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Chemical studies of the degree of decomposition and dissolution in microwave digests

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The Pennsylvania State University
The Graduate School
Department of Engineering Science and Mechanics
RADIATIVE TRANSFER APPROACH TO DESIGN
THE ELECTROMAGNETIC RESPONSE
OF
MICROWAVE CHIRAL COMPOSITES
A Thesis in
Engineering Science and Mechanics
by
Neil Rhodes Williams
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
May 1995
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UMI Number: 9532052
UMI Microform 9532052
Copyright 1995, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized
copying under Title 17, United States Code.
UMI
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We approve the thesis of Neil Rhodes Williams.
Date of Signature
Vijdy K. Varadan
Distinguished Alumni Professor of Engineering
Science and Mechanics and Electrical Engineering
Thesis Co-Advisor
Chair of Committee
Vasundara V. Varadan
Distinguished Alumni Professor of Engineering
Science and Mechanics and Electrical Engineering
Thesis Co-Advisor
Bernhard R. Tittmann
Kunkle Professor of Engineering Science and
Mechanics
Lynn A. Carpenter
Associate Professor of Electrical Engineering
Richard F. McNitt
Professor of Engineering Mechanics
Head of the Department of Engineering
Science and Mechanics
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iii
ABSTRACT
There has been much interest in the last decade to use chiral materials for
controlling microwave propagation characteristics. Previous research has found that
materials can be made chiral by embedding chiral microgeometries, such as helices,
into dielectric volumes. Researchers have also shown experimentally that chirality in
a material alters a propagating wave’s polarization, reflection, and transmission
characteristics. Since then, experimental research has been concentrated into two
areas: 1) to understand the role of helical geometries on wave propagation
characteristics and 2) observe and use chiral related attenuation in thin flexible
composites for possible radar absorbing material (RAM) and electromagnetic
interference (EMI) applications.
This thesis presents the first thorough examination of wave attenuation in thin,
flexible chiral composites using extinction and the equations of radiative transfer. The
effective extinction cross section for helices of varying geometries has been measured
and entered into a data base within the context of a scalar transport theory. Using
the equations of radiative transfer and the extinction database, the attenuation
characteristics of chiral composites are predicted.
Chiral samples have been made by embedding helical geometries of varying
geometries into RTV silicone rubber. All the samples are approximately 3 mm thick
and vary by metal volume concentrations and helical size parameters.
A free-space measurement system using specially constructed test fixtures has
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iv
been used to characterize the samples from 8.2 GHz to 40 GHz. The test fixtures are
designed to give the thin, flexible composites mechanical rigidness, to prevent
sagging or bulging. By being more rigid and planar, the sample’s electrical properties
can be obtained accurately.
Using the reflection and transmission coefficients of the samples, effective
extinction cross sections of helices of varying diameters, turns, and pitches have been
obtained, using a single scattering approximation and scalar radiative transfer
equations. The justification for using a simple transport model was validated for the
samples by demonstrating that the electromagnetic rotation of the field is low. In
addition, multiple scattering is negligible by keeping the thickness of the sample to
less than a half a wavelength (at 30 GHz).
It was found by using this approximation, the extinction could be reliably
found for long wavelengths (ka < 1) and low helix concentrations. The diameter of
the helix greatly affected the extinction cross section more than the turn number or
pitch. The diameter of the helix was proportional to the extinction. In addition, the
extinction cross section of a sphere with the same diameter as a measured helix was
calculated. Thorough analysis showed that the extinction of the sphere was similar
to a helix with the same diameter. However, the sphere’s metal volume is several
times of that of the helix. This suggests that the helices can give a cross section and
transmission loss that is equivalent to a sphere’s, but at a fraction of the metal
volume and weight.
By relating the helical geometries to the extinction cross sections, attenuation
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within a chiral sample can be characterized. The equations of radiative transfer along
with measured helix extinctions were used to predict transmission of composites
containing single sized and mixed sized helices. Through the measurement of
extinction of individual helices, attenuation of samples containing two and three
different sized helices were predicted accurately. Attenuation of samples of different
concentrations, but with the same size helix are modeled effectively where ka < 1.
This study has provided a practical and simple scheme for predicting transmission
loss through thin samples of chiral composite structures, which will have useful
applications to EMI and RFI shielding.
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vi
TABLE OF CONTENTS
LIST OF T A B L E S .................................................................................................x
LIST OF F I G U R E S ...........................................................................................xi
ACKNOWLEDGMENTS..................................................................................... xv
CHAPTER 1. IN T R O D U C T IO N ............................................................... 1
1.1 Objective............................................................................................1
1.2 Scattering and Absorption by P a r tic le s .........................................3
1.2.1 Extinction Examples: Spherical
and Nonspherical S c a tte r e r s .........................................6
1.3 C hirality............................................................................................9
1.3.1 Origins of C h ir a lity ............................................................... 10
1.3.2 Twentieth Century R e s e a r c h ..............................................15
1.3.3 Constitutive Relations and
Propagation in a Chiral M e d iu m ........................................ 18
1.4 Thesis O r g a n iz a tio n ..........................................................................23
CHAPTER 2. PROPAGATION CHARACTERISTICS OF
WAVES IN CHIRAL M E D I A ................................................... 26
2.1 Reflection and Transmission of
a Chiral Slab..................................................................................... 26
2.2 Computing Electromagnetic P ro p e rtie s ..............................................36
CHAPTER 3. EFFECTIVE EXTINCTION CROSS SECTIONS
AND EQUATIONS OF RADIATIVE TRANSFER
. . .
40
3.1 Introduction......................................................................................... 40
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vii
3.2 Transport T h e o r y ............................................................................... 41
3.3 Extinction Cross S e c t i o n .................................................................... 42
3.4 Scattering and Extinction in a slab of
Many P a r tic le s ................................................................................47
3.5 Specific I n t e n s i t y ............................................................................... 49
CHAPTER 4. FREE-SPACE MEASUREMENT SYSTEM
AND SAMPLE P R E P A R A T IO N ............................................. 56
4.1 Introduction
..................................................................................... 56
4.2 Description of the Measurement System..............................................57
4.3 Sample Preparation................................................................................60
4.4 Experimental P ro ce d u res.................................................................... 62
CHAPTER 5. SAMPLE THICKNESS EFFECTS ON HELIX
EXTINCTION CROSS SECTION
CALCULATIONS.......................................................................... 70
5.1 Introduction........................................................................................... 70
5.2 Calculation of the Effective Total Extinction
Cross S e c t i o n ............................................................................... 72
5.3 Transmission and Extinction Characteristics
of Thin and Thick S a m p l e s .........................................................77
5.3.1 Thin Sample Results ( 7 4 2 ) ..............................................
78
5.3.2 Thick Sample Results (6 2 5 )..............................................
80
CHAPTER 6. SPRING CONCENTRATION EFFECTS ON
MEASURED EXTINCTION CHARACTERISTICS . . .
88
6.1 Introduction......................................................................................... 88
6.1.1 Coherent I n t e n s i t y .............................................................89
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viii
6.2 Spring 742, 1% and 2% Volume Concentration
R e s u l t s ......................................................................................91
6.2.1 Extinction and Optical Depth ( 7 4 2 ) .............................
93
6.2.2 Extinction of Sphere ( 7 4 2 ) ..............................................
95
6.3 Spring 993, 1% and 2% Volume Concentration
R e s u l t s ........................................................................................... 100
6.3.1 Extinction and Optical Depth ( 9 9 3 ) .............................
100
6.3.2 Extinction of Sphere ( 9 9 3 ) ..............................................
103
CHAPTER 7. EFFECT OF SPRING DIMENSIONS ON
PROPAGATION C H A R A C T E R IST IC S .................................. 108
7.1 Introduction........................................................................................... 108
7.2 Thickness and Spring Concentration
of S a m p l e s ..................................................................................... 110
7.3 Springs with Different Number
of T u r n s ........................................................................................... I l l
7.4 Springs with Different P i t c h ...............................................................116
7.5 Springs with Different Diam eters......................................................... 116
CHAPTER 8. EXTINCTION AND TRANSMISSION
CHARACTERISTICS OF MIXED SPRING
SIZE S A M P L E S ..........................................................................124
8.1 Introduction...........................................................................................124
8.2 Single Spring Size, Different Concentration........................................126
8.3 Samples with Two Spring Sizes.............................................................. 130
8.4 Three Spring Size Mixed S a m p l e s ................................................... 134
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ix
CHAPTER 9. C O N C L U S IO N .......................................................................... 138
BIBLIO G R A PH Y .................................................................................................143
A PPE N D IX ............................................................................................................ 148
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x
LIST OF TABLES
Table
Page
4.1 Spring D im ensions...................................................................................... 61
5.1 Thin and Thick Sample C haracteristics.................................................... 78
6.1 742 and 993 Sample C haracteristics..........................................................88
7.1 Spring Sample Characteristics.......................................................................... 110
8.1 Mixed Spring Sample Characteristics............................................................... 126
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LIST OF FIGURES
Figure
Page
1.1
Extinction and scattering from spherical (top) and
nonspherical (bottom) particles............................................................... 7
1.2
Optical rotatory dispersion and circular dichroism of
transmitted wave............................................................................................ 13
2.1
Linearly polarized wave incident on chiral slab of
thickness d......................................................................................................27
2.2
Multiple reflections and transmissions of LCP and RCP
incident waves................................................................................................32
3.1
A plane wave incident on particles in the i direction,
producing a scattered wave in the o direction at a
distance R...................................................................................................... 44
3.2
Geometric construction for definition of specific intensity..........................50
3.3
Scattering of specific intensity incident on volume ds from
direction §’ into direction §........................................................................... 52
4.1
Free Space measurement system for characterization of
microwave materials......................................................................................58
4.2
a) Bulging of RTV sample using plexiglass holder with
15.24 cm opening, b) minimized bulging due to smaller
opening, c) small opening foam holder for 8.2 GHz to 12.4
GHz band.......................................................................................................65
4.3
Permittivity and permeability of 3.33 mm RTV sample using
a sample holder with a 15.24 cm and a holder with a 7.62 cm
opening...........................................................................................................67
4.4
Permittivity and permeability of Teflon using holder with
7.62 cm opening.............................................................................................68
5.1
Rotation angle* for the 2% 742 and 993 chiral composites
s a m p le s .......................................................................................................76
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xii
LIST OF FIGURES (CONTINUED)
5.2
tog: Transmitted intensity of 2% 742 sample with interface
reflections using Eqs. (5.12) & (5.13), and without interface
reflections using Eqs. (5.7)-(5.11). bottom: Reflection and
transmission of 2% 742 sample.................................................................... 79
5.3
top: Transmitted intensity of 1.6% L-625 sample with
interface reflections using Eqs. (5.12) & (5.13), and without
interface reflections using Eqs. (5.7)-(5.11). bottom:
Reflection and transmission of 1.6% L-625 sample....................................81
5.4
tog: Transmitted intensity of 1.6% R-625 sample with
interface reflections using Eqs. (5.12) & (5.13), and without
interface reflections using Eqs. (5.7)-(5.11). bottom:
Reflection and transmission of 1.6% R-625 sample................................... 82
5.5
tog: Transmitted intensity of 1.6% M-625 sample with
interface reflections using Eqs. (5.12) & (5.13), and without
interface reflections using Eqs. (5.7)-(5.11). bottom:
Reflection and transmission of 1.6% M-625 sample...................................84
5.6
top: Extinction of L-625 sample from reflection and
transmission measurements using Eqs. (5.11) & (5.7),
and from transmission using Eqs. (5.12) & (5.13). bottom:
Same as top frame but with R-625 sample..................................................85
5.7
top: Extinction of M-625 sample from reflection and
transmission measurements using Eqs. (5.11) & (5.7),
and from transmission using Eqs. (5.12) & (5.13). bottom:
Extinction of L-625, R-625, and M-625 using Eqs. (5.11)&(5.7). .
.
87
6.1
Reflection and transmission of 1% and 2% 742 samples............................ 92
6.2
The extinction cross section (top) and optical depth
(bottom) vs. frequency for the 1% and 2% 742 samples............................94
6.3
Extinction cross section of, 1% and 2% 742 spring, large
sphere with radius equaling 742, and small radius sphere......................... 96
6.4
Transmitted intensities for 1% and 2% 742 springs and
spheres........................................................................................................... 98
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LIST OF FIGURES (CONTINUED)
6.5
Reflection and transmission of 1% and 2% 993 samples.............................101
6.6
The extinction cross section (top) and optical depth
(bottom) vs. frequency for the 1% and 2% 993 samples............................102
6.7
Extinction cross section of, 1% and 2% 993 spring, large
sphere with radius equaling 993, and small radius sphere..........................105
6.8
Transmitted intensities for 1% and 2% 993 springs and
spheres........................................................................................................... 106
7.1
Extinction cross sections and power absorption
coefficients for the 742 and 743 springs whose number of
turns are 3 and 5, respectively......................................................................112
7.2
Extinction cross-sections and power absorption
coefficients for the 942 and 990 springs whose number of
turns are 3 and 2, respectively......................................................................114
7.3
Extinction cross-sections and power absorption
coefficients for the 991 and 993 spring whose number of
turns are 2 and 3, respectively...................................................................... 115
7.4
Extinction cross-sections and power absorption
coefficients for the 941 and 942 springs whose pitch
differs............................................................................................................. 117
7.5
Extinction cross-sections and power absorption coefficients
for the 942 and 993 springs whose diameters differ....................................118
7.6
Extinction cross-sections and power absorption coefficients
for the 990, 991, and 992 springs whose diameters differ...........................120
7.7
Extinction cross-sections and power absorption coefficients
for the 625, 742, and 941 springs whose diameters differ...........................121
7.8
Extinction cross-sections and power absorption coefficients
for the 625, 742, 942, 990, 991, and 992 springs..........................................122
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xiv
LIST OF FIGURES (CONTINUED)
8.1
Theoretical and experimental transmitted intensity for the
2% 742 sample (top graph) and the 1% 742 sample
(bottom graph).............................................................................................. 127
8.2
Theoretical and experimental transmitted intensity for the
2% 993 sample (top graph) and the 1% 993 sample
(bottom graph).............................................................................................. 129
8.3
Theoretical and experimental effective extinction and
transmitted intensity for the 2% 742/991 mix sample................................131
8.4
Theoretical and experimental effective extinction and
transmitted intensity for the 2% 742/993 mix sample................................133
8.5
Theoretical and experimental effective extinction and
transmitted intensity for the 2% 742/991/625 mix sample.........................135
8.6
Theoretical and experimental effective extinction and
transmitted intensity for the 2% 742/993/625 mix sample.........................137
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1
CHAPTER 1
INTRODUCTION
1.1 Objective
Since the discovery of chirality, researchers have sought to gain a better
understanding of its nature. Theoretical work throughout the early 1900s has set the
cornerstones for experimental analysis of chiral media. The latter part of the 1980s
witnessed extensive research concerning the measured properties of these materials
at microwave frequencies. Within this decade, much work has been devoted to
modeling the material properties of chiral composites for possible applications. Such
applications include absorbing and polarizing materials.
The Center for the Engineering of Electronic and Acoustic Materials
(CEEAM) situated at the Pennsylvania State University, has led research in material
characterization of chiral composites since the late 1980s. The center’s research has
been focused on designing a facility to fabricate and analyze composites in the
microwave realm. Researchers at the center have predicted the possibility of tailoring
reflection and absorption of electromagnetic waves using chiral composite materials.
They found that the introduction of chirality, via the chirality parameter in a
dielectric volume, drastically alters its scattering and absorption characteristics. Using
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thick (1 cm), non-flexible, samples the center researchers showed experimentally that
chirality in a material, not only affects the polarization characteristics of a
propagating wave, but the reflection, transmission, and attenuation characteristics.
Discovery of these properties due to helical geometries has spurred research
by CEEAM into two areas: 1) understanding the role of the geometry of a helix in
the attenuation of an electromagnetic wave transversing through a chiral composite,
and 2) observing chiral related attenuation in thin flexible composites for possible
anti-reflection and electromagnetic interference (EMI) applications.
This thesis will contribute to these areas by studying wave attenuation in thin,
flexible chiral composites using extinction and the equations of radiative transfer. The
primary focus is to study the extinction cross sections of helical inclusions at
microwave frequencies, and draw conclusions about the effect of extinction on
attenuation. By using the equations of radiative transfer along with extinction
measurements, the attenuation within the chiral composites can be modeled.
Extinction and radiative transfer have been used extensively in modeling
mediums filled with scattering inclusions. However, there have been no prior
investigations analyzing the extinction of helical geometries and applying the
information in radiative transfer equations to chiral composites.
To understand a material’s transmission losses, it is necessary to understand
an inclusion’s interaction with electromagnetic (EM) waves. A simple approach to
this problem is to examine the extinction cross section of the inclusion. Extinction
describes the amount of energy taken out of an incident beam due to absorption and
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scattering. By relating the helical geometries to extinction cross sections, transmission
losses can be predicted and tailored. Consequently, anti-reflection and EMI materials,
which depend on internal attenuation can be manufactured to specified parameters.
Therefore, several objectives are met in this thesis: 1) construct 3 mm thick,
flexible chiral composites, 2) design test fixtures and experimental procedures to
accurately measure material properties, 3) understand transmission behavior by
analyzing extinction cross sections of helices of varying diameters, turns, and pitches,
and 4) use the equations of radiative transfer to predict microwave transmission of
composites containing single-sized and mixed-sized helices.
1.2 Scattering and Absorption by Particles
The scattering of electromagnetic (EM) waves is related to the homogeneity
of the medium being illuminated. If an electromagnetic wave transverses a
homogeneous medium, it is not scattered within the medium. Only inhomogeneities
cause scattering. It can be pointed out that every material is inhomogeneous on a
molecular level where in each molecule can be considered as a scatterer.
No matter what kind of material is being illuminated, the underlying idea is
that an atom, molecule, or larger scatterer (or particle) is illuminated by an
electromagnetic wave, where electric charges in the particle are polarized by the
incident wave. The polarization is a displacement of the electric charges of the
particle, which then radiate an additional electromagnetic field. The field that is
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generated due to the displacement of the charges of the particle is the particle’s
scattered field.
Scattering is often accompanied by absorption. When a particle reradiates
electromagnetic energy, the movement of the particle’s charges sometimes transforms
part of the incident electromagnetic energy into some other form of energy such as
heat. An example of absorption would be microwave energy incident on carbon
particles. Carbon particles possess a degree of electron mobility that will absorb
incident electromagnetic energy (such as microwave energy) and convert that energy
into heat.
The process by which scattering and absorption remove energy from a beam
of EM energy transversing a medium is called "extinction." Extinction is the
attenuation of the incident beam, which is defined as
Extinction = Scattering + Absorption
The mechanism of extinction will be studied in more detail in subsequent sections.
There is always scattering of electromagnetic energy even on a molecular level.
But the final scattering result depends on the "orderly arrangement" of the scattering
obstacles. That is, the net effect of all the scatters must be considered. If the total
scattering effect is dependent on the phase relations between the waves scattered by
neighboring particles, then this is a problem of dependent scattering. As a final note,
the terms scatterers, particles, or inclusions refer to objects that scatter
electromagnetic energy; the three terms will be used interchangeably throughout this
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5
thesis.
Waves scattered by different particles from the same incident beam will still
have a phase relationship and may still interfere (constructive and destructive
interference). But the idea of independent scattering assumes that there is no
systematic phase relationships between particles. There may be slight phase
deviations from one particle to the next, but the net effect is that intensities scattered
by the various particles must be added without regard to phase. Thus, scattering by
different particles without regard to phase is called coherent scattering.
A thin, tenuous medium containing M scattering particles will remove a total
intensity equal to the extinction of one particle times M. This is called single
scattering. Single scattering assumes that the incident beam from a source reaches
a receiver after encountering very few particles. The received scattered wave is
assumed to be due to single scattering by a collection of particles, and that all double
or multiple scattering effects are insignificant.
Multiple scattering is predominant when each particle is exposed to waves
scattered by other particles. A thicker or highly dense medium may take the original
beam of energy and scatter it several times before it emerges to be detected. In other
words a particle will scatter an incident wave, but the wave incident on this particle
has previously been attenuated by scattering and absorption. The scattered wave from
this particle will continue to be scattered and absorbed by other particles before it
reaches a detector. There will be some scatterers within the medium that may receive
no direct energy, but only diffuse scattered energy. Therefore, simple proportional
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energy loss of single scattering will not be predominant in a multiple scattering
scenario.
In order to confirm the predominance of single scattering over multiple
scattering, several criteria should be met. Single scattering can first be tested by
illuminating two samples. One sample should contain twice as many inclusions as the
other sample. If the resultant intensity is decreased by one-half by doubling the
inclusions, then single scattering will dominate. A second method to confirm the
dominance of single scattering, is to calculate the optical depth. The optical depth
is defined as r = p o/d. The term p is the concentration of scatterers, a, is the total
cross section (extinction) and d is the thickness of the media. The intensity of a beam
is attenuated by extinction to e"7of its original value. If r < < 0.5, single scattering
is predominant. For larger values of r, double or multiple scattering may have to be
considered.
