# Synthesis of spatial filters and broadband microwave absorbers using micro-genetic algorithms

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Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The Pennsylvania State University The Graduate School College o f Engineering SYNTHESIS OF SPATIAL FILTERS AND BROADBAND MICROWAVE ABSORBERS USING MICRO-GENETIC ALGORITHMS A Thesis in Electrical Engineering by Sourav Chakravarty © 2001 Sourav Chakravarty Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2001 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3014607 UMI UMI Microform 3014607 Copyright 2001 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. W e approve the thesis o f Sourav C hakravarty. D ate o f S ign atu re Raj Mittra Professor o f Electrical Engineering Thesis AdvisoK'ChairJbf Com m ittee ^ / . (aI siamQ/^ 3 /a?/of Werner Ass Professor o f Electrical Engineering V * 1 / 0 1 ___________ Lynrf A. Carpente A ssociate Professor o f Electrical Engineering Professor.of Electrical Engineering M ^ d U 2 - 5 , 2 -QO' ikhfles"tiLcjJiftWkia ^^pfessonsf Engineering Science and echanics t Vijay K. (jraradan Distinguished A lum ni Professor o f Engineering Science and M echanics and Electrical Engineering 3 J z 6/a/ W. Kennet,h,Jenkins Professor o f Electrical Engineering Head o f the Department o f Electrical Engineering Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Over the years, Frequency Selective Surfaces (FSSs) have found frequent use as radomes and spatial filters in both commercial and military applications. The FSSs are ofren embedded within dielectric or magnetic materials that may be lossy or lossless. Past research has concentrated on the synthesis of broadband microwave absorbers and spatial filters using multilayered dielectrics via the application of Genetic Algorithms (GAs). Typically, these problems can be categorized, relatively speaking, as being computationally inexpensive. A comprehensive research effort has not been made to apply the GA to synthesize broadband microwave absorbers and spatial filters with embedded FSS screens, which can be termed as a computationally expensive problem. To enhance the computational efficiency of the GA, a variant of the Conventional Genetic Algorithm (CGA), referred to as the Micro-Genetic Algorithm (MGA), is introduced here. The MGA is applied to optimize various parameters, viz., the thickness and relative permittivity or permeability o f each layer, the FSS screen design and materials, the x- and y-periodicities of the FSS screen, and their placement within the dielectric composite. The result is a multilayer composite that simultaneously provides a maximum reflection or transmission of both TE and TM waves for a prescribed range of frequencies and incident angles, while automatically placing an upper bound on the total thickness o f the composite. Three basic types of problem geometries have been considered to illustrate the numerical efficiency of the MGA: (i) composites with no FSS Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. screens embedded in them; (ii) composites with a single embedded FSS screen; and (iii) two FSS screens embedded in the composite. For the first problem, a recursive formulation for layered inhomogeneous media is utilized to calculate the reflection and transmission coefficients of the composite. The second problem is analyzed by using a frequency domain Electric Field Integral Equation (EFIE) formulation in conjunction with a spectral domain Method of Moments (MoM) solution. Finally, for the third problem, the Generalized Scattering Matrix (GSM) approach is used to cascade multiple FSS screens and generate reflection and transmission coefficients. MGA, which is a variant of the CGA, uses a very small population base for the optimization process, and this serves as the key to its numerical efficiency. CGAs perform poorly with small population sizes due to insufficient information processing and they converge prematurely to non-optimal results. The MGA has two major advantages over the CGAs: (i) it works with a small population base for each generation; and (ii) it reaches near-optimal regions faster than the CGAs that work with a large population base. The general choice o f population size for the CGAs can range between 100 and 10000, while the MGAs typically work with a population size between 5 and 50. Numerical experiments show that using the MGA can decrease the computational run time by 50%, even for “best-case” problems for the CGAs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V TABLE O F CONTENTS LIST OF TABLES........................................................................................................ viii LIST OF FIGURES...................................................................................................... ix ACKNOWLEDGEMENTS............................................................................................ xiv Chapter I INTRODUCTION......................................................................................... 1 1.1 Research Objective and General Technique............................................... 1.2 Thesis Outline............................................................................................... 3 6 Chapter 2 THE OPTIMIZATION ALGORITHM....................................................... 9 2.1 Genetic Algorithm Fundamentals................................................................ 2.1.1 GA Terminology........................................................................... 2.1.2 Overview o f Genetic Algorithms................................................ 2.1.3 GA Operators................................................................................ 2.1.3.1 Selection Operator........................................................ 2.1.3.2 Crossover Operator....................................................... 2.1.3.3 Mutation Operator........................................................ 2.1.4 Selection Schemes........................................................................ 2.1.4.1 Population Decimation................................................ 2.1.4.2 Roulette Wheel Selection............................................ 2.1.4.3 Tournament Selection................................................... 2.1.5 Crossover Schemes....................................................................... 2.1.5.1 Single Point Crossover................................................ 2.1.5.2 Dual Point Crossover.................................................... 2.1.5.3 Uniform Crossover....................................................... 2.1.6 Mutation Schemes........................................................................ 2.1.7 Elitism............................................................................................ 2.1.8 Fitness Function............................................................................ 2.1.9 Classification o f Genetic Algorithms......................................... 2.2 The Conventional Genetic Algorithm (CGA)........................................... 2.2.1 Population Sizing for the CGA.................................................... 2.2.2 Drawbacks o f the CGA................................................................. 2.3 Micro-Genetic Algorithm (MGA).............................................................. 2.3.1 The MGA Optimization Procedure............................................ 2.3.2 Advantages of the MGA............................................................... 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Further reproduction prohibited without permission. 9 9 11 12 13 13 13 14 14 15 16 17 17 19 21 22 22 23 24 25 26 27 28 29 31 vi Chapter 3 SYNTHESIS OF COMPOSITES COMPRISING ONLY DIELECTRIC AND MAGNETIC MATERIALS.................................................................................. 3.1 Problem Geometries..................................................................................... 3 .2 Formulation o f Reflection and Transmission Coefficients for Inhomogeneous Layered Media........................................................................ 3 .3 MGA Formulation for the Spatial Filter and Broadband Microwave Absorber Problems............................................................................................. 3.3.1 Spatial Filter Synthesis................................................................. 3.3.2 Numerical Results for Spatial Filter Design............................... 3.3.3 Synthesis o f Broadband MicrowaveAbsorbers........................ 3 .3 .4 Numerical Results for the Synthesis o f Broadband Microwave Absorbers............................................................................................... 3 .3 .5 Numerical Results for the Synthesis o f Broadband Microwave Absorbers Using Lossy Carbon Fiber and Lossless Dielectric Materials................................................................................................. Chapter 4 SYNTHESIS OF COMPOSITES COMPRISING A SINGLE FREQUENCY SELECTIVE SURFACE (FSS) EMBEDDED IN DIELECTRIC AND/OR MAGNETIC MEDIA..................................................................................... 4.1 Problem Geometries..................................................................................... 4.2 Analysis o f Doubly-Periodic, Planar, Infinite FSS Screens Embedded in Inhomogeneous Layered Media........................................................................ 4.2.1 Numerical Solution Employing the Method of Moments (MoM).................................................................................................... 4.2.2 Evaluation of Reflection and Transmission Coefficients 4.3 MGA Formulation for the Spatial Filter and Broadband Microwave Absorber Problems............................................................................................. 4.3.1 Broadband Microwave Absorber Synthesis............................... 4.3.1.1 Numerical Results for Single FSS Screen Embedded in Lossy or Lossless Dielectrics............................................... 4.3.1.2 Numerical Results for Single FSS Screen Embedded in Dielectric and Magnetic Media that can Either be Lossy or Lossless.................................................................................. 4.3.1.3 Numerical Results for a Single FSS Screen Embedded in Lossy Carbon Fiber and Lossless Dielectric Material....................................................................................... 4.3.2 Spatial Filter Synthesis................................................................. 4.3.2.1 Numerical Results for Spatial Filter Synthesis 4.3.2.2 The Domain Decomposition Approach Applied to the Design o f Spatial Filters..................................................... 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Further reproduction prohibited without permission. 34 34 36 40 40 43 44 48 64 68 69 71 87 96 98 99 104 126 129 132 132 136 vii Chapter 5 SYNTHESIS OF COMPOSITES COMPRISING MULTIPLE FREQUENCY SELECTIVE SURFACES (FSSs) EMBEDDED IN DIELECTRIC AND/OR MAGNETIC MEDIA.................................................................................... 5.1 Problem Geometry....................................................................................... 5.2 Analysis of Multiple FSS Screens Embedded in Dielectric and Magnetic Medium.............................................................................................. 5.3 MGA Formulation for Spatial Filters and Broadband Microwave Absorber Designs.............................................................................................. 5.3.1 Broadband Microwave Absorber Synthesis............................... 5.3.1.1 Numerical Results for Two FSS Screens Embedded in Lossy or Lossless Dielectrics............................................... 5.3.1.2 Numerical Results for Multiple FSS Screens Embedded in Dielectric and Magnetic Medium that is Either Lossy or Lossless...................................................................... 5.3.2 Spatial Filter Synthesis................................................................ 5.3.2.1 Numerical Results for Spatial Filter Synthesis Chapter 6 THE PARALLEL IMPLEMENTATION OF THE MICRO-GENETIC ALGORITHM (MGA).................................................................................................. 6.1 Reasons for Parallelization......................................................................... 6.2 Micro-Grain MGA....................................................................................... 6.3 Design of Polarization Selective Surfaces (PSSs) for Dual Reflector Applications........................................................................................................ Chapter 7 VALIDATION OF THE RESULTS GENERATED BY THE MICROGENETIC ALGORITHM (MGA)................................................................................ 142 143 145 157 157 163 180 186 186 191 193 195 199 205 7.1 The FDTD Method...................................................................................... 7.2 The Direct Method of Moments (MoM) Approach to Analyze Multiple FSS Screens........................................................................................................ 7 .3 Application of the FDTD Method to Analyze Single Screen FSS Embedded in Inhomogeneous Media............................................................... 7.4 Application of the Direct-MoM to Analyze Multiple FSS Screens Embedded in an Inhomogeneous Composite................................................... 205 Chapter 8 SUMMARY, CONCLUSIONS, AND FUTURE WORK......................... 226 8.1 Improving the Efficiency of the Optimization Process............................. 8.2 Accuracy o f the MGA Optimized Results.................................................. 8.3 Suggestions for Further Research............................................................... 226 228 229 BIBLIOGRAPHY........................................................................................................... 231 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 210 211 216 viii LIST OF TABLES 3-1 GENETIC ALGORITHM PARAMETER SEARCH SPACE 43 3-2 e\ AS A FUNCTION OF FREQUENCY FOR THE LOSSY DIELECTRIC MATERIAL.......................................................................... 61 e” r AS A FUNCTION OF FREQUENCY FOR THE LOSSY DIELECTRIC MATERIAL.......................................................................... 62 3-4 GA PARAMETER SEARCH SPACE......................................................... 63 3-5 MATERIAL DISTRIBUTIONS AND THICKNESS (MM) OF COMPOSITE FOR THE THREE C A SES................................................... 63 3-6 PERFORMANCE COMPARISION OF CGA AND MGA 64 4-1 PARAMETERS SELECTED BY THE GA FOR THE 7 CA SES 120 4-2 MGA PARAMETER SEARCH SPACE....................................................... 121 5-1 MGA PARAMETER SEARCH SPACE FOR ONE SUB-COMPOSITE 179 5-2 PARAMETERS SELECTED BY THE MGA FOR THE TWO CASES 180 3-3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ix LIST OF FIGURES Single point crossover with p ^ Peross ....................... ..................................... Single point crossover withp ^ Pcross. .............................. Dual point crossover withp > pcro« . ............................................................. Dual Point Crossover with p < Pcross. ............................................................. Uniform Crossover with p > / w * . ................................................................ Procedural flowchart o f the CGA................................................................... Procedural flowchart o f the MGA.................................................................. Synthesis o f spatial filters with lossless dielectrics. A/-layered composite with a TE polarized uniform plane wave incident at an arbitrary angle. For TM polarization the E and H fields are interchanged along with a phase shift of 180 degrees introduced in the H field................................... 3.2 Synthesis o f broadband microwave absorbers with lossy or lossless dielectrics and lossy magnetic materials, ^-layered composite with a TE polarized uniform plane wave incident at an arbitrary angle. For TM polarization the E and H fields are interchanged along with a phase shift o f 180 degrees introduced in the H field....................................................... 3.3 Three-layered medium with TE polarized wave incident at an arbitrary angle................................................................................................................. 3.4 Low pass filter design. The weights used for the MGA were 1.3 for 12.2 to 18.0 GHz and 1.0 for the rest of the band. The desired cutoff is at 13 .0 GHz.................................................................................................................. 3.5 High pass filter design. The weights used for the MGA were 1.3 for 14.0 to 18.0 GHz and 1.0 for the rest of the band. The desired cutoff is at 14.0 GHz.................................................................................................................. 3.6 Band stop filter design. The weights used for the MGA were 1.5 for 1.0 to 11.0 GHz and 15 .0 to 18.0 GHz, and 1.0 for the rest of the band. The desired cutoffs are at 10.0 and 15.0 GHz...................................................... 3.7 Band pass filter design. The weights used for the MGA were 1.3 for 1.0 to 11.0 GHz and 15 .0 to 18.0 GHz, and 1.0 for the rest of the band. The desired cutoffs are at 10.0 and 15.0 GHz...................................................... 3 .8 Real part o f permittivity o f magnetic material................................................ 3.9 Imaginary part of permittivity of magnetic material..................................... 3.10 Real part of permeability of magnetic material.............................................. 3.11 Imaginary part o f permeability o f magnetic material.................................... 3.12 (a) Worst- and best-case reflection coefficient levels for the composite with only lossless and lossy dielectric materials.................................................. 3.12 (b) Design o f composite with only lossy and lossless dielectric materials 3.13 (a) Worst- and best-case reflection coefficient levels for the composite with only lossless dielectric and lossy magnetic materials.................................. 2.1 2.2 2.3 2.4 2.5 2.6 2.7 3.1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 19 20 20 21 27 33 35 36 37 45 45 46 46 52 53 54 55 56 56 57 X 3 13(b) Design o f composite with only lossy magnetic and lossless dielectric materials.......................................................................................................... 3.14 (a) Worst- and best-case reflection coefficient levels for the composite with interspersed lossy or lossless dielectric and magnetic materials................ 3 .14 (b) Design of composite with both lossy and lossless magnetic and dielectric materials.......................................................................................................... 3.15 Evolution of the average and best fitness values of the CGA and MGA for Case-1........................................................................................................ 3.16 Evolution of the average and best fitness values of the CGA and MGA for Case-2........................................................................................................ 3.17 Evolution of the average and best fitness values of the CGA and MGA for Case-3........................................................................................................ 3.18 The permittivity o f carbon fiber material as a function of frequency: (a) real part; (b) imaginary part........................................................................... 3 .19 Case-1: (a) composite design; (b) reflection coefficient vs. frequency 3.20 Case-2: (a) composite design; (b) worst- and best-case reflection coefficient vs. frequency................................................................................ 4.1 A'-layered composite with a TE polarized uniform plane wave incident at an arbitrary angle. For TM polarization the E and H fields need to be interchanged along with a phase shift of 180 degrees introduced in the H field................................................................................................................. 4.2 Synthesis of broadband microwave absorbers with a single FSS screen embedded in lossy or lossless dielectric and magnetic material with PEC backing. jV-layered composite with a TE polarized uniform plane wave incident at an arbitrary angle. For TM polarization the E and H fields need to be interchanged along with a phase shift of 180 degrees introduced in the H field................................................................................ 4.3 The composite structure to be analyzed: (a) side view; (b) top view 4.4 The original problem is split into two sub-problems..................................... 4.5 Surface equivalence principle applied to Problem-II: (a) equivalence principle applied to Region-I; (b) perfect Magnetic conductor (PMC) used to short out the surface magnetic current density; (c) image theory applied to the Region-I problem................................................................... 4.6 The transmission line equivalent of the FSS scattering problem for Region-I.......................................................................................................... 4.7 The admittance seen by the current induced on the FSS screen for R superstates...................................................................................................... 4.8 x- and y-components of the unit amplitude rooftops..................................... 4.9 FSS cell discretized into 16 x 16 pixels with the shaded part corresponding to metal................................................................................... 410 Grouping the infinite summation over 0 ^ into M segments of M 4.11 elements. The like terms o f each segment are summed, forming an array A, composed o fM elements. In this caseM =M=4.................................... Multilayered composite with single embedded FSS screen and uniform Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 58 58 59 59 60 65 66 67 69 70 72 74 74 80 81 90 92 93 99 xi 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 4.35 4.36 5.1 5 .2 5.3 5.4 A 16 x 16 pixel FSS unit cell design represented in the form of “ l ’s” and “0’s” ................................................................................................................ Design for Case-1......................................................................................... Design for Case-2......................................................................................... Design for Case-3......................................................................................... Case-3: Reflection coefficient as a function of the elevation angle with frequency as a parameter............................................................................... Design for Case-4......................................................................................... Design for..Case-5.......................................................................................... Design for..Case-6......................................................................................... Case-6: Reflection coefficient as a function of the elevation angle with frequency as a parameter............................................................................... Design for..Case-7.......................................................................................... Fitness value vs., the number of generations for Case-7.............................. Population distribution of the MGA in the 1st, 40th, and 75th generations for Case-7........................................................................................................ Frequency response o f the composite shown in Fig. 4.21 with a FSS screen having two-fold or top-bottom symmetry embedded in it............... Frequency response o f dielectric composite with a single FSS screen possessing eight-fold symmetry embedded in it........................................... Worst- and best-case reflection coefficients over the frequency band for composite with FSS screen in Fig. 4.25....................................................... Design for composite with both dielectric and magnetic materials Population distribution for the optimization carried out for the case with lossy magnetic layers in the composite......................................................... Fitness value vs. the number of generations for case with magnetic losses in the composite............................................................................................. The permittivity of carbon fiber material as a function of frequency: (a) real part; (b) imaginary part.......................................................................... Design o f carbon fiber composite with FSS screen embedded in it: (a) FSS unit cell design (white - metal, black - free space); (b) composite design; (c) frequency response of composite in (b)..................................... Idealized response of spatial fi Iter.................................................................. Spatial filter design with single FSS screen embedded in lossless dielectric medium.......................................................................................... Sensitivity study of spatial filter design with single FSS screen embedded in it................................................................................................................... 64 x 64 pixel modified DSL FSS screen embedded in dielectric composite........................................................................................................ Frequency response o f composite in Fig. 4.35.............................................. Problem geometry............................................................................................ Schematic representation of a two-port system............................................. The scattering matrix structure and contents for composites comprising multiple FSS screens embedded in layered media........................................ System division into sub-systems................................................................... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 108 109 110 112 113 114 115 117 118 119 119 123 124 125 127 128 129 130 131 133 135 136 140 141 144 145 152 154 xii 5.5 5.6 5.7 (a) 5.7(b) 5.8 (a) 5.8 (b) 5 .9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5 .21 5.22 5.23 5.24 5 25 6.1 6.2 6.3 6.4 7.1 Composite system........................................................................................... FSS unit cell with either two-fold, four-fold, or eight-fold symmetry Composite design for Case-1......................................................................... Composite design for Case-2......................................................................... FSS unit cell design for Case-1...................................................................... FSS unit cell design for Case-2...................................................................... Frequency response o f composite in Fig. 5.7 (a).......................................... Worst- and best-case reflection coefficients of composite in Fig 5.7 (b): (a) TE polarization; (b) TM polarization; (c) TE and TM polarization Performance o f the MGA: (a) Case-1; (b) Case-2....................................... Population distribution w.r.t. total thickness of composite and reflection coefficient for Case-1..................................................................................... Population distribution w.r.t. total thickness of composite and reflection coefficient for Case-2..................................................................................... Invariance o f the frequency response with azimuthal angle for composite inCase-1......................................................................................................... Invariance o f the frequency response with azimuthal angle for composite in Case-2: (a) 0 = <J>= 0 degrees, 0 = 0 and <J>= 45 degrees; (b) 0 = 45 and <(>= 0 degrees, 0 = <J>= 45 degrees................................................................. Effect of the change in the number o f harmonics included in the scattering matrix on the frequency response: (a) Case-1, 0 = 45, <f>= 0; (b) Case-2, 0 = 45, <J>= 0 ............................................................................... Plot of the worst-case higher order harmonics vs. frequency for Case-1: (a) TE polarization; (b) TM polarization................................................... Plot of the worst-case higher order harmonics vs. frequency for Case-2: (a) TE polarization; (b) TM polarization................................................... FSS unit cell design for the two FSS screens embedded in the composite in Fig. 5.20...................................................................................................... Synthesized composite for two FSS screens embedded in layered media............................................................................................................... Worst- and best-case reflection coefficient of composite in Fig 5.20: (a) TE polarization; (b) TM polarization; (c) TE and TM polarization Population distribution with evolving generations........................................ MGA performance to optimize composite in Fig. 5.20................................. Idealized response of spatial filter.................................................................. Spatial filter synthesis: (a) composite design; (b) FSS unit cell design; (c) frequency response......................................................................................... Pseudocode for a micro-grain micro-genetic algorithm................................ Problem geometry for PSS optimization........................................................ FSS unit cell for PSS design application........................................................ Frequency response of PSS: (a) 0 = 0, <J>= 90; (b) 0 = 10, <j>= 90; (c) 0 = 20, (j>= 90; (d) 0 = 30, 4» = 90; (e)0 = 40, <f>= 90....................................... The Yee cell with the six electric and magnetic field components all offset by half a space step.............................................................................. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 160 167 168 168 169 169 171 172 173 173 174 175 176 177 178 181 182 184 185 185 187 189 197 200 202 204 207 xiii 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 FSS screen for example in Fig. 4.25: (a) MGA-generated FSS unit cell; (b) after processing...................................................................................... Comparison between FDTD and MoM results for example 1 in Fig. 4.25 MGA-generated spatial filter design: (a) composite; (b) FSS unit cell design............................................................................................................... Comparison between FDTD and MoM results for spatial filter design with four lossless dielectric layers and one FSS screen............................... Comparision between the GSM-MGA and the Direct-MoM for Case-1: (a) TE polarization; (b) TM polarization...................................................... Comparision between the GSM-MGA and the Direct-MoM for Case-2: (a) TE polarization; (b) TM polarization...................................................... Comparision between the GSM-MGA technique and the Direct-MoM for Case-3: (a) normal incidence-TE polarization; (b) normal incidenceTM polarization; (c) 45° from broadside-TE polarization; (d) 45° from broadside-TM polarization............................................................................ Comparision between the GSM-MGA and the Direct-MoM for Case-1 with 9 harmonics included in the scattering matrix: (a) TE polarization; (b) TM polarization......................................................................................... Comparision between the GSM-MGA and the Direct-MoM for the composite with low-loss dielectric layers and two FSS screens................. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 213 214 215 216 218 219 222 223 225 xiv ACKNOWLEDGEMENTS I am dedicating this thesis to my parents, for their patience, support and unbounded love throughout my graduate studies. Without their encouragement over the years, the completion of this thesis would not have been possible. 1 would like to also thank my master’s thesis adviser Dr. M. N. Roy for encouraging me to pursue doctoral studies. 1 am indebted to my adviser. Dr. Raj Mittra for his valuable guidance and support throughout the course of this research. I also thank my committee members for patiently reviewing my thesis and administering the Comprehensive and Final Oral Examinations My deepest appreciation for the support and advice from all the members of the Electromagnetic Communications Laboratory at Penn State, especially Dr. Wenhua Yu, Dr. Ji-Fu Ma, Dr. Nader Farahat and Dr. Slava Bulkin. Their helpful suggestions and assistance are gratefully acknowledged. Finally, I would like to thank Dr. Neil R. Williams at W. L. Gore and Associates Inc., and Dr. V. K. Varadan for providing me with the lossy dielectric and magnetic material values. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 INTRODUCTION In recent years, Genetic Algorithms (GAs) [1], which are robust and efficient optimization techniques, have received widespread attention of the engineering community. Since its inception by Holland [2], GAs have come a long way towards solving real world problems, especially where other conventional optimization schemes appear to have failed. GAs and its variants have been applied to solve many design problems in electromagnetics, involving spatial filters [3-5], array antenna configurations [6-7], and wire antennas [8], Though Conventional GAs (CGAs) [1] have been applied to many optimization, synthesis, and inverse problems with enormous success, only a few of these can be categorized as computationally intensive. To date, the problems solved by the CGAs have either been formulated by using simple analytic functions [3-5], or by employing numerical Green’s function for fixed geometries. Furthermore, CGAs have not been applied to computationally intensive problems, since they require a large number of objective function evaluations to achieve convergence. This is because the CGAs are inherently stochastic search processes and they perform the search in a large solution space [1]. The objective of this effort is to synthesize broadband microwave absorbers and spatial filters using a composite comprising a combination of dielectric and magnetic materials with or without doubly periodic, multiple, Frequency Selective Surfaces (FSSs) [9] embedded within it. Throughout this thesis, only isotropic materials are considered. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FSS structures, also referred to as spatial filters, are counterparts of filter circuits used in the microwave frequency range and they find widespread applications as radomes [10] and reflectors [11]. Considering the synthesis problem as stated above, it is evident that with no FSS screens within the composite, the objective function has a simple recursive form [12], and evaluating it is computationally inexpensive. However, in the presence of FSS screens, we require the Method of Moments (MoM) to solve for the unknown current coefficients on the metalized portions of the FSS structures, and this is a very time-consuming, computationally intensive procedure. Since the CGAs typically search from a large space o f potential solutions, and the matrix solution has to be repeated for each iteration step, this leads to an impractical run time to achieve convergence, and prompts one to seek more advanced and numerically efficient optimization techniques. A possible solution to decreasing the run-time is to reduce the size of the search space of the potential solutions. However, the caveat is that the CGAs cannot be employed as an optimizer if this is done, because they exhibit very poor convergence to global extrema with reduced search spaces. Hence, to work with such reduced search spaces, we need a version o f the GA whose convergence to global extrema is not affected as the search space is reduced. A good candidate for this type of GA is the Micro-genetic Algorithm (MGA) [13], which is a small-search-space-based GA with good convergence properties. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.1 Research Objective and General Technique The objective of this thesis is to develop a numerically efficient optimization technique for synthesis of spatial filters or broadband microwave absorbers. The generic problem geometry of interest is a composite comprising of multiple FSS screens embedded in lossy or lossless dielectric or magnetic materials, with or without a Perfect Electric Conductor (PEC) backing. The known quantities for the synthesis problem are the frequency response, the allowed total thickness of the composite, the range o f values for relative permittivities and permeabilities o f lossless dielectric/magnetic material, and the frequency dependent real and imaginary parts of relative permittivity and permeability o f lossy dielectric/magnetic material. The synthesized quantities are the thickness o f each layer, their constitutive parameters, the design and material of the FSS screen, the position o f this screen within the composite, and the total thickness of the composite. The MGA is initially applied to the synthesis of composites without FSS screens. This is done to familiarize oneself with the various parameters of the MGA and their effect on the optimization results. A recursive formulation for heterogeneous layered media [12] is employed to calculate the reflection coefficient of the composites. Both spatial filters and broadband microwave absorbers are designed. The MGA optimizes the thickness and constitutive parameters for each layer resulting in a multi-layered composite that provides maximum absorption o f both TE and TM waves simultaneously for a prescribed range of frequencies and incident angles. A vast improvement in both the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 computational efficiency and convergence is observed in the MGA as compared to the CGA for both of these sets of problems. The first problem geometry that is expected to be computationally intensive involves the optimization o f a single lossy or lossless FSS screen embedded in dielectric or magnetic materials that may be lossy or lossless. Periodicity in the FSS screen is handled by applying Floquet’s theorem [14], according to which the surface currents induced over the infinite FSS due to an incident plane wave can be expressed in terms of the current on a single unit cell. Next, the spectral domain MoM [15] technique is applied to evaluate the unknown current coefficients on the metalized portion of the FSS unit cell. Once the induced current on the infinite FSS screen is known, the electric field scattered from the FSS screen is easily evaluated [15]. Finally, transmission line techniques in conjunction with the spectral immitance approach [16] are employed to determine the reflection coefficient of the FSS-dielectric combination. The MGA optimizes the thickness and constitutive parameters for each layer, the design, material and position of the FSS screen within the composite. This results in a multi-layered composite that provides maximum absorption of both TE and TM waves simultaneously for a prescribed range of frequencies and incident angles. For designs requiring optimization at very high frequencies, e.g., 95.0 GHz, the FSS unit cell resolution has to be increased to account for the fine geometrical features. This leads to a large number of unknowns to be solved for in the MoM, and results in a drastic increase in the computation time for a single objective function evaluation, prompting us to seek an alternate approach to the optimization process. The domain Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 decomposition [10] technique is employed to handle these problems, wherein the dielectric composite is synthesized by using the MGA, while conventional FSS screen design techniques are employed to design the FSS screen. Finally, the two separate designs are combined to satisfy the design criteria. The second problem geometry considered consists of multiple FSS screens embedded within the composite, it has been found [39] that two or more layered FSS screens are needed to provide the necessary degrees of freedom for meeting stringent design specifications, which requires the mutual coupling between the screens to be modeled accurately. This, in turn, requires that the surface currents on each screen be generated, and then translated to scattered electric fields, which have to be made incident on all other FSS screens within the composite to generate the total scattered fields. However, using the direct method to model the currents on each FSS screen leads to a large number o f unknowns to be solved for, and this precludes the application of the MGA to the synthesis problem. Consequently, we employ an alternate “circuit theory” approach called the Generalized Scattering Matrix (GSM) [17-18] technique. In this method we derive, as a first step, the generalized scattering matrices of the individual screens by using the MoM, and of the dielectric layers by following the procedure we described in the previous page. These matrices can be subsequently used to generate a composite scattering matrix for the entire system. Both spatial filters and broadband microwave absorbers were designed successfully with the MGA using the GSM technique. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 1.2 Thesis Outline Chapter 2 provides a background for the optimization technique employed for the synthesis. First, the basic GA fundamentals are discussed and a brief discussion is presented on the stochastic operators used, the fitness or objective function employed, and the classification of the GA. Next, the CGA is described with the aid of a flowchart, and a discussion on population sizing as well as the drawbacks of the CGA is presented. Finally, the MGA is described with the help of a flowchart along with a discussion on population sizing, population restart strategy, and the advantages of the MGA. The proposed MGA technique is validated in Chapter 3 by applying it to certain composites without any FSS screens. First, the problem geometry is defined, and this is followed by the formulation for the reflection coefficient of inhomogeneous-layered media. Next, the details of the application of the MGA to design high-pass, low-pass, band-pass, and band-stop filters [19] are discussed. Finally, the MGA is applied to design three types of broadband microwave absorbers, viz., composites comprised of either dielectric or magnetic materials only, or a combination of dielectric and magnetic materials. This is followed by a detailed discussion and interpretation of the results. Chapter 4 deals with the synthesis of composites comprising a single FSS screen embedded in dielectric or magnetic materials, which may be lossy or lossless. First, the problem geometry is defined, followed by the formulation for the spectral domain MoM to evaluate the induced electric currents on the metalized portions of the FSS screen and the reflection coefficient o f the FSS/dielectric combination. Next, the details of the application of the MGA to the design of spatial filters and broadband microwave Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. absorbers are discussed followed by an interpretation o f the results. Finally, the domain decomposition approach is described for geometries that require the problem to be partitioned into two or more domains to maintain numerical efficiency. Chapter S deals with the synthesis of composites comprising multiple FSS screens embedded in lossy/lossless dielectric/magnetic materials. First, the problem geometry is defined, followed by the formulation and limitations of the Generalized Scattering Matrix technique, which extends the solution from single to multiple screens. Finally, the details of the application of the MGA to the design of spatial filters and broadband microwave absorbers are discussed, followed by an interpretation of the results generated by the MGA. In Chapter 6 a parallel MGA is implemented to decrease the computational expense o f the serial implementation. First the reasons for parallelization are outlined. Next, the Micro-grain MGA model is discussed and a psuedocode is presented. Finally, the computational expenses involved in the serial and parallel implementations of the MGA are compared by considering the problem of designing a polarization selective surface (PSS) for dual reflector applications. In Chapter 7, specific composites designed by the MGA in Chapters 4 and 5 are selected, and the results are validated by using the Finite Difference Time Domain (FDTD) technique and the direct Method of Moments (MoM). First, the limitations of the MoM and FDTD technique for a single FSS screen are discussed, and this is followed by an identification of drawbacks on using the scattering matrix approach vis-a-vis the MoM applied to multiple FSS screens. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 8 presents a summary and conclusions of the research and suggests future avenues o f investigation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 THE OPTIMIZATION ALGORITHM 2.1 Genetic Algorithm Fundamentals Genetic Algorithms (GA) fall under a special category of optimization techniques that are robust stochastic search methods, loosely modeled on Darwin’s principles of natural selection and survival of the fittest. The GA - as an optimizer - is useful for solving complex combinatorial problems. It is particularly effective in searching for global extrema in a multi-dimensional and multi-modal functional domain. The GA excels when the problem can be cast in a combinatorial form. It simultaneously processes a population of points in the optimization space, and uses stochastic operators to transition from one generation of points to the next, resulting in a decreased probability of them being trapped in local extremas. 2.1.1 GA Terminology Several definitions, borrowed from the descriptions in the natural world are frequently utilized in the literature on GA optimization, and are also used throughout the thesis. Some o f these words are defined here for the sake o f clarity, and are further embellished during the course of this thesis in subsequent chapters. Allele. A single binary bit that can be either a zero or a one. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 Gene: A sequence o f bits or alleles. This represents an encoded parameter in the optimization space. The number of parameters to be optimized determines the dimensionality of the optimization problem. Chromosome: Also referred to as an individual, is a sequence o f genes/parameters. Each individual represent a potential design/solution. Cardinality: Number of possibilities in a single allele, e.g., for a binary bit (allele) the cardinality is 2, as it can assume two values, either 0 or 1. Population: A group of individuals or chromosomes that the GA utilizes to search for the optimum solution. Generation: Each GA iteration is called a generation. The application o f GA operators continues until a new generation replaces the original generation. Parents: The individuals that are selected for reproduction to create offspring/children are called parents. The selection process can be either deterministic or probabilistic. Children: Simple stochastic operators are applied to the selected parents to generate individuals that are termed “children”. They constitute the population of the next generation and replace the current generation. Fitness: The function that defines the optimization goal is called the fitness or objective function. The objective function is applied to each decoded individual to generate a real number, which provides a measure of the “goodness” of the trial solution. This is the only link between the physical problem and the GA optimization process. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.1.2 Overview of Genetic Algorithms Numerical search techniques can be divided into two broad groups, viz., local and global. Conjugate Gradient (CG) methods can be considered as good candidates for the former technique, while the GAs fall under the latter category. The local and global techniques can be distinguished from each other by the fact that the former produce results that are highly dependent on the initial guesses, while the latter are largely independent of the starting points. Local techniques are tightly coupled to the solution domain, resulting in fast convergence to local extremums. Furthermore, the tight coupling to the solution space also places constraints, such as differentiability and continuity in the solution domain, which are difficult to handle. In contrast to the local techniques, the global ones are loosely coupled to the solution domain and place very few constraints on it. This means that these techniques are better equipped to deal with solution spaces that have constrained parameters, discontinuities, large number of dimensions and large number of potential local extrema. The GA-based combinatorial optimization technique offers several advantages over existing approaches: (i) it succeeds in designing broadband microwave absorbers and spatial filters consisting o f only a few layers, and, therefore, almost always leads to a physically-realizable structure; (ii) it is considerably simpler to implement than gradientbased search procedures [20-21], The binary GA operates on a coding of the parameters to be optimized. The coded representation o f the problem geometry consists of sequence of bits or alleles that contain information regarding each parameter, represented by a string of bits called the gene, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 whose length is determined by the allowed range of real values and the discretization step to be implemented. Potential design geometries can be represented by the concatenation of all the genes in the design space and are referred to as chromosomes or individuals. Each individual represents a separate design, and a group of them constitutes a GA population. The GA, which optimally chooses each parameter, is an iterative optimization procedure. It starts with a randomly selected population of potential solutions, and gradually evolves towards improved solutions via the application of the genetic operators. These genetic operators mimic the processes o f procreation in nature. The GA begins with a large population Po, comprising of an aggregate of sequences, with each sequence consisting o f a randomly selected string o f bits. It then proceeds to iteratively generate a new population P,~i, derived from P„ by the application of selection, crossover, and/or mutation operators. 2.1.3 GA Operators The GA utilizes three stochastic operators to transition from one generation of points to the next, viz., selection, crossover and mutation. The first two are the primary operators, while the last one is considered a secondary operator. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 2.1.3.1 Selection Operator For successful procreation, individuals belonging to a particular generation need to be selected from the population base to act as parents and produce children that become members o f the next generation. The process of selecting particular individuals while rejecting others is called selection. Different schemes have been proposed to carry out this process. Popular among them are population decimation, roulette wheel selection and tournament selection. These processes are discussed further in later sections. 2.1.3.2 Crossover Operator Crossover is a primary operator that creates offspring from selected parents Several methods have been proposed to generate offspring from parents, viz., single point, multi-point, and uniform crossover [20], The effect of crossover is to produce fitter individuals with better genetic make-up than the parent individuals. The commonly used schemes for crossover are discussed in a later section. 2.1.3.3 Mutation Operator Mutation is a secondary operator that aids the GA in exploring possible solution spaces that are not included within the genetic makeup of the present generation. In the case o f binary coded GAs, the mutation operator randomly changes a bit from “ 1” to " 0 ”, to prevent the GA from converging prematurely to local maxima. The probability of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 mutation is usually kept very low to ensure that it does not prohibit the beneficial actions of selection and crossover. 2.1.4 Selection Schemes With research on GAs moving at a rapid pace, several schemes have been proposed for performing the selection. Three popular schemes are discussed below. 2.1.4.1 Population Decimation In this selection scheme, individuals are ranked according to their fitness values. A cutoff for fitness is decided either arbitrarily or by tailoring it to the problem, and all individuals satisfying the cutoff requirements are considered parents. Crossover and mutation are applied repeatedly to the selected parents until the population quota is filled for the next generation. This is a deterministic selection scheme with simplicity as its advantage. However, the disadvantage of this scheme is that important genetic traits of an individual are lost forever due to the decimation of individuals, resulting in a loss of diversity, which occurs long before the beneficial properties of the decimated individual are recognized by the evolutionary process. The unique characteristics of a decimated individual can be reintroduced into the evolutionary process by using mutation, which adds new genetic materials, and assists the GA to explore portions of search space that otherwise it would not have. However, this is a very inefficient method for adding a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 specific genetic material, and the detrimental effects of premature losses o f beneficial characteristics has led to the development of stochastic selection strategies. 2.1.4.2 Roulette Wheel Selection This scheme is also called proportionate selection. It is a stochastic scheme and works on the principle of generation replacement. For a particular generation G;, (j = I Ngen), where Mgen is the number of generations, the probability of selection for each individual is calculated as follows [20 ]: a „ £ / ( parent,) i=i where / ( parent,) is the fitness of the i,h parent in the population base and N pop is the number o f individuals in a generation. From (2.1) it is evident that the probability of selection is dependent on the normalized fitness of each individual. The larger the value of pseiect, the higher the probability of an individual being selected as a parent for reproduction. The obvious distinction between population decimation and Roulette Wheel selection is that the latter provides a finite probability for currently unfit individuals to be selected for the reproduction phase; hence it maintains the diversity in the genetic makeup. Proportionate selection is prone to stochastic errors for small population sizes. The drawbacks of this method are as follows: (i) problems involving large run-times for single function evaluation require enormous computational resources if a large population base is selected; (ii) highly fit individuals tend to dominate the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 population base at the initial stage of optimization; (iii) at the final stages of the optimization the scheme is unable to differentiate between best individuals, leading to convergence to local extremums; (iv) the fitness value has to be maintained positive. The Roulette Wheel selection scheme can be implemented in three steps. First, a random number is generated between 0 and 1. Next, the normalized fitness value of each individual is calculated by using (2.1). Finally, a loop aggregates the normalized fitness values until the sum exceeds the value of the random number selected in step 1. At this point, the loop counter variable gives the number of the selected individual. This process is repeated until the required number of individuals is selected. 2.I.4.3 Tournament Selection The most effective o f the three schemes for a wide range of applications is tournament selection [22]. In this scheme, a sub-population of M individuals is randomly selected from the population base. Next, these individuals are made to compete against each other on the basis o f their fitness. Finally, the individual with the best fitness is judged to be the winner of the tournament, and is selected as a parent. The other M-I individuals are placed back into the general population, and this process is repeated until the population base to be used for crossover is filled. The number M can be arbitrarily chosen. In this thesis, binary tournament selection is applied with M = 2. First, a random number is generated between 1 and the number of individuals constituting the population base. This number indicates the individual to participate in the tournament. For a tournament size of M, this process is repeated M times until the sub Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 population of M individuals is filled. Next, a simple comparison sort of individual fitness filters out the best individual, which is selected as a parent. Finally, the remaining M -l individuals are placed back in the original population. The process is repeated until the tournament sub-population is filled. This scheme does not rule out the possibility of multiple pairing, the probability increasing with convergence to an optimal solution. Tournament selection shows a better convergence at the initial stages o f the optimization, and has a faster execution time than proportionate selection. The execution time of tournament selection is O(n), as compared to 0(n2) for proportionate selection. 2.1.5 Crossover Schemes With the selection of parent individuals completed, offspring or children need to be created to replace the current generation with the next generation. Children are created from parents by employing crossover with probability / W j- Upon applying crossover, the genes rearrange with the objective of producing better combinations, thus resulting in fitter individuals. The probability of crossover depends strongly on the crossover scheme employed as discussed below. 2.1.5.1 Single-Point Crossover The single-point crossover scheme was first suggested by Holland [2]. In this scheme, first a random number p with a value lying between 0 and 1 is generated. As Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 shown in Fig. 2.1, if p > p Cross, crossover is applied by selecting random locations in both parents-1 and -2. The alleles to the left of the selected locations in parent-1/parent-2 are copied into the corresponding positions in child- l/child-2 , respectively, while alleles to the right of the selected location in parent- 1/parent-2 are copied into the corresponding positions in child-2/child-l, respectively. If p < p cr0ss, then, as shown in Fig. 2.2, child-1 and child-2 are made exact copies of parent-1/parent-2, i.e., crossover is not applied. This process is repeated for each selected parent individual. For most optimization problems a value between 0.6 and 0.8 for pcross has been found suitable [2 0 ], Parent-1 Parent-2 Chi d -1 Chi d -2 Crossover point Fig. 2 . 1. Single point crossover with p > p cr0ss. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 Parent-1 1 0 Parent-2 1 II 0 1 0 0 1 1 1 1 0 1 1 \ 0 Chi « 1 . I 0 V I 1/ / 0 A J \ J V J W Chi d-2 0 1 1 Crossover point Fig. 2.2. Single point crossover with/? pcross. 2.1.5.2 Dual-Point Crossover In the dual-point scheme, we have two crossover points instead of a single one. Similar to the single point crossover, a random number p with a value between 0 and 1 is generated. As shown in Fig. 2.3, ifp > p cr0ss, the alleles between the two crossover points of parents-1 and -2 are copied into child-2 and -1, respectively. The remaining alleles are copied into the corresponding positions from parents- 1 and -2 to child- 1 and -2 respectively. If p < p cr0ss, then as shown in Fig. 2.4, parents-1 and -2 are copied into child- 1 and -2 , respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 Parent-1 Parent-2 C h ild -1 Child-2 crossover point-2 crossover point- 1 Fig. 2.3. Dual point crossover forp > p c Parent 2 Parent 1 1 0 0 1 1 0 Chi 1 0 °\ 1 > crossover point- 1 4 0 1 1 0 i ^ / / Chi « X X *0/ '/ i crossover point-2 Fig. 2.4. Dual point crossover forp <Pc Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i i 21 2.1.5.3 Uniform Crossover The logical extreme o f the dual point crossover is the uniform crossover [23], First, a random number p with value lying between 0 and 1 is generated. Next, a mask comprising alleles and having the same chromosome length as the parent individual is created, the alleles being distributed in accordance with a uniform probability distribution. Finally, for p > p Cmu, a suitable boolean function, viz., AND, OR, NOR, is applied to the mask and the parent chromosome to generate child chromosomes as shown in Fig. 2.5; while, if p < p cross, the parent chromosome is copied into the child similar to single and dual point crossovers. Conventionally, the probability of uniform crossover is maintained either at 0.5 or 1.0. parent chromosome 1 0 0 1 1 0 child chromosome Boolean Operator 0 1 1 0 1 1 1 1 1 1 0 0 Mask chromosome with a certain probability distribution Fig. 2.5. Uniform crossover for p > p cr0ss. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 2.1.6 Mutation Schemes The application o f the mutation operator in GAs is secondary in nature. It provides the means to explore those parts of solution surfaces that otherwise would not have been searched by introducing a new genetic makeup into the current population. It also prevents the GA from convergence to local extrema. There is one scheme that is predominantly used for binary-coded GAs and can be implemented as follows: A random number p is generated, and compared with the stored value of p mut (probability of mutation). Ifp > p mut, a random allele is selected in the parent chromosome and inverted, i.e., 0 to 1 or I to 0, while, mutation is not applied if p < p mut. The process is repeated for all the selected parent individuals. The mutation probability is usually kept low, between 0.01 - 0 . 1, becuase a higher value typically hampers convergence and overshadows the beneficial effects o f crossover and selection. 2.1.7 Elitism De Jong [24] introduced the concept of elitism as an extension to the basic GA. Being stochastic in nature, the GAs do not assure that the best individual in one generation betters its counterpart in the preceding generation, in the same manner as the elitist operator, which is an extension of the basic GA. If the best individual in the current generation has a fitness value lower than its counterpart in the preceding generation, then we can proceed one of the following three ways: (i) the best individual in the current generation is replaced with its counterpart in the preceding generation; (ii) the best Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 individual is passed on from one generation to the next as the (N* J)th individual; and, (iii) the best individual in the preceding generation replaces the least fit individual in the current generation. Elitism ensures that a monotonic increase in the individual fitness is maintained throughout the optimization process. 2.1.8 Fitness Function Also termed as the objective function, the fitness function is the only link between the physical problem being optimized and the GA. To utilize the fitness function, each individual inthepopulation proportional to the base is assigned a fitness value, which istypically goodness o f the trial solution. Fitness functions can be broadly categorized as single-objective and multi-objective types. In the former, the contribution to the value o f fitness comes from one objective function being evaluated; in a multi objective function, it is from more than one such function that are being evaluated simultaneously. The following equations are an example for single- and multi-objective fitness functions: fitness = m in(g) (2 .2 ) fitness = m in(g + a t) (2.3) where, g and t are the two objective functions and a is a weight coefficient for the latter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 2.1.9 Classification of Genetic Algorithms Depending on the type o f optimization, Genetic Algorithms can be broadly classified into two groups, viz., constrained or unconstrained optimization. Each of these can be further classified as stationary and non-stationary, depending on the type of function being optimized. The modality of the function results in a final classification into unimodal and multimodal functions. Suppose we have a function y = f { x ) for <5; < x< 8, (2.4) and we need to find the global minimum ofy. We can define an objective function as, obj = m in ^ ] for Sx < x < (2.5) Equation (2.4) is an example o f an unconstrained function optimization. An example for a constrained optimization function is obj = minf vl for 8, <x< 8, r 1 ‘ = max [y] for 8Z < x < 8l (2 6) where 61 < 82 < 83. In this research proposal, spatial filter design comes under the category of constrained optimization, while broadband microwave absorber design is considered as an unconstrained optimization problem. Stationary functions are well-defined entities that do not change at a rate faster than the GA can reach a global optimum. The functions dealt with in this work fall into this category. Contrastingly, the non-stationary functions can be defined as those that evolve at a rate faster than the GA can reach a global maximum. A typical example of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 non-stationary function is pursuit and evasion, i.e., the function to be optimized is redefined depending on what the evader does. Finally, the modality classification can be explained with the following examples [25], A unimodal function is one that has a single maximum, as for instance, the following: / ( x , y ) = (x + y - l l ) : +(x + .y -7 ): (2.7) This function is called Himmelblau’s function, the search space is confined to x > 0 andy < 6 , a single minimum point is at (3,2) with a function value of zero. In contrast, a multi-modal function may be written as 1 = 10 / (*| *lo) = 10O + 2 > r - 10 cos ( 2 ;rx,) 1=1 ( 2 .8 ) This iscalledRastrigin’s function in which each variable lies in the range (-6 , 6 ), and has a globalminimum at x, = 0, where the function value also equals tozero. All the functions optimized in this work are categorized as massively multi-modal functions. 2.2 The Conventional Genetic Algorithm (CGA) The Conventional Genetic Algorithm (CGA), also referred to as the Simple Genetic Algorithm (SGA), is the first and simplest of GAs proposed by Holland [2]. It utilizes a serially implemented binary-coded GA with three basic operators, viz., selection, crossover and mutation. The CGA is illustrated via a flowchart in Fig. 2.6 [20], First, a random set of individuals that constitutes the initial population is created. Then the selection operator is applied to generate a sub-population o f individuals that act as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. parents. Next, the crossover and the mutation operator are repeatedly applied with probability p cross and p mut respectively, to pairs of individuals in the sub-population to generate offspring constituting the population base o f the next generation. This process is repeated until convergence is reached. Finally, the termination criterion is applied, which is arbitrary and mostly problem-specific for a GA. Any of the schemes described in the earlier sections can be applied for selection, crossover and mutation. The choice of parameters for the CGA is usually based on studies by De Jong [24] and Grefenstette [26], The termination criterion can be a fixed number of generations, or convergence of the fitness function to a predetermined value. 2.2.1 Population Sizing for the CGA The question of what is an appropriate population size needs to be addressed before any GA calculations can be run. For CGAs, sizing the population is problemspecific and a strong function of the length of the chromosome and the cardinality of the allele. An estimate of the population size for the CGA is made using the following equation [27-28]: n pop where Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.9) 27 Initialize Population Evaluate Fitness Selection r i Until Crossover ^ T A population^ full Replace Population Evaluate Fitness n Termination criteria met Yes Fig. 2.6. Procedural flowchart of the CGA. / = length of the chromosome or individual k= 1 V" V G, ; Npar = number of parameters or genes; G, = length of the ilh gene Npar tr X = number of possibilities for each allele or bit = 2 for binary coding 2.2.2 Drawbacks of the CGA Equation (2.9) leads us to conclude that the sizing of population for the CGA is dependent on the length of the chromosomes, the number of parameters or genes that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 determine the dimensionality of the optimization problem and the type of encoding employed. The number o f alleles in each gene is dependent on the range of parameters and the implemented step-size. A multi-dimensional search space leads to a large number of parameters to be optimized by the CGA, and results in an increase in the chromosome size for each individual. Utilizing (2.9) and estimating the population size under the conditions mentioned above leads to a large population base, which requires several generations to achieve convergence, and hence places a considerable burden on computational time and resources. To reduce the computational burden, population sizes less than that estimated can be employed by the CGA. However, this results in a premature convergence to nonoptimal results due to insufficient information processing by the CGA. 2.3 Micro-Genetic Algorithm (MGA) The drawbacks o f the CGA in handling computationally intensive problems prompted the search for a small-population-based GA, coupled with efficient convergence properties. The ideal candidate is the Micro-Genetic Algorithm (MGA) [13], which is a non-mutation-based GA, i.e., it does not employ the mutation operator, instead uses a population restart strategy to avoid local extremas. It also makes clever use of the elitist operator to ensure the migration of the best individual from one generation to the next. The MGA is a variant o f the SGA, wherein a small population size is utilized for the optimization procedure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 The poor performance o f GAs for small population sizes can be attributed to insufficient information base and convergence to local extremums. As the population size increases, so does the number of function evaluations. In real-world problems, more often than not, the computation time for a single function evaluation is excessive, and this necessitates the use o f enormous computational resources to complete the optimization. The need to minimize the number of function evaluations while maintaining the global optimality o f the GA has led to the MGA as one possible solution. Outlined below is a scheme proposed for GAs with small population sizes [28]: 1. Randomly generate a small population. 2. Perform genetic operations until nominal convergence is reached (as measured by bit wise convergence or some other reasonable measure). 3. Generate a new population by transferring the best individuals of the converged population to the new population and then generating the remaining individuals randomly. 4. Go to step-2 and repeat. 2.3.1 The MGA Optimization Procedure Based on the scheme outlined above, the MGA procedure utilized in this thesis is presented below and is illustrated by the flowchart shown in Fig. 2.7. 1. Randomly generate N individuals to constitute the first generation. A random number generator with uniform probability distribution is utilized. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 2. Evaluate the fitness of each individual by applying the problem-specific objective function. Select the individual with the best fitness value and place it as the N01 individual in the next generation. Next, generate the remaining N-1 individuals for the next generation. 3. To select the individuals constituting the mating pool, binary tournament selection is applied as explained in Sec. 2.1.4.3. Hence, (N-l) 2 individuals with better fitness values are selected as parents and are passed on to the next generation. 