application of a microgenetic algorithm (mga) to the ...callen/hi-z_surfaces/chakravarty... · mga...

284 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 50, NO. 3, MARCH 2002

Application of a Microgenetic Algorithm (MGA) tothe Design of Broad-Band Microwave Absorbers

Using Multiple Frequency Selective SurfaceScreens Buried in Dielectrics

Sourav Chakravarty, Raj Mittra, Life Fellow, IEEE, and Neil Rhodes Williams

Abstract—Over the years, frequency selective surfaces (FSSs)have found frequent use as radomes and spatial filters in bothcommercial and military applications. In the literature, theproblem of synthesizing broadband microwave absorbers usingmultilayered dielectrics through the application of genetic algo-rithms (GAs) have been dealt with successfully. Recently, spatialfilters employing multiple, freestanding, FSS screens have beensuccessfully designed by utilizing a domain-decomposed GA. Inthis paper, we present a procedure for synthesizing broadbandmicrowave absorbers by using multiple FSS screens buried ina dielectric composite. A binary coded microgenetic algorithm(MGA) is applied to optimize various parameters, viz., thethickness and relative permittivity of each dielectric layer; theFSS screen designs and materials; their - and -periodicities;and their placement within the dielectric composite. The resultis a multilayer composite that provides maximum absorption ofboth transverse electric (TE) and transverse magnetic (TM) wavessimultaneously for a prescribed range of frequencies and incidentangles. This technique automatically places an upper bound onthe total thickness of the composite. While a single FSS screen isanalyzed using the electric field integral equation (EFIE), multipleFSS screens are analyzed using the scattering matrix technique.

Index Terms—Frequency selective surfaces (FSSs), genetic algo-rithm, microwave absorbers, multilayered media, scattering ma-trices.

I. INTRODUCTION

GENETIC ALGORITHMS (GAs) are robust stochasticsearch methods modeled on the principles of natural

selection. The powerful heuristic of the GA, as an optimizer,is useful for solving complex combinatorial problems. It isparticularly effective in searching for near-global maxima indomains that are both multidimensional and multimodal. TheGA simultaneously processes a population of points in the opti-mization space, and uses stochastic operators to transition fromone generation of points to the next, resulting in a decreasedprobability of their being trapped in local extrema.

Manuscript received May 28, 2001.S. Chakravarty was with The Pennsylvania State University, University Park,

PA 16802-2707 USA. He is now with Intel Corporation, Hillsboro, OR 97124USA.

R, Mittra is with The Pennsylvania State University, University Park, PA16802-2707 USA (e-mail: [email protected]).

N. R. Williams was with The Pennsylvania State University, University Park,PA 16802-2707 USA. He is now with W. L. Gore & Associates, Advanced Elec-tromagnetic Products, Newark, DE 19714–9236 USA.

Publisher Item Identifier S 0018-926X(02)02615-7.

Numerical search techniques can be divided into two broadgroups, viz., local and global. They can be distinguished fromeach other by the fact that the local techniques produce resultsthat are highly dependent on the initial guesses, while theglobal methods are largely independent of these starting points.Local techniques are tightly coupled to the solution domain,resulting in fast convergences to local maxima. Furthermore,the tight coupling to the solution space also places constraintson the solution domain, such as differentiability, continuity,etc., which are difficult to handle. In contrast to the localtechniques, the global ones are loosely coupled to the solutiondomain and place very few constraints on it. This means thatthese techniques handle ill-behaved solution spaces efficientlyand are better equipped to deal with solution spaces thathave constrained parameters, discontinuities, large numberof dimensions and large number of potential local maxima.The GA-based combinatorial optimization technique offersseveral advantages over the existing approaches: i) it succeedsin designing broad-band microwave absorbers consisting ofonly a few layers, and therefore, almost always leads to aphysically-realizable structure; and ii) it is considerably lesscomplex to implement than gradient-based search procedures[4], [5].

