RECTANGULAR WAVEGUIDE MATERIAL CHARACTERIZATION: ANISOTROPIC PROPERTY EXTRACTION AND MEASUREMENT VALIDATION By Benjamin Reid Crowgey A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Electrical Engineering – Doctor of Philosophy 2013 ABSTRACT RECTANGULAR WAVEGUIDE MATERIAL CHARACTERIZATION: ANISOTROPIC PROPERTY EXTRACTION AND MEASUREMENT VALIDATION By Benjamin Reid Crowgey Rectangular waveguide methods are appealing for measuring isotropic and anisotropic materials because of high signal strength due to field confinement, and the ability to control the polarization of the applied electric field. As a stepping stone to developing methods for characterizing materials with fully-populated anisotropic tensor characteristics, techniques are presented in this dissertation to characterize isotropic, biaxially anisotropic, and gyromagnetic materials. Two characterization techniques are investigated for each material, and thus six di↵erent techniques are described. Additionally, a waveguide standard is introduced which may be used to validate the measurement of the permittivity and permeability of materials at microwave frequencies. The first characterization method examined is the Nicolson-Ross-Weir (NRW) technique for the extraction of isotropic parameters of a sample completely filling the cross-section of a rectangular waveguide. A second technique is proposed for the characterization of an isotropic conductor-backed sample filling the cross-section of a waveguide. If the sample is conductor-backed, and occupies the entire cross-section, a transmission measurement is not available, and thus a method must be found for providing two su ciently di↵erent reflection measurements.The technique proposed here is to place a waveguide iris in front of the sample, exposing the sample to a spectrum of evanescent modes. By measuring the reflection coe cient with and without an iris, the necessary two data may be obtained to determine the material parameters. A mode-matching approach is used to determine the theoretical response of a sample placed behind the waveguide iris. This response is used in a root-searching algorithm to determine permittivity and permeability by comparing to measurements of the reflection coe cient. For the characterization of biaxially anisotropic materials, the first method considers an extension of the NRW technique for characterization of a sample filling the cross-section of a waveguide. Due to the rectangular nature of the waveguide, typically three di↵erent samples are manufactured from the same material in order to characterize the six complex material parameters. The second technique for measuring the electromagnetic properties of a biaxially anisotropic material sample uses a reduced-aperture waveguide sample holder designed to accommodate a cubical sample. All the tensor material parameters can then be determined by measuring the reflection and transmission coe cients of a single sample placed into several orientations. The parameters are obtained using a root-searching algorithm by comparing theoretically computed and measured reflection and transmission coe cients. The theoretical coe cients are determined using a mode matching technique. The first technique for characterizing the electromagnetic properties of gyromagnetic materials considers requires filling the cross-section of a waveguide. The material parameters are extracted from the measured reflection and transmission coe cients. Since the crosssectional dimensions of waveguides become prohibitively large at low frequencies, and it is at these frequencies that the gyromagnetic properties are most pronounced, su ciently large samples may not be available. Therefore, the second technique uses a reduced-aperture sample holder that does not require the sample to fill the entire cross section of the guide. The theoretical reflection and transmission coe cients for both methods are determined using a mode matching technique. A nonlinear least squares method is employed to extract the gyromagnetic material parameters. Finally, this dissertation introduces a waveguide standard that acts as a surrogate material with both electric and magnetic properties and is useful for verifying systems designed to characterize engineered materials using the NRW technique. A genetic algorithm is used to optimize the all-metallic structure to produce a surrogate with both relative permittivity and permeability near six across S-band, and with low sensitivity to changes in geometry to reduce the e↵ects of fabrication errors. Copyright by BENJAMIN REID CROWGEY 2013 For the women in my life: Martha, Stacey, and Elliana. v ACKNOWLEDGMENTS There are so many people who deserve my appreciation. I first must thank my committee members who so graciously agreed to help me with the completion of my PhD. Thank you, Dr. Leo Kempel, for all the years of advice and support. Though you have become increasingly busy over the years with administrative duties, you have always found the time to meet with me to discuss research, give opinions, or just see how I’m doing. It has really meant a lot to me, and it’s amazing to think how far I’ve come since sitting in on your cell phone freshmen seminar. To Dr. Shanker Balasubramaniam, I have appreciated the many years of challenging coursework, humorous advice, and helpful insight on the material discussed in this dissertation. I enjoyed posting you up on the basketball court and you putting me back in my place in the classroom. Maybe one day soon I will take up running like you so frequently say I should. Another special thanks is extended to Dr. Prem Chahal, who has been available many times to answer questions in times of confusion and to give general advice from the industry perspective. Thanks to Dr. Andrew Christlieb for agreeing to participate in my graduate committee and for his recommendations on this work. Recognition must be extended to those at AFOSR since the majority of this work was performed under Subcontract Agreement FA 9550-09-1-0182. Additionally, thank you to Dr. Tayfun Ozdemir for supporting me for a couple of semesters while I worked on the very interesting UAV project. I had a terrific time performing this research, and I hope to work with you again. Thank you to General Electric for allowing me to participate on the conductor-backed material characterization project. A special thanks to Craig Baucke and Glen Hilderbrand for their helpful insight into this project. It was great working with you. Dr. Mike Havrilla deserves my appreciation for all the years of advice and insightful conversations. Additionally, thank you for letting us use your S-band waveguides, which still have yet to be returned to you. Thanks also to Captain Milo Hyde, IV, from AFIT for sharing measurement data with me. vi Thanks to all of my Ben Crowgey associates, fellow students, o ce mates, and friends. Since I’ve been here for so many years, I’ve gotten to know many of you. I can say each and every one of you made this an enjoyable journey and I will forever remember the years here at MSU. Special thanks goes out to Dr. Raoul Ouedraogo for his many years of help with anything related to EM. You are a good friend and I hope to stay in contact with you for many years to come. Also thanks go to Junyan Tang for your numerous amount of hours spent in the lab measuring the di↵erent types of material this dissertation uses for characterization. I will miss your Tang-isms and helpful weight loss advice. My appreciation to Collin Meierbachtol for all your help. Anytime I had a random question you were there with the answer, you were my go-to guy. Thank you Josh Myers for helping me disengage from everyday work and enjoy the time spent in our o ce. To Korede Akinlabi-Oladimeji and Andrew Temme, thank you for the years of friendship and assistance with measurements, coursework, and everyday issues. Very special thanks go to Dr. Edward Rothwell for being an exceptional and knowledgeable advisor. The amount of time you spent in your o ce assisting and guiding me through both school and life is time I will always cherish. Words seriously can not express how much I have appreciated everything you’ve done for me over the years. I am sure if I did try to express these feeling in words, I’d make you pull your hair out because of my many grammatical and logic writing errors. I know the writing of this dissertation has been a journey for both of us, but I really do appreciate all the time you’ve spent reviewing it. All said and done, I know I could not have asked for a better advisor. Thank you. I am indebted to my parents John and Martha Crowgey for their love and support. You’ve been there from the beginning and I know I would not be where I am now if it wasn’t for you. Especially to my mother, who is and was always a phone call away. You made my life during graduate school easier and I can’t begin to tell you how much that has meant to me. To my sisters Sarah and Julie, thank you being there when I needed you like big sisters should. To my whole family, you have pushed and applauded me, and for all of that vii I am thankful. Thank you Granddad for always knowing my potential and making sure I achieved my goals. I did it. I miss you, but I always know you are with me. Finally I want to thank my wife, Stacey, for all of her support, motivation, encouragement, and patience while I completed my education. You spent numerous hours looking over this dissertation before I sent it to Dr. Rothwell for final review. I know he appreciates those hours as much, if not more, as me. We met while I was completing my masters here at MSU, moved in together during the second year of my doctorate, got married when I was in the middle of the program, and welcomed a little girl into the world when I was finishing my degree. Needless to say, these years with you have been eventful and memorable, and I am sure we will never forget this time in our lives when I was completing my PhD. It is now time to move on to the next phase of our lives, and I can’t wait. viii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv CHAPTER 1 INTRODUCTION AND BACKGROUND . . . . . . . . . 1.1 Electromagnetic Material Characterization . . . . 1.2 Rectangular Waveguide Material Characterization 1.3 Biaxially Anisotropic Material Characterization . 1.4 Gyromagnetic Material Characterization . . . . . 1.5 Overview of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 2 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fields in a Bianisotropic Material . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Decomposition of Fields and Sources into Transverse and Longitudinal Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Decomposition of Maxwell’s Equations into Transverse and Longitudinal Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ 2.2.3 H⇢ in Terms of Eu , Hu , and Sources . . . . . . . . . . . . . . . . . . ~ 2.2.4 E⇢ in Terms of Eu , Hu , and Sources . . . . . . . . . . . . . . . . . . 2.2.5 TE/TM Decomposition of the Fields in a Bianisotropic material . . . 2.2.6 Explicit Expressions for the Transverse Field Components . . . . . . 2.2.6.1 TE Decomposition . . . . . . . . . . . . . . . . . . . . . . . 2.2.6.2 TM Decomposition . . . . . . . . . . . . . . . . . . . . . . . 2.2.7 Homogeneous Wave Equations for the Longitudinal Field Components 2.2.7.1 TE Homogeneous Wave Equation . . . . . . . . . . . . . . . 2.2.7.2 TM Homogeneous Wave Equation . . . . . . . . . . . . . . 2.3 Specialization to Rectangular Waveguides . . . . . . . . . . . . . . . . . . . 2.3.1 Isotropic Material Filled Guide . . . . . . . . . . . . . . . . . . . . . 2.3.1.1 Solution to the TEz Wave Equation for Isotropic Material Filled Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1.2 Solution to the TMz Wave Equation for Isotropic Filled Guide 2.3.2 Biaxial Material Filled Guide . . . . . . . . . . . . . . . . . . . . . . 2.3.2.1 Solution to the TEz Wave Equation for Biaxial Material Filled Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Gyromagnetic Material Filled Guide . . . . . . . . . . . . . . . . . . 2.3.3.1 Solution to the TEz Wave Equation for Gyromagnetic Material Filled Guide . . . . . . . . . . . . . . . . . . . . . . . ix 1 1 2 6 8 10 13 13 14 15 18 23 25 28 29 30 31 32 33 33 34 35 36 42 46 48 53 55 CHAPTER 3 ISOTROPIC MATERIAL CHARACTERIZATION . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 NRW Characterization for Isotropic Materials . . . . . . . . . . . . . . . . . 3.2.1 NRW Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.1 Field Structure in a Waveguide Filled with Isotropic Material 3.2.1.2 Solution for Reflection and Transmission Coe cients . . . . 3.2.2 Validation of Theoretical Analysis . . . . . . . . . . . . . . . . . . . . 3.2.3 Extraction Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Phase Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Measurement of the Electromagnetic Properties of a Conductor-backed Material Using a Waveguide-Iris Technique . . . . . . . . . . . . . . . . . . . . . 3.3.1 Theoretical Reflection and Transmission Coe cients . . . . . . . . . . 3.3.1.1 Reflection Coe cient with Iris . . . . . . . . . . . . . . . . . 3.3.1.2 Reflection Coe cient with No Iris . . . . . . . . . . . . . . . 3.3.2 Validation of Theoretical Analysis . . . . . . . . . . . . . . . . . . . . 3.3.3 Extraction Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Iris Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 59 60 60 61 62 68 69 73 77 78 78 81 82 100 100 101 105 112 133 CHAPTER 4 BIAXIAL MATERIAL CHARACTERIZATION . . . . . . . . . . . . . . . . . . . . . 134 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.2 NRW Material Characterization Extended to Biaxially Anisotropic Materials 135 4.2.1 Theoretical Reflection and Transmission Coe cients . . . . . . . . . . 136 4.2.1.1 Field Structure in a Waveguide Filled with Biaxial Material 136 4.2.1.2 Solution for Reflection and Transmission Coe cients . . . . 138 4.2.2 Validation of Theoretical Analysis . . . . . . . . . . . . . . . . . . . . 143 4.2.3 Extraction Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.2.4 Measurement Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.2.5 Error and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . 150 4.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.3 Material Characterization of Biaxial Material Using a Reduced Aperture Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.3.1 Theoretical Transmission and Reflection Coe cients Using Mode-Matching Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.3.1.1 Field Structure in a Waveguide Filled with Biaxial Material 164 4.3.1.2 Solution for S-Parameters Using Modal Expansions . . . . . 165 4.3.2 Specification of Computational Accuracy . . . . . . . . . . . . . . . . 168 4.3.3 Validation of Theoretical Analysis . . . . . . . . . . . . . . . . . . . . 169 4.3.4 Extraction Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.3.5 Error and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . 178 x 4.3.6 4.3.7 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 CHAPTER 5 GYROMAGNETIC MATERIAL CHARACTERIZATION . . . . . . . . . . . . . . . 201 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.2 Characterization of Gyromagnetic Material Filling the Cross-Section of a Rectangular Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.2.1 Theoretical Transmission and Reflection Coe cients Using Mode-Matching Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.2.1.1 Field Structure in a Waveguide Filled with Gyromagnetic Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 5.2.1.2 Solutions for S-Parameters Using Modal Expansions . . . . 207 5.2.2 Validation of the Mode-Matching Analysis . . . . . . . . . . . . . . . 213 5.2.3 Extraction Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 5.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.3 Material Characterization of Gyromagnetic Material Using a Reduced Aperture Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 5.3.1 Theoretical Transmission and Reflection Coe cients Using Mode-Matching Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 5.3.1.1 Field Structure in a Waveguide Filled with Gyromagnetic Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5.3.1.2 Solution for S-Parameters Using Modal Expansions . . . . . 223 5.3.2 Validation of Theoretical Analysis . . . . . . . . . . . . . . . . . . . . 226 5.3.3 Extraction Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 5.3.4 Error and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . 234 5.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 CHAPTER 6 VERIFICATION STANDARDS FOR MATERIAL CHARACTERIZATION . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Material Characterization Procedure . . . . . . . . . . . . . . . . . . 6.3 Waveguide Verification Standard Design . . . . . . . . . . . . . . . . 6.4 Computation of S-Parameters of adopted Waveguide Standard using Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Mode Matching Analysis for the Waveguide Standard . . . . . 6.4.2 Validation of the Mode-Matching Analysis . . . . . . . . . . . 6.5 Error and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Theoretical Material Parameters for the Fabricated Standard . 6.5.2 Measured Material Parameters for the Fabricated Standard . . 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 246 248 251 262 262 270 273 274 278 285 CHAPTER 7 CONCLUSIONS AND FUTURE STUDIES . . . . . . . . . . . . . . . . . . . . . . . 288 xi APPENDICES . . . . . . . . . . . . . . . . . . . . . Appendix A: Useful Identities . . . . . . . . . . . Appendix B: De-embedding S-parameters . . . . Appendix C: Calculation of Waveguide Transverse Appendix D: Analysis of NRW Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 292 294 300 321 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 xii LIST OF TABLES Table 3.1 Nominal material parameters for FGM125. These data were obtained using the waveguide NRW method. Data provided by Captain Milo Hyde IV form the Air Force Institute of Technology , Wright-Patterson AFB, Dayton, OH [46] . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Table 3.2 Nominal material parameters for FGM40. These data were obtained using the waveguide NRW method. Data provided by Captain Milo Hyde IV form the Air Force Institute of Technology , Wright-Patterson AFB, Dayton, OH [46] . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Table 4.1 Material parameters for a fictitious biaxial material. . . . . . . . . . . 144 Table 4.2 Material parameters for a fictitious biaxial material. . . . . . . . . . . 173 Table 6.1 Real parts of relative material parameters for the waveguide verification standard optimized for S-band using WR-284 waveguides found using ` r mode matching. Dimensions of the standard are: y1 = y1 = 5.064 mm, ` r y2 = y2 = 23.86 mm, ` = r = 1/8 inch (3.175 mm), and s = 1/2 inch (12.7 mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Table 6.2 Measured dimensions of fabricated waveguide standard. . . . . . . . . 275 xiii LIST OF FIGURES Figure 2.1 Cross-section view of a rectangular waveguide used in derivations for isotropic and gyromagnetic materials. . . . . . . . . . . . . . . . . . 34 Cross-section view of rectangular waveguide used in derivations for biaxially anisotropic materials. . . . . . . . . . . . . . . . . . . . . . 47 Rectangular waveguide with isotropic sample placed in cross-sectional region with waveguide extensions attached. . . . . . . . . . . . . . . 70 Figure 3.2 S-parameters computed for a FGM125 test material. . . . . . . . . . 70 Figure 3.3 Extracted permittivity using HFSS generated S-parameters. . . . . . 74 Figure 3.4 Upper plot shows phase shift through material while lower plot shows extraction of permittivity from S-parameters using NRW method where n = 0 and n = 1. For interpretation of the references to color in this and all other figures, the reader is referred to the electronic version of this dissertation. . . . . . . . . . . . . . . . . . . . . . . . 76 Real parts of relative permittivity extracted from 5 sets of measurements. Center solid line is the average of the measurements. Upper and lower lines show the 95% confidence intervals. Dotted line shows permittivity extracted from the S-parameter measurements performed at AFIT with a di↵erent sample than the one used at MSU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Figure 2.2 Figure 3.1 Figure 3.5 Figure 3.6 Real parts of relative permeability extracted from 5 sets of measurements. Center solid line is the average of the measurements. Upper and lower lines show the 95% confidence intervals. Dotted line shows permeability extracted from the S-parameter measurements performed at AFIT with a di↵erent sample than the one used at MSU. 80 Figure 3.7 Conductor-backed material with iris and waveguide extension attached. 83 Figure 3.8 Top view of reduced step junction. . . . . . . . . . . . . . . . . . . . 88 Figure 3.9 Top view of expanded step junction. . . . . . . . . . . . . . . . . . . 88 Figure 3.10 Top view of pinched-down iris. . . . . . . . . . . . . . . . . . . . . . 97 Figure 3.11 Top view of material-loaded guide. . . . . . . . . . . . . . . . . . . . 97 xiv Figure 3.12 S-parameters computed for a conductor-backed FGM125 test material placed behind an iris. . . . . . . . . . . . . . . . . . . . . . . . . 102 Figure 3.13 S-parameters computed for a conductor-backed FGM125 test material.103 Figure 3.14 Extraction of permittivity and permeability from S-parameter generated using HFSS with d = 0 mm and d = 1 mm in the model. . . . . 106 Figure 3.15 Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the parameter sweep assuming w2 = 11.43 mm and d = 0 mm at 8.2 GHz. . . . . . . . . 109 Figure 3.16 Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the parameter sweep assuming w2 = 11.43 mm and d = 0 mm at 10.09 GHz. . . . . . . . 110 Figure 3.17 Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the parameter sweep assuming w2 = 11.43 mm and d = 0 mm at 12.4 GHz. . . . . . . . . 111 Figure 3.18 Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the d parameter sweep assuming w2 = 11.43 mm and = 0.1 mm at 8.2 GHz. . . . . . . . 113 Figure 3.19 Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the d parameter sweep assuming w2 = 11.43 mm and = 0.1 mm at 10.09 GHz. . . . . . . 114 Figure 3.20 Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the d parameter sweep assuming w2 = 11.43 mm and = 0.1 mm at 12.4 GHz. . . . . . . 115 Figure 3.21 Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the w2 parameter sweep assuming d = 0 mm and = 0.1 mm at 8.2 GHz. . . . . . . . . . . 116 xv Figure 3.22 Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the w2 parameter sweep assuming d = 0 mm and = 0.1 mm at 10.09 GHz. . . . . . . . . . 117 Figure 3.23 Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the w2 parameter sweep assuming d = 0 mm and = 0.1 mm at 12.4 GHz. . . . . . . . . . . 118 Figure 3.24 Comparison of the error propagated to constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the parameter sweep assuming w2 = 11.43 mm and d = 0 mm at 8.2 GHz. . . . . . 119 Figure 3.25 Comparison of the error propagated to the constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the parameter sweep assuming w2 = 11.43 mm and d = 0 mm at 10.09 GHz. . . . . 120 Figure 3.26 Comparison of the error propagated to the constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the parameter sweep assuming w2 = 11.43 mm and d = 0 mm at 12.4 GHz. . . . . 121 Figure 3.27 Comparison of the error propagated to constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the d parameter sweep assuming w2 = 11.43 mm and = 0.1 mm at 8.2 GHz. . . . 122 Figure 3.28 Comparison of the error propagated to constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the d parameter sweep assuming w2 = 11.43 mm and = 0.1 mm at 10.09 GHz. . . 123 Figure 3.29 Comparison of the error propagated to constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the d parameter sweep assuming w2 = 11.43 mm and = 0.1 mm at 12.4 GHz. . . . 124 Figure 3.30 Comparison of the error propagated to constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the w2 parameter sweep assuming d = 0 mm and = 0.1 mm at 8.2 GHz. . . . . . . . 125 xvi Figure 3.31 Comparison of the error propagated to constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the w2 parameter sweep assuming d = 0 mm and = 0.1 mm at 10.09 GHz. . . . . . 126 Figure 3.32 Comparison of the error propagated to constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the w2 parameter sweep assuming d = 0 mm and = 0.1 mm at 12.4 GHz. . . . . . . 127 Figure 3.33 Manufactured iris with waveguide extension, sliding short sample holder, and FGM125 sample. . . . . . . . . . . . . . . . . . . . . . . 129 Figure 3.34 Relative permittivity and permeability extracted using the iris technique and the air-backed/conductor-backed method. . . . . . . . . . 130 Figure 3.35 I Comparison of measured and forward problem generated S11 . . . . . 131 Figure 3.36 N Comparison of measured and forward problem generated S11 . . . . . 132 Figure 4.1 Rectangular waveguide with biaxial sample placed in cross-sectional region with waveguide extensions attached. . . . . . . . . . . . . . . 145 Figure 4.2 S-parameters computed for a biaxial test material. . . . . . . . . . . 145 Figure 4.3 Permittivity extraction using HFSS generated S-parameters. . . . . 151 Figure 4.4 Permittivity extraction using HFSS generated S-parameters. . . . . 152 Figure 4.5 Permeability extraction using HFSS generated S-parameters. . . . . 153 Figure 4.6 Permeability extraction using HFSS generated S-parameters. . . . . 154 Figure 4.7 Real relative permittivities for a fictitious material extracted using 100,000 random trials. Center line is the average of the trials. Upper and lower bars show the 95% confidence interval. . . . . . . . . . . . 156 Figure 4.8 Real relative permeabilities for a fictitious material extracted using 100,000 random trials. Center line is the average of the trials. Upper and lower bars show the 95% confidence interval. . . . . . . . . . . . 157 Figure 4.9 Imaginary relative permittivities for a fictitious material extracted using 100,000 random trials. Center line is the average of the trials. Upper and lower bars show the 95% confidence interval. . . . . . . . 158 xvii Figure 4.10 Imaginary relative permeabilities for a fictitious material extracted using 100,000 random trials. Center line is the average of the trials. Upper and lower bars show the 95% confidence interval. . . . . . . . 159 Figure 4.11 Waveguide cubical sample holder with waveguide extensions attached. 162 Figure 4.12 Cross-sectional view of reduced-aperture waveguide. . . . . . . . . . 162 Figure 4.13 Side view of reduced aperture waveguide. . . . . . . . . . . . . . . . 163 Figure 4.14 Magnitude and phase of reflection and transmission coe cients for teflon sample completely filling reduced-aperture waveguide sample regions at 2.6 GHz. These coe cients are plotted vs. 1/N . . . . . . 170 Figure 4.15 Magnitude and phase of reflection and transmission coe cients for teflon sample completely filling reduced-aperture waveguide sample regions at 3.275 GHz. These coe cients are plotted vs. 1/N . . . . . 171 Figure 4.16 Magnitude and phase of reflection and transmission coe cients for teflon sample completely filling reduced-aperture waveguide sample regions at 3.95 GHz. These coe cients are plotted vs. 1/N . . . . . . 172 Figure 4.17 S-parameters computed for a biaxial test material. . . . . . . . . . . 174 Figure 4.18 Extracted permittivity using HFSS generated S-parameters. . . . . . 179 Figure 4.19 Extracted permeability using HFSS generated S-parameters. . . . . 180 Figure 4.20 Relative permittivities for a fictitious material extracted using 500 random trials. Center line is the average of the trials. Upper and lower lines show the 95% confidence interval. . . . . . . . . . . . . . 181 Figure 4.21 Relative permeabilities for a fictitious material extracted using 500 random trials. Center line is the average of the trials. Upper and lower lines show the 95% confidence interval. . . . . . . . . . . . . . 182 Figure 4.22 Cubical sample holder and teflon sample. . . . . . . . . . . . . . . . 184 Figure 4.23 Relative permittivities extracted from 10 sets of teflon measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval for ✏A . . . . . . . . . . . . . 186 Figure 4.24 Relative permeabilities extracted from 10 sets of teflon measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval for µA . . . . . . . . . . . . 187 Figure 4.25 Cubical sample holder and alternating-layer dielectric sample. . . . . 188 xviii Figure 4.26 Relative permittivity ✏A extracted from 10 sets of measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval. . . . . . . . . . . . . . . . 191 Figure 4.27 Relative permittivity ✏B extracted from 10 sets of measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval. . . . . . . . . . . . . . . . 192 Figure 4.28 Relative permittivity ✏C extracted from 10 sets of measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval. . . . . . . . . . . . . . . . 193 Figure 4.29 Relative permeability µA extracted from 10 sets of measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval. . . . . . . . . . . . . . . . 194 Figure 4.30 Relative permeability µB extracted from 10 sets of measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval. . . . . . . . . . . . . . . . 195 Figure 4.31 Relative permeability µC extracted from 10 sets of measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval. . . . . . . . . . . . . . . . 196 Figure 4.32 Average permittivity from 10 sets of measurements extrapolated using a fifth order polynomial. . . . . . . . . . . . . . . . . . . . . . . 198 Figure 4.33 Average permeability from 10 sets of measurements extrapolated using a fifth order polynomial. . . . . . . . . . . . . . . . . . . . . . . 199 Figure 4.34 Relative permittivity extracted from 10 sets of measurements assuming the sample is nonmagnetic. Center line is the average of the measurements. Upper and lower lines show the 95% confidence interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Figure 5.1 Rectangular waveguide with gyromagnetic sample placed in crosssectional region with waveguide extensions attached. . . . . . . . . . 206 Figure 5.2 Side view of gyromagnetic material filled waveguide. . . . . . . . . . 206 Figure 5.3 Permeability tensor entries determined using (5.3) and (5.4). . . . . 214 Figure 5.4 S-parameters computed for a gyromagnetic test material. . . . . . . 216 Figure 5.5 Material parameters characterized using FEM generated S-parameters.218 xix Figure 5.6 Reduced-aperture waveguide sample holder with waveguide extensions attached. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Figure 5.7 Cross-sectional view of reduced-aperture waveguide. . . . . . . . . . 221 Figure 5.8 Side view of reduced aperture waveguide. . . . . . . . . . . . . . . . 222 Figure 5.9 Magnitude of S-parameters computed for a gyromagnetic test material. FEM simulation uses 100000 unknowns. . . . . . . . . . . . . . 228 Figure 5.10 Phase of S-parameters computed for a gyromagnetic test material. FEM simulation uses 100000 unknowns. . . . . . . . . . . . . . . . . 229 Figure 5.11 Magnitude of S-parameters computed for a gyromagnetic test material. FEM simulation uses 400000 unknowns. . . . . . . . . . . . . . 230 Figure 5.12 Phase of S-parameters computed for a gyromagnetic test material. FEM simulation uses 400000 unknowns. . . . . . . . . . . . . . . . . 231 Figure 5.13 Relative permittivity parameters characterized using FEM generated S-parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Figure 5.14 Relative permeability parameters characterized using FEM generated S-parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Figure 5.15 Relative permittivity extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 12.02 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permittivity values used to generate the S-parameters employed in the Monte Carlo simulation. . . . . . . . . . . . . . . . 236 Figure 5.16 RRelative permittivity extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 24.04 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permittivity values used to generate the S-parameters employed in the Monte Carlo simulation. . . . . . . . . . . . . . . . 237 Figure 5.17 Relative permittivity extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 36.06 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permittivity values used to generate the S-parameters employed in the Monte Carlo simulation. . . . . . . . . . . . . . . . 238 xx Figure 5.18 Relative permittivity extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 48.08 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permittivity values used to generate the S-parameters employed in the Monte Carlo simulation. . . . . . . . . . . . . . . . 239 Figure 5.19 Relative permittivity extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 60.10 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permittivity values used to generate the S-parameters employed in the Monte Carlo simulation. . . . . . . . . . . . . . . . 240 Figure 5.20 Relative permeability values extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 12.02 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permeability values determined using (5.3) and (5.4). 241 Relative permeability values extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 24.04 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permeability values determined using (5.3) and (5.4). 242 Relative permeability values extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 36.06 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permeability values determined using (5.3) and (5.4). 243 Relative permeability values extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 48.08 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permeability values determined using (5.3) and (5.4). 244 Relative permeability values extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 60.10 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permeability values determined using (5.3) and (5.4). 245 Figure 5.21 Figure 5.22 Figure 5.23 Figure 5.24 xxi Figure 6.1 Waveguide material measurement system showing presence of material sample (top), and waveguide verification standard surrogate ` r material (bottom). Adopted surrogate has ` = r , y1 = y1 , and ` r y2 = y2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Figure 6.2 E↵ective material parameters extracted from symmetric and asymmetric layered dielectric surrogate materials. . . . . . . . . . . . . . 252 Figure 6.3 Matlab to HFSS flow chart. . . . . . . . . . . . . . . . . . . . . . . . 255 Figure 6.4 Example of brass-post surrogate design. . . . . . . . . . . . . . . . . 256 Figure 6.5 Characterization results from a GA analysis of a brass-post surrogate design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Figure 6.6 X = Figure 6.7 Real parts of material parameters extracted from HFSS simulation of optimized verification standard. . . . . . . . . . . . . . . . . . . . . 260 Figure 6.8 X = Figure 6.9 Computed S-parameters of the verification standard. . . . . . . . . . 271 Figure 6.10 Comparison of material parameters extracted from mode matching and HFSS simulated S-parameters. . . . . . . . . . . . . . . . . . . . 272 Figure 6.11 Relative constitutive parameters found when randomly varying ` , s , and r extracted using 500 trials. Center line in the upper plot is the average of real relative parameters of the trials, while the upper and lower dashed-lines show the 95% confidence intervals. The average of the imaginary relative parameters is near zero, while the lower plot shows +2 values. . . . . . . . . . . . . . . . . . . . . . . 275 Figure 6.12 ` Relative constitutive parameters found when randomly varying y1 , ` r r y2 , y1 , and y2 extracted using 500 trials. Center line in the upper plot is the average of real relative parameters of the trials, while the upper and lower dashed-lines show the 95% confidence intervals. The average of the imaginary relative parameters is near zero, while the lower plot shows +2 values. . . . . . . . . . . . . . . . . . . . . . . 276 S11r for optimized brass-post surrogate. Extracted material |S11 |2 parameters become complex over regions where X 2  1. . . . . . . . 258 S11r of optimized verification standard. Extracted material |S11 |2 parameters are real over regions where X 2 > 1. . . . . . . . . . . . . 261 xxii Figure 6.13 Relative constitutive parameters found when randomly varying ` , s , r , y ` , y ` , y r , and y r extracted using 500 trials. Center line 1 2 1 2 in the upper plot is the average of real relative parameters of the trials, while the upper and lower dashed-lines show the 95% confidence intervals. The average of the imaginary relative parameters is near zero, while the lower plot shows +2 values. . . . . . . . . . . . . . 277 Figure 6.14 Brass aperture plates forming the fabricated waveguide verification standard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Figure 6.15 Real parts of relative permittivity and permeability for optimized geometry (solid line) and fabricated geometry (dotted line). . . . . . 280 Figure 6.16 Imaginary parts of relative permittivity and permeability for optimized geometry (solid line) and fabricated geometry (dotted line). . 281 Figure 6.17 Real parts of relative permittivity and permeability extracted from 10 sets of measurements. Center solid line is the average of the measurements. Upper and lower lines show the 95% confidence intervals. Dotted line shows the material parameters extracted from the modematching S-parameters generated using the measured geometry. . . 283 Figure 6.18 Imaginary parts of relative permittivity and permeability extracted from 10 sets of measurements. Center solid line is the average of the measurements. Upper and lower lines show the 95% confidence intervals. Dotted line shows the material parameters extracted from the mode-matching S-parameters generated using the measured geometry. 284 Figure 6.19 Real parts of the relative permittivity and permeability extracted from 10 sets of measurements (solid lines). Center solid line is the average of the measurements. Upper and lower solid lines show the 95% confidence intervals. Dotted lines show mode-matching results for 500 random trials. Center dotted line is the average and upper and lower dotted lines show the 95% confidence intervals. . . . . . . 286 Figure 6.20 Imaginary parts of the relative permittivity and permeability extracted from 10 sets of measurements (solid lines). Center solid line is the average of the measurements. Upper and lower solid lines show the 95% confidence intervals. Dotted lines show mode-matching results for 500 random trials. Center dotted line is the average and upper and lower dotted lines show the 95% confidence intervals. . . 287 Figure B.1 Measurement and sample plane modal coe cients for a sample holder.295 xxiii Figure B.2 Measurement and sample plane modal coe cients with sample inserted into the waveguide extension attached to port 1. . . . . . . . 295 xxiv CHAPTER 1 INTRODUCTION AND BACKGROUND 1.1 Electromagnetic Material Characterization Electromangetic material characterization has long been an interest of the research community. The goal is to determine permittivity (✏) and permeability (µ), also known as the constitutive parameters, as accurately as possible [1]. The constitutive parameters describe the e↵ect externally applied electric and magnetic fields have on a material. Many disciplines rely on the knowledge of the electromagnetic material properties, including stealth and integrated circuits. In stealth technology, the constitutive parameters describe how e↵ectively a particular materials absorbs a radar signal [2]. Integrated circuits rely on new materials, and subsequently accurate knowledge of constitutive parameters, to increase the ability to transmit higher bandwidth signals as clock speeds in electronic devices continue to increase [3]. Engineered materials, which have gained increasing importance in a variety of applications at microwave frequencies, often have anisotropic electromagnetic characteristics. The use of engineered materials in the design of radio frequency (RF) systems requires an ac- 1 curate knowledge of material constitutive parameters. Some recently synthesized materials include the use of graphene nanoribbons or metallic inclusions in miniaturization of electronic components [4]-[5], ferrite loaded polymers to increase EMI shielding [6], cellular materials such as honeycomb structures to decrease radar cross section [7]-[8], and anisotropic materials used to enhance antenna operation [9]-[10]. Since the properties of these materials are often hard to accurately predict (due to modeling uncertainties and variability in the manufacturing process), they are usually measured in a laboratory. The theoretical implications of anisotropic materials are well described in the literature [11], and several techniques for the characterization of these materials have been introduced. These include free space methods [12]-[13], resonant cavity methods [14]-[17], open-ended waveguide probe methods [18]-[19], and rectangular waveguide methods [20]-[22]. Free-space methods are unattractive for many materials because the sample size must be large compared to the footprint of the interrogating beam to minimize edge di↵raction, and samples are generally limited in size. Similarly, contact probe methods require sample sizes larger than the probe aperture. Cavity methods usually involve small samples but are inherently narrowband, and thus characterization of materials over a wide frequency range is di cult. This dissertation concentrates on the use of rectangular waveguide applicators for anisotropic material characterization. 1.2 Rectangular Waveguide Material Characterization Rectangular waveguide methods are popular for measuring the electromagnetic properties of materials because of ease of sample preparation, high signal strength, and wide bandwidth. A variety of characterization techniques use rectangular waveguide applicators. In these techniques, samples can be placed inside the guide, against an open-ended flange, or even pressed against a slot cut into a waveguide wall. The most common method is to machine a sample to fill the cross-section of the waveguide and measure the transmission and reflection 2 coe cients. These measurements provide the two necessary data to find both material parameters of an isotropic sample. One of the most widely used extraction algorithms is the Nicolson-Ross-Weir (NRW) method [23]-[24]. The attraction of the NRW characterization method results from the availability of closed-form expressions for µ and ✏. The NRW extraction can be used with rectangular waveguides [25], coaxial applicators [26]-[27], freespace methods [28], and stripline measurements [29]. Unfortunately, a closed form expression is not available for most types of anisotropic materials. Typically, these parameters are obtained using iterative solvers, such as Newton’s method [30]-[31] or a least squares approach [32]-[33] by comparing theoretically computed and measured reflection and transmission coe cients. Rectangular waveguide characterization techniques can be limited by sample size restrictions or poor signal-to-noise ratio (SNR) for the transmission. To maintain dominant mode excitation, the cross section of a waveguide becomes larger as the operational frequency of the guide decreases. In [34], the authors make use of a reduced-waveguide technique to characterize isotropic materials with small electrical size at low frequencies, avoiding the large waveguides ordinarily required. The reflection and transmission coe cients are measured to determine the scalar permittivity and permeability. The use of waveguide applicators may also result in extracted material parameters that are highly sensitive to error propagation from measurement uncertainties. This is especially prominent when measuring high loss or highly reflective samples where there is poor SNR for the transmission coe cient measurement. In [35] the authors present a partially-filled waveguide method that enhances transmission quality and accuracy of the material characterization. It was demonstrated that this method accurately characterizes samples with low-loss and high-loss parameters. To calculate the theoretical reflection and transmission coe cients, both of these techniques use a mode-matching technique to accommodate the high-order mode excitation resulting from the waveguide discontinuities. The extraction of material parameters was performed in [34] using a nonlinear least squares algorithm, while [35] used a Newton root search method. 