The Pennsylvania State University The Graduate School Department of Engineering Science and Mechanics

Permittivity Measurement of Circular Shell Using a Spot-Focused Free-Space System and Reflection Analysis of Open-ended Coaxial Line Radiating into a Chiral Medium

AThesisin Engineering Science and Mechanics

by Kai Du

Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

August, 2001 We approve this thesis of Kai Du. Date of Signature

Vasundara V. Varadan Distinguished Professor of Engineering Science and Mechanics Thesis Advisor Co-Chair of Committee

Vijay K. Varadan Distinguished Alumni Professor of Engineering Mechanics and Electrical Engineering Co-Chair of Committee

Raj Mittra Professor of Electrical Engineering

Douglas H. Werner Associate Professor of Electrical Engineering

Jose A. Kollakompil Senior Research Associate

R. P. McNitt Professor of Engineering Science and Mechanics Head of the Department of Engineering Science and Mechanics Abstract

The present thesis is concerned with electromagnetic simulations for material char-

acterization. The content can be divided into two parts. The first part addresses

questions related to a spot-focused free-space microwave measurement system. The

goal is to establish an inversion procedure for non-planar samples. The second part is

focused on the problem of an open-ended coaxial line radiating into a chiral medium.

The objective is to evaluate the feasibility of using the open-ended probe method for

characterization of chiral materials.

The free-space system under study consists mainly of two horn-lens antennas and a vector network analyzer. Permittivity and permeability of a sample under test are determined from its reaction to beam waves generated by the antennas.

The network analyzer provides measurement output in the form of S-parameters for a linear two-port network. This system has been designed and successfully tested for planar slab samples, where a uniform plane-wave model has been used in the inversion algorithm. Extension to curved objects calls for a new model in which the wave scattering needs to be numerically simulated, and the measured S-parameters have to be defined carefully so that their (mathematical) expressions conform with the physics. To simplify the problem, the incident beam is assumed to be Gaussian, and samples with the simple geometry of a circular cylindrical slab are considered.

Existing theories for Gaussian beam scattering by simple shapes and antenna near-

field scanning are combined together to obtain an improved model. The improvement lies in the fact that characteristics of the antenna are now manifested in formulation iii Abstract iv of the measured response.

The model is first applied to simulate free-space mono-static measurement of pla- nar slab samples. Numerical results are presented to validate the plane-wave approx- imation used in the inversion. Next, a three dimensional formulation is derived for the bi-static setup. Calculation shows that, depending on the sample properties and thickness, the difference between the reflection coefficients of a Gaussian beam and a plane wave might be much smaller than the measurement error. Thus inversion of measured data using the plane wave model is appropriate even without correction for the defocusing effect. Finally, a simple technique is presented to estimate the reflec- tion of two dimensional Gaussian beam by a circular shell. The corresponding data inversion problem is studied with several approaches, including an optimization pro- cedure using only magnitude data and a curvature correction procedure. The results are carefully evaluated and possible improvements are discussed.

Extensive experiments have been performed using the free-space system. The fo- cus is on the calibration method and settings of some important network analyzer parameters. For the free-space bistatic setup, it is found that an offset-short method reported previously is inappropriate. Therefore a simple two-tier calibration proce- dure is proposed. Inversion of measured data for Teflon, Plexiglas and glass slab samples shows that this procedure produces permittivity values within 10% differ- ence from published data. Curved Plexiglas samples of several radii of curvature have also been measured. The data are compared with theoretical prediction and potential causes of the deviation are identified. A practical technique is proposed for estimation of the time gating error in network analyzer measurement. This technique can be used to find out desirable minimum gate span for free space measurement.

The open-ended coaxial line method has been extensively studied by other re- Abstract v

searchers. However, its potential application to chiral material is an interesting prob-

lem which has yet to be investigated. A spectral-domain moment method solution is presented in this thesis. The formulation obtained can also be to study the perfor- mance of aperture antennas covered with chiral medium, which has been proposed

for controlling the polarization property. Contents

List of Figures ix

List of Tables xv

Acknowledgments xvi

1 Introduction 1 1.1Motivations...... 3 1.2OverviewoftheIssuesStudiedinThisThesis...... 6 1.3SummaryofContributions...... 16 1.4OrganizationoftheThesis...... 18

2 Theoretical Model for Measuring the Dielectric Properties of Planar Slabs Using a Spot-Focused Free-Space System 20 2.1IntroductoryRemarks...... 21 2.2 Description of the Spot-focused Free-space Measurement System . . . 27 2.3ModelingAssumptions...... 31 2.4Meaningof“ReflectionCoefficient”forBeams...... 39 2.5 2—D Gaussian beam scattering by a dielectric slab at normal incidence 46 2.6 2—D Numerical Results and Comparison with Measurement ...... 48 2.7 Formulation of 3—D Gaussian Beam Obliquely Incident on a Slab . . 52 2.7.1 SpecifyingtheIncidentBeam...... 54 vi CONTENTS vii

2.7.2 SpectrumFunctionfortheReflectedField...... 58 2.7.3 “ReflectionCoefficient”for3—DGaussianBeam...... 60 2.8DirectCouplingintheBistaticSetup...... 63 2.9 Experimental Results for Planar Slab Using Bistatic Setup ...... 66 2.10Conclusion...... 79

3 Techniques for Obtaining Dielectric Properties of Curved Slab Using Only Reflection Measurement 80 3.1IntroducingtheCurvedSlabProblem...... 80 3.2 Literature on the Scattering of Shaped Beam by Curved Objects . . . 82 3.3AModificationoftheFourierExpansionmethod...... 84 3.4TheReflectedFieldandReflectionCoefficient...... 87 3.5 Numerical Result for 2—D Gaussian Beam Reflection from circular shell 91 3.6ExperimentsandDataInversionforCircularShells...... 96 3.7Conclusion...... 104

4 Open-ended Coaxial Line Radiating into a Chiral Medium 107 4.1Introductoryremark...... 108 4.2Basicideaforsolvingthehalfspaceproblem...... 110 4.3Formulation...... 112 4.3.1 FieldRepresentationintheChiralHalf-Space...... 112 4.3.2 Electromagnetic Fields in the Coaxial Waveguide ...... 114 4.3.3 SpectralDomainSurfaceAdmittance...... 116 4.3.4 IntegralEquation...... 118 4.3.5 Method of Moment—Derivation of Matrix Equations ...... 119 4.4NumericalResults...... 122 4.5Conclusion...... 125

5 Summary 131 Table of Contents viii

Appendix A: Gaussian Beam 134

Appendix B: Near Field and Far Field 140

Appendix C: Time Gating in 8510C 143

M − M + Appendix D: Derivation of the reaction ml and ml 154

Bibliography 164 List of Figures

1.1 Magnitude plot for E-polarized uniform plane wave scattered by a per- fectlyconductingcylinder...... 9

1.2 Phase plot for E-polarized uniform plane wave scattered by a perfectly conductingcylinder...... 10

1.3 Result of real ray tracing for a Gaussian beam scattering by a conduct- ingcylinder...... 11

2.1 Magnitude of plane wave reflection coefficient as a function of permit- tivity ε foradielectricslab...... 24

2.2 Illustration of a spot-focused free-space measurement system...... 28

2.3 A picture of the spot-focused free-space measurement system used in thisstudy...... 29

2.4Farfieldpatternofthehorn-lensantennaat11GHz...... 33

2.5 Field amplitude distribution over a 16cm × 16cm rectangular area on thefocalplane...... 35

2.6 Phase distribution of the focal region field measured at 10.92GHz using acustom-madedipole...... 36

2.7 The E-plane 1/e half beam width c as a function of frequency. . . . . 38

2.8 The H-plane 1/e half beam width e as a function of frequency. . . . . 39

ix LIST OF FIGURES x

2.9ThecouplingbetweentwoHermite-Gaussianbeams...... 41

2.10 Diagram for defining the coupling coefficient between two antennas. . 42

2.11 Magnitude of reflection coefficient versus frequency for a 2-D E-polarized Gaussian beam reflected by a dielectric slab at normal incidence. . . . 50

2.12 Magnitude of reflection coefficient versus frequency for a 2-D E-polarized Gaussian beam reflected by a dielectric slab at normal incidence. . . . 51

2.13 Geometry and coordinate systems for the modeling of bistatic free spacemeasurement...... 53

E˜i k , k θ k θ k ,k /k k /k 2.14 Plot of y( x 0) = ( z cos + x sin )Φ( x y) z v.s. x 0 for dif-

ferent incident angle θ and beamwidth w0x...... 58

2.15 A schematic diagram showing the basic idea of the theoretical formu- lation...... 63

2.16 Geometry and coordinate system used for calculating the direct cou- pling between the antennas in a free-space bistatic setup...... 64

2.17 Magnitude of the directly coupled signal versus the incident angle for differentbeamwidths...... 66

o 2.18 Log magnitude of the 40 bistatic “reflection” coefficient (S21 measured by the network analyzer) for three “samples”: metal plate, sample holder,freespace...... 68

2.19 Time domain response of the 40o bistactic setup calculated from 2GHz— 18GHzfrequencydomaindata...... 69

2.20 Linear magnitude of the S21 measured for metal plate, glass, plexiglas andteflonslabsamplesaftertheSLOTcalibration...... 70

2.21 Phase of the S21 measured for metal plate, glass, plexiglas and teflon slabsamplesaftertheSLOTcalibration...... 71

o 2.22 Magnitude of the 40 bistatic “reflection” coefficient S21 for the glass sampleobtainedfrommeasurementandsimulation...... 72 LIST OF FIGURES xi

o 2.23 Phase of the 40 bistatic “reflection” coefficient S21 for the glass sample obtainedfrommeasurementandsimulation...... 72

o 2.24 Magnitude of the 40 bistatic “reflection” coefficient S21 for the plexi- glas sample obtained from measurement and simulation...... 73

o 2.25 Phase of the 40 bistatic “reflection” coefficient S21 for the plexiglas sampleobtainedfrommeasurementandsimulation...... 73

o 2.26 Magnitude of the 40 bistatic “reflection” coefficient S21 for the teflon sampleobtainedfrommeasurementandsimulation...... 74

o 2.27 Phase of the 40 bistatic “reflection” coefficient S21 for the teflon sample obtainedfrommeasurementandsimulation...... 74

2.28 S11 measured after the SLOT calibration for a series of offset shorts. 75

2.29 Magnitude of the “reflection coefficient” for offset-short (planar perfect conductor)asafunctionoftheoffsetdistance...... 75

2.30 Phase of the “reflection coefficient” for offset-short (planar perfect con- ductor)asafunctionoftheoffsetdistance...... 76

2.31 Applying the Savitzky-Golay smoothing to the magnitude of S21 for theglasssample...... 77

2.32 Applying the Savitzky-Golay smoothing to the phase of S21 for the glasssample...... 78

2.33 Real part ε of permittivity inverted from bistatic measurements. . . . 78

2.34 Imaginary part ε of permittivity inverted from bistatic measurements. 79

3.1 Gaussian beam scattering by a circular cylindrical Slab–problem geom- etry...... 84

3.2 Amplitude of the reflected field for a 2-D Ez-polarized Gaussian beam scatteringatacirculardielectricinterface...... 93 LIST OF FIGURES xii

|Es| 3.3 Phase distribution of the reflected electric field ( z )alongthebeam waist for the same geometry as in Fig. 3.2. Solid lines are obtained with modifiedfieldexpansion,whilethedottedcurvesarefromcomplexray tracing...... 94

3.4 Amplitude of the reflection coefficients for curved plexiglas sample of thickness0.53cmversusfrequency...... 96

3.5 Phase of the reflection coefficients for curved plexiglas sample of thick- ness0.53cmversusfrequency...... 97

3.6Apictureofthecurvedsamplemountedintheholder...... 98

3.7 Repeatability of the Measurement–|S11| for plexiglas circular shells obtainedontwooccasions...... 98

3.8 Repeatability of the Measurement–Phase of S11 for plexiglas circular shellsmeasuredontwooccasions...... 99

3.9 Repeatability of the Measurement–Magnitude of S11 for cylindrical coppersheetsmeasuredontwooccasions...... 99

3.10 Repeatability of the Measurement–Phase of S11 for cylindrical copper sheetsmeasuredontwooccasions...... 100

3.11 1/e half beam width calculated from measured |S11| of copper sheets. 101

3.12 Real part of permittivity ε inverted from the estimated beam width

and magnitude of reflection coefficients S11 for curved plexiglas slabs. 101

3.13 Curvature-corrected |S11| fortheplexiglassamples...... 102

3.14 Phase of curvature-corrected S11 for the plexiglas samples...... 103

3.15 Real part (ε ) of permittivity inverted from the curvature-corrected S11 fortheplexiglassamples...... 103

3.16 Imaginary part (ε) of permittivity inverted from the curvature-corrected

S11 fortheplexiglassamples...... 104 LIST OF FIGURES xiii

4.1 Geometrical configuration and coordinate system for the problem: an open-ended coaxial line radiating into a chiral half space...... 111

4.2 Amplitude of TEM mode reflection coefficient versus frequency(GHz)

and real part of chiral parameter ≡|kc|Re(β)...... 125

4.3 Phase of TEM mode reflection coefficient v.s. frequency(GHz) and

real part of chiral parameter ≡|kc|Re(β)...... 126

4.4 Amplitude of TEM mode reflection coefficient v.s. frequency(GHz)

and real part of chiral parameter ≡|kc|Re(β)...... 126

4.5 Phase of TEM mode reflection coefficient v.s. of frequency(GHz) and

real part of chiral parameter ≡|kc|Re(β)...... 127

4.6 Amplitude of TEM mode reflection coefficient v.s. frequency(GHz)

and imaginary part of chiral parameter ≡|kc|Im(β)...... 127

4.7 Phase of TEM mode reflection coefficient v.s. frequency(GHz) and

imaginary part of chiral parameter ≡|kc|Re(β)...... 128

4.8 Amplitude of TEM mode reflection coefficient v.s. frequency(GHz)

and imaginary part of chiral parameter ≡|kc|Im(β)...... 128

4.9 Phase of TEM mode reflection coefficient v.s. frequency(GHz) and

imaginary part of chiral parameter ≡|kc|Im(β)...... 129

4.10 Amplitude of TE01 mode reflection coefficient v.s. frequency(GHz)

and real part of chiral parameter ≡|kc|Im(β)...... 130

A.1ComparisonofuniformplanewaveandGaussianbeam...... 137

A.2 Gaussian beam can be thought of as a bundle of complex rays. . . . . 139

C.1Bandpassimpulseshapesforthethreewindowtypes...... 146

C.2MeaningofthegateshapeparametersintableC.2...... 148

C.3 Selection of gate shape and gate span for isolating the desired response fromtheunwantedresponses...... 148 LIST OF FIGURES xiv

C.4 Relative error ((|S11gate|−|S11ideal|)/|S11ideal| in percentage) in the mag- nitude of reflection coefficients caused by time gating under MINIMUM WindowtypeandNORMALgateshape...... 150

C.5 Relative error ((|S11gate|−|S11ideal|)/|S11ideal| in percentage) in the mag- nitude of reflection coefficients caused by time gating under NORMAL windowtypeandNORMALgateshape...... 150

C.6 Change in the phase of reflection coefficients caused by time gating under MINIMUM window type and NORMAL gate shape...... 151

C.7 Change in the phase of reflection coefficients caused by time gating under NORMAL window type and NORMAL gate shape...... 151

C.8 Linear magnitude of the reflected impulse calculated by the HP8510C analyzer for the ideal frequency domain measurement of a slab sample (ε =2.59,thickness=0.5348cm)...... 152

C.9 Zoom-in view (t=0.4ns-1ns) for some details of figure C.8...... 153 List of Tables

2.1 1/e half beam width w on the aperture of the X-band horn lens antenna estimatedfromthefarfieldpattern...... 34

2.2 Minimum 1/e half beam width w0 on the focal plane (beam waist) of the X-band horn lens antenna estimated from the aperture field and a quadraticphasefrontassumptionfortheaperturefield...... 34

2.3 Results of fitting the field amplitude on the focal plane to the 2-D Gaussiandistribution...... 37

2.4 Comparison of two definitions for the “reflection coefficient”...... 45

3.1Thicknessofthecurvedsamplesarenotuniform...... 95

4.1 Amplitude and phase of the TEM mode reflection coefficient of a flangedopen-endedcoaxialline...... 124

C.1 Technical parameters for the window type in HP8510C network ana- lyzerspecifiedinitsdatasheet...... 145

C.2 Properties of the gate shapes (time filters) available in HP8510C net- workanalyzer...... 147

xv Acknowledgments

I am indebted to many teachers and friends for their generous advice. It has been a privilege to listen to their lectures and talk with them about small issues. I am also indebted to those researchers whose work I have only read about from books and journal papers. The knowledge they gave me extends far beyond what is reflected from this thesis, and my appreciation cannot be fully expressed by any language. It is a great pleasure to put some of my deepest gratitude into record.

I thank all of the committee members for their guidance and criticism in complet- ing this thesis. Especially:

Thanks are due to my thesis co-advisors Prof. Vasundara V. Varadan and Prof. Vijay V. Varadan for their direct oversight and detailed guidance on this research, and their unfailing supports.

Thanks to Dr. K. A. Jose for his expertise in measurement and instrumentation.

Thanks to Prof. Raj Mittra for reviewing this work and his insightful comments.

Thanks to Prof. D. H. Werner for his careful review of this work and insightful comments.

I also thank my family members, especially my parents. They have inspired me over the years with their hard work and desire for knowledge. Although they cannot understand English, I would like to dedicate this thesis to them.

xvi Chapter 1

Introduction

This thesis is concerned with material characterization. The term “material charac- terization” is used to refer to the quantification of electrical property of material, and

“electrical property” means the parameters–permittivity, permeability, loss tangent

and a few others–used to represent the macroscopic behavior of a substance in in-

teracting with electromagnetic fields. These parameters are needed in a variety of

applications, such as the design of microstrip antennas and radar absorbing mater-

ial(RAM).

Material characterization is also an interesting field to work in. It provides a lot of

intellectual stimulus. To illustrate this, the most simple “material”–the vacuum–

can be used as an example. Its permittivity ε0 and permeability µ0 are intimately

1 related to the velocity of light c by c = √ .Thevaluesofε0 and µ0 are quite µ0ε0 fundamental in view of the holiness of c in modern physics. Material parameters measured in common laboratory environment obviously can not be more precise than

ε0 and µ0. It is interesting to find out their accuracy. However, just by looking at the numbers and following a simple logic, one would run into a question whose answer involves some complicated issues.

1 §1.0 Prelude 2

−7 −2 The value µ0(= 4π × 10 NA ) is the one to ponder. Having learned about the

unit of force Newton(N)andtheunitAmpere(A) for electric current at high school, we naturally think the value 4π × 10−7 was determined from some experiment. On

the other hand, it is noticed that π is a mathematical constant and its value can be

calculated with any desired accuracy. Herecomesthequestion:isthemeasuredvalue

−7 for µ0 exactly 4π × 10 or just very close to it? Either case would be too much of a

coincidence.

Physicists and metrologists seem to have a different perspective. In his book

the Fundamental Physical Constants and the Frontier of Measurement,B.W.Petley

wrote:

“...... the desired simplicity of Maxwell’s equations is ensured by simultaneously

−7 −2 defining the permeability of free space µ0 as 4π × 10 NA ) which, after the Henry

is defined (to keep the energy units the same as ... ), allows us to use different units

−1 for µ0,namelyHm and the impedance of free space is also simply µ0cΩ.”

According to the above statement, µ0 is a defined constant in the international

system(SI) of units. Similarly, the length unit Meter (m) has been defined such that

the speed of light in vacuum is exactly 299792458ms−1, and the permittivity in vac-

2 uum ε0 is therefore exactly 1/(µ0c ). Although somewhat intriguing, this explanation is understandable as long as the relevant units have been defined consistently, so that, for example, the unit Newton has the same meaning for both electric force and me- chanical force. However, the question remains: are we sure that µ0 and c are constant, independent of frequency, temperature and direction? This question should not be dismissed as being too academic as we know that precision value for the velocity of light is often critical for many applications. §1.1. Motivations 3

The above discussion demonstrates a point that measurement of physical parame-

ters is often based upon a theory which reflects our understanding or interpretation

of the world. As our knowledge improve, technologies advance and new applications

are envisioned, existing techniques are constantly re-evaluated and possible improve-

ments are proposed. This point is also reflected in the evolving area of material

characterization.

1.1 Motivations

The few questions addressed in this thesis are obviously not about velocity of light.

Instead, they are related to the measurement of material parameters at microwave

frequencies. Specifically, we will examine the theory and/or experimental technique

(calibration) in two measurement methods. The focus will be on a few issues which

might affect the precision of measurement results. The first method to be studied is

called “spot-focused free-space system”. The issue identified here is the plane wave

assumption used in existing model. We replace it with a Gaussian beam assumption

and examine whether more accurate results can be obtained. As an application

of the new model, properties of a few curved samples are measured and compared

with theoretical prediction. The second method is called “open-ended coaxial probe”

method. The basic that we ask is whether this method can be used in evaluating the

property of chiral materials. To put the various questions in the right perspective, it

is helpful to describe the historical background under which they were originated.

During the last two decades, a topic called electromagnetic chirality has attracted substantial attention from researchers around the world. There have been two books §1.1. Motivations 4

devoted to this subject (Lakhtakia et al. [2] and Lindell et al. [3]) where theoretical

results obtained prior to 1994 were summarized. These authors presented the point

of view that complex materials such as chiral and bi-isotropic media represents one of

the important directions in microwave technology in this 21st century. This viewpoint

is supported by many other experts, as is evident by the number of papers published

since 1994. The inclusion of a chapter [4] on bianisotropic composites in the book

Frontier in Electromagnetics and a chapter on chirality [5] in the book Direction in

electromagnetic wave modeling further indicates the importance of this topic. An

number of interesting applications have been envisioned [54], some of them have

materialized.

Theoretical interest in chiral and complex media can be simply explained. Just

similar to the idea that the vacuum might not be isotropic, not all of the (linear

homogeneous) materials are necessarily isotropic or anisotropic. Their properties

might be more appropriately modeled with new constitutive relations such as this

one: D = ε(E + β∇×E), and B = µ(H + β∇×H), where β hasbeencalledthe

chiral parameter. The intrinsic coupling between electric and magnetic fields is im-

mediately evident from these equations. This intrinsic coupling does not exist in a

simple isotropic medium. Consequently chiral or other complex media can produce

novel effects on electromagnetic fields, forming the basis for potential applications.

Researchers have observed that those effects are very weak in natural materials and

studied the artificial fabrication of composites with the desired “complex” properties.

And naturally comes the twin problem of how to measure these properties. These two has been the focus of experimental studies on chiral and complex media.

Although the first demonstration of man-made chirality was effected some 90 years §1.1. Motivations 5 ago [3], measurement of chirality started to receive wide spread consideration only around 1990. In this aspect, the spot-focused free-space measurement system, one of the subjects of the present thesis, has played an important role. It was utilized to achieve “the first complete experimental study of chiral composite materials at microwave and millimeter wave frequencies” in Ro’s Ph.D. dissertation [6]. Simi- lar systems have been constructed elsewhere for more or less the same purpose(see e.g. [7]). The present study is largely a continuation of previous researches on the free-space measurement system. The goal, however, is no longer directly related to chirality measurement, but to study, both theoretically and experimentally, the mea- surement accuracy of such a method and to extend it to measuring the properties of curved objects.

As always, predicting the future seems to be complicated. Popular commercial application of chiral and complex materials remain to be discovered, and the most reliable and practical method for characterizing complex properties is a matter for research and debate. More generally, as pointed out in [1], measurement of funda- mental constants is “an area where theory and experiment are pushed and tested to the limit, where the fallibility of man is often all too apparent and where one’s best is only just good enough”. The same comment applies to material characterization.

Depending on the requirements of a specific application, a best method might be simply not good enough. Recognizing that the free-space method is somewhat incon- venient at frequencies lower than 5GHz, the idea of using open-ended coaxial probe for chirality measurement was also considered for study. This is the motivation for the second work presented in this thesis. §1.2. Overview of the Issues Studied in This Thesis 6 1.2 Overview of the Issues Studied in This Thesis

The discussion so far has emphasized that research on material characterization is

indispensable for fabrication of composites with complex and useful properties. Pre- cision measurement of material properties is also useful in many other applications, such as microwave non-destructive evaluation of defects and diagnosis of deceased tis- sues. The basis for these applications can be understood by observing that physical properties are interdependent. And the permittivity or permeability might change depending on factors as simple as the temperature, or as complicated as the pres-

ence of an abnormal protein. Thus a change in these factors can be determined by

measuring the electrical properties.

A variety of methods have been studied. They can be classified by the test setup

being used: cavity resonators, waveguides, open-ended coaxial probes, and free space

methods. The relevant literature is copious and the task of comprehensive review

should be left to experts (see e.g. Asfar et al. [8], Varadan et al. [13], Birch et

al. [9] and Stuchly et al. [65]). It is obvious that the research has achieved a

certain level of maturity. Commercial products are provided by vendors like Agilent

Technologies1 and HVS Technologies2. On the other hand, there are certainly many

remaining problems that deserve detailed investigation. For example, the open-ended

coaxial probe is generally considered to be less accurate than the cavity resonator and

waveguidemethods.Thusimprovingitsaccuracyhasbeenthefocusofcontinuing

research in recent years [65]. Similar comments also apply to the free space method.

An overview of the issues examined in this thesis is presented in the following.

1Open-ended coaxial dielectric probe is available from Agilent Technologies(product 85070). 2Complete free-space measurement system is available from HVS Technologies. §1.2. Overview of the Issues Studied in This Thesis 7

The first subject of this thesis is a spot-focused free-space measurement system.

Electromagnetic(EM) waves are used to sense properties of materials. “free-space”

means the wave is propagating in free space, as oppose to being guided by transmission lines. The term “spot-focused”, meaning that the wave is focused on a spot, signifies that the system is a near field one in contrast to remote sensing. That is, the material is located close to the probing antenna. The word “system” indicates it has several modular components which can be classified by their functions: source, radiator, signal processing etc. It should be noted that not all of the existing free-space systems

are the same. The difference lies mainly in the mechanism of focusing component.

Thus conclusion drawn for one system should not be arbitrarily applied to other

systems without examining the details.

As explained in previous section, the free space method has been established and the ultimate interest of this work is the characterization of curved objects of general shapes. Meaningful applications can obviously be derived once this is achieved. A natural first step is to study its use for simple curved objects. A single layer circular shell seems to be an appropriate choice3. As one begins to look at the problem– probing the property of a circular shell using microwave beams–available theory and

measurement practice should be carefully examined and potential difficulties fully

anticipated.

The particular free-space system [14] under study consists of two conical horn-lens

antennas. They are connected to two test ports of a vector network analyzer which

functions as transceiver. Microwave beam fields sent out (transmitted) by the an-

3A circular cylinder might be the most simple one but is not appropriate here. Because the size of the sample should not be too small so as to avoid complicated edge diffraction. The sample should not be too big either so that it can fit in between the antennas. See the detail of the free space setup in chapter 2. §1.2. Overview of the Issues Studied in This Thesis 8

tennas interact with material under test(MUT), generating reflected and transmitted

fields. These fields are again picked up (received) by the antennas. Measured data,

in the form of scattering parameters for a linear two-port network, are processed by

an inversion algorithm, translating the S-parameters to material properties for the

MUT. In existing theory [14], the inversion algorithm is based on the simple model

of a uniform plane wave incident upon a planar slab. Since uniform plane wave never

exist in practice, one might raise the question:

Issue 1: Howgoodistheuniformplanewavemodelinrepresentingthefocused

beam? Will this cause any error in the resulting material parameters that

are measured?

This question appears too academic at first look. Indeed, although “uniform plane

wave” exists only in theoretical sense, it is applied to solve engineering problems

day to day. People seldom worry about whether the radio signals are uniform or

not. However, for the problem of measuring the material property of curved object,

the uniform plane wave model runs into some difficulty. Consider the scattering of a

(uniform) plane wave by a perfect conducting cylinder, the scattered field is obviously

no longer a plane wave. Fig. 1.1 shows the amplitude distribution of the reflected

field along a line tangential to the cylinder. The reflected field does not look like a

plane wave if the radius is less 20λ0. The phase distribution, illustrated in Fig. 1.2,

shows further deviation from a plane wave. Therefore we need to consider what is

the “reflection coefficient” for cylinders of small radii of curvature(< 20λ0).

