1 (x1: x29 x3)

F (I) = f2 (x1, x2, x3) (5.4.3)

 

 

[3 (x1, x21x3)d

and

NV
II
R
N

(5.4.4)

 

 

Then the Jacobian matrix can be evaluated as

:35 a a
8x1 8x2 8x3
1(3) = 31:2 % iffi . (5.4.5)
3x1 8x2 3x3
% % 31:3.
3x1 8x2 3x3

 

 

With this definition, we have, as the analogue of eq. (5.4.1),

-1
3c“, = 3,,— (Jk) Fk (5.4.6).

154

I . .
where F k = F 8‘11) , Jk = J ($11) 1111111,, denotes the k” iterative values of 1%.

In the programming implementation, the Jacobian matrix is carried out by using a

finite difference approximation for each component of J k . That is,

 

 

(5.4.7)

f,-(?r)-f,-(3c') 1113(3)
(Jk) — k k ~[ )—

. o — ’ ’ ~
1,, (x1), - (1,1)}. 319-
where (xk —xk.) = 8 denotes apreset difference.

To solve n equations for n unknowns, this Newton’s iterative method can be regarded as
an n-dimensional analogue of the secant method. Substituting eq. (5.4.7) in eq. (5.4.6),
the (k+ 1)’h iterative value of 3 is calculated. If 19(1):,” 1) 20 is satisfied at some
preset accuracy then the problem is solved with 11k“ to be the roots of equation
F (it) = O and the iterative process is terminated. Otherwise, a new approximation will

be generated to continue the iterative procedure defined by the recursion formula (5.4.6).

As developed earlier, since the reflection coefficient or the input admittance of the
waveguide probe is an implicit function of the material parameters such as permittivity 8 ,
permeability 11 and the thickness of the material layer, etc., the inversion of material
parameters can be achieved by the Newton’s iterative method. Let in, be a vector
representing the n different measured reflection coefficients or the input admittance of the
waveguide probe with the unknown material layer, and 131;. be the corresponding vector
for the n theoretical reflection coefficients or the input admittances at the probe aperture.

The nonlinear equation, F (3c) = 0 , with n unknowns is constructed as
9 9 9 '>
F(x) = R1). (x) -Rm = 0 (5.4.8)

With an appropriate set of guess values 5:0 and the use of the recursion formula given
in eq. (5.4.6), the correct material parameters, ;, can be determined inversely that lead to

the n measured reflection coefficients or the input admittances.

155

With the assumption of free-space permeability (p. = 110) for all measured materials
discussed in the preceding sections, the complex permittivities of these materials are
inversely determined from the measured waveguide input admittances. For an isotropic
material layer, one measurement of the waveguide input admittance is needed to inversely
determine the complex permittivity of the material layer via the inverse technique, while,
for an anisotropic material layer, three measurements of the waveguide input admittance
will be required to inversely determine three complex permittivities with respect to the

three principal axes of the material layer.

In Fig. 5.19, we show the permittivities of the open space inversely determined from
the measured input admittances of the waveguide probe as shown in Fig. 5.8. Two sets of
results obtained by the inverse technique based on two different methods, the Hertzian
potential method and the transverse field method, which have been addressed in Chapter 3
and 4, are shown in this figure. The marks with ’*’ represent the results which are
inversely determined from technique based on Hertzian potentials, and the marks with ’0’
represent that from the technique based on the transverse field method. Both methods

give good and consistent results for the complex permittivities of free space.

Since the computation time of the inverse procedure based on the transverse field
method is longer than that based on the Hertzian potential method, the inverse results of
the measured complex permittivities of the following isotropic materials were determined
by employing the technique based on Hertzian potentials only. On the other hand, the
measured permittivities of the anisotropic materials were determined by using the

technique based on the transverse field method.

Figure 5.20 shows the measured complex permittivity of an acrylic layer with a
thickness of 0.06 inch. The results indicate 8,2 2.5 ~ 2.7 and Biz—0.1 ~0 for the
most part of frequency range. These results are quite consistent with that obtained by Li

[46].

156

 

 

 

 

1—-. S2 9 1329-: I» 9 931a 9.111.111.111411111191114115 a —
0.8... ...................................... _1

2:

IE

E o.6~ -

Q)

a.

D

.2

304,. ........................................................................... _

T)

a:

5

302- ~

E

o

D . L I . . ' .
o—--1a~sa saga I! Q syn sane-1125529 mnefian -----

_02 .1.1111..
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5

Frequency ( GHz)

Figure 5.19 The inverse results of the complex relative permittivities of free space
determined by employing two different methods. The ’*’ marks represent the results
based on the Hertzian potential method, while the ’o’marks represent that based on
transverse field method. The upper curVes show the real part of the complex relative
permittivities, while the lower curves show the imaginary part.

157

 

25-.. ......... ‘3 ......... : .......... waxiefi ...... _.

Q 2.. ................ .4
.2
.E
6:15- .......... a
Q)
.2
E
d.) 1- ...... _
M
X
3
is"
o 05_ ........................................................................... _4
U
0_ ................................................................................................. _

 

 

 

_0.5 l l l l l l l l l
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)

Figure 5.20 The inverse results of complex relative permittivity of a material layer
(acrylic) which has a dielectric constant of Sr = 2.5~2.7 and a thickness of 0.06 inch.
The upper curve shows the real part of complex relative permittivity, while the lower
curve shows the imaginary part.

158

Figure 5.21 shows the measured complex permittivity of a plexiglass layer with a
thickness of 0.06 inch. The results indicate er z 2.5 ~ 2.7 and 8'. = —O.1 ~ 0 which are

close to the published results.

Figure 5.22 shows the measured complex permittivity of an tefion layer with a
thickness of 0.51625 inch. The results indicate 8r 2 2.05 and 8(- z 0 for the most part of

the frequency range. These results are considered to be quite accurate.

