A Generalized Diffraction Synthesis Technique for High Performance Reflector Antennas

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IEEETRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL 43, NO. 1, JANUARY 1995 21

A Generalized Diffraction Synthesis Techniquefor High Performance Reflector A ntennasDah-Weih Dum, Member, IEEE, and Yahya Rahmat-Samii, Fellow, IEEE

Abstract-Stringent requirem ents on reflector antenn a perfor-mances in m odern apprications such as direct broad cast satellite(DBS) communications,radar systems, and radio astronomy havedemanded the developmentof sophisticated synthesis techniques.Presented in this paper is a generalized diffraction synthesistechniqw for single- and dual-reflector antennas fed by eithera single feed or an array feecl. High versatility and accuracy areachieved by combining optimization procedures and diffractionanalysis such as physical optics (PO) and physical theory ofdiffraction (PTD). With this technique, one may simultaneouslyshape the refleetor surfaces and adjust the positions, orientations,and excitations ofan ar bitrarily configured array feed to producethe specified radiation characteristics such as high directivity,contoured patterns, and low sidelobe levels, etc. The shapedreflectorsare represented by a set of orthogonal global expansiontunetions (the Jaeobi-Fourier expansion), and are characterizedby smo oth surfaces, well-defined (sup erqua dric) circum ferences,an d Eootirmoussurface derivatives. Th e sample applications ofcontouredbeam antenna designs and reflector surface distortioncompensation are given to illustrate the effectiveness of thisdm actio n synthesis technique.I. INTRODUCTION

HE challenging requirements on the radiation perfor-T ance and transmission capability of reflector antennasfor modem apjdications such as direct broadcast satellite(DBS) communications, radar systems, and radio astronomyhave called upon the development of sophisticated antennasynthesis techniques. A synthesis technique is deeme d usefulif it can be used to design these high performance antennas inan effective manner. F urthermore, the many choices of antennacomponents and configurations also suggest that a techniquemust be generaland lexible enough that the repeated efforts ofdeveloping specialized tools can be minimized. In this paper,a generalized diffraction synthesis technique for single- anddual-reflector antennas fed by either a single feed or an arrayfeed is presented.A . Geometrical Op tics Shaping

Many techniques have been devised to solve the problemof syn thesizing reflector antenna systems. Most of the se tech-niques are based on the principles of geometrical optics (GO).GO-shaping of circularly symmetric dual-reflector antennaswas formulated in terms of simultaneous nonlinear ordinarydifferential equations [13- 131. For offs et dual-r efle ctor ant en-

Manuscript received January 6, 993; revised August 19, 1994. Work wassupported in part by Contract SS-274389-K-A03 with Space Systems/Loral.The authors are with the Departmentof Electrical Engineering, Universityof California. Los Angeles. CA 90024-1594 USA.

nas, the shaping problem can be descr ibed by a se t of partialdifferen tial equatio ns [4]. Ano ther formulatio n of G O-shapingis based on the apparatus of complex coordinates 151, 161.Other variations of these GO -shaping procedures can also befound in the literature [7]-[ll].A major limitation intrinsic to the GO-based techniques isthat diffraction effects are not incorporated in the process ofreflector shaping. The ignored diffraction effects include thediffraction from the surface and the edge of the reflectors, thenear-field effect between the feed and the reflector, and thatbetween the main and the subreflector. The radiation patternsof a GO-shaped reflector antenna, when evaluated by diffrac-tion analysis techniques, may at times de viate from the desiredpattern to the extent that stringent specifications such as verylow sidelobe levels are v iolated. Due to this d eficiency, GO-shaping techniques are primarily applicable to large antennasystems where ray-tracing is an acceptabIe approximation. Forsmall antenna systems, more accurate synthesis proceduresmust be used. Another difficulty associated with the GO-shaping techniques is that array and aperture-type feeds (suchas horns) may not be easily incorporated in the sy nthesisprocedure. Althou gh modifications of the GO -shaping method shave been proposed t o overc ome this difficulty [121, [131, theyare mainly used as the initial designs for subsequent accuratediffraction synthesis.It is noticed that in GO algorithms one typically synthesizesthe ape rture field, from w hich the far-field patterns are inferred.This is sometimes referred to as the indirect approach,in contrast to methods in which the far-field radiations aresynthesized directly. The aperture field and the radiated far-field can be related by m ethods such as clased-form formulasin pencil beam designs [14], and optimization algorithmsin contoured beam designs [15], [16]. A shaped reflectorresulted from a GO algorithm is typically characterized bya set of discrete points, which may render a surface that hasdiscontinuities and irregular boundary 1151. The interpolationprocedure that is required before fabrication may furtherintroduce errors that degrade the radiation pattems.B. Previous DiJcSraction Shaping

To overcome the limitations of G O shaping, several meth-ods have been attempted. In an early work [17], reflectorshaping by diffraction synthesis was carried out using themethod of spherica l wave expan sion (SWE).The more generalframework of reflector surface expansion and coefficientsoptimization was first effectively applied to synthesizing the

IEEE Log Number&O7702. aperture field using a GO method [18], and later employed0018-926X/95$04.00 0 1995 IEEE

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Fig. I . An overviewI I I Icof the eenera fized diffraction svnthesis techniaue. The unknown coefficients (Cam,D,they antenna system, are to be determined using optimization algOrithms.

in m ethods that incorporate both GO algorithm and physicaloptics (Po) nalysis such as [19] an d [20] (the INDIRECTmethod). This concept was further extended to methods thatcarry out full PO ynthesis (shaping) of reflector antennas withcircular aperture and single feed, such as the single-reflectorcontoured beam antennas in [20] (the DIRECT method), andthe dual-reflector pencil-beam antennas in [19], [21]. Th eexpansion functions used in these w orks, such as the Fourierseries and polynom ials, are usually heuristically chosen. In thePO diffraction synthesis (shaping) method introduced in [22],a set of orthogonal global expansion functions was appliedeffectively to represent reflectors with circular apertures. Thisorthogonal representation will be used and generalized in thediffraction synthesis technique presented in this paper.C . The Generalized Dif iaction SynthesisThe primary goal of this paper is to develop a generalizeddiffraction synthesis technique that can be used to synthesizesingle- and dual-reflector antennas with a single feed or anarray feed, as shown in Fig. 1.There are several features that distinguish this techniquefrom others. First of all, the previous notion of reflectorsurface expansion and coefficients optimization has beengeneralized to antenna system cha racterization and parame -ters optimization, in which the reflector surface expansionbecomes a portion of the characterization process, and theexpansion coefficients consist of a subset of the antennaparameters to be optimized. This conceptual generalizationallows us to solve the problems of i) reflector shaping withgeneral array feed and ii) simultaneous synthesis of reflectorsand feeds, which are not addressed in the PO synthesismethods published so far.Another feature that makes the current technique powerfulis that the global surface expansion [22] is used in this paper,and generalized to reflector aperture boundaries that may becircular (elliptical), squa re (rectangular), or any intermediaterounded-corner shapes described by the superquadric unctions

PENCILBEAMG

CONTOURED BEAnrs

COSECANTSQUAREDm ..

fn , wbich characterize

[23]. The shaped reflectors are characterized by having smoothsurfaces, well-defined c ircumferences, and continuous surfacederivaEives.Also unique in our method is that the PO/Physical The ory ofDiffraction (PTD) technique is used to analyze the antennas.The PTD fringe field [24], [25] is particularly useful whenPO analysis is not sufficiently accurate for very stringentrequirements on, for example, sidelobe levels at far-angularregions and cross-polarization control. The edge diffractionand near-field effects are included automatically in the courseof diffraction synthesis.The generalized diffraction synthesis technique can be ap-plied to produce various radiation patterns such as pencilbeams and contoured beams, and to improve the radiationcharacteristics of an existing canonical or distorted system. Itis believed that the scope of the application of this techniqueembraces the majority of commonly used reflector antennaconfigurations. The formulation and implementation of thegeneralized diffraction synthesis technique will be describedin Section 11, and sample applications will be presented inSection III.