1.2.1 Extinction Examples: Spherical and Nonspherical Scatterers
Extinction is mainly dependent on the geometiy and composition of a
scattering object, and as a function of wavelength. Figure 1.1 (top graph) shows three
extinction curves for water droplets in air of three different radii. The extinction is
shown as the extinction efficiency (defined as Qejtt — a, I no 2) as a function of inverse
wavelength. The 1.0 /im droplet is seen to exhibit three distinct features between 0.5
/a o i ' 1
and 5 / a m'1. The first feature is a series of regularly spaced broad maxima and
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4
a ■ I.Oum
lo ■0.2 jim
3
a ■ .0 5 |im
j
aS2
i
i
2
3
4
5
IN V E R SE
7
6
WAVELENGTH
e
9
10
II
( ^ m * ')
10
m ■ 1.55
X ■0.6328 fim
£u e
o
6
K
u2
3 a
§
■s
□
2
° 2
O
O
A SY M PT O T IC
/
LIM IT
2
3
D IA M ET ER
4
5
6
( |i m )
Figure 1.1: Extinction and scattering from spherical (top) and nonspherical (bottom)
particles. The bottom frame is the scattering cross section of normally incident light
polarized parallel (solid) and perpendicular (dotted) to the axis of an infinite cylinder
in air (Bohren and Huffman 1983).
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minima known as the interference structure. The second characteristic is the ripple
structure shown as irregular fine structures. The third feature is the Rayleigh region,
which shows extinction monotonically increasing with decreasing wavelength when a
< A'1 (radius is less than a wavelength). The interference structure, also known as the
Mie or resonance region, is due to the interference between the incident and forward
scattered wave. The ripple structures, are usually seen in extinction curves of weakly
absorbing spheres. Figure 1.1 shows strong damping for the ripple and interference
structures where the inverse wavelength is greater than 6 /Am'1. This is a region of
large absorption. If the particle is small compared to the wavelength, bulk absorption
peaks will occur. This is evident for the 1.0 pm and 0.05 pm radius droplets at 0.03
p m 1 and 6 pm'1, respectively. For very small particles relative to wavelength, rising
extinction toward shorter wavelengths is observed. This is known as reddening and
is characteristic of small nonabsorbing scatterers such as the 0.05 pm water droplet
in Fig. 1.1.
Extinctions of spherical geometries are much easier to compute than
nonspherical scatterers. However, there are many instances where scattering from
geometries other than spheres are present every day. From the atmosphere to
interstellar dust, cylindrical, spheroid, and helical geometries exist which depolarize
incident waves giving different extinctions (Bohren and Huffman 1983, Whittet 1992).
Figure 1.1 (bottom graph) depicts the scattering cross sections per unit particle
volume for an incident wave, polarized parallel and perpendicular to a cylinder axis
as a function of diameter. The cylinder is considered infinite and nonabsorbing with
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a refractive index of 1.55, and is illuminated by a wavelength of 0.6328 fim. From the
figure several interesting points are evident. At cylindrical diameters of 5 fim or
greater, there is little difference between scattering cross sections due to an incident
wave that is parallel or perpendicularly polarized. At smaller diameters, this is not
so. The parallel polarization has a larger scattering cross section than the
perpendicular polarization. The difference between the two is the greatest at
diameters close to 0.2 fim. The greatest amount of scattering occurs at a diameter of
0.5 fim.
13 Chirality
A chiral medium is characterized by either left-handedness or righthandedness in its microstructure. In the optical frequency range, many organic
molecules exhibit chirality (optical activity). Since the discovery of optical activity
early in the 19th centuiy, physicists, chemists and biologists have been studying this
phenomenon in their respective fields. Through experimentation, optical activity is
found to lie in the molecular or ciystal structure. A chiral substance exists in two
different forms that are identical in chemical and physical properties, but one’s
spatial geometiy is a mirror image of the other. This image cannot coincide with its
mirrored partner. Physical chemists have developed polarimetric techniques to
investigate handed molecular and ciystal structures. However, chirality exists in other
forms other than at the molecular level. Helices, sea-shells, and a person’s hands are
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10
common examples.
Twentieth centuiy investigation of chirality has gained momentum due to
acceptable constitutive equations for chiral media. Extensive theoretical and
experimental research has been done to investigate a wave’s polarization, reflection,
and absorption in chiral media relative to achiral media (non-chiral). Extensive
literature surveys have been done by Barron (1982), Applequist (1987), Lakhtakia
(1990), and Lakhtakia, Varadan and Varadan (1989) concerning past and present
research. The next few sections will briefly cover the origins of chirality, relevant
twentieth centuiy research, and classical electrodynamics in chiral media.
1.3.1 Origins of Chirality
The discovery of chirality, or optical activity, had its origins when the
phenomenon of double refraction was first described. In 1690, Erasmus Bartholinus
found that when a beam of light passes through calcite (Iceland spar), the beam splits
into two parallel rays. He discovered, by placing the crystal over the image of a dot,
a double image occurred. By rotating the ciystal, one dot remains stationary while
the other appeared to move in a circle. The ray that gave the stationary image
obeyed the law of refraction; the other did not.
Huygens (1690) had applied his wave theory to explain many aspects of double
refraction in calcite. He referred to the ray that obeyed the law of refraction as the
ordinary ray (o-rays); the other he called extraordinary ray (e-rays). He noticed when
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a ray passes along the optic axis of the calcite, the o-rays and e-rays travel with the
same speed. In other directions, the e-rays travel faster than the o-rays.
An experiment on this property was performed by Malus (1809). He reflected
a beam of light from a glass sheet into calcite. When light was reflected at a certain
angle, one of the images disappeared for certain positions of the calcite. Malus said
the reflected light was polarized. The reflecting angle that gave the maximum effect
was called the polarizing angle.
A further discovery was made by Arago (1811) in the early 1800s. He
polarized light (by reflection) and passed the light through a plate of quartz, then
through calcite. Two images were produced whose color would change by rotating the
calcite. Arago explained that the plane of polarization of the light is rotated on
passing through the quartz, and that the amount of rotation is different for different
colors (optical rotation).
The nature of light that made polarization and optical rotation possible was
not understood until Fresnel (1866) developed his conception of transverse waves.
He postulated that a wave’s direction of oscillation is perpendicular to the direction
of travel. Fresnel showed that a linearly polarized ray (LPR) can be broken down into
two circularly polarized rays rotating in opposite directions. The right-circularly
polarized (RCP) and left-circularly polarized (LCP) components of a LPR, traveling
through a medium at different velocities, will rotate the linear polarized wave. He
concluded that quartz occurs in two forms: dextrorotatory for a LPR rotated clockwise
through quartz, and levorotatory for a LPR rotated counterclockwise. For brevity a
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12
medium having right-handed (left-handed) character is d-rotatory (l-rotatory).
The rotation of the light’s plane of polarization by quartz was investigated
extensively by Biot (1817) after Arago’s experiments. Biot found that the amount of
rotation was proportional to the thickness of the quartz plate and inversely
proportional to the square of the wavelength. This effect is now called optical rotatory
dispersion ([ORD). Biot also found that ORD could be observed in organic substances
such as oils of turpentine and sugar. He concluded that optical activity was inherent
in the molecular arrangement of a substance.
Around 1950, Louis Pasteur discovered molecular dissymmetry to rotate light
left or right. He found that crystallized sodium ammonium tartrate came in two
forms. Both forms had the same molecules, but different symmetry; they were mirror
images of each other. One form of the substance in a solution rotated light to the
left, while its mirror image rotated it to the right.
Cotton (1895) discovered that LCP and RCP waves had a different amount
of absorption when traveling through an optically active material such as tartaric acid.
He termed this observation as circular dichroism (CD). The variation of rotation and
circular dichroism in the region of optically active absorption bands is called the
Cotton Effect.
Rotatory dispersion and circular dichroism can be illustrated by Figure 1.2. A
plane polarized wave passes into a chiral medium where an LCP and an RCP wave
are produced, but are absorbed to different extents while passing through the
medium. The transmitted result is an elliptically polarized wave.
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13
(Minor Axis)
Major Axis
Figure 1.2: Optical rotatoiy dispersion and circular dichroism of transmitted wave.
The top figure shows a transmitted wave whose polarization has rotated while passing
through a chiral medium. The bottom figure is the ellipticity of the transmitted wave.
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14
If the medium has a thickness d, the angle through which
the major axis of the ellipse is rotated with respect to the incident polarization is
expressed as:
a = ^ ( n ' L- n k = ^ { k 'L -k%
A,q
( 1.1)
where nL > nR is and nR > nL characterize a d-rotatoiy (right-handed) and 1-rotatoiy
(left-handed) medium, respectively. The refractive indices nL’ and nR , and
wavenumbers kL’ and kR’ are the real parts for the LCP and RCP waves. The freespace wavelength is given by k a. If the amplitudes of the LCP and RCP waves are
given as E L and Er, then the major axis of the ellipse is given as (ER + El), and the
minor axis by (ER - El). The ellipticity of the ellipse or the ratio of the minor to
major axis is expressed as:
(1.2)
The ellipticity can also be expressed in terms of the imaginary refractive indices and
wavenumbers in the chiral medium as:
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15
where nL" and nR" are the absorption refractive indices, and kL" and kR are the
wavenumbers for the LCP and RCP waves respectively.
13 2 Twentieth Centuiy Research
The 20th centuiy opened up for chirality when Drude (1900) assumed in a
chiral medium, electrons move back and forth along spiral paths. He proposed that
rotation of the plane of polarization can be predicted if Maxwell’s equations provided
a polarization term proportional to the curl of the electric field. Bom (1915), Oseen
(1915), and Gray (1916) set forth ground breaking results in analyzing a substance’s
molecular and atomic structure with respect to electromagnetic fields. Their
contributions set a foundation for physical chemists to characterize molecular
structures.
On a macroscopic scale, Lindman (1920) reported ORD for randomly oriented
helices in the
8 .8
GHz to 25 GHz frequency band. He found that the Cotton effect
was maximum when helices of length L were X ^ 2L. His claim was disputed by
Winkler (1956) who said Lindman’s observations were due to anisotropic scattering.
However, Tinoco and Freeman (1957) concurred with Lindman’s results by studying
optical activity of a foam matrix filled with copper helices having a radius = 0.25 cm
and L = 1 cm. This worked proved that optical activity can be observed at
frequencies lower than the optical realm. This research also gave a basic framework
for the development of chiral composite media.
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16
To develop a more general treatment of composite chiral media, Bohren
(1974, 1978) devised a transformation to solve scattering by chiral spheres and
cylinders. This method decomposes an electromagnetic field in chiral media into two
circularly polarized fields. These fields then satisfy the homogeneous wave equation
in chiral media.
Throughout the 70’s and 80’s there have been many theoretical works
concerning the modeling, and possible applications of chiral materials. In 1979,
Jaggard, Mickelson and Papas (1979), modeled electromagnetic wave interaction with
chiral conductors embedded in a dielectric. Lakhtakia et al. (1985), Varadan et al.
(1987), and Varadan et al. (1987) have done several works on scattering from:
nonspherical chiral objects, helically arranged point-polarizable scatterers, and beaded
helices. Further modeling has included reflection and refraction between chiral and
non-chiral interfaces. Such problems have been investigated by several authors
including Silverman (1986), Bassiri et al. (1988), and Lakhtakia et al. (1986).
Lakhtakia presented data that shows that reflected power from a chiral interface is
controlled by the handedness of the medium. This was further proven through
numerical computations by Varadan et al. (1987). He showed that chiral coatings
could reduce power reflection over a frequency range of 50 GHz to 300 GHz. Liu
and Jaggard (1992) have also presented theoretical evidence that chiral magnetic and
dielectric screens can enhance broadband reflection reduction.
Until the latter 1980s, there was little experimental research on chiral
composites. Guire (1990) measured the ORD of metal springs embedded in a low
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17
loss dielectric. He reported that the rotation angle was proportional to the volume
concentration of the helices. Guire had also measured the composites’ reflection
when backed by a metal plate. In 1990 Hollinger (1990,1991) and Pelet and Engheta
(1990) studied eigenmodes and wave propagation in waveguides filled with chiral
materials. Hollinger’s work included circular waveguides, while Pelet and Engheta
worked with parallel plate waveguides.
The chiral parameter /3, which accounts for the handedness of a chiral
medium, was measured for the first time by Ro (1991). In addition, Ro characterized
spring-embedded composites using normal reflection and transmission data to
ascertain the permittivity and permeability. He also predicted macroscopic properties
such as the LCP and RCP wavenumbers kL and kR, respectively, for chiral media of
different volume concentrations. His theory was derived from Varadan et al. (1990),
who related the microscopic rotabilities of the helices, to the macroscopic properties
kL, kR and (3. The relations are similar to the Clausius-Mossotti relations but describe
circularly polarized fields in a chiral medium. Ro calculated the rotatabilities of a
given sample, then solved for the material properties of a sample with different
spring concentrations. The effective properties of chiral composites have also been
investigated by Sihvola and Lindell (1990,1992) using a generalized Maxwell-Gamett
mixing formula. Guerin (1992) fully characterized composites made with randomly
dispersed ceramic helices embedded in an epoxy resin. Guerin et al. (1994) have also
offered modeling theories to predict the effective properties of helix-loaded
composites. Guerin showed satisfactory agreement between computed and
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18
experimental material properties of a chiral medium.
1 3 3 Constitutive Relations and Propagation in a Chiral Medium
Electric and magnetic fields are generally vector quantities that have both
magnitude and direction. The relations and variations of electric and magnetic fields
associated with electromagnetic waves are governed by Maxwell’s equations. In
differential form, Maxwell’s equations for any general media can be written as
V xH = — +J
(1.4)
dt
Vx E = -
3B
dt
(1.5)
V D =p
( 1.6)
V B =0
(1.7)
Equation (1.4) is derived from Amperes’ law, and relates the magnetic field H (A/m)
to the electric displacement D (C/m2) and conduction current J (A/m2) density,
respectively. From Faraday’s law, (1.5) relates the electric field intensity E (V/m), to
the magnetic flux density B (Wb/m2). Equation (1.6) is derived from Gauss’ law and
relates D to a volume’s charge density. The fourth relation (1.7) specifies that
magnetic field is devoid of sources or sinks (divergenceless).
In a medium free of sources, (1.4)-(1.7) in phasor form can be written as:
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19
V xH= - J oD
( 1 .8 )
V x E = i o)B
(1.9)
V -D= 0
(1.10)
V -B = 0
(1.11)
Equations (1.8) and (1.9) are derived based on a time-harmonic dependence of the
form e''“' .
When a material’s charged particles interact with electromagnetic field vectors,
the particles produce currents that modify wave propagation through the media. To
account on a macroscopic scale for the presence and behavior of these charged
particles, constitutive relations are given to relate the field vectors. The constitutive
equations for an isotropic, homogeneous material are
D= e E
(1.12)
B = ju.H
where e = €0er = (8.85 x 10 12)(er) F/m, and /jl = fijir = (4ir x 10'7)(p,r) H/m, are the
permittivity and permeability, respectively. If a medium has loss, the permittivity and
permeability are complex and are written as
€ = e 1+ i e "
(1.13)
fi =/// + i n "
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20
Using (1.8)-(1.11) and (1.12) together with the vector identity
V x ( V x F ) =V(V-F) - ^ F
(1-14)
yields the vector Helmholtz equation
V>E + k 2E
=0
(1.15)
V2 H+ ir 2 H = 0
where
k = oVcTT =
“ y/eriir
(1.16)
= k '+ i k "
is the wavenumber (units of 1/m) and c is the wavespeed in vacuum. From (1.13), the
wave number is complex if the media is lossy. The real part of (1.16) is used to
derive the phase velocity v = w/fc’, while the imaginary part describes a wave’s
attenuation through a medium.
The constitutive equations in (1.12) are acceptable for most isotropic media,
but are unacceptable in describing field/material interaction in chiral media. Drude
(1900) proposed that the intrinsic polarization field P in an optically active medium,
must include the circulation of E, (V x E). Bom (1915) expounded on this idea in
further studies. However, it was not until 1959 that Fedorov (1959a,b) and Bokut’
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and Federov (1959) modified Drude and Born’s equations to give the reciprocal
constitutive equations:
D= e [E + /3 V xE|
B = /i[H+/3VxH|
(1.17)
in which (3 is the chirality parameter having the dimensions of meters. The chirality
parameter can be complex, /3 = /3’ + i/3", where the real part is positive (negative)
for right-handed (left-handed) media. In order for a chiral medium to exhibit its
handedness, it must interact with time-vaiving fields. Other wise, (1.17) will reduce
to the equations of (1.12). Other equivalent constitutive equations have been
proposed (Post 1962), but the present study will use (1.17).
Consider a chiral region governed by (1.17). To find the wave equation,
several steps are taken. The constitutive equations (1.17) are combined with
Maxwell’s Eqns. (1.8) and (1.9) to give
V XE = i coj/i,H+ /3/llV Xl^
(1.18)
V XH= - / (x[e E+ /3e V xE]
(1.19)
Substitute (1.18) into (1.19) and (1.19) into (1.18), and take the curl of both
equations. After some algebraic manipulation, the following wave equation emerges:
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22
E( 1 - k 22(3) + V x E ( 2/3) k 2 + E £ 2 = 0
(1-20)
V2^ 1 - A-2 2/3) + V xH( 2/3) k 2 + Yik2 = 0
(1.21)
where fc2 = oz/ i f . For a material that exhibits no handedness or )3 = 0, then (1.20)
and (1.21) reduce to the vector Helmholtz equations in (1.15). Following Bohren
(1974) the electromagnetic field is transformed to a linear combination of left (LCP)
and right (RCP) circularly polarized fields by
E = Q + aRQ,
( 1.22)
+ Qi
where aL = -i
and aR = -i
The LCP and RCP fields, QL and QR, in
( 1 .2 2 ), must satisfy the conditions
^ (W iQ ^ o
(1.23)
^(V ^O ^O
along with
vxQ=*,-Q;
v-Q =o
V x C k — ^ Q ,;
V-Q*=0
(1.24)
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23
The curl terms in (1.24) are circulation equations given by Chen (1983). They
confirm that QL and QR are circularly polarized. The LCP and RCP wavenumbers
in (1.23) and (1.24) are expressed as
k =
A
L 1 - kfi
(1.25)
'■R
(1.26)
Equations (1.25) and (1.26) are distinct wavenumbers resulting from the chirality
term /3. These terms cause the LCP and RCP waves to travel at different phase
velocities and undergo differential attenuation, thus causing polarization rotation and
circular dichroism. LCP and RCP waves will rotate counterclockwise (CCW) and
clockwise (CW),respectively, along the direction of propagation.
1.4 Thesis Organization
Subsequent chapters have been broken down into eight areas to include
theoretical considerations and experimental results. Experimental results are broken
down into four chapters for enhanced readability and analysis.
Chapter 2 provides the theoretical framework to compute the reflection and
transmission coefficients of an isotropic chiral slab. The methodology to find the
material properties from the coefficients is also given.
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24
Chapter 3 explores mathematical properties of scattering by inclusions. A basic
introduction to scattering and extinction cross sections with respect to a slab of
randomly oriented particles is described. Also, the equation of transfer is derived to
explain absorption and scattering in an inclusion filled medium.
Chapter 4 details the measurement set up and sample preparation methods.
A description of the free-space system and its calibration procedures is outlined. A
method is given to construct and test thin flexible RTV samples. Experimental data
is provided to verify that the calibration and test methods yield accurate material
property results.
Chapter 5 compares the thickness of the samples to variances in calculated
extinction results. Transmission, reflection and extinction data are given for thin and
semi-infinite samples. The effect of including and omitting internal boundary
reflections on extinction measurements is examined.
Chapter 6 investigates helix inclusion concentrations with respect to extinction
measurements. Co- and cross-polarized measurements are compared at 1% and 2%
metal volume concentrations. Extinction measurements are derived as well as the
optical depth for these samples. The theoretical extinction for a sphere is computed
and plotted for comparison. Theoretical and experimental transmission results are
analyzed using the sphere and helix cross sections.
Chapter 7 analyzes a helix’s geometrical dimensions and its impact on
extinction cross sections. A spring’s number of turns, pitch, and diameter are
analyzed. In addition, the apparent power absorption for different spring geometries
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25
are examined.
Chapter 8 compares the extinction of samples containing mixed spring sizes.
Transmitted intensity plots are shown for samples with one, two, and three different
sized springs mixed together. The plots compare theoretical transmissions calculated
from extinction of single springs, to experimental data.
Chapter 9 concludes the study by summarizing the work. Future work in the
areas of modeling, fabrication, and testing are proposed.