Instead of looking at the beam as a uniform plane wave, one can think of it in terms of plane wave ray. According to the theory of geometrical optics, the ray is a

plane wave locally (but not uniformly). Its reflection and transmission given by the §1.2. Overview of the Issues Studied in This Thesis 9

20λ 1.0 0 10λ 0 5λ 15λ 0 0 0.9 4λ 0 3λ 0.8 0

2λ 0.7 0 (y)| z E | 0.6 along this line y radius=λ 0 0.5 E z x 0.4

0.3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 y (unit:λ ) 0

Figure 1.1: E-polarized uniform plane wave scattered by a perfectly conduct- ing cylinder. Curves shown are magnitude distribution of the re- flected field |Ez| along the line (x = 0) tangential to the cylinder. Different curves correspond to different radii. The reflected field does not look like a uniform plane wave if the radius of cylinder is less 20λ0,inwhichcasetheuniformplanewavemodelruns into difficulty answering the question of what is the “reflection coefficient”. usual Fresnel formulas. Therefore the plane wave model is justified to some extent if the radius of the cylinder is big enough (so that one can model the beam as just one ray). Yet how big is big enough? How high is high frequency? This kind of question doesnotseemtohavebeenansweredinasymptotictheory.Andtheremightnotbe a general solution for all types of geometry.

Figure 1.3 might shed better light on the above discussion of one ray versus one beam consisting of a bundle of rays. Consider the scattering of a Gaussian beam by §1.2. Overview of the Issues Studied in This Thesis 10

150

100

50 5λ 0 (degrees) z 4λ E 0 0 2λ -50 radius=λ 0 0 3λ 0 Phase of -100 15λ 10λ 0 0 -150 20λ 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 y (unit:λ ) 0

Figure 1.2: Plot of phase distribution for the reflected field Ez along the line (x = 0) tangential to the cylinder. (see the geometry in Fig. 1.1)

a perfectly conducting cylinder. Referring to the geometry in Fig. 1.3, one can trace

the ray (phase path) passing through the half beam width point (0,w) and finds the intersection point (0,wr) of its reflected path with the beam waist. In general wr >w

because the reflected beam is divergent and their difference wr − w will be bigger for smaller radius of curvature. If the change is small, then one can say the reflected

beam keeps the shape of incident beam and therefore the beam can be thought of as

one ray. The ratio (wr − w)/w hasbeenplottedinFig.1.3asafunctionoftheradius

for different w’s. It can be seen that for a Gaussian beam of half beam width 1.3λ0,a radius of 14λ0 is big enough, if the tolerance is 98% agreement between the reflected beam and incident beam. It is tempting to conclude that if a focused beam of 2.6λ0

spot size is used to measure the property of a curved object with radius of curvature

bigger than 14λ0, then no curvature correction is needed. And one can continue to §1.2. Overview of the Issues Studied in This Thesis 11 use the uniform plane wave model. On the other hand, it should be recognized that wave scattering by a penetrable object is more complicated than the non-penetrable case. Further analysis is needed in order to draw a convincing conclusion.

0.7

0.6

(0,wr) 0.5 w

)/ 0.4 (0,w) -w r 0.3 (w λ w =3.0 0 waist 0.2 2.0 1.7 0.1 w=1.3

0 5 10 15 20 λ radius of conducting cylinder ( 0)

Figure 1.3: Result of real ray tracing for a Gaussian beam scattering by a conducting cylinder: trace the ray passing through the half beam width point (0,w) and find the intersection point (0,wr)ofits reflected path with the beam waist. As an indication of the sim- ilarity between the reflected beam and incident beam, the ratio (wr − w)/w is plotted versus the radius of cylinder for different w (unit: λ0). The result provides some sort of an estimation on whether one can consider the Gaussian beam as one ray, rather than a bundle of rays. The inhomogeneous wave tracking method of Choudhary and Felsen [39] has been used.

Issue 2: What exactly is the reflected signal picked up by the antenna?

If the receiving antenna is far away from the cylinder, it can be considered as a point receiver (a dipole) and the reflected signal is simply proportional to the reflected §1.2. Overview of the Issues Studied in This Thesis 12

field amplitude at that point. The reflection coefficient is something similar to the radar cross section of the cylinder. Unfortunately, for the free space system under study, the MUT is located in the near field region of the antenna (see Appendix B).

Therefore in order to formulate the reflected signal, the assumption of uniform plane wave should be replaced with a bounded beam.

The focused beam is most accurately simulated by using measured data of actual near field produced by the lens antenna. Such an approach is numerically intensive and the result is subject to error in the near field measurement. A Gaussian beam is used in this thesis mainly because it can be specified by analytical formula (see

Appendix A). In addition, Gaussian beam has been studied extensively in optical systems. Many of the existing results are directly applicable for the microwave free space system.

Once a decision is made on how to represent the bounded beam, modeling of the free space measurement process is turned into a series of wave scattering problems.

Gaussian beam scattering by a planar slab has been analyzed by many researchers.

Most of the available results are on 2-D problems, i.e., the beam and the geometry are invariant along one direction. 3-D case has also been touched on. Therefore the planar slab can be handled by following the existing approaches. On the contrary, the circular shell might bring some difficulties.

Issue 3: Simulation of Gaussian beam scattering by a circular shell?

A number of references have been found to be instructive for the above issue. The

first category is on Gaussian beam scattering by a layered cylinder, where a Fourier expansion method has been used (Kojima et al. [40] and Kozaki [42]). The second category uses the (real) ray tracing for plane wave ray (e.g. Lee et al. [49]). The third §1.2. Overview of the Issues Studied in This Thesis 13

category uses the complex ray tracing for a Gaussian beam (e.g. Einziger and Felsen

[50]). Finally both the FEM and MoM methods have used to analyze the problem of

plane wave scattering by a circular shell(e.g. Peterson et al. [11]. These numerical

methods are directly applicable to the Gaussian beam problem. It is realized that

for the purpose of calculating material properties, an analytical or semi-analytical method for the forward problem is desirable. Therefore in this work attention has been focused on complex ray tracing and the Fourier expansion method.

After calculating the scattered fields, signals measured by the analyzer should be formulated to complete the model. The classical Friis transmission formula for far

field “coupling” between two antennas is not applicable. Thus the answer to the following question is not immediately obvious.

Issue 4: Relationship between the scattered field and the measured signal?

Asocalled“beammodecouplingformula”wasfirstpickedupinaliteraturesearch.

This formula was proposed for calculation of coupling and conversion coefficients in optical systems(Kogelnik [28]). Examination of the details shows that the theory starts with a TEM assumption, which is acceptable for optical beams4. The formu-

lation is based solely on mathematical reasoning5. On the contrary, the theory of plane wave scattering matrix (PWSM) is seen to give a much better physical picture on the coupling mechanism. The PWSM theory was documented in an NBS6 note

(Kerns [29]). Although not widely used, it has an important application in the probe

4As a matter of fact, whether or not the TEM assumption is acceptable depends on the beam parameters (or its spectrum). In optical systems, the beamwidth is relatively big. 5the coupling coefficient is defined mathematically from the first term in a series expansion of one beam in terms of beams of another beamwidth. 6National Bureau of Standards,now NIST(National Institute of Standards and Technology). §1.2. Overview of the Issues Studied in This Thesis 14 correction problem in antenna near field measurements (Hansen et al. [12] chapter

6). The model presented in this thesis is based upon the PWSM theory.

At this point, the main issues examined in this thesis that are related to the mod- eling of spot-focused free-space measurement have been described. However, the most difficult one has been left untouched. It is the multiple reflection between the antenna and the material sample. Because they are located close to each other, experiments have shown that multiple interactions can not be completely ignored. Incorporating them into the forward scattering model seems impossible at this stage. Therefore the problem has been tackled with an experimental technique: time gating feature of the network analyzer. This technique has been found useful for characterizing the discontinuities in high speed circuits (transmission lines) and measuring the radiation pattern of broadband antenna (time domain antenna measurement). Its use in the spot-focused free-space measurement system was established when the method was originally proposed. The basis of time gating is not a matter of electromagnetic the- ory, but rather a subject in digital signal processing. The essential idea is to isolate the desired response by filtering the time domain signal. Therefore how to apply the time gating is a problem of filter design. Selection of filter parameters do require a clear understanding of wave scattering and should not be considered as a trivial problem. Furthermore, the time gating as applied in the HP8510C network analyzer is achieved through numerical computation. Error are introduced even if the mea- sured data in the frequency domain perfectly matches the physics. There has not been a study on the magnitude of this error, partially due to the fact that algorithms and technical parameters of the time gating filter haven not been published in the analyzer’s user manual or other literature. Some preliminary result will be presented §1.2. Overview of the Issues Studied in This Thesis 15

in this thesis.

Issue 5: Time gating error and selection of gate shape parameters?

Another experimental issue is the network analyzer calibration, which is commonly explained as an error correction procedure. The error refers to the discontinuities in the internal microwave circuits of the analyzer and in the coaxial feed lines and

adapters of the antennas. Calibration of the free-space setup for normal incidence

measurement has been established with the TRL method, where the “TRL” refers to

three standards (Thru, Reflect, and Line). Calibration of bistatic free-space is an issue

which has not been completely settled. A two tier calibration and its corresponding

results will be presented in this thesis.

The second topic of this thesis is a study of the open-ended coaxial probe and

its potential application to chiral material characterization. The result presented is

purely theoretical. Essentially what has been solved is the excitation of TEmn modes in the coaxial line when terminated with a chiral medium. These higher order modes

are not excited if the sample is simply isotropic. It is recognized that not all of the

theoretical aspects of this problem have been pursued to completion. An example is

the convergence property.

Implementation of this method is hindered by several practical problems. The fist

is the inhomogeneity of the available chiral samples. Because the probe is restrictedly

small for the frequency range of interest, it is not sensing an average chiral effect as

prescribed by the design method of the chiral composites. The second problem is

related to the calibration of the probe. The present accuracy is judged as insufficient

for detection of the relatively weak chiral effect. Nevertheless, it is too soon to

conclude that the method is of no value for characterizing a chiral medium. §1.3. Summary of Contributions 16 1.3 Summary of Contributions

The contribution of this thesis can be divided into three parts according to the mea-

surement setup being studied.

Part 1: Justification of the plane wave assumption used in inverting the material properties of planar slabs using a focused beam system:

• Existing theory for 3-D Gaussian beam scattering by simple shapes and antenna

near-field scanning is combined together to obtain an improved model for the

spot-focused free space measurement system. The improvement lies in the fact

that characteristics of the antenna are now manifested in formulation of the

measured response. The model is intended for use in estimating potential error

caused by the deviation of actual incident beam from the assumed uniform plane

wave.

• The model is applied to simulate free-space measurement of planar slab samples.

Numerical results are presented to validate the plane wave approximation used

in the inversion process.

• The model is further employed to study the free-space bistatic measurement

setup. Three dimensional formulation is used so as to account for possible

complication caused by focal shift and angular shift of the reflected beam. Cal-

culation shows that, depending on the sample properties and thickness, the

difference between the reflection coefficients of Gaussian beam and plane wave

might be much smaller than the possible measurement error. Thus inversion of

measured data using the plane wave model is appropriate even without correc-

tion for the defocusing effect. §1.3. Summary of Contributions 17

• Bistatic measurements are performed with a two-tier calibration7. First, a full

two-port coaxial calibration is performed so that the measurement plane is set

at the coaxial ports immediately before the adapters of the lens antenna. This

is followed by a simple “response” calibration so that the measurement plane

is moved to the surface of the sample. Inversion of measured data for Teflon,

Plexiglas and glass slab samples shows that this procedure produces permittivity

values within 10% difference from published data.

Part 2: Inversion of dielectric properties of curved slabs using only reflection measurement performed with a focused beam system:

• An ad hoc technique is presented for estimating the reflection of two dimensional

Gaussian beam by a circular shell. This technique is compared with the complex

ray tracing method for the special case of a circular dielectric interface.

• Experimental results for spot-focused free-space measurement of curved Plex-

iglas samples of several radii of curvature are obtained. Measured reflection

coefficients in the frequency range of 8GHz-13.7GHz are compared with the-

oretical predictions calculated with a two dimensional Gaussian beam model.

Close agreement in magnitude is observed, while the phase difference is within

150.

• A practical technique is proposed for estimation of the time gating error in

network analyzer measurement. This technique can be used to find out desirable

minimum gate span for free space measurement.

7It is found that the offset-short calibration method [10] is un-suitable without accounting for the effect of multiple reflections. In contrast with the normal incidence setup, time-domain gating has not been implemented in the bistatic measurement. §1.4. Organization of the Thesis 18

• Inversion for permittivity of curved slab sample using measured reflection coef-

ficients is studied with several approaches, including an optimization procedure

using only magnitude data and a curvature correction procedure. The results

are carefully evaluated and possible improvements are discussed.

Part 3: A model for estimating the reflection coefficient of coaxial line radiating into a chiral half space is presented. The model has potential applications in develop-

ing a coaxial line method for measurement of chiral parameter, and also in studying

the property of aperture antenna covered by chiral media.

1.4 Organization of the Thesis

This thesis is organized into five chapters and four appendices. Chapter 1 (this

chapter) is a general introduction on the nature of the research, its primary goal

and relevant theoretical and experimental issues. Chapter 2 is concerned with the

spot-focused free-space measurement system. Discussion is presented in the order of

normal incidence measurement, bistatic measurement. Chapter 2 is concerned with

the characterization of curved slabs. For each of these measurements, relevant theo-

retical formulation is presented first, followed by experimental results and appropriate

comparison. Chapter 4 presents a theoretical treatment of open-ended coaxial line terminated by a chiral medium. Chapter 5 provides a summary and some outlook for future research.

The first two appendices discuss the concepts of Gaussian beam, near field and far field of antenna radiation. These concepts are important for understanding the §1.4. Organization of the Thesis 19 models in chapter 2. Appendix C describes the selection of time gate parameters for the experiments reported in this thesis. It also includes a technique for estimating time gating errors and finding the minimum gate span required for a certain accuracy.

Appendix F contains details of mathematical derivation for two equations in chapter

3. Chapter 2

Theoretical Model for Measuring the Dielectric Properties of Planar Slabs Using a Spot-Focused Free-Space System

A spot-focused free-space measurement system is examined in this chapter. We will

first explain the details of the system is described, with emphasis on the fact that material sample is located at the near field of the antennas. We then clarify the meaning of reflection and transmission coefficients measured in such a system. The plane-wave scattering matrix theory of antenna-antenna interaction is utilized to ex- press the coefficient mathematically. This is considered the cornerstone of the model.

The assumptions made in the derivation are discussed.

To estimate the errors induced by a plane wave model, Gaussian beams are used to represent the antenna radiation. The beam reflection coefficient is calculated and compared with those of uniform plane wave. For commonly seen material and thick- ness, the difference is found to be generally much smaller then potential measurement error. This demonstrates the validity of theuniformplanewaveassumptionusedin the inverse problem for obtaining material parameters (permittivity and permeabil-

20 §2.1. Introductory Remarks 21 ity) from the measured S-parameters. Experimental results are presented for both monostatic and bi-static setup.

Characterization of curved slab samples is investigated both theoretically and experimentally. An simple technique is presented for modeling a Gaussian beam scattering by a circular shell. Inversion of measured data is approached in several ways. The results are carefully examined and possible improvements proposed.

2.1 Introductory Remarks

The free-space material measurement method employing focused microwave beams has attracted increasing attention in recent years [14]-[31]. The method is based on a simple idea: material parameters can be inverted from measured S-parameters for focused beams interacting with a slab sample. Focused beams are used mainly to reduce the overall size of the system so that it can be conveniently setup inside a common laboratory without the need of an expensive anechoic chamber, and to min- imize the effect of edge diffraction so that samples with small transverse dimensions can be measured. Other advantages of this method includes its non-destructive and contactless nature([15] [16]), the ability to measure complex permittivity, complex permeability and other constitutive parameters (e.g. chiral parameter [18]), and the capacity to cover a wide frequency bandwidth (8GHz-110GHz, [17]) and non-planar samples [13].

Representative researches on the free-space method include but are not limited to the following. To reduce the cost of ownership, Matlacz and Palmer has built a system using commercially available reflector antennas [31]. Pushing for a wider §2.1. Introductory Remarks 22 bandwidth, Hollinger et al. reported a system operating from 8GHz to 100GHz [17], and Friedsam et al. described a system in 75GHz to 95GHz range. Varadan et al. obtained the permittivity of curved slab sample using the transmission measurement

[13]. The method has also been applied to determine chiral parameter of reciprocal bi-isotropic materials (Varadan [18] and Cloete [33]). The method is proved to be quite versatile, accommodating simultaneous determination of both complex permit- tivity and complex permeability [15], temperature dependence of material properties

(Varadan, [16]). To increase measurement accuracy, Umari et al. investigated the de- focusing effect for bistatic setup [34], and established a bistatic calibration technique using offset shorts. More importantly, Ghodgaonkar et al. presented the implementa- tion of TRL (Thru-Reflect-Line) calibration method in free-space system [14]. With the TRL calibration, accuracy better than ±2% was reported in reference [15] for permittivity measurement.

As already noted in chapter 1, to apply the free-space method on curved objects,

field distribution along the transverse directions should be considered. Understanding the accuracy of the results for planar samples also calls for a study on the use of uniform plane wave for modeling the focused beam, which seems to have been a common practice in the above mentioned papers. With the plane wave assumption, the inverse problem is relatively simple because analytical formulas are available for both the reflection and transmission coefficients. This assumption deserves careful consideration because of a dilemma. For the sake of argument, accept temporarily that the focused beam can be represented by a Gaussian beam. One observes that both the field amplitude variation in space and its spectral density are given by the

Gaussian distribution. In order for the spectral density to come close to a Dirac §2.1. Introductory Remarks 23 delta function, which corresponds to a uniform plane wave, the beam width needs to be large compared with wavelength. On the other hand, to ensure the relatively small sample size in the transversal direction–one of the advantages of the free space method–the beam width should be kept as small as possible while preserving a planar phase front. The same observation applies for system with any focused beam shape.

It is therefore interesting to find out whether a compromise has to be made so as to balance the need for compactness of the system (smaller beamwidth) and the need for simplicity in the inverse problem (plane wave model). The criteria is obviously accuracy of the measurement results.

In the past, confidence in the plane wave assumption was obtained experimentally by comparing results for a number of well-characterized materials with published data

(Ghodgaonkar [14]). However, as a general rule in electrical parameter measurement, one cannot expect the same accuracy for all samples with different properties and thickness. This fact can be observed by plotting the magnitude of plane wave reflec- tion coefficient for a slab of fixed thickness but changing permittivity ε.Asshownin

Fig. 2.1, because the slope changes with ε, the same magnitude of measurement error in the reflection coefficient Γ will induce different error in the resulting ε.Computer simulation of the measurement process can provide guidelines on how to set up a measurement appropriately. For example, thickness of the sample material can be chosen to obtain a higher sensitivity.

The focused beam field can be more accurately represented as a Gaussian beam, whose interaction with dielectric slab has been studied extensively. For example, Riesz and Simon [19] discussed the origin of focal shift and angular shift for Gaussian beam reflection from a dielectric slab. Shah and Tamir [22] studied the abnormal absorption §2.1. Introductory Remarks 24

} 0.6 1 same error in |Γ| 0.5 0.4 | Γ | 0.3 } 2 0.2 0.1 small error in ε big error in ε } 2} 4 6 8 10 Permittivity ε

Figure 2.1: Magnitude of normally incident plane wave reflection coefficient as a function of permittivity ε for a dielectric slab of fixed thick- ness (5.334mm). The same measurement error at point 1 and 2 will induce different error in the resulting ε. of Gaussian beams incident upon a lossy layered media. Lee and Marcuvitz [23] gave a different perspective on the lossy layer problem. These analysis are based upon the

Fourier transform method. Using the beam mode expansion method, Ooya et al. [20] analyzed the transmission and reflection of a Gaussian beam at normal incidence on a dielectric slab. The same method is applied to multilayered slab by Tanaka et al. [21].

Scattering of Gaussian beam at a planar dielectric interface has also been tackled with the complex ray concept (Ra et al. [25]) and other high frequency techniques such as the steepest descent method (Lu et al. [24]). The Gaussian beams are assumed as two dimensional in all of the studies mentioned in this paragraph.

Recognizing the importance of a beam assumption in free space measurement, Luk and Cullen [37] have undertaken the analysis of three-dimensional Gaussian beam reflection from a short-circuited isotropic ferrite slab at oblique incidence. Their analysishasbeengearedtowardananalytical treatment, with primary results being §2.1. Introductory Remarks 25

approximated formulas for the 3-D beam shift and depolarization. Assumption has

been made of a large beamwidth, which is not necessary if numerical calculation is

resorted to. Examination of their derivation also shows that not all of the steps

have been treated appropriately, as will be pointed later in an improved version of

their formulation. In addition, their analysis stopped short of giving an expression

for the reflection coefficient, which is the important second step, after formulation of

scattered field, for the modeling of free-space system.

In contrast to a plane wave model, the concepts of reflection and transmission co-

efficients are not well defined for the focused-beam system. Since the scattered fields

have been formulated in above mentioned papers, the paramount issue at hand is

therefore the formulation of these coefficients as measured by the network analyzer at

the coaxial feed of the horn-lens antennas. Literature survey shows that researchers

have defined these coupling coefficients in at least two different ways [26, 27]. In an-

alyzing a focused-beam system for measuring the scattering cross section of a small

cylinder, Shavit et al. [26] resorted to the beam coupling formula for defining the transmission signal. Although their problem is different from material characteri- zation, this reference provides the first light on the current issue. Smith et al.,in discussing the tolerance in the measurement of RAM1 reflectivity [27], provided a formula for the reflection coefficient which is conceptually different from the beam coupling formula. Numerical results from these definitions do not agree with each other, raising the question of which one is more appropriate. While Smith’s defin- ition has been given without much explanation, the origin of beam-mode coupling formula can be traced to Kgelnik [28] in his analysis of conversion coefficients for

1Radar Absorbing Material §2.1. Introductory Remarks 26

optical modes. His analysis starts with a TEMassumptionandreliesonscalarwave

equation, which is common for optical system. The derivation seems to base solely on

mathematical reasoning, where the basic idea is that optical beam with beamwidth

w1 can be expanded into series of beams of beamwidth w2 = w1.Theconversion

coefficient is just the (first) expansion coefficient. Generalizing this idea into vector

fields requires some further insight into the physics of the coupling process. Luckily

this is provided by Kerns [29] in his plane-wave scattering matrix (PWSM) theory of

antenna-antenna interaction, which has been extensively used for the probe-correction

problem in antenna near field measurement. Because due respect is paid to math-

ematical rigor, the derivation in [29] seems quite tedious. The essential concept,

however, is the Lorenz reciprocity theorem, and a straightforward presentation can

befoundfromHansen[30].UsingthePWSM theory, reflection and transmission coefficients can be easily defined for the free-space system, and a complete theoretical framework for its modeling is obtained.

The theory is first applied to the case of free space measurement at normal in- cidence. The reflection coefficient of 2-D Gaussian is calculated and compared with those of uniform plane wave. Numerical results show good agreement for moderate beam radius (1.5 wavelength) and commonly seen samples with ε in the range of 2 8 and thickness < 0.5 wavelength. This validates the uniform plane wave assumption used in the inversion process.

Secondly, the bistatic free-space measurement setup is analyzed in section 2.8, with focus on two of the potential sources of error. The first is the direct coupling from the transmitting antenna to the receiving antenna when no sample is present.

This direct coupling can be thought of as a leakage effect and was neglected in previ- §2.2. Description of the Spot-focused Free-space Measurement System 27

ous study (Umari, [34]). Secondly, the de-focusing effect is re-formulated. The term

de-focusing, as defined in Umari’s thesis [34], refers to partial overlapping between the

focal areas of transmitting antenna and receiving antenna. There is some ambiguity

in this definition because it assumes that field is uniform within the boundary defined

by the -6dB amplitude point and is zero outside such an area. This ambiguity is re-

moved by using the Gaussian beam assumption and the PWSM theory. 3-D Gaussian

beam is used in the formulation and two conclusions are obtained from theoretical calculation. First, the angle subtended by the two antennas in bistatic setup should be as small as possible to reduce the direct coupling. Secondly, depending on the properties of sample under test, the defocusing effect might be much smaller than potential measurement errors. Under this circumstance, inversion of measured data without correction for the defocusing effect is justified. This point is demonstrated by experimental results obtained with a two-tier calibration procedure.

2.2 Description of the Spot-focused Free-space Mea- surement System

The free-space measurement system used in this study was developed in late 1980’s

[14]. It has been successfully applied to accurate measurement of the complex per-

mittivity, complex permeability, and chiral properties of microwave composite ma-

terials [13]-[18]. The system components are illustrated in Fig. 2.2. A photograph

of the system is shown in Fig. 2.3. A vector network analyzer is connected to two

conical horn-lens antennas via coaxial cables and coax-waveguide transitions. High

frequency oscillation synthesized by the network analyzer is feed to antenna through

these waveguide components. For normal incidence measurement, the antennas are §2.2. Description of the Spot-focused Free-space Measurement System 28 mounted facing each other and the material under test, typically in the shape of a rectangular slab, is placed in the center. The antennas send out linearly polarized

(nominal) microwave beams focusing on the surfaces of the slab sample. Reflection and transmission fields are coupled into the antennas, and the measurement output is in the form of scattering parameters for a linear two-port. These data are then used in a computer program to calculate the material parameters. The program im- plements an inverse scattering process based on the model of a homogeneous plane wave interacting with a slab extending to infinity in the transverse direction [14].

Computer Network Analyzer

CH1 S11 1 U FS C LOG or MA G PHA SE Hi DEL d AY SMI CHATH RT POL HP-IB AR LIN MA G SW R MO RE

START .300 000N MHZ

Horn Lens Antennae Sample Material

Figure 2.2: Illustration of a spot-focused free-space measurement system.

The vector network analyzer consists of three modules: an HP83651B synthesized sweeper capable of generating radio frequency (RF) signals of 10MHz to 50GHz; an

HP8516A S-parameter test set providing signal separation in frequencies 45MHz to

40GHz, and an HP8510C network analyzer for IF signal processing and measurement control. Performance of this combination is not provided in the HP8510C datasheet found in the manufacturer’s website. However performance of the following system §2.2. Description of the Spot-focused Free-space Measurement System 29

1 2

6

3 4

5 8 7

9 8

Figure 2.3: A picture of the spot-focused free-space measurement system used in this study. The components are: 1—Vector Network Analyzer(HP8510C); 2—Horn-lens Anten- nas; 3—Waveguide adapter(Rectangular to circular); 4—Adapter (Coaxial line to rectangular waveguide); 5—Coaxial cable; 6— Sample holder; 7—Sample holder fixture; 8—Antenna fixture; 9— Circular slots on the aluminum table for the bistatic setup. hardwares can be used as a reference: 8517B S-parameter test set, 83651B synthesized sweeper and 8510C network analyzer. Under a full two-port calibration with sliding loads using the 85056A 2.4mm calibration kit, the worst case uncertainty in reflection

o measurement (S11 at 2GHz-20GHz) is as follows: ±0.012 for the magnitude and ±1.5

o for the phase at |S11| =0.5; ±0.025 for the magnitude and ±1.1 for the phase at

|S11| =1.0. These data are of interest in deciding which model to be used in the inversion process for material characterization.

Theantennasaremountedonmovablebasesatopanaluminumtable.Theycan be translated so that the distance between them can be adjusted, so is the distance §2.2. Description of the Spot-focused Free-space Measurement System 30

between the antenna and the sample under test. The antennas can be rotated with

respect to two axes: one is the antenna axis (cone axis) and the other is the center line

perpendicular to the aluminum table. Consequently, cross-polarized component of

transmitted signal can be measured, and the antennas can be arranged into a bistatic setup. The minimum angle subtended by the antenna axes that can be achieved in

the bistatic setup is about 40o. Diameter of the circular aperture is about 30.5cm,

which is equal to the nominal focal distance. The latter refers to the distance between

thefocalplaneandsurfaceofthelens.Ingeneral, the focal planes are adjusted on

the surface of sample during measurement. This means the sample is located in the

near field region of the antenna radiation2.

The prominent advantage of the above free-space setup lies in the use of spot- focused beam. Because most of the energy is concentrated on the central region of

the material sample, diffraction effects from the edges are minimized. Therefore, the

size of the sample is reasonably small, and the sample can also be enclosed inside an

oven for controlling the temperature [16].