In this paper, the GA is employed to solve a computation-ally-intensive design problem. In the past, GAs have met withmoderate success in solving these problems within a practicaltime frame. A design problem is typically categorized as com-putationally intensive when a single function evaluation takesa significant amount of computation time. The problem dealtwith in this paper partly involves the analysis of scattering fromfrequency selective surface (FSS) screens, which is achievedby using the Method of Moments (MoM). The main contrib-utors to the large computation time for a single function eval-uation are the fill and inversion times of the MoM matrix bothof which are dependent on the complexity of the FSS screendesign. The number of function evaluations in a single genera-tion depends on the population base used for the optimization.For conventional GAs, sizing the population is problem-specificand a strong function of the length and cardinality of the chro-mosome [6], [7]. For most optimization problems, the length ofthe chromosome is a function of the number of parameters to beoptimized, the individual parameter range, and the step size tobe implemented. Hence, for a multidimensional search space,

0018-926X/02$17.00 © 2002 IEEE

CHAKRAVARTY et al.: APPLICATION OF AN MGA TO THE DESIGN OF BROAD-BAND MICROWAVE ABSORBERS 285

Fig. 1. Multiple FSS screens embedded in dielectric composite.

a large population base and several generations are required toachieve optimal or near optimal results if the conventional GAis used, and this places a considerable burden on the computa-tional time and resources. One possible approach to solving thistype of problem is to employ a parallel implementation of theGA. However, in this work we avoid the complexity of the par-allel-GA implementation and solve the problem efficiently byusing a serially implemented version referred to as the microge-netic algorithm (MGA).

It is well known that conventional serial GAs perform poorlywith small population sizes due to insufficient information pro-cessing, and they converge prematurely to nonoptimal results.To circumvent this difficulty, a serially-implemented GA with asmall population base and efficient convergence properties is re-quired. Goldberg [8] has suggested that the key to success with

small population sizes is to use the MGA. We follow this sug-gestion and employ the above algorithm to optimize the problemat hand.

The MGA has two major advantages: i) small populationbase for each generation; and ii) it reaches near-optimal regionsquicker than the conventional GAs that deal with a large popula-tion base. The general choice of population size for conventionalGAs can range between 100 and 10000, while the MGAs typi-cally work with a population size between 5 and 50. Numericalexperiments show that using the MGA can decrease the compu-tational run time by 50%, even for the “worst-case” problemsfor the conventional GAs.

The individual FSS screens are analyzed by using theelectric field integral equation (EFIE) in the spectral domain, asdescribed in [9]. The integral equation is solved via Galerkin’s


TABLE IMGA PARAMETER SEARCH SPACE FORONE SUBCOMPOSITE

method applied directly in the spectral domain. The scatteringmatrix technique [9], [10] is used to analyze multiple FSSscreens, which has been shown to obviate the difficultiesassociated with full wave techniques without compromising onaccuracy, which is necessary for most engineering applications.

II. GENETIC ALGORITHM FORMULATION

Fig. 1 shows a multilayered multiscreen composite structurewhose parameters we wish to optimize, with a view to realizinga specified frequency response. The composite is divided intosubcomposites, each comprising(number of layers in thethsubcomposite) dielectric layers. The parameters for each sub-composite are generated separately by the MGA. Assume thatwe are given a set of different materials with frequencydependent permittivities . For any singlesubcomposite, the GA determines the following: i) material pa-rameters of layers; ii) design of the FSS cell element; iii) cellperiodicity of the FSS; iv) position of the FSS screen within thedielectric subcomposite; and v) the FSS screen material. For,such subcomposites, the same process is repeated such that the

Fig. 2. The FSS cell structure defined in terms of 1’s and 0’s.

combined subcomposites exhibit a low reflection coefficient fora prescribed set of frequencies and incidentangles , simultaneously for both the transverse


Fig. 3. The coordinate system used: (a) periodic FSS screen and (b) uniform plane wave propagating in thek direction incident on a freestanding FSS screen.

electric (TE) and transverse magnetic (TM) polarizations. In thecontext of the present problem, the magnitude of the largest re-flection coefficient is minimized for a set of angles, for both TEand TM polarizations, and for a selected band of frequencies.Hence, the fitness function can be written as

(1)

where is the material parameter of theth layer in the thsubcomposite, is the thickness of theth layer in the th sub-composite, is the unit cell design of the FSS screen em-bedded in theth subcomposite, and are the periodici-ties of the FSS screens in the- and -directions, respectively,for the th subcomposite, determines the placement of theFSS screen in theth subcomposite, and isthe reflection coefficient as a function of polarization, incidentangle and frequency.