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26
CHAPTER 2
PROPAGATION CHARACTERISTICS OF WAVES IN CHIRAL MEDIA
2.1 Reflection and Transmission of a Chiral Slab
A linearly polarized wave is normally incident on a chiral slab as shown in
Figure 2.1. Free-space regions 1 and 3, defined by z < 0 and (d < z), do not exhibit
chirality. Region 2, (0 < z < d), is occupied by a chiral slab. The incident electric
and magnetic fields are given by:
Einc = ( ^ V ^ A )
X
V0
(2 .1)
. ^ -(V X E J -
e**
(2 .2 )
'Io
where r)0=kj<sifx0 and ka are the free space impedance and wavenumber, respectively.
If Eyj = 0 and
or
& 0, or vice versa, the incident polarization is linear. If E^ = -i Eyi
= i Eyi, the incident wave is left-circularly (LCP) or right-circularly polarized
(RCP), respectively. Once a linearly polarized wave enters a chiral medium (medium
2), it is decomposed into positive and negative LCP and RCP plane waves. These
fields are given as:
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27
x
Eo F 0
M-o
w m m m m
Incident
• : • : • : • .* • .* • .* • .• • : • .
W W W -
Transmitted
•v.* •••.
. . . . . .
m m m m
Reflected
•V
TT^FT.*•..* •..**•v ••v '
•♦V*•v *■v
•.*i .*••• •.*».*♦•• • .• •.*• .
v.-v.-
Free
Space
z=0
Free
Space
z=d
Figure 2.1: Linearly polarized wave incident on chiral slab of thickness d.
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28
Ql =
Q r
(2.3)
Au («y - * * ) e * 1-2 + ^ ( « y + ^ x )
A R 1 (® y +
e
** +
(^y
^x ) e
^
(2.4)
where y4tl, A L2, A m, and >4^ are unknown coefficients. Also, the terms kL and kR are
the wavenumbers in the chiral media for the LCP and RCP waves, respectively. The
first terms in equations (2.3) and (2.4) are LCP and RCP waves going out to z = d,
while the second terms are LCP and RCP waves going toward z = 0. Using the
Bohren decomposition in Chapter 1 (eq. 1.22), the electric and magnetic fields in the
chiral slab are:
(2.5)
( 2 .6 )
where
aR = -i \l(\ile) = -irj and aL = -i \/(e/ti) = -Hy
The fields transmitted in region 3 (free space) are:
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E „ - if.Eu * E ^ y -
" in - j
(2.7)
e**
(2 .8 )
<lEu -E Rp y - K B ^ E ^ e * s
The reflected fields in region 1 (free space) have the form
*W = ( i . E ^ E J t , * J(EU -E a ) i j «-*•'
(2.9)
e**
H„r =
(2.10)
The transmission and reflection coefficients Eu , E rp ^Ln and E » are solved as part
of a boundary value problem.
To solve the boundaiy problem, boundary conditions (BC) are enforced at z
= 0, and z = d. At z = 0, the BCs are
ex .[ E inc + Eref- E ch] = 0
(2.11)
6x - [ Hinc +
H ^-H ^O
(2.12)
ey - [ Eiac + E«f - E uJ = 0
(2.13)
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30
ey .[ H inc + H ref- H ch] = 0
(2.14)
and at z = d
* [Ed, - Etr] = 0
(2.15)
• [Hch - Htr] = 0
(2.16)
• [Ech - Etr] = 0
^ .1
®y • [Hd, - Htr] = 0
/<\ -i !-l\
!)
(2.18)
By substituting (2.1)-(2.10) into (2.11)-(2.18), a system of eight equations is used to
solve for the unknown coefficients in (2.5)-(2.10).
A linear polarized wave can be decomposed into LCP and RCP waves. Since
a chiral medium sustains only LCP and RCP waves, it is convenient to break up a
linear incident wave into parts. By considering separately LCP and RCP incident
waves, the unknown coefficients can be found. Then adding both contributions will
yield reflection and transmission coefficients for a linear polarized wave. It can be
shown for an incident LCP plane wave, the coefficients A m, A L2, ER„ and Eu
disappear, leaving only waves with ^ - i e* polarizations traveling in either the
positive or negative z direction. For an RCP incident wave, the coefficients A L1, A R2,
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31
E lp and E r, will disappear leaving ^ + / e* polarized waves. It should be pointed out,
that reflected waves will undergo a mode conversion (Lakhtakia et al. 1989). That is,
an incident LCP wave will have an RCP reflected wave and vice versa.
The problem can also be solved by considering multiple reflections and
transmissions of an incident wave. Consider an incident LCP wave on a chiral slab
(Figure 2.2). When the LCP wave strikes the interface of region 1 and 2, a partial
reflected wave of amplitude ri2is produced. A transmitted wave of amplitude T\z eikLd
is then incident on the interface of region 2 and 3. Part of this is reflected back
toward region 1 to give a wave of amplitude r a Tn e,(kR+ “, while the other part of
the wave is transmitted to region 3. This transmitted wave has an amplitude of
Ti2 T23eikLd. The reflected wave traveling toward region 1, crosses the region 2/1
interface and results in a reflection of T2l Tl2 T^ e'(te +^ d. While going from region
2 to 1, a portion of this wave is reflected back toward region 3, and is transmitted
through to region 3. The transmitted wave’s amplitude is Tn T23
T21 e'(te + 2kL)d.
Figure 2.2 illustrates the first few of the infinite number of reflections and
transmissions that occur.
The total reflected wave, Sn, is the sum of all the partial waves transmitted
across the region 2/1 interface. The sum is given by:
c
_ p
^ l i z _ 1 12
+ T
21
T F
12
23
4
T T Y
F2
1 2\ - 'l 2 1 21i 2 3 e
(2.19)
= r,2 + Tn Tn
(r21ra
V
n=0
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1
LCP
2
T12
1
e i k Ld
e
32
3
■T T eikL
d
12 23
L
L
12
T
T„r
T T e ^ kL+ k R^d
21 12 23
-t
t
r r p*(^k rf kR?d
12 2T 23T21
T21T,2r 2,r232ei2<kL+ ^
W
'Ti2T23ri ri ei(3k£r2kR)d
Chiral Slab
Free Space
z =
RCP
ir > W
Free Space
z =d
0
Tn e^rf1
1
12
T
*21
T
T e'^kL+ k R*d
2 ,
KT12r 2
V V 2 3 ? l ^ L+2kR-
T T r r 2e i2(kL+
21 12 21 23
W
*
*
1* 2 V
k R)d
Tn r2 & m t k <£J
p 2 p 2 i(2k^-3k^)d
12 23 23 21
Chiral Slab
Free Space
z =
0
Free Space
=d
Figure 2.2: Multiple reflections and transmissions of LCP and RCP incident waves.
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33
This geometric series is in the form of
i
n=o
r" - r z
t r
(2-20)
where the sum is given as:
T*01 •T*1 0 A
Toi &
■
Invoking the following equalities given free space in regions 1 and 3
r = (77-O/O7+O = r ,2 = -r2i = -r^
(2.22)
TX2 = 1 + r I2, Tn — 1 + r21, Tn = 1 + r n
and after some algebraic manipulation the total reflection from the chiral slab due
to an LCP incident wave is:
_ r (i -
s
ll£
1 - r 2 e dki ' kktd
By following the same derivation for an incident RCP wave, the reflection is given
by:
r(i
11/? 1 - r 2 e i{kL' kR)d
-
(2.24)
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34
Adding the reflected contributions for an LCP and RCP wave, yields the total
reflection for an incident linearly polarized wave
11
i _ p2
2
11L
11/?
v
>
The reflection coefficients, (2.23) and (24,) for the LCP and RCP waves are
the same. This indicates there is no polarization change and that the reflection
polarization is the same as the incident.
The transmission of the chiral slab can be found by summing up all the partial
transmitted wave amplitudes into region 3 (Fig. 2.2). The summation of the
transmitted wave with an incident LCP wave is given as:
c
21L
-
t
12
t
23
t
12
T V Y
.
23 23 21
y
nr Y^ Y2,
y i2 7 231 231 2 1 e
—
(Z.ZO )
oo
- T T
1 V2 23
jL
it=0
<Y T
V 23 21
/
Since (2.26) is a geometric series in the form of (2.20), the transmission is expressed
as:
T n T23e *ld
I
^ 23 21
( 2
' 2 7 )
Using the equalities of (2.22) and arranging (2.27) yields:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Equation (2.28) is the transmission through a chiral slab due to an incident LCP
wave. Transmission for an incident RCP is solved in a similar way and is expressed
as:
(2.29)
Adding the LCP and RCP transmission contributions gives the total transmission
through a chiral slab due to an incident linear polarized wave.
'21
2
2
l - r2 e‘^L*lck)cl
(2.30)
It can be seen from equations (2.28) and (2.29) the transmission coefficients for the
LCP and RCP wave are different. A wave traveling through a chiral slab will have
two wavenumbers kL and kR. The resulting transmitted wave will be elliptically
polarized due to circular dichroism and optical rotatory dispersion.
As a final note, equations (2.25) and (2.30) can be used to describe the
reflection and transmission of a non-chiral slab. In a non-chiral medium, /3 = 0,
leaving only one wavenumber k = kL = kR. Therefore, equations (2.25) and (2.30)
reduce to:
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36
U
(2.31)
*1 - r2
tv?
_ ( l - r 2) a ^
! - r2
(2.32)
2 2 Computing Electromagnetic Properties
Experimentally, to find the three material parameters of a chiral sample, e, fi,
and /3, three measurements must be taken. For the first two measurements, the
antennas are in co-polarized positions. The reflection and transmission S-parameters
5Uco and S21co are then measured. A third measurement, S2ie is taken by rotating the
receiving antenna by an angle 6 from the co-polarized position. As mentioned earlier
a linear polarized wave can be mathematically decomposed into LCP and RCP
waves. The reflection and transmission characteristics of these waves’ different
polarization with respect to measured S-parameters are given as:
S... =
= ^219
21 cross
21c/0* “
*^1
sin6l
(2.33)
(2.34)
(2.35)
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37
^2\L
^21co ^21 cross
(2.36)
Measuring 5lleo, S21eo, and S2ie, and substituting the terms into (2.33)-(2.36) will yield
the reflection and transmission coefficients for the LCP and RCP waves.
The rotation angle and ellipticity of the transmitted field are calculated from
(2.33)-(2.36) and equation (2.29)
i(kK-k£)d _ $21R
e
"
C
132\L
(2.37)
where kR = kR’ + ikR" and kL - kL’ + ikL".
From (2.37) the rotation angle and ellipticity are written as:
{k'L- k ^ d
a = ------------
(2.38)
(k'[- k n )d
tan 4> = tanh-------------
(2.39)
The logarithm of a complex function is multivalued and can have ±2irn
possibilities. Theoretically from (2.37) and (2.38), there could be an infinite number
of solutions for a. However, none of the samples fabricated in this study nor in
others at CEEAM have exceeded a rotation angle of ± v (Ro 1991 and Guerin
1992).
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38
To solve for kL and kR (2.25) is first arranged as follows:
iik,d
s nL
r
T e i{kR' kdd(\ - 5n i r )
^2'40^
Then (2.40) is substituted into (2.28) which yields (after some algebra)
a t 1 + bT + a = 0
(2.41)
where a = SllL and b = S2lR S2lL - (1 + (SUL)2).
Since (2.41) is quadratic, there will be two solutions for T. Based on the
definition of (2.22), the root satisfying IT I <
solved for
17 given
1
is chosen. At this point (2.22) can be
the value of T
V=
T +1
(2.42)
Equation (2.40) is then solved by using T and the value of (2.37). Taking (2.40)’s
results, the complex LCP and RCP wavenumbers, kL and kR are found by:
i2 k Ld = In | e i2kLd\ + i(2 kLd + 2irm )
(2.43)
j\k R - k j d = In | e Kk*~kdd\ + i[(kR - k j d+ 2 vn)]
(2.44)
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where m and n = 0, ±1, ±2, etc. The determination of the imaginary parts of (2.43)
and (2.44) will give the real parts of kL and kR, which relate to the wavespeeds of
LCP and RCP waves, respectively. However, from (2.43) and (2.44) it is seen there
exists an infinite number of solutions for the real parts of the propagation constants.
A procedure by Ro (1991), the approximate phase velocity concept, determines the
values of m and n to find a unique solution for the LCP and RCP wavenumbers.
Details of this procedure are found in Chapter 4.
Once kL, kR, and
77
are found, the material properties /3, k, e, and /x are
derived from:
(2.45)
2
k =
1
+
1
(2.46)
(2.47)
It is noted that Equation (2.46) relates kRand kLto an average propagation constant
of the medium. By using the measured transmission and reflection coefficients and
the inversion procedure outlined above, beta, the permittivity and the permeability
can be solved from (2.45)-(2.47).
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40
CHAPTER 3
EFFECTIVE EXTINCTION CROSS SECTIONS
AND EQUATIONS OF RADIATIVE TRANSFER
3.1 Introduction
In the last chapter, the propagation of a wave through a homogeneous chiral
medium was discussed. Macroscopic properties such as the permittivity, permeability,
and beta are derived from the reflection and transmission coefficients of a sample.
This chapter will address the factors that affect the reflection and transmission
through a medium containing a distribution of particles. When a wave propagates
through a material containing many particles, energy is absorbed and scattered. The
loss of energy due to absorption and scattering directly affects the transmitted and
reflected intensity of a sample. The transmission and reflection then influence the
macroscopic properties of the material. Therefore, to understand how helices in a
chiral composite alter propagation characteristics, the transport of energy through a
medium containing scattering particles must be understood.
It should be noted that the extinction cross sections and radiative transfer
equations discussed in subsequent sections, are based on homogeneous non-chiral
mediums. A non-chiral material will have one propagation constant while a chiral
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41
medium will have two constants (LCP and RCP). As will be shown in Chapter 5, the
effective propagation constant (K) of a non-chiral medium of particles is related to
the cross section of the individual particles. Therefore, for a chiral medium, there are
two distinct extinction cross sections for a chiral particle. From equation (2.46), the
average (effective) propagation constant of a chiral medium is related to kL and kR.
This average constant will then yield an effective cross section of a chiral inclusion.
So keep in mind that the non-chiral extinction and radiative transfer equations
discussed, are applied to thin chiral composites by using an effective extinction cross
section.
3.2 Transport Theory
Two main theories have been used to model the propagation of
electromagnetic waves through media containing random particles. They are the
analytical and transport theories. Analytical theory starts by using Maxwell’s
differential equations or the wave equation. The scattering and absorption of the
particles are then introduced, and finally differential or integral equations for
statistical quantities are introduced (such as variances and correlation functions).
Transport theoiy involves the transport of energy through a medium of
particles. The theory does not start with the wave equation, and is not as
mathematically demanding as the analytical theory. Transport theory is also called
radiative transfer theory. The basic equation in this theoiy is also the Maxwell-
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42
Boltzman collision equation. It has been used in neutron transport theory and the
kinetic theoiy of gasses. Transport theory has also been used to solve numerous
problems such as atmospheric visibility, optics of papers, and radiant energy
propagation in the atmosphere of planets, stars and galaxies.
The basic equation for radiative transfer of a medium depends on many
factors including absorption and scattering properties of a wave in the presence of
randomly distributed particles. The extinction or total cross section relates the
amount of power removed from an incident beam of energy due to absorption and
scattering. Extinction by a particle is necessary to apply radiative transfer theory to
a medium of inclusions.
3.3 Extinction Cross Section
Energy intercepted by a particle may be either absorbed or scattered in all
directions, or a combination of both mechanisms. When a beam of light transverses
a medium, it is attenuated when energy is removed from the beam by scattering and
absorption. This attenuation is called extinction. In composite materials, the
attenuation may be related to the number of particles and the extinction cross section
of each particle. This condition will hold provided the scattering adds coherently (no
multiple scattering).
Consider a linearly polarized wave of the form,
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E - .t o
= 6 j ^ o e x p ( / * i r )
(3.1)
impinging upon a non-chiral particle from the direction i (Figure 3.1). The
magnitude, Ea, is of unit magnitude, and carries the unit of volts/meter. At a distant
observation point r, in the direction o, the scattered field behaves as a spherical wave
(Ishimaru 1978):
Eg(r) = f(6,i) {eikR/R )
for R > D 2/ X
(3.2)
where D is roughly the particle diameter, and X is the wavelength of the field. The
term f(o,i) is the amplitude junction. This function represents the amplitude, phase
and polarization of the scattered wave in the direction, o, produced by a wave
incident on the particle from the i direction. The time averaged poynting vector of the
incident and scattered power are given as:
S, = | ( E i x H 1- ) . ( | i . f / 2 r , 0)i,
S , - I ( E , x H i- ) - ( I 4 I !/ 2 ^ 6
(3.3)
where t70=(/i0/e0) is the characteristic impedance of the medium.
The scattered intensity can be defined as the power flux density confined in
a solid angle d fl in a particular direction. This is shown as (Papas 1988):
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44
Particle
8O9rO
,Ll
A
E.(r)
❖
Transmit
Degrees
(Forward)
Figure 3.1: A plane wave incident on particles in the i direction, producing a
scattered wave in the o direction at a distance R. Extinction is measured in the
forward direction (0 degrees).
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where dCl = sinO dO d(f> is the element of solid angle. The incident power flux can be
regarded as an intensity since the magnitude is squared, |E, |2. However, in this case
the incident intensity is not confined to a solid angle, since it is a plane wave.
Looking at the scattered power in the direction 6, in relation to the incident
power, the differential cross section of the particle is:
Od(6,1) = lim [(i?2^ )/^ .] = |/(o,i) |2 = (pl/iT r)p(d,l)
(3.5)
7?—>00
where p(o,i) is the phase function (dimensionless), and R 2Ss = /s. The term ad has the
dimensions of area per solid angle. It can be defined as the cross section of a particle
that would scatter power over one steradian (1 sr) in the direction o.
If power is scattered at all angles around the particle, then the scattering cross
section can be expressed as:
a, =
1.
|f(ftl) 12
^
j> ( « 4 ) <«
(3.6)
where the solid angle is integrated over all directions. Therefore, the total energy
scattered in all directions is equal to the energy of the incident wave falling on the
area os. The energy absorbed inside a particle may also be put equal to the energy
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46
incident on the area oa. For a dielectric body, the absorption cross section is the
volume integral of the loss inside the particle (Ishimaru 1978):
(3.7)
Equation (3.6) is useful in many practical situations to determine approximate cross
sections for particles with complex shapes. The energy removed from the original
beam is put equal to the energy incident on the area a,. The law of conservation of
energy requires the sum of the scattering and absorption cross sections to equal the
total cross section a, or the extinction cross section:
(3.8)
The ratio of the scattering cross section to the total cross section is the albedo,
W0, of a single particle and is given as:
When WQ = 1, energy is conserved. That is, there is pure scattering and no
absorption.
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47
3.4 Scattering and Extinction in a Slab of Many Particles
The extinction cross section, a,, represents power loss due to scattering and
absorption of a particle. This loss is related to scattering in the forward direction (0
= 0). Forward direction means the receiving antenna is 180 degrees from the
transmitting antenna, and is measuring radiation from a slab between the two
antennas (refer back to Figure 3.1). Put another way, the direction of received
scattering (from a particle) is characterized by the angle 9, which it makes with the
direction of propagation of the incident beam and an azimuth angle (p. For forward
scattering, the incident and scattered beams will be in the same azimuthal plane. The
extinction cross section of a particle and forward scattering is related through the
optical theorem, and is given as (Ishimaru 1978):
ot = (4ir/A) Im[f(i,I)] • 6;
(3 .10 )
where o, is related to the imaginary part of the scattering amplitude (f(i,i) = f(0=O))
in the forward direction.
Therefore, consider a thin slab containing a cloud of particles where the
incident intensity, Ia, is approximately the same for each particle. It is assumed that
single scattering by independent particles exist within the medium (see section 1.2).
That is, the single scattering approximation will apply to a thin medium containing
a small concentration of scatterers. The scattered intensity for each particle is given
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48
as (van de Hulst 1981):
4 = r c w ) / / 2K
(3.11)
S A U ) - |fi(8,«)l2
where i is the index for each particle. The particles do not need to be similar. The
amplitude function, f;(0,<p ) in (3.11) is a function of scattering angles, 0 and (f>. The
square of the amplitude function in the forward direction (6 = 0) is 5(0). By
summing up the total amplitude function, and consequently 5, (0) for each particle,
the square of the total amplitude function is:
(3.12)
Then, from the optical theorem (3.10) for a single particle, and (3.11)-(3.13) for a
cloud of particles, the extinction cross sections of all particles are additive,
(3.13)
Equation (3.13) implies that the cross sections of each particle can be added to give
a slab of particles a bulk extinction cross section. Finding the cross section of each
particle means finding the particles scattered intensity. Therefore, the intensities of
each particle can be added to yield a total forward intensity. This concept is very
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49
useful when a medium contains different sized inclusions.
3.5 Specific Intensity
To understand transport theory, the power flow, dP, in a frequency interval
(v, v + dv ) transported through an element of area within a unit solid angle must
be considered. This power flow, or power flux density in a given direction, is the
specific intensity, /(r,§) measured in W m'2 sr'1 Hz 1 (sr = steradian = unit solid angle).