It is important to note that the network analyzer actually measure the modal

voltages of TEM modes that are induced in the coaxial lines. So the essential task of

the simulation is to relate these modal voltages to the fields in the free space region

surrounding the material sample.

2See appendix B for a discussion of near field and far field §2.3. Modeling Assumptions 31 2.3 Modeling Assumptions

The basic objective of a theoretical model is to estimate the scattering parameters, namely the reflection coefficients and transmission coefficients measured at the test ports of the network analyzer. This can be achieved in two steps: first, calculate the reflected and transmitted fields; second, estimate the coupling of the fields into the network analyzer. To simplify the analysis and capture the essence of the problem, the following assumptions have been made:

• The horn-lens antennas are linear and the sample under test responds linearly

to electromagnetic excitation. Therefore the principle of superposition is ap-

plicable. This assumption is used in formulation of the coupling coefficients.

Network analyzer measurement is also basedontheassumedlinearityofthe

network.

• The spot-focused microwave beam sent out by the antennas can be represented

as a linearly polarized fundamental-mode Gaussian beam. This assumption is

intended to bypass the laborious task of calculating the scattering of a quite

arbitrary field, often given as antenna field mapping at discrete locations(the

methodology is applicable in the latter case).

• Transverse dimension of the sample is large so that diffraction from the edges

can be ignored. For curved samples, it is further assumed that complicated

phenomenon such as excitation of creeping wave can be ignored.

• The antennas are reciprocal. Multiple scattering between the antennas or be-

tween the antenna and the material sample can be ignored. This assumption

is essential for the theoretical model that will be presented. Although multiple §2.3. Modeling Assumptions 32

reflections can be formally included in the model, no quantitative result can be

obtained due to the complexity of the scattering process. In practice, multiple

reflections are removed with the time gating capacity of the 8510C network

analyzer.

• A 2-D model is used for circular cylindrical slab. Thus effects such as the depo-

larization associated with the 3-D nature of the real beam have been ignored.

For the free-space bistatic measurement of a planar slab, 3-D Gaussian beam is

used and cross-polarized components are accounted for.

Since the Gaussian beam assumption directly affects the accuracy of the simulation results, it is interesting to find out how closely this represent the true field distribution.

In the following, antenna far-field pattern will be used to estimate the near field distribution. The result is compared with a near field measurement using a custom made dipole antenna.

A simple and useful relationship exists between the far-field pattern and the plane wave spectrum(PWS) of an antenna. In the far zone of the antenna

j −jkr x, y, z exp( )k θ k θ ϕ, k θ ϕ E( )= r cos A( sin cos sin sin ) (2.1) where (r, θ, ϕ) is a spherical coordinate system centered at x = y = z =0,which is the location of the antenna. Function A(kx,ky) is the PWS of the antenna near field and k is the free space wavenumber. The PWS of a Gaussian beam also has a −x2/w2 ⇒ −k2w2/ Gaussian shape: exp( ) exp( x 4). Therefore by examining the PWS, one can determine whether the near field is Gaussian or not. §2.3. Modeling Assumptions 33

E-plane and H-plane far field patterns of the horn-lens antennas were provided

by the manufacturer(or some independent service provider) when the system was

developed. Typical shape of the antenna pattern is shown in Fig. 2.4, which was

obtained by digitizing a scanned image of the original pattern for 11GHz. Some

sidelobes seen in the original document have been omitted. The patterns are visually

found to be fairly Gaussian. After a division by the factor cos θ as required in eq. (2.1)

and fitting the curves as functions of sin θ by a Gaussian distribution, the resulting

values for the 1/e half beams width w of the antenna aperture field are shown in table

2.1.

1.0 E-Plane H-Plane

0.8 6.8o(H) o 5.5 (E) -3dB beam width 0.6

|E| 0.4

0.2

0.0

-20 -15 -10 -5 0 5 10 15 20 Elevation Angle θ (in degrees)

Figure 2.4: Far field pattern of the horn-lens antenna at 11GHz provided by the manufacturer(some sidelobes have been omitted).

Notice that no phase distribution information can be obtained from the far field

pattern, and the beam widths in table 2.1 are estimations on the antenna aperture.

To find the minimum beam width (on the focal plane), phase distribution of the PWS §2.3. Modeling Assumptions 34

Frequency 9GHz 10GHz 11GHz E-plane 3.38λ0 3.76λ0 4.17λ0 H-plane 2.58λ0 2.80λ0 3.32λ0

Table 2.1: 1/e half beam width w on the aperture of the X-band horn lens antenna estimated from the far field pattern. λ0 is the free space wavelength.

Frequency 9GHz 10GHz 11GHz E-plane 1.01λ0 1.00λ0 1.00λ0 H-plane 1.32λ0 1.34λ0 1.25λ0

Table 2.2: Minimum 1/e half beam width w0 on the focal plane (beam waist) of the X-band horn lens antenna estimated from the aperture field and a quadratic phase front assumption for the aperture field.

is needed. A reasonable guess is that the aperture field has a quadratic phase front

3 with a radius of curvature R0 equal to the focal distance . If this is the case, the

minimum beam width w0 is related to w and R0 by (p.164, Ishimaru [85]):

λ R w 0 0 0 = πw (2.2)

4 The results are listed in table 2.2, where R0 =35.56cm has been assumed .Notice

that from the above equation the beam spot size 2w0 becomes smaller as the original

beam size 2w increase for given wavelength λ0 and focal distance R0. This means in

order to realize a smaller focal spot size, relatively bigger antenna is needed.

3This should have been the objective of the lens design. 4The number 35.56cm is the sum of focal length (30.48cm) mentioned above and the thickness of the lens, i.e. the aperture is taken at the open end of the conical horn. §2.3. Modeling Assumptions 35

It is worth mentioning that the -3dB beam widths cited by the manufacturer are

1.153in and 1.426in for the E-plane and H-plane respectively. These numbers are seen

to be given by the formula R0α,whereα is the far field beam width (an angle, see

Fig. 2.4) in radian. The underlying assumption is that the focal plane is in the far

field region, which is not the case for the current focus system. Nevertheless, if these

-3dB beam widths are correct, the corresponding 1/e half beamwidths are calculated

at 11GHz as 0.912λ0 in the E-plane and 1.13λ0 in the H-plane. Assuming a Gaussian

distribution, the -3dB beam width w3dB is related to the 1/e half beam width w0 by √ the relation w0 = w3dB ln 4.

1 | 0.5 y |E

0 0 3 5 8 10 13 y ( 15 ) uni mm t: 5 18 t: 5 mm 20 uni ) 23 x ( 25 28 30

Figure 2.5: Field amplitude distribution over a 16cm×16cm rectangular area on the focal plane, measured at 10.92GHz using a custom-made dipole. A Gaussian shape is observed. §2.3. Modeling Assumptions 36

20 20 12 12 4 4 -4 phase

-4 phase -12 -12 -20 -20 18 18 17 16 y (unit:16 5mm) 15 14 14 13 12 12 11 10 x (unit: 5mm) 9 10

Figure 2.6: Phase distribution of the focal region field measured at 10.92GHz using a custom-made dipole. It is seen to be non-uniform. But this might be largely an indication of the measurement error.

The antenna near field distribution on the focal plane has also been measured with a custom-made dipole antenna over a 16cm×16cm rectangular area with a uniform spacing of 0.5cm along each direction5. The field amplitude measured at 10.92GHz is shown in Fig. 2.5. The corresponding phase distribution over a smaller area surround- ing the focal point (amplitude maxima) is shown in Fig. 2.6, where the measured data have been interpolated to plot a denser mesh. It is seen that the amplitude distribu- tion is fairly Gaussian. But the phase distribution is much different from the desired uniform pattern(a planar phase front). This is judged as an indication of measure- ment error due to the difficulty in positioning the dipole antenna. Similar data are obtained for 20 other frequencies ranging from 7.8GHz to 13GHz. The amplitude

5Thanks to Dr. Kollakompil of ESM department at PSU, and Mr. A. Tellakula of HVS Tech- nologies, State college, PA, for providing the data of field mapping §2.3. Modeling Assumptions 37 data are fitted by a two dimensional Gaussian distribution:

x − b 2 y − d 2 a − − object function = exp c e (2.3)

The resulting values for the fitting parameters a, b, c, d, e are listed in table 2.3, along with the standard error of fit.

f(GHz) a b(5mm) c(5mm) c(λ0) d(5mm) e(5mm) e(λ0) StdErr 7.8 1.196 13.525 8.994 1.169 15.445 8.580 1.115 0.05865 8.06 1.162 13.563 9.329 1.253 15.528 8.768 1.178 0.07314 8.32 1.226 13.532 7.993 1.108 15.389 7.679 1.065 0.08289 8.58 1.351 13.673 8.273 1.183 15.327 7.897 1.129 0.07638 8.84 1.307 13.874 8.377 1.234 15.256 7.947 1.171 0.06398 9.1 1.261 13.703 7.842 1.189 15.193 7.627 1.157 0.06811 9.36 1.289 13.709 7.443 1.161 15.188 7.207 1.124 0.06753 9.62 1.315 13.802 7.316 1.173 15.042 7.067 1.133 0.07316 9.88 1.296 13.826 7.171 1.181 15.101 6.967 1.147 0.07281 10.14 1.308 13.918 7.142 1.207 15.129 6.872 1.161 0.06324 10.4 1.331 13.786 6.769 1.173 15.092 6.560 1.137 0.06208 10.66 1.309 13.737 6.793 1.207 15.103 6.575 1.168 0.05814 10.92 1.350 13.744 6.349 1.156 15.138 6.189 1.126 0.06571 11.18 1.390 13.653 6.200 1.155 15.283 6.115 1.139 0.06925 11.44 1.455 13.640 6.229 1.188 15.247 6.038 1.151 0.07291 11.7 1.448 13.559 6.261 1.221 15.347 5.917 1.154 0.05996 11.96 1.434 13.636 6.001 1.196 15.294 5.919 1.180 0.06551 12.22 1.432 13.670 5.914 1.204 15.412 5.919 1.205 0.06592 12.48 1.538 13.377 5.814 1.209 15.555 5.571 1.159 0.07385 12.74 1.558 13.429 5.912 1.255 15.347 5.678 1.206 0.07577 13 1.629 13.495 5.552 1.203 15.546 5.272 1.142 0.06035

Table 2.3: Results of fitting the field amplitude on the focal plane to the 2 2 a − x−b − y−d 2-D Gaussian distribution: exp c e .Notethat the values of parameter c and e are given in two units: 5mm(the sample spacing) and λ0 (the free space wavelength). §2.3. Modeling Assumptions 38

Observe from table 2.3 that the half 1/e beam widths c for E-plane have an average value of 1.19λ0, while the e’s for H-plane have an average of 1.15λ0.These should be compared with the 1.00λ0 and 1.3λ0 estimated above from the far field pattern. The standard error in the last column of table 2.3 shows that the field amplitude is over 91% Gaussian, in a statistical sense. Fig. 2.7 and 2.8 further show the variation of c and e with respect to the frequency, along with results of linear

fitting. The beam width parameters obtained in this section are used as reference in later calculations. It should be emphasized that these estimations are very rough. A higher level of confidence can only be achieved by analyzing the percentage error in near field measurement data. This has not been pursued in the present study.

1.26 c 1.24 linear fit 1.22 1.20 1.18 1.16 1.14

half beam width (E-plane) linear fit y=A+Bx parameters:

e 1.12

1/ A=1.1268, B=0.006239. 1.10

7 8 9 10 11 12 13 Frequency(GHz)

Figure 2.7: The E-plane 1/e half beam width c as a function of frequency. The result of a linear fit is also shown. §2.4. Meaning of “Reflection Coefficient” for Beams 39

1.22 e 1.20 linear fit

1.18

1.16

1.14

1.12

1.10 half beam half width (H-plane) e linear fit (y=A+Bx) parameters: 1/ 1.08 A=1.0493, B=0.009671 1.06

7 8 9 10 11 12 13 Frequency(GHz)

Figure 2.8: The H-plane 1/e half beam width e as a function of frequency. The result of a linear fit is also shown.

2.4 Meaning of “Reflection Coefficient” for Beams

Of paramount importance in simulation of the free-space system is the definition of reflection and transmission coefficients. This is not an issue for existing model where a plane-wave assumption is made and planar slab sample is considered. It is noted that there is an equivalent transmission line description for homogeneous plane wave transmission through and reflection from a plane-parallel slab. Essential to this network representation is the similarity between the distribution of the incident fields and the scattered fields. In fact, they are uniform along the transverse directions so that the only variable in their field expressions is the complex amplitude. This complex amplitude is called a “voltage” or “current” in network theory because it satisfies a transmission line equation. The reflection and transmission coefficients for this linear network are mathematically well defined and physically meaningful. For a focused beam propagating in free space or through a dielectric slab, there is no §2.4. Meaning of “Reflection Coefficient” for Beams 40

equivalent network representation. That is, the usual transmission line model is not

appropriate for describing the propagation of a beam. The basic reason is that the

beam becomes divergent or de-focused as it propagates. The field distribution in the

transverse direction is dependent upon the propagation distance. It is also affected

by the presence of a material interface. For the free-space system, since what the

network analyzer actually measures is the TEM mode voltage in the coaxial feed of

the antenna, to come up with a definition for the scattering parameters, it is necessary

to consider how the beam field in the open region is coupled into the coaxial line.

The beam mode coupling formula first presented in [28] for optical or quasi-optical systems has been used to define the received signals in antennas [26]. With reference to Fig. 2.9, the coupling or conversion coefficient from the incoming beam into the system beam is defined as[28]:

∞ c u y u∗ y dy mm¯ = ¯m¯ ( ) m( ) (2.4) −∞

whereu ¯m¯ and um are the potential functions for two Hermite-Gaussian beams at the common reference plane. For two fundamental Gaussian beams, the above formula reduces to:

4 κ = (2.5) 2 2 (W0 W¯ 0 + W¯ 0 W0) + λ(X + X¯) (πW0W¯ 0)

for the coupling of power. A perfect conversion (κ=1) is realized only for matched

beam widths (W0 = W¯ 0) and overlapping beam waists(X + X¯ =0).

In terms of incident field distribution Ei and scattered field distribution Es,the

coupling formula can be written as an integral over a reference plane A perpendicular §2.4. Meaning of “Reflection Coefficient” for Beams 41

y Incoming beam System beam mode uym () uym ()

W W0 W W0

X X reference plane

Figure 2.9: The coupling between two Hermite-Gaussian beams with mini- mumhalfbeamwidthsW0 and W 0 and beam waist located at X and -X from the reference plane. W and W¯ are the correspond- ing half beam width at the reference plane. to the beam propagation direction [26]:

κ = EiEs∗dA. (2.6) A

Note that Ei actually corresponds to the system beam mode, and Es corresponds to the incoming beam in Fig. 2.9. If Es is substituted by the reflected or transmitted

field, then κ is the reflection or transmission coefficient for the current problem. Su- perscript * in eq. 2.6 denotes complex conjugation. Considering the physical meaning of κ, location of the reference plane A is quite arbitrary. For example, it can be the focal plane in the free space system. The only requirement is that it should separate the components that are generating the incoming beam and the system that is sensing this beam. The latter generates the system beam mode in Fig. 2.9. However, it can be shown that the coupling coefficient κ defined by eq. 2.6 is not strictly independent of the reference position. Variation in the phase can be observed if the incoming beam or the system beam contains significant longitudinal field components. §2.4. Meaning of “Reflection Coefficient” for Beams 42

A close inspection reveals that this definition is essentially based upon a mathe-

matical consideration. Due to the orthogonality of Hermite-Gaussian beam modes, a

general incoming beam(not necessarily Gaussian) can be expanded into a superposi-

tion of fundamental-mode Gaussian beam and higher order Hermite-Gaussian beams.

The expansion coefficient corresponding to the fundamental-mode Gaussian beam is

then assigned as the coupling coefficient, under the assumption that this is exactly

the system beam. No physical reasoning has been given as to why this is exactly the signal an antenna or, for the case of an optical system, a lens would pick up.

A physically more meaningful definition can be obtained by borrowing the “plane- wave scattering matrix (PWSM) theory of antennas and antenna-antenna interac- tions” developed by David M. Kerns [29]. The PWSM theory has been used exten- sively in probe correction problems related to antenna near field measurementss [30].

The treatment by Kerns is very detailed but mathematically quite involved. A more

concisedescriptioncanbefoundinreference[30].Themainideaispresentedbelow,

with appropriate modifications into a two dimensional problem.

y Transmitted Ez(y) R (k ) Receiving field T0(ky) p y Characteristic

Sa Sp a b 0 x p

Test Antenna Probe Antenna Reference Plane (x=0)

Figure 2.10: Diagram for defining the coupling coefficient between two antennas.

Referring to the geometry for coupling between the two antennas in Fig. 2.10, the §2.4. Meaning of “Reflection Coefficient” for Beams 43

incident field can be expressed by the plane-wave expansion:

a ∞ Ei x, y 0 T k −jk x − jk y dk , z( )= 0( y)exp( x y ) y (2.7) 2π −∞

where a0 is the amplitude of the coaxial line mode that is feeding the horn-lens antenna, T0 is the transmitting spectrum given by:

∞ T k 1 Ei x ,y jk y dy, 0( y)= z( =0 )exp( y ) (2.8) a0 −∞

and     k2 − k2, when |k |≤k k y y . x =   (2.9)  −j k2 − k2, |k | >k y when y

The receiving characteristic Rp(ky) of the probe antenna is defined to be the output of the probe when it is located at the point (x =0,y = 0) and is irradiated by an incident plane wave (1/2π)exp(−jkxx − jkyy). This means the output of the

probeduetothesameplanewave(1/2π)exp(−jkxx − jkyy) when it is located at

(x = x0,y =0)isRp(ky)exp(−jkxx0). The output bp of the probe antenna, when illuminated by the incident field is therefore given by:

∞

bp = a0 Rp(ky)T0(ky)exp(−jkxx0)dky. (2.10) −∞

If the probing antenna is reciprocal, its receiving characteristic Rp(ky)isrelatedto §2.4. Meaning of “Reflection Coefficient” for Beams 44

the transmitting characteristic Tp by:

1 kx Rp(ky)= Tp(−ky), (2.11) η0 ηk

where η and η0 are the characteristic impedance of free space and the propagating

mode in the waveguide feed of the probe antenna. Note that the second term on

the hand side of the above equation is the characteristic admittance of a plane wave

exp(−jkxx − jkyy) viewed as a propagating mode in x direction. This signifies that,

while the Tp corresponds to the spectral density of E field, the Rp corresponds to the spectral density of H field. Written in terms of a convolution in the spatial domain,

the spectral domain integral in eq. (2.10) is equivalent to:

a ∞ b 0 Ei x ,y Hp x ,y dy. p = z( =0 ) y ( =0 ) (2.12) ηη0 −∞

Ei z in the above equation represents the electric field generated by the transmitting Hp antenna. y is the magnetic field generated by the probing antenna in its trans- mitting mode. Details of derivation for equations (2.11) and (2.12) and an excellent

description of the underlying physics can be found in reference [29]. It should be

noted that the result of eq. (2.12) is truly independent of the position of the reference

plane. To apply the above idea in defining the reflection/transmission coefficients for

the free-space system, one can think of the reflected or transmitted fields as being

generated by a fictitious antenna under test. The desired coefficient is then given by

the coupling between the real and fictitious antenna, under the assumption that the multiple interactions between the two antennas can be ignored.

For comparison purpose, the two definitions of reflection coefficient resulting from §2.4. Meaning of “Reflection Coefficient” for Beams 45

E-E definition E-H definition (beam mode expansion coefficient) (antenna-antenna coupling coefficient)

 ∞  ∞ Γ= Es(y)Ei∗(y)dy Γ= Es(y)Hi (y)dy  −∞ z z  −∞ z y ∞ Es y −y2 w2 dy ∞ Es y ε −y2 w2 = −∞ z ( )exp( / 0) = −∞ z ( ) µ exp( / 0) 1 2y2 (1 − 2 2 + 2 4 )dy k w0 k w0

Table 2.4: Comparison of two definitions for the “reflection coefficient”. equations (2.10) and (2.12) are listed together in table 2.4, where it has been assumed that the receiving characteristic of the antenna is given by a E-polarized Gaussian

E z Es beam(the only field component is in the direction). Thus z in table 2.4 is Ei Hi the scattered (reflected) field that needs to be calculated, z and y correspond to the Gaussian beam generated by the reciprocal probing antenna. w0 is the 1/e half beam width of the Gaussian beam, and k is the wavenumber in free space. The reference plane has been chosen at the beam waist (x = 0). The formulas are in the form of a line integral along the y direction because the problem being considered is two dimensional(invariant along the z-direction). Note that the result has the

“dimension”(unit) of a time average power passing through a strip that is infinitely long along the y-direction and has a unit length along the z-direction. To obtain a reflection coefficient with magnitude less than 1, a proper normalization needs to be applied. This can be done, for example, by assigning a unit power for the Gaussian beam.

It is noted that the two definitions are different in that one is the “reaction” calculated from two E fields, the other is the “reaction” calculated from an E field and a H field. In general, the results of these two definitions are different. For §2.5. 2—D Gaussian beam scattering by a dielectric slab at normal incidence 46

special cases where scattered field Es hassomekindofsymmetryasafunctionofy,

1 it is possible that the integrations corresponding to the two extra terms(− 2 2 and k w0 2y2 2 4 )intheE-H definition will cancel out. For example, this is the case for planar k w0 perfect conductor, both definitions give the same result which is equal to the time

average power carried by the Gaussian beam. For optical frequencies, the beam field

is considered as a TEM wave, and the H field is approximately proportional to the

E field (by a factor of wave impedance in free space), the E-H definition is reduced to the E-E one. However, depending on the beamwidth, the TEM assumption is not always appropriate for the Gaussian beam. The beam field will have significant longitudinal component if the 1/e half beam width is about a wavelength. The E-H definition is more appropriate because it incorporates the reciprocity theorem.

2.5 2—D Gaussian beam scattering by a dielectric slab at normal incidence

Having clarified the meaning of the reflection coefficient measured in the free-space

measurement system, we now apply the theory to the case where the material un-

der test is a slab. The first step is to calculate the reflected field. The reflection of a

Gaussian beam from a slab has been studied extensively, with focus on the phenomena

of lateral shift (Goos-Hanchen effect), focal shift, angular shift and abnormal absorp-

tions ([19]—[25]). For the case of normal incidence, where the above phenomena are

not the issue and excitation of evanescent waves can be safely ignored, the reflected

field can be easily calculated using the plane wave expansion method. Corresponding §2.5. 2—D Gaussian beam scattering by a dielectric slab at normal incidence 47

to an incident beam given by

Ei ,y −y2/w2 , z(0 )=exp[ 0]  , ε y2 (2.13) Hi ,y 0 −y2/w2 − 1 2 y(0 )= exp[ 0](1 2 2 + 2 4 ) µ0 k0w0 k0w0 where k0 is the wavenumber in free space and w0 is the 1/e half beam width, the reflected field can be obtained as: w ∞ Es ,y √0 r k −k2w2/ − jk y dk z (0 )= ⊥( y)exp[ y 0 4 y ] y 2 π −∞ (2.14) w k0 ≈ √0 r k −k2w2/ − jk y dk , π ⊥( y)exp[ y 0 4 y ] y 2 −k0

using the plane wave spectrum expansion method. The term r⊥(ky)denotesthe reflection coefficient of perpendicular polarized plane wave incident at the angle

arcsin(ky/k0)ontotheslab:

− − jk d r k r k 1 exp( 2 1x ) , ⊥( y)= 01( y) 2 (2.15) 1 − r01 exp(−2jk1xd)

with r01(ky) being the reflection coefficient of the plane wave incident upon a planar

interface separating the free space (ε0,µ0)andthematerial(ε1,µ0):

r01(ky)=(k0x − k1x)/(k0x + k1x), (2.16)

k0x and k1x are the projections of wavevectors k0 and k1 on the x-direction(propagation direction of the beam).

After obtaining the reflected field, the “reflection coefficient” for the Gaussian §2.6. 2—D Numerical Results and Comparison with Measurement 48

beam can be easily calculated as:

1 ∞ Es y −y2/w2 − / k w 2 y2/ k2w4 dy, Γ= z ( )exp( 0)[1 1 ( 0 0) +2 ( 0 0)] (2.17) Px −∞

where the factor Px is a normalization term corresponding to the reflected signal for a perfect conduct sample:

 π P − 1 w x =(1 2 2 ) 0 (2.18) 2k0w0 2

Px can also be thought of as the total incident power passing through the x=0 plane. Similarly, expression for the “transmission coefficient” of the Gaussian beam pass-

ing through a planar slab can be obtained if we replace r⊥(ky) by the plane wave

transmission coefficient t⊥(ky).

2.6 2—D Numerical Results and Comparison with Measurement

As a numerical example, consider the case in which a 2—D E-polarized6 Gaussian beam is reflected by a dielectric (εr=2.685) slab of thickness 0.53cm at normal incidence. We plot the magnitude of the reflection coefficient in eq. (2.17) as a function of frequency

in Fig. 2.11. The half beam-width w0 is assumed to be linearly proportional to wavelength λ0 in free space. It is seen that the as w0 increases from 1.3λ0 to 10.0λ0,

the results approach those of homogeneous plane wave (denoted by w0=∞). The

spectral density of the Gaussian beam is plotted in the right hand side of Fig. 2.11,

6E-field direction is perpendicular to the direction of Gaussian distribution, while both of them are normal to the beam propagation direction. §2.6. 2—D Numerical Results and Comparison with Measurement 49 along with the reflection coefficient |r⊥(ky)| for perpendicularly polarized plane wave incident obliquely at angles specified by ky. The spectral density of the reflected field ˜ ˜ is given by Ei(ky)r⊥(ky). As w0 increases, Ei(ky) become more concentrated around ky=0, the reflected field spectrum is more closely approximated by E˜i(ky)r⊥(0), giving the result for normal incident plane wave. Since |r⊥(ky)|≥|r⊥(0)| for perpendicular plane waves, it is therefore reasonable to see in Fig. 2.11 that the reflection of Gaussian beam is larger than a homogeneous plane wave. It is also seen that a larger beam width in the free space system would give measured results closer to the plane wave result. However, a larger beam width is not desirable because the material sample must have a correspondingly larger size. Nevertheless, a small difference of about

0.003 (linear magnitude) is observed for a moderate beam radius of 1.3λ0.Sucha difference is of the same order as the potential measurement error. Therefore the uniform plane wave assumption is justified. Repeatability of experimental results for a planar plexiglas slab of thickness 5.334mm is presented in Fig. 2.12 to support this argument.

In Fig. 2.12, the smooth curve is the magnitude of reflection coefficient for the plexiglas slab obtained from measure No. 1. Its y-axis is plotted on the left handed side. Subsequently the same experimental procedure is repeated for three times, each with a new calibration on a different day. The difference of these measurements from the first one (|Γi|−|Γ1|) are plotted as the symbolled curves, with their y-axis on the right handed side. It is seen that the deviation is of the same order of magnitude as the difference between the theoretical results for Gaussian beam and plane wave shown in Fig. 2.11. Notice that the unit for the right y-axis is 100 times smaller than the left y-axis. Therefore, unless the measurement can be performed with a higher §2.6. 2—D Numerical Results and Comparison with Measurement 50

0.46 ∞ w0= λ w0=1.3 0 0.45 λ w0=2.0 0 λ w0=5.0 0 0.44 w =10.0λ ~ 0 0 Ei(ky) 0.43 y |r⊥(ky)| 0.42

| x Γ | 0.41 0.53cm thick 0.4 εr=2.685 0.39

0.38 w0 :beam radius E along z-axis ky 0.37 8 9 10 11 12 Frequency (GHz)

Figure 2.11: Magnitude of reflection coefficient versus frequency for a 2-D E-polarized Gaussian beam reflected by a dielectric slab at nor- mal incidence. The difference between the plane wave reflection coefficient and those of a 1.3λ0 Gaussian beam is about 0.003, which is of the same order as the potential measurement er- ror(see Fig. 2.12). Thus the use of uniform plane wave assump- tion in the inversion process is justified. Shown on the right is the spectrum function for the Gaussian beam (also a Gaussian) and the magnitude of plane wave reflection coefficient at oblique incidence. The beam reflection can be thought of as the aver- age of a bundle of plane waves incident at different angle, with a Gaussian as the weight in the averaging process. order of accuracy(repeatability), the use of uniform plane wave model is justified for

planar slab samples. It is also seen that the measurement errors tend to be larger on

the two edges of the frequency band. This is an indication of the time gating error.