U. IMPLEMENTATION OF THE GENERALIZEDDIFFRACTION SYNTHESISTECHNIQUEA . Coordinate Systems and Their Transformations

A local coordinate system is erected for each component ofthe antenna system as shown in Fig. 1. Explicitly, they arethe main reflector coordinate system C, = {i,,~,,im}.the subreflector coordinate system C, = { P S I S , s } ,hefeed array coordinate system Cf= {Pf er,i f},nd a feedelement coordinate system Ce,= { k e a ,e,, &,}, i = 1, . . ,nfor each of the n elements. These coordinate systems arechosen such that the geometry and the radiation pattern ofthe associated component can be described most conveniently.Eulerian angles and translation vectors [26], [27] are usedto relate these coordinate systems; the vector field quantities



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DUAN AND RAHMAT-SAMII DIFFRACTION TECHNIQUE FOR.HIGH PERFORMANCE ANT ENNAS 29

and the position vectors can therefore be transformed in asystematic manner.B. The Feed Array

way of computing these quantities for various reflectors isformulated in the next section.Th e PO current (2) provides a good approximation of theactual surface current (except near the ed ge) when the reflectorThe feed array in our implem entation may have generalconfiguration in the sense that the position and orientation of

the ith feed element is specified independently by cho osinga translation vector and a set of Eulerian angles that relatethe coordinate system Cet to Cf. The excitation coefficientand the type of radiation (acosq 6 feed [28], a G aussian pointsource, a microstrip patch, a sectoral horn [29], or a set ofmeasured data, etc.) may also vary from element to element.The total feed field Ef is the sum of the element fields

where it is assumed that the Ee, s have been properly trans-formed to the feed array coordinate system Cf. The totalradiated power of the feed m a y is obtained by integratingthis composite field over a spherical surface in the far-field.Notice that if in (1) one modifies the pattem and excitationof each feed element to account for the array environment,mutual coupling effect can be partly taken into consideration.C . The Difiaction Analysis

The diffraction technique of Po s used to analyze theradiated fields of the reflectors in this paper. For reflectorswith perfect electric conducting surfaces, it is assumed in POthat the current at a point r on the reflector surface is(2)

where f i is the un it vector normal to the reflector and pointingto the feed, and Hi s the incident magnetic field. With theconvention ejwt fo r the time-h armo nic field, the radiated fieldof the PO current (2) at an observation point r can be written as

Jp o = 2A x Hz

EPo(r) = -jk&(3)

e - j kRHPo(r)= -jk 93 ( R x Jpo) m d x1 1

3 . 3g 2 = 1 - - 1g 3 = 1 - j - kR

( k R ) 2-

(4)

(7)

with k the free space wave number and 20 he free spaceimpedance. Equations (3) an d (4) are exact radiation integralsfor the approximate current Jpo, and are applicable to obser-vation points both n the near- and he far-field regions. N oticethat to evaluate the radiation integrals in (3)and (4),one has tocalculate for each reflector point the position vector r, the unitnormal vector A, and the surface element dC. A systematic

is rather smooth, and more than a few wavelengths in size.Th e PO field is known to be very accurate in the main beamand the first few sidelobe regions [28]. It is sufficient indiffraction synthesis to use PO analysis alone if the goal isto im prove the antenna gain or suppress the near-in sidelobeslevels. For observatio ns made in the far-ang ular regions or inthe predictio n of the cross-po larized fields, howe ver, the edgediffracted field may contribu te significan tly,and hence the POfield must be modified. In this paper, the edge diffraction istaken into account using the PTD, in which the total scatteredfield consists of the PO field and a fr inge field [30], [31]

(9)The PTD fringe field corrects the PO field in an effectivemanner, and is for the first time implemented for dual-shapedreflector antennas with array feed. In the implementation ofPTD, how ever, care must be exercised to properly handle theaperture- type sources. The details of PTD implem entation isnot included in this paper due to page limitations. The readeris referred to the literature [30], [31] for the formulation.It must be mentioned that in general other diffractiontechniques such as geom etrical theory of diffraction (GTD)[32], [33] and its uniform versions [34]-[36], the method ofmoments, and various hybrid techniques can also be used indiffraction synthesis.A comparative study of these diffractionanalysis techniques can be foun d in 1301.

EPTD = Epo +Efr.

D. eflectors With Superquadric AperturesFor convenience in evaluating the surface element dC overa curved surface E, a reflector is usually represented as afunction of the parameters defined on the planar aperture A,

as shown in Fig. 3.With the aperture parameters t an d $ [23],[28], for example, one may write the coordinates of a pointr = (x,, 2) on the reflector as

XI = x(t,?)), y = y ( t ,$ ) , 2 = Z ( t , $ ) (10)O < t < l , 0 5 ? ) 5 2 ? r (11)

/ l + - . d C+ / L + . .A( t , $ ) d t d ? ) (12)where J A ( t ,$) is the transformation Jacobian. The paramet-ric representation (10) facilitates systematic construction ofJA(t ,?)) and the unit normal vector A

(13)dC

A=*- (14)

(15)(16)

and rearrange the integrals in (3 ) an d (4) as

J A( t 7 j )z-&d$ - $1ri x r&Ir: x r&l



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initial variable set(OQShaped. conic, ...)reffectorsurfaces*EeedoonfiguraLion...etc

i Idiffractionanalysis

Mockageeffectnear-field effect-edge&f f 9 allayfeed

radiation

asetof[ I variablevaluesearchnew i- no ( m i n i m i z e c i ? w optim ized I:U i I variables IFig. 2. Application of OptirniZation techniques to antenna synthesis.

Fig. 3. Antenna geometry of an offset single-reflectorantenna.