Figure 3.2 shows the amount of power at a point r transported across an element of
area da (oriented with the outward normal, §0), and confined in the solid angle dw,
in the direction §Qis given by, (Chandrasekhar 1960):
dP = /(r, §) cosfl da d o dv
(watts)
(3-14)
where 0 is the angle between §„ and §. The specific intensity describes the radiation
characteristic of the flux emitted from a surface.
Equation (3.14) gives the power which flows across an element area
da and is confined to an element of solid angle dfi. The flux is given by integrating
(3.14) over a solid angle 2ir in the forward range (O<0<ir/2) and can be written as
F+ da where F+, the forward flux density, is defined by:
= / (2ir)t 7(r ’§) S • So ^
(315)
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JZJ>
50
Figure 3.2: Geometric construction for definition of specific intensity. Radiation is
emitted by source and passes radially outward through solid angle dfl.
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Also, a backward flux density F. is defined for the flux flowing through da in the (§„) direction. This is given by:
= / M - / ( r,S)S • (-SJrfn
(3.16)
The specific intensity can be used to describe the way energy is scattered and
absorbed in a medium of random particles. To describe this interaction, the specific
intensity is included in a differential equation called the equation of transfer.
To begin, take a cylindrical volume with a cross sectional area of unity (A =
1 cm2) and a length ds shown in Figure 3.3. A specific intensity /(r,§) is incident on
the volume, which contains p Ads particles, where p is the density of particles in a
unit volume. The extinction cross of each particle is ar The fraction of the unit area,
A, that is occupied by the particles is given by (p ds a,A)IA = p ds ot. This quantity
is the total effective area of all the particles. Therefore, as the energy transverses the
medium in the direction §, scattering and absorption will take place decreasing the
specific intensity d/(r,s). This is described as:
<//(r,§) = - p dsot I
(3.17)
But since the specific intensity is incident on many particles, reradiation of this
energy can be redistributed and redirected in the direction of s. This redirecting of
energy will add to the intensity /( r,§).
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52
dQ
dQ
I(r,s)
Figure 33: Scattering of specific intensity incident on volume ds from direction s’
into direction s.
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53
For example, consider a wave in the §’ direction, incident on a particle in the
volume Ads. The incident flux density through a solid angle dCl’ is given by 5,- = /(r,s)
dCV (equation (3.16)), which is incident on a particle in the volume. The power flux
scattered by the particle is redirected in the direction §. The scattered power flux is
given by
(3.18)
where f(3,§’) is the scattering amplitude. Therefore the scattered specific density in
the direction § due to St is:
Sr R 2 = |f(§,§0|2 ^
= |f(§,§')|2 /(r,§ /) f l^
(3.19)
Equation (3.19) relates the reflected flux due to the incident flux on one particle. To
complete the derivation, the incident flux on the particle from all directions, and each
particle in the volume must be considered in order to find the specific intensity
scattered into the direction §. This is shown as:
(3.20)
From this equation, the incident flux is added from all directions (integrated over the
solid angle of a sphere), and then multiplied by p ds particles in the volume.
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54
Equation (3.20) can be rewritten to include the phase function, expressed
p iS J 1) = — \f(& £)\2
(3.21)
The phase function, is a function of direction, which incorporates the total
cross-section of a particle and describes the redistribution of energy from s’ to s.
Finally, if there is an internal source within the volume which increases the
specific intensity, the quantity
<*e(r,§)
(3.22)
will indicate the power radiation per unit volume per unit solid angle in the direction
s.
By adding equations (3.17), (3.20), and (3.22), the equation of transfer is given
by:
as
= - pa /(r,S) + - j p f p(Sfi ) / ( r,S ) dw ‘ + e(r,S)
4tt J4«-
(3.23)
Equation (3.23) is a first order differential equation whose general solution
is:
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55
7(r,S) = 7oexp(-r) + exp(-T) J (2(s)exp(r) ds
where
Q{s) = ( p a ./iv ) f
•Mtt
(3.23)
7 (r,S ) d t t + e (r,S)
The term r is the optical depth given as:
r = J p at ds
(3.24)
This equation describes to what extent an incident wave encounters particles in a
medium. In scattering theory, a medium exhibiting single scattering characteristics
will have r < l.
Equation (3.23) gives the transmitted total intensity for a beam incident on
a slab containing random particles. The first term on the right hand side of the
equation gives the intensity that has underwent scattering and absorption as it
propagates through a slab (extinction). The second term of (3.23) shows the
contribution to the transmitted intensity when scattered energy from other directions
are re-directed in the forward direction by particles in the given volume. The
extinxtion cross-section for a helix will be an effective cross section due to two
propagation constants in a chiral medium. The main scope of this study is to use the
first term to describe the extinction of a beam as it passes through a distribution of
helices.
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CHAPTER 4
FREE-SPACE MEASUREMENT SYSTEM AND SAMPLE PREPARATION
4.1 Introduction
Free-space methods for measuring the electrical properties of materials are
non-destructive and contactless. Free-space characterization has been done by several
investigators who needed to find the dielectric and magnetic properties, e*, p,*, of
certain materials (Redheffer 1966, Joseph et al. 1987). Researchers at The
Pennsylvania State University have developed a more accurate and compact means
of measuring the dielectric constant and magnetic properties of materials using
focused horn antennas (Ghodgaonkar et al. 1989, 1990).
Conventional methods of finding e* and p* measured complex reflection
coefficients of short-circuited and open-circuited (quarter-wavelength short-circuited
line) samples (Von Hippie 1954, Amin and James 1981). Measurements using this
method are difficult and time consuming because the open-circuited sample has to
be established at each measurement frequency.
There are other measurement systems such as cavity or waveguide methods
that are commonly used, but free-space techniques are preferred for several reasons:
1) Microwave composite materials are inhomogeneous due to variations in
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57
manufacturing. Because of this, unwanted higher order modes are excited at the
air/dielectric interface in waveguides, coaxial media and cavities but not in free space.
2) The free-space measurement of materials is nondestructive and contactless.
Because of this, the free-space medium is preferred to coaxial or waveguide
techniques for measurements, especially under high and low temperature conditions.
3) In coaxial and waveguide media, the sample must be machined to fit the
testing device with negligible air gaps. This requirement tends to limit the accuracy
of measurements for materials that cannot be machined precisely. Conversely, there
is little sample preparation required for free space measurements.
4) Free-space measurements allow the use of planar samples with large or
small cross-sectional areas (within limits).
4.2 Description of the Measurement System
Figure 4.1 is the free-space system used to measure the samples under study.
The configuration consists of three principle parts: 1) signal source and analyzer, 2)
experimental hardware including focused antennas and 3) data acquisition and
processing.
The integrated signal source and analyzer used is the HP8510B vector network
analyzer (VNA) system, which operates from 0.5 to 40 GHz and consists of a
synthesized sweeper, S-parameter test set, and a IF/Detector and Display/Processor
unit. The network analyzer transmits microwave power to the focused antennas.
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58
Synthesized
Frequency
Sweeper
(0.01 to 40 GHz)
HP 9836
Technical
Computer
with Hard Disk
HP 8510B
Microwave
Network
Analyzer
HP 7440A
S-Parameter
Test Set-Up
(40 MHz to 40 GHz)
HP 82906A
Port 1
Coaxial Cable
Mode
Transitions
Plotter
Printer
Port 2
Coaxial Cable
Mode
Transitions
Figure 4.1: Free Space measurement system for characterization of microwave
materials.
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59
The transmit and receive antennas are custom designed spot-focusing horn
lens antennas that are mounted on a precision machined aluminum table (1.83 m x
1.83m). The ratio of the focal distance to diameter (F/D) of the lens is unity where
D is approximately 30.5 cm. The depth of focus for the horn lens antennas is
approximately ten wavelengths. The antennas’ scalar feeds produce primary patterns
with nearly equal E- and H-plane beamwidths. The 3 dB, 10 dB, and 20 dB E-plane
beamwidths are approximately equal to k 0, 1.9
and 3X0 respectively. A series of
different mode transitions and adapters is used to cover the frequency bands: 8.2-12.4
GHz, 12.4-18 GHz, 18-26.5 GHz, and 26.5-40 GHz. A specially designed sample
holder, mounted on a micrometer-driven carriage, is placed at the common focal
plane of the two antennas. It can hold samples with transverse cross sections ranging
from 15.25 cm by 15.25 cm to 61 cm by 61 cm. The transmit and receive antennas
are mounted on a carriages where the distance between them can be varied to an
accuracy of 25 mm.
Transmitted and reflected power is collected by the focused antennas and is
routed to the network analyzer using circular to rectangular waveguide adapters,
rectangular waveguide to coaxial adapters, and coaxial cables. The information is
acquired by an HP9836 computer where it is processed and stored for future use.
A two-port TRL calibration technique is used with time domain gating to
remove the errors associated with any multiple reflections. Ghodgaonkar, Varadan,
and Varadan (1989, 1990) found the TRL (thru, reflect, line) calibration to produce
the highest quality calibration available for the free-space system. This technique is
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60
accomplished by assuming that the electromagnetic fields in the common focal plane
of the two antennas are uniform plane waves. For normal incidence measurements,
the two port TRL calibration uses three different standards. A thru standard is
obtained by maintaining a distance between the two antennas that is equal to twice
the focal distance (60.96 cm). A reflect standard is obtained by placing a metal plate
at the focal plane of the transmit and receive antenna. The line standard is achieved
by separating the focal planes by a distance equal to a quarter of the free-space
wavelength at the center of the frequency band of interest. The TRL calibration is
unable to fully correct for multiple reflections between the antennas and sample.
Therefore, time domain gating is used to eliminate multiple reflections. Detailed
discussions regarding TRL and time domain gating for the free-space setup are given
by Ghodgaonkar et al. (1989, 1990).
4 3 Sample Preparation
All the samples under study were made by embedding metal springs into a
RTV silicone matrix. The silicone is Dow Coming 3110 RTV Silicone (Dow Coming
Corporation, Midland, MI). The matrix’s complex dielectric constant is er = 2.8 +
i0.04. The metal springs are copper coated and are all right-handed. Table 4.1 lists
the physical dimensions of each spring used in the study.
Eighteen samples were made, each with different concentrations, spring sizes
or a mix of different sizes. Every sample had an area cross section of
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61
Table 4.1: Spring Dimensions
SPRINGS
DIA.
(mm)
PITCH TURNS
(mm)
625
1.32
0.483
3
742
1.98
0.483
3
743
1.98
0.483
5
941
2.24
0.483
3
942
2.24
0.787
3
990
2.24
0.787
2
991
2.49
0.787
2
993
2.49
0.787
3
992
2.74
0.787
2
15.24 cm x 15.24 cm (6 " x 6 "). The thickness of each sample was targeted to be 3 mm.
However, imperfect sample preparation gave varying thicknesses (this is discussed in
later chapters). All the samples prepared had a 1% or 2% by metal volume spring
concentration. The inclusions occupied the entire sample in order to intercept low
frequency incident waves adequately.
The samples were prepared by first mixing a catalyst to liquid RTV. The
silicone/catalyst matrix was poured into a mold, and inclusions dispersed randomly
throughout the mix. A metal plate was placed on the composite and pressure applied
until the sample was fully cured.
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62
4.4 Experimental Procedures
Once the samples were fabricated, free-space measurements were taken from
8.2 GHz to 40 GHz. Measurements are swept through four frequency ranges: 8.2-12.4
GHz; 12.4-18 GHz, 18-26.5 GHz, and 26.5-40 GHz. Chiral sample measurements and
procedures have been done extensively by Ro (1991). Following his procedures, chiral
and non-chiral (plain RTV) samples were measured. Several S~parameter
measurements were taken to characterize each sample. Because of non-uniformities
in spring dispersion, each S-parameter was measured for four sample orientations.
For example, the co-polarized reflection and transmission were measured for a
sample in one position. The sample was rotated 90 degrees and the same
measurements were taken. This was done two more times. The S-parameters
measured for each position were then averaged.
Each time a S-parameter was measured, time domain gating was implemented
to remove post-calibration errors due to residual mismatches between the antennas
and sample. This was achieved by taking the inverse Fourier transform of the
frequency-domain data to give a time-domain response. Gating is applied over the
time-domain response including the main reflection (transmission) and multiple
peaks. A Fourier transform is performed on the gated time-domain response yielding
the reflection or transmission frequency response of the sample. This manipulation
is performed with the HP 8510B VNA.
In a chiral sample, a wave’s plane of polarization rotates as it transverses
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63
through the medium. Therefore, in order to characterize a chiral medium fully, three
S-parameters must be measured. When the transmitting and receiving antenna are
co-polarized the Sllc0 and S21co coefficients are measured. To find the cross-polarized
transmission component, S2Wis measured. S21fl is measured by rotating the receiving
antenna by 9 (degrees). The antenna rotation for this study was 15 and -15 degrees
for right and left-handed springs, respectively.
To find values of kL’ and kR’ of a sample Ro (1991) has devised an algorithm
which finds a unique wavespeed and time propagation for each frequency. From
Chapter 2, it was pointed out that solving for kL and kR involved an infinite number
of values due to a complex logarithm. This problem is solved first by measuring the
average time delay through the sample under test using the time domain feature on
the network analyzer. Then taking the thickness of the sample, the average wave
veloity, v = d/t is found. Using measured reflection and transmission data, n=0,l,2..
roots are found for the wavespeed to / k’ (Chapter 2). This average velocity is
compared to several roots calculated for the wavespeed. The correct value for n is
chosen so that it has the closest value to the measured wavespeed. Once the
wavespeed is known, k is found and consequently kL' and kR .
The samples used in this investigation are thin, flexible panels. Consequently,
when they are placed in the free-space sample holder, they tend to bulge giving
erroneous magnitude and phase information. Lack of sample rigidity is a major
problem in attaining accurate complex dielectric and magnetic values. Ghodgaonkar
et al. (1990) had used two quartz plates that sandwiched a flexible sample. By
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64
knowing the plates’ thickness and dielectric properties, an inversion program was
used to give the sample’s permittivity and permeability. The inversion algorithm used
three ABCD matrixes for both the quartz plates and the investigated sample. A
matrix inversion was performed to provide the S-parameters of the flexible sample.
Thus, the material properties could be found. However, the matrix inversion still
incorporates some computational error due to matrix multiplication. Also, the
resulting sample properties can be thrown off if there are thickness variations in the
quartz plates. This is a problem especially at higher frequencies.
In this study no quartz plates were used. Instead, specialized sample holders
were constructed to impart ridged properties to a flexible sample. The normal sample
holder made of plexiglass has an opening which is 15.24 cm x 15.24 cm (Figure. 4.2a).
Another plexiglass holder which has a 7.62 cm x 7.62 cm opening was made and used
to support the flexible samples (Fig. 4.2b). The holder allows enough energy to probe
the sample, and at the same time minimize sample sagging or bulging. Ro (1991) and
Ghodgaonkar et al. (1990) have shown using the spot focusing horn antennas, that
diffraction effects at the edges of a sample are negligible if the minimum transverse
dimension of the sample is greater than three times the wavelength at the lowest
frequency. Therefore, the holder with the smaller opening (large holder surface area)
is adequate from 12.4 GHz (3A0=7.26 cm) to 40 GHz (3A.0=2.25 cm). The opening
is large enough to decrease diffraction problems. From 8.2 GHz to 12.4 GHz the
plexiglass holder can still be used, but at low frequencies, diffraction from the
plexiglass is a problem. To correct for this, foam plates were made with a 7.62 cm x
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65
Plexiglass
Sample
Holder
15.24 cm
15.24 cm
Sample
(a)
7.62 cm
\-
7.62 cm
(b)
7.62 cm Opening
15.24 cm Opening
Plexiglass Holder
X
X
(C)
Figure 4.2: a) Bulging of RTV sample using plexiglass holder with 15.24 cm opening,
b) minimized bulging due to smaller opening, c) small opening foam holder for 8.2
GHz to 12.4 GHz band.
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66
7.62 cm opening (Fig. 4.2c). The foam (ers 1.03) is more transparent to microwaves
than the plexiglass (era 2.59), thus lower diffraction results. The foam plates were
placed in the 15.24 cm x 15.24 cm holder as shown in Figure 4.2.
The complex relative permittivity and permeability found from measured Sparameters of plain, 3.23 mm thick, RTV silicone is shown in Figure 4.3. A
comparison can be made between the measurements using the 7.62 cm x 7.62 cm
opening and the 15.24 cm x 15.24 cm opening. The region between 20 GHz and 35
GHz is where the sample’s half-wavelength, thickness resonance occurs. The
resonance points have been removed, but there are still discrepancies around 20 GHz
and 35 GHz due to the resonance response. It is apparent both e* and p.* improve
dramatically with the smaller holder opening. The imaginary part of the permittivity
approaches 0 as expected for a lossless medium. For a non-magnetic material, the
real and imaginary parts of the permeability improve by approaching
1
and
0,
respectively. This is evident at the higher frequencies where bulging is significant for
smaller wavelengths (larger phase variations).
To verify the calibration of the system using the smaller opening, a rigid slab
of Teflon (3.33 mm thick) was measured from 8.2 GHz to 40 GHz. The complex
dielectric and magnetic properties are shown in Figure 4.4. Accuracy of the dielectric
constant is better than ± 0.5% at 10 GHz compared to Ghodgaonkar el al. (1990)
who had er’=2.0. According to Von Hippel (1954), the dielectric constant for Teflon
at 10 GHz should be er =2.08 + i7.7xl04, using slotted line methods (more accurate).
There is a ±4% difference in e ’ in this study compared to Von Hippel’s
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67
3
Real Ep.
7.62 cm Opening
RTV
2
Real Ep.
15.24 cm Opening
RTV
1
Imag. Ep.
7.62 cm Opening
RTV
ptS 0
Imag. Ep.
15.24 cm
Opening
RTV
5
10
15
20
25
30
35
40
35
40
F r e q u e n c y (G H z)
1.25
Real MU
15.24 cm Opening
RTV
Real MU
0.75
CO
a->
7.62 cm
Opening
RTV
B
CL>
CL,
er
C3
&o
s
0.5
CO
Imag. MU
CO
0.25
15.24 cm Opening
-
RTV
Imag. MU
7.62 cm Opening
RTV
-
0.25
5
10
15
20
25
30
F r e q u e n c y (G H z)
Figure 43: Permittivity and permeability of 3.33 mm RTV sample using a sample
holder with a 15.24 cm and a holder with a 7.62 cm opening. The half-wave
resonance values between 20 GHz and 35 GHz are removed.
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68
3
-------------------------------------------------------------------------------------------------Real Ep.
7.62 cm Opening
Teflon
Real/Imaginary Permittivity
/
1 - ............................................................................................................
Imag. Ep
7.62 cm Opening
/
0
- 1
^
* — i—
5
i— i—
i—
i—
i— i—
10
Teflon
_
i—
i— i—
i—
15
. ..
i—
i— i—
r * —1—
20
1—
1— *— i—
>— 1—
1—
25
-|—i—i— 1i—— ri---1
—r~ i—
•—
1— i— f * - '—
30
35
1—
1— '—
40
F r e q u e n c y (G H z)
1.25
Real MU
7.62 cm Opening
Real/Imaginary Permeability
1
0.75
0.5
Imag. MU
0.25
7.62 cm O pening
Teflon
0
l—I—I—l—I—i—l—I—l—1—i—i—I—I—1—l—I—l—i—|—I—i—l—i—I—l—•—l—i—I—I—i—l—r
-0 .2 5
5
10
15
20
25
30
35
40
F r e q u e n c y (GH z)
Figure 4.4: Permittivity and permeability of Teflon using holder with 7.62 c m
opening.
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69
measurement. For low loss dielectric materials such as Teflon, the loss tangent’s
maximum error is approximately ±0.06. Since Teflon’s actual loss tangent is less than
±0.06, the dielectric loss cannot be measured accurately on the free-space system.
Ideally the permeability should be /ir = 1 + iO for Teflon. From Figure 4.4, the
agreement is good. However, there is some error from 34 GHz and higher due to the
thickness resonance of the slab.
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70
CHAPTERS
SAMPLE THICKNESS EFFECTS ON
HELIX EXTINCTION CROSS SECTION CALCULATIONS
5.1 Introduction
A homogeneous material slab can be characterized in free-space by
transmitting a linearly polarized, uniform plane wave onto the material’s surface. The
complex reflection and transmission coefficients Sn and S2X of the sample relate the
amount of power reflected, transmitted and stored within the sample. From these
coefficients, the material properties such as the complex propagation constant can be
found. Sn occurs at the front of the sample and consists of the initial reflection
between the first air/sample interface, and the secondary reflections caused by the
second interface (sample to air). Refer to Chapter 2 for illustration. For a material
of finite thickness d, Sn can be expressed as:
( 4 - u,)
( 4 + Hi)
(5.1)
where
;
k= k ' + i k "
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71
r 2 is the reflection from the sample’s second interface, t]2 is the sample’s intrinsic
impedance, d is the thickness of the sample, and rql is the impedance of free-space.