4. In step-3, prior to copying the parents selected to the next generation, uniform crossover (Sec. 2.1.5.3), is applied to generate one offspring from each selected parent with pcross = 0.5. This generates the remaining (N-l) 2 individuals required to fill the next generation. 5. The process is repeated for each generation until convergence is reached. Convergence is achieved when every individual in a particular generation differs by less than 5% - on a bit-by-bit comparison - with the best individual in that generation. 6. Once convergence is achieved, the best individual is replicated and passed to the next generation. The remaining N -l individuals are generated randomly. This is the start-restart or population restart strategy employed by the MGA. 7. Repeat process from step-2 until termination criteria are met. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 The key factor that contributes to the ability of the MGA to solve a computationally intensive problem efficiently is its employment of a small population base, while the population restart strategy ensures that the algorithm does not get trapped in local extrema. 2.3.2 Advantages of the MGA In this thesis the MGA is employed to solve a computationally intensive design problem. In the past, CGAs have met with minimal success in solving such problems at least within a practical time frame. A design problem is typically categorized as computationally intensive when a single function evaluation takes an appreciable amount of computation time. Some o f the design problems dealt with in this proposal involve the analysis o f scattering from FSS screens, which is achieved by using the Method of Moments (MoM). The main contributors to the large computation time for a single function evaluation are the matrix fill time and inversion, which are dependent on the complexity of the FSS screen design. The number of function evaluations in a single generation depends on the population base used for the optimization. To attain a high numerical efficiency, we need to work with a small population base and this is provided by the MGA, which also has good numerical convergence properties. Furthermore, the MGA reaches near-optimal regions quicker than the largepopulation-based CGAs. The general choice of population size for CGAs can range between 100 and 10000, compared to the MGAs, which typically utilize a population size Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 ranging between 20 and 50. It is easily estimated [33] that even worst-case considerations lead to a decrease by 50% in computational run time. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 Initialize Population Selection Evaluate Fitness * Select best individual (Elitism) Crossover Evaluate Fitness Evaluate Fitness Until temporary population is full Restart population Select best individual (Elitism) Replace Population Check convergence Termination criteria met? Select best individual (Elitism) Fig 2.7. Procedural flowchart of the MGA. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 SYNTHESIS OF COMPOSITES COMPRISING ONLY DIELECTRIC AND MAGNETIC MATERIALS Equipped with the Micro-Genetic Algorithm (MGA), which was described in the previous chapter, we proceed now to apply it to a variety of synthesis problems in this chapter. For instance, we consider multilayer composites consisting of dielectric, magnetic or combinations of interspersed dielectric and magnetic materials. First, we synthesize spatial filters by using lossless dielectrics to obtain specified frequency responses. Next, we study the effect of combining dielectric and/or magnetic materials, which may be lossy or lossless, to design broadband microwave absorbers. Finally, we synthesize a broadband absorber consisting of lossy carbon fiber and lossless dielectric materials. 3.1 Problem Geometries Figure 3.1 shows a /V-layered composite structure whose parameters we wish to optimize with a view to realizing a specified frequency response for spatial filters, while Fig. 3.2 shows the same structure except for a Perfect Electric Conductor (PEC) backing that must be included in the synthesis o f broadband microwave absorbers. We assume a plane wave to be incident on the composite at any arbitrary angle and polarized as either Transverse Electric (TE) or Transverse Magnetic (TM). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 S \, d \ Fig. 3.1. Synthesis o f spatial filters with lossless dielectrics. Nlayered composite with a TE polarized uniform plane wave incident at an arbitrary angle. For TM polarization the E and H fields are interchanged along with a phase shift of 180 degrees introduced in the H field. Given the range of allowed values for the constitutive parameters, in this case s ' (real part of the relative permittivity), the minimum thickness of each layer and the total allowed thickness o f the composite, the MGA synthesizes a AMayer spatial filter that has the desired frequency response for a given incident angle and polarization. To synthesize broadband microwave absorbers the MGA optimizes the thickness of each layer, and the type of material in the layer, which may be either lossy dielectric, lossy magnetic or lossless dielectric, such that the composite exhibits a low reflection coefficient for a prescribed set of frequencies and incident angles simultaneously for both TE and TM polarizations. The MGA accesses the values of the frequency Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 dependents’ (imaginary part o f the relative permittivity) a n d ( i m a g i n a r y part of the relative permeability) of lossy dielectric and magnetic materials via a lookup table. £\, UNt d \ ----------- ^► PEC backing Fig. 3 .2. Synthesis of broadband microwave absorbers with lossy or lossless dielectrics and lossy magnetic materials. yV-layered composite with a TE polarized uniform plane wave incident at an arbitrary angle. For TM polarization the E and H fields are interchanged along with a phase shift of ISO degrees introduced in the H field. 3.2 Formulation of Reflection and Transmission Coefficients for Inhomogeneous Layered Media Consider an inhomogeneous, A'-layered medium, as shown in Figs. 3.1 and 3.2 with the material inhomogeneity along the z-direction. To ease the effort of developing an expression for the reflection and transmission coefficients of the AMayered media a simpler three layered media as shown in Fig. 3.3 is considered. Due to the recursive Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. nature of the expressions to be developed, the formulation can be easily extended from the J-layered to ^-layered media [12]. Alternate formulations for the reflection and transmission coefficients for an AMayered inhomogeneous media [29-30] can also be employed. Region-1 Region-2 Z = -d 2 Region-3 Fig. 3.3. Three-layered medium with TE polarized wave incident at an arbitrary angle. Considering Fig. 3.3, the Ey component in Region-1 can be expressed as the sum of a downward- and upward-going wave as follows: (31) The upward-going wave has contributions from both the wave reflected from the first and the second layers, and transmitted through the first layer. In (3.1) R is referred to as the generalized reflection coefficient, which includes the effect of reflections from the first interface and the subsurface. Similarly, the fields in the Regions-2 and -3 can be expressed as (3.2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 E,y = A ,e lk- \ (3.3) where R 23 in (3.2) is the Fresnel reflection coefficient expressed as p < N, q < N +1 Mpkq: + V i * -1 p = N ,q = N + \ k p z = k po cos 0 p i,kn po (3.3a) = G)Ju \ r * Ps P for a iV-layered medium, kpz is the wave number along z in layer p, and pp and ep are the permeability and permittivity, respectively, of the p lh layer. The second equation in (3 .3a) is used when the composite is terminated with a Perfect Electric Conductor (PEC). Equation (3.3) has only one term since Region-3 extends to infinity in the -z direction, and, hence, there is no reflected wave. The four unknowns Ai, A A 3, and /^,in (3.1), (3.2), and (3.3) need to be evaluated to arrive at the expression for the reflection and transmission coefficients for the three-layered medium. The wave traveling downward in Region-2 is the sum of two waves; (i) transmitted from Region-1 to Region-2; and, (ii) reflected from Region-1. Thus, at z = -d], we have the following equality: A2e,k'-A = Ate:k'A Ti: + R2lA2R2/ - ,k'-A'-,k'-A (3.4) where i?,, is the Fresnel reflection coefficient as defined in (3.3a) and 7J, is the Fresnel transmission coefficient which can be expressed as 7* = ! + /?„, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (34b) 39 Similarly, the upward traveling wave in Region-1 can be written as a combination of the wave reflected of Region-1 while incident from Region-1 and the wave transmitted through Region-1 while incident from Region-2. This can be expressed as follows: A,R,ze'k'A = Rl2Ale*lA + (3.5) From (3.4), A: can be solved for in terms o f A/, yielding T A ~k'- ld' ,36) Substituting Aj into (3.5) we arrive at the expression for the generalized reflection coefficient for the three-layered medium, which includes the effect of subsurface reflections as well as the reflection from the first interface. It can be expressed as (3.7) Rr = R r + : 11 If a fourth layer is added to the original three then the only change required in (3.7) will be to change the Fresnel reflection coefficient R?s to the generalized reflection coefficient incorporating subsurface re fle c tio n s^ . Therefore, for a general ^-layered medium (3.7) can be extended as follows: I R ( R * Furthermore, using (3.4b) and the identity Rn = -R v , (3.8) can be simplified to R,,.= R, . , + £ , i ,n 5 J-------r dj) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.9) 40 A similar expression can be derived for the TM polarized wave. For the special case of a PEC backed composite, the reflection from the PEC backing (see (3.3a)) is taken to be - I , and the recursive formulation given in (3.9) can still be employed to determine the overall reflection and transmission coefficients of an iV-layered inhomogeneous composite. 3.3 MGA Formulation for the Spatial Filter and Broadband Microwave Absorber Problems The MGA is applied to optimize various parameters pertaining to the two design problems of interest. Binary tournament selection (see Sec. 2.1.4.3) is employed along with uniform crossover (see Sec. 2.1.5.3) with a crossover probability of 0.5. Elitism is utilized along with a population restart strategy that is applied when the bitwise difference between the best and other individuals/chromosomes in the current generation is less than 5%. 3.3.1 Spatial Filter Synthesis For the synthesis o f spatial filters [19] we are given the range of allowable values of the real part o f permittivity for the lossless dielectric layers, the minimum thickness of each layer and the total permissible thickness of the composite. The MGA needs to synthesize a composite incorporating jV layers of dielectric such that it has the desired frequency response for a given incident angle 0, The design process therefore entails Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 making choices for the optimal values of e o i each layer and its thickness. For the present problem, we define the fitness function to be: j=l ' ' + £;=1a - K ^ f ’ (3I0) where M and N are the number of frequencies, respectively, for which the transmission and reflection are maximized, and T is the reflection coefficient evaluated using (3.9). In (3.10), a, s are the coefficients that can be used to weigh the relative importance of the filter performance for different frequencies, and or is a parameter that controls the trade off between the ripple in the stop and pass bands, as well as the rejection ratio. The relative values o f the weight coefficients can be tailored to yield the desired frequency response and can be used to trade-off the filter performance in one part of the frequency spectrum against that in another part. A smaller ripple in the transmission band can be achieved at a cost of increased reflection, by assigning larger weight coefficients to the sampling points in the transmission band, and vice versa. The number of sampling points also plays an important role in the optimization procedure. Especially near the edges of the frequency band of interest, it may be preferable to choose several points with moderate weight coefficients, rather than a few points with large coefficients. This could result in a design that matches the desired characteristic at the specified frequency points perfectly, but has a large ripple in the neighborhood o f the transition. Specification of the desired response at more points in the transition region leads to designs with smaller ripples. However, this increases the computational cost associated with a single function evaluation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 The MGA operates on a chromosome defined in (3.14), which represents a combination o f the parameters in a coded form. The coded representation of the coating consists of a sequence of bits that contain information regarding each of the parameters. Given the allowed range o f values and the choice of step size to be implemented for the real part of permittivity o f the layers in the composite, the number of possibilities for a single layer, (NP), can be represented as follows: (f,.) -(fr) r* v l/mui (£r‘ (3.11) The material choice for layer / is represented by a sequence M, composed of y bits, as shown below: 2s <(NP) <2r (312) M , = mlm :.......... m; The higher power o f 2, i.e., y determines the number of bits in the parameter. The thickness o f the ith layer can be encoded by applying (3 .11) with er replaced with d, and represented by a sequence D, consisting of /? bits as Dt = d)d;.......... d f (3.13) The entire Allayer composite can be represented by a sequence C, which represents the ilh chromosome, as given below: C, = (A/,A/; M SL ,L l s. )_ (3.14) Each quantity in (3.14) is defined in (3.12) and (3.13) and each sequence C, consists of (y-P) bits. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 3.3.2 Numerical Results for Spatial Filter Design The MGA has been successfully applied to the design of various filters in the frequency range o f 8.0 - 18.0 GHz. The number of dielectric layers was fixed at five and is surrounded by air, though the algorithm is generally applicable to any number of layers and to any type of termination (e.g., PEC backing), as will be explained in the next section when we deal with broadband microwave absorbers. The angle of incidence is 45 degrees and the optimization is performed only for the TE polarized incident wave. Four basic types of filter design, viz., low-pass, high-pass, band-pass and band-stop are attempted. The parameter a employed in (3.10) is chosen to equal 1.0 for all the cases investigated. Table 3-1 shows the GA parameter search space for this particular problem. The reflection coefficient in dB vs. frequency for both TE and TM polarizations is plotted for the synthesized filter in Figs. 3.4 through 3.7. The reflection coefficents for the TMpolarized wave are obtained by using the MGA optimized parameters for the TEpolarized wave in (3 .9). The cutoff frequencies for these filters can be easily identified in the reflection coefficient plots. While only lossless dielectrics have been considered in all the cases, the method itself can easily be used to handle lossy dielectrics with both electric and magnetic losses, as is shown in the next section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 TABLE 3-1 GENETIC ALGORITHM PARAMETER SEARCH SPACE Parameters 1st layer thickness (cm) 2nd layer thickness (cm) 3rd layer thickness (cm) 4th layer thickness (cm) 5th layer thickness (cm) er of 1st layer sr of 2nd layer er of3rd layer sr of 4th layer er of 5th layer Range of Increment parameters 0.35-3.0 0.01 0.35-3.0 .0.01 0.35-3.0 0.01 0.35-3.0 0.01 0.35-3.0 0.01 1.03-10.0 0.1 1.03-10.0 0.1 1.03-10.0 0.1 1.03-10.0 0.1 0.1 1.03-10.0 Number of possibilities 512 512 512 512 512 128 128 128 128 128 Number of binary digits 9 9 9 9 9 7 7 7 7 7 Total# of bits 80 3.3.3 Synthesis of Broadband Microwave Absorbers The MGA is employed to optimize various parameters of the composite, viz., the thickness of each layer, and the type of material in the layer, which may be either a lossy dielectric, lossy magnetic or lossless dielectric. We assume that we are provided with a set of lossy dielectric materials Em with frequency dependent permittivities s,(f), (i= I E ^ , as well as with a set of lossy magnetic materials M m, with frequency dependent permittivities e,(f), (i= I (i = 1 Mm), and frequency dependent permeabilities M m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 -10 <D-15 TM TE £T* 1 .4 S ,.d ^ A » . -35 -40 Frequency in GHz Fig. 3.4. Low pass filter design. The weights used for the MGA were 1.3 for 12.2 to 18.0 GHz and 1.0 for the rest of the band. The desired cutoff is at 13.0 GHz. TM TE E -15 . Ef5? 'd *r L 5 .' • ‘ • C -25 -30 -35 -40 Frequency in GHz Fig. 3.5. High pass filter design. The weights used for the MGA were 1.3 for 14.0 to 18.0 GHz and 1.0 for the rest of the band. The desired cutoff is at 14.0 GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TM TE -10 -20 -25 -35 -40 -45 Frequency in GHz Fig. 3.6. Band stop filter design. The weights used for the MGA were 1.5 for 1.0 to 11.0 GHz and 15.0 to 18.0 GHz, and 1.0 for the rest o f the band. The desired cutoffs are at 10.0 and 15 .0 GHz. TE TM 00 •o c § -10 o £u o / W -15 c o o a ^ *20 00 -25 a 9 10 11 12 13 14 15 16 17 18 Frequency in GHz Fig. 3.7. Band pass filter design. The weights used for the MGA were 1.3 for 1.0 to 11.0 GHz and 15.0 to 18.0 GHz, and 1.0 for the rest o f the band. The desired cutoffs are at 10.0 and 15.0 GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 For the composite shown in Fig. 3.2, the MGA determines the following: (i) type o f material for the N layers; (ii) material parameters of N layers; and, (iii) thickness of the N layers such that the composite exhibits a low reflection coefficient for a prescribed set of frequencies f (/ = I Fj) and incident angles 0, (i = I Ad, simultaneously for both the TE and TM polarizations. In the context of the present problem, the magnitude o f the largest reflection coefficient is minimized for a set of angles, for both the TE and TM polarizations, and for a selected band of frequencies. Hence the fitness function can be written as [31] F {m ^d u where ms ,d s ) = -m ax jr*1*™ (0,, / , )J (3.15) miand dt are the material and thickness parameter of the ith layer in the TE. , , composite, respectively, and T n{ (0,, f s ) is the reflection coefficient. Thelatter is a function o f polarization, incident angle and frequency, and is computed using (3 .9). The MGA operates on a chromosome, as defined in (3.19), which represents a combination o f the parameters in a coded form. The coded representation of the coating consists o f a sequence of bits that contain information regarding each of the parameters. Given a database containing Lm= 2 ^ different materials, the material choice for layer / is represented by a sequence M, of bits as follows: M t - m\mf............................(3.16) The thickness o f the iih layer can be encoded by a sequence D, o f bits and represented as shown below: D ,= d]d;................................................................(3.17) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 Additional bits are needed to aid the MGA in selecting lossy dielectric layers, lossy magnetic layers and lossless dielectric layers. A two-bit sequence is needed to represent the three choices. Hence, the type of material for the ilh layer can be encoded by a sequence (CL), made up of two bits and represented as (.CL\=(cl)](cl ); (3.18) The entire AMayer composite can be represented by the sequence C,, which is the ith chromosome or individual, given below: C, ={(A/,A/: ..... M SL ,L L ,( C L \ (CZ)v)}_ (3.19) Each quantity in (3.19) has been defined in (3.16) through (3.18) and each sequence in (3.19) consists of (2~Lmb~Ldb) bits. 3.3.4 Numerical Results for Synthesis of Broadband Microwave Absorbers The reflection coefficient evaluation procedure given in Sec. 3.2 is used in conjunction with the MGA algorithm described in Sec. 3.3 to synthesize broadband microwave absorbers in the frequency range of 19.0-36.0 GHz, with the number of layers fixed at four. These layers are surrounded by air on one side and terminated on the other side by a PEC backing as shown in Fig. 3.2. The measured values of s ’ and s ’ of ten lossy dielectric materials and s r , s ’ , and n ’ of six lossy magnetic materials are considered and a database of these values as a function of frequency is created. Figures 3.8 through 3.11, plot the parameters of the lossy magnetic material. The real and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 imaginary parts o f the permeability and permittivity of the lossy dielectric material are not plotted because, unlike the lossy magnetic materials, they are relatively slowly varying functions of frequency, and they are catalogued in Tables 3-2 and 3-3 instead. Linear interpolation is employed to generate the values of s 'r and e" as functions o f frequency for odd-numbered material types, while the even-numbered material types are averaged. For example, values of the “Type 2” material at 19.0 GHz is obtained by averaging the material values of “Type 1” and “Type 3” at 19.0 GHz, and the same procedure is repeated for the lossy magnetic materials. Thus, in effect, we use 19 types of lossy dielectric materials and 11 types of lossy magnetic ones. Small values of losses are added to the layers designated as lossless, because it is not practically feasible to fabricate a perfectly lossless dielectric. Hence, the e ’ value of the first layer is usually 0.01 if it is tagged as lossless by the MGA, and fixed at 0.1 for the rest of the lossless layers. The MGA simultaneously optimizes the absorber for elevation angles varying from 0 to 60 degrees (increment of 15 degrees) for both TE and TM polarizations over the frequency band ranging from 19.0 to 36.0 GHz. The population size and the number of generations for all the three cases are 50 and 1000 respectively. The following material combinations are investigated. (i) Case-1: lossless or lossy dielectric (ii) Case-2: lossless dielectric or lossy magnetic (iii) Case-3: lossless or lossy dielectric, or lossy magnetic. The parameter search space for the MGA is illustrated in Table 3-4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 The MGA-optimized composites and their frequency responses are shown in Figs. 3.12 through 3.14 for the three cases, respectively. The results can be explained as follows: over the band of frequencies, the range of elevation angles of interest, and for both the TE and TM polarizations, the maximum and the minimum values of the reflection coefficients correspond to the worst and the best results expressed in dB. Mathematically, this can be expressed as , T ™ J (in dB) (worst case) (in dB) (best case) (3.20) (3.21) where Ff and A$ give the number of frequencies and elevation angles over which the MGA optimization is carried out. We observe that, for Cases-1 and -2, the MGA fails to maintain the largest reflection coefficient below —15 .0 dB over a broad range of elevation angles, whereas it succeeds in maintaining the worst-case reflection coefficient below -15 dB for Case-3. Thus Case-3 provides an improvement of -2.6 dB over Case-1, and -5.0 dB over Case-2. Table 3-5 lists the types of material selected by the MGA for each case. The parameters used in the CGA are: population size = 500; number of generations = 1000; probability of creep mutations = 0.0153; probability of jump mutations = 0.002. Uniform crossover is applied along with Elitism and Niching [32], We note that the population size used in the CGA optimization is ten times larger, than that employed by the MGA, because the CGA does not converge efficiently for smaller population sizes, typically handled by the MGA. This results in a ten-fold increase in the number of function evaluations, and increases the run time appreciably. Table 3-6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 compares the performance o f the MGA and CGA algorithms. It is seen that the CGA is not able to reach the best fitness value attained by the MGA. This leads us to deduce an important conclusion that the CGA needs more individuals in its population base than does the MGA to converge to an optimal solution. Depending on the chromosome size as shown in Table 3-4, an estimate of the population size can be made for this particular problem by using (2.9) in Sec. 2.2.1. From Table 3-4, we see that, for binary coding we have / = 93, x = 2 and the average length o f the schema (this is the average length of the genes that make up one parameter) of interest is (l/12)(10+13+l3+13+2+9+2+9+2+9+2+9) = 8. Using (2.9), we estimate the population size for this case to be (93/8)(28) = 2976, although we have only used a population size which is l/6th of the estimated one. This could be one reason for the poor performance of the CGA. On the other hand, if we do use the estimated population size in the CGA, the computational expense to achieve the same results as derived by the MGA would be enormous. This leads us to conclude that it is preferable to use the MGA over the CGA, to take advantage of its superior convergence properties and lower computational expense. Figures 3.15, 3.16, and 3.17 illustrate the average and best fitness values for the MGA and the CGA for the three cases. In Fig. 3.15, we note that there are well-defined undulations in the curve for the average fitness value until we reach the 450th generation for the MGA, and this is attributable to the population restart strategy (Sec. 3.3). The random variations in the curve for the average fitness value after the 450th generation is due to the inability o f the MGA to restart the population when following the 5% criteria. The same characteristics are observed in Figs. 3.16 and 3.17. Thus, Table 3-6 and the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 curves in Figs. 3.15 through 3-17, leads us to definitely conclude that the convergence of MGA is superior to that of the CGA, and that the former requires less computational time as well. 15 —« — •••■A-- 14 aS^B' a|> ^B 13 m■ m* typel type3 typeS type7 type9 type11 i 12 o £■11 0 1 CL 75 o * * 9 8 18 20 22 24 26 28 30 32 34 36 F re q u e n c y (GHz) Fig. 3.8. Real part of relative permittivity of magnetic material. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 - # -W ■ typel type3 type5 type7 - - type9 typel 1 6.4 .§*6.2 > p 6.0 E 5.8 8 . 5,6 O 5.4 t: 5.2 & !5.0 g 4.8 ' 5 ) 4.6 (0 E 4.4 4.2 18 20 22 24 26 28 30 32 34 36 F re q u e n c y (GHz) Fig. 3.9. Imaginary part of relative permittivity of magnetic material. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 1-6r -W — Real part of p e rm e a b ility 1.5 - typel type3 type5 type7 type9 typel 1 1.4 1.3 - 1.2 h 1.1 1.0 0.9 _ l __________ I__________ I__________ I__________ I__________ I__________ I__________ I__________ I__________ L_ 18 20 22 24 26 28 30 32 34 36 F re q u e n c y (GHz) Fig. 3.10. Real part of relative permeability of magnetic material. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 Imaginary pert of p e rm e a b ility 0.7 # typel type3 type5 — “ • * type7 - W - - type9 typel 1 ,...A... 0.6 0.5 0.4 0.3 0.2 ’"A-. 0.1 0.0 18 20 22 24 26 28 30 32 34 36 F re q u e n c y (GHz) Fig. 3.11. Imaginary part of relative permeability of magnetic material. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 -1Z I ■ I------------------1 ♦ I T *~ ' ~ 1-----------------1 -----------1 - t Reflection coefficient in dB -14*4'*'......... ...... -1 6 - + -16- + t + + -2C + - ♦ + 22- worst case best case 4* '2 18 20 * 1 22 24 26 28 Frequency in GHz i 30 i 32 34 Fig. 3.12. (a) Worst- and best-case reflection coefficient levels for the composite with only lossless and lossy dielectric materials. Sr = 1.29, Sr = 0.1, = 1.0, |ir =0.0 d = 3.65 mm Sr = 4.85, Sr = 1.526, fir = 1.0, M-r = 0.0, d = 1.25 mm (Type 2, lossy dielectric material) Sr = 21.37, Sr"= 16.77, Hr' = 1.00, (ir " = 0.0, d = 0.55 mm (Type 18, lossy dielectric material) Sr = 1.03, Sr =0.1, Hr = 1.0, |ir =0.0, d = 0.05 mm Fig. 3 .12. (b) Design o f composite with only lossy and lossless dielectric materials Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 w -1Q ♦ « t ♦ ♦ <f► ..............1► ........... t" . ♦ + ► Reflection coefficient in dB -15♦ <► ♦ H ► ♦ 4 -- ♦ + -2C - + ■b f + 4- -25 ‘ F............. + - + H- -3C - •+ • •3? T 20 22 24 26 ♦ + + 1 28 1 30 worst case best case * 32 » 34 36 Frequency in GHz Fig. 3.13. (a) Worst- and best-case reflection coefficient levels for the composite with only lossless dielectrics and lossy magnetic materials Er = 1.74, er = 0.1, pr = 1.0, |ir =0.0 d = 2.54 mm Er = 8.25, Sr = 5.09, |ir = 1.016, Hr = 0.11, d = 0.85 mm (Type 6, lossy magnetic material) Er = 9.5 1, Er = 5.35, |ir = 1.007, (ir = 0.25, d = 0.05 mm (Type 10, lossy magnetic material) Er = 2.44, Er = 0.1, pr = 1.0, (ir = 0.0, d = 0.05 mm Fig. 3.13. (b) Design of composite with only lossy magnetic and lossless dielectric materials Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 I -14T ± -16' I I V I I ♦ + ♦ .* ... ♦ t Reflection coefficient in dB I ♦ ; ♦ f * r * ’ 4- ~f.... -18' .. 4 - + - 20 ' + - 22 ' -24' ♦ + T" worst case best case -26......... -28 f -30' 4 8 20 22 24 26 28 30 32 34 Frequency in GHz Fig. 3.14. (a) Worst- and best-case reflection coefficient levels for the composite with interspersed lossy or lossless dielectric and magnetic materials. Br = 1.59, E r = 0.1, | ! r = 1.0, |ir =0.0 d = 2.75 mm Sr Er = 4.85, E r = 1.52, ^lr = 1.0, f i r = 0.0, d = 1.25 mm (Type 2, lossy dielectric material) = 24.77, E r = 18.98, ( i r = 1.0, (ir = 0.0, d = 0.35 mm (Type 19, lossy dielectric material) 8r = 9.0, Er = 6.25, |ir = 1.0, (ir = 0.039, d = 0.27 mm (Type 11, lossy magnetic material) Fig. 3.14. (b) Design of composite with both lossy or lossless magnetic and dielectric materials. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 59 -0.1 -0 .2 ' r Average fitness MGA IX best fitness MGA Average fitness CGA best fitness CGA -0. Number of Generations Fig. 3.15. Evolution of the average and best fitness values of the CGA and MGA for Case-1. - 0.2 _ Average fitness MGA ' ~ best fitness MGA Average fitness CGA hest fitness CGA 0 I 100 I_____ L .. ■■ .1 1 1 200 300 400 500 600 700 Number of Generations L » 800 900 1000 Fig. 3.16. Evolution o f the average and best fitness values of the CGA and MGA for Case-2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 Fitness value - 0.2 -o.e -0.7 Average fitness MGA best fitness MGA Average fitness CGA best fitness CGA -0.9 800 900 Number of Generations Fig. 3 .17. Evolution of the average and best fitness values of the CGA and MGA for Case-3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1000 61 TABLE 3-2 e’r AS A FUNCTION OF FREQUENCY FOR THE LOSSY DIELECTRIC MATERIAL Material type F (GHz) 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 Type 1 Type 3 Type5 Type 7 Type 9 4.48257 4.48299 4.48331 4.48353 4.48367 4.48377 4.48383 4.48388 4.48390 4.48392 4.48393 4.48394 4.48395 4.48395 4.48395 4.48396 4.48396 4.48396 5.21093 5.21152 5.21190 5.21216 5.21232 5.21244 5.21251 5.21256 5.21259 5.21261 5.21262 5.21263 5.21264 5.21265 5.21265 5.21265 5.21265 5.21265 5.83968 5.84028 5.84070 5.84098 5.84117 5.84130 5.84138 5.84143 5.84147 5.84149 5.84151 5.84152 5.84153 5.84153 5.84154 5.84154 5.84154 5.84154 7.08171 7.08190 7.08231 7.08264 7.08288 7.08304 7.08314 7.08321 7.08326 7.08329 7.08331 7.08333 7.08334 7.08334 7.08334 7.08335 7.08335 7.08335 22.41061 22.43457 22.44027 22.44174 22.44219 22.44237 22.44247 22.44252 22.44256 22.44258 22.44260 22.44261 22.44262 22.44262 22.44262 22.44263 22.44263 22.44263 Material type Type ii Type 13 Type 15 Type 17 Type 19 F (GHz) 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 9.83625 9.83703 9.83769 9.83816 9.83848 9.83870 9.83884 9.83893 9.83899 9.83904 9.83906 9.83908 9.83909 9.83910 9.83911 9.83911 9.83912 9.83912 11.86623 11.86821 11.86921 11.86979 11.87017 11.87041 11.87057 11.87068 11.87075 11.87080 11.87083 11.87085 11.87086 11.87087 11.87088 11.87088 11.87088 11.87088 12.73006 12.73291 12.73412 12.73477 12.73516 12.73541 12.73557 12.73568 12.73575 12.73580 12.73583 12.73585 12.73586 12.73587 12.73588 12.73588 12.73589 12.73589 17.96287 17.96867 17.97069 17.97164 17.97217 17.97249 17.97271 17.97285 17.97295 17.97300 17.97305 17.97308 17.97309 17.97310 17.97311 17.97312 17.97312 17.97312 24.76098 24.76884 24.77163 24.77292 24.77365 24.77411 24.77440 24.77459 24.77473 24.77481 24.77487 24.77491 24.77493 24.77495 24.77497 24.77497 24.77497 24.77497 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 TABLE 3-3 < AS A FUNCTION OF FREQUENCY FOR THE LOSSY DIELECTRIC MATERIAL Material type F (GHz) 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 Type 1 Type 3 Type 5 Type 7 Type 9 1.84577 1.86013 1.86536 1.86726 1.86794 1.86819 1.86828 1.86831 1.86832 1.86833 1.86833 1.86833 1.86833 1.86833 1.86833 1.86833 1.86833 1.86833 1.18222 1.18308 1.18340 1.18351 1.18355 1.18357 1.18358 1.18358 1.18358 1.18358 1.18358 1.18358 1.18358 1.18358 1.18358 1.18358 1.18358 1.18358 1.65893 1.65913 1.65922 1.65925 1.65927 1.65928 1.65928 1.65928 1.65928 1.65928 1.65928 1.65928 1.65928 1.65928 1.65928 1.65928 1.65928 1.65928 2.31634 2.31605 2.31597 2.31595 2.31594 2.31594 2.31594 2.31594 2.31594 2.31594 2.31594 2.31594 2.31594 2.31595 2.31595 2.31595 2.31595 2.31595 8.03474 8.03710 8.03804 8.03841 8.03856 8.03862 8.03865 8.03866 8.03866 8.03867 8.03867 8.03867 8.03867 8.03867 8.03867 8.03867 8.03867 8.03867 Material type Tvpe 11 Tvpe 13 Type 15 Type 17 Type 19 F (GHz) 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 4.98416 4.96560 4.95882 4.95636 4.95547 4.95515 4.95504 4.95500 4.95499 4.95498 4.95498 4.95498 4.95498 4.95498 4.95498 4.95498 4.95498 4.95498 9.74786 9.73054 9.72429 9.72205 9.72125 9.72097 9.72087 9.72084 9.72083 9.72082 9.72082 9.72082 9.72082 9.72082 9.72082 9.72082 9.72082 9.72082 8.17483 8.14305 8.13155 8.12741 8.12593 8.12540 8.12521 8.12515 8.12512 8.12512 8.12511 8.12511 8.12511 8.12511 8.12511 8.12511 8.12511 8.12511 14.62385 14.59117 14.57935 14.57511 14.57359 14.57306 14.57287 14.57281 14.57279 14.57278 14.57277 14.57277 14.57277 14.57277 14.57277 14.57277 14.57277 14.57277 19.05948 19.01088 18.99331 18.98702 18.98477 18.98397 18.98369 18.98359 18.98356 18.98355 18.98354 18.98354 18.98354 18.98354 18.98354 18.98354 18.98354 18.98354 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 TABLE 3-4 GA PARAMETER SEARCH SPACE Parameters Total thickness (mm) 1st layer thickness (mm) 2nd layer thickness (mm) 3rd layer thickness (mm) Choice of type o f material for 1st layer er of 1st layer Choice of type of material for 2nd layer er of 2nd layer Choice o f type of material for 3rd layer er of 3rd layer Choice o f type of material for 4th layer er of 4th layer Number of possibilities 590 5950 5950 5950 Number of binary digits 10 13 13 13 1 3 2 0.1 497 9 1 3 2 0.1 497 9 1 3 2 0.1 497 9 1 3 2 0.1 497 9 Range of Increment parameters 0.01 0.1-6.0 0.05-6.0 0.001 0.05-6.0 0.001 0.001 0.05-6.0 0-3 1.03-10.0 0-3 1.03-10.0 0-3 1.03-10.0 0-3 1.03-10.0 Total# of bits 93 TABLE 3-5 MATERIAL DISTRIBUTIONS AND THICKNESS (MM) OF COMPOSITE FOR THE THREE CASES Total thickness Dielectric Layer Types Case 1 Case 2 Case 3 Layer 1 Layer 2 Layer 3 Layer 4 Lossless dielectric d = 3.65 Lossless dielectric D = 2.54 Lossy dielectric d = 1.25 Lossy magnetic D = 0.85 Lossy dielectric d = 0.55 Lossy magnetic D = 0.05 Lossless dielectric d = 0.05 Lossless dielectric D = 0.05 Lossless dielectric d = 2.75 Lossy dielectric d = 1.25 Lossy dielectric d = 0.35 Lossy magnetic d = 0.27 5.5 3.49 4.62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 TABLE 3-6 PERFORMANCE COMPARISION OF CGA AND MGA Cases Case 1 Case 2 Case 3 MGA (populaltion size = 50) Best fitness Computation value reached time (secs) after 1000 generations -0.234 299.0 -0.314 199.0 -0.174 232.0 CGA (population size = 500) Best fitness Computation value reached time (secs) after 1000 generations 3312.0 -0.32 2294.0 -0.33 2336.0 -0.297 3.3.5 Numerical Results for the Synthesis of Broadband Microwave Absorbers Using Lossy Carbon Fiber and Lossless Dielectric Materials In this section we optimize a composite consisting of either lossy carbon fiber or lossless dielectric material. Figs. 3 .18 (a) and (b) plot the real and imaginary parts of the permittivity o f the five (m=lg, 3g, 7g, 9g and 11.25g) carbon fiber materials as a function of frequency. The MGA optimization is performed in the X-band (8.2-12.4 GHz) and linear interpolation is utilized for the carbon fiber materials, in a manner similar to that applied to the lossy dielectric and magnetic materials in the previous sections. The two cases investigated are as follows: (i) normal incidence - TE and TM polarization (ii) 0 varying from 0 to 45 degrees - TE and TM polarization The population size and the number of generations are fixed at 5 and 1000, respectively, and the frequency resolution is chosen to be 0.42 GHz. The results of the MGA optimization o f the reflection coefficient in dB are presented in Figs. 3 .19 and 3 .20. For both the cases the azimuthal angle is fixed at zero degrees. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 Carbon Fiber Composite •— • — • — • — • — • — •— • — • — • — • 18 9 10 11 12 13 12 13 Frequency (GHz) Carbon Fiber Composite 10 11 Frequency (GHz) Fig. 3.18. The relative permittivity of carbon fiber material as a function of frequency , (a) real part; (b) imaginary part. We see from Fig 3.19 that the frequency response obtained by using the MGA resembles, as expected, the familiar filter response derived by using the Tchebyscheff equal ripple optimization procedures. For Case-1, the worst-case reflection coefficient is maintained below -25.0 dB. Figure 3.20 shows the worst- and best-case reflection Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 coefficients realized by the MGA, while optimizing simultaneously for both the TE and TM polarizations overthe range of elevation angles of interest (0=0.0 to45.0degrees). Imposing an additional burden of optimizing over a range of elevation angles, causes the worst-case reflection coefficient value to degrade further to -16.4 dB. The total thicknesses o f the composites are maintained at 6.0 and 5.96 mm for the two cases, respectively. er’=4.55, er” =0.1, d = l.55 mm m = 5g, d= 1.45 mm m = 11.25g, d = 2.25 mm m = 11,25g, d = 0.75 mm (a) -24 -26 -28 S -30 c -32 K -34 -36 -38 8 8.5 9 9.5 10.5 10 Frequency in GHz 11 11.5 12 12.5 (b) Fig. 3.19. Case-1: (a) composite design; (b) reflection coefficient vs. frequency. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 er’=6.0, er” =0.1, d=1.15 mm m = 2g, d= 1.45 mm m = 11.25g, d = 2.85 mm m = 11.25g, d = 0.51 mm (a) -16 -18 Reflection coefficient in dB -20 -22 -24 -26 -28 -30 worst case best case -32 -34 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 Frequency in GHz (b) Fig. 3.20. Case-2: (a) composite design; (b) worst- and best-case reflection coefficient vs. frequency. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 SYNTHESIS OF COMPOSITES COMPRISING A SINGLE FREQUENCY SELECTIVE SURFACE (FSS) EMBEDDED IN DIELECTRIC AND/OR MAGNETIC MEDIA In Chapter 2 we have successfully demonstrated the enormous advantages of the MGA over the CGA when applied to problems that are computationally non-intensive in nature. The optimization problems in Chapter 2 were investigated with the sole purpose of gaining insight into the way the MGA performs, and in the subtle manner by which the MGA parameters affect its efficiency and convergence properties. Armed with this knowledge, we now proceed to the next step, viz., that of tackling a computationally intensive optimization problem via the application of the MGA. The first problem we consider is that of synthesizing a broadband microwave absorber comprising a single FSS screen embedded in dielectric and/or magnetic media-that may be lossy and/or lossless to operate simultaneously for both TE and TM polarizations and for a specified range of incidence angles [33]. Next, we turn to the problem of designing a spatial filter with a single FSS screen embedded in lossless dielectric layers that exhibits good performance for different angles o f incidence and polarizations. Finally, the domain decomposition method is applied to synthesize a dual-band radome consisting of a single FSS screen embedded in lossless dielectric medium, and operating at a high off-normal incident angle [10], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 4.1 Problem Geometries Figure 4.1 shows a AMayered composite structure whose parameters we wish to optimize with a view to realizing a specified frequency response for spatial filters. Figure 4.2 depicts the same structure except for a Perfect Electric Conductor (PEC) backing, which is introduced to test the effectiveness of broadband microwave absorbers designed for shielding. We assume a plane wave is incident upon the composite at an arbitrary angle and is polarized either Transverse Electric (TE) or Transverse Magnetic (TM) Sn Mn V ? - W z dn Lossy or lossless FSS screen Fig. 4.1. iV-layered composite with a TE polarized uniform plane wave incident at an arbitrary angle. For TM polarization the E and H fields are interchanged along with a phase shift of 180 degrees introduced in the H field. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 PEC backing Lossy or lossless FSS screen Fig. 4.2. Synthesis of broadband microwave absorbers with a single FSS screen embedded in lossy or lossless dielectric and magnetic material with PEC backing. AMayered composite with a TE polarized uniform plane wave incident at an arbitrary angle. For TM polarization the E and H fields are interchanged along with a phase shift of 180 degrees introduced in the H field. To synthesize both spatial filters and broadband microwave absorbers, the MGA optimizes the thickness and constitutive parameters of the layers in the composite, and also determines the FSS screen design, its material content, position within the composite, and the x- and y-periodicities. An upper bound is imposed on the total thickness o f the composite. As mentioned in Chapter 2, the MGA selects the values of frequency dependent e r and /ur of lossy dielectric and magnetic materials via lookup tables shown in Tables 3-2 and 3-3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 4.2 Analysis of Doubly-Periodic, Planar, Infinite FSS Screens Embedded in Inhomogeneous Layered Media In this section, we present a quantitative analysis of planar FSS systems [15], [17,18], [34] to illustrate the computationally intensive nature of the problem. The geometry to be analyzed is shown in Fig. 4.3. The structure consists o f N stratified planar layers, infinite in both the x- and y-directions, with arbitrary constitutive parameters and thicknesses s„, fun, and d, (i=l, 2........ N), respectively, for each layer in the composite, which also has an infinite, planar, doubly periodic, and infinitesimally thin FSS screen embedded in it. The assumptions regarding the thickness and infinite nature of the FSS in the x-y plane has been made to significantly reduce the computational complexity of the problem. We assume that there are R superstrates and S substrates in the structure, i.e., R~S = N, where N denotes the total number of layers in the composite. FSS Screen SrR+S SrR+S-l SrR+l | £r2 Sri Y MrR+S MrR+S-1 dR+S dR+S-l -z • • • fJrRH 1 1 ftrR ■ dR+i | d,R • • • Vr2 d2 d, 1 (a) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 ▲y Reference cell arbitrary cell (b) Fig. 4.3. The composite structure to be analyzed: (a) side view; (b) top view. The steps for analyzing the composite structure, shown in Fig. 4.2, are now given. A uniform plane wave incident on the structure induces currents on the metalized portions of the screen that reradiate to form the scattered fields. Knowledge of the currents induced on the screens can be used to calculate the scattered fields. These, in turn, can be utilized to calculate the reflection and transmission coefficients of the entire structure. In view of the infinite and periodic nature of the FSS screen, the currents induced on any one unit cell will be identical to those on any other, with the exception of a phase progression introduced by the incident plane wave. This is the basic premise of Floquet’s theorem, which is invoked in the analysis of periodic structures, to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 mathematically relate the induced current density J { x , y ) in an arbitrary cell to the current J 0 (x,_y)in the reference cell (see Fig. 4.3). Utilizing this theorem, we can write [35]: £ Y . S ( x - m T x) S ( y - n T y) e ^ z' ^ >] (4.1) where, m and n are arbitrary integral multiples of the x- and y-periodicities of the FSS screen; = /?0 sin 0 cos^ and p '* = p o sin 0 sin where /?„ = (OyJ/unstl is the free space wave number; 0 and ^are the angles in elevation and azimuth respectively; and ® is the convolution operator. Equation (4.1) implies that the surface current densities need be calculated only for a single unit cell and this drastically reduces the computational burden o f the problem at hand. We should mention that all the FSS geometries dealt with in this work are periodic along a rectangular grid and (4.1) is valid only for this case, though it can be generalized for a skewed grid [15], Next, we present the formal derivation employed to obtain the currents induced on the metallic portions o f the screen and the scattered fields in the entire composite. We begin by splitting the original problem into two sub-problems (see Fig. 4.4), viz.. the calculation o f the incident and scattered fields. The former exist in the structure in the absence o f the metallic screen, while the latter arise from the currents induced on the screen, and summing the two yields the total fields. The first part of the problem (Problem-I) evaluates the incident fields in the absence of the metallic screen by following a procedure similar to that described in Sec. 3.2, while for the second pan Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 (Problem-II) an integral equation is formulated in terms of the unknown surface currents. The procedure for deriving this equation is outlined below. E t, Z = Zr Q Z -!Z r z = Zr t Et, H t E ,H Original problem, total fields Problem-I, incident fields Problem-II, scattered fields Fig. 4.4. The original problem is split into two sub-problems. Z = ZR Region-1 Region-0 z = Zr Region-I j Region-0 z = Zr Region-I Region-0 Js E=0 0 = 0 Es Os pmc Ms (a) lI i i t t (b) (c) Fig. 4.5. Surface equivalence principle applied to Problem-II: (a) equivalence principle applied to Region-I; (b) perfect magnetic conductor (PMC) used to short out the surface magnetic current density; (c) image theory applied to the Region-I problem. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 First, the Surface Equivalence principle [29] is applied to Problem-II by replacing the metallic conductor with equivalent electric and magnetic current densities as shown in Fig. 4.5 (a). These currents produce the original fields Es and H s in Region-I and are expressed as Js = zxH s (4.2) M , = 0 on metal surfaces . = - z x , otherwise (4.3) Next, Region-II is replaced with a Perfect Magnetic Conductor (PMC), which shorts out M s as shown in Fig 4.5 (b). Finally, to eliminate the presence of the PMC, we employ the image theory [29] to derive an equivalent problem shown in Fig. 4.5 (c), leaving only the unknown electric currents radiating into Region-I.The same procedure is repeated to evaluate the unknown electric currents radiating into Region-II. Superposition o f the currents radiating into Region-I and -II yields the total currents from which the scattered fields in the whole composite is evaluated. The scattered field in the media above and below the plane containing the FSS screen can be expressed [29] in terms of the magnetic and electric vector potentials, A and F , respectively, as Es = - j ( o A — ^ - V ( v i ) - - ( V x F ) tofie v 7 sv 7 (4.4) Hs =-ja)F— ^ - V ( V F ) + - ( V x i) iofis v M (4.5) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 For thescatteredfieldwe select the z-component of the vectorpotentials A =zA(x,y,z) andF = z F(x,y,z). Thesepotentials satisfy the source-free, scalarHelmholtz wave equations V 2A; +/?2A; =0 (4.6) V 2FZ+ 0 2F: =O (4.7) where /? = coV^s is the wave number in the medium. The spectral domain approach [36] has been found to be very useful for solving infinite, planar, periodic geometries. To apply this approach to our problem we define the Fourier transform pair (4.8) /(A . A ) = I J /(*■>’) =7rV J / f(P „ P r) > A^ ' M dP,Pr [ 27Cj (4.9) -cc Applying the method of separation o f variables, we express the solution of (4.6) and (4.7) in the spatial domain as Transverse Electric (TE) and Transverse Magnetic (TM) E and H fields as: Este = r ^ a e dy xr - - j c tA z . h (o/ ie dxdz te co/us dxcz H m _ _ cA. M cy jj d2A: d A . . ^TE _ j dutm 1 cA. (4.10) ’ JV f \_ 1 TV o)/u£ cydz o)fi£ cydz /J. c x f ^2 ( m J C 02 E t J= 0 - E :‘ ‘ = ^ t + p A e -.h ? =0 onus dz J eofds c: £Te _ 1 dF. s dx £.ni _ ’ TV Substituting / (x ,y ) as F:(x,y) and Az(x,y) in (4.8) and (4.9), we get Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 ~jPA(P^Py^) aft (A. A-*) -jPf:(P*,Py^) (4.11) ~JP,A: ( P ^ P ^ - ) dx cA: ( p x,Py,z) dy - j P y A : (Px,Py,z) Using (4.11) in (4.10), the E and H fields in the spectral domain for TE and TM modes are expressed as e tJ (A. A-*) = ^ f t (A.A.--);£“ (A. A.--) = e ■ cojus cz (A. A.--)=- hco/j£ - cA A hczih A £? (A .A.*)=-— ft (A. A e J ( c2 £?(A.A.-) = 0;£"'(A,A.-) = - — -rr-A J A(A,A.-') ry/ic^cz (4.12) «?(A . A--) = <y//f fl? (A. A ' - (A , cz ) A?(A.A.-) = —COfXS ■ r *' %CZ - :?W A.--) = A(A. A.-) (A. A- -') ° ■ — A (A. A--) M |ft(A.A-•);«“ (A.A.-) =0 In (4.10) and (4.12) the scattered fields are decomposed into TE components (transverse to z -directed electric field) that are expressed in terms o f F only, and TM components (transverse to z -directed magnetic field) that are written in terms of A only. To calculate the total scattered fields, the superposition theorem is applied to sum the TE and TM components. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 Referring to Fig. 4.5(c), and using (4.2) and (4.12), the transforms o f the x- and ycomponents o f the surface current at z = zRcan be related to the transforms of the vector potentials as (4 13) (Ofxs dz cojue dz It is evident n — n (4.14) that the transforms of the x- and y-componentsof the surface representcoupled equations. To current decouple these equations, thefollowingcoordinate transformation, as suggested by the spectral domain immittance approach [16], is applied v = xcos#+> 'sin 0 ; x - vcostf+wsin# u = jrsin# —>>cos0 ; y = v sin 0 -w co s0 (4.15) sinfl = - 7=-^-— , cos 6 = This results in a new Cartesian coordinate system rotated by an angle 6 relative to the original x-y coordinates. It is evident from the last equation in (4.15) that the rotation angle 6 is different for each spectral component. Applying the transformation in (4.15) to (4.13) and (4.14), we get J. (4.,6) (DUE * L , . while inserting it into (4.12) yields Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (417) We note that (4.16), (4.17), and (4.18) are decoupled and the solutions to (4.6) and (4.7) in each of the homogeneous layers of the FSS structure can now be obtained. It is possible to derive an alternate representation for the stratified structure by using the transmission line analogy, as portrayed in Fig. 4.6, and applying the transmission line solution to the transform of the electric and magnetic vector potentials. Both A. and F; can be written, as shown below in (4.19) and (4.20) as superpositions of negative- and positive-going wave potentials associated with the individual layers within the composite: A. = A~e' y: + A . e ' r: (4.19) F = F . V r- '+ F .V y; (4.20) where chosen to satisfy the radiation condition. Equations (4.19) and (4.20) can be written as (4.21) (422) where, T u (z) = - — e2y: and TE (z) = — e~r: A' F~ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 J S|K.V) J S lu .v i I Fig. 4.6. The transmission line equivalent of the FSS scattering problem for Region-I. Taking the first derivative o f (4.21) and (4.22) we get dA 5z (4.23) and dz = - y F : e ^ [ \ - Y E(z)] (4.24) Using (4.18), (4.22), and (4.24) we obtain the impedances for the TE modes, which may be expressed as Z E(z) = ^ = - ^ V ' H.. Y f Likewise, the expression for i+rE(z) 1- r E(z) (4.25) impedances for the TM modes can be derived by using (4.18), (4.21), and (4.23). It reads 7 u (-\= ^ - H..u 1 Y:u 0 'l +r"(r)‘ 1-r"(r) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.26) 81 V J/i)P where, Y E = —— and Tjw = - — , are the characteristic impedances of the individual ja>M Y transmission line segments. Employing the relationships given above, and using the fact that the tangential components o f the electric field at the interface between the layers are continuous above and below the FSS screen position, we can evaluate the reflection coefficients and the transmission line admittances at any point along the line. To evaluate the scattered fields in Region-I, we need to find the admittances Y ' [E'Kt), as seen from the left o f the current source at z = zr, (see Fig. 4 .7). *(E. \1), no*) ( zr ) no-) free space Line Line 2 Line R -1 Line R z=0 z = Z\ z = Z; z- Z r -2 Z - Z r -1 z- Zr Fig. 4.7. The admittance seen by the current induced on the FSS screen for R superstrates. A recursive procedure, outlined below, may be employed to calculate the admittance seen from the left o f the current source. The procedure is outlined below. (i) Initialize the admittance at z = 0 as r (£. u ) = ^ . u , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (427) 82 (ii) Calculate I+ r < 1( ; = ,v , ) coth { r , {Y, - V ,)}+ r 'SM'(: = -V.) Yf, = - r 1— ; c = — ) Z- • r, = jm p Ya y ■ ' . Mr, = . . - JA,. (4.28) recursively, until p = R. Once the impedances for the TE and TM modes are known, the transverse components of the scattered fields can be easily related to the surface currents as follows: e ; =(z‘'-7;)•-•A (4.29) £; =(z“v;) where the (+) superscript indicates the currents and impedances of the Region-I problem. A relationship which is similar to (4.29), between the surface currents and the scattered electric fields in Region-II, can be expressed as e; =(z* I ). (4.30) e; = (z “v ;) To evaluate the electric and magnetic impedances for Region-II, we apply the same procedure as we did for Region-I. The total surface current on the screen at z = zr is given by •7“ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. H-31* 83 where J (* v) and J,~ v) are the currents that produce the fields in Regions-I and -II, respectively. The total currents can be related to the scattered fields above and below the FSS screen at z = zr as follows: (4.32) where, Z(EM1 = * U)_ is a parallel combination of the upper- and lower-half impedances. Using the coordinate transformations of (4.15) in (4.32), and then implementing some mathematical manipulations, we arrive at the following expressions for the transforms of the scattered electric fields in terms of the transforms of the currents: Ex =Gxx J x +Gxy J v (4.33) Ey =G va: J x +G» ' J v where - fcZ " + p ;Z E P l +P; - p ;Z E+ p;Z M -1 T W ~ ^ (434) M r (2 " - 2 ‘ ) Lr„. = Lr,_ = --------- —;--- ---------- P;+P; The G ’s in (4.34) can be interpreted as elements of a Green’s tensor that is symbolically represented as G . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 Since the expressions in (4.33) are in the spectral domain, we need to apply the inverse Fourier transform to revert to the spatial domain, so that we can impose the boundary condition Es - - E x on the metalized portions of the FSS screen. This leads to the following Electric Field Integral Equation (EFIE): | -E'*(x,y) -OC *<E i& lL JX e ^ (0'x^ yy)d p xd p v (4.35) G >y G >* where the incident fields - E™[ x , y ) and -E'™(x,y), the dyadic Green’s functions Ga , Gn , G„ , and Giy, are known, and Jx and J v,are the unknowns. From (4.1), we note that the surface currents on a periodic screen can be expressed in terms of the currents on a single unit cell. Utilizing the fundamental properties of Fourier transforms [37], viz., frequency shifting and the Fourier transform of a periodic train of delta functions spaced Tx centimeters apart, consists of another set of delta functions weighted by the factor I Tx and regularly spaced I Tx radians/cm apart along the spectral axis, we arrive at the following expression for the induced surface current on the FSS screen, -x —X X l7t) 1 T T m=-<cn=-cc 1 x v .( 2 a-) •/,(a . a W ,.( a ,a )VTx Tfv - »1r= —* 1n = - « - a .) (4.36) 5(a - a .M a - a .) where (4.37) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 are the Floquet wave numbers. Substituting (4.36) into (4.35) we get j Tx Ty m=~<c n=-co A, (A . A ) I In=-<x ■ U A .A ) (4.38) where, Tx and Ty are the periodicities o f the FSS screen in the x- and y-directions, respectively. We note that the dyadic Green’s function or the Green’s tensor as given in (4.34) is no longer a continuous function but assumes discrete values due to the periodic nature of the geometry. Equation (4.38) is the final form of the EFIE that is well suited for numerical computations. It is useful to point out that: (i) only the currents on the unit cell are used to evaluate the scattered feilds, which makes the computation tractable; (ii) the Green’s function is dependent upon P ^ and p , which are functions of the incident field, and this implies that the associated EFIE matrix must be re-computed for each incidence angle. Equation (4.38) is enforced at s = zr, but the incident fields, E'"c (*,>’) and £^(x,_y)are known only at z = 0. To evaluate the incident fields at z = zr, we utilize transmission line techniques as shown in Figs. 4.6 and 4.7, along with the recursive procedure as outlined in (4.27) and (4.28) for the N =R~S layers. To facilitate this step, we need to separate the TE and TM components of the incident fields. Towards this end we express the electric and magnetic vector potentials for the incident fields as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 I =eA^ y)e ^ - (4.39) p =eAtr*-t>?y)er«; Substituting (4.39) into (4.10) we obtain the x- and y-components of the electric fields for the TE and TM modes J fiy g tn c T E _ p . pm cTE _ j Px p e (4.40) inc n in e ~incTM _ Px / (OHS ineTSt a :- e : A. coh£ Having derived the expressions for the incident fields and the dyadic Green’s function, we can now apply the Method of Moments (MoM) to (4.38), and evaluate the unknown induced surface currents on the metallic portions of the FSS screen. When the screen has a finite conductivity, the boundary condition on its surface maybe written as = Es - ZSJ S, where a surface of finite thickness and given loss tangent is approximated by an infinitely thin surface with complex surface impedanceZ s measured in . For the finite conductivity case (4.38) becomes I xv m=*<®n~-<o <U a . a ) (4.41) AP>mX~ where the additional term on the right side corresponds to the product of the complex surface impedance of the FSS screen and the induced surface currents in the spatial domain. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 4.2.1 Numerical Solution Employing the Method of Moments (MoM) To apply the MoM [38] to (4.38) and (4.41), we need to represent the unknown currents in terms of a complete set of basis functions as follows: M .V P = --< F ~ "-I u (4.42) ■> s p= -— 1 = ~ where the c^ ’s are the unknown complex coefficients of the known basis set b.\"i)*/Qi, (*p, q, x, y)*. The inner product integral used in Galerkin’s testing procedure is defined as Ty T, where, t(x rt)( r,j,x ,y )is the testing function which has the same form as the basis function b{Xay )( p , q, x, y) and * is the complex conjugation operator. Substituting (4.42) into (4.38) and applying (4.43), we arrive at a set of M x N equations in terms of the M x N unknowns, which can be expressed in a matrix form as v f ,nrjx,,(r,s) v^nr-™>(r,s) Z~ { P ^ S) Zytip, q>r ’s ) Zv i W ' s) Zyy( p, q, r , s ) (4.44) Cy0 ( P ’ <?) where Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 _ , a!nc ' ? m = J £ £2 - K ( r' s< K '’P ? ) s tnc nine (ope v - ' inc a r n c C w = —cOfie ^ - > ^ (' r , * . / r . / > r )7 | *<c *-<c Z xx{ p , q , r , s ) = - — ^ ' Z G BC{ m , n ) b Xa( p , q , m , n ) ^ ( r , i ,m ,/ i ) x v m=-<c«=-« 1 Zxy{p,q, r, s) = - — '£t £ Gr ,(w,/i) 6Vo(/>,?,m,n) £ ( r ,5,m,«) x v w=-<e /r=-<c = *vC 1 -<c Z G A m ' n ) b* ( P ' £l ' m ' n ) i ^ { r , s , m , n ) m=-xi n--« J -<x «t Zw ( p, q, r, s) = — Z Z ^vr K " ) \ x v x v m = n= -< c (p,q,/n,n) Ka {r,s,m,n) (4.45) U M ., --------< N o , and, andA p, q, r, andA s are integers whose ranges a r e M < p , r < ----1 ? ? ? AT 5 < — - 1 . When the FSS screen has a finite conductivity, we modify (4 44) as follows: v^TE.nt)^ ^ C r f n n (r,s) ^ (p ,< 7 ,r,5 ) Z „( p, q, r , s ) Z„( p, q, r , s ) Zyy(p, q,r,s) (4.46) Z« (/>,?,/■, 5 ) 0 0 Z[y ( p , q , r , s ) where the new matrix on the right side of (4.46) is called the adjunct impedance matrix. Its elements can be expressed as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 .h-JL 1 ■> zUP’<i’r’s)= J J c^ R ) K ( p ^ x'y) C(r’s’Jf’>’) ^ Jr-h . 2 2 J±,l1 2 (4.47) 2 ^ (P, R,r,s) = | J c>o (p, q) AVn(/>, <7, x,.y) t’yo (r, s,x ,y) dxdy Ijl-L. 2 2 The integrals in (4.47) can be evaluated analytically since they are simple multiplication operations Z^. instead of convolutions. The expressions for Z a’ (p,q,r,s) and will be provided after the sub-domain basis functions have been defined. With the formulation o f the EFIE for the unknown surface currents now complete, we proceed next to define an appropriate basis set. The basis functions should be chosen to ensure that they portray the physical behavior of the surface currents accurately and efficiently. Depending on the FSS-screen geometry, we either choose the entire domain or the sub-domain basis functions. However, the former are tailored to specific unit cell geometries for which the physical form o f the currents are known a priori, and their use is not convenient for us because the MGA carries out the optimization of the FSS screen in real time. For this reason we opt for the sub-domain basis functions and discretize the currents to span a small fraction of the domain. These functions can accurately represent the currents in arbitrary unit cell geometries; hence, they are versatile. However, the drawback of using these functions is that, in many cases, a very large number of basis functions are needed to accurately represent the surface currents. It is useful to select the basis set whose Fourier transform can be derived analytically, which is needed in (4.45). Furthermore, the transforms of the basis set Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 should be such that the product, G(m,n) b ( p , q , m, n ) t ' ( r , s , m , n ) , decays rapidly for increasing orders o f the Floquet’s harmonics, and /? . This allows us to truncate the double summation in (4.45) by retaining only the lower order of harmonics, improving the rate of convergence o f the summations, which is the principal factor in determining the speed with which we solve the EFIE. x = pAx x = pAx x = (p+l)Ax Ay y = qAy- Ay Ax •y = qAy 4 Ax bx(p,q,x,y) Ax by(p,q,x,y) Fig. 4.8. x- and y-components of the unit amplitude rooftops. The selected sub-domain basis functions, called “rooftops”, are shown in Fig. 4.8 and expressed in the spatial domain as bg( p, q, x, y) = \ ( p , x ) I l ( q , y ) by { p ^ x , y ) = n { p , x ) \ { q , y ) (4.48) where u- Au A (/, m) = Am n(/,w) = .. Am .. 3Am /Am --------< m < /Am h-----2 2 elsewhere Am A// /Am -------- < m < /Am 4---2 2 elsewhere Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.49) 91 In the spectral domain the transforms of the rooftop basis functions in (4.48) and (4.49) are given by \r bx{p,q,m,n) = (AxAy)sine2 & — sine ^ 2 ) -j(ft.m p^v) (4.50) s, {p,q,m,n) = ( AxAy)sinc2 where sinc(//) = \ f 0^ ^ J s in c [ i \ Ax ^ e 2j ,R Av 1 f t >n ^ sin(w) To apply the rooftop basis functions just defined, the FSS unit cell needs to be discretized into an M x N grid, as shown in Fig. 4.9 for a square patch and M = N =16. Next, we place x - and y -directed basis functions everywhere on the metalized portions of the FSS screen. If necessary, the density of the grid can be increased to provide a more accurate representation of the surface currents. In (4.44) and (4.46), the doubly infinite summations can be efficiently handled when the FSS screen is represented as described above. Since these summations cannot in general be evaluated in closed forms, they must be computed numerically; after truncation. The rooftop basis functions possess very good convergence properties [15] and their transforms are well suited for summation using the Fast Fourier Transform (FFT) algorithm. Utilizing the FFT and a special partitioning scheme for the summations, a large number of terms can be summed in a highly efficient manner. Substituting (4 .50) into the last four equations in (4.45) we get Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 Fig. 4.9. FSS cell discretized into 16 x 16 pixels with the shaded part corresponding to metal. -<x Za ( p, q, r, s) = £ *ac ' £ G a f ; P ]( p, q) P; (r, s) *l=-<xi n = -< c Z„{p, q, r, s) = £ nt— Z yr(w ,s)= £ ;L ^ Iyf xf yP\ { P^) P' {r , s ) e —<c £ 6 >x/ x/ ^ (/>,?)/}*(/•,*)<? (B=-<Cffr-H® Zvy( p ,^ ,r,5 )= £ £ G ^ /; ^ ( p ,< 7) ^ * (r ,j) where f / Ax" , AxAy . , / x= — sin e P*m sine T TV V V ^ J / ArAy . 2 Ax" Ay"* sine sine A. /v = A . T ; 7T T, V V * X * Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.51) 93 The infinite summation in (4.51) can be rearranged as follows. First, we subdivide the spectrum into M segments consisting of M harmonics along /&, and N segments consisting of N harmonics along $ , where M and N are the number of segments into which the FSS unit cell has been divided. Upon doing this, we effectively truncate the summation to ( M x M) x ( N x N) elements. Next, we generate an array consisting of the sum o f corresponding elements from each of the M and N segments (see Fig. 4.10), and use (4.37) to express the summations in (4.51) as ±L, (4.52) where , Ax -jZ.T - l [ p - r Av; ( < ? - 11 !- — Fz = e Segment I Segment 2 Segment 3 A2 A3 Segment 4 -10 -9 A A1 Fig. 4.10. Grouping the infinite summation over A4 into M segments of M elements. The like terms o f each segment are summed, forming an array A, composed o f M elements. In this case M =M=-f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 Following the same procedure, we can also express Zxy, Z>x, and Zyy to define the T T remaining terms in the impedance matrix of (4.46). Next, by letting Ax = — and Ay = — , M N - jZ x we can show that P{(m'M,n'N) = 1 and Px{m",n’) - e mV ' — { n p - r y — {$-*) u v Applying these expressions to (4.52) we get •"-i X X<5J ; Z„ = FFT m \1 (4.53) , .V n- — where the FFT is used to implement the discrete Fourier transform of the type T -1 ^ / ( f'*7)= S m. T-> M '« • «• ' F i m ^ n ^ e 1 -u v' (4.54) .V n. =— The summation in (4.54) can be efficiently evaluated by using the FFT for a large number o f matrix elements. To achieve the highest efficiency, the values of M and N must be powers o f 2. We should also point out that the use of the FFT also requires that the unit cell geometry be discretized in a rectangular grid. With the basis functions and its transforms defined in (4.48) through (4.50), we can now derive expressions for the elements o f the adjunct matrix. Substituting (4.48) and (4.49) into (4.47) we obtain the following equations [34] for these elements: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 -S{p-r)S(q-s) = AxAy S ( p - r - l ) S ( q - s ) / + S ( p - r +\ ) S ( q - s ) f 2 + (4.55) y £ (p -r)< % -.s) Z w = AxAy + 8 ( p - r ) 8 { q - s - \ ) f ^ 8 { p - r ) S { q - s +\)fi where for p = -1 and r = - M (4.55a) otherwise for p ~ fz M M and r = ------ 1 = 0 /,= otherwise e y for q = ----- 1 and 5= ----2 2 otherwise 0 e (4.55b) y y for q = 2 /> = 0 and 5 = ----- 1 2 (4.55c) (4.55d) otherwise The phase terms in (4.55a) through (4.55d) are necessary because the basis functions are selected such that at the edge of the unit cell they overlap the adjacent unit cell. With all the tools ready to compute the matrix elements efficiently and accurately, (4.44) and (4.46) can be expressed as v = Zc r_ _ _ .-i v = | Z + ZSZ Ic Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.56) 96 The solution to (4.56) requires the inversion of a matrix to obtain the unknown current coefficients, which is typically performed by using the LU decomposition algorithm with partial column pivoting. 4.2.2 Evaluation of Reflection and Transmission Coefficients Once we have computed the surface currents, the reflection and transmission coefficients can be easily calculated from the tangential scattered fields at r = 0 and z = :r~ s, expressed in the spectral domain. The scattered fields are calculated from (4.38) or (4.41), depending on whether the FSS screen is a PEC or has a finite conductivity, and then transferred to z = 0 and z = zr-s by repeatedly applying (4.27) and (4.28) The incident fields can be evaluated by using (4.40) and similarly transferred to z = 0 and z = zr-s . The sum o f the scattered and incident fields yields the total fields at z = 0 and z = Z r-s- The scattered field at z = 0 can be expressed [39] as a superposition of Floquet’s harmonics: (4.57) where -fa . The scattered fields can also be written in terms of the vector potentials A1 and F 5as Es = -ja )A s =£— V ( V - / H - — ( V x F 1) co ^sa v 7 v where the potentials are expressed as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4 58) 97 •<r wr I O V « * = < > p=-<0 (4.59) F ’ =z £ f p=-«?=- at : = 0 _ t?j(v * V ) e "*. Next, applying (4.10) to (4.59), we get = and WO jP y * «: = I E •CO —x \ »<o p --< t3 C p d te , 'K 'V ym . D r.u y e° jP*, dte W *<> -X ^ + Pyjm p m , . 1. . r. W ■o ¥ pqi (4.60) V pq y Multiplying both sides of (4.60) by ^*„ and integrating over the unit cell, we obtain •<C j <bdy = X umt ctil -WO *■<*• *** I 1 j3 umt cell JHx? ftTE + w B y w gr\t w • f p = -* c j= - umt ctil VpqVm dxdy j PH '0 v. P =~,c- f JPy. Dnr , P xp. Y pm ^r.u Z (4.61) V py mn dxdy umt cell The left sides o f (4.61) are the Fourier transforms of E\ and E*y evaluated at fi and /? . The double summations appearing on the right sides can be eliminated by using the orthogonality properties of Floquet’s harmonics. Applying these properties we get C = je n Py. {£ K ’ Py. ) + } ~ P , { % { P , •Py. ) + ^ S4 Pl+Pl. (4.62) Rmr = 0 )^en A {£ (a . . a..)-1-£;*_} +a {*; (a . ■a . ) ■*5 a ,} /-(A + A ) where ^ is equal to unity only for m = n = 0, which corresponds to the reflected field term. Following the same procedure, the scattered fields at 2 = zR~s can be derived, and the following expressions for the transmission coefficients can be obtained: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 TTE mn = 7is *0 P i * o ly« (4.63) 'rTM mn A . { a ( A ..A .K A .} + ^ . { 5 K A ) + f iA ,} j ~ ^ M l^ O y~.{oi+oi.) In (4.62) and (4.63) the reflected and transmitted fields are (£; ) TE =- ^ -k ,(£; )™= £•„ ( £ ;) rc J -h a - r ^ e ; )™ = ^ ^ L mA> \ TE (S) - -/? <y//n£-0 r (4.64) jfiy 0 j e -rn\R-S) ( r< \ n i _ 0,1 ■Te G)^0£0 -y^R-S) Qi^uSn 4.3 MGA Formulation for the Spatial Filter and Broadband Microwave Absorber Problems In this section we apply the MGA to optimize various parameters pertaining to the two design problems of interest. Binary tournament selection (Sec. 2.1.4.3) is employed along with uniform crossover (Sec. 2.1.5.3) with a crossover probability of 0.5. Elitism is utilized along with a population restart strategy that is applied when the bitwise difference between the best and other individuals/chromosomes in the current generation is less than 5%. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 4.3.1 Broadband Microwave Absorber Synthesis Figure 4.11 shows a multi-layered composite structure whose parameters we wish to optimize with a view to realizing a specified frequency response. PEC backing FSS screen Fig. 4.11. Multilayered composite with single embedded FSS screen and uniform plane wave incident at an arbitrary angle. We assume that we are given a set of different materials M m with frequency dependent permittivities s,(f) (/ = 1, M m). Our design goal is to determine a coating consisting of N different layers, an FSS cell design generated by the MGA, the FSS cell periodicity, its position within the dielectric composite and the FSS screen material, such that the coating exhibits a low reflection coefficient for a prescribed set of frequencies f (/ = / , Ff) and incident angles 9, (i = 1 A q/, for both the I E and TM polarizations. In the context of the present problem, the magnitude of the largest reflection coefficient is minimized for a set of angles, for both TE and TM polarizations, and for a selected band of frequencies. Hence the fitness function can be written: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 F (mx, dx, ....... ms , ds , 5 ^ , tx, ty, 5 ^ ) = - max {r^™ (0,, / ; ) J (4.65) where, m, and d, are the material parameter and thickness of the i,h layer, respectively; S'j" is the unit cell o f the periodic FSS screen generated by the MGA; tz and ty are the xand y-periodicities of the FSS screen; TE, / is the position of the FSS screen within the v dielectric composite; and, T nt [ 0 , , / j ) is the reflection coefficient, which is a function of polarization, incident angle and frequency. The MGA operates on a coding of the parameters. The coded representation of the coating consists of sequence of bits that contain information regarding each parameter. Given a database containing Mm = 2Klmt different materials, the material choice for layer / is represented by a sequence L, o fM ^ bits as follows: L, =/,'/;.................................................................... (4.66) The thickness of the i,h layer can be encoded by a sequence D, of bits and represented as A ..................................................................(4.67) The periodicity o f the FSS screen can be encoded depending on its allowed range of values and the step-size (see Table 4-2) utilized by the MGA. It can be represented by a sequence Tx of M t b bits for the x-directed periodicity, and similarly by Ty of M t J>bits for the y-directed one as given below: Tx =t1 / ; ............ r y w T = /‘r *y y y ............ t * v» Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.68) 101 For an N layer composite, the FSS screen can be placed in the composite at N-l positions. For example, for a four-layer composite, the FSS screen can be placed at the following three positions: (i) between the first and the second layers; (ii) between the second and third layers; and, (iii) between the third and fourth layers. For this case, we would need 2 bits to encode the three possibilities. Thus, the position of the FSS screen within the composite can be encoded by a sequence P of two bits as P = p lp 2 (4.69) The MGA designs the FSS cell structure automatically The pan of the code analyzing the FSS screen embedded in dielectric media utilizes a 16 x 16 discretization (32 x 32 and 64 x 64 discretizations can also be handled) of the periodic structure unit cell in the form o f 1’s (one) and 0’s (zeros); the l ’s corresponding to PEC or lossy metal and the 0’s to free space. The MGA randomly generates this 16 x 16-gridded structure filled with l ’s and 0’s as explained next. Figure 4.12 shows a 16 x 16 matrix filled with l ’s and 0’s. The MGA considers each row in the FSS cell as a parameter. For each row, the MGA generates a random number between 0 and 2Ucb-J, where MCb is the number of columns in the FSS cell matrix. These random numbers are then converted to a binary format for each row. These binary numbers are combined into an array, which is ready to be analyzed by the FSS code. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 R, 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 0 R2 0 1 1 1 1 1 0 0 1 1 I 1 1 1 0 1 R3 0 1 1 1 0 0 1 1 1 I 1 1 1 1 1 0 1100110011001000 1 1 0 0 1 1 00 1 I 00 1 0 0 0 01 1 I 0 0 1 1 1 I I 1 1 1 1 0 0 I 1 1 1 1001 1 1 1 1 10 1 0 0 0 0 10 1 1 1 I 1 10 1 10 0 0 0 0 10 1 1 1 I 1 10 1 10 01 1 1 1 1 0 0 1 1 1 1 I 1 0 1 0 1 1 1 00 I 1 I I 1 1 1 1 1 0 1 1 00 I 1 00 1 1 00 1 0 0 0 1100110011001000 01 1 1 0 0 1 1 1 I 1 1 I 1 1 0 0 I 1 1 1 10 0 1 1 1 1 1 10 1 Ri6 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 0 y A Fig. 4.12. A 16 x 16 pixel FSS unit cell design represented in the form of “ l ’s” and “O’s”. Two-fold symmetry is introduced into the FSS structure by making the rows 5 through 8 mirror images o f the rows 1 through 4, and rows 9 to 16 the images of rows 1 through 8. This effectively reduces the number of MGA parameters needed to design the FSS cell from 16 to 4, and makes it considerably more efficient than when no symmetry is imposed. Thus the FSS cell can be designed by encoding each row into a sequence {CD)j of M Cb bits as (iCD)j = (CD)\ ( C D ) ' ................. (C D )"- (4.70) where j is the number of rows considered as parameters by the MGA; j max is the total number of rows in the FSS cell structure. Additional bits are needed in the MGA optimization process if we select lossy or lossless dielectric layers, and a lossy or PEC FSS screen. As each selection offers two Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 choices, we need one bit for each, i.e., a total of two bits. These two parameters can be represented by the sequences (CL), o f 1 bit and (CS) of 1 bit as follows: -(CL)I (4.71) (CS)=(CS)' If a lossy FSS screen is selected, the real and imaginary parts of the complex sheet impedance appear as two new parameters, to be optimized by the MGA. The range of these parameters can be encoded depending on their permissible range of values and the step-size (see Table 4-2) introduced by the MGA. The real and imaginary parts of the loss can be represented by a sequence (FLR) of A//W)bits and (FLI) of M lhbbits, respectively, yielding (FLR)-(flr) (fir)' (Jlrf' (4.72) (F L I)-(flij(jliy (ftif» The entire composite can be represented by a sequence C, which is referred to as a chromosome: C = L,....Ls Dv . . . D „ ( C L \ . . . . ( C L ) J J yP(FLR){FLI) (CS)(CD\....(CD)j (4.73) Each quantity in (4.73) has been defined in (4.66) through (4.72) and each sequence in (4.73) consists of (2~Mmb- M db~Mt t b ~2~j*Mcb-Mfirb~Mfi,b) bits. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 4.3.1.1 Numerical Results for Single FSS Screen Embedded in Lossy or Lossless Dielectrics The MGA has been successfully applied to the synthesis of broadband microwave absorbers in the frequency range o f 19.0 - 36.0 GHz. The number of dielectric layers was fixed at four, was surrounded by air on one side, and terminated on the other side by a PEC backing. Though only dielectric layers with electric loss are considered in this section, we show later that the method can be extended to handle both electric and magnetic losses. The measured values of s ’ and e* of ten different lossy materials were considered and a database of these values as a function of frequency was created, as shown in Tables 3-2 and 3-3, respectively. A linear interpolation was used to generate the values of s ’ and s ' as functions of frequency for intermediate material types to obtain the values of the real and imaginary parts of the permittivity for even numbered material types at a particular frequency, while the material parameters of the odd-numbered ones were averaged. For example, values for the “Type 2” material at 19.0 GHz was obtained by averaging the material values for “Type 1” and “Type 3” at 19.0 GHz. Thus, effectively, a total of 19 different types of materials were used in this study. For lossless layers, small values of losses were added, since it is not feasible in practice to fabricate a perfectly lossless dielectric. Hence, if the MGA selects the first layer as lossless, its er value is deliberately fixed at 0.01, while for all the other layers the corresponding value o f ^ is 0.1. A total of seven cases, listed below were investigated Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 (i) Normal Incidence (0 = 0.0,<J>=0.0), TE polarization (ii) Oblique Incidence (0 = 45.0, <j>= 0.0), TE polarization (iii) 0 varying from 0 to 45 degrees, TE polarization (iv) Normal Incidence (0 = 0.0,<J>=0.0), TM polarization (v) Oblique Incidence (0 = 45.0, <J>= 0.0), TM polarization (vi) 0 varying from 0 to 45 degrees, TM polarization (vii) 0 varying from 0 to 45 degrees, TE and TM polarizations The population size and the number o f generations were fixed at 50 and 405, respectively. The periodicity of the FSS screen in the x- and y-directions were made equal. The loss in the FSS screen material was restricted to real values by forcing the imaginary part to be zero and the frequency resolution was chosen to be 1.0 GHz. To obtain the values of reflection coefficients at intermediate frequency points, a spline interpolation in frequency was carried out by using MATLAB™. The optimized parameters from the MGA output were used in a separate code, which handled FSS screens embedded in lossy dielectric media. To verify the intermediate values of the reflection coefficients, the FSS code was run with the same frequency resolution as the spline interpolation. If the reflection coefficient values at the intermediate frequencies generated by these two methods were found to disagree within a certain tolerance, the MGA was re-run at a finer frequency resolution, using the population of the last generation, to obtain the correct values. The results of the MGA optimization o f the reflection coefficient in dB are presented in Figs. 4.13 through 4.21. In the FSS cell structure white represents metal, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 while black corresponds to free space. For all the seven cases, a two-fold symmetry is introduced in the FSS unit cell design and the azimuthal angle is fixed at zero degrees (<J> = 0.0, x-z plane). The parameters selected by the MGA for each case is listed in Table 41. We see from Figs. 4.13, 4.14, 4.17 and 4.18 that the frequency response obtained by using the MGA resembles, not unexpectedly, the familiar filter response derived by using the Tchebyscheff equal ripple optimization procedures. For all the four cases, the worst-case reflection coefficient is maintained below -18.0 dB. In Figs. 4.15 and 4.19, the worst and best-case reflection coefficient can be interpreted as follows. For a single frequency, over the elevation angles of interest (0 = 0.0 to 45.0 degrees), the maximum and the minimum values of the reflection coefficients correspond to the worst and the best results expressed in dB. Imposing an additional burden of optimizing over a range of elevation angles, causes the worst-case reflection coefficient value to degrade further. Figures 4.16 (a) through (d) and 4.20 (a) through (d) present the behaviors of the reflection coefficient as functions of the elevation angle for the TE and TM polarization, respectively, with frequency as a parameter. Figure 4.21 shows the worst- and best-case reflection coefficients realized by the MGA, while optimizing simultaneously for both the TE and TM polarizations over the range of elevation angles o f interest (0=0.0 to 45.0 degrees). We see that the MGA maintained the worst-case reflection coefficient below 15 .0 dB for this case. Figures 4.22 and 4.23 illustrate certain details of the MGA optimization procedure for Case-7, which may be regarded as a representative example. Figure 4.22 depicts the variation o f the fitness value as given by (4.65), vs. the number of generations. Two Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 important observations can be made about this result: (i) the MGA restarts the population at the 50th generation as seen from the sudden dip in the curve for average fitness value; (ii) the best value of fitness (which is not necessarily the global maximum, but satisfies our design requirements) is achieved by the 75th generation. Figure 4.23 shows the population distribution at the 1st, 40th, and 75th generation, respectively. Table 4-2 lists the search space of the MGA parameters for this problem. If the MGA selects a PEC FSS screen with lossy dielectric layers, the number of parameters and bits in the chromosome correspond to 18 and 168, respectively. However, if the MGA selects a lossy FSS screen together with lossy dielectric layers, the number of parameters increases to 19, and the chromosome length becomes 194 bits. These two cases constitute the best- and worst-case scenarios in terms of the population sizes for the problem at hand. It is possible to obtain an estimate of the population size for a CGA by using (2 .9) given in Sec. 2.2.1. From Table 4-2 we observe that for the best-case we have / = 168, x = 2 and the average length o f the schema of interest was (1/18)(64+10+52+18+2+1+1+20) = 9. This is the average length of the gene that makes up one parameter Correspondingly, for the worst-case, the parameters were: / = 194, x = 2, while the average length of the schema of interest was (l/19)(64+10+52+18+2+l+l0+l+36) = 10. Applying equation (2.9), the population sizes for the best- and worst-cases is estimated to be 9560 and 19866, respectively. The relative advantage of the MGA over the CGA in terms of the required population size provides adequate justification of its use to address our design optimization problem. It is evident that we gain substantial savings in computational time and resources when the MGA is used. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 \' o * — :______ ...... V ........................................................... Original points Interpolated points -40 -42 -------- 1-----------1-----------1— 18 20 22 24 _ -• • ____ O i \ p \ I \t , rt o -36 uu e4) -38 DC i' \ V u c - ^ — ■- t 26 — |- 3 4 / / i 1 4) r • >• \ ’ !■■■ f c -32 -r r 00 -30 T3 ■ o — -28 ~~ ■■■ i - 28 30 32 34 36 Frequency in GHz er = 1.03, d = 0.45 s r - 1.65, d = 1.85 Type 1, d == 1.25 K Type 2, d = 0.6 Fig. 4.13. Design for Case-1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 £-22 2 *26 5 -28 o Orieinal Doints Interpolated points •r Freauencv in G H z er = 1.03, d = 1.65 er = 2.38, d = 1.75 Type 14, d = 0.25 Type I, d = 0.77 Fig. 4.14. Design for Case-2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 i -12 i -14 T CQ -o c -16 i ♦ ♦ ■r i i 30 32 t ♦ ’.........♦' c <u -18 o fc 8u -20 c o -22 o 4> c -24 it -26 ♦ -28 18 20 22 worst case best case 24 26 28 34 36 Frequency in GHz Er = 1.03, d = 3.15 6r = 1.65, d = 0.05 Er = 2.09, d = 1.75 Type 16, d = 0.6 Fig. 4.15. Design for Case-3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reflection coefficient in dB 111 -10 -15 -25 -30 Angle Of Incidence m degrees 19.0GHfe - * 20.0GHz — * —21.0GHz — * -2 2 .0 G H z — * -23.0G H : Reflection coefficient in dB (a) -10 -15 -20 Angle Of Incidence in degrees 24.0 Gffe— ■— 25.0 GHz- ~tr - 26.0 GHz — * -2 7 .0 GH-— * - 28.0 Ghfe (b) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Rellcction coeflicienl in dB 112 -10 -15 -20 -25 -30 ^ Angle Of Incidence in degrees ■29.0 Gto — ■— 30.0 GHz- -tr -31.0GI-te — * -3 2 .0 GHz—• * -3 3 .0 GHz Reflection coefficient in <JB (C) -20 -25 Angle Of Incidence m degrees ■34.0 Ghte — ■— 35.0 GHz- -*- -36.0 GHz (d) Fig. 4.16. Case-3: Reflection coefficient as a function of the elevation angle with frequency as a parameter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 -20 i ; CQ -21 T3 t : \\ 1; I: _C cu -22 '3 £ gu ' 23 : c o -24 U V cV P © f\ -V - t / ; A- ■ x < 2 ° ~b~ -25 20 t - W -26 18 •4 Original points Interpolated points 22 24 26 28 30 32 34 36 Frequency in GHz s r = 2.67, d —1.45 Type 1, d = 0.05 Type 7, d = 0.95 Type 16, d = 3.55 Fig. 4.17. Design for Case-4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O' -36 er 18 I I i 1 20 22 24 26 O 11 28 O riainal nom ts tnieroolaied noinis . ........................... 30 32 34 36 Frenuenov in G H 7 er = 2.67, d = 1.35 Type 1, d == 1.45 Type 11, d = 1.05 Type 17, d = 0.96 Fig. 4.18. Design for Case-5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 I I I ! ♦ ♦ CO T3 c-20 ♦ + ♦ c V E- 25 ♦ worst case best case c •2-30 u c:u ^-351 18 20 22 24 26 28 30 32 34 36 Freauencv in GHz Er = 2.67, d = 1.25 er = 2.5, d = 0.35 Type 6, d = 0.95 c Type 15, d = 0.75 Fig. 4.19. Design for Case-6. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 10 15 25 20 30 40 45 Reflection coefficient in dB -5 -10 -15 -40 - ........................................................................................................... Angle o f Incidence m degrees 19.0GHz —■*— 20.0 Ghte - -Ar -2 1 ,0GHz — * - 2 2 .0 GHz — * -23.0G H z (a) Reflection coefficient in dB -10 -15 -20 -25 -30 -35 -40 -45 Angle o f Incidence in degrees ♦ — 24.0 Ghfe— » — 25.0 GHz - -A r - 26.0 Ghfe— * -2 7 .0 Gffe— * - 28.0 GHz (b) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 117 .1 -15 a a » -20 e2 5a -25 -30 -35 -40 Angle of Incidence in degrees '29.0 GHz - —• • ' ‘‘•30.0 GHz ‘ 31.0 GHz — 32.0 GHz * " 33.0 GHz (C) e -io C I -15 1 -20 Ia * a ca £ -30 -35 -40 Angle o f Incidence in degrees *34.0 GHz' “ 35.0 GHz “ * * 36.0 GHz (d) Fig. 4.20. Case-6: Reflection coefficient as a function of the elevation angle with frequency as a parameter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 15 ■ ffl -a c ♦ + 16 ■ worst case best case 17 c u '3 18 £ 8o 19 c 20 o *■3 ou c:u 21 a: + 22 •23 18 20 22 24 26 28 +• + 30 32 34 36 Freauencv in GHz Fig. 4.21. Design for Case-7. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 -0.1 - 0.2 Fitness value -0.3 average fitness value best fitness value -0.4 -0.5 - 0.6 -0.7 - 0.8 -0.9 0 50 100 150 200 250 300 350 400 Number of Generations Fig. 4.22. Fitness value vs., the number of generations for Case-7. SsnsKHtpn Generation 40 G e n e ratio n 7.-> -12 -10 -8 -6 -4 Reflection coefficient in dB Fig. 4.23. Population distribution of the MGA in the 1st, 40th, and 75th generations for Case-7. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 TABLE 4-1 PARAMETERS SELECTED BY THE MGA FOR THE 7 CASES Dielectric Layer types FSS screen properties Layer 1 Layer 2 Layer 3 Layer 4 Case-1 Lossless Lossless Lossy Lossy Case-2 Lossless Lossless Lossy Lossy Case-3 Lossless Lossless Lossle ss Lossy Case-4 Lossless Lossy Lossy Lossy Case-5 Lossless Lossy Lossy Lossy Case-6 Lossless Lossless Lossy Lossy 3rd and 4th 1st and 2nd Case-7 Lossless Lossless Lossy Lossy 1st and -jnd Position between Cell size (mm) 3rd and 4th 3rd and 4th 2nd and 3rd 1st and 2nd 17.11 x 17.11 11.26 x 11.26 11.16 x 11.16 9.92 x 9.92 31.78 x 31.78 26.70 x 26.70 8.767 x 8.767 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Surface Resista nee Q square 137.08 164.45 394.16 988.5 646.36 980.67 447.93 121 TABLE 4-2 MGA PARAMETER SEARCH SPACE Range of parameters Step size Number of possibilities (nearest 2n) Number of binary digits 4 0 -6 5 5 3 5 1 65536 16 bits x 4 rows = 64 1 0.1 -6 .0 0.01 1024 10 di, d2, d3, d» 4 0 .0 5 -6 .0 0.001 8192 Txi, Tyi 2 1.0-50.0 0.1 512 1-4 1 4 13 bits x 4 layers = 52 bits 9 bits x 2 = 18 bits 2 1-3 1 2 1 10.0- 1000.0 1 1024 10 1 2 1 1.03-6.0 0.01 512 9 bits x 4 layers = 36 1 - 19 1 32 5 bits x 4 layers = 20 One row Total 4 rows Total thickness of composite (mm) FSS position 1 Choice of FSS screen 1 material Re(Zs) 1 Choice of layer 1 1 material Erl-, Sr2i Sr3i Sr4 4 for lossless dielectric Lossy dielectric 4 for all four layers Lowest total parameters Highest total parameters o r»-i 1 O Number of parameters Parameters 18 19 Lowest total bits = 168 Highest total bits = 194 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 We note from Figs. 4.13 through 4.21 that the MGA introduces a two-fold (or top-bottom) symmetry in the FSS screen designs. As explained earlier, the MGA optimizes, simultaneously, for a range o f frequencies, elevation angles, and for both the TE and TM polarizations. However, it is necessary to maintain the invariance of reflection coefficient not only with frequency and the elevation angle 9, but also with the azimuthal angle <j>. For composites that are formed by using only dielectric layers, the reflection coefficient is independent of the azimuthal angle <j>and is not considered during the optimization process. However, for composites with an FSS screen embedded within the dielectric layers, the reflection coefficient is a strong function of <f>and has to be accounted for during the process of optimization. The ^-variation can be introduced into the optimization process either explicitly or implicitly. In the former case, we incorporate an additional loop into the numerical process, which accounts for each azimuthal angle to be optimized. For the implicit case we introduce higher orders of symmetry into the FSS screen design to reduce the strong dependence of the reflection coefficient on <f>. The drawback of the explicit optimization scheme is that it leads to an increase in the computational time and resources. In contrast, as shown below, the implicit scheme carries out the optimization with respect to the <fb variation efficiently, and without additional burden on the computational expense. For this reason, the implicit scheme is preferable to the explicit one in our work. For the implicit scheme, the MGA incorporated eight-fold symmetry in the FSS screen design. Case-7 was first optimized for a composite with an FSS screen possessing two-fold symmetry. Thereafter, the same case was repeated with an FSS screen Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 incorporating eight-fold symmetry. The composite containing the FSS screen, shown in Fig. 4.21, is simulated for 0= 45, <j>= 0 and 45 degrees, and for both the TE and TM polarizations. The variation of the reflection coefficient vs. frequency is plotted in Fig. 4.24 for the four cases mentioned above. It is evident that the frequency response shows an considerable amount o f variation as we change the azimuthal angle <p. -1Q -15 CQ -o 'C eu o sa 8 -2 5 .... c \ 0 \ 1 H -3C........ aL -35 ++ TE - theta = 45, phi = 0 TM - theta = 45, phi = 0 TE - theta = 45, phi = 45 TM - theta = 45, phi = 45 -4(7 Frequency in GHz Fig. 4.24. Frequency response of the composite shown in Fig. 4.21 with a FSS screen having two-fold or top-bottom symmetry embedded in it. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 Reflection coefficient in dB -161 TE-phi=0 TM-phi=0 TE-phi=45 TM-phi=45 -24 Frequency in GHz e r - 1.02, d = 0.35 e r = 2.272, d = 1.85 Type 6, d = 1.05 Type 14, d = 2.28 Fig. 4.25. Frequency response of dielectric composite with a single FSS screen possessing eight-fold symmetry embedded in it. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 -19 ♦ 4 4► ♦ - 4 ♦ * .....▼.... 18- H c ♦ + -k + + : + ce u -2C ' '5 £ 8u 2 2 I u 'S -24 - t +• CQ ■o - > ♦ ♦ ► t + ’ 4- + ♦ + w o rs t best .................J L........... 4-......... + 26T - -2 ff 8 20 22 24 26 28 30 Frequency in GHz 32 34 36 Fig. 4.26. Worst- and best-case reflection coefficients over the frequency band for composite with FSS screen in Fig. 4.25. Figure 4.25 depicts the frequency response of a dielectric composite with an FSS screen possessing eight-fold symmetry embedded in it. The x- and y-periodicities of the FSS screen are equal to 1.671 mm, and its surface resistance is 31.5 Q/square Comparing Figs. 4.24 and 4.25, we observe that the frequency response for the latter varies relatively little with 0. Thus, we have achieved the invariance of reflection coefficient with 0 via the implicit scheme without placing additional burden on the computational resources or time. We point out that the dependence of the reflection coefficient on the azimuthal angle can be eliminated completely only if we incorporate radial symmetry in the FSS screen design and choose the x- and y-periodicities to be equal. Figure 4.26 shows the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 worst- and best-case reflection coefficients for the composite and FSS screen shown in Fig. 4.25. 4.3.1.2 Numerical Results for Single FSS Screen Embedded in Dielectric and Magnetic Media that can Either be Lossy or Lossless In the previous section, we designed a composite structure comprising either lossy or lossless dielectric layers, with a single FSS screen embedded within the layer. In this section we will optimize a composite consisting of dielectric or magnetic layers that could either be lossy or lossless. The magnetic materials, if selected by the MGA, are always assumed to be lossy. The lossy dielectric materials that we choose from are unchanged from the previous section, and Figs. 3.8 through 3.11 show the frequency dependence o f £r, s r’ ,jur, a n d //’ for the six lossy magnetic materials utilized. Linear interpolation is utilized for the magnetic materials, in a manner similar to that applied to the lossy dielectric materials in the previous section. All the optimization parameters remain the same as in the previous section, and Case-7 is repeated with the lossy magnetic materials included in the optimization, and with an eight-fold symmetry incorporated in the FSS screen design. Figure 4.27 shows the composite with an FSS screen whose x- and y-periodicities are both 8.767 mm. For a PEC FSS screen, the figure also shows the worst- and best-case reflection coefficients over the frequency band of interest. Comparing Figs. 4.26 and 4.27, we see that the worst-case reflection coefficient in both the cases are within 1 dB of each other, and that the composite thickness is 5.53 mm for the former design, and 2.95 for the latter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 I Reflection coefficient in dB -151 11" ' I ♦ 4- I + ♦ -2C ' + -25 + -3C ’ t + - + + w o rs t b est + ■+* -35 -40 8 20 22 24 26 28 30 Frequency in GHz 32 34 e r = 1.873, e = 0.01, d = 1.95 H r = 1.0, H r = 0.0, lossless dielectric material " r e r = 8.19, e r = 4.8, d = 0.85 H r = 1.0093, h’V = 0-195 (36.0 GHz), Type-7-lossy magnetic material 8r = 1.75, 8 r = 0.1, d = 0.05 H r = 1.0, H r = 0.0, lossless dielectric material er = 5.543, e r = 0.1, d = 0.107 H r = 1.0, H r = 0.0, lossless dielectric material Fig. 4.27. Design for composite with both dielectric and magnetic materials. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 128 As expected, the employment of lossy magnetic material leads to a thinner composite [40], albeit at the expense of increased weight due to the high concentration of iron in the lossy magnetic materials. Figures 4.28 and 4.29 illustrate the population distribution and the fitness values, respectively, as the generations evolve during the MGA optimization. We note that at the beginning of the optimization the population is randomly distributed with respect to the total thickness of the composite. At the 40th generation the individuals are all lined up at a constant value o f the total thickness. Eventually, at the 75th generation, the effect of the Elitist operator is observed, as only one individual having the same thickness as the others but with a superior value of reflection coefficient is filtered out of the population. 1 ----------------- 1 O c • D ° ?15 c/1 c/1 O O O O. -5 0 O Generation 40 ■ 5‘ o o o o 1 cn u e u Is H -10 O °o$% I i aooo me o -5 o E u c o Generation 1 0 1 O Generation 75 - 5o ~ oo ® o o --------------------------------- 1-------- -------------------------1--------------------------------- -15 -10 -5 Reflection coefficient in dB Fig. 4.28. Population distribution for the optimization carried out for the case with lossy magnetic layers in the composite. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 -0.1 § -0.4 J > (A A g -0.5 c u. - A verage b e st -0.9 3 100 Number of Generations 150 Fig. 4.29. Fitness value vs. the number of generations for case with magnetic losses in the composite. 4.3.1.3 Numerical Results for a Single FSS Screen Embedded in Lossy Carbon Fiber and Lossless Dielectric Material In this section, we optimize a composite consisting of either lossy carbon fiber or lossless dielectric material with a single FSS screen embedded within the layer. Figures 4.30 (a) and (b) plot the real and imaginary parts of the permittivity o f the five carbon fiber materials as a function o f frequency. The MGA optimization is performed in the Xband (8.2-12.4 GHz) and linear interpolation is utilized for the carbon fiber materials, in a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 manner similar to that applied to the lossy dielectric and magnetic materials in the previous sections. Case-7 is repeated with the carbon fiber materials included in the optimization, and with an eight-fold symmetry incorporated in the FSS screen design. Carbon Fiber Composite 18 9 10 11 12 13 Frequency (GHz) Carbon Fiber Composite -e— • — • — e 8 9 10 11 12 13 Frequency (GHz) Fig. 4.30. The permittivity o f carbon fiber material as a function of frequency: (a) real part; (b) imaginary part. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 Figure 4.31 depicts the composite with an FSS screen whose x- and yperiodicities are both 4.44 cm. For a PEC FSS screen, the figure also shows the best- and worst-case reflection coefficients over the frequency band of interest. We observe that the worst-case reflection coefficient is maintained below —14.0 dB and the total thickness of the composite is equal to 5.23 mm. sr’=4.34, Er” = 0 .1, d=2.25 mm m = 11.25g, d=2.85 mm m = 5g, d = 0.05 mm sr’=1.03, 8r” =0.1, d=0.08 mm (a) -12 (b) • r• » r -14 ♦ CD -16 ■o <u a £O) o u e 0 1 c« 0£ i ♦; i ■i t ^ -t ■1■" ■ ♦ ♦ -18 ♦ worst case best case -20 -22 + •r i -24 'r * -26 -28 -30 1+ — - *■ 8.5 » 9 t i 9.5 10 10.5 11 11.5 ____+ 12 12.5 Frequency in GHz (C) Fig. 4.31. Design o f carbon fiber composite with FSS screen embedded in it: (a) FSS unit cell design (white - metal, black - free space); (b) composite design; (c) frequency response of composite in (b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 4.3.2 Spatial Filter Synthesis To synthesize spatial filters, we follow the same procedure as described in Sec. (3.3.1), except that we now embed a single FSS screen within dielectric layers that are lossless, and remove the PEC backing. The FSS unit cell periodicities, its position within the composite, its design, and material content are encoded in binary by the MGA and are given by equations (4.68) through (4.71), respectively. The fitness function employed can be expressed as *4j £ a; (max(r(<ffi),rraln(tffi))) (4.74) F = - (dB))) where T and T are the reflection and the transmission coefficients, respectively, A and B are additional weight coefficients that are problem-specific and are introduced to equalize the two summation terms. Also, M and M are the number of frequencies for which the reflection and transmission coefficients are minimized to fixed, problem-specific values, corresponding to and Tmin, respectively. 4.3.2.1 Numerical Results for Spatial Filter Synthesis The objective of the spatial filter synthesis is to design a composite comprising lossless dielectric layers and a single FSS screen that satisfies the following specifications: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 (i) incident wave polarization: TM (ii) pass band: 960 MHz - 1.215 GHz for 0 = 70 degrees and <J>= 0 and 90 degrees simultaneously. (iii) stop band: 2.0 - 18.0 GHz for 0 = 70 degrees and 4> = 0 and 90 degrees simultaneously. (iv) pass-band transmission: Transmission better than -0.5 dB. (v) stop band attenuation: Transmission less than —15.0 dB. (vi) total composite thickness: 12.7 mm. (vii) minimum layer thickness: 0.05 mm. Four layers o f dielectric are employed in the filter design along with a single FSS screen. Small losses are introduced in the lossless dielectric layers by assigning s r =0.1 to simulate practical materials. The MGA is given the choice of selecting a lossy or lossless material for the FSS screen. 0 dB ■20 dB start fend Fig. 4.32. Idealized response of spatial filter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 The desired frequency response for the transmission coefficient is displayed in Fig. 4.32, where f c u t i , f c u t 2 , f sta rt, and f end are 1.2, 2.0, 0.8, and 18.0 GHz, respectively. The frequency increments are 0.2 GHz and 0.025 GHz for frequencies ranging between 0.8-2.0 GHz and 2.0-18.0 GHz, respectively. The values of M and N are 6 and 40, respectively, while, rmin=Tmin= -20.0 dB. Finally, the weighting coefficients are selected as follows: A = 5 0 0 ,5 = 1 , and 1.0 forj=l,2 a,= 1.1 forj=3, 4, 5,6 1.2 forj=7 to 47 The number of generations and the population size for this filter design are 300 and 5, respectively. Figure 4.33 shows the optimized composite, the FSS screen design, and the frequency response of the filter. Except for a single peak near the 8.0 GHz point, the transmission coefficient values over the frequency band meets the design criteria. The FSS screen has a periodicity of 13.18 mm in both the x- and y-directions. The loss in the FSS screen has identical resistive and inductive components equal to 1.73 Q/square. It is to be noted that the MGA was optimized at 0 = 70 and <j>= 90 degrees. The design criterion requires that the frequency response of the composite be identical at both <j>= 0 and 90 degrees. This is achieved by employing the implicit scheme mentioned in Sec. 4.3.1.1, and also by observing that at <J>= 0 and 90 degrees the FSS screen appears the same to the incident plane wave due to the imposed eight-fold symmetry. Finally, a sensitivity study of the designed composite is performed to check the feasibility of fabricating the MGA optimized design in practice. The frequency responses o f the four cases studied are listed below. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 (i) composite in Fig. 4.33 with the er = 0 instead of 0.1 - solid line in Fig. 4-34. (ii) original composite in Fig. 4.33 - dashed line in Fig. 4-34. (iii) material values changed to that practically available - dotted line in Fig. 4-34. (iv) material values changed to that practically available but the thickness of the first layer is decreased by the same ratio (square root) by which the er increases dotted-dashed line in Fig. 4-34. •o -1 C -15 -25 -30 Frequency in GHz Fig. 4.33. Spatial filter design with single FSS screen embedded in lossless dielectric medium. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 We observe that the frequency response does not change appreciably for any of the four designs even with an appreciable variation in the constitutive parameters and thicknessess o f the layers. This leads us to conclude that the MGA optimized design for the spatial filter is robust, dependable, and can be fabricated in practice. CQ •a c .1 o - 10 - s 8 c _o ’35 ai E c -20t2 E-25 -3Q Frequency in GHz Fig. 4.34. Sensitivity study of spatial filter design with single FSS screen embedded in it. 4.3.2.2 The Domain Decomposition Approach Applied to the Design of Spatial Filters The domain decomposition approach is utilized to design both spatial filters and broadband microwave absorbers when the required geometry resolution of the FSS unit Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 cell is very fine, and a 16x16 pixel resolution, shown in Fig. 4.12, is unable to resolve the geometrical details that are required to achieve the design specifications. This typically occurs when the wavelength of operation lies in the millimeter regime and a larger pixel resolution, viz., 32x32 or 64x64, is required to describe the FSS unit cell and accurately model the surface current distribution. For the same geometry, the numerical procedure for 16x16, 32x32, and 64x64 pixel resolutions requires about 0.65, 60, and 3780 seconds, respectively, to compute the response for a single frequency. So, with the cpu time increasing rapidly when a fine pixel resolution is used, it is not feasible in practice to utilize the MGA in conjunction with the MoM code to design an FSS screen because of the heavy burden imposed on the computational resources. To circumvent this difficulty, a two-step approach is utilized to meet the design requirements. The first step involves the development of a free-standing FSS screen, which is later augmented by a multi-layer dielectric composite backing designed to embellish the performance o f the FSS. The complexity of the task prompts us to employ the MGA to develop the design o f the five-layer dielectric composite. In the second step, the freestanding FSS screen is designed by using well-established concepts for single and multiband FSS screen designs [11], [41-42]. Finally, the two separate designs are combined to synthesize a composite structure that satisfies the design criteria. The domain decomposition approach is applied to design a dual band radome operating at high off-normal incidences using a composite with a single FSS screen embedded in lossless dielectric. The radome simultaneously transmits frequencies between 2.0 to 18.0 GHz and also at 95.0 GHz. The design to be presented has three unique features: (i) broad passband ranging from S to Ku band (2.0 - 18.0 GHz) at the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 low frequency end of the spectrum of interest and a narrow millimeter wave band at 95.0 GHz; (ii) wide separation (77.0 GHz) between passbands; and (iii) good performance at wide elevation angles, e.g., 70 degrees. To design a single dual-band FSS screen that simultaneously transmits 2.0 to 18.0 and 95 .0 GHz, we require a variant of the Double-Square-Loop (DSL) patch element (see inset in Fig. 4.36). This element exhibits resonances at two frequencies, determined primarily by the dimensions of the larger (lower frequency) and the smaller (higher frequency) loops. As shown in the inset of Fig. 4.36, the FSS cell consists of 9 square loops. Two of these are small square loops, which resonate around 95.0 GHz, and are inserted into a large square loop. The latter helps to pass through signals over a wide frequency band ranging from 2.0 to 18.0 GHz. Three sets of these loops are stacked with the ends of the three outer loops joined at the intersections. Accurate modeling of the induced surface currents at 95.0 GHz requires the use of a high-resolution discretization of the FSS element, comprising 64x64 pixels. This, in turn, increases the computational burden substantially, but yields improved results over coarser discretizations, e.g., 16x16 or 32x32. However, the MGA is not directly applied to optimize the composite radome. comprising a combination o f an FSS screen and a multilayer dielectric substrate, because of the excessive computational expense as mentioned previously. The MGA is used to optimize the thickness and relative permittivity of each layer simultaneously, while maintaining the total thickness of the composite below a certain limit. The upper limit for the total composite thickness is 1.75 cms. In virtually all multi layer radome designs, thick low-density layers i.e., layers with Sr <1.3 (foam or core material) alternate with thin high density laminates (layers with Sr > 2.2) to provide for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 good bond lines. Bond lines are required between foam or core materials (er <1.3) to provide a structural wall as materials with low density have relatively small bonding surfaces. The range o f parameters used in the MGA optimization is governed by the limits mentioned above. Due to mechanical considerations, the er of the first and second layer is held constant at 3 .2 and 1.24, respectively. The thickness of the first layer is also fixed at 0.1016 cms. Honeycomb or foam material (sr = 1.03) is used for the fourth layer. The thickness and er o f the remaining layers are now optimized by the MGA to satisfy the design criteria. After designing the dielectric composite and the freestanding FSS, the last and crucial step in the optimization procedure is to optimally position the FSS screen in the dielectric composite. The placement of the FSS screen is important because it determines whether combining the two degrades the individual responses of both the dielectric composite and the FSS screen and leads to a poor response. Since mechanical considerations rule out the possibility of introducing the FSS screen between either the 1st and 2nd, or the 2nd and 3rd layers, there are only two allowed positions, viz., either between the 3rd and 4th, or 4th and 5th layers. Both of these positions are individually analyzed by using an FSS code capable o f handling the presence of both dielectric superstates and substrates [15], Positioning the FSS screen between the 4th and the 5th layers was found to give the best results. Figure 4.35 shows the dielectric composite with the modified DSL/FSS screen introduced between the 4th and 5th layers. The values of 6r and thickness of each dielectric layer is determined by the MGA. The 64 x 64 pixel DSL/FSS screen is shown in the inset Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 o f Fig. 4.36. The cell size o f the FSS screen is 0.18 x 0.43 square cms. Figure 4.36 plots the frequency response of the FSS screen embedded in the dielectric composite for TE and TM polarizations. Note that, in line with the specifications, the transmission coefficient is better than -3.0 dB over the bands of interest. It is observed that the insertion o f the FSS screen into the dielectric composite has a greater influence at the higher passband (95.0 GHz) than it does at the lower one (2-18 GHz). s r=3.2, d=0.1016 cm er=l.24, d=l.O cm sr=3.0, d=0.00508 cm s r=1.03, d=0.489 cm FSS superstrate er=2.2, d=0.00508 cm FSS substrate Fig. 4.35. 64 x 64 pixel modified DSL FSS screen embedded in dielectric composite. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 0 Specular transmission coefficient in dB -5 -10 -15 -20 -25 -30 -35 -40 -45 0 10 20 30 40 50 60 Frequency in GHz 70 80 90 Fig. 4.36. Frequency response of composite in Fig. 4.35. TE TM Inset: Single cell structure of 64 x 64 pixel modified DSL FSS screen. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 Chapter 5 SYNTHESIS OF COMPOSITES COMPRISING MULTIPLE FREQUENCY SELECTIVE SURFACES (FSSs) EMBEDDED IN DIELECTRIC AND/OR MAGNETIC MEDIA In the previous chapter we have dealt with the design of composite type of absorbers that incorporated a single FSS screen embedded in various types of layered dielectric media. The technique applied in that chapter is very efficient for analyzing single screen FSSs, but suffers from some drawbacks when dealing with multiple screens, owing to a rapid increase in the number of unknowns as we increase the number of screens. For a single FSS screen with N unknowns, the size of the matrix to be inverted is JV* For M screens the number o f unknowns increase to ( N i-N ? -......... ~N\0 and the size of the matrix becomes ( N i-N ? - rendering the direct method impractical for all but a moderately small M. One solution to this problem is to use the Conjugate Gradient (CG) or other iterative techniques to solve for the unknowns, thus circumventing the need to invert a matrix. However, the disadvantage of using iterative methods is that they frequently require an exceedingly large computer run time to yield accurate solutions. One approach to obviating these difficulties is to employ the Generalized Scattering Matrix (GSM) [17-18] technique. In this method, also referred to as the Scattering Matrix Technique (SMT), the solution to an FSS screen analysis is expressed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 in terms o f the generalized scattering parameters. In this approach, we derive, as a first step, the generalized scattering matrices of the individual screens by using the MoM, and of the dielectric/magnetic layers by following the procedure described in the previous chapter for a single FSS screen. These matrices are subsequently used to generate a composite scattering matrix for the entire system by using cascading techniques, as explained in detail later. Initially, broadband microwave absorbers with two FSS screens embedded in lossy and/or lossless dielectric and/or magnetic media are synthesized for different angles of incidence and polarizations. Finally, a spatial filter with two FSS screens embedded in lossless dielectric layers is designed. S.l Problem Geometry A ’ Fig. 5.1 shows a structure incorporating S sub-composites, £ N, layers, where N, i= I denotes the number of layers in the ith sub-composite, and S corresponds to the number of FSS screens embedded in it. The PEC backing is introduced to evaluate the effectiveness of the absorber but is removed when we design a spatial filter, where transmission through the filter is o f interest. The MGA is applied to optimize various parameters, for instance the thickness and constitutive parameters of each layer, the FSS screen designs and materials, their x- and y-periodicities and their placement within the composite. The result is a multilayer composite that provides maximum absorption o f both TE and TM waves simultaneously for a prescribed range of frequencies and incident angles. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 144 MGA automatically places an upper bound on the total thickness of the composite. For lossy dielectric and magnetic material values the MGA accesses the lookup tables in Chapter 3. 1st FSS screen 1st sub-composite with Ni dielectric layers mN,I’ d\,l mt:, 4t; < ni22, 422 2nd FSS screen 2nd sub-composite with Ni dielectric layers mN,2, d N%, J S'11FSS screen S'11sub-composite with Ns dielectric layers N ,- I ) S ’ U (N ,-1 |S ZL PEC backing Fig. 5.1. Problem geometry. X Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 5.2 Analysis of Multiple FSS Screens Embedded in Dielectric and Magnetic Medium From the fundamentals of circuit theory we know that a two-port system can be represented schematically as shown in Fig. 5.2. reference plane 1 reference plane 2 Z = Zi Z = Z2 ai a2 Two-port system bi b2 Fig. 5.2. Schematic representation of a two-port system. The complex amplitudes of the incoming and outgoing wave can be expressed in terms of the currents and voltages at the two reference planes as [43] a = x -4- / Z" N 7 2f t _ ^ ( r (1.2)’/ ) 7 ^(*(1.2)’*) 2& 2 ^ ( 5 j) where V and I are the voltages and currents evaluated at the reference planes rn ;i, respectively, / is the time, and Zo is the characteristic impedance of the network. The S- parameters for the two-port network shown in Fig. 5.2 can be expressed in terms of the a ’s and the b's as follows: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 A — *1*^11 "^"*2*^12 (52) b-. — Alternatively (5.2) can be expressed in matrix form as b = S a, where a= a\I fA l ,b = i b2 az ,s = 'Sn Ai Spl I_ s ::_ Using the linear systems approach we obtain S„ = V & ,= (5.3) 5,2 = 'A , ' 'A ' s,,= 1*2 / J, =0 1*2 / ^=0 When extended to the w-port case, the above analysis results in an n x n scattering matrix S with vectors a and b containing n elements. As explained in Chapter 4, the spectral domain approach can be applied to determine the electromagnetic fields at any plane z = z„ for a plane wave incident on a single-screen FSS structure embedded in an inhomogeneous medium. The fields are expressed in terms o f Floquet’s harmonics, where the nlh harmonic corresponds to the nth port in an n-port system. A set of generalized scattering parameters incorporating the vector nature o f the electromagnetic fields for an w-port system can be defined as follows [17-18]: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 TE S™TE( m ,n J ,j ) = A x . ’ A>„ ’ A j ’ A y , 1 *L 7 <-.TE A x . . A y . - A t , >A y , 7 * £ Sun, (m ,n \i,j) = J P n s ( iJ ) r r ( * .- S™e ( m , n j , j ) = inf (w ,/i;/,y) = SgTE(m ,rr,ij) = ym T \t V TE TE S\2TU Ax.,Ay„,AV A y,,^ Ax.>Ay.,A,,Ay,,^ A x . ’ A y . ' A x, ’ A y , »-R A x . ’ A y , ' A t, ' A y , ' “ R ~ iO i) yi-.-) S™E{m ,n\i,j) = ym A x . ' A v . >A t , ' A y , i - R w ) y i - .- l yTX! ini A x .. A y . , A i . A y , ^ « = S™TB( m , n J j ) = TE y<-.~ Slmt { m j r , i j ) = y TE vm A x . ’ A y . ’ A t! 1 A y , >- £ Ax.’Ay.’Ax^Ay,^ Ax..Av,,Av Ay,,-, jT £ {i.J) y \~ .~ S S k i{ m w ,j) = ynr A x . > A y . » A s > A v , ' Z £. V o u ) Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. (5.4) 148 -ri V ’ ’ ’-// ” r ;- - y ..A .4 .A ,- - - « ) / ,//k R u ) (5.5) JW o J ) ^22TM J)“ V o u j In Fig. 5 .1 ,r = r/? and 2 = 2*. correspond to the upper and lower interfaces of the S'* sub composite, respectively, and ’K . A .„ - ^ ; vn i " ’ ( A . *A . - A - A, ' - ) = 7 .A , K '* -A - ) ‘i-1 1 , : ) = a (m ,n , /, j , 2 )l " \ l PTE ( m < « ) (56) 1 ,]Pn i (m,n) are the normalized Floquet voltage waves. Referring to the coordinate system in Fig. 5 1 we observe that the (±,±) notation in (5.6) denotes the waves traveling in the positive and negative-2-directions, respectively. The first element in the parentheses corresponds to the scattered waves and the second to the incident. Thus, can be interpreted as the (m,n)‘hharmonic reflected along the +2 direction from the leftmost reference plane when the (/,y)rt plane wave is incident from the -2 direction on this plane. The other terms in (5.4) and (5 .5) can be interpreted along similar lines. From (5.6) we see that the voltage waves are expressed in terms o f the transforms of the electric and magnetic vector potentials, f ( m , n , i , j , z ) and a ( m ,n ,i,j,z ) , respectively. To compute the scattering parameters these potentials need to be evaluated at the reference planes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 located at the extreme right and left of the FSS. Thus, the vector potentials f { m ,n , i ,j , z ) and a (m ,n ,i,j,z ) can be expressed as / (m, n, /, j , z)[z'z} = F (im, n, /, j ) 1z'~1e/m": a ( m ,n ,ij,z ) ~ ~ = A (m ,n ,i,j)" ~ «?'-• (57) Xmn( : ) = - j K +P i - P i for o r , v _________ = +y l K +P k ~ P l for z < z R (5.8) where and the sign o f the radical is chosen to satisfy the radiation condition. Two approaches can be utilized to evaluate the scattering parameters of an FSS screen embedded in the layered media. In the first approach, one computes the scattering matrix for each layer and the FSS screen separately, and then cascades them by using the scattering matrix technique to generate the scattering parameters for the entire composite. In the second approach, one computes the scattering matrix for an FSS screen embedded within a number of superstates and substrates, rather than just for the screen by itself. The reference planes for the latter approach are taken to be at the extreme left and right of the entire sub-composite. The secondapproach is oftenconsidered to be preferable because it requires a smaller number o f harmonics to be included in the scattering matrix, especially when individual layers are thin and the first approach becomes vulnerable to errors unless a large number of harmonics are included in the matrix. In (5.4) and (5.5) the normalization factors are expressed as follows: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 / £ > . » ) = (A - A ) 1'™ « ( < » .» ) = (A* A) „ (59) 7M mn where _ J®£ y mn Ymn JVM and the other terms are defined in (4.37) and (4.1). The scattering matrix only includes the propagating harmonics and they constitute a finite subset of the infinite spectrum of Floquet’s harmonics. Applying (4.10), (4.11) and (4.12) to (5 .7), and expressing it in terms of the total electric fields in the spectral domain, we obtain [ 15] MAA. - AAJ (5.10) E~\ can be found as follows: Aa.(A.A)-C,. £ ,jA .A ) -£ n where E ' x -y>( r t f .tram ) - a Ma - a ) - a Wa - a ) (5.11) are the reflected and transmitted electric fields in the layered media with the FSS screens removed, evaluated at z = J zr and z = zi (see Fig. 5.1). As the reflected and transmitted fields in a layered media follow Snell’s law o f reflection and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 transmission, these fields are added to the total fields only when the incident harmonic is equal to the scattered harmonic, as indicated by the Kronecker’s delta functions in (5.11). Efxy) is the scattered harmonic evaluated at z = zr and z = zl, following the approach outlined in Chapter 4, for analyzing a single FSS screen embedded in layered media. Substituting (5.10), (5.9), and (5.6) into (5.4) and (5.5) leads us to the final expressions for the scattering parameters of a single sub-composite. The expressions for Sn are given below. SurE( m , n j , j ) - r— ------ ^ , -r vK- p i + P i M ' K ) VK •/»;)(*•*) (5.12) S ^ ( m ,n ,i ,j ) ^ r / mn or.w / • 1 1 v\ ymnT X lvYijT E m „P I17™' ij /• mn \y™ I mn ( P E~ +3 t >iam}) p l + X M + K ) The other scattering parameters can be derived in a similar manner. It is worthwhile at this point to comment on the details o f the structure and the contents of the scattering matrix. It is evident from the analysis in Chapter 4 that a periodic FSS screen scatters an incident plane wave into a discrete spectrum of Floquet’s harmonics. In a multi-screen system, the fields incident on one screen are the harmonics scattered from the other. Thus, the scattering matrix must contain the scattered field representations due to an entire spectrum o f incident field harmonics. The scattering matrix elements correspond to the expressions given in (5.4), (5.5), and (5.12). First, the composite scattering matrix is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 divided into four quadrants, one each for Su, S 12 , S21 , and S 2 2 . Next, each quadrant is further sub-divided into four quadrants, corresponding to the two co-polarized ( Sf* and S™ ) and cross-polarized ( S™, and S™ ) components. Each of these sub quadrants consists of a two-dimensional array of elements corresponding to the scattered fields due to each incident harmonic. The complete description is pictorially presented in Fig. 5.3. c r£ , « J U 7£-„ ■ II 5= te„ c _ i>\, .1 L -1 S .j 22 ' “ r» scat * 1 inc«l r» sea l»1 ^ m c ff2 r»jaaf®2 inc»l a scat *2 me *2 • sS r u «» nTe E «„ l_ ,lr • Sc nt~ u rr.v/«. -v,« .. • • s sca'l' ne*S 1 • sc a t» .V inc*\ Fig. 5.3. The scattering matrix structure and contents for composites comprising multiple FSS screens embedded in layered media Theoretically speaking, we need to include an infinite number of harmonics in the scattering matrix to derive rigorous results. However, this is not feasible in practice, and we must truncate the matrix by retaining only a finite number of harmonics, say N. It is not altogether too obvious how this number N is to be determined, so that we have a good balance between accuracy and computational efficiency. In general, the scattering parameters o f FSS structures decay as exponentially damped sinusoids for increasing and Az . Consequently, the scattering parameters can be expressed in terms of the envelope o f the decay (exponentials) as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 W - e " ' - 4* From (5.13) we note that, for a fixed A r, we can find a value of ym„ = p (5.13) such that the scattering parameters Smn are essentially negligible as are those for which ymn > y Furthermore, we observe that y decreases with an increase in Az, implying that fewer harmonics need be included in the scattered field representation as we move further away from the screen (obviously, the converse is true when Azis decreased). As mentioned earlier, in the first approach used, the reference plane is often chosen to coincide with the plane o f the FSS screen (Az=0) when creating the scattering matrix and to achieve desirable accuracy levels it may be necessary to include an inordinately large number of harmonics in the scattering matrix. In contrast, the second approach does not suffer from this problem, and is therefore the preferred scheme for creating the scattering matrix. The next question of interest is: what is the cutoff-level for |5mn| below which we can approximate it as being numerically zero? A rule of thumb has been established on the basis of several numerical experiments in which the composite solutions, generated by using different number o f harmonics were compared with a benchmark version ([43], [34]). This criterion can be stated as follows: if the magnitude of the higher-order harmonic is -22 dB or higher, then it and all the lower-order harmonics - even if these harmonics have a magnitude less than -22 dB - should be included in the scattering matrix. Once the higher-order harmonics have permanently decayed to a value of -22 dB or less, and none is greater than -22 dB, the matrix can be truncated. After the formation o f the scattering matrix we need to combine the matrices to form the composite scattering matrix. For this we need to follow the linear systems Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 approach to cascading the sub-systems. Shown in Fig. 5.4 is a two-sub-system composite. We will apply the cascading technique to derive the expressions for the scattering parameters of the composite system shown in Fig. 5.5 in terms of the scattering parameters of the individual sub-systems, where the ar’s and the b's correspond to those in Fig. 5.2 with the superscripts denoting either the sub-system-1 or -2. a\ a; 4 <----Sub-system-1 Sub-system-2 <------b: — b; ----- <------b\ b; Fig. 5.4. System division into sub-systems. a; ------- ► < -------b; ___ a. Composite System ----- ^ b\ Fig. 5.5. Composite system. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 155 The matrix form corresponding to sub-system-1, -2, and the composite system can s 2 1 "«r _S;, S ;2 j _a\ ' b\~ b: IV J n Js ic: J a ! X . s :c: j _a; , i j t 1 ~s2 So S l j l a\ I 1 A r„, X b\ where the expressions for ___ 1 r.,-i i be expressed, respectively, as: (5.14) in terms of S‘ and Sr need to be derived. Also, from Fig. 5.4 we have the auxiliary relations b\=a{,b; =a\ (5.15) The first two matrix equations for the sub-systems in (5.14) can be re-expressed to read A,1 = 5 '1a 1l + 5 ,‘:a i (5.16) b\ = (5.17) b2 = S 2xa; + S 22a; (5.18) b; = S;xa 2 + S;2a; (5.19) Substituting (5.15) and (5.18) in (5.17) we obtain of = TS^al + TS\2S 22a: (5.20) where, T = [l - 5 ^ 5 , ] '. Using (5.20) in (5.19) leads to b; = [5 ;17’5’i1]a 11+ [S;JS\2S;2 + S;2]a: & Following the same procedure an expression for A,1 can be derived as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.21) 156 (5.22) ■v where, R = [l - ] The matrix form for the composite system (last equation in (5.14)) can also be expressed as (5.23) (5.24) Comparing (5.23) with (5.22), and (5.24) with (5.21), the scattering parameters for the composite system in terms of the scattering parameters of sub-systems-1 and -2 can be expressed as (5.25) If more than two sub-systems make up the multi-layered structure, then the composite system is formed by repeatedly cascading the additional systems to the composite one until all the layers have been added. For both the spatial filter and broadband microwave absorber designs, equal periodicities for the FSS screens have been assumed. This restriction imposed on the optimization process is due to the GSM technique and is not inherent to the MGA. Unequal periodicities can be handled by the GSM technique only when we can identify a common period in the entire system. More often than not, the system period is either equal to the larger of the two sub-system periods or an integral multiple of the same. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 157 FSS unit-cell design is also assumed to be identical for the two FSS screens in order to reduce the number of parameters to be optimized by the MGA. Neither of these restrictions compromised the optimization process, and the MGA successfully generated composite designs that satisfied the specified design criteria quite well. 5.3 MGA Formulation for Spatial Filters and Broadband Microwave Absorber Designs We will now apply the MGA to optimize various parameters pertaining to the two design problems of interest. Binary tournament selection (Sec. 2.1.4.3) will be employed along with uniform crossover (Sec. 2.1.5.3) with a crossover probability of 0.5. Elitism will be utilized along with a population restart strategy that will be applied when the bitwise difference between the best and other individuals/chromosomes in the current generation is less than 5%. 5.3.1 Broadband Microwave Absorber Synthesis Figure 5.1 shows a multi-layered multi-screen composite structure whose parameters we wish to optimize, with a view to realizing a specified frequency response. The composite is divided into S sub-composites, each comprising of N, (number of layers in the s'* sub-composite) dielectric layers. The parameters for each sub-composite are generated separately by the MGA. Assume that we are given a set of different materials M m with frequency dependent permittivities e,(f) (/ = / M m). For any single sub Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 158 composite, the MGA determines the following: (i) material parameters of N layers; (ii) design o f the FSS cell element; (iii) cell periodicity of the FSS; (iv) position of the FSS screen within the dielectric sub-composite; and, (v) the FSS screen material. For S, such sub-composites, the same process is repeated such that the combined sub-composites exhibit a low reflection coefficient for a prescribed set of frequencies f ( i = / , ........... F,) and incident angles 9, (i = / , A q), simultaneously for both the TE and TM polarizations. In the context of the present problem, the magnitude of the largest reflection coefficient is minimized for a set of angles, for both TE and TM polarizations, and for a selected band o f frequencies. Hence the fitness function can be written as ^\\’ F \ S I >• • J » .Vl 1 ’ ’ • ■• S’ t \ ! ’ •• t ’ ■ -‘ x j ’ t \ S ’ t J y , > g , •• S’p o j , ° 1 Sp a j j = - max {T ni («../,)} <526> ’ • • •'J where mtJ is the material parameter of the i,h layer in the f h sub-composite; d,} is the thickness o f the ith layer in the f h sub-composite; SdeSi is the unit cell design of the FSS screen embedded in the ilh sub-composite; t and tv are the periodicities of the FSS screens in the x- and y- directions, respectively, for the ith sub-composite; S determines TE the placement of the FSS screen in the ilh sub-composite; and, T ni (# ,,/,) is the reflection coefficient as a function of polarization, incident angle and frequency. The MGA operates on a binary coded version of the parameters. The coded representation of the coating consists of a sequence of bits, containing information regarding each parameter. Given a database of M m = 2Umb different materials, the material choice for layer / in the f h sub-composite is represented by a sequence LtJ of A /^ bits as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 A = /;/,; -C (5.27) The thickness of the ith layer in the f h sub-composite can be encoded and represented by a sequence Dt] of bits as shown below: (5.28) The periodicity of the ilh FSS screen can be encoded depending on its permissible range of values and the step size (see Table 5-1) utilized by the MGA. It can be represented by a sequence T of M t b bits for the x-directed periodicity, and similarly by T of M t b bits for the y-directed one, as follows: A =1^ .............................................................. (5 29) (5 30) For an N layer sub-composite, the FSS screen can be placed within the composite at N -l positions. For example, for a four-layer sub-composite, the FSS screen can be inserted at the following three positions; (i) between the first and the second layers; (ii) between the second and third layers; and, (iii) between the third and fourth layers. Thus, the position o f the FSS screen within the ith sub-composite can be encoded by a sequence P, oiMpb bits and represented as P,=P!P,2.............P :1* (5.31) The MGA designs the FSS cell structure automatically by choosing the geometry pattern of the unit cell. The code analyzing the FSS screen embedded in dielectric media Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 160 utilizes a 16 x 16 discretization (32 x 32 and 64 x 64 discretizations can also be handled) o f the periodic structure unit cell in the form of l ’s (ones) and 0’s (zeros), where the 1’s correspond to PEC or lossy metal, and the 0’s to free space. As explained next, the MGA randomly generates this 16 x 16 gridded stricture filled with l ’s and 0’s. Figure 5.6 shows such a cell geometry. 0 0 0 1 0 1 1 1 1 1 1 0 110 0 11 1111110 1 111111I0 0 1 1 1 ON) 1 1 0 0 1 1 0 0 1 0 0 0 1 1 0 ONI 10 0 1 1 0 0 1 0 0 0 0 111 0\) 1 1 I 1 1 1 1 I 1 0 0 1 0 1 1 1 1 IX) 0 I I n 0 0 0 0 I 0 l \ |l 1 1 I0 b’o'ooToTiM 1110 1 0 o 1 1 1 1 0 oil 11 I 1 1 0 1 o 1 1 B ill 11 1 1 1 1 0 oil 1 0 0 1 0 0 0 11 o d 1 1 0 ( 1 1 0 oil 1 0 0 1 0 0 0 0 1 1 1 0 0 1 ill 1I 1 1 1 I 0 0 11111 ooll 11 I I 10 1 Rl6 0 0 0 0 1 0 1 1 jl 1 _ 1 ] 0 _ 1 1 Oj ya Fig. 5.6 FSS unit-cell with either two-fold, four-fold, or eight-fold symmetry The MGA considers each row in the FSS cell to be a parameter. For each row, the MGA generates a random number between 0 and 2Ucb-l, where Mcb is the number of columns in the FSS cell matrix. The random numbers are converted to binary format for each row. These binary numbers are combined into an array, which is ready to be analyzed by the FSS code. The number of columns considered depends on the type of symmetry introduced into the FSS screen geometry. The regions A, B and C in Fig. 2 represent the sections considered for eight-fold, four-fold and two-fold symmetry, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 respectively. This reduces the effective number of MGA parameters needed to design the FSS cell, (see Table 5-1) resulting in efficient optimization. Thus, the FSS cell in the i,h sub-composite can be designed by encoding each row into a sequence (CD),, of MCb bits as (C D \= (C D )'ti(C D )l................. (C D )"- (5.32) where j is the number of rows considered as parameters by the MGA, and j max is the total number o f rows in the FSS cell structure. Additional bits are needed to assist the MGA in selecting lossy or lossless dielectric layers and a lossy or PEC FSS screen. As there are two choices in each selection, we need one bit for each, totaling two bits. These two parameters can be represented by sequences (CL),, of 1 bit and (CS), of 1 bit as follows: (CL)„ = (CLt, (5 33) (C S)r (5.34) and ( C S \' If a lossy FSS screen is selected, the real and imaginary parts of thecomplex sheet impedanceappear as two new parameters, whose values are to beassigned by the MGA. These parameters can be encoded in accordance with their allowed range of values and the step-size (see Table 5-1) utilized by the MGA. The real and imaginary parts of the loss can be represented by a sequence (FLR)j of M Jlrbbits and (FU), of M m bits, respectively, as shown below: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 162 ( F L R l^ f lr l'iJ l r ) ; (5.35) (F U \ (.fli) / " “ (5.36) The entire composite, can be represented by the sequence C, which is referred to as a chromosome: C = CmCDCTxCTyCPCCDCCLCcsCFWCFU (5.37) A ,=A>- ....Av, 1A 2.... -Av,: ..... A ,.... •AvjS (5.38) C o = A i .....A VA : ..... Av .... A s- (5.39) where />•..< A V= A ..... r (5.41) (5.42) 11 (5.40) •P Q .= A ,..... A J _!"0 ............ C co=(C D )n . \ C D \ ( C D l r . ( C O ) ... .(CD) ii i CL),ACL)r. • (C C ) ,,:' £ II .... (CS)„ Cra?=(FZJ?), ( FLR)S =(*£/),(FU\ (5.45) (5.46) (5.47) Each term in (5.37) has been defined in (5.38) through (5.47) and each sequence in (5.37) consists of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 l A ,, J , (W ..+ A /,» + 2) (5.48) + [(S )* (M Iit +M lyl + +A-/,.,)] bits, where all terms are as defined in (5.27) through (5.36). 5.3.1.I Numerical Results for Two FSS Screens Embedded in Lossy or Lossless Dielectrics The MGA has been successfully applied to the synthesis o f broadband microwave absorbers in the frequency range of 19.0 - 36.0 GHz [44], The composite comprising of two (the algorithm can handle any number o f layers) sub-composites is surrounded by air on top and terminated by a PEC backing at the bottom. The number of dielectric layers in each sub-composite is fixed at four for this exercise, though this number is flexible. The FSS cell design and periodicities are maintained constant for both the sub-composites. Though only dielectric layers with electric loss are considered, the method can be further extended to handle both electric and magnetic losses as will be shown in the next section. Table 5-1 shows the parameter search space for the MGA for a single sub-composite It is evident that the number o f parameters and the chromosome length is dependent on the type of symmetry used for the FSS screen. The total number of parameters and chromosome length (total number of bits) for the entire composite is 42 and 310, respectively, when eight-fold symmetry is imposed. The measured values of er and s ' of ten different lossy dielectric materials are considered and a database o f these values as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 164 a function o f frequency are shown in Tables 3-2 and 3-3, respectively. Linear interpolation is used to generate the values of £r and s ' as functions of frequency for intermediate material types; thus we effectively use 19 types of materials. To obtain the values o f the real and imaginary parts of permittivity for even numbered material types at a particular frequency, the odd-numbered material types are averaged. For example, values for material “Type 2” at 19.0 GHz are obtained by averaging the material values for “Type 1” and “Type 3” at 19.0 GHz. For lossless layers, small values of losses are added, as it is not practical to find a perfectly lossless dielectric. Hence, if the MGA selects the first layer as lossless, its er value is fixed at 0.01, while for all the other layers the corresponding value o ff, is 0.1. Two cases were investigated, as listed below: (i) Oblique Incidence (0 = 45.0, <j>= 0.0), TE and TM polarization (ii) Normal and Oblique Incidence (0 = 0.0 and 45.0, <J>= 0.0), TE and TM polarization The population size and the number o f generations were fixed at 50 and 100, respectively. The periodicity of the FSS screen in the x- and y-directions were chosen to be equal. Only the resistive component of the complex surface impedance of the FSS screen was considered and the frequency increment was chosen to be 1.0 GHz in the MGA optimization. Following the guideline mentioned in Sec. 5.2, the scattering matrix for each sub-composite was truncated with 25 harmonics. This number was obtained after several numerical experiments to maintain the trade-off between computational speed and accuracy. The simulation showed that the magnitudes o f the higher order harmonics were less than -40 dB; hence, our truncation criterion was met satisfactorily. Eight-fold Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 165 symmetry was applied to the FSS cell design to take advantage of the rotational symmetry in the azimuth plane. The MGA-optimized composites are shown in Figs. 5.7 (a) and (b) for the two cases o f interest. Figures 5.8 (a) and (b) show the MGA-generated FSS unit cell design. For both Cases-1 and -2, the x- and y-periodicities for the FSS cells are 8.863 and 23 .44 mm, respectively. The reflection coefficient in dB is plotted vs. frequency in Fig. 5.9 for Case-1. The dotted curve in Fig. 5.9 gives the worst-case reflection coefficient realized by the MGA while optimizing simultaneously for both the TE and TM polarizations and for each frequency in the band of interest at 0=45 degrees. The worst-case reflection coefficient can be mathematically expressed as follows: V rn ax (rrE, r m )(indB ) (5 49) where Nf is the number o f frequencies in the band of interest. For Case-2, the worst- and best-case reflection coefficient values in Figs. 5.10 (a), (b), and (c) can be interpreted as follows. Over the frequencyband,the range of elevation angles of interest, and both the TE and TM polarizations, the maximum and the minimum values of the reflection coefficients correspond to the worst and the best results expressed in dB. Mathematically, the worst-case reflection coefficients can be written as V m a x [rnr(0 = O ),rnr(0 = 45)] (indB) s, V m a x [r™ {0 = 0 ) ,r m {d = 45)] (indB) V maxjY7^ (6 = 0 ) ,Tre(0 = 45),f™ (0 = 0 ),I™ (6 = 45)] (in dB) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.50) 166 where Nf is the same as defined earlier in (5.49). The worst-case reflection coefficients for Cases-1 and -2 are -19.57 and -19.2 dB, respectively, and the total thickness of the composites are 6.5 and 6.48 mm. Figures 5.11 (a) and (b) illustrate the performance of the MGA search process by plotting the variation of average and best fitness value vs. the number o f generations for the two cases. The dips in the curves for average fitness value indicate the generations at which the MGA performs the population restart, as explained in Chapter 2. The population distribution for Cases-1 and -2, are shown in Figs. 5.12 and 5.13, respectively, as we move from the first to the last generation. The additional burden o f optimizing over a range o f elevation angles results in a slight degradation of the worstcase reflection coefficient values for Case-2. Case-1 and -2 have run times of 21 and 42 hours, respectively. It takes the MGA approximately 21 hours to optimize for a single elevation angle, simultaneously, for both the TE and TM polarizations. In the figures showing the FSS cell structure, white represents metal and black corresponds to free space, and an eight-fold symmetry is evident in the FSS cell structures for both the cases. (All dimensions are in millimeters). Table 5-2 lists the parameters selected by the MGA for each case. For both the cases, the azimuthal angle is fixed at zero degrees (<|> = 0.0, x-z plane) in the MGA optimization program. From Figs. 5.14, 5.15 (a), and (b) we see that the frequency response of the structure remains relatively invariant of the azimuthal angle. The angular independence is achieved via the use o f eight-fold symmetry imposed on the FSS element. Figures 5.16 (a) and (b) show that, for 0 = 45 and <j) = 0.0 and for both Cases-1 and 2, increasing the number o f harmonics from 25 to 49 in the MGA has little or no effect on the result. This confirms that an adequate number of harmonics were included in the scattering matrix for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 167 the MGA optimization procedure. To verify the rule of thumb mentioned in Sec. 5.2, the magnitude o f the reflection coefficient in dB, of the worst-case higher order harmonic, for each frequency is plotted in Figs. 5.17 (a) - (b) and 5.18 (a) - (b), respectively, for the two cases. We note that the magnitudes of the higher order harmonics are well below the -22.0 dB limit. e r = 1.75, e r = 0.01, d =1.825 FSS screen I e r = 2.526, e r = 0.1, d = 0.525 V Sub- composite I Type 1, d = 0.475 s r = 4.65, e r - 0.1 , d = 0.493 Type 14, d =2.025 e r = 1.93, e r = 0.1, d = 0.125 FSS screen 2 Sub- composite 2 s r = 3.612, e r = 0.1, d = 0.475 Type 14, d = 0.56 PEC backing Fig. 5.7. (a) Composite design for Case-1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 168 e r= 1.99,e r = 0.01,d =1.875 FSS screen 1 Type 2, d = 0.975 \ Sub-composite 1 Type 12, d = 0.025 &r = 3.26, 6 r - 0.1 ,d = 0.119 Type 13, d =1.625 FSS screen 2 6 r = 1.685, 6 r = 0.1, d = 0.175 y Sub-composite 2 6 r = 3.34, 6 r = 0.1, d = 0.775 Type 8, d = 0.918 PEC backing Fig. 5.7. (b) Composite design for Case-2. Fig. 5.8. (a) FSS unit cell design for Case-1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 Reflection coefficient in dB Fig. 5.8. (b) FSS unit cell design for Case-2. ;yJ •30- TE and TM TE only TMonjy 32- Frequency in GHz Fig. 5.9. Frequency response o f composite in Fig. 5.7 (a). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 170 -20 co ■a su o £ 8o s oa Cu ae! worst case best case -28 -30 -34 -36 18 20 22 24 26 28 30 Frequency in GHz 32 34 36 (a) -18- cn •o - 20 - - 22 '♦ •*“ - -24S ZJ 3 £ -268o so -28ou co -30ac -32- worst case best case + -34- ± + i . ....... •f -36' 18 I_____ 20 - i ------------------------- 1--------------------------1________________ i________________ ■ 22 24 26 28 30 32 ■ 34 Frequency in GHz (b) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 171 Reflection coefficient in dB -18 - 20 - - 22 - ♦ * *. * .♦ ♦ t ± j + -24 -26-28♦ + -30- worst case best case -32-34 8 20 22 24 26 28 30 32 34 Frequency in GHz (c) Fig. 5. 10. Worst- and best-case reflection coefficients of composite in Fig 5.7 (b): (a) TE polarization; (b) TM polarization; (c) TE and TM polarization. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 172 Fitness value -0.1 -0.4 -0.5 Average fitness best fitness - O .f f 100 Number of generations (a) - 0.1 1 -0.4 C/S 8 I -0.5 - 0.6 Average fitness best fitness - 0.8 100 Number of generations (b) Fig. 5.11. Performance of the MGA: (a) Case-1; (b) Case-2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 173 ° 0 -18 -16 -14 -12 -10 -8 -6 -4 -2 on nPVHDflP T1rifmnrnnr o ° Generation 25 ---------1--------- \---------1---------1--------- 1-----L ------1---------1-------- 1 -J£___ 4 * ___ A?___ A$____ -? -? -4 -? O Thickness in cm o c p o q y p g p iL iiiii Generation 1 O OD — )C3) 0.5 OO Generation 50 t p — --4o 0.5 A?— -J4— -J£. O O O (3D AfL -4- 4- CD M i i B f l f ) n m n i i m o 4~ 0 (3 ) GfineratiQnJl 0 ?p 4 8 ___ -A£___ 4 4 ___ A£. 40- 4 ------4 ------ 4 ------ 4 Generatinn 1001 - 0.5 -20 -18 -16 -14 -12 -10 -8 -6 -4 Reflection coefficient in dB Fig. 5.12. Population distribution w.r.t. total thickness of composite and reflection coefficient for Case-1. 11 1 | T11 1 o 0.5 ^20 0.5 -14 \ 1 Generation 1 -18 -16 -12 -10 OO G D D 4B H B OCBD O -18 -16 -8 .. ° E c -2Q -14 -12 -10 -6 -8 -4 -2 Generation 25 ■ - 6 - 4 - 2 . w O O J n rn n m in n iii) n Generation 50 ■ g 0.5 O VI V] 0 fl---- =1£-----A£---- 44------lfL . o o Generation 75 0.5 0 0----- -18 -14 -12 -10 1 0 -20 -18 -16 -14 -12 -10 -? O O O OO O l'IW IM IIBIII -8 1 ) 0 1 0.5 -16 4 -8 -6 ° -4 -2 Generation lOf 1 . -6 -4 -2 Reflection coefficient in dB Fig. 5.13. Population distribution w.r.t. total thickness of composite and reflection coefficient for Case-2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 174 -18 -20 Reflection coefficient in dB -22 -24 -28- -32 -34 -36 Frequency in GHz Fig. 5.14. Invariance o f the frequency response with azimuthal angle for composite in Case-1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 175 -15 -20 -25 € c -30 TE .0 = 0. = 0 TM. e = 0. ♦ = 0 TE. 9 = 0. <t>= 45 TM. 9 = 0. ♦ = 45 -35 eo a eo a -40 -45 -50 (a) -18 -20 -22 ! -24 I -26 £ § *28 os - -30 e3 TE. 9 = 45. <t>= 0 TM. 9 = 45. 4>= 0 TE. 9 = 45. <t>= 45 TM. 9 = 45. * = 45 -34 -36 Frequency in GHz (b) Fig. 5.15. Invariance of the frequency response with azimuthal angle for composite in Case-2: (a) 0 = <J>= 0 degrees, 0 = 0 and <J>= 45 degrees; (b) 0 = 45 and <t>= 0 degrees, 0 = <t>= 45 degrees. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 176 -18 -20 c -24 a -26 § -28 oa -34 TE-25 harmonics TM-2S harmonics TE-49 harmonics TVI-49 harmonics -36 Frequency m GHz (a) -1 8 1 -20 c e a -24 TE-25 harmonics TV1-25 harmonics TE-49 harmonics TM-49 harmonics a -30 -34 -36 Frequency in GHz (b) Fig. 5.16. Effect of the change in the number of harmonics included in the scattering matrix on the frequency response: (a) Case-1, 0 = 45, (f) —0; (b) Case-2, 0 = 45, <J>= 0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 177 -50 -55 Reflection coefficient (dB) -60 -65 x O -70 sub-com posite 1 sub-com posite 2 -75 -80 O 0 0 0 -85 -90 © - -95 20 22 24 26 28 30 32 34 36 Frequency (GHz) (a) -4 5 ----------- 1 1 i---------- 1— ■■■ i----------- 1----------- 1----------- 1----------- -50 - .......... ;................................ .. « i^ X Xx X x „ X - .... --............ *.....*......-..... *.....- x x * , * X - Reflection coefficient (dB) -55 - ..........i............................... X - -............-............. -..............- -60 • x oo <> o sub-composite-1 sub-composite-1 sub-composite-2 posite-2 sub-com : - 7 0 --------- JO... ® o " O -75 ------------•.......... = o - -............ -............. ...............- © * o - ° -80 --- ------- •.............. •..............•.............................. - ...- *.............................* ..C l O— Q --- — O o -8 5 1 --------- i--------- --------- i---------- --------- --------- ---------- ---------r 1 20 22 24 26 1 1 1 1 28 30 32 34 Frequency (GHz) (b) Fig. 5.17. Plot o f the worst-case higher order harmonics vs. frequency for Case-1: (a) TE polarization; (b) TM polarization. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 178 -40------------i---------- 1 ■ r ........... i i i i ............................ A. ... -45 -.......................................................................... - .. i---------X 1 Reflection coefficient (dB) -50 •....................................................................... - .............-............................- 5 5 ................. » ' X -*•.................... X ■ - ••......... - ............................. -.............-.............-.................... * J , -60 - * - « « « x x - „ -60 - - - x O - I -70 -............................................................... o ° o sub-composite-1 sub-composite-2 -.... •• 0 - • ......................... O oO oO o( oo O © °Q O ©......................................................... -75 -.......................... © O .Q O 18 1 i 20 22 O O oo O O i---------------1 24 oo oo i 26 28 Frequency (GHz) i i 30 32 i-------------- 34 36 (a) -45 -50 ' * * x x x x x x Reflection coefficient (dB) -55 -60 sub-composite-1 sub-composite-2 o -65 -70 -75 o o o O o o o -80 -85 18 20 22 24 26 28 Frequency (GHz) 30 32 34 36 (b) Fig. 5.18. Plot of the worst-case higher order harmonics vs. frequency for Case-2: (a) TE polarization; (b) TM polarization. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 179 TABLE 5-1 MGA PARAMETER SEARCH SPACE FOR ONE SUB-COMPOSITE Parameters One row Total 16 rows Range of Number of Increment parameters possibilities Two-fold symmetry (rectangular section in Fig. 2) 0 -2 5 5 1 Number of binary digits 256 1 6x8= 128 four-fold symmetry (square section in Fig. 2) One row Total 8 rows Row 1 Row 2 Row 3 Row 4 Row 5 Row 6 Row 7 Row 8 Total thickness of one composite (mm) d„ da dji d4i Tx, FSS position Choice of FSS screen material Re(Z.) Choice of layer 1 material B,(U) Choice of layer 2 material 8*2,1) Choice of layer 3 material £*3,1) Choice of layer 4 material e*4,l) 0 -2 5 5 I 256 Eight-fold symmetry (triangular section in Fig. 2) 0 -2 5 5 1 256 0 -1 2 7 1 128 0 -6 3 1 64 0 -3 1 1 32 0 -1 5 1 16 0 -7 1 8 0 -3 1 4 0 -2 1 2 8 x 8 = 64 | 8 7 6 5 4 3 2 1 0 .0 -3 .0 0.01 512 9 0.025-3.0 0.025-3.0 0.025-3.0 0.025-3.0 1.0-50.0 1.0-4.0 0.001 0.001 0.001 0.001 0.1 1 2974 2974 2974 2974 512 4 12 12 12 12 9 2 1.0-3.0 1 2 1 10.0 - 1000.0 I 1024 10 1.0-3.0 1 2 1 1.03-6.0 0.01 512 9 1.0-3.0 I 2 1 1.03-6.0 0.01 512 9 1.0-3.0 1 2 I 1.03-6.0 0.01 512 9 1.0-3.0 1 2 1 1.03-6.0 0.01 512 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 180 TABLE 5-2 PARAMETERS SELECTED BY THE MGA FOR THE TWO CASES Sub-composite 1 Dielectric Layer types Layer I Case 1 Case 2 Layer 2 Layer 3 FSS screen 1 properties Layer 4 Position between Lossless Lossless Lossy Lossless 1* 2nd Lossless Lossy Lossy Lossless 1st, 2nd Cell size (mm) Surface Resistance <%> 8.863 x 8.863 23 .44 x 23.44 714.8 667.87 Sub-composite 2 Dielectric Layer types Layer 1 Case I Case 2 Layer 2 Layer 3 FSS screen 2 properties Layer 4 Position between Lossy Lossless Lossless Lossy 2nd ^rd Lossy Lossless Lossless Lossy 1st, 2nd Cell size (mm) Surface Resistance 8.863 x 8.863 23.44 x 23.44 <%> 842.85 956.23 5.3.1.2 Numerical Results for Multiple FSS Screens Embedded in Dielectric and Magnetic Medium that is Either Lossy or Lossless In the previous section we designed a composite comprising either lossy or lossless dielectric layers with two FSS screens embedded in it. In this section we will optimize a composite consisting of layers that could either be dielectric or magnetic material and may be lossy or lossless. The magnetic materials, if selected by the MGA, are always assumed to be lossy. The lossy dielectric materials used are the same as in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 previous section, and, Figs 3.8 through 3.11 plot the frequency dependence of er,e r,n r, and jur for the six lossy magnetic materials utilized. Linear interpolation, similar to that applied to the lossy dielectric materials in the previous section, is again utilized for the magnetic materials. All the optimization parameters remain the same as in the previous section. Case-2 is repeated with the lossy magnetic materials included in the optimization, and an eight-fold symmetry is incorporated in the FSS screen design. Figure 5.19 shows the FSS unit-cell design (identical for both the screens), having x- and y-periodicities of 5.03 mm. The surface resistances in the upper and lower FSS screens are 658.1 and 330.62 ohms/square, respectively. Fig. 5.19. FSS unit cell design for the two FSS screens embedded in the composite in Fig. 5.20. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 182 e r - 1.5, e r = 0.01, d =2.375 mm, lossless dielectric FSS screen 1 A e r = 3.0, s r = 0.1, d =0.275 mm, lossless dielectric Sub-composite 1 Total thickness = 2.73 mm e r = 15.35, e r = 11.35, d=0.025 mm,Type 16, lossy d ielectric |*—4.77, E r 0.1, d = 0.055 mm, lossless dielectric 6 FSS screen 2 e r = 2.26, e r = 0.1, d =1.175 mm, lossless dielectric < 6 r=7.81,E r=5.33,d=0.525 mm, p r= l . l l (p r= 0.027,Type 4,lossy magnetic > e r = 5.57,e r = 0.1, d = 0.025 mm, lossless dielectric Sub-composite 2 Total thickness = 2.135 mm e r = 9.0, e r = 6.25, d = 0.41 mm, Hr = 1.0,(i r = 0.04,T ype9,lossy magnetic PEC backing j Fig. 5.20. Synthesized composite for two FSS screens embedded in layered media. Figure 5.20 shows the composite with two FSS screens embedded in it. The total thickness o f the composite is 4.865 mm - a 25% reduction in the overall thickness as compared to 6.48 mm for Case-2 in the previous section, which dealt with a composite consisting o f lossy and lossless dielectrics only. Figures 5.21 (a), (b), and (c), show the MGA optimized worst- and best-case values o f reflection coefficient over the frequency band o f interest for TE, TM, and both TE and TM polarizations. Comparing 5.21 (a), (b), Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 183 and (c) with 5.10 (a), (b), and (c), we see that the inclusion of lossy magnetic materials in the composite design results in an improvement of -4.0 dB in the worst-case reflection coefficient while simultaneously reducing the overall thickness of the composite. -24 -26” ♦ 00 -a c -26 ” +...+ ••+ - + 3 -3C ‘ 4I> o0 -32‘ | 1c -34” o cC + worst case best case -36” -38 ” -4Qr 8 20 22 24 26 28 30 Frequency in GHz 32 34 (a) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 184 -24 ♦ 26h ... ♦'.... .t co 28 .. ■a e 3q— ......*....+--V' 0 + 1 -32h ± ....± ■+ - - * 0 -34 1 i -36- -......*cu O' -38.......... - ♦ + worst case best case -40 ~ -42' -441-------- 1-------- '-------- 1-------- 1-------- 1-------- 1— 18 20 22 24 26 28 30 32 34 36 Frequency- in GHz (b) -24 r*** -zq 1-------------* — + — !------------!------------- r . . - ♦ ................. dir... -i— + ~r~ * CO ■a -28 ■ f* ■g -30T £1) 8 32" I -34 ! + f i + ; * + worst case best case 5= U at, -36I" -38- + + i. t i---------1 i -4C 8 20 22 24 26 28 30 32 34 36 Frequency in GHz (c) Fig 5.21. Worst- and best-case reflection coefficient of composite in Fig. 5.20: (a) TE polarization; (b) TM polarization; (c) TE and TM polarization. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 185 ---- 1---- — o — 45— ------ o a B o 0.5I Thickness in cm - ? n ____ — 4 5 — C 1--- 1 .o n Generation. 1 ------------- Generation 25 ---------4 0 --------- ----- -S------------- -0 9 OOO OQDODBRD flDQDQDO CD u Generation 50 . eration 75 tmn 1Q0 -2 0 -1 5 -1 0 Reflection coefficient in dB Fig 5.22. Population distribution with evolving generations. - 0.1 -0.G- - Averaee fimess best fimess 0.8 Number of Generations Fig. 5.23. MGA performance to optimize composite in Fig. 5.20. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 186 5.3.2 Spatial Filter Synthesis We employ the same procedure as described in Sec. (3.3.1) for designing spatial filters, except we now embed multiple FSS screens in lossless dielectric layers and remove the PEC backing. The FSS unit cell periodicities, its position within the composite, its design, and the material content are encoded in binary forms by the MGA, and are given in equations (5.29) through (5.34), respectively. The fitness function employed can be expressed as F =- (551) where T and T are the reflection and the transmission coefficients, respectively. A and B are additional weight coefficients that are problem specific, and are introduced to equalize the two summation terms. Also, M and N are the number of frequencies for which the reflection and transmission coefficients are minimized to fixed, problemspecific values corresponding to r mn and Tmin, respectively. 5.3.2.1 Numerical Results for Spatial Filter Synthesis We now present the results for a composite comprising lossless dielectric layers and a single FSS screen that satisfied the following constraints: (i) incident wave polarization: TM Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (ii) pass band: 960 MHz - 1.215 GHz for 0 = 70 degrees and <j>= 0 and 90 degrees simultaneously. (iii) stop band: 2.0 - 18.0 GHz for 0 = 70 degrees and <j> = 0 and 90 degrees simultaneously. (iv) pass-band transmission: Transmission better than -0.5 dB. (v) stop band attenuation: Transmission less than -15.0 dB. (vi) total composite thickness: 12.7 mm. (vii) minimum layer thickness: 0.025 mm. Eight layers of dielectric were employed, along with two FSS screens, to design the composite. Small losses were introduced in the dielectric layers by assigning £v=0.1, as fabricating a perfectly lossless dielectric is not feasible in practice. The MGA was given the choice o f selecting a lossy or lossless material for the FSS screens. 0 dB ■20 dB start fend f (G H z) Fig. 5.24. Idealized response of spatial filter Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 188 The ideal frequency response for the transmission coefficient is shown in Fig. 5.24, where fcuti, fcuti, /start, and f end are 1.2, 2.0, 0.8, and 18.0 GHz, respectively The frequency increment was 0.2 GHz and 0.025 GHz for frequencies ranging between 0.8 2.0 GHz and 2.0 - 18.0 GHz, respectively. The values o f M and N were 6 and 40, respectively, while, = Tmn = -20.0 dB. Finally, the weighting coefficients were selected as follows: A = 500, B = 1, and 1.0 for j= 1,2 a. = 1.1 forj=3, 4, 5, 6 1.2 forj=7 to 47 The number o f generations and the population size are 219 and 5, respectively. Figure 5.25 (a), (b), and (c) shows the optimized composite, the FSS screen design, and the frequency response of the filter. e r = 3.1, e r = 0.1, d =0.4825 s r = 4.0, e r = 0.1, d = 0.0225 e r = 3.0, e r = 0.1, d = 0.0025 \ Sub- composite 1 FSS screen 1 e r = 6.6,8 r = 0.1, d = 0.0025 8 r = 2.3, 8 r = 0.1, d =0.5875 < 8 r = 6.0,8 r = 0.1, d = 0.0025 8 r = 4.0, 8 r - 0.1, d = 0.0025 y Sub- composite 2 FSS screen 2 8 r = 6.8, e r = 0.1, d = 0.0025 (a) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CQ ■o Io -io> 3 | C/3 52 -15- sC/3 i -2G- -2S Frequency in GHz (C) Fig. 5.25. Spatial filter synthesis: (a) composite design; (b) FSS unit cell design; (c) frequency response. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 190 The two FSS screens have a periodicity of 79.7 mm in both the x- and ydirections, and are lossless. The total thickness o f the composite is 11.05 mm, which is less than the specified maximum. Comparing Fig. 5.25 (c) with the frequency response in Fig. 4.30, we see that there is an overall improvement in the values of the transmission coefficient, for both the pass and the stop bands, when two FSS screens are utilized instead of one. However for the multiple FSS screen case, a lower minimum thickness is needed for the dielectric layers, which may be difficult to realize in practice. We note that the MGA was optimized at 0 = 70 and <t> = 90 degrees. The design criterion requires that the frequency responses of the composite be identical at both <J>= 0 and 90 degrees. This was achieved by the implicit scheme mentioned in Sec. 4.3.1.1, and also by observing that the eight-fold symmetry imposed on the FSS screen design makes it appear to be identical to the incident plane wave at <t>= 0 and 90 degrees. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 THE PARALLEL IMPLEMENTATION OF THE MICRO-GENETIC ALGORITHM (MGA) As described in Chapter 2, the genetic algorithm (GA) is a probabilistic search technique, which navigates through the search space by making random choices, instead of following a specific pattern. The steps we follow in the basic genetic algorithm is quite simple: (i) begin with a random set (population) of points, where each point (chromosome) represents a coding of a unique set of the problem's parameters; (ii) evaluate the chromosomes individually to see how well they satisfy an object (fitness) function; (iii) select individuals probabilistically to “breed” (selection) and exchange portions o f their parameter encoding (crossover); (iv) occasionally mutate the chromosomes to allow the exploration of new areas of the domain; finally, (v) repeat steps (i) through (iv) until some member of the population satisfies a termination criteria (i.e., the fitness is above a certain level) or a specified number of iterations (generations) is reached. GAs are highly successful in optimizing complex systems for several reasons. First, they start with a population of initial guesses rather than a single one, and, hence, they explore many regions o f the search domain simultaneously. This widespread search also causes the GA to be less prone to (although not totally immune from) settling into a local optimum during the search. GAs are a “weak” search method, meaning they do not Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 192 rely on a knowledge o f the domain in performing the search, so the algorithm is less likely to be led astray by inconsistent or noisy domain data as is common with hillclimbing or domain specific heuristics. Although there are no rigorous guidelines available for deciding when to use a genetic algorithm for a given optimization problem, there are several useful "rules of thumb.” For example, a GA is most applicable when the search space of a problem is large, multi-modal, and not necessarily smooth. A GA is also useful when the evaluation (fitness) function is noisy or when a detailed knowledge of the characteristics of the domain are not available. Perhaps the most useful application of the GA is to problems that are multi-variate. Many heuristic algorithms are uni-variate, meaning they attempt to optimize the solution one parameter at a time. In multi-variate problems, parameters interact non-linearly, so a simple “one at a time” approach will not be successful. Another important consideration is that a GA does not necessarily guarantee an optimal solution, so a GA may not be suited to any problem where an optimal solution is strictly required. Recently there has been a great deal of study on methods for parallelizing genetic algorithms [45-46], The GA algorithm has a structure which is ideal for parallelization. The process o f searching numerous points in the domain simultaneously can be carried out very naturally in a parallel environment, rather than “one at a time” as is done in the serial methods. Many o f the parallel GA techniques hold advantages beyond the simple speedup of the basic GA algorithm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 193 6.1 Reasons for Parallelization There are many reasons for parallelizing the genetic algorithm. The most obvious one, o f course, is the speed of computation. The GAs are computationally more expensive than the more deterministic forms of search and optimization schemes. This is due, in part, to the probabilistic approach taken by the GA, which requires it to explore over a larger area than is needed in a deterministic algorithm. This is related to the fact that numerous points in the domain are searched simultaneously, and this attribute makes the GAs ideal for parallelization. In some cases, the GA can be parallelized such that each individual chromosome is assigned to its own processor to perform necessary computations. This essentially reduces the run time for each generation to that required to perform the genetic operations on just one individual. Speed is not the only reason for parallelizing a genetic algorithm. Some implementations o f parallel GAs have a significantly higher cost than the serial ones, though the former increase the likelihood of finding the optimal solution. A more intuitive argument for parallelizing a GA is simply that if nature is parallel, then an algorithm that models systems in nature should also be parallel. Genetic algorithms are built on the foundation that nature evolves solutions to hostile environments all the time by recombining current patterns to make newer and better patterns. If we are going to model a natural system, perhaps one that mimics nature will be the best performer. There are several additional mathematical reasons for parallelizing the GA. One is to reduce the likelihood of premature convergence. Premature convergence occurs when a few individuals with high fitness begin to dominate the population and cause all of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 194 other chromosomes to become much too alike each other [45], When crossover is used with such chromosomes, no new pattern is created, and this causes the search to grind to a halt. Although, one purpose of the mutation operator is to prevent premature convergence, this can still occur. This is due to the low probability of mutation and the fact that mutated chromosomes may not necessarily survive the next selection process Many of the parallel GA methods allow multiple populations to be kept, and this enables them to explore different areas in the search domain. The concept of using multiple populations has another saluting feature. In some cases, the objective function for a problem is so complicated that attempting to simultaneously optimize the entire function at once is difficult. Each sub-population can focus on a different area of the objective function, and later combine them to reach the overall optimum. There are several possible ways that one might run the genetic algorithm in a distributed environment. For example, when speed is the most important factor, a global GA is the simplest way is to distribute chromosomes without going through the effort of segmenting the population and coordinating several separate GAs. Conversely, if an optimal or near-optimal solution to a problem that has proven difficult for a simple genetic algorithm is required, some of the more advanced techniques that use several populations may be necessary. Similarly, some problems may be very sensitive to the parameters of the GA. In this case, attempting to find the correct combination by trial and error may be difficult, so a multi-layered GA that self-adjusts may be the only viable solution. Other factors that influence the decision of which model to use is the number of processors available and the processor to chromosome ratio. The type of parallel system Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 195 available is also important {i.e., multi-processor - all processors on the same mother board - versus a network o f physically distant machines). It has already been demonstrated in Chapters 1 through 5 that the MGA is not only a numerically efficient scheme, but that it also shows very good convergence properties leading to solutions that satisfy the design criteria for most problems of interest. However, we note that most of the FSS screens designed by using the MGA has a pixel resolution of 16x16. Using a higher pixel resolution enables us to better approximate of the induced surface current on the FSS screens. However, as mentioned in Sec. 4.3.2.2, the time it takes to evaluate the reflection and transmission coefficients increase dramatically with increasing resolution. To maintain the run times of the MGA optimization such that they do not depend on the resolution of the FSS screen, a parallel implementation of the MGA is required. Thus, as mentioned earlier, for simple speed-up requirements, the global or Micro-grain GA [46] is the model best-suited for this problem. We discuss this approach in the next section. 6.2 Micro-Grain MGA The micro-grain micro-genetic algorithm (mgMGA), also referred to as a global MGA, is probably the simplest form of parallel GA. Principally, a master process maintains a single population. This master process generally performs the selection, crossover and mutation operations while assigning all o f the fitness evaluations to other processors acting as slaves. The task of evaluating a chromosome is usually more Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 196 expensive than the others, allowing the majority of the work in the mgGA to be done outside the master process. Ideally, there should be one processor for each individual in the population, but the chromosomes can be distributed among any number of slave processors. Speedup associated with the mgMGA for a population size of N and p processors is given by: S (p 'N ) = ^ ~ W a +— P <6I) where a represents the time required for the master process to perform selection, crossover and mutation and b represents the amount of time required to perform an evaluation by the slave process. Maximum speedup can only be attained when each of the slaves receives an equal amount of work. In some instances, this means that sending an equal number of individuals to each slave will result in sub-optimal performance, as some evaluations may be more expensive than others in the same population. This is especially true in our case because each composite has a different FSS screen design and this causes the run times to be different. Sub-optimal performance could also occur when slave processors do not run at a uniform speed, therefore, balancing the load among slave processes is an issue with the mgMGA. The complete algorithm is given in Fig. 6.1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 197 Master Process population nt pop_size, generation; float p_cross, p_mutation; . old_pop 4. /* prob. o f crossover & m utation*! = initial random population 2. for each sta ve p ro cess p 3. I* current and n ext populations*! old_pop, new_pop; I* evaluate each chrom osom e *1 se n d a su b se t o/old_pop to p ro cess p wait until all sla ves have returned their results 5. while(generation < MAX_GEN) I* se le c t N m em bers o f old_pop *1 6. new_pop = select(N, old_pop) 7. new_pop = crossover(p_cross, new_pop) 8. old_pop = mutate(p_mutation, new_pop) 9. for each sla ve p ro cess p 10. I* evaluate each chrom osom e *1 se n d a su b set <?Aold_pop to p ro cess p 11. wait until all sla ves have returned their results 12. generation = generation + 1 13. endWhile 14. re\um (individual with g rea test fitness) Slave Processes population int locaLpop; local_size; 1. local_pop = receive one or m ore chrom osom es from m aster 2. evaluate(old_pop) I* calculate fitn ess o f individuals *1 3. sen d locaLpop back to m aster Fig. 6.1. Pseudocode for a micro-grain micro-genetic algorithm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 198 The greatest advantage to the mgMGA method is its simplicity. This method does not require a particular network topology, although a highly connected network is desirable for reducing the communication overhead. The algorithm is roughly equivalent to the serial MGA; hence, the serial MGA is usually a good indicator of whether a particular problem will map well to the mgMGA. This is not generally the case with the other distributed GA methods. Differing implementations of the GA network often cause some methods to fail while others succeed. The mgMGA is generally a good method for gaining speed with the genetic algorithm without introducing extra complexity issues. There are also some disadvantages with the mgMGA compared to some of the other distributed GA schemes. Perhaps the most important is that the mgMGA is not a solution to the premature convergence problem, though this is not a problem we encountered since the serial MGA always converged to solutions that satisfied the specified design criteria. In our case only one population was maintained and selection was performed by a single processor, so all o f the chromosomes interacted in exactly the same manner as they did in the serial MGA. Also, it should be mentioned that the distributed network environment plays a significant role in determining the speedup associated with this method. For example, a multi-processor may minimize the communication overhead between the master and the slaves, but a network of physically-distributed machines may suffer from wasted cycles because o f errors in synchronization and imperfect communication. The mgMGA relies heavily on the master process; each slave communicates exclusively with the master, and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 199 each one passing through the MGA loop must be synchronized. This causes the master process to become a bottleneck in the mgMGA model. 6.3 Design of Polarization-Selective Surfaces for Dual Reflector Applications The serial and parallel MGA will now be applied to the design of a polarization selective surface (PSS) for dual reflector applications. Figure 6.2 depicts the problem geometry whose parameters were optimized by both the serial and parallel versions of the MGA. The design required a freestanding FSS screen to transmit TE polarized waves to within -0.01 dB and reflect TM polarized waves with isolation better than -20.0 dB, for elevation angles ranging from 0 to 40 degrees, and for frequencies extending from 33.0 to 36.0 GHz. The problem was first optimized by the serial MGA with the population size and number of generations equal to 5 and 100, respectively. The MGA designed the unit cell of the freestanding FSS screen with 16x16 pixel resolution for 0 = 1 0 ° and <J) = 90° over the frequency band of interest simultaneously for both the TE and TM polarizations. The objective function employed by the optimization is shown below: F ( ^ . < ,. < , ) = - m a x { ^ * r rc( » „ / , ) + a * T w (« „ /,)} where A and B are the relative weight coefficients; (6.2) is the unit cell of the periodic FSS screen generated by the MGA; tx and ty are the x- and y-periodicities of the FSS screen; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 200 and r , T are the reflection and transmission coefficients as a function of incident angle and frequency. The coding for the three parameters, viz., tx, ty, and Sdes are represented by (4.68) and (4.70), respectively. The entire system is represented by a sequence C similar to that in (4.73) as C = TxTy {CD\ .(CD)j (6.3) Each quantity in (6.3) has been defined in (4.68) and (4.70), and each sequence in (6.3) consists of (Mab~Mtyb~j*MCb) bits. Air superstate ► FSS Air substrate Fig. 6.2. Problem geometry for PSS optimization. The serial MGA was executed on a 500 MHz DEC-ALPHA™ workstation with 512 MB RAM, and took 15 seconds to complete one generation. Figure 6.3 (white and black represent metal and free space, respectively) shows the FSS screen design made of PEC with unequal x- and y-periodicities. Figures 6.4 (a) through (e) plots the transmission coefficient o f the freestanding FSS screen as a function of frequency with elevation angle as a parameter for both the TE and TM polarizations. It was observed that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 201 the design constraints were satisfied for both the TE and TM polarized waves and the TM response degraded for higher elevation angles. The MGA formulation for the parallel MGA remained exactly the same as the serial version and the optimized results obtained were identical. The parallel version of the MGA was executed on a CRAY™ T3E, with 512 ALPHA™ processors, each with a 450 MHz cpu, local RAM o f 128 MB and a theoretical machine peak speed of 460 GFLOPS. For our optimization, six processors were utilized for the execution, with one processor assigned as the master and the remaining five as slaves, thus assigning one individual per slave processor. The execution time for one generation using the parallel version of the MGA was 5.0 seconds, which was three times faster than the serial MGA. Theoretically, a speed-up o f five times (100 % efficiency) was expected, as five processors were utilized. However, this was not achieved in our case (60 % efficiency) because the serial MGA code was only partly converted to a parallel MGA one. The master process that performed the selection and crossover still remained a serial code. Only with the complete parallelization of the serial code can higher efficiencies be achieved as per Amdahl’s law [47], The speed up achieved using the parallel version of the MGA provided the opportunity to generate FSS unit cell designs with higher resolutions in the optimization process. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.3. FSS unit cell for PSS design application. o TE TM ■5 10 15 ■20 •25 •30 •35 -40 -45 -50 33 34 35 36 37 38 39 40 41 42 43 Frequency in GHz (a) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 203 TE TM c E S o m c 7o(0> c £ -30 -35 -40 -45 Frequency in GHz (b) TE TM jj -15 9 -20 | -25 -30 -35 Frequency in GHz (C ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 204 TE TM ■ - -10 -15 -20 -25 -30 33 34 35 36 37 38 39 Frequency in GHz 40 41 42 43 (d) TE TM •o S -10 -15 -20 -25 -30 33 34 35 36 37 38 39 Frequency in GHz 40 41 42 43 (e) Fig. 6.4. Frequency response of PSS: (a) 0 = 0, <J>= 90; (b) 0 = 10, 4> = 90; (c) 0 = 20, <t>= 90; (d) 0 = 30, <t>= 90; (e) 0 = 40, <J>= 90. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 7 VALIDATION OF THE RESULTS GENERATED BY THE MICRO-GENETIC ALGORITHM (MGA) In this chapter, the results generated earlier by the MGA are validated by comparing them against those obtained via the Finite Difference Time Domain (FDTD) method for single embedded screens in Chapter 4. The MGA-generated results for two FSS screens embedded in a composite, given earlier in Chapter 5, are verified against a direct MoM solution for cascaded screens as opposed to the approximate Generalized Scattering Matrix (GSM) technique used in the MGA optimization. A brief overview of the two methods is presented along with some numerical results. 