3 Other rectangular waveguide techniques make use of only the reflection measurements because of the simplification of the measurement configuration, or because a transmission measurement is not available. Two common types of reflection-only characterization techniques are underlay and overlay methods [36]. Underlay methods make use of changing the backing of the material under test (MUT). The air/conductor backed method [37] measures a sample first with the MUT backed by a conductor and then with the MUT backed by air. In a rectangular waveguide these measurements are performed with the MUT backed by a short circuit and a match load. In the layer-shift method [38] the reflection is measured with a conductor placed at two positions behind the MUT. For a rectangular waveguide measurement system this is done using two di↵erent o↵set shorts. Overlay methods are commonly used when the backing of the MUT cannot be altered. Magnetic radar-absorbing materials (MagRAM) are often applied to the conducting surfaces of air vehicles to reduce radar cross section. To ensure proper aircraft design, the electromagnetic properties of these materials must be accurately characterized. Unfortunately, the process of bonding the MagRAM to the conductor produces a chemical reaction that alters the intrinsic electromagnetic properties of the absorber in unpredictable ways, and thus it is crucial that the absorber be characterized while still attached to the conductor backing. If the sample is conductor backed, and occupies the entire cross section of a waveguide, a transmission measurement is not available, and thus a method must be found for providing two su ciently di↵erent reflection measurements. Overlay methods are prominent in contact probe characterization techniques. In [39], the authors achieve the characterization of ✏ and µ of a conductor-backed sample using a rectangular open-ended waveguide probe applicator. Two independent reflection measurements are obtained using either two thicknesses of the same material or one measurement with one material and a second measurement with a known material as a top layer. The theoretical reflection coe cients necessary for characterization are determined using a variational method to develop expressions of the waveguide impedance. The approach using two thicknesses is useful in the lab where two samples of the 4 MUT are available, but it is not practical in the field where the MUT is typically a xed to a conductor backing. However, material characterization using the two-layer method in the field is possible since a known layer of material can be placed on top of a conductor-backed sample without having to remove the conductor backing. Another contact probe method is described in [40], where the authors discuss the use of a two-iris waveguide probe technique to characterize lossy, conductor-backed materials. A open-ended rectangular waveguide is applied to the MUT, and the reflection coe cient is measured under two conditions. First, the reflection is measured when the open-ended waveguide aperture is unobstructed, and a second measurement is obtained when an iris is placed in the aperture of the guide. This provides the two necessary data to characterize the constitutive parameters. The theoretical reflection coe cients are determined using a combination of a modal expansion in the waveguide and aperture regions with a magneticfield integral equation (MFIE). It is found that this technique is not as accurate as the two-thickness method but is better than the two-layer method. Since the two-thickness method cannot be employed in the field, the two-iris technique provides a favorable contact probe method. A method is describe in [41] where conductor backed material samples are characterized through the use of a transverse slot cut in the bottom wall of a rectangular waveguide. The reflection and transmission coe cients of the sample pressed against the waveguide slot are measured. The theoretical coe cients were determined using a MFIE technique. A Newton’s complex root search algorithm is then employed to iteratively solve for the constitutive parameters by comparing measurements to theory. This technique was partially successful using the transmission and reflection measurements for characterization. However, it was determined that using measurements of two separate thicknesses of the same material was more encouraging for characterization of ✏ and µ. This dissertation investigates a characterization technique for determining scalar ✏ and µ of a conductor-back material when two thicknesses of a sample are not available. In the 5 present case, there is no need for measurements to be conducted in the field; therefore a sample can be manufactured for placement inside a rectangular waveguide for laboratory measurements. An iris approach, like that discussed in [40], is used to provide the two necessary reflection measurements for characterization. 1.3 Biaxially Anisotropic Material Characterization The tensors for biaxial materials have six nonzero entries, corresponding to six complex material parameters, and therefore a minimum three di↵erent sets of complex reflection and transmission coe cient measurements are required. The typical inverse problem consists of first determining what measured data are su cient for unequivocal determination of the constitutive parameters, and secondly, what experimental setup is preferred to collect the needed data. Some methods consider only characterizing the permittivity tensor, which reduces the number of unknown constitutive parameters to three. One such method uses a cavity perturbation technique to determine the permittivity parameters of a sample [17]. Here, the authors present the methodology to characterize low-loss engineered material and naturally uniaxial dielectrics. In the case of uniaxial dielectric materials there are only two unknown material parameters, because two dielectric material parameters are identical and the third is di↵erent. Multiple samples are manufactured and measured with the axes of the material tensor aligned with the geometry of the cavity. Using the multiple measurements, the cavity perturbation technique is combined with a finite element (FEM) solver to determine the permittivity tensor and loss tangents. This method proved to produce highly accurate results for characterizing the loss tangents. However, like most cavity methods, the characterization is inherently narrow-band. Another method that considers characterizing only the dielectric material properties is discussed in [42]. The authors use a modified free-space focused beam system to measure the co- and cross-polarized transmission coe cients. Plane waves probe the material to provide 6 the adequate number of measurements for characterization. This approach uses a hybrid global-local optimization to obtain the material parameters by minimizing the di↵erence between theoretically computed and measured transmission coe cients. The theoretical coe cients are determined using a computational electromagnetic (CEM) modeling tool. Using CEM modeling for calculation of the theoretical coe cients can be computationally expensive. Thus, the overall extraction of the material parameters is computationally intensive when a lengthy optimizer is used. Analytical solutions for the theoretical coe cients are more attractive since the time required for convergence of the solutions is typically serval orders of magnitude smaller than the time required for CEM modeling tools. In [18], the authors describe a waveguide probe system for characterizing both the electric and magnetic biaxial material properties. Here the authors extended the open-ended waveguide probe method to anisotropic materials. This method makes use of higher-order modes produced at the probe aperture. These higher-order modes have fields with components in multiple directions, and thus the fields are related to all of the entries in the biaxial tensors. To determine the material parameters, multiple measurements with the probe oriented in multiple directions is required. Two coupled electric field integral equations (EFIEs) for the aperture field are solved numerically using a method of moments. The electric field is then expressed in terms of the material parameters. A Newton-Raphson method is employed to minimize the di↵erence between theoretically computed and measured reflection coe cients. Like with most waveguide probe applicators, the size of the required sample must be larger than the probe aperture for accurate characterization. Placement of a biaxial sample in the cross-section of a rectangular waveguide is considered in [20]. The necessary measurements required for characterization are obtained by using TE10 and TE20 mode excitations. Closed-form solutions for three out of the six material parameters are derived assuming these two di↵erent excitations. By rotating and measuring the sample again, the additional material parameters can be extracted in closed form. Here four measurements are obtained to characterize three material parameters. The additional 7 measurement greatly simplifies the steps needed to solve the inverse problem. A complicated measurement fixture is proposed in this paper where the two di↵erent modes are excited; however, no actual measurements of a biaxial sample were carried out. The authors do not go into detail on how to rotate the sample such that the fields interrogate all the material parameters. Since the shape of the sample must be rectangular to fill the cross section of the guide, it is assumed that a separate sample must be manufactured with the material parameters aligned di↵erently. This dissertation investigates two biaxial material characterization techniques. The first is an extension of the NRW method, where solutions for all six material parameters can be obtained in closed form. Much like in [20], multiple samples are required. However, a complicated waveguide applicator is not needed for characterization. To characterize a biaxial material using a single sample, the second method considers a sample holder which can accommodate a cubical sample. Here the single sample is interrogated in multiple orientations. This reduced-waveguide technique is similar to the isotropic characterization method discussed in [34]. However, the goal in [34] was to accommodate samples of small electrical size at low frequencies. Thus, sample shapes were typically not cubical, and a single sample was measured under one orientation to determine the scalar permittivity and permeability. In this dissertation the multiple measurements of the single sample provide the necessary measurements for full biaxial material characterization. 1.4 Gyromagnetic Material Characterization Gyromagnetic materials are commonly used in microwave devices such as circulators, phase shifters, and tunable filters [43]. As a step towards full anisotropic tensor characterization, this dissertation considers the extraction of diagonal and o↵-diagonal tensor entries. Gyromagnetic materials have scalar permittivity and anisotropic permeability. The permeability tensor has two equivalent (µg ) and one di↵erent (µ0 ) diagonal entries, and two o↵-diagonal 8 entries. The two o↵-diagonal tensor entries are determined using the same parameter . Thus characterizing gyromagnetic materials require the extraction of three complex unknowns: ✏, µg , and . In [22] and [44] the authors describe a method for measuring the complex scalar permittivity and permeability tensor components of a gyromagnetic material using a partially filled waveguide technique. A sample partially fills the cross-section of the guide in the broad dimension while extending fully in the shorter dimension. The sample is placed o↵ center about the broad dimension. A known isotropic material is inserted next to the gyromagnetic sample for support. A mode-matching technique is used to derive the theoretical reflection and transmission coe cients. With this anisotropic material o↵ center, the waveguide system is nonreciprocal, which means S11 = S22 and S21 6= S12 . The nonreciprocity of the system permits the characterization of the permeability tensor and scalar permittivity from a signal set of measurements. The material parameters are extracted using an optimization method which seems to minimize the di↵erence between theoretically computed and measured S-parameters. In this technique, the calculation of the theoretical S-parameters needs to deal with a complicated field structure in the sample regions. This complicated field structure includes finding the modal propagation constants by solving a transcendental equation. Thus the implementation of this technique is numerically complicated. In this dissertation, two methods are investigated for the characterization of gyromagnetic materials. The first approach considers completely filling the cross-section of a rectangular waveguide while the second technique uses a reduced aperture sample holder to accommodate samples of small electrical size. The calculation of the theoretical reflection and transmission coe cients for both methods are analytically less complex than the approach described in [22] and [44]. Thus the implementation of the techniques described in this dissertation are numerically less complicated. 9 1.5 Overview of Research The motivation for this dissertation is to explore the possible methods for characterization of anisotropic materials. As a stepping stone to developing methods for characterizing materials with fully-populated anisotropic tensor characteristics, techniques are presented in this dissertation to characterize isotropic, biaxially anisotropic, and gyromagnetic materials. Two characterization techniques are investigated for each of these materials, and thus six di↵erent techniques are described. Additionally, a waveguide standard is introduced which may be used to validate the measurement of the permittivity and permeability of materials at microwave frequencies. Chapter 2 provides the mathematical background required for the modal analysis used in anisotropic material characterization techniques discussed in this dissertation. A bianisotropic medium containing sources is considered for transverse and longitudinal direction decomposition. Following this, a TE/TM decomposition is performed and the resulting differential equation is solved using a Fourier transformation method. Next the geometries of the materials considered in this dissertation considers are restricted to fit into a rectangular waveguide applicator and the appropriate wave equations arenderived. These wave equations are then solved using separation of variables in conjunction with the application of boundary conditions on the waveguide conducting walls. Finally, expressions for the modal fields for various anisotropic materials placed inside a rectangular waveguide are formulated. These can then be used in the proposed characterization techniques. The third chapter presents two methods for characterizing isotropic materials. The first technique is the classical NRW extraction method for a sample filling the cross section of a rectangular waveguide. The second technique assumes the sample is conductor-backed and occupies the entire cross section. Here a transmission measurement is not available, and thus a method must be found for providing two su ciently di↵erent reflection measurements. The technique proposed investigates placing a waveguide iris in front of the sample, thereby ex- 10 posing the sample to a spectrum of evanescent modes. By measuring the reflection coe cient with and without an iris, the necessary two data may be obtained to determine ✏ and µ. A mode-matching approach is used to determine the theoretical response of a sample placed behind a waveguide iris. This response is used in a root-searching algorithm to determine ✏ and µ using measurements of the reflection coe cient. A Monte Carlo analysis is used to characterize the sensitivity of the technique to typical measurement uncertainties, thereby providing a basis for the design of the waveguide applicator. Measured results are presented using commercially-available absorbers to establish the e cacy of both techniques. Chapter 4 discusses two techniques for measuring the electromagnetic properties of a biaxially anisotropic material sample. The first approach uses three samples machined from the same material with the axes of the permittivity and permeability tensors aligned with the geometry of the waveguide. The six complex material parameters can be extracted from the measured reflection and transmission coe cients of these three samples using closedform expressions. The second technique uses a reduced-aperture waveguide sample holder designed to accommodate a cubical sample. All of the tensor material parameters can then be determined by measuring the reflection and transmission coe cients of a single sample placed into several orientations. The theoretical reflection and transmission coe cients necessary to perform the material parameter extraction are obtained using a modal analysis technique. A Newton’s method is used to perform the extraction by comparing theoretically computed and measured reflection and transmission coe cients. A Monte Carlo analysis is used to investigate the e↵ects of the propagation of random error inherent to a vector network analyzer (VNA) used in measurement for both techniques. S-band material parameters are extracted for a biaxial sample placed in the reduced-aperture waveguide sample holder. The fifth chapter describes two characterization techniques used to determine the electromagnetic properties of gyromagnetic materials. The first technique examined considers completely filling the cross-section of a waveguide. The material parameters are extracted from the measured reflection and transmission coe cients. Unfortunately, the available sample 11 size for gyromagnetic materials is too small to completely fill the waveguide cross section. To address this issue, the second technique describes the use of a reduced-aperture sample holder that does not require the sample to fill the entire cross section. Modal analysis is used in both methods to determine the reflection and transmission coe cients of the dominant mode. A non-linear least squares method is employed to extract the gyromagnetic material parameters. Details on the methodology for obtaining the theoretical reflection and transmission coe cients and a comparison to a full wave FEM solver are presented for both techniques. A Monte Carlo analysis is used to characterize the sensitivity of the reduced aperture approach to typical measurement uncertainties for di↵erent sized apertures. The sixth chapter in this dissertation introduces a waveguide standard that acts as a surrogate material with both electric and magnetic properties and is useful for verifying systems designed to characterize engineered materials using the NRW technique. A genetic algorithm was used to optimize the all-metallic structure of the standard to produce a surrogate with both relative permittivity and permeability near six across S-band, and with low sensitivity to changes in geometry to reduce the e↵ects of fabrication errors. A mode-matching approach allows the user to predict the material properties with high accuracy, and thus compensate for di↵erences in geometry due to machining inaccuracy or limited availability of component parts. The mode-matching method also allows the user to design standards that may be used within other measurement bands. An example standard is characterized experimentally, the errors due to uncertainties in measured dimensions and the experimental repeatability are explored, and the usefulness of the standard as a verification tool is validated. The final chapter discusses any concluding remarks and suggestions for future work. 12 CHAPTER 2 Mathematical Background 2.1 Introduction This chapter presents the theory required for the formulation of the waveguide modal analysis for anisotropic materials filling a rectangular waveguide. The field structures are first examined for a bianisotropic medium containing electric and magnetic sources. The sources and fields are decomposed along transverse and longitudinal directions. Next a TE/TM decomposition of the fields is performed, and the transverse fields are related to the longitudinal fields in terms of a di↵erential equation. The di↵erential equation is solved using a Fourier transform method. Then the specific materials of interest in this dissertation are considered for the derivation of the TE/TM wave equations. The solution to the wave equation determines the longitudinal fields. Additionally, expressions are derived that explicitly relate the transverse fields to the longitudinal fields. The wave equation is then solved using separation of variables and the application of boundary conditions at the rectangular waveguide conducting walls. 13 2.2 Fields in a Bianisotropic Material The characterization of the electromagnetic properties of a material is largely dependent on the physical behavior of the material in the presence of an electromagnetic field. Maxwell’s equations and the constitutive relations describe the overall relationship between the electromagnetic field and the media through which the field propagates. Consider a homogeneous ~ r bianisotropic medium containing phasor electric current sources, J (~), and magnetic current ~ r sources, Jm (~). Maxwell’s equations in the frequency domain are given by ~ r r ⇥ E (~) = ~ r j! B (~) ~ r Jm (~) ~ r ~ r ~ r r ⇥ H (~) = j! D (~) + J (~) (2.1) (2.2) ~ r·D = ⇢ (2.3) ~ r · B = ⇢m . (2.4) The constitutive relationships are expressed as ~ r ~ r ~ r D (~) = ✏ · E (~) + ⇠ · H (~) (2.5) ~ r ~ r ~ r B (~) = ⇣ · E (~) + µ · H (~) . (2.6) ~ ~ ~ Here E is the electric field intensity, H is the magnetic field intensity, D is the electric flux ~ density, and B is the magnetic flux density. Additionally, ✏ is the permittivity dyadic and µ is the permeability dyadic. The quantities ⇠ and ⇣ are coupling dyadics, and ⇢ and ⇢m are the electric and magnetic charge density, respectively. In (2.1) - (2.6) it is assumed that µ and ✏ do not depend on applied field strengths or spatial coordinates and thus the material is linear and homogeneous. 14 2.2.1 Decomposition of Fields and Sources into Transverse and Longitudinal Components The first step for field decomposition is obtaining equations that relate the transverse field to the longitudinal fields. For brevity, the dependence on ~ in (2.1) - (2.6) will be left implicit. r The sources and fields are decomposed into their respective transverse and longitudinal parts. ~ The longitudinal component of A is defined as uAu , where u is a constant but otherwise ˆ ˆ arbitrary direction. Here Au = u · A. ˆ ~ (2.7) The transverse component in turn is defined as ~ ~ A⇢ = A u Au , ˆ (2.8) ~ such that u · A⇢ = 0. Using this notation, the longitudinal component of B is given by ˆ ~ Bu = u · B. ˆ ~ (2.9) Using the constitutive relationship for bianisotropic media, (2.6), and the distributive identity (A.2), (2.9) becomes h i h i ~ ~ Bu = u · ⇣ · E + u · µ · H . ˆ ˆ (2.10) From the dyadic identity (A.2), it can be shown that ~ ~ ~ ~ Bu = ⇣u · E + µu · H, 15 (2.11) ~ where ⇣u = u · ⇣ and µu = u · µ. The electric and magnetic fields are then decomposed into ˆ ~ ˆ their respective transverse and longitudinal components, resulting in h i h i ~u · E⇢ + uEu + µu · H⇢ + uHu . ~ ~ Bu = ⇣ ˆ ~ ˆ (2.12) ~ ˆ Using the distributive property again and defining ⇣uu = ⇣u · u and µuu = µu · u, the ~ ˆ ~ longitudinal component of B is given as ~ ~ ~ Bu = ⇣u · E⇢ + ⇣uu Eu + µu · H⇢ + µuu Hu . ~ (2.13) With the longitudinal component determined, the transverse component is expressed using (2.8), yielding ~ ~ ~ B⇢ = ⇣ · E + µ · H h i ~ u ⇣u · E⇢ + ⇣uu Eu + µu · H⇢ + µuu Hu . ˆ ~ ~ ~ (2.14) ~ ~ Decomposing E and H into their transverse and longitudinal parts and rearranging (2.14) gives h i ⇣ ⌘ ~ ~ B ⇢ = ⇣ · E ⇢ + uE u ˆ u ⇣u · E⇢ ˆ ~ ~ u (⇣uu Eu ) ˆ h i ⇣ ⌘ ~ ~ +µ · H⇢ + uHu ˆ u µ u · H⇢ ˆ ~ u (µuu Hu ) . ˆ (2.15) Using the distributive property again results in ~ B⇢ = h i h i ~u · E⇢ + ⇣u u⇣uu Eu ~¯ ˆ ~ ⇣ u⇣ ˆ ⇥ ⇤ ~ + µ uµu · H⇢ + [~ u uµuu ] Hu , ˆ~ µ¯ ˆ 16 (2.16) ~¯ where ⇣u = ⇣ · u and µu = µ · u. This is simplifed further by defining ˆ ~¯ ˆ ⇣⇢ = ⇣ u⇣ u ˆ~ (2.17) µ⇢ = µ uµu ˆ~ (2.18) ~ ~¯ ⇣⇢u = ⇣u u⇣uu ˆ (2.19) µ⇢u = µu ~ ~¯ uµuu . ˆ (2.20) ~ The expression in (2.16) for the transverse component of B is then reduced to ~ ~ ~ ~ B⇢ = ⇣ ⇢ · E⇢ + ⇣⇢u Eu + µ⇢ · H⇢ + µ⇢u Hu . ~ (2.21) ~ Following a similar procedure, the electric flux density, D, can be decomposed into transverse and longitudinal components. This decomposition results in ~ ~ Du = ~u · E⇢ + ✏uu Eu + ⇠u · H⇢ + ⇠uu Hu ✏ ~ (2.22) ~ ~ ~ ~ D⇢ = ✏⇢ · E⇢ + ~⇢u Eu + ⇠ ⇢ · H⇢ + ⇠⇢u Hu , ✏ (2.23) where ~u = u · ✏, ⇠u = u · ⇠, ~u = ✏ · u, ⇠u = ⇠ · u, ✏⇢ = ✏ u~u , ⇠ ⇢ = ⇠ ✏ ˆ ~ ˆ ✏¯ ˆ ~¯ ˆ ˆ✏ ~ ~¯ and ⇠⇢u = ⇠u u⇠u , ~⇢u = ~u ˆ~ ✏ ✏¯ u✏uu , ˆ u⇠uu . ˆ The bianisotropic medium considered here contains phasor electrical and magnetic current sources; these sources must also be decomposed into their respective transverse and longitudinal parts. The decomposition of the sources are carried out in a manner similar to the decomposition of the fields, resulting in ~ ~ Jm = Jm⇢ + uJmu ˆ (2.24) ~ ~ J = J⇢ + u Ju , ˆ (2.25) where Jmu = u · Jm and Ju = u · J. ˆ ~ ˆ ~ 17 2.2.2 Decomposition of Maxwell’s Equations into Transverse and Longitudinal Components ~ ~ Now that B and D are decomposed into transverse and longitudinal components, the next step is decomposing the operators in Maxwell’s equations into transverse and longitudinal parts. First note that @ ⌘ u · r, ˆ @u (2.26) and that the transverse del operator is defined as r⇢ ⌘ r u ˆ @ . @u (2.27) Using these definitions, the vector curl can be decomposed into transverse and longitudinal components. The transverse component of the vector curl is determined using the identity (A.4) as ⇣ ~ r⇥A ⌘ ⇢ = ⇣ ⌘ ~ . u⇥u⇥ r⇥A ˆ ˆ (2.28) Using the identity (A.6), (2.28) becomes ⇣ ⌘ ~ = r⇥A ⇢ ⇣ ~ u ⇥ u ⇥ r ⇢ ⇥ A⇢ ˆ ˆ ⌘ u⇥u⇥ ˆ ˆ " ~ @ A⇢ u⇥ ˆ @u r⇢ Au #! . (2.29) Here the first term on the right side is equal to zero by property (A.7). Using the identity in (A.9) the second term on the right side can be rewritten as ( u u· ˆ ˆ " ~ @ A⇢ u⇥ ˆ @u r⇢ Au #!) " ~ @ A⇢ + (ˆ · u) u ⇥ u ˆ ˆ @u 18 r⇢ Au #! . (2.30) The first of these terms in (2.29) is analyzed. The expression " ~ @ A⇢ u⇥ ˆ @u u· ˆ r⇢ Au #! (2.31) is rearranged using the property (A.10), yielding u· ˆ " ~ @ A⇢ u⇥ ˆ @u r⇢ Au #! " ~ @ A⇢ = @u # r⇢ Au · (ˆ ⇥ u) . u ˆ (2.32) Here (ˆ ⇥ u) = 0, and thus the transverse component of the vector curl is given by u ˆ ⇣ " ⌘ ~ @ A⇢ ~ r⇥A =u⇥ ˆ @u ⇢ # r⇢ Au . (2.33) Using the property (2.8), the longitudinal part is just the di↵erence between the curl and its transverse component. Thus ⇣ ⌘ ~ ~ u u · r ⇥ A = r⇢ ⇥ A⇢ . ˆ ˆ (2.34) With the curl operation decomposed, Maxwell’s equations can be decomposed into transverse and longitudinal parts, which results in u⇥ ˆ " ~ @ E⇢ @u ~ r⇢ ⇥ E ⇢ = # r⇢ E u = j!ˆBu u ~ j! B⇢ uJmu ˆ ~ Jm⇢ ~ r⇢ ⇥ H⇢ = j!ˆDu + uJu u ˆ # ~ @ H⇢ ~ ~ u⇥ ˆ r⇢ Hu = j! D⇢ + J⇢ . @u " (2.35) (2.36) (2.37) (2.38) Now (2.21) is substituted into (2.36), giving u ⇥ r⇢ E u + u ⇥ ˆ ˆ ~ @ E⇢ = @u h i ~ ~ ~ j! ⇣ ⇢ · E⇢ + ⇣⇢u Eu + µ⇢ · H⇢ + µ⇢u Hu ~ 19 ~ Jm⇢ . (2.39) ~ Then substituting the expressions for the longitudinal component of D, (2.23), into (2.38), results in u ⇥ r ⇢ Hu + u ⇥ ˆ ˆ h i ~ @ H⇢ ~ ~ ~ ~ = j! ✏⇢ · E⇢ + ~⇢u Eu + ⇠ ⇢ · H⇢ + ⇠⇢u Hu + J⇢ . ✏ @u (2.40) Taking the cross product of u and the expressions in (2.39)and(2.40) results in ˆ ⇣ ⌘ ⇣ ⌘ ~ ~ j!ˆ ⇥ ⇣ ⇢ · E⇢ u j!ˆ ⇥ ⇣⇢u Eu u ⌘ ~ j!ˆ ⇥ µ⇢ · H⇢ u j!ˆ ⇥ µ⇢u Hu u ~ u ⇥ Jm⇢ , ˆ ~ u ⇥ u ⇥ r⇢ E u + u ⇥ u ⇥ ˆ ˆ ˆ ˆ ~ @ E⇢ = @u ⇣ (2.41) and ⇣ ⌘ ~ @ H⇢ ~ ⇢ + j!ˆ ⇥ ~⇢u Eu u ⇥ u ⇥ r ⇢ Hu + u ⇥ u ⇥ ˆ ˆ ˆ ˆ = j!ˆ ⇥ ✏⇢ · E u u ✏ @u ⇣ ⌘ ⇣ ⌘ ~ ~ +j!ˆ ⇥ ⇠ ⇢ · J⇢ + j!ˆ ⇥ ⇠⇢u Hu + u ⇥ J⇢ , u u ˆ ~ (2.42) respectively. ~ To simplify (2.41) and (2.42), the longitudinal component of A is considered. Using the property (A.4) and (2.8), it can be shown that ⇥ ⇤ u ⇥ u ⇥ r ⇢ Au = ˆ ˆ ⇥ r⇢ Au r⇢ Au ✓ u u · r⇢ Au ˆ ˆ ⇤ . (2.43) Using (2.27), (2.43) may be expressed as ⇥  ⇤ u ⇥ u ⇥ r⇢ Au = ˆ ˆ ⇢ u u · rAu ˆ ˆ @Au u ˆ @u ◆ . (2.44) From (2.26), this expression reduces to ⇥ ⇤ u ⇥ u ⇥ r ⇢ Au = ˆ ˆ ⇥ r⇢ Au u (ˆ · {rAu ˆ u 20 ⇤ u (ˆ · rAu )}) . ˆ u (2.45) Using the property (A.1) this expression becomes ⇥ ⇤ u ⇥ u ⇥ r⇢ Au = ˆ ˆ ⇥ r ⇢ Au u (ˆ · rAu ˆ u u · rAu ) ˆ ⇤ (2.46) or ⇥ ⇤ u ⇥ u ⇥ r ⇢ Au = ˆ ˆ r⇢ Au . (2.47) Similarly, using the property (A.4) and (2.8), it can be shown that " # = " " # = @ h~ A⇢ @u ~ @ A⇢ u⇥ u⇥ ˆ ˆ @u ~ @ A⇢ @u !# (2.48) ⇣ ⌘i u u · A⇢ . ˆ ˆ ~ (2.49) ~ @ A⇢ u u· ˆ ˆ @u or ~ @ A⇢ u⇥ u⇥ ˆ ˆ @u Then from property (A.5), this expression becomes " ~ @ A⇢ u⇥ u⇥ ˆ ˆ @u # = ~ A⇢ . (2.50) Using the relationships in (2.47) and (2.50), (2.41) and (2.42) reduce to r⇢ E u r⇢ Hu ~ @ E⇢ = @u ⇣ ~ j!ˆ ⇥ ⇣ ⇢ · E⇢ u ⌘ ⇣ ~ j!ˆ ⇥ ⇣⇢u Eu u ⌘ ⇣ ~ j!ˆ ⇥ µ⇢ · H⇢ u ⌘ u ⇥ Jm⇢ ˆ ~ ⇣ ⌘ ⇣ ⌘ ~ @ H⇢ ~ ~ = j!ˆ ⇥ ✏⇢ · H⇢ + j!ˆ ⇥ ~⇢u Hu + j!ˆ ⇥ ⇠ ⇢ · E⇢ u u ✏ u @u ⇣ ⌘ ~ +j!ˆ ⇥ ⇠⇢u Eu + u ⇥ J⇢ . u ˆ ~ j!ˆ ⇥ µ⇢u Hu u ~ (2.51) (2.52) These expressions can be simplified further by considering the dyadic P , where P ⇢ = 21 ⇣ P ⌘ ~ uPu and Pu = u · P . Using the identity from (A.3) it can be shown that ˆ~ ˆ ⇣ ⌘ ⇣ ⌘ ~ ~ u ⇥ P ⇢ · A⇢ = u ⇥ P ⇢ · A⇢ ˆ ˆ (2.53) ⇣ (2.54) or ⌘ ⇣ h ~⇢ = u ⇥ P u ⇥ P⇢ · A ˆ ˆ u Pu ˆ~ i⌘ ~ · A⇢ . Since u ⇥ u = 0, this expression becomes ˆ ˆ ⇣ ⌘ ⇣ ⌘ ~ ⇢ = u ⇥ P · A⇢ . ~ u ⇥ P⇢ · A ˆ ˆ ⇣ ~ ~ ~¯ Additionally, using the dyadic P and the transverse component of A, where P⇢u = Pu ~¯ Pu = P · u, and Puu = u · P · u, it can be shown that ˆ ˆ ˆ ⇣ ⌘ ⇣ h ~ ~¯ u ⇥ P⇢u Au = u ⇥ Pu ˆ ˆ i ⌘ uPuu Au ˆ (2.55) ⌘ uPuu , ˆ (2.56) or ⇣ ⌘ ⇣ h i ~ u ⇥ P⇢u Au = u ⇥ P · u Au ˆ ˆ ˆ h i ⌘ u ⇥ u u · P · u Au . ˆ ˆ ˆ ˆ (2.57) Since u ⇥ u = 0, this expression becomes ˆ ˆ ⇣ ⌘ h i ~ u ⇥ P⇢u Au = u ⇥ P · u Au . ˆ ˆ ˆ (2.58) Using the identity (A.3) then results in ⇣ ⌘ ⇣ ⌘ ~ u ⇥ P⇢u Au = u ⇥ P · uAu . ˆ ˆ ˆ 22 (2.59) Employing the properties from (2.55) and (2.59), (2.51) and (2.52) become ~ @ E⇢ = @u r⇢ E u r⇢ Hu ⇣ ⌘ ~ j! u ⇥ ⇣ · E⇢ ˆ ⇣ ⌘ j! u ⇥ ⇣ · uEu ˆ ˆ ~ j! u ⇥ µ · H⇢ ˆ j! u ⇥ µ · uHu u ⇥ Jm⇢ ˆ ˆ ˆ ~ ⇣ ⌘ ~ @ H⇢ ~ ~ = j! u ⇥ ✏ · E⇢ + j! u ⇥ ✏ · uEu + j! u ⇥ ⇠ · H⇢ ˆ ˆ ˆ ˆ @u ⇣ ⌘ +j! u ⇥ ⇠ · uHu + u ⇥ J⇢ . ˆ ˆ ˆ ~ (2.60) (2.61) By defining ⇣ ux = u ⇥ ⇣, µux = u ⇥ µ, ✏ux = u ⇥ ✏, and ⇠ ux = u ⇥ ⇠, (2.60) and (2.61) can ˆ ˆ ˆ ˆ be expressed as r⇢ E u ~ @ E⇢ = @u ~ j!⇣ ux · E⇢ j!⇣ ux · uEu ˆ ~ j!µux · H⇢ u ⇥ Jm⇢ ˆ ~ j!µux · uHu ˆ r⇢ Hu 2.2.3 ~ @ H⇢ ~ ~ = j!✏ux · E⇢ + j!✏ux · uEu + j!⇠ ux · H⇢ ˆ @u +j!⇠ ux · uHu + u ⇥ J⇢ . ˆ ˆ ~ (2.62) (2.63) ~ H⇢ in Terms of Eu , Hu , and Sources ~ ~ ~ The H⇢ field will first be related to the Eu , Hu , and the sources by substituting (2.63) into (2.62). Before the substitution, (2.63) is rearranged to become ~ @ H⇢ + r⇢ Hu @u ~ j!✏ux · E⇢ = j!✏ux · uEu ˆ ~ j!⇠ ux · H⇢ u ⇥ J⇢ . ˆ ~ j!⇠ ux · uHu ˆ Taking the dot product of ✏ux ~ E⇢ = ✏ux 1· ( (2.64) 1 and (2.64) and diving by j! yields ~ 1 @ H⇢ 1 + r⇢ Hu j! @u j! ✏ux · uEu ˆ ⇠ ux · uHu ˆ 23 ~ ⇠ ux · H⇢ 1 u ⇥ J⇢ . ˆ ~ j! (2.65) Here ✏ux 1· ✏ ux = I. This expression is now substituted into (2.62) and results in ( ~ ~ @ H⇢ 1 @ 2 H⇢ 1 @Hu @Eu + r⇢ ✏ux · u ˆ ⇠ ux · j! @u2 j! @u @u @u ) ~ @ J⇢ @Hu 1 ~ ⇠ ux · u ˆ u⇥ ˆ = j!⇣ ux · uEu j!µux · H⇢ j!µux · uHu ˆ ˆ @u j! @u ( ~ 1 @ H⇢ 1 ~ u ⇥ Jm⇢ j!⇣ ux · ✏ux 1 · ˆ ~ + r⇢ Hu ✏ux · uEu ⇠ ux · H⇢ ˆ j! @u j! r⇢ E u ✏ux 1· ⇠ ux · uHu ˆ 1 u ⇥ J⇢ . (2.66) ˆ ~ j! Taking the dot product of ✏ux and (2.66) yields ~ ~ @ H⇢ 1 @ 2 H⇢ 1 @Hu @Eu r⇢ + ✏ux · u ˆ + ⇠ ux · j! @u2 j! @u @u @u ⇣ ⌘ ⇣ ⌘ ~ @ J⇢ @Hu 1 ~ +⇠ ux · u ˆ + u⇥ ˆ = j!✏ux · ⇣ ux · uEu ˆ j!✏ux · µux · H⇢ @u j! @u ( ⇣ ⌘ ~ 1 @ H⇢ j!✏ux · µux · uHu ˆ ✏ux · u ⇥ Jm⇢ ˆ ~ · E j! @u ✏ux · r⇢ Eu + + 1 r⇢ Hu j! ✏ux · uEu ˆ ⇣ where E = j! ✏ux · ⇣ ux · ✏ux uct of E in (2.67) gives ~ ⇠ ux · H⇢ ⇠ ux · uHu ˆ 1 u ⇥ J⇢ , ˆ ~ j! (2.67) ⌘ 1 . Expanding the expression associated with dot prod- ~ 1 @ 2 H⇢ 1 @Hu @Eu ✏ux · r⇢ Eu + r⇢ + ✏ux · u ˆ 2 j! @u j! @u @u ⇣ ⌘ ~⇢ ~⇢ @H @J @Hu 1 +⇠ ux · + ⇠ ux · u ˆ + u⇥ ˆ = j!✏ux · ⇣ ux · uEu ˆ @u @u j! @u ⇣ ⌘ ⇣ ⌘ ~ j!✏ux · µux · H⇢ j!✏ux · µux · uHu ˆ ✏ux · u ⇥ Jm⇢ ˆ ~ ⇣ ⌘ ~ 1 @ H⇢ 1 + E· · r⇢ Hu + j!✏ux · ⇣ ux · uEu ˆ j! @u j! E ✓ ◆ ⇣ ⌘ ⇣ ⌘ 1 ~ + E · ⇠ ux · H⇢ + E · ⇠ ux · uHu + E · ˆ u ⇥ J⇢ . ˆ ~ j! 24 (2.68) ⇣ ⌘ The two terms of j!✏ux · ⇣ ux · uEu on the right-hand side cancel out. Rearranging ˆ the expression to have the transverse components on the left-hand side and longitudinal components on the right-hand side, and multiplying through by j! results in " # ~ ~ ~ @ H⇢ @ 2 H⇢ @ H⇢ ~ ~ + j!⇠ ux · k E · H⇢ E · @u + j!⇠ ux · H⇢ @u @u2 ✓ ◆ ~ @ J⇢ @Eu @Hu @Hu = j!✏ux · r⇢ Eu + u ˆ + r⇢ j!⇠ ux · u ˆ u⇥ ˆ @u @u @u @u ⇣ ⌘ h i +k E · uHu j!✏ux · u ⇥ Jm⇢ + E · j!⇠ ux · uHu r⇢ Hu + u ⇥ J⇢ , ˆ ˆ ~ ˆ ˆ ~ where k E = ! 2 ✏ux · µux . From (2.27), the first term of the right side reduces to (2.69) j!✏ux · ~ rEu . Finally, the transverse components of H may now be related to the longitudinal components and sources through the di↵erential equation ~ ~ @ 2 H⇢ @ H⇢ + j!⇠ ux · @u @u2 = 2.2.4 " ~ @ H⇢ ~ · + j!⇠ ux · H⇢ E @u # ~ k E · H⇢ ~ @ J⇢ @Hu @Hu j!✏ux · rEu + r⇢ j!⇠ ux · u ˆ u⇥ ˆ + k E · u Hu ˆ @u @u @u ⇣ ⌘ h i j!✏ux · u ⇥ Jm⇢ + E · j!⇠ ux · uHu r⇢ Hu + u ⇥ J⇢ . ˆ ~ ˆ ˆ ~ (2.70) ~ E⇢ in Terms of Eu , Hu , and Sources ~ ~ ~ In a similar set of steps the field E⇢ may be related to the fields Eu , Hu and the sources through the substitution of (2.62) into (2.63). First (2.62) is rearranged as ~ j!µux · H⇢ = ~ @ E⇢ @u r⇢ E u ~ j!⇣ ux · E⇢ j!⇣ ux · uEu ˆ j!µux · uHu ˆ 25 u ⇥ Jm⇢ . ˆ ~ (2.71) Taking the dot product of µux ~ H⇢ = µux 1· ( 1 and (2.71) and then diving by j! results in ~ 1 @ E⇢ j! @u 1 r⇢ E u j! ~ ⇣ ux · E⇢ µux · uHu ˆ Here µux 1·µ ⇣ ux · uEu ˆ 1 u ⇥ Jm⇢ . ˆ ~ j! (2.72) ux = I. This expression is now substituted into (2.62) and produces ( ~ ~ @ E⇢ 1 @ 2 E⇢ 1 @Eu @Eu r⇢ ⇣ ux · ⇣ ux · u ˆ j! @u2 j! @u @u @u ) ~ @ Jm⇢ @Hu 1 ~ µux · u ˆ u⇥ ˆ = j!✏ux · E⇢ + j!✏ux · uEu + j!⇠ ux · uHu ˆ ˆ @u j! @u ( ~ 1 @ E⇢ 1 ~ +ˆ ⇥ J⇢ + j!⇠ ux · µux 1 · u ~ r⇢ Eu ⇣ ux · E⇢ ⇣ ux · uEu ˆ j! @u j! r⇢ Hu µux 1· µux · uHu ˆ 1 u ⇥ Jm⇢ . ˆ ~ j! (2.73) Taking the dot product of µux and (2.73) results in ~ ~ @ E⇢ 1 @ 2 E⇢ 1 @Eu @Eu + r⇢ + ⇣ ux · + ⇣ ux · u ˆ j! @u2 j! @u @u @u ⇣ ⌘ ~ @ Jm⇢ @Hu 1 ~ +µux · u ˆ + u⇥ ˆ = j!µux · ✏ux · E⇢ + j!µux · ✏ux · uEu ˆ @u j! @u ( ⇣ ⌘ ⇣ ⌘ ~ 1 @ E⇢ ~⇢ + +j!µux · ⇠ ux · uHu + µux · u ⇥ J ˆ ˆ · H j! @u µux · r⇢ Hu 1 r⇢ E u j! ~ ⇣ ux · E⇢ ⇣ ux · uEu ˆ 26 µux · uHu ˆ 1 u ⇥ Jm⇢ , ˆ ~ j! (2.74) h where H = j! µux · ⇠ ux · µux product of H in (2.74) gives i 1 . Expanding the expressions associated with the dot ~ ~ @ E⇢ 1 @ 2 E⇢ 1 @Eu + r⇢ + ⇣ ux · j! @u2 j! @u @u ⇣ ⌘ ~ @ Jm⇢ @Eu @Hu 1 ~ +⇣ ux · u ˆ + µux · u ˆ + u⇥ ˆ = j!µux · ✏ux · E⇢ @u @u j! @u ⇣ ⌘ ⇣ ⌘ +j!µux · ✏ux · uEu + j!µux · ⇠ ux · uHu + µux · u ⇥ J⇢ ˆ ˆ ˆ ~ ⇣ ⌘ ~ 1 @ E⇢ 1 ~ + H· · r⇢ E u · ⇣ ux · E⇢ H j! @u j! H ✓ ◆ ⇣ ⌘ ⇣ ⌘ 1 ~m⇢ . ˆ j!µux · ⇠ ux · uHu ˆ ˆ H · ⇣ ux · uEu H · j! u ⇥ J µux · r⇢ Hu ⇣ The two terms of j!µux · ⇠ ux · uHu ˆ ⌘ (2.75) on the right-hand side cancel out. Rearranging the expression to have the transverse components on the left-hand side and longitudinal components on the right-hand side and multiplying through by j! results in " # ~ ~ ~ @ 2 E⇢ @ E⇢ @ E⇢ ~ ~ j!⇣ ux · + E· j!⇣ ux · E⇢ k H · E⇢ @u @u @u2 ✓ ◆ ~ @ Jm⇢ @Hu @Eu @Eu = j!µux · r⇢ Hu + u ˆ + r⇢ + j!⇣ ux · u ˆ +u⇥ ˆ @u @u @u @u ⇣ ⌘ h i ~⇢ + ~m⇢ , +k H · uEu j!µux · u ⇥ J ˆ ˆ ˆ ˆ H · j!⇣ ux · uEu + r⇢ Eu + u ⇥ J (2.76) where k H = ! 2 µux · ✏ux . From (2.27), the first term of the right side reduces to j!µux · ~ rHu . Finally, the transverse components of E may be expressed in terms of the longitudinal components and sources, through the di↵erential equation ~ @ 2 E⇢ @u2 +k H · uEu ˆ " ~ ~ @ E⇢ @ E⇢ j!⇣ ux · + E· @u @u ~ j!⇣ ux · E⇢ # ~ k H · E⇢ ~ @ Jm⇢ @Eu @Eu = j!µux · rHu + r⇢ + j!⇣ ux · u ˆ +u⇥ ˆ @u @ui ⇣ ⌘ h @u j!µux · u ⇥ J⇢ + H · j!⇣ ux · uEu + r⇢ Eu + u ⇥ Jm⇢ . ˆ ~ ˆ ˆ ~ 27 (2.77) 2.2.5 TE/TM Decomposition of the Fields in a Bianisotropic material To analyze materials inside a rectangular waveguide, typically the electromagnetic fields are first decomposed into TE and TM fields. TE/TM field decomposition is the source-free ~ ~ special case of longitudinal/transverse decomposition. Setting J = 0 and Jm = 0, (2.70) and (2.77) become ~ ~ @ 2 H⇢ @ H⇢ + j!⇠ ux · @u @u2 = " ~ @ H⇢ ~ · + j!⇠ ux · H⇢ E @u j!✏ux · rEu + r⇢ @Hu @u # ~ k E · H⇢ @Hu j!⇠ ux · u ˆ + k E · uHu ˆ @u h i + E j!⇠ ux · uHu r⇢ Hu ˆ (2.78) and ~ @ 2 E⇢ @u2 " ~ ~ @ E⇢ @ E⇢ j!⇣ ux · + H· @u @u = j!µux · rHu + r⇢ ~ j!⇣ ux · E⇢ # ~ k H · E⇢ @Eu @Eu + j!⇣ ux · u ˆ + k H · uE u ˆ @u @u h i + H · j!⇣ ux · uEu + r⇢ Eu , ˆ (2.79) respectively. The electromagnetic field is now completely specified in terms of two scalar fields Eu and Hu . To solve the di↵erential equations, it is beneficial to use superposition since the right-hand side of the equation has two forcing terms. These equations can be solved by considering only one forcing term at a time and then adding the results. If Eu = 0 then the field is termed TE to the u direction. When Hu = 0 the fields are termed TM to ˆ the u direction. The relationships between transverse and longitudinal fields in the TE case ˆ 28 are given as " # ~ @ H⇢ ~ ~ k E · H⇢ E · @u + j!⇠ ux · H⇢ h i + k E · uHu + E j!⇠ ux · uHu r⇢ Hu ˆ ˆ (2.80) # ~ ~ j!⇣ ux · E⇢ k H · E⇢ = j!µux · rHu , (2.81) ~ ~ @ 2 H⇢ @ H⇢ + j!⇠ ux · @u @u2 ~ @ 2 E⇢ @u2 @Hu @Hu j!⇠ ux · u ˆ @u " @u ~ ~ @ E⇢ @ E⇢ j!⇣ ux · + H· @u @u = r⇢ and in the TM case as # ~ @ H⇢ ~ ~ k E · H⇢ = j!✏ux · rEu (2.82) E · @u + j!⇠ ux · H⇢ " # ~ ~ ~ @ 2 E⇢ @ E⇢ @ E⇢ ~ ~ j!⇣ ux · + E· j!⇣ ux · E⇢ k H · E⇢ @u @u @u2 h i @Eu @Eu = r⇢ + j!⇣ ux · u ˆ + k H · uEu + H · j!⇣ ux · uEu + r⇢ Eu . (2.83) ˆ ˆ @u @u ~ ~ @ 2 H⇢ @ H⇢ + j!⇠ ux · @u @u2 2.2.6 " Explicit Expressions for the Transverse Field Components The di↵erential equations in (2.80) - (2.83) can be solved using the Fourier transform pairs F (~, ) = ⇢ Z 1 f (~, u) e j u du ⇢ 1 Z 1 1 f (~, u) = ⇢ F (~, ) ej u d , ⇢ 2⇡ 1 (2.84) (2.85) to obtain explicit expressions for the transverse field components. In (2.84) and (2.85) is the transform variable. Standard transform notation is f (~, u) $ F (~, ) , ⇢ ⇢ 29 (2.86) where $ designates a Fourier transform pair. The transform of the derivative of f (~, u) is ⇢ found by di↵erentiating (2.85) which gives Z 1 @ 1 f (~, u) = ⇢ @u 2⇡ 1 j F (~, ) ej u du ⇢ (2.87) and so @ f (~, u) $ j F (~, ) . ⇢ ⇢ @u 2.2.6.1 (2.88) TE Decomposition Performing the Fourier transform of (2.80) and (2.81) results in h 2 H (~, ) + ! ⇠ · H (~, ) + j ~⇢ ⇢ ux ~ ⇢ ⇢ E· = 2 E (~, ) ~⇢ ⇢ ~ ⇢ ! ⇣ ux · E⇢ (~, ) i ~ ⇢ ~ ⇢ ~ ⇢ H⇢ (~, ) + !⇠ ux · H⇢ (~, ) + k E · H⇢ (~, ) h ~ ⇢ E⇢ (~, ) j r⇢ Hu (~, ) ! ⇠ ux · uHu (~, ) k E · uHu (~, ) ⇢ ˆ ⇢ ˆ ⇢ h i · j!⇠ ux · uHu (~, ) r⇢ Hu (~, ) ˆ ⇢ ⇢ (2.89) E j E· = i ~ ⇢ ~ ⇢ !⇣ ux · E⇢ (~, ) + k H · E⇢ (~, ) ⇥ ⇤ j!µux · r⇢ Hu (~, ) + j uHu (~, ) . (2.90) ⇢ ˆ ⇢ These expressions may also be written as ~ ⇢ ⌘ · H⇢ (~, ) = ~ ⇢ ⌧ · E⇢ (~, ) = j r⇢ Hu (~, ) ⇢ ! ⇠ ux · uHu (~, ) ˆ ⇢ h k E · uHu (~, ) ˆ ⇢ ˆ ⇢ E · j!⇠ ux · uHu (~, ) ⇥ ⇤ j!µux · r⇢ Hu (~, ) + j uHu (~, ) , ⇢ ˆ ⇢ 30 r⇢ Hu (~, ) ⇢ i (2.91) (2.92) where ⌘= h 2I + ! ⇠ + j ux E + j! E ⇠ ux + k E i (2.93) and ⌧= h 2I ! ⇣ ux i + j! H ⇣ ux + k H . H j (2.94) The expressions for the transverse field components for the TE case are ~ ⇢ H⇢ (~, ) = ⌘ 1 · n ! ⇠ ux · uHu (~, ) k E · uHu (~, ) ˆ ⇢ ˆ ⇢ h io · j!⇠ ux · uHu (~, ) r⇢ Hu (~, ) ˆ ⇢ ⇢ (2.95) E ⇥ ⇤ ~ ⇢ j!µux · r⇢ Hu (~, ) + j uHu (~, ) . E⇢ (~, ) = ⌧ 1 · ⇢ ˆ ⇢ (2.96) 2.2.6.2 j r⇢ Hu (~, ) ⇢ TM Decomposition Performing the Fourier transform of (2.82) and (2.83) yields 2 H (~, ) + ! ⇠ · H (~, ) + j ~⇢ ⇢ ux ~ ⇢ ⇢ E· h 2 E (~, ) ~⇢ ⇢ h ~ ⇢ ! ⇣ ux · E⇢ (~, ) = j E· i ~ ⇢ ~ ⇢ ~ ⇢ H⇢ (~, ) + !⇠ ux · H⇢ (~, ) + k E · H⇢ (~, ) ⇥ ⇤ = j!✏ux · r⇢ Eu (~, ) + j uEu (~, ) ⇢ ˆ ⇢ (2.97) ~ ⇢ E⇢ (~, ) i ~ ⇢ ~ ⇢ !⇣ ux · E⇢ (~, ) + k H · E⇢ (~, ) j r⇢ Eu (~, ) + ! ⇣ ux · uEu (~, ) k H · uEu (~, ) ⇢ ˆ ⇢ ˆ ⇢ h i · j!⇣ ux · uEu (~, ) + r⇢ Eu (~, ) . (2.98) ˆ ⇢ ⇢ H 31 Using the definitions given in (2.93) and (2.94), (2.97) and (2.98) become ⇥ ⇤ ~ ⇢ ⌘ · H⇢ (~, ) = j!✏ux · r⇢ Eu (~, ) + j uEu (~, ) ⇢ ˆ ⇢ ~ ⇢ ⌧ · E⇢ (~, ) = (2.99) j r⇢ Eu (~, ) + ! ⇣ ux · uEu (~, ) k H · uEu (~, ) ⇢ ˆ ⇢ ˆ ⇢ h i · j!⇣ ux · uEu (~, ) + r⇢ Eu (~, ) . ˆ ⇢ ⇢ H (2.100) Finally, the expressions for the transverse field components for the TM case are ⇥ ⇤ ~ ⇢ H⇢ (~, ) = ⌘ 1 · j!✏ux · r⇢ Eu (~, ) + j uEu (~, ) ⇢ ˆ ⇢ n ~ ⇢ E⇢ (~, ) = ⌧ 1 · j r⇢ Eu (~, ) + ! ⇣ ux · uEu (~, ) ⇢ ˆ ⇢ h io · j!⇣ ux · uEu (~, ) + r⇢ Eu (~, ) . ˆ ⇢ ⇢ H 2.2.7 (2.101) k H · uEu (~, ) ˆ ⇢ (2.102) Homogeneous Wave Equations for the Longitudinal Field Components The homogeneous wave equations for the longitudinal field components are obtained using the expressions from the longitudinal decomposition of Maxwell’s equations. Substituting (2.13) into (2.35) and setting the source to zero results in ~ ⇢ r⇢ ⇥ E⇢ (~, ) = h j!ˆ ⇣u · E⇢ (~, ) + ⇣uu Eu (~, ) u ~ ~ ⇢ ⇢ i +~ u · H⇢ (~, ) + µuu Hu (~, ) . µ ~ ⇢ ⇢ (2.103) Similarly, substituting (2.22) into (2.37) and eliminating the sources yields h i ~ ~ ⇢ ~ ⇢ r⇢ ⇥ H⇢ (~, ) = j!ˆ ~u · E⇢ (~, ) + ✏uu Eu (~, ) + ⇠u · H⇢ (~, ) + ⇠uu Hu (~, ) .(2.104) u ✏ ~ ⇢ ⇢ ⇢ 32 2.2.7.1 TE Homogeneous Wave Equation The decomposition of TE to the u-direction requires Eu (~, ) = 0. Thus (2.103) becomes ⇢ ~ ⇢ r⇢ ⇥ E⇢ (~, ) = j!ˆ⇣u · E⇢ (~, ) u~ ~ ⇢ ~ ⇢ j!ˆµu · H⇢ (~, ) u~ j!ˆµuu Hu (~, ). u ⇢ (2.105) Now substituting (2.95) and (2.96) into (2.105) yields the homogeneous wave equation for Hu (~, ) ⇢ = h ⇥ ⇤ i r⇢ ⇥ ⌧ 1 · j!µux · r⇢ Hu (~, ) + j uHu (~, ) ⇢ ˆ ⇢ h ⇥ ⇤ i j!ˆ⇣u · ⌧ 1 · u~ j!µux · r⇢ Hu (~, ) + j uHu (~, ) ⇢ ˆ ⇢ j!ˆµuu Hu (~, ) u ⇢ h n j!ˆµu · ⌘ 1 · u~ j r⇢ Hu (~, ) ! ⇠ ux · uHu (~, ) k E · uHu (~, ) ⇢ ˆ ⇢ ˆ ⇢ ⇣ ⌘oi ˆ ⇢ ⇢ . (2.106) E · j!⇠ ux · uHu (~, ) r⇢ Hu (~, ) 2.2.7.2 TM Homogeneous Wave Equation The decomposition of TM to the u-direction requires Hu (~, ) = 0. Thus (2.104) becomes ⇢ ~ ~ ⇢ ~ ⇢ r⇢ ⇥ H⇢ (~, ) = j!ˆ~u · E⇢ (~, ) + j! ⇠u · H⇢ (~, ) + j!⇠uu Hu (~, ). u✏ ~ ⇢ ⇢ (2.107) Substituting (2.101) and (2.102) into (2.107) gives the homogenous wave equation for Eu (~, ) ⇢ as h ⇥ r⇢ ⇥ ⌘ 1 · j!✏ux · r⇢ Eu (~, ⇢ h ⇥ ⇤ = j!ˆ⇠u · ⌘ 1 · j!✏ux · r⇢ Eu (~, ) + j uEu (~, ) u~ ⇢ ˆ ⇢ h n +j!ˆ~u · ⌧ 1 · u✏ j r⇢ Eu (~, ) + ! ⇣ ux · uEu (~, ⇢ ˆ ⇢ ⇣ ˆ ⇢ H · j!⇣ ux · uEu (~, 33 ⇤ i ) + j uEu (~, ) ˆ ⇢ i + j!ˆ✏uu Eu (~, ) u ⇢ k H · uEu (~, ) ˆ ⇢ ⌘oi ) + r⇢ Eu (~, ) ⇢ . (2.108) ) y y=b ϵ,µ X x=a Figure 2.1: Cross-section view of a rectangular waveguide used in derivations for isotropic and gyromagnetic materials. y 2.3 Specialization to Rectangular Waveguides y=b With expressions for the homogeneous wave equations for TE and TM fields in a bianisotropic material determined, the special cases of less complex media completely filling a rectangular waveguide are considered. The resulting homogeneous wave equations will be solved using ϵ,µ a separation of variables technique in conjunction with the application of boundary conditions of the transverse field at the conducting surfaces of the guide. An expression for the longitudinal field may be determined and subsequently the transverse fields can be derived X using (2.95) and (2.96) for fields TE to the u-direction and (2.101) and (2.102) for fields TM y=-a 2 y= a 2 to the u-direction. 34 2.3.1 Isotropic Material Filled Guide Isotropic material are considered first. Those materials have the tensor entries 2 3 2 3 6 1 0 0 7 6 7 ✏ = ✏ 6 0 1 0 7 = ✏I 6 7 4 5 0 0 1 and 6 1 0 0 7 6 7 µ = µ 6 0 1 0 7 = µI. 6 7 4 5 0 0 1 (2.109) (2.110) Additionally, cross coupling between the electric and magnetic fields does not exist, and so ⇣ = 0 and ⇠ = 0. In order to derive the homogenous wave equation for isotropic material, the general bianisotropic homogenous wave equations for TE fields, (2.106), and for TM fields, (2.108), are reduced to their respective isotropic forms. These equations are given by = h i 1· r⇢ ⇥ ⌧ j!µˆ ⇥ r⇢ Hu (~, ) u ⇢ h i j!µˆHu (~, ) j!µˆu · ⌘ 1 · u ⇢ uˆ j r⇢ Hu (~, ) ⇢ (2.111) for TE decomposition and = h i r⇢ ⇥ ⌘ 1 · j!✏ˆ ⇥ r⇢ Eu (~, ) u ⇢ h i j!✏ˆEu (~, ) + j!✏ˆu · ⌧ 1 · u ⇢ uˆ j r⇢ Eu (~, ) ⇢ (2.112) for TM decomposition. Here ⇣ ⌘ ⇣ ⌘ 2 I + ! 2 µ✏ u ⇥ I · u ⇥ I = ⌘. ⌧= ˆ ˆ 35 (2.113) Consider the case of fields TE to z for the filled waveguide shown in Figure 2.1. The equation (2.111) can be written as  =  ⇢  @ @ @Hz (~, ) ⇢ @Hz (~, ) ⇢ 1· x ˆ +y ˆ ⇥ ⌧ j!µ y ˆ x ˆ @x @y @x @y  ⇢ ✓ ◆ @Hz (~, ) ⇢ @Hz (~, ) ⇢ 1· z j!µHz (~, ) z z j!µ · ⌘ ˆ ⇢ ˆˆ j x ˆ +y ˆ @x @y , (2.114) where ⌧ 1 = ⌘ 1 = xx ˆˆ 1 + yy ˆˆ ! 2 ✏µ 1 1 + zz . ˆˆ 2 ! 2 ✏µ (2.115) Thus the wave equation for TEz fields reduces to @2 @2 2 + + kc 2 @y 2 @x ! Hz (~, ) = 0, ⇢ (2.116) where the wavenumber, kc , is 2 kc = ! 2 µ✏ 2. (2.117) Through a similar set of steps, the TMz wave equation for isotropic materials is found to be @2 @2 2 + + kc @x2 @y 2 ! Ez (~, ) = 0, ⇢ (2.118) where the wavenumber is described in (2.117). 2.3.1.1 Solution to the TEz Wave Equation for Isotropic Material Filled Guide To solve the TEz wave equation, (2.116), a separation of variables is utilized, in conjunction with applying boundary conditions on the tangential electric field at the interfaces between the isotropic material and the perfectly conducting waveguide surfaces. Using the product 36 solution of Hz (~, ) = X(x, )Y (y, ), (2.116) becomes ⇢ Y (y, ) @ 2 X(x, ) @ 2 Y (y, ) 2 + X(x, ) + kc X(x, )Y (y, ) = 0. 2 2 @x @y (2.119) Dividing through by X(x)Y (y) gives 1 @ 2 X(x, ) 1 @ 2 Y (y, ) 2 + + kc = 0. 2 X(x, ) Y (y, ) @y 2 @x (2.120) By observing that each term in (2.120) must equal a constant, this equation can be expressed as 2 2 2 kc = k x + ky , (2.121) where 2 kx = 1 @ 2 X(x, ) X(x, ) @x2 (2.122) 2 ky = 1 @ 2 Y (y, ) . Y (y, ) @y 2 (2.123) and These are two separated ordinary di↵erential equations, @ 2 X(x, ) 2 + kx X(x, ) = 0 2 @x 2 Y (y, ) @ 2 + ky Y (y, ) = 0. 2 @y (2.124) (2.125) It will be shown through the application of boundary conditions on the tangential fields that takes the form of discrete eigenvalues. Additionally, kx and ky in (2.122) and (2.123) take 37 on discrete values. The eigenfield associated with these eigenvalues is Hz (~, z) = [An sin( kxn x )+Bn cos (kxn x)] [Cn sin( kyn y +Dn cos kyn y ⇢ ⇤ ±j z n .