The MGA operates on a coding of the parameter. The codedrepresentation of the coating consists of a sequence of bits thatcontain information regarding each parameter. Each parameteris represented by a string of bits, and the length of each stringis determined by the allowed range of real values and the dis-cretization step to be implemented. The entire composite can berepresented by the sequence, given in (2), and is referred toas a chromosome

(2)

where

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Next, the encoding of the various parameters optimized by theMGA is briefly discussed. The MGA chooses a dielectric layerto be either lossy or lossless. For the latter, the MGA is provideda range of values [1.03 (Styrofoam) to 6.0 (Lead glass orMica)] (see Table I) to select from, while for the former, theMGA chooses a material type from a given database containing

different lossy materials. The material choice forlayer in the th subcomposite is represented by a sequenceof bits as,

(13)

Similarly, in (4)–(6), is a sequence of bits representingthe thickness of layer in the th subcomposite, and and

are strings of length and representing the -and -periodicities of the th FSS screen, respectively. Also,in (7)–(12), the string consists of two bits representing theposition of the th FSS screen; and are strings of unitlength that represent the choice of selecting a lossy or losslessdielectric for the th layer in the th subcomposite and lossy orPEC material for theth FSS screen, respectively; and,and are strings of length and that represent thereal and imaginary parts of the surface impedance of theth FSSscreen. The formation of sequence is explained below inmore detail.

The MGA designs the FSS cell structure automatically. Thepart of the code analyzing the FSS screen embedded in dielectricmedia utilizes a 16 16 discretization (32 32 and 64 64 dis-cretizations can also be handled) of the periodic structure unitcell in the form of 1’s (ones) and 0’s (zeros) (see Fig. 2); the1’s correspond to PEC or lossy metal and the 0’s correspond tofree space. As explained next, the GA randomly generates this16 16 gridded structure filled with 1’s and 0’s.

The MGA considers each row in the FSS cell to be a param-eter. For each row, the MGA generates a random number be-tween 0 and , where is the number of columnsin the FSS cell matrix. The random numbers are converted tobinary format for each row. These binary numbers are com-bined into an array, which is ready to be analyzed by the FSS


code. The number of columns considered depends on the typeof symmetry introduced into the FSS screen geometry. The re-gions A, B and C in Fig. 2 represent the sections considered foreight-fold, four-fold, and two-fold symmetry, respectively. Thisreduces the effective number of MGA parameters needed to de-sign the FSS cell, (see Table I) resulting in efficient optimiza-tion. Thus, the FSS cell in theth subcomposite can be designedby encoding each row into a sequence of bits as

(14)

where is the number of rows considered as parameters by theMGA, and is the total number of rows in the FSS cellstructure.

The number of bits contained in the sequence represented by(2) is as shown as follows:

(15)

where all terms are as defined in (3)–(12).The GA, which optimally chooses each parameter, is an iter-

ative optimization procedure, which starts with a randomly se-lected population of potential solutions, and gradually evolvestoward improved solutions, via the application of the genetic op-erators. These genetic operators mimic the processes of procre-ation in nature. The GA begins with a large population, com-prising an aggregate of sequences, with each sequence (similarto the one represented in (2)) consisting of a randomly selectedstring of bits. It then proceeds to iteratively generate a new pop-ulation , derived from , by the application of selection,crossover, and mutation operators.

For this particular problem, a variant of the conventionalGA, called the MGA, is used. It has been shown that MGAsavoid premature convergence and show faster convergenceto the near-optimal region compared to the conventional GAfor multidimensional, multimodal problems [6], [11], [12].The MGA starts with a random and small population, whichevolves in a conventional GA fashion and converges after afew generations. At this point, keeping the best individual fromthe previously converged generations (elitist strategy), a newrandom population is chosen and the evolution process restarts.In our case, population convergence occurs when the differencein bits between the best and other individuals is less than 5%.

Among the many types of selection strategies that can beused to suit a particular application, the one used for the presentproblem is the tournament selection. In this method, a subpopu-lation of individuals is randomly chosen from the populationand made to compete on the basis of their fitness values. The in-dividual in the subpopulation with the highest fitness value winsthe tournament, and is thus selected. The remaining membersof the entire subpopulation are then put back into the generalpopulation and the process is repeated. This selection scheme ispreferred because it converges more rapidly and has a faster ex-ecution time compared to many other competing schemes [13].

