# Radiative transfer approach to design the electromagnetic response of microwave chiral composites

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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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 " Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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’ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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): Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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,§). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 dQ dQ I(r,s) Figure 33: Scattering of specific intensity incident on volume ds from direction s’ into direction s. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 " Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 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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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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