(2.126) e In the exponential, a negative sign corresponds to a wave traveling in the +z direction while a positive sign corresponds to a traveling in the wherever a z direction. For the following expressions appears a sign is included to designate the directions of the wave. For brevity, the dependence on ⇢ and z will be left implicit. ~ To determine the transverse field components for TE modes in isotropic materials, (2.96) and (2.95) first must be reduced from bianisotropic to isotropic form. These expressions are described by ~ E⇢ = ⌧ 1 · ⇥ ⇤ j!µ u ⇥ r⇢ Hu ˆ ~ H⇢ = ⌘ 1 · ⌥j n r⇢ Hu . (2.127) (2.128) Referring to the geometry in Figure 2.1 and assuming TEz modes, (2.127) and (2.128) are expressed as ~ E⇢ = x ˆ ~ H⇢ = ⌥ˆ x j!µ @Hz 2 µ✏ @x ! (2.129) j n @Hz j n @Hz ⌥y ˆ . 2 2 2 2 n ! µ✏ @x n ! µ✏ @y (2.130) 2 n j!µ @Hz 2 µ✏ @y ! y ˆ 2 n Now with general expressions for the transverse fields, the solution to the wave equation, (2.126), can be substituted into (2.130) and (2.129) which results in the transverse electric 38 field components expressed as j!µ [A sin( kxn x )+Bn cos (kxn x)] ⇥ 2 n n ⇥ ⇤ Cn kyn cos kyn y Dn kyn sin kyn y e±j n z j!µ Ey = [A k cos( kxn x ) Bx kxn sin (kxn x)] ⇥ 2 x xn ! 2 µ✏ n ⇥ ⇤ Ay sin kyn y + By cos kyn y e±j n z , Ex = ! 2 µ✏ (2.131) (2.132) and the transverse magnetic field components expressed as Hx = ± j n [A k cos( kxn x ) Bn kxn sin (kxn x)] ⇥ 2 n xn ! 2 µ✏ n ⇥ ⇤ Cn sin kyn y + Dn cos kyn y e±j n z j n [A sin( kxn x )+Bn cos (kxn x)] ⇥ 2 n 2 µ✏ ! n ⇥ ⇤ Cn ky cos kyn y Dn ky sin kyn y e±j n z . (2.133) Hy = ± (2.134) The next step is applying the boundary condition on the tangential components of the electric field at the perfectly conducting surfaces of the rectangular waveguide walls. At the interface y = 0, the tangental electric field boundary condition requires j!µkyn [A sin (kxn x) + Bn cos (kxn x)] Cn e±j n z = 0. 2 n 2 µ✏ ! n (2.135) The tangential electric field boundary condition at x = 0 requires ⇤ j!µkxn [Cn sin( kyn y +Dn cos kyn y An e±j n z = 0. 2 ! 2 µ✏ n (2.136) Therefore, the coe cients An and Cn equal zero. Defining Bn Dn = Fn , then Hz = Fn cos (kxn x) cos kyn y e±j n z . 39 (2.137) At x = a the tangential electric field boundary condition for Ey requires j!µkxn F sin (kxn a) cos kyn y e±j n z = 0. 2 n 2 µ✏ ! n (2.138) This means kxn a must be an integer multiple of ⇡ and therefore kxn = un ⇡ where un is an a integer. Similarly, applying the tangential electric field boundary conditions for Ex at y = b requires j!µkyn F cos (kxn x) sin kyn b e±j n z = 0, 2 n 2 µ✏ ! n (2.139) which leads to kyn = vn ⇡ where vn is an integer. Since the modal analysis in this disb sertation for isotropic materials use both TE and TM modes the variables un and vn are introduced to distinguish between the di↵erent types of modes. Here the variables un and vn correspond to the indices (u, v) for a TEuv wave (if the nth mode is TEz ) or a TMuv wave (if the nth mode is TMz ). As was mentioned previously, takes on discrete values and the propagation constant for mode n is given by q n = ! 2 µ✏ 2 kcn . (2.140) Here kcn is the cuto↵ wave number of the nth mode and is given by kcn = q 2 2 kxn + kyn . (2.141) The electric field components then reduce to Ex = Ey = j!µkyn Fn cos (kxn x) sin kyn y e±j n z 2 kcn j!µkxn Fn sin (kxn x) cos kyn y e±j n z , 2 kcn 40 (2.142) (2.143) and the magnetic field components become j n kxn Fn sin (kxn x) cos kyn y e±j n z 2 kcn j n kyn Hy = ⌥ Fn cos (kxn x) sin kyn y e±j n z , 2 kcn Hx = ⌥ (2.144) (2.145) where Fn is the modal amplitude coe cient. By defining the variable Kn = j!µ , 2 kcn (2.146) (2.142) - (2.145) reduce to Ex = Fn Kn kyn cos (kxn x) sin kyn y e±j n z (2.147) e±j n z (2.148) e±j n z (2.149) e±j n z , (2.150) Ey = Fn Kn kxn sin (kxn x) cos kyn y Kn Hx = ⌥Fn kxn sin (kxn x) cos kyn y Zn Kn Hy = ⌥Fn kyn cos (kxn x) sin kyn y Zn where Zn = !µ/ n is the TE modal wave impedance. Since (2.147) - (2.150) all share the common variable Kn , a new variable can be defined as Cn = Fn Kn which results in the electric and magnetic transverse field equations for TEz modes reducing to ~ r E⇢ (~) = Cn~n (~) e±j n z e ⇢ (2.151) ~ r H⇢ (~) = ⌥Cn~ n (~) e±j n z . h ⇢ (2.152) Here ~n (~) and ~ n (~) are the transverse electric and magnetic modal fields, respectively, e ⇢ h ⇢ 41 and are given by ~n (~) = xkyn cos (kxn x) sin kyn y e ⇢ ˆ y kxn sin (kxn x) cos kyn y , ˆ ˆ e ⇢ ~ n (~) = z ⇥ ~n (~) = h ⇢ Zn ⇥ ⇤ 1 xkxn sin (kxn x) cos kyn y + y kyn cos (kxn x) sin kyn y . ˆ ˆ Zn (2.153) (2.154) Note that the definitions of ~n (~) and ~ n (~) in (2.153) and (2.154) are di↵erent than those e ⇢ h ⇢ given in [45] and [46]. 2.3.1.2 Solution to the TMz Wave Equation for Isotropic Filled Guide To solve the TMz wave equation, (2.118), a separation of variables is utilized in conjunction with applying the boundary conditions on the tangential electric field at the perfectly conducting waveguide walls. Using the product solution of Ez (~, ) = X(x, )Y (y, ) and ⇢ following the similar steps outlined for the TEz case, results in @ 2 X(x, ) 2 + kx X(x, ) = 0 2 @x 2 Y (y, ) @ 2 + ky Y (y, ) = 0. @y 2 (2.155) (2.156) As it was shown for the TEz case, with the application of boundary conditions, , kx , and ky take the form of discrete eigenvalues. The eigenfield associated with on these eigenvalues is Ez = [An sin( kxn x )+Bn cos (kxn x)] [Cn sin( kyn y +Dn cos kyn y ⇤ ±j z n . e (2.157) The transverse field components for the T M modes in the isotropic material are reduced from the general expressions for bianisotropic material, (2.101) and (2.102), thus resulting 42 in ⇥ ⇤ ~ H⇢ = ⌘ · j!✏ u ⇥ r⇢ Eu ˆ ~ E⇢ = ⌘ 1 · ⌥j n r⇢ Eu . (2.158) (2.159) Referring to the geometry in Figure 2.1 and assuming TMz modes, (2.158) and (2.159) become j n @Ez j n @Ez ⌥y ˆ 2 2 2 2 n ! µ✏ @x n ! µ✏ @y j!✏ @Ez j!✏ @Ez ~ H⇢ = x ˆ y ˆ . 2 ! 2 µ✏ @y 2 ! 2 µ✏ @x n n ~ E⇢ = ⌥ˆ x (2.160) (2.161) Now with general expressions for the transverse fields, the solution to the wave equation, (2.157), can be substituted into (2.160) and (2.161). The transverse electric field components are thus given by Ex = ± j n [An kxn cos( kxn x ) Bn kxn sin (kxn x)] ⇥ 2 kcn ⇥ ⇤ Cn sin kyn y + Dn cos kyn y e±j n z j n [An sin( kxn x )+Bn cos (kxn x)] ⇥ 2 kcn ⇥ ⇤ Cn kyn cos kyn y Dn kyn sin kyn y e±j n z , (2.162) Ey = ± (2.163) and the transverse magnetic field components are j!✏ [An sin( kxn x )+Bn cos (kxn x)] ⇥ 2 kcn ⇥ ⇤ Cn kyn cos kyn y Dn kyn sin kyn y e±j n z Hx = Hy = j!✏ [An kxn cos( kxn x ) Bn kxn sin (kxn x)] ⇥ 2 kcn ⇥ ⇤ Cn sin kyn y + Dn cos kyn y e±j n z . 43 (2.164) (2.165) Following the procedure outlined in the TE case, the boundary conditions on the tangential field components are applied at the perfectly conducting walls. Through these steps, the longitudinal electric field component reduces to Ez = Fn sin (kxn x) sin kyn y e±j n z . (2.166) The transverse electric field components are given by j n kxn Fn cos (kxn x) sin kyn y e±j n z 2 kcn j n kyn Ey = ± Fn sin (kxn x) cos kyn y e±j n z , 2 kcn Ex = ± (2.167) (2.168) and the transverse magnetic field components are Hx = Hy = j!✏kyn Fn sin (kxn x) cos kyn y e±j n z 2 kcn j!✏kxn Fn cos (kxn x) sin kyn y e±j n z , 2 kcn (2.169) (2.170) where Fn is the modal amplitude coe cient. By defining the quantity Ln = j n , 2 kcn (2.171) (2.167) - (2.170) reduces to Ex = ±Fn Ln kxn cos (kxn x) sin kyn y e±j n z (2.172) Ey = ±Fn Ln kyn sin (kxn x) cos kyn y e±j n z (2.173) Ln kyn sin (kxn x) cos kyn y e±j n z Zn Ln Fn kxn cos (kxn x) sin kyn y e±j n z , Zn Hx = F n Hy = 44 (2.174) (2.175) n where Zn = !✏ is the TM modal wave impedance. Since the transverse field equations all share the variable Ln , a new variable Dn = Fn Ln is defined which results in a simplified form of the electric and magnetic transverse field equations for TMz modes: ~ r E⇢ (~) = ⌥Dn~n (~) e±j n z e ⇢ (2.176) ~ r H⇢ (~) = Dn~ n (~) e±j n z . h ⇢ (2.177) Here ~n (~) and ~ n (~) are the transverse electric and magnetic modal fields, respectively, e ⇢ h ⇢ and are given by ~n (~) = e ⇢ xkxn cos (kxn x) sin kyn y ˆ y kyn sin (kxn x) cos kyn y , ˆ 1 ⇥ xkyn sin (kxn x) cos kyn y ˆ T Zn M ˆ e ⇢ ~ n (~) = z ⇥ ~n (~) = h ⇢ T Zn M ⇤ kxn cos (kxn x) sin kyn y . (2.178) (2.179) To help with subsequent modal analysis, it is beneficial for the TE and TM field expressions to have the same form. By substituting Dn = ⌥Cn , (2.176) and (2.177) becomes ~ r E⇢ (~) = Cn~n (~) e±j n z e ⇢ (2.180) ~ r H⇢ (~) = ⌥Cn~ n (~) e±j n z . h ⇢ (2.181) Since this substitution only a↵ects the modal amplitude constants, as long as this change is implemented in both equations, the results are the same. Consequently, the general expressions for the isotropic transverse fields for TE and TM modes are of the same form, but with di↵erent transverse electric and magnetic modal fields. 45 2.3.2 Biaxial Material Filled Guide The next type of material to be considered is a biaxially anisotropic material. Biaxial materials are anisotropic and have six non-zero complex permittivity and permeability tensor entries, 2 6 ✏x 0 0 6 ✏ = 6 0 ✏y 0 6 4 0 0 ✏z and 2 6 µx 0 0 6 µ = 6 0 µy 0 6 4 0 0 µz 3 7 7 7 7 5 (2.182) 3 7 7 7. 7 5 (2.183) In order to derive the homogenous wave equation for biaxial material, the general bianisotropic homogenous wave equations for TE waves, (2.106), and for TM waves, (2.108), are reduced to their respective anisotropic form. The TE decomposition is given by = h ⇥ r⇢ ⇥ ⌧ 1 · j!µux · r⇢ Hu ( ) + j uHu ( ˆ h n j!ˆµuu Hu ( ) j!ˆµu · ⌘ 1 · u u~ j r⇢ Hu ( ) k E · u Hu ( ˆ ) ⇤ i ) oi , (2.184) and the TM decompositions is expressed as h ⇥ r⇢ ⇥ ⌘ 1 · j!✏ux · r⇢ Eu ( ) + j uEu ( ˆ h n = j!ˆ✏uu Eu ( ) + j!ˆ~u · ⌧ 1 · u u✏ j r⇢ E u ( ) k H · uE u ( ˆ ) ⇤ i ) oi . (2.185) Measurements of a biaxial sample with finite thickness typically uses empty waveguide extensions connected to a sample holder. A vector network analyzer (VNA) measures the reflection and transmission coe cients of a biaxial sample completely filling the cross-section of the sample holder. In the waveguide extension, the first mode that propagates is TE10 . 46 x=a y y=b ϵ,µ X x=-w 2 x=w 2 Figure 2.2: Cross-section view of rectangular waveguide used in derivations for biaxially anisotropic materials. It will be shown further on in this dissertation that when this mode is incident on a biaxial material there is no coupling between orthogonal field components and therefore no excitation of higher order modes. All of the characterization techniques for biaxial materials completely filling the cross-section described in this dissertation assume the presence of only TE modes. Consider a waveguide completely filled with a biaxial material, as shown in Figure 2.2 . For TEz modes, the expressions in (2.184) become  ⇥ ⌧ 1· = ⇢ x ˆ @ @ +y ˆ @x @y @Hz @Hz µy x y · x ˆˆ ˆ +y ˆ + j z Hz ˆ @x @y  ⇢ ✓ ◆ @Hz @Hz 1· j!ˆ (ˆµz ) · ⌘ z z j x ˆ y ˆ @x @y oi 2 yy✏ µ ! ˆ ˆ x y x x ✏ y µx · z H z . ˆˆ ˆ j! µx y x ˆˆ j!ˆµz Hz z   47 (2.186) Here ⌧ 1 = xx ˆˆ 1 + yy ˆˆ ! 2 ✏ y µx 1 1 + zz ˆˆ 2 ! 2 ✏ x µy (2.187) ⌘ 1 = xx ˆˆ 1 + yy ˆˆ ! 2 ✏ x µy 1 1 + zz . ˆˆ 2 ! 2 ✏ y µx (2.188) and The TEz wave equation for Hz simplifies to 2 4 2.3.2.1 @2 @x2 + µy µx ⇣ ⇣ 2 2 ! 2 µx ✏ y ⌘ @2 ⌘ 2 µy ✏x @y 2 ! µz ⇣ 2 µx ⌘ 3 µx ✏y 5 Hz = 0. (2.189) Solution to the TEz Wave Equation for Biaxial Material Filled Guide In addition to describing a characterization technique for biaxial material completely filling the cross-section of a guide, this dissertation considers a reduced-aperture waveguide approach for single-sample characterization purposes. The reduction of the of the waveguide is symmetric about the width. Therefore, although a single TE10 mode is incident from an empty waveguide region onto the sample region, because of mode conversion at the discontinuity with the reduced aperture, an infinite number of waveguide modes are reflected back into the empty guide. Also, there are an infinite number of modes present in the reduced-aperture material region and the rectangular waveguide receiving extension. However, because the aperture is symmetric about the width of the guide, only rectangular waveguide modes of structure TEn0 (n = 1, 2, 3, . . .) are supported in the waveguide extensions and in the sample region. For TEn0 modes in a rectangular waveguide there is no y-dependence of the fields and the wave equation reduces to @2 2 + kc 2 @x ! Hz (~, ) = 0 ⇢ 48 (2.190) where 2 kc = µz ⇣ 2 ! µx ✏ y µx 2 ⌘ (2.191) is the cuto↵ wavenumber. To solve (2.190) a separation of variables technique is used in conjunction with the application of boundary conditions on the tangential electric field at the perfectly conducting waveguide walls. As was shown when considering isotropic materials, with the application of boundary conditions and kcn take the form of discrete eigenvalues. The eigenfield associated with these eigenvalues is Hz (x, z) = [An sin (kcn x)) + Bn cos (kcn x)] e±j n z . (2.192) To determine the transverse field components for TE modes in a biaxial material, (2.95) and (2.96) first must be simplified to anisotropic form. This gives n o ~ H⇢ = ⌘ 1 · ⌥j n r⇢ Hu k E · uHu ˆ ⇥ ⇤ ~ E⇢ = ⌧ · j!µux · r⇢ Hu + j uHu . ˆ (2.193) (2.194) Then using the tensors described in (2.182) and (2.183) and assuming TEz modes, (2.193) and (2.194) become j!µy @Hz y ˆ 2 n µy ✏x @y j n @Hz ~ H⇢ (x, z) = ⌥ˆ x ⌥y ˆ 2 µx ✏y @x n ~ E⇢ (x, z) = x ˆ 49 j!µx @Hz 2 n µx ✏y @x j @Hz . 2 µy ✏x @y n (2.195) (2.196) Since only TEn0 modes are of interest, the transverse field equations reduce to j!µx @Hz 2 2 n ! µx ✏y @x j n @Hz Hx (x, z) = ⌥ . 2 ! 2 µx ✏y @x n Ey (x, z) = (2.197) (2.198) The axial field, (2.192), can now be substituted into (2.197) and (2.198) giving j!µx kcn [A cos (kcn x) Bn sin (kcn x)] e±j n z 2 n 2 µx ✏ y ! n j n kcn Hx = ⌥ [A cos (kcn x) Bn sin (kcn x)] e±j n z . 2 n ! 2 µx ✏ y n Ey = (2.199) (2.200) The next step is applying the boundary condition on the tangential component of the electric field at the perfectly conducting waveguide walls. At the interfaces x = ± w , the tangental 2 electric field boundary condition requires  ✓ ◆ j!µx kcn wkcn An cos 2 2 ! 2 µx ✏ y n Bn sin ✓ Bn sin ✓ wkcn 2 ◆ e±j n z = 0. (2.201) and  ✓ ◆ j!µx kcn wkcn An cos 2 2 ! 2 µx ✏ y n wkcn 2 ◆ e±j n z = 0. (2.202) These can be expressed in matrix form as 2 ⇣ ⌘ cos kcn w 6 ⇣ 2 ⌘ 4 cos kcn w 2 3 2 3 ⌘ 32 kcn w sin 76 A 7 6 0 7 ⇣ 2 ⌘ 54 5 = 4 5. kcn w sin B 0 2 ⇣ (2.203) This system of homogeneous linear equations has a non-trivial solution, if its determinant is zero [47]. Therefore, the matrix in (2.203) has an infinite number of solutions if its 50 determinant is equal to zero. The determinant is given by cos ✓ wkcn 2 ◆ sin ✓ wkcn 2 ◆ + cos ✓ ◆ sin ◆ = 0. wkcn 2 ✓ wkcn 2 ◆ = 0, (2.204) or 2 cos ✓ wkcn 2 ◆ sin ✓ wkcn 2 (2.205) From (2.205) it is determined that kcn = n⇡ . The variable kcn is the cuto↵ wavenumber w for the nth TEn0 mode in the sample region. As was stated previously, takes on discrete values and thus the propagation constant for mode n is given by r n = ! 2 µx ✏ y µx 2 k . µz cn (2.206) The equations for the tangential electric field boundary condition, (2.201) and (2.202), also gives the expression ⇣ ⌘ Bn sin kcn w 2 ⇣ ⌘ . An = kcn w cos 2 (2.207) Substituting (2.207) into the general expression for the longitudinal magnetic field component, (2.192), results in h ⇣ 0 cos k Hz (x, z) = Bn cn x w ⌘i ±j n z e , 2 (2.208) where 0 Bn = B ⇣ n ⌘. cos kcn w 2 51 (2.209) This new expression for the longitudinal magnetic field component also ends up altering the expressions for the transverse field components, which are now given by h ⇣ j!µx kcn sin kcn x 2 ! 2 µx ✏ y n h ⇣ j n kcn 0 Hx (x, z) = ±Bn sin kcn x 2 ! 2 µx ✏ y n w ⌘i ±j n z e 2 w ⌘i ±j n z e . 2 0 Ey (x, z) = Bn (2.210) (2.211) These equations can be simplified using (2.206), which results in h ⇣ j!µz kcn sin kcn x 2 kcn h ⇣ 0 j n kcn sin k Hx (x, z) = ⌥Bn cn x 2 µx kcn Ey (x, z) = w ⌘i ±j n z e 2 w ⌘i ±j n z e 2 0 Bn (2.212) (2.213) or h ⇣ 0 Bn Kn sin kcn x h ⇣ 0 Kn sin k Hx (x, z) = ⌥Bn cn x Zn w ⌘i ±j n z e 2 w ⌘i ±j n z e . 2 Ey (x, z) = (2.214) (2.215) 2 Here Kn = j!µz /kcn and Zn = !µx / n is the modal wave impedance. By defining the 0 variable Cn = Bn Kn , the transverse field equations reduce to Ey (x, z) = Cn eyn (x) e±j n z (2.216) Hx (x, z) = ⌥Cn hxn (x) e±j n z , (2.217) where the modal transverse field equations are given by h eyn (x) = ⇣ kcn sin kcn x h ⇣ z ⇥ eyn (x) ˆ kcn hxn (x) = = sin kcn x Zn Zn 52 w ⌘i 2 w ⌘i . 2 (2.218) (2.219) 2.3.3 Gyromagnetic Material Filled Guide The final type of material considered is a gyromagnetic material magnetized along the y-axis of Figure 2.1. The permittivity dyadic is given by ✏ = ✏I, (2.220) and the permeability dyadic is expressed as 2 6 µg 0 6 µ = 6 0 µy 6 4 j 0 3 j 7 7 0 7. 7 5 µg (2.221) Using the constitutive parameter tensor and the general bianisotropic wave equations for TE fields, (2.106), and for TM fields, (2.108), the homogenous equation for anisotropic material is given by = h ⇥ ⇤ i r⇢ ⇥ ⌧ 1 · j!µux · r⇢ Hu + j uHu ˆ h n oi j!ˆµuu Hu j!ˆµu · ⌘ 1 · u u~ j r⇢ Hu k E · u Hu , ˆ (2.222) and h ⇥ ⇤ i r⇢ ⇥ ⌘ 1 · j!✏ux · r⇢ Eu + j uEu ˆ h n oi 1· = j!ˆ✏uu Eu + j!ˆ~u · ⌧ u u✏ j r⇢ E u k H · uE u , ˆ (2.223) respectively. Consider a sample of gyromagnetic material filling the cross-section of the rectangular waveguide shown in Figure 2.1, with finite thickness, and magnetized along the y-axis. When the TE10 propagates inside an empty waveguide extension and this mode is incident on the 53 sample, a coupling exists between orthogonal field components in the sample. This coupling results in an excitation of higher-order modes. It will be shown in this dissertation that when a TE10 is incident on this sample, and the sample is magnetized along the y-axis, only TEn0 modes are supported. The homogeneous wave equation for TEz fields is given by  ⇥ ⌧ 1· = ⇢ j! µg y x ˆˆ  x ˆ @ @ +y ˆ @x @y @Hz @Hz µy x y · x ˆˆ ˆ +y ˆ + j z Hz ˆ @x @y  ⇢ ✓ ◆ @Hz @Hz 1· j!ˆz µg · ⌘ zˆ j x ˆ +y ˆ @x @y oi ! 2 y y ✏µy xx✏µg + j xz ✏ · z Hz , ˆˆ ˆˆ ˆˆ ˆ j yz ˆˆ j!ˆµg Hz + !ˆx z zˆ  (2.224) where ⌘ 1= 2 1 ! 2 µg ✏ ! xx + ˆˆ 1 ! 2 µy ✏ 2 0 @ 2 ⇣ ! yy + ˆˆ 1 zz ˆˆ 2 1 j!✏ ⌘ A xz , ˆˆ 2 ! 2 µg ✏ (2.225) and ⌧ 1= 2 1 ! 2 µy ✏ ! xx + ˆˆ 2 1 ! 2 µg ✏ ! yy + ˆˆ 1 zz. ˆˆ 2 (2.226) The wave equation simplifies to µy 2 @2 + @x2 µg 2 µg ✏! 2 @ 2 2 + kc µy ✏! 2 @y 2 ! Hz = 0, (2.227) where the cuto↵ wavenumber is given by 2 2 k c = ! 2 µg ✏ 1 µ2 g 54 ! 2. (2.228) 2.3.3.1 Solution to the TEz Wave Equation for Gyromagnetic Material Filled Guide For TEn0 modes in a rectangular waveguide there is no y-dependence of the fields, and the wave equation in (2.227) reduces to @2 2 + kc 2 @x ! Hz (x) = 0. (2.229) As was shown for the isotropic and biaxial material filled guides, through the application of boundary conditions, and kcn will take the form of discrete eigenvalues. The eigenfield associated with these eigenvalues is Hz (x, z) = [An sin (kcn x) + Bn cos (kcn x)] e±j n z . (2.230) To determine the transverse field components for TE modes in a gyromagnetic material, (2.95) and (2.96) are reduce to anisotropic form, which results in n o ~ H⇢ = ⌘ 1 · ⌥j n r⇢ Hu k E · uHu ˆ ⇥ ⇤ ~ E⇢ = ⌧ · j!µux · r⇢ Hu + j uHu . ˆ (2.231) (2.232) Considering the waveguide geometry shown in Figure 2.1 and the material tensors in (2.182) and (2.183), the expressions in (2.231) and (2.232) become " # j n @Hz j! 2 ✏ ~ H⇢ (x, z) = x ⌥ ˆ + H ⌥y ˆ 2 µg ✏! 2 @x 2 µg ✏! 2 z n " n j!µy j!µg @Hz @Hz ~ E⇢ (x, z) = x ˆ y ˆ + 2 2 2 2 n µy ✏! @y n µg ✏! @x j n @Hz 2 2 n µy ✏! @y # j! n H . 2 2 z n µg ✏! (2.233) (2.234) Since in this dissertation only higher order TEn0 modes are analyzed, the transverse fields 55 are " # j n @Hz j! 2 ✏ ~ Hx (x, z) = x ⌥ ˆ + H 2 µg ✏! 2 @x 2 µg ✏! 2 z n " n # j!µg @Hz j! n ~ Ey (x, z) = y ˆ ± H . 2 µg ✏! 2 @x 2 µg ✏! 2 z n n (2.235) (2.236) The axial magnetic field (2.230), can be substituted into (2.236), resulting in ~ Ey (x, z) = j! 2 n µg ✏! 2 ⇥ µg kcn An cos (kcn x) n µg kcn Bn sin (kcn x)±  n An sin (kcn x) ±  n Bn cos (kcn x)] e±j n z . (2.237) The unknown eigenvalues may be determined by applying the boundary conditions on Ey at the interfaces between the material and the conduction waveguide walls. At x = 0, the boundary condition on the tangential electric field requires ⇥ ⇤ µg kcn An ±  n Bn e±j n z = 0 µg ✏! 2 j! 2 n (2.238) or An = ⌥ Bn n . µg kcn (2.239) Substituting the expression for An into (2.237) yields, Ey (x, z) = " # 2 2 µ2 kcn 2 B + n sin (kcn x)e±j n z . 2 2 n µg kcn  µg ✏! j! 2 n (2.240) Next, inserting the expression for An into (2.230) gives  Hz (x, z) = Bn cos (kcn s) ⌥  n sin (kcn x). e±j n z µg kcn 56 (2.241) This expression for the solution to the wave equation is substituted into (2.236) and simplified, giving  j nB kcn Hx (x, z) = ⌥ sin (kcn x) ⌥ cos (kcn x) e±j n z . 2 2 µg ✏ µg n ! n (2.242) At z = a, the boundary condition on the tangential electric field requires j!Bµg 2 ! 2 µg ✏ n 1 2 µ2 g ! sin (kcn z)e±j n z = 0 (2.243) or sin (kcn a) = 0. (2.244) From (2.244) it is determined that kcn = n⇡/a. Here kcn is the cuto↵ waveguide for the nth TEn0 mode in the sample region. The propagation constant for mode n is found from (2.228) v u u n = t ! 2 µg ✏ 1 2 µ2 ! 2 kcn . (2.245) Next, using (2.245) the transverse fields in (2.240) and (2.242) become ! 2 Ey (x, z) = 1 sin (kcn x)e±j n z 2 µg  j nB kcn Hx (x, z) = ⌥ sin (kcn x) ⌥ cos (kcn x) e±j n z . kcn µg n j!Bn µg kcn (2.246) (2.247) By defining 2 Kn = j!µg /kcn , 57 (2.248) (2.246) and (2.247) reduce to ! 2 Ey (x, z) = Bn Kn kcn 1 sin (kcn x)e±j n z 2 µg  Kn kcn kcn Hx (x, z) = ⌥Bn sin (kcn x) ⌥ cos (kcn x) e±j n z , Zn µg n (2.249) (2.250) where and Zn = !µg / n is the modal wave impedance. By defining Cn = Bn Kn , the transverse field equations reduce to ! 2 Ey (x, z) = Cn kcn 1 sin (kcn x)e±j n z µ2 g  Cn kcn kcn Hx (x, z) = ⌥ sin (kcn x) ⌥ cos (kcn x) e±j n z . Zn µg n 58 (2.251) (2.252) CHAPTER 3 ISOTROPIC MATERIAL CHARACTERIZATION 3.1 Introduction This chapter presents two methods for characterizing isotropic materials using rectangular waveguides. The first technique described is the classical “Nicolson-Ross-Weir” (NRW) extraction applied to an isotropic material completing filling the cross-section of a rectangular waveguide [23]-[24]. This method extracts the material parameters using closed form expressions from the measurement of reflection and transmission coe cients. The convenience of the NRW method, and its insensitivity to propagation of measurement uncertainties, commonly makes it a first choice for characterization of isotropic materials. It is for these reasons that this dissertation makes use of the NRW extraction method and extends the technique to biaxially anisotropic materials (Section 4.2). In addition, the developed waveguide verification standard described in this dissertation (Chapter 6) assumes the use of this classical extraction algorithm. The second technique is a reflection-only method for conductor-backed absorbing materials using rectangular waveguides. If the sample is conductor backed, and occupies the 59 entire cross-section of the guide, then a transmission measurement is not available, and thus a method must be found for providing two su ciently di↵erent reflection measurements. The technique proposed here is to place a waveguide iris in front of the sample, exposing the sample to a spectrum of evanescent modes. By measuring the reflection coe cient both with and without an iris the necessary two measurements may be obtained to determine ✏ and µ. 3.2 NRW Characterization for Isotropic Materials The classic NRW extraction algorithm employs the measured reflection and transmission coe cients for a waveguide section completely filled by the material. The attraction of the NRW characterization method results from the availability of closed-form expressions for µ and ✏. This contrasts with methods requiring an iterative solver such as Newton’s method [30]-[31] or a least squares approach [32]-[33]. The closed-form expressions are wellconditioned for most of the material parameters of interest, except when the material thickness approaches multiples of a half wavelength. The NRW extraction can be used with rectangular waveguides [25], coaxial applicators [26]-[27], free-space methods [28], and stripline measurements [29]. This section presents the method for isotropic material characterization using this classical technique with rectangular waveguides. 3.2.1 NRW Derivation Assume the sample placed in the cross-sectional plane of the waveguide is linear and homogeneous as well as isotropic. The permittivity and permeability are represented by the constitutive material tensors 2 6 ✏r 0 0 6 ✏ = ✏0 6 0 ✏r 0 6 4 0 0 ✏r 60 3 7 7 7 7 5 (3.1) and 2 6 µr 0 0 6 µ = µ0 6 0 µr 0 6 4 0 0 µr 3 7 7 7, 7 5 (3.2) where tensor entries are complex quantities: ✏r = ✏0 + j✏00 , µr = µ0 + jµ00 . r r r r Figure 3.1 shows the experimental configuration used in the NRW method. A sample with unknown properties is placed into a sample holder occupying the region 0  z  d in a rectangular waveguide system. Waveguide extensions are then attached to guarantee only the dominant mode is present at the measurement ports. The S-parameters S11 and S21 are measured and used to determine the sample propagation constant and the interfacial reflection coe cient which then can be used to extract ✏r and µr . 3.2.1.1 Field Structure in a Waveguide Filled with Isotropic Material Assume a TE10 mode is incident from the transmitting empty waveguide (z < 0). This field will couple into the TE10 material filled waveguide mode since the field structure in both regions is identical. In the empty and sample waveguide region, it can be shown from Section 2.3.1.1 that the TE10 mode is the dominant mode (n = 1), which means u1 = 1 e,s e,s e,s and v1 = 0. Therefore, kx1 = ⇡/a, ky1 = 0, and thus kc1 = ⇡/a is the cuto↵ wavenumber for the first order mode. Now, using (2.140) the propagation constant in the empty guide is given by e 1= r ⇣ 2 k0 e kc1 ⌘2 , (3.3) and the propagation constant in the sample region results in s 1= r 2 k 0 ✏ r µr 61 ⇣ s kc1 ⌘2 , (3.4) p where k0 = ! µ0 ✏0 . The transverse fields from (2.151) - (2.152) for both regions are given by e,s ±j 1 z e,s e,s Ey (x, z) = C1 ey1 (x) e e,s ±j 1 z e,s e,s Hx (x, z) = ⌥C1 hx1 (x) e . (3.5) (3.6) e,s e,s Here ey1 and hx1 are the transverse electric and magnetic modal fields in the empty guide and sample regions, and are expressed as ⇣ ⌘ e,s e,s kx1 sin kx1 x , e,s ⇣ ⌘ kx1 e,s e,s hx1 = e,s sin kx1 x , Z1 e,s ey1 = (3.7) (3.8) e where the modal wave impedance in the empty guide and sample region is given by Z1 = e s e !µ0 / 1 and Z1 = !µ0 µr / 1 , respectively. 3.2.1.2 Solution for Reflection and Transmission Coe cients The transverse fields in the sample region and in the waveguide extensions can be represented using the modal fields, with the modal amplitudes determined through the application of boundary conditions. The dominant-mode transverse fields for the transmitting extensions, z < 0, may thus be written as e j 1z e e e Ai kx1 sin kx1 x e Ar kx1 sin e e e i kx1 sin k e x e j 1 z Ar kx1 sin Hx (x, z) = A e e x1 Z1 Z1 Ey (x, z) = 62 e j 1z (3.9) e j 1z . (3.10) e kx1 x e e kx1 x e Similarly, the dominant-mode transverse fields in the sample region 0 < z < d may be written as s j 1z s s s B + kx1 sin kx1 x e B kx1 sin s B+ s j 1z B s s Hx (x, z) = s kx1 sin kx1 x e s k sin Z1 Z1 x1 Ey (x, z) = s j 1z (3.11) s j 1z , (3.12) s kx1 x e s kx1 x e while those in the receiving waveguide extension, z > d, may be expressed as e C t kx1 sin ke Hx (x, z) = C t x1 sin e Z1 Ey (x, z) = e kx1 x e e j 1 (z d) (3.13) e kx1 x e e j 1 (z d) . (3.14) Here, Ai is the amplitude of the incident TE10 wave. The modal amplitudes Ar , B + , B , and C t may be determined by applying the boundary conditions on Ey and Hx at the interfaces between the isotropic sample and the empty guides. At the interface z = 0, the tangential electric field boundary condition requires h i h i i + Ar k e sin k e x = B + + B s s A kx1 sin kx1 x , x1 x1 (3.15) s e and thus since kx1 = kx1 , h i h i i + Ar = B + + B A . (3.16) The tangential magnetic field boundary condition at z = 0 requires h Ai Ar i ke h x1 sin k e x = B + e x1 Z1 63 B i ks x1 sin k s x , s x1 Z1 (3.17) or h i Zs h Ar 1 = B + e Z1 Ai B i . (3.18) At z = d, the boundary condition on the tangential electric field requires  B+e s j 1d j sd s s e e + B e 1 kx1 sin kx1 x = C t kx1 sin kx1 x , (3.19) or  B+e s j 1d j sd + B e 1 = C t, (3.20) while the boundary condition on the tangential magnetic field at z = d results in  s e s j 1 d kx1 s x = C t kx1 sin k e x , B e s sin kx1 e x1 Z1 Z1 s j 1d B+e (3.21) or  B+e s j 1d B e s j 1d 1 t 1 s = C Ze . Z1 1 (3.22) The reflection coe cient, R = Ar /Ai , is determined by first adding (3.16) to (3.18), yielding, 2B + = Ai s Z1 e +1 Z1 ! Ar s Z1 e Z1 ! 1 , (3.23) and then subtracting (3.16) from (3.18), producing 2B = Ai s Z1 e Z1 1 ! 64 + Ar ! s Z1 e +1 . Z1 (3.24) Adding (3.20) and (3.22) results in " # s s + = C t Z 1 + 1 ej 1 d , 2B e Z1 (3.25) while subtracting (3.20) from (3.22) gives 2B = " s t Z1 C e Z1 # s j 1d . (3.26) # s e s Z 1 Z1 j2 1 d , s + Ze e Z1 1 (3.27) 1 e Then dividing (3.26) by (3.25) produces B = B+ " which can also be written as B = B+ P 2. (3.28) Here the interfacial reflection coe cient is established as s e Z1 Z1 = s e, Z1 + Z1 (3.29) and the propagation factor is defined as P =e s j 1d . (3.30) Next, dividing (3.24) by (3.25) gives B = B+ ✓ s e◆ Z1 Z1 e Z1 ✓ s e◆ i Z1 +Z1 A e Z1 Ai 65 ✓ s e◆ Z1 +Z1 e Z1 ✓ s e ◆. r Z1 Z1 A e Z1 Ar (3.31) e Multiplying the numerator and denominator by Z1 /Ai yields ⇣ ⌘ s Ze Z1 1 ⇣ ⌘ s e Z1 + Z1 B = B+ Ar Ai Ar Ai ⇣ ⌘ s + Ze Z1 1 ⇣ ⌘, s e Z1 Z 1 (3.32) s e and dividing this expression by (Z1 + Z1 ) produces B = B+ R . R 1 (3.33) The reflection coe cient is now determined by setting (3.28) equal to (3.33), which results in 1 R . R 1 P2 P2 = (3.34) Solving for R then gives R= ⇣ ⌘ 2P 2 1 . (3.35) Next, the transmission coe cient, T = C t /Ai , can be derived by first rearranging (3.20): " # sd B j2 s d j 1 1 = Ct B+e 1 + +e B (3.36) Substituting (3.28), this expression becomes B+e s j 1d [1 + ] = C t . (3.37) Then (3.23) can be substituted, yielding " 1 Ai 2 s e Z1 + Z1 e Z1 ! Ar s Z1 e Z1 e Z1 66 !# [1 ]e s j 1d = C t. (3.38) e Factoring Ai /Z1 and rearranging the equation produces  C t j sd 1 s e e 1 = e Z 1 + Z1 2Z1 Ai Ar s Z1 Ai e Z1 [1 ], (3.39) or Te s Factoring out Z1 s 1 ⇥ s j 1d e = e Z1 + Z1 2Z1 s R Z1 e Z1 ⇤ [1 ]. (3.40) # s e Z1 Z 1 R s e [1 Z1 + Z1 ], (3.41) e Z1 gives 1 j sd s e Te 1 = e Z1 + Z 1 2Z1 " 1 which can also be written as Te s 1 j 1d s e = e Z + Z1 [1 2Z1 1 R ] [1 ]. (3.42) R ], (3.43) Substituting (3.29) then yields Te e s 2Z1 1 j 1d s e = Z1 + Z1 e s e [1 2Z1 Z1 + Z 1 and through addition algebraic manipulation, the transmission coe cient becomes T = (1 R ) P. (3.44) Finally, using the equation for the reflection coe cient, (3.35), the transmission coe cient is expressed as T = ⇣ P 1 1 2 ⌘ 2P 2 67 . (3.45) Note that S11 and S21 are typically measured at the ports of the waveguide extensions. With properly applied phase shift, these may be transformed to the faces of the material sample at z = 0 and z = d. In that case S11 = R and S21 = T . This process is called de-embedding and is described in detail in Appendix B. 3.2.2 Validation of Theoretical Analysis It is useful to validate the theoretical reflection and transmission coe cients derived in the previous section. The theoretical model is validated against the computational finite element (FEM) solver HFSS with an Eccossorb FGM125 test sample. FGM125 has approximate material parameters of ✏r = 7.3197 j0.0464 and µr = 0.5756 j0.4842 at 10.09 GHz. These values were obtained using the waveguide NRW method and were provided by Captain Milo Hyde IV of the Air Force Institute of Technology; they are tabulated in [46]. Though the material properties for FGM125 vary with frequency, for this validation the material parameters are assumed to be frequency independent. This provides a general validation of the theory. The derived theoretical expressions were used to compute the reflection and transmission coe cients for the FGM125 material paced in the cross-section of an X-band waveguide system. The dimensions of the X-band system are a = 22.86 mm, b = 10.16 mm, and d = 3.175 mm. Figure 3.2 shows S11 and S21 at the faces of the sample computed using (3.35) and (3.45). Also shown are the values of the reflection and transmission coe cient computed using HFSS. The waveguide extensions were explicitly modeled in the EM solver and were of significant length to ensure that only the fundamental TE10 mode propagates; these lengths were chosen to be 50 mm. The convergence in HFSS was specified to a maximum delta S of 0.01, which is defined as the absolute di↵erence between all S-parameters from two succeeding iterations computed at a chosen “solution frequency” of 12.4 GHz (this is HFSS terminology and does not imply that the problem is only solved at one frequency). The Sparameters were computed at 31 frequency points over the X-band range of 8.2 68 12.4 GHz using an HFSS discrete frequency sweep. Choosing a “solution frequency” of 12.4 GHz ensures convergence using a fine mesh since this is the highest frequency analyzed. Excellent agreement is obtained between the theoretical equations and HFSS, thus validating the computation of the theoretical reflection and transmission coe cients. 3.2.3 Extraction Process The S-parameters are measured using a vector network analyzer (VNA) attached at the end of the waveguide extensions shown in Figure 3.1. The S-parameters are then mathematically transformed to obtain the S-parameters at the sample planes, S11 and S21 , as is discussed in Appendix B. In this case S11 = R and S21 =. These sample-plane S-parameters are used s to determine the sample propagation constant 1 and the interfacial reflection coe cient , which may in turn be used to find ✏r and µr . Define V1 = S21 + S11 (3.46) V2 = S21 (3.47) S11 . Substituting (3.35) and (3.45), these variables then become P 1+ P P V2 = . 1 P V1 = (3.48) (3.49) From (3.48), the propagation factor is given by P = V1 1 69 V1 . (3.50) µ0 , ε 0 µ, ε Δ Δ Δ 𝑠 𝑟 Figure 3.1: Rectangular waveguide with isotropic sample placed in cross-sectional region with waveguide extensions attached. Magnitude (dB) µ0 , ε 0 µ0 , ε 0  y2 S -2 11 -4 -6 y1 -8 S21 y b Analytical PEC HFSS -10 8 9 PEC -80 Phase (°) ,ε µ0 , ε 0 y2 -100 10 11 Frequency (GHz) µ , ε0 S21 0 y1 -120 PEC -140 0 Analytical HFSS a x S11 -160 12 -180 8 9 10 11 Frequency (GHz) Figure 3.2: S-parameters computed for a FGM125 test material. sday, June 6, 13 70 12 Next, (3.49) is multiplied by 1 P yielding V2 (1 P) = P . (3.51) Then inserting (3.50) into (3.51) gives V2 Multiplying through by 1 V2  V1 1 V1 = V1 1 V1 . (3.52) P V1 and simplifying the expression results in 2 2 X + 1 = 0, (3.53) 1 V1 V2 . V1 V2 (3.54) where X= The solution to (3.53) is q = X ± X2 1. (3.55) Here the appropriate sign is chosen such that | |  1. The propagation constant is then determined from (3.30) as s 1= ln(P ) ± j2n⇡ , jd (3.56) where n depends on the thickness of the sample in terms of a wavelength. Proper choice of n is discussed in Section 3.2.4. Next, rearranging (3.29) gives s Z1 1+ , e =1 Z1 71 (3.57) and the expressions for the modal impedances can then be substituted yielding s e Z1 = 1 µr . e s Z1 1 (3.58) Combining (3.57) and (3.58) gives µr = 1+ 1 s 1, e 1 (3.59) thus providing a closed-form expression for the permeability of the isotropic sample. Using (3.4), an expression for ✏ can also be derived. First, k0 is brought outside the square root yielding v u u s = k t✏ µ 0 r r 1 s kx1 k0 !2 (3.60) or ! s 2 1 = ✏ r µr k0 s kx1 k0 !2 . (3.61) Next, inserting (3.59) and simplifying gives 2 ! s 2 ✏r = 4 1 + k0 ! 3 s 2 e kx1 5 11 s1+ . k0 1 (3.62) Thus, by measuring S11 and S21 for an isotropic sample filling the cross-section of a rectangular waveguide, two equations, (3.59) and (3.62), can be used to extract the two material parameters µr and ✏r . It is important to validate the inversion method before proceeding with the characterization of unknown materials. Therefore, the FGM125 material parameters outlined in Section 3.2.2 are used in HFSS to generate S-parameters with the sample completely filling the 72 cross-section of an X-band waveguide, and the NRW extraction method is performed. The HFSS simulation uses the same model as was used to generate the data for Figure 3.2. The results of the characterization are shown in Figure 3.3. As expected, the characterized material parameters are in accordance with the parameters used in the HFSS simulation. This extraction gives confidence in the characterization technique before moving forward with measurements. 3.2.4 Phase Ambiguity An ambiguity exists when the electrical length of the material reaches a half wavelength and every subsequent full wavelength, where the value n in (3.56) must be incremented. The correct value of n must be determined for accurate characterization. This ambiguity results from the VNA measuring the phase of the S-parameters from 180 to +180 . The phase shift through the material is determined from the propagation factor, which is computed using the measured S-parameters via (3.50). Since the VNA measures the phase of the Sparameters from 180 to +180 , the computed phase shift will wrap around when these measurement boundaries are met. Thus when measuring a sample of material, the phase shift can actually wrap around multiple times and this needs to accounted for in (3.56). This is done by incrementing n every time the phase shift wraps around. The 1-way propagation factor for a wave traveling through the sample region is given by jd P =e ⇣ s s 1r +j 1i ⌘ (3.63) where the propagation constant in the material is split into its real and imaginary components. This expression can also be written as ⇣ P =e ⌘ sd 1i e(j ) , 73 (3.64) Relavtive Material Parameters 8 ϵ'r 6 4 2 ϵ''r μ'r 0 μ''r -2 8 9 10 11 12 Frequency (GHz) Figure 3.3: Extracted permittivity using HFSS generated S-parameters. 74 where = s 1r d is the phase shift through the material. The wavelength in the material is given by s = 2⇡ . s 1r (3.65) Thus phase shift through the material is given by = The first wrap around occurs when = = 2⇡ s d. (3.66) 180 or ⇡= = 2⇡ s d. ⇡, where (3.67) Thus the first phase wrap happens when the thickness of the sample is a half-wavelength, or d = s /2. An example of phase wrap can be seen in Figure 3.4. Here an FGM125 sample was s simulated using (3.66) where 1 was determined using (3.4). The phase shift was determined with the FGM125 sample placed inside an X-band waveguide and simulated from 8.2 GHz to 12.4 GHz. The thickness of the FGM125 sample was chosen to be6.35 mm. From Figure 3.4, it can be seen that n should be zero from 8.2 GHz to approximately 11.1 GHz. Around 11.1 GHz, or when = ⇡ radians, the phase wraps around and n should change to one. The next phase wrap doesn’t occurs until = 3⇡ radians. Using (3.66) it is found that, n = 1 for s /2 < d < 3 s /2. Figure 3.4 also shows the typical sort of error that results when using the NRW method to extract the permittivity when the wrong value of n is used over certain portions of the frequency band. Though not shown, the permeability follows a similar trend. The ambiguity causes problems when measuring an unknown sample, since the values of ✏r and µr are yet to be determined, and so s is not known a priori. Thus the number of times the phase shift wraps around is unknown. A typical solution is to fabricate samples that 75 200 ϕ (°) 100 n=1 n=0 0 -100 -200 9 10 11 12 Permittivity Frequency (GHz) 25 20 15 10 5 0 -5 -10 n=0 n=1 ϵ'r ϵ''r 8 9 10 11 12 Frequency (GHz) Figure 3.4: Upper plot shows phase shift through material while lower plot shows extraction of permittivity from S-parameters using NRW method where n = 0 and n = 1. For interpretation of the references to color in this and all other figures, the reader is referred to the electronic version of this dissertation. 76 are thin and assume the half-wavelength boundary does not occur within the measurement band. 3.2.5 Experimental Results To experimentally validate the NRW technique at X-band, the S-parameters from a sample of Eccosorb FGM125 were measured and the material parameters extracted. The measurement system consisted of two 152 mm long section of WR-90 commercial X-band waveguide, which acted as extensions, with coaxial transitions attached at the ends. Since a sample holder was not manufactured to the specific thickness to the FGM125 sample, as an alternative, the sample was placed inside the waveguide extension connected to Port 1. The S-parameters were then mathematically de-embedded to the faces of the sample. This type of sample insertion is the second case discussed in Appendix B, where the mathematical equations used for the de-embedding are given. Measurements of the S-parameters of the FGM125 sample placed into the waveguide extensions were made using an Aglient E5071C VNA. The VNA was calibrated using an X7005M Maury Microwave X-Band calibration kit, consisting of two o↵set shorts and a load. The measurements were made with VNA settings of 5 dBm source power, 64 averages, and an IF bandwidth of 5 kHz. Finally, the material parameters were extracted using the measured values of S11 and S21 . The measurement repeatability error was assessed by measuring the FGM125 sample 5 separate times, with the VNA calibrated at the start of each set of measurements. The permittivity parameters extracted from the 5 measurements are shown in Figure 3.5, while the permeability parameters are shown in Figure 3.6. The center solid line in these figures represents the mean of the extracted values while the upper and lower solid lines define the 95% confidence levels, or ± two standard deviations. The dotted lines in Figure 3.5 and Figure 3.6 show the material parameters of a di↵erent sample of FGM125 extracted from S-parameter measurements performed at AFIT using the NRW method. These material 77 parameters are tabulated in [46]. A strong agreement exists between the characterization of the two di↵erent FGM125 samples; however, the extraction of the permeability parameters are slightly di↵erent. This is mostly due to material parameters of FGM125 varying among stock. Though the comparison of the results from the two extractions vary slightly, it still gives confidence the NRW characterization method is outlined properly in this dissertation. 3.2.6 Summary The NRW method for the characterization of isotropic materials using rectangular waveguides is described. Closed-form expression are given to characterize the two complex material parameters. A discussion is presented on the ambiguity that exists when the electrical length of the material reaches a half wavelength and subsequent full wavelengths. Validation of the technique was performed using S-parameters generated using HFSS for an FGM125 sample filling the cross-section of an X-band waveguide. Additionally, measurements of the S-parameters for a sample of FGM125 were used for further validation of the technique. 3.3 Measurement of the Electromagnetic Properties of a Conductor-backed Material Using a WaveguideIris Technique Magnetic radar-absorbing materials (MagRAM) are often applied to the conducting surfaces of air vehicles to reduce radar cross-section. To ensure proper aircraft design, the electromagnetic properties of these materials must be accurately characterized. Unfortunately, the process of bonding the MagRAM to the conductor produces a chemical reaction that alters the intrinsic electromagnetic properties of the absorber in unpredictable ways, and thus it is crucial that the absorber be characterized while still attached to the conductor backing. As outlined in Section 3.2, a sample is typically placed into the cross-section of a wave- 78 Realative Permittivity ϵ'r 6 4 7.4 7.35 7.3 7.25 7.2 10 10.1 10.2 10.3 10.4 0.1 0.05 0 -0.05 -0.1 10 10.1 10.2 10.3 10.4 2 0 ϵ''r 9 10 11 12 Frequency (GHz) Figure 3.5: Real parts of relative permittivity extracted from 5 sets of measurements. Center solid line is the average of the measurements. Upper and lower lines show the 95% confidence intervals. Dotted line shows permittivity extracted from the S-parameter measurements performed at AFIT with a di↵erent sample than the one used at MSU. 79 Realative Permeability 0.6 0.4 0.2 0 -0.2 -0.4 μ'r 0.58 0.56 0.54 0.52 0.5 10 10.1 10.2 10.3 10.4 -0.38 -0.42 -0.46 -0.5 10 10.1 10.2 10.3 10.4 μ''r -0.6 -0.8 9 10 11 12 Frequency (GHz) Figure 3.6: Real parts of relative permeability extracted from 5 sets of measurements. Center solid line is the average of the measurements. Upper and lower lines show the 95% confidence intervals. Dotted line shows permeability extracted from the S-parameter measurements performed at AFIT with a di↵erent sample than the one used at MSU. 80 guide and the transmission and reflection coe cients are measured, providing the two necessary data to find both the permittivity and permeability of the sample. If the sample is conductor-backed, and occupies the entire cross-section, a transmission measurement is not available, and thus a method must be found for providing two su ciently di↵erent reflection measurements. The technique proposed here is to place a waveguide iris in front of the sample, exposing the sample to a spectrum of evanescent modes. By measuring the reflection coe cient with and without an iris, the two necessary data may be obtained to determine ✏ and µ. A mode-matching approach is used to determine the theoretical response of a sample placed behind a waveguide iris. This response may be used in a root-searching algorithm to determine ✏ and µ using measurements of the reflection coe cient. 