Further discussion of time gating error can be found in the appendix.

One defect of the 2-D model presented above is its inability to account for the

cross-polarization. As seen from the field mapping in Fig. 2.5, the beam is bounded

in both of the transverse directions. The beam widths along these two directions §2.6. 2—D Numerical Results and Comparison with Measurement 51

1 0.425 Experiment #1 0.9 0.4 0.8

0.7 0.375 #4−#1 0.6 − 0.35 Experiment #2 #1 |(unit:0.01) | Γ

Γ 0.5 | 0.325 0.4

0.3 0.3

#3−#1 0.2 0.275 Deviationin| 0.1 0.25 0 8 9 10 11 12 13 Frequency(GHz)

Figure 2.12: Magnitude of reflection coefficient measured with the free space system for a plexiglas slab of thickness 5.33mm. The same mea- surement procedure is repeated four times with a new calibra- tions. The deviations (|Γi|−|Γ1|) from the first measurement are plotted in the second y-axis. It is seen that the difference between two measurements has the same order of magnitude as the difference between the theoretical results for Gaussian beam and plane wave. Therefore unless a higher order of accuracy (re- peatability) can be achieved in the experimental results, the use ofplanewavemodelisjustified. §2.7. Formulation of 3—D Gaussian Beam Obliquely Incident on a Slab 52

are almost the same. Therefore there is an ambiguity in defining the polarization of

this 3-D beam. If the beam is modelled as a 2-D H-polarized Gaussian beam, the

predicted reflection coefficient will be smaller than that of the uniform plane wave

for the above plexiglas slab sample. Therefore a 3—D model is considered in the next

section. And the analysis is extended to obliquely incidence so that the potential

error is a bistatic measurement can be predicted.

2.7 Formulation of 3—D Gaussian Beam Obliquely Incident on a Slab

Consider the free-space bistatic measurement system applied on a slab sample. Three

rectangular coordinate systems are used in analyzing the problem (Fig. 2.13): the

sample coordinates (x, y, z), the transmitting antenna coordinates (xi,yi,zi), and the

receiving antenna coordinates (xr,yr,zr). A slab material with relative permittivity εr

and relative permeability µr is placed between z=h and z=h + t.Regionzh+t could be either free space or a perfect conductor. Two identical antennas are located somewhere within the region z>0. In their transmitting mode, both antennas radiate focused microwave beams, propagating in the direction zˆi and

−zˆr respectively. These two axes are tilted an angle θ from −zˆ. The antennas are assumed to be reciprocal so that their receiving characteristics can be deduced from the transmitting characteristic. A time dependent factor exp(jωt) is understood and suppressed in the formulation. Under the assumption that multiple scattering between the three components of the system–two antennas and a sample–can be neglected, the analysis is divided into three steps:

• Specify the incident beam and derive its spectrum function with respect to the §2.7. Formulation of 3—D Gaussian Beam Obliquely Incident on a Slab 53

reference plane z =0.

• Derive the spectrum function for the reflected beam by superposing perpendic-

ular and parallel polarized plane wave components.

• Apply the plane wave scattering matrix theory to express the response measured

by the receiving antenna.

z zr zi

xr t ε µ r, r

h θ θ y=yi=yr x

xi

Figure 2.13: Geometry and coordinate systems for the modeling of bistatic free space measurement.

We follow the formulation given by Luk and Cullen [37], but the formulation given below is not exactly the same. Two subtle differences should be noted. The

first is the way by which the incident beam is specified. This step is important because it essentially gives the characteristic of the antennas. If the formulation in

[37] is used, the antenna will be different for different incident angle (see eq. (5b) in [37]). This is presumably an oversight. The second difference is that the large §2.7. Formulation of 3—D Gaussian Beam Obliquely Incident on a Slab 54

beam-width assumption employed in [37] will be avoided here. Thus the scattered

fields are calculated numerically and the accuracy is not compromised at small beam

width. The analysis is proceeded in several steps as indicated by the titles of the

following sub-sections.

2.7.1 Specifying the Incident Beam

There are several ways in defining a Gaussian beam. The amplitude of total field

vector or its component along a preferred direction may be chosen to have a Gaussian

distribution. Polarization of the beam also needs to be specified. By the surface

equivalence principle, it suffices to assign tangential electric or magnetic field vector

at the focal plane zi =0.Thuslet:

| G x ,y Et zi=0 = ( i i)yˆi (“perpendicular” polarization) (2.19) | G x ,y Ht zi=0 = ( i i)yˆi (“parallel” polarization)

where G(xi,yi) is the Gaussian distribution given by:

  −x2 −y2 G x ,y 1 i i ( i i)=πw w exp w2 + w2 (2.20) 0x 0y 0x 0y

and w0x and w0y are the beam width parameters. Two remarks should be mentioned. First, although the terms perpendicular and parallel polarization are used, the beam is not strictly a linearly polarized one. It will be apparent (see eq. (2.24)) that the

“perpendicular” polarized Gaussian beam defined above has Ezi component, while

the “parallel” polarized beam has Hzi components. Secondly, while two different polarization state can be achieved physically by rotating the antenna 900 with respect §2.7. Formulation of 3—D Gaussian Beam Obliquely Incident on a Slab 55

to the beam axis, the two beams defined in eq. (2.19) are not related in the same way.

Nevertheless the longitudinal components Ezi or Hzi is quite small compared to the transverse component. So the two issues mentioned here are simply mathematical.

In the following the “perpendicular” polarization will be considered explicitly. The

results for “parallel” polarization can be obtained by similar procedure or by duality

considerations. Applying the plane-wave spectrum representation, the components of

incident beam can be expressed as:

E¯i x ,y,z , xi ( i i i)=0 (2.21) ∞ 1 E¯i (x ,y,z)= Φ(¯ k ,k )exp[−j(k x + k y + k z )]dk dk (2.22) yi i i i 4π2 xi yi xi i yi i zi i xi yi −∞ −1 ¯ = F {Φ(kxi,kyi)exp[−jkzizi]}, (2.23) i   k E¯i x ,y,z F −1 − yi ¯ k ,k −jk z , zi ( i i i)= i Φ( xi yi)exp[ zi i] (2.24) kzi where an overbar “-” signifies that the function is defined in the incident coordinate x ,y,z F −1 ( i i i); the symbol i denotes the inverse Fourier transform with respect to

wavenumbers kxi and kyi;and

  2 2 2 2 −k w −k w0 Φ(¯ k ,k )=exp xi 0x + yi y . (2.25) xi yi 4 4

Note also that:

k2 k2 k2 k2. xi + yi + zi = 0 (2.26)

√ where k0 = jω ε0µ0 denotes the wavenumber in free space. To find the reflected §2.7. Formulation of 3—D Gaussian Beam Obliquely Incident on a Slab 56

field, it is convenient to express the incident beam in the sample coordinates. Since

the reference plane in the bistatic free-space measurement is at z = 0, the PWSM

theory also requires the spectrum to be expressed in the sample coordinate. To this

end, substitution of xi = x cos θ − z sin θ, yi = y, and zi = x sin θ + z cos θ reduces eq. (2.23) to:

∞ k θ k θ Ei x, y, z 1 k ,k −j k x k y k z z cos + x sin dk dk y( )= 2 Φ( x y)exp[ ( x + y + z )] x y 4π kz −∞

−1 kz cos θ + kx sin θ = F {Φ(kx,ky)exp[−jkzz] } kz (2.27)

where a change of variable (kxi → kx := kxi cos θ + kzi sin θ, kyi → ky) has been

applied. The newly defined kx and ky happen to be the spectral variable in the sample

−1 coordinate, so thatF is the inverse Fourier transform with respect to (kx,ky), and k −k θ k θ k2 k2 k2 k2 the z(:= xi sin + zi cos ) satisfies the relation x + y + z = 0. In arriving at eq. (2.27) the following relations have been used:      ∂k ∂k   xi/∂k xi/∂k  k cos θ + k sin θ dk dk =  x y  dk dk = z x dk dk , (2.28) xi yi  ∂k ∂k  x y k x y  yi/ yi/  z ∂kx ∂ky and: 2 2 kx cos θ − kz sin θ kyw0y Φ(k ,k )=exp − w0 − . (2.29) x y 2 x 2

Similar operation can be applied on eq. (2.24), resulting at a complete plane wave §2.7. Formulation of 3—D Gaussian Beam Obliquely Incident on a Slab 57

spectrum representation for the incident beam.

  i −1 −jkzz E (x, y, z)=F [−ky sin θxˆ +(kz cos θ + kx sin θ)yˆ − ky cos θˆz] e Φ(kx,ky)/kz (2.30)

The corresponding magnetic field vector is needed for expressing the receiving char-

acteristic of the antenna, it is:

 Hi(x, y, z)= 1 F −1 (−k2 cos θ − k2 cos θ − k k sin θ)xˆ +(−k k sin θ + k k cos θ)yˆ η0 y z x z z y x y  k k θ k2 θ k2 θ e−jkzz k ,k / k k , +( x z cos + x sin + y sin )ˆz Φ( x y) ( 0 z) (2.31)

where η0 is the impedance in free space. It is noted that the field vectors given by plane

wave representation satisfy Maxwell’s equation exactly, so no paraxial approximation

has been applied. If such an assumption were used, then kzi ≈ k0, dkx ≈ dkxi cos θ, and the results in [37] are recovered with one exception. In contrast to eq. (5) of [37],

the electric field has nonzero x-component as given in eq. (2.30).

It is interesting to visualize the effect of varying incident angle θ on the spectrum

E˜i k ,k k function. Fig. 2.14 shows the spectrum function y( x y)at y = 0 for different 0 0 0 0 incident angle θ =0, 30 , 45 , or 60 , and different beamwidth w0x =2λ0, or λ0,

where λ0 is the wavelength. It is seen that for obliquely incidence, the maximum of

the spectrum function lies at nonzero kx not exactly equal to k0 cos θ. This means the plane wave component carrying maximum energy might not be the one propagating along the beam axis direction. Notice also that the spectrum functions for different angles are not simply related by a translation along kx (the angular direction). In fact, the Gaussian shape for normal incidence is lost at oblique incidence. This is §2.7. Formulation of 3—D Gaussian Beam Obliquely Incident on a Slab 58

E i (,kk=0 ) λ y xy w0x=2 0 2 w =λ 0x 0 θ=60o

1.5 θ=45o θ=30o θ o 1 =0

0.5

-0.5 -0.25 0.25 0.5 0.75 1 kx /k0

E˜i k , k θ k θ k ,k /k k /k Figure 2.14: Plot of y( x 0) = ( z cos + x sin )Φ( x y) z v.s. x 0 for different incident angle θ and beamwidth w0x.

different from the large beamwidth situation. Another important point is that as the

angle θ increases, a significant portion of the plane wave spectrum moves into the

invisible(evanescent) region (kx >k0). This happens faster for smaller beamwidth as seen by a comparison of the two curves for θ =600. Therefore the incident

angle for free-space bistatic measurement should be chosen carefully according to the

beamwidth. If surface wave is excited significantly, diffraction from the edge of sample

might introduce much bigger error.

2.7.2 Spectrum Function for the Reflected Field

The reflected field can be expressed by first separating each plane wave component

of the incident beam into two parts either perpendicular or parallel to the plane of §2.7. Formulation of 3—D Gaussian Beam Obliquely Incident on a Slab 59

incidence. The unit vectors for these two components are given by:

k k k k −k2 − k2 −kyxˆ + kxyˆ x zxˆ + y zyˆ +( x y)ˆz ˆa⊥ =  , ˆa =  , (2.32) k2 k2 k k2 k2 x + y 0 x + y

foraplanewavealongthedirection(kx,ky,kz). Note that the plane of incidence is defined by the wave vector k and normal directionz ˆ. If these two vectors are parallel

to each other (as is the case for the normally incident component), the plane is not

uniquely defined and special attention is needed. The total incident electric field is

i i i E = E⊥ + E,with   k2 sin θ + k k cos θ + k2 sin θ i −1 y x z x −jkzz E = F ˆa⊥  Φ(k ,k )e , (2.33) ⊥ k k2 k2 x y z x + y

  k k θ i −1 0 y cos −jkzz E = F ˆa  Φ(k ,k )e . (2.34)  k k2 k2 x y z x + y

The corresponding reflected electric field is given by:

  −k k r −1 yxˆ + xyˆ 2 2 jkzz E = F Γ⊥(k ,k ) (k sin θ + k sin θ + k k cos θ)Φ(k ,k )e , ⊥ x y k k2 k2 y x x z x y z( x + y) (2.35)

  −k k θ −k2 cos θ −k θ r −1 x y cos y y cos jkzz E = F Γ(k ,k ) xˆ + yˆ + ˆz Φ(k ,k )e .  x y k2 k2 k2 k2 k x y x + y x + y z (2.36) §2.7. Formulation of 3—D Gaussian Beam Obliquely Incident on a Slab 60

r Notice the sign change in the unit vector for E, and the reflection coefficient Γ must be expressed accordingly. The total reflected electric field is:

 r r r −1 r jkzz E = E⊥ + E = F E˜ (kx,ky)e (2.37)

r where the vector spectrum function E˜ (kx,ky)isgivenby:

 2 2 −ky(k + k )Γ⊥(kx,ky)sinθ − kxkykz cos θ[Γ⊥(kx,ky)+Γ(kx,ky)] E˜r(k ,k )= xˆ x y + x y k k2 k2 z( x + y) 2 2 2 2 kx(k + k )Γ⊥(kx,ky)sinθ + kz cos θ[k Γ⊥(kx,ky) − k Γ(kx,ky)] yˆ x y x y + k (k2 + k2)  z x y −ky cos θΓ(kx,ky) ˆz Φ(kx,ky) (2.38) kz

2.7.3 “Reflection Coefficient” for 3—D Gaussian Beam

Similar to the 2-D case, the PWSM (Plane Wave Scattering Matrix) theory [29]

is again used to calculate the reflection coefficient. Applying an important result

(eq. (3.1-1), p.87 [29]) of the theory, the equivalent voltage Vr developed at the coaxial feed of the receiving antenna due to the reflected field found in the previous section

is proportional to (eq. (D-1), p.110 [29]):

∞ V ∝ ˜ r k ,k · ˜ (re) −k , −k × − dk dk , r Et ( x y) Ht ( x y) ( ˆaz x y) (2.39) −∞

r In this equation E˜ t is the transverse component of the electric field spectrum in ˜ (re) eq. (2.38). The term Ht , expression of which will be given shortly, corresponds to the spectrum function of the magnetic field of the receiving antenna. Physical

meaning of the above equation can be understood by first writing the integrand as §2.7. Formulation of 3—D Gaussian Beam Obliquely Incident on a Slab 61

˜r k ,k −jk z · ˜ (re) −k , −k jk z × − ˜ r k ,k −jk z Et ( x y)exp( z ) Ht ( x y)exp( z ) ( ˆaz). Then Et ( x y)exp( z ) ˜ (re) −k , −k jk z is viewed as a plane wave component of the reflected field, and Ht ( x y)exp( z ) gives the voltage in the receiving antenna due to an incident plane wave. The total

voltage is a superposition (integral) of response due to all the incident plane waves.

The above result incorporates the reciprocity principle and a number of fundamental

principles in electromagnetics. The similarity of the integrand to a power density

expression is to be noted. Notice also that the expression gives not only the received

power, but also the phase of the received signal. The latter is crucial in determining

property of the material under test.

˜ (re) k ,k The function Ht ( x y) in eq. (2.39) is the transverse (with respect to ˆaz)com-

(re) ponent of H˜ (kx,ky) which, in accordance with the reciprocity theorem, is related to the magnetic field H(re)(x, y, z = 0) radiated by the receiving antenna in its trans-

mitting mode(eq. (D-4), p.111 [29]):

H(re)(x, y, z)= 1 F −1{H˜ (re)(k ,k )exp[j(−k )z]} (2.40) η0 x y z

Since we assume that the receiving antenna is the same as the transmitting antenna.

(re) The only differece is that the field H (x, y, z = 0) propagates energy along −ˆazi di- ˜ (re) −k , −k rection. Therefore Ht ( x y) can be written from the spectrum function within the right handed side of eq. (2.31) by the following steps: 1) take z =0;2)letθ →−θ

to account for the different orientation of the receiving antenna; 3) let kx →−kx and

ky →−ky.Theresultis:

˜ (re) −k , −k −k2 θ − k2 θ k k θ Ht ( x y)=[( y cos z cos + x z sin )xˆ (2.41)

+(kykz sin θ + kxky cos θ)yˆ]Φ(kx,ky)/(k0kz), §2.7. Formulation of 3—D Gaussian Beam Obliquely Incident on a Slab 62

so that the induced voltage Vr in eq. (2.39) can be reduced to:

∞ − k2 k2 2 θ k2 2 θ ( x + y)Γ⊥ sin −kxΓ⊥ sin 2θ yΓ cos Vr ∝ + + + k0kz k0 k0kz −∞ (2.42) 2 2 2  kz cos θ(k Γ − k Γ⊥) + y x Φ2(k ,k )dk dk k k2 k2 x y x y 0( x + y) where it is understood that the plane-wave reflection coefficients Γ⊥ and Γ are func-

tions of kx and ky(corresponding to incident angle). It should be emphasized again

that the sign of Γ must conform to the unit vector used in eq. (2.36). In particular,

Γ(kx =0,ky =0)=−Γ⊥(0, 0) for the plane wave component at normal incidence.

Therefore when kx = ky = 0 the factor inside the squared brackets is not singular but

2 reduces to Γ⊥(0, 0) cos θ.

To estimate the scattering parameters measured by the network analyzer, the result in eq. (2.42) needs to be normalized. Consider the received voltage when the sample under test is a planar perfect conductor. Since Γ⊥ = −Γ ≡−1, the integral

in eq. (2.42) reduces to:

∞ k2 k θ k θ 2 y +( x sin + z cos ) 2 Vc = Φ (kx,ky)dkxdky (for conductor) (2.43) k0kz −∞

This is also the incident energy of the beam along the zˆ direction, as can be calculated

from eq. (2.30). Multiplying the right handed side of eq. (2.42) by a normalization

factor of −1/Vc, the beam “reflection coefficient” can therefore be defined as:

2 2 2 2 2 2 −1 −Γ⊥[(k + k )sinθ + kxkz cos θ] +Γk k0 cos θ Γ = x y y Φ2(k ,k )dk dk beam V k k k2 k2 x y x y c 0 z( x + y) −∞ (2.44) §2.8. Direct Coupling in the Bistatic Setup 63

As a sanity check of the above definition, let the beam width w0x and w0y go to infinity, the plane wave reflection coefficient should be recovered. Indeed, substitution

2 of Φ (kx,ky)byδ(kx − k0 sin θ)δ(ky)intheaboveequationverifiesthetrivialresult that (−1/Vc)Vr =Γ⊥(k0 sin θ, 0) for perpendicular plane wave incidence.

The main idea of the formulation is summarized in the schematic diagram in

Fig. 2.15. The output Vr(kx,ky) denotes the integrand in eq. (2.42). From a circuit analysis point of view, both the sample under test and the receiving antenna function

as spatial frequency filters. The “transfer function” for the sample are determined

from its material property and thickness of each slab. The “transfer function” for

the receiving antenna can be derived from its transmitting property, under the as-

sumption that the antenna is reciprocal. For non-reciprocal antenna, the transmitting

characteristic of its adjoint antenna should be used.

E i (,)kk i^ ⊥ xy a⊥ (,)kkxy Γ⊥ (,)kkxy

i r ()re E (,)kk E (,)kkxy iH (,)kk Transmitting xy + xy Sample Under Test ^ Vkkrxy(,) antenna ×az E i (,)kk i^ xy Receiving a(,)kk Γ (,)kk xy xy Antenna

Figure 2.15: A schematic diagram showing the basic idea of the theoreti- cal formulation. Both the sample under test and the receiving antenna function as spatial frequency filter.

2.8 Direct Coupling in the Bistatic Setup

Another interesting aspect of the bistatic measurement setup is the direct coupling

between the transmitting and receiving antennas. This direct coupling means the §2.8. Direct Coupling in the Bistatic Setup 64

signal received without the existence of a sample under test (or one can say the

sample is the free space). It is seen that if θ =900, the antennas are directly facing each other, and (almost) all of the transmitted energy will couple into the receiving antenna. As θ becomes smaller, the direct coupling is reduced but not identically zero.

Similar to the reflection coefficient formulated in the previous section, the magnitude of direct coupling term also depends upon the spatial bandwidth of the incident beam and might be significant enough to affect the measurement result. To account forsuchaneffect,anisolationtermshould be included in the error model during the calibration. To avoid this complication in the calibration process, it is generally desirable to perform bistatic measurement with a small incident angle θ. However this is subject to constraints from the size of antenna. Furthermore, sometimes a relatively large value of θ (say 60o) is more useful because, for example, the reflection

of plane wave incident at such an angle is minimum. As shown below, the direct

coupling term can be easily estimated if we know the radiating characteristics of the

antennas (assumed as Gaussian here).

z

zi

θ y=yi=yr θ x

zr x r xi

Figure 2.16: Geometry and coordinate system used for calculating the direct coupling between the antennas in a free-space bistatic setup. §2.8. Direct Coupling in the Bistatic Setup 65

To model the direct coupling between the antennas, the plane wave spectrum

analysis is performed with respect to the configuration shown in Fig. 2.16. Note that

2θ is the angle subtended by the axes of two antennas. The same analysis procedure

as in the previous section can be followed. The only difference is that the reference

plane is now set on z = 0 of Fig. 2.16, which lies between the two antennas. The

resulting expression for the directly coupled voltage Vd is:

∞ k2 k2 2 θ − k2 k2 2 θ ( x + y)cos ( y + z )sin Vd(θ) ∝ Φi(kx,ky)Φr(kx,ky)dkxdky (2.45) k0kz −∞ with: 2 2 kx sin θ − kz cos θ kyw0y Φ (k ,k )=exp − w0 − (2.46) i x y 2 x 2 2 2 kx sin θ + kz cos θ kyw0y Φ (k ,k )=exp − w0 − (2.47) r x y 2 x 2

Numerical results for the directly coupledsignalisillustratedinFig.2.17,where

0 the voltage Vd(θ) has been normalized by its maximum at θ=90 . Three curves are plotted for different beam radius. As expected, the voltage becomes smaller as the

angle θ is reduced. For fixed angle θ, the voltage is smaller for larger beam radius. This

is reasonable because as the beam radius goes higher, the Gaussian beam becomes

more similar to a uninform plane wave. Plane wave propagating in one direction will

not couple into a (uniform) plane wave in other directions.

The minimum θ that is physically possible for the free space system used in this

thesis work is 40o. Judging with the calculated result in Fig. 2.17, the direct coupling

at this angle is too small to cause any concern. Such an observation is also verified §2.9. Experimental Results for Planar Slab Using Bistatic Setup 66 experimentally. Results of bistatic measurement at 40o are presented in the next section.

θ o 1 Vd( )/Vd(90 ) wox=woy=λo 0.8 wox=woy=1.5λo 0.6 wox=woy=2.0λo

0.4 θ θ 0.2

o o o o o o 90o 30 40 50 θ 60 70 80

Figure 2.17: Magnitude of the directly coupled signal versus the incident an- gle for different beamwidths. Direct coupling is the transmitted signal picked up by the receiving antenna in absence of a mate- rial under test, thereby the signal is coupled directly without a reflection.

2.9 Experimental Results for Planar Slab Using Bistatic Setup

Free-space bistatic measurement has been studied before. Umari, in his thesis ([34]), implementedanoffset-shortcalibrationprocedure. The transmission coefficient mea- sured by the network analyzer was re-defined as a “reflection coefficient” so that the error model of a one-port network can be applied. Such an error model is not an exact representation of the physical arrangement, because the pathes of incident and reflected signal are different in the bi-static setup, whereas a one-port error model assumes that the pathes are the same. Another defect of the analysis in [34] is that §2.9. Experimental Results for Planar Slab Using Bistatic Setup 67

the sample under test is always assumed to have infinite thickness.

The bistatic setup is re-evaluated in this work using a different calibration proce-

dure. First, a full two port calibration is performed using the SLOT7 method with

commercial standards. After this calibration, the measurement plane is set at the

end of the coaxial cable immediately before the coax-waveguide adapters of the horn

lens antennas. Secondly, a copper plate is measured with the bistatic setup. The

result will be used as a normalization factor in processing the measured data for the

material samples. Such a simple operation is termed “Response” calibration in the

network analyzer manual. Thus the method can be called a two-tier calibration, with

the first one (SLOT) accounting for the discontinuities in the coaxial connection and

inside the network analyzer, while the second one (Response) representing the effect

of all the components of lens antenna (adapters, horn and lens).

To observe the bandwidth of the horn lens antennas, 40o bistatic measurement in the frequency range of 2GHz-18GHz is performed. Fig. 2.18 shows the transmission coefficient in Log Magnitude format(dB) for three cases: no sample holder, sample holder only, and metal plate (plus sample holder). These results are obtained after the SLOT calibration without doing a normalization(the response calibration). A noisy and oscillating pattern is seen in all three curves. Observe from the top curve for metal plate that the radiating bandwidth of the antenna range from 8GHz to

13.7GHz. This frequency band will be used in later measurements. For frequency lower than 7GHz, the antenna is basically non-radiating. From the no-sample holder curve, there also seem to a passband. This is an indication of the direct coupling discussed above. However its magnitude is about 50dB down from that of the metal

7SLOT refers to the standards Short, Load and sliding load, Open, and Through. §2.9. Experimental Results for Planar Slab Using Bistatic Setup 68 plate, therefore isolation can be omitted in the bistatic transmission measurement.

Now observe from the difference between the sample holder curve and no-sample holder curve that the reflection caused by sample holder is nonzero. On the other hand, the signal magnitude for sample holder is about 30dB down from the metal plate result. For the purpose of this study, they can be treated as systematic noise.

Thus the dynamic range of this bistatic measurement is about 30dB. In view of the noisy behavior of the top curve, the response from a material sample should clearly differentiate from the metal plate result and the noise floor(around -20dB in this

figure).

-10 metal plate -20 -30 sample holder -40 -50 -60 | (dB) 21

|S -70

-80 no sample holder (free space) -90 -100 -110 -120 5 10 15 Frequency(GHz)

o Figure 2.18: Log magnitude of the 40 bistatic “reflection” coefficient (S21 measured by the network analyzer) for three “samples”: metal plate, sample holder, free space. §2.9. Experimental Results for Planar Slab Using Bistatic Setup 69

Time domain response corresponding to the above frequency domain data are

shown in Fig. 2.19. The results are calculated by a program internal to the network

analyzer under the setting of MINIMUM window type. A peak due to the reflection

from metal plate is easily identified. The rising edge of this peak is very sharp, whereas

the falling edge contains many “ringings”. The signal goes away completely only after

20ns later. This is an indication of the multiple reflections inside the antenna and

possibly between the antenna and the metal plate.

0.002

sample holder metal plate 0.0015 0.10

0.001 Linear Mag

No holder (free space)

Linear Mag 0.0005 0.05

0 1E-08 2E-08 3E-08 4E-08 Time(seconds)

0.00 0 1E-08 2E-08 3E-08 4E-08 Time(second)

Figure 2.19: Time domain response of the 40o bistactic setup calculated from 2GHz—18GHz frequency domain data.

Next, the network analyzer is set to the frequency range of 8GHz—13.7GHz and the bistatic response of three planar slab samples (glass, plexiglas, and teflon) are

measured. Linear magnitudes of the S21 after the SLOT calibration are shown in §2.9. Experimental Results for Planar Slab Using Bistatic Setup 70

Fig. 2.20. Corresponding phase results are shown in Fig. 2.21, where the difference between the samples are not identifiable in the chosen scale.

0.8

0.7 metal plate 0.6 glass (4.6863mm thick) 0.5

| plexiglas (5.334mm thick) 21

|S 0.4

0.3 teflon (6.48mm thick) 0.2

0.1

8 9 10 11 12 13 Frequency(GHz)

Figure 2.20: Linear magnitude of the S21 measured for metal plate, glass, plexiglas and teflon slab samples after the SLOT calibration.