The sign in (14) is so chosen that .L points to the illuminatedside of the reflector.One very useful class of aperture shapes defined by thesuperquadric function[(E)]. [ = 1

has been recently employed to describe circular, elliptical,and rounded-com er aperture boundaries [23]. In (17), a and bare th e radii along the z-axis and y-axis (see Fig. 3). Th esuperquadric aperture boundary has the feature that with asingle parameter, v, an elliptical (circular) aperture boundaryca n be deformed to a rectangular (square) one through theintermediate rounded-comer shapes in a systematic manner.Many of the antennas used in practical applications haveaperture boundaries fall into this ge neral category. It was found[23] that the pa rametric representation

d( t ,11) = a t cos,$ - T ( $ )y ( t , $ ) = b t s in $ - T(+)

produces the most un iform subgridding when the ratio a / b de -viates much from unity. Notice that the sup erqua dric boundaryis exactly represented by the parametric curve t = 1. Thisfeature is important for PO analysis because the inaccuracycaused by the irregular boundary of a nonexac t subgriddingscheme is avoided. Fu rthermore, in PTD an d GTD, a well-repres ented reflector boundary is in dispen sable for effectiveevalua tion of the edge- diffracte d fields.E. Global Surface Expansion Using Orthogonal Functions

The key role of the diffraction synthesis technique presentedin this paper is to describe the shape d reflector surfaces by thefollowing expansionN M

n=O m=Owhere C,, and D ,, are the e xpansion coefficients, andF g ( t ) s the modified Jacobi polynomials defined by (27) inAppendix A. This expansion can be applied to each reflectorin terms of the corresponding local coordinate system asshown in Fig. 1. The modified Jacobi polynomials are relatedto the circle polynomials of Zernike, which w ere previouslystudied in the classification of optical aberrations [37]. Th erelationships between the modified Jacobi polynomials andthe circle polynomials of Zem ike are studied in Appendix B,and it is found that b esides the norm alization constant, theydiffer only in the index sche mes. The derivatives of the mod-ified Jacobi polynom ials, which are needed in com puting thederivative (15), are formulated in A ppendix A. Combinationsof the modified Jacobi polynomials and the harmonics cos n$,sin n$ consist of a complete set of orthogonal basis functionsin the unit circle region (11) of the t-11,plane [37].The expansion (21) is global in the sense that each basisfunction acts in the entire domain of (l l) , n contrast to alocalized function which usually has nonzero (or significant)values only within a subgridded patch. The formula (21)ensures that the reflector surface is continuous, and so arethe de rivatives of all orders. Since reflector shaping is carriedout by ad justing the expa nsion coefficients, the syn thesize dreflectors have smooth surfaces and welldefined circumfer-ences. The fact that the global expansion of (21) providesan effective representation for reflector surfaces can be bestmanifested by showing that many commonly used reflectorsurfaces can be exactly described by only several lowest orderexpansion functions, and the expansion coefficients can beobtained in closed-form using (42). Several examples are givenin Appendix C . It is inter esting at this point to notice thatfor a circularly symmetric antenna, it suffices to use the CO,coefficients alone in (21) to represent the shaped reflectors. Insuch cases, the number of the expansion coefficients s reducedfrom a two-dimensional (2-D) set to a onedimensional set.The two steps in applying the optimization techniques todiffraction synthesis, characterization and optimization, aredescribed in the following sections.



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2" , II I

-1" 1w-4" -3" -2" -1" 0" 1" 2" 3" 4"

A z-2"

Fig. 4. The CONUS coverage map. A f O 1 degree pointing error is con-sidered. The three gain zone s are designed for compensating rain attenuation.

F . CharacterizationFrom the mathematical point of view, the unknowns inthe problem of reflector shaping are the 2-D functions thatdescribe the reflector surfaces. These 2-D fu nctions are to bedetermined in continuous region s (over the reflector apertures)instead of on a set of discrete points. An effective approach tosolve this difficult problem is to represent a reflector surface

by a series of functions such as in (21). In this situation, theshape of the reflector is determine d (or, "characterized") bythe expansion coefficients, which become the unknowns of theshaping problem. Notice that although theoretically an infiniteset of coefficients is needed to represent an arbitrary reflectorshape, in practice only a finite number of coefficients can beprocessed by the computers. The answer to the question ofhow m any coefficients must be used f or a particular problem,depends on the size of the antennas, the desired radiationcharacteristics, and the required design precision.The transformation from an unknown tw o-variable functionto a set of discrete coefficients is part of the characterizationof the antenna systems as shown in Fig. 1. The characteri-zation procedure can be extended essentially to any antennacomponents such as an array feed, for which the excitationcoefficients, positions, and orientations of the feed elementsmay be taken as the unknowns (see Fig. 1). It may also beapplied repeatedly, sampling the frequency band of interest.Notice that one may use any combination of the antennaparameters as the unknowns of the synthesis problem.G. Optimization

Since the design problems are usually over-specified, theremay not be exact solutions. In this situation, one needs ameans to search for the "best" (or the "optimum") solution interms of some criterion. In our approach, the best solution isdetermined by the optim ization techniques as shown in Fig. 2,and the criterion is the minim ization of an object (penalty, cost)function. Having different forms for different applications,the object function is designed to reflect the deviation ofthe performance of the current design to the desired antennaradiation ch aracteristics. Upon the m inimization of the objectfunction, the goal of searching for the optimum variable valuesis deemed accomplished.As shown in Fig. 2, the initial values of the optimizationvariables can be chosen according to one's best knowledge or

convenience. For ex ample, the surface expansion coefficientscan be those obtained from GO-shaping, or those of the conic-section reflectors. The initial excitation coefficients of the feedcan be a uniform distribution, or a set of properly taperedexcitations. A judicious choice of the initial values leads to,usually within few steps of iterations, a converged solution thatis close to the physical limitations. In the loop enclosed in thedashed box of Fig. 2, diffraction analysis is applied to com-puting the radiation characteristics of interest. The computedand the desired values are compared, and the differences arerepresented by the object function, which is to be minimized.The iteration ends if the result of comparison is satisfactory.Otherwise, new variable values that produce a smaller objectfunction value are determined by the optimization technique.The updated variables are then used in the next iteration.In view of the optim ization science, the problem of d iffrac-tion synthesis of reflector antennas can be classified into thecategory of nonlinear multivariable problems with intermedi-ate scale, and further characterized by that the evaluation of theobject function may be time-consuming, and the derivative ofthe object function is in general not available in closed-form.Based on these considerations, a set of computer programs thatperform safeguarded Newton's methods are implemented inthis paper. Scaling of the variables, termination criterions, andnumerical differentiation are carefully implemented in theseprograms.A question often raised is whether the searched solutionis a local or global minimum. To the best knowledge of theauthors, there is no optimization algorithms that generallyguarantee the global minimum. Also, the solution space ofthese multidimensional problems may possess many localminima w here the optimizer may be trapped. The optimizationprograms used in this paper does not exempt from thisgeneral problem. There is a distinction between the purelymathema tical optimization problem, however, and the antennasynthesis problems that may help alleviate this difficulty.That is, the radiation characteristics of a synthesized antennasystem can be compared with the physically ideal situations,based on which the goodness of the solution can be as-sessed. Furthermore, according to the authors' experiences,the optimization iteration usually converges rapidly toward asatisfactory solution for most of the attempted antenna designproblems provided that the initial variable values and theobject function are properly chosen.One last comment on optimization is that in practice some-times it is convenient to perform optimization in steps fordifferent subsets of the variables. This helps reduce the sizeof the problem, and identify the sensitivity of the antennaperformance to each subset of variables. One is also cautionedthat optimization in steps is m ore vulnerable to local m inima.In a recent study, the utilization of genetic-based algorithmsis also being investigated [38].