Once a pair of individuals is selected as parents, the basiccrossover operator creates an offspring by recombining the

Fig. 4. A Single FSS screen embedded in dielectric layers.

chromosomes of its parents. The mutation operator is not uti-lized in our problem, i.e., we put . Uniform crossoveris preferred to single point crossover, as it has been found thatMGA convergence is faster with the uniform crossover [6],[11]. The value of is used. An elitist strategy[14] is also employed wherein the best individual from onegeneration is passed on to the next generation.

III. SINGLE FSS SCATTERING FORMULATION

In this section, we consider the formulation of scattering froma FSS in the spectral domain. As this subject has received ex-tensive treatment in the literature [9], [10], [15]–[17], and theobjective of this paper is to show a successful MGA implemen-tation using the FSS, only a few important equations will be in-cluded here. A MoM-based computer code has been employedto perform the electromagnetic simulations of the FSS screen.The numerical analysis follows the well-established procedureof solving the EFIE for the current distribution on perfectly con-ducting patches, derived by enforcing Floquet’s periodicity con-dition in an elementary cell. As shown in Fig. 3, we consider ascreen lying in the - plane with cell periodicities andalong the - and -directions, respectively. Then we can cast theEFIE in the form

(16)

where

discrete spectral Green's function.

where

Identity tensor

Equation (16) can now be solved by using Galerkin’s pro-cedure for the unknown currents that are expressed in terms ofsubdomain basis functions (rooftops)as

(17)

where are the unknown coefficients to be determined. Sub-stituting (17) into (16) and using as testing functions, (16)is transformed into a matrix equation [9], [10] that is solved to


TABLE IIPARAMETERS SELECTED BY THE MGA FOR THETWO CASES

TABLE IIIMEASUREDVALUES OF " AS A FUNCTION OF FREQUENCY

obtain the values of . The values of the induced currents areobtained from (17) once are known.

For a finite-surface conductivity, the total electric field nolonger vanishes on the surface of the screen and it becomes nec-essary to modify (16) to satisfy the impedance type boundarycondition, which is expressed as

(18)

where a surface of finite thickness and given loss tangent is ap-proximated by an infinitely thin surface with a complex sheetimpedance measured in /square.

Employing (18), (16) is modified as

(19)


TABLE IVMEASUREDVALUES OF " AS A FUNCTION OF FREQUENCY

The formulation given above can be readily extended tothe case of an FSS structure with a dielectric superstrateand a substrate (see Fig. 4) by simply replacing the spectraldyadic Green’s function in (16) and (19) with a compositeGreen’s function, which accounts for the presence of both thesuperstrate and the substrate. Since such a composite Green’sfunction for layered dielectric media can be easily found in theliterature [9], [10], [17], the details will be omitted. Replacingthe spectral dyadic Green’s function with the compositeGreen’s function in (16) and (19), we get

(20)

(21)

where

composite discrete spectral dyadic

Green's function for layered dielectric media

IV. M ULTIPLE FSS SCATTERING FORMULATION

To analyze multiple FSS screens, one can resort to an ap-proach that deals with the entire structure simultaneously anduse the MoM technique to determine the unknown current distri-bution on all the FSS screens concurrently. However, such a pro-cedure places a very heavy burden on the computer resources,

and interfacing such a MoM code with the MGA becomes im-practical. To overcome this difficulty, we, instead, resort to thescattering matrix technique [9], [10]. In this approach, we de-rive, as a first step, the generalized scattering matrices of the in-dividual screens by using the MoM, and of the dielectric layersby following the procedure described in Section III. These ma-trices can be subsequently used to generate a composite scat-tering matrix for the entire system by using the following rela-tionships:

(22)

where , and thesuperscripts are associated with the two subcomposites.

One requirement for this cascading procedure is that we mustuse screens with the same periodicities and include an adequatenumber of harmonics in creating the-matrices, ensuring thatthe magnitude of the highest harmonic is less than dB.