The curves in Fig. 2.20 and 2.21 can be not easily interpreted. However, after normalization (response calibration), the data start making sense, as can be seen in

the magnitude graphes in Figs. 2.22(glass), 2.24(plexiglas), 2.26(teflon) and the phase

graphes in Figs. 2.23(glass), 2.25(plexiglas), Fig. 2.27(teflon). Also shown in these

figures are the theoretical prediction calculated for a 3-D Gaussian beam of 1/e beam

o radius 1.3λ0 at oblique incidence of 40 . The permittivity ε used in the calculation

are collected from published results and therefore might not reflect the true values

for these specific samples. At this point, it is judged that the noisy behavior of

the measured curves are due to the multiple reflections between the sample and the

antenna. Also notice that the measured magnitudes tend to be smaller than the §2.9. Experimental Results for Planar Slab Using Bistatic Setup 71

150

100 21 50

0

-50 Phase of bistatic S

-100

-150

8 9 10 11 12 13 Frequency(GHz)

Figure 2.21: Phase of the S21 measured for metal plate, glass, plexiglas and teflon slab samples after the SLOT calibration. theoretical prediction. This is considered as a systematic error induced by the simple response calibration.

Before performing the inversion to find the permittivity, the noise in the measured data should be removed. As mentioned above, the noise could be due to multiple

reflections between the antenna and the sample. It might also be an indication of the

defect of the response calibration. If the latter is the case, a more complete calibration

procedure such as the offset short method should be used. Unfortunately, it is found

that the noise could not be simply explained in this way. The reason will be discussed

with reference to Fig. 2.28.

After a SLOT calibration, the S-parameter S11 obtained from the bistatic static

setup for offset shorts are plotted in Fig. 2.28. The horizontal axis is the real part,

while the vertical axis is the imaginary part. Three series of points correspond to §2.9. Experimental Results for Planar Slab Using Bistatic Setup 72

0.8 Theory Measured 0.6 |

21 0.4 |S

0.2

0.0

8 9 10 11 12 13 14 Frequency(GHz)

o Figure 2.22: Magnitude of the 40 bistatic “reflection” coefficient S21 for the glass sample, along with prediction of the beam theory for a slab of ε=7.0, thickness 4.6863mm.

100 theory 80 measure 60 40 )

21 20 0

Phase(S -20 -40 -60 -80 -100 8 9 10 11 12 13 14 Frequency(GHz)

o Figure 2.23: Phase of the 40 bistatic “reflection” coefficient S21 for the glass sample, along with the beam theory for a slab of ε=7.0, thickness 4.6863mm. §2.9. Experimental Results for Planar Slab Using Bistatic Setup 73

0.58

0.56

0.54 0.52 | 21

|S 0.50 measure 0.48 theory 0.46

0.44

8 9 10 11 12 13 14 Frequency(GHz)

o Figure 2.24: Magnitude of the 40 bistatic “reflection” coefficient S21 for the plexiglas sample, along with prediction of the beam theory for aslabofε=2.59(1 + 0.0069i), thickness 5.334mm.

20 measured 10 theory

0

-10 Phase 21 S -20

-30

-40 8 9 10 11 12 13 14 Frequency(GHz)

o Figure 2.25: Phase of the 40 bistatic “reflection” coefficient S21 for the plexiglas sample, along with prediction of the beam theory for aslabofε=2.59(1 + 0.0069i), thickness 5.334mm. §2.9. Experimental Results for Planar Slab Using Bistatic Setup 74

0.50 0.48 measured theory 0.46 0.44 0.42 |

21 0.40 |S 0.38 0.36 0.34 0.32 8 9 10 11 12 13 14 Frequency(GHz)

o Figure 2.26: Magnitude of the 40 bistatic “reflection” coefficient S21 for the teflon sample, along with prediction of the beam theory for a slab of ε=2.0, thickness 6.48mm.

20 measured 10 theory

0 ) 21 -10

Phase(S -20

-30

-40

8 9 10 11 12 13 14 Frequency(GHz)

o Figure 2.27: Phase of the 40 bistatic “reflection” coefficient S21 for the teflon sample, along with prediction of the beam theory for a slab of ε=2.0, thickness 6.48mm. §2.9. Experimental Results for Planar Slab Using Bistatic Setup 75

10.01GHz 0.04 0.05 10.0313GHz 0.16 0.02in 0.01in 10.0525GHz 0.14 0.03 d=0in 0.4in 0.1in 0.12

) 0.10 h 11 0.15 h n 0.08 n

Im(S n h 0.3 0.5in 0.06 0.2 0.04

0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Re(S ) 11

Figure 2.28: S11 measured after the SLOT calibration for a series of offset shorts.

1

λ wox=woy=2.0 o 0 .9

0 .8

λ conductor wox=woy=1.5 o 0 .7 beam

Γ h

0 .6 o 45 λ wox=woy= o

0 .5

0 .4 0 0 .2 5 0 .5 0 .7 5 1 λ h ( 0)

Figure 2.29: Magnitude of the “reflection coefficient” for offset-short (planar perfect conductor) as a function of the offset distance. §2.9. Experimental Results for Planar Slab Using Bistatic Setup 76

150

100

50 m a e b Γ 0

Phase of -50

-100

-150

0 0.25 0.5 0.75 1 λ h ( 0)

Figure 2.30: Phase of the “reflection coefficient” for offset-short (planar per- fect conductor) as a function of the offset distance. three frequencies. The offset distances d (in inches) are identified for the first series.

The points denoted by symbols “n” and “h” are the results for “no sample holder” and “holder only”. They can be treated as the response of a matched load. Now if a linear two port network8 is used to model the discontinuities lying in between the end of coaxial cable and the surface of metal plate, the locus of S11 should be part of a circle. The center of the circle is the response of matched load(point “n” or point “h”).

This kind of pattern is not observable from the three curves in Fig. 2.28. Accounting for the defocusing effect due to the metal plate being moved away from the focal plane, the circle is degenerated into a spiral with decreasing radius of curvature. If the position denote by d=0 is not the true focal plane, the radius of curvature will

first increase and then decrease. But the pattern of change should be synchronized for different frequencies, assuming the focal plane remains the same. Again, this

8Refer to appendix D for an explanation of a linear error model §2.9. Experimental Results for Planar Slab Using Bistatic Setup 77

kind of pattern is not reflected in Fig. 2.28. Thus it is concluded that an offset-short

calibration alone will not give a better result. The noise in Figs. 2.22 through 2.27

will be treated as random, and the Savitzky-Golay method is used to smooth the

magnitude and phase separately. The results of smoothing are illustrated in Figs.

2.31(magnitude) and 2.32(phase) for the glass sample.

0.9

0.8 two-tier calibration Savitzky-Golay smoothing 0.7 0.6 0.5 |

21 0.4 |S 0.3 0.2 0.1 0.0 8 9 10 11 12 13 14 Frequency(GHz)

Figure 2.31: Applying the Savitzky-Golay smoothing to the magnitude of S21 for the glass sample.

Theoretical results in Figs. 2.22 through 2.27 have also been compared with the

reflection coefficients of a uniform plane wave incident at 40o. The difference are found to be within 1%, so that the explicit formula for the plane wave is used to invert the measurement data after the Savitzky-Golay smoothing. The resulting permittivity

ε is shown in Figs. 2.33 (real part ε ) and 2.34(imaginary part ε). A reasonable

accuracy is achieved for the real part. §2.9. Experimental Results for Planar Slab Using Bistatic Setup 78

80 two-tier calibration 60 Savitzky-Golay smoothing

40

20

0

-20 Phase (degree) 21 S -40

-60

-80 8 9 10 11 12 13 14 Frequency(GHz)

Figure 2.32: Applying the Savitzky-Golay smoothing to the phase of S21 for the glass sample.

8

7

6 Teflon plexiglas 5 ’ glass ε 4

3

2

8 9 10 11 12 13 14 Frequency(GHz)

Figure 2.33: Real part ε of permittivity inverted from bistatic measure- ments. §2.10. Conclusion 79

0.07 Teflon 0.06 Plexiglas Glass 0.05 0.04 " ε 0.03 0.02

0.01

0.00

8 9 10 11 12 13 14 Frequency(GHz)

Figure 2.34: Imaginary part ε of permittivity inverted from bistatic mea- surements.

2.10 Conclusion

A theoretical model for the spot-focused free space measurement system configured at both normal incidence (mono-static) and bi-static setups is developed. Radiating and receiving characteristics of the horn-lens antenna has been incorporated into the model. This has not been achieved in previous studies. Numerical computation and experimental data are presented to justify the use of uniform plane wave assumption in the inversion of measured data for the permittivity.

It should be noted that a three dimensional Gaussian beam has been used in the calculation, and the multiple reflections between the antenna and the sample has been ignored. These are the main issues to be considered if further improvement is deem necessary. Chapter 3

Techniques for Obtaining Dielectric Properties of Curved Slab Using Only Reflection Measurement

3.1 Introducing the Curved Slab Problem

The rest of this chapter is concerned with the modeling and measurement of circular shell sample. The applicability of spot-focused beam for characterization of non- planar objects can be explained by the following physical reasoning. First of all, the focused beam system has been tested on planar slab samples and it is found that the response can be predicted by geometrical optics with reasonable accuracy. That is, for samples with infinite radii of curvature, the reflection and transmission of the beam are similar to those of a locally uniform plane wave. Next, consider samples with a radius of curvature that is finite but is still large compared to the wavelength and the spot-size of the beam. Obviously the response can still be approximated by the geometrical optics, and the scattering of the beam can be modelled as one ray passing through a planar slab at normal incidence, rather than a bundle of rays arriving at a curved object at different incident angles. The curvature effect might be ignored and the existing measurement method and inversion algorithm would still work. For radii of curvature less than a wavelength, a significant portion of the incident beam 80 §3.1. Introducing the Curved Slab Problem 81

is not intercepted by the sample. The effect effect and the possible excitation of

creeping wave must be accounted for. It would be very difficult, if not impossible,

to invert the measurement data. However, if the radius of curvature is within some

intermediate range where the geometrical optics approximation is in error but can be

remedied with some higher order correction, one can still obtain meaningful material

parameters using the free space method. It is the last situation that is of interest in

the present study.

For non-magnetic material, the complex permittivity can be calculated from either

the reflection or transmission measurement only. It has been found previously [13]

the transmission response for curved samples is very close to the planar sample,

and the permittivity inverted from the transmission data is quite accurate [13]. In

the same study, the reflection response of curved sample is found to be significantly

different from a planar sample. Therefore the experimental part of this study have

be concentrated on the measurement of reflection coefficients. The inversion problem

is tackled with two approaches.

For the modeling part, the basic question is the analysis of Gaussian beam reflec-

tion from a convex circular cylindrical slab. In what follows, relevant results in the

open literature will be reviewed and a modification of the Fourier expansion method

will be presented for the circular shell. It is helpful to summarize the basic idea of this modification before going into the details. Gaussian beam scattering by a closed cylindrical shell can be formulated rigorously as a summation of cylindrical harmon- ics. The same procedure can be used for scattering by a circular shell, but with only

incoming waves in the field expansion in the inner region. Physically this means an

infinite sink (perfect absorber) is placed on the axis of the cylindrical shell so that the §3.2. Literature on the Scattering of Shaped Beam by Curved Objects 82 reflection from half of the shell is eliminated. For the scattering of Gaussian beam by a circular dielectric interface, such a modification can be validated by comparison with the complex ray tracing method.

Similar to planar slab problem, the reflected fields are used to calculate an equiv- alent “reflection coefficient” as measured in a free space material characterization system using spot-focusing antennas. Numerical results are compared with experi- ments and a fair agreement is observed.

3.2 Literature on the Scattering of Shaped Beam by Curved Objects

This section summarizes the existing results on analyzing the interaction of shaped beams with curved objects. One of the problems that have been extensively studied is the scattering of Gaussian beam by an infinitely long dielectric cylinder [38]-[48].

Alexopoulos and Park [38] have investigated the scattering by cylinders due to waves with normal amplitude distribution. Choudhary and Felsen [39] developed the theory of inhomogeneous wave tracking and applied it to high frequency scattering of two- dimensional Gaussian beam by conducting cylinder. Kojima and Yanagiuchi [40] were the first to apply the Fourier series expansion method for analysis of offset Gaussian beam scattered by a dielectric cylinder. Using a somewhat different formulation of the Fourier expansion method, Kozaki [41] obtained a simple expression for scatter- ing of Gaussian beam by conducting cylinder and compared the scattering coefficients with an experiment. The Fourier expansion method has also been extended to in- homogeneous cylinder [42] and high frequency scattering [43] by Kozaki. Yokota, et. al. [44] have dealt with the scattering of Hermite-Gaussian beam by a ferrite-coated §3.2. Literature on the Scattering of Shaped Beam by Curved Objects 83 conducting cylinder. Zimmermann, et. al. [45] applied the scalar theory to analyze a refractive-index sensor at optical frequencies. For three-dimensional problems, Lan- glois, et. al. [46] have studied the diffraction of a transversal Gaussian beam, where the fields have Gaussian intensity distribution along the axis of cylinder. Lock [47] has formulated the problem of Gaussian beam obliquely incident on a cylinder. Gouesbet,

[48] et. al. have considered the case where the Gaussian beam has an amplitude cut- off along the cylinder axis [48]. It is noted that in these analysis, the cylinder being considered is closed, instead of an open geometry (see Fig. 3.1). With wave scat- tering as the primary concern, researchers have dealt mainly with far-field patterns.

For problems related to an open curved slab (Fig. 3.1), the following problems have analyzed. Lee, et. al. studied point source field propagating through spherical shell

[49], using the geometrical optics theory. Einziger and Felsen [50] obtained a hybrid formulation for ray fields transmitted from a line source through a two-dimensional radome. Treating the Gaussian beam as a bundle of complex rays, Gao and Felsen

[51] has applied the complex ray tracing method to access the radiation characteristics of radome covered antenna. More recently, Maciel and Felsen [52] have used Gaussian beam as the basis element to calculate the transmission of a general aperture field through a circular cylindrical slab. The primary concern in these studies has been antenna performance. For characterizing the material properties of a curved slab, it is necessary to first obtain the reflected fields on a planar surface near the slab. Ray tracing technique might not be desirable because it is time-consuming to compute

field quantities at many observation points. It is found that a simple modification of the available Fourier expansion method serves this purpose very well. The detail is presented in the next section. §3.3. A Modification of the Fourier Expansion method 84 3.3 A Modification of the Fourier Expansion method for Analyzing the Reflection of a 2—D Gaussian Beam from a circular shell

For 2—D Gaussian beam scattering by a cylinder, the formulation given by Kojima

and Yanagiuchi [40] is summarized in the following.

ε ,µ 0 0 ε ,µ d 0 ε ,µ ε µ 0 0 0 , 0 ε ,µ d 0 ε µ 0 , 0 a b y y x=x a 0 x=x0 3 3 2 2 x 1 1

y=−y − d 0 x y= y0

(a) Cylinder (b) Curved Slab

Figure 3.1: Gaussian beam scattering by a circular cylindrical Slab– problem geometry.

Consider an infinite long dielectric cylindrical shell with outer radius a and inner

radius b interacting with a 2-D Gaussian beam of the fundamental mode (Fig. 3.1a).

A rectangular coordinate system (x, y, z) is chosen such that the axis of the cylinder

coincides with the z-axis. The Gaussian beam is propagating in the +y direction,

with a beam axis located on the line (x = x0,y = z = 0). The beam waist is along the line (y = −y0,x = z =0).The1/e beam radius is denoted as w0. Define

also a cylindrical coordinate system (r, f, z), where z-axis is the same as that of the rectangular coordinate system, and φ(0 ≤ φ<2π) is measured from positive x-axis. §3.3. A Modification of the Fourier Expansion method 85

For an E-polarized incident wave with time dependence exp(−jwt)[40]:

Ez(ρ, φ)=u(ρ, φ)=

1/2 2 2 =(w0/w) exp{−Ir(ρ cos φ − δ) + j[θ − Ii(ρ cos φ − δ) − (ρ sin φ + γ)]}

(3.1) where:

1/2 2 2 w0/w = Q ; Ir = α /Q; Ii =2α (ρ sin φ + γ)/Q

Q α4 ρ φ γ 2 θ 1 α2 ρ φ γ =1+4 ( sin + ) ; = 2 arctan[2 ( sin + )] (3.2)

ρ = kr, α =1/(kw0),δ= kx0,γ= ky0

The functions u and its derivative with respect to ρ canbeexpandedinFourierseries with respect to φ:

∞ u(ρ, φ)= An(ρ)exp(jnφ) n=−∞ ∂ ∞ u ρ, φ B ρ jnφ ∂ρ ( )= n( )exp( ) n=−∞ 2π (3.3) 1 A (ρ)= u(ρ, φ)exp(−jnφ) dφ n 2π 0 2π 1 ∂ B (ρ)= u(ρ, φ)exp(−jnφ) dφ n 2π ∂ρ 0 §3.3. A Modification of the Fourier Expansion method 86

The general forms of scattered fields in the outer region 1 are:

+∞ Es ρ, φ C H(2) ρ −jnφ z ( )= n n ( )exp( ) =−∞ n (3.4) +∞ Hs ρ, φ C −j ∂ H(2) ρ −jnφ φ( )= n( ∂ρ n ( )) exp( ) n=−∞

The total field in the dielectric region 2 can be expressed as:

+∞ E(2) ρ, φ E J ρ F H(2) ρ −jnφ z ( )= [ n n( d)+ n n ( d)] exp( ) =−∞ n (3.5) +∞ H(2) ρ, φ −jY E J ρ F H (2) ρ −jnφ φ ( )= ( d)[ n n( d)+ n n ( d)] exp( ) n=−∞ √ with ρd = kr εd.

For the full cylinder problem in Fig. 3.1a, the fields in inner region 3 have expres- sions similar to (3.5), except that ρd should be replaced by ρ, Yd by Y .Forthecurved slab problem in Fig. 3.1b, we modify the expansion as:

+∞ E(3) ρ, φ G H(1) ρ −jnφ z ( )= n n ( )exp( ) =−∞ n . (3.6) +∞ H(3) ρ, φ G −jH(1) ρ −jnφ φ ( )= n( n ( )) exp( ) n=−∞

That is, instead of standing waves represented by Jn(ρ) and out-going waves corre- (2) (1) sponding to Hn (ρ), only incoming waves (Hn (ρ) terms) are used to represent the

fields inside. Since the Hankel functions are singular at the origin, strictly speaking the above expansion is not valid everywhere for the representation of the true trans- mitted fields in the original problem (Gaussian beam scattering by a circular shell). §3.4. The Reflected Field and Reflection Coefficient 87

However, we note that we will use the above field representation eq. (3.6) only near

the inner surface of the circular shell, far away from the origin. Our motivation for

using this artifice is to mimic the situation for the case of the truncated shell (our

original problem), which does not generate the outgoing waves because the back part

of the shell, which should have produced such fields, is not present there. To further

justify this approach of treating the open shell problem, we will present in section §3.5 the results derived for a related problem by using the complex ray tracing technique, and demonstrate that indeed the two are quite similar (see Fig. 3.2 on page 93 and

Fig. 3.3 on page 94).

3.4 The Reflected Field and Reflection Coefficient

The reflected field can be determined from the expansion coefficient Cn which should be chosen to satisfy the boundary conditions. For comparison purpose, the results

for several different situations are provided in the following. For perfectly conducting

cylinder:

C −A ρ /H(2) ρ ρ ka. n = n( a) n ( a)witha = (3.7)

For a dielectric cylinder of radius a (b =0):

Y A ρ J ρ − YB ρ J ρ d n( a) n( da) n( a) n( da) C = n YJ ρ H (2) ρ − Y J ρ H(2) ρ n( da) n ( a) d n( da) n ( a) (3.8) √ √ with ρda = ka εd, Yd = Y εd. §3.4. The Reflected Field and Reflection Coefficient 88

For a circular cylindrical interface of radius a:

Y A ρ H (1) ρ − YB ρ H(1) ρ C d n( a) n ( da) n( a) n ( da) n = (1) (2) (1) (2) (3.9) YHn (ρda)Hn (ρa) − YdHn (ρda)Hn (ρa)

For a dielectric cylindrical shell of outer radius a and inner radius b with a perfectly

conducting core:

Y A ρ J ρ − H (2) ρ K − YB ρ J ρ − H(2) ρ K d n( a)[ n( da) n ( da) ] n( a)[ n( da) n ( da) ] C = n YH (2) ρ J ρ − H(2) ρ K − Y H(2) ρ J ρ − H (2) ρ K n ( a)[ n( da) n ( da) ] d n ( a)[ n( da) n ( da) ]. J ρ K n( db) = (2) Hn (ρdb) (3.10)

For a dielectric cylindrical shell of outer radius a and inner radius b with inside empty:

Y A ρ J ρ H (2) ρ G − YB ρ J ρ H(2) ρ G C d n( a)[ n( da)+ n ( da) ] n( a)[ n( da)+ n ( da) ] , n = (2) (2) (2) (2) YH (ρ )[J (ρ )+H (ρ )G] − Y H (ρ )[J (ρ )+H (ρ )G] n a n da n da d n a n da n da (3.11) YJ ρ J ρ − Y J ρ J ρ √ n( db) n( b) d n( db) n( b) G = with ρdb = kb ε Y J ρ H (2) ρ − YJ ρ H(2) ρ d d n( b) n ( db) n( b) n ( db)

For a circular cylindrical slab of outer radius a and inner radius b:

Y A ρ J ρ H (2) ρ G − YB ρ J ρ H(2) ρ G d n( a)[ n( da)+ n ( da) ] n( a)[ n( da)+ n ( da) ] C = , n YH(2) ρ J ρ H(2) ρ G − Y H(2) ρ J ρ H (2) ρ G n ( a)[ n( da)+ n ( da) ] d n ( a)[ n( da)+ n ( da) ] (3.12) YJ ρ H (1) ρ − Y J ρ H(1) ρ √ n( db) n ( b) d n( db) n ( b) G = withρ = kb ε Y J ρ H (2) ρ − YJ ρ H(2) ρ db d d n( b) n ( db) n( b) n ( db)

Once the scattered field is calculated, the reflected signal can be determined by using the PWSM theory [29]. The main result relevant to the current problem is §3.4. The Reflected Field and Reflection Coefficient 89

given by the following formula([29], p.112):

−1 ψ ,z × a − ,z − d • d (P)= [E(R 1) H2 (R P 1 ) ez] R (3.13) a0a0η0 which essentially gives the antenna coupling function ψ as a convolution of the elec- tric field of one antenna(antenna under test) and the magnetic field of the other antenna(probe antenna, denote by the prime ). The domain of integration is a plane

z=z1 separating the two antennas. R is a vector denoting the 2-D coordinates orthog- onal to z-axis. Scalar d is the distance along z-axis between the two antennas. Vector

P denotes the relative position of the antennas in the transverse plane. Quantities without the prime are corresponding to one antenna, those with the superscript

to the other. a0 denotes amplitude of the waveguide mode in the antenna feed. η0 is

the mode impedance. Superscript “a” denotes the quantity is related to a conjugate

antenna, which is relevant only when either one of the antennas is non-reciprocal.

The main difference between the above formula and what was used in the planar

slab problem is that field quantities, rather than their spectrum functions, are now

used to define the received signal. This is convenient for the problem geometry where

incident Gaussian beam is analytically specified in a rectangular coordinate system,

while the scattered fields are formulated in terms cylindrical coordinates.

To derive an expression for the antenna response for the current problem, a virtual antenna is added to the system. This antenna has a radiation described exactly by the scattered field produced by the cylinderuponwhichaGaussianbeamisincident.

Since the present problem is 2-D, the response (coupling) should be expressed as a 1-

D integral. The transversal displacement vector P is out of the picture because only

the coupling between two antennas at a fixed relative position (P = 0) is needed. §3.4. The Reflected Field and Reflection Coefficient 90

Since it is easier to the scattered E field than the H field for the E-polarized case, the

virtual antenna will be treated as the antenna under test, and the antenna generating

the Gaussian beam will be treated as the probe antenna. With these considerations,

the response can be expressed within a constant multiplication factor as:

ψ EsHi dy = z x (3.14) where ψ, now a scalar, denotes either the reflection or the transmission signal; su- perscript “s” denotes the scattered field; superscript “i”denotesthattheH field is corresponding to the Gaussian beam “probing” field. Note that, for calculating the transmission coefficient, this “probing” field is not the Gaussian beam exciting the cylinder. The line integral is done over the beam waist. However, theoretically the same value should be obtained when the integral is defined over another line parallel to the beam waist. A proper normalization must be chosen so as to obtain a reflection coefficient with magnitude less than one, for example:

∞ 1 ES x −x2/w2 − / k w 2 x2/ k2w4 dx Γ= z ( )exp( 0)[1 1 ( 0 0) +2 ( 0 0)] (3.15) Py −∞

where the factor Py is a normalization term corresponding to the reflected signal for a perfect conduct sample:

 π P − 1 w y =(1 2 2 ) 0 (3.16) 2k0w0 2

Py can also be thought of as the total incident power passing through the y =0plane. §3.5. Numerical Result for 2—D Gaussian Beam Reflection from circular shell 91 3.5 Numerical Result for 2—D Gaussian Beam Re- flection from circular shell

Computer programs have been written inFORTRANandMathematicatoevaluate

the reflected fields and the equivalent “reflection coefficients”. Numerical examples are

considered to ascertain: 1) the validity of the modification represented by equations

(3.9) and (3.12); 2) The extend of agreement between the 2—D Gaussian beam theory

and the actual measurement.

Thefirstcasetobeconsideredisthereflected fields of Gaussian beam incident normally on a circular dielectric interface. Ruan and Felsen [53] have formulated this problem using the complex ray tracing method. They also gave the magnitude of reflection and refraction fields for various combinations of parameters. The reflected

fields obtained from their formulation are used here as a reference for checking the validity of eq. (3.9). The reflected field distribution along the transverse direction on the beam waist are illustrated in Fig. 3.2(magnitude) and 3.3(phase). The parameters have been chosen as follows: dielectric constant εd =5;beamwaistradiusw0 =

1.56λ0; radius of curvature of the circular interface a increases from 5λ0 to 40λ0;

y0 = a(i.e. beam waist locates on the plane tangential to the interface); x0 = 0(i.e.

beam axis is normal to the interface). Dotted curves are obtained from a full complex

ray tracing, rather than the paraxial approximation given in [53]. However, these ray

tracing results have been checked with the paraxial approximation and no significant

difference can be found. Solid curves are calculated from equations (3.4) and (3.9).

It is seen that both methods give answers very close to each other, except in the

phase behavior for a =30λ0 and 40λ0. While the change in magnitude is not easily

distinguishable for different radii of curvature, the phase distribution shows larger §3.5. Numerical Result for 2—D Gaussian Beam Reflection from circular shell 92 deviation as the radius increases. In fact, the magnitude distribution can be closely approximated as the incident Gaussian distribution times the reflection coefficient of plane wave, which has been plotted in the figure as the curve denoted by |EiΓplane|. However, the phase distribution does not come close to those of plane wave reflected

0 from a planar conductor (180 uniformly), for radius of curvature as big as 40λ0.

Therefore it can be concluded that the finite radius of curvature affects mainly the phase distribution. While the energy density at any particular point remains the same for decreasing radius of curvature, the variation in phase distribution will still induce a change in the received signal. Furthermore, both magnitude and phase of the received signal will be affected.

Cautions must be used in applying the complex ray tracing method [51] due to some mathematical issues related to the choice of branch cut(see appendix A). In defining the Gaussian beam as a bundle of complex rays, a branch cut is set along the beam waist. In the above numerical example, the beam waist is tangential to the curved interface and the reflected fields along the same line are calculated. To ensure a correct answer, it is important to choose the branch properly for all of the multiple √ valued complex functions( ·, sin, cos, etc.). Gao et al. provided a good reference

[51].

It is noted that the paraxial approximation in [53] also gives accurate reflected

fields. For the special case of Gaussian beam incidence normally, the departure angles for the axial ray is easily determined, resulting in an analytical expression for the reflected fields. Advantage can be taken of this analytical expression to avoid the summation of series in (3.4) and obtain a more efficient inversion algorithm. §3.5. Numerical Result for 2—D Gaussian Beam Reflection from circular shell 93

Highest: a/λ0=5,10,15,20,25,30,40(this study) λ 0.35 middle: a/ 0=5, 20,15, 20,25,30,40(ray tracing) lowest: |Ei Γplane|

0.3

0.25 5λ0 10λ0 | z 0.2 15λ E 0 | 25λ0 along this line x λ 0.15 30 0 20λ0 40λ0 0.1 a |Ei Γplane|

0.05 ε d=5

0 0 5 10 15 k0x

Figure 3.2: 2-D Ez-polarized Gaussian beam scattering at a circular dielec- tric interface. Amplitude distribution of reflected electric field |Es| ( z ) along the beam waist. Solid lines are obtained with the modified field expansion. Dotted curves are from complex ray tracing. Relative permittivity ε = 5. Radius of the interface a i changes from 5λ0 to 40λ0. Also shown is the curve |E (x)Γplane|, where Γplane is the reflection coefficient of plane wave incident normally on a planar dielectric interface with ε =5. §3.5. Numerical Result for 2—D Gaussian Beam Reflection from circular shell 94

40λ 150 30λ0 0

100

20λ0 50 25λ0 )] x

( λ z a=10 0 E 0 a=5λ0

Phase[ x -50 a=15λ0

-100

-150

0 5 10 15 k0x

|Es| Figure 3.3: Phase distribution of the reflected electric field ( z )alongthe beam waist for the same geometry as in Fig. 3.2. Solid lines are obtained with modified field expansion, while the dotted curves are from complex ray tracing.