The variable k is the complex propagation constant of the medium, which contains
the attenuation and phase (k ", k’) information of the sample. If the material is semi­
infinite, there is very little energy transmitted back to the front of the sample
(depending on k" and d) . Consequently, T2 grows smaller and (5.1) can simplify to:
where Tj is the initial reflection off the front interface of the sample (air/material).
52i depends on the amount of power attenuated in the sample, the reflection
r2, and the total reflection (5n). It can be written as:
(i + r 2) ( i + 5u) e ik d
'2
If a wave transversing inside the sample greatly attenuates due to the thickness or
loss of the sample, there will be veiy little energy reflected back to the front of the
sample (depending on k" and d). Therefore, T2 in the bottom of (5.3) will approach
zero faster depending on the -i2kd attenuation in the sample. Using equation (5.2),
(5.3) will reduce to:
S , = (i* r2) (i * s „ ) = (i * r2) (i + r, ) e i u - r 2 j
;
(5 .4)
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72
7\ and T2 are the transmission coefficients at the sample’s first and second interface,
respectively. The result of (5.4) is similar to the first order result of (2.26) in Chapter
2. If r 2 and T, are much less than 1, and attenuation and sample thickness is large,
then (5.4) can simplify to:
= 1 e ikd
(5.5)
Equation (5.5) states that a wave with unit amplitude impinging on a sample, with
negligible interface and internal reflections, will yield a final transmission affected by
the
complex propagation constant and thickness of the sample. The results of
equations (5.2) and (5.5) apply to samples that have high loss, large thickness, or
K
wr\fV
v t u> • P m i o f i n t i c
(^
11 \
n /v1t ( ^
uo u
y «* «i ’ U
m
♦/>
n nu ov ir u
c ou m
c a
l i oi ut nt i nu r^t u
f i nmi f w
A
t v lt lui t m
u in
| / li o
vu
1 r \c c u
on
u rul
prominent interface reflections.
5.2 Calculation of the Effective Total Extinction Cross-Section
The reduced coherent intensity for line of site propagation in a non-chiral
medium containing a given concentration of scattering inclusions p is (M aet al. 1990):
7 = Ioe - 2K"d = Io e * ° ' d
and
(5.6)
-j- = {S2lf = l e - 2K"d = l e - pa-d
O
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where K" is the imaginary part of the effective propagation constant (EPC), K, a, is
the effective total extinction cross-section of a single particle, and d is the thickness
of the medium. K should not be confused with the propagation constant of a
homogeneous medium,T'. Also, from Chapter 3’s introduction, it was discussed that
an effective extinction cross section is used to describe a chiral particle (helix). This
effective cross section is then used to describe the intensity attenuation through a
chiral medium. Therefore, the cross section , a„ used to describe helical extinction
in this study will also be known as the effective extinction cross section. Note that
equation (5.6) is similar to (5.5) in that reflections are discounted, and intensity decay
is due to loss and thickness of the sample. From (5.6), it is deduced that:
9 KT" = rn n
■
-t’
rr
-f
---nr" ’/ rn
=9
which indicates that twice the imaginary part of the effective propagation constant,
is equal to the product of the concentration of scatters and total cross-section.
If the samples are thin and interface reflections are prevalent, the Sn and the
S2i measurement must be taken to find the complex EPC. The imaginary part of the
effective propagation constant is then used to find the total cross-section of a
scatterer. The reflected and transmitted left and right-handed circularly polarized
fields for a single slab chiral material can be given as (see Chapter 2):
T(1 - e**L*k*)d)
(5.8)
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74
J _ p2 g'^L **R>d
(5.9)
(5.10)
where T is the interface reflection coefficient, kR and kL are the right and left-handed
propagation constants respectively, and d is the thickness of the slab. Experimentally,
Sn, S2lL, and S2m can be found from the co and cross polarized antenna
measurements. Through inverse manipulation of (5.8), (5.9), and (5.10), kL and kR
can be found. Consequently, the effective propagation constants can be found from:
Substituting the imaginary part K' of (5.11) into (5.7) will provide the effective
extinction cross section. The total transmitted intensity of the chiral medium can also
be found from (5.9) and (5.10) to give:
o
(5.12)
It was pointed out that equation (5.5) is mostly effective if the medium under
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75
consideration is semi-infinite. That is, the equation does not consider internal or
interface reflections of the media (air/sample-sample/air interfaces). Here, the
imaginary EPC and the effective extinction cross section can be found by substituting
the single transmission in (5.12), into (5.6) to yield:
pd
pd
(5, 3)
v
,
Equation (5.13) will yield predictable results for the total cross-section of a
particle if interface reflections of the sample under study are minimized or accounted
for. A thicker sample will seem like a semi-infinite medium by reducing internal
reflections, thus minimizing the total reflection. If minimum reflection is achieved
from a thick sample, ?nd the intensity decreases due to attenuation, then (5.12) and
(5.13) will be sufficient to approximate the total cross-section. Otherwise the
imaginary EPC and cross section, of a material whose reflections are prominent, must
be found first using equations (5.8) - (5.11), and (5.7).
A characteristic of the thin chiral samples investigated is that the rotation of
the incident wave is veiy small. From equation 2.38, it is shown that the rotation of
a linear polarized wave depends on the LCP and RCP wave propagation constants,
and the thickness of the chiral medium. If the thickness of the sample is very thin,
rotation is very little. A small rotation indicates that most of the intensity received
is co-polarized. Typical rotations of the samples in this study are shown in Figure 5.1.
The graph of the figure is the rotaion for the sample containing 2% by metal volume
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8 ----2% 742
-1 0
5
10
15
20
25
30
35
40
35
40
F r e q u e n c y (GHz)
2% 993
-8
-
-10
5
10
15
20
25
30
F r e q u e n c y (GHz)
Figure 5.1: Rotation angles for the 2% 742 and 993 chiral composites samples.
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77
of the 742 helix. The bottom graph is the rotation for the sample containing 2% by
metal of the 993 helix. A 3.23 mm thick RTV sample (no chiral inclusions) exhibited
a rotation angle equal to 0 degrees (± 0.2). This small rotation error indicates
rotation accuracy in this measurement system from 8.2 GHz to 40 GHz. Both the
samples reach a maximum of 8 degrees rotation. The 742 sample’s maximum occurs
at a higher frequency than the 993 sample. If the chiral samples were thicker, the
cross-polarized component would increase while the co-polarized component would
decrease. Therefore, thickness induced rotation is inconsequential for the thin
composites in this study and justifies the use of the simple scalar transport model.
The following sections will show the cross-section results of thick and thin
samples. Both the approximate ((5.12) & (5.13)) and more rigorous ((5.7)-(5.11))
solutions to K" will be applied to the samples.
5.3 Transmission and Extinction Characteristics of Thin and Thick Samples
A thin sample examined contains 2% (by metal volume) concentration of the 742
sized spring, and embedded in a Dow Corning RTV rubber matrix (et * 2.8 + 0.04).
The 3 thick samples tested are a right, left, and a mix of right and left, 2%
concentration of the 625 spring. The host medium of the thick samples is Eccogel
1365-90 produced by Emerson and Cuming (er a 2.7 + i0.06). Table 5.1 lists the
spring and spring sample characteristics. The thick samples analyzed were the same
as those used in Varadan et al. (1994).
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78
Table 5.1: Thin and Thick Sample Characteristics
TURNS
SPRING
CONC.
p(sps/cm3)
THICK
(mm)
(mm)
#
1 TURN
L
(mm)
Rt. 742
2%
1.98
0.483
3
6.239
54.64
3.23
Rt. 625
1.6%
1.32
0.483
3
4.175
70.5
11.81
Lt. 625
1.6%
11
It
11
II
70.4
11.84
Mx. 625
1.6%
tf
II
It
If
70.5
35.25 L/R
11.94
SAMPLES
DIAMETER PITCH
(mm)
5.3.1 Thin Sample Results (742)
Figure 5.2 (top graph) shows two power transmission curves for the 742 spring
sample. The dark solid curve is the experimental power received using the lc
measurement (5.12). The thin curve is the power received when the imaginaiy part
of (5.11) was used in (5.6). The darker trace includes the attenuation and the
interface reflections of the sample. The thin curve shows the intensity transmitted
when the interface reflections are not contributing factors. Clearly the two curves
converge at the higher frequencies (from 23 GHz to 40 GHz). Figure 5.2 (bottom
graph) shows a plot of the reflection and transmission of the 742 spring sample. It
can be seen that the reflection is around -4 dB at 8.2 GHz and decreases below -6
dB at 23 GHz. This is the point where the two curves of Figure 5.2 converge. As the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2% 742
No Interface
-2
Reflections
- •
ca
zn
-4
CO
2% 742
With Interface
-6
-
Reflections
1=3
ca
et:
-8
-10
5
10
15
20
25
35
30
40
Frequency (GHz)
2% 742
Transmission
With Interface Reflections
co
-4 -
-10
-
-12
-
_i4
...
2% 742
"a3
cts
Reflection .
-1 6 -1 8 -20
5
10
15
20
25
30
35
40
Frequency (GHz)
Figure 5.2: top: Transmitted intensity of 2% 742 sample with interface reflections
using Eqs. (5.12) & (5.13), and without interface reflections using Eqs. (5.7)-(5.11).
bottom: Reflection and transmission of 2% 742 sample.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
80
losses increase at the higher frequencies, the internal reflections reduce. This
reduction manifests itself in the convergence of the thick and thin curves.
5.3.2 Thick Sample Results (625)
Using the same procedure described above for the 742 sample, 3 thick samples
were analyzed. Figure 5.3 (top graph) shows two traces of the transmitted intensity
of the left-handed sample (L-625). The plots were derived in the same way as in
Figure 5.2. The thin line is the transmission using K ' ((5.11) & (5.6)), while the dark,
solid line is the experimental transmission with interface reflections included (Eq.
(5.12)). It is seen that the curves show good agreement, with minor fluctuations at
23 GHz and 31 GHz. Figure 5.3 (bottom graph) shows the reflection (Su)2, and
transmission Ic of the sample. Minimum reflection is near -15 dB, while the maximum
is about -6.2 dB. The low dB values indicate the interface reflections did not play a
major role in this thicker sample. Notice at approximately 17 GHz, the non-reflection
transmission crosses over the other plot. This correlates to the transmission curve, in
the bottom graph, crossing over the reflection at about 17 GHz. At this frequency,
scattering and absorption losses have increased, keeping internal reflections low. This
is why the sample’s overall reflection is fairly constant.
The top frame of Figure 5.4 shows the transmission curves for the
right-handed sample (R-625). The traces are almost identical suggesting that the
interface reflections were minimal. This is verified by the bottom graph of Figure 5.4,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-2
-
-4
1.6% L-625
W ith Interface
Reflection
-1 2
-
1.6% L-625
- 1 6 -■
No Interface
Reflections
-1 8 -
-20
5
10
15
20
25
30
35
40
35
40
F r e q u e n c y (G H z)
-4 1.6% L-625
-6 -8
Reflection
-
jS -10 -12
-
1.6% L-625
Transmission
-1 6 -
With Interface Reflections
-1 8 -
-2 0
5
10
15
20
25
F r e q u e n c y (G H z)
30
Figure 53: tog: Transmitted intensity of 1.6% L-625 sample with interface reflections
using Eqs. (5.12) & (5.13), and without interface reflections using Eqs. (5.7)-(5.11).
bottom: Reflection and transmission of 1.6% L-625 sample.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-2
-4
1.6% R-625
-8
”
With Interface Reflections
-1 0
-1 2
2 -14
-1 6
1.6% R-625
No Interface Reflections
-1 8
-2 0
5
10
15
20
25
F r e q u e n c y (G H z)
-2
35
40
30
35
40
1.6% R-625
Transmission
-4
pea
T 3
30
With Interface Reflections
-6
-8
o
-12
1.6% R-625
Reflection
-1 8
-2 0
5
10
15
20
25
F r e q u e n c y (G H z)
Figure 5.4: tog: Transmitted intensity of 1.6% R-625 sample with interface reflections
using Eqs. (5.12) & (5.13), and without interface reflections using Eqs. (5.7)-(5.11).
bottom: Reflection and transmission of 1.6% R-625 sample.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
83
which shows the reflection of the right-handed sample to be a maximum of -7 dB.
Again notice the increased transmission loss where the cross over point for both
graphs is 21 GHz.
Figure 5.5 (top) shows the transmission curves for the mixed sample. The
results are similar to the two previous samples. Both traces show the attenuated
intensity governed primarily by the loss and thickness of the sample. Figure 5.5
(bottom) shows the reflection of the sample to be smaller than -6 dB from 8.2 GHz
to 40 GHz. The cross over points for this sample occurred at approximately 18 GHz.
The reflection, except for minimal variation, remains constant.
As mentioned earlier, The extinction cross section was calculated two ways:
1) using just the Ic transmission measurement in (5.13), and 2) using both reflection
and transmission measurements to find (5.11). Plots using both calculations were
made for the thick samples listed above.
Figure 5.6 (top graph) is the extinction plots for the left-handed sample. The
thin trace is the cross-section calculated from the sample’s reflection and transmission
measurements ((5.11) & (5.7)). The dark trace was calculated from the sample’s
transmission only ((5.12) & (5.13)). Both plots are close. The extinction calculated
from the reflection and transmission varies from the other curve at about 23 GHz
and 31 GHz. This variation is due to a fluctuating K".
Figure 5.6 (bottom) shows the extinction cross-section plots for the
right-handed sample. The approximate and exact calculation of the cross-section
seems to be very close. There is a small deviation of the two traces at 23 GHz and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-2
1.6% M-625
' n
£
—8
With Interface Reflections
-------
- 1 0 .....
~
OJ>
^
-1 2
...
2 -14
1.6% M-625
- 1 6 -■
No Interface Reflections
-1 8
-2 0
5
10
15
20
25
Frequency (GHz)
30
35
40
35
40
1.6% M-625
-6 -
Reflection
-8 2 -1 0 -
E—i
1.6% M-625
T milsmiBaiOri
With Interface Reflections
-1 8 -
-20
5
10
15
20
25
Frequency (GHz)
30
Figure 5.5: top: Transmitted intensity of 1.6% M-625 sample with interface
reflections using Eqs. (5.12) & (5.13), and without interface reflections using Eqs.
(5.7)-(5.11). bottom: Reflection and transmission of 1.6% M-625 sample.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.05
L-625
0.045
Reflection & Transmission
M easurement
S 0.035
C_>
^
0.025
L-625
Transmission
<3 0.02 -
Measurement Only
0.015 -•
0.005 -
5
10
15
20
25
F r e q u e n c y (G H z)
30
35
40
35
40
0.05
0.045 75
R-625
Reflection & Transmission
0.04 -
Measurement
s
0.035 -
C->
^
81
o
3
0.025 0.02
R-625
Transmission
-
Measurement Only
o
^
0.015 -
0.005 -
5
10
15
25
20
F r e q u e n c y (G H z)
30
Figure 5.6: tog: Extinction of L-625 sample from reflection and transmission
measurements using Eqs. (5.11) & (5.7), and from transmission using Eqs. (5.12) &
(5.13). bottom: Same as top frame but with R-625 sample.
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86
31 G H z .
Figure 5.7 (top) shows the plots for the mixed sample. There is somewhat
good agreement across the frequency range. However, there are slight variations at
about 23 GHz and 31 GHz.
Finally, Figure 5.7 (bottom) is a plot of the extinction cross-sections of the
right, left, and mixed sample. The traces were calculated using both reflection and
transmission measurements ((5.7)-(5.11)). The cross-sections seem very close until 22
GHz. After 22 GHz, the cross-sections of each sample deviate. It seems that all three
samples oscillate at about 23 GHz and 31 GHz. Again, this is due to the imaginary
part of the EPC fluctuating.
From the previous analysis, it is apparent the extinctions for the thicker
samples can be approximated by using just a single transmission measurement.
Although it is clear from Figures 5.6 and 5.7 that there are still variations between
the approximate and exact solutions for the extinction. The transmission results for
the thinner samples clearly indicate that the interface reflections must be taken into
account by using reflection and transmission measurements. This means reflection
and transmission measurements will be used to calculate all extinction cross sections
in this study using equations (5.11) and (5.7).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.05
0.045 -
M-625
Reflection & Transmission
Measurement
g 0.035 -
^
0.025 -
M-625
Transmission
=-3 0.02 e
Measurement Only
0.015 -
0.005 -
5
10
15
20
25
Frequency (GHz)
30
35
40
0.05
0.045 R -625
7U5 1 0.04 -
L-625
S 0.035 -
cj
^
0.025 -
=73 0.02 0.015 -
From
Reflection & Transmission
Measurement
Eqs. (11) & (7)
0.005 -
5
10
15
20
25
30
35
40
F r e q u e n c y (G H z)
Figure 5.7: tog: Extinction of M-625 sample from reflection and transmission
measurements using Eqs. (5.11) & (5.7), and from transmission using Eqs. (5.12) &
(5.13). bottom: Extinction of L-625, R-625, and M-625 using Eqs. (5.11) & (5.7).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
88
CHAPTER 6
SPRING CONCENTRATION EFFECTS
ON MEASURED EXTINCTION CHARACTERISTICS
6.1 Introduction
This chapter will examine extinction cross-sections of two different spring sizes in 1%
and 2% metal volume concentrated samples. Table 6.1 shows the spring and spring sample’s
characteristics.
Table 6.1: 742 and 993 Sample Characteristics
1 TURN
L
(mm)
SPRING
CONC.
p(sps/cm3)
THICK
3
6.239
30.01
2.94
0.483
3
6.239
54.64
3.23
2.49
0.787
3
7.862
23.31
3.00
2.49
0.787
3
7.862
44.80
3.12
DIAM.
PITCH
(mm)
(mm)
742 (1%)
1.98
0.483
742 (2%)
1.98
993 (1%)
993 (2%)
SAMPLES
TURNS
(mm)
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89
6.1.1. Coherent Intensity
The reduced coherent intensity for line of sight propagation in a medium
containing a given concentration of scattering inclusions p is (Ma et al. 1990):
/ = I0 exp (-2 K 'd ) = Ia exp (-p a, d)
(6.1)
where K ' is the imaginary part of the effective propagation constant K, a, is the
extinction cross-section term (from Chapter 3), and d is the thickness of the medium.
From (6.1), it is deduced that
2K" = p a, ; a, = 2K" / p
(6.2)
which indicates that twice the imaginaiy part of the effective propagation constant,
is equal to the product of the concentration of scatters and total cross-section.
The product of the arguments of the exponential in equation (6.1) is known
as the optical depth (dimensionless). In scattering theory, if the optical depth is less
than 1, then coherent transmittance is dominant.
The imaginary part of the effective propagation constant is used to find the
total cross-section of a scatterer. Experimentally, the effective propagation constant
can be found in a chiral medium by considering the left and right-handed fields
within the sample. The reflected and transmitted left and right-handed circularly
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
90
polarized fields in a single slab chiral material can be given as:
f 1(1
-
(6.3)
| _ jp2
^R)d
(6.4)
(6.5)
where T is the interface reflection coefficient, kR and kL are the right and left
-handed propagation constants respectively, and d is the thickness of the slab.
Experimentally, Su, S2lL, and S2lR can be found from the co and x polarized antenna
measurements. Through inverse manipulation of (6.3), (6.4), and (6.5), kL and kRcan
be found. Consequently, the effective propagation constant K can be found from:
( 6 .6 )
Substituting the imaginaiy part (X") of (6.6) into (6.2) will provide the extinction.
The next few sections will focus on the 742 and 993 springs. Physical
dimensions of the springs are given in Table 6.1. A 1% and 2% (volume of metal)
sample was made for each spring size (four total). Parameters such as extinction
(total cross-section) and optical depth have been calculated for all the samples based
on applying experimental data to equation’s (6.1) and (6.2).
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91
6.2 Spring 742, 1% and 2% Volume Concentration Results
Two spring samples were made using the 742 spring size. The first sample
has a 1% by volume spring concentration, while the second sample has a 2% by
volume spring concentration. Both samples were measured as described in the
experimental setup (Chapter 4).
Reflection and transmission (no metal back) for the 1% and 2% 742 samples
are shown in Figure 6.1. The S21cotransmission is the measurement taken while both
antennas are co-polarized. The S21co + S 2lcma = S21 coefficient is the transmission
when the rotated field’s cross polarization component is taken into account. It is seen
for the 1% and 2% samples that both transmission measurements are nearly
coincident except from 12.4 GHz to 18 GHz. At this point the rotation is the greatest
for both samples. A wave’s polarization rotation through a chiral medium will depend
on the thickness of the sample. Since the 1% and 2% samples are 2.94 mm and 3.23
mm thick, respectively, the amount of rotation is very small.
The 1% 742 sample has more power transmitted due to a smaller spring
concentration. This sample has a lower reflection and higher transmission from 8.2
GHz to 17 GHz. In this frequency range, the 2% sample’s reflection is slightly larger
and its transmission is smaller. However, above 17 GHz, the 2% sample’s reflection
and transmission drops, due to attenuation of internal reflecting waves. The 2%
sample has a reflection close to that of the 1% sample, but its transmission is lower
as seen from the figure.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
S21 Co + S21 Cross
-2
co
T3
-■
1% 742 Trans.