V. NUMERICAL RESULTS

The algorithm described in Section II is successfully appliedto the synthesis of broadband microwave absorbers in the fre-quency range of 19.0–36.0 GHz. In this section, unless other-wise specified, all dimensions are in millimeters. The compositecomprising two (the algorithm can handle any number of layers)subcomposites is surrounded by air on top and terminated bya PEC backing at the bottom. The number of dielectric layersin each subcomposite is fixed at four for this exercise, thoughthis number is flexible. The FSS cell design and periodicitiesare identical for both the subcomposites. Though only dielec-tric layers with electric loss are considered, the method can befurther extended to handle both electric and magnetic losses.


(a)

(b)

Fig. 5. The optimized composites: (a) Case-1 and (b) Case-2.

Table I shows the parameter search space for the MGA for asingle subcomposite. It is evident that the number of param-eters and the chromosome length is dependent on the type ofsymmetry used for the FSS screen. The total number of param-eters and chromosome length (total number of bits) for the entirecomposite is 42 and 310, respectively, when eightfold symmetryis imposed. The measured values ofand of ten differentlossy materials are considered and a database of these valuesas a function of frequency are shown in Tables III and IV. Wenote that, to choose from a database of ten different lossy di-electric materials using a binary coding scheme, we will needat least four bits to represent the database, i.e., in (13) hasto be equal to four. Thus, we introduce six additional indexingterms, which are redundant. To deal with the redundancy, themaximum and minimum value of all the parameters used in theMGA are fixed at 0.0 and 1.0, respectively, and then scaled to theactual maximum and minimum parameter values of the problemat hand as follows:

for (23)

whereparameter value to be generated;

actual minimum value of the parameter;

actual maximum value of the parameter;

number of parameters;

floating point number between 0.0 and 1.0,whose resolution is dependent on that de-sired by the user for a particular param-eter.As an example, for the choice of lossydielectric materials from the database (Ta-bles III and IV) is equal to1.0 and , thus re-stricting the MGAs choice for this param-eter to be within 1.0 and 10.0 and rejectingthe values between 11.0 and 16.0 (the re-dundant ones).

Small values of losses are added to the layers designated aslossless, because it is not practically feasible to fabricate a per-fectly lossless dielectric. Hence, thevalue of the first layer isusually 0.01 if it is tagged as lossless by the MGA, and is fixed


(a)

(b)

Fig. 6. The FSS unit cell designed by the MGA: (a) Case-1 and (b) for Case-2.

Fig. 7. The frequency response of the composite in Case-1.

at 0.1 for the rest of the lossless layers. Two cases are investi-gated, as follows:

i) oblique incidence , TE and TMpolarization;

ii) normal and oblique incidence ( and), TE and TM polarization.

Fig. 8. The worst- and best-case reflection coefficients in dB for the compositein Case-2—TE and TM polarization.

(a)

(b)

Fig. 9. The performance of the MGA: (a) Case-1 and (b) Case-2.

The population size and the number of generations are fixed at50 and 100, respectively. The periodicity of the FSS screen in the


Fig. 10. The population distribution w.r.t. the total thickness of the composite for Case-1.

Fig. 11. The population distribution w.r.t. the total thickness of the composite for Case-2.

- and -directions are made equal, eight-fold symmetry is ap-plied to the FSS cell designs, and the frequency resolution of theMGA optimization is fixed at 1.0 GHz. Following the guidelinementioned in Section IV, the scattering matrix for each subcom-posite is truncated with 9 harmonics. This number is obtainedafter several numerical experiments to maintain the tradeoff be-tween computational speed and accuracy. Upon simulation, itis found that the higher order harmonics have magnitudes lessthan dB; hence, our truncation criterion is met satisfacto-rily. However, we note that it is possible to reduce the numberof harmonics significantly only when the composite is made upof materials with high losses, and the propagating higher order

harmonics rapidly attenuate. It has been found that such a re-duction is not possible when the composite is fabricated witheither lossless or very low-loss materials.

The MGA-optimized composites are shown in Fig. 5(a) and(b) for the two cases of interest. Fig. 6(a) and (b) show theMGA-generated FSS cell designs with eight-fold symmetry andblack representing metal and white corresponding to free space.The periodicities of the FSS screens for Cases-1 and -2 are 14.04and 10.1 mm, respectively. The reflection coefficients in dB areplotted vs. frequency in Figs. 7 and 8 for Cases-1 and -2. Thecurves for the worst- and the best-case correspond to the max-imum and minimum values of reflection coefficient in dB, over


a band of frequencies, range of elevation angles, and both theTE and TM polarizations. Mathematically, the previous state-ment can be expressed as

(worst-case)

(best-case) (24)

where and are the number of frequencies and elevationangles, respectively, over which the design parameters are opti-mized by the MGA.

