# Design of guided mode resonant filters at microwave and millimeter wave frequencies using genetic algorithms

код для вставкиСкачатьTHE CATHOLIC UNIVERSITY OF AMERICA Design o f Guided Mode Resonant Filters at Microwave and Millimeter Wave Frequencies Using Genetic Algorithms A DISSERTATION Submitted to the Faculty o f the Department of Mechanical Engineering School o f Engineering O f The Catholic University o f America In Partial Fulfillment o f the Requirements For the Degree Doctor o f Philosophy © Copyright All Rights Reserved By Joseph R. Krycia. Washington, D. C. 2005 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3169861 Copyright 2005 by Krycia, Joseph R. All rights reserved. INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3169861 Copyright 2005 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest 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. Design o f Guided Mode Resonant Filters at Microwave and Millimeter Wave Frequencies Using Genetic Algorithms Joseph R. Krycia, Ph.D. Director: Mark S. Mirotznik, Ph.D. For decades engineers have exploited the structural properties o f composites to develop material systems with desirable mechanical properties. Only recently have investigators applied similar concepts towards the development o f composite materials with "designer" electromagnetic properties. These designer materials are called metamaterials and are generally defined as a composite o f two or more materials producing an "effective" electromagnetic material with desirable properties. Examples o f metamaterials includes the very popular research topics o f photonic crystals and left handed materials. One application o f metamaterials that is o f particular interest is the design of frequency selective materials or surfaces (FSS). An FSS is in general, a single material or material structure that transmits (or reflects) only specific frequencies. Most FSS are designed using a periodic array of metallic patterns (e.g. dipoles, crosses, patches ...). Analytical as well as numerical algorithms have been developed for producing FSS structures based on the size and shape o f the metallic arrays. For very narrow band applications, however, losses in the metal surfaces can be significant enough to necessitate the search for a different FSS design methodology. Moreover, for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. applications that have stringent out-of-band signature requirements it may be necessary to reduce the amount of metal in the FSS structure. As a result, an all dielectric FSS is o f interest. Fortunately, a design approach has been developed using the concepts o f metamaterials to produce all dielectric FSS structures. This approach, called the Guided Mode Resonant Filter (GMRF) method, constructs a periodic pattern o f wavelength scale dielectrics o f different permittivity. The overall goal o f this project was to study the application o f GMRF at lower frequencies (i.e. microwave and millimeter wave hands) and large scale applications (e.g. radomes) requiring in some cases 2D GMRF. To this end, the following specific goals were accomplished; (1) Rigorous ID and 2D analysis codes were developed using the rigorous coupled wave (RCW) method. (2) Design algorithms based on genetic algorithms were developed for the synthesis o f ID and 2D GMRF transmission filters. (3) Specific designs requiring only a single low dielectric material and air were demonstrated. (4) Specific designs with relatively low dielectric constant materials were demonstrated. (5) Finite element analysis was used to validate ID and 2D GMRF designs. (6) Experimental test samples were fabricated at both the microwave and millimeter wave frequency band. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This dissertation by Joseph R. Krycia fulfills the dissertation requirements for the doctoral degree in Mechanical Engineering approved by Mark S. Mirotznik, Ph.D., as Director, and by John J. McCoy, Sc.D., Steven J. Russell, Ph.D., and John A. Judge, Ph.D. as Readers. Mark S. Mirotznik, PhD, Director John J. McCoy, Sc.D., Reader ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To my beautiful and loving wife Kristin Krycia, without her endless patience through this long journey I would not have finished. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C on ten ts List o f Figures vi 1 Introduction 1.0.1 M otivation................................................................................. ....................................... 1.0.2 Problem Statement . 1 1 4 1.0.3 1.0.4 Literature R eview .................. New C ontributions ............................................. 5 8 1.0.5 Overview of this dissertation 9 .......................................... 2 A nalytical M ethods: R igorous C oupled W ave (R C W ) A nalysis 2.1 Overview of RCW T h eo ry . 2.2 Geometry of 2D Grating Stack.............................................................. 2.3 Fields in the Incident and Exit regions . .......... 2.3.1 Definition of Fields in the Incident Region ................. 2.3.2 Definition of Fields in the Exit Region .................... 2.4 Fields in the Grating and Homogeneous la y ers.............. 2.5 Boundary C onditions ................. 2.6 Formulation of Matrix Equations ................ 2.7 Code Development and Validation. ............................ 11 11 . 12 14 14 16 17 22 25 29 3 A nalytical M ethods: Itera tive D esign o f G uided M ode R eso nant F ilters 30 3.1 Design Approach Considerations and Assumptions . . . . . . . . . . . 32 3.2 Iterative Design Approach . .............................. 34 3.2.1 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2.2 Direct Search 39 .............. .............................................. ... . 3.2.3 GMRF Design Algorithm ............................. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 4 5 A n aly tical R esu lts 4.1 One-Dimensional and Two-Dimensional GMRF Designs at Microwave ............... and Millimeter Wave Frequencies 4.1.1 E xam ple#!: One-dimensional GMRF Band pass Filter Design at 14GHz' . ..................... 4.1.2 Example #2: One-dimensional GMRF Band pass Filter Design at 38 G H z .................. 4.1.3 Example #3: Two-dimensional GMRF Band pass Filter De sign at 36 GHz and 42GHz . . . . . . . . . . . . . . . . . . . . 45 E x p e rim e n ta l M eth o d s 5.1 T est. Article Fabrication ............. 5.2 Description of Measurements ............................ 5.2.1 University of Delaware Millimeter Wave M easurement............ 5.2.2 SPAWAR Systems Center Dual Anechoic Chamber Spatially 69 69 71 73 Averaged Transmission Microwave Measurement . . . . . . . . 6 E x p e rim e n ta l R e su lts 48 47 53 63 76 81 6.1 Experimental Results: Microwave Guided Mode Resonant Filter at 14 G H z ........................................ Experimental Results: Millimeter Wave Guided Mode Resonant Filter 82 6.2 at 36 GHz 85 ............... 7 D iscussion a n d C onclusions 89 7.0.1 7.0.2 7.0.3 Significant Accomplishments ........................................ Theoretical Discussion . . . . . . . . . . . . . . . . . . . . . . Experimental D iscussion................. 90 91 93 7.0.4 7.0.5 Future Work .................... C onclusions.......................................... 94 96 B ibliography 98 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L ist o f Figures 2.1 Generic two-dimensional grating structure ..................................... ... . 13 3.1 3.2 3.3 GMRF design and evaluation process . . . . . . . . . . . . . . . . . . Generation of children in next generation for genetic algorithm . . . . Mesh pattern for direct search ................................................................. 31 38 41 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 RCW predcition of 14 GHz design performance .................................. Objective function evaluation for 14GHz design ............ Dimensioned schematic of 14 GHz design ............................................ Comparison of design with HFSS at 14 GHz ..................................... 36 GHz Design . . .................. Objective function evalaution by iteration .................. Dimensioned schematic of 36 GHz design ...................... Comparison of Design with HFSS predicition at 36G H z............... Loss sensitivity comparison........................................... ... . .................. Incident angle sensitivity comparison ........................ Two Dimensional dual pass band design ............... Dimensioned schematic of two-dimensional 36GHz and 39GHz filter . Comparison of 2 dimensional dual band design with HFSS . . . . . . 49 50 51 52 55 56 57 58 60 62 65 66 67 .................................. Conceptual experimental setup Schematic of University of Delaware experiemtal setup . . . . . . . . University of Delaware millimeter wave meaurement s y s t e m ............ Schematic of SPAWAR measurement facility .................. Inside SPAWAR chamber looking at the back of the horn toward the test sample ............... 5.6 Inside SPAWAR chamber showing the aperature for tranmission through the GMRF .................. 72 74 75 77 5.1 5.2 5.3 5.4 5.5 6.1 Comparison of the predicted and measure data for the 14 GHz guided mode resonant filter. ................... vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 79 83 6.2 Test article within aperature surrounded by radar absorbing materia! 6.3 Comparison, of the measured vs the predicted data for the 36 GHz guided mode resonant filter . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Image on the left shows the amplitude across the lens, the image on the right shows the phase ............... vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 86 87 A cknow ledgem ents There are so many pepole I wish to thank for their support and encouragement. First, I would like to thanks my advisor and friend, Dr. Mark Mirotznik for not only his invaluable guidance, but for Ms persistent faith in my abilities. I would also like to thank Dr. John Meloling and David Hurdsman at SPAWAR Systems Center and Dr. Dennis Prather for their expertise in measurement systems. I would like to thank my parents for stressing the importance of a good education. Most of all, I would like to thank my family. My wife Kristin who has been with me through this entire journey, my daughter Erin who joined us at the very beginning, my son Nichloas who joined us during the journey, and my daughter Juliette who came to us near the end. I can not express enough gratitude for their scarifices of time and the encouragement they gave me. Thank You. viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 1 In trod u ction 1.0.1 M o tiv atio n For decades engineers have exploited the structural properties of composites to develop material systems with desirable mechanical properties. [1] Only recently have investigators applied similar concepts towards the development of composite materi als with “designer” electromagnetic properties. In fact, this idea of ’'engineering" the electromagnetic properties of composite structures has become so popular that the research community has coined the term "metamaterials" to refer to such material systems. A metamaterial is generally defined as a composite of two or more materials producing an “effective” electromagnetic material with desirable properties. This rel atively recent interest in metamaterials is primarily due to advances in both efficient computer design codes and the availability of state-of-the-art micro-fabrication facili- 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 ties. Examples of metamaterials includes the very popular research topics of photonic crystals and left handed materials. These "man made" composites exhibit very un usual electromagnetic properties not achievable with naturally found materials. One application .of metamaferiais that is of particular interest is the design of frequency selective materials or surfaces (FSS). [2]An FSS is in general, a single material or material structure that transmits (or reflects) only specific frequencies. Most FSS are designed using a periodic array of metallic patterns (e.g. dipoles, crosses, patches . . . ) . Analytical as well as numerical algorithms have been developed for producing FSS structures based on the size and shape of the metallic arrays. For very narrow band applications, however, losses in the metal surfaces can be significant enough to necessitate the search for a different FSS design methodology. Moreover, for applica tions that have stringent out-of-band signature requirements it may be necessary to reduce the amount of metal in the FSS structure. As a result, an all dielectric FSS is of interest. Fortunately, a design approach has been developed using the concepts of metamaterials to produce all dielectric FSS structures. This approach, called the Guided Mode Resonant Filter (GMRF) method, constructs a periodic pattern of wavelength scale dielectrics of different permittivity. Due to the periodic nature of the grating structure the diffracted reflection and transmission orders are a result of the interactions of the incident energy and the resonant modes of the grating. At the resonant frequency the grating structure can exhibit very strong transmission and reflection coefficients approaching unity for a narrow frequency band. This combina Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tion of th e efficient