-4
S21 Co
-6
- •
CO
s
P - 1 0 -•
CO
-1 2
- •
1% 742 Ref.
-1 6 - 1 8 -■
-2 0
5
10
15
20
25
30
35
40
F r e q u e n c y (G H z)
o
S21 Co + S21 Cross
-2
-4
2% 742 Trans.
-6
-8
S21 Co
P -10
-1 8
-1 8
-2 0
5
10
15
25
20
Frequency (GHz)
30
35
40
Figure 6.1: Reflection and transmission of 1% and 2% 742 samples. Both the co
(S21c0) and co + cross (S21) polarized transmissions are shown.
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93
6.2.1 Extinction and Optical Depth (742)
The extinction for the 742 spring was calculated from experimental data using
equation (6.2) and the 742, 1% and 2% samples. Figure 6.2’s top graph shows both
samples’ extinction traces from 8.2 to 40 GHz. Aside from slight variations, both
curves seem to follow each other until 26 GHz. At 27 GHz the curves start to be
dissimilar. At 30 GHz, both traces diverge with the 1% extinction curve increasing
more than the 2% extinction curve. From equation (6.2), the extinction was derived
from the effective propagation constant (EPC). The EPC is responsible for the
attenuation per centimeter in a sample. It is apparent the 2% sample’s EPC
decreases, compared to the 1% sample’s EPC. In other words, the quotient 2K"/p
remains linear for both samples up to 26 GHz.
From equation (6.1), a decrease in the received intensity relates to an increase
in the EPC, which is related to an increase in the extinction cross-section. Conversely,
an increase in received intensity relates to smaller extinction. From Figure 6.2, the
2% sample’s extinction falls off due to an increase in received intensity. The receiving
antenna seems to collect more intensity in the forward direction. A reason for this
would be more incoherent scattering at the higher frequencies. This incoherent
scattering would contribute to the coherent field at the receiving antenna.
In the experimental set-up section, it was pointed out that different antennas
and transitions had to be used to cover the 8.2 to 40 GHz range. In Figure 6.2, there
are discontinuities in the data at 12.4, 18, and 26.5 GHz. These jumps represent
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.14
94
1% 742
0.12
-
0 . 0 6
-
ca
co
£ 0.06 -
2% 742
c_>
^
0.04 -
0.02
-
0
T—r
5
10
T
T—r
15
t—i—i—i—|—I—r t—i—r
t—r
20
25
30
35
t—r
40
35
40
t—r t—I—r
Frequency (GHz)
2% 742
<a
1% 742
5
10
15
20
25
30
Frequency (GHz)
Figure 6.2: The extinction cross section (top) and optical depth (bottom) vs.
frequency for the 1% and 2% 742 samples.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
95
transitions from one frequency band to another.
The optical depth
t
,
is obtained by taking the product of equation (6.1)’s
exponential argument. Both the 1% and 2% 742 samples’ results are shown in the
bottom graph of Figure 6.2. The 2% trace reaches 1 at 23.5 GHz, while the 1%
trace reaches 1 at 37 GHz. Scattering theory usually dictates for r >1, that multiple
scattering may be prevalent. If multiple scattering exists, then equation (6.1) is not
valid to describe scatter/wave interaction. However, from the extinction curves, both
the 1% and 2% sample’s cross sections are in good agreement up to 26 GHz.
Therefore, the proportionality in Eq. (6.1) holds (in this case) for r < 1.3.
6.2.2 Extinction of Sphere (742)
The extinction of a sphere with two different diameters was calculated. An
extinction cross section was calculated using a sphere radius equaling the radius of
the 742 spring (a = 0.0990 cm). It should be noted that a sphere with this radius will
have a greater metal volume than one 742 spring. The radius was plugged into a
program (Appendix I) which computed the extinction cross section of a perfectly
conducting sphere. This same radius was used to compute the circumference of the
spring and sphere, measured in wavelengths (27ra IX = ka). The extinction as a
function of ka and frequency is shown in Figure 6.3.
A second extinction cross section was calculated using a smaller radius than
the 742 spring radius (asma„ = 0.0433 cm). Metal volume of this small diameter sphere
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced
with permission
0.14
1% 742
of the copyright owner.
0.12
s
0.08
GO
^
0.06
Further reproduction
0.04
Sphere Radius
= Spring Radius
0.02
Sphere Radius < Spring Radius
10
15
20
25
30
35
40
prohibited without p e r m is s io n .
F r e q u e n c y (GHz)
0.2
0.4
0.6
O.fl
1.2
1.4
ka
Figure 63: Extinction cross section of, 1% and 2% 742 spring, large sphere with radius equaling 742,
and small radius sphere. The ka axis pertains to the 742 spring, and large radius sphere only.
VO
Ov
97
equals the metal volume of one 742 spring. The extinction of the small diameter
sphere as a function of frequency is plotted in Figure 6.3.
It should be pointed out that the ka axis in Figure 6.3 pertains to the sphere
and spring radius of a = 0.0990 cm, and not the smaller sphere’s radius. The
extinction cross section of the 742 spring for the 1% and 2% sample, as a function
of ka and frequency is shown in Figure 6.3.
As seen from the figure, the large radius sphere’s cross section is less than
the cross section of the 742 spring. The sphere’s extinction is due entirely to
scattering since it is perfectly conducting. The 742’s extinction is higher than the
sphere’s extinction due to an increase in scattering or absorption, or both.
Several conclusions can be made from Figure 6.3. First, a sphere of an
equivalent spring radius, but greater metal volume will have an extinction close to
that of the spring. A sphere with an equivalent spring metal volume, but smaller
radius will offer a smaller cross section than the spring. Also from Figure 6.3, the 1%
and 2% springs cross section traces begin to diverge before ka = 1, which is
immediately before the sphere’s extinction peaks at ka = 1. The region where the
size of the sphere is small compared with the wavelength (ka < 1) is the Rayleigh
region. Concluding, equations (6.1) and (6.2) are valid for the 742 spring at ka < 1
and r < 1.3 (Figure 6.2).
Figure 6.4 shows the theoretical transmitted intensities for a medium
containing spheres. The intensities were calculated using equation (6.1). The top and
bottom graphs show four cases relating to the 1% and 2% 742 samples respectively.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
98
1=0
A
eu
^
Sphere Rad. = Spring Rad.
ff Spheres = # Springs
-6
B
C/2
Sphere Rad. < Spring Rad.
ff Spheres = ff Springs
1=3
CO
c
-8 - D
-1 0
Sphere Rad. = Spring Rad.
■# Spheres < # Springs
Exp. 1% 742 Spring
-j—j— i—i—>—i—i—r—i— i—|—i—i—«— i—|—i—i—i—r~
—
i—i—j—i—|—i—i—i—i |
10
15
20
25
30
35
40
Frequency (GHz)
-2
-
co
A
Sphere
if Spheres ~ # Springs
S
C/3
B
Sphere Rad. < Spring Rad.
ff Spheres = if Springs
CO
—8
C
Sphere Rad. = Spring Rad.
if Spheres < # Springs
D Exp. 2% 742 Spring
-10
-i—i—r r—i—i—i—i—i—|—i—i—i—>T
10
15
20
-i—j—i—i—i—i—r
25
30
35
40
F r e q u e n c y (G H z)
Figure 6.4: Transmitted intensities for 1% and 2% 142 springs and spheres. Curve
A is transmitted intensity through a theoretical sample with total sample metal
volume of 12.19% (top graph) and 22.18% (bottom graph).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
99
The first case is a medium containing spheres with the same radius and
number of springs in the 1% (2%) 742 sample (marked A). However, the total metal
volume content within the medium is 12.19% (22.18%) due to a larger volume
sphere.
The second case (marked B) is a medium containing spheres of the same
metal volume as one spring, but smaller radius. The number of spheres equals the
number of springs in the 1% (2%) sample. The total metal content equals the metal
content of the 1% (2%) sample.
The third case (marked C) is a medium containing a 1% (2%) metal volume
of spheres. The spheres have a radius equal to the 742 spring. But, the number of
spheres has been reduced to attain a 1% (2%) total metal concentration, as opposed
to 12.19% (22.18%).
Finally the fourth transmitted intensity trace is the experimental 1% (2%) 742
sample. This is marked as D.
Several observations can be made about Figure 6.4. Samples containing
spheres with the same radius and number of springs (curve A), have a transmitted
intensity close to the experimental 742 samples. However the metal content within
the theoretical sample is 12 and 11 times more than the 1% and 2% samples,
respectively. If the radius is kept the same, but the number of scatterers decreased
to 1% (2%) metal volume concentrations, the transmitted intensity increases to less
than 0.25 dB (curve C). If the number of spheres and metal volume is kept the same
as the 1% (2%) sample’s, but decrease the radius, the intensity is close to 0 dB
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
100
(Curve B). The sample is almost transparent. Evidently, the 742 spring will yield
lower transmitted intensities than spheres with equivalent volumes or diameters.
63 Spring 993, 1% and 2% Volume Concentration Results
Two 993 samples were made. One sample contains a 1% by metal volume of
springs while the other contains 2%. The spring diameter is one of the largest in this
study (2.49 mm). Both samples were measured from 8.2 GHz to 40 GHz.
Figure 6.5 shows the reflection and transmission for the 1% and 2% samples.
The S21co and S2lco + S2lcma = S21 transmission curves are nearly coincident
throughout the frequency band. This is due to the thickness’ of the samples, which
are small enough that little rotation takes place. Figure 6.5 also shows that the
reflections from both samples depend on the internal attenuation of the samples. The
lower transmissions and lower reflections point to the fact that energy is being taken
out of the system. Since there are no reflection resonances, but a wideband reflection
reduction, there is high loss in the sample. This loss could be due to scattering or
absorption, or both. This topic will be covered in later chapters.
63.1 Extinction and Optical Depth (993)
Figure 6.6 shows the plotted extinction curves of the 1% and 2% 993 samples.
Both curves were calculated from equation (6.2). The 1% curve has an oscillatory
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-2
S21 Co + S21 Cross
-4
co
1% 993 Trans.
-6
-8 —
S21 Co
-1 0
-12
- -
1% 993 Ref.
-1 6 —
-1 8
-2 0
5
10
15
-2 --
20
25
F requency (GHz)
30
35
40
S21 Co + S21 Cross
-6 -CO
CO
E
S
2% 993 Trans.
-8 --
CO
-1 0
- -
-1 2
- -
S21 Co
2% 993 Ref.
S -1 4 - -1 6 —
-1 8 —
-2 0
5
10
15
20
25
Frequency (GHz)
30
35
40
Figure 6.5: Reflection and transmission of 1% and 2% 993 samples. Both the co
(S21co) and co + cross (S21) polarized transmissions are shown.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.3
102
0.28
0.26
1% 993
E 0-22
" e ” 02
M 0.18
M 0-16
k 0.14
0.12
2% 993
. S 0.08 - 2 5 0.06 - 0.04 - -
0.02
5
10
15
20
25
F r e q u e n c y (G H z)
30
35
40
35
40
2.5
2% 993
c/a
O
Q
C
=
l
o
co
«a?
s
1% 993
0.5
5
10
15
25
20
F r e q u e n c y (G H z)
30
Figure 6.6: The extinction cross section (top) and optical depth (bottom) vs
frequency for the 1% and 2% 993 samples.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
103
trend throughout the frequency range, while the 2% sample is smoother. Aside from
the variance, the 1% trace seems to follow the 2% curve until 25.5 GHz. At this
point the two curves diverge with the 1% curve attaining higher extinction than the
2%. This behavior is similar to the extinction curves of the 742 samples analyzed
earlier. The divergence of the 993 extinction curves occurs at a lower frequency than
the742 samples. As mentioned earlier for the 742 sample’s, more incoherent
scattering occurs at the higher frequencies, which contributes to the coherent field
at the receiving antenna. A larger intensity yields a smaller extinction cross-section.
The optical depth r for the 1% and 2% 993 samples is shown at the bottom
of Figure 6.6. It is evident that r = 1 is reached at a lower frequency for the 993
sample than the 742, due to a larger diameter and cross section. Another conclusion
is that using (6.1) and (6.2) for 2% metal volume samples will give rough, but
predictable results for the extinction cross section of the 993 spring at r < 1.75.
63.2 Extinction of Sphere (993)
The extinction cross sections shown have been experimentally derived. For
comparison, the theoretical extinction cross section of a perfectly conducting sphere,
with the same diameter as the 993 spring was generated. The radius of the 993 spring
and theoretical sphere is 0.1245 cm. The radius is designated as a and is used to
compute the sphere’s cross section, and its circumference measured in wavelengths
(liralX = ka). As with the 742 case mentioned earlier, the metal volume of the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
104
sphere is greater than the metal volume content of one spring. Therefore another
cross section was calculated for a sphere that has the same metal volume content, but
smaller radius (asmaU = 0.0468 cm). Cross sections of the 993 for the 1% and 2%
sample, and the spheres are plotted in Figure 6.7 as a function of ka. Again, ka refers
only to the spring and large diameter sphere.
Since the theoretical sphere is perfectly conducting, the extinction cross
section will be totally due to scattering and not absorption. As seen from Figure 6.7,
the sphere’s total cross section is smaller than the measured total cross section of the
993 spring. The 993’s increased cross section, could be due to larger scattering or
absorption cross sections.
Similar conclusions about the 742 spheres can be made for the 993 spheres.
A larger metal volume sphere with the same radius as the 993 spring will have an
almost equal cross section due to scattering. By reducing the radius of the sphere to
the same metal content as one spring, will yield a much smaller cross section. Also
from the figure the Rayleigh of the spring corresponds to 22 GHz and below. This
region also corresponds to r < 1.75 from Figure 6.6. Therefore, equations (6.1) and
(6.2) can be valid for the 993 spring at ka < 1, and r < 1.75.
Figure 6.8 shows the theoretical transmitted intensities for mediums containing
spheres. The intensities were calculated using equation (6.1). The top and bottom
graphs show four cases relating to the 1% and 2% 993 samples respectively.
The first case is a medium containing spheres with the same radius and
number of springs in the 1% (2%) 993 sample (marked A). However, the total metal
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
lission of the copyright owner.
0.3
1% 993
0.25
0.2
2% 993
Further reproduction
Sphere Radius
= Spring Radius
0.05
Sphere Radius < Spring Radius
10
15
20
25
35
30
40
prohibited without p e r m is s io n .
F r e q u e n c y (GHz)
i -)-------1-------1------ 1-------1-------1-------1-------1-------1-------1-------1—
0.2
0.4
0.6
0.8
1
1.2
— I----------1----------1----------r
1.4
1.6
1.8
ka
Figure 6.7: Extinction cross section of, 1% and 2% 993 spring, large sphere with radius equaling 993,
and small radius sphere. The ka axis pertains to the 993 spring, and large radius sphere only.
oVi
106
Transmitted Intensity (dB)
-2
A
Sphere Rad. = Spring Rad.
#Spheres = # Springs
_ g
_b
Sphere Rad. < Spring Rad.
# Spheres = # Springs
C
Sphere Rad. = Spring Rad.
# Spheres < # Springs
D
Exp. 1% 993 Spring
10
Transmitted Intensity (dB)
5
A
15
20
25
Frequency (GHz)
30
35
40
Sphere Rad. = Spring Rad.
tt Spheres = # Springs
B
Sphere Rad. < Spring Rad.
tt Spheres = it Springs
_c _ Sphere Rad. = Spring Rad.
tt Spheres < it Springs
D
Exp. 2% 993 Spring
-10
5
10
15
25
20
Frequency (GHz)
30
35
40
Figure 6.8: Transmitted intensities for 1% and 2% 993 springs and spheres. Curve
A is transmitted intensity through a theoretical sample with total sample metal
volume of 19.07% (top graph) and 36.66% (bottom graph).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
107
volume content within the medium is 19.07% (36.66%) due to a larger volume
sphere.
The second case (marked B) is a medium containing spheres of the same
metal volume as one spring, but smaller radius. The number of spheres equals the
number of springs in the 1% (2%) sample. The total metal content equals the metal
content of the 1% (2%) sample.
The third case (marked C) is a medium containing a 1% (2%) metal volume
of spheres. The spheres have a radius equal to the 993 spring. But, the number of
spheres has been reduced to attain a 1% (2%) total metal concentration, as opposed
to 19.07% (36.66%). Finally the fourth transmitted intensity trace is the experimental
1% (2%) 993 sample. This is marked as D.
Several conclusions can be made about Figure 6.8. The medium with the same
radius and number of springs as 993 (curve A), has a slightly larger transmitted
intensity than the 993 samples. However, the metal content within the theoretical
sample is 19 and 18 times more than the 1% and 2% samples, respectively. If the
radius is kept the same, but the number of scatterers decreased to 1% (2%) sample
metal volume concentrations, the transmitted intensity increases to less than 0.25 dB
(curve C). If the number of spheres and metal volume is kept the same as the 1%
(2%) sample’s, but decrease the radius, the intensity is close to 0 dB (curve B). The
power passes through the sample with little scattering. Obviously the 993 spring will
yield lower transmitted intensities than spheres with equivalent volumes or diameters.
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108
CHAPTER 7
EFFECT OF SPRING DIMENSIONS ON PROPAGATION CHARACTERISTICS
7.1 Introduction
From the previous chapters, the total cross-section was expressed as:
a, — 2K" / p
(7.1)
where K" is the imaginary part of the effective propagation constant (EPC), and p is
the concentration of springs (springs/cm3). The effective propagation constant came
from an inverse algorithm using the co- and cross-polarization measurements done
on the samples investigated. The reduced coherent intensity in a medium of thickness
d, containing a number of scattering inclusions p is:
/ = /„ exp (-2 K" d) = Ia exp (-p a,d)
(7.2)
The power absorbed in a homogeneous material is usually due to dielectric
losses. The attenuation in a sample is accounted for by the imaginary part of the
dielectric constant. However, a composite with scattering inclusions removes energy
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109
from the system not only by conventional losses, but by scattering as well. Even if
there are no losses, the imaginary part of the dielectric constant can be nonzero. The
power absorption for the samples analyzed may be considered as effective power
absorption due to the uncertain role of scattering. The effective power absorbed was
found by finding the power transmitted from the co- and cross-polarization
measurements:
ic = i/io
= o.5 [(s2lLy + (s2lRf ]
(7.3)
Ic is expressed as the total transmitted power (interface reflections included). The
reflection power, Sn2, is found from the sample’s co-polarization measurements.
Power absorbed is expressed as:
Power Absorbed = 1 - [ (S n)2 + Ic ]
(7.3)
To reiterate, equation (7.3) is valid for homogeneous materials. For the samples
under study, the power removed by the incident beam, could have been due to
scattering as well as absorption.
The spring size and characteristics for the samples tested are listed in Table
7.1. From top to bottom, the spring diameters increase with variations in turns and
pitch. The smallest diameter spring is the 625 (1.32 mm), and the largest spring is
992 (2.74 mm). The following analysis begins by analyzing the total cross section and
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110
Table 7.1: Spring Sample Characteristics
SAMPLE DIA.
PITCH
TURNS
(mm)
(mm)
#
1 TURN
L
(mm)
625
1.32
0.483
3
4.175
82.61
3.20
26.44
742
1.98
0.483
3
6.239
54.64
3.23
17.65
743
1.98
0.483
5
6.239
30.68
3.45
10.58
941
2.24
0.483
3
7.054
49.14
3.18
15.62
942
2.24
0.787
3
7.081
54.44
2.87
15.62
990
2.24
0.787
2
7.081
80.27
2.92
23.44
991
2.49
0.787
2
7.862
60.81
3.45
20.98
993
2.49
0.787
3
7.862
44.80
3.12
13.98
992
2.74
0.787
2
8.644
62.85
3.05
19.17
SPRING THICK
CONC.
p(sps/cm3) (mm)
pd
power absorption of the spring samples for the following cases:
1) Springs with different number of turns, but the same diameter
and pitch.
2) Springs with different pitch, but the same diameter and number
of turns.
3) Springs with different diameters, but the same pitch and number of turns.
7.2 Thickness and Spring Concentration of Samples
The thickness for each sample was originally targeted to be 3 mm, with a
sample volume of 69.68 cm3. But, the sample preparation made it difficult to keep
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ill
tight thickness tolerances. Consequently, Table 7.1 displays the thickness variations
for each sample. The spring concentration p was calculated by dividing the number
of springs within the sample by the final sample volume dimensions. The final volume
dimension was calculated using the final sample thickness (which was not 3 mm
exactly). This meant that a thickness variation changed spring concentration. If one
parameter changed, the other would as well. The product pd, in Table 7.1 was
calculated to yield a non-vaiying number for each sample. That is, if a sample varied
in thickness or spring concentration, pd would remain the same value. In all of the
samples, the thickness variation was too small to greatly affect concentration. The
p d value is mostly a reflection of a sample’s spring concentration.