Next, we compare the theoretical prediction with experimental results1 in Figs.

3.4(magnitude) and 3.5(phase). For the calculation, the Gaussian beam is incident normally onto the curved slab. The beam waist is assumed to be tangential to the convex surface of the slab. 1/e half beam width is 1.3λ0. The relative permittivity is taken from [13] ε =2.68. Thickness of the slab is 0.53cm. Solid lines are calculated results, while the dotted curves correspond to a experiment. A fair agreement is

1Thanks to Dr. Kollakompil of ESM department at PSU for providing the experimental results in Figs. 3.4 and 3.5 at early stage of this study. §3.5. Numerical Result for 2—D Gaussian Beam Reflection from circular shell 95

Sample spot #1 #2 #3 #4 #5 20cm 5.322 5.461 5.537 5.297 5.359 30cm 5.334 5.359 5.575 5.334 5.628 40cm 5.347 5.537 5.461 5.357 5.359

Table 3.1: Thickness (mm) of the curved samples measured at five spots.

observed. The discrepancy could be related to several factors. First of all, a 2-D

model has been used while the measurement system actually generates 3-D focused

beam. Secondly, the actual beam is only about 93% Gaussian, as shown by the last

column of table 2.3. A planar phase front isassumedinthecalculation, while the

field mapping obtained with a dipole antenna shows some 20o phase variation over the focal spot. Thirdly, the thickness of each sample is not uniform and variation as much as 0.17mm is observed. Table 3.1 shows five measurements of the thickness obtained at different spot of the sample. Heat treatment and stress were applied in making the curved samples. This might cause different change in the permittivity for samples of different radii. Lastly, mechanism for precise positioning of the curved sample is not yet available. As shown in Fig. 3.6, a holder designed for planar sample has been used and it is difficult to control and verify the location and orientation of the curved shell. The exact location of the focal plane of the system remains unknown. A small displacement might induce significant change in the measurement results, especially the phase(about 7.2o for 0.3mm displacement at 10GHz). The same comments apply

to the measurement results reported in the next section. §3.6. Experiments and Data Inversion for Circular Shells 96

-7 2-D theory -7.5 Plane-slab Measured

-8 a=40cm

-8.5 a=30cm -9 a=20cm

| (dB) -9.5 11 |S -10

-10.5 a

-11 plexiglas sample -11.5 of thickness d=0.53cm -12 9 101112 Frequency(GHz)

Figure 3.4: Comparison of theoretical prediction and experimental results for focused beam reflected from circular cylindrical slabs: ampli- tude of the reflection coefficients for plexiglas sample of thickness 0.53cm versus frequency.

3.6 Experiments and Data Inversion for Circular Shells

In spite of the above mentioned difficulties, much effort has been made to improve the accuracy of the measurement results. Among the factors investigated are: precision oftheTRLcalibrationmethodusedinthefreespacesystem,andthepotentialerror involved with the time gating process for removal of multiple reflections. Some of the findings are discussed in appendix D and E. Since improper setting of network analyzer parameters such as the gate shape and window type would render incorrect measurement results, a standard experimental procedure has been followed. §3.6. Experiments and Data Inversion for Circular Shells 97

180 2-D theory Measured

170 Plane-slab

160 a=40cm

150 a=30cm Phase (Degree)

a=20cm 140

130 9101112 Frequency (GHz)

Figure 3.5: Comparison of theoretical prediction and experimental results for focused beam reflected from circular shells: phase of the reflec- tion coefficients for plexiglas sample of thickness 0.53cm versus frequency.

Measured S11 for circular plexiglas slabs are shown in Figs. 3.7(magnitude) and

3.8(phase). Measured S11 for perfectly conducting cylinders obtained by attaching copper sheets on top of the curved sample are shown in Figs. 3.9(magnitude) and

3.10(phase). The same experiment has been repeated many times with new calibra- tion. The results from two different days are shown in these figures to illustrate the repeatability of the measurement. It is seen that the magnitude has a better repeata- bility than the phase. Deviation of the results might be explained by the reasons given at the end of last section.

To test the consistency of the Gaussian beam theory, magnitudes of measured S11 §3.6. Experiments and Data Inversion for Circular Shells 98

Figure 3.6: A picture of the curved sample mounted in the holder originally designedforplanarsamples.Ithasnotbeenpossibletoprecisely adjust and verify the location and orientation of the sample.

0.40 50cm 40cm Day 1 0.38 Day 2

0.36 30cm 0.34

| 20cm 11 0.32 |S 0.30 0.28 0.26 0.24

8 9 10 11 12 13 14 Frequency(GHz)

Figure 3.7: Repeatability of the Measurement–|S11| for plexiglas circular shells obtained on two occasions. §3.6. Experiments and Data Inversion for Circular Shells 99

180 50cm: day 1 day 2 40cm: day 1 day 2 170 30cm: day 1 day 2 20cm: day 1 day 2 160 11 150 Phase S 140

130

120

8 9 10 11 12 13 14 Frequency(GHz)

Figure 3.8: Repeatability of the Measurement–Phase of S11 for plexiglas circular shells measured on two occasions.

0.94 50cm 0.92

0.90 40cm

| 0.88 11 |S 0.86 30cm

0.84 20cm Day 1 0.82 Day 2 0.80 8 9 10 11 12 13 14 Frequency(GHz)

Figure 3.9: Repeatability of the Measurement–Magnitude of S11 for cylin- drical copper sheets measured on two occasions. §3.6. Experiments and Data Inversion for Circular Shells 100

176 40cm 174

172 50cm 11

170

Phase S Phase 20cm 30cm 168

166 Day 1 Day 2 164 8 9 10 11 12 13 14 Frequency(GHz)

Figure 3.10: Repeatability of the Measurement–Phase of S11 for cylindrical copper sheets measured on two occasions. for the copper sheets are used to estimate the 1/e half beam width w. A computer program is written to find the value w within the range λ0 and 2λ0 for which the calculated reflection coefficient has the same magnitude as the measured result at each frequency point. The result is presented in Fig. 3.11. Although these values are much different from the 1.2λ0 obtained by fitting the 3-D antenna field distribution

measured with the dipole, some sort of consistency is observed for the 30cm, 40cm

and 50cm copper sheet. Subsequently, these 2-D beam widths are used as the input

parameter in inverting the measured |S11| for circular plexiglas slabs. That is, beam

widths estimated from 20cm copper sheet are used for 20cm plexiglas slab; 30cm copper sheet results are used for 30cm plexiglas slab; etc. The resulting real part of permittivity ε are presented in Fig. 3.12.

A second approach has been tested for the inversion problem. Measured S11 for the §3.6. Experiments and Data Inversion for Circular Shells 101

2.0 1.9 )

0 1.8 λ 1.7 1.6 1.5 1.4 50cm 1.3 40cm 1.2 30cm 20cm

1/e half beam 1/e (unit: beam half width 1.1 1.0 8 9 10 11 12 13 14 Frequency(GHz)

Figure 3.11: 1/e half beam width calculated from measured |S11| of copper sheets.

50cm 2.8 40cm 30cm 20cm slab

2.6 ’ ε

2.4

8 9 10 11 12 13 14 Frequency(GHz)

Figure 3.12: Real part of permittivity ε inverted from the estimated beam width and magnitude of reflection coefficients S11 for curved plexiglas slabs. Also shown is the output from HVS free-space material measurement software for a planar plexiglas slab sam- ple. §3.6. Experiments and Data Inversion for Circular Shells 102

plexiglas samples are normalized by the negative of S11copper measured for the copper

sheet of the same radius. The results are treated as the reflection coefficient of planar

samples of the same thickness. The underlying idea is that −1.0/S11copper can be ap-

proximately treated as a curvature correction factor for all samples of the same radius

of curvature. The results after this normalization are shown in Fig. 3.13(magnitude),

and 3.14(phase) for the plexiglas samples. The permittivity inverted from these cur-

vature corrected data are shown in Figs. 3.15(ε) and 3.16(ε).Itcanbeseenthat

the ε results are in reasonable agreement with the output of the commercial program

for a planar plexiglas slab. The imaginary part ε are found to be too high for two

of the samples. This is a defect of the reflection method. Generally speaking, even

for planar slab sample, it is very difficult to obtain accurate ε result for low loss materials from one reflection measurement.

0.44 slab 50cm 40cm 0.40 30cm

| 20cm 11

0.36

0.32 Normalized |S Normalized

0.28

8 9 10 11 12 13 14 Frequency(GHz)

Figure 3.13: Curvature-corrected |S11| for the plexiglas samples. §3.6. Experiments and Data Inversion for Circular Shells 103

200

190 50cm

11 40cm 180 30cm 20cm 170 slab 160

150

Phase of normalized S 140

130

120 8 9 10 11 12 13 14 Frequency(GHz)

Figure 3.14: Phase of curvature-corrected S11 for the plexiglas samples.

2.8

2.4

2.0 ’ ε

slab 50cm 1.6 40cm 30cm 20cm

1.2 8 9 10 11 12 13 14 Frequency(GHz)

Figure 3.15: Real part (ε) of permittivity inverted from the curvature- corrected S11 for the plexiglas samples. §3.7. Conclusion 104

0.4 slab 50cm 0.3 40cm 30cm 0.2 20cm " ε 0.1

0.0

-0.1 8 9 10 11 12 13 14 Frequency(GHz)

Figure 3.16: Imaginary part (ε) of permittivity inverted from the curvature- corrected S11 for the plexiglas samples.

3.7 Conclusion

This chapter first presents a theoretical model for estimating the reflection coefficient measured by a spot-focused free-space system for curved samples in the shape of a covex circular shell. Within this model, a modification of the field expansion has been proposed for calculating the reflected field of a Gaussian beam scattering by a circular shell. Although the modification is not rigorous mathematically, heuristic arguements and numerical comparison are provided in justification of its use for the current problem and the intended application. The plane wave scattering matrix theory has been applied to formulate the reflection coefficient. Although the radiating and receiving characteristics of the antennas have been modeled by a Gaussian beam, the same procedure is applicable for a shaped beam whose near-field pattern has been accurately measured. §3.7. Conclusion 105

Experimental results for several circular shell samples have also been presented.

Reasonably good results are obtained for the real part of the permittivity for low loss plexiglas samples. To achieve high accuracy, precise sample positioning is required, which is not possible with the availalbe sample fixture. For comparison, it should be noted that reflection measurement (without applying metal backing) is not suitable for determining the loss tangent of low loss material even in a traditional waveguide setup.

Therefore the results presented in this chapter has demonstrated the applicability of spot-focused free-space system for characterizing curved samples through reflection

measurement. The fact that only circular shell sample has been studied does not

prevent us from suggesting the following two different procedures for obtaining the

permittivity of a general curved sample.

Method 1: Measure the reflection coefficient Γs for the sample; measure or cal- culate the reflection coefficient Γm for a conductor of the sample radii of curvature; apply the inversion algorithm for a planar sample to the curvature corrected reflection

coefficient −Γs/Γm to obtain the permittivity.

Method 2: Measure the reflection coefficient Γs for the sample; write a computer program for calculating the reflection coefficient of curved samples, where the input

parameters include radii of curvature, thickness, permittivity and near-field pattern

of the interrogating antenna; search for the optimum value of permittivity so that the

calculated reflection coefficient matches the measured result.

Method1iseasytoimplementandtherefore more practical. However, it is ap- plicable for a limited range of sample thickness and permittivity. Method 2 is theo- retically more sound, but involves two difficult issues which have not been studied in this thesis. The first issue is accurate measurement of antenna near field. The other §3.7. Conclusion 106 issue is the computation complexity inherent in the optimization problem. These should be interesting topics for future study. Chapter 4

Open-ended Coaxial Line Radiating into a Chiral Medium

In this chapter1, a spectral-domain moment method solution is presented for estima- tion of the reflection coefficients of an open-ended coaxial waveguide radiating into a chiral half-space. The electromagnetic fields in the waveguide region are expanded in the normal waveguide modes, while those in the half-space are formulated in terms of plane wave spectrum using a two dimensional Fourier transform. A spectral domain admittance matrix is derived through which an electric field integral equation can be obtained by matching the tangential field components on the open end interface. This integral equation is solved with the Galerkin method. The coefficients of the matrix equation are reduced to one-dimensional integrals to facilitate numerical calculation.

The problem is different from the case of isotropic non-chiral medium in that chi- rality causes excitation of TE modes, whose effect is accounted for rigorously in the formulation. This is in contrast to plane wave reflected from a chiral media at nor- mal incidence, where no de-polarization effect can be observed. Numerical results are presented to illustrate the idea that the reflectivity of TEM mode carries information

1A paper containing most of the materials in this chapter will appear in the May 2001 issue of Journal of Electromagnetic Waves and Applications

107 §4.1. Introductory remark 108 on the presence of chirality. The analysis has a potential application in developing an open-ended waveguide method for the characterization of chiral materials using only reflection measurements including depolarization effects.

4.1 Introductory remark

Apart from its theoretical value within the framework of electromagnetic theory, re- search on chiral material in the last few years has also been driven by its potential applications in the microwave frequency range. These applications, as conceived in a number of theoretical explorations, include radar absorbing material [54], waveguide

filter and coupler [55], polarization transformer for aperture antennas [56], chirostrip antenna [57], etc. Of fundamental importance in realizing these applications is an accurate method for the characterization of chiral property. Two measurement meth- odshavebeenreportedinthepublicliterature: the free space method [58] and the circular waveguide method [63]. The purpose of this paper is to present a theoretical analysis of the reflection property of open-ended coaxial line terminated by a chiral half-space. The problem has not been analyzed before. It is treated here as a first step toward developing an open-ended waveguide method for the characterization of chiral materials.

The use of an open-ended coaxial line as a material sensor has been studied for over two decades(see, for example [65], [66], [71]). It was observed that the coaxial probe has its advantages over other sensors, most importantly being its non-destructive na- ture and its usefulness over a broad frequency bandwidth. Although the method was originally designed for measuring high permittivity dielectrics like liquid and biologi- §4.1. Introductory remark 109 cal tissues, it has been generalized to the measurement of both complex permittivity and permeability [72]. It is also used for finite thickness composite sheets [73] and in high temperature environment[74]. The working principle of coaxial probe sensor is that the reflection coefficient will carry information about the material properties of its termination. Therefore the problem of this paper is to determine the effect of chiral parameter on the reflection property of an incident TEM mode in the coaxial line. We will formulate the problem rigorously by following the conventional steps of a spectral domain moment method and applying the necessary modification to account for the presence of chirality. It should be mentioned that the modification is not trivial. The available moment method formulation for the achiral problem does not apply directly to the chiral case. This is due to the fact that the chiral medium cannot support individual transverse electric(TE) or transverse magnetic(TM) modes, but the con- ventional formulation such as those in reference [72] uses a continuous spectrum of

TM modes to express the fields in the open space region. The coupling between TM modes and TE modes must be considered in order to observe the effect of chirality.

As will be seen in the main text of this paper, this can be achieved by using the two dimensional Fourier transform to express the electromagnetic fields in the chiral medium, and defining a spectral domain surface admittance matrix to characterize the wave interacting in the half-space. This last step, from a mathematical point of view, reduces the problem to solving one integral equation for the aperture electric

field only, eliminating the equation for aperture magnetic field. Application of the

Galerkin technique transforms the integral equation into a set of linear equations for the unknown modal expansion coefficients(reflectioncoefficients).Inaccordancewith the ideas in reference [77], the elements of the coefficient matrix can be reduced to one-dimensional integrals, by writing the Fourier transforms in term of polar coordi- §4.2. Basic idea for solving the half space problem 110 nates and exchanging the integration sequence. The mathematical analysis is outlined insection2withthedetailsgiveninsection 3 and an appendix. Numerical results are illustrated in section 4 along with some general discussion about the potential of using coaxial probe in chiral material characterization.

4.2 Basic idea for solving the half space problem

The geometry of the problem is shown in Fig. 4.1: a flanged open-ended coaxial probe is placed against a homogeneous chiral media. The chiral media is assumed to be filling the half space on the right. The waveguide is excited so that only the axisymmetric TEM mode is incident upon the aperture. The reflected field in the waveguide will consist of both the propagating TEM mode and higher order axisymmetric modes. In contrast to the case of an isotropic medium, where only higher order TM0n modes are excited, the TE0n modes also exist for the chiral case. The reflection coefficients of these waveguide modes are to be determined.

The problem can be analyzed using the spectral domain moment method. The main ideas of the analysis procedure is summarized in the following:

• The electromagnetic(EM) field in the coaxial waveguide is expanded as a sum-

mation of normal modes, with the unknown amplitudes as the coefficients. The

amplitudes are proportional to the reflection coefficients of the normal modes.

• The EM fields in chiral half space consists of plane waves with continuous spec-

trum, therefore they can be readily expressed through two dimensional rectan-

gular Fourier transformations. §4.2. Basic idea for solving the half space problem 111

x

ε µ 0, 0 ε, µ, β 2a 2b z y

Figure 4.1: Geometrical configuration and coordinate system for the prob- lem. A flanged open-ended coaxial waveguide with inner radius b and outer radius a isradiatingintoachiralhalf-spacewith permittivity ε, permeability µ and chiral parameter β. The per- mittivity and permeability of the waveguide material TEM mode isexcitedinthewaveguideandincidentfromtheleft.ε0 and µ0 denote the permittivity and permeability of the waveguide medium(not necessarily air).

• A spectral domain surface admittance can be derived, giving the relation be-

tween the tangential spectral magnetic field and spectral electric field on the

open end boundary.

• The tangential fields are continuous on the boundary. By the Fourier transfor-

mation, The spectral EM fields on the aperture are in turn expressed through

the unknown amplitudes. An integral equation about these unknown ampli-

tudes is obtained. §4.3. Formulation 112

• Using the eigenmode functions as the test functions, a set of linear equations are

derived using the Galerkin method. The linear equations are solved numerically

giving the required reflection coefficients

4.3 Formulation

The mathematical details of the analysis is given in this section. The coaxial probe has

inner radius a and outer radius b. The material inside the coaxial line has permittivity

ε0 and permeability µ0. ε0 and µ0 areassumedtobereal.Thehalfspacez>0is taken to be homogeneous chiral media with permittivity ε , permeability µ and chiral parameter β . The flange is assumed to extend to infinity in the xy plane. The time harmonic factor ejωt is suppressed in all field representation.

4.3.1 Field Representation in the Chiral Half-Space

The Drude-Born-Fedorov notation for the chiral constitutive relations is:

D = ε(E + β∇×E), (4.1) B = µ(H + β∇×H).

Substituting the constitutive relations (1) into the Maxwell’s equations, we obtain:

      ω2µεβ −jωµ E 1   E ∇× = 2 . (4.2) H 1 − ω2µεβ jωε ω2µεβ H §4.3. Formulation 113

To uncouple the matrix equations, we apply the wave splitting technique and define two vectors U and V by:       E 11 U =   , (4.3) H j/η −j/η V

 µ where η = ε . The curl equations for vectors U and V can be easily verified as:

      U k1 0 U ∇× =   (4.4) V V 0 −k2 where:

√ k1 = kc/(1 − kcβ),k2 = kc/(1 + kcβ),kc = ω µε. (4.5)

It can be verified that U and V are divergenceless in a source free region, and they also satisfy the vector wave equations. Upon defining a two dimensional Fourier transform over the tangential (xy) plane, the wave equations reduce to:

∂2 U% +(k2 − k2 − k2)U% =0. ∂z2 1 x y (4.6) ∂2 V% +(k2 − k2 − k2)V% =0. ∂z2 2 x y where the Fourier transform and its inverse transform are defined as: % jkxx+jkyy U(kx,ky,z)= U(x, y, z) e dx dy; A ∞ (4.7) 1 U(x, y, z)= U% (k ,k ,z) e−jkxx−jkyydk dk . 4π2 x y x y −∞ §4.3. Formulation 114

The general solution to the differential equations (4.6) are given by:

% % −jk1zz % jk1zz U(kx,ky,z)=U+(kx,ky) e + U−(kx,ky) e . (4.8) % % −jk2zz % jk2zz V(kx,ky,z)=V+(kx,ky) e + V−(kx,ky) e

where k1z and k2z (with Im(k1z) ≤ 0, Im(k2z) ≤ 0) are defined by:

k2 k2 − k2 − k2 k2 k2 − k2 − k2. 1z = 1 x y; 2z = 2 x y (4.9)

% % For the current problem, U−(kx,ky)andV−(kx,ky)areequaltozerosinceEMfields in the chiral half-space is propagating in the positive z direction only.

The decomposition in (4.3) remains true for spectral domain values, therefore

∞ 1 E(x, y, z)= U% + V% (k ,k ,z) e−jkxx−jkyydk dk ; 4π2 x y x y −∞ . ∞ (4.10) j H(x, y, z)= U% − V% (k ,k ,z) e−jkxx−jkyydk dk 4π2η x y x y −∞

4.3.2 Electromagnetic Fields in the Coaxial Waveguide

Consider the electromagnetic fields inside the coaxial waveguide when a TEM mode is incident from the left. Because of the rotation symmetry of the problem, only higher order TE0n and TM0n modes are excited from the discontinuity at the open end. This should be compared with the case of isotropic media, where only the TM0n modes are present. The reason is that the TE and TM waves are not eigenwaves for a chiral media.IftheEMfieldsinthechiralregionweretobeexpandedintermsofTEand TM waves, these waves are coupled with each other. For the problem of non-chiral isotropic media, EM fields in both the waveguide and the open regions are of the TM §4.3. Formulation 115

type, in the sense that the electric field has only a radial component and a z-directed component, the magnetic field has only azimuthal component. The incident TEM mode will not be coupled to any TE waves. This can be proved mathematically.

In view of the above discussion, the tangential EM fields in the coaxial waveguide can be written as:

∞   − γn2 −j e jk0z − R ejk0z ∇ ψ jR eγn1z ×∇ψ R eγn2z∇ ψ Et = ( 0 ) t 0 + n1 aˆz t n1 + n2 t n2 k0 n=1

(4.11) −j −jk0z jk0z Ht = (e + R0e )aˆz ×∇tψ0 η0 ∞   γn1 −j R eγn1z∇ ψ R eγn2z ×∇ψ + n1 t n1 + n2 aˆz t n2 (4.12) k0η0 η0 n=1 √ where k0 = ω µ0ε0 is the wavenumber in vacuum, η0 is the wave impedance, R denotes the reflection coefficients, a subscript 0 denotes corresponding values for TEM

mode, n1forTE0n mode, n2forTM0n mode, ψ is the transverse wave potential. γ’s are z propagation factors:

 2 2 γn1 = (kn1/a) − ω ε0µ0 for TE0n modes;  (4.13) 2 2 γn2 = (kn2/a) − ω ε0µ0 for TM0n modes.

kn1 is the n-th nonzero solution to the equation:

J0(k)N0(k/c) − N0(k)J0(k/c)=0, (4.14)

where the prime denote derivative with respect to the argument, and kn2 is the n-th §4.3. Formulation 116

nonzero solution to the equation:

J0(k)N0(k/c) − N0(k)J0(k/c)=0, (4.15) with the constant c = a/b.

The transverse potential functions are given by:

ψ0 =lnr ψ N k /c J k r/a − J k /c N k r/a n1 = 0( n1 ) 0( n1 ) 0( n1 ) 0( n1 ) (4.16) ψ N k /c J k r/a − J k /c N k r/a n2 = 0( n2 ) 0( n2 ) 0( n2 ) 0( n2 )

4.3.3 Spectral Domain Surface Admittance

The spectral domain surface admittance gives the relation between the spectrum of % % tangential magnetic field Ht and tangential electric field Et. By equations (4.4) and (4.7):

% −jkxx−jkyy % −jkxx−jkyy ∇×[U(kx,ky,z)e ]=k1U(kx,ky,z)e , (4.17) % −jkxx−jkyy % −jkxx−jkyy ∇×[V(kx,ky,z)e ]=−k2V(kx,ky,z)e .

Writing the tangential components in terms of the longitudinal component, we have:

 %  %   U+ U+ −jk1k − k k1 x = z y x z (4.18) % k2 k2 jk k − k k U+y x + y 1 x y 1z

% Similarly, changing k1 to −k2, k1z to k2z gives the corresponding relation for V :

 %  %   V+ V+ jk2k − k k2 x = z y x z (4.19) % k2 k2 −jk k − k k V+y x + y 2 x y 2z §4.3. Formulation 117

The spectral values of tangential electric field can then be expressed in terms of % % Uz and Vz :    %   %  E 1 −jk1ky − kxk1z jk2ky − kxk2z U+ x =   z (4.20) % k2 k2 % Ey + V+z x y jk1kx − kyk1z −jk2kx − kyk2z

   %   %  H j −jk1ky − kxk1z −jk2ky + kxk2z U+ x =   z (4.21) % k2 k2 η % Hy ( + ) V+z x y jk1kx − kyk1z jk2kx + kyk2z

So that:       H% j −AD − BC 2AB E% x =   x (4.22) H% η E% y −2CD AD + BC y where:

A = −jk1ky − kxk1z B = −jk2ky + kxk2z (4.23) C = jk1kx − kyk1z D = jk2kx + kyk2z

The spectral domain admittance can be defined by:         H% Y Y E% ↔ E% x  xx xy x x % = % ≡ Y · % (4.24) Hy Ey Ey Yyx Yyy §4.3. Formulation 118

with

−j(AD + BC) Y = −Y = xx yy (BC − AD)η 2jAB Y = (4.25) xy (BC − AD)η −2jCD Y = yx (BC − AD)η

For a non-chiral isotropic half-space, β =0,k1z = k2z = kz,k1 = k2 = k,

A = −jkky − kxkz,B= −jkky + kxkz, (4.26) C = jkkx − kykz,D= jkkx + kykz.

The spectral domain surface admittance reduces to:

−kxky Yxx = −Yyy = kkzη −k2k2 − k2k2 Y = y x z xy kk k2 k2 η (4.27) z( x + y) k2k2 + k2k2 Y = x y x yx kk k2 k2 η z( x + y)

4.3.4 Integral Equation

The boundary condition for EM fields at z =0reads:

Et|z=0− = Et|z=0+ , Ht|z=0− = Ht|z=0+ (4.28) §4.3. Formulation 119

Using the Fourier transform and spectral domain admittance relation (4.24), Ht can be written as:

∞ 1 % + −jkxx−jkyy H | =0− = H | =0+ = H (k ,k ,z =0 ) e dk dk t z t z 4π2 t x y x y  −∞  ∞ 1 ↔ = Y ·  E (x, y, z =0−) ejkxx+jkyydx dy e−jkxx−jkyydk dk 4π2 t x y −∞ A (4.29)

This gives an integral equation for the unknown reflection coefficients in views of eqns. (4.11) and (4.12).

4.3.5 Method of Moment—Derivation of Matrix Equations

Using the transverse mode functions as the test functions, the integral equation is transformed to the following set of linear equations by the Galerkin’s method: − ∗ ∗ ml M ≡ × | − · ds × | + · ds ≡ M ml [eml Ht z=0 ] aˆz = [eml Ht z=0 ] aˆz + A A (for m, l =0, 1, 2,...)