reflection and transmission properties coupled with the frequency selectivity can provide very efficient pass band and stop band filters. Additionally, since the GMRF design does not employ any metallic structures it can be designed with good out-of-band signature control as well as a minimal amount of loss. Since their inception nearly all applications of GMRF have focused on optical and infrared wavelength scale devices. In fact, GMRF have been commercially employed as effi cient dielectric ,mirrors in laser systems and other optical components. Additionally, most investigators have concentrated on one-dimensional GMRF since they can pre cisely control the state of polarization in their applications and thus do not require complicated 2D designs. The overall goal of this project was to study the application of GMRF at lower frequencies (i.e. microwave and millimeter wave bands) and large scale applications (e.g. radomes) requiring in some cases 2D GMRF. To this end, the following specific goals were accomplished; • Rigorous ID and 2D analysis codes were developed using the rigorous cou pled wave (RCW) method. « Design algorithms based on genetic algorithms were developed for the syn thesis of ID and 2D GMRF' transmission filters. • Specific designs requiring only a single low dielectric material and air were demonstrated. By using only a single dielectric material, other than air, the fabri cation is considerably simplified. • Specific designs with relatively low dielectric constant materials were demon Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. strated. This is important since for applications requiring large areas at microwave and millimeter wave frequencies it Is difficult to find inexpensive high dielectric con stant materials. • Finite element analysis was used to validate ID and 2D GMRF designs. • Experimental test samples were fabricated at both the microwave and mil limeter wave frequency bands. • Experimental validation was conducted for both microwave and millimeter wave test samples. The results, presented in detail later in this thesis, will show that the combination of rigorous coupled wave code and a well thought out iterative design algorithm can be used to efficiently synthesize GMRF at microwave and millimeter wave frequencies. I will also show that these designs can be fabricated using common materials and validated both numerically and experimentally. 1.0.2 Problem Statem ent All-dielectric narrow-band frequency selective surfaces (FSS) are useful for a num ber of applications of interest to security and defense. The design of such structures can be challenging and for most applications requires the development of rigorous computational electromagnetic analysis and synthesis algorithms. The goal of this work was to develop a methodology for the practical design of all-dielectric narrow Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 band FSS at microwave and millimeter wave frequencies. To this end, I developed a methodology for the design of ID and 2D all-dielectric FSS structures based on the concept of guided mode resonant filters (GMRF), Specifically, I developed an imple mentation of the Rigorous Coupled Wave (RCW) method in MATLAB that can be used for multiple arbitrary layers in one or two dimensions. I also developed and applied an optimization techniques based on a genetic algorithm and direct search to find candidate designs. I numerically validated the design algorithm using a finite element model. Samples designs were fabricated and experimentally measured as a further validation. 1.0.3 Literature R eview A large amount of research has been conducted in frequency selective surfaces (FSS). At microwave and millimeter wave frequencies this research had been focused on the development of metallic elements that create the desired FSS behavior. [2] Although the periodic metal patch FSS technology is maturing to the point in which there are fielded applications, there are a couple of shortfalls with the metallic element approach. The losses in the metal patches can often create, a wider than desired pass band, consequently very narrow pass bands can be difficult to achieve. Also, the out of band characteristics of the metallic structure can create undesirable scattering from the surface at frequencies removed from the desired pass band. Specifically with finite sized metallic FSS arrays, there are grating lobes at higher frequencies Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and propagation of surface wav® at lower frequencies that contribute to the out of band scattering. [3] An alternative to the traditional metallic element FSS, is an all dielectric option called a .guided mode resonant filter (GMRF). Rather than relying upon.the periodic metallic patterns, GMRFs rely upon a periodic modulation in the dielectric constant of the materials. When the period of this modulation is near the wavelength, the resonant modes of the grating couple with the diffracted and transmitted inodes of the grating. [4] [5] This resonant behavior is strongly dependent upon the frequency and consequently it can be used to design very efficient narrow band filters known as GMRF. [6] [7] GMRF theory has been developed to show their theoretical ability to produce high efficiency reflection and transmission filters. [8] [9] (10) [11] [12] [13] [14] [15] [18] [17] [18] [19] [20] Wang et. ai. had shown that the frequency response of these filters can be con trolled by varying the geometry of the grating structure. Lemach et, al. and others also showed that by dictating the geometric configuration of the filter, including pe riod, fill factor, material properties, and number of layers, the spectral response can be controlled.[21] [22] [23] [24][25] [26] The early GMRF work was all done for one dimensional gratings until Peng and Morris further developed the theory to two di mensional gratings and theoretically investigated the scattering properties and later foliwed by others. [46] [28] [29] Applications of GMRFs were considered for optical wavelengths with a primary focus upon using GMRF to replace homogeneous layers Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 thin film reflection filters. The advantages of the GMRFs was an increases frequency selectivity through the inherent efficient.and narrow filter properties.[30][31][32][33] Further work has been done to show the effectiveness of cascaded reflection filters. Magnusson et. al. showed the effect of cascading filters gave even lower side bands and even narrower reflection bands. [34] Norton et. a l conducted experimental work that investigated the effect of varying one of the grating feature dielectric constants and the resultant effect upon reflection efficiency and frequency. [35] It should be noted, however, th at the vast majority of this work has been done at optical frequen cies with few studies aimed at the millimeter and microwave frequency bands. The exception to this is a study done by Tibuleac et. al. that conducted experiments on multilayer gratings with high dielectric contrasts at microwave frequencies. [36] To the best of my knowledge no prior studies have been conducted on the design, fabrication or testing of GMRF at millimeter wave frequencies. For the analysis of GMRF, a number of effective analytical -method have been de veloped. Beyond typical numerical techniques such as method-of-moments (MOM), finite element methods (FEM), and the finite-difference time-domain (FDTD), the rigorous coupled wave (RCW) algorithm has been particularly useful for the solu tion of periodic planar structures such as GMRF.[37] [38] [39] It is considered rigorous because if directly applies Maxwell’s equations with no .approximations unlike the numerical techniques. RCW has been extended to three dimensions to analyze many classes of gratings th at include GMRF. [40] [41] [42] [43] Methods of increas- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 lag the stability of the method and increase its efficiency have been researched and validated. [44][45] [42] [47] [48] [49]. These previous RCW algorithms, particularly the work of Noponen et. al. [40] were extended to multiple dielectric layers in this study and applied towards the analysis of GMRF'. While a reasonable .amount of. work has been reported on the development of rigor ous analysis algorithms little research has been published on efficient iterative design algorithms. Most of the reported GMRF designs seem to have been .found through a "trial and error" design approach. The only reported use of modem optimization algorithms towards the design of GMRF was presented by Zufada et. al.[50]. Thus, I believe the work presented here on the use of a genetic algorithm/pattern search approach towards GMRF design is quite novel and fills a hole in the literature. 1.0.4 N ew Contributions As mentioned earlier the principal goals of this study were to (1) analyze the elec tromagnetic characteristics of GMRFs at microwave and millimeter wave frequencies, (2) develop a methodology to predict and design practical GMRFs, and (3) to develop and apply an experimental protocol to validate the analytical and numerical models applied in this study. Of specific interest in this research is the development of filters with single grating layers made from a low contrast dielectric. The advantage of a low contrast Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 dielectric is th at at the frequencies of interest high dielectric materials are difficult to find and work with in addition to being prohibitively expensive for practical designs. Major new work I present in this dissertation is: Novel iterative design algorithm for GMEF design based on a genetic algo rithm and direct pattern search Two measurement techniques for GMRF characterization at millimeter wave and microwave frequencies was applied Low dielectric contrast GMRF designs were obtained Experimental data for both microwave and millimeter wave GMEF designs was obtained and used to validate designs 1.0,5 Overview of this dissertation The organization of this dissertation is as follows. In Chapter 2, I discuss the theoretical models used to analyze the transmission properties of GMRFs. The primary method used was a rigorous coupled wave (RCW) technique. This method was used for both design and analysis of the GMRFs. A full three dimensional RCW method was developed to analyze two-dimensional gratings of finite thickness and a two-dimensional method was developed for the analysis and design of one-dimensional gratings. design. Chapter 3 addresses the iterative design methods developed for GMRF The iterative design scheme uses both a genetic algorithm and a pattern search method to find a design solution. Is Chapter 4 ,1 present analytical results for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. both microwave and millimeter GMRF designs. This chapter also provides numerical validation of these designs using the finite element method. In Chapter 5, I present the experimental methods. This includes a description of two distinct experimental methods used to characterize samples. Namely, a spatially averaged measurement method th at was used for the microwave experiments and a collimated lens system that was used for the millimeter wave experiments. In Chapter 6 ,1 present the results and analysis of three experiments. Finally, in Chapter 6, I present a discussion of the results, concluding remarks and a discussion of future research directions based on the results of this work. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 2 A n alytical M ethods: R igorous C oupled W ave (R C W ) A n alysis In this chapter I present the analytical method used to analyze GMRF. I give a detailed derivation of the Rigorous Coupled Wave analysis applied to a multilayer planar grating problem. I .also provide the formulation of the matrix equations I used for encoding in MATLAB. 2,1 O verview o f R C W T h e o ry Rigorous Coupled Wave.Analysis (RCWA) is a rigorous numerical .method for solving Maxwell’s equations for planar periodic structures. The term rigorous.is used to denote th at no hypothesis or simplification of Maxwell’s equations axe used. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1*2 In this dissertation, I developed RCWA codes for both one- and two-dimensional rectangular binary gratings that includes multiple grating/homogenous layers. G e o m e try o f 2D G ra tin g S ta c k 2.2 An example of a generic structure is shown in Figure 2.1. The generic two di mensional grating structure in Figure 2.1 contains an incident region, grating layer, a single homogeneous layer and exit region. It should be noted that in general the code can actually handle an arbitrary number of layers sandwiched between the inci dent and exit regions. However, for the designs developed during this study only a single grating and homogenous layer were needed. In the grating region, where 0 < z < h, the periodicity of the structure is given by A x and Ay in the x and y direction respectively. In this grating region, the permittivity is expressed as a doubly periodic function e(x,y). The incident and exit regions are semi-infinite isotropic dielectric media with permittivity of ei (z < 0) and €4 (z>h2). The unit polarization vector of the incident wave is denoted in terms of 6 , 6 , and ip. Here 9 denotes the elevation angle, 0 denotes the azimuthal angle, and ip denotes the angle between the E field vector and the incident plane defining the polarization. For my analysis, the incident and exit regions were set to free space. As an overview, in RCWA the fields inside all interior layers are expanded in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 z=oI J ■.Tl .'it ill J ! t U I >ij Z=h. ;Jl £4 i-tii'iii r." E-it ttxtkm Figure 2.1: Generic two-dimensional grating structure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. terms of a sum of spatial harmonics. This expansions can be expressed and solved as an eigenvalue problem for all of the harmonic orders. At the interfaces between each interior layer and at material interface separating the incident and exit regions (denoted in the Figure), electromagnetic boundary conditions are enforced. Once the all of the electric and. magnetic fields axe matched, a set of linear simultaneous equations result. The solution of. those equations is used to calculate the fields within the incident, exit and all interior regions. Based on those solutions the reflected and transmitted fields for all diffractive orders can be determined. It should be noted that since the goal of a GMRF device is to act as a narrow band filter that the generation of propagating diffractive orders, other than the zeroth order, is normally undesirable. Thus all the grating structures are normally subwavelength and all reflective and transmitted orders are evanescent with the exception of the zeroth order. In the next section the fields within each of the regions (i.e. incident, exit and grating layers) are defined. I then define the boundary conditions th at must be satisfied at the interface. 2.3 F ield s in th e In c id e n t and E x it region s 2.3.1 Definition of Fields in the Incident Region The incident field is assumed to be a monochromatic plane wave with unit ampli tude, a free-space wavelength of Xa, and arbitrary polarization. Mathematically this Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 is written as Emc(x, y, z) = «exp(—s k inc • r*) (2.1) The time dependence exp(iut) is assumed throughout this thesis and omitted everywhere. The wave vector in the incident region, k inc, is given by k inc — k()ti2 ^siu 9 cos "1“ sin 0 sin tpy ~f~cos 0 z) — oY,.x + (30y + r0z where n* is the index of refraction of the incident layer and kQ = (2.2) The polarization vector u may be written in terms of direction cosines as u — (cos ip cos 9 cos <p —sin ip sin o)x -f (cos ip cos 9 sin <p+ sinip cos <p)y — (cos ip sin 6)z (2.3) The unit polarization vector of the incident wave is shown in Figure 2.1 in terms of 8, p, and ip. The electric field in the incident region is the sum of the incident plane wave and all reflected orders including those that may be evanescent. Thus the total electric and magnetic field distributions within the incident region can be written in the form of Rayleigh expansions where the coefficient R mn denotes the reflection coefficient of the mn-th reflected orders Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 E j — E in c 4~ ^ ( R-rrm J k Ijnn " T j (2.4) m = —oo r = —oo oo oo — E in c 4" ^ ^ ^ ^ ^?mn®XP [ J (^ira® 4~ kynV j (2.5) m = —oo ra=—oo If/ = — Aync x J ^ rec w/i0 J ■ oo ^ oo (2.8) ^ ^ ^ ^ ^ i,mn ^ R jn n CXp | j k j,TOn * T J (2.7j ’X^ ° m = —oo r = —oo The wave vectors for the incident fields axe given by the following kxm =a m = a 0 + (2.8) **» = A » = & + ( ^ ) (2J) kz,mn = t'rrm = \ji™A ) 2 - G?m - /3^ *£mn = rVr», = ~ J \ / + 0m~ (»l*3o) if if (bi&0)2 > + 0m (n l ^ f < ° 4 + 0L (2.10) (2-U ) where Ax and Ay are the grating periods in the x and y directions respectively as shown in Figure 2.1. 2.3.2 Definition of F ie ld s in the Exit Region The electric and magnetic fields in the exit region is the sum of the mn-th trans mitted orders, T m n propagating out from the final grating layer. Similar to 2.4 it Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 can be expressed in terms of a Rayleigh expansion: OO ~Ee = OO ^ T mne x p ^ ~ j k E^m n - ( y - h2z ) ^ (2.12) m = —oo n = —oo oo oo 5 ] T mn exp { —j [k^nX + kyny + k f mn (z - h2)]} = £ m = —oo n ——oo ^ oo oo H where E /1 2 (2.13) ^ ~ ~ *y ^ ^ k OJHo■ m = —o o » = —oo E ,m n T rn„eXp ( j k E ,m n ' { T" h^Z) ^ (2.14) is the total thickness of the grating and homogeneous layers. The wave vectors for the exit fields are given by the following OLm f 2Ttm\ OLq + | J (2.15) vyn — f i n ~ fio + ( ? ) <“ Kmn = tmn = y («4&o)2 ~ d 2m ~ f i n if («***) > kf,mn = tOT.„ = - j y - (n4fco) if + fin ( n A ) 2 < ^ 4 + dn > (2.17) (2.18) In my numerical implementation of 2.1 and 2.12 the infinite series is truncated such that there are Lx orders in the x direction and Ly orders in the y direction for a total of Lt — Lx Ly orders. 2.4 F ield s in th e G ratin g a n d H o m o g en eo u s layers Inside the region 0 < z < h% shown in 2.1 all of the grating and homogeneous layers are contained. These fields are defined by Maxwell’s curl equations Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 V x E = iojji0H (2.19) V x ff - -iw e E (2.20) When these equations are projected upon a Cartesian coordinate system, they lead to a set of six coupled equations dEz dEx dz dEy dx dHz dy dHx dz dHv dx dE y dz dE z dx dEx dy dHy dz dH z dx dHx dy (2.21) ibJIlHy (2.22) iwfj,Hz (2.23) —iojeEx (2.24) —iuieEy (2.25) —iuieEz (2.26) To eliminate the z component we can solve 2.26 and 2.23 for E z and H z respec tively. *■ - - « - ( £ - £ ) Substituting Equations 2.27 and 2.28into Equations 2.21, 2.22, 2.24, and 2.25 results in the following set of four coupled partial differential equations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In deriving Equations 2.29, 2.30, 2.31, and 2.32 it was assumed that the material properties (e and y) can vary only in the x and y direction within any planar layer (i.e. uniform in z). If it is also assumed that the material layers are non-magnetic (i.e. y = y Q) then the equations above simplify to 5 -sr . j o « -'( 5 - f ) ) - m dz " » uiy0dy Vdx ,3.36, dy J In theECW formulation the materialproperties within thegrating are assumed to vary periodically in the x and y directionswith periodsof A* and Ay respectively. Due to the grating’s periodic structure the material properties can be expanded into a Fourier series as given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 p~ oo « x ,v ) q—oo = < . £ £ fp,,e)2' £ V 2'* » (2.37) p ~ —oo q= ~ o o 1 «*-*> - i q—oo •vt— q=oo "i.—^ * 3 T 5 . E p = —o o q = —oo (2.38) In addition, due to the presence of a doubly periodic medium the field distributions within the grating layer can be expanded, via Floquet, into a sum of harmonic inodes given by — E x(x, y. z) — E v(x, y, z) — H j x , y, z) _ Hy{x, y , z) — m—oo n=oo V V F J (amx+Pny+jz} / / ^ juarmnc yri=—oo n=~-oo m=oo n=oo EymneP(-amX+^nV+lz^ 77?==—oc n=—OO m=oo n=oo Hxmnerta™x+®nV+'yz'> i7i=—oon=— oo n=oo Hvmn^ ^ m=—oc n=—oo (2.39) (2.40) (2.41) 371=00 (2.42) Substitution of■Equations 2.39, 2.40, 2.41, and 2.42 into Equations. 2.33, 2.34, 2.35, and 2.36 performing all partial derivatives results in the following set of coupled linear equations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 9= L g P = Ia s SjJ€(f ) ,E xmn = ^ “) k H y mn S,m -p,n-q (p tp^ypq 0 q E Xpq) (2.43) ~~ ftqH xpq) (2.44) |3=—Ij£ £^——£jy p=Lx ■UJCajEymn = —fe H ^ “] ^ n — Qn p= -Lx p=Lx UJflolHxmn = -A:2 E Cro-p.M.-g {°hp-^ypq «=-£» q—Lv E Cm— jmx~-§ "FOcm M p » - P » ) (2.45) •p=x—Lx q——Ly p~£*x E E w p~~Lx g=—Ly - qExpq "t" Pn ipt-mEymn 0 n-Exmn) (2-46) Equations 2,43,2.44, 2.45, and 2.48 .are of the form of an eigenvalue problem. Note that the infinite summation of modes has been truncated in Equations 2.47, 2.48, 2.49, and 2.50 to 2Lt modes where L t = L xLy. The solution of the eigenvalue problem results in the propagation constants amplitudes E ^ a , Eymni, eigenmode I. -jl (i.e. eigenvalues) and corresponding complex and Hymrd (i.e. eigenvectors), for each distinct Once determined the fields within the grating layers are calculated as a superposition of each mode as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 2 Lt. E x(x ,y ,z) £ m=i>* {Ae> B ie-3 E E xrmd^ amlB+^ v)- (2.47) £ ?72.—~~Lx, ft— .2 L t E y(x,y, z) = m=L-r, £ M £ 1=1 ~ k *° 1£= 1 2Lt Hy{x, y, z) , £ (2.48) m = ~Lx n= ~Lv j m =Lx n=L v £ £ Hxmnlej(amX+0^ A 9 ) 2Lt Hx(x,y, z) n— !% fm + B ie ~ j'>l<'z ~h'! Ate ^ z + B i e - ^ z~k) 1 m - ~ L* n=—Ly 7Tl~L/x. —*-*lj £ £ m = —L z n = —L,. The only unknowns in Equations 2.47, 2.48, 2.49, and 2.50 are the complex coeffi cients At and B[. These coefficients are determined by applying boundary conditions at the interface between each layer. This is described in the next section. 2.5 B oundary C on d ition s Maxwell’s, equations, require that at the boundary between any two dielectric ma terials the tangential component of electric .and magnetic be continuous. For the geometry described in 2.1 the boundary condition is written mathematically at the grating boundary (z=0) as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 2Lt ■F R x m n — 'y ~F B i fsx p ^ iy g h i)jE & fjfg n x ( 2 .5 1 ) ~F B t ( 2 .5 2 ) e=i 2Lt U yS fj^ S fiQ -j- R y rn n = g h i j j E ym ,ni 1=1 (J3q V,z = TQoUy ) 8 m$)8 raO &n R gm n + F m nR ym n , 2Lt /■ftp k \ i€„, ^ Ai *' e=i {fo o U x — a o u z ) 8 m odno ( 2 .5 3 ) (2.54) ^m nR xm n &m R zm n ( 2 .5 5 ) 21, = fci / — y ^[A f - B i e x p ( ijehi))H ymni V e° *=i (2.56) A t z = hi, interface separating the grating region and homogenous region, the equations describing the boundary conditions are as follows (homogeneous layer goes from hi to h^) [note that the appended subscript h denotes values from the homogeneous layer] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 (2.57) y^A |?exp(i 7 ^ i ) + B t e x p (ijt (hi - h2)}Exmra G_-i K.----A. 2L t = J ^ [A a exp(i7a h 1) + B ih e x p ( - t j eh(hi - h2))j£a™ntt 8_1 K - X 2 Lt '^T{Aeexp(i'y£hi) -f B ee x p (ije(hx - h2)]Emini (2.58) (2.59) G_1 2Lt = T i A ehexp (i'ythhi) 4- B the x p i- i') th{hi - h2))]Eymnlh (2.60) 8_"i1 £— 2L t ]T [A f j?_-i £—A (2.61) exp(*7A) - B £e x p (ije(hi - h%)]Hxmni 21, = J 2 { A th e x p (ijmhi) - S a e x p (-’i7&(h1 - h2))\Hxmjdh f _IJ. 2 ^ [ A ^ e x p f ^ h i) - B ie x p ( ijg(hi - h2)]Hyrrml (2.62) (2.63) /?,_— _-jA K 2L t (2.64) - Bgh ex.p(-E feh(hi fc=i At z = h2, interface separating the homogenous region from the exit region, they are as follows 2L t y ^ )lAihexp(i'Yihh2) + B ih]Ex,mnih = Txmn e=i (2.65) ^ ■&=! 2Lt (2.66) fc\j V j k%f exp(yyift^2) A B tfjjE pm nlh = T-yrrvn 6Xp{yygfjh^j) -flxroraife — 0 nh^'z^n ^mnT'-ytrm (2.67) exp (ij(hh2) Hynrnih == ^rtvnXxmri ^mh^'zmn (2.68) £=i 2Lf €n0 i=i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 Note for any additional grating and/or homogenous layers, the boundary condi tions will take the form of the z = hi boundary conditions. 2.6 F o rm u la tio n o f M a trix E q u a tio n s By applying the 12 boundary conditions described in the following section (4 for each of the 3 material interfaces) to the electric and magnetic field distributions defined for each of the 4 regions (i.e. incident, homogenous, grating and exit regions) a large system of linear equations can be constructed. This system of equations when cast as a matrix problem can then be numerically solved for the unknown field The final matrix equation will have form distribution in each of the four regions. shown below. C2 0 0 P Cs c 4 0 0 Q Cr 2 -W i Di d Ds d4 d5 -ir3 -W 2 A 0 -W 4 B 0 Dq -W 5 - W 9 Dr Dg -W 7 -W 8 1 T J 0 0 0 0 Ft f 2 0 0 0 f 3 f 4 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.69) where, C i, C 2 , Cs and CAjare {2Lt x Li) submatrices derived from applying bound ary conditions 2.51 to 2.55, D \ —D$ and W \ —W% are {Lt x Li) submatrices derived from applying boundary conditions 2.57 to 2.63 and F ^ F ^ , F% and i^ a re {2Lt x L t) submatrices derived from applying boundary conditions 2.65 to 2.68. Also A, B , R and 5 are (2Lt x 1) vectors representing the unknown coefficients of each of the spa tial harmonics in the grating and homogenous layers respectively and P and Q are (Lt x 1 } excitation vectors. The values for each of the submatrices given in Equation 2.69 is provided below. Applying boundary conditions 2.51 and 2.55 with the substitution of Equations 2.47 and 2.50 results in Clnl rj $ n O cm f coIj f ^ o Uxnl---------Arol m + J i'mti ■n' G^nl ki i — Hxni — C3nl k JIK — Hyrd + | r, 1 (c r„ ^ Eynl (2.70) f'nm/ Exnl ( r™”+ if e ) aW 2l jp E" exp(i7#/ii) , Pna iri rp Z-'xnl "i & yn l (2.71) (2.72) ' C*4ni k\ j Hyrd "F j T-nm OZ Fxal (3n Oim Eyni exp(*7 ^ 1 ) (2.73) Applying boundary conditions 2.57 and 2.63 with the substitution of Equations Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 2.47 and 2.50 results in Di„i = E m lexp(i'yehi) D 2ni = E xnie x p ( - i^ £(hi D 3ni = Eyni exp(«7|fti) (2.74) h2)} e x p ( - ^ (fei - fe2)) (2.75) (2.76) (2.77) D 5ni = k ^ H ^ e M iJ e h i) (2-78) Deni = - k xf ^ H xrde K p (-il l (hl - /*,)) V ^0 (2.79) Dm; = k xf ^ H ynlexp(i'yeh l ) (2.80) V Aw = —k y ~ ^ H ynlexp(—i'yi (hi —h2)) (2.81) W Xni = E„d exp{i^th t) (2.82) W 2rd = E xn(&q>{-i'ye (hi - h^)} (2.83) W3nl = Dy„iexp(i7£.fe1) (2.84) W 4ni = D jp u e x p f-^ C ^ i-/la )) (2.85) Wsni- = k ^ H mdexp(i'yth l) (2.86) W'eni = - k x [ ^ H xru exp (—i'yt(h1 - h 2)) V £e (2.87) WWi = k J ^ H snlexpii-fth) (2.88) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 Applying boundary conditions 2.85 and 2.88 with the substitution of Equations 2.47 and 2.