73 Springs with Different Number of Turns
The following spring sets have different number of turns, but the same
diameter and pitch: 742 & 743, 942 & 990, 991 & 993. The number of turns and
other physical characteristics is shown in Table 7.1.
Figure 7.1 shows the comparison of the total cross sections and power
absorptions of the 742 and 743 set. The cross-sections of the two springs are very
close. However, the Absorption of the 742 spring is greater than the 743, especially
at the higher frequencies. Although there is approximately an equal metal
concentration between the two samples, the 742 sample has a larger spring
concentration (or pd). This higher concentration, and given cross-section inserted into
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112
0.25
0.2
—
Different Num ber o f Tom s
_SC-J
■M
°-15
CJ>
QJ
oo
C
to/2
O
cu
o
0.1
743 (5 Turns)
-•
c£3 0.05
742 (3 T unis)
10
15
20
25
30
35
40
35
40
F r e q u e n c y (G H z)
1
0.8
742 (3 Turns)
Different N um ber o f Turns
o 0.6
0.4
743 (5 Turns)
0.2
0
5
10
15
25
20
F r e q u e n c y (G H z)
30
Figure 7.1: Extinction cross sections and power absorption coefficients for the 742
and 743 springs whose number of turns are 3 and 5, respectively.
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113
equation (7.2) will result in a larger attenuation within the 742 sample. The higher
attenuation will be evident in the larger effective power absorption shown in Figure
7.1. For this spring pair, it seems that a larger spring concentration is more important
than the cross-sections.
Figure 7.2. is the extinction cross-sections and power absorptions of the 942
and 990 set. The cross-sections of the two springs are similar until the upper
frequencies. At 21 GHz, the two curves diverge, and the 942 spring attains a higher
cross-section. Both spring samples have approximately the same metal concentration.
The thickness of each sample is negligible (0.05 mm). Though the 990 spring has
lower cross-section values compared to the 942, the 990 sample has a larger spring
concentration (and larger pd). This higher concentration appears to be the factor that
gives the 990 sample higher power absorption.
The total cross sections and power absorption coefficients are shown in Figure
7.3 of the 991 and 993 springs. This group of springs has a larger diameter than the
last two groups. Both extinction curves are close with intermittent overlapping
throughout the frequency band. The power absorption between the two curves is
close, but clearly the 991 sample dominates over the 993 sample across the frequency
band. The sample thickness of the 991 sample is larger (0.33 mm difference), but its
large pd is mainly due to the spring concentration. Larger power absorption is mainly
due to the high spring concentration.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.25
GO
Different Number o f Tunis
S
CJ>
P
oS
942 (3 Turns)
0.15
&S 0.05
990 (2 Turns)
5
0.8
10
15
20
25
F r e q u e n c y (GH z)
30
35
40
-
990 (2 Turns)
Different Number o f Tlima
°
0.6
-
J S 0 .4 -
0.2
942 (3 Turns)
-
5
10
15
20
25
Frequency (GHz)
30
35
40
Figure 7.2: Extinction cross-sections and power absorption coefficients for the 942
and 990 springs whose number of turns are 3 and 2, respectively.
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115
0.25
S
0.2
......... - .............
Different N um ber o f T am s
ao
993 (3 Tunis)
. 2 0.15 -
CLJ>
>
CCO
so
of=3
991 (2 Turns)
0.05 -
15
20
25
30
Frequency (GHz)
0.8
991 (2 Tunis)
es
OJ
Different Number o f Tunis
993 (3 Turns)
0.4
0.2
5
10
15
25
20
Frequency (GHz)
30
35
40
Figure 73: Extinction cross-sections and power absorption coefficients for the 991
and 993 spring whose number of turns are 2 and 3, respectively.
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116
7.4 Springs with Different Pitch
Figure 7.4 shows the extinction cross-section and power absorption for the 941
and 942 spring set. The 942 spring has a larger pitch compared to the 941. The figure
shows the 941 spring has a greater cross section than the 942 spring, and that there
is a divergence between the two springs from 12 GHz to 40 GHz. Also, pd is the
same for both samples. The power absorption is larger for the 941 as well. This is
due to a larger cross-section that gives higher attenuation within the sample. The
increased attenuation contributes to the over all loss of power in the system.
7.5 Springs with Different Diameters
The following spring sets have the same turns and pitch, but different
diameter: (942 & 993), (992, 991, 990), and (625, 742, 941). Table 7.1 shows the
spring’s diameters and other physical data associated with the samples.
Figure 7.5 shows the cross sections and power absorptions of the 942 and 993
spring set. Both springs have the same pitch (0.787 mm) and number of turns (3).
The 993 spring has a larger diameter and cross-section than the 942. However, from
Table 7.1, the 993 spring has a smaller pd. This would indicate that the spring’s crosssection is primarily responsible for the larger attenuation, hence greater absorption.
The power absorption graph shows increased absorption of 993 over 942 between 14
GHz and 36 GHz. Looking at the cross-sections of the two springs, the 14 GHz to
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
0.25
0.2 -
^
CO
Different Pitch
941 (0.483 mm
Pitch)
. 2 0.15 -
942 (0.787 mm Pitch)
c S 0.05 -
0
5
10
15
20
25
30
35
40
F r e q u e n c y (G H z)
941 (0.483 mm Pitch)
Different Pitch
942 (0.787 iran Pitch)
-o
GL>
m
o
O-,
5
10
15
20
25
F r e q u e n c y (GH z)
30
35
40
Figure 7.4: Extinction cross-sections and power absorption coefficients for the
and 942 springs whose pitch differs.
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118
0.25
-ZZ 0.2 Different Diameters
993 (2.49 mm Dia.)
. 2 0.15 C->
Q
J
CO
£3
t=
t
o
0.1
-
942 (2.24 mm Dia.)
0.05 -
20
25
F r e q u e n c y (GH z)
0.8
-
Different Diameters
993 (2.49 mm Dia.)
o
<L
J>
o
M 0.4 -
0.2
942 (2.24 ru n Dia.)
-
V
5
10
15
25
20
F r e q u e n c y (GHz)
30
35
40
Figure 7.5: Extinction cross-sections and power absorption coefficients for the 942
and 993 springs whose diameters differ.
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119
36 GHz range is where 993 greatly increases.
Figure 7.6 displays the extinction cross-section and power absorption curves
for the 990, 991, and 992 spring set. Spring sizes are shown in Table 7.1. The crosssection graph indicates that the larger diameter spring 992, has the largest crosssection. The 991 spring has the next smallest diameter and cross-section. The 990
spring has the smallest diameter of all three, and consequently has the smallest crosssection. From viewing the prf’s in Table 7.1, concentration (or thickness variations)
obviously has little effect on power absorption. In fact, the pd increases from the
largest to the smallest spring respectively. The power absorption is the greatest for
the largest diameter spring, and decreases as the spring diameter grows smaller.
Figure 7.7 shows the extinctions and power absorptions of the spring set 625,
742, and 941. Spring 941 has the largest diameter, 742 the next largest, and 625 has
the smallest diameter. From the figure, the 941 spring has the largest cross section.
The 742 spring (next largest diameter) has a cross section smaller than the 941, but
larger than the 625 spring. A similar trend is evident in the cross section traces
displayed in the power absorption graph. The 941 spring which has the largest
diameter, has the largest power absorption coefficient. The 742 spring the next
largest. Finally the 625 has the smallest absorption coefficient of all three spring
samples. By looking at the three springs pd values, it is clear they had little effect on
power absorption. The 941 spring which had the largest absorption had the smallest
pd. The 625 spring had the smallest absorption, but the largest pd. The diameter of
each spring had a significant effect on the cross-sections, and the effective power
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.25
992 (2.74 mm Dia.)
0.2 ....
991 (2.49 mm Dia.)
Different Diameters
§ 0.15
C
J>
a>
co
CO
990 (2.24 mm Dia.)
CO
dS 0.05
10
5
15
20
25
30
40
35
Frequency (GHz)
1
o.a
Different Diameters
<S 0.6
2.74 m
Dia.
&
2.49 mm
2.24 mm
Dia.
Dia.
M 0.4
992
991 ----- 990
0.2
0
5
10
15
20
25
30
35
40
Frequency (GHz)
Figure 7.6: Extinction cross-sections and power absorption coefficients for the 990,
991, and 992 springs whose diameters differ.
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121
0.25
0.2
^
m
Different Diameters
ca
941 (2.24 mm Dia.)
0.15 —
742 (1.98 mm Dia.)
.S
62S (1.32 mm Dia.)
0.05 -
0
5
"• r
T
10
15
20
25
30
35
40
F r e q u e n c y (GH z)
Different Diameters
<3 0.6 1=3
0.4 1.98 nan
1.32 mm
625
5
10
15
20
25
2.24 mm
. Dia.
i Dia.
30
. Dia.
742
943
35
40
F r e q u e n c y (GH z)
Figure 7.7: Extinction cross-sections and power absorption coefficients for the 625,
742, and 941 springs whose diameters differ.
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122
0.25
2.74 mm Dia.
991
690
■2.49 mm Dia.
2.24 mm Dia.
,2.24 mm Dia.
742
1.98 mm Dia.
626
C3
O
cj
c:
1.32 mm
Dia.
w 0.05 -
0
Tr
5
10
tr
15
■> r
T r
20
25
i—i—|—i—i—r
30
35
40
Frequency (GHz)
992
0.8
691
990
/ = 0.6
742
0.4
625
0.2
5
10
15
20
25
30
35
40
F r e q u e n c y (GH z)
Figure 7.8: Extinction cross-sections and power absorption coefficients for the 625,
742, 942, 990, 991, and 992 springs. The figure shows the influence of the diameter
on the extinction cross sections.
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123
absorption.
Finally, Figure 7.8 show the general trends of the cross sections and
absorption coefficients of the following selected springs (starting from the largest to
the smallest diameter): 992, 991, 990, 942, 742, and 625. As seen from the figure, the
larger diameter springs tend to have larger cross-sections and absorption coefficients.
From the figure an interesting trend is observed. The cross-section for the smallest
diameter spring rises from 5.85 GHz to about 24 GHz and levels out. The larger
diameter spring’s cross-sections, however, start leveling out at higher frequencies.
The 992 spring, which has the largest diameter of all the springs, has its cross section
leveling out at 30 GHz. This is due to the wavelength approaching the circumference
of the spring diameters. The leveling effect would occur at a ka (2 tt a/X ) of about
1. As the diameters grow smaller, the leveling out of the cross-sections will appear
at lower frequencies.
From the data reviewed so far the total cross section and power absorption
is affected by the diameter of the spring rather than the amount of turns. From the
figures shown, a larger diameter spring provides a larger total cross section
(extinction) than a smaller diameter spring. However, the cross sections of the 941
and 942 spring, indicate that pitch plays a role as well in larger cross-sections. A
smaller rather than a larger pitch yields a larger cross-section.
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124
CHAPTER 8
EXTINCTION AND TRANSMISSION CHARACTERISTICS
OF MIXED SPRING SIZE SAMPLES
8.1 Introduction
This chapter investigates the attenuated transmission of single and mixed
spring size samples using the extinction of a particular spring. The extinction of a
spring size can be found in a sample (sample 1) containing a given concentration of
the particular spring. Using this information, the transmission characteristics of
another sample (sample 2) containing a different concentration of the same spring
size can be found. The attenuated coherent intensity for sample 2 of thickness d2 can
be found by:
h = Io2 exp(-p2 an d2)
(8.1)
where aa is the extinction of a particular spring from sample 1, and p2 is the spring
concentration of sample 2.
Prediction of this kind can be taken a step further by calculating the intensity
for a sample containing several different spring sizes. To predict the transmission for
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125
a sample containing more than one spring size, the extinction of each spring size
must be found from previous single spring size samples. The extinction information
for each spring size is substituted into the following equation to yield the predicted
intensity.
I2 = /„ exp[-rf2 (p12 an + p22 oa + p32 aa +... p^ a,,)]
(8.2)
= Ia exp[-d2 a]
The subscript z'2 refers to the f 1spring in sample 2, and a is the attenuation constant
of the sample. The predicted extinction can be found from:
at = a / pe
(8.3)
where p, is the effective spring concentration in the media. The effective spring
concentration of the mixed size sample is found by combining the number of springs
of each size and dividing by the volume of the sample.
The next few sections will show predicted and experimental transmitted
intensities for: 1) samples containing a single spring size, but varied concentrations,
and 2) samples containing mixed spring sizes. Table 8.1 lists each spring sample’s
characteristics.
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126
Table 8.1: Mixed Spring Sample Characteristics
PITCH TURNS 1 TURN L
(mm)
(mm)
SAMPLES
DIA.
(mm)
625
1.32
0.483
3
742
1.98
0.483
991
2.49
993
2.49
SPRING
CONC.
p(sps/cm3)
THICK
(mm)
4.175
82.61
3.20
3
6.239
54.64
3.23
0.787
2
7.862
60.81
3.45
0.787
3
7.862
44.80
3.12
742-991
—
27.58 / 32.79
3.20
742-993
—
24.45 / 19.37
3.61
21.10 / 25.08 /
31.59
2.79
18.68 / 14.81 /
27.99
3.15
742-991625
742-993625
—
—
742(1%)
—
30.01
2.94
993(1%)
—
23.31
3.00
8.2 Single Spring Size, Different Concentration
Two samples were made using the 742 spring size, a 1% and a 2% metal
volume concentration. The extinction cross-section, a„ was found from the 1%
sample, and substituted along with the 2% sample’s spring concentration into
equation (8.1) to yield the theoretical attenuation. Both the theoretical and
experimental curves for the 2% 742 sample are shown in the top graph of Figure 8.1.
Both curves seem to follow each other until about 30 GHz. The divergence after
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
«
-3
-4
Exp. 2% 742
a>
-6
—
-7
-
Theo. 2% 742 Using 1% Sample’s
Extinction Cross Section
-1 0
5
10
15
20
25
30
35
40
Frequency (GHz)
Theo. 1% 742 Using 2% Sample’s
.Extinction Cross Section
O
Q
T3
-3 -
55
QJ
-4 -
Exp. 1% 742
- 7 -■
-8
-
-1 0
5
10
15
20
25
30
35
40
F r e q u e n c y (G H z)
Figure 8.1: Theoretical and experimental transmitted intensity for the 2% 742 sample
(top graph) and the 1% 742 sample (bottom graph).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
128
30 GHz is due to the 1% sample’s extinction being larger than the 2% sample’s
extinction. Explained in earlier chapters, this divergence is due to increased
incoherent scattering in the 2% sample. The increased scattering contributes to more
received power. Since all experimental extinction cross-sections are derived from
power measurements at the receiving antenna, an increase in power yields a smaller
cross-section. Thus, the 2% sample’s transmitted intensity will be less due to the
smaller extinction.
The bottom graph of Figure 8.1 shows the predicted (from the a, of the 2%
sample) and experimental trace for the 1% 742. The 1% sample’s transmitted
intensity is logically less than the 2% due to a smaller concentration of inclusions.
Again both the experimental and theoretical curves agree very well until 30 GHz.
The 2% sample’s extinction is smaller than the 1% from 30 GHz. Therefore, the
theoretical transmitted intensity will be less than the experimental.
Figure 8.2 shows the theoretical and experimental transmitted intensities for the
2% (top graph) and 1% (bottom graph) 993 spring samples. The 2% sample's theoretical
and experimental curves do not agree as well as the previous 742 curves. The 993 has a
larger diameter than the 742 spring. The larger diameter contributes to a larger degree of
scattering. At the lower frequencies the wavelength is larger than the diameter of the
spring, thereby giving better agreement between the two curves. Above 15 GHz however,
the wavelength approaches the diameter of the spring, and the curves start diverging due
to increased scattering. As pointed out in the 742 samples, the 2% extinction is
smaller than the 1% extinction. The smaller extinction results in less transmitted
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-4
Exp. 2% 993
o>
•S
- j o
--
o>
| - 12-
GO
P -14 --16 Theo. 2% 993 Using 1% Sample’s
Extinction Cross Section
-1 8 -■
-2 0
5
10
15
20
25
Frequency (GHz)
30
35
40
Theo. 1% 993 Using 2% Sample’s
Extinction Cross Section
-4 -
'53
—8
-
M
- 5 -1 0
Exp. 1%993
-
£ -1 4 -1 6 -1 8 -
-20
5
10
15
20
25
30
35
40
F r e q u e n c y (G H z)
Figure 8.2: Theoretical and experimental transmitted intensity for the 2% 993 sample
(top graph) and the 1% 993 sample (bottom graph).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
130
intensity.
The bottom graph of Figure 8.2 shows the theoretical and experimental
transmitted intensity for the 1% 993 sample. The agreement between the two curves
is not as good as the 742 samples, but it is better than the 2% 993 curves. Since the
concentration is lower in the 1% sample, the amount of scattering is less. This result
is shown as a smaller dB variation (smaller than the 2% sample) between the
theoretical and experimental curves. The gap that opens up between the two curves
at 25 GHz, is due to the 2% sample’s extinction being smaller than the 1% sample’s
extinction.
83 Samples with Two Spring Sizes
A 2% 742 sample was measured, and its extinction was calculated. The same
procedure was done to a 2% 991 sample. Taking the extinction from each sample,
and the concentration of each spring in the mixed sample, a theoretical transmitted
intensity was calculated using equation (8.2). Also, an effective theoretical extinction
was calculated for the mix sample using equation (8.3). The mix sample’s extinction
cross-seciion’s are understood to be effective, since the extinction depends on
multiple springs instead of one.
Figure 8.3 shows the theoretical and experimental effective extinction crosssection and transmitted intensity for the sample containing an equal mix of the 742
and 991 spring. The total metal volume concentration is 2% for the sample (1% 742
and 1% 991).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
131
0.25
Extinction Cross Section (cm.sq.)
742
0.2
991
Exp. Mix 7 42/991
0.15
Theo. Mix 7 4 2 /9 9 1
0.1
0.05
0
10
5
15
20
25
30
35
40
30
35
40
Transmitted Intensity (dB)
F r e q u e n c y (G H z)
-8
-
-1 0
-
742
- 1 4 —-
991
-1 6 - Exp. Mix 7 4 2 /9 9 1
- 1 8 - -Theo. Mix 7 4 2/991
-20
5
10
15
20
25
F r e q u e n c y (G H z)
Figure 83: Theoretical and experimental effective extinction and transmitted intensity
for the 2% 742/991 mix sample.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
132
The effective extinction of the mix sample shows both the theoretical and
experimental curves to be very close. For reference, the extinction for the 742 and
991 spring (from 2% samples) is shown. Clearly the extinction curves for the mix take
on values that are between the values of the 742 and 991 values. Single spring
features are washed out if springs of different sizes are present. Since the extinction’s
of the theoretical and experimental curves are close, the transmitted intensity of the
mix sample follows the same trends. At about 26 GHz, the theoretical and
experimental curves diverge. This is where incoherent scattering is increased giving
a lower extinction than the theoretical.
Figure 8.4 shows the theoretical and experimental effective extinction crosssection and transmitted intensity for the 742 and 993 mix sample. The sample
contained an even metal volume mix (1% 742 and 1% 993) of the two spring sizes,
giving a metal volume concentration of 2%. The theoretical transmitted intensity and
effective extinction were calculated from equations (8.2) and (8.3) respectively using
data from the 2% 742 and 2% 993 samples measured. The effective extinction
shown in Figure 8.4 depicts a close comparison between the theoretical and
experimental curves. A deviation is noted at 31 GHz for the experimental trace. The
transmitted intensity of the two curves is veiy close as well. Again, the experimental
curve deviates at 31 GHz due to extinction variation. The mix samples effective
extinction and intensity curves take on values that are between the 2% 742 and 2%
993 extinction and intensity values. For a sample containing different size inclusions,
single scattering theory predicts this type of trend. Of course single scattering theory
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
133
0.25
748
993
Exp. Mixed 7 4 2 /9 9 3
. 2 0.15
COJ
T2
Theo. Mixed 7 4 2 /9 9 3
C_>
c S 0.05 -•
0
T
5
T
10
t—
i— |— i— i— i— i— r
15
20
T
t—
i— I— i— r
t—
25
i— i—
|—
i—
r
t—i—|—i—r
T
T
30
35
40
30
35
40
F r e q u e n c y (G H z)
-2
—
-4 —
- B
- -
cu>
^ -12 +••
e
=Q
5
I
&
I -14
993
-1 6
Exp. Mixed 7 4 2 /9 9 3
-IB — Theo. Mixed 7 4 2 /9 9 3
-2 0
5
10
15
20
25
F r e q u e n c y (G H z)
Figure 8.4: Theoretical and experimental effective extinction and transmitted intensity
for the 2% 742/993 mix sample.
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134
is not perfect for the samples shown, since there are still fluctuations in effective
extinction caused by multiple scattering.
8.4 Three Spring Size Mixed Samples
Figure 8.5 depicts the theoretical and experimental extinction and transmitted
intensity of a mix sample containing 3 spring sizes: 742,991, and 625. Each spring has
an equal metal concentration (0.667% x 3) to give a total metal volume concentration
of 2%. Equations (8.2) and (8.3) along with the extinctions from the 742, 991, and
625 samples, were used to find the theoretical curves.