Figure 5.23 shows the complex permittivity of a layer of distilled water inversely
determined from the measured input admittances of the waveguide probe as shown in Fig.
5.13. Two set of results are obtained from two measurements of distilled water with two
different thicknesses of 0.2344 inch and 0.3125 inch. Both cases give very consistent
results on the real and imaginary components of the permittivities. At 10 GHz, the
measured results indicate er z 59 and Si z —17 which agree very well with the published
results of Metaxas and Meredith et al. [47] who gave er 2 59 and 8'. z -—20 ~ —30. While
there is a good agreement between the real components of the permittivity, the
discrepancy of the imaginary components of the permittivity is probably due to different

degrees of pureness of distilled water used in the experiments.

Figure 5.24 shows the measured complex permittivity of a layer of acetone with a
thickness of 0.345 inch. The results indicate er z 6 ~ 8 and El. == 0 for the most part of

the frequency range. These results are considered to be reasonable.

The measured complex tensor permittivities of a layer of anisotropic material, an
epoxy/glass-fiber supplied by Composite Material Laboratory of Michigan State
University, with a thickness of 0.1105 inch , are shown in Fig. 5.25. In the figure, the
real components of three complex relative permittivities in the three principal directions
are shown in the upper graph (a), while the imaginary components of the corresponding

permittivities are in the lower graph (b). Both graphs contain two sets of results; the solid

159

 

 

 

 

3 T 1 l l I 1 fi m l
* **¥ x x x xxx x x * j
: i ; . *** X x * **x
2.52.. ....... ........ i. ”£316.”...
5 2,_ ....... ..
IE
E‘LSF .............. ..
On
D
.Z
‘65
3 1.. ................................................................... _.
04
X
.9.
D-
EO.5-— ''''''' n
O
U
o_**x** ****.. .i ....... ; ......... , .......... , .......... 1 ....... _.
x * : : * * * *** x x x xxx x *
4,5 . . . g . . ; ..
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5

Frequency ( GHz)

Figure 5.21 The inverse results of complex relative permittivity of a material layer
(plexiglass) which has a dielectric constant of 8’, = 2.59 and a thickness of 0.06 inch.
The upper curve shows the real part of complex relative permittivity, while the lower
curve shows the imaginary part.

160

 

 

 

 

2-5 7 T I l ' 7. T 7 l
2_ H. x*x * * *.***H*.*.*,***.*H*.*.****,X -
X ¥ . I i ' .

3‘
.3: 1.5,. ............ -
E
0
On
3 1
--- l-
3 1
0
Of.
5
30.5- ...... -
E
O
U

0,_.,., * * **x ** * **** * *.****** - ........ ..

7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5

Frequency ( GHz)

Figure 5.22 The inverse results of complex relative permittivity of a material layer
(teflon) which has a dielectric constant of 8, = 2.046 and a thickness of
0.51625 inch. The upper curve shows the real part of complex relative permittivity,
while the lower curve shows the imaginary part.

l6l

 

“’0 ‘ F ? f f T f ? T
60 ......... 3 .......... ....... ....... -

Complex Relative Permittivitv

Qflgfiflfiflflifleg.‘ E
: R39 52 9a

 

 

 

4,0 ; ; ; ; r it ; ; ;
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)

Figure 5.23 The inverse results of complex relative permittivities of two material
layers (distilled water) with two different thickness’s of 0.2344 inch and
0.3125 inch. The upper curves of the graph show the real part of complex relative per-
mittivity, while the lower curves show the imaginary part.

162

 

 

 

 

10 I I I I I 1 7 I I
8" ....................... ......... -
. ; x. .
6,... 3K ,, ... _
é.“ *‘ x
.2. i
“g 4_ ...................................................................................... ..
b
o.
3
.5 2... ........................................................ —1
w
3
a: .
>< .
£0- I. .. X* * * * * .xx * * x * * * .*..x ..... _.
E I 1 x 4
o . .x
u_2__ Ni. x. _
{ x
-4_ .2. _
3K
_6 i i I i i . I i i
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)

Figure 5.24 The inverse results of complex relative permittivity of a material layer
(acetone) which has a thickness of 0.345 inch. The upper curve shows the real part of
complex relative permittivity, while the lower curve shows the imaginary part.

163

 

 

Real part of complex relative permittivities

 

 

 

‘1 ..A
(’1
m , .
CD
0'1
‘0

 

 

 

 

 

9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)
(3)

U: 0.5 I I I I I I I I I
.2
">'
'5
.E 0.— --.
Q)
,,>_-o.5- -
E.
8
X
2 -1- —
D.
E
o
O
t...
o—1.5" "‘
1:
<6
G‘ , i I
be.
m I I l
E-2.5 i L l 1 1 l l l 1

7.5 a 8.5 9 9.5 10 10.5 11 11.5 12 12.5

Frequency ( GHz)
(b)

Figure 5.25 Measured complex tensor permittivities of an anisotropic material layer
(epoxy/glass-fiber) with a thickness of 0.1105 inch. The solid lines represent the mea-
sured results inversely determined by employing three measurements of the probe input
admittances with the probe orientations of 0, 45 and 90 degrees with respect to a princi-
pal axis of the material, while the dash lines represent the corresponding results with the
probe orientation of 0, 30 and 90 degrees.

164

lines represent the measured permittivities obtained by employing three different
measurements of the probe input admittances with the probe orientations of 0, 45 and 90
degrees with respect to a principal axis of the material, and the dash lines represent those
with the probe orientations of 0, 30 and 90 degrees. As shown in the figure, both cases
give quite consistent results on the real and imaginary components of the permittivities. It
is also important to note that these measured permittivities are inversely determined by
assuming a dominant TE 10 mode only for the aperture electric field in the solution of the
matrix equation. As discussed in Chapter 4, since the effect of the higher order modes in
the aperture electric field is insignificant when the waveguide probe is attached to an
anisotropic material layer, the inverse results with only a dominant mode consideration

should be sufficiently accurate.