-50 results in Fo tj | Hxnl + — J o &nm Flnl — Finl = — k i f — H zn i + V €e fin \ fijZ m £^ x n l, . i '■nm. F$nl ~ . k i l — Hynl — (4 1 «nm e0 \ Flnl — -Jfe4 ynl (t I hn m \ j \ i , \ x? I -&xnl / / &xnl p Jvnwi Z / Eynl B2 +j ,• n J e-Ljyn tllTO/ W B„a„ 271 exp(i7£h2) (2.90) (2.91) , ^ *-*yn l ^nm Bnotm . E'ynl exp(?>/fh2) (2.92) (2.93) Applying boundary conditions 2.512.55 and 2.652.68 with the substitution of Equations 2.4 and 2.12 results in = a^ - n x - ^2r0 + o£\ ( 2r' % + Bquz JnS i aoBa -Uy - a Quz (2.94) (2.95) After substitution of Equations 2.70 through 2.95 into Equation 2.69 and solving the resultant matrix equation the reflected and transmitted diffracted orders can be determined by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 2 Lt R xm n — T , ' A (2.96) i= \ 2Lt Ryrnn — Txmn ~ -f- B ( GXp{i,y £ h i)^ E p mni *=i 2Lt Y ^ iR e v x p iiJ ih ) + Se)Esmnl e=i Uy$mQ$nQ (2.97) (2.98) 2L t Tyum = 2.7 exp(n'i^2) + s e)Eymni (2.99) C ode D evelop m en t and V alidation I implemented this numerical formulation of RCWA in MATLAB. The code I developed resulted in two separate MATLAB .m files, also known as MATLAB scripts. I had one file for the one dimensional GMRJFs and a second for the two dimensional GMRFs. The two dimensional .m file was a direct implementation of the numerical development I outlined above. The one dimensional .m file was a simplification that eliminated either the x or y components dependent upon the polarization of the incident field. Once I had the developed code, I used published GMKF results to verify the code. [46] [51] Both my one dimensional and two dimensional codes did a very good job of duplicating published results of GMEFs. In the next chapter I will describe the iterative design algorithm developed during the course of this project. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 3 A n alytical M ethods: Itera tiv e D esign o f G uided M ode R esonant F ilters In this chapter I will present the design methodology developed for the synthesis of GMRFs. Overall the design process is an iterative approach in which the ROW algorithm, presented in Chapter 2, is used to analyze various test structures until a desired response is found. Figure 3.1 shows the design process flow graphically. As an overview, an initial design was selected based upon wavelengths .and design constraints. The initial design was used as input to the optimization algorithm. This chapter describes the design assumptions, iterative design approach, and grating designs including predicted performance. The output of the design algorithm was 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 IVaiacg ! forGMRF r ---- y-j 'TffiSiVi.- 2W‘V : vxii A'idc?; ^jxari &V<vr»L Beslggt AlgaritiHti u 1afc:*Crr.fim»v. M«asaraiiejsS X s im e ru a } & £ x |i « r i c i e > i t s f ! V'aiTjlHBon Step *L Evaluate «bjeirtiv*fimctie»wi& RCW mode! for iuisial value Step #2: Apply genetic algorithm using RCW model to evaluate each child in the objective function to begin mixuntizatiou Step #& Apply direct search method to refiae andfm aike uaramizatuni of objective function Step M i Ensure fabrication constraints are met, i on e "with n-/v c o n s tr a in t. if not return t littU i Anai^<w Figure 3.1: GMRF design and evaluation process numerically evaluated not only with the RCW developed for this research, but also against a commercially available finite element code. The fabrication, experimental measurement and analysis will be covered in following chapters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 3.1 D esign A p p ro ach C o n sid e ra tio n s a n d A ssu m p tio n s Once the.RCW code was verified, I developed a methodology to design .a GMRF with a desired pass band response. The RCW7' code allowed me to change both the geometric and material property parameters of the GMRF including (1) grating pe riod, (2) grating fill factor, (3) grating thickness, (4) homogeneous layer thickness, (5) number of layers either grating or homogeneous, (6) grating layer(s) material prop erties, and (7) homogeneous material layer(s) properties. By appropriately changing these seven properties, I could completely control the response characteristics of the GMRF. Although the RCW code can handle the complete range of parameters and calculate the response, real world experimental constraints had to be considered such as fabrication capabilities, available materials, and measurement methods and capa bilities. The fabrication method used was a computer numerically controlled (CNC) industrial router. The CNC router provided the capability to precisely fabricate the lamellar grating structure within excellent tolerances, approximately 1 mil. With this fabrication method comes constraints on the ratio of the fill factor to grating thickness. If the fill factor was too high ( i.e. only a small amount of material in width to be machined away) and the grating was thick the grating could not be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 fabricated with this method. This is due to the constraint on the cutting bit profile that the total depth of cut could be no more than 3 times the width of the bit used. There is no machining constraint on the homogeneous layer other than a. reasonable thickness for handling, measurement, .and cost. The material selected for this experiment was REXOLITE®. REXOL1TE® is a thermoset rigid plastic produced by cross linking polystyrene with divinylbenzene. The resultant translucent plastic has several excellent properties for this experiment. It has a dielectric constant of 2.53 from virtually 0 GHz to 500 GHz with a very low loss factor. The low loss factor matches my material assumptions in the RCW code for a near lossless material and the dielectric of 2.53 large enough to design reasonably good GMRF. Additionally, this rigid plastic has excellent machining properties that allow the use of the CNC router with standard carbide tipped tools for fabrication. The final design constraint to consider is the measurement techniques and capabil ities. Two measurement methods at different frequencies were used to experimentally validate the designs. One technique was a collimated lens system for millimeter wave (32-50GHz) at the University of Delaware and the other was a sliding phase elimina tion system for microwave (6-18Ghz) at SPAWAR Systems Center in San Diego. The details of these experimental systems are described in Chapter 4. The constraints of these systems limited the frequency bands that the experimental GMRFs had to operate in. Given the constraints of fabrication, material, and measurement I designed two Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 one-dimensional GMRF test articles, one at microwave frequencies and one at mil limeter wave frequencies for fabrication. I also designed a two dimensional GMRF at millimeter wave frequencies in a material other than REXOLITE®. As de scribed later in this document, it was found to be very difficult to design a good two-dimensional GMRF in a low dielectric contrast material such as REXOLITE®. As a consequence, a design, was shown and validated numerically using the FEM for a material with a higher dielectric constraint than REXOLITE®, however it was not fabricated due to the expense of the high dielectric materials. For the designs that were fabricated I ensured that the aspect ratio of the profile fell within the fabrication constraints and that the overall layer thickness was a reasonable thickness that could be readily purchased. 3.2 Iterative D esign A pproach In Chapter 2 ,1 presented a method based on RCW for analyzing the transmission of a GMRF given an incident plane wave. This a straight forward function evaluation problem. The inverse problem, however, is of greater significance for filter design. The inverse problem is to find the GMRF parameters that produce, as closely as pos sible, the desired filter characteristics. This inverse problem can be easily approached as an iterative optimization calculation illustrated in Figure 3.1. Put simply, given Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 some desired output, mathematically described by an objective or cost function, I iteratively analyze test structures using the RCW algorithm until an acceptable de sign is found. Clearly, the brute force method, of testing all possible combinations of design parameters (i.e. all grating periods, fill factors, ...) is not practical due to the infinite number of perturbations. Thus, a more intelligent optimization algorithm must be identified and used. I found th at a combination of two techniques provided good designs in a short period of time. These were a genetic algorithm coupled with a direct search method. The genetic algorithm is run initially to find a subspace in the entire design space in which a good solution exits. The direct search is then used to refine the genetic algorithm search until it locates the final design. In the next sections the details are these algorithms are presented. 3.2.1 G enetic Algorithm The genetic algorithm is a method for solving optimization problems that is based on natural selection. This is the process that drives biological evolution. The genetic algorithm repeatedly modifies a population of individual evaluations of an objective function. At each step or generation, the genetic algorithm selects individuals from the current population to be parents and integrates their most attractive features to produce children for the next generation. Over successive generations, the population "evolves" towards an optimal solution. The initial population is randomly generated, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 while the succeeding generations evolve from the parent population. Each succeeding generation is referred to as the children of the previous generation. [52]. Occasionally, a parameter is randomly varied, referred to as a mutation, to prevent the algorithm from being, stuck in a local minimum. The following outlines ,the genetic algorithm process: The algorithm is initiated by creating a random initial population of test samples. A single test sample, for example, might be a vector of design parameters that includes the grating period, fill factor and grating thickness. Each of these design parameters is a single number in the test vector and is called a gene in genetic algorithm terminology. The algorithm then evolves a sequence of new populations of test samples, or generations. At each step, the algorithm uses the individuals in the current generation to create the next generation. To create the new generation, the algorithm perforins the following steps: (1) scores each member of the current population by evaluating its objective function value, (2) scales the objective function evaluations to convert them into a more usable range of fitness values, (3) selects parents based on their fitness scores, (4) generates children from the parents. Children are produced either by making random changes to a single parent - mutation - or by combining the vector entries of a pair of parents - crossover, and (5.) replaces the current population with the children to form the next generation. The algorithm stops when one of the stopping criteria is met, which include a stable best fitness value and/or exceeds the maximum number of allowable generations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To create the next generation at each step, the genetic algorithm uses the current population to create the children that make up the next generation. The algorithm selects a group of individuals in the current, population, called parents, who contribute their genes - the entries of their vectors - to their children. The algorithm selects individuals that have better fitness values as parents. The genetic algorithm creates three types of children for the next generation: (1) Elite children are the individuals in the current generation with the brat fitness values. These individuals automatically survive to the next generation, (2) Crossover children are created by combining the vectors of a pair of parents and (3) Mutation children are created by introducing random changes, or mutations, to a single parent. Crossover children are created by combining pairs of parents in the current population. At each coordinate of the child vector, the default crossover function randomly selects an entry, or gene, at the same coordinate from one of the two parents and assigns it to the child. Mutation children are created by randomly changing the genes of individual parents by adding a random vector from a Gaussian distribution to the parent. Figure 3.2shows the children of the initial population, that is, the population at the second generation, and indicates whether they are elite, crossover, or mutation children. My implementation of the genetic algorithm is sometimes referred to as a microgenetic algorithm. The micro prefix is added because the implementation I selected Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 Current Generation Next Generation < ! Crossover Child j Mutation Child / Figure 3.2: Generation of children in next generation for genetic algorithm Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 has both a reduced population size and limited number of generations. 1 implemented the algorithm to quickly evolve the solution to an area in the domain where a design solution was likely to exist, I then passed that final elite child to a direct search methods to rigorously search the design space for the best solution. 