There is very close agreement between the theoretical and experimental curves
for the effective extinction and transmitted intensity of the mixed sample. It should
be noticed that the effective extinction curve for the 742/991 mix sample reviewed
earlier, had values between the 742 and 991 extinction values. For the 742/991/625
mix sample, the effective extinction is lower and coincides with the 742 extinction
trace. The addition of the 625 spring has "pulled down" the effective extinction for
the 3 spring sized mix sample. Also, the 2 spring sample showed significant
divergence between the theoretical and experimental curves. With the 3 spring
sample, there is no significant divergence. The transmitted intensity of the
742/991/625 sample is larger than the 742/991 sample. This is due to a smaller
effective extinction for the 2 spring sized sample.
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135
0.25
742
^
0.2
991
625
Exp. Mix 7 4 2 /9 9 1 /6 2 5
Theo. Mix 7 4 2 /9 9 1 /6 2 5
3
0.05 -
10
5
20
15
25
30
35
40
30
35
40
F requency (GHz)
-2
-
CL>
-10 -
742
-1 4 625
-1 6 - Exp. Mixed 7 4 2 /9 9 1 /6 2 5
-1 8 Theo. Mixed 7 4 2 /9 9 1 /6 2 5
-2 0
5
10
15
20
25
F r e q u e n c y (G H z)
Figure 8.5: Theoretical and experimental effective extinction and transmitted intensity
for the 2% 742/991/625 mix sample.
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136
Figure 8.6 shows the theoretical and experimental effective extinction and
transmitted intensity for the mixed sample, 742/993/625. The effective extinction
curve reflects the effect of adding the concentration of 625 springs. Compared to the
2 spring sized sample 742/993, the effective extinction of the 742/993/625 is lower.
Also, the 742/993 sample had larger variations between the theoretical and
experimental curves. The 742/993/625 curves still have variations, but not as dramatic
as the 742/993 sample. It seems a larger range in inclusion size dampens any effective
extinction fluctuations within a given sample. Because of a smaller effective extinction
in the 742/993/625 sample, the transmitted intensity is greater than the 742/993
sample.
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137
0.25
742
^
0.2
993
625
. 2 0.15
Exp. Mix 7 4 2 /9 9 3 /6 2 5
Theo. Mix 7 4 2 /9 9 3 /6 2 5
0.1
C3
cS 0.05
0
10
5
15
20
25
30
35
40
30
35
40
Frequency (GHz)
-2
- -
-4 —
-S -10 -- 742
-o
aj
-12
—
-1 4 --1 6 —
993
625
Exp. Mix 7 4 2 /993/625
- 1 8 -Theo. Mix 7 4 2 /993/625
-2 0
5
10
15
20
25
Frequency (GHz)
Figure 8.6: Theoretical and experimental effective extinction and transmitted intensity
for the 2% 742/993/625 mix sample.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
138
CHAPTER 9
CONCLUSION
The main objective of this study was to design, fabricate, and do extensive
analysis of thin, flexible chiral materials. Three millimeters thick samples were made
with an RTV matrix embedded with helices. Certain aspects were involved when
carrying out this task. One was to design sample holders and test procedures to
evaluate flexible samples’ wave propagation characteristics in free-space. Another
consideration was to examine the extinction cross section characteristics of the
helices, which was necessaiy to understand the transmission losses in the composite
samples. This was achieved by using a simplified version of the equation of radiative
transfer. By cariying out this investigation, several conclusions have precipitated in the
area of testing and evaluation.
Flexible samples pose a major problem when probed by electromagnetic plane
waves. The composites tend to bulge within a holder that provides minimal support.
Bulging manifests itself in erroneous permittivity and permeability measurement
results. As the wavelength decreases the errors are more pronounced. Specially
designed holders were constructed to impart mechanical rigidity to the samples.
Consequently, the composites were more planar giving acceptable material properties
for RTV and Teflon slabs.
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139
A measuring procedure was devised to properly examine the extinction
properties of the helices. Thick, semi-infinite samples offer small internal interface
reflections due to the absorption and thickness of the medium. To find the extinction,
transmission measurements for such samples would suffice. But since this study’s
samples are 3 mm thick, and internal losses are low, a reflection measurement is
needed. By using the reflection and transmission coefficients of a sample, the
effective propagation constant (EPC) and extinction is found. This procedure was
used on all samples in the research.
Some conclusions can be made about transmission and extinction results for
thin samples having different spring concentrations. It was found that thin chiral
samples limited the amount of polarization rotation in samples with 1% and 2%
metal volume concentration. The co-polarized, and co + cross polarized
measurements bore insignificant differences. Also, the extinction for the 1% and 2%
742 samples is in good agreement where ka < 1. This is true of the 1% and 2% 993
samples as well.
The theoretical extinction cross section of a sphere was compared to the 742
and 993 springs as a function of ka. A sphere with the same radius as the 742 or 993
spring gave similar extinction results. However, the spheres metal volume is several
times larger than the springs’. This means that the helices can give a cross section
and transmission loss that is equivalent to a sphere’s, but at a fraction of the metal
volume and weight.
The effect of the geometrical dimensions of a helix on extinction cross sections
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140
and apparent power absorption were investigated. The cross sections of springs with
different number of turns, or pitch showed small differences in extinction. However,
helical diameters are proportional to the extinction. A large diameter helix has a
larger extinction than a small diameter helix. By measuring the total reflection and
transmission of the composite the apparent power absorption is found. That is, the
amount of energy that is lost in the system due to scattering and absorption. The
diameters of the springs have a greater effect on the apparent power absorption as
well as extinction. Large diameter springs had higher extinctions, which yielded a
larger attenuation and transmission loss. The high transmission losses contributed to
the apparent power absorption.
A simplified approach to using the equation of radiative transfer to predict
transmission losses within a chiral medium was developed. Two applications of the
theory were put forth. Based on the extinction of one spring size, the transmission
of another sample of different metal concentration can be predicted. A second
application of the theory is modeling transmission losses of composites with several
spring sizes. By knowing the extinction of different sized springs through
experimentation, a sample can be tailored for an overall extinction. By knowing the
overall extinction, the transmission loss within the media is known. This modeling is
extremely easy and useful for thin chiral composite construction. Given varying spring
extinctions, weight and metal volume constraints, a material can be constructed
exhibiting desired transmission losses in a particular frequency band.
Although this study has achieved the directives outlined initially, there is still
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141
a wealth of research that must be done. Much of the future work is broken down into
three areas: modeling, testing, and fabrication.
A more rigorous calculation of extinction cross sections is to be done for
future work. A chiral medium consists of LCP and RCP waves with one wave
preferentially absorbed. The absorbed component will have a different intensity than
the other wave. These two different intensities will yield two different extinctions for
the same particle. Vector radiative transfer theory must be implemented for this
purpose. This study involved samples so thin that a good approximation of the
extinction was found. Future fabrication materials, however, may be thicker
warranting a detailed description of the cross section. A second area to be worked
on in modeling is to find the amount of scattering and absorption that contribute to
the extinction of a helix. This study found the extinction, but there was no direct way
of discerning the amount of scattering or absorption.
To help solve for the scattering and absorption, different testing methods must
be employed. An independent measurement of either scattering or absorption is
required. Total scattering measurements must be done over all directions of a helix
(spherical integration). Subtracting the scattering cross section from the extinction
will yield the absorption cross section.
Finally, refinement of composites for real world applications is needed. The
samples fabricated in this study are only the first step. The morphology of the
samples can be improved by finding helical materials that are lighter in weight such
as carbon. Carbon helices would be lighter and offer more absorption due to high
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142
lossiness. Unification of scattering and absorption in a helix is a perfect choice for
electromagnetic interference materials. Fillers such as carbon, nickel, or other lossy
material could be added to the composite to enhance absorption of scattering from
the helices. The matrix material can be made from thermoplastics that are more
suited for commercial applications. There are also thermoplastics and thermosets
available with higher dielectric constants capable of helping to absorb larger
wavelengths.
The results found in this research have given some insights to possible
applications of thin chiral materials. With the advances in modeling, measurement
techniques, materials and fabrication, effective commercial shielding using chiral
inclusions is eminent.
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143
BIBLIOGRAPHY
Amin, M.B., and James, J.R., "Techniques for utilization of hexagonal ferrites
in radar absorbers, Part 1, Broadband planar coatings," Radio Electron. Eng., Vol.
51, pp.209-218, 1981.
Applequist, J., "Optical Activity: Biot’s Bequest," American Scientist, Vol. 75, pp.5968, 1987.
Arago, F., "Memoire sur une Modification Remarquable qu’eprouvent les rayons
lumineux dans leur passage a travers certains corps diaphanes, et sur queiques
autres nouveaux phenomenes d’optique," Memoires des la Classe des Sciences
Mathematiques et Physiques de I’Instilut Imperial de France, Part 1, Vol. 1, pp.
93-134, 1811.
Barron, L.D., Molecular Light Scattering and Optical Activity, Cambridge University
Press, Cambridge, U.K., 1982.
Bassiri, S., Papas, C.H., and Engheta, N., "Electromagnetic Wave Propagation
through a Dielectric-Chiral Interface and Throgh a Chiral Slab," Journal o f the
optical Society o f America, Vol. 5-A, No. 9, pp. 1450-1459, 1988.
Biot, J.B., Memoire sur les rotations que certains substances impriment aux axes de
polarisation des rayons lumineux, Memoires de VAcademie royale des sciences de
Vlnstitut de France, Vol. 2, pp.41-136, 1817.
Bohren, C.F.,"Light scattering by an optically active sphere," Chemical Physics
Letters, Vol. 29, pp. 458-462, 1974.
Bohren, C.F., "Light Scattering by an Optically Active Sphere," Chemical Physics
Letters, Vol. 29, pp. 458-462, 1974.
Bohren, C.F., "Scattering of Electromagnetic Waves by an Optically Active
Cylinder," Journal o f Colloid and Interface Science, Vol. 66, pp. 105-109, 1978.
Bohren, C.F., and Huffman, D.R ., Absorption and Scattering o f Light by Small
Particles, John Wiley & Sons, New York, 1983.
Bokut’, B.V., and Federov, F.I.,"Reflection and Refraction of Light in an Optically
Isotropic active Media," Optics and Spectroscopy, Vol. 9, pp. 334-336, 1959.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
144
Born, M., "Uber die naturliche optische Aktivitat von Flussigkeiten und Gasen," Phys.
Z , Vol. 16, pp. 251-258, 1915.
Chandrasekhar, S., Radiative Transfer, Dover, New York,pp. 1-3, 1960.
Chen, H.C., Theory o f Electromagnetic Waves, Me Graw-Hill, New York, 1983.
Cotton, A., absorption inegale des rayons circulaires droit et gauche dans certains
corps actifs, Comptes rendus hebdomadaires des seances de I’Academie des sciences,
Vol. 120, pp.989-991, 1895.
Drude, P., Lehrbuch der Optik, Leipzig: S. Hirzel, 1900.
Fedorov, F.I., "On the Theoiy of Optical Activity in Ciystals. I. The Law of
Conservation of Energy and the Optical Activity Tensors, Optics and
Spectroscopy, Vol. 6, pp. 49-53, 1959a.
Fedorov, F.I., "On the Theory of Optical Activity in Ciystals. II. Crystals of Cubic
Symmetry and Plane Classes of Central Symmetry," Optics and Spectroscopy, Vol.
6, pp. 237-240, 1959b.
Fresnel, A., Oeuvres completes, Paris: Imprimerie imperiale, 1866.
Ghodgaonkar, D.K., Varadan, V.V., and Varadan, V.K., "A free-space method
for measurement of dielectric constants and loss tangents at microwave
frequencies," IEEE Trans. Instrum. Meas., Vol. 38, pp. 789-793, 1989.
Ghodgaonkar, D.K., Varadan, V.V., and Varadan, V.K., "Free-Space
Measurement of Complex Permeability of Magnetic Materials at Microwave
Frequencies," IEEE Trans. Instrum. Meas., Vol. 39, pp. 387-394, 1990.
Gray, F., "The optical Activity of Liquids and Gases," Physical Review, Vol. 7, pp.
478-488, 1916.
Guerin, F., "Experimental Study of the Properties of Ceramic Chiral
Composites at Microwave Frequencies," Master Thesis, 1992, Engineering
Science and Mechanics, The Pennsylvania State University, University Park, PA.
Guerin, F., Mariotte, F., Leroy, B., Bannelier, P., "Modeling Chiral
Composites at CEA-CESTA and Thomson-CSF: from RCS to Effective
Properties Computation," Proceedings o f Chiral 94’, pp. 59-69, Perigueux, France,
1994.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
145
Guire, T., Varadan, V.V., Varadan, V.K., "Influence of Chirality on the
Reflection of EM Waves by Planar Dielectric Slabs," IEEE Transaction on
Electromagnetic Compatibility, Vol. 32, No. 4, pp. 300-303, 1990.
Hollinger, R.D., "Electromagnetic Wave Propagation in Circular Waveguide
Containing Isotropic Chiral Media," M.S. Thesis, 1990, Engineering Sdenoe, The
Pennsylvania State University, University Park, PA.
Hollinger, R., Varadan, V.V., and Varadan, V.K., "Eigenmodes in a Circular
Waveguide Containing an Isotropic Chiral Medium," Radio Science, Vol. 26, No.
5, pp. 1335-1344, 1991.
Huygens, C., Treatise on Light, Dover Publications, New York, 1962 (1690).
Ishimaru, A., Wave Propagation and Scattering in Random Media, Academic, New
York, 1978.
Jaggard, D.L., Mickelson, A.R., and Papas, C.H., "On Electromagnetic Waves in
Chiral Media," Applied Physics, Vol. 18, pp. 211-216, 1979.
Joseph, J.C., Jost R.J., and Utt, E.L., "Multiple angle of incidence
measurement technique for the permittivity and permeability of lossy materials
at millimeter wavelengths," IEEE AP-S Int. Symp. Dig., pp. 640-643, 1987.
Lakhtakia, A., Varadan, V.K., and Varadan, V.V., "Scattering and Absorption
Characteristics of Lossy Dielectric, Chiral, Nonspherical Objects," Applied Optics,
Vol. 24, No. 23, pp. 4146-4154, 1985.
Lakhtakia, A., Varadan, V.V., and Varadan, V.K., "A Parametric Study of
Microwave Reflection Characteristics of a Planar Achiral-Chiral Interface," IEEE
Transactions on Electromagnetic Compatibility, Vol. EMC- 28, No. 2, pp. 90-95,
1986.
Lakhtakia, A., Varadan, V.K., and Varadan, V.V., Time-Harmonic
Electromagnetic Fields in Chiral Media, Springer-Verlag, New York, 1989.
Lakhtakia, L. (editor), Selected Papers on Natural Optical Activity, SPIE Optical
Engineering Press, Bellingham, Washington, 1990.
Lindman, K.F., Annalen der Physik, Vol. 63, p. 621, 1920.
Liu, J.C., and Jaggard, D.L., "Chiral Layers on Planar Surfaces," Journal o f
Electromagnetic Waves and Applications, Vol. 6, No. 5/6, pp. 651-667, 1992.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
146
Ma, Y., Varadan, V.K., Varadan, V.V., "Enhanced Absorption Due to
Dependent Scattering," Transactions o f the ASME, Vol. 112, pp. 402-407, 1990.
Malus, E.L. "Sur une Propriete de la Lumiere Reflechie," Memoires de la Societe
d ’Arceuil, pp. 143-158, 1809.
Oseen, C.W., "Uber die Wechselwirkung zwischen Zwei Elektrischen Dipolen und
uber die Drehung der Polarisationsebene in Kristallen und Flussigkeiten,Mnna/en
der Physik, Vol. 48, pp. 1-56, 1915.
Papas, C.H., Theory o f Electromagnetic Wave Propagation, Dover, New York, pp. 117120, 1988.
Post, E.J., Formal Structure o f Electromagnetics, Amsterdam, North-Holland, 1962.
Redheffer, R.M., "The measurement of dielectric constants," in Techniques of
Microwave Measurements, C.G. Montgomery, Ed., Vol. 2 New York: Dover, pp.
591-657. from FS Com. Perm,mag\deepak, 1966.
Ro, R., "Determination of the Electromagnetic Properties of Chiral
Composites Using Normal Incidence Measurements," Ph.D. Thesis, 1991,
Engineering Science and Mechanics, The Pennsylvania State University,
University Park, PA.
Sihvola, A.H., and Lindell, I.V., "Chiral Maxwell-Gamett Mixing Formula,"
Electronic Letters, Vol. 26, No.2, pp. 118-119, 1990.
Sihvola, A.H., and Lindell, I.V., "Analysis on Chiral Mixtures," Journal o f
Electromagnetic Waves and Applications," Vol. 6, No. 5/6, pp. 553-572, 1990.
Silverman, M.P., "Reflection and Refraction at the Surface of a Chiral Medium:
Comparison of Gyrotropic Constitutive Relations Invarient and Noninvarient
under a Duality Transformation," Journal o f the Optical Society o f America, Vol.
3, No. 6, pp. 830-837, 1986.
Tinoco, I., and Freeman, M.P., "The optical Activity of Oriented Copper Helices: I.
Experimental," Journal o f Physical Chemistry, Vol. 61, pp. 1196-1200, 1957.
van de Hulst, H.C., Light Scattering by Small Particles, Dover,New York, pp. 15-16,
1981.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
147
Varadan, V.K., Lakhtakia, A., Varadan, V.V.,"Scattering by Beaded Helices:
Anisotropy and Chirality," Journal o f Wave-Material Interaction, Vol. 2, No.2, pp.
154-159, 1987.
Varadan, V.K., Varadan, V.V., and Lakhtakia, A., "On the Possibility of Designing
Anti-Reflection Coatings Using Chiral Composites," Journal o f Wave-Material
Interaction, Vol. 2, No.l, pp.71-81, 1987.
Varadan, V.V., Lakhtakia, A., and Varadan, V.K., "Equivalent Dipole Moments
of Helical Arrangements of Small, Isotropic, Point Polarizable Scatters:
Application to Chiral Polymer Design," Journal o f Applied Physics, Vol. 63, No.
2, pp.280-284, 1988.
Varadan, V.V., Lakhtakia, A., and Varadan, V.K., "Microscopic Circular
Polarizabilities (Rotabilities) and the Macroscopic Properties of Chiral Media,"
Radio Science, Vol. 26, No. 2, pp. 511-516, 1991.
Varadan, V.V., Ro, R., Varadan, V.K., "Measurement of the Electromagnetic
Properties of Chiral Composite Materials in the 8-40 GHz Range," Radio
Science, Vol. 29, No. 1, pp. 9-22, 1994.
Von Hippel, A.R., Dielectric Materials and Applications, Wiley, New York, p.332.
1954.
Whittet, D.C.B., Dust in the Galactic Environment, Institute of Physics Publishing,
New York, 1992.
Winkler, M.H., "An Experimental Investigation of Some Models for Optical
Activity," Journal o f Physical Chemistry, Vol. 60, pp.1656-1659, 1956.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
148
APPENDIX
s t a r f := 26.5
s t o p f := 4 0
a := 0.0990
i : = 0 , 1.. 40
.
freqi
sto p f - sta rf
r
400
fireq. := sta rf + f r e q i i - 10
to
, .
310
t e i r . :'
ffe q .-l-1 0 9
Xair.
te m p . := -----ri
I—
V 2.8
.
2-jt
pk. :=
y '
te m p .
,
x. := a pk.
'
1
n := 1 , 2 .. 10
__sin(x.)
J 0,i
1
X.
X.
3(x.)
sin^x.j
1 fcos(x.J
y0, i :=
X.
to]
1 *
X.
sin
to )
/2 * n + l \ .
l,i
^ n —l ,i
I\
Vi
i
2 -n -t-1 \
y n + l . i := ‘ yn - l , i +
— X.
1 II
h:=j+y-i
hdr
. =-h
n —l , i
.+
n,i
- — -1 -h
1 x
’ n”
.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
149
* n - l,i
n —l,i
\2
X - jd r
»J
n —l,i
' n — l,i
. . -hi
n —l,i J n - l , i
x. h d r
i
, . -t-h
n —l , i
, .
n —l , i
2-71
* ’K )!
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
VITA
Neil Rhodes Williams was bom in Tunkhannock, Pennsylvania, on Januaiy 5,
1965. He received a B.S. in Electrical Engineering in 1987 from Wilkes University.
In the fall of 1987, he entered The Pennsylvania State University in the Department
of Electrical Engineering. There he was employed as a research assistant investigating
lossy conducting polymers, RAM and EMI materials. The author received an M.S.
in the summer of 1989.
In 1990, the author transferred to the Department of Engineering Science and
Mechanics where he began work on his Ph.D. During 1990 and 1994, he served as
a design and development engineer at HVS Technologies, while finishing his
dissertation. The author is a member of Sigma Xi and the Institute of Electrical and
Electronics Engineers.
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