Figure 5.26 shows the measured complex tensor permittivities of a layer of anisotrOpic
material, a dielectric-fiber manufactured by Boeing Airplane Company, with a thickness
of 0.053 inch. The results are determined inversely by employing three measurements
of the probe input admittances with the probe orientations of 0, 30 and 90 degrees with
respect to a principal axis of the material. In the figure, we only show the real and
imaginary components of the principal permittivities in the transverse directions, while the
permittivity in the perpendicular direction is omitted because it is found to be of highly
unstable giving a value between 10 to 10000 for both real and imaginary components.

The difficulty encountered in this case will be discussed in the next section.

5.5.Analysis on the Effects of Material Parameters

The reflection coefficient or the input admittance of a waveguide probe placed against
a layer of material has been shown in the preceding chapters to be a function of the EM
parameters of the material, such as permittivity, permeability, conductivity and sample

thickness. In our measurements, with the assumption of having free-space permeability

165

 

4.3 I I I I I I 1 I fl

Real part of complex relative permittivities

 

 

 

 

10 10.5 11 11.5 12 12.5

.4...
U"
m
a)
U1
‘0

 

 

 

 

 

9.5
Frequency ( GHz)
(21)
m -3 I I I fl I I I I I
.2 »
’S
'5 -3.5-.
a -4~
U
¢¢>Ifl
33' —4.5~
O
l—
X
2 -5--
a.
E
o
o -5.5~
h—
o
E
:1. ‘5h
,5 -6.5~ ]
N)
«I .I
E _7 I I I I I I I I I
75 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Frequency ( GHz)
0))

Figure 5.26 Measured complex tensor permittivities of an anisotropic material layer
(dielectric-fiber) with a thickness of 0.053 inch. The results are determined inversely
by employing three measurements of the probe input admittances with the probe orienta-
tions of 0, 30 and 90 degrees with respect to a principal axis of the material. The com-
plex permittivity in the direction perpendicular to the waveguide aperture is determined
to be unstable and is omitted.

166

and fixed known thickness for the material layer, there is no difficulty in determining the
permittivity and conductivity of isotropic materials inversely; however, for anisotropic
materials, there exist some constraints. In this section we will discuss the effects of the
material parameters on the input admittance of the probe and the constraints one needs to

consider in the measurements of anisotropic materials.

As discussed in the preceding section, we have difficulty in the inverse determination
of the EM parameters of a layer of anisotropic dielectric-fiber manufactured by Boeing
Airplane Company. In that measurement, the conductivity in the direction perpendicular
to the waveguide aperture was found to be highly unstable. To find out the reason for

causing this problem, we develop the following scheme.

Let’s assume that a waveguide probe is placed against an anisotropic material layer
with three principal complex permittivities of £1 = 5.4 -j5.3, 82 = 5.8 —j6.4 and
£3 = 3.8 —j5.7. For this kind of materials, we can first plot the input admittance of the
waveguide probe as a function of the material thickness. If we further change one of the
imaginary components of three principal complex permittivities in the theoretical
calculation, the effects of the specific conductivity associated with the thickness on the

probe input admittance can be sought.

Figure 5.27 shows the real and imaginary components of the input admittance of a
waveguide probe as functions of the material thickness when the probe is placed against a
layer of anisotropic material at an operating frequency of 10 GHz. In the figure, the
anisotropic material is assumed to have three principal complex permittivities of
£1 = 5.4 —jx, £2 = 5.8 —j6.4 and 83 = 3.8 —j5.7 , wherextakes the values of 0.3, 10.3
and 1000.3 in three different cases. Since the aperture electric field is dominated by the
dominant TE 10 mode which has an electric field in the y-direction only, changing the
material conductivity in the x direction does not change the induced current in the material

layer, leading to an insignificant change in the probe input admittance. This phenomenon

167

 

 

 

 

 

 

 

 

5.5 I I I I
5~ - 05¢, r a
_ ‘9 (15¢
. <3
o 4.5— it my» -
0 ; ‘Q
5 ii : 12
a 4* a ' ’5' 1
’3 In
'2 ’5 b
E: p tgg’amcwoemo‘a
g 3-, . ’I’ .............. _.
z: 97
a I
E 2.5— 0 - I _
o I , *
2 ’¢ : - 5.4—1x 0 O . 0,3 .. - -
2- ‘7, E = 0 5.8—j6.4 0 I x =~ I103 000‘ -
O 0 3.8 -j5.7 10003 +++
d .
/ .
1 as I I I I
0 0.05 0.1 . 0.15 0.2 0.25
Thickness ( inch)
(a)
1.5 I I I T
Co .
9“ ”E
0.
1... . .4
If E
:3 ‘ I Is
0.. \
d)
0
U3) 0_ § .. .1
-o 5
a \
:: —0.5-‘ . . -
E a?
\
o d" _
2 _1L. 6‘ ..... -
\ 69.90
0\ 9°}
_15_ a ............ , ........ awn ..... ................ 1
;b 00 I
: 0\__,6w‘°
“@009 .
_2 I 1 l 1
0 0.05 0.1 0.15 0.2 0.25
Thickness( inch)
0))

Figure 5.27 Input admittance of a waveguide probe as a function of the material
thickness at 10 GHz when the probe is placed against an anisotropic material layer sup-
plied by Boeing Airplane Company with various conductivities in the x direction.

168

can be verified from the figure that shows almost identical curves for the three different

cases of x.

At the same operating frequency of 10 GHz, Fig. 5.28 shows the real and imaginary
components of the input admittance of a waveguide probe as functions of the material
thickness when the probe is placed against a layer of anisotropic material with three
principal complex permittivities of 8] = 5.4—15.3 , 82 = 5.8 —jx and e3 = 3.8 —j5.7 ,
where x is assumed to have values of 0.4, 10.4 and 1000.4 in three different cases. Since
the electric field of the dominant TE 10 mode is in the y-direction, the change in the
material conductivity in the y direction should cause a great change in the probe input
admittance. We can observe in this figure that the probe input admittances for three
different cases are highly distinguishable over a range of material thickness. It is also
noted that when the measured anisotropic material is highly conductive in the y-direction,
the probe input admittance becomes independent of the material thickness because the

material layer essentially becomes a short circuit.