3.3.1 Theoretical Reflection and Transmission Coe cients The conductor-backed material under test is assumed to be linear and homogeneous, with tensor constitutive parameters 2 6 ✏s 0 0 6 ✏ = ✏0 6 0 ✏s 0 6 4 0 0 ✏s and 2 6 µs 0 0 6 µ = µ0 6 0 µ s 0 6 4 0 0 µs 3 7 7 7 7 5 3 7 7 7, 7 5 (3.68) (3.69) where tensor entries are complex quantities: ✏s = ✏0 + j✏00 , µs = µ0 + jµ00 . s s s s The reflection coe cient is measured under two conditions: with and without an iris placed in front of the conductor-backed material. Thus, the theoretical reflection coe cients must accurately be determined under these two conditions. 81 3.3.1.1 Reflection Coe cient with Iris For the calculation of the theoretical reflection coe cient with the iris, consider Figure 3.7. The conductor-backed material with thickness thickness is placed a distance d away from an iris of and a window opening width of w2 . The iris itself consists of two waveguide junctions: a reduced step and an expanded step. The iris is centered about the width of the guide. If a single TE10 mode is assumed to be incident on the iris boundary from the transmitting region, then because of mode conversion at the discontinuity with the iris an infinite number of waveguide modes are reflected back into the transmitting region. Additionally, an infinite number of waveguide modes are transmitted into the iris region, which are then incident on the discontinuity at the expanded-step boundary, and thus a spectrum of modes are also reflected back into the iris region and transmitted into the guide region containing the material under test. Since the electric field of the incident TE10 mode is even about the width of the transmitting guide, and because the iris is also symmetric about the width of the guide, only modes with electric fields even about the width of the guide will be excited. Thus, only TEn0 modes with odd values of n are needed to describe the fields in each of the waveguide regions. To simplify the mode matching technique, each of the iris junctions are analyzed separately. Consider a general reduced-step junction shown in Figure 3.8, where wB < wA . The isotropic transverse field equations from (2.151) and (2.152) can be used in each of these regions where the waveguide is centered about the y-axis, as long as the sinusoidal functions are o↵set by either wA /2 or wB /2. Since only TEn0 modes are excited, v = 0 for all excited modes. This leads to kyn = 0, reducing the transverse field equations to A,B A,B A,B Ey (x, z) = Cn eyn (x) e±j n z A,B A,B A,B Hx (x, z) = ⌥Cn hxn (x) e±j n z . (3.70) (3.71) A,B A,B Here eyn and hxn are the transverse electric and magnetic modal fields, respectively, and 82 a y ϵS,μS ϵg,μg PEC ϵ0,μ0 x b w2 d Δ z δ PEC Figure 3.7: Conductor-backed material with iris and waveguide extension attached. are found from (2.153) and (2.154) to be A,B eyn (x) =  ✓ A,B A,B kxn sin kxn x A,B hxn (x) = wA,B ◆ , 2 A,B eyn (x) , A,B Zn (3.72) (3.73) A,B A,B A,B A,B where kxn = n⇡/a and Zn = !µ0 µA,B / n . Here n is determined from (2.140) as A,B = n q (k A,B )2 A,B (kxn )2 , (3.74) p where k A,B = k0 µA,B ✏A,B . Using only odd values of n to describe the fields can be confusing when programing the mode matching technique. Therefore, a change to the indexing of n can be implemented, where n will now order the odd modes instead of all the modes. This is done by first 83 A,B A,B examining the sinusoidal functions in the expressions for en and hn :  ✓ A,B sin kc,n x " wA,B ◆ n⇡ = sin wA,B 2 ✓ x wA,B ◆ 2 # n = 1, 3, 5 . . . . (3.75) (3.76) This can also be expressed as  ✓ A,B sin kc,n x wA,B ◆ 2 " ✓ 2n 1 = sin ⇡ x wA,B wA,B ◆ 2 # n = 1, 2, 3 . . . . (3.77) (3.78) Expanding the product and using a trigonometric identity results in  ✓ A,B sin kc,n x wA,B ◆ 2 = " #  2n 1 2n 1 cos ⇡x sin ⇡ wA,B 2 n = 1, 2, 3 . . . , (3.79) or  ✓ A,B sin kc,n x wA,B ◆ 2 = ( 1)n cos " 2n 1 ⇡x wA,B # n = 1, 2, 3 . . . . (3.80) Hence (4.80) and (4.81) become A,B eyn (x) = h i A,B n cos k A,B ⇡x , kc,n ( 1) c,n A,B eyn (x) A,B hxn (x) = e , Zn (3.81) (3.82) A,B 2n where kc,n = w 1 . A,B The transverse fields in regions A and B can be expanded in an infinite sum of modal fields, with modal amplitudes to be determined through the application of appropriate 84 boundary conditions. The transverse fields in region A, z < 0, are given by N X A aA eA (x)e j n z + bA eA (x)ej n n n n n=1 n=1 N N X X A A hA (x)e j n z Hy (x, z) = an n bA hA (x)ej n n n=1 n=1 Ey (x, z) = N X A nz (3.83) A n z, (3.84) B nz (3.85) B n z. (3.86) and the transverse fields in region B, z > 0, are described as N X B aB eB (x)e j n z + bB eB (x)ej n n n n n=1 n=1 N N X X B B hB (x)e j n z Hy (x, z) = an n bB hB (x)ej n n n=1 n=1 Ey (x, z) = N X The unknown modal amplitude coe cients aA , aB , bA , and bB may be determined by n n n n applying the boundary conditions on Ey and Hx at the interfaces between regions A and B. At z = 0, the boundary condition on tangential electric field requires 8 P w > PN < aB eB (x) + N bB eB (x), |x| < B n n n n n=1 n=1 2 aA eA (x) + bA eA (x) = n n n n wB (3.87) > 0, : |x| > , n=1 n=1 2 N X N X while the boundary condition on tangential magnetic field requires N X n=1 aA hA (x) n n N X n=1 bA hA (x) = n n N X n=1 aB hB (x) n n N X bB hB (x). n n (3.88) n=1 The system of functional equations (3.87)-(3.88) may be transformed into a system of linear equations by applying appropriate testing operators. First, the equation resulting from the tangential electric field boundary condition, (3.87), is multiplied by eA (x) and integrated m wA /2  x  wA /2. Then, the equation resulting from the tangential magnetic field boundary condition, (3.88), is multiplied by hB (x) and integrated over wB /2  x  wB /2. m over 85 Here 1  m  N . This results in N X aA n n=1 wA Z2 eA (x)eA (x)dx + n m N X bA n wA Z2 eA (x)eA (x)dx = n m n=1 wA wA 2 2 wB wB 2 Z Z2 N N X X B B (x)eA (x)dx + B an en bn eB (x)eA (x)dx m n m n=1 n=1 wB wB 2 2 N X aA n n=1 N X aB n n=1 wB Z2 hA (x)hA (x)dx n m wB 2 wB Z2 N X bA n n=1 hB (x)hA (x)dx n m wB 2 wB Z2 (3.89) hA (x)hA (x)dx = n m wB 2 wB Z2 N X B bn hB (x)hA (x)dx. n m n=1 wB 2 (3.90) By defining new quantities, these expressions results in N X n=1 N X n=1 aA Cmn + n aA Emn n N X n=1 N X n=1 bA Cmn = n bA Emn = n N X n=1 N X n=1 86 aB Dmn + n aB Fmn n N X n=1 N X n=1 bB Dmn n (3.91) bB Fmn , n (3.92) where Cmn = Emn = wA Z2 wA 2 wB Z2 wB 2 eA (x)eA (x)dx, m n eB (x)eA (x) m n dx, A Zn Dmn = Fmn = wB Z2 wB 2 wB Z2 wB 2 eA (x)eB (x)dx, m n (3.93) eB (x)eB (x) m n dx. B Zn (3.94) This yields the 2N ⇥ 2N matrix equation 2 6 D 4 F 32 D 76 54 F 3 2 aB 7 6 C 5=4 B b E 32 C 76 54 E 3 aA 7 5, A b (3.95) where each block of the matrix comprises N ⇥ N submatrices. These submatrix entries are specified in Appendix B.1. Next, the expanded step junction, as shown in Figure 3.9, where wA < wB , is analyzed. The transverse field equations (3.83) - (3.86) are the same as with the reduced step junction, while the expressions resulting from the implementation of the boundary conditions change. At z = 0, the tangential electric field boundary condition requires 8 PN wA > PN A A A A < n=1 an en (x) + n=1 bn en (x), |x| < 2 (3.96) B eB (x) + B eB (x) = an n bn n w > 0, : |x| > A , n=1 n=1 2 N X N X while the tangential magnetic field boundary condition requires N X n=1 aA hA (x) n n N X n=1 bA hA (x) = n n N X n=1 aB hB (x) n n N X bB hB (x). n n (3.97) n=1 As was done for the reduced step junction, the system of functional equations (3.96)-(3.97) may be transformed into a system of linear equations by applying the appropriate testing 87 Region A wA Region B A an ! a ! B n wB b !ϵB,μB b ! A Region n A B n ϵA,μA wA A an ! b ! A Figure 3.8: n ϵA,μA wA Region B x a ! A n x ϵA,μA b ! ϵA,μA b ! A n B an ! z wB z wB z b !B ϵB,μB B Region n Region A wA wB b !ϵ ,μB Region B A n A an ! a ! B n B Top view of reduced step junction. n B Region A z x B an ! b ! ϵB,μB B n x Figure 3.9: Top view of expanded step junction. 88 operators. The equation resulting from the tangential electric field boundary condition, (3.96), is multiplied by eA (x) and integrated over m wB /2  x  wB /2. Then the equation resulting from the tangential magnetic field boundary condition, (3.97), is multiplied by hB (x) and integrated over m N X wB Z2 aB n n=1 wB 2 N X aA n n=1 N X aA n n=1 N X n=1 aB n wA Z2 wA /2  x  wA /2. Once again 1  m  N . This results in eB (x)eB (x)dx + n m N X wB Z2 bB n n=1 wA Z2 eA (x)eB (x)dx + n m N X bA n n=1 wA 2 hA (x)hA (x)dx n m wA 2 wA Z2 wB 2 N X bA n n=1 hB (x)hA (x)dx n m wA 2 wA Z2 eB (x)eB (x)dx = n m wA Z2 eA (x)eB (x)dx n m (3.98) wA 2 hA (x)hA (x)dx = n m wA 2 wA Z2 N X B bn hB (x)hA (x)dx. n m n=1 wA 2 (3.99) By defining new quantities, these expressions become N X aA Lmn + n n=1 N X aA Omn n n=1 N X bA Lmn = n N X aB Mmn + n n=1 n=1 N N X X AO bn mn = aB Wmn n n=1 n=1 89 N X bB Mmn n (3.100) n=1 N X bB Wmn , n n=1 (3.101) where Lmn = wB Z2 eB (x)eB (x)dx, m n Mmn = wB 2 wA Z2 A em (x)eB (x) n Omn = dx, B Zn wA 2 Wmn = wA Z2 wA 2 wA Z2 wA 2 eA (x)eB (x)dx, m n (3.102) eA (x)eA (x) n m dx. A Zn (3.103) This yields the 2N ⇥ 2N matrix equation 2 6 L 4 O 32 L 76 54 O 3 2 aB 7 6 M 5=4 B b W 32 M 76 54 W 3 aA 7 5, A b (3.104) where L, W , M , and O are N ⇥ N submatrices with entries specified in Appendix B.1. With general expressions for the two di↵erent step junctions determined, a pinched-down iris is analyzed. Consider the waveguide setup shown in Figure 3.10. The first junction is a reduced step. Thus using (3.95), it can be shown that 2 6 C 4 E 32 aA 3 2 C 76 7 6 D 54 5=4 E bA F 32 aB 3 D 76 7 54 5, F bB (3.105) where wA = w1 , wB = w2 , ✏A = ✏0 , ✏B = ✏i , µA = µ0 , and µB = µi . Next, the expanded step junction can be analyzed using (3.104) such that 2 6 L 4 O 32 aC 3 2 L 76 7 6 M 54 5=4 O bC W 32 3 D M 76 a 7 54 5. W bD (3.106) Here wA = w2 , wB = w1 , ✏A = ✏i , ✏B = ✏0 , µA = µi , and µB = µ0 . In (3.106), since the width of the guides on either side of the iris are equivalent, the submatrices reduce to 90 L = C, M = D, O = E, and W = F , and thus 2 32 6 D 4 F D 76 54 F 3 2 aC 7 6 C 5=4 C b E 32 C 76 54 E 3 aD 7 5. D b (3.107) Now, the modal amplitude coe cients at Port A are shifted to Port D using 2 6 4 aC 3 2 7 6 P 5=4 bC 0 32 aB 3 0 76 7 54 5, P+ bB (3.108) where i ± Pmn = mn e±j n . (3.109) i Here mn is Kronecker delta and n is the propagation constant in the iris region. Now, (3.108) can also be expressed as 2 6 4 aC bC 3 " # 2 7 6 5= P 4 aB bB 3 7 5, (3.110) or aC = P a B (3.111) bB = P bC . (3.112) Next, substituting (3.111) into (3.105) and (3.112) into and (3.107), results in CaA + CbA = DaB + DP bC 91 (3.113) EaA EbA = F aB F P bC (3.114) DP aB + DbC = CaD + CbD (3.115) F P aB (3.116) F bC = EaD EbD . Multiplying (3.113) and (3.115) by C 1 , and (3.114) and (3.116) by F 1 yields aA + b A = U a B + U P b C V aA V bA = aB (3.117) P bC (3.118) U P aB + U bC = aD + bD P aB bC = V aD (3.119) V bD , (3.120) where U = C 1 D and V = F 1 E. Now, rearranging (3.118) produces aB = P bC + V aA V bA , (3.121) which can then be substituted into (3.117), (3.119), and (3.120), resulting in aA + bA = U P bC + U V aA U P P bC + U P V aA P P bC + P V aA U V bA + U P bC (3.122) U P V b A + U b C = aD + b D (3.123) P V bA bC = V aD V bD . (3.124) Rearranging (3.124) and factoring bC gives [I P P ] bC = P V aA P V bA 92 V aD + V bD , (3.125) or bC = [I P P ] 1 P V aA [I [I P P ] 1 V aD + [I P P ] 1 P V bA P P ] 1 V bD . (3.126) Here I is the identity matrix. Substituting (3.126) into (3.122) yields aA + bA = 2U P [I 2U P [I P P ] 1 P V aA P P ] 1 V aD + 2U P [I 2U P [I P P ] 1 P V bA P P ] 1 V bD + U V aA U V bA . (3.127) This is then rearranged, resulting in n I UV 2U P [I n n o A + I + U V + 2U P [I P P ] 1 P V bA = a o n o P P ] 1 V aD + 2U P [I P P ] 1 V bD . (3.128) PP] 1 PV 2U P [I o The expression U V + 2U P [I P P ] 1 P V, (3.129) which appears multiple times in (3.128), can be reduced by factoring U and V and simplifying, resulting in U V + 2U P [I P P ] 1 P V = U SV, (3.130) where Smn = 2 1 + Pmn . 2 1 Pmn (3.131) Note that since P is diagonal, the resulting S matrix is also diagonal. Thus inserting (3.130) 93 into (3.128) gives U SV ] aA + [I + U SV ] bA = o n o P P ] 1 V aD + 2U P [I P P ] 1 V bD . [I n 2U P [I (3.132) Through similar simplification, the expressions on the right-hand side of (3.132) reduces to [I U SV ] aA + [I + U SV ] bA = [ U T V ] aD + [U T V ] bD , (3.133) where Tmn = 2Pm n . 2 1 Pmn (3.134) The matrix T is also diagonal. Next, substituting (3.126) into (3.123) and simplifying results in [U T V ] aA + [ U T V ] bA = [I + U SV ] aD + [I U SV ] bD . (3.135) These two equations, (3.133) and (3.135), can now be written in matrix form as 2 6 K 4 J where K = I 32 ¯ K 76 54 J aA bA 3 2 7 6 5=4 J ¯ K 32 aD 3 J 76 7 54 5, K bD (3.136) ¯ U SV , K = I + U SV , and J = U T V . This reduces the system of equations from four equations with 4N unknowns, (3.113) - (3.114), to a system of two equations with 2N unknowns, (3.136). Now assuming aA is known, the remaining modal amplitude coe cients, aD , bA and 94 bD , can be determined by first defining bD = RaD , (3.137) where R is called the reflection matrix. This reflection matrix depends on what type of load is being attached to Port D. A myriad of loads can be attached to the iris region and the subsequent reflection matrix derived. In this dissertation, the expressions for the theoretical reflection coe cient for an iris placed in front of a conductor-backed material sample are derived. Substituting (3.137) into (3.133) and (3.135) results in ¯ KaA + KbA = JaA JaD + JRaD (3.138) ¯ JbA = KaD + KRaD (3.139) or ¯ KaA + KbA = J (R JaA I) aD (3.140) ¯ JbA = K + KR aD . (3.141) This can be written as a matrix equation 2 6 J (R I) 4 ¯ K + KR 32 ¯ K 76 54 J 3 2 3 aD 7 6 KaA 7 5=4 5, A A b Ja (3.142) which, upon solution, gives the desired reflection coe cient for a terminated iris. The reflection matrix for a material-loaded guide is required in (3.142) to find the desired reflection coe cient. This may be determined as follows. Considering Figure 3.11, the 95 transverse fields for the region d < z < 0, are written as N N g X g g X g g an en (x)e j n z + bn en (x)ej n=1 n=1 N N g X g g X g g j nz Hx (x, z) = an hn (x)e bn hn (x)ej n=1 n=1 Ey (x, z) = g nz (3.143) g nz , (3.144) s nz (3.145) s nz . (3.146) while those in the sample region, 0 < z < , are N X s as es (x)e j n z + bs es (x)ej n n n n n=1 n=1 N N X X s s hs (x)e j n z Hx (x, z) = an n bs hs (x)ej n n n=1 n=1 Ey (x, z) = N X Here, en (x) and hn (x) are the transverse electric and magnetic modal fields, respectively, and as shown in (4.80) and (4.81) are given by g,s en (x) = g,s where kxn = (2n h i g,s g,s kxn ( 1)n cos kxn x g,s e (x) g,s hn (x) = n g,s , Zn (3.147) (3.148) g,s g,s g,s 1)⇡/w1 and Zn = !µ0 µg,s / n . The propagation constant n determined from (2.140) as g,s n = r (k g,s )2 ⇣ ⌘ g,s 2 kxn , (3.149) p where k g,s = k0 µg,s ✏g,s . g g Expressions for the unknown modal amplitude coe cients an , bn , as , and bs may be n n determined by applying the boundary conditions on transverse electric and magnetic fields at the interfaces between the guide and sample region and between the sample region and conducting surface. At the interface z = 0 the tangential electric field boundary condition 96 Port A w1 Port D Δ B aD ! w n 1 D B B B an ! w aC ! n A an ! b ! b ! B n A n ϵ0,μ0 x 2 C B bn ! bn ! ϵi,μi z ϵ0,μ0 Figure 3.10: Top view of pinched-down iris. Port D δ an ! an ! B s B g an ! A T w1 A T bn ! ϵg,μg b ! B g n x z s B n b ! ϵs,μs z=0 z=-d Figure 3.11: Top view of material-loaded guide. 97 z=δ Short! Circuit! Wall requires N N N N X g g X g g X X an en (x) + bn en (x) = as es (x) + bs es (x), n n n n n=1 n=1 n=1 n=1 (3.150) g or, since en (x) = es (x), n g g an + b n = as + b s . n n (3.151) The tangential magnetic field boundary condition requires N X g g an en (x) n=1 N N X g g X bn hn (x) = as hs (x) n n n=1 n=1 N X bs hs (x) n n (3.152) n=1 or g an g Zn At z = g bn as = n g s Zn Zn bs n. s Zn (3.153) the boundary condition on the tangential electric field requires N X N X s s s es (x)e j n + an n bs es (x)ej n = 0 n n n=1 n=1 (3.154) s s as e j n + bs ej n = 0. n n (3.155) or Rearranging (3.155) and substituting into (3.151) and (3.153) gives h i g g an + b n = as 1 e 2 n n g h i Zn s g g 2 , an + g n = s an 1 + e n Zn 98 (3.156) (3.157) s where en = e j n . Dividing (3.156) by (3.157) results in g g an + b n Zs = n Qn , g g g an + b n Zn (3.158) where Qn = e2 n. 2 1 + en 1 (3.159) This expression is recognized as a tangent function, and can be written as Qn = j tan g Now, multiplying (3.158) by Zn ⇣ s n . (3.160) ⌘ g g an + bn yields g g g g g g Z n an + Z n b n = Z n Q n an g g Zn Q n b n . (3.161) This expression is rearranged to give g g bn Z s Q n Zn = n g g. s an Zn Q n + Zn (3.162) Now, as is shown in Appendix B, the modal field coe cients can be shifted from one interface to another. Thus the expressions to shift the modal field coe cients from the interface at z = 0 to z = d are g g an = aT e j n d n g T = bg e j d d . bn n (3.163) (3.164) These expressions can then be inserted into (3.162). Thus the reflection vector, rn = bT /aT , n n 99 is determined to be g Z s Qn Z g n. rn = e j n 2d n s Qn + Z g Zn n (3.165) This in turn means the reflection matrix, [R] from (3.138), is a diagonal matrix with entries Rmn = mn rn . Once the modal coe cient bA is found by solving the matrix equation from (3.142), the reflection coe cient at Port A is given by A I = b1 . S11 aA 1 3.3.1.2 (3.166) Reflection Coe cient with No Iris The characterization of ✏s and µs is also dependent on the measurement of the reflection coe cient from the conductor-backed sample with no iris present. The theoretical reflection coe cient for no-iris case is determined in this section. Because the no-iris setup is the same as shown in Figure 3.11, (3.165) can be used directly. The reflection coe cient at the measurement plane z = d is given by T N = b1 = r . S11 1 aT 1 (3.167) With no iris present, only the TE10 mode is implicated, thus g s g N = e j 1 2d Z1 Q1 Z1 . S11 g s Z1 Q 1 + Z 1 3.3.2 (3.168) Validation of Theoretical Analysis Once again, the theoretical analysis is validated before using it for parameter extraction. As was done for the NRW method in Section 3.2.2, the theoretical model is validated against 100 HFSS with an FGM125 test sample. Using the FGM125 material parameters given in Section 3.2.2, the theoretical analysis was used to compute the reflection coe cient from the sample placed in an X-band waveguide with dimensions a = 22.86 mm and b = 10.16 mm. The iris waveguide dimensions are d = 1 mm, = 1 mm, and w2 = 7.62 mm. The results of I this theoretical analysis are shown in Figure 3.12. Absolute tolerances of 0.1 dB for |S11 | I and 0.01 for \S11 were chosen so that the accuracy of the computed modal analysis is better than the expected measurement accuracy of the HP 8510C vector network analyzer. These tolerances were obtained using the HP 8510 Specifications & Performance Verification Program. For the FGM 125 material considered, the S-parameters typically converge to the specified tolerance with N = 140 terms. I Also shown in Figure 3.12 is S11 computed using HFSS. The waveguide port was once again model with a length of 50 mm. The convergence tolerance in HFSS was specified as a maximum delta S of 0.01 at the solution frequency 12.4 GHz. Excellent agreement is obtained between the modal analysis and HFSS, thus validating the modal computation. Figure 3.13 shows the reflection coe cient for the no-iris case computed using the theoretical analysis and HFSS. The theoretical analysis was used to determine the reflection coe cient from a sample of FGM125 sample with conductor-backing. The HFSS results were obtained with similar port dimensions and convergence criteria used for the HFSS iris simulation. Once again excellent agreement is achieved, which provides confidence before implementation of the forward problem in the extraction routine. 3.3.3 Extraction Process N With the measurements of the reflection coe cients with no iris, S11 , and with an iris, I S11 , the permittivity and permeability can be extracted. A complex root solver such as a Newton’s method can be used to solve for two complex unknowns from two independent measurements. Experience has shown that with typical values of experimental error it can sometimes be di cult to find solutions to this system of equations. This is why closed101 Magnitude (dB) 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6 Analytical HFSS S11 8 9 10 11 12 Frequency (GHz) Phase (°) 175 170 Analytical HFSS S11 165 160 155 8 9 10 11 12 Frequency (GHz) Figure 3.12: S-parameters computed for a conductor-backed FGM125 test material placed behind an iris. 102 Magnitude (dB) -5 -6 -7 Analytical HFSS S11 -8 -9 -10 9 10 11 12 Frequency (GHz) Phase (°) 200 100 0 Analytical HFSS S11 -100 -200 9 10 11 12 Frequency (GHz) Figure 3.13: S-parameters computed for a conductor-backed FGM125 test material. 103 form equations such as those in the NRW method or the Air-backed/Conductor-backed [37] method are preferred. An alternative approach allows a complex root-solver to determine k s from one measurement and then closed-form expressions can be used to compute µs and s ✏s . First, Z1 is solved for in (3.168) resulting in ˜N s Q = Z g 1 + S11 = ↵, Z1 1 1 1 SN ˜ 11 (3.169) g N j 2d ˜N S11 = S11 e 1 . (3.170) where N This ↵ value is a known quantity since S11 is measured and the impedance of the guide depends on known values of µg and ✏g . Now, (3.169) can also be expressed as !µs s Q1 = ↵, 1 (3.171) s 1 ↵. µs = Q1 ! (3.172) and solving for µs results in s The only unknown values in (3.172) are Q1 and 1 . From (3.160) it is seen that Q1 only s s depends on 1 and the known value of . Therefore, the only unknown value is 1 , which p depends on k s = k0 µs ✏s . A secant method can then be used to extract k s from the I s measurement of S11 . Knowing k s , 1 can be computed, which in turn can be used to s compute Q1 . With 1 and Q1 determined µs is found using (3.172). Finally using µs and k s , ✏s can be determined. Once again the extraction process is validated with the characterization of a known material. The FGM125 material parameters are extracted from the HFSS generated S- 104 parameters shown in Figure 3.12 and Figure 3.13. Here the distance between the iris and the material is d = 1 mm. Figure 3.14 shows the results of the characterization. Also shown in Figure 3.14 are the material parameters extracted from S-parameters generate in HFSS with d = 1 mm in the model. Note that the material parameters used to generate the S-parameters in HFSS assume frequency independent values of ✏s = 7.3197 and µs = 0.5756 j0.0464 j0.4842. The extracted constitutive parameters shown in Figure 3.14 are frequency dependent and not consistent with the values used in HFSS. The material parameter ✏00 is especially inconsistent since the values become positive. This is a result of s the S-parameter error propagating through to the extraction. Notice that the constitutive parameters are more consistent with the values used in HFSS when the distance between the material and iris is decreased. The e↵ects of the propagation of S-parameter uncertainty on the characterization of the material parameters is explored in greater detail in Section 3.3.4. 3.3.4 Iris Design This characterization method for conductor-back materials can be customized for a particular material by optimizing the iris-measurement system so that it has minimal sensitivity to Sparameter uncertainty. To determine the optimal iris geometry a Monte Carlo technique was used to study e↵ects of the propagation of the random error inherent to the VNA used in the measurements. Depending on the material under test, the geometrical parameters of the iris measurement ( , d, and w2 in Figure 3.7) are optimized to have low sensitivity to uncertainties of the measured S-parameters. A single parameter sweep was used to perform an exhaustive search of each geometrical parameter assuming the remaining two parameters were fixed. The fixed parameters in each of the parameter sweeps were determined through initial observations of random configurations of the iris-measurement system. The Monte Carlo analysis of the propagation of VNA uncertainty was used for each of the individual configurations in the parameter sweep and the amount of propagated error was calculated. The uncertainties of the measured S-parameters used in the Monte Carlo error analysis 105 Permittivity 8 ϵ's 6 d=1mm d=0mm 4 ϵ''s 2 0 9 10 11 12 Frequency (GHz) Permeability 1 μ's 0.5 0 d=1mm d=0mm μ''s -0.5 -1 8 9 10 11 12 Frequency (GHz) Figure 3.14: Extraction of permittivity and permeability from S-parameter generated using HFSS with d = 0 mm and d = 1 mm in the model. 106 were chosen to be smaller than those for an HP8510C network analyzer system, due to the strong sensitivity of this technique to S-parameter uncertainty. The uncertainties for the HP8510C network analyzer were determined using the software package HP 8510 Specifications & Performance Verification Program provided by Hewlett Packard. Although the VNA measurement uncertainty is dependent on S-parameter amplitudes, for the range of amplitudes encountered in this dissertation the VNA measurement uncertainty can be assumed to be amplitude and frequency independent. Using the software package, the statistical variance of S11 is specified linearly in amplitude and phase as values of A = 0.004 and 11 = 0.8 , respectively, for the HP8510C network analyzer. It should be noted that this is 11 not the analyzer used in subsequent measurements; however, these values give a worse case set of uncertainties to gauge in the iris-measurement design optimization. Unfortunately, these uncertainty values prove to be too high for the iris characterization technique and the resulting extracted parameters are very inaccurate. Therefore, the statistical variance on phase is reduced to = 0.1 . 11 The materials under test originally planned for experimental validation were Eccosorb FGM125 and FGM40. The geometry of the FGM125 sample is w1 = 22.86 mm, b = 10.16 mm, and = 3.175 mm. The geometry of the FGM40 sample is w1 = 22.86 mm, b = 10.16 mm, and = 1.016 mm. The forward problem was solved at 8.2 GHz, 10.09 GHz, and 12.4 GHz using the geometrical parameters determined from the parameter sweep. The material parameters of FGM125 and FGM40 at these frequencies are shown Table 3.1 and Table 3.2, respectively. White Gaussian noise was then added to each of the S-parameter sets, and the noisy data was used to extract the material parameters. Five hundred trials were used in the Monte Carlo analysis in each of the parameter sweeps, and the average values of the material parameters and the standard deviations were calculated. The first material used in the iris-measurement design is FGM125. The e↵ects of changing the thickness of the iris ( ) were studied first. In these simulations the iris opening was set to half the width of the X-band waveguide, w2 = 11.43 mm (0.45 inches), and the distance 107 Table 3.1: Nominal material parameters for FGM125. These data were obtained using the waveguide NRW method. Data provided by Captain Milo Hyde IV form the Air Force Institute of Technology , Wright-Patterson AFB, Dayton, OH [46] Frequency (GHz) 8.2 10.9 12.4 Relative Permittivity 7.258 j0.056 7.320 j0.046 7.374 j0.025 RelativePermeability 0.600 j0.761 0.576 j0.484 0.615 j0.212 Table 3.2: Nominal material parameters for FGM40. These data were obtained using the waveguide NRW method. Data provided by Captain Milo Hyde IV form the Air Force Institute of Technology , Wright-Patterson AFB, Dayton, OH [46] Frequency (GHz) 8.2 10.9 12.4 Relative Permittivity 21.864 j0.390 22.036 j0.332 22.244 j0.166 RelativePermeability 2.088 j2.537 1.668 j2.354 1.245 j2.110 between the iris and the material was set to d = 0 mm. The forward problem was solved at 21 iris thicknesses evenly distributed from 0.1 mm to 4 mm at the three frequencies. To better gauge the e↵ectiveness of this characterization technique, the results of the Monte Carlo error analysis are compared against the two-thickness method, which is another reflectiononly measurement technique with relatively low sensitivity to propagation of measured Sparameter error [27]. The second measurement in the two-thickness technique assumes a thickness of 2 . Instead of plotting the average values and 2 error bars, as will be done in subsequent Monte Carlo error analyses in this dissertation, the values obtained by dividing the error of the iris technique ( iris ) by the error from the two-thickness technique ( 2-Thick ) are plotted. This gives a sense of how many times worse the iris technique is compared to the two-thickness approach. Figures 3.15 - 3.17 show the results of the parameter sweep for the FGM125 sample. It is seen that the thinner the iris is, the less sensitive the extracted material parameters are to the propagation of measured S-parameter uncertainty. With the knowledge that a thin iris is best, the distance between the iris and the sample 108 80 μ''r σIris____ σ2-Thick 60 40 ϵ'r ϵ''r 20 μ'r 0 0 1 2 3 4 Δ (mm) Figure 3.15: Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the parameter sweep assuming w2 = 11.43 mm and d = 0 mm at 8.2 GHz. 109 30 25 μ''r σIris____ σ2-Thick 20 ϵ''r 15 μ'r 10 ϵ'r 5 0 0 1 2 3 4 Δ (mm) Figure 3.16: Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the parameter sweep assuming w2 = 11.43 mm and d = 0 mm at 10.09 GHz. 110 7 6 ϵ''r σIris____ σ2-Thick 5 μ'r 4 μ''r 3 ϵ'r 2 1 0 0 1 2 3 4 Δ (mm) Figure 3.17: Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the parameter sweep assuming w2 = 11.43 mm and d = 0 mm at 12.4 GHz. 111 was analyzed. Here d is swept from 0 mm to 2 mm at 0.1 mm steps. In these simulations the iris opening was once again set to w2 = 11.43 mm (0.45 inches) and as determined from the previous parameter sweep, a narrow iris thickness was used, specifically, the thickness was set to = 0.1 mm. The results of this parameter sweep are shown in Figures 3.18 through 3.20. From these figures it is determined that the least error occurs when the FGM125 material sample is placed against the iris. Finally, the last parameter sweep analyzes the e↵ects of changing the width of the iris opening while keeping d = 0 mm and = 0.1 mm fixed. The iris width, w2 , is swept from 5.08 mm (0.2 inches) to 16.51 mm (0.65 inches) at 21 evenly spaced steps. Figures 3.21 through 3.23 show the results of the parameter sweep. It was determined that the optimal iris opening width is 7.62 inches (0.3 inches) for measurements of an FGM125 sample. A similar iris measurement design was implemented for measurements of FGM40. Figures 3.24 through 3.26 show the results of the parameter sweep from 0.1 mm to 4 mm with w2 = 11.43 mm and d = 0 mm fixed. The results of the d parameter sweep from 0 mm to 2 mm are shown in Figures 3.27 through 3.29. Here the parameters and w2 are fixed to be 0.1 mm and w2 = 11.43 mm, respectively. Finally, Figures 3.30 through 3.32 show the results of the w2 parameter sweep from 2.55 mm (0.1 inches) to 15.24 mm (0.6 inches) with = 0.1 mm and d = 0 mm. Similar to the FGM125 analysis, the optimal thickness of the iris is to be as thin as possible and the optimal position of the sample is at d = 0 mm. It was also determined that the best iris opening width is 5.08 mm (0.2 inches) for the measurement of FGM40. 3.3.5 Experimental Results To experimentally validate the proposed technique at X-band, an iris waveguide insert was machined from 360 brass. The iris was made for the characterization of FGM125. The specified inner dimensions of the iris are w2 = 0.3 inches by b = 0.9 inches, and measurement of the constructed iris using precision calipers showed that the iris was constructed with a 112 40 μ'r σIris____ σ2-Thick 30 20 ϵ''r ϵ'r 10 μ''r 0 0 0.5 1 1.5 2 d (mm) Figure 3.18: Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the d parameter sweep assuming w2 = 11.43 mm and = 0.1 mm at 8.2 GHz. 113 50 σIris____ σ2-Thick 40 μ''r ϵ''r 30 μ'r 20 ϵ'r 10 0 0 0.5 1 1.5 2 d (mm) Figure 3.19: Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the d parameter sweep assuming w2 = 11.43 mm and = 0.1 mm at 10.09 GHz. 114 30 25 ϵ''r μ''r σIris____ σ2-Thick 20 μ'r 15 10 ϵ'r 5 0 0 0.5 1 1.5 2 d (mm) Figure 3.20: Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the d parameter sweep assuming w2 = 11.43 mm and = 0.1 mm at 12.4 GHz. 115 30 25 σIris____ σ2-Thick 20 ϵ''r 15 μ''r 10 μ'r ϵ'r 5 0 4 6 8 10 12 14 16 18 w2(mm) Figure 3.21: Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the w2 parameter sweep assuming d = 0 mm and = 0.1 mm at 8.2 GHz. 116 30 25 σIris____ σ2-Thick 20 ϵ''r 15 μ''r 10 μ'r ϵ'r 5 0 4 6 8 10 12 14 16 18 w2(inches) Figure 3.22: Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the w2 parameter sweep assuming d = 0 mm and = 0.1 mm at 10.09 GHz. 117 14 12 σIris____ σ2-Thick 10 μ''r 8 6 ϵ''r μ'r 4 ϵ'r 2 0 4 6 8 10 12 14 16 18 w2(mm) Figure 3.23: Comparison of the error in the constitutive parameters of FGM125 sample due to S-parameter error using the iris technique and two thickness method. This shows the results of the w2 parameter sweep assuming d = 0 mm and = 0.1 mm at 12.4 GHz. 118 400 μ''r σIris____ σ2-Thick 300 200 ϵ'r ϵ''r 100 μ'r 0 0 1 2 3 4 Δ (mm) Figure 3.24: Comparison of the error propagated to constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the parameter sweep assuming w2 = 11.43 mm and d = 0 mm at 8.2 GHz. 119 200 μ''r 150 σIris____ σ2-Thick μ'r 100 ϵ''r ϵ'r 50 0 0 1 2 3 4 Δ (mm) Figure 3.25: Comparison of the error propagated to the constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the parameter sweep assuming w2 = 11.43 mm and d = 0 mm at 10.09 GHz. 120 300 250 μ''r σIris____ σ2-Thick 200 μ'r 150 ϵ''r 100 50 0 ϵ'r 0 1 2 3 4 t (mm) Figure 3.26: Comparison of the error propagated to the constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the parameter sweep assuming w2 = 11.43 mm and d = 0 mm at 12.4 GHz. 121 200 μ''r σIris____ σ2-Thick 150 100 ϵ''r ϵ'r μ'r 50 0 0 0.5 1 1.5 2 d (mm) Figure 3.27: Comparison of the error propagated to constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the d parameter sweep assuming w2 = 11.43 mm and = 0.1 mm at 8.2 GHz. 122 300 μ''r 250 μ'r σIris____ σ2-Thick 200 150 ϵ''r 100 ϵ'r 50 0 0 0.5 1 1.5 2 d (mm) Figure 3.28: Comparison of the error propagated to constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the d parameter sweep assuming w2 = 11.43 mm and = 0.1 mm at 10.09 GHz. 123 400 μ''r σIris____ σ2-Thick 300 μ'r 200 ϵ'r 100 ϵ''r 0 0 0.5 1 1.5 2 d (mm) Figure 3.29: Comparison of the error propagated to constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the d parameter sweep assuming w2 = 11.43 mm and = 0.1 mm at 12.4 GHz. 124 180 160 140 ϵ''r σIris____ σ2-Thick 120 100 μ''r 80 μ'r 60 40 ϵ'r 20 0 2 4 6 8 10 12 14 16 w2(mm) Figure 3.30: Comparison of the error propagated to constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the w2 parameter sweep assuming d = 0 mm and = 0.1 mm at 8.2 GHz. 125 200 μ''r 150 σIris____ σ2-Thick ϵ'r 100 ϵ''r 50 0 μ'r 2 4 6 8 10 12 14 16 w2(mm) Figure 3.31: Comparison of the error propagated to constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the w2 parameter sweep assuming d = 0 mm and = 0.1 mm at 10.09 GHz. 126 350 300 μ''r 250 σIris____ σ2-Thick μ'r 200 150 ϵ'r 100 ϵ''r 50 0 2 4 6 8 10 12 14 16 w2(mm) Figure 3.32: Comparison of the error propagated to constitutive parameters of FGM40 sample from S-parameter error using the iris technique and two thickness method. This shows the results of the w2 parameter sweep assuming d = 0 mm and = 0.1 mm at 12.4 GHz. 127 manufacturing error of less than 5 mils. The iris was inserted between two 6 inch long commercial X-band waveguide extensions of cross-section 0.4 inch by 0.9 inch. A sliding short with a cross-section of 0.4 inch by 0.9 inch was used as a sample holder to provide the conductor backing of the FGM125 sample. A sample of FGM125 was cut from a sheet of Eccosorb FGM125. Figure 3.33 shows a picture of the iris, FGM125 sample, waveguide extension, and sliding short sample holder. The sample was cut approximately 0.01 mm larger than the inner dimensions of the sliding short so that when inserted it would compress slightly and eliminate air gaps between the FGM125 sample and the sliding short walls. The thickness of the sample is 2.96 mm. Measurements of the S-parameters of the FGM125 sample with and without the iris were made using an Agilent E5071C VNA. The VNA was calibrated using a Maury Microwave X7005M X-band calibration kit, consisting of two o↵set shorts and a load. The measurements were made with VNA settings of 5 dBm source power, 64 averages, and an IF bandwidth of 5 kHz. Finally, the material parameters were extracted using the measured values of I N S11 and S11 . The results of the characterization are shown in Figure 3.34. Also shown is the characterization of the same sample using the air-backed/conductor-backed method. There exists good agreement between the two methods for the extraction of the permeability. However, the characterization of the permittivity is poor using the iris technique. The error in the extraction of the permittivity is a direct result of the propagation of error from the S-parameter measurements. Figure 3.35 and Figure 3.36 show the comparison of I N I N the measured S11 and S11 , respectively, to the computed values of S11 and S11 generated using the material parameters extracted with the air-back/conductor-backed method. The deviation of the measurement from theory, and the strong oscillations in the measured values, cause a dramatic e↵ect on the permittivity characterization. The most notable deviation is the extraction of ✏00 , where the values are positive across the frequency band. Thus, it can r be concluded that while this technique can be used to accurately characterize permeability, the characterization of permittivity is not reliable. 128 Waveguide extension! Sample! Iris! Sliding short! Figure 3.33: Manufactured iris with waveguide extension, sliding short sample holder, and FGM125 sample. 129 Relative Material Parameters 8 ϵ'r 6 4 Iris Method Air-backed/Conductor-backed 2 ϵ''r μ'r 0 μ''r -2 8 9 10 11 12 Frequency (GHz) Figure 3.34: Relative permittivity and permeability extracted using the iris technique and the air-backed/conductor-backed method. 130 Magnitude (dB) Measurement Forward Problem -0.6 -0.8 -0.52 -0.54 -0.56 -0.58 -0.6 8.2 8.3 8.4 8.5 8.6 -1 -1.2 -1.4 8 9 10 11 12 10 11 12 Phase (°) 170 165 Measurement Forward Problem 170.2 169.8 169.4 169 160 155 8.2 8.3 8.4 8.5 8.6 150 8 9 Frequency (GHz) I Figure 3.35: Comparison of measured and forward problem generated S11 . 131 Measurement Forward Problem Phase (°) Magnitude (dB) -4 -5 -6 -8 -10 -10 -12 -15 -14 -16 -20 -18 -20 -25 200 8 100 Theoretical - AFIT -5.6 Theoretical - GE1 -5.8 Theoretical - GE2 -6 20130507 Measurement -6.2 20130522 Measurement 8.2 8.3 8.4 8.5 8.6 20130524 Measurement 8 9 9 10 10 11 11 12 11 12 177 176.9 176.8 176.7 176.6 8.2 8.3 8.4 8.5 8.6 0 -100 -200 8 9 10 Frequency (GHz) N Figure 3.36: Comparison of measured and forward problem generated S11 . 132 3.3.6 Summary A reflection-only measurement technique is outlined for characterizing conductor-backed materials. The measurements of the reflection coe cients for a conductor-backed material with and without the presence of an iris are used to characterize the permeability and permittivity. A mode-matching technique is outlined for computing the theoretical reflection coe cient from an iris in front of a conductor-backed sample. The dimensions of the iris measurement system were optimized to reduce the amount of S-parameter error propagated through to the characterization. Using these dimensions, an iris was fabricated and the technique was validated using laboratory measurements of a FGM125 sample. It was found that the technique is very sensitivity to S-parameter error, especially for the characterization of ✏00 . The technique performs very well for the characterization of the permeability. r 133 CHAPTER 4 BIAXIAL MATERIAL CHARACTERIZATION 4.1 Introduction This chapter presents two methods for characterizing biaxially anisotropic materials using rectangular waveguides. The first technique uses an extension of the NRW method (Section 3.2) applied to biaxial materials. These materials react di↵erently depending on the incident wave polarization, but do not establish coupling between orthogonal field components. The tensors for biaxial materials have a total of six nonzero entries, corresponding to complex material parameters, and therefore three di↵erent sets of reflection and transmission measurements are required. This technique yields closed form solutions for all six material parameters. Due to the rectangular nature of the waveguide, typically three separate samples are manufactured from the material for placement in the cross-sectional plane of the waveguide. These samples are manufactured so the axes of the permittivity and permeability tensors align with the geometry of the waveguide. The second technique uses a reduced-aperture, square-waveguide sample holder, such that a cubical sample can be used to characterize. This provides the minimum three reflection and 134 transmission measurements required for characterization and allows all material parameters to be determined from one sample. The sample holder is positioned between two empty sections rectangular waveguides to guarantee only the dominant mode is present at the sample holder, and the S-parameters are measured. Mode matching is used to determine the theoretical reflection and transmission coe cients of the dominant mode. Measurements of the reflection and transmission coe cients for various orientations of the same sample within the sample holder produce multiple transcendental equations. The inverse problem is then solved using a Newton’s method to find the material parameters. 4.2 NRW Material Characterization Extended to Biaxially Anisotropic Materials Of the techniques proposed for measuring properties of isotropic materials, the most accurate are those that expose a sample of the material to an electromagnetic wave and use measurements of the reflection and transmission of the wave to determine the permittivity and permeability. Confinement of the wave produces good signal strength, so an appealing approach is to put a layer of the material into a rectangular waveguide and use a network analyzer to measure the reflection and transmission coe cients as a function of frequency. Using simple guided wave theory, closed-form inverse expressions for permittivity and permeability may be obtained (see Section 3.2). These expressions are well-conditioned for most of the material parameters of interest, except, as was discussed previously, when the material thickness approaches multiples of a half wavelength. This section presents an extension of the classical waveguide technique for biaxial anisotropic materials and will describe the waveguide technique along with exploring parameter sensitivity by performing a Monte Carlo error analysis. 135 4.2.1 Theoretical Reflection and Transmission Coe cients The sample placed in the cross-sectional plane of the waveguide is assumed to be linear and homogeneous. The permittivity and permeability has the constitutive material tensors 2 6 ✏x 0 0 6 ✏ = ✏0 6 0 ✏y 0 6 4 0 0 ✏z and 2 6 µx 0 0 6 µ = µ0 6 0 µy 0 6 4 0 0 µz 3 7 7 7 7 5 3 7 7 7, 7 5 (4.1) (4.2) where tensor entries are relative complex quantities: ✏x = ✏0 + j✏00 , µx = µ0 + jµ00 and so x x x x on. The waveguide system, shown in Figure 4.1, consist of empty waveguide sections attached to the sample region. It is assumed that the empty waveguide sections are of adequate length such that a single TE10 rectangular waveguide mode is incident on the sample from the transmitting section (z < 0) and that single mode is obtained at the receiving extension (z > d). 4.2.1.1 Field Structure in a Waveguide Filled with Biaxial Material The TE10 mode incident from the transmitting empty waveguide couples into the TE10 biaxial filled waveguide mode since the field structure in both regions is identical. This means the fields of the TE10 mode do not couple into orthogonal field components and therefore higher order modes are not excited. This can be seen by first specializing the field equations for the isotropic material filled guide (Section 2.3.1.1) and the biaxial filled guide (Section 2.3.2.1) to the waveguide setup shown in Figure 4.1. The field equations for the 136 isotopic material can be used to represent the propagating TE10 mode in the transmitting region, and the field equations for the biaxial material can be used to determine the TE10 mode in the sample region. In the empty waveguide region, it can be shown from Section 2.3.1.1 that the TE10 mode is the dominant mode (n = 1), which means u1 = 1 and v1 = 0. Therefore, e e ky1 = 0, and kc1 = ⇡/a is the cuto↵ wavenumber for the first order mode. e kx1 = ⇡/a, Now, using (2.140) the propagation constant is given by e 1= r ⇣ 2 k0 e kc1 ⌘2 (4.3) p where k0 = ! µ0 ✏0 . The transverse field equations from (2.151) - (2.152) then results in e ±j 1 z e Ey (x, z) = C1 ee (x) e y1 e ±j 1 z e Hx (x, z) = ⌥C1 he (x) e . x1 (4.4) (4.5) Here ee and he are the transverse electric and magnetic modal fields in the empty guide, y1 x1 respectively, and are expressed as ee = y1 e e kx1 sin kx1 x , ke e he = x1 sin kx1 x , e x1 Zn (4.6) (4.7) e e where the modal wave impedance in the empty guide is given by Z1 = !