111. SA M P L E P P L IC A T IO N STwo diverse applications of the generalized diffraction syn-thesis techniqu e are illustrated in this section . In the firstapplication, three types of contoured beam satellite antennas



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EL

EL

Az (a) Initial status.

EL

(b) After 9 iterations.C

Az (c) After 30 iterations.

"MblH U06 5 4 3 2 KAz (d) After 60 iterations.

Fig. 5. The initial, intermediate,and optimized contour patterns produced by a shaped,reflector (with 27 expansion coefficients) and a single feed. Thecrosses indicate citie s that outline the three gain zone s. Each expansion coefficient is represented by a "tile," in w hich the s u e of the center dark squareis proportional to loglo lCnml (or loglo lDnml), ormalized in the range l o o toare designed : a shape d single-reflector antenn a, a sh aped dual-reflector antenna with circular main reflector aperture, anda shaped dual-reflector antenna with elliptical m ain reflectorape-. ne aspects of these designs are the useof a single feed. In the second application, two schemes forcompensating reflector surface distortion are studied: compen-sation using an array feed, and compensation using a shapedsubreflector fed by a single feed.

These examples by no means exhaust the applications towhich the diffraction synthesis technique is applicable. Instead,they are selected to illustrate the nature of the methodology

such as the evolution of optimization and the coefficients-controlled surfaces, and the variety of solutions that one mayassess using this technique. 0 t h design examples may befound in the iterature. For instance, simultaneou s optim izationOf reflector and feed was demonstrated in [391y and synthe-sis of dual-offset reflector antenna fed by hom arrays wasdocumented in [40].A. Contoured Lkam satellite A nt" UZ

Contoured beam antennas find many applications in mod-em satellite communication systems. These antennas radiate



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deviation from the initial paraboloid110

-110

0

01 o 0.5 0.0 0.5 1 o

tFig. 6. The shaped reflector for CONUS coverage and its deviation to theoriginal paraboloid, (Q = D/2 is the radius of the reflector aperture).shaped beams that cov er prescribed service regions in whichthe highest possible antenna gain is required.Contoured beam reflector antennas (CBRA) can have vari-ous configurations.A typical CBRA consists of a paraboloidalreflector and a multihom feed. The shaped radiation is pro-duced by adjusting the positions and excitations of the feedhorns. Due to the complexity of the beam-forming network(BFN) associated with a large array feed, interest has beenrecently focused on reducing the number of feeds withoutdegrading the quality of the con toured beam. One is challengedwith, in the extreme case, using a single feed to illuminate areflector. In this situation, the reflector must be shaped to gen-erate the desired contoured beams. If high cross-polarizationdiscrimination s required, one has to consider yet other CBRAconfigurations such as Gregorian type dual-offset reflectorantennas. A discussion on CB RAs configurations for variousmission requirements can be found in [41].For purpose of illustration, we present the design of threetypes of CBRA s in the following: i) a single-reflector an-tenna, ii) a du al-reflector antenna with cir cular main reflectoraperture, and iii) a dual-refle ctor antenna with elliptical mainreflector aperture, all with a single feed and the same coverageregion. The goal is to produce a contoured beam that coversthe contiguous United States (CONUS). To compensate forfactors such as rain atten uation, gain correction s are specifiedin differen t areas. or sites. T his results in a w eighted (or,nonuniform) plateau profile. For the current application, thedesired contoured beam pattem is depicted in Fig. 4, in whichthere are three prescribed gain zones in the contoured beam.A f0 .1 degrees pointing error has been considered in Fig. 4.The locations of the observation sites (represented by dots inthe figure) sample the gain regions in a uniform manner, and

X

Fig. 7. Geometry of a Gregorian type dual-offse t reflector antenna (with sin-gle feed) for CONUS-contouredbeamapplication.D = 1.524 m,H = 0.838m, FJD= 0.8, e = 0.333, p = 1 3 - 8 4degrees, inter-foci distance = 0.3995m.

are separated by about half of the half-power beam width ofthe pencil beam radiated by a paraboloidal main reflector.Le t N , be the num ber of obser vation sites, and Di thecom puted directivity at the ith site modified by gain correction .The ob ject function 3 s constructed as

where D is the average directivity (including gain correc-tions), SD r epresents the root-mean-square gain ripple, and 20is a weighting coeff icient with a typic al value of unity. Theoperation frequency is 11.95 GHz. T he object function (22)ca n be further modified for cross-polarization improvem ents.A circular offset reflectorwith D = 2a = 1.524 m, F = 1.506 m, H = 1.245 m,and f3f = 42.77 degrees (Fig. 3, with v = 1) is used. Thefeed is an 2-polarized (cos 9)q source with -12 dB edge taper( q = 14.28).The reflector is represented by 28 terms in theJacobi-Fourier expansion (2 1). They are chosen according tothe index scheme of the circle polynomials of Zemike. (C learlyone could start with the Zemike p olynomials as an altemative.)The first (i.e., the con stant) coefficien tCOO 0.849 (meters)is left intact to keep the shaped reflector in proxim ity to theoriginal paraboloid, and the other 27 coefficientsare optimizedwith the initial values derived fim a paraboloid. Samples ofthe optimization process are depicted in Fig. 5, in which theinitial, intermediate, and o ptimized contour pattems and thecorresponding expansion coefficients are displayed. In Fig. 5each exp ansion coefficient is represented by a tile, in whichthe size of the center dark square is proportional to log,, lCnml(o r log,, ID,,I), normalized in the range 10 to

Single ReJectorlSingle Feed:



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34 IEEE TRANSACTIONS ON ANTENNAS AN D PROPAGATION,VOL. 3, NO. 1, JANUARY 1995

COO CO1 Cl0 (720a 0H8&F 4F-- -2H + a-- F + - -Jz 8 d 5 F

TABLE ICLOSED-FORMACOBI-FOUIUERXPANSIONOEFFICIENTSOR TYPICALEFLECTORURFACESITH ELLIPTICALPERTURES(a AN D b AR E HE RADIIALONG HE 2- AND Y-AXIS, ESPECTIVELY).LL OT HER OEFFICIENTS REZEROES

single-reflectordual-reflector, circulardual-reflector, elliptical

&I ~ 6 0 cross-pol33.86 dE3i 0.58 dB 12.9dBi33.80 dBi 0.87 dB 0.5 dBi33.89 dBi 0.74 dB 4.2 dBi

1 2 H 2+ (a + b 2 ) / 2 ( a a+ ba) ) /2 aH aa - baelliptical aperture F 8&F 4F 8&F