The worst-case reflection coefficient for Cases-1 and -2 areand dB, and the total thickness of the compos-

ites are 5.7 and 5.65 mm, respectively. Fig. 9(a) and (b) illustratethe performance of the MGA search process by plotting the vari-ation of average and best fitness value vs. the number of genera-tions for the two cases, respectively. The dip in the curves for av-erage fitness value indicates the generations at which the MGAperforms the population restart as explained in Section II. Thepopulation distribution for Cases-1 and -2, as we move from thefirst to the last generation, are shown in Figs. 10 and 11, respec-tively. The additional burden of optimizing over a range of ele-vation angles results in the slight degradation of the worst-casereflection coefficient value for Case-2. Cases-1 and -2 have runtimes of 15 and 31 hours, respectively. The MGA approximatelytakes 15 hours to optimize at a single elevation angle, simulta-neously, for both the TE and TM polarizations. Table II liststhe parameters selected by the MGA for each case. For both thecases, the azimuthal angle is fixed at zero degrees (

- plane) in the MGA optimization program.It is observed in Figs. 12, 13(a), and (b) that the frequency

response of the structure remains relatively invariant of the az-imuthal angle . The angular independence is achieved via theuse of the eight-fold symmetry imposed on the FSS element.Fig. 14(a) and (b) show that, for both Cases-1 and -2, increasingthe number of harmonics from 9 to 25 or 49, in the MGA for

and has little or no effect on the result. Thisconfirms that an adequate number of harmonics are included inthe MGA optimization procedure. To verify the rule of thumbmentioned in Section IV, the magnitude of the reflection coef-ficient in dB, of the worst-case higher order harmonic, for eachfrequency is plotted in Figs. 15 and 16, for cases-1 and -2, re-spectively. It is observed that the magnitudes of the higher orderharmonics are well below the dB limit.

VI. CONCLUSION

A novel approach, which is based on a binary coded micro-genetic algorithm, has been developed to optimize a broadbandmicrowave absorber, which employs multiple FSS screens em-bedded in dielectric media. Given the total number of dielectriclayers, the total thickness of the composite, and the range of per-mittivity values for each layer, the MGA iteratively constructsa composite whose frequency response closely matches the de-sired response. The MGA also optimizes the FSS cell design,its - and -periodicities, its position within the dielectric com-posite, and the surface impedance of the FSS screen. The major

Fig. 12. Invariance of the frequency response with� for the composite inCase-1.

(a)

(b)

Fig. 13. Invariance of the frequency response with� for the composite inCase-2: (a)� = � = 0 ; � = 0 and� = 45 . (b) � = 45 and� =

0 ; � = � = 45 .

advantages of the present approach are that it is simple and thatit demands far less computational time and resource than does


(a)

(b)

Fig. 14. Effect of the change in the number of harmonics included in thescattering matrix on the frequency response: (a) Case-1,� = 45 ; � = 0 .(b) Case-2,� = 45 ; � = 0 .

Fig. 15. Plot of the worst-case higher order harmonic versus frequency forCase-1: (a) TE polarization and (b) TM polarization.

the conventional GA to solve the same problem. We have shownthat the MGA technique can successfully handle multidimen-

Fig. 16. Plot of the worst-case higher order harmonic versus frequency forCase-2: (a) TE polarization and (b) TM polarization.

sional and multimodal optimization problems by using smallpopulation sizes, which gives it an edge over conventional GAs.Furthermore, the MGA technique does not require a crude pre-liminary design to ensure convergence. This is due mainly tothe fact that the MGA is not a gradient-based search procedure,and, therefore, it does not get easily trapped in local maxima.All the designs presented are obtained on a DEC-ALPHA 500workstation employing a machine-optimized math library spe-cially designed for the LAPACK subroutines used to invert theFSS-MoM matrix.

ACKNOWLEDGMENT

The authors gratefully acknowledge the technical supportprovided by J. F. Ma in implementing the LAPACK subroutinesand increasing the computational efficiency of the program.They are also grateful to W. L. Gore and Associates, Inc.,for providing us with the measured values of the frequencydependent material parameters.