µ0 / 1 . From Section (2.3.2.1) a TE10 mode in the biaxial material filled sample region implies s that n = 1, and so kc1 = ⇡/a is the cuto↵ wavenumber for the first order mode. Thus the propagation constant from (2.206) is given by s 1= r 2 k 0 µx ✏ y 137 µx 2 k . µy c1 (4.8) The transverse field equations from (2.216)-(2.217) become s ±j 1 z s Ey (x, z) = C1 es (x) e y1 s ±j 1 z s Hx (x, z) = ⌥C1 hs (x) e , y1 (4.9) (4.10) where the modal fields are given by es (x) = y1 s kc1 sin s s (x) = kc1 sin hx1 s Z1 s kc1 (4.11) s kc1 . (4.12) s s Here the modal wave impedance in the empty guide is given by Z1 = !µ0 µx / 1 . It is clearly seen from the transverse field equations that the field structures in the empty guide and in the sample region are identical; this results in direct coupling of the first order mode from the empty region to the biaxially filled sample region. Thus a mode matching analysis does not need to be used to calculate the reflection and transmission coe cients. 4.2.1.2 Solution for Reflection and Transmission Coe cients The transverse fields in the sample region and in the waveguide extensions can be represented using the modal fields, with the modal amplitudes determined through the application of boundary conditions. For the empty waveguide region z < 0 the fields are e j 1z e e e H i kx1 sin kx1 x e H r kx1 sin e ke ke j 1z e Hx (x, z) = H i x1 sin kx1 x e H r x1 sin e e Z1 Z1 Ey (x, z) = 138 e j 1z (4.13) e j 1z . (4.14) e kx1 x e e kx1 x e Here, H i is the amplitude of the incident TE10 wave, and is assumed to be known during analysis. In the sample region, 0 < z < d, the transverse fields are given by s j 1z s s s H + kc1 sin kc1 x e H kc1 sin s s kc1 ks j 1z s Hx (x, z) = H + c1 sin kc1 x e H s s sin Z1 Z1 Ey (x, z) = s j 1z (4.15) s j 1z . (4.16) s kc1 x e s kc1 x e In the receiving waveguide extension, z > d, the fields are e H t kx1 sin ke Hx (x, z) = H t x1 sin e Z1 Ey (x, z) = e kx1 x e e j 1 (z d) (4.17) e kx1 x e e j 1 (z d) . (4.18) The modal amplitudes H r , H + , H , and H t may be determined by applying the boundary conditions on Ey and Hx at the interfaces between the biaxial sample and empty guide. Applying the boundary condition on the tangential electric field at z = 0 results in h i h i i + H r k e sin k e x = H + + H s s H kc1 sin kc1 x , x1 x1 (4.19) s e and since kc1 = kx1 , then h i h i i + Hr = H+ + H H . (4.20) The tangential magnetic field boundary condition at z = 0 requires h Hi e i h e k e H r kx1 x1 sin kx1 x = H + e Z1 H i s s k s kc1 c1 sin kc1 x , s Z1 (4.21) i (4.22) or h Hi Hr i Zs h 1 = H+ e Z1 139 H . At z = d, the boundary condition on the tangential electric field requires  H +e s j 1d j sd s s e e + H e 1 kc1 sin kc1 x = H t kx1 sin kx1 x , (4.23) or  H +e s j 1d j sd + H e 1 = H t, (4.24) while the boundary condition on the tangential magnetic field at z = d results in  H +e s j 1d H e s s ke j 1 d kc1 s e sin kc1 x = H t x1 sin kx1 x , s e Z1 Z1 (4.25) or  H +e s j 1d H e s j 1d 1 t 1 s = H Ze . Z1 1 (4.26) Now, to determine the reflection coe cient, R = H r /H i , (4.20) is added to (4.22) yielding 2H + = H i s Z1 e +1 Z1 ! Hr s Z1 e Z1 ! 1 , (4.27) while subtracting (4.20) from (4.22) produces 2H = Hi s Z1 e Z1 1 ! + Hr ! s Z1 e +1 . Z1 (4.28) Similarly, adding (4.24) and (4.26) together results in " # s Z1 j sd 2H + = H t +1 e 1 , e Z1 140 (4.29) and subtracting (4.24) from (4.26) gives 2H " s t Z1 H e Z1 = # s j 1d . (4.30) # s e s Z1 Z 1 j2 1 d e , s e Z1 + Z1 (4.31) 1 e The next step is to divide (4.30) by (4.29) producing " H = H+ which can also be written as H = H+ P 2. (4.32) Here the interfacial reflection coe cient is established as s e Z1 Z1 = s e, Z1 + Z1 (4.33) and the propagation factor is defined as s j 1d . P =e (4.34) Next (4.28) is divided by (4.29) giving H = H+ ✓ s e◆ Z1 Z 1 Ze ✓ s 1 e◆ Z1 +Z1 Hi e Z1 Hi ✓ s e◆ Z1 +Z1 Ze ✓ s 1 e ◆. Z1 Z1 Hr e Z1 Hr (4.35) e Multiplying the numerator and denominator by Z1 /H i yields H = H+ ⇣ ⇣ s Z1 e Z1 s e Z1 + Z1 ⌘ ⌘ 141 ⇣ ⌘ Hr Zs + Ze 1 Hi ⇣ 1 ⌘, Hr Zs Ze 1 1 Hi (4.36) s e and dividing through by (Z1 + Z1 ) produces H = H+ R . R 1 (4.37) Now setting (4.32) equal to (4.37) results in P2 = R , R 1 (4.38) and through algebraic manipulation the reflection coe cient is determined to be R= ⇣ 1 P2 ⌘ 2P 2 1 . (4.39) Next, the transmission coe cient, T = H t /H i , can be derived by first rearranging (4.24): " # s H j2 s d j 1d 1 = H t. H +e 1 + +e H (4.40) Using (4.32), this equation becomes H +e s j 1d [1 + ] = H t . (4.41) Then (4.27) can be substituted, yielding " 1 Hi 2 s e Z1 + Z1 e Z1 ! Hr s Z1 e Z1 e Z1 !# [1 ]e s j 1d = H t. (4.42) e Factoring H i /Z1 and rearranging the equation produces  H t j sd 1 s e e 1 = e Z1 + Z1 i 2Z1 H Hr s Z1 i H 142 e Z1 [1 ], (4.43) or Te s 1 ⇥ s j 1d e = e Z1 + Z1 2Z1 s R Z1 e Z1 ⇤ [1 ]. (4.44) # s e Z1 Z 1 R s e [1 Z1 + Z1 ], (4.45) Simplifying further gives 1 j sd s e Te 1 = e Z + Z1 2Z1 1 " 1 also which can be written as Te s 1 j 1d s e = e Z + Z1 [1 2Z1 1 R ] [1 ]. (4.46) R ], (4.47) Then (4.33) can be substituted, yielding e s 2Z1 1 j 1d s + Ze Te = e Z 1 Z s + Z e [1 2Z1 1 1 1 and through addition alalgebraic manipulation, the transmission coe cient becomes T = (1 R ) P. (4.48) Now, using the equation for the reflection coe cient, (4.39), the transmission coe cient is expressed as T = 4.2.2 ⇣ P 1 1 2 ⌘ 2P 2 . (4.49) Validation of Theoretical Analysis Before parameter extraction is performed, it is very important to validate the theoretical reflection and transmission coe cients of a biaxial sample completely filling the cross-section 143 of a waveguide. Table 4.1 shows a fictitious biaxial test sample which is used to validate the theoretical model against the computational FEM solver HFSS. Because a fictitious material is being considered, each of the material parameters can be chosen such that none of the diagonal tensor entries are the same. This provides a general validation of the theory. Table 4.1: Material parameters for a fictitious biaxial material. Parameter ✏x ✏y ✏z µx µy µz Value 2.0 2.35 3.50 2.75 2.25 5 Theoretical analysis was used to compute the reflection and transmission coe cient for the fictitious material paced in the cross-section of an S-band waveguide system. The dimensions of the S-band system are a = 72.136 mm, b = 34.036 mm, and d = 10 mm. Figure 4.2 shows R and T computed using Figure 4.39 and Figure 4.49, respectively, with the orientation of the sample axes aligned with the geometry of the waveguide. Also shown in Figure 4.2 are the values of the reflection and transmission coe cient computed using HFSS. The waveguide extensions were explicitly modeled in the EM solver and were of significant length to ensure that only the fundamental TE10 mode propagates; these lengths were chosen to be 100 mm. The convergence in HFSS was specified to a maximum delta S of 0.001, which is defined as the absolute di↵erence between all S-parameters from two succeeding iterations at the solution frequency of 3.95 GHz. Excellent agreement is obtained between the theoretical equations and HFSS, thus validating the computation of the theoretical reflection and transmission coe cients. 144 d µ0 , ε 0 y z b µ0 , ε 0 µ, ε x a Magnitude (dB) Figure 4.1: Rectangular waveguide with biaxial sample placed in cross-sectional region with waveguide extensions attached. 0 S21 -5 -10 Analytical HFSS S11 -15 -20 -25 Wednesday, September 25, 13 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Phase (°) 200 S11 100 0 Analytical HFSS S21 -100 -200 2.6 2.8 3 3.2 3.4 3.6 Frequency (GHz) Figure 4.2: S-parameters computed for a biaxial test material. 145 3.8 4.2.3 Extraction Process The interfacial reflection coe cient, , and the propagation factor can be extracted form the measurements of R and T using an NRW method [23] - [24]. It is known that the measured m m S-parameters, S11 and S21 , are equal to the reflection and transmission coe cients, respecs tively. These S-parameters are used to determine the sample propagation constant 1 and the interfacial reflection coe cient , which may in turn be used to find permittivities and permeabilities. Thus, as was done in Section 3.2.1, the S-parameters are combined into two auxiliary variables, m m V1 = S21 + S11 m m V2 = S21 S11 . (4.50) (4.51) Now using (4.39) and (4.49) these variable become P 1+ P P V2 = . 1 P V1 = (4.52) (4.53) Rearranging (4.52) to determine the propagation factor yields P = V1 1 V1 . (4.54) Similarly rearranging (4.53) and then inserting (4.54) gives V2 Multiplying through by 1 V2  V1 1 V1 = V1 1 V1 . (4.55) P V1 and simplifying the expression results in 2 2 X + 1 = 0, 146 (4.56) where X= 1 V1 V2 . V1 V2 (4.57) Using the quadratic equation, the solution to (4.56) is q = X ± X2 1, (4.58) where the appropriate sign is chosen such that | |  1. Next, the propagation constant is determined using (4.34), such that s 1= ln(P ) ± j2n⇡ , jd (4.59) where n depends on the thickness of the sample in terms of wavelengths. This n value is the same n value discussed in Section 3.2.4. Next (4.33) can be rearranged such that s Z1 1+ . e =1 Z1 (4.60) Using the expressions for the modal impedances, it can be shown that s e Z1 1µ . e = s x Z1 1 (4.61) Combining (4.60) and (4.61) gives 1+ µx = 1 s 1, e 1 (4.62) which is the first material parameter extracted from the measurements. Now using (4.8) an 147 p expression for the next material parameter can be derived. First, factoring k0 µx yields v u u s = k pµ t ✏ 0 x y 1 1 µz s kc1 k0 !2 (4.63) or 2 ! s 2 1 = µx 4 ✏ y k0 1 µz s kc1 k0 !2 3 5. (4.64) Next, inserting (4.62) and simplifying gives s e 1 11 ✏y = 2 k0 1 + 1 + µz s kc1 k0 !2 . (4.65) m m Thus, by measuring S11 and S21 from a biaxial sample filling the cross-section of a rectangular waveguide, two equations, (4.62) and (4.65), can be used to extract the two material parameters µx and ✏y . To determine the other four unknown material parameters, additional measurements are needed. A measurement procedure to determine all six parameters is discussed in the following section. 4.2.4 Measurement Procedure Due to the rectangular shape of the waveguide, enough measurements for full tensor characterization can not be determined from one sample using the method discussed above. Multiple samples are required to obtain the additional measurements needed for characterization. At a minimum, three samples with the material tensor rotated in orthogonal orientations is required. To simplify the measurement procedure, consider the material tensors expressed 148 as 2 and 0 6 ✏A 0 6 ✏ = ✏0 6 0 ✏B 0 6 4 0 0 ✏C 2 0 6 µA 0 6 µ = µ0 6 0 µB 0 6 4 0 0 µC 3 7 7 7 7 5 3 7 7 7. 7 5 (4.66) (4.67) The determination of the six relative biaxial material parameters (✏A , ✏B , ✏C , µA , µB , µC ) requires a minimum of six independent measurements. The parameters can be determined by measuring S11 and S21 with each of the axes (A, B, C) individually aligned along the y-direction. The most straightforward approach is to measure the transmission and reflection coe cients for the samples placed with the A, B, C, axes under the following three orientations, (A, B, C) ! (x, y, z) ) Sample 1, (4.68) (A, B, C) ! (z, x, y) ) Sample 2, (4.69) (A, B, C) ! (y, z, x) ) Sample 3. (4.70) Using (4.62), the material parameters µA , µB , and µC can be extracted from the Sparameter measurements for Sample 1, Sample 2 and Sample 3, respectively. With all the permeabilities characterized, the permittivities, ✏A , ✏B , and ✏C , can be determined using (4.65), from the S-parameter measurements for Sample 3, Sample 1, and Sample 2, respectively. The extraction process is validated using the fictitious material parameters listed in Table 4.1. These parameters are used in HFSS to generate S-parameters for the multiple samples with their material axes aligned in the three orientations previously described. The HFSS 149 analysis uses the same waveguide model as was used to simulate the data in Figure 4.2. Figure 4.3 and Figure 4.4 show the results of the permittivity characterization, while Figure 4.5 and Figure 4.6 show the permeability extraction. The results are in accordance with the values listed in Table 4.1 and give reassurance of the characterization technique before proceeding with the error and sensitivity analysis. 4.2.5 Error and Sensitivity Analysis The error in the extracted material parameters may be placed into two categories. The first results from inaccuracies in the theoretical reflection and transmission coe cients, either because an imperfect mathematical model of the experimental apparatus is used, or because of errors in the numerical solution to the theoretical problem. The present analysis uses closed-form expressions for the reflection and transmission coe cients and therefore this type of error is not an issue. The second source of error is due to measurement inaccuracies and can be divided into systematic error and random error. Systematic error mostly arises from the imperfect construction of the sample holder and fabricated sample. Construction inaccuracies can produce gaps between the sample and waveguide walls and cause uncertainty in geometric parameters such as sample length. Random error includes sample alignment (which may change from experiment to experiment) and measurement uncertainty inherent in the VNA. It is difficult to model the errors produced by uncertainty in the geometrical parameters, due to the analysis assumption that the sample completely fills the cross-section of the guide, and does not provide an accessible means for including air gaps or oversized samples. When measurements are carried out, attempts must be made to reduce this error to the greatest extent possible by using the smallest available manufacturing tolerances and ensuring that the sample fits tightly within the holder. It is expected that these errors should be similar to those encountered with other waveguide methods [27]-[48]. In comparison, propagation of the random error inherent to the VNA may be easily studied using Monte Carlo techniques, 150 4 εC' Relative Permittivity 3.5 3 εB' 2.5 εA' 2 1.5 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.3: Permittivity extraction using HFSS generated S-parameters. 151 ε''A 0.01 0.005 0 -0.005 -0.01 2.6 2.8 3 3.2 3.4 3.6 3.8 3.6 3.8 3.6 3.8 Frequency (GHz) ε''B 0.01 0.005 0 -0.005 -0.01 2.6 2.8 3 3.2 3.4 Frequency (GHz) ε"C 0.01 0.005 0 -0.005 -0.01 2.6 2.8 3 3.2 3.4 Frequency (GHz) Figure 4.4: Permittivity extraction using HFSS generated S-parameters. 152 5.5 μC' Relative Permeability 5 4.5 4 3.5 μA' μB' 3 2.5 2 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.5: Permeability extraction using HFSS generated S-parameters. 153 0.01 0.005 μ"A 0 -0.005 -0.01 2.6 2.8 3 3.2 3.4 3.6 3.8 3.6 3.8 3.6 3.8 Frequency (GHz) μ"B 0.01 0.005 0 -0.005 -0.01 2.6 2.8 3 3.2 3.4 Frequency (GHz) 0.01 0.005 μ"C 0 -0.005 -0.01 2.6 2.8 3 3.2 3.4 Frequency (GHz) Figure 4.6: Permeability extraction using HFSS generated S-parameters. 154 and simple error bounds established. The uncertainties of the measured S-parameters were determined for an HP8510C network analyzer system using the software package HP 8510 Specifications & Performance Verification Program provided by Hewlett Packard. Although VNA measurement uncertainty is dependent on S-parameter amplitudes, for the range of amplitudes encountered in this work the VNA measurement uncertainty can be assumed independent of amplitude and frequency. Statistical variance of S11 is specified linearly in amplitude and phase, and for the equipment used in the experiment, values of A = 0.004 and = 0.8 were indicated. 11 11 Variance of S21 is specified logarithmically in amplitude and linearly in phase, and values of A = 0.04 dB and = 2.0 were found to be appropriate. 21 21 The fictitious test material describe in Table 4.1 was used in the the Monte Carlo analysis of the propagation of VNA uncertainty. The geometry of the sample is a = 72.136 mm, b = 34.036 mm, and d = 10 mm. The forward problem was solved at 31 frequency points over the portion of S-band from 2.6 to 3.95 GHz with the axes of the material tensor aligned in the three orientation outlined in Section Figure 4.2.4. White gaussian noise was then added to each of the S-parameter sets, and the noisy data was used to extract the material parameters. The standard deviations used to generate the additive noise are those indicated by the HP 8510 Specifications & Performance Verification Program. This Monte Carlo analysis used 100, 000 trials, and the average values of the material parameters were calculated, along with the standard deviations. Figure 4.7 - 4.10 show the results of the Monte Carlo error analysis. In these figures, the center of each triplet of lines is the average value of the 100, 00 trials, while the two surrounding lines indicate the 95% confidence interval of ±2 standard deviations. The error in the extracted material parameters due to S-parameter noise is small when comparing to other material characterization techniques in this dissertation. This is to be expected since this technique is an extension of the NRW method, which is often used due to its insensitivity to measurement error. 155 Relative Permittivity 3.5 εC' 3 εB' 2.5 εA' 2 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.7: Real relative permittivities for a fictitious material extracted using 100,000 random trials. Center line is the average of the trials. Upper and lower bars show the 95% confidence interval. 156 Relative Permeability 5 μC' 4.5 4 3.5 μA' 3 μB' 2.5 2 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.8: Real relative permeabilities for a fictitious material extracted using 100,000 random trials. Center line is the average of the trials. Upper and lower bars show the 95% confidence interval. 157 0.2 0.1 ε''A 0 -0.1 -0.2 2.6 2.8 3 3.2 3.4 3.6 3.8 3.6 3.8 Frequency (GHz) 0.2 0.1 ε''B 0 -0.1 -0.2 2.6 2.8 3 3.2 3.4 Frequency (GHz) 0.2 0.1 ε"C 0 -0.1 -0.2 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.9: Imaginary relative permittivities for a fictitious material extracted using 100,000 random trials. Center line is the average of the trials. Upper and lower bars show the 95% confidence interval. 158 0.2 0.1 μ"A 0 -0.1 -0.2 2.6 2.8 3 3.2 3.4 3.6 3.8 3.6 3.8 Frequency (GHz) 0.2 0.1 μ"B 0 -0.1 -0.2 2.6 2.8 3 3.2 3.4 Frequency (GHz) 0.2 0.1 μ"C 0 -0.1 -0.2 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.10: Imaginary relative permeabilities for a fictitious material extracted using 100,000 random trials. Center line is the average of the trials. Upper and lower bars show the 95% confidence interval. 159 4.2.6 Summary An extension of the NRW method to biaxially anisotropic materials was introduced. Closedform expressions were given to characterize the six complex material parameters. A step toward validation was gained using the HFSS generated S-parameters for characterization of the fictitious test sample. The performance of the technique was established using a Monte Carlo error analysis based on network analyzer error. Total validation of the technique was not completed since no measurements of biaxial material were conducted. The major flaw of this approach is requiring three distinct samples completely filling the cross-section of the waveguide. To eliminate this issue, the following section describes a technique in which biaxial material can be characterized using a single sample. 4.3 Material Characterization of Biaxial Material Using a Reduced Aperture Waveguide This section presents a convenient waveguide method that allows full biaxial tensor characterization using a single sample, in comparison to the multiple samples needed in the method described in the previous section. The theory and results described here are also published in [49]. A sample manufactured in the shape of a cube is placed into a reduced-aperture square-waveguide sample holder, and the transmission and reflection coe cients are measured with the cube in several orientations. Inversion is accomplished by minimizing the di↵erence between the theoretical reflection and transmission coe cients and the measured S-parameters. Ideally, only three orientations are required, but it has been found that by measuring the S-parameters at a fourth orientation the numerical complexity of parameter extraction is reduced significantly. Because a carefully computed forward problem is needed for accurate characterization, a mode-matching approach is used to obtain the theoretical reflection and transmission coe cients. This allows the error in the forward problem to be 160 easily controlled by specifying an appropriate number of modes. 4.3.1 Theoretical Transmission and Reflection Coe cients Using Mode-Matching Analysis The material under test (MUT) is assumed to be linear and homogeneous, with a permittivity and a permeability that are biaxial along the orthogonal axes A, B, and C. The MUT thus has the tensor constitutive parameters 2 0 6 ✏A 0 6 ✏ = ✏0 6 0 ✏B 0 6 4 0 0 ✏C and 2 0 6 µA 0 6 µ = µ0 6 0 µB 0 6 4 0 0 µC 3 7 7 7 7 5 3 7 7 7, 7 5 (4.71) (4.72) where the values ✏A , µA , etc., are relative parameters, and are complex quantities: ✏A = ✏0 + j✏00 , µA = µ0 + jµ00 , etc. A A A A The reduced aperture waveguide dimensions, shown in 4.11, consists of empty waveguide extensions connected to a sample holder that is completely filled by the MUT. A cross-section view of the reduced-aperture guide is shown in 4.12. The MUT and associated sample holder are cubical, allowing the same sample to be inserted in 24 di↵erent orientations. The width and length of the sample holder are chosen to be identical to the height of the waveguide extensions. It is assumed that the lengths of the extensions are such that a single TE10 rectangular waveguide mode is incident on the sample holder from the sending (z < 0) extension, and that a single mode appears at the end of the receiving (z > d) extension. Since the constitutive material parameters are determined by minimizing the di↵erence between the measured and the theoretically computed reflection and transmission coe 161 ε 0 , µ0 ε 0 , µ0 ε,µ y b x z a d Figure 4.11: Waveguide cubical sample holder with waveguide extensions attached. y b µ, ε aturday, January 26, 13 0 a − 2 x − w w a 2 2 2 Figure 4.12: Cross-sectional view of reduced-aperture waveguide. y 162 cients, accurate material characterization depends on careful modeling of the measurement apparatus. This is accomplished by computing the S-parameters of the reduced-aperture waveguide system using a mode-matching technique, which accommodates the higher order modes excited at the interface between the empty waveguide extensions and the sample holder. In this manner, the modeling error can be controlled in a predictable fashion. Although a single TE10 rectangular waveguide mode is incident on the sample holder, as shown in 4.13, because of mode conversion at the discontinuity with the reduced aperture an infinite number of waveguide modes are reflected into the transmitting extension, while an x infinite number of waveguide modes are transmitted into the sample region 0 < z < d. The transmitted modal fields are incident on the y discontinuity at z = d, and thus a spectrum b of modes is also reflected back into the sample region, and transmitted into the rectangular waveguide receiving extension. However, because the electric field of the incident TE10 mode µ, ε is even about x = 0, and because the aperture is symmetric about x = 0, only modes with electric fields even about x = 0 will be excited. Thus, only TEn0 modes with odd values x 0 of n are needed to describe the fields in each of the waveguide regions. Note that since the a w w a − − fields in the waveguide section filled with biaxial material are of similar structure to those 2 2 2 2 in the empty waveguide extensions, modes in all sections can be numbered identically. y µ0 , ε 0 µ0 , ε 0 µ, ε TE10 TE n0 TE n0 TE n0 TE n0 z 0 d Figure 4.13: Side view of reduced aperture waveguide. Saturday, January 26, 13 163 4.3.1.1 Field Structure in a Waveguide Filled with Biaxial Material The material sample can be placed into the cubical sample holder in a number of orientations. Assume that an orientation has been chosen such that the material is biaxial along the directions x, y and z of 4.11. The wave equation for Hz (x, y) for TEz fields in the sample p region is given by (2.189) where µx , ✏x , etc., are relative parameters, and k0 = ! µ0 ✏0 is the free-space wavenumber. For TEn0 modes in a rectangular waveguide there is no y dependence of the fields and the wave equation reduces to (2.190). This wave equation is written again in this section to reiterate what material parameters are used in the calculation of the forward problem: where " # @2 s s + kcn 2 Hz (x) = 0 2 @x µz h 2 s kcn 2 = k µx ✏ y µx 0 s 2 n (4.73) i (4.74) is the cuto↵ wavenumber. This equation is solved using separation of variables and applying the boundary conditions on the tangential electric field at the perfectly conducting waveguide walls in Section 2.3.2.1. This leads to the following expressions for the transverse fields s s Ey (x, z) = Cn es (x) ej n z yn s s Hx (x, z) = ⌥Cn hs (x) e±j n z , yn (4.75) (4.76) where the transverse modal fields are es (x) = yn h ⇣ s s kcn sin kcn x w ⌘i , 2 hs (x) = xn es (x) yn s . Zn (4.77) s Here w = b is the width of the guide, kcn = n⇡/w is the cuto↵ wavenumber for the nth s s TEn0 mode (n = 1, 2, . . .), and Zn = ⌘0 µx (k0 / n ) is the modal wave impedance with p s s ⌘0 = µ0 /✏0 the free-space intrinsic impedance. Note that n is determined from kcn 164 using (4.74). The isotropic transverse field equations from (2.151) and (2.152) can be used in this empty guide region for the present technique where the width of the waveguide is centered about the y-axis, as long as sinusoidal functions will be o↵set by a/2. Also, since only TEn0 modes are excited, v = 0 for all excited modes. This leads to kyn = 0, reducing the transverse field equations to e e Ey (x, z) = Cn ee (x) e±j n z yn e e Hx (x, z) = ⌥Cn he (x) e±j n z . xn (4.78) (4.79) Here ee and he are the transverse electric and magnetic modal fields in the empty guide, yn xn respectively, and are shown from (2.153) and (2.154) to be ee (x) = yn h ⇣ e e kxn sin kxn x he (x) = xn a ⌘i , 2 ee (x) yn e , Zn (4.80) (4.81) e e e e e where kxn = n⇡/a and Zn = ⌘0 (k0 / n ), with n determined from kcn through e n= 4.3.1.2 q 2 k0 e (kcn )2 . (4.82) Solution for S-Parameters Using Modal Expansions The transverse fields in the sample holder and in the waveguide extensions can be expanded in an infinite sum of modal fields, with modal amplitudes to be determined through application 165 of appropriate boundary conditions. For the empty waveguide extension region z < 0, 1 e e j 1z X Ey (x, z) = A+ ee (x)e + An ee (x)ej n z yn 1 y1 n=1 (4.83) odd e j 1z Hx (x, z) = A+ he (x)e 1 y1 1 X e An he (x)ej n z . yn (4.84) n=1 odd Here A+ is the amplitude of the incident TE10 wave, which is taken to be known during 1 analysis. In the sample holder, 0 < z < d, the transverse fields are Ey (x, z) = 1 Xh n=1 odd Hx (x, z) = 1 Xh s s i + Bn e j n z + Bn ej n z es (x) yn (4.85) s i Bn ej n z hs (x). yn (4.86) s + Bn e j n z n=1 odd Finally, in the waveguide extension z > d the fields are Ey (x, z) = 1 X e + Cn ee (x)e j n (z d) yn (4.87) e + Cn he (x)e j n (z d) . yn (4.88) n=1 odd Hx (x, z) = 1 X n=1 odd + + The modal amplitudes An , Bn , Bn and Cn may be determined by applying the boundary conditions on Ey and Hx at the interfaces between the two waveguide extensions and 166 the sample holder. At z = 0 the boundary condition on tangential electric field requires A+ ee (x) + 1 y1 1 X An ee (x) yn n=1 odd 8 1 h i > X w > + > Bn + Bn es (x), 0 < |x| < > yn < 2 n=1 = > odd > > > w < |x| < a : 0, 2 2 (4.89) while the boundary condition on tangential magnetic field requires A+ he (x) 1 y1 1 X An he (x) = yn n=1 odd 1 Xh + Bn n=1 odd i Bn hs (x), yn 0 < |x| < w . 2 (4.90) At z = d the boundary condition on tangential electric field requires 1 X + Cn ee (x) = yn n=1 odd 8 1 h i > X s s w > + > Bn e j n d + Bn ej n d es (x), 0 < |x| < > yn < 2 n=1 > odd > > > w < |x| < a : 0, 2 2 (4.91) while the boundary condition on tangential magnetic field requires 1 X + Cn he (x) = yn n=1 odd 1 Xh n=1 odd s + Bn e j n d s i w Bn ej n d hs (x), 0 < |x| < . yn 2 167 (4.92) To convert the system of functional equations (4.89)-(4.92) to a system of linear equations, the infinite summations are each truncated at N terms and the following testing operations are applied. First, (4.89) is multiplied by ee (x) and integrated over a/2 < x < a/2. ym s Second, (4.90) is multiplied by Zm hs (x) and integrated over w/2 < x < w/2. Third, xm (4.91) is multiplied by ee (x) and integrated over a/2 < x < a/2. Lastly, (4.92) is ym s multiplied by Zm hs (x) and integrated over w/2 < x < w/2. Here 1  m  N . This xm yields the 4N ⇥ 4N matrix equation 2 Dmn 6 6 6 Pmn 6 6 6 0 6 4 0 Fmn 0 Qmn 0 Fmn en Qmn en Dmn Pmn Fmn en 32 An + Bn + Cn 76 76 Qmn en 7 6 76 76 76 Fmn 7 6 54 Qmn bn 3 2 Dm1 3 7 6 7 7 6 7 7 6 P 7 7 6 m1 7 7 = A+ 6 7. 1 6 7 7 0 7 7 6 5 4 5 0 (4.93) s Here the indices m and n are unique to each submatrix, en = exp{ j n d}, and bn = Bn /en . Note that bn is introduced to avoid overflow during computation. In (4.93), Dmn , Qmn , Pmn and Fmn are N ⇥ N sub-matrices with entries specified in Appendix B.2. Once the modal coe cients are found, the S-parameters of the system are given by A S11 = 1 , A+ 1 4.3.2 + C1 S21 = + . A1 (4.94) Specification of Computational Accuracy To assess the performance of the extraction procedure, it is important to have a clear measure of the accuracy of the computed theoretical S-parameters. With the mode-matching approach this is easily done by terminating the modal series at N terms and properly choosing N . Rather than merely truncating the modal series when a specified accuracy has been achieved, the extrapolation technique described in [50] is used. It is found that the Sparameters may be extrapolated to the limit as 1/N ! 0, with N the number of modes used 168 in both the sample region and in the waveguide extensions. To show how the S-parameters converge as a function of 1/N a sample of teflon placed inside an S-band reduced aperture sample holder is considered. The teflon sample is taken to be a lossless dielectric with real relative permittivity of 2.1. The dimensions for the S-band system are a = 72.136 mm and b = w = d = 34.036 mm. Figure 4.14 shows the reflection and transmission coe cients at 2.6 GHz. It is desirable to estimate the e↵ect of using an infinite number of modes without the use of a large number of modes. By examining the magnitude and phase of the S-parameters vs. 1/N , a trend may exist in which the value for the magnitude and phase can be extrapolated to estimate their values when 1/N = 0, which corresponds to N = 1; this produces an improved estimate of the reflection and transmission coe cients. Several di↵erent extrapolation methods exist, including linear extrapolation, polynomial extrapolation, and least squares. Examining Figure 4.14 shows there is a linear trend to the data. This linear trend suggests that a simple linear extrapolation can be used to estimate modal series as N approaches 1. A similar trend is seen when examining the other frequencies across the band. The linear trend may also be seen for coe cients computed in the middle of the frequency band of interest, 3.275 GHz, in 4.15, and may also be seen for the upper end, 3.95 GHz, in Figure 4.16. The series is terminated when the extrapolated values of the magnitude and phase of the S-parameters have reached specified tolerances for M contiguous values of N . Experience shows that a value of M = 5 is su cient to guarantee convergence to desired accuracy in the method presented here. It is found that by using this extrapolation technique, the S-Parameters may be e ciently computed to high tolerance with many fewer terms in the modal series compared to truncation based on using the value of the series. 4.3.3 Validation of Theoretical Analysis It is important to validate the theoretical model before employing it in parameter extraction. To do this, a fictitious biaxial test material is considered, with material parameters shown 169 0.9942 -2.826 ∠S11 (radians) Magnitude 0.99415 -2.828 0.99405 -2.83 | S11| (dB) 0.9941 0.994 Phase -2.832 0.99395 -2.834 0.9939 0.11 0 0.01 0.02 0.03 0.04 0.05 1.884 ∠S21 (radians) Phase | S21| (dB) 0.1095 1.882 0.109 1.88 0.1085 1.878 0.108 0.1075 Magnitude 0 0.01 0.02 0.03 1.876 0.04 0.05 1/N Figure 4.14: Magnitude and phase of reflection and transmission coe cients for teflon sample completely filling reduced-aperture waveguide sample regions at 2.6 GHz. These coe cients are plotted vs. 1/N . 170 -1.1 Magnitude -1.105 0.618 -1.11 0.616 -1.115 Phase 0.614 -1.12 0.612 -1.125 0.61 0.792 0 0.01 0.02 Phase 0.03 0.04 -2.68 0.79 | S21| (dB) -1.13 -2.675 0.05 ∠S21 (radians) S | 11| (dB) 0.62 ∠S11 (radians) 0.622 -2.685 0.788 -2.69 0.786 -2.695 -2.7 0.784 0.782 Magnitude 0 0.01 0.02 0.03 -2.705 0.04 -2.71 0.05 1/N Figure 4.15: Magnitude and phase of reflection and transmission coe cients for teflon sample completely filling reduced-aperture waveguide sample regions at 3.275 GHz. These coe cients are plotted vs. 1/N . 171 ∠S11 (radians) 0.594 S | 11| (dB) -2.81 Magnitude -2.812 0.592 -2.814 -2.816 Phase 0.59 -2.818 0.588 0.81 0 0.01 0.03 0.04 -2.82 -1.24 0.05 ∠S21 (radians) Phase 0.809 | S21| (dB) 0.02 -1.242 0.808 0.807 -1.244 0.806 -1.246 0.805 0.803 -1.248 Magnitude 0.804 0 0.01 0.02 0.03 0.04 -1.25 0.05 1/N Figure 4.16: Magnitude and phase of reflection and transmission coe cients for teflon sample completely filling reduced-aperture waveguide sample regions at 3.95 GHz. These coe cients are plotted vs. 1/N . 172 in Figure 4.2. By using a fictitious material, each of the parameters can be chosen to have di↵erent values, allowing for the most general validation of the theory. Table 4.2: Material parameters for a fictitious biaxial material. Parameter ✏A ✏B ✏C µA µB µC Value 2-j0.1 4-j0.5 3-j0 1-j0.2 2.5-j0 2-j1 The mode matching technique was used to compute the S-parameters for the fictitious material placed into a cubical sample holder in an S-band waveguide system. Figure 4.17 shows S11 and S21 computed using the modal series with the orientation (A, B, C) ! (x, y, z), meaning that the A-axis of the sample is aligned with the x-axis of the waveguide, etc. Absolute tolerances of 0.1 dB for |S| and 0.01 for \S were chosen so that the accuracy of the computed series is significantly better than the expected measurement accuracy of the HP 8510C vector network analyzer (VNA) used in subsequent experiments. For the material considered, the S-parameters typically converge to the specified tolerances within N = 81 terms. Also shown in Figure 4.17 are the values of the S-parameters computed using the commercial EM solver HFSS. The waveguide extensions must be explicitly modeled in HFSS, and must be su ciently long to ensure that only the fundamental TE10 mode propagates at the waveguide ports. Here the extensions were chosen to be 82.98 mm in length. The convergence tolerance in HFSS was specified as a maximum delta S of 0.001, which is the absolute di↵erence between S-parameters from two iterations at the solution frequency of 3.95 GHz. It is seen that excellent agreement is obtained between the modal analysis and HFSS, validating the modal computation of the theoretical S-parameters. Note that computational time is significantly shorter using the modal analysis compared to the HFSS finite 173 Magnitude (dB) -4 S11 -6 -8 -10 Mode Matching HFSS S21 -12 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Phase (°) 200 S11 100 0 Mode Matching HFSS S21 -100 -200 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.17: S-parameters computed for a biaxial test material. 174 element method solver. For a typical set of parameters, the modal analysis programmed in Fortran is several orders of magnitude faster than FEM, and thus provides a good basis for performing parameter extraction. 4.3.4 Extraction Process Determination of the the six biaxial material parameters (✏A , ✏B , ✏C , µA , µB , µC ) requires a minimum of six independent measurements. However, if it is known a priori that the material is nonmagnetic (i.e., µA = µB = µC = 1), then the extraction process is simplified greatly. The parameters ✏A , ✏B , and ✏C can be determined independently by measuring S21 with each of the axes (A, B, C) individually aligned along the y-direction, since only ✏y is implicated in (4.74). For instance, if the A direction is chosen to coincide with y, then ✏A may be found as the solution to the equation thy S21 (✏A ) meas = 0, S21 (4.95) which is easily determined using Newton’s method. For the generally biaxial case the most straightforward approach is to measure the transmission and reflection coe cients for the sample placed with its A, B, and C axes under three orientations, as follows: (A, B, C) ! (x, y, z) (A, B, C) ! (z, x, y) (A, B, C) ! (y, z, x) h i meas , S meas , ) S11,1 21,1 h i meas meas ) S11,2 , S21,2 , h i meas meas ) S11,3 , S21,3 . (4.96) (4.97) (4.98) The material parameters are then found by solving the following system of six nonlinear 175 complex equations in the six complex unknown material parameters: thy S11,n (✏A , ✏B , ✏C , µA , µB , µC ) meas S11,n = 0, n = 1, 2, 3, (4.99) thy S21,n (✏A , ✏B , ✏C , µA , µB , µC ) meas S21,n = 0, n = 1, 2, 3. (4.100) Experience shows that with typical values of experimental error it can be di cult to find solutions to the system of equations (4.99)-(4.100), even with good initial guesses. An alternative approach allows a subset of the six material parameters to be first determined, and these parameters to be subsequently used in the computation of the remaining parameters. This approach is based on the observation from (4.74) that only three of the six material parameters are implicated in a measurement under any one orientation. That is, only ✏y , µx , and µz appear in (4.74). Refer to Figure 4.11. First, measurements are made with these orientations: (A, B, C) ! (x, y, z) (A, B, C) ! ( z, y, x) h i meas meas ) S11,1 , S21,1 , h i meas , S meas . ) S11,2 21,2 (4.101) (4.102) These measurements only implicate the parameters ✏B , µA , and µC . Using three out of the four complex measurements yields the following system of equations with three unknowns: thy S11,n (✏B , µA , µC ) thy S21,n (✏B , µA , µC ) meas S11,n = 0, n = 1, (4.103) meas S21,n = 0, n = 1, 2, (4.104) thy which can be solved using a Newton’s method. Note that the di↵erent between S11,2 and meas S11,2 could be used in place of one of the transmission coe cient equations, however, experience has shown more e cient characterization using (4.103) and (4.104). Additionally, a least squares approach could be used in place of a Newton’s method and use all four meas measurements instead of eliminating the use of S11,2 . Next, a measurement is made under 176 the orientation h i meas meas ) S11,3 , S21,3 . (A, B, C) ! (y, x, z) (4.105) This measurement implicates ✏A , µB , and µC . However, µC is known from solving (4.103)(4.104), and thus the system of equations thy S11,3 (✏A , µB ) meas S11,3 = 0, (4.106) thy S21,3 (✏A , µB ) meas S21,3 = 0, (4.107) can be solved for ✏A and µB . Finally, a last measurement is made under the orientation (A, B, C) ! (z, x, y) h i meas , S meas . ) S11,4 21,4 (4.108) This measurement implicates ✏C , µA , and µB . However, at this point only ✏C is unknown, and thus the single complex equation thy S21,4 (✏C ) meas S21,4 = 0 (4.109) is solved using a Newton’s method. The drawback to this approach is that an additional measurement is required. However, experience shows that reducing the number of unknowns improves the extraction process to the extent where the extra experimental e↵ort is advantageous. It is important to validate the inversion method before proceeding with the characterization of unknown materials. Therefore, the fictitious material parameters from Table 4.2 are used in HFSS to generated S-parameters with the cube in four orientations, and the fourorientation extraction method is performed. The HFSS simulation uses the same model as was used to generate the data for Figure 4.17. The results of the characterization are shown in Figure 4.18 and Figure 4.19. As expected, the characterized material parameters are in accordance with the parameters shown in Table 4.2. This extraction gives confidence in the 177 characterization technique before moving forward with error analysis and measurements. 4.3.5 Error and Sensitivity Analysis As done with previous characterization techniques, and described in Section 4.2.5, a Monte Carlo technique was used to study e↵ects on the propagation of the random error inherent to the VNA used in the measurements. Once again, the uncertainties of the measured Sparameters were determined for an HP8510C network analyzer system using the software package HP 8510 Specifications & Performance Verification Program provided by Hewlett Packard and the values used for the statistical variance of S11 and S21 as outlined in Section 4.2.5. A Monte Carlo analysis of the propagation of VNA uncertainty was undertaken using the fictitious test material with the parameters given in Table 4.2. The geometry of the sample is a = 72.136 mm and b = w = d = 34.036 mm. The forward problem was solved at 101 frequency points over the portion of S-band from 2.6 to 3.95 GHz under the four orientations described in Section 4.3.4. White gaussian noise was then added to each of the S-parameter sets and the noisy data used to extract the material parameters according to the fourmeasurement process of Section 4.3.4. The standard deviations used to generate the additive noise are those indicated by the HP 8510 Specifications & Performance Verification Program. The process was repeated 500 times, and the average values of the material parameters were calculated, along with the standard deviations. Results are shown in Figure 4.20 and Figure 4.21. In these figures, the center of each triplet of lines is the average value of the 500 trials, while the two surrounding lines indicate the 95% confidence interval of ±2 standard deviations. These results for VNA uncertainty will be compounded by any systematic errors present in the experiment, and have values typical of those encountered with other material extraction methods, such as [51] and [40]. Note that for low loss materials with small ✏00 and µ00 , the relative error in the measured imaginary parts can be quite severe, even though the absolute error may be less than those for ✏0 and µ0 . 178 Relative Permittivity 4 ε'B 3 ε'C 2 ε'A 1 ε''A ε'' ε''C B 0 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.18: Extracted permittivity using HFSS generated S-parameters. 179 Relative Permeability 3 μ'B 2 μ'C 1 μ'A μ''A μ''B μ''C 0 -1 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.19: Extracted permeability using HFSS generated S-parameters. 180 ϵ'B Relative Permittivity 4 ϵ'C 3 ϵ'A 2 1 ϵ''A ϵ''C ϵ''B 0 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.20: Relative permittivities for a fictitious material extracted using 500 random trials. Center line is the average of the trials. Upper and lower lines show the 95% confidence interval. 181 Relative Permeability 3 μ'C 2 μ'B 1 μ'A 0 μ''A μ''C μ''B -1 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.21: Relative permeabilities for a fictitious material extracted using 500 random trials. Center line is the average of the trials. Upper and lower lines show the 95% confidence interval. 182 4.3.6 Experimental Results To experimentally validate the proposed technique at S-band, a reduced-aperture waveguide was machined from 360 brass. The specified inner dimensions of the cubical sample holder are b = w = d = 34.036 mm, and measurement of the constructed sample using precision calipers showed that the holder was constructed with tolerances of less than 0.04 mm. The holder was aligned with two 152.4 mm-long commercial S-band waveguide extensions of cross-section dimension 34.036 mm by 72.136 mm and holes for precision alignment pins were drilled in the sample holder to guarantee high repeatability of the alignment of the holder and the extensions. This is necessary since the extensions must be removed to calibrate the system and to insert and remove the sample. Two samples were constructed to test the system. The first is a simple isotropic teflon cube. The properties of teflon are well-established, and since the material is isotropic, it can be used to characterize the repeatability of measurements as the sample is rotated into the required four orientations. Figure 4.22 shows a picture of the sample holder and the teflon sample. The sample was constructed approximately 0.01 mm larger than the inner dimensions of the sample holder so that when inserted it would compress slightly and eliminate air gaps between the MUT and the sample holder walls. Measurements of the S-parameters of the teflon sample placed into the four required orientations were made using an HP 8510C VNA. The VNA was calibrated using the ThruReflect-Line (TRL) algorithm internal to the 8510C, with an aluminum plate used as the reflection standard, and a 33.53 mm-long section of commercial waveguide used as the line standard. As with the sample holder, precision alignment pins were used to ensure repeatability of the alignment of the guides. All measurements were made with a 10 dBm source power, 32 averages, and a 25 ms dwell time. Teflon is a low-loss dielectric with a real relative permittivity of approximately 2.1. To assess repeatability error, the teflon sample was measured 10 separate times, with the system 183 Figure 4.22: Cubical sample holder and teflon sample. 