- - -

It is observe d in Fig. 5 that in the early stages the optimizerspreads out the energy contained in the initial pencil beam, andthen po lishes the shaped pattem until it precisely delineatesthe gain regions with minimum gain ripples. Profile of theresultant shaped reflector is depicted in Fig. 6.The deviationfrom the initial paraboloid is within (-0.4X, 0.6X).Dual Rejector, Single Feed, Circular Aperture: The cross-polarized field of the above single-reflector antenna as shownin Fig. 8(a) is excessively high for stringent requirement on thediscrimination evel (30dB, or example). To reduce the cross-polarized field level, we choose a Gregorian type dual-offsetreflectoi antenna with optimum tilted angles [42] as shown inFig. 7.The feed is an x-polarized (cose)* source with -1 2 dBedge taper ( q = 25.98). Notice that for purpose of comparison,the sam e main reflector diameter is used, and the offset heightand focal length are reduced to keep the total volume of theantenna similar to that of the single-reflector case.To obtain the CONUS-shaped beam, the main reflector andthe subreflector are shaped simultaneously. Each reflector isrepresented by 28 expansion terms. The optimization startswith parahloid/ellipsoid, and there are totally 54 coefficients(27 for each reflector) to be adjusted. The resultant co-polarized and cross-polarizedpattems are com pared with thoseproduced by the single-reflector antenna Fig. 8. It is seenthat the c ross-polar ized field is significantly reduced, with theintegrity of the CO-polarizedcontours maintained.Dual Re ject or, Single Feed, Elliptical Aperture: Reflectorantennas with elliptical apertures find many applications inmodem satellite communications and radar systems. Utiliza-tion of an elliptical aperture is usually motivated by therequired radiation pattems, and the reduced weight and cost.It is a challenging task though to redirect the field from acircularly symm etric feed to reflecto rs with elliptical apertures.This is particularly so when a com plex radiation pattern suchasa contoured-beam is to be generated. The key to such designproblems is reflector shaping.It is observed in Fig. 4 that the CONUS coverage regionis elongated in the azimuthal direction. This suggests thatan antenna with elongated aperture (in a direction that isperpendicular to that of the contoured pattem) may produce asimilar far-field pattem if the reflec tors are prop erly shaped. Tojustify this concep t, we perform dif fraction synthesis (shaping )on an elliptical antenna as shown in Fig. 8(c), which has anaspect ratio of 1 :0.75 but otherwise identical to the previouscircular antenna. After properly shaping both reflectors, atypical resultant contoured beam is depicted in Fig. 8(c). Itis seen that a contoured pattem with similar average gain

TABLE I1COMPARISON OF cows ANTENNA ESIGNSI average gain gain ripple maximum

and gain ripple has been produced, with slightly decreasedresolution n the azimuthal plane due to the reduced reflectorsize. The level of the cross-polarized field is raised about 4dB compared to the circular case shown Fig. 8(b). Table I1summarizes the three CBRA designs.B . Reflector Su$ace Distor tion Com pensa tion

Very larg e reflector antennas have been widely used in mod-ern satellite com munications systems because these antennasproduce high gaidlow noise radiations and provide enhanceddata transmission capacity [43]. For both ground and spaceantennas, however, large reflectors may suffer from systematicsurface distortion due to thermal or gravitational effects.Additionally, for nonrigid reflector surfaces such as thoseused in unfurlable or inflatable [43], [44] antenna systems,distortion may be resulted from the mechanical constructionof th e reflector. The disto rted reflector surface typically causesaperture phase errors and degraded antenna performance.

In this section, the diffraction synthesis technique is em-ployed to investigate different compensating systems: i) arrayfeeds and ii) a shaped subreflector fed by a single feed. Forpurpo se of illustration, we use the example problem depicte d inFig. 9, in which the disto rted paraboloidal reflector is mod eled

( E + H )2 + y2 +Fd4F= - F +Fd = 0 . 0 1 1 ( ~ ) ~ ~ ~ ~ 2 4meters)

Yp = d m , $ = rc t an -where a = 0 / 2 is the radius of the reflector aperture, F is thefocal length of the ideal paraboloid, and H is the offset heightof the reflector aperture center. The fun ction Fd as depicted inFig. 9represents a typical slowly varying thermal distortion.It is seen tha t this distortion has severely deteriorated the far-field patterns, reducing the peak directivity from 42.5 dB to39.0 dB.



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DUAN AN D RAHMAT-SAMII: DIFFRACTION TECHNIQUE FOR HIGH PERFORMANCE ANTENNAS

FEED

EL

Az

EL

Az

EL

Az

0SUBREFLECTOR EL

0SUBREFLECTOR

EL

EL

Az

35

Fig. 8. Comparison of the CO -and cross-polarized CONUS pattems using (a) a single-reflector antema, (b) a d ual-reflector antenna with circu lar mamreflector,and (c) a dual-reflector antenna with elliptical main reflector. In all cases, only a single feed is used. The frequency is 11.95 GHz.



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D=

L F=1.832m4(frequency=8.45 GHz)

(a)Fd

-6.0dB-9.4 dB0-10.3 dB -302'800

-77*60 -3.1 dB0-5.5 dB -21 30-92*80 -3.9 dB0

0-225.30-9.4 dB0-2.6 dB -302'80-10.3dB0 0

00.0 dB -21'30 -5.5 dB0 0-323.4' -3.1 dB -77.6'

0.0' -3.9 dB -92.8'0 0

0 0 0-46.0' -9.2dB -318.9" -9.2 dB -46.0'0 0

-12.6 dB -3.1 dB -'3'40 -12.6 dB

-301.2" -7.7 dB -301.2"0-226.6'

(a)

Fig. 10. Distortion compensation using a 19-element array feed with aninterelement spacing of 1.06X. (a)The optimum feed excitation s. (b) Far-fieldpattems. (Solid lines: 4 = 0 degrees, dashed lines: 4 = 90 degrees.)

Fig. 9. A distorted paraboloidal reflector antenna. (a) Antenna geometry. (b)'zhe surface distortion, F d . (c) The distorted far-field pattems. Ideal pattemsrefer to those produced by the paraboloid. (Solid lines:d, = 0 degrees, dashedlines: 4 = 90 degrees.)Focal Plane Array Feedr: One way of achieving distor-

tion compensation is to use an array feed w ith proper exci-tations, which may be obtained by the method of conjugatefield matching [45]. In this paper, instead, we use the diffrac-tion synthesis technique to determine the optimum arrayexcitations. As an example, let us consider a representative19element array as depicted in Fig. 10(a). In the design, theoptimization variables are the real and imaginary parts of theexcitation coefficients, with reference to the center element.