Similarly, Fig. 5.29 shows the real and imaginary components of the input admittance
of a waveguide probe as functions of the material thickness when the probe is placed
against a layer of anisotropic material with three principal complex permittivities of
£1 = 5.4 —j5.3 , £2 = 5.8 -j6.4 and £3 = 3.8 —jx, where x assumes the values 0f 0.7,
10.7 and 1000.7 in three different cases. In the figure, we found that when the anisotropic
material is highly conductive (such as $10.7 and 1000.7) in the direction perpendicular to
the waveguide aperture, the probe input admittance remains nearly constant for the
material thickness smaller than 0.07 inch. This phenomenon is maybe due to the fact that
the induced current in the perpendicular direction of a thin conducting layer is quite small

and is independent of the conductivity in the perpendicular direction.

From Fig. 5.26, we found that three principal permittivities of the dielectric-fiber from

Boeing Airplane Company are approximately of £1 = 3.5 —j5, 82 = 3.2 -j6 and 83

169

 

 

 

 

 

 

 

 

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+
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+ + i + .
030.. . +,+++,+++++++++++++++++++++++++ ++++++++++ _
Q _
c
8
o
:325- -1
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c
O
()20_ q.. .H. "‘_ M ... .. ... H__ I a
'13 : 5.4-j5.3 0 0 ’ 0.4 3‘ ---
5,5 0 48-ij o {101,900,
g 0 0 > 3.8 —j5.7 10004: +++
O
Z101— —1
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00000. ’l’ IOOOOOoooo6oooqqpooooooo
0998-*";” i i -‘1"
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0 0.05 0.1 0.15 0.2 0.25
Thickness ( Inch)
(a)
5 I I I I
O-GOOOGO.OOOOO I \ x x —1
. 0o ‘s‘ ’—,_,.
' 0000Oooooooooooodboddboooooo
g _5_+ H .. .. .. __ .. . .. .. _ ............... -
D-
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m
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:.-:-15—1 ~~~~~~~~~~ ~
E
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Z-20- —
+ . +
_25_..H++++++++++++++++++++H+++++f++.++++++++++ ......... q
+ t ‘ + ‘
_30 1 l l l
0 0.05 0.1 . 0.15 0.2 0.25
Th1ckness( Inch)
(b)

Figure 5.28 Input admittance of a waveguide probe as a function of the material
thickness at 10 GHz when the probe is placed against an anisotropic material layer sup-
plied by Boeing Airplane Company with various conductivities in the y direction.

170

 

 

 

 

 

 

 

 

6 fl +++ f I '
. + . .
+ "\
+ , [I \
+ 00 +- \
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5,. .. 0' 100;; I .\_\ _.
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I 0 7 0 3-3-jx 1000.7 +++
0 . I I I
0 0.05 0.1 0.15 0.2 0.25
Thickness ( inch)
(20
3 I _‘ I— I I
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oo 0 Q ++ i
_2». . 000000 +.+ ......W- ............. .-
i +,
+1 ++ : 5
f+++++ E
_3 I l l l
0 0.05 0.1 . 0.15 0.25
Th1ckness( inch)
('3)

Figure 5.29 Input admittance of a waveguide probe as a function of the material
thickness at 10 GHz when the probe is placed against an anisotropic material layer sup-
plied by Boeing Airplane Company with various conductivities in the z direction.

171

with an unstable value of between 10 to 10000, which is just the same order of quantities
used in the calculation of the results shown in Figs. 5.27-5.29. The thickness of this
dielectric-fiber is 0.053 inch and at this thickness the conductivity in the z direction causes
very small change in the probe input admittance. Therefore, it is very difficult to inversely

determine the 83 (in the z-direction) from the measured probe input admittance.

Employing the similar scheme, the measurement using a waveguide probe placed
against a layer of epoxy/glass-fiber supplied by MSU Composite Material Laboratory can
also be analyzed. From Fig. 5.25, we found that three principal permittivities of the
epoxy/glass-fiber at 10 GHz are approximate of 81 = 5.1—j0.2, £2 = 5.3 -j0.3 and
£3 = 2.5 — j l .7. From a plot of the probe input admittance as a function of the material
thickness, if we further change one of the imaginary components of three principal
complex permittivities in the theoretical calculation, the effect of the specific conductivity

associated with the thickness on the probe input admittance can be determined.

Figure 5.30 shows the real and imaginary components of the input admittance of a
waveguide probe as functions of the material thickness when the probe is placed against a
layer of anisotropic material at an operating frequency of 10 GHz. The anisotropic
material is assumed to have three principal complex permittivities of £1 = 5.4 —jx,
£2 = 5.8 -j0.4 and 83 = 3.8 —jl.7 , where x takes the values of 0.3, 10.3 and 1000.3 in
three different cases. As mentioned earlier, since the aperture electric field is dominated
by the dominant TE 10 mode which has no electric field in the x direction, the change in
the material conductivity in the x direction has insignificant effect on the probe input

admittance, as shown in the figure.