184 recalibrated between sample measurements. The results of the characterization are shown in Figure 4.23 and Figure 4.24, where the center solid lines represent the mean extracted values of the 10 measurement sets and the upper and lower circles show the 95% confidence interval (± two standard deviations). Although teflon is isotropic, the measurements were conducted assuming a biaxial material sample, and so six material parameters were computed for each measurement set. The mean extracted values for each of the three permittivities and each of the three permeabilities overlap closely in Figure 4.23 and Figure 4.24, as expected for an isotropic material. Note that to avoid clutter, only the confidence intervals for ✏A and µA are shown in Figure 4.23 and Figure 4.24. The confidence intervals for the remaining material parameters are very similar. The extracted mean values show excellent agreement to the expected material parameters of teflon and the confidence intervals closely resemble the error bars shown in Figure 4.20 and Figure 4.21, suggesting that the dominant repeatability error is due to S-parameter measurement uncertainty. To test the characterization procedure using an anisotropic material, a cube was constructed by using layers of alternating dielectrics as shown in Figure 4.25. This produces a material with anisotropic (uniaxial) dielectric properties, but isotropic magnetic properties. Since commercial, o↵-the-shelf anisotropic dielectric and anisotropic magnetic material is not readily available, a uniaxial cube was constructed by gluing together Rogers RO3010 circuit board (the light layers in Figure 4.25) and Rogers RT/duroid 5870 circuit board (dark layers). The 3010 board has a thickness of t1 = 1.27mm, a dielectric constant of ✏0 = 10.2 r1 and a loss tangent of tan 1 = 0.0022. The 5870 board has a thickness of t2 = 3.4mm, a dielectric constant of ✏0 = 2.33 and a loss tangent of tan 2 = 0.0012. If the B direction is r2 chosen to be aligned perpendicular to the layer interfaces, then it is expected that ✏A and ✏C should be identical, but di↵erent from ✏B . The advantage to using this structure is that simple formulas exist for estimating ✏A , ✏B , and ✏C . Note that at the highest frequency considered in the measurements, the free-space elecq trical length of the stack period is k0 (t1 + t2 ) = 0.387. Since k0 ✏0 (t1 + t2 ) ⌧ 2⇡, the r1 185 Relative Permittivity 2 ϵ'A ϵ'B ϵ'C 1.5 1 0.5 ϵ''A ϵ''B ϵ''C 0 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.23: Relative permittivities extracted from 10 sets of teflon measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval for ✏A . 186 Relative Permeability 1 μ'A μ'B μ'C 0.8 0.6 0.4 0.2 μ''A μ''B μ''C 0 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.24: Relative permeabilities extracted from 10 sets of teflon measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval for µA . 187 Figure 4.25: Cubical sample holder and alternating-layer dielectric sample. 188 following approximate formulas may be used to determine the biaxial material constants [59], ✏B =  1 ✏r2 ✏r1 ✏r2 t1 ✏r1 ✏r2 t1 + t2 ✏A = ✏C = ✏2r + (✏1r ✏2r ) 1 , (4.110) t1 , t1 + t2 (4.111) where ✏r1 = ✏0 (1 j tan 1 ) and ✏r2 = ✏0 (1 j tan 2 ). Substituting the board parameters r1 r2 gives ✏B = 2.95 j0.0038 and ✏A = ✏C = 4.47 j0.0081. It is expected that the formula for ✏A and ✏C will be less accurate than the formula for ✏B , since the presence of internal reflections within the layers is most significant when the cube is oriented such that the interfaces lie normal to the wave propagation. Results given in [53] for a layered sample in a non-reduced rectangular guide suggest that a deviation of as much as 10-20 percent may be expected unless the number of layers is significantly large. Although these results cannot be directly used with the reduced-aperture guide, they suggest that a deviation of 10 percent would not be unexpected. Also note that the circuit boards themselves are slightly anisotropic due to the alignment of the glass fibers used in their construction, and thus it is expected that ✏A and ✏C should be slightly di↵erent [54]. However, no attempt is made to quantify this di↵erence theoretically. Similar to the measurements of the teflon cube, the layered cube was measured 10 separate times. The results of the extraction are shown in Figure 4.26 through Figure 4.31, where the solid lines represent the averaged extracted material parameter and the upper and lower circles show the 95% confidence intervals for the 10 measurement sets. In each of these figures, gaps are seen in the extracted parameters over certain ranges of frequency. This is due to the well-known di culty of extracting material parameters near frequencies where the sample thickness approaches one half of a guided wavelength. This problem is inherent to all guided-wave techniques in which both permittivity and permeability are determined (including the Nicolson-Ross-Wier closed-form method for isotropic materials [23]-[24]), and 189 is due to the propagation of measurement uncertainties. It can be seen in Figure 4.26 that as the frequency nears 3.7 GHz, the uncertainties begin to grow dramatically. within the indicated gap, the uncertainties become so large that extraction is completely unreliable. This is a drawback of using a cubical sample holder, since the thickness of the material cannot be reduced to avoid these half-wavelength gaps when the material parameters are su ciently large. Experience has shown that the a frequency range within approximately ±5% of the half-wavelength frequency should be avoided, and data within that range is not displayed in the figures. Across other portions of the frequency band, the averaged extracted values are reasonably close to those predicted from (4.110)-(4.111), and the confidence intervals are consistent with the error due to S-parameter uncertainty described in Section 4.3.5. As discussed above, it is not surprising to find that there is a deviation of about 10% in ✏A and ✏C from the result predicted using the simple formula (4.111). The process of extracting the various material parameters in stages, with ✏B , µA , and µC determined first, produces an unfortunate cascading of half-wavelength gaps in the extracted data. Figure 4.27, Figure 4.29 and Figure 4.31 only show gaps near 3.7 GHz since the orientations used to determine these parameters produce just a single half-wavelength point in the measurement band. Similarly, the orientations used to determine ✏A , ✏C , and µB produce only a single half-wavelength point near 2.8 GHz, and thus gaps at this frequency can be seen in Figure 4.26, Figure 4.28 and Figure 4.30. However, since the previously extracted values of ✏B , µA , and µC are used to find ✏A , ✏C , and µB , the gaps at 3.7 GHz in those data are also present in Figure 4.26, Figure 4.28 and Figure 4.30. Because the material parameters do not vary strongly with frequency, their values within the gap regions may be approximated through interpolation. A simple approach is to fit a polynomial function to the mean data, and evaluate the polynomial in the gap. For the materials considered here, it is found that a fifth order polynomial fits the data well over the entire measurement band. Figure 4.32 and Figure 4.33 show the interpolating polynomials evaluated across the measurement band. These interpolated results show good agreement 190 Relative Permittivity 5 ϵ'A 4 3 2 ϵ''A 1 0 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.26: Relative permittivity ✏A extracted from 10 sets of measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval. 191 Relative Permittivity 3 ϵ'B 2 1 ϵ''B 0 -1 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.27: Relative permittivity ✏B extracted from 10 sets of measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval. 192 Relative Permittivity 5 ϵ'C 4 3 2 ϵ''C 1 0 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.28: Relative permittivity ✏C extracted from 10 sets of measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval. 193 Relative Permeability 1.2 μ'A 1 0.8 0.6 0.4 0.2 μ''A 0 -0.2 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.29: Relative permeability µA extracted from 10 sets of measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval. 194 1.4 μ'B Relative Permeability 1.2 1 0.8 0.6 0.4 μ''B 0.2 0 -0.2 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.30: Relative permeability µB extracted from 10 sets of measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval. 195 μ'C Relative Permeability 1.2 1 0.8 0.6 0.4 μ''C 0.2 0 -0.2 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.31: Relative permeability µC extracted from 10 sets of measurements. Center line is the average of the measurements. Upper and lower circles show the 95% confidence interval. 196 with the expected values from the simple formula outlined previously. Note that there is a small amount of unphysical oscillation in the permeabilities, due to the variations in these parameters near the edges of the gap regions. This oscillation can be reduced by using a lower order polynomial, or by taking slightly wider gaps. Because it is known a priori that the layered cube is nonmagnetic, the permittivity parameters ✏A , ✏B , and ✏C can be determined independently from three measurements of S21 using (4.95). Figure 4.34 shows the extracted values using the same 10 measurements used to generate Figure 4.28-4.31. It is clearly seen that the di culties at half-wavelength frequencies do not occur in this case, and thus there are no gaps in the data. This is because only transmission coe cients are used in the extraction, and the extracted material parameters are less sensitive to uncertainty in S21 than in S11 . The results are very close to those of Figure 4.32 found using the extrapolating polynomial. Although the use of a non-magnetic uniaxial material does not fully explore the ability of the proposed technique to extract all the parameters of a biaxial sample, the agreement between ✏A and ✏C gives good evidence that the technique is viable. 4.3.7 Summary A reduced aperture waveguide method is introduced for measuring the permittivity and permeability of biaxially anisotropic materials. Only a single cubical sample is required to completely characterize the material under test. This contrasts with the three distinct samples required when using the standard waveguide technique in which a sample completely fills the rectangular waveguide cross section. The performance of the technique is established using an error analysis based on network analyzer uncertainty, and validation is provided by laboratory measurements of a teflon and a stacked dielectric layered sample. It is found that the technique performs very well for regions of the frequency band where the electrical length of the sample does not approach a half wavelength, and that an interpolating polynomial may be used to supply the data near the half-wavelength points 197 Relative Permittivity 5 4 ϵ'C ϵ'A ϵ'B 3 2 1 ϵ''A ϵ''B ϵ'' C 0 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.32: Average permittivity from 10 sets of measurements extrapolated using a fifth order polynomial. 198 Relative Permeability 1 μ'A 0.8 μ'B μ'C 0.6 0.4 0.2 μ''B μ''A μ''C 0 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.33: Average permeability from 10 sets of measurements extrapolated using a fifth order polynomial. 199 ϵ'A Relative Permittivity 5 ϵ'C 4 ϵ'B 3 2 1 ϵ''A ϵ''B ϵ''C 0 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 4.34: Relative permittivity extracted from 10 sets of measurements assuming the sample is nonmagnetic. Center line is the average of the measurements. Upper and lower lines show the 95% confidence interval. 200 CHAPTER 5 GYROMAGNETIC MATERIAL CHARACTERIZATION 5.1 Introduction This chapter presents two methods for characterizing gyromagnetic materials using rectangular waveguides where the sample is magnetized perpendicular to the broad dimension of the guide. The first technique considers a gyromagnetic material completely filling the crosssection of a waveguide. Gyromagnetic materials react di↵erently depending on the incident wave polarization and the wave couples into orthogonal field components. A mode-matching technique is used determine the theoretical reflection and transmission coe cients needed for material parameter characterization. The parameters are obtained using a nonlinear least squares method that seeks to minimize the di↵erence between theoretically computed and measured reflection and transmission coe cients. The second characterization technique uses a reduced-aperture sample holder such that the gyromagnetic sample can be machined to have a smaller width than the cross-section of the guide. The sample holder is positioned between two empty sections of rectangular waveguides to guarantee only the dominant mode is present at the sample holder. The S- 201 parameters are then measured. Mode matching is used to determine the theoretical reflection and transmission coe cients of the dominant mode. The inverse problem is then solved using a non-linear least squares method to determine the material parameters. 5.2 Characterization of Gyromagnetic Material Filling the Cross-Section of a Rectangular Waveguide This section concentrates on the extraction of the e↵ective permittivity and permeability of a sample of gyromagnetic material completely filling the cross-section of a rectangular waveguide. It is customary for optimizers or root solvers (such as the Newton’s method used in Section 4.3.4) to be used to determine the constitutive parameters of materials in rectangular waveguides. In these methods, the reflection and transmission coe cients from a material placed in the cross-sectional plane of a waveguide are measured and compared to theoretical S-parameters. It is essential to have accurate formulations of the theoretical S-parameters for valid characterization of materials. A modal matching method is used to determine the reflection and transmission coe cients of the dominant mode from a sample of gyromagnetic material filling the cross section of a rectangular guide. Details on the methodology for obtaining the reflection and transmission coe cients are presented along with a comparison to a finite element full wave solver. Additionally, the characterization method is tested by extracting the material parameters from the FEM generate S-parameters. 5.2.1 Theoretical Transmission and Reflection Coe cients Using Mode-Matching Analysis The material under test (MUT) is assumed to be linear and homogeneous with isotropic permittivity and a permeability that is uniaxial along the orthogonal axes and has o↵ diagonal 202 parameters. The MUT has the tensor constitutive parameters 2 6 ✏r 0 0 6 ✏ = ✏0 6 0 ✏r 0 6 4 0 0 ✏r 3 7 7 7 7 5 (5.1) and 2 6 µg 0 6 µ = µ0 6 0 1 6 4 j 0 3 j 7 7 0 7, 7 5 µg (5.2) where the values ✏r , , and µg are relative complex quantities: ✏r = ✏0 + j✏00 ,  = 0 + j00 r r and µg = µ0 + jµ0 . This permeability tensor is for a material biased in the y-direction, or g g the along the height of the waveguide. Here the o↵-diagonal parameter  is given by = !!m , 2 !0 ! 2 (5.3) and the diagonal element µg is described as µg = ! !m 1+ 0 2 !0 ! 2 ! . (5.4) In (5.3) and (5.4), ! = 2⇡f is the operating frequency, !0 = µ0 H0 is the Larmor frequency, and !m = µ0 Ms . Here H0 is the strength of the internal static biasing magnetic field in oersteds and Ms the saturation magnetization typically expressed as 4⇡Ms gauss. The ratio of spin magnetic moment to the spin angular momentum for an electron is called the gyromagnetic ratio and is given by = 1.759 ⇥ 1011 C/kg. The Lamor frequency can be expressed as f0 = !0 /2⇡ = (2.8 MHz/oersted)(H0 oersted), and fm = !m /2⇡ = (2.8 MHz/oersted)(4⇡Ms gauss) [55]. 203 Additionally, in order to account for magnetic losses inherent in the magnetic materials, the linewidth H is in- cluded in the calculation of the Larmor frequency. Thus, !0 = µ0 H0 + jµ0 H/2, or f0 = (2.8 MHz/oersted)(H0 oersted + j H/2 oersted). During measurements the applied biasing magnetic field will be applied external to the sample using permanent magnets and it is therefore important to understand the relationship between the external and internal biasing fields. When the applied field is parallel to the broad face of the gyromagnetic sample, which is the case considered in this dissertation, the continuity of the magnetic field at the surface of the sample results in the external field being equal to the internal field. The waveguide system, shown in Figure 5.1, consists of empty waveguide extensions connected to a sample holder with the same cross-section as the empty guide. This sample holder is completely filled by the MUT. It is assumed that the lengths of the extensions are such that a single TE10 rectangular waveguide mode is incident on the sample holder from the transmitting section (z < 0) and that a single mode is present at the receiving extension (z > d). The material parameters are obtained using optimization methods that seek to minimize the di↵erence between theoretically computed and measured reflection and transmission coefficients. As a result, accurate formulation of the theoretical scattering parameters is critical for valid characterization of gyromagnetic materials. This is accomplished by computing the S-parameters of the gyromagnetic material filled sample holder using a mode-matching technique, which accommodate the higher order modes excited from the coupling between orthogonal field components inside the sample region. A single TE10 rectangular waveguide mode is incident on the sample, as shown in Figure 5.2, and, because of mode conversion at the interface been the empty guide and sample an infinite number of waveguide modes are reflected into the transmitting extensions, while an infinite number of waveguide modes are transmitted into the sample region, 0 < z < d. The transmitted modal fields are incident on the next interface at z = d, and thus a spectrum of modes is reflected into the sample region and transmitted into the empty waveguide 204 extensions. Since the TE10 mode is invariant in the y-direction and even about x = a/2 and the gyromagnetic material is magnetized along the y-axis, then all excited higher order modes are also y-invariant and even about x = a/2. Thus, only TEn0 modes are needed to describe the fields in each of the waveguide regions. 5.2.1.1 Field Structure in a Waveguide Filled with Gyromagnetic Material The wave equation for Hz (x, y) for TEz fields in the sample regions is given by (2.227). For TEn0 modes in a rectangular waveguide there is no y-dependence of the fields and the wave equation reduces to (2.229) and is expressed as " # @2 s 2 H (x) = 0, + kc z @x2 (5.5) where s 2 k c 2 = k 0 µg ✏ r 1 2 µ2 g ! 2 (5.6) is the cuto↵ wavenumber and k0 = ! 2 µ0 ✏0 is the free-space wavenumber. The transverse fields are given by (2.251) and (2.252) and are shown to be ! 2 s s s Ey (x) = Cn kcn 1 sin kcn x e±j µ2 g s  s Cn kcn k s s s Hx (x) = ⌥ sin kcn x ⌥ cn cos kcn x e±j s s Zn µg n s nz (5.7) s nz . (5.8) s Here kcn = n⇡/a is the cuto↵ wavenumber for the nth TEn0 modes (n = 1, 2, 3 . . .) and s s s s Zn = !µ0 µg / n is the modal wave impedance. Note that n is determined from kcn using (5.6). The isotropic transverse field equations from (2.151) and (2.152) are used to represent the fields in the empty guide region for the present technique. Here only TEn0 modes are 205 H0 ϵ0,µ0 y ϵ,µ ϵ0,µ0 x z x a d b b y Figure 5.1: Rectangular waveguide with gyromagnetic sample placed in cross-sectional region with waveguide extensions attached. µ, ε 0 a − 2 x − w w a 2 2 2 H0 y µ0 , ε 0 µ0 , ε 0 µ, ε TE10 TE n0 TE n0 TE n0 TE n0 z 0 d Figure 5.2: Side view of gyromagnetic material filled waveguide. 206 excited,. This means v = 0 for all excited modes. This leads to kyn = 0, which reduces the field equations to e Ey (x, z) = e Cn kxn sin ke e Hx (x, z) = ⌥Cn xn sin e Zn e e kxn x e±j n z e e kxn x e±j n z . (5.9) (5.10) s e e Here kcn = n⇡/w is the cuto↵ wavenumber, and Zn = !µ0 / n is the modal wave impedance. e Note that n is determined using (2.140). 5.2.1.2 Solutions for S-Parameters Using Modal Expansions The transverse fields in the sample region and in the empty waveguide extensions can be expanded in an infinite sum of modal fields where the modal amplitude coe cients are determined through the enforcement of boundary conditions. In the empty waveguide extensions, z < 0 and z > d, the transverse fields are expressed as Ey (x, z) = e j 1z e e a+ kx1 sin kx1 x e 1 e e + kx1 sin k e x e j 1 z Hx (x, z) = a1 e x1 Z1 1 X e e e an kxn sin kxn x ej n z (5.11) n=1 1 X e ke e an xn sin kxn x ej n z , e Zn n=1 (5.12) and 1 X e e e c+ kxn sin kxn x e j n (z d) n (5.13) n=1 1 X ke e e Hx (x, z) = c+ xn sin kxn x e j n (z d) , n Ze n n= (5.14) Ey (x, z) = 207 respectively. In the sample region, 0 < z < d, the transverse fields are given by Ey (x, z) = 1 X h s s i s kcn b+ e j n z + bn ej n z 1 n n=1  1 X ks cn b+ sin k s x + Hx (x, z) = cn s n n= Zn  1 X ks cn b s sin kcn x s Zn n n= 2 µ2 g ! s sin kcn x s kcn s s cos kcn x µg n (5.15) s e j nz s kcn s s cos kcn x µg n s ej n z . (5.16) In (5.11) and (5.12), a+ is the amplitude of the incident TE10 wave, which is assumed to 1 be known during analysis. The remaining modal amplitude coe cients an , b+ , bn , and n c+ may be determined by applying boundary conditions on Ey and Hx at the interfaces n between the empty waveguide and sample regions. At z = 0 the boundary condition on the tangential electric field requires e e a+ kxn sin kx1 x + 1 1 X n=1 1 X e e an kxn sin kxn x = n=1 i s b+ + b kcn n 1 n h 2 µ2 g ! s sin kcn x , (5.17) while the boundary condition on the tangential magnetic field requires 1 e X ke + kxn sin k e x e a1 e an xn sin kxn x = e x1 Z1 Zn n=1  1 s s X k cn b+ sin k s x + kcn cos k s x cn cn s Zs n µg n n=1 n  1 s X ks kcn cn b s x s s n sin kcn s cos kcn x . Z µg n n=1 n 208 (5.18) Similarly, imposing the tangential boundary conditions on the electric and magnetic fields at z = d gives 1 X e e c+ kxn sin kxn x = n n=1 1 X n=1 h s s i s kcn b+ e j n d + bn ej n d 1 n 2 µ2 g ! s sin kcn x , (5.19) and  1 e X ks cn b+ sin k s x + + kxn sin k e x = cn e xn cn Zn Zs n n=1 n=1 n  1 X ks cn b s s n sin kcn x Z n=1 n 1 X s kcn s s cos kcn x µg n s kcn s s cos kcn x µg n s e j nd s ej n d , (5.20) respectively. To convert the system of functional equations (5.17)-(5.20) to a system of linear equations, the infinite summations are each truncated at N terms and the following testing operations e are applied. First, (5.17) is multiplied by sin (kxm x) and integrated over 0 < x < a. This results in N X e a e a a+ kx1 m1 + an kxn mn = 1 2 2 n=1 ! N h i X 2 a s kcn b+ + bn 1 mn m = 1, 2, 3 . . . N, n µ2 2 g n=1 (5.21) or am + b + m 1 2 µ2 g ! + bm 1 2 µ2 g ! 209 = a+ m1 1 m = 1, 2, 3 . . . N. (5.22) s Then, (5.18) is multiplied by sin (kcm x) and integrated over 0 < x < a. This produces e + kx1 a a1 e Zm 2 m1  N e s X ks kxn a kcn a cn b+ a an e mn = mn + mn s s Zn 2 Zn n 2 µg n 2 n=1 n=1  N s s X k a kcn a cn b m = 1, 2, 3 . . . N, mn mn s s Zn n 2 µg n 2 n=1 N X (5.23) or  N e X ks kxm k s cn b+ am e + + cn mn mn s Zm Zs n µg n n=1 n  N e s X ks kcn xn b + kx1 mn s s mn = a1 Z e m1 Zn n µg n m n=1 m = 1, 2, 3 . . . N. (5.24) e Next, (5.19) is multiplied by sin (kxm x) and integrated over 0 < x < a. This results in N X n=1 e a c+ kxn mn = n 2 + N X N X n=1 s bn kcn n=1 s b+ kcn n 1 2 µ2 g ! 1 2 µ2 g ! s a j nd mn e 2 a j sd mn e n 2 m = 1, 2, 3 . . . N, (5.25) or c + = b+ m m 1 2 µ2 g ! s e j m d + bm 1 2 µ2 g ! s ej m d m = 1, 2, 3 . . . N. (5.26) s Finally, (5.20) is multiplied by sin (kcm x) and integrated over 0 < x < a. This produces  N e X ks s k s a cn b+ a + kxn a j nd cn e = + cn mn s n 2 mn µg s 2 mn e Zn 2 Z n n=1 n=1 n  N s X ks a kcn a cn b j sd m = 1, 2, 3 . . . N, mn mn e n s s Zn n 2 µg n 2 n=1 N X 210 (5.27) or  N e s X ks s kcn cn b+ + kxm = j nd cm e n mn + µ s mn e Zm Zs g n n=1 n  N s X ks s kcn cn b j nd m = 1, 2, 3 . . . N. s n mn µg s mn e Zn n n=1 (5.28) In (5.21) - (5.28), mn = 8 > 0, < m > 4 m , : ⇡ m2 n2 n even m n odd. (5.29) Note that these integrals are derived in Appendix B.4. These equations can be simplified by s ± s s defining Ln = kcn /(µg n ), C = (1 2 /µ2 ), and Pn = e⌥j n d , which results in g N e X ks kxm cn b+ [ mn e a + s Zm m Zn n n=1 N X ks cn b+ [ mn s Zn n n=1 + Ln mn ] Pn am + Cb+ + Cbm = m1 a+ (5.30) m 1 N s X k cn b [ Ln mn ] mn + Ln mn ] s Zn n n=1 ke = xm m1 a+ e 1(5.31) Zm + Cb+ Pm + Cbm Pm = c+ (5.32) m m N e X ks kxm + cn b [ mn + Ln mn ] Pn = e cm . s Zn n Zm n=1 (5.33) Next, rearranging (5.30) and substituting into (5.31) yields N ⌘ X ks cn b+ [ a+ + mn Ln mn ] m1 1 s Zn n n=1 N e X ks kxm cn b [ + mn + Ln mn ] = e m1 a1 . s Zn n Zm n=1 e kxn ⇣ + e Cbm + Cbm Zm (5.34) The modal amplitude coe cients, b+ and bn , resulting from the substitution can be brought n 211 into the summations by multiplying the coe cients by mn . This results in N X b+ n n=1 ⇢  e N X kxn + bn mn C e Zn n=1 ⇢  ke C xn + mn e Zn s kcn s Zn s kcn s Zn s kcn s Ln mn Zn s kcn s Ln mn Zn = e 2kxm a+ . e Zm m1 1 (5.35) Similarly, substituting (5.32) into (5.33) results in N X ⇢  e s kxn kcn ks + + cn Ln mn Pn mn C e s s Zn Zn Zn n=1 ⇢  e N s s X kxn kcn kcn + bn mn C e + s + s Ln mn Pn = 0. Zn Zn Zn n=1 b+ n (5.36) Now, (5.35) and (5.36) can be represented in a 2N ⇥ 2N matrix equation 2 32 b+ 6 D F 76 4 54 H J b 3 2 3 7 6 K 7 5, 5=4 0 (5.37) where D, F , H, and J are N ⇥ N sub-matrices and K is a vector with N terms. Once the modal coe cients, b+ and bn , are found by solving the matrix equation, the n remaining unknown modal coe cients an and c+ may be determined using (5.30) and (5.32), n respectively. The sample-plane S-parameters are then given by a S11 = 1 a+ 1 + c S21 = 1 . a+ 1 (5.38) (5.39) The modal series is terminated when the magnitude and phase of the S-parameters reach specified tolerances for M contiguous values of N . As was the case in Section 4.3.2, a value 212 of M = 5 is su cient to guarantee convergence to desired accuracy in the method presented here. 5.2.2 Validation of the Mode-Matching Analysis The mode-matching technique is validated by considering the gyromagnetic material G1010 made by Trans-Tech, Inc. A sample of the material with 5 mm thickness is simulated inside the cross-sectional plane of WR-650 L-band waveguide with dimensions a = 6.50 inches by b = 3.25 inches. The physical material parameters have a saturation magnetization of 4⇡Ms = 1000 gauss, H = 25 oersteds, and an applied static magnetic field H0 = 1000 oersteds. In this simulation a value of H = 0 oersteds is considered. The value of H0 was chosen because the resulting Ferromagnetic resonance (FMR) frequency of 2.8 GHz is above the operating band of the L-band waveguide. FMR is also known as gyromagnetic resonance and occurs when the permeability parameters, (5.3) and (5.4), become infinite or when the operational frequency equals the Larmor frequency. An example of FMR is shown in Figure 5.3, where, using these physical parameters,  and µg are determined from (5.3) and (5.4), respectively. FMR just above the measurement band is beneficial because the extracted µg and  have interesting values, i.e. µg > 1 and  6= 0. This is demonstrated in the inset of Figure 5.3, where the permeability parameters are shown over the L-band frequency range. When FMR occurs within the measurement band, the number of modes, N , needed for convergence of the S-parameters becomes exceedingly high and the computational time required for the mode matching analysis becomes prohibitively large. Thus, when FMR occurs outside the measurement band, the mode matching approach is faster, making lengthy optimizations far more feasible. When FMR is below the frequency range of interest, either  ⇡ 0 and µg ⇡ 1, or  < 0 and µg < 0. These conditions are not of interest for the characterization technique discussed in this dissertation. The mode matching technique was simulated with absolute tolerances of 0.1 dB for |S| and 0.01 for \S. These values were chosen so that the accuracy of the computed series 213 Permeability Tensor Entries 150 100 FMR = 2.8 GHz Frequency 50 μg 0 κ -50 4 3 2 1 0 1.2 -100 -150 1 1.5 1.5 1.8 2 2.5 3 3.5 Frequency (GHz) Figure 5.3: Permeability tensor entries determined using (5.3) and (5.4). 214 4 is significantly better than the expected measurement accuracy of the HP 8510C VNA. The accuracy of the S-parameters computed by the mode matching analysis was tested by comparing the simulated data to those obtained using a full-wave FEM solver developed at Michigan State University. As seen in Figure 5.4, the FEM simulated S-parameter match the results of the mode matching analysis very well across the L-band frequency range, thus validating the modal computation of the theoretical S-parameters. 5.2.3 Extraction Process To characterize a gyromagnetic sample with one set of measurements at discrete frequencies requires extracting three complex unknown parameters (µg , , and ✏r ) from two complex measurements (S11 and S21 ). This results in an underdetermined system. Therefore, a nonlinear least squares inversion method [56] is utilized to extract the frequency-independent parameters, Ms and H0 , along with permittivity ✏r = ✏0 r j✏00 which is assumed to be r frequency independent, although it is not. For the types of gyromagnetic materials that are of interest, the permittivity changes very little over the measurement band, and therefore the assumption of a frequency-independent permittivity is su cient for accurate characterization. This method uses the transmission and reflection coe cients measured (or simulated) at a number of frequencies in the desired band, and employs a least squares method to solve the over-determined problem. The squared error between the measured and theoretical S-parameters is defined as v u u Nf  uX 2 2 thy u meas meas =t S11,r (fj ) S11,r (fj ) + S11,i (fj ) S11,i (fj )thy j=1 v u u Nf  uX 2 2 thy thy u meas meas +t S21,r (fj ) S21,r (fj ) + S21,i (fj ) S21,i (fj ) , j=1 215 (5.40) Magnitude (dB) 0 -10 -20 S21 S11 Mode Matching FEM -30 -40 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.7 1.8 Frequency (GHz) Phase (°) 200 100 Mode Matching FEM 0 S11 S21 -100 -200 1.2 1.3 1.4 1.5 1.6 Frequency (GHz) Figure 5.4: S-parameters computed for a gyromagnetic test material. 216 where fj is the j th frequency point, and r and i denote the real and imaginary parts of the S-parameters, respectively. Here, S thy are the theoretical S-parameters calculated using modal analysis, and S meas are the measured (or simulated) values. To test the extraction method, simulated S-parameters are used as a substitute for measured S-parameters. This current technique does not characterize H, since initial imple- mentation assumed lossless permeability. Although characterization of H was originally planned, obtaining large gyromagnetic samples to fill the cross-section of an L-Band or SBand waveguide was not possible. Therefore, the second technique in this chapter discusses use of a reduce-aperture sample holder for characterizing smaller samples and subsequently H is included in the analysis. To validate the current extraction technique, the FEM simulated S-parameters shown in Figure 5.4 are used, and the characterization technique is tested by extracting the material parameters from the simulated data. Figure 5.5 shows the extracted and theoretical material parameters, and it is apparent that the developed extraction routine is capable of accurately characterizing G1010 material. Note that the theoretical material parameters are determined using (5.3) and (5.4) with the physical parameter outlined in Section 5.2.2. 5.2.4 Summary A technique for characterizing gyromagnetic materials has been developed and described in this section. The technique uses the least-squares method to find the material parameters from measured or simulated S-parameters. The technique has been tested using simulated S-parameters obtained using an in-house FEM code that includes a full tensor model of permeability. It was shown that the method is capable of accurately extracting the material parameters of the gyromagnetic material G-1010. This method has limitations since the sample size required for characterizing gyromagnetic samples with the use L-band or S-band waveguides is not feasible due to sample size restrictions. 217 Relative Permittivity 16 14 12 10 8 6 4 2 0 ϵ'r Theoretical Extraction G1010 ϵ''r 1.2 1.3 1.4 1.5 1.6 1.7 1.8 2.5 2.5 μ'g 2 Theoretical Extraction 1.5 2 G1010 1.5 𝜿 1 1 0.5 0.5 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Frequency(GHz) Figure 5.5: Material parameters characterized using FEM generated S-parameters. 218 Relative 𝜿 Relative Permeability Frequency (GHz) 5.3 Material Characterization of Gyromagnetic Material Using a Reduced Aperture Waveguide The focus of this section is the extraction of the permeability tensor and isotropic permittivity of a gyromagnetic material using a reduced aperture sample holder. The cross-sectional dimensions of waveguides become prohibitively large at low frequencies, and it is at these frequencies that the gyromagnetic properties are most pronounced. However, su ciently large samples may not be available. Therefore, the use of a reduced aperture sample holder is proposed which does not require the sample to fill the entire cross section of the guide. Modal analysis is used to determine the reflection and transmission coe cients of the dominant mode. A non-linear least squares method is employed to extract the gyromagnetic material parameters 5.3.1 Theoretical Transmission and Reflection Coe cients Using Mode-Matching Analysis The MUT is assumed to be linear, homogeneous, and biased perpendicular to the broad dimension of the waveguide which results in the permittivity and permeability tensor given in (5.1) and (5.2). The waveguide system, shown in Figure 5.6, consists of empty waveguide extensions connected to a reduced aperture sample holder completely filled by the MUT. A cross-section view of the reduced aperture is shown in Figure 5.7. It is assumed that the lengths of the extensions are such that a single TE10 mode rectangular waveguide mode is incident on the sample holder from the transmitting extension (z < 0), and a single mode is obtained at the end of the receiving extension (z > d). As with every characterization technique discussed in this dissertation, accurate formulation of the theoretical S-parameters is essential for reliable characterization of the MUT. The constitutive material parameters are determined by minimizing the di↵erence between mea- 219 sured and theoretically computed reflection and transmission coe cients. The theoretical Sparameters of the reduced-aperture waveguide system are determined using a mode-matching technique which accommodates the higher order modes excited from the discontinuity with the reduced aperture and the coupling between orthogonal field components. The single TE10 mode incident on the sample holder results in an infinite number of waveguide modes reflected back into the transmitting extension. Additionally, an infinite number of waveguide modes are transmitted into the sample region and incident on the discontinuity at z = d. Thus a spectrum of modes is also reflected back into the sample region and transmitted into the waveguide extension. Since the electric field of the TE10 mode incident on the sample holder, as shown in Figure 5.8, is even about x = 0, and because the aperture is even about x = 0 and the gyromagnetic material is biased along the y-axis, only modes with electric field even about x = 0 will be excited. Thus, only TEn0 modes are needed to describe the fields in each of the waveguide regions. 5.3.1.1 Field Structure in a Waveguide Filled with Gyromagnetic Material Assume the gyromagnetic material is magnetized along the y-axis, such that the material parameters are described in (5.1) and (5.2). For TEn0 modes in a gyromagnetic filled rectangular waveguide region the wave equation from (2.227) reduces to (5.5), where the cuto↵ wavenumber is expressed in (5.6). As was shown previously in Section 2.3.3.1, the wave equation is solved using a separation of variables and enforcing the boundary conditions on the tangential electric field at the perfectly conducting waveguide walls. Since the reducedaperture sample region is centered about x = 0 as seen Figure 5.6, the sinusoidal function in the transverse electric and magnetic field equations from (2.251) and (2.252) are o↵set by 220 ϵ0,µ0 ϵ0,µ0 ϵ,µ y b x z w a d Figure 5.6: Reduced-aperture waveguide sample holder with waveguide extensions attached. y b ϵ,µ 0 a − 2 x − w w a 2 2 2 Figure 5.7: Cross-sectional view of reduced-aperture waveguide. y ϵ µ 221 ϵµ 0 a − 2 − w w a 2 2 2 y ϵ,µ ϵ0,µ0 TE10 TE n0 TE n0 ϵ0,µ0 TE n0 TE n0 z d 0 Figure 5.8: Side view of reduced aperture waveguide. w/2. This results in the following transverse field equations: s Ey (x, z) = s Cn kcn s Cn kcn s Hx (x, z) = ⌥ s Zn ! h ⇣ 2 w ⌘i ±j s z s n 1 sin kcn x e 2 µ2 g ⇢ h ⇣ ⌘i h ⇣ s s x w ⌥ kcn cos k s x sin kcn cn s 2 µg n (5.41) w ⌘i ±j s z n .(5.42) e 2 s s s Here kcn = n⇡/w is the cuto↵ wavenumber for the nth TEn0 mode, Zn = !µ0 µg / n , and s n is determined from (5.6). The transverse field equations for the empty waveguide extensions may be determined using (5.9) and (5.10), as long as, once again, the sinusoidal functions are o↵set. For the empty regions, the sinusoidal functions are o↵set by a/2. This yields the following transverse field equations: h ⇣ e e Dn kxn sin kxn x h ⇣ ke e e Hx (x, z) = ⌥Dn xn sin kxn x e Zn e Ey (x, z) = a ⌘i ±j e 2 a ⌘i ±j e 2 e nz (5.43) e nz . (5.44) s e e Here kcn = n⇡/w is the cuto↵ wavenumber, and Zn = !µ0 / n is the modal wave impedance. e Note that n is determined using (2.140). 222 5.3.1.2 Solution for S-Parameters Using Modal Expansions Mode-matching analysis is utilized to first expand the transverse field in the sample holder and in the waveguide extensions in an infinite sum of modal fields. Then with the application of boundary conditions, the modal amplitudes can be determined. First, in the empty waveguide extension region, z < 0, the transverse fields are h ⇣ e e A+ kx1 sin kx1 x 1 1 h ⇣ X e sin k e x An kxn xn n=1 e h ⇣ A1 kx1 e Hx (x, z) = sin kx1 x e Z1 1 h ⇣ X A ke n xn sin k e x xn e Zn n=1 Ey (x, z) = a ⌘i j e 2 a ⌘i j e 2 e 1z e nz (5.45) a ⌘i j e z 1 e 2 a ⌘i j e z e n . 2 (5.46) Here A+ is the amplitude of the incident TE10 wave and is assumed known during analysis. 1 Next, in the sample holder region, 0 < z < d, the transverse fields are 1 X ! w ⌘i j s z n e 2 n=1 ! 1 h ⇣ X 2 w ⌘i j s s Bn kcn 1 sin kcn x e 2 µ2 g n=1 1 + s ⇢ h ⇣ ⌘i h ⇣ ⌘i s X B k n cn sin k s x w + kcn cos k s x w Hx (x, z) = e j cn cn s s Zn 2 µg n 2 n=1 1 ⇣ ⌘i k s h ⇣ ⌘i X B ks ⇢ h n cn sin k s x w cn cos k s x w ej cn cn s s Zn 2 µg n 2 n=1 Ey (x, z) = + s Bn kcn 1 2 µ2 g h ⇣ s sin kcn x 223 s n z (5.47) s nz s n z . (5.48) Finally, in the waveguide extension z > d the fields are given by Ey (x, z) = 1 X h ⇣ + e e Cn kxn sin kxn x n=1 1 + e h ⇣ X C k n xn sin k e x Hx (x, z) = xn e Zn n=1 a ⌘i j e (z d) n e 2 a ⌘i j e (z d) n e . 2 (5.49) (5.50) By applying the boundary conditions on Ey and Hx at the interfaces between the two + waveguide extensions and the sample holder, the modal amplitude coe cients An , Bn , + Bn , and Cn may be determined. The boundary condition on the tangential electric field at z = 0 requires 8 > > > > > > > > < > > > > > > > > : 1 h ⇣ ⌘i X h ⇣ ⌘i + k e sin k e x a + e sin k e x a = A1 x1 An kxn xn x1 2 2 n=1 ! 1 h ⇣ ⌘i X 2 +ks s x w Bn cn 1 sin kcn 2 µ2 g n=1 ! h ⇣ ⌘i 2 P s s + 1 Bn kcn 1 2 sin kcn x w , 0 < |x| < w n=1 2 2 µg w < |x| < a , 0, 2 2 (5.51) and the boundary condition on the tangential magnetic field requires 1 e h ⇣ h ⇣ e A1 kx1 a ⌘i X An kxn e e sin kx1 x sin kxn x e e Z1 2 Zn n=1 1 ⇣ ⌘i h ⇣ s X B +ks ⇢ h n cn sin k s x w + kcn cos k s x cn cn s s Zn 2 µg n n=1 1 ⇣ ⌘i k s h ⇣ X B ks ⇢ h n cn sin k s x w cn cos k s x cn cn s s Zn 2 µg n n=1 224 a ⌘i = 2 w ⌘i 2 w ⌘i . 2 (5.52) Additionally, the boundary condition on the tangential electric field at z = d requires 1 X h ⇣ + e e Cn kxn sin kxn x n=1 8 ! 1 h ⇣ > X 2 w ⌘i j s d > + s s > n > Bn kcn 1 sin kcn x e > > 2 > µ2 g > n=1 < ! h ⇣ ⌘i 2 P1 s s s > + n=1 Bn kcn 1 2 sin kcn x w ej n d , > 2 > µg > > > > > : 0, a ⌘i = 2 0 < |x| < w 2 (5.53) w < |x| < a , 2 2 while the boundary condition on the tangential magnetic field requires 1 ⇣ X B +ks ⇢ h n cn sin k s x cn s Zn n=1 1 ⇣ X B ks ⇢ h n cn sin k s x cn s Zn n=1 1 h ⇣ ⌘i X C +ke n xn sin k e x a = xn e Zn 2 n=1 h ⇣ ⌘i s s w ⌘i kcn s x w + cos kcn e j nd s 2 µg n 2 w ⌘i 2 h ⇣ s kcn s cos kcn x s µg n w ⌘i j s d e n . 2 (5.54) The system of functional equations (5.51) - (5.54) are converted to a system of linear equations by first truncating the infinite summations to N terms and then applying the following testing operations. The equations resulting from applying the tangential electric field e boundary conditions, (5.51) and (5.54), are multiplied by sin [kxm (x over a/2  x  a/2. Then, (5.52) and (5.54), which result from applying the tangential s magnetic field boundary conditions, are multiplied by sin [kcm (x over a/2)] and integrated w/2)] and integrated w/2  x  w/2. Here 1  m  N . This yields the 4N ⇥ 4N matrix equation 2 G 6 6 6 K 6 6 6 6 0 4 0 J L+ JP L P JP J L+ P L 0 32 A 76 76 0 7 6 B+ 76 76 76 G 76 b 54 K C+ 225 3 2 g 3 7 6 7 7 6 7 7 6 7 7 +6 k 7, 7 = A1 6 7 7 6 7 7 6 0 7 5 4 5 0 (5.55) where G, J, K, and L± are N ⇥ N submatrices, and g and k are vectors of length N . These entries specified in Appendix B.4. Here P is a diagonal matrix with entries Pmn = s j n d , with mn e mn the Kronecker delta. Note that bn is introduced to avoid overflow during computation. The modal coe cients are found by solving the matrix equation in (5.55). The Sparameters of the system are given by A1 A+ 1 + C S21 = 1 . A+ 1 S11 = (5.56) (5.57) Similar to the first technique discussed in this Chapter, the modal series is terminated when the magnitude and phase of the S-parameters reach specified tolerances at M contiguous values of N . Here M = 5 is su cient to guarantee convergence to desired accuracy. 5.3.2 Validation of Theoretical Analysis The mode matching technique was used to compute the S-parameters for a sample of G1010 placed into a reduced aperture sample holder in an L-band waveguide system. The dimensions of the L-band system are a = 6.50 inches and b = 3.25 inches. The sample thickness was chosen to be 0.50 inches, while the width of the aperture was set to 1.75 inches. Figure 5.9 and Figure 5.10 show S11 and S21 computed by using the modal series with the applied biasing magnetic field oriented along the y-axis of the waveguide. This biasing field value was set to 900 oersteds and the line width, H, to zero. The resulting FMR occurs at 2.52 GHz. Absolute tolerances of 0.1 dB for |S| and 0.01 for \S were chosen so that the accuracy of the computed series is similar to the other techniques discussed in this dissertation. For the material and H0 considered here, the S-parameters typically converge to the specified tolerances within N = 136 terms. 226 Also shown in Figure 5.9 and Figure 5.10 are the values of the S-parameters computed using the full-wave FEM solver developed at Michigan State University. The FEM simulation used approximately 100000 unknowns with a specified mesh size. Excellent agreement was obtained between the modal analysis and FEM for certain portions of the frequency band. However, near resonances there is a discrepancy between the two techniques. Initial thoughts were to implement higher-order basis functions in the FEM code. However, by the completion of this dissertation, the higher-order FEM code had not been tested for this problem. Therefore, to give confidence that this mode matching technique is providing accurate results, the mesh size in the FEM simulation was increased; which resulted in the FEM simulation having 400000 unknowns. There was a dramatic increase in FEM computational time for this many unknowns. Figure 5.11 and Figure 5.12 show the comparison of the mode matching technique and the FEM simulation using 400, 000 unknowns. It is seen that FEM results are trending closer to the mode matching S-parameters. This gives confidence that the theoretical S-parameters are computed properly. 5.3.3 Extraction Process The same non-linear least squares method was used to determine the frequency independent physical parameters H0 , 4⇡Ms , ✏0 , ✏00 , and r r H as was described in Section 5.2.3. The squared error between the measure and theoretical S-parameters is defined in (5.40). The extraction method was tested by using simulated S-parameters in lieu of measured Sparameters. The FEM generated S-parameters shown in Figure 5.11 and Figure 5.12 are used and the extraction process carried out. Figure 5.13 and Figure 5.14 show the extracted parameters of the G1010 material filling the cross-section of the reduced aperture guide. It is apparent that the significant amount of error present in the FEM simulations propagates to the characterized material parameters, causing inaccurate results. Note that the theoretical material parameters are determined using (5.3) and (5.4) with the physical parameter outline in Section 5.2.2. 