(4.30)

where a superscript ∗ denotes complex conjugation, and    ×∇ψ l aˆz t m1 for TE mode =1 eml =  (4.31) ∇ ψ l t m2 for TM mode =2 §4.3. Formulation 120

M − M + ml and ml are often called reactions. After some tedious but straightforward algebra, the reactions are can be expressed as:

− l − M =(2− ιn +(−1) Rml)K where ml  ml  −j η 2π ln c TEM mode (m =0)  0    2 (4.32) ( ) − γm1 4 J0 km1/c − m> ,l K ≡ η ( ) ( ) 1 TE modes ( 0 =1) ml  k0 0 π J0 km1     2  − 4 ( )  j ( ) J0 km2/c − 1 TM modes (m>0,l =2) η0 π J0(km2)

2 ∞ 2 (k0a) M + = (2 − ι +(−1)i−1R )K+ ml πη n ni ml,ni n=0 i=ιn (4.33) ∞ K+ ≡ a Y u F u F u udu where ml,ni ni ml,ni( ) ml( ) ni( ) 0

where ι0 = 1 (TEM), ιn =2(n>0), R02 ≡ R0,also:

a02 ≡ a0 = −j (TEM); an1 = j (TE); an2 = −γn2/k0 (TM). (4.34)

   0(k0au =0) F02(u) ≡ F0(u)= (4.35)  π[J0(k0au/c) − J0(k0au)]/(k0au)(k0au =0)

F 1(u)= n    1 1 J1(k 1)J0(k 1/c) − J1(k 1/c)J0(k 1) (k0au = k 1) J1(kn1) c n n n n n   2kn1 1 2 2 [J1(k0au/c)J1(kn1) − J1(k0au)J1(kn1/c)] (k0au = kn1) [(k0au) −kn1] J1(kn1) (4.36) §4.3. Formulation 121

F 2(u)= n    1 1 J1(k 2/c)J0(k 2) − J1(k 2)J0(k 2/c) (k0au = k 2) J0(kn2) c n n n n n   2k0au 1 2 2 [J0(k0au)J0(kn2/c) − J0(k0au/c)J0(kn2)] (k0au = kn2) [(k0au) −kn2] J0(kn2) (4.37)

  4k1zk2z  (for TE-TE coupling l =1,i=1) k1 k2 + k2 k1  z z  k2zk1 − k1zk2 2j (for TE-TM or TE-TEM coupling l =1,i=2) k1zk2 + k2zk1 Yml,ni(u)=  k2 k1 − k1 k2 2j z z (for TM-TE or TEM-TM coupling l =2,i=1)  k k k k  1z 2 + 2z 1  4k1k2  (TM-TM, TEM-TM or TM-TEM l =2,i=2) k1zk2 + k2zk1 (4.38)

Notice that the integrand in (4.33) does not have any real singular point when β =0.

It has a removable singularity at k1z = k2z =0whenβ =0.

Equations (4.30) can now be written as an infinite set of linear equations about the unknown reflection coefficients:

· · [Aml,ni] [Tni]=[Bml,ni] [Rni] 2 (k0a) A = K+ − K− δ δ ml,ni πη ml,ni ml mn li   2 (4.39) (k0a) A =(−1)i K+ + K− δ δ ml,ni πη ml,ni ml mn li

where vector T denotes the amplitude of incident modes, vector R has the reflection §4.4. Numerical Results 122

coefficients as its elements. With R0 as the first element of R, T is:

T =[1, 0, 0, ··· , 0]t. (4.40)

By selecting appropriate vector T and propagating factor γ, equation (4.39) can be

used to calculate the reflection coefficients when several higher order TE0n or TM0n modes are incident from the left in the coaxial waveguide.

4.4 Numerical Results

The numerical calculation is straightforward. The matrix elements defined in eqn. (4.39) are computed, after which a matrix inversion and matrix multiplication are performed. This gives the reflection coefficient vector R which contains the reflection coefficient of TEM mode. The infinite integral in eqn. (4.33) requires some special care. The adaptive integration routine DQAGS in [82] has been used. The -algorithm implementedin[82]withthezerosofBessel functions as the break points was used to decide the truncation point and estimate the integral tail.

To verify the theoretical formulation and the computer program, the TEM mode reflection coefficient of an open-ended coaxial line terminated by a dielectric half-space is first computed. Table I shows the results along with data published in [83]. In this calculation a nonzero yet very small value(10−10) is chosen for the chiral parameter β, so that the infinite integral in eqn. (4.33) does not contain any singularity. For the sake of checking the program, a relatively large number of 30 modes (including both TE and TM) are used in the field expansion. The geometry parameters of the coaxial line is: outer radius of inner conductor(b =1.4364mm), inner radius of outer conductor(a =4.725mm).Thedielectrichalf-spaceandthecoaxiallinematerial §4.4. Numerical Results 123

both have relative permittivity 2.05. This specific set of parameters have been used in several papers. The fact that TE modes are not excited by the simple dielectric half-space is used as a sanity check of the computer program. The computed results shows that TE mode reflection coefficients are indeed very small. It is also observed that the TEM mode reflection coefficient remains the same after removing the TE mode components from the matrix equation (4.39). In table I, a relatively bigger discrepancy is seen for high frequency values. This is reasonable because the chiral parameter is fixed at 10−10 and the chiral effect is bigger for higher frequencies. In thefollowing,wewillusekβ instead of β as an independent variable. Figures 4.2-4.9 demonstrate the behavior of the TEM mode reflection coeffi- cient(amplitude and phase) as a function of frequency and the chiral parameter β for a small coaxial line(a =4.725mm, b =1.4364mm). To avoid improper values for the chiral parameter, one of the independent variables is chosen as either |kc|Re(β) −6 or |kc|Im(β), with their values varying between 10 and 0.5. Since a small coaxial probe is not suitable for measuring low permittivity material, two different values are chosen for the permittivity (ε=5 and 50). Some conclusions can be drawn by com- paring these figures: A) Consider the change of TEM mode reflection coefficient as a function of frequency, it follows the same pattern for small and high values of chiral parameters. With respect to the small probe used in these computation, very little change can be observed for low frequency(1GHz-5GHz) and low permittivity materi- als. Therefore a bigger probe is needed to measure small permittivity chiral material at low frequencies. B) The chiral parameter affects TEM mode reflection coefficient. This effect, although generally small at low frequency, become more evident as the dielectric constant ε goes higher.

Figure 4.10 shows the reflection coefficient of TE01 mode for the TEM mode incident case. As can be expected, its amplitude goes higher as the chiral parameter becomes larger. However, as a function of frequency, it reaches a maximum value at §4.4. Numerical Results 124

frequency Amplitude Phase(degrees) (GHz) This work Ref[83] This work Ref[83] 1 0.999983 0.99998 -4.84594 -4.8689 1.5 0.999912 0.99992 -7.28700 -7.3210 2 0.999725 0.99972 -9.74894 -9.7978 2.5 0.999337 0.99929 -12.23744 -12.298 3 0.998646 0.99860 -14.75706 -14.834 3.5 0.997539 0.99740 -17.31123 -17.405 4 0.995890 0.99568 -19.90201 -20.012 4.5 0.993576 0.99324 -22.53010 -22.660 5 0.990472 0.98997 -25.19478 -25.344 5.5 0.986462 0.98576 -27.89428 -28.064 6 0.981445 0.98049 -30.62526 -30.817 6.5 0.975333 0.97408 -33.38368 -33.597 7 0.968062 0.96646 -36.16445 -36.400 7.5 0.959587 0.95760 -38.96190 -39.219 8 0.949888 0.94747 -41.76990 -42.048 8.5 0.938970 0.93609 -44.58203 -44.881 9 0.926856 0.92350 -47.39188 -47.710 9.5 0.913593 0.90973 -50.19298 -55.530 10 0.899244 0.89487 -52.97951 -53.333

Table 4.1: Amplitude and phase of the TEM mode reflection coefficient of a flanged open-ended coaxial waveguide(a =4.725mm, b = . mm, ε . ,µ . 1 4364 r =205 r =10) terminated by a dielectric half- ε . ,µ . space( cr =205 cr =10) some particular frequency(about 10GHz for the chosen set of parameters) and starts decreasing afterwards. Nevertheless, this suggests the use of TE mode reflection coefficients as independent measurement data for the inversion of chiral parameters. §4.5. Conclusion 125

1

0.75

|R 0.5 0 | 0.25 0 0 5 ) 0.1 Hz 0.2 10 (G |k ncy c Re 0.3 ue (β 15 req )| 0.4 f 0.5 20

Figure 4.2: Amplitude of TEM mode reflection coefficient v.s. fre- quency(GHz) and real part of chiral parameter ≡|kc|Re(β). The Chiral half-space has relative permittivity εcr =5.0 − 0.1j µ . − . j and relative permeability cr =20 0 02 , The dielectric in the waveguide has εr =2.05,µr =1.0,Im(β)=0.0, The geometry of the coaxial waveguide is: a =4.7250mm, b =1.34364mm. 4.5 Conclusion

A spectral domain moment method has been used to analyze the open-ended coaxial line terminated by a chiral half-space. The reflection coefficients of TEM and other higher order modes are numerically calculated, and their characteristics as functions of frequency and chiral parameters are discussed. Numerical results show that chirality affects the TEM mode reflection coefficient. It is therefore possible to use the coaxial probe as a chiral material sensor. It is also observed that even a small value of chirality will cause TE mode to be excited. A simple dielectric material will not have such an effect. If one could operate the coaxial probe in a multimode state and §4.5. Conclusion 126

0

−50 Arg(R

−100 0 ) 0 −150 0 5 0.1 ) GHz 0.2 10 y( |k enc c R 0.3 qu e(β 15 fre )| 0.4 0.5 20

Figure 4.3: Phase of TEM mode reflection coefficient v.s. frequency(GHz) and real part of chiral parameter ≡|kc|Re(β). Parameters are the same as in figure 4.2.

1 0.9 0.8 |R 0

| 0.7 0.6 0 0.5 5 0.1 z) 10 GH 0.2 y( |k enc c R 0.3 u e(β 15 req )| 0.4 f 0.5 20

Figure 4.4: Amplitude of TEM mode reflection coefficient v.s. fre- quency(GHz) and real part of chiral parameter ≡|kc|Re(β). Parameters are the same as in figure 4.2 except that permittivity is bigger(εcr=50.0-1.0j). §4.5. Conclusion 127

0

−50

Arg(R −100 0 ) −150 0 0 5 0.1 z) GH 0.2 10 y ( |k enc c R 0.3 qu e(β 15 fre )| 0.4 0.5 20

Figure 4.5: Phase of TEM mode reflection coefficient v.s. of frequency(GHz) and real part of chiral parameter ≡|kc|Re(β). Parameters are the same as in figure 4.4.

1

0.75

|R 0.5 0 | 0.25 0 0 5 ) 0.1 Hz 0.2 10 (G |k ncy c I 0.3 ue m(β 15 req )| 0.4 f 0.5 20

Figure 4.6: Amplitude of TEM mode reflection coefficient v.s. fre- quency(GHz) and imaginary part of chiral parameter ≡ |kc|Im(β). Other parameters are the same as in figure 4.2 ε . − . j, µ . − . j, Re β . ,ε except( cr =50 0 1 cr =20 0 02 ( )=00 r = 2.05,µr =1.0,a=4.7250mm, b =1.34364mm). §4.5. Conclusion 128

0

−50 Arg(R

−100 0 ) 0 −150 0 5 0.1 z) GH 0.2 10 y ( |k enc c I 0.3 qu m(β 15 fre )| 0.4 0.5 20

Figure 4.7: Phase of TEM mode reflection coefficient v.s. frequency(GHz) and imaginary part of chiral parameter ≡|kc|Re(β). Parame- ters are the same as in figure 4.6.

1

0.8 |R 0 | 0.6 0 0 5 ) 0.1 Hz 0.2 10 G |k ncy( c I 0.3 ue m(β 15 req )| 0.4 f 0.5 20

Figure 4.8: Amplitude of TEM mode reflection coefficient v.s. fre- quency(GHz) and imaginary part of chiral parameter ≡ |kc|Im(β). Parameters are the same as in figure 4.6 except that ε . − . j, µ . − . j, Re β permittivity is bigger( cr =500 1 0 cr =20 0 02 ( )= 0.0,εr =2.05,µr =1.0,a=4.7250mm, b =1.34364mm). §4.5. Conclusion 129

0

−50

Arg(R −100 0 ) −150 0 0 5 0.1 z) GH 0.2 10 y ( |k enc c I 0.3 qu m(β 15 fre )| 0.4 0.5 20

Figure 4.9: Phase of TEM mode reflection coefficient v.s. frequency(GHz) and imaginary part of chiral parameter ≡|kc|Im(β). Parame- ters are the same as in figure 4.8.

measure the TE01 mode reflection coefficient effectively, the chirality parameter can

be related to this information and possibly inverted directly. However, there remain

many practical issues affecting the feasibility of open-ended coaxial probe method.

For example, artificial chiral materials currently available tend to be inhomogeneous.

It is necessary to use a large coaxial probe in order to obtain an average chiral

property. Alternatively, one can apply the probe at different locations of the chiral

sample and then average the measured results through some data processing. It is noted that inhomogeneity also causes difficulty in the waveguide method [63]. The main contribution of the current study therefore lies in the fact that we have shown the influence of chirality on the reflection coefficient of TEM mode inside a coaxial line.

For further research, several issues can be addressed. The first is to study the reflection properties of coaxial probe terminated by layered chiral structures. This §4.5. Conclusion 130

0.2

|R 0.1 TE01 |

0 20 0.4 15 0.3 |k 10 ) c Re( 0.2 (GHz β ency ) 5 u | 0.1 freq 0 0

Figure 4.10: Amplitude of TE01 mode reflection coefficient v.s. fre- quency(GHz) and real part of chiral parameter ≡|kc|Im(β). ε . − . j, µ Parameters are the same as in figure 4.2( cr =50 0 1 cr = 2.0−0.02j, Im(β)=0.0,εr =2.05,µr =1.0,a=4.7250mm, b = 1.34364mm). will give theoretical prediction for three independent measurements for simultaneously obtaining the three material parameters. Secondly, in order to efficiently invert the material parameters, it is desirable to derive some simplified model for the forward problem. To this end, one can study the accuracy of using a small number of modes for the modal expansion, or one can look for efficient method for estimation of the infinite integral. With some modification, it is also possible to use the moment method formulation to study the polarization characteristic of aperture antennas covered with chiral material. Chapter 5

Summary

In this thesis, we have considered a few modeling problems related to the charac-

terization of electromagnetic properties of materials. Research of this kind has been

driven by a desire for better accuracy and ease of implementation. The latter, eas

of implementation, is often an important consideration in many engineering appli-

cations. In the respect, the first setup considered in this thesis, the spot-focused

free-space measurement system, is superiorovermanyothermethods.Thismethod also provides very good accuracy if implemented with proper calibration and time-

domain gating. The second method considered in this thesis, the open-ended coaxial

line method, has been extensively applied in biological applications. It is arguable

that chirality is inherent in some biological tissues. Thus the model report in chapter

4 has a potential application in measurement of chiral parameter, although this has

notbeenconfirmedexperimentally.

While there is a tendency to categorize a research work as either theoretical or

experimental, material characterization certainly requires efforts in both aspects. Pre-

cision of any measurement is dependent on many factors, some of which might be im-

possible to capture in a model of any imaginable complexity. Likewise, a theoretical

model is often based upon numerous assumptions, some of which might not reflect the 131 §5. Summary 132 physical reality. In fact, the instrument itself is built on theories of energy conversion, signal transformation and propagation, etc. Therefore, we should restrain ourselves from judging the correctness of a model or accuracy of a measurement method by simply comparing a few outputs. Simulation is better used as a tool in understanding the behavior of a measurement system and to identify the source of error. From this point of view, the essential message in chapter 2 is that a Guassian beam (of approx- imately 1.5λ01/e beamwidth) is not much different from a uniform plane wave when used to interrogate the permittivity of planar samples. If the difference is of concern, it can be predicted using the calculation procedure provided. For reflection from curved samples, we do see in chapter 3 a difference between the Gaussian beam and uniform plane wave. The intuitive explanation is the divergence of the reflected field

(an amplitude effect), while calculation shows that the phase distribution has a higher order impact. The horn-lens antenna in the spot-focused free-space system plays the role of averaging the fields. The reflection coefficient measured by the system is the weight and averaged effect of the reflected fields distributed over the focal spot area.

Experimental results in chapter 3 indicates that the geometrical parameters (radii of curvature) might be the main cause of the difference, while the material parameters

(permittivity) might have a lower effect. Thus the idea of doing a curvature correction using the reflection coefficient of conductor sample is proposed. After multiplying the curvature correction factor with the measured reflection coefficient for a penetrable sample under test, inversion algorithm for planar samples can be used to obtain the material parameters. This method is easy to implement and therefore might be a practical way for characterizing a curved sample.

As discussed in previous chapters, there are several interesting issues that deserve §5. Summary 133

further study. Some of them are repeated here. First of all, although a comparison

with complex ray tracing is a positive indication, the correctness of the modified field

expansion used in chapter 3 requires a detailed investigation. Secondly, to obtain

a better confidence in the measured data for curved samples, new fixture should be

designed so that the location and orientation of the sample can be precisely measured

and controlled. Three dimensional beam or an accurate measurement of the antenna

near-field is preferable in predicting the reflection coefficient. Lastly, it is interesting

to study the result of changing the sequence of TRL calibration and time-domain

gating in the free-space measurement procedure. Presently, the effect of multiple reflections are not accounted for (or removed) in the calibration process. Appendix A: Gaussian Beam

The concept of a Gaussian beam is frequently used in the analysis of optical (laser)

systems and so called “quasi-optics” (millimeter wave) systems [84]. The focal re-

gion fields in front of parabolic reflector antennas and dielectric lens antennas are

often assumed to be Gaussian. Obviously, the word “Gaussian” here means the field

amplitude or power density follows a Gaussian distribution as a function of spatial coordinates. It should be immediately noted that there has not been an agreement on the phase distribution. So the term “Gaussian beam” actually refers to a num- ber of different objects in the literature. For example, some authors assume that the amplitude of the total (electric) field is Gaussian, while others define that only a tangential component of the electric field is Gaussian. Some authors specify that the phase is uniform on the beam waist–so called planar phase front1, while others

allow for some sort of variation in the phase on the focal plane. As always, better

understanding can be obtained by studying the various procedures from which the

concept was derived mathematically. This appendix summarizes these developments,

with an emphasis on the underlying physical ideas.

As the first viewpoint, a Gaussian beam is a solution of the parabolic approx-

imation to the wave equation [84]. In accordance with an idea in asymptotic field

analysis, one can write a formal solution of the scalar wave equation (∇2 + k2)u =0

1“Beam waist” refers to the focal plane where the spot-size is minimum.

134 Appendix A: Gaussian Beam 135

as:

u =exp(−jkz) · Ψ(x, y, z). (A.1)

With a time dependence exp(jωt), the exponential term in the above equation indi-

cates the wave is propagating in +z direction. An expression for Ψ is sought under

the assumption that the beam is well collimated, which is essentially saying that Ψ

changes slowly with z. It is readily verified that Ψ(x, y, z) should exactly satisfy

∂ ∇2 − jk . ( 2 ∂z)Ψ=0 (A.2)

Now if the variation of Ψ over a wavelength is negligibly small, i.e.

                ∂Ψ ∆Ψ ∆Ψ Ψ   ∼   <      ∼|kΨ| . (A.3) ∂z ∆z λ λ

so that |∂2Ψ/∂z2|2k|∂Ψ/∂z|, and the following approximate equation is obtained[85]:

∂2Ψ ∂2Ψ ∂Ψ + − 2jk =0 (A.4) ∂x2 ∂y2 ∂z

Its solutions are called Hermite-Gaussian beam modes in rectangular coordinates and Laguerre-Gaussian beam modes in cylindrical coordinates (p.170 [85]). Since the

Hermite polynomials and Laguerre polynomials involved in the expressions of these beam modes satisfy some orthogonal relation, it is said that a general beam field can be expanded into an infinite number of beam-wave modes. Only the fundamental

Gaussian mode, which is axis-symmetric, is commonly used in analyzing focused beam systems. Its expression can be readily derived after some purely mathematical Appendix A: Gaussian Beam 136 manipulation without appealing to other physical reasoning:

   A x2 y2 u x, y, z 2 0 1 − + ( )= 2 exp 2 π w0 1 − jλz/(πw ) w − jλz/π   0 0  2 2 2 2 (A.5) 2 A0 x + y π(x + y ) = exp − − jkz − j + jφ π w w2 λR

where A0 is a constant(the amplitude) and:

& λz 2 w w = 0 1+ 2 πw0 πw2 2 R z 1 0 (A.6) = + z λ λz φ = arctan 2 πw0

A graphical comparison between plane wave and the fundamental Gaussian beam in presented in Fig. A.1, The term “phase front” in this figure refers to equal phase contour(i.e. the electric fields have the same phase along the points on this contour).

The “phase paths” are a set of curves orthogonal to the phase front. They give

(approximately) the energy propagation directions [39]. It should be noted that the

Gaussian beam depicted in Fig. A.1 has a (uniform) planar phase front on the beam waist (plane z=0),butthisisnotastrictrulethathasbeenfollowedbyallauthors.

Equation (A.4) is also called the parabolicwaveequation.Therelevantassump- tionisbaseduponaspatialdomainconsideration.Inthespectraldomain,since a focused beam can be considered as a superposition of plane waves, an equivalent statement is the para-axial approximation:

k2 + k2 k =(k2 − k2 − k2)2 ≈ k − x y (A.7) z x y 2k Appendix A: Gaussian Beam 137

y 2 2 Ekyw|z=0 =− exp[ /0 ] Plane Wave: Intensity Distribution

z

Phase Paths

Phase Fronts Phase Paths Phase Fronts

Figure A.1: Comparison of uniform plane wave and Gaussian beam

which is essentially saying that the wavenumbers kx and ky along the transverse directionsareverysmallforallofthemajorplanewavecomponents.Toderivethe

Gaussian beam using the above assumption, a procedure similar to antenna near field

measurement is used. The starting point is the field distribution measured on a plane

(say, z = 0) in front of the antenna. In this case the antenna behaves so perfectly

that the field is found to be Gaussian, and the spectrum is concentrated on a small

region around kx =0andky = 0. Thus the plane wave components of the field are mainly propagating along the z direction and eq. (A.7) can be invoked. Since this

idea has been used extensively in the main text of this thesis, mathematical details

(p. 162 [85]) are not repeated here.

YetanotherviewpointisthattheGaussian beam can be thought of as the ra-

diation of so called “complex sources” –a point source (3-D) or line source (2-D) Appendix A: Gaussian Beam 138

located at the complex space [86]. Complex space is an extension of the real space by

assigning complex numbers for the coordinates. Complex sources and complex rays

are powerful concepts in both asymptotic analysis [25] and numerical computation

[87] of electromagnetic scattering.

The derivation of a 3-D Gaussian beam starts with the potential function exp(−ikR)/R

for a point source located at (x0,y0,z0). R is the distance to the observation point

(x, y, z):

2 2 2 2 R =(x − x0) +(y − y0) +(z − z0) . (A.8)

Even if x0,y0 and z0 are complex, the potential function is still an exact solution of

the scalar wave equation. Suppose that x0 = y0 =0andz0 = −ib with b positive

real, then

R = {ρ2 +(z + ib)2}(1/2), (A.9)

where ρ =(x2 + y2)(1/2). A branch cut is introduced in z =0,ρ ≤ b to render R

single valued, and the branch is chosen such that R = z + ib on ρ =0whenz>0

and R = −(z + ib)whenz<0. Within the paraxial region where ρ2  z2 + b2 and

z>0,   −ikR ρ2 exp( ) ≈ 1 −ik z ib 1 R z ib exp ( + + z ib +  2 + 1 1 ρ2 ≈ exp −ikz 1+ (A.10) z2 b2 1/2 z2 b2 ( + ) 2 +  1 kbρ2 b − + kb − i arctan , 2 z2 + b2 z Appendix A: Gaussian Beam 139

This represents a 3-D axis-symmetric Gaussian beam. The branch cut is illustrated in

Fig. A.2. This figure also emphasizes the fact that the Gaussian beam approximation is valid only inside the paraxial region.

y

intensity phase path b distribution

z Paraxial Region

Branch Cut -b (equivalent source)

Figure A.2: Gaussian beam can be thought of as a bundle of complex rays. Appendix B:

Near Field and Far Field

In deciding what methodological simplification to be used in modeling the free-space

measurement system, a critical question is whether the antennas and sample should

be considered together as in-separable parts of a system subject to a full-wave simu-

lation. If this is the case, the incident fields generated by the lens antennas are much different with and without the presence of the material sample. modeling the wave scattering in such a system, even not taking into account the purpose of material

measurement, would present insurmountable difficulties. Questions of similar nature

arises for aperture antennas. For example, in the open-ended coaxial probe problem,

it is quite common to assume that the fields at the open-end interface is given by

the TEM mode, while a full wave model would take into account the existence of

higher order modes. For the free-space system considered, a full-wave model seems

out of the question. Nevertheless, the validity of assuming a field distribution for the

interrogating beam needs to be justified to some extent. An obvious consideration is

the distance between sample and the antenna. If the sample is located far away, then

it is reasonable to assign the incident field. But how far is far enough? This appendix

summarizes the concepts of far field and near field of antenna radiation. The essen-

tial purpose is to present the argument that, since the sample is located outside the

reactive near-field region of the antenna radiation, the incident fields should remain

almost the same regardless of the sample properties.

Exterior fields of a linear radiating antenna are commonly divided into near-field

andfar-fieldregions[89].Thefar-fieldregion is the region of space where radial 140 Appendix B: Near Field and Far Field 141

dependence of electric field and magnetic fields varies approximately as inverse pro-

portional to the distance (1/r). Under the assumption that the antenna is not an

extraordinarily highly reactive radiator 1, a rough criteria for the far field region can

be estimated from the general integral for the vector potential and is given by [90]:

2 r>2D λ0,r>5D, and r>1.6λ0 (B.1)

where r is the distance of the observation point from the antenna, D is the maximum linear dimension of the antenna and λ0 is the wavelength in vacuum. For the free-

space measurement system discussed in this thesis D ≈ 30cm, which corresponds to a

far boundary at r =600cmforλ0 =3cm(10GHz).Thedistancefromsamplesurface

to the nearest point on the antenna lens is about 30.48cm, much smaller than the

radius of far field boundary.

The near-field region extends from the surface of the antenna to the inner bound-

ary of far field. It is further divided into two subregions–the reactive and radiating

near field [89]. The reactive near-field region is characterized by a substantial amount

of evanescent component of the electromagnetic energy. Evanescent energy density

decays rapidly with distance, so is the fieldamplitude.Intheradiatingnearfield

region, the energy is mainly propagating, energy density and field amplitude change

slowly with distance. Experience with near-field measurements indicates that a dis-

tance of one wavelength (λ0) or so would form a reasonable outer boundary for the

reactive near field. For the free space measurement system, the material sample is

definitely located outside the reactive near field region.

1see, e.g. Skigin, Veremey, and Mittra[93] for an example of super directivity. Appendix B: Near Field and Far Field 142

Reference [89] also discusses the optical terms “Fresnel region” and “Fraunhofer region”. The term Fraunhofer region can be used synonymously with the far-field region, or to refer to the focal region of an antenna focused at a finite distance. The

Fresnel region is a region inside the radiating near-field region where a quadratic phase approximation can be used in the vector potential integral. One rule of thumb

2 (1/3) formula for the inner boundary of Fresnel region is (D/2λ0) D/2+λ0.

1/3 2 r D D D The following formula is also seen in the literature: = 4 + 2 λ0 Appendix C:

Time Gating in 8510C

The purpose of this appendix is to discuss the choice of window type and gate shape

for the free space measurement reported in this thesis. A technique is described for

estimation of the time gating error corresponding to a chosen value of gate span.

The HP8510C network analyzer is a frequency domain instrument. Time domain response displayed by the analyzer is mathematically calculated using the inverse

Fourier transform (IFT). According to the HP8510C user manual, the calculation

is done by the Cirp-Z transform (CZT) computation technique. Using the CZT

algorithm one can efficiently evaluate the z-transform of a sequence of N samples

at M points in the z-plane which lie on circular or spiral contours beginning at any

arbitrary point in the z-plane [97]. The angular spacing of the points is also arbitrary.

For the usual fast Fourier transform (FFT), those M points lie on the unit circle with

uniform spacing and fixed starting point. Thus the essential purpose of using CZT in

HP8510C might be to accommodate arbitrary START and STOP time as specified

by the user. The CZT reduces to FFT if calculated at the same N time sample points,

without enhancement in accuracy.

HP8510C also provides several windowing options and gate shape options. The

windowing options are used in the process of calculating the IFT to control spectral

leakage [98]. The gate shape options provides choice among several time filters in

the gating process. The difference in gate shape is essentially a result of different

time-domain window being applied for the purpose of reducing “cepstral” leakage:

143 Appendix C: Time Gating in 8510C 144

component of the “impulse” 1 at one time leaks into the vicinity of another time.