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[17] T. Itoh, “Spectral domain immitance approach for dispersion character-istics of generalized printed transmission lines,”IEEE Trans. MicrowaveTheory Tech., vol. MTT-28, pp. 159–164, Mar. 1975.

Sourav Chakravarty received the B.Tech. degreein electronics and telecommunications from the Re-gional Engineering College, Kurukshetra, India, theM.E. degree in telecommunications from JadavpurUniversity, Calcutta, India, and the Ph.D. degree inelectrical engineering from the Pennsylvania StateUniversity, University Park, in 1992, 1997, and2001, respectively.

From 1992 to 1995, he was a Senior Antenna De-sign Engineer at Superline Microwave Pvt. Ltd., Ban-galore, India. He worked as a Research Assistant in

the Electromagnetic Communication Laboratory, Pennsylvania State Univer-sity, University Park, from 1997 to 2001. He is currently a Senior CAD En-gineer at Intel Corporation, Hillsboro, OR. His research interests include com-putational electromagnetics with emphasis on probabilistic optimization tech-niques and the applications of MoM and FDTD techniques to predict delay andcrosstalk in interconnects.

Raj Mittra (S’54–M’57–SM’69–F’71–LF’96) is a Professor in the ElectricalEngineering Department of the Pennsylvania State University (Penn State),University Park. He is also the Director of the Electromagnetic CommunicationLaboratory, which is affiliated with the Communication and Space SciencesLaboratory of the Electrical Engineering Department. Prior to joining PennState, he was a Professor in Electrical and Computer Engineering at theUniversity of Illinois, Urbana Champaign. He has been a Visiting Professor atOxford University, Oxford, U.K., and at the Technical University of Denmark,Lyngby, Denmark, and is currently the President of RM Associates, which is aconsulting organization that provides services to industrial and governmentalorganizations, both in the U.S. and abroad. His professional interests includethe areas of Communication Antenna Design, RF circuits, computationalelectromagnetics, electromagnetic modeling and simulation of electronicpackages, EMC analysis, radar scattering, frequency selective surfaces,microwave and millimeter wave integrated circuits, and satellite antennas. Hehas published over 600 journal papers and more than 30 books or book chapterson various topics related to electromagnetics, antennas, microwaves, andelectronic packaging. He also has three patents on communication antennas tohis credit. For the last 15 years, he has directed, as well as lectured in, numerousshort courses on Computational Electromagnetics, Electronic Packaging, andWireless antennas, both nationally and internationally. He currently serves asthe North American editor of the journalAEÜ.

Dr. Mittra is a Past-President of the Antennas and Propagation Society,and has served as the Editor of the IEEE TRANSACTIONS ONANTENNAS AND

PROPAGATION. He won the Guggenheim Fellowship Award in 1965, the IEEECentennial Medal in 1984, and the IEEE Millennium Medal in 2000.

Neil Rhodes Williams received the B.S. degreein electrical engineering in 1987 from WilkesUniversity, Wilkes-Barre, PA, the M.S. and Ph.D.degrees from The Pennsylvania State University(Penn State), University Park, in 1987, 1989, and1994, respectively. His dissertation was focusedon a “Radiative Transfer Approach to Design TheElectromagnetic Response of Microwave ChiralComposites.”

From 1990 to 1994, he served as a design anddevelopment engineer at Penn State, Research Park.

There, he worked on advanced electrical composites and electrical materialcharacterization systems. From 1994 to 1996, he served as an Engineer/Scien-tist in the field of GPS Ground Systems for Lockheed Martin. There he workedon signal processing techniques for atmospheric signal propagation, as wellas new architecture development for GPS. Research included manipulation ofGPS in military arenas. In 1996, he joined W. L. Gore & Associates, Newark,DE, where he is currently Engineering Director for Advanced ElectromagneticProducts. His work includes flexible Magnetic Resonance Imaging (MRI)surface coils for MRI scanner systems. His research responsibilities alsoinclude microwave and optical fabrics and electronic material characterizationsystems. Some of the products which have precipitated from this research areaare radome and electromagnetic manipulating fabrics.

application of a microgenetic algorithm (mga) to the ...callen/hi-z_surfaces/chakravarty... · mga...

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