3.2.2 Direct Search Direct search is a method for solving optimization problems that does not require any information about the gradient of the objective function, it is strictly a numerical evaluation of the objective function at a discrete set of points. As opposed to more traditional optimization methods that use information about the gradient or higher derivatives to search for an optimal point, a direct search algorithm searches a set of points around the current point, looking for one where the value of the objective function is lower than the value at the current point. A direct search is useful to solve problems for which the objective function is not differentiable, or even continuous. The special class of direct search method I used is a pattern search algorithm. The pattern search algorithm computes a sequence of points that get closer and closer to the optimal point. At each step, the algorithm searches a set of points, called a mesh, around the current point - the best point computed at the previous step of the algorithm. The algorithm forms the mesh by adding the current point to a scalar multiple of a fixed set of vectors called a pattern. If the algorithm finds a point in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 the mesh that improves the objective function at the current point, the new point becomes the current point at the next step of the algorithm. Each set of points, the mesh, is called a poll. The points of the mesh for the current poll are selected by adding and subtracting the current mesh size to the set of variables that were the best point of the previous poll, see Figure 3.3. If the poll was successful, i.e. one of the new points had a better fitness value for the objective function, the new point becomes the center of the next mesh and the mesh size is expanded. If the poll is unsuccessful, the current best point is retained and the mesh size is reduced. This algorithm can be stopped in several ways including: mesh size below a threshold, a set maximum number of polls, and a threshold value of change for 2 successful polls. I choose to incorporate both the mesh size criteria and maximum number of polls. The maximum number of polls ensured that the computational time would not become unreasonably long, while the mesh size stopped the algorithm when the parameter changes were much smaller than the constraints previously identified. 3.2.3 G M RF D esign Algorithm The complete algorithm I used combined these two optimization techniques, ge netic algorithm and direct search. The genetic algorithm is used to initiate the search and was given the geometric constraints of the problem along with the algorithm con straints of a maximum number of generations. The best fitness values variables were Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 n u Poll Point initial Point Poll Point Pol! Point Poll Point J “L Figure 3.3: Mesh pattern for direct search Reproduced with permission of the copyright owner. Further reproduction prohibited without permission 42 then passed to the pattern search algorithm. The best fit results of the pattern search algorithm was then compared to the desired filter properties. The final piece of the design optimization problem was to develop a reliable objec tive function. The objective function for the transmission filter had to combine the following criteria of good narrow band pass characteristics at the desired frequency with good band stop characteristics away from the band pass frequency. The objec tive function was constructed to use the RCW formulation to calculate the transmis sion characteristics of a grating structure for a given frequency band. The frequency band was centered on the desired pass band frequency denoted as P(A)with a band width of j3w.. At the pass band frequencies the goal was to maximize transmission while minimizing the transmission throughout the out-of-band portion of the fre quency range, denoted as £(Aj) for the lower side band and S(XU) for the upper side band. This lead to an objective function th at calculated one transmission coefficient at the deign pass band frequency and one transmission coefficient at each of the out of band frequencies. (3.1) Where A is the design center frequency, j3w is the bandwidth of the pass band, A* Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 is the lower bound of the lower stop band stop band, and Xu is the upper bound of the upper stop band. Depending upon the bandwidth of the stop band regions above and below the pass band, this can be many frequency points. Given the nature of guided mode resonant filter’s characteristic of a narrow sharp passband, dense sampling in the stop band is required to ensure there is not a undesirable second pass band within the design range. This need to balance the requirement of one band pass evaluation point with many, typically numbering in the hundreds, stop band evaluation points leads to weighting the passband evaluation such that it has an equivalent impact on the objective function as the many stop band points. A ^ p. (3.2) The value of the weighting coefficient was determined through numerical experi mentation An initial guess was to have a value for W equal to the total number of lower and upper stop band points. This value gave equal weight to the pass band frequency relative to all of the stop band frequency. The equal weighting sometimes lead to a solution where the stop band was acceptable but the desired pass band was weak to nearly nonexistent. After conducting a series of numerical experiments, I found that a value for W that was 25% greater than the total number of stop band frequency points yielded consistently good results. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 la the next chapter I present analytical results for the design of GMRF using the genetic algorithm presented in this chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 4 A n alytical R esu lts In this chapter I present several analytical results of GMRF designed at microwave and millimeter wave frequencies using the iterative design algorithm described in the previous chapter. In addition to the RCW predictions I provide numerical validation by comparing the results to a commercial finite element program HFSS. I will follow this chapter with experimental results that provide further validation. 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 4.1 O ne-D im ensional a n d T w o -D im en sio n al G M R F D esig n s a t M icrow ave a n d M illim eter W ave Frequencies Gratings were designed for both microwave and millimeter wave frequencies. These particular designs were selected to account for the previously identified fabrication and material property constraints. The designs were achieved by using the de sign algorithm as detailed in the previous chapter, including both genetic and direct search algorithms. Two designs were done as one dimensional gratings and one de sign was done as a two dimensional grating. One-dimensional GMRF designs were achieved at both microwave frequencies and millimeter wave frequencies while the two-dimensional design was accomplished at millimeter wave frequencies. For all of my designs, I validate them with a separate numerical technique, namely, a commercially available Finite Element Method (FEM) code marketed by ANSOFT as High Frequency Structural Simulator (HFSS). FEM is a well understood numer ical technique that can be used to conduct rigorous electromagnetic analysis. In fact, the ANSOFT HFSS package is an industry leading standard for high frequency electromagnetic simulation codes. It is a frequency domain technique and therefore must be run at each individual frequency. One of the inputs that drive the accuracy is the size of elements used in the FEM mesh. The resonant structure of GMRFs Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 have characteristically rapidly changing fields within the structure which drive a small element size in the mesh. The small element size is needed to accurately capture the effects of those rapidly changing fields. 'This in turn creates a significantly long run time for the analysis - more elements require more memory and CPU time to solve. Convergence of the HFSS results was ensured by refining the mesh until the GMRF performance curves converged. In the next several sections I will provide three different GMRF designs that serve as examples of the analysis and design methods developed during the course of this project. 4.1.1 E xam ple # 1 : O ne-dim ensional G M R F B an d p ass F il te r Design at 14GHz In this section, I present the design of a one-dimensional GMRF’ with a passband at 14GHz to demonstrate a design a microwave frequencies. This design is done is REXOLITE with few other constraints on the design. I allowed the period, fill factor, and both grating and homogeneous layer depths to vary. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 D esign analysis for exam ple # 1 The objective of this example was to design a one-dimensional GMR transmission filter with a. center passband of 14 GHz and a fractional bandwidth of less than 1% . The initial design parameters chosen for this design were a grating period A = 2cm, fill factor =0.5, and a grating thickness of each layer was 1cm. The design range was given as: 1 < A < 4, 0.2 < fill factor < 0.8, and both the grating and homogeneous layer thicknesses 0.75 < A* < 3. The stop band was calculated at 125 frequency points above and 125 frequency points below the stop band frequency. Using the objective function 3.2 the weighting factor for the stop band frequency value was 325, or slightly more than 125% of the total number of stop band frequency points. I then ran through the algorithm as described in Chapter 3. As shown in Figure 4.1 an good design was achieved with a clear pass band at 14GHz and fractional bandwidth of much less than 1%. The initial guess was a rather inefficient design. This can be seen in Figure 4.2. The genetic algorithm portion of the design process produced a substantially better design while the direct search algorithm further refined the design. The initial guess shown as the top point on the y axis produced a value for the objective function of approximately 480. Upon completion of the genetic algorithm (shown in the graph as the drop from the initial point on the y axis to the next point at iteration Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 Transmission Designed 14 GHz P ass Band 1.35 1.4 Frequency 1.45 x 10 Figure 4.1: RCW predcition of 14 GHz design performance Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 Best Function Vitos: 235.1797 §00? m *23 « m > c © 130 200 0 100 200 300 €30 iteration Figure 4.2: Objective function evaluation for 14GHz design #1), the objective value functions was reduced to approximately 245. The direct search algorithm was then used to refine the design to a final value of 235. design was used to fabricate a test article. This Final values from the design algorithm were: A = 1.5, fill £actor=0.5, the thickness of the grating hi = 1.8cm, and thickness of the homogeneous layer h2 = 0.85cm. A schematic of the output design is shown Figure 4.3. It shows one-dimensional grating view with the dimensions used, for development of both the numerical valida tion model and the experimental test article. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 Example#! one-dimensional 14 GHz 0.75 0,75 — ■siiii 2.65 i i Profile view ABiM mmmmsfgreim eemiimisiem' Figure- 4.3: Dimensioned schematic of 14 GHz design Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 HFSS Validation of 14 GHz P ass Band w so £m as -6 j- -10 j — * — D esign Prediction I • - • Q- - H FSS Prediction -12 1.25 1.3 1.35 1.4 Frequency 1.45 1.5 1.55 Figure 4.4: Comparison of design with HFSS at 14 GHz H F S S v alid atio n for exam ple # 1 The results from this 14 GHz design as predicted by the RCW formulation were numerically validated by comparison to HPSS. As shown in Figure 4.4 the HFSS prediction clearly is in agreement with the RCW prediction. It should be noted, that when the predicted curve in Figure 4.4 is compared with the curve in Figure 4.1, there Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 is a noticeable difference in the magnitude of the passband of the RCW prediction. The reason for this difference is that the two RCW predictions are done at different frequency points. Fewer frequency points were predicted and plotted with HFSS than the RCW. This reduced set of points was used because the run time requirements of HFSS would rapidly become excessive with more points. Even with a reduced frequency point set, here 40 HFSS frequency points vs. 1001 RCW frequency points, the HFSS would run overnight to produce the result while the RCW would run in approximately 5 minutes. With this numerical validation, I found the 14GHz design to be acceptable for fabrication. 4.1.2 E xam ple # 2 : O ne-dim ensional G M R F B an d p ass F il te r D esign at 36 GHz This section presents the design of a transmission filter with a center passband of 36 GHz to demonstrate a millimeter wave filter. I put additional design constraints on the thickness of the material to better conform to economically available material. D esign analysis for ex am ple # 2 With 36 GHz as the center band frequency the initial values used were a A = 7mm and a fill factor =0.5. For the thickness of the grating and homogeneous layers I added additional constraints. The motive for these additional constraints were two fold. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. One was to apply additional constraints due to material availability and fabrication constraints. A total thickness of 19mm was used because this was the thickness of material available off the shelf. A grating depth of 6mm was used because this corresponded to the maximum depth of the smallest groove that could be removed by the CNC milling machine - the smallest bit available is 2mm and so the constraint on the depth was placed at 3 times the bit diameter, for a 6 mm maximum depth at the smallest feature size. The second motive was to test the design algorithm to see if an acceptable design could be found with only the period and fill factor as variables. The design range was given as: 5 < A < 9 and 0.2 < fil l factor < 0.8. The stop band was calculated at 125 frequency points above and 125 frequency points below the stop band frequency. Using the objective function 3.2 the weighting factor for the stop band frequency value was 325, or slightly more than 125% of the total number of stop band frequency points. I then ran through the algorithm as identified in Chapter 3. The result is shown in Figure 4.5. The result was an efficient response with a nearly symmetrical line shape below -2db. My initial guess for the 36GHz design was not much better than the 14GHz design with an initial value of nearly 382. In this case, however, the genetic algorithm reached its maximum number of generations before a significantly better design had been achieved, see Figure 4.6. The direct search algorithm took over and did an excellent job of refining the design as seen by reducing the value of the objective Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 Designed 36 GHz P ass Band ‘ "T Transmission -5 - -10 h -15 -20 - -25 L3.4 3.45 3.5 3.55 3.8 3.85 Frequency 3.? 3 .7 5 3.8 x1G1° Figure 4.5: 36 GHz Design Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 390 r Objective Function Evaluation 380 ^ 370 - 360 - 350 - 340 - 330 - 320 s- 10 20 30 40 50 Iteration Figure 4.6: Objective function evalaution by iteration function to 327.5. Final values from the design algorithm were: A = 7.8, fill factor=0.7, the thickness of the grating hi = 6mm, and thickness of the homogeneous layer = 13mm. A schematic of the output design is shown Figure 4.7. It shows one-dimensional grating, view with the dimensions used for development of both the numerical valida tion model and the experimental test article. H FSS validation for ex am ple # 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 Example #2 one-dimensional 36 GHz 5.45 •2.35 \ \ -— 5 , 4 5 - 4 - 2.35— i mm mm ■ ill ■aBBi mam ■SB! mmm. I j IBB mSm IS S rt’l ■Bi iiiiigii — f 8 t j 1 19 Ml 13 Pkii mew Profile view ANBimmmmis m e m mtSimetm's Figure 4.7: Dimensioned schematic of 38 GHz design Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 Transmission HFSS Validation of 36 GHz P ass Band -10 r -12 -14 h -16 — * — Design Prediction j HFSS Prediction i -18 -20 3.4 3.45 3 .5 3.55 3 .6 3.65 3.7 3.75 Frequency Figure 4.8: Comparison of Design with. HFSS predication at 3SGHz Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 Figure 4.8 shows the HFSS prediction has reasonably good agreement with the design value for the 36 GHz design. The location and magnitude of the passband were nearly identical. As with the 14 GHz design, I found this numerical validation to be acceptable and proceeded to fabricate the 36 GHz design. S en sitiv ity o f D esign to M aterial Loss For this design, I also conducted a numerical study with ECW on the effects of material loss. The objective was to investigate if even a small amount of material loss would have a significant effect on the filter’s performance. To this end, I varied the loss tangent of the material from 0 to 0.01. The loss tangent is defined as the ration of the imaginary component of the complex permittivity to the real component (i.e. loss tangent = A-). The results are shown in Figure 4.9. As can be seen in the data, even small amounts of material loss can have a noticeable impact on the magnitude of the filter performance while the location of the passband remains substantially unchanged. The pass band drops off rapidly with increasing loss. A very slight widening of the pass band can also be seen, but that effect is small compared to the impact on the magnitude. This effect is not too surprising. Since the GMRF response is a very narrow band resonance it would be very sensitive to any material loss. It should be noted that the loss tangent of REXOLITE is known to be approximately 0.006. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 L oss Sensitivity Comparison Loss Tan 0.01 -7.6 8 db c -10 0 1 E sro Loss Tan 0 .0 0 6 -9.4 5 db Loss Tan 0 .0 0 3 -12.18 db t— -15 No Loss -20 No Loss -20.76 db L oss Tan 0 ,0 0 3 — L oss Tan 0 .0 0 6 • — - L oss Tan 0.01 -25 3 .55 3.6 3 .65 Frequency 3 .7 Figure 4.9: Loss sensitivity comparison Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 .75 61 S en sitiv ity o f D esign to Sm all Changes in Incident A ngle For this design, I also conducted a numerical study with RCW on the effects of small changes in the incident angle measured. For all the designs it was assumed th at a plane wave is normally incident on the GMRF. The objective of this numerical experiment was to see if even small changes of incident angle off normal would have an impact on the filter’s.performance. To this end, I varied the incident angle from 0 (normal incidence) to 1.5 degrees off normal. The results are shown in Figure 4.10. As can be seen in the plot, the effect of incident angle is significantly more profound than the effect of material loss previously shown. The first striking difference is the separation of the single passband into 2 pass bands, one higher and one lower in frequency. This is seen on the 0.5 degree incident data, where a decrease in the magnitude can also be seen for both of the new pass bands. As the incident angle increases, the two pass bands move farther way from the original pass band. Clearly this example illustrates that GMRF that are designed for a given incident angle are unlikely to work at other incident angles. It also, however, provides a potential means of producing good calibration targets for applications that require precise system alignments. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 Angie Sensitivity Comparison s 0 D©Qf©© 0.5 Degree 1 Degree 1.5 Degree 0.5 Degree 15.1 db 36.79 GHz 1.5 Degrees -17.4 db 36.04 GHz 1.5 Degrees 18.63 db 37.11 GHz 1 Degree -19,4 db 36.21 GHz 0 Degree -23.8 db 36.44 GHz 0.5 Degree -22.3 db 36.36 GHz .............. .................. - J 1 j I 1 Degree -21.5 db 36.95 GHz -25 x 10 Figure 4.10: Incident angle sensitivity comparison Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 4,1.3 E xam ple # 3 '. T w o-dim ensional G M R F B an d p ass F il te r D esign a t 36 G H z and 42G H z In this section I will present the design analysis and numerical validation of a two-dimensional filter design with two band pass frequencies. The purpose of the two-dimensional filter is to have a polarization independent design and to show that multiple band passes are possible. It was determined that to achieve a good 2D design th at a material with a higher dielectric constant than REXOLITE was needed. D esign analysis for exam ple # 3 The objective of the two-dimensional design was to produce a transmission filter with center passbands of 36 GHz and 42 GHz and fractional bandwidths of less than 1% at each passband. Initial attempts at this design employed the same material as the one dimensional designs (i.e. REXOLITE) but it was found that for these structures that a dielectric contrast ratio higher than 2.5:1 was needed for an ac ceptable design. As a result, the dielectric constant of the material was included as an additional variable to the design algorithm. The initial values of this design were chosen as, periods in the x and y directions Az = Av = 6mm, fill factors in the x and y directions Sx —ffy—Q.S, initial dielectric constant e = 5.0, and a grating thickness of each layer was restricted to the same value as the 36 GHz and 40 GHz Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 designs. The grating depth of 6mm was still a valid constraint and the total thick ness was carried as still a standard stock quantity. The design range was given as: 4 < AXty < 9, 0.2 < fill factor < 0.8, and both the grating and homogeneous layer dielectric 4 < e < 7. I then ran through the algorithm as described in Chapter 3 and the results are shown in Figure 4.11. The design has a deep pass band at 36 GHz and a second good passband at 39 GHz. Both pass bands have notable side lobes higher in frequency from the pass band. A schematic of the output design is shown Figure 4.12. It shows the two- dimensional grating view with the dimensions used for development of both the nu merical validation model. H FSS v alid atio n for exam ple # 3 The comparison of the 14 GHz design as predicted by the RCW formulation with the HFSS prediction shows reasonably good agreement as shown In Figure 4.13. There is a good match in the relative magnitudes and locations of both of the pass bands. In this Chapter I presented three examples illustrating of the design of GMRF using the iterative design algorithm presented in Chapter 3. The results were val- R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 65 Designed 2 dimensional dual p assband of 36 GHz and 39 GHz -10 Transmission -1 5 -20 -2 5 b -30 -35 -40 3.4 3.5 3.6 3.8 Frequency 3.7 3.9 4.1 x 10 Figure 4.11: Two Dimensional dual pass band design R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 10 88 Example #3 two-dimensional 36 & 39 GHz - i«- 4.S- * . "H2.3j— :llllllll 1I lM ^ Wmm | -*|2.5|— LJ LJ U -------------------------------- 125 4.8 ■2.3- ----- ------------------------------ 1.25 □ j . 19 LJ LJ 13 1 i 1i P i j . view Profile view AH MmmMmtM m e m miliimieiem Figure 4.12: Dimensioned schematic of two-dimensional 36GHz and 39GHz filter R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 87 Designed 2 dimensional dual passband of 36 GHz and 39 GHz -10 $ £ -20 i/3 C H E -25 - -30 Design Prediction j -X- • HFSS Prediction ! -35 j- -40 3.4 3.5 3.6 3.7 3.8 Frequency 3.9 4 4.1 Figure 4.13: Comparison of 2 dimensional dual band design with HFSS R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. idated using an independent numerical algorithm (FEM). Additionally, for one of the designs numerical experiments were conducted to determine its sensitivity to ma terial less and incident angle. The results of these experiments indicate that the performance of GMRF are indeed very sensitive to both material noise and incident angle. In the next chapter I will present the experimental methods th at were used to. fabricate and characterize the examples presented in this chapter. R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. C hapter 5 E xp erim en tal M eth od s In this chapter I will discuss the experimental methods used during the course of this project. Specifically, I will provide a description of the methods used to fabricate test articles as well as provide a detailed description of two different mea surement techniques used to characterize the samples. Specifically, a collimating lens measurement technique for the millimeter wave designs, located at The University of Delaware, and a spatial averaging technique for the microwave designs, located at Space Warfare .(SPAWAR) Systems Center in San Diego, CA . 