227 0 S11 Magnitude (dB) -2 -4 -6 Modal Analysis In-House FEM -8 S21 -10 -12 -14 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Frequency (GHz) Figure 5.9: Magnitude of S-parameters computed for a gyromagnetic test material. FEM simulation uses 100000 unknowns. 228 200 S21 S11 Phase (°) 100 0 Modal Analysis In-House FEM -100 -200 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Frequency (GHz) Figure 5.10: Phase of S-parameters computed for a gyromagnetic test material. FEM simulation uses 100000 unknowns. 229 0 S11 Magnitude (dB) -2 -4 Modal Analysis In-House FEM -6 -8 S21 -10 -12 1.2 1.3 1.4 1.5 1.6 1.7 Frequency (GHz) Figure 5.11: Magnitude of S-parameters computed for a gyromagnetic test material. FEM simulation uses 400000 unknowns. 230 200 S21 S11 Phase (°) 100 0 -100 -200 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Frequency (GHz) Figure 5.12: Phase of S-parameters computed for a gyromagnetic test material. FEM simulation uses 400000 unknowns. 231 16 ϵ'r Relative Permittivity 14 12 10 Extraction Theoretical 8 6 4 2 ϵ''r 0 -2 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Frequency (GHz) Figure 5.13: parameters. Relative permittivity parameters characterized using FEM generated S- 232 3.5 1 μ'r 0.8 2.5 2 0.6 Extraction Theoretical 1.5 𝜿'r 1 0.5 𝜿''r 0 -0.5 1.1 1.2 1.3 1.4 1.5 0.4 μ''r 1.6 Relavtive 𝜿 Relative Permeability 3 0.2 1.7 0 Frequency (GHz) Figure 5.14: Relative permeability parameters characterized using FEM generated Sparameters. 233 5.3.4 Error and Sensitivity Analysis A sensitivity analysis can be used to determine the e↵ects of aperture size on error propagation from S-parameter uncertainties. To determine the sensitivity, a Monte Carlo technique was used to study the e↵ects of propagation of random error inherent to the VNA. Depending on the material under test, the width of the aperture (w in Figure 5.6) can have a dramatic e↵ect on the sensitivity to uncertainties of the measured S-parameters. Five di↵erent aperture widths were analyzed: w = 12.02 mm, 24.04 mm, 36.06 mm, 48.08 mm, and 60.10 mm. The uncertainties of the measured S-parameters used in the Monte Carlo analysis are the same as were employed in the similar analysis done for the iris technique in Section 3.3.4. Once again, the VNA measurement uncertainty is assumed to be independent of amplitude and frequency. Statistical variance of S11 is specified linearly in amplitude and phase with values of A = 0.004 and = 0.8 . Variance of S21 is specified logarithmically in 11 11 amplitude and linearly in phase with values A = 0.04 dB and = 2.0 . 21 21 The G1010 test sample described in Section 5.2.2 was used in the Monte Carlo analysis of the propagation of VNA uncertainty. The geometry of the sample holder is a = 72.136 mm, b = 34.036 mm, and d = 5 mm. The biasing magnetic field is H0 = 2000 oersteds, while the line width is This H = 25 oersteds. This H0 results in a FMR occurring at 5.60 GHz. H is the approximate value specified by Trans Tech, Inc. The forward problem was solved at 51 frequency points over the portion of S-band from 2.6 GHz to 3.95 GHz. White Gaussian noise was then added to the S-parameters, and the noisy data was used to extract the material parameters. One hundred trials were used in the Monte Carlo analysis, and the average values of the material parameters and the standard deviations were calculated for each aperture width examined. Figures 5.15 - 5.19 show the e↵ects on the permittivity characterization with di↵erent apertures widths. Figures 5.20 - 5.24 similarly show the e↵ects on the permeability characterization for the five di↵erent aperture widths. Note that 234 the red lines in these figures are the theoretical values of G1010 used to generate the Sparameters employed in the Monte Carlo analysis. It is seen that the wider the aperture is, the less sensitive the extracted material parameters are to the propagation of measured S-parameter uncertainty. Additionally, the results of the sensitivity analysis have values typical of those encountered with other material extraction methods, such as [51] and [40]. This is encouraging for characterization of measured data, which by the completion of this dissertation has not been carried out and is left for future work. 5.3.5 Summary A reduced aperture waveguide method is introduced for measuring the permittivity and permeability of gyromagnetic materials. The technique is capable of characterizing sample sizes less than the cross-section of a rectangular waveguide. The performance of the technique is established using a sensitivity analysis based on network analyzer uncertainties. It is found that the technique performs well using simulated data, however, further validation is needed using laboratory measurements. 235 Relative Permittivity 15.75 15.5 15.25 15 14.75 14.5 14.25 14 13.75 ϵ'r 2.6 2.8 3 3.2 3.4 3.6 3.8 4 3.8 4 Relative Permittivity Frequency (GHz) 0.40 ϵ''r 0.20 0.00 -0.20 -0.40 2.6 2.8 3 3.2 3.4 3.6 Frequency (GHz) Figure 5.15: Relative permittivity extracted from 100 random trials of simulated Sparameters for a G1010 sample filling a reduced aperture of width 12.02 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permittivity values used to generate the S-parameters employed in the Monte Carlo simulation. 236 Relative Permittivity 15.75 15.5 15.25 15 14.75 14.5 14.25 14 13.75 ϵ'r 2.6 2.8 3 3.2 3.4 3.6 3.8 4 3.8 4 Relative Permittivity Frequency (GHz) 0.40 ϵ''r 0.20 0.00 -0.20 -0.40 2.6 2.8 3 3.2 3.4 3.6 Frequency (GHz) Figure 5.16: RRelative permittivity extracted from 100 random trials of simulated Sparameters for a G1010 sample filling a reduced aperture of width 24.04 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permittivity values used to generate the S-parameters employed in the Monte Carlo simulation. 237 Relative Permittivity 15.75 15.5 15.25 15 14.75 14.5 14.25 14 13.75 ϵ'r 2.6 2.8 3 3.2 3.4 3.6 3.8 4 3.8 4 Relative Permittivity Frequency (GHz) 0.40 ϵ''r 0.20 0.00 -0.20 -0.40 2.6 2.8 3 3.2 3.4 3.6 Frequency (GHz) Figure 5.17: Relative permittivity extracted from 100 random trials of simulated Sparameters for a G1010 sample filling a reduced aperture of width 36.06 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permittivity values used to generate the S-parameters employed in the Monte Carlo simulation. 238 Relative Permittivity 15.75 15.5 15.25 15 14.75 14.5 14.25 14 13.75 ϵ'r 2.6 2.8 3 3.2 3.4 3.6 3.8 4 3.8 4 Relative Permittivity Frequency (GHz) 0.40 ϵ''r 0.20 0.00 -0.20 -0.40 2.6 2.8 3 3.2 3.4 3.6 Frequency (GHz) Figure 5.18: Relative permittivity extracted from 100 random trials of simulated Sparameters for a G1010 sample filling a reduced aperture of width 48.08 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permittivity values used to generate the S-parameters employed in the Monte Carlo simulation. 239 Relative Permittivity 15.75 15.5 15.25 15 14.75 14.5 14.25 14 13.75 ϵ'r 2.6 2.8 3 3.2 3.4 3.6 3.8 4 3.8 4 Relative Permittivity Frequency (GHz) 0.40 0.20 ϵ''r 0.00 -0.20 -0.40 2.6 2.8 3 3.2 3.4 3.6 Frequency (GHz) Figure 5.19: Relative permittivity extracted from 100 random trials of simulated Sparameters for a G1010 sample filling a reduced aperture of width 60.10 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permittivity values used to generate the S-parameters employed in the Monte Carlo simulation. 240 2.5 Relative Material Parameter 2.25 μ'g 2 1.75 1.5 1.25 1 0.75 𝜿' 0.5 μ''g 𝜿'' 0.25 0 -0.25 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Frequency (GHz) Figure 5.20: Relative permeability values extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 12.02 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permeability values determined using (5.3) and (5.4). 241 2.5 Relative Material Parameter 2.25 2 μ'g 1.75 1.5 1.25 1 0.75 𝜿' 0.5 0.25 μ''g 𝜿'' 0 -0.25 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Frequency (GHz) Figure 5.21: Relative permeability values extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 24.04 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permeability values determined using (5.3) and (5.4). 242 2.5 Relative Material Parameter 2.25 2 μ'g 1.75 1.5 1.25 1 0.75 𝜿' 0.5 0.25 μ''g 𝜿'' 0 -0.25 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Frequency (GHz) Figure 5.22: Relative permeability values extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 36.06 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permeability values determined using (5.3) and (5.4). 243 2.5 Relative Material Parameter 2.25 2 μ'g 1.75 1.5 1.25 1 0.75 𝜿' 0.5 0.25 μ''g 𝜿'' 0 -0.25 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Frequency (GHz) Figure 5.23: Relative permeability values extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 48.08 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permeability values determined using (5.3) and (5.4). 244 2.5 Relative Material Parameter 2.25 2 μ'g 1.75 1.5 1.25 1 0.75 𝜿' 0.5 0.25 μ''g 𝜿'' 0 -0.25 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Frequency (GHz) Figure 5.24: Relative permeability values extracted from 100 random trials of simulated S-parameters for a G1010 sample filling a reduced aperture of width 60.10 mm. Center black line is the average of the trials. Upper and lower black lines show the 95% confidence interval. Red lines are the theoretical permeability values determined using (5.3) and (5.4). 245 CHAPTER 6 VERIFICATION STANDARDS FOR MATERIAL CHARACTERIZATION 6.1 Introduction The use of engineered materials in the design of radio frequency (RF) systems requires an accurate knowledge of material constitutive parameters. Some recently synthesized materials include graphene nanoribbons or metallic inclusions for use in miniaturization of electronic components [4]-[5], ferrite loaded polymers to increase EMI shielding [6], cellular materials such as honey comb structures to decrease radar cross-section [7]-[8], and anisotropic materials used to enhance antenna operation [9]-[10]. Since the properties of these materials are often hard to accurately predict (due to modeling uncertainties and variability in the manufacturing process), they are usually measured in a laboratory. As seen in this dissertation, rectangular waveguide applicators are commonly used to extract the electromagnetic properties of materials. Benefits over alternative methods, such as free-space systems, include the availability of analytical expressions to describe sample interrogation, high signal strength, and the simplicity of manufacturing rectangular samples [20]-[22]. The S-parameters measured with a material sample placed in the cross-sectional 246 plane of a waveguide may be used to determine the complex values of the permittivity, ✏ = ✏0 (✏0 + j✏00 ), and the permeability, µ = µ0 (µ0 + jµ00 ), from closed-form expressions r r r r described in Section 3.2. The accuracy of a material characterization system is often established by measuring a standard material with known properties. A desirable standard material is easy to machine, is stable thermally and chemically, is readily available, and has values of constitutive parameters numerically close to those of samples of interest with only minor variations within the measurement band. If the permittivity of materials is of primary interest, a dielectric sample such as Rexolite R may be used to provide highly-predictable parameters in the microwave spectrum with a value characteristic of many plastics [35] and [57]. The recent introduction of engineered materials with magnetic as well as dielectric properties has made it more difficult to find standard test materials whose constitutive parameters are known with great accuracy and have appropriate values within the microwave spectrum. Fortunately, it is possible to create a surrogate material sample that can act as the standard test material in the sense that its use in a material measurement system will produce extracted constitutive parameters with predictable, highly-accurate, and appropriate values. The surrogate need not resemble an actual material, and in fact can be inhomogeneous and non-magnetic. The only requirement is that when inserted in place of a test material, the surrogate provides proper S-parameters that may be used in the extraction process. Although some previous work has been done to create waveguide standards using circuits, obstacles, or components [58], these fabricated structures are only meant to validate the reflection and transmission coe cients, not the extracted material parameters. It is crucial that a material standard produce highly accurate, stable, and repeatable values of extracted permittivity and permeability across the measurement band. It must be reproducible for the majority of users, and thus should be of simple design and made from readily available materials. The values of the extracted constitutive parameters should not be overly sensitive to the geometrical dimensions of the standard, so that errors in fabrication 247 or changes in the operating environment (temperature, humidity, etc.) do not cause unpredictable or unacceptable results. Additionally, the values of permittivity and permeability must be predictable to high accuracy using standard, reproducible analytic techniques. For these reasons, the design adopted for the surrogate described here uses two simple metallic waveguide windows, or apertures, separated by a spacer. Given just four geometrical parameters, mode-matching techniques can be used to accurately compute the S-parameters of the surrogate material, and from these the extracted constitutive parameters may be determined. A description of the design process used to obtain an optimized geometry and a typical design for S-band using standard WR-284 dimensions are given below. Details of the theoretical analysis are also given, so that any user may design an appropriate surrogate. The measured results for an example verification standard constructed from materials on hand are shown and discussed. 6.2 Material Characterization Procedure A verification standard provides confidence in the operation of an experimental system by producing known results under appropriate operating conditions. In the case of material characterization, the verification standard is placed into the system and known values of µ and ✏ are extracted using a specific algorithm. The standard described in this dissertation assumes the use of the classic “Nicolson-Ross-Weir” (NRW) extraction algorithm, outlined in Section 3.2, that employs the measured reflection and transmission coe cients for a waveguide section completely filled by the material [23]-[24]. Other algorithms are available that use, for instance, only reflection measurements [36]; these would require a di↵erent standard. The attraction of the NRW characterization method results from the availability of closedform expressions for µ and ✏. This contrasts with methods requiring an iterative solver such as Newton’s method [30]-[31] or a least squares approach [32]-[33]. The convenience of the NRW method, and its insensitivity to propagation of measurement uncertainties, commonly 248 makes it a first choice for material characterization. Since NRW extraction can be used with rectangular waveguides [25], coaxial applicators [26]-[27], free-space methods [28], and stripline measurements [29], the concept of the waveguide surrogate described here has wide applicability, although the details of the structural design vary from system to system. Figure 6.1 shows the experimental configuration used in the NRW method. A sample with unknown properties is placed into a sample holder occupying the region 0  z  d in a rectangular waveguide system. Waveguide extensions are usually employed to guarantee only the dominant mode is present at the measurement ports. The S-parameters S11 and S21 are measured using a vector network analyzer (VNA) attached at these ports, and the Sparameters are then mathematically transformed to obtain the S-parameters at the sample planes. These sample-plane S-parameters are used to determine the sample propagation constant and the interfacial reflection coe cient , which may in turn be used to find ✏ and µ. The waveguide standard is placed into the measurement system in a manner identical to an actual unknown material; see Figure 6.1. The length of the surrogate sample is z3 = d and thus the S-parameter planes are taken at z = 0 and z = d as in the case of an actual sample. The NRW method assumes that the material sample is isotropic and homogeneous, and thus the measured S-parameters obey S11 = S22 and S12 = S21 . Although it is not necessary that the waveguide standard be homogeneous, it is helpful if the S-parameters of the surrogate obey the same relationships as do those of an actual material. Thus, the waveguide standard should be symmetric in the longitudinal (z) direction. This allows the standard to be interrogated from either direction with identical results, and if the standard is constructed from lossless (or perfectly conducting) materials, it also allows the permittivity and permeability of the surrogate extracted using the NRW method to be purely real (corresponding to a lossless material). The condition under which this occurs is identified in Appendix C. Interestingly, if the waveguide standard is made from lossless materials but is not symmetric along z, the NRW method may return complex values for µ and ✏ with 249 µ0 , ε 0 Δ µ, ε µ0 , ε 0 µ, ε Δ𝑠 Δ𝑟 µ0 , ε 0 µ0 , ε 0  y2 y1 y b PEC Figure 6.1: Waveguide material measurement system showing presence of material sample (top), and waveguide verification standard surrogate material (bottom). Adopted surrogate ` r ` r has ` = r , y1 = y1 , and y2 = y2 . y2 y1 0 PEC µ0 , ε 0 PEC a x 250 positive imaginary parts, indicating gain, which is not a property of a useful surrogate. To demonstrate this, consider two surrogate materials, each constructed from alternating layers of perfect (lossless) dielectrics, as shown in Figure 6.2. The sample plane S-parameters of each system may be easily computed using cascaded matrices [59], and the permittivity and permeability of the surrogate extracted using the NRW method; see Figure 6.2. When the properties of the symmetric material (Sample A) are extracted, the permittivity is found to be purely real (characteristic of a lossless material). However, when the properties of the asymmetric material (Sample B) are extracted, the imaginary part of the permittivity is found to be nonzero and positive (characteristic of an active material). The extracted permeability of the asymmetric material is found to have a small negative imaginary part and a real part less than unity. This contrasts the extraction of the symmetric sample where the imaginary permeability is found to be zero. Note that in both cases, the real parts of the extracted permittivity and permeability are frequency dependent, even though the dielectric constants of the material layers are independent of frequency. This frequency dependence is expected of any inhomogeneous surrogate, and leads to the drawback that the material parameters do not obey the Kronig-Kramers relations [60]. However, an inhomogeneous surrogate is still valuable as a verification standard since its material parameters can be tailored to a desired range and, if lossless, provide a quick check of the credibility of the measurements. 6.3 Waveguide Verification Standard Design It is important to develop a waveguide standard that is easily fabricated using materials readily available to most users. For this reason the surrogate was chosen to be constructed from purely metallic parts, and in the following analysis is assumed to be perfectly conducting. Assemblies of structures such as loops, posts, and both circular and rectangular apertures were considered, with the goal to produce a surrogate material with both a permittivity and 251 6 Material Parameters 5 ϵ'r 4 3 Extraction of Sample A Extraction of Sample B 2 μ'r 1 ϵ''r 0 μ''r 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 6.2: E↵ective material parameters extracted from symmetric and asymmetric layered dielectric surrogate materials. 252 a permeability as near to six as possible across S-band. (Note that materials with large but equal values of permittivity and permeability are useful in the construction of microwave antennas with enhanced radiation properties; values near 6 are a good goal.) As discussed in the previous section, by making the structure longitudinally symmetric, the extracted material parameters may be purely real and correspond to a lossless material. However, as also discussed, the parameters are frequency dependent, and thus a genetic algorithm (GA) was employed to find a structure with slowly varying parameters as near to a target value of six as possible. It is also important that the extracted material parameters be relatively insensitive to small changes in the geometry of the structure, so that tight manufacturing tolerances are not required. To find structures that satisfy the design requirements, a combination of a GA and a Monte Carlo error analysis routine was employed. For each candidate structure, the commercial solver HFSS was used to determine the S-parameters, and the NRW method was used to find ✏r and µr . A Matlab to HFSS interface was implemented, where all exported data from HFSS was stored in Matlab and analyzed during post processing and GA execution. Figure 6.3 shows a flow chart of the Matlab to HFSS interface. An initial design of the waveguide surrogate (loops, posts, apertures, etc.) is programmed into Matlab, which in turn generates application programming interface (API) functions that are used to talk between Matlab and HFSS. The Matlab program then creates and executes an HFSS script. Using this script, HFSS draws and analyzes the structure and then exports the specified solution data back to Matlab. Then Matlab generates a new set of API functions using the next randomly determined geometrical modification (such as post radius, aperture size, sample thickness) to the surrogate design. Using the stored results from HFSS it is possible to take a small population of possible designs, analyze the permittivity and permeability, and determine if with fine tuning this type of surrogate is viable for a waveguide standard. Figure 6.4 shows a surrogate design of brass posts that was analyzed for a possible waveguide standard candidate. In this simulation, the number of posts was fixed, but the GA was used 253 to determine the individual radius of each post, whether the posts came from the bottom or top of the guide, and the length of each post. It was found that this type of surrogate would not meet the objectives for a waveguide standard. Typical results from the GA analysis of the brass-post surrogate design are shown in Figure 6.5. The surrogate design has highly varying ✏0 and µ0 , and thus is not a usable waveguide standard design. Additionally, the ✏r r r and µr become complex of certain portions of the frequency band. Since the surrogate design is symmetric in the longitudinal dimension and made from lossless materials, the portions of the frequency band in which the material parameters are complex result from Case 2 in Appendix C, where X 2  1 with X = S11r /|S11 |2 . This can be seen in Figure 6.6, where the inset shows an expanded vertical region of a plot of X 2 . Comparing Figure 6.5 with Figure 6.6, it can be seen that the frequencies where ✏r and µr are complex correspond with frequencies where X 2  1. Thus, avoiding regions where Case 2 is in e↵ect results in an acceptable surrogate design for use as a verification standard. Once a possible surrogate design was determined, the geometry of each candidate structure was perturbed randomly multiple times and the error that propagated to the extracted material properties was computed. After many lengthy searches, several structures were found with acceptable properties, some exhibiting highly complex geometries. One structure in particular was found to be quite promising due to its simple geometry, and was adopted as the waveguide standard analyzed and presented in this dissertation. It consists of two identical rectangular apertures of thickness `= r separated by a spacer of thickness s, as shown in Figure 6.1. The simple geometry of the adopted waveguide standard has several important advantages over more complex structures. First, it may be easily machined from simple metal sheets. Second, an easily implemented mode-matching technique may be used to analyze the structure quickly and with great accuracy, and thus commercial solvers are not required to determine the material properties of the surrogate. By controlling the accuracy of the analysis, the errors propagated to the extracted material parameters from the uncertain- 254 MATLAB HFSS Start Create Initial Parameters Generate API Functions Export Output Optimization (GA) Analyze Structure Generate HFSS Script Draw Structure Execute Script Execute Script Figure 6.3: Matlab to HFSS flow chart. Wednesday, August 8, 12 255 S-Band Waveguide Brass Posts Figure 6.4: Example of brass-post surrogate design. 256 Material Parameters Material Parameters 40 ε'r 20 0 -20 ε''r -40 -60 2.6 2.8 3 3.2 3.4 3.6 3.8 3.6 3.8 Frequency (GHz) 120 100 80 60 40 20 0 -20 μ''r μ'r 2.6 2.8 3 3.2 3.4 Frequency (GHz) Figure 6.5: Characterization results from a GA analysis of a brass-post surrogate design. 257 20000 1 0.8 0.6 15000 0.4 X2 0.2 0 10000 2.6 2.8 3 3.2 3.4 3.6 3.8 4 5000 0 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) S11r for optimized brass-post surrogate. Extracted material parameters |S11 |2 become complex over regions where X 2  1. Figure 6.6: X = 258 ties in the dimensions of the fabricated parts and from the uncertainties of the measured S-parameters may be appropriately calculated. Third, if the standard is manufactured using materials of thicknesses di↵erent than those described in the optimized design, or if the manufactured apertures are somewhat o↵set from those of the optimized design, mode matching may be used to determine the theoretical properties of the manufactured standard, and the standard may still be used to verify the accuracy of a material measurement system. The adopted design was further tuned using an additional GA. This GA varied the ` r ` r vertical window positions (positions y1 = y1 and y2 = y2 in Figure 6.1), horizontal window positions, and the thickness of the apertures and waveguide spacer. The thicknesses of each aperture and spacer were limited to the values of United States standard brass stock (multiples of 1/16 inch, or 1.5875 mm), to simplify the fabrication process. To produce a symmetric verification standard, the apertures were assigned identical openings. The final optimized design consists of apertures with widths identical to the waveguide width and ` r ` r with vertical positions y1 = y1 = 5.064mm, y2 = y2 = 23.86mm. The thicknesses of the optimized design are the standard stock thicknesses ` = r = 1/8 inch (3.175mm), and s = 1/2 inch (12.7mm). The material parameters extracted from the S-parameters of the optimized design are shown in Figure 6.7. As expected, the imaginary part of the extracted parameters are zero (to computational accuracy) and the real parts are dependent on frequency. Note that the optimizer has produced a trade o↵ between frequency variation, and values near to six. In contrast with the brass-post surrogate design, Figure 6.8 shows that X 2 > 1 for the entire frequency band and thus the material parameters are real. Precise values for the material parameters produced by the optimized waveguide standard may be found by applying the mode-matching technique (see Section 6.4 for details of the mode-matching theory.) A su cient number of modes was used (N = 500) to guarantee a minimum of five digits of precision in the material parameters extracted using the NRW method. The resulting values are listed in Table 6.1. Note that only the real parts of ✏r and µr are shown, since the imaginary parts are less than 10 5 . These values can be used to 259 8 Material Parameters 7.5 μ'r 7 6.5 ϵ'r 6 5.5 5 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Frequency (GHz) Figure 6.7: Real parts of material parameters extracted from HFSS simulation of optimized verification standard. 260 150 X2 100 50 0 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) S11r of optimized verification standard. Extracted material parameters |S11 |2 are real over regions where X 2 > 1. Figure 6.8: X = 261 verify the operational accuracy of waveguide material parameters measurement systems in Sband. Similar standards may be devised for other waveguide bands using the mode-matching method. 6.4 Computation of S-Parameters of adopted Waveguide Standard using Mode Matching Useful implementation of the waveguide standard requires that the user has access to an accurate method for computing the S-parameters. For this reason, a detailed description of the mode-matching method applied to the adopted standard is presented here. Consider the geometry of the adopted waveguide standard shown in Figure 6.1. Perfectly conducting apertures are positioned at z = 0 and z = z2 with thickness ` and r, respectively. Although the adopted standard is longitudinally symmetric, the thicknesses of the apertures are allowed to be di↵erent so that the e↵ects of manufacturing errors may be evaluated. The width of the apertures are the same as the width of the empty wave` ` guide, a, while the vertical openings are describe by the height positions y1 and y2 for the r r left aperture, and y1 and y2 for the right aperture. The waveguide extensions are assumed to be of su cient length such that, even though a full spectrum of higher order modes is produced at the apertures, only a TE10 mode is received at the measurement ports. The waveguide used is standard S-band WR-284 with dimensions a = 2.84 inch by b = 1.34 inch (72.136 ⇥ 34.036 mm) and a designated operational band of 2.6 to 3.95 GHz. 6.4.1 Mode Matching Analysis for the Waveguide Standard Assume a TE10 mode is incident from the transmitting extension z < 0. Interaction of this field with the aperture at z = 0 generates an infinite spectrum of modes in each of the waveguide sections. However, because of the symmetry of the incident field, only the TE1,v and TM1,v modes are excited with nonzero amplitude [61]. The transverse fields for the 262 Table 6.1: Real parts of relative material parameters for the waveguide verification standard optimized for S-band using WR-284 waveguides found using mode matching. Dimensions ` r ` r of the standard are: y1 = y1 = 5.064 mm, y2 = y2 = 23.86 mm, ` = r = 1/8 inch (3.175 mm), and s = 1/2 inch (12.7 mm). Frequency (GHz) 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 263 ✏r 6.0356 6.1182 6.1797 6.2232 6.2509 6.2650 6.2670 6.2584 6.2403 6.2137 6.1795 6.1383 6.0908 6.0374 5.9785 5.9145 5.8457 5.7723 5.6944 5.6120 5.5254 5.4343 5.3389 5.2389 5.1342 5.0245 4.9096 4.7889 µr 7.8526 7.5486 7.2848 7.0541 6.8508 6.6707 6.5105 6.3676 6.2399 6.1258 6.0238 5.9330 5.8523 5.7811 5.7188 5.6650 5.6194 5.5817 5.5518 5.5298 5.5157 5.5098 5.5123 5.5238 5.5449 5.5764 5.6194 5.6754 transmitting extension, z < 0, may thus be written as the modal series N f j 1z X f f ~ r Et (~) = a+~1 (~)e e ⇢ + an ~n (~)ej e ⇢ 1 n=1 N f X f ~ (~) = a+~ f (~)e j 1 z Ht r an ~ n (~)ej h ⇢ 1 h1 ⇢ n=1 f nz, (6.1) f nz, (6.2) where the series has been truncated at N modes for computational expediency. Similarly, the transverse fields in the region z1 < z < z2 may be written as ~ r Et (~) = ~ r Ht (~) = N " X # f f f c+ e j n (z z1 ) + cn e j n (z2 z) ~n (~), e ⇢ n (6.3) # f f cn e j n (z2 z) ~ n (~), h ⇢ (6.4) n=1 N " X f c+ e j n (z z1 ) n n=1 while those in the receiving extension, z > z3 , may be expressed as ~ r Et (~) = ~ r Ht (~) = N X f + ef ⇢ fn ~n (~)e j n (z z3 ) , (6.5) f + hf ⇢ fn ~ n (~)e j n (z z3 ) . (6.6) n=1 N X n=1 Here a+ is the known amplitude of the incident TE10 mode, and the ej!t time convention 1 is assumed. From (2.140) the propagation constant for mode n is given by f n= r 2 k0 f (kcn )2 , (6.7) where kcn is the cuto↵ wave number for the nth mode expressed as f kcn = r f (⇡/a)2 + (kyn )2 , 264 (6.8) f with kyn = vn ⇡/b. The index n orders the modes according to cuto↵ frequency. If the nth mode is TEz the variable vn corresponds to a TE1,vn wave; if the nth mode is TMz this variable corresponds to a TM1,vn wave. The fields in the aperture regions are expanded similarly as ~ r Et (~) = ~ r Ht (~) = N X ` ` b+ e j n z + bn e j n (z1 z) ~` (~), en ⇢ n n=1 N X ` b+ e j n z n (6.9) ` bn e j n (z1 z) ~ ` (~), hn ⇢ (6.10) n=1 in the left aperture region, 0 < z < z1 , and ~ r Et (~) = ~ r Ht (~) = N Xh i r r d+ e j n (z z2 ) + dn e j n (z3 z) ~r (~), en ⇢ n (6.11) i r j n (z3 z) ~ r (~), dn e hn ⇢ (6.12) n=1 N Xh r d+ e j n (z z2 ) n n=1 in the right aperture region, z2 < z < z3 . Note that the propagation constants in the apertures may be di↵erent, depending on the aperture sizes, and are thus written as `,r n = r 2 k0 `,r (kcn )2 . (6.13) Here `,r kcn = r `,r (⇡/a)2 + (kyn )2 , ` r ` with kyn = vn ⇡/w` and kyn = vn ⇡/wr , where w` = y2 (6.14) ` r y1 , and wr = y2 r y1 . As in the other waveguide sections, the modes in the apertures are ordered according to cuto↵ frequency. In the above expressions, ~n (~) and ~ n (~) are the transverse electric and magnetic modal e ⇢ h ⇢ 265 fields, respectively, and as shown in (2.153) and (2.154) are given by ⇣ ⇡x ⌘ h ⇣ ⌘i f,`,r f,`,r f,`,r f,`,r ~n e = xkyn cos ˆ sin kyn y a ⇣ ⇡x ⌘ h ⇣ ⌘i ⇡ f,`,r f,`,r , y sin ˆ cos kyn y a a f,`,r ˆ e ~ f,`,r = z ⇥ ~n hn , f,`,r Zn (6.15) (6.16) for the TE1,vn modes and from (2.180) and (2.181) are expressed as f,`,r ~n e = ⇣ ⇡x ⌘ h ⇣ ⌘i ⇡ f,`,r f,`,r x cos ˆ sin kyn y a a ⇣ ⇡x ⌘ h ⇣ ⌘i f,`,r f,`,r f,`,r , cos kyn y y kyn sin ˆ a f,`,r ˆ e ~ f,`,r = z ⇥ ~n hn , f,`,r Zn (6.17) (6.18) ` r f,`,r = !µ / f,`,r is the TE for the TM1,vn modes. Here f = 0, ` = y1 , r = y1 , Zn 0 n f,`,r f,`,r wave impedance, and Zn = n /(✏0 !) is the TM wave impedance. + The unknown modal amplitudes an , b+ , bn , c+ , cn , d+ , dn , and fn may be detern n n ~ ~ mined by applying the boundary conditions on Et and Ht at the interfaces between the full waveguide and the aperture regions. At the interface z = 0, the tangential electric field boundary condition requires N X f f a+~1 (~) + e ⇢ an ~n (~) = e ⇢ 1 n=1  8 ` > PN + j n ` ~` (~), ⇢ 2 ⌦ < en ⇢ ~ 1 n=1 bn + bn e , > : 0, ⇢ 2 ⌦1 ~ 266 (6.19) while the tangential magnetic field boundary condition requires f a+~ 1 (~) 1h ⇢ N X b+ n N X f an ~ n (~) = h ⇢ n=1 ` ` ` ~ (~), bn e j n hn ⇢ n=1 ⇢ 2 ⌦1 . ~ (6.20) ` ` Here ⌦1 designates the aperture extending from y = y1 to y = y2 , while ⌦1 designates ` ` the remaining conducting surfaces occupying 0 < y < y1 and y2 < y < b. At z = z1 the boundary condition on the tangential electric field requires N " X # f s f c+ + cn e j n ~n (~) = e ⇢ n  n=1 8 ` > PN + j n ` + b ~` (~), ⇢ 2 ⌦ < n en ⇢ ~ 1 n=1 bn e , > : 0, ⇢ 2 ⌦1 ~ (6.21) while the boundary condition on the tangential magnetic field requires N " X c+ n n=1 N X ` ` b+ e j n n # f s f ~ (~) = cn e j n hn ⇢ bn ~ ` (~), hn ⇢ n=1 ⇢ 2 ⌦1 . ~ (6.22) Similarly imposing the tangential boundary conditions on the electric and magnetic fields at z = z2 gives N " X # f s f c+ e j n + cn ~n (~) = e ⇢ n n=1 8 h r ri r > PN < d+ + dn e j n ~n (~), ⇢ 2 ⌦2 e ⇢ ~ n=1 n , > : 0, ⇢ 2 ⌦2 ~ 267 (6.23) and N " X f s c+ e j n n # f cn ~ n (~) = h ⇢ n=1 N Xh n=1 d+ n r ri r ~ (~), dn e j n hn ⇢ ⇢ 2 ⌦2 , ~ (6.24) r r respectively. Here ⌦2 designates the aperture region between y = y1 and y = y2 , while ⌦2 r r designates the conducting surfaces occupying 0 < y < y1 and y2 < y < b. Finally, at z = z3 the boundary conditions on the tangential electric and magnetic fields require N X + ef ⇢ fn ~n (~) = n=1 8 i PN h + j r r > < n + dn ~r (~), ⇢ 2 ⌦2 en ⇢ ~ n=1 dn e , > : 0, ⇢ 2 ⌦2 ~ and N Xh n=1 r r d+ e j n n N i X + hf ⇢ ~ dn ~ r (~) = hn ⇢ fn ~ n (~), ⇢ 2 ⌦2 , n=1 (6.25) (6.26) respectively. The system of functional equations (6.19)-(6.26) may be transformed into a system of linear equations by applying appropriate testing operators as follows. First, the equations resulting from applying a tangential electric field boundary condition, (6.19), (6.21), (6.23) f and (6.25), are multiplied by ~m (~) and integrated over the cross-section of the waveguide e ⇢ designated by 0  x  a, 0  y  b. Then, (6.20) and (6.22), which result from applying the tangential magnetic field boundary conditions on the left aperture, are multiplied by ~ ` (~) hm ⇢ and integrated over ⌦1 . Similarly for the right aperture, (6.24) and (6.26) are multiplied by ~ r (~) and integrated over ⌦2 . The result is a system of linear equations that may be hm ⇢ 268 written in terms of an 8N ⇥ 8N partitioned matrix. Note that all of the integrals may be computed in closed form, and that many matrix entries are repeated or zero. This allows the matrix to be filled rapidly and leads to the overall e ciency of the mode-matching approach compared to more generic numerical electromagnetic techniques such as finite elements. The matrix equation may be written in block form as 2 3  ` `  6 A B 7 + d . 4 5 c = a1 r Ar B (6.27) Here A` , B ` , B r , are Ar are partitioned into the 4N ⇥ 4N sub-matrices 2 J `,r G 6 6 6 `,r L`,r `,r = 6 K A 6 6 J `,r P `,r 6 0 4 0 L`,r P `,r J `,r P `,r 3 0 L`,r P `,r 0 J `,r G L`,r K `,r 7 7 7 7 7 7 7 5 (6.28) and 2 6 6 6 6 B `,r = 6 6 6 4 0 0 0 0 0 0 0 0 0 0 0 GQ 0 0 0 K `,r Q 3 7 7 7 7 7, 7 7 5 (6.29) where G, J `,r , K `,r , and L`,r are N ⇥ N sub-matrices with entries specified in Appendix `,r `,r `,r B.5. Here P and Q are diagonal matrices with entries Pmn = mn e j n and Qmn = f s j n , with mn e mn the Kronecker delta. The unknown vector is partitioned into eight h i + subvectors each of length N , and is given by cT = an , b+ , bn , c+ , fn , dn , d+ , cn , while n n n the right hand side is similarly partitioned as dT = [Gm1 , Km1 , 0, 0, 0, 0, 0, 0]. Once the modal coe cients are found by solving the matrix equation, the sample-plane 269 S-parameters of the verification standard are given by a S11 = 1 a+ 1 + f S21 = 1 . a+ 1 6.4.2 (6.30) (6.31) Validation of the Mode-Matching Analysis It is important to validate the mode-matching technique before using it to compute the Sparameters of a standard design, or to establish the errors introduced by measurement or fabrication uncertainties. This is accomplished by comparing to results from the commercial EM solver HFSS. Figure 6.9 shows the values of S11 and S21 computed for the waveguide standard used to generate Table 6.1. To obtain the HFSS-computed S-parameters shown in Figure 6.9 the waveguide extensions were explicitly modeled with a length su cient to ensure that only the fundamental TE10 mode is important at the waveguide ports. Here the extensions were chosen to be 55.475 mm long. This is an arbitrary value of su cient length. The convergence in HFSS was set to a maximum S of 0.0005, which is the absolute di↵erence between the magnitudes of the S-parameters at successive iterations computed at the solution frequency 3.95 GHz. This very tight convergence criterion, which requires significant computer resources, is needed to achieve good agreement with the mode-matching S-parameters at the higher frequencies. Excellent agreement between the S-parameters obtained by the two methods is seen, giving confidence to the mode matching results. Additional validation is provided by comparing the permittivity and permeability extracted from each set of S-parameters; these results are shown in Figure 6.10. Note that the computational time required for HFSS is approximately 60 times larger than the time required for mode matching with equivalent accuracy on a 3.5 GHz Intel quad core processor with 24 GB of RAM. Thus, the faster mode-matching approach makes lengthy optimizations far more feasible, and is used in the computationally-intensive Monte Carlo 270 Magnitude (dB) 0 S21 -5 Mode Matching HFSS -10 S11 -15 -20 -25 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Phase (°) 200 S11 Mode Matching HFSS 100 S21 0 -100 -200 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 6.9: Computed S-parameters of the verification standard. 271 8 Material Parameters 7.5 μ'r Mode Matching HFSS 7 6.5 ϵ'r 6 5.5 5 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Frequency (GHz) Figure 6.10: Comparison of material parameters extracted from mode matching and HFSS simulated S-parameters. 272 error analysis presented in the next section. 6.5 Error and Sensitivity Analysis One of the objectives for the waveguide verification standard is to have an easily manufactured design that can be accurately characterized. Therefore, the sensitivity of the standard needs to be minor when small changes to the design geometry occur. These geometry changes can result from inaccuracies in the manufacturing process. To access this sensitivity error, a Monte Carlo error analysis was used to explore the dependence of the extracted material parameters on small changes to the geometry. These manufacturing inaccuracies are systematic errors, which include the uncertainties in the thickness of the apertures and spacer ( ` , s , and r ), the vertical window positions ` ` r r for both apertures (y1 , y2 , y1 , and y2 ), the uniformity of the thicknesses, and uniformity of the widow width and height. Another possible type of error is random error, which can result from the alignment of the verification standards (this may change from experiment to experiment) and measurement uncertainty inherent in the VNA. It is di cult to model all the di↵erent types of error produced by uncertainties in the geometrical parameters, since the mode-matching technique assumes uniform thicknesses and vertical window positions. However, the other systematic errors may be easily studied using Mote Carlo techniques, since the mode-matching technique is capable of analyzing di↵erent thicknesses and vertical window positions. This allows for error bounds to be established due to small inaccuracies in the manufacturing processes. A Monte Carlo analysis on the propagation of uncertainties in the geometrical parameters was undertaken using the designed parameters outlined in Section 6.3. The forward problem was solved at 28 frequency points over the portion of S-band from 2.6 to 3.95 GHz. Dimensions were generated randomly using a Gaussian distribution with a mean value set equal to designed parameters, and a standard deviation equal to the manufacturing accuracy outlined 273 by the Department of Physics and Astronomy Machine Shop at Michigan State University: 5 mils for uncertainty in thickness and 2 mils for uncertainty in the aperture heights. The mode matching technique analyzed these dimensions and the erroneous S-parameter data was used to extract the material parameters. This process was repeated 500 times, and the average values and the standard deviations of the extracted material parameters were calculated. Figure 6.11 shows results of the Monte Carlo error analysis on varying the thicknesses `, s , and r . Figure 6.12 shows results from varying the vertical window positions y ` , 1 ` r r y2 , y1 , and y2 . Finally, Figure 6.13 shows the combination of uncertainties in thicknesses and vertical window positions. In these figures, the center of each triplet of lines is the average value of the 500 trials, while the two surrounding lines indicate the 95% confidence intervals of ±2 standard deviations. For the imaginary values the mean of the 500 trials is near zero, therefore only the error is shown. Note that complex material parameters result from the asymmetric trials being analyzed in the Monte Carlo analysis. These results for geometrical uncertainties exhibit low sensitivity to possible manufacturing error, leading to the decision to fabricate and further test this surrogate design. 6.5.1 Theoretical Material Parameters for the Fabricated Standard The thicknesses and aperture positions of the constructed inserts were measured using precision calipers with a manufacturer-specified accuracy of ±0.02 mm. Measurements were made at 15 positions on the inserts and the mean and standard deviation computed. The mean is taken to be the estimated dimension, while the standard deviation is taken to be the uncertainty in that dimension. The results are shown in Table 6.2, along with the specified values for the optimized standard. Note that since the fabricated standard is not perfectly symmetric along z it is anticipated that measured values of µ and ✏ will have small imaginary parts. Although the dimensions of the fabricated standard di↵er somewhat from those of the 274 Material Parameters Error 8 7.5 7 6.5 6 5.5 5 4.5 0.08 0.07 0.06 0.05 0.04 0.03 0.02 μ'r ϵ'r 2.6 2.8 3 3.2 3.4 3.6 3.8 3.6 3.8 Real ϵ andμ ϵ''r 2.6 2.8 3 3.2 μ''r 3.4 Frequency (GHz) Figure 6.11: Relative constitutive parameters found when randomly varying ` , s , and r extracted using 500 trials. Center line in the upper plot is the average of real relative parameters of the trials, while the upper and lower dashed-lines show the 95% confidence intervals. The average of the imaginary relative parameters is near zero, while the lower plot shows +2 values. Table 6.2: Measured dimensions of fabricated waveguide standard. Dimension ` r s ` y1 ` y2 r y1 r y2 Optimized Value (mm) Fabricated Value (mm) 3.175 3.175 12.7 5.064 23.86 5.064 23.86 3.622 ± 0.028 3.571 ± 0.038 12.347 ± 0.071 5.183 ± 0.057 23.907 ± 0.111 5.276 ± 0.071 23.494 ± 0.105 275 Material Parameters 8 μ'r 7.5 7 6.5 6 ϵ'r 5.5 5 4.5 2.6 2.8 3 3.4 3.6 3.8 3.6 3.8 Real ϵ andμ 0.1 Error 3.2 0.08 ϵ''r 0.06 μ''r 0.04 2.6 2.8 3 3.2 3.4 Frequency (GHz) ` ` r Figure 6.12: Relative constitutive parameters found when randomly varying y1 , y2 , y1 , and r y2 extracted using 500 trials. Center line in the upper plot is the average of real relative parameters of the trials, while the upper and lower dashed-lines show the 95% confidence intervals. The average of the imaginary relative parameters is near zero, while the lower plot shows +2 values. 