Figure 5.31 shows the real and imaginary components of the input admittance of a
waveguide probe as functions of the material thickness at 10 GHz when the probe is
placed against a layer of anisotropic material with three principal complex permittivities

of 81 = 5.4 —j0.3, 82 = 5.8 -jx and £3 = 3.8 —jl.7, where x assumes the values of

172

 

 

 

 

 

 

 

 

6 F I F T
555
25 "‘5
’ \e
6’ ‘
5_ I. V ._ . t _. . _
c: a
I ;
g ‘b 3..
,.
U / \
:s p - ex
'2 - G
o if i o
Ua~ _¢_.. a». . _
'O I Q\
d) 6) 0b
.E . ,5 . , b3,
0 i :
Z . : ;
a : 5.4-1.x 0 O l 0.3 ..--
1_ Go ..... s: ”.0. '5.8-'j0.4‘ 0....’ "'x‘=' 103 ..... 000.-
0 0 3.8-j]..7 1000.3 H‘F
o 4 l l l
o 0.05 o. 1 o. 15 0.2 0.25
Thickness ( inch)
(a)
3.5 Y F I T
3- -~ 6556+ —
. a ‘\
\
25_ .. f .. 35 4
o :1: W6
3 ¢ 1 \
S 2r. . a \ _4
a. F . ¢\
0 9
815- . .9. .i _
I?) ,5“ ‘
'8 1w “t .
.91 a. ‘
E305» ..‘i‘. _
o a\
Z 0-. ......\ -4
9\
—o.5- '9‘ p -
,. 9‘ .. _ ' no? _
. 99 fig?
_15 J l l QQQQ
' o 0.05 0.1 o. 15 o. 2 0.25
Th1ckness( inch)
(b)

Figure 5.30 Input admittance of a waveguide probe as a function of the material
thickness at 10 GHz when the probe is placed against an anisotropic material layer sup-
plied by MSU Composite Material Laboratory with various conductivities in the x direc-
tion.

173

 

 

 

 

 

 

 

 

40 + I I I r
‘ +
+
35.. + . V. .. .. . 7+ .. _ ,.. ., _
+++++++++++++++++++ + +++++++++++ ++++++
+ l l
830,. ...... J
c:
c:
H
325*
'D
c
o I
320— ,. . .......... l
8 : 54—103 03 0 04 : ---
£15 5 = O 5.8-j): O , x = { 10.4 g 000
g 0 0 38—11-7 1000.4. +++
o .
Z 10.. ,,,,,,,,, .4
_ ... ‘oooooooo: ——,-—~; . . ,_ d
5 000°C , , 996000000000‘009099oooooo
000 i , ’ ’ '. 3 ‘ ‘-‘ —
Q09 _____ :" I : . .
O l l J l
o 0.05 0.1 . 0:15 0.2 0.25
Thickness ( ll’lCh)
(a)
5 T I I I
0.530006023600000 a . \ \ \ x . l l . d
00 i ‘ ‘ ‘ ______ ._ .—
OOOOoooooooooooooooooooooooo
Q) ; '
o
_ + : l, . . , , V , s V . -
g 5+ : _ .
D.
a)
8
5—10" _
m
“U
S
;_-::-15-' -
E
O
z_20_...r + ...... .4
+
_25,_. .+.++.+++-l-+.+.+f+++++'k+,+ ,+.t+++++++f++++.++++++..., 4
+ . . V .
_ + f
_30 1 g l l
0 0.05 0.1 0.15 0.2 0.25
Thickness( inch)

0))

Figure 5.31 Input admittance of a waveguide probe as a function of the material
thickness at 10 GHz when the probe is placed against an anisotropic material layer sup-
plied by MSU Composite Material Laboratory with various conductivities in the y direc-
tion.

174

0.4, 10.4 and 1000.4 in three different cases. As shown in the figure, the probe input
admittances for three different cases are very different over a range of material thickness
because the change in the conductivity in the y-direction will cause significant change in

the induced current in the material layer.

Figure 5.32 shows the real and imaginary components of the input admittance of a
waveguide probe as functions of the material thickness at 10 GHz when the probe is
placed against a layer of anisotropic material with three principal complex permittivities
of £1 = 5.4 —j0.3, £2 = 5.8 —j0.4 and 83 = 3.8 —jx, where x has the values of 1.7,
10.7 and 1000.7 in three different cases. In the figure, we found that when the anisotropic
material becomes highly conductive in the direction perpendicular to the waveguide
aperture, the effect of the conductance on the probe input admittance become significant if
the material layer is thicker than about 0.07 inch. Since the thickness of the epoxy/glass-
fiber is 0.1103 inch which is thicker than 0.07 inch, we can then inversely determine three

principal complex permittivities of the epoxy/glass-fiber without any difficulty.

We also analyzed the measurement using a waveguide probe placed against a layer of
isotropic material with a complex permittivity of e = 2.046 -j0.1 . In Fig. 5.33, we show
the real and imaginary components of the probe input admittance as functions of the
material thickness at 10 GHz when the probeis placed against a layer of isotropic material
with a complex permittivity of 8 = 2.046 —jx , where x has the values of 0.1, 0.2 and 0.5
in three different cases. We found that those probe input admittances of three cases are
clearly distinguishable. It is also noted that when the material is highly conductive, the
probe input admittance becomes independent of the material thickness, which is the same

phenomenon as in the anisotropic materials and is not shown in this figure.

Figure 5.34 shows the real and imaginary components of the probe input admittance as
functions of the material thickness at 10 GHz when the probe is placed against a layer of

isotropic material with a complex permittivity of e = x— jO.1 , where x assumes the

175

 

 

 

 

 

 

 

 

10 I T T T
+
9_ '+ : 5.4—10.3 0 0 _,
+ 8 = 0 5.8—j0.4 0
8— 0 O - ‘ 3.8“}..r -!
8 .
c 7__ + 1.7 -- - _4
a ++
z; x = { 10.7 000
:3 ;
g 6.. + .... 1000‘.7...+++ _,
U 5- O .2./00 \~- _
'U + / oO \
a O I + 00 j \
= 4- _ .. ’. .......... +00 .. \ _J
N 6 /’ -+-+ .o \\
+ O \
E 0 I, ++ O \\
C 3" 1. + \ -
Z i QQ/ ++ o 5‘
Q / ++ 000
2,. ._ .. Q’J . _ +++f000 ........ T
r +
99- ‘+++
o°9 '
1b—WO - —1
0 r i i i
0 0.05 0.1 _ 0:15 0. 0.25
Thickness ( inch)
(a)
5 7 I I I
4- » +++ -
+ +
+ 1
+ .
3" 000 — " " “
8 +SOII‘OO +\.\
g \
*3: 2_ 699/ 0 ...... \\. a
\
_ ' \, _
a) 1 aw“ 9 ‘
'6 Jo \ ,
o - \ .
N 0.. .o... ..\ ...... _
= O \ 1
CB : OO \ E
E . 1.. 00 \x ..... $99 d
0‘1 j " 0 "CL; 653 ' '
z ; 90000000++1
_2._ .....++ ............................ ..
+ +f++
+ + ‘
-3' ‘1‘ '+' g T
s + i
_4 l I l I
0 0.05 0.1 0.15 0.2 0.25
Th1ckness( inch)
(b)