5.1 T e st A rticle F a b ric a tio n In this section I will describe the materials and methods used to fabricated 89 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 70 test samples. All test articles were fabricated out of REXOLITE®, a thermoset plastic material with both excellent machining qualities and good dielectric proper ties. REXOLITE® is easily worked with standard carbide tipped machine tools. For my fabrication I used a computer numerically controlled (CMC) router at The Catholic Universfty of America. The CMC router was able to mill the gratings into REXOLITE® with an accuracy of 1 mil (“0.025 mm). REXOLITE’s® dielectric properties are very consistent and well known. It has a dielectric constant of er =2.53 and a loss tangent of approximately 0.0066. These values are consistent across both of the frequency bands of interest in this dissertation. The REXOLITE® was pur chased in 12”xl2" squares which were of sufficient size for both measurement systems. The CNC router used to fabricate the test articles was a computer controlled three axis machine that had a 1 mil tolerance in fabrication. The geometry of the grating was inputted into the computer using a program called BOBCAD. Within this soft ware, the two dimensional structure of the grating was drawn using the BOBCAD interface. Using the geometric information and machine movement speeds deter mined by the REXOLITE®, the software automatically generated the machine code used to drive the CNC router. R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 71 5.2 D escrip tion o f M easurem ents Experiments were conducted at both microwave and millimeter wave frequen cies. The.microwave frequency measurements were conducted at the Space Warfare (SPAWAR) Systems Center in San Diego, CA and the millimeter wave frequency measurements at the University of Delaware. Department of Electrical Engineering. The two measurement approaches were different in both their operational frequency range as well as the details in their construction. Conceptually, however, both mear surement techniques are similar. The basic idea is that the transmission through a sample can be measured by simply placing a sample between two horns on either side of the test article. One horn is used to transmit energy and the other horn is used as a detector. A vector network analyzer is used to both supply energy to the transmit side and detect energy on the detector side. This basic concept is illustrated in the figure below Figure 5.1. However, the actual experiment proves to be much more complex than described in Figure 5.1. The resonance phenomena in the GMRF depends upon a flat phase front, i.e. a plane wave, incident upon the material. Typically available horns have a curved wave front at reasonable horn separation distances. In addition, the conceptually simple measurement set up is complicated by the finite size of the panel and interactions from within the room. The finite panel size introduces edge effects while the room can introduce additional scattering from other objects. R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. The two 72 HSPNetwwk Art*fys«r D C i T5*¥t«*»'l¥towa / Receiving horn Transmitting t i : \ horn x- REXOLITE Test sample Figure 5.1:. Conceptual experimental setup R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 73 measurement techniques take separate approaches to eliminating the edge effects and the other scattering from non-test article objects. The following two sub-sections will describe in detail the different measurement apparatus used as well as the calibration procedures. 5.2.1 U niversity o f Delaware M illim eter Wave M easurem ent The millimeter wave measurements were conducted at the University of Delaware. The experimental setup, shown in Figure 5.3, integrated a vector network analyzer (HP 8530) with two millimeter wave horns configured to radiate through a pair of microwave lenses. GHz. This system works well with frequencies ranging from 30-110 The test article was placed between the lenses where the beam was to have the maximum degree of collimation. The desired result is a nearly flat phase front beam impinging on the center of the test article. In this manner, the curved wave front was converted to a flat wave front by the lens and the beam width of the wave front was smaller than the test article. This achieved two of the goals of the measurement system: (1) a nearly incident plane wave, and (2) elimination of edge effects and scattering from within the room. The network analyzer was swept through the desired frequency range obtaining transmission, measurements at each frequency. In addition to measurements taken with the test article in place I also conducted measurement with no sample (i.e. free-space) to use as a calibration sample. Since the free-space measurement should, in theory, provide a transmission coefficient of R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 74 N«te*Ariaiyier x Horns with /" " Collimating Jens 1 k\! Test Sample Figure 5.2: Schematic of University of Delaware experiemtal setup 1.0 for all frequency it can be used to determine correction factors. Errors that are corrected using this type of calibration are; (1) finite beam widths of the horns, (2) frequency dependency of the horns, cables and connectors and (3) any polarization mismatch between the two horns. The Figure 5.3 shows the setup with a sample placed between the collimating lenses. 5.2.2 SPAW AR S ystem s C en ter D u al A necholc Cham ber Spa- R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 75 Network sna=> t3s" Test Article Receive Horn with Coiiimating iens i ransmit Horn with ji Coilimatmg lens ■ Figure 5.3: University of Delaware millimeter wave meaurement system R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 76 tially A veraged Transm ission M icrow ave M easurem en t A second set of measurements were conducted at microwave frequencies at the 3PAWAR Systems Center. This system employs the Dual Aaechoic Chamber Spa tially Averaged Transmission Measurement technique. This method has several significant differences from the UDEL system. The most important difference is the ability to translate the transmit and receive horns closer to and farther away from the test article and the anechoic chambers under computer control. By doing so spatial averaging can be used to remove a number of noise sources and other artifacts from the measurement. See Figure 5.4. The measurement process is controlled using a PC running LabView. Lab View is a data acquisition software package that interfaces with the HP 85 IOC Network An alyzer and the stepper motor th at moves the transmit horn. Both the transmit and receive horns are placed within a tapered anechoic chamber to minimize scattering from anything other than the te s t.article which is placed between the horns. One complete measurement is achieved by making separate measurements of the back ground, an empty test fixture, and the test article. Each background and test article measurement actually consists of multiple measurements and several distances to take advantage of spatial averaging. The receive horn is then moved and another mea surement is taken. This cycle is repeated until the desired number of measurements has been taken, normally 49. When the measurement cycle has been completed for each background and test article, the data is processed. The processing consists of R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 77 1 % l .1 1 HPtM nmit A t«Jp*r o c T» ¥ ^ C ¥ , direct path a / ! / transmitting horn f \ receiver translation ■receiving hero indirect patii Figure 5.4: Schematic of SPAWAR measurement facility R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 78 Figure 5.5: Inside SPAWAR chamber looking at the back of the horn toward the test sample dividing the test article measurements by the background measurements, scaling the data to the empty aperture. The scaling is done for every frequency at each position at which a measurement was made. The results of these divisions is then averaged, i.e. spatial averaging. [53] The advantage of this technique is to eliminate the scattering due to edges of the test article and other objects by averaging out the phase variations. This elimination occurs because the scaled direct transmission path energy measured through the filter R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 79 Figure 5.6: Inside SPAWAR chamber showing the aperature for tranmission through the GMRF R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 80 is unchanged and each distance, however, the indirect path energy, such as that from the edges, changes with the distance. The change in distance due to the indirect path has a corresponding phase shift that is different for every distance measured. When the results are averaged over a sufficient number of distances, the effects of the edges and other non-direct path scattering are effectively, averaged out. is illustrated graphically in Figure 5.4. This The result is a measurement that contains transmission data from the direct path only effectively eliminating the edge effects of the aperture. [54] [53] In the next chapter I will describe some experimental results obtained using the experimental methods described in this chapter. R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. C hapter 6 E xp erim en tal R esu lts In this chapter I present results from my experimental measurements. I compare the these experimental results to the analytical predictions designs from the RCW model and HFSS. These comparisons were conducted for both of the one dimensional GMRF designs. Unfortunately, I was not able to fabricate the two-dimensional design for experimental validation. Materials for this experiment were prohibitively expensive .and difficult to specify the exact dielectric constant required. Therefore the numerical validation with HFSS is the filial validation for the two-dimensional design. 6.1 E x p e rim e n ta l R e su lts: M icrow ave G uided M o d e 81 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 82 R esonant F ilter a t 14 G H z The 14 GHz microwave design was experimentally measured using the previously described dual anechoic chamber spatially averaged apparatus at SPAWAR Systems Center in San Diego, GA. A comparison of the measure results with the predicted design are shown in Figure 6.1. In this prediction the material loss (tangent = 0.0006) of REXOLITE® was included. As shown in Chapter 4 even small amounts have material loss can have a significant impact on the performance of the GMRF at its resonance frequency. As shown in the figure the RCW prediction are in good agreement with the measurement. The pass band is centered at 14 GHz for both the prediction and the experiment and the magnitudes are quite similar. The oscillations seen in the measured data are due most likely to noise in the measurement system. A broadening of the passband can also be seen in the measurement. The widening of the measured transmission notch can be attributed, I believe, to the finite aperture that was actually illuminated in the measurement- Figure 6.2 shows the illuminated aperture window with the test article in place. The theoretical predictions assumed that both the incident plane wave and the filter grating were infinite in two dimensions. Obviously neither the test article nor the plane wave are infinite in the experiment, but the plane wave is significantly larger than the actual illuminated area of the filter grating. In this case there are approximately 14 periods illuminated from corner to comer in the aperture. As can be seen in Figure 6.2, the R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 33 14 GHz Measured vs Predicted SPAWAR Systems Center I. f i n * I . . r'jUvtAlu fii* V II® | 18 i %n i ?• I®.-*. * a 5 ■Measured ■Predicted (w/ loss) -10 1.2 1.25 1.3 1.35 1.4 Frequency 1.45 1.5 1.55 1.6 x 10 10 Figure 6.1: Comparison of the predicted and measure data for the 14 GHz guided mode resonant filter.. R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 84 Ifi Figure 6.2: Test article within aperature surrounded by radar absorbing material aperture cover effectively eliminates the edges of the test article, but sacrifices the .size of the illuminated area. The spatial averaging techniques help reduce the effects of the edges of the aperture, however, the number of periods illuminated still is quite finite. The limited illuminated area has been shown to impact the performance of the GMEF. Bendickson et. ah and Boye and Kostuk investigated both the effects of finite grating sizes of GMRFs and finite incident beams. [55] [56] Both papers showed R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 85 that when the illuminated area of the grating is finite the response both the width and efficiency of the pass band are effected. Specifically they found the efficiency decreased as the size of the illuminated area decreased. In addition to the decreased efficiency, the pass band broadened in frequency. This is seen, in the data of the 14 GHz design measurement with the noted decrease in efficiency and the broadening of the passband in the low frequency. 6.2 E xp erim en tal R esu lts: M illim eter W ave G uided M ode R esonant F ilter at 36 G H z The 36 GHz design was experimentally measured in the millimeter wave labo ratory at the University Of Delaware in Newark, DE using the collimated beam system previously described. A comparison of the measured data with the predicted data is shown in Figure 6.3. Since the designed GMRF actually had an additional passband at a frequency that was also measured, I extended the frequencies range in the plot for comparison. As with the 14 GHz design, I added the actual REXOLITE® loss to the prediction. The design pass band as measured was slightly lower in frequency, but the second pass band at 34 GHz also had a good match. Measurement data in this system had some significant noise that could not be eliminated. For the main pass band the magnitude of the measured data was reduced by approximately 2 dB Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 36 GHz Measured vs Predicted University of Delaware .