These variables are initially set to values (1 for real parts and0 for imaginary parts) that correspond to uniform excitations.The object function is

3= Do (25)where DO s the boresigh t directivity. The resu lts of optim iza-tion is show n Fig. 10(b), in which it is show n that the anten napattem has been effectively restored, with 41.8 dB boresightdirectivity (the actual antenna gain may be somewhat lowerdue to losses in the array). The optimum excitations resultedfrom the diffraction synthesis technique as show n in Fig. 10(a)are similar to those obtained by the method of conjuga te fieldmatching [45].A Shaped Subrejector and a Single Feed: To avoid the in-creased complexity, loss, and weight of a BFN, anothercompensation scheme that employs a feed system consistingof a shaped subreflector and a single feed may be c onsidered[46]. A design is carried out for the distorted paraboloid asshown in Fig. 1 (a). The feed is described by a (cos e ) q modelwith q1 = q2 = 70. The surface of the deform able subreflector



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is determined by the diffraction synthesis (shaping) techniquewith the object function %3 = - D o + w - R (26)

where w is a weigh ting coefficient, and R is the root-mean-square ripple of the directivities at, for example, 13 = 1.3degrees (the -10 dB beamwidth) in the q5 = 0 degrees, 90degrees, 180 degrees, and 270 degree planes. The term w . Ris included to control asymm etry in the patterns. As a result ofoptimization, effective compensation is achieved as shown inFig. 1 (c), with 41.3 dB bo resight directivity. The d eviation ofshaped subreflector to the original hyperboloid is depicted inFig. 1 (b). Mechanical structures that facilitate reconfigurablereflector surfaces represented by the Jacobi-Fourier expansionsare recently being investigated.

IV . CONCLUSIONA generalized diffraction synthesis technique that can beused to solve the problem s of synthesizing array-fed single-and dual-reflector antennas is developed. This technique has

the following unique features:Due to the application of global surface expansion byorthogonal functions, the synthesized reflectors alwayshave smooth surfaces, well-defined circumferences, andcontinuous surface d erivatives.In particular, the resultantreflectors are ready for fabrication (by computerizedmachining, for exam ple) without requiring interpolation.The diffraction analysis techniques of PO and PTD areapplied to take into account the diffraction from thesurface and the edge of the reflectors.Antennas of general configurations can be synthesized.For example, there is no limitation on the number ofreflectors. The apertu res of the reflectors may h ave bound -aries varying from an ellipse (circle) to a rectangle(square) using the superquadric function.The feed system may consist of a single feed or anarray of general configuration. Depend ing on the require-ments of the problem, the feed field can be calculatedusing model formulas, measured data, or sophisticateddiffraction analysis programs.Any antenna parameter can be used as an optimizationvariable. For exam ple, the reflector surfaces expansioncoefficients,the position s, orientation s, and ex citations ofthe feed elem ents, and the antenna dimensions can all beoptimized simultaneously or separately.Performance of the synthesized antennas is evaluated bydiffraction analysis in the course of optimization itera-tions. Therefore , unlike the GO -related metho ds, there isno need for a final diffraction analysis.Using properly designed object functions, this diffractionsynthesis technique can be applied to optimize the an-tenna performance over a frequency band and a range ofscanning angles, with the CO- and cross-polarized fieldsproper ly taken into co nsideration . It is also capable ofsynthesizing various types of antennas such as high gainantennas, small-aperture low-noise ground stations, con-

e(C)

Fig. 11. Distortion compen sation using a deformable-shap ed ubreflectoranda single feed. (a) Antenna geometry. Subreflector:D,= 0.5 m, H , = 0.363m, eccentricity = 2.3. Feed: located at 1.415 m from the foca l point,Of = 17.58 degrees. (b) Deviation of the shaped subreflectorto the originalhyperboloid. (c) Far-field pattems. (Solid lines: 9= 0 degrees, dashed lines:@ = 90 degrees.)

toured beam satellite antennas, and multibeam antennas.Furthermore, it can be used to modify a portion of anexisting antenna system for the purpose of distortioncompensation or performance upgrade.With this method, limitations associated with the GO-shapingmethods are overcome to a large extent. The versatility andaccuracy of this method, however, are obtained at the price ofmore computation time. It is particularly so for very largemultiple-reflector antennas due to the time-consuming POintegrations. This hindrance is alleviated to some extent bythe thriving growth in computer speed and parallel compu-



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tation technology. At the same time, methods leading towardimproved com putation efficiencyare being investigated. Theseinclude advanced optimization techniques and algorithms thataccelerate the Po ield evaluation.APPENDIX

A . The ModiJiedJacobi Po lynom ials and Their Derivativ esDefinition of the modified Jacobi p olynomials, and formulasthat are convenient for numerical evaluation of the derivativesdF;( t) /dt are presented in this appendix.The modified Jacobi polynomials, F z ( t ) , are defined asWI, 1481

F z ( t )= J2(n+ 2m+ 1 ) P?*)(I - t 2 ) tn,O < t < l (27)

where P?) is a Jacobi polynomial. The Jacobi polynomials,generally denoted by Pp( (z) , are defined as [49]

They can be calculated most efficiently using the recurrencerelationsPoP)(z)1 (29)

Ppp(.) = -(a +p + 2). + s ( a - PI (30)2

In the diffraction analysis of reflector antennas, one typicallyha s to find the derivatives such as (15). For a reflector thatis represented by the m odified Jacobi polynomials F z ( t ) asshown in (21), the derivative dFG( t ) / d t must be calculated.Fo r this purpose, we use the recurrence relation for thederivatives of the Jacobi polynomials [49] to obtain that ofthe m odified Jacobi polynomialsd n m 1d t t n + 2m t(1- t 2 )-[FG(t)]= -F2 ( t ) - *-

Equation (32) is not convenient for computer program im-plementation because of the apparent singularities at t = 0,t = f l , n = m = 0, an d n + 2m - 1 = 0. To overcomethis difficulty, a stu dy on (32) for various combinations of theindices m an d n is conducted. As a result, it is found thatthe apparent singularities in (32) can be avoided using theformulas (33H36). These formulas consist of an algorithmwhich can be directly transformed into com puter code.When n = 0 an d m = 0,

dd tF Z ( t ) = 0 (33)

When n = 0 an d m = 1,2,3, .- . ,dd tF Z ( t ) = -4-

When n = 1 , 2 , 3 , . . . an d m = 0,(35)d- F z ( t ) = d m . tn-ld t

Whenn=1 , 2 , 3 , . . . a ndm=1 , 2 , 3 , . . . ,

B . Relationship Between the Modified Jacobi Polynomialsand the Circle Polynomials of ZernikeThe modified Jacobi polynom ials are closely related to the

circle polynom ials of & mike, R,(p),ha t are defined as [37]

(37)It can be shown that (the right arrows read corresponds to)

F $ ( t )+ ( - l ) - & T T . R$(p) (38)if the substitution

(39)n - mis made. Conversely, one may write

n + m , m + - I t + P

as a result of the substitutionn - m

2+ n , - m , p + t .It is obvious from these equations that, besides the no rmaliza-tion constant, the modified Jacobi polynomials and the circle



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DUAN AN D M A T - S A M 1 D IF FRA CT IO N T EC HN IQ UE FOR HIGH PERFORMANCE NTENNAS 39

polynomials of Zemke differ only in their index scheme.According to the authors experience, both index schemescan be applied to reflector antenna synthesis. Since the circlepolynomials of Zernike together with sinusoidal harmonicsform a com plete set of orthogonal basis functions for a circularregion, it follows that the surface expansion (21) is also anorthogonal global expansion.C . Closed-FormExpansion Coefficientscients can be determined byFor a given reflector surface ~ ( t ,), the expansion coeffi-

(43)Evaluation of these integrals is generally done numerically.For several typical reflector surfaces, however, one can findthe closed-form Jacobi-Fourier expansion coefficients,and twoexamples are given in Table I.