276 Material Parameters 8 μ'r 7.5 7 6.5 6 ϵ'r 5.5 5 4.5 0.14 2.6 2.8 3 3.4 3.6 3.8 3.6 3.8 Real ϵ andμ 0.12 Error 3.2 0.1 0.08 μ''r 0.06 ϵ''r 0.04 2.6 2.8 3 3.2 3.4 Frequency (GHz) Figure 6.13: Relative constitutive parameters found when randomly varying ` , s , r , ` ` r r y1 , y2 , y1 , and y2 extracted using 500 trials. Center line in the upper plot is the average of real relative parameters of the trials, while the upper and lower dashed-lines show the 95% confidence intervals. The average of the imaginary relative parameters is near zero, while the lower plot shows +2 values. 277 optimized standard used to establish Table 6.1, the fabricated standard is still useful for its intended purpose. Because the optimization process included minimizing the sensitivity of extracted µ and ✏ to small changes in the geometrical parameters, the resulting values of µ and ✏ extracted from measurements of the fabricated standard are only slightly di↵erent than those for the optimized standard. Also, due to the availability of the mode-matching method, the dimensions of the fabricated standard may be used to establish the expected values of µ and ✏, and these parameters used for verification of the measurement system. The mode matching method was used to compute the theoretical S-parameters for the fabricated geometry with the mean dimensions shown in Table 6.2. These S-parameters were then used to compute the values of µ and ✏ associated with the fabricated standard. Figure 6.15 compares the real parts of these parameters with those found using the optimized design, while the imaginary parts are compared in Figure 6.16. (Recall that the optimized design is longitudinally symmetric and thus produces zero imaginary parts.) While the di↵erence in geometry does produce a shift in the material parameters, this shift is not excessive and the values still retain their desired properties of being near 6 while varying slowly across the measurement band. 6.5.2 Measured Material Parameters for the Fabricated Standard Measurements of the S-parameters of the verification standard were made using an Agilent E5071C VNA. The verification standard was sandwiched between two 6 inch (152.4 mm) long sections of WR-284 commercial S-band waveguide, to act as extensions, with coaxial transitions attached at the ends. The VNA was calibrated at the ends of the waveguide extensions using the Through-Reflect-Line (TRL) method. Alignment pins were used for the di↵erent assemblies to ensure high repeatability of the measurements. All measurements were made with VNA settings of 5 dBm source power, 64 averages, and an IF bandwidth of 70 kHz. Finally, the material parameters were extracted using the measured values of S11 and S21 , assuming the average values of the insert dimensions shown in Table 6.2. 278 Figure 6.14: Brass aperture plates forming the fabricated waveguide verification standard. 279 8 Material Parameters 7.5 μ'r 7 6.5 6 ϵ'r 5.5 5 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 6.15: Real parts of relative permittivity and permeability for optimized geometry (solid line) and fabricated geometry (dotted line). 280 Material Parameters Material Parameters 0.008 0.006 0.004 ϵ''r 0.002 0 2.6 2.8 3 3.2 3.4 3.6 3.8 3 3.2 3.4 3.6 3.8 0 -0.002 μ''r -0.004 -0.006 -0.008 2.6 2.8 Frequency (GHz) Figure 6.16: Imaginary parts of relative permittivity and permeability for optimized geometry (solid line) and fabricated geometry (dotted line). 281 The measurement repeatability error was assessed by measuring the verification standard 10 separate times, with the VNA calibrated at the start of each set of measurements. The real parts of the material parameters extracted from the 10 measurements are shown in Figure 6.17, while the imaginary parts are shown in Figure 6.18. The center solid line in these figures represents the mean of the extracted values while the upper and lower solid lines define the 95% confidence intervals (± two standard deviations). The dotted lines show the theoretical material parameters from Figure 6.15 and Figure 6.16, found using the measured dimensions of the waveguide standard. Although the theoretical values for the imaginary parts lie within the 95% confidence intervals, the real parts lie just slightly outside these intervals, at least at some frequencies. This is shown more clearly in the insets, which zoom in on a chosen narrow region of the measurement band. There are several possible reasons for this, including calibration error, misalignment of the waveguide sections, imperfect electrical connections, variability in the thickness of the inserts, errors in the machining of the apertures, etc. One error that can be modeled is produced by the uncertainty in the measured values of several of the geometric parameters. These include the thicknesses of the aperture plates and spacer, and the sizes and vertical positions of the aperture openings. Unfortunately, the mode-matching approach does not allow the modeling of possible rotation of the apertures or spatial variations in thickness or opening size. A Monte Carlo analysis was undertaken to determine the e↵ects of geometry uncertainty on the extracted material parameters. Dimensions were generated randomly using a Gaussian distribution with a mean value set equal to the average of the measured values shown in Table 6.2, and a standard deviation equal to the standard deviation of the measured values shown in the table. The material parameters were extracted and the process repeated 500 times. The average values and the standard deviations of the extracted material parameters were calculated and are shown in Figure 6.19 and Figure 6.20. In these figures the measured data and their 95% confidence interval (solid triplet of lines) is shown along with the average Monte Carlo data and their 95% confidence interval (dashed triplet of lines). Clearly the 282 Material Parameters 7.5 μ'r 7 6.5 5.6 5.55 5.5 5.45 5.4 ϵ'r 6 6.25 6.22 6.19 6.16 6.13 6.1 2.75 5.5 5 2.6 2.8 2.8 2.85 3 3.82 3.86 3.9 3.94 2.9 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 6.17: Real parts of relative permittivity and permeability extracted from 10 sets of measurements. Center solid line is the average of the measurements. Upper and lower lines show the 95% confidence intervals. Dotted line shows the material parameters extracted from the mode-matching S-parameters generated using the measured geometry. 283 Material Parameters Material Parameters 0.06 0.04 ϵ''r 0.02 0 -0.02 0.02 2.6 2.8 3 3.2 3.4 3.6 3.8 3 3.2 3.4 3.6 3.8 0 -0.02 μ''r -0.04 -0.06 -0.08 2.6 2.8 Frequency (GHz) Figure 6.18: Imaginary parts of relative permittivity and permeability extracted from 10 sets of measurements. Center solid line is the average of the measurements. Upper and lower lines show the 95% confidence intervals. Dotted line shows the material parameters extracted from the mode-matching S-parameters generated using the measured geometry. 284 measured data (including the 95% confidence interval) lie within the uncertainty interval of the Monte Carlo data for both real and imaginary parts of µ and ✏, and the usefulness of the standard to predict the operability of the measurement system is demonstrated. 6.6 Summary A waveguide standard is introduced to provide a surrogate material useful for verifying material characterization systems in the microwave spectrum. A surrogate is needed since no convenient materials are available that have slowly-varying, predictable and accurately reproducible magnetic characteristics at microwave frequencies. The standard is constructed from all metal parts, allowing for easy fabrication, and is straightforward to design using a mode-matching method for analysis. Specific dimensions are provided for an S-band standard to give relative permittivity and permeability near six when using the NicolsonRoss-Weir method. The example standard is optimized using a genetic algorithm such that the extracted material properties are not highly sensitive to changes in the geometrical parameters, and thereby reduce the need for tight manufacturing tolerances. An example standard is measured and the errors introduced by uncertainty in the dimensions of the machined parts is characterized. By showing that the measurement precision defined by the repeatability of the experiments is consistent with the error due to propagation of geometrical uncertainties, the usefulness of the standard for verifying the proper operation of a waveguide material measurement system is demonstrated. 285 Material Parameters 7.5 μ'r 7 6.5 5.6 5.55 5.5 5.45 5.4 ϵ'r 6 5.5 6.3 6.25 6.2 6.15 6.1 2.75 5 2.8 2.85 3.82 3.86 3.9 3.94 2.9 4.5 2.6 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) Figure 6.19: Real parts of the relative permittivity and permeability extracted from 10 sets of measurements (solid lines). Center solid line is the average of the measurements. Upper and lower solid lines show the 95% confidence intervals. Dotted lines show mode-matching results for 500 random trials. Center dotted line is the average and upper and lower dotted lines show the 95% confidence intervals. 286 Material Parameters Material Parameters 0.1 ϵ''r 0.05 0 -0.05 -0.1 2.6 0.1 2.8 3 3.2 3.4 3.6 3.8 2.8 3 3.2 3.4 3.6 3.8 μ''r 0.05 0 -0.05 -0.1 2.6 Frequency (GHz) Figure 6.20: Imaginary parts of the relative permittivity and permeability extracted from 10 sets of measurements (solid lines). Center solid line is the average of the measurements. Upper and lower solid lines show the 95% confidence intervals. Dotted lines show modematching results for 500 random trials. Center dotted line is the average and upper and lower dotted lines show the 95% confidence intervals. 287 CHAPTER 7 CONCLUSIONS AND FUTURE STUDIES Major contributions to the research community discussed in this dissertation are the new methods for the characterization of anisotropic materials. Additionally, the waveguide standard is introduced which may be used as a surrogate material useful for verifying material characterization systems in the microwave spectrum. A general description of the NRW method is outlined for isotropic material characterization using a rectangular waveguide. An iris technique is investigated for the characterization of conductor-backed isotropic materials. A mode-matching method is used to calculate the theoretical reflection and transmission coe cients. These theoretical coe cients are used in the extraction of isotropic material parameters using a secant method. This technique was shown to be very sensitive to measurement uncertainties, especially for the characterization of ✏00 . By the completion of this dissertation accurate characterization of a conductor-backed material using this technique has not been accomplished. Possible future work includes manufacturing a sample holder for the FGM125 sample such that a shorting plate can be connected to the back of the sample. Here the use of sliding short is not necessary. Use of the shorting plate might decrease the amount of error in the measured S-parameters. 288 Previous work was not found on the characterization of biaxial material using closed form solutions. This dissertation described the use of the NRW method for biaxial materials without the use of an iterative solver. This technique produced accurate results using simulated S-parameter data in lieu of measurements. Future work might include the verification of this technique using measurements of biaxial or uniaxial materials. If three samples are not available for characterization of a biaxial material, then this dissertation introduced a reduced aperture waveguide method for measuring the permittivity and permeability of biaxially anisotropic materials. Only a single cubical sample is required to completely characterize the material under test. This contrasts with the three distinct samples required when using the NRW extension technique. A mode-matching technique is used to accommodate the higher-order mode excitation resulting from the waveguide discontinuity. Then a Newton’s method is employed to extract the six complex material parameters. The reduced aperture waveguide technique was used to characterize isotropic and uniaxial properties of two samples. However, as with many characterization techniques, there are limits on the accuracy of the extraction when the electrical length of the sample approaches a half wavelength. Possible future work includes exploring the use of additional measurements to ameliorate this issue. Also it is possible to construct an electrically-biaxial sample by using layers of alternating strips, as suggested in [59]. This could be accomplished by slicing a layered cube, rotating the slices by ninety degrees, and gluing the slices together. Additionally, a magnetic uniaxial sample could be constructed from alternating layers of dielectric and FGM125 or FGM40. The usefulness of using such a sample as a biaxial or magnetic uniaxial surrogate is left for future exploration. As a next step towards full anisotropic material tensor characterization, gyromagnetic materials were considered. Previous work the characterization of gyromagnetic materials using rectangular waveguides was limited. This dissertation described two characterization methods using a completely filled cross section or reduced aperture sample holder. Here the reduced aperture sample holder is used to characterize samples with electrical size smaller 289 than the cross-section of a waveguide. Details on the mode matching technique used for obtaining the theoretical reflection and transmission coe cients and a comparison to a full wave FEM solver are presented for these two characterization techniques. The extraction method using a nonlinear least squares algorithm was tested by using simulated S-parameters in lieu of measurements. The sensitivity analysis for measurement uncertainties performed for the reduced-aperture technique demonstrates promise for characterization using measurements. The measurement uncertainties used in the Monte Carlo technique represent a worst case error. Since the results of the sensitivity analysis in Section 5.3.4 demonstrate that the extraction techniques is not highly sensitive to measurement uncertainties, it is hopeful that accurate characterization will result from measurements. Extraction of gyromagnetic material parameters from experimental results is left for future study. The usefulness of the waveguide standard for verifying the proper operation of a waveguide material measurement system is demonstrated in this dissertation. Additionally a mode-matching approach is described which allows the user to predict the material properties with higher accuracy, and thus compensates for manufacturing inaccuracies. Surrogate materials of similar geometries should be useful for verifying the performance of other types of material measurements systems that employ the NRW method, such as coaxial and stripline applicators. Design of these standards is left for future study. 290 APPENDICES 291 Appendix A: Useful Identities ⇣ ⌘ ~ ~ ~ ~ ~ ~ ~ A· B+C =A·B+A·C ⇣ (A.1) ⌘ ⇣ ⌘ ~ ~ ~ ~ ~ ~ A·P ·B =A· P ·B =A·P ·B ⇣ (A.2) ⌘ ⇣ ⌘ ~⇥P ·B =A· P ⇥B ~ ~ ~ A ⇣ ⌘ u⇥ u⇥A = ˆ ˆ ~ (A.3) ~ A⇢ (A.4) u · A⇢ = 0 ˆ ~ (A.5) " ~ @ A⇢ ~ ~ r ⇥ A = r ⇢ ⇥ A⇢ + u ⇥ ˆ @u r ⇢ Au # ⇣ ⌘ ~ u ⇥ r⇢ ⇥ A⇢ = 0 ˆ ⇣ ⌘ ~ =0 u· u⇥A ˆ ˆ ⇣ ⌘ ⇣ ⌘ ~ ~ ~ ~ ~ ~ A⇥ B⇥C =B A·C 292 (A.6) (A.7) (A.8) ⇣ ⌘ ~ ~ ~ C A·B (A.9) ⇣ ⌘ ⇣ ⌘ ~ ~ ~ ~ ~ ~ A· B⇥C =C · A⇥B P =P ·P n 1 0 P =I 293 (A.10) (A.11) (A.12) Appendix B: De-embedding S-parameters S-parameters of a waveguide system are measured by a VNA using waveguide to coaxial line adaptors and waveguide extensions. The extensions are used to guarantee that only the dominant waveguide mode is present at the measurement ports. A calibration is performed at the ends of the waveguide extensions and calibration planes are established there. When characterizing a material sample, the S-parameters are measured with the sample positioned somewhere in the waveguide system. Typically, the sample is placed inside a sample holder, and the holder is positioned in between the two calibration planes, as shown in Figure B.1. Another common sample placement is to insert the sample inside the waveguide extension as is shown in Figure B.2. The characterization techniques described in this dissertation make use of sample plane S-parameters. Sample plane S-parameters are defined at the faces of the material sample while the calibration plane S-parameters are measured by the VNA at the calibration planes. The sample plane S-parameters can be obtain by properly applying corrections to the Sparameters measured by the VNA. This process is called de-embedding. De-embedding procedures for the two di↵erent sample insertion cases are described in this appendix. In Figure B.1 and Figure B.2, the modal amplitude coe cients are shown at both the calibration and sample planes. The modal coe cients can be shifted from one plane to the other by multiplying by e j D in the direction of the wave propagation. Here, D is the 294 Calibration Plane d1 c a1 Port 1 s a1 c 1 s 1 b b Calibration Plane d1 d2 S A M P L E Calibration Plane s b2 c b2 s 2 c 2 a Sample Holder a d2 Port 2 Calibration Plane Calibration Calibration c s Plane Plane Figure B.1: Measurement and sample plane modal coe cients for a sample holder. S s aΔ 1 c a1 Port 1 Port 1 c 1 b a s 1 s 1 b S s A 1 M b P L E d A M Ps L b2 E a sc 21 Port 1 s 1 b S A M P L E ba c 1 c 2 a a s 2 b c c 2 2 a Port 2 Port 2 Sample Holder d Δ s a1 b2 b2 Calibration Calibration Plane Plane s b2 c a1 c b2 s 2 c 1 c 2 a b a Port 2 Figure B.2: Measurement and sample plane modal coe cients with sample inserted into the waveguide extension attached to port 1. 295 distance of the shift, and is the propagation constant of the empty waveguide extension which can be found using (3.3). The calibration plane S-parameters are defined as bc 1 ac 1 bc c S21 = 2 ac 1 bc c S12 = 1 ac 2 c b c S22 = 2 , ac 2 c S11 = (B.1) (B.2) (B.3) (B.4) while the sample plane S-parameters are defined as bs 1 as 1 s b s S21 = 2 as 1 bs s S12 = 1 as 2 bs s S22 = 2 . as 2 s S11 = (B.5) (B.6) (B.7) (B.8) Consider the case shown in Figure B.1. The calibration and sample plane modal coe cients are related as follows: as = ac e 1 1 bc = bs e 1 1 as = ac e 2 2 bc = bs e 2 2 j d1 (B.9) j d1 (B.10) j d2 (B.11) j d2 . (B.12) Here d1 and d2 are the distances between the sample and the ends of the waveguide ex- 296 tensions attached to port 1 and port 2, respectively. To express the sample plane modal coe cients in terms of calibration plane modal coe cients, (B.10) is divided by e j d1 and (B.12) is divided by e j d2 , yielding b s = b c ej d 1 1 1 b s = b c ej d 2 . 2 2 (B.13) (B.14) s c Now, S11 can be expressed using the measured S11 , by first substituting (B.9) and (B.13) into (B.5), which results in c j d1 s = b1 e S11 . ac e j d 1 1 (B.15) Using (B.1) and simplifying, this becomes s c S11 = S11 ej2 d1 . (B.16) The remaining sample plane S-parameters can be derived by a similar set of steps producing s c S21 = S21 ej (d1 +d2 ) s c S12 = S12 ej (d1 +d2 ) s c S22 = S22 ej2 d2 . (B.17) (B.18) (B.19) The next case involves inserting the sample inside the waveguide extension as shown in 297 Figure B.2. In this setup the calibration and sample plane modal coe cients are related by ac = as e 1 1 bs = bc e 1 1 as = ac e 2 2 bc = bs e 2 2 j (d+ ) (B.20) j (d+ ) (B.21) j d (B.22) j d. (B.23) Here d is the distance between the sample and the calibration plane and is the thickness of the sample. The sample plane modal coe cients are expressed in terms of calibration plane modal coe cients by dividing (B.20) by e j (d+ ) and (B.23) by e j d . This gives as = ac ej (d+ ) 1 1 b s = b c ej d . 2 2 (B.24) (B.25) The expressions for the sample plane modal coe cients are substituted into definitions for the sample plane S-parameters, (B.5) - (B.8), which yields the following: s c S11 = S11 e s c S21 = S21 e s c S12 = S12 e s c S22 = S22 e j2 (d+ ) (B.26) j (B.27) j (B.28) j2 d . (B.29) In both cases of de-embedding, since the sample plane S-parameters result from a reciprocal and symmetric system, it is known that s s S21 = S12 s s S11 = S22 . 298 (B.30) (B.31) However, the calibration plane S-parameters, or the S-parameters measured by the VNA, have the following relationships: c c S21 = S12 c c S11 6= S22 . (B.32) (B.33) This can be deduced from (B.16) and (B.19) for the first case, and from (B.26) and (B.29) c c for the second case. Here, since (B.31) holds true, in general S11 6= S22 . This means the VNA measurements of the reflection coe cient at port 1 will in general be di↵erent than c c the reflection coe cient at port 2. However, a special case does exist when S11 = S22 . This occurs in the first case when the sample is centered in the sample holder (d1 = d2 ). 299 Appendix C: Calculation of Waveguide Transverse Mode Integrals C.1: Iris Integrals The derivation of the transverse mode integrals for the mode matching technique used in the calculation of the reflection coe cient from an iris in front of a conductor-backed material is outlined here. The integrals for the reduced-aperture junction, wB < wA , are Cmn = wA Z2 eA (x)eA (x)dx, m n (C.1) eA (x)eB (x)dx, m n (C.2) eB (x)eA (x) m n dx, A Zn (C.3) wA 2 Dmn = wB Z2 wB 2 Emn = wB Z2 wB 2 300 Fmn = wB Z2 wB 2 eB (x)eB (x) m n dx, B Zn (C.4) while the integrals for the expanded-aperture, wA < wB are Lmn = wB Z2 eB (x)eB (x)dx, m n (C.5) eA (x)eB (x)dx, m n (C.6) wA Z2 eA (x)eB (x) m n dx, B Zn (C.7) wA Z2 eA (x)eA (x) n m dx. A Zn (C.8) wB 2 Mmn = wA Z2 wA 2 Omn = wA 2 Wmn = wA 2 First the integrals for the reduced step are evaluated. Cmn is expressed as A A Cmn = kxn kxm ( 1)n+m wA Z2 cos wA 2 301 ✓ ◆ ✓ ◆ 2n 1 2m 1 ⇡x cos ⇡x dx. wA wA (C.9) Using the integral identity Z cos (ax) cos (Bx) dx = 1 sin [(a b)x] 1 sin [(a + b)x] + , 2 a b 2 a+b (C.10) Cmn becomes i h i 9 wA 8 h ⇡x (2n 2m) ⇡x (2n + 2m 2) = 2 < sin w sin w A k A ( 1)n+m A A Cmn = kxn xm + (C.11) . ⇡ ⇡ (2n + 2m 2) ; : w (2n 2m) wA wA A 2 When m 6= n, Cmn = 0. (C.12) When m = n, (C.9) becomes Cmn = ⇣ ⌘ A 2 kxn wA Z2 wA 2  2 2n 1 ⇡x dx. cos wA (C.13) Through the use of integral integral identity, Z cos2 (ax) dx = 2ax + sin(2ax) , 4a (C.14) (C.13) becomes ⇣ ⌘ A 2 wA . Cmn = kxn 2 302 (C.15) Therefore, or 8 ⇣ ⌘ > < k A 2 wA , m = n xn 2 Cmn = > : 0, m 6= n ⇣ ⌘ A 2 wA Cmn = kxn mn . 2 (C.16) (C.17) The integral for Fmn can be evaluated in a similar set of steps and is given by ⇣ ⌘ B 2 wB Fmn = kxn . B mn 2Zn (C.18) Next, the integral for Dmn is expressed as B A Dmn = kxn kxm ( 1)n+m wB Z2 wB 2   2n 1 2m 1 cos ⇡x cos ⇡x dx. wB wA (C.19) Using the integral identity (C.10), Dmn becomes ⌘i 8 h ⇣ < sin ⇡x 2n 1 2m 1 wB wA B A ⇣ ⌘ + Dmn = kxn kxm ( 1)n+m : 2⇡ 2n 1 2m 1 wB wA h ⇣ ⌘i 9 w B sin ⇡x 2n 1 + 2m 1 = 2 wB wA ⇣ ⌘ , ; wB 2⇡ 2n 1 + 2m 1 wB wA 2 303 (C.20) or h w i sin 2n2 1 ⇡ 2m 1 ⇡ wB 2 B A ⇣ ⌘A + Dmn = kxn kxm ( 1)n+m 4 2n 1 2m 1 ⇡ w wA B h w i3 sin 2n2 1 ⇡ + 2m 1 ⇡ wB 2 ⇣ ⌘ A 5. 2n 1 + 2m 1 ⇡ w wA B 2 (C.21) Using a trigonometric identity, it can be shown that sin  2n 1 2   2m 1 wB 2n 1 2m 1 wB ⇡± ⇡ = sin ⇡ cos ⇡ , 2 wA 2 2 wA (C.22) or sin  2n 1 2 2m 1 wB ⇡± ⇡ = 2 wA  n cos 2m 1 ⇡ wB . ( 1) 2 wA (C.23) Substituting (C.23) into (C.21) yields Dmn =  B k A ( 1)n+m ( 1)n cos 2m 1 ⇡ wB 1 kxn xm 2 wA ⇡ 2 3 1 1 4 5, + 2n 1 2m 1 2n 1 + 2m 1 wB wA wB wA (C.24) or ⇣ ⌘ 2n 1 wB 2m 1 wB 2 B A Dmn = kxn kxm (1)m+1 cos ⇡ ⇣ ⌘2 ⇣ ⌘ . 2 wA ⇡ 2n 1 2m 1 2 wB wA  304 (C.25) The case of 2n 1 = 2m 1 must be handled separately. When this occurs, (C.19) becomes wB wA wB Z2 B A Dmn = kxn kxm ( 1)n+m wB 2  2 2n 1 ⇡x dx. cos wB (C.26) This integral can be evaluated in the manner for Cmn , and results in w B A Dmn = kxn kxm ( 1)n+m B . 2 (C.27) Finally, the integral for Emn can be derived in a similar set of steps and given as Emn = Dnm . A Zn (C.28) Following similar procedure to that outlined for the reduced-aperture junction integrals, the integrals for the expanded step are derived as follows: ⌘2 w B mn 2 ⇣ ⌘2 w A A Wmn = kxn A mn 2Zn Lmn = ⇣ B kxn (C.29) (C.30) Mmn = Dmn (C.31) A$B Mnm Omn = . B Zn (C.32) A Note to obtain Mmn using Dmn , all instances of wA and kxn must be changed to wB and B B A kxn , and all instances of wB and kxn must be changed to wA and kxn . 305 C.2: Reduced Aperture Waveguide for Biaxial Materials In the calculation of the matrix elements for biaxial material characterization using a reduced aperture waveguide, it is necessary to evaluate the integrals associated with the transverse waveguide fields. These are Dmn = Fmn = Pmn = Qmn = Z a 2 e e (x)ee (x)dx, yn a ym 2 Z w 2 s e (x)es (x)dx, yn w ym 2 Z w 2 s s Zm h (x)he (x)dx, xn w xm 2 Z w 2 s s Zm h (x)hs (x)dx. xn w xm 2 (C.33) (C.34) (C.35) (C.36) First Dmn is expressed as Z a a h ⇣ 2 e e e R e eym (x)ee (x)dx = kxm kxn 2 a sin kxm x yn a 2 2 Using the change of variables, where u = a 2 e e Dmn = kxm kxn ⌘i h ⇣ e sin kxn x a 2 ⌘i dx. C.37) ( x, this becomes Z a 0 a 2 e e sin kxm u sin kxn u du. (C.38) e e From (2.3.2), it is shown that kxn = n⇡/a and similarly kxm = m⇡/a. Therefore Z ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ mn⇡ 2 a Dmn = sin u sin u du. a a a2 0 (C.39) Because the integrand is even about x, this integral can be expressed as mn⇡ 2 Dmn = 2a2 Z a ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ sin u sin u du a a a 306 (C.40) If m = n, then the identity Z sin2 (ax) dx = x 2 1 sin 2ax 4a (C.41) can be used. If m 6= n then Dmn = 0. Therefore 8 > n2 ⇡ 2 < 2a , m = n Dmn = > : 0, m 6= n or Dmn = (C.42) n2 ⇡ 2 mn . 2a (C.43) The integral for Qmn can be evaluated similarly and is given by Qmn = n2 ⇡ 2 s mn . 2wZn (C.44) Next, the integral for Fmn is expressed as e s Fmn = kxm kcn Z w h ⇣ 2 e sin kcm x w 2 h ⇣ a ⌘i s x sin kcn 2 w ⌘i dx. 2 (C.45) This integral will be derived generally, since integrals of similar form will be needed for Chapter 6. Therefore, Fmn is written as e s Fmn = kxm kcn B , mn (C.46) where B = mn Z ⌫ 2 ⌫1 sin [↵m (x )] sin [ n (x 307 ⌫1 )]dx. (C.47) e Here ↵m = kcm , s n = kcn , ⌫2 = w/2, ⌫1 = w/2, and  = a/2. Expanding the arguments of the sine functions gives B = mn Z ⌫ 2 ⌫1 sin (↵m x ↵m ) sin ( n x n ⌫1 )dx. (C.48) This definite integral now can make use of the expression [62] Z ↵ 2 sin [(A Therefore C) x + (B 2 (A C) ↵1 D)] sin (Ax + B) sin (Cx + D)dx = (C.49) sin [(A + C) x + (B + D)] ↵2 . 2 (A + C) ↵1 (C.50) B can be expressed as mn B = sin [(↵m mn n ) x (↵m  2 (↵m n) n ⌫1 )] sin [(↵m + n ) x (↵m  + ⌫1 )] ⌫2 . C.51) ( 2 (↵m + n ) ⌫1 Expanding this further gives n ) sin (↵m 1 ) B = sin (↵m 2 mn 2 (↵m n) where sin (↵m 2 + n ) sin (↵m 1 ) , 2 (↵m + n ) = ⌫2 ⌫1 , 1 = ⌫1 , and 2 = ⌫2 . Multiplying through by (↵m (C.52) n ) (↵m + n ) to get a common denominator and combining like elements gives B = mn ⇣ 1 2 2 2 ↵m n + n [sin (↵m 2 ⌘ {↵m [sin (↵m 2 n ) sin (↵m 2 + n )] (C.53) sin (↵m 2 + n )] + 2 n sin ( m 1 )} . (C.54) 308 n ) Using the trigonometric identies ✓ ◆ ✓ ◆ u+v u v sin u + sin v = 2 sin cos 2 2 ✓ ◆ ✓ ◆ u+v u v sin u sin v = 2 cos sin , 2 2 simplifies B = mn (C.55) (C.56) B further to mn ↵m cos (↵m 2 ) sin ( n ) + n sin (↵m 2 ) cos ( n ) 2 2 ↵m n n sin (↵m 1 ) , (C.57) or B = n [sin (↵m 2 ) cos ( n ) mn sin (↵m 1 )] 2 2 ↵m n ↵m cos (↵m 2 ) sin ( n ) . A special case exists when ↵m = n = 0. This results in 8 > < B = mn > : (C.58) B = 0. Therefore, mn 0, ↵m = n = 0 . n [sin(↵m 2 ) cos( n ) sin(↵m 1 )] ↵m cos(↵m 2 ) sin( n ) , otherwise(C.59) 2 2 ↵m n Finally the integral for Pmn can be evaluated in the manner done for Fmn and is given by Pmn = Fnm s . Zn (C.60) C.3: Gyromagnetic Material Filled Waveguide Crosssection Closed-form expressions for integrals associated with the transverse waveguide functions used in Section 5.2.1.2 are derived in this appendix. The integrals associated with (5.21) 309 and (5.25) are Z a Z 0a 0 e e sin kxm x sin kxn x dx (C.61) e s sin kxm x sin kcn x dx, (C.62) while the integrals associated with (5.23) and (5.27) are Z a s e sin kcm x sin kcn x dx (C.63) s s sin kcm x sin kcn x dx (C.64) e s sin kcm x cos kcn x dx. (C.65) Z0a Z0 a 0 e s First, it is known that kxm = m⇡/a and kcn = n⇡/a. Thus (C.61) - (C.64) become Z a ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ x sin x dx, a a (C.66) ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ sin x cos x dx. a a 0 (C.67) 0 sin while (C.65) is given by Z a Now, since sine is an odd function and the multiplication of two odd functions is an even function, (C.66) can be expressed as Z a Z ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ 1 a sin x sin x dx = sin x sin x dx. a a 2 a a 0 a (C.68) If m = n, then the identity from (C.41) can be used. If m 6= n then this integral is equal to 310 zero. Therefore, 8 > a < , m=n ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ 2 sin x sin x dx = > 0, m 6= n a a 0 : Z a or Z a 0 sin (C.69) ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ a x sin x dx = mn . a a 2 (C.70) Next, using a substitution of variables, where u = ⇡/ax, (C.67) becomes Z a Z ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ a ⇡ sin x cos x dx = sin (mu) cos (nu) du. a a ⇡ 0 0 (C.71) The trigonometric identity sin(x) cos(y) = 1 [sin(x + y) + sin(x 2 y)] (C.72) is employed, yielding Z a Z ⇡ ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ a sin x cos x dx = sin [(m + n) u] + sin [(m a a 2⇡ 0 0 n) u] du. (C.73) If m = n, then Z a Z ⇡ ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ a sin x cos x dx = sin (2mu) du, a a 2⇡ 0 0 (C.74) which when evaluated is equal to zero. If m 6= n then ⇢ ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ a cos [(m + n)u] sin x cos x dx = a a 2⇡ m+n 0 Z a 311 cos [(m n)u] m n ⇡ . 0 (C.75) Now if m n is even then Z a 0 and if m sin ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ x cos x dx = 0, a a (C.76) n is odd  ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ a 1 1 sin x cos x dx = + . a a 2⇡ m + n m n 0 Z a (C.77) Thus Z a 0 sin ⇣ m⇡ ⌘ ⇣ n⇡ ⌘ a x cos x dx = mn , a a 2 (C.78) where mn = 8 > 0, < m > 4 m , : ⇡ m2 n2 312 n is even m n is odd. (C.79) C.4: Reduced Aperture Waveguide for Gyromagnetic Materials Closed-form expressions for the matrix entries in (5.55) are derived in this section. The entries are Z a h ⇣ h ⇣ ⌘i a ⌘i 2 e e x a dx sin kxm x sin kxn a 2 2 2 !Z w h ⇣ h ⇣ 2 a ⌘i w ⌘i 2 s e s Jmn = kcn 1 sin kxm x sin kcn x dx w 2 2 µ2 g 2 Z w h ⇣ h ⇣ ⌘i e kxn w ⌘i 2 s e x a dx Kmn = e sin kcm x sin kxn w Zn 2 2 2 h ⇣ s Z w w ⌘i n h s ⇣ w ⌘i 2 ± = kcn s Lmn sin kcm x sin kcn x s w Zn 2 2 2 h ⇣ ⌘i s kcn s x w ± dx. s cos kcn µg n 2 e Gmn = kxn (C.80) (C.81) (C.82) (C.83) First, Gmn is extracted in a similar procedure as the entry Dmn from Appendix C.2. Thus, Gmn is expressed as e a Gmn = kxn mn . 2 (C.84) Next, the integral for Jmn is extracted similarly as was done for Fmn in Appendix C.2, and results in Jmn = 8 > < ↵m = n = 0 (C.85) > Q n [sin(↵m 2 ) cos( n ) sin(↵m 1 )] ↵m cos(↵m 2 ) sin( n ) otherwise, : 2 2 ↵m n e where ↵m = kcm , ⇣ ⌘ s kcn 1 2 /µ2 . g 0, s n = kcn , 1 = w/2 + a/2, 2 = w/2 + a/2, 313 = w, and Q = Now, the integral for Kmn can be evaluated by once again following the steps done for Fmn in Appendix C.2, and is evaluated as k e Jnm Kmn = xn e . QZn (C.86) Finally, the integral for Lmn is evaluated in steps. The first part of the integral examined is s kcn s Zn Z w h ⇣ 2 s sin kcm x w 2 h ⇣ w ⌘i s sin kcn x 2 w ⌘i dx. 2 (C.87) This is a similar integral evaluated to obtain the entry Qmn from Appendix C.2, and may be expressed as s kcn s Zn Z w h ⇣ 2 s sin kcm x w 2 h ⇣ w ⌘i s sin kcn x 2 s w ⌘i wkcn dx = s mn . 2 2Zn (C.88) The second part of the integral is expressed as k s k s ± cn cn s s Z n µg n Z w h ⇣ 2 s sin kcm x w 2 h ⇣ w ⌘i s x cos kcn 2 ⇣ ⇡ Using the substitution of variables u = w w 2 k s k s ± cn cn s s Z n µg n x w ⌘i dx. 2 ⌘ Z w h ⇣ h ⇣ w ⌘i w ⌘i 2 s s sin kcm x cos kcn x dx = w 2 2 2 s s Z kcn kcn ⇡ sin (mu) cos (nu) du. ⌥ s s Z n µg n 0 314 (C.89) (C.90) Then performing similar steps as was done to evaluate the integral in (C.71), (C.90) becomes k s k s ± cn cn s s Z n µg n Z w h ⇣ 2 s sin kcm x w 2 h ⇣ ⌘i w ⌘i s x w dx = cos kcn 2 2 k s k s w ⌥ cn cn mn , s s Z n µg n 2 (C.91) where mn = 8 > 0, < m > 4 m , : ⇡ m2 n2 n is even m n is odd. (C.92) Therefore the entry L± is expressed as mn ◆ s ✓ s kcn w ± = kcn w Lmn mn ⌥ mn . s s Zn 2 µg n 2 (C.93) Additionally, the vectors g and k are given by gm = Gm1 km = Km1 . 315 (C.94) C.5: Waveguide Standard Integrals Closed-form expressions for the matrix entries in (6.27) are derived in this appendix. The entries are Gmn = `,r Jmn = `,r Kmn = `,r Lmn = Z bZ a f f ~m (~) · ~n (~)dxdy, e ⇢ e ⇢ 0 0 Z y `,r Z a f `,r 2 ~m (~) · ~n (~)dxdy, e ⇢ e ⇢ `,r y1 0 Z y `,r Z a 2 ~ `,r (~) · ~ f (~)dxdy, hm ⇢ hn ⇢ `,r y1 0 Z y `,r Z a 2 ~ `,r (~) · ~ `,r (~)dxdy. hm ⇢ hn ⇢ `,r y1 0 (C.95) (C.96) (C.97) (C.98) To simply the derivations, only the integrals for the left aperture are derived in this appendix. This means to get the entries J r , K r , and Lr , use the equations for J l , K l , and Ll , and change all instances of ` to r. The entries depend on whether modes m and n are TE or TM, and are thus split into the following four cases: Case 1: TEm , TEn First Gmn is expressed as Z bZ a Z bZ ah ⇣ ⇡x ⌘ ⇣ ⌘ f f f f ~m (~) · ~n (~)dxdy = e ⇢ e ⇢ xkym cos ˆ sin kym y a 0 0 0 ⌘i h 0 ⇣ ⇡x ⌘ ⇣ ⇣ ⇡x ⌘ ⇣ ⌘ ⇡ f f f y sin ˆ cos kym y · xkyn cos ˆ sin kyn y a a ⇣ ⇡x ⌘ a ⇣ ⌘i ⇡ f y sin ˆ cos kyn y dxdy. a a 316 (C.99) Expanding the dot product, this becomes Z a ⇣ ⌘ Z b ⇣ ⌘ ⇣ ⌘ f f f f 2 ⇡x dx Gmn = kym kyn cos sin kym y sin kyn y dy + a 0 0 Z a⇣ ⌘ ⇣ ⇡x ⌘ Z b ⇣ ⌘ ⇣ ⌘ ⇡ 2 2 f f sin dx cos kym y cos kyn y dy. a 0 a 0 (C.100) Using the integral identities  2 (Ax) dx = x + sin (2Ax) cos 2 4A  x sin (2Ax) sin2 (Ax) dx = , 2 4A (C.101) (C.102) it can be shown that the integrals over x both become a/2. Thus f f a Gmn = kym kyn 2 Z b ⇣ ⌘ ⇣ ⌘ f f sin kym y sin kyn y dy + 0 2Z b ⇡ 2a 0 ⇣ ⌘ ⇣ ⌘ f f cos kym y cos kyn y dy. (C.103) If m = n, then the identities in (C.101) and (C.102) can be used again. If m 6= n then Gmn = 0. Therefore or 8 ⇣ ⌘2 > ab , m = n < kf cn 4 Gmn = > : 0, m 6= n ⇣ ⌘ f 2 ab Gmn = kcn mn . 4 (C.104) (C.105) f A special case does exist when m = n = 1, which means ky1 = 0. When this occurs ⇣ ⌘ f 2 ab Gmn = kcn mn . 2 317 (C.106) Therefore a general expression is given by ⇣ ⌘ f 2 ab Gmn = kcn mn , ⇠ (C.107) f f where ⇠ = 2 for kym = kyn = 0, and otherwise ⇠ = 4. The integral for L` can be derived in a similar set of steps and is evaluated as mn L` mn = ` kcn ` Zn !2 aw` mn , ⇣ (C.108) ` ` where ⇣ = 2 for kym = kyn = 0, and otherwise ⇣ = 4. ` Next, the integral for Jmn is expressed as Z y` Z a h ⇣ ⇡x ⌘ ⇣ ⌘ ⇣ ⇡x ⌘ ⇣ ⌘i ⇡ f f f 2 ` = Jmn xkym cos ˆ sin kym y y sin ˆ cos kym y · ` a a a y1 0 h ⇣ ⇡x ⌘ ⇣ ⌘ ⇣ ⇡x ⌘ ⇣ ⌘i ⇡ ` ` ` ` xkyn cos ˆ ` sin kyn (y y1 ) y sin ˆ cos kyn (y y1 ) dxdy. (C.109) a a a Expanding the dot product gives Z a ⇣ ⌘ Z y` ⇣ ⌘ ⇣ ⌘ f ` f 2 2 ⇡x dx ` ` Jmn = kym kyn cos sin kym y sin kyn (y y1 ) dy + ` a 0 y1 Z a⇣ ⌘ Z y` ⇣ ⌘ ⇣ ⌘ ⇡ 2 2 ⇣ ⇡x ⌘ f 2 ` ` sin dx cos kym y cos kyn (y y1 ) dy. ` a 0 a y1 (C.110) Once again using (C.101) and (C.102) results in Z y` ⇣ ⌘ ⇣ ⌘ f 2 ` ` sin kym y sin kyn (y y1 ) dy + ` y1 Z ` ⇣ ⌘ ⇣ ⌘ ⇡ 2 y2 f ` ` cos kym y cos kyn (y y1 dy. 2a y ` 1 f ` a Jmn = kym kyn 2 318 (C.111) The general expression for B = mn B given in Appendix B.2 can now be used, where mn Z ⌫ 2 ⌫1 sin [↵m (x )] sin [ n (x ⌫1 )]dx. (C.112) It was shown that 8 > < B = mn > : 0, ↵m = n = 0 (C.113) n [sin(↵m 2 ) cos( n ) sin(↵m 1 )] ↵m cos(↵m 2 ) sin( n ) otherwise. 2 2 ↵m n Following the same steps outlined in Section B.2, a similar general integral for A = mn Z ⌫ 2 ⌫1 cos [↵m (x )] cos [ n (x ⌫1 )]dx, (C.114) can be derived. The result is 8 > < A = mn > ↵m [sin(↵m 2 ) cos( n ) sin(↵m 1 )] : 2 2 ↵m n , ↵m = n = 0 (C.115) n cos(↵m 2 ) sin( n ) otherwise. ` Using (C.113) and (C.115), Jmn reduces to ` Jmn = ↵m n a B ⇡2 A + , 2 mn 2a mn (C.116) f ` ` ` where ↵m = kym , n = kyn , = ` , 2 = y2 , and 1 = y1 . ` The integral for Kmn can be derived in a similar set of steps and is evaluated as ` Kmn = ` Jnm . ` f Zm Z n (C.117) Following steps similar to those used for Case 1, the rest of the integrals are derived, and given below. 319 Case 2: TMm , TMn ⌘ f 2 ab kcn mn ⇠ ! ` 2 aw` kcn L` mn mn = ` ⇣ Zn a ⇡2 B ` Jmn = ↵m n A + 2 mn 2a mn ` Jnm ` Kmn = ` f Zm Zn Gmn = ⇣ (C.118) (C.119) (C.120) (C.121) (C.122) Case 3:TEm , TMn Gmn = 0 (C.123) L` mn = 0 (C.124) ⇡ A ⇡ B mn ↵m 2 mn 2 1 h ⇡ A ⇡ B i ` Kmn = ↵n m 2 nm 2 nm ` f Zm Zn ` Jmn = n (C.125) (C.126) Case 4: TMm , TEn Gmn = 0 (C.127) L` mn = 0 (C.128) ⇡ B ⇡ A mn + ↵m 2 mn 2 1 h ⇡ B ⇡ A i ` Kmn = ↵n + m 2 nm 2 nm ` f Zm Zn ` Jmn = n 320 (C.129) (C.130) Appendix D: Analysis of NRW Method Consider an isotropic, lossless, symmetric 2-port network with S-parameters S11 , S12 , S21 and S22 . The goal of this appendix is to establish the conditions under which values of µ and ✏ extracted from the S-parameters using the NRW equations will be real. The S-parameters of a lossless 2-port network satisfy the unitary condition [S]T [S ⇤ ] = [U ] or 2 32 3 3 2 ⇤ S⇤ 6 S11 S21 7 6 S11 12 7 6 1 0 7 4 54 5=4 5. ⇤ ⇤ S12 S22 S21 S22 0 1 (D.1) Writing out the matrix product gives the four equations |S11 |2 + |S21 |2 = 1, (D.2) ⇤ ⇤ S11 S12 + S21 S22 = 0, (D.3) ⇤ ⇤ S12 S11 + S22 S21 = 0, (D.4) |S12 |2 + |S22 |2 = 1. (D.5) If the network is reciprocal, S12 = S21 . Making this substitution and taking the complex conjugate of (D.4) results in both (D.3) and (D.4) becoming ⇤ ⇤ S11 S21 + S21 S22 = 0. 321 (D.6) If the network is also symmetric, such that S11 = S22 , then the allowed values of the Sparameters are subject to two restrictions. The first restriction results from (D.2) and (D.5) both becoming |S11 |2 + |S21 |2 = 1. (D.7) The next restriction is derived by first substituting S11 for S22 in (D.6) yielding ⇤ ⇤ S11 S21 + S21 S11 = 0. (D.8) ⇤ ⇤ S11 S21 + S11 S21 ⇤ = 0, (D.9) A + A⇤ = 0, (D.10) This can be rewritten as or ⇤ where A = S11 S21 . The addition of a complex variable with its complex conjugate is just twice the real part of the variable. Thus (D.9) becomes, ⇤ Re S11 S21 = 0, (D.11) or, by splitting the S-parameters into the real and imaginary parts, Re n S11r + jS11i o S21r + jS21i ⇤ = 0. (D.12) This restriction on the allowed values of the S-parameters reduces to S21r S11r + S21i S11i = 0. (D.13) Examining these restrictions further revels a relationship between S11 and S21 . From 322 (D.13), it can be shown that S21r = S S21i 11i S11r (D.14) S21i = S S21r 11r . S11i (D.15) and Rearranging (D.7) yields |S21 |2 = 1 |S11 |2 , (D.16) or 1 2 2 |S11 |2 = S21r + S21i . (D.17) Then substituting (D.15) results in 2 2 2 S |S11 |2 = S21r + S21r 11r . 2 S11i 1 (D.18) 2 Factoring out S21r and simplifying gives 1 2 |S11 |2 = S21r 2 2 S11i + S11r 2 S11i ! , (D.19) or 1 2 |S11 |2 = S21r 323 |S11 |2 2 S11i ! . (D.20) Solving for S21r yields S21r = ±S11i Q, (D.21) where Q= s 1 |S11 |2 . |S11 |2 (D.22) A similar process of substituting (D.14) into (D.17), gives an expression for S21i : S21i = ±S11r Q. (D.23) Using (D.21) and (D.23), it is easily seen that S21 = ±jQS11 (D.24) ⇤ S21 = ±jQS11 . (D.25) or Two possible values of S21 result since the signs on (D.21) and (D.23) do not have to be the same. However, (D.7) and (D.13) must still hold true. By substituting either (D.24) or (D.25) into (D.7) will result in the expressions equaling 1. (D.13) holds true when substituting (D.24), however, substituting (D.25) will not work. Therefore only (D.24) is a valid relationship between S11 and S21 . Substituting this relationship into the NRW equations, (3.46) and (3.47), gives V1 = S11 (1 ± jQ) (D.26) V2 = S11 ( 1 ± jQ), (D.27) 324 which satisfy |V1 | = |V2 | = 1. From (3.55), q = X ± X2 1, (D.28) where X= 1 V1 V2 . V1 V2 (D.29) Substituting (D.26) and (D.27) into (D.29) yields X= 2 S11 [1 ± jQ] [ 1 ± jQ] 1 2S11 , (D.30) or X= h i 2 1 + S11 1 + Q2 2S11 . (D.31) Using (D.22) gives " # 2 1 |S11 | 2 1 + S11 1 + |S11 |2 X= , 2S11 (D.32) or 2 S11 1+ |S11 |2 X= . 2S11 (D.33) Multiplying both sides of the equation by |S11 |2 produces |S11 |2 X = 2 |S11 |2 + S11 2S11 325 , (D.34) and S11 can now be split into real and imaginary parts yielding 2 2 2 S 2 + S11i + S11r S11i + 2jS11r S11i |S11 |2 X = 11r . 2S11 (D.35) This can be simplified to S 2 + jS11r S11i |S11 |2 X = 11r . S11 (D.36) Then by factoring S11r gives S + jS11i |S11 |2 X = S11r 11r , S11 (D.37) |S11 |2 X = S11r . (D.38) or Thus it is found X= S11r , |S11 |2 regardless of the sign chosen in (D.24) and it can be seen that X is real. Evaluating (D.39) from (D.28) then leads to two possible cases: Case 1: X 2 > 1. In this case is real. Examining (3.50) and splitting V1 into real and imaginary parts yields P = V1r + jV1i . 1 V1r + jV1i 326 (D.40) The magnitude of P is expressed as |P | = (V1r (1 2 )2 + V1i , 2 V1r )2 + 2 V1i (D.41) or 2 2 V1r + 2 2 V1r + V1i |P | = . 2 2 1 + 2 V1r 2 V1r + 2 V1i (D.42) Now substituting (3.46) gives (S21r + S11r )2 + 2 2 (S21r + S11r ) + S21i + S11i 2 |P | = . 1 + 2 (S21r + S11r )2 2 (S21r + S11r ) + 2 S21i + S11i 2 (D.43) By expanding the products and using (D.13) gives |P | = |S21 |2 + |S11 |2 + 2 2 (S21r + S11r ) ⇣ ⌘ . 1 + 2 |S21 |2 + |S11 |2 2 (S21r + S11r ) (D.44) With the knowledge of (D.7), the magnitude of the propagation factor is determined to be |P | = 1 (D.45) P = 1ej , (D.46) Thus P can be expressed as and from (3.56), it is determined that s 1= 2n⇡ , d which is a real value. Finally, with the knowledge that 327 (D.47) s and 1 are both real and from (3.59) and (3.62) it is found that both µ and ✏ are real. Case 2: X 2  1. In this case = A±jB, where B = p 1 A2 . This yields | | = 1, which would represent total reflection in an actual material measurement scenario. Thus, this condition should be avoided when designing a material surrogate. Regardless, there might still exist a special case where ✏ and µ are real. In (3.59), occurs in the ratio 1+ . R=1 Therefore, start by substituting (D.48) = A ± jB into (D.48). This results in (1 + A) ± jB R = (1 A) ⌥ jB . (D.49) Next, multiplying the top and bottom by the complex conjugate of the denominator results in R= [(1 + A) ± jB] [(1 A) ± jB] (1 A)2 + B 2 (D.50) or R= ⇣ 1 ⌘ 2 ± jB (1 A 2 (1 This simplifies to A + 1 + A) . A) R = ±j r 1 R = ±j r 1+A 1 A or B B A (D.51) (D.52) (D.53) and is thus imaginary. Now as was done in Case 1, when examine the propagation factor, 328 (3.50) and are split in real and imaginary parts. This results in P = V1r + jV1i 1 r +j i r +j i . V1r + jV1i (D.54) or P = V1r r + j V1i i . V1r + i V1i j i V1r + r V1i r 1 (D.55) Now multiplying the top and bottom by the complex conjugate of the denominator results in P = ⇥ V1r ⇤ ⇤⇥ 1 r + j V1i r V1r + i V1i + j i V1r + r V1i i , 1 V1r + i V1i 2 + i V1r + r V1i 2 r P = ⇥ V1r r + j V1i (D.56) or i ⇤⇥ 1 r V1r + i V1i + j D i V1r + r V1i ⇤ , (D.57) where D= 1 r V1r + i V1i 2+ 2 i V1r + r V1i . (D.58) Writing P = Pr + jPi , the imaginary part of P is examined. Multiplying both sides of the (D.58) by D and analyzing the imaginary part yields DPi = V1i i 1 r V1r + i V1i + (V1r r) i V1r + r V1i . (D.59) Expanding and simplifying the expression results in h DPi = V1i 1 i h | |2 + i |V1 |2 329 i 1 . (D.60) Using | | = 1 and |V1 | = 1, this expression reduces to DPi = 0, (D.61) which proves that P is a real quantity. Now, examining the real part of P shows DPr = (V1r r) 1 r V1r + i V1i + V1i i i V1r + r V1i , (D.62) and through similar simplification process that was done for the imaginary part of P , it is determined that DPr = 2 (V1r r) . (D.63) Expanding the products in D and canceling like terms results in D=2 1 r V1r + i V1i . (D.64) Finally, the propagation constant for Case 1 can be expressed as P = V1r r , r V1r + i V1i 1 (D.65) <{V1 <{1 (D.66) or P = } . V1 } Case 2 can now be split into 2 cases: Case A: P > 0 In this case P = |P |ej0 330 (D.67) and thus ln P = ln |P |, (D.68) which is real. So (3.56) then becomes s ln |P | ± j2n⇡ , 1= jd (D.69) s which is complex unless n = 0. In this case 1 is imaginary. In (D.53) it is shown that R s is imaginary, thus when 1 is complex, which occurs for all values of n except n = 0, µ will be complex. If µ is complex, then from (3.62), it can be deduced that ✏ will be complex. s Now when n = 0, 1 is imaginary and R is imaginary, thus µ will be real. If µ is real, then from (3.62), it can be deduced that ✏ will be real. Case B: P < 0 In this case P = |P |e±j⇡ (D.70) ln P = ln |P | ± j⇡. (D.71) s ln |P | ± j⇡ ± j2n⇡ , 1= jd (D.72) and thus Then from (3.56), s and is complex, since ±j⇡ can never cancel with ±j2n⇡. Thus if 1 is complex both µ and ✏ will be complex. So, in summary if the 2-port network being analyzed is lossless and symmetric then 331 X = S11r /|S11 |2 and two cases exist to determine if µ and ✏ are real or complex. First it was determined if X 2 1 then µ and ✏ are real. Next, if X 2 < 1 then 1+ 1 = ±j r 1+A 1 A (D.73) and is thus imaginary. Then with some e↵ort it was shown that P = <{V1 <{1 } V1 } (D.74) s and is thus real. So, if P is positive and n = 0 is chosen in (3.56), 1 will be imaginary and again µ and ✏ will be real. 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