§ .1 E S t-5 -10 -12 Measured predicted (wI toss) -14 -16 3.3 3.4 3.5 i | 3.6 Frequency 3.7 3.8 x 10™ Figure 6.3: Comparison of the measured vs the predicted data for the 36 GHz guided mode resonant filter and was shifted down in frequency 0.17 GHz. For the second pass band, the magni tude was reduced by 1.5 dB and was shifted down in frequency 0.08 GHz. Given the large amount of noise in the system and the previously seen sensitivity in incident angle, I consider this a reasonably good match. The frequency shifts of the measurement may be traced to the experimental setup. The millimeter wave experimental setup used a collimating lens to produce a plane wave incident upon the test article. The collimating lens system was charac terized at Delaware using a near field probe scanned, over the lens to measure both amplitude and phase. To replicate a plane wave, the amplitude and phase should be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 Amplitude Phase ta f -100 -75 -50 -25 0 25 50 75 100 125 xjmmS .. ■ 1??25-1£» -75 -50 -25 0 25 50 75 100 125 x{mm) Figure 8.4: Image on the left shows the amplitude across the lens, the image on the fight shows the phase constant across the lens. Figure 6.4 shows the amplitude and phase characteristics of the collimating lens. Examining the images there is a constant amplitude as expected across the lens. However, there is a variation in phase across the lens system. This variation, or phase .tilt, varies nearly 380 degrees across the lens. This tilt in phase can be thought of as caused by a plane wave at a non-normal angle of incidence. The impact of this on the measurement can have.substantial effects, as demonstrated in Chapter 4, including a shift in the pass band center frequency and/or a decrease in the amplitude Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 of the passband. For the 36GHz design there appears to be a combination of these two phenomena. Additionally, much the same as the 14GHz design, there were a finite number of periods illuminated that lead to a decrease in the amplitude of the passband notch. The phase tilt of the lens can have the impact of moving the center frequency. This effect was seen in the sensitivity data presented earlier. However, even considering these experimental artifacts, the comparison between experiment and theory were reasonable good and, I believe, serve as conclusive validation of the GMRF design algorithm developed in this project. In the next chapter I will present a brief discussion and conclusion of the results presented here as well as some comments on potential fixture work. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 7 D iscu ssion and C onclusions The principal objective of this project was to develop a practical method of de signing guided mode resonant filters at microwave and millimeter wave frequencies. From a theoretical standpoint I developed rigorous analysis code based on the rigor ous coupled wave method. I also developed an iterative design algorithm for GMRF' synthesis using a combination of genetic algorithms and a direct search. These iter ative design algorithms allowed the design of guided mode resonant filters that met performance requirements within design constraints. Several examples using the al gorithms developed in this project were presented. These examples were numerically validated using a commercially available finite element package (HFSS). The exper imental research focused on developing methods for fabricating and characterizing GMRFs for transmission passband applications at microwave and millimeter wave frequencies. I demonstrated that GMRF at both microwave and millimeter wave 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 frequencies could be fabricated and characterized using two different measurement techniques. Although the two techniques were at different frequencies, 1 was able to compare and contrast, them for their suitability for the measurement of transmis sion guided mode resonant filters. In the next sections I will briefly outline the significant accomplishments and in the following two sections I will provide a brief discussion/summary of the major finding of this project from both a theoretical and experimental perspective. 7.0.1 Significant Accomplishments In the course of this dissertation I had several significant accomplishments: 1. I implemented a three-dimensional RCW code in MATLAB that can ana lyze one and two-dimensional GMRF with an arbitrary number of layers in an arbi trary order. 2. I developed a design algorithm for GMRF using genetic algorithms and direct search methods. 3. I developed three unique GMRF designs. One microwave design in one dimension and two millimeter wave designs, a one-dimensional design and a dual band pass two dimensional design. 4. I used two different measurement techniques for GMRF, the dual anechoic chamber spatial averaging system and a collimating lens system. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 5. I experimentally validated the two one dimensional designs, one in each of the different measurement systems. 6. I demonstrated simple filter designs of single grating layers with low con trast dielectrics (2.5:1). 7- I designed and measured a GMRF at millimeter wave, no literature exists in this frequency regime. 7.0.2 T h eo retical D iscussion With the use of REXOLITE® and air as the two contrasting media in the grating structure I was able to demonstrate a simple guided mode resonant filter using low dielectric contrast materials. This simple design consisted of using a single grating layer with a single homogeneous layer for a 2 layer structure. My unique contribution was to use these low contrast materials for applications in both the microwave and millimeter wave frequency regimes. There has only been a few published papers of research at microwave frequencies and none at millimeter wave frequencies. My research showed that REXOLITE® with a dielectric of 2.53 and air make an excellent material system for a grating. The advantage of a guided mode resonant filter with a modulated air and REXOLITE® grating layer on top of a homogeneous layer is that it is a quite straightforward to fabricate. The ability to use the a combination optimization algorithm of genetic algorithm and pattern search techniques demonstrated the ability to solve the backward design Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 problem for guided mode resonant filters in a reasonable amount of time. I used this algorithm to design three filters, one at microwave frequencies and two at millimeter wave frequencies. Thus algorithm was developed to find designs that were not only two layer designs, but also fell within the material fabrication constraints identified. The design algorithm provided the capability to use stock material sizes and standard machine tools and practices to build custom guided mode resonant transmission fil ters. The microwave design at 14 GHz demonstrated the ability to use the developed design algorithm for a transmission filter that met all of the assumption and con straint criteria. The 2 millimeter designs at 36 GHz and the two-dimensional design demonstrated the robustness and versatility of the design algorithm to tailor the filter performance to the required needs. All of the designs were in good agreement with the HFSS predicted performance which provide a numerical validation of the designs. The 36 GHz design illustrated the sensitivities of the GMRF design to material loss and incident angle. The effect of changing the incident angle even small amounts was quite profound. The center of the passband was shown to split into two nearby passbands that were shifted both up and down in frequency. increased as the incident angle increased. And the separation Adding loss made the magnitude of the pass band rapidly decrease to the point in which the filter was marginally effective. 7.0.3 E x p erim en tal Discussion Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The initial experimental setup seemed quite simple but turned out to be more challenging than anticipated. My original plan was to use a focused lens system for the microwave measurements but that proved to be a failure. The problem was the illuminated area from the focused lens was only approximately 1.5 inches at the frequency of interest. This illuminated too small of an area to measure the filter re sponse because too few periods of the structure were illuminated. Additional research showed there were two options, (1) a another more conventional collimating beam ap proach and (2) a novel spatial averaging approach. The University of Delaware had an appropriate collimating beam system at millimeter wave that I was able to use. This system proved to be a good experimental setup for measuring GMRFs. One of the few difficulties was aligning the test sample perfectly perpendicular to the uniform phase front of the collimated beam. The characteristic tilt to the phase front need to be accounted for in the physical orientation of the test articles. The SPAWAR dual anechoic chamber with spatial averaging was an excellent method for measuring the characteristics of the GMRF. This technique eliminated all of the problems including edge effects and background scattering while only slightly reducing the illuminated area. This reduced area results in a decreased measurement of the amplitude of the pass band filter. Overall the experimental data validated the predicted performance of the designed GMRFs. The dual chamber spatial averaging technique was demonstrated to an excellent new measurement technique to compliment the more traditional collimating Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 lens approach I used for the millimeter wave measurements. For both measurement techniques, the effects of finite size had an impact relative to the efficiency. This effect was apparent at 14 GHz and very pronounced in the 40 GHz data. There was .a slight degrade in the 36 GHz data but it was not nearly as significant. It was understandable to see this effect at 14 GHz because given the wavelength, the illuminated are was rather small, on the order of 6 wavelengths across. One hypothesis to explain the large efficiency decrease in the 40 GHz measurement is the very narrow nature of the predicted filter. This predicted response may be much more sensitive to a finite illuminated area than the more symmetric responses seen in the 14 GHz and 36 GHz design. 7.0.4 Future Work This work provides a good foundation for application of GMRFs at microwave and millimeter wave frequency bands. Further research with GMRF can fall into two cat egories: (1) advanced designs and design issues, and (2) experimental characterization improvements. Advanced designs with two-dimensional GMRF could include different shaped gratings. My design in this dissertation used periodic square elements for the grat ing. Round elements could be used to produce a GMRF with a response that is independent of the polarization orientation of the incident field. Also, a two di Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 mensional design could be developed that has different frequency response dependent upon the polarization and could achieve this using rectangular elements. Finally, the rectangular elements could be converted to ovals to introduce some polarization tolerance to the filter. For the experimental measurements, my data showed that the spatially averaging techniques is an excellent method to measure these filters. Extending this technique to millimeter wave would be beneficial. Another option for the millimeter wave mea surement is to carefully analyze and adjust the collimating lens system to eliminate the measurable phase tilt and reduce the noise in the system to a more acceptable level. One GMRF application of a is to serve as a calibration standard for other mea surement systems. Given the distinct frequency response that is directly dependent upon incident angle, the measured output of a well characterized GMRF could be used to align a test article within a collimating lens system. Additionally, a well characterized polarization dependent GMRF could be used to find the uniform phase plane of the system by examining the location and amplitudes of the transmission peaks and nulls. This frequency movement of the pass band with changes in the incident angle were shown in the 36GHz design. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 C onclusions 7.0.5 The goals of my research were: (1) to analyze the electromagnetic characteris tics of GMRFs, (2) define a methodology to predict and design practical GMRFs for fabrication, and (3) develop and apply an experimental protocol to validate the analytical and numerical models applied in this study. In this dissertation I showed the following: 1. A RCW method for one and two dimensional gratings with an arbitrary number of layers. The RCW I implement in MATLAB was validated both numeri cally and experimentally as an excellent method to predict GMRF performance. 2. I developed a novel design techniques using a genetic algorithm and direct search method for GMRFs. I was able to put arbitrary requirements on a GMRF and develop designs with this algorithm that were validated numerically and with experimentation. 3. I conducted two different types of measurements of GMRFs. At mi crowave I showed the use of a unique spatial averaging techniques to experimentally characterize the filters. And at millimeter wave I showed to utility of a collimating lens approach to the measurement. 4. I combined the analysis and design techniques with the demonstrated experimental protocol to show a validated design technique for GMRFs. Additionally, I contributed to the field of GMRFs by designing and developing efficient GMRF with minimal number of layers out of low contrast dielectric materials. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The unique combined genetic algorithm and direct search design method has shown that a simple design of one grating layer design utilizing only 2 different dielectric materials such as the 14 GHz design exists and can be designed. The two dimensional design I developed had 2 designed band passes with both of them nearly symmetrical in shape and were efficient. I added experimental data at microwave and millimeter frequencies for transmission filters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B ibliography [1] S. 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