REFERENCES[ I] B. Y. Kinber, On two-reflector antennas, Radio E ng. EIec. Phys., pp.914-921, June 1962.[2] V. Galiido-Israel, Design of dual-reflector antennas with arbitraryphase and amplitude distributions, IEEE Trans. Antenn. Propuga t., vol.AP-12, no. 4, pp. 403-408, July 1964.[3] W. F. W illiams, High efficiency antenna reflector, Microwave J., vol.8, no. 7, pp. 79-82, July 1965.[4] V. Galiio-Israel, R. Mima, and A. G. Cha, Aperture amplitude andphase con trol of offset dual reflectors, IEEE Trans. Antenn. Propagut.,vol. AP-27, no. 2, pp. 154-164, Mar. 1979.[5 ] B. S. Westcott, F. A. Stevens, and F. Brickell, GO ynthesis of offsetdual reflectors,IEEProc., vol. 128, Pt.H, no. I, pp. 11-18, Feb. 1981.[6] B. S. Westcott, ShapedReflectorAntenna Design. London: ResearchStudies Press Ltd., 1983.[71 S. von Hoemer, Minimum-noise maximum gain telescopes and relax-ation method for shaped asymmetric surfaces, IEEE Trans. Antenn.Propagat., vol. AP-26, no. 3, pp. 464-471, May 1978.[SI J. J. Lee, L. I. Parad, and R. S. Chu, A shaped offset-fed dual-reflector antenna, IEEE Trans. Antennas Propugat., vol. AP-27, no.2, pp. 165-171, Mar. 1979.[9] M. Mehler, S. Tun, and N. Adatia, Direct far-field GO ynthesis ofshaped beam reflector antennas, IEE Proc. H, vol. 133, no. 3, pp.21>2u), June 1986.

[ IO] B. S. Westcott, F. Brickell, and I. C. Wolton, Crosspolar control infar-field synt hesis of dual offset reflectors, IEE Proc. H, vol. 137, no.I , pp. 31-38, Feb. 1990.

[ I I ] P.-S. Kildal, Synthesis of m ultireflector antennas by kinematic anddynamic ray tracing, IEEE Trans. Antenn. Propaga t., vol. 38 ,no. IO ,[I21 G. Bjantegaard and T. Pettersen, An offset dual-reflector antenna

shaped from near-field measurements of the feed hom: Theoreticalcalculations and measurements, IEEE Trans. Antenn. Propugat., vol.AP-31, no. 6, pp. 973-977, Nov. 1983.[I31 B. S. Westcott and F. Brickell, Geometric-optics synthesis of dual-reflector antennas with distributed sources, IEE Proc. H, vol. 136, no.

[14] D. W. Duan and Y. Rahmat-Samii, A generalized three-parameter (3 -P) aperture distribution for antenna applications, IEEE Trans. Antenn.Propagar., vol. 40, o. 6, pp. 697-713, June 1992.[151 A. R. Cherrette, S. W. Lee, and R. J. Acosta, A method for producinga shaped contour radiation pattem using a single shaped reflector and a

pp. 1587-1599, Oct. 1990.

5, pp. 361-366, Oct. 1989.

single feed, IEEE Trans.Antenn. Propagat., vol. 37 ,no. 6, pp. 698-706,June 1989.[I61 R. A. Shore and A. D. Yaghjian, Incremental diffraction coefficientsfor plane conformal strips w ith application to bistatic scattering from thedisk, J. Electromagn. Waves Appl., vol. 6, no. 3, pp. 359-396, 1992.[171 P. J. Wood, Reflector profiles for the pencil-beam Cassegrain antenn a,Marconi. Rev., vol. 35 , no. 185, pp. 121-138, 2nd quarter, 1972.[I81 F. Watanabe and Y. Mizugutch, An offset spherical tri-reflector an-tenna, Trans. IECE Japan, vol. E66,no. 2, pp. 108-115, Feb. 1983.[I91 S. Nomoto and F. Watanabe, Shaped reflector design for small-sizeoffset dual reflector antennas, Electron. C o m u n . Japan, vol. 72 , Pt.1, no. 1 1 , pp. 11-18, Nov. 1989.[20] J. Bergmann, R. C. Brown, P. J. B. Clarric oats, and H. Zhou, Synthesisof shaped-beam reflector antenna pattems, IEE Proc. H, vol. 135, no.

I , pp. 48-53, Feb. 1988.[21] B. Schlobohm and F. Amdt, Small earth station antenna synthesizedby a direct Po method, Space Commun., vol. 7, pp. 621-628, 1990.[22] Y. Rahma t-Samii and J. M umford, Reflector diffraction synthesis usingglobal coefficients optimization techniques, in IEEE AP-S Intl. Symp.,1989, pp. 1166-1169.[23] D. W. Duan and Y. Rahmat-Samii, Reflector antennas withsuperquadric aperture boundaries, IEEE Trans. Antenn. Propagu t.,vol. 41 , no. 8, Aug. 1993.[24] A. Michaeli, Elimination of infinities in eq uivalent edge cu rrents, part I:Fringe current components, IEEE Trans.Antenn. Propagat., vol. AP-34,no. 7, pp. 912-918, July 1986.[25] P. Y. Ufimtsev, Elementary edge waves and the pdiffraction, Electromagn., vol. 11 , no. 2, pp. 125-160[26] Y. Rahma t-Samii, Useful coordinate transformationscations, IEEE Trans. Antenn. Propaga t., vol. AP-27, W. 4,July 1979.[27] D. W. Duan and Y. Rahmat-Samii, Novel coordinate system androtation transformations for antenna applications, Electromagn., Jan.1995.[28] Y. Rahm at-Samii, Reflector antennas, in Antenna Handbook, Y. T . Loand S. W. Lee, Eds., Chapter 15. New Y o k Van Nostrand, 1988.[29] C. A. Balanis, Antenna Th eory Analysis ana Design. New York Harper& Row, 1982.[30] D. W. Duan, Y. Rahmat-Samii, and J. P. Mahon, Scattering from acircular disc: A comparative study of FTD and GTD echniques, IEEEProc., vol. 79 , no. 10, pp. 1472-1480, Oct. 1991.[31] D. W. Duan and Y. Rahmat-Samii, Axial field of a symmetricparaboloidal antenna: a POPTI) solution, in IEEE AP-S Intl. Symp.,Seattle, WA, June 1994.[32] J. B. Keller, Geometrical theory of diffraction,J. Opt. Soc. of America,vol. 52 , no. 2, pp. 116-130, Feb. 1962.[33] J. B. Keller, Diffraction by an aperture, J. Appl. Phys., vol. 28, pp.4 2 6 4 4 4 , Apr. 1957.[34] R. G. Kouyoumjian and P. H. Pathak, A uniform geometrical theory ofdiffraction for an edge in a perfectly conducting surface, IEEE Proc.,[35] D. S . Ahluwalia, R. M. Lewis, and J. Boersma, Uniform asymptotictheory of diffraction by a plane screen, SIAM J. Appl. Math., vol. 16,pp. 783-807, 1968.[36] S-W Lee and G. A. Deschamps, A uniform asymptotic theory ofelectromag netic diffraction by a curved wedge, IEEE Trans. Antenn.Propugat., vol. AP-24, no. 1, pp. 25-34, Jan. 1976.[37] M. Bom and E. Wolf, Principles of Optics, 6th ed. New York:Pergamon Press, 1980.[38] M. Johnson and Y. Rahm at-Samii, Genetic algorithm optimization andits application to antenna design, in IEEE AP-S Intl. Symp., Seattle,WA, June 1994.[39] D. W. Duan and Y. Rahmat-Samii, Reflector/feed(s) synthesis chal-lenges for sate llite contour beam applications, in IEEE AP-S Intl. Symp.,Chicago, IL, July 1992, pp. 297-300.[40]Y.Rahmat-Samii, D. W. Duan, D. Gin, and L. Libelo, Canonicalexamples of reflector antennas for high-power microwave applications,