Since the FFT result is an approximation of the continuous Fourier transform and

any one of the window types represents a compromise between impulse width and

side lobe level, it might be necessary to study their effect on the frequency domain

data and determine which option will give the best result. Unfortunately no technical

data is available on how the different type of gate shapes in HP8510C are generated.

This makes it very difficult to accurately simulate the time gating process as applied

inside the analyzer. In the following, settings used in the experiments reported in

this thesis are explained.

Table C.1 lists the technical parameters for the three window type copied from the

HP8510C network analyzer data sheet. Notice that the impulse width depends on the

frequency span of a particular measurement and ambiguously listed as “minimum”.

Approximate formula is given in the user manual as:

1.20 Impulse Width(50%) = (for minimum window) (C.1) Frequency Span

where the number 1.20 seems to be coming from the solution fspant =0.603 of the following equation:

sin(πf t) sin(πf t) span ≈ span =0.5(C.2) npo int s sin(πfspant/npoints) πfspant

where fspan is the bandwidth of the frequency domain data, npoints is the Number of Points in the measurement. Left hand side of the above equation is exactly the

1Since the frequency domain data is band-limited, the impulse is not a δ function but has some width. Appendix C: Time Gating in 8510C 145

impulse shape for rectangular window type.

Window Type Kaiser Bessel Impulse Width Sidelobes Parameter (relative to peak) Minimum 0 Minimum -15dB Normal 6 1.5 × Minimum -50dB Maximum 13 2.5 × Minimum -90 dB

Table C.1: Technical parameters for the window type in HP8510C network analyzer specified in its data sheet.

For the other two window types, the impulse shape can be approximated by the following function [98]:  1 sinh β2 − (πf t)2  span (C.3) I β 2 2 0( ) β − (πfspant)

where β is the Kaiser-Bessel parameter, and I0 is the modified Bessel function of the

first kind. Solving for the -6dB point gives an impulse width of 1.954/fspan for β =6,

2.7754/fspan for β = 13, or 1.63 and 2.31 times the minimum, which are somewhat different from the specification in table C.1. Nevertheless, the above exercise shows

that the first zero-crossing impulse widths are about 2.0/fspan,4.0/fspan and 8.0/fspan respectively for the three window types (see Fig. C.1). These parameters are useful

in deciding the gate span for the free space measurement. The -6dB impulse widths

are more useful for qualitatively resolving two close abrupt discontinuities.

Table C.2 lists the gate shape properties as provided in the HP8510C user manual.

Meanings of related parameter are illustrated in Fig. C.2. Of interest here are the

passbandrippleandtheminimumgatespan.Inthefreespacesystem,thesample

is at a distance of 30.5cm from the antenna. Ideally this corresponds to a 2ps flight Appendix C: Time Gating in 8510C 146

1

0.8

0.6 (-6dB point minimum window) 0.6 0.98(-6dB point normal window)

1.39(-6dB point maximum window) 0.4

0.2 zero crossing points

t (unit:1/f ) -4 -2 -1 1 2 4 span

Figure C.1: Bandpass impulse shapes for the three window types time between successive impulses caused by multiple reflection between the sample and the antenna. The purpose of time gating is to remove these multiple reflections.

The passband ripple (±0.4dB) for the minimum gate shape is obviously too high for material characterization. To find out which one of the rest gate shapes is appro- priate, some calculation is needed. The experimental results reported in this thesis is obtained with the frequency range of 8GHz-13.7GHz(fspan =5.7GHz). The max- imum gate shape is seen to require a too big minimum gate span. With reference

to Fig. C.3, the desired reflection should be covered by the passband of the time

filter and the unwanted reflection should be inside the stopband of the time filter,

leaving transition region between them. Taking into account the expansion caused by

interaction with the sample, the desired impulse width should be 3/fspan to 4/fspan. Under these consideration, the cutoff time for wide shape is too high. Therefore in

the experimental results reported in this thesis, the normal gate shape and minimum Appendix C: Time Gating in 8510C 147 window type have been used. The gate center is always set to a local maximum of the linear amplitude near 0ps. The gate span is dependent on the sample under test.

A rough criteria is that with gate on the trace in frequency domain should be smooth and fit into the noisy trace without time gating. In general, the gate span is smaller when measuring reflection from perfect conduction and bigger when measuring ma- terial samples. Furthermore, different settings are required for processing the four

S-parameters of the same sample.

Gate Shape Passband Ripple Sidelobes Level Cutoff Time Minimum T2=T3 Gate Span T1 Minimum ±0.40dB -24dB 0.6/fspan 1.2/fspan Normal ±0.04dB -45dB 1.4/fspan 2.8/fspan Wide ±0.02dB -52dB 4.0/fspan 8.0/fspan Maximum ±0.01dB -80dB 11.2/fspan 22.4/fspan

Table C.2: Properties of the gate shapes (time filters) available in HP8510C network analyzer. Passband ripple for the MINIMUM gate shape is too big for material characterization. For fspan =5.7GHz, and an estimated delay of 2ns for the first multiply reflected impulse, the gate span for MAXIMUM and WIDE gate shapes are judged to be too high. Thus measurement results reported in this thesis are obtained with NORMAL gate shape.

To get an idea on the potential errors caused by the time gating process, a com- puter program is written to simulate an ideal frequency domain measurement. Calcu- lated reflection coefficients are transferred into the raw data memory of the network analyzer. With correction turned off, window types and gate spans are set to different values. After turn on the gating, the resulting frequency domain data are compared with the ideal calculated values. The program calculates the reflection coefficients for a plane wave at normal incidence onto a dielectric slab of thickness 0.5348cm and permittivity ε =2.59. Thus the following discussion applies for such an ideal sample Appendix C: Time Gating in 8510C 148

Passband ripple

-6dB point -6dB point

Cutoff time T2 T3

Gate Span T1 Gate Start Gate Stop

Figure C.2: Meaning of the gate shape parameters in table C.2. Note that none of the gate shapes available in HP8510C has an ideal rec- tangular shape.

Time gate to be applied Desired reflection

Leakage Unwanted reflection

This is is about 2ns Transition region(2T2)

Figure C.3: Selection of gate shape and gate span for isolating the desired response from the unwanted responses. Desirable gate shape should have a smooth passband and large stopband loss. Abrupt transition might not be desirable. Appendix C: Time Gating in 8510C 149 only.

Figure C.4 shows the relative errors in magnitudes of reflection coefficients for the

MINIMUM window type and NORMAL gate shape with different gate spans. The gate center is fixed at 27.5ps which is the maximum point for a time range setting

2 of -1.0ns to 1.0ns . The relative error is defined as (|S11gate|−|S11ideal|)/|S11ideal| and the values shown are in percentage. It is seen that a gate span of 350ps would result in an error as much as 20%. However, as the gate span is increased to 1400ps, the relative error is less than 1.5%. Examination of the numbers further shows that the relative errors is less than 0.2% over the frequency range of 8.5GHz-13GHz. The values of gate spans are chosen from the first several local minimums of the impulse shape (shown in Fig. C.8).

Figure C.5 illustrates the potential time gating errors in magnitude of reflection coefficients with the NORMAL window type and NORMAL gate shape. The gate center is now fixed at 25ps. The values of gate span are again chosen from the first several local minimums of the reflected impulse (see Fig. C.9). It is seen that the relative errors are less than 3% for all of the gate spans.

Figure C.6 shows the changes in phase of reflection coefficients after time gating for the MINIMUM window type. The errors are less than 0.7o forallofthegatespans used. Fig. C.7 is the corresponding result for the NORMAL window type. It can be concluded that the phase errors caused by time gating are relatively smaller than the magnitude errors.

It should be noted that the above error analysis does not consider multiple in- teractions. If multiply reflected impulses are taken into account, the error should

2it is observed that the maximum point changes slightly with different time range settings. Appendix C: Time Gating in 8510C 150

20 f =350ps span f =700ps span f =1050ps span 15 f =1400ps span

10

5 Magnitude Error(percent) 0

-5 8 9 10 11 12 13 Frequency(GHz)

Figure C.4: Relative error ((|S11gate|−|S11ideal|)/|S11ideal| in percentage) in the magnitude of reflection coefficients caused by time gating with the window type set to MINIMUM and gate shape set to NORMAL. Different curve corresponds to different value of gate span.

3.0 f =789ps span f =991.5ps span 2.5 f =1258.5ps span f =1565ps span 2.0

1.5

1.0

0.5 Magnitude Error (percent)

0.0

-0.5 8 9 10 11 12 13 Frequency(GHz)

Figure C.5: Relative error ((|S11gate|−|S11ideal|)/|S11ideal| in percentage) in the magnitude of reflection coefficients caused by time gating under NORMAL window type and NORMAL gate shape. Appendix C: Time Gating in 8510C 151

1.0 f =350ps span f =700ps 0.8 span f =1050ps span f =1400ps 0.6 span

0.4

0.2

0.0 Phase Error (degrees)

-0.2

-0.4

8 9 10 11 12 13 Frequency(GHz)

Figure C.6: Change in the phase of reflection coefficients caused by time gating with the window type set to MINIMUM and gate shape set to NORMAL. Curves correspond to different values of gate span.

0.2 f =789ps span f =991.5ps span f =1258.5ps span f =1565ps span

0.0 Phase Error(degrees)

8 9 10 11 12 13 Frequency(GHz)

Figure C.7: Change in the phase of reflection coefficients caused by time gating with the window type set to NORMAL and gate shape set to NORMAL. Curves correspond to different values of gate span. Appendix C: Time Gating in 8510C 152

Minimum Window 0.4 Normal Window Minimum Window with Gate(f =350ps) span 0.3

0.2 Linear Magnitude

0.1

0.0

-1.0 -0.5 0.0 0.5 1.0 Time (ns)

Figure C.8: Linear magnitude of the reflected impulse calculated by the HP8510C analyzer for the ideal frequency domain measurement of a slab sample (ε =2.59, thickness=0.5348cm). Curves cor- respond to MINIMUM window type, NORMAL window type, and MINIMUM window type with TIME GATING(NORMAL gateshape with 350ps gate span centered at 27.5ps). Appendix C: Time Gating in 8510C 153

3.5E-3

3.0E-3

2.5E-3

2.0E-3

1.5E-3

1.0E-3 Linear Magnitude

0.5E-3

0.0E-3

0.4 0.5 0.6 0.7 0.8 0.9 1.0 Time (ns)

Figure C.9: Zoom-in view (details of figure C.8 for the time interval 0.4ns- 1ns) of the linear magnitude of the reflected impulse calculated by the HP8510C analyzer. The curve corresponds to NORMAL window type without time gating. be bigger because their sidelobes overlap with (leak into) the main reflected impulse

(Fig. C.3). In practice, it has been observed that the unwanted responses supposedly caused by multiple reflection do not have a clean impulse shape as shown in Fig. C.1.

This is as expected because after reflection from the lens, the field is no longer fo- cused. Nevertheless, the above analysis procedure should provide a certain level of confidence for the use of time gating technique in free space measurement.

A final comment is that multiple reflections also exist during the calibration process. Understanding their effect on calibration is an interesting subject for further study. Appendix D: − + Derivation of the reaction Mml and Mml

By mode orthogonality: ∗ × · ds δ δ ∗ × · ds, (eml ht,ni) aˆz = mn li (eml ht,ml) aˆz A A

− it is seen that the reaction M0 on z =0− is nonzero only when mode (ml) and mode (ni)isthesame.Whenm = n and l = i, it is found that: − −j ∗ M0 = (1 + R0) [(∇tψ0) × (aˆz ×∇tψ0)] · aˆz ds η0 A (D.1) −j a = (1 + R0)2π ln η0 b

−γ M − m1 −R ×∇ψ ∗ ×∇ψ · ds m1 = ( m1) [(aˆz t m1) t m1] aˆz k0η0 A   (D.2) γ − J k /c 2 m1 R 4 0( m1 ) − = m1 1 k0η π j (k 1) 0 0 m −j M − R ∇ ψ ∗ × ×∇ψ · ds m2 = m2 [( t m2) (aˆz t m2)] aˆz η0 A   2 (D.3) −j 4 J0(km2/c) = Rm2 − 1 η0 π J0(km2) where use has been made of the Green’s first identity in two dimensional form: ' ∂ψ ∇ ψ ·∇ψds ψ d − ψ∇2ψds t t = ∂n l t (D.4) A ∂A A

ψ ∇2ψ k /a 2ψ . and that ml satisfies: t ml +( ml ) ml =0

154 M − M + Appendix D: Derivation of the reaction ml and ml 155  ψ2 ds ψ The integral A ml can be obtained by first noting that ml is a solution of: d dψ r ml k /a 2rψ . dr dr +( ml ) ml =0 (D.5)

− rψ k /a 2rψ . Therefore: ( ml) =( ml ) ml

1 1 − rψ rψ dr k /a 2 rψ rψ dr ( ml)( ml) =( ml ) ( ml)( ml) (D.6) b/a b/a

1   k /a 2 ψ2 rdr k /a 2 1r2ψ2 1 1 rψ 2 1 ( ml ) ml =( ml ) ml r=b/a + ( ml) r=b/a (D.7) b/a 2 2

So that: 1 2π k2 ψ2 rdr m1 m1 = b/a 0  πk2 N k /c J k − j k /c N k 2 = m1 [ 0( m1 ) 0( m1) 0( m1 ) 0( m1)] 2 2 −(1/c) [N0(km1/c)J0(km1/c) − j0(km1/c)N0(km1/c)]   N k J k /c 2 (D.8) (4.14) πk2 0( m1) 0( m1 )J k − J k /c N k = m1 0( m1) 0( m1 ) 0( m1) J0(km1) 2 2 −(1/c) [N0(km1/c)J0(km1/c) − J0(km1/c)N0(km1/c)]    J k /c 2 4 0( m1 ) − = 1 π J0(km1) where the following Wronskian relation has been used:

W J αz ,N αz J αz N αz − J αz N αz 2 [ p( ) p( )] = p( ) p( ) p( ) p( )=παz (D.9)

+ ∗ ∗ ∗ M ≡ × | + · ds e H − e H ds ml (eml Ht z=0 ) aˆz = ( x,ml y y,ml x) (D.10) A A M − M + Appendix D: Derivation of the reaction ml and ml 156   ∞ 1 = e∗ H% (k ,k , 0+) e−jkxx−jkyydk dk  x,ml 4π2 y x y x y A −∞  ∞  1 −e∗ H% (k ,k , 0+) e−jkxx−jkyydk dk ds y,ml 4π2 x x y x y −∞ +∞ ∗ 1 = H% (k ,k , 0+) e (x, y) ejkxx+jkyydx dy dk dk 4π2 y x y x,ml x y −∞ A +∞ ∗ 1 − H% (k ,k , 0+) e (x, y) ejkxx+jkyydx dy dk dk 4π2 x x y y,ml x y −∞ A +∞ 1 ∗ = e (x, y) ejkxx+jkyydx dy dk dk 4π2 x,ml x y −∞ A  ∞ 2  a − ι − i−1R Y e x, y ejkxx+jkyydx dy  ni(2 n +( 1) ni) yx x,ni( ) =0 = n i ιn A  Y e x, y ejkxx+jkyydx dy dk dk + yy y,ni( )  x y A +∞ 1 ∗ − e (x, y) ejkxx+jkyydx dy dk dk 4π2 y,ml x y −∞ A  ∞ 2  a − ι − i−1R Y e x, y ejkxx+jkyydx dy  ni(2 n +( 1) ni) xx x,ni( ) n=0 = n i ι A  Y e x, y ejkxx+jkyydx dy dk dk + xy y,ni( )  x y A ∞ 2 (ka)2 = a (2 − ι +(−1)i−1R ) 4π2 ni n ni n=0 i=ιn 2 ∞ 2 1  π  π ∗ ududφ e (ρ, ϕ) ejk0auρ cos(φ−ϕ)ρdρdϕ a2  x,ml 0 0 0 b/a M − M + Appendix D: Derivation of the reaction ml and ml 157

 2π 1 jk0auρ cos(φ−ϕ) Yyx ex,ni(ρ, ϕ) e ρdρdϕ 0 b/a  2π 1  +Y e (ρ, ϕ) ejk0auρ cos(φ−ϕ)ρdρdϕ yy y,ni  0 b/a ∞ 2 (ka)2 − a (2 − ι +(−1)i−1R ) 4π2 ni n ni n=0 i=ιn 2π∞ 2π 1 ∗ ududφ e (ρ, ϕ) ejk0auρ cos(φ−ϕ)ρdρdϕ a2  y,ml 0 0 0 b/a  2π 1 jk0auρ cos(φ−ϕ) Yxx ex,ni(ρ, ϕ) e ρdρdϕ 0 b/a  2π 1  +Y e (ρ, ϕ) ejk0auρ cos(φ−ϕ)ρdρdϕ xy y,ni  0 b/a ∞ 2 (ka)2 = a (2 − ι +(−1)i−1R ) 4π2 ni n ni n=0 i=ιn 2π∞ ∗ ∗ a2 F% Y F% F% Y F% x,ml yx x,ni + x,ml yy y,ni 0 0 ∗ ∗ −F% Y F% − F% Y F% ududφ y,ml xx x,ni y,ml xy y,ni (D.10 ) where the following new variables are used to normalize the integration range:

kx = k0u cos φ, ky = k0u sin φ, x = aρ cos ϕ, y = aρ sin ϕ. (D.11) jkxx + jkyy = jk0auρ(cos φ cos ϕ +sinφ sin ϕ)=jk0auρ cos(φ − ϕ) M − M + Appendix D: Derivation of the reaction ml and ml 158 also:

2π 1 % jk0auρ cos(φ−ϕ) F x,ml = ex,ml(ρ, ϕ) e ρdρdϕ

0 b/a (D.12) 2π 1 % jk0auρ cos(φ−ϕ) F y,ml = ey,ml(ρ, ϕ) e ρdρdϕ

0 b/a

Expressions of F% for TEM, TE and TM modes are derived separately as follows, using the following important relation:   k auρ +∞ jk0auρ cos(φ−ϕ) 0 j(φ−ϕ) 1 τ jτ(φ−ϕ) e =exp je − = J (k0auρ)j e 2 jej(φ−ϕ) τ τ=−∞ (D.13)

For TEM mode:

∇ ψ 1 1 ϕ ϕ t 0 = r aˆr = ρa(aˆx cos + aˆy sin )(D.14)

2π 1 cos ϕ F% ejkauρ0 cos(φ−ϕ)ρdρdϕ x,02 = ρa 0 b/a (D.15) 1 je−jφ e2jφ F u = a (1 + ) 02( )

2π 1 sin ϕ F% ejkauρ0 cos(φ−ϕ)ρdρdϕ y,02 = ρa 0 b/a (D.16) 1je−jφ j − je−jφ F u = a ( ) 02( ) M − M + Appendix D: Derivation of the reaction ml and ml 159

For TE modes:

kn1 ∇ ψ ˆ ϕ ˆ ϕ N1 k 1/c J1 k 1ρ − J1 k 1/c N1 k 1ρ t n1 = a (ax cos + ay sin )[ ( n ) ( n ) ( n ) ( n )] (D.17)

2π 1 F% ×∇ψ  ejk0auρ cos(φ−ϕ)ρdρdϕ x,n1 = aˆz t n1 x (D.18) 0 b/a 2π 1 kn1 kn1 kn1 = (− sin ϕ)[N1( )J1(k 1ρ) − J1( )N1(k 1ρ)] a c n c n 0 b/a +∞ τ jτ(φ−ϕ) Jτ (k0auρ)j e ρdρdϕ τ=−∞ 1 k π n1 j ejφjJ k auρ e−jφjJ k auρ = a [ 1( 0 )+ −1( 0 )] b/a k k N n1 J k ρ − J n1 N k ρ ρdρdϕ [ 1( c ) 1( n1 ) 1( c ) 1( n1 )] 1 k π k k n1 jφ −jφ n1 n1 = (−e + e ) J1(k0auρ)[N1( )J1(k 1ρ) − J1( )N1(k 1ρ)]ρdρ a c n c n b/a

   jφ −jφ kn1π (−e + e ) kn1 kn1 = k 1J1(k0au) J1(k 1)N1( ) − J1( )N0(k 1) a k au 2 − k2 n n c c n ( 0 ) n1   1 kn1 kn1 kn1 kn1 − k 1 J1(k0au/c) J0( )N1( ) − J1( )N0( ) n c c c c c   k k k auJ k au J n1 N k − J k N n1 + 0 0( 0 ) 1( c ) 1( n1) 1( n1) 1( c )   1 kn1 kn1 kn1 kn1 −k0au J0(k0au/c) J1( )N1( ) − J1( )N1( ) c c c c c  k k π −ejφ e−jφ J n1 n1 ( + ) 1( c ) = k 1J1(k0au) [J0(k 1)N1(k 1) − N0(k 1)J1(k 1)] a k au 2 − k2 n J k n n n n ( 0 ) n1 1( n1)  k k k k − k J k au/c J n1 N n1 − N n1 J n1 n1 1( 0 ) 0( c ) 1( c ) 0( c ) 1( c ) M − M + Appendix D: Derivation of the reaction ml and ml 160   k jφ −jφ  n1  k 1π (−e + e ) −2 k 1J1(k0au)J1( ) 1 2 n n c k J k au/c = 2 2 + n1 1( 0 ) a k au − k πk J k c k 1  ( 0 ) n1 n1 1( n1) π n   c jφ −jφ kn1 (−e + e ) 2 kn1 = J1(k0au/c)J1(k 1) − J1(k0au)J1( ) a k au 2 − k2 J k n c ( 0 ) n1 1( n1) 1 je−jφ −j je2jφ F u = a ( + ) n1( )(D.18)

2π 1 F% ×∇ψ  ejk0auρ cos(φ−ϕ)ρdρdϕ y,n1 = aˆz t n1 y 0 b/a 2π 1 k n1 ϕ N k /c J k ρ − J k /c N k ρ = a cos [ 1( n1 ) 1( n1 ) 1( n1 ) 1( n1 )] 0 b/a +∞ τ jτ(φ−ϕ) Jτ (k0auρ)j e ρdρdϕ τ=−∞ 1 je−jφ e2jφ F u = a (1 + ) n1( )(D.19)

For TM modes:

kn2 ∇ ψ ˆ ϕ ˆ ϕ J0 k 2/c N1 k 2ρ − N0 k 2/c J0 k 2ρ t n2 = a (ax cos + ay sin )[ ( n ) ( n ) ( n ) ( n )] (D.20)

2π 1 F% ∇ ψ  ejk0auρ cos(φ−ϕ)ρdρdϕ x,n2 = t n2 x (D.21) 0 b/a 2π 1 kn2 kn2 kn2 = cos ϕ[J0( )N1(k 2ρ) − N0( )J1(k 2ρ)] a c n c n 0 b/a +∞ τ jτ(φ−ϕ) Jτ (k0auρ)j e ρdρdϕ τ=−∞ M − M + Appendix D: Derivation of the reaction ml and ml 161

1 k π n2 ejφjJ k auρ − e−jφjJ k auρ = a [ 1( 0 ) −1( 0 )] b/a k k J n2 N k ρ − N n2 J k ρ ρdρdϕ [ 0( c ) 1( n2 ) 0( c ) 1( n2 )] 1 k π k k n2 jφ −jφ n2 n2 = j(e + e ) J1(k0auρ)[J1( )N0(k 2ρ) − N0( )J1(k 2ρ)]ρdρ a c n c n b/a   jφ −jφ kn2π j(e + e ) kn2 kn2 = k 2J1(k0au) J0( )N0(k 2) − N0( )J0(k 2) a k au 2 − k2 n c n c n ( 0 ) n2   kn2 kn2 kn2 kn2 kn2 − J1(k0au/c) J0( )N0( ) − N0( )J0( ) c c c c c   k k k auJ k au J n2 N k − N n2 J k − + 0 0( 0 ) 0( c ) 1( n2) 0( c ) 1( n2)   k au k k k k 0 J k au/c J n2 N n2 − N n2 J n2 c 0( 0 ) 0( c ) 1( c ) 0( c ) 1( c )    k 2  k π j ejφ e−jφ k au J k au J n k au n2 ( + ) 2 0 0( 0 ) 0( c ) 1 2 0 = − J0(k0au/c) a k au 2 − k2  πk J k c k  ( 0 ) n2 n2 0( n2) π n2 c jφ −jφ k0au j(e + e ) 2 = [J0(k0au)J0(k 2/c) − J0(k0au/c)J1(k 2)] a k au 2 − k2 J k n n ( 0 ) n2 0( n2) 1 je−jφ e2jφ F u = a (1 + ) n2( )(D.21)

2π 1 F% ∇ ψ  ejk0auρ cos(φ−ϕ)ρdρdϕ y,n2 = t n2 y 0 b/a 2π 1 kn2 kn2 kn2 = sin ϕ[J0( )N1(k 2ρ) − N0( )J1(k 2ρ)] a c n c n 0 b/a +∞ τ jτ(φ−ϕ) Jτ (k0auρ)j e ρdρdϕ τ=−∞ 1 je−jφ j − je2jφ F u = a ( ) n2( )(D.22) M − M + Appendix D: Derivation of the reaction ml and ml 162

∞ 2 (ka)2 M + = a (2 − ι +(−1)i−1R )(D.23) ml 4π2 ni n ni n=0 i=ιn 2π∞  2jφ −jφ ∗ 2jφ −jφ (cml + dmle je Fml Yyx (cni + dnie )je Fni 0 0 ∗ 2jφ −jφ 2jφ −jφ + (cml + dmle )je Fml Yyy (jcni − jdnie )je Fni ∗ − (jc − jd e2jφ)je−jφF Y (c + d e2jφ)je−jφF − ml ml ml xx ni ni ni ∗ 2jφ −jφ 2jφ −jφ (jcml − jdmle )je Fml Yxy (jcni − jdnie )je Fni ududφ ∞ 2 (ka)2 = a (2 − ι +(−1)i−1R ) 4π2 ni n ni n=0 i=ιn 2π∞ F ∗ F c∗ c Y d∗ d Y c∗ d Y e2jφ d∗ c Y e−2jφ ml ni ml ni yx + ml ni yx + ml ni yx + ml ni yx 0 0 c∗ c jY − d∗ d jY − c∗ d jY e2jφ d∗ c jY e−2jφ + ml ni yy ml ni yy ml ni yy + ml ni yy c∗ c jY − d∗ d jY c∗ d jY e2jφ − d∗ c jY e−2jφ + ml ni xx ml ni xx + ml ni xx ml ni xx −c∗ c Y − d∗ d Y c∗ d Y e2jφ d∗ c Y e−2jφ ududφ ml ni xy ml ni xy + ml ni xy + ml ni xy ∞ 2 (ka)2 = a (2 − ι +(−1)i−1R ) 4π2 ni n ni n=0 i=ιn 2π∞ F ∗ F c∗ c Y − Y c∗ d e2jφ Y Y − jY jY ml ni ml ni( yx xy)+ ml ni ( yx + xy yy + xx) 0 0 d∗ c Y jY − jY Y e−2jφ d∗ d Y − Y ududφ + ml ni( yx + yy xx + xy) + ml ni( yx xy) ∞ 2 (ka)2 = a (2 − ι +(−1)i−1R )2π 4π2η ni n ni n=0 i=ιn ∞  k k k k k k − k k − k k k k F ∗ F c∗ c 2( 1 2 + 1z 2z) c∗ d 2( 1 2 1z 2z 1 2z + 2 1z) ml ni ml ni + ml ni k1k2z + k2k1z k1k2z + k2k1z 0  k k − k k k k − k k k k k k d∗ c 2( 1 2 1z 2z + 1 2z 2 1z) d∗ d 2( 1 2 + 1z 2z) udu + ml ni + ml ni k1k2z + k2k1z k1k2z + k2k1z

(D.23) M − M + Appendix D: Derivation of the reaction ml and ml 163 where:     −j (TE, l=1) +j (TE, l=1) c = d = ml  ml  +1 (TM or TEM, l=2) +1 (TM or TEM , l=2)

equation (4.33) is obtained after some algebraic manipulation. Bibliography

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Kai Du was born in a small town in southern China in 1970. He graduated from

GaoZhou High School in GaoZhou county, GuangDong province, China. He received a B.S. degree in Radio Electronics in 1992 from University of Science and Technology of China (USTC); a M.S. degree in Electrical Engineering in 1995 from USTC; a M.S. in Mathematics in 1997 from Purdue University; and a Ph.D. degree in Engineering

Science from the Pennsylvania State University. His graduate study has been mainly related to electromagnetic fields and microwave technologies.