Figure 5.32 Input admittance of a waveguide probe as a function of the material
thickness at 10 GHz when the probe is placed against an anisotropic material layer sup-
plied by MSU Composite Material Laboratory with various conductivities in the z direc-
tion.

176

 

 

 

 

 

2

1.8%“ -
Q)
0
§
3 1.6% —
3
1:
C.
0
U 1.4r -
-o
0
.55.
"a
E 1.2» —
0
Z .

1- A
0.8 l l r 1 ‘1
o 0.1 0.2 0.3 0.4 0.5 0.6
Thickness ( inch)
(a)

 

1.2 _r T r T I

P
on
1

Normalized Susceptance
p o
b 0’
I I

0.2" '

 

 

 

 

o 0.1 0.2 0.3 0.4 0.5 0.6
Thickness( inch)
0))

Figure 5.33 Input admittance of a waveguide probe as a function of the material
thickness at 10 GHz when the probe is placed against an isotropic material layer with
various conductivities.

177

 

Normalized Conductance
e .- .-‘ N .N N
& O, a: N N b O)
I I I I I I
T
.4
I I

.5
N
I

 

 

 

0.8
0 0.6

 

(a)

 

1.4

0.8- ..

Normalized Susceptance

0.2 ~

 

 

 

 

o 0.1 0.2 0.3 0.4 0.5 0.6
Thickness( inch)
(b)
Figure 5.34 Input admittance of a waveguide probe as a function of the material

thickness at 10 GHz when the probe is placed against an isotropic materiallayer with
various permittivities.

178

values of 2.046, 2.146 and 2.546 in three different cases. In this figure, the probe input
admittances clearly show distinguishable differences between three different cases, even

when the change in the permittivity is of small quantity.

To summarize this analysis, we found that for the measurement of the EM properties
of isotropic materials the thickness of the material is not important factor; while for
anisotropic material, the material layer needs to be thicker than some value if accurate

results on the EM parameters are sought.

 

CHAPTER 6

CONCLUSIONS

The purpose of this research was to determine the EM properties of materials
nondestructively using a waveguide probe system over the X-band of 8 to 12 GHz. In this
chapter, we review the techniques we have developed as well as the limitations for the use

of this waveguide probe system. Finally, suggestions for future studies are addressed.

There are two major contributions from this research: two proposed theoretical
techniques for the measurement of the EM properties of isotropic and anisotropic
materials using a waveguide probe system, and the development of the experimental

procedure for this system to achieve the nondestructive measurement of materials.

To measure the EM properties of a layer of material, a nondestructive measurement
can be achieved by placing a waveguide probe system, which consists of an open-ended
rectangular waveguide terminated on a flange and a network analyzer, against the material

layer

In Chapters 3 and 4, we have conducted theoretical and numerical analyses of the
waveguide probe system attached to a layer of assumed known material. In these

analyses, the reflection coefficient and other relevant quantities such as the input

179

I80

admittance and the EM fields of the probe aperture were formulated as functions of the

assumed EM parameters of the material layer.

The theoretical analyses of the system were based on two coupled integral equations
developed for the aperture electric field by matching the tangential EM fields across the
probe aperture. The method of moments was applied to solve these electric field integral
equations numerically. After the Galerkin’s method was used to convert these EFIE’s into
a matrix equation, the unknown aperture electric field can be determined numerically.
Then, the reflection coefficient or the input admittance of the probe aperture can be

calculated.

The technique presented in Chapter 3 employed the EM fields expressed in terms of
Hertzian potentials to construct two EFIE’s for the electric field at the waveguide aperture.
Since the EM fields expressed in terms of Hertzian potentials are only suitable in an
isotropic medium, this technique was then limited to the measurement of the EM
parameters of isotropic materials. The numerical results on the input admittances of the
waveguide probe obtained with this method were compared with the existing results
published by other workers and they showed a good agreement between them. It was also
found that a good convergence in the numerical calculation can be achieved when only the

first four waveguide modes (TE 10 , TE 30 , TE 12 and TM 1 2) were used.

The technique presented in Chapter 4 used the EM fields determined by the transverse
field method to construct two EFIE’s for the aperture electric field. Since the transverse
field method can deal with a general case of the EM fields inside an anisotropic medium
and a degenerate case of the EM fields inside an isotropic medium, this technique is
suitable for the measurement of the EM parameters of both isotropic and anisotropic
materials. The numerical examples illustrated in Chapter 3 for isotropic materials were
also recalculated by the transverse field method and compared with the existing results.

The results showed that our numerical results using four modes for both techniques agreed

l81

well to each other, as well as to other published results. We also conducted the theoretical
calculation of the input admittance of the waveguide probe placed against a layer of
assumed known anisotropic material. The numerical results indicated that the aperture
electric field is dominated by the dominant TE 10 mode when the waveguide probe is
attached to an anisotropic material layer, while using a dominant TE 10 mode plus one
higher order mode TE20 in the numerical calculation, we can obtain a good convergence

on the probe input admittance.