7.1 The FDTD Method The FDTD method is a widely used electromagnetic modeling technique, first proposed by Yee [48] in 1966. In this method the differential form of Maxwell’s equation are transformed to central difference equations, and then discretized for computer solution. The equations are explicit in nature, which implies that the field quantities at a particular instant o f time are only dependent on their values at the preceding instant. The equations can thus be solved by using a leapfrog scheme in which the electric fields are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 206 solved at one instant in time followed by the solution of the magnetic fields at the next instant. The process is repeated over and over again. The differential form o f Maxwell’s curl equations can be written as 5HZ 1 (dEy cE. dy dHv 1 dt My l 5Z ( BE. ^ dx 1 (dEx c t /A oEx dz dH. dt Mz I dEy dx dEx __ 1 'dH. CHy ct & 5Ey 1 ct dE. 8 t rcHz fv y CZ _ 1 ( dHy dx CZ cH. dx dHx dy A first step to applying the FDTD for numerically modeling a given problem is to establish a computational domain and to discretize it. Yee [48] suggested that this be done in the form of small cubes, commonly referred to as the “Yee cells” (see Fig. 7.1). The material within each individual cell in the computational domain is then specified, thus enabling one to model arbitrarily inhomogeneous structures. In approximating a differential equation with central differences it is essential that the cell size be chosen small enough to assure that the numerical inaccuracies introduced by the discretization step is within the acceptable limits. In FDTD, this nominal cell size lies between A710 and X/20, where X is the smallest wavelength in the computational domain. In certain cases, the criterion mentioned above is superseded by one imposed by the geometrical features Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 207 of the object being modeled. If a resolution lying between X/10 and X/20 is not sufficient to accurately model the fine geometrical features of the object, then the cell size is determined by the geometry instead. z Hi Ex Hx O.i.K) .'l* X Fig. 7.1. The Yee cell with the six electric and magnetic field components all offset by half a space step. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 208 Approximating the differential equations in (7.1) by central differences, and using the Yee cell distribution of the fields, we can express (7 .1) as At * v A '.J.k) * r s ( u * ) - * r s (u * )+ E:(ij+ik)-E:(ij,k) E :{i+ \j,k)-E :{ij,k) My ( i J , k ) ' Ax E zn { i J , k + l ) - E : ( i , j , k ) Ac Erx { i j + \ , k ) - E nx { i j , k ) H !’ ~lz{ i J , k ) = H /n ' 2( i J , k ) + E" (#' + \ , j , k ) ~ E”(/, y, k ) (7.2) r Hn : l'-{ij,k)-H :n' ' - { i j - i k ) E r(ij,k)= E :(u ,k)+ At Ay j f Hn ;''-{ij,k)-Hn ; K- { i j , k - 1) At At 12 (/ - 1, y, at) Ax h j,k)+ "; ' - ( i j . k ) - H : - ( i - u , k ) Ax H ? ' - { i J , k ) - H ;n l' - ( i J - l k ) Ay where At is the maximum time step whose value is determined by the Courant stability [48] condition and can be expressed in terms of the cell sizes as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 209 r • (73) \ ( to)! +(Av): V ' f where v is the maximum velocity of propagation in any region within the computational domain and Ax, Ay and Az are the cell sizes in the x-, y-, and z-directions, respectively. It is necessary to impose the above stability condition in the FDTD algorithm because of its explicit nature. Once the computational domain and the properties o f the material media within this domain have been established, we move on to specify the nature of the excitation source. The source commonly used in FDTD is a Gaussian pulse with a specified 3-dB frequency point. However, for our FDTD simulations we employ a sine-modulated Gaussian pulse with specified 3-dB and modulating frequencies. Results valid over a wide frequency band can be derived in a single simulation by using the FDTD, and choosing the frequency characteristic of the source excitation appropriately. The source can be an impinging plane wave, a current on a wire, or a voltage between two plates, depending on the problem to be modeled. As the spectral-MoM used to analyze the periodic structures assumes an incident plane wave, the FDTD simulations also utilize the same type of excitation. For numerical purposes, the computational domain must be truncated at some point, even if we are dealing with an open region problem. In our case, recognizing the periodic and symmetrical nature of the structure being modeled, we utilize appropriately placed Perfect Electric Conductor (PEC) and Perfect Magnetic Conductor (PMC) walls to truncate the x- and y-boundaries (assuming the FSS structure lies in the x-y plane), but Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 210 employ either a Perfectly Matched Layer (PML) [49-50] type of Absorbing Boundary Condition (ABC) or a Mur-type Outer Radiation Boundary Condition (ORBC) [51] at the z-boundaries. Six layers o f unsplit-uniaxial-anisotropic-PML, proposed by Gedney [50] are employed in our FDTD modeling. In certain cases, the PEC and PMC can be replaced by a Periodic Boundary Condition (PBC) [52], which can be easily implemented by using the conventional FDTD update equations, though only for the normal incidence case. Thus we only validate our MoM results by comparing them to those obtained via the FDTD for normal incidence, because, we only have access to a periodic-FDTD code that can handle normal incidences. The material content of the conducting patches in the FSS screens is assumed to be PEC in the FDTD simulation. 7.2 The Direct Method of Moments (MoM) Approach to Analyzing Multiple FSS Screens In Chapter 5, the Generalized Scattering Matrix (GSM) was introduced as an alternate and approximate technique for analyzing multiple FSS screens embedded in a composite. The numerically rigorous approach to simultaneously solving for the unknown current coefficients for all of the FSS screens, though computationally expensive, are nonetheless useful for validating the results generated by the MGA, which incorporates the GSM technique to analyze multiple FSSs. The problem geometry is the same as shown in Fig. 5.1. Equation (4.41) pertaining to a single FSS screen can be expressed in a matrix form as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 211 i -E T (x,yj -Er(*-y) =— »<e -**<o <m z z fTT=—«3 m=-cc a ,a ) <m a a )' •UA..A.)' G„(A..A.) <UA„.A,.) A. (A.. A.) ’A^Py.y) - Z . (7.4) \{*>y) Jyofry) where the definition o f all the terms were given in (4.34) through (4.40) in Chapter 4. For a multilayered FSS composite containing S conducting screens, the scattered fields or the induced currents on each of the conducting surfaces are obtained by modifying (7.4) as S •<*> T Tv *<c Gi(A..A.) g ;( a .,/? J rr= -<e g; ( a .. a .) <%(a . . a .) I I I y = 1 m= 'K { P ^ P y .) y.) ->(7.5) J, ^ y ) J ,y» ( x ^ y ) where, the subscript / (i = 1, 2 S) corresponds to the i'h conducting surface. The left hand side in (7.5) corresponds to the sum of the scattered fields due to the currents J, on the j h conducting surface, where j varies from I through 5. Equation (7.5) is in a form that is amenable to solution via the use of the spectral-Galerkin technique, which was described earlier in Sec. 4.2.1. 7.3 Application of the FDTD Method to Analyze Single Screen FSS Embedded in Inhomogeneous Media The first example to be validated by using the FDTD is a composite whose configuration is the same as shown in Fig. 4.25 in Chapter 4. Note that this composite Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 212 contains lossy dielectric materials, whose accurate simulation over a broad band of frequencies requires the use of a version of the FDTD that can handle dispersion. To overcome this difficulty, the FDTD is simulated at six separate frequencies, i.e., 19.0, 22.0, 25.0, 28.0, 34.0, and 36.0 GHz and the appropriate values o f conductivity corresponding to the lossy layers are used at each of these frequencies. To carry out a good comparison between the two methods, viz., the MoM and the FDTD, for screens designed by using the MG A, we need to ensure that their simulation environments are close to each other, and this necessitates some changes in the unit cell design of the FSS. First we note that, in the MoM technique we apply sub-domain basis functions only to those regions in which the l ’s are consecutive in both the x- and y-directions. Consequently, the current on isolated pixels that are ignored by the MoM formulation but are included in the FDTD method, can lead to different results. To avoid this difficulty, all single pixels from the FSS unit cell in Fig. 4.25 are removed when using the FDTD simulation. Next, the sub-domain basis functions in the MoM are applied only in the xand y-directions, but not diagonally. Hence, the currents are not modeled in the MoM at the point of intersection of the comers of pixels where the metallic patches make contact, though they are in the FDTD. Figure 7.2 shows the FSS unit cell before and after processing the geometry by removing all such intersecting metal comers. We also observe in Fig. 7.2 that the FSS unit cell has metal patches penetrating into the x- and yboundaries. We find that for such structures the PEC and PMC truncation of the x- and yboundaries lead to a formation of psuedo-cavities that introduce late time ringing in the FDTD simulations, and, in turn, affect the low frequency response of the composite. To Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 213 overcome this difficulty, we use the PBC to truncate the x- and y-boundaries for all the examples that have this attribute. (a) (b) Fig. 7.2. FSS screen for example in Fig. 4.25: (a) MGA-generated FSS unit cell; (b) after processing. The boundaries at zmm and zmax are truncated with a PEC and the PML, respectively. Also, one pixel (there are 16 pixels in both the x- and y-directions when the discretization is 16x16) in the FSS unit cell in the MoM is regarded as being the equivalent o f four cells in the FDTD. The FDTD cell sizes in the x-, y-, and z-directions are 0.025, 0.025, and 0.0875 mm, respectively, which effectively results in a domain size o f 64 x 64 x 64 cells. The FDTD is run for 16384 time steps in the absence o f any object in the computational domain to provide the incident fields. As mentioned earlier, the FDTD is simulated six times to cover the frequency range to account for the change in conductivity with frequency due to the presence of lossy materials in the composite. Figure 7.3 presents the comparison between the FDTD and MoM results and they are seen to agree very well with each other. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 214 -14i -16- - s o MoM FDTD - 22- -26- -32- Frequency in GHz Fig. 7.3. Comparison between FDTD and MoM results for example 1 in Fig. 4.25. In the next example, the results for a spatial filter designed by the MGA are validated by using the FDTD. The design requires that for TE and TM polarizations the filter reflects all frequencies between 19.0 and 30.0 GHz with a reflection coefficient greater than -1.0 dB, and transmits all frequencies between 31.0 and 40.0 GHz with a transmission coefficient greater than -5.0 dB at normal incidence. The maximum thickness o f the composite is restricted to be within 2.5 mm and four lossless dielectric layers with a single FSS screen are used for the optimization. The population size and the number of generations are 5 and 300, respectively. The MGA selects the FSS screen to be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 215 lossless with x- and y-periodicities chosen to equal 4.26 mm. The MGA-generated composite and the FSS unit cell are shown in Fig. 7.4. er = 4.786, d = 0.55 mm er= 11.57, d = 0.25 mm er= 6.053, d = 0.75 mm er= 2.125, d = 0.59 mm (a) ri y (b) Fig. 7.4. MGA-generated spatial filter design: (a) composite; (b) FSS unit cell design. The parameters used in the FDTD simulation for this example are: Ax = Ay = 0.0665625, and As = 0.05 mm; domain size of 64 x 64 x 143 cells; PML absorbing boundary condition at both zmm and zmax; and, a plane wave source located 20 cells away from the PML. The FDTD simulation is run for 16384 time steps for both the incident and total fields. Due to the frequency-independent (absence of lossy dielectric materials in the composite) property o f this problem, the FDTD simulation for the total field is nan only once to generate accurate results for all the frequencies of interest. The FDTD results are shown in Fig. 7.5 along with those obtained from the MoM. We observe that the frequency response obtained by using the FDTD shows a slight shift toward higher frequencies, and the worst-case difference is about 1.4 %. The difference in the level of the reflection coefficient at resonance can be improved by running the FDTD for a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 216 greater number o f time steps. Also, we found that the results converge towards the MoM values [53] when the mesh density in the FDTD simulation is increased. os .£ c<u *5- 1 10 - " 8O J -15" O 0> c: <2 -2C ■ FDTD MoM -25" Frequency in GHz Fig. 7.5. Comparison between FDTD and MoM results for spatial filter design with four lossless dielectric layers and one FSS screen. 7.4 Application of the Direct-MoM to Analyze Multiple FSS Screens Embedded in an Inhomogeneous composite The four cases studied in Chapter 5 will now be validated by comparing the frequency responses generated by using the GSM-incorporated MGA method for multiple FSS screens, and the direct-MoM as described in Sec. 7.2. The composite design Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 217 and the structure o f the FSS unit cell are shown in Figs. 5.7 (a) and 5 .8 (a), respectively, for Case-1. As explained in Chapter 5, 25 harmonics are included in the scattering matrix for the GSM-MGA method. The parameters optimized by the MGA are utilized in a separate code incorporating the direct-MoM to generate the frequency response of the composites for the four cases. The frequency responses generated by the two methods are compared for both TE and TM polarizations at 0 = 45°, and <j) = 0°, and are shown in Fig. 7.6. - 20- - 22- -24 -26-28-3C GSM Technique-25 harmonics direct-MoM -32 Frequency in GHz (a) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 218 -ia - GSM Technique-25 harmonics direct-MoM 20- ■o -26- -32-34 -3© Frequency in GHz (b) Fig. 7.6. Comparision between the GSM-MGA and the direct-MoM for Case-1: (a) TE polarization; (b) TM polarization. The same comparison is performed for Case-2 for which the composite structure and the FSS unit cell design are shown in Figs. 5.7 (b) and 5.8 (b), respectively. For this case the MGA simultaneously optimizes for both TE and TM polarizations, at 0 = 0° and 45°. The frequency responses are compared only for 0 = 45°, for both the TE and TM polarizations, and are presented in Fig. 7.7. For Case-3, we have multiple FSS screens embedded in lossy or lossless dielectric and magnetic media. The FSS unit cell and the composite design were earlier shown in Figs. 5.19 and 5.20, respectively. The two methods are used to analyze both the TE and TM polarization incidences with 0 = 0° and 45°, and <J>= 0°. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reflection coefficient in dB 219 GSM Technique-25 harmonic? direct-MoM ______ Frequency in GHz (a) GSM Technique-25 harmonics direct-MoM_______________ c O ‘G £ -26eo oo e & Frequency in GHz (b) Fig. 7.7. Comparision between the GSM-MGA and the direct-MoM for Case-2: (a) TE polarization; (b) TM polarization. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 220 We arbitrarily choose the constitutive parameter values for the lossy magnetic materials used in the validation at a frequency of 36.0 GHz. Thus, the reflection coefficient values are accurate only at 36.0 GHz, and are maintained below the worstcase level optimized by the MGA in Chapter 5. To derive accurate results at other frequencies in the band of interest, we must insert the corresponding constitutive parameter values in the program. This is particularly true for lossy magnetic media, whose constitutive parameters vary more rapidly with frequency than they do for lossy dielectric media. -12i -14 cu ‘3 £ 8 GSM Technique-25 harmonics direct-MoM -18- - 20 - § S -22- c:o ac -2 4 -26- Frequency in GHz Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 221 -12i GSM Technique-25 harmonics direct-MoM -14 .2 8 - 18 - - 20- - 22- -24 -26- Frequency in GHz (b) -10. — GSM Technique-25 harmonics direct-MoM -1500 ■o c S u 0 -2 0 - E 1 -25co -35- •4Qr Frequency in GHz (c) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 222 -1S - 20- 25- 8 -30GSM Technique-25 harmonics direct-MoM -40- Frequency in GHz (d) Fig 7.8. Comparision between the GSM-MGA technique and the direct-MoM for Case-3: (a) normal incidence-TE polarization; (b) normal incidence-TM polarization; (c) 45° from broadside-TE polarization; (d) 45° from broadside-TM polarization. We observe from Figs. 7.6, 7.7, and 7.8 that the agreement between the two methods is excellent. This, in turn, leads us to conclude that an adequate number of harmonics were included in the scattering matrices for each of the sub-composites. After further study, we found that the 25 harmonics used by the MGA-GSM technique was more than was necessary. This observation is based on the results shown in Fig. 7.9, where only 9 harmonics were included in the scattering matrix for each of the sub composites in Case-1 and we see that the results agree very well with the direct-MoM. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 223 Obviously, reduction o f the number of harmonics included in the scattering matrix speeds up the MGA optimization process for multiple FSS screens. -18i -2C - -24 GSM Technique-9 harmonics direct-MoM -32* Frequency m GHz (a) -18i -2C - V GSM Technique-9 harmonics direct-MoM -24 £ -26-28 -30 Frequency in GHz (b) Fig. 7.9. Comparision between the GSM-MGA and the directMoM for Case-1 with 9 harmonics included in the scattering matrix: (a) TE polarization; (b) TM polarization. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 224 However, we note that it is possible to reduce the number of harmonics significantly only when the composite is made up of materials with high losses, and propagating higher order harmonics attenuate rapidly. As we will see in the next example, such a reduction is not possible when the composite is fabricated either with lossless or very low loss materials. For the final example, we consider a spatial filter designed by the MGA discussed earlier in Sec. 5.3.2. The composite and the FSS unit cell design are shown in Figs. 5.25 (a) and (b), respectively. The composite consists of low-loss dielectric materials and 49 harmonics were included in the scattering matrix for each sub-composite during the MGA optimization. The results generated by the two methods are compared in Fig. 7.10 for the TM-polarized wave incident at angle of 0 = 70° off broadside, with <J>= 90°. It was observed that although the frequency responses exhibit the same general trends, the results do not match as well as they did for the previous three cases. We attribute this to the fact that the number of harmonics included in the scattering matrix in the MGA-GSM optimization process were inadequate. To achieve better accuracy, this number should be increased, albeit with an increase in the computational expense of the optimization process. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 225 Transmission coefficient in dB GSM Technique-49 harmonics direct-MoM -1C -15- -2C -2 9 Frequency in GHz Fig. 7.10. Comparision between the GSM-MGA and the direct-MoM for the composite with low-loss dielectric layers and two FSS screens. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 8 SUMMARY, CONCLUSIONS, AND FUTURE WORK This thesis has presented a general stochastic technique for the efficient design and optimization o f spatial filters and broadband microwave absorbers. In the past, various stochastic optimization techniques have been utilized to successfully optimize problems that were computationally non-intensive. The Micro Genetic Algorithm (MGA) utilized in this thesis has successfully optimized designs that can be categorized as computationally intensive. The optimization problem becomes computationally intensive because o f the presence of doubly periodic, infinite Frequency Selective Surfaces (FSSs) that must be analyzed by using the Method of Moments (MoM) in the spectral domain. 8.1 Improving the Efficiency of the Optimization Process There are two ways to decrease the computational expense of the optimization process: (i) increasing the efficiency of the MoM matrix solution process, which plays a key role in the optimization process; and, (ii) improving the efficiency and convergence properties of the conventional genetic algorithms (CGA). The FSS screens can be efficiently analyzed by working in the spectral domain to avoid costly convolution operations (Chapter 4) and by using the Fast Fourier Transforms (FFTs) for efficient evaluation o f the double summations present in the matrix elements (Chapter 4). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 227 Furthermore, highly efficient matrix inversion subroutines can be utilized to invert the matrix (Chapter 4). To improve the efficiency of the optimization algorithm, we can use the MGA instead o f the CGA, because it demonstrates superior convergence (Chapter 3) properties for multimodal (dominant property of the problem at hand) problems. Furthermore, the computational efficiency of the MGA can be attributed to its ability to work with small population sizes (Chapter 3). Therefore, combining the MGA with the efficient spectral-MoM enables us to optimize complicated geometries efficiently and to maintain the computational expense within practical limits. We need an alternate approach to optimize a design when FSS screens with higher resolutions are embedded in the layered media, because it becomes computationally very expensive to deal with them by using conventional methods. Such high resolution in FSS screens is required for designs in which the frequencies of interest are in the millimeter wave band. To avoid excessive computational expenses, a domain decomposition approach is utilized by splitting the problem domain into two, viz., the dielectric composite without the FSS screen, and the freestanding FSS screen. The two structures are designed separately by using the approach discussed in Chapter 4, and then combined in a suitable manner to meet the design criteria. Chapter 5 addressed the problem of designing composites with embedded multiple FSS screens. The analysis for the single FSS screen can be extended to multiple screens by using the MoM, as explained in Chapter 7. The direct-MoM can be computationally expensive, but this difficulty can be circumvented by employing the generalized scattering matrix (GSM) approach. In the latter method, we begin by dividing the composite structure into sub-composites, each comprising a FSS screen embedded in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 228 layered media. Next, each sub-composite is assumed to be an Alport system and the scattering parameters for each system is generated and assembled in a matrix. We have demonstrated in Chapter 5 that it is necessary to include an adequate number of harmonics in the scattering matrix to maintain the accuracy. Finally, we combine the scattering matrices o f the sub-composites to generate the matrix for the composite from which the reflection and transmission coefficients can be evaluated. As shown in Chapter 7, for composites incorporating high-loss materials the number of harmonics included can be as low as 9, compared to 49 harmonics needed when the composite consists of lowloss or lossless materials. To further enhance the efficiency of the MGA, a parallel MGA has been implemented using the master-slave model (micro-grain MGA), and Message Passing Interface (MPI) protocols. The parallel version has been executed on a CRAY™ T3E™, a parallel architecture with 512 ALPHA™ processors working at clock speeds of 450 MHz. The parallel MGA population was equally distributed on 5 processors, while the sixth was used as the master process. The slave processors performed the reflection and transmission coefficient calculations corresponding to each individual, while the master process carried out the selection, crossover, and the population restarts. The parallel code achieved a speed-up of three times, as compared to the serial version of the MGA. 8.2 Accuracy of the MGA Optimized Results In Chapter 7, a few designs from Chapters 4 and 5 were selected to validate the accuracy o f the results generated by the MGA-MoM optimization technique. The FDTD Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 229 and the direct-MoM were employed to verify the results generated by the MoM/MGA method for composites with single and multiple FSS screens embedded in layered media, respectively. The FDTD was found to faithfully capture the trend of the MGA/MoM results, except for a small shift in the frequency response characteristics. The results generated by the direct-MoM and the MGA/ MoM-GSM methods agreed very well for composites consisting of materials with high-loss, and closely reproduced the characteristics o f the results for composites fabricated with low-loss materials. 8.3 Suggestions for Further Research Improving the efficiency of the MGA-MoM to analyze FSS screens with higher unit cell resolutions can extend the capabilities of the optimization procedure presented in this thesis. Either improving the optimization algorithm or enhancing the matrix solution procedure by using, for example, Model Order Reduction (MOR) [54] techniques to effectively reduce the matrix size can possibly increase the efficiency. While a very simple parallel MGA model has been presented in this thesis, several advanced models [45-46] have been proposed in the literature that can further improve the efficiency of the optimization algorithm. The most important advantage of the parallel GAs is that, in many cases, they provide better performance than the single-population-based GA algorithms, even when the parallelism is simulated on conventional serial machines. This is because multiple populations permit speciation, a process by which different populations evolve in different directions, i.e., towards different optima. Thus, parallel Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 230 GAs are not only an extension o f the tradition sequential GA, but they represent a new class o f algorithms in that they search the solution space differently. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BIBLIOGRAPHY [1] D. E. Goldberg, “Genetic Algorithms in Search, Optimization, and Machine Learning,” Addison-Wesley Publishing Company Inc., Reading, Massachusetts, 1989 [2] J. H. Holland, “Adaptation in Natural and Artificial Systems,” University of Michigan Press, Ann Arbor, MI, 1975. [3] E. Michielssen, S. Ranjithan, and R. Mittra, “Optimal multilayer filter design using real coded genetic algorithms,” IEE Proc. Part J 139, pp. 413-420, 1992. [4] D. S. Wiele, E. Michielssen, and D. E. Goldberg, “Genetic algorithm design of Pareto optimal broadband microwave absorbers,” IEEE Trans. Electromagn. Compat. vol. 38, pp. 518-524, 1996. [5] E. Michielssen, J. M. Sajer, S. Ranjithan, and R. Mittra, “Design of lightweight, broadband microwave absorbers using genetic algorithms,” IEEE Trans. Microwave Theory Tech. 41, pp. 1024-1031, 1993. [6] R. L. Haupt, “Thinned Arrays using genetic algorithms,” IEEE Trans. Antennas Propagat. vol. 42, pp. 993-999, 1994. [7] D. S. Wiele, and E. Michielssen, “Integer coded Pareto genetic algorithm design of antenna arrays,” Electronic Lett. 32, pp. 1744-1745, 1996. [8] D. S. Linden, “Automated design and optimization of wire antennas using genetic algorithms,” Ph. D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1997. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 232 [9] B. A. Munk, “Frequency Selective Surfaces, Theory and Design,” John Wiley & Sons, Inc., New York, 1999. [10] D. Lee, S. Chakravarty, and R. Mittra, “Design of Dual Band Radomes for High Off-normal Incidence Using Frequency Selective Surfaces Embedded in Dielectric Media,” Electronics Lett., vol. 36, pp. 1551-1552, Aug 2000. [11] V. D. Agrawal and W. A. Imbriale, “Design of Dichroic Cassegrain Subreflector,” IEEE Trans. Antennas Propagat., vol. 27, pp. 466-473, 1979. [12] W. C. Chew, “Waves and Fields in Inhomogeneous Media,” Van Nostrand Reinhold, New York, 1990. [13] E. G. Johnson and M. A. G. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A 12(5), pp. 1152-1160, 1995. [14] A. Ishimaru, “ Electromagnetic Wave Propagation, Radiation, and Scattering,” Prentice Hall, Englewood Cliffs, NJ, 1991. [15] R. Mittra, C. H. Chan and T. Cwik, “Techniques for analyzing frequency selective surfaces - A review,” Proc. IEEE, vol. 76, pp. 1593 - 1615, 1988. [16] T. Itoh, “Spectral domain immitance approach for dispersion characteristics of generalized printed transmission lines,” IEEE Trans, on Microwave Theory and Tech., vol. MTT-28, pp. 159-164, March 1975. [17] T. A. Cwik, “Scattering from General Periodic Screens,” Ph. D Dissertation, University o f Illinois, Urbana, IL, 1986. [18] T. Cwik and R. Mittra, “The Cascade Connection of Planar Periodic Surfaces and Lossy Dielectric Layers to Form an Arbitrary Periodic Screen,” IEEE Trans. Antennas Propagat., vol. 35, pp. 1397-1405, 1987. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 233 [19] S. Chakravarty and R. Mittra, “Design of Microwave Filters Using a Binary-Coded Genetic Algorithm,” Microwave and Optical Technology Letters, vol. 26, no. 3, pp. 162166, August 2000. [20] J. Michael Johnson and Yahya Rahmat-Samii, “An introduction to genetic algorithms, “Electromagnetic Optimization by Genetic Algorithms, Y. Rahmat-Samii and E. Michielssen, Ed., New York:John Wiley & Sons, pp. 1 - 27, 1999. [21] D. S. Wiele and E. Michielssen, “Genetic algorithm optimization applied to electromagnetics: A review,” IEEE Trans, on Antennas and Propagat., vol. 45, pp. 343 353, March 1997. [22] D. E. Goldberg and K. Deb, “A Comparative Analysis of Selection Schemes Used in Genetic Algorithms,” in Foundations of Genetic Algorithms, Ed. G. J. E. Rawlins, Morgan Kaufmann, pp. 69-93, 1991. [23] G. Sywerda, “Uniform Crossover in Genetic Algorithms,” in Proceedings o f the Third International Conference on Genetic Algorithms, Ed. J. D Schaffer, Morgan Kaufmann, Los Altos, CA, pp. 2-9, 1989. [24] K. A. De Jong, “An Analysis of the Behavior of a Class of Genetic Adaptive Systems,” Doctoral Dissertation, University of Michigan, Dissertation Abstracts International, 36(10), 5140B, University Microfilms No. 76-9381, 1975. [25] Kalyanmoy Deb and Samir Agrawal, “Understanding Interactions Among Genetic Algorithm Parameters,” in Foundations of Genetic Algorithms, Ed. Wolfgang Banzhaf and Colin Reeves, Morgan Kaufmann, San Francisco, CA, pp. 268-269, 1999. [26] J. J. Grefenstette, “Optimization of Control Paramaters for Genetic Algorithms,” IEEE Trans, on Systems, Man, and Cybernetics, vol. 16, no. 1, pp. 122-128, 1986. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 234 [27] D. E. Goldberg, K. Deb, J. H. Clark, “Genetic Algorithms, Noise, and Sizing of Populations,” Complex Systems, vol. 6, Complex System Publications, pp. 333-362. [28] D. E. Goldberg, “Sizing Populations for Serial and Parallel Genetic Algorithms,” in Proceedings o f the Third International Conference on Genetic Algorithms, Ed. J. D Schaffer, Morgan Kaufmann, Los Altos, CA, 1989. [29] C. A. Balanis, “Advanced Engineering Electromagnetics”, John Wiley & Sons, New York, 1992. [30] D. T. Paris and F. K. Hurd, “Basic Electromagnetic Theory”, McGraw-Hill, New York, 1969. [31] S. Chakravarty, R. Mittra, and N. R. Williams, “Synthesis of broadband microwave absorbers using dielectric and magnetic materials via the application of a genetic algorithm,” Submitted to I EE Proceedings, August 2000. [32] J. M. Johnson and Y. Rahmat Samii, “Genetic algorithms in electromagnetics,” Proc. Int. Symp. on IEEE Antennas and Propagation., Baltimore, MD, pp. 1480-1483, 1996. [33] S. Chakravarty, R. Mittra, and N. R. Williams, “Application of a Genetic Algorithm to the Design of Broadband Microwave Absorbers Using Frequency Selective Surfaces Embedded in Dielectric Media,” accepted for publication in IEEE Trans. Microwave Theory and Tech. [34] J. D. Vacchione, “Techniques for Analyzing Planar, Periodic, Frequency Selective Surface Systems,” Ph. D. dissertation, University of Illinois, Urbana, Illinois, August 1990. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 235 [35] K. O. Merewether, “Spectral-domain Analysis of Finite Frequency Selective Surfaces,” Ph. D. dissertation, University o f Illinois, Urbana, Illinois, August 1989. [36] C. H. Tsao, “Spectral-domain approach for analyzing scattering from frequency selective surfaces,” Ph. D. dissertation, University of Illinois, Urbana, Illinois, 1981. [37] Simon Haykin, “Communication Systems,” John Wiley and Sons, Inc., 1978. [38] R. F Harrington, “Field Computation by Moment Methods,” Robert E. Krieger Publishing Company, Inc., Malabar, FL, 1968. [39] C. H. Chan, “Frequency Selective Surface and Grid Array,” John Wiley & Sons, Inc., New York, 1995. [40] Eugene F. Knott, John F. Scaeffer, Michael T. Tuley, “Radar Cross Section,” Artech House Inc. Dedham, MA, 1985. [41] G. H. Schennum, “Frequency Selective Surfaces for Multiple Frequency Antennas,” Microwave J., vol. 16, pp. 55-57, 1973. [42] T. K. Wu, “Four Band Frequency Selective Surface with Double-Square-Loop Patch Elements,” IEEE Trans. Antennas Propagat., vol. 42, pp. 1659-1663, 1994. [43] J. D. Vacchione, “Frequency Selective Surface and Grid Array,” T. K. Wu, Ed., John Wiley & Sons, Inc., New York, 1995. [44] S. Chakravarty, R. Mittra, and N. R. Williams, “Application of a binary coded genetic algorithm to the design of broadband microwave absorbers using multiple frequency selective surface screens buried in dielectrics,” Submitted to IEEE Trans. Antennas Propagat. June 2000. [45] M. L. Maudlin, “Maintaining diversity in genetic search”. Proceedings of the National Conference on Artificial Intelligence, AAAI-84, 1984. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 236 [46] M. Nowostawski and R. Poli, “Parallel genetic algorithm taxonomy”, Proceedings of the Third International Conference on Knowledge-Based Intelligent Information Engineering Systems, Adelaide, Australia, August 1999. [47] Pittsburgh Supercomputing Research Center, unpublished seminar notes, Pennsylvania State University, PA. [48] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat., vol. 14, pp. 302-307, 1966. [49] J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Computational Physics, vol. 114, pp. 185-200, 1994. [50] S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation o f FDTD lattices,” IEEE Trans. Antennas Propagat., vol. 44, pp. 1630-1639, 1996. [51] G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat., vol. 23, pp. 377-382, 1981. [52] Wen-Juinn Tsay and D. M. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Micraw. Guided Wave Lett., vol. 3, pp. 250252, August 1993. [53] P. Harms, R. Mittra, and Wai Ko, “Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures,” IEEE Trans. Antennas Propagat., vol. 42, pp. 1317-1324, September 1994. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 237 [54] D. S. Wiele and E. Michielssen, “ The use o f the domain decomposition algorithms exploiting model reduction for the design of frequency selective surfaces,” Computer Methods in Applied Mechanics and Engineering, vol. 186, pp. 439-458, 2000. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VITA Sourav Chakravarty was bom in West Bengal, India on the 14th of April 1971. He enrolled in the Electronics and Communications program at Regional Engineering College, Kurukshetra, India in 1988 and received his B. Tech. degree in May 1992. From 1992 to 1995 he was employed as an Antenna Design Engineer at Superline Microwave Pvt. Ltd., at Bangalore, India. He received his M. E. degree in Electronics and Telecommunication in December 1997 at Jadavpur University. He began his doctoral studies in Electrical Engineering at the Pennsylvania State University in fall of 1997 and has held a research assistantship at the Electromagnetic Communication Laboratory at Penn State. His current research interests are in the area of antennas, computational electromagnetics with an emphasis on stochastic optimization techniques, numerical methods, and the finite difference time domain (FDTD) technique. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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