IEEE Truns. Electromagn. Compat., vol. 34 , no. 3, pp. 197-205, Aug.1992.[41] R. A. Pearson, Y. Kalatidazeh, B. G. Driscoll, G. Y. Philippou, B.Claydon, and D. J. Brain, Application of contoured beam shapedreflector antennas to mission requirements, in Intl. Conf. Antenn.Propugat., 1993, pp. 9-13.[42] W. V. T. Rusch, Jr. A. Prata, Y. Rahmat-Samii, and R. A. Shore,Derivation and application of the equivalent paraboloid for classi-cal offset Cassegrain and Gregorian antennas, IEEE Trans. Antenn.Propugat., vol. 38 , no. 8, pp. 1141-1 149, Aug. 1990.

vol. 62 , pp. 1448-1461, NOV.1974.



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[43] A. G. Roederer and Y. Rahmat-Samii, Unfurlahle satellite antennas:a review, Annales des Telecommunications, vol. 44 , no. 9-10, pp.475-488, Nov. 1989.[44] . E. Freeland and G. Bilyeu. IN-STEP inflatable antenna experiment,in 43rd Congress Intl. Astronautical Federation, 1992.[45] Y. Rahmat-Sam ii, Array feed s for reflector surface distortion compen-sation: concepts and implementation, IEEE AP-S Magazine, vol. 32,no. 4, pp. 2C26, Aug. 1990.[46] S . von Hoerner, The design of correcting secondary reflectors, IEEETrans. Antenn. Propagat., vol. AP-24, no. 3, pp. 336340, May 1976.[47] V. Galindo-Israel and R. Mittra, A new series representation for theradiation integral with application to reflector antennas, IEEE Trans.Antenn. Propagat., vol. AP-25, no. 5, pp. 631-641, Sept. 1977.[48] Y. Rahmat-Sam ii et al., Computation of Fresnel and Fraunhofer fieldsof planar aperture and reflector ante nna by Jacohi-Bessel series-Areview, Electromagnetics, vol. 1, no. 2, pp. 155-185, Apr.-June 1981.[49] G. Szego, Orthogonal Olynomials, American Mathematical Society,1959.

Dah-Weih Duan (S88-M92) was born in Taipei,Taiwan on September 11, 1962. He received theB.S. degree in electrical engineering in 1985 fromNational Taiwan University, Taipei, Taiwan, andthe M.S. and Ph.D. degrees in electrical engineer-ing from University of Califomia, Los Angeles(UCLA), in 1988 and 1992, respectively.He joined International Business Machines Corp.,Thomas J. Watson Research Center, New York,in 1993, where he has worked on the researchand development of wireless systems for futureapplications. His current research interests are in antenna designs with amobile system perspective, antenna-device interactions, and computationalelectromagnetics.Dr. Duan is a membe r of the IEEE Antennas and Propagation Society.He has authored and co-authored over 25 journal and symposium papers. Hehas made contributions to the theory and design of reflector antenna systems,high frequency diffraction techniques, general aperture distributions, and highpower microwave antennas. In 1993,he received the Outstanding Ph.D. Awardfrom the School of Engineering and Applied Science, UCLA.

Yahya Rahmat-Samii (S73-M75-SM79-F85)received the M.S. and Ph.D. degrees in electn-cal engineering from the University of Illinois,Champaign-Urbana.He is a Professor of Electrical Engineering at theUniversity of California, Los Angeles. He has been aSenior Research Scientist at N ASAs Jet PropulsionLaboratory, California Institute of Technology since1978, where he contributed to antenna technologyfor space programs. He was a Guest Professor atthe Technical University of Denmark (TUD) in thesummer of 1986. He has also been a consultant to many aerospace companies.Dr. Rahmat-Samiis research interests include developments in near-fieldplane-polar and hi-polar antenna measurements, microwave holographic di-agnostics, mobile satellite communication antennas, reflector surface com-pensation, multireflector antenna diffraction analysis and synthesis, scatteringand radiation form complex objects, RCS com putations, singularity in dyadicGreens function, high power m icrowave (HPM ) antennas, EM P and aperturepenetration, the spectral theory of diffraction (STD) and GTD. For thesecontnbutions, he has received numerous NASA Certificates of Recognitionand recently earned the JPL Team NASAs Distinguished Group AchievementAward.He is a Fellow of IAE (1986) and was the 1984 recipient of the HenryBooker Award of URSI. He was appointed an IEE E Antennas and PropagationSociety Distinguished Lecturer and presented lectures internationally. Hewas an elected IEEE AP-S AdCom member for the second term andhas been an Associate Editor of the IEEE TRANSACTIONSN ANTENNASAN D PROPAGATIONnd the societys magazine. He IS currently the electedPresident of IEEE AP-S. He was the Chairman of the IEEE Antennas andPropagation Society of Los Angeles from 1987-1989. In 1989, his chapterreceived the Antennas and Propagation Best Chap ter Award from the APSociety. He is one of the three Intemational Editors of the IEE book serieson Electromagnetics and Antennas. H e is also one of the Editors of theJOURNALF ELECTROMAGNETICAVES ND APPLICATIONS.e is one of theDirectors of Antenna Measurements Technique Association (AMGA) and theElectromagnetics Society. He is listed in Whos Who in America, W hos Whoin Frontiers of Science and Technology, and Whos W ho in Engineering.Dr. Rahmat-Samii has authored or co-authored over 300 technical journalarticles and conference papers and has written chapters in 13 books. In 1992,he was the recipient of the Best Application Paper Award (Wheeler Award) fora paper published in the Trans actions in 1991. In 1993 and 1994, two of hisPh.D. students were named the Most Outstanding Ph.D. Student at UCLAsScho ol of Engineering and Applied Sc ience and another received the BestStudent Paper Award at the 1993 IEEE AP-S Symposium. He is a memberof Commissions A, B, and J of USNCKJRSI, Sigma Xi, Eta Kappa Nu , andthe Electromagnetics Academy.

A Generalized Diffraction Synthesis Technique for High Performance Reflector Antennas

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Transcript of A Generalized Diffraction Synthesis Technique for High Performance Reflector Antennas