Chapter 5 described the experiments and the inverse procedure for the nondestructive
measurement of EM properties of materials. A series of experiments were conducted to
measure the input admittances of the waveguide probe system attached to various
materials, which included solid and liquid materials, as well as isotropic and anisotropic
materials over the frequency range of 8 to 12 GHz. Two calibration procedures for the
waveguide probe system were developed and discussed in detail. In the calibration
procedure, we found that an oscillation of less than 5% amplitude along the measured data
occurred probably due to the inaccuracy in the hardware and software of the calibration
method. Therefore, a better calibration technique is needed for improving the accuracy of
the measurement. It was suggested that a better method may be to use the HP waveguide

calibration kits with an HP network analyzer to calibrate the waveguide probe system.

An inverse procedure to determine the EM parameters of the measured materials from
the measured input admittances of the waveguide probe was described in Chapter 5. The
measured probe input admittances were used to determine the conductivity and
permittivity of the measured materials inversely by this numerical inverse procedure. The
inverse results showed that the inverse procedure based on the Newton’s iterative method
provided a stable and efficient method to evaluate the EM properties of isotropic
materials; while for anisotropic materials, the measured EM parameters are sensitive to

the guess values given in the computer program. This was due to the increasing

182

complexity in the inverse procedure for determining three unknown complex
permittivities of the anisotropic material simultaneously. The analysis of this difficulty

was also presented detailedly in Chapter 5.

As mentioned in section 5.3., since we assume that the material layer to be measured is
backed by free space in our configuration, an unintentional air gap may occur if the
material layer was too thin or too light to secure a tight contact with the aperture. This
may cause considerable inaccuracy because the EM fields excited inside the material layer
were found to localize around the probe aperture. Therefore, the extension of the analysis
of the waveguide probe system with a stratified material layer is suggested to further

improve the accuracy of the measurement.

Finally, the forward procedures presented in Chapters 3 and 4 showed that the
reflection coefficient and the input admittance at the probe aperture can be expressed as
functions of the assumed EM properties such as the conductivity, permittivity,
permeability and thickness of the material layer. Thus, it is possible to determine these
quantities of interest simultaneously. However, it may require a faster computer to solve

more quantities simultaneously.

APPENDIX

APPENDIX A

2-1) FOURIER TRANSFORM OF SINUSOIDAL
FUNCTIONS OVER WAVEGUIDE APERTURE

In this appendix the two-dimensional Fourier transforms of the sinusoidal functions
are derived analytically. The simplified expressions of the notations are used to facilitate
the implementation of a computer program. The shorthand notations of the transforms
given in Chapter 4 are first rewritten in eqs. (A.1) to (AA). They are based on the mode
indices of the basis and testing functions represented as [ p, q] and [1, r] , respectively.
On the other hand, to use these notations in Chapter 3, the mode indices are replaced by
[ (2p - 1), (2q — 2) ] and [(21—1), (2r— 2) ] , respectively, for the basis and testing

functions.

Isin [1%] [cos (kxx) +jsin (kxx) 1de x

b
[‘[cos [Cg-X] [cos (kyy) +jsin (kyy) ] dy]

E [311(kx) +512 (1%)] [313 (ky) +514 (1%)] (A-l)

183

184

a

l
= [Isin [35):] [cos (kxx') -jsin (kxx') ] dx'] x

b .
[loos [(11%] [cos (kyy') -jsin (kyy') ] dy']

E [321 (kx) + 522 (kg) [523 (ky) + 524 (19)] (A2)

Icos [1%] [cos (kxx) +jsin (kxx) ] dx] x
b . my . .
(ism [7] [cos (kyy) +jSlI1 (kyy) ] dy]

E [331(kx) +532(kx)] [S33 (ky) +534 (18)] (A3)

= [Icos [p111] [cos (kxx') -jsin (kxx') ] dx'] x

b .
( I[sin [II—71?] [cos (kyy’) -J‘ sin (kyy') l dy']

a [341(k) +s4z(kx)] [543(5) 4.544(5)] (AA)

The following derivations list the analytical results of the single finite integrals which
represent sub-items of each inversion. Each result divides into two conditions as shown in

the following.

 

511(kx)

312 (kx)

$13 (ky)

a
a t! sin [11%] cos (kxx) dx

art! (1 + cosakx)

22 2 2
ln—akx

 

O

.a . [fix .
5]"; sm [7] sm (kxx) dx

artl sin akx

22 22
ln-akx

 

=j
a
511—508)]

 

b
_ ”‘)’
=t[cos [7] cos (kyy) dy
bzk ' bk
— ysrn y
= J rzrtz—bzky2
b
28*.

 

b
_ . my .
=1gcos [—b ]srn (kyy) dy

’ 2
—b ky (l — cosbky)

22 22
=jJ rrt—bky

 

O

 

185

11:
for kx¢z
In:
for kx = g
In
for kxlf-‘Z
In
for kx = 3
fit
for k $3
_ ”I
for ky - 3
fit
for 18$?
for k, = it

(A.5)

(A.6)

(A.7)

(A.8)

186

=" - m . .
s21(kx) _Jsrn[ a ]cos(kxx)dx

a1tp( l + cosakx)

 

 

 

 

 

 

: p21t2_azkx2
0
a RX.
522 (k) 5 fl“! sin [”7] sin (kxx') dx'
anpsinakx
2 2 2 2 for
_ —j p It —a k}.r
a
§[1—5(kx)] for
b qnyi
523 (ky) alcos [7] cos (kyy') dy'
, —b2k sinbk
2 2y 2 2 for
= J q 1: —b k),
b
58k), for
b qny.
524 (ky) s—j ‘[ cos [T] sin (kyy') dy'
r bzk (1 bk )
— — cos
y y for
2 2 2 2
= _jl q 1t —b ky
O for

p11:
kx¢7
(A.9)
_ PTt
kx — 7
p11:
kx¢7
(A.10)
_ PTc
kx — 7
fl
k)¢ b
(A.ll)
_ ‘11
ky — b
21‘
ky: b
(A.12)
k =

_ a [fix
53.1 (kg) =lcosl: a ]cos (kxx) dx

S32 (kx)

s33 (ky)

S34 (ky)

l
2 .
a kxsrnakx

_ J lzrtz—azk2

- X

 

a
sjlcos [1%t] sin (kxx) dx

[ —a2kx (l + cosakx)

22 22
=jJ [TC-altI

 

 

O

b
t[sin [$1] cos (kyy) dy

brtr (1 — cosbk),)

2 2 2
—bky

 

2
r1!

0

 

b
sjgsin [2)] sin (k ,y) dy
b )
—b1trsinbkv
. r21:2 — bzk ,2
= J .l

b
511—6091]

187

In
for kx #5 Z
(A.l3)
In
for kx = Z
11:
for kx at -a—
(A.l4)
In
for kx = 2
fit
for ky 7': ?
(A.lS)
fit
for ky — f
m
for ky ¢ b
(A.l6)

s41 (kx)

S42 (kx)

S43 (ky)

s44 (ky)

a 1
a fig cos [p%] sin (k xx') dx'

([COS [p—ZJE] cos (kxx') dx'

 

'J

J

2 .
a kxsmakx

 

2 2

22
prt—akJr

f

 

28k.

—a2kx (l + cosakx)

22 22
pn-akx

 

O

b .
_ ' qny 1 1
= t[srn [7] cos (kyy ) dy

brtq (l — cosbky)

(121:2 _ bZkyZ

 

O

b .
—jt[8in [7%] sin (kyy') dy'

-f

—b1tq sin bky

q2n2 _ b2ky2

 

b
2[1-5(k)')]

188

31‘
for k¢b

#—
fOI k E

(A. l 7)

(A.18)

(A. l9)

(A20)

189

As given in Chapters 3 and 4, these inversion functions are used to define the

following notations such as

¢12u(kx) E [51108) +512(kx)] [52103) +522(kx)]

«112,031 a [513(k,.) +s,,<k_,.>1 623(k).) +s24(k,.)1

to be used in Dag, and

¢34u(kx) E [531081) +532(ka [541(kx) +542(kx)]

¢34v(ky) E [533 (ky) +534(ky)] [S43 (ky) +344(ky)]

in D313, and
“P32110895 [531(kx) +532(kx)] [52108) +322081)]

¢32v(k).) E [333 (ky) +334 (ky)] [523 (ky) +324(k).)]
in D33, and
¢14u(kx) E [511(kx) +512(kx)] [54108) +342(kx)]

¢,.,<k,) a (snag) +s..(k,)1 1s,,(k,.> +s,.,(k,>1

. yx
1n DaB°

(A.21)

(A.22)

(A.23)

(A.24)

(A.25)

(A26)

(A27)

(A28)

If we further denote S f) as the finite integral result for the multiplication of two sinusoidal

functions with different arguments and 5:), as the result with the same argument, where

xy is the index of the above sub-inversion functions, then eqs. (A21) to (A.28) can be

tabulated in Table A.l to be used in Chapter 4 and Table A2 in Chapter 3 if the symmetric

190

Table A.1 Summary for simplified expressions of notations used in Chapter 4.

 

yx xy
D0113 D0113 D0113
¢l4u (kx) ¢32u (kx) (b34u (kx)

111A ... (53.57,). (55.53). (531.2,). (55.2,).
x a (55“5‘22) (5401+532) (5201+522) (541+532)

“‘37!” =th (511+5‘112JSI22 (511+5‘112JSZ (531+5‘312JS’2’2 (50+S;2)5b

 

 

 

 

 

 

 

 

 

k1: gAkx=p7n 572532 5113521 $15122 $21521
yx xy XX
Dan 00:13 Dan

 

 

q)l4v (ky) ¢32v (ky) ¢34v (ky)

laggTA/(gtfl (S':+Slll4)x ($103+574JX (533+5a34JX ($303+Sa34)x
.1 a .1 a (553+534) (5.6-+5044) ($203+S‘2'4) (543+Sa44)

 

 

k)-¢%[Ak1~=q7n (312+5‘1’4JSl23 (513+5‘114JSL (533+5l314JSI2’3 (532+5034JSZ4

 

 

 

 

 

 

 

 

 

191

Table A.2 Summary for simplified expressions of notations used in Chapter 3.

 

 

)‘y )‘x x )' xx
0 ”0,70% D013 0043 DUB D045
0 , 0 0
od’l‘joo (')qu (kx) ¢104u (kx) ¢32u (kx) ¢3:411 (kx)

 

 

[1t p11: a a a a a a a a a
kx¢zAkx¢7 511521+572522 511522+5l112541 S31522+St§2521 S31541+S325f12

 

 

 

 

 

 

I a a
WE”. = I? 511522 5112531 5:2522 531521
I a a
kx = '3" kx ¢ p7“ $2521 5113522 531522 531541
.45...- was; 0 o sass.
y) yx xy xx
p1) , DaB DorB DaB D018
of
000d- [0’7 c o o e
10.00 ¢12v (ky) ¢l4v (ky) $3212 (ky) $34v (ky)

 

1'7! (In a a a a a a a a
k)“ 7 A k)‘ 1‘ 7 513523 + 574524 5135044 + 514543 533524 + 5034523 533543 + 5345044

 

 

 

I")! It
k)‘ ’t '5' A I“ = ()7; 3103523 5114524 5034533 5303524
__ r1: qrt
ky " 7 A k)‘ at $ 51135203 £3524 $34524 $13,454?
_ m _ ‘17I
k). - 2- A k), - 7 51:35.33 0 0 534524

 

 

 

 

 

 

 

Note that in this table I, p, r and q represent (21-1), (2p-I), (Zr-2) and (2q-2),
respectively.

192

properties of these‘notations are considered. Note that in Table A.2 the mode indices need
to be replaced by [(2p— 1), (2q-2)] and [(21—1),(2r-2)] instead of [p, q]
and [1, r] .

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