Novel Gaussian beam method for the rapid analysis … IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 6, JUNE 2001 Novel Gaussian Beam Method for the Rapid Analysis of

880 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 6, JUNE 2001

Novel Gaussian Beam Method for the Rapid Analysisof Large Reflector Antennas

Hsi-Tseng Chou, Prabhakar H. Pathak, Fellow, IEEE, and Robert J. Burkholder, Senior Member, IEEE

Abstract—A relatively fast and simple method utilizingGaussian beams (GBs) is developed which requires only a few sec-onds on a workstation to compute the near/far fields of electricallylarge reflector antennas when they are illuminated by a feed witha known radiation pattern. This GB technique is fast, because itcompletely avoids any numerical integration on the large reflectorsurface which is required in the conventional physical optics (PO)analysis of such antennas and which could take several hourson a workstation. Specifically, the known feed radiation fieldis represented by a set of relatively few, rotationally symmetricGBs that are launched radially out from the feed plane and withalmost identical interbeam angular spacing. These GBs strike thereflector surface from where they are reflected, and also diffractedby the reflector edge; the expressions for the fields reflected anddiffracted by the reflector illuminated with a general astigmaticincident GB from an arbitrary direction (but not close to grazingon the reflector) have been developed in [1] and utilized in thiswork. Numerical results are presented to illustrate the versatility,accuracy, and efficiency of this GB method when it is used foranalyzing general offset parabolic reflectors with a single feed oran array feed, as well as for analyzing nonparabolic reflectorssuch as those described by ellipsoidal and even general shapedsurfaces.

Index Terms—Gaussian beams, reflector antennas.

I. INTRODUCTION

A FAST procedure based on a novel application of Gaussianbeams (GBs) is described in this paper for predicting the

near and far fields of electrically large reflectors, with slowlyvarying surface properties, when they are illuminated by a givenfeed antenna or a feed array. The conventional procedure forcomputing the fields of reflectors requires a numerical integra-tion of the physical optics (PO) integral over the reflector sur-face; this can become computationally very slow and highly in-efficient for large reflectors. Computational speed is particularlyessential when performing reflector antenna synthesis whereinthe reflector radiation pattern needs to be computed during eachiterative step of the synthesis algorithm until the desired radia-tion pattern is obtained to within some prescribed bounds. Thus,in such a synthesis procedure, the numerical PO method may be-come nearly intractable for large reflectors. The present GB pro-cedure for predicting the near and far fields of the reflector an-

Manuscript received November 19, 1996; revised March 6, 2000. This workwas supported in part by the Joint Services Electronics Program and the NationalScience Council of Taiwan.

H.-T. Chou is with Yuan-Ze University, Department of Electrical Engi-neering, Chung-Li, Taiwan.

P. H. Pathak and R. J. Burkholder are with The Ohio State University Elec-troScience Laboratory, Department of Electrical Engineering, Columbus, OH43212 USA.

Publisher Item Identifier S 0018-926X(01)03586-4.

tenna completely avoids any time-consuming numerical PO in-tegration on the reflector surface. As a result, the present methodis extremely fast as will be demonstrated for the analysis of largereflector antennas. A wide variety of applications of this GBmethod are presented here to illustrate its efficiency, versatility,and accuracy by analyzing not only offset parabolic reflectorswith a single feed or an array feed, but also nonparabolic reflec-tors made up, for example, of ellipsoidal and even more generalshaped surfaces.

Specifically, theelectromagnetic (EM)radiation fromaknownfeed, or from a known feed array, is represented by a setof relatively few GBs. The number of GBs required and thewaists of the GBs at their launch points, as well as their axialdirection at launch, are selected according to some simplecriteria. An efficient point matching procedure, or a least squaresmethod, can be employed for accurately determining the GBexpansion coefficients (i.e., the initial amplitudes of the GBsat their points of launching). Once these GBs are launchedfrom a single point, or in special cases from a very smallnumber of points in the feed plane, they strike the reflectorsurface where they undergo reflection and diffraction by thesurface of the parabolic reflector and its edge, respectively.Explicit, closed-form expressions for the fields associated withthe reflection and diffraction of any given incident GB by asmooth reflector containing an edge, which have been derivedin detail in a previous paper [1], are employed here. It isimportant to note that the closed-form expressions for the fieldsassociated with the reflection and diffraction of any single GBwhich is incident on the reflector have been developed in [1],via an asymptotic evaluation of the corresponding radiationintegral that is approximated therein by PO, for the singleGB illumination case. Thus, the GB analysis presented herefor predicting the total reflector near/far fields provides anasymptotic closed-form analytical solution of the PO integralfor reflectors; this is in contrast to the conventional numericalevaluation of the PO integral. Indeed, the present closed-formPO-based GB analysis is shown here to have essentially thesame level of accuracy as the numerical PO solution. It is wellknown that the PO approximation can be improved by usingan edge diffraction correction based on the physical theoryof diffraction (PTD) [2], [26], [3]; such a correction can beincorporated in a straightforward manner. Clearly, since thepresent GB analysis actually represents a closed-form solutionof the PO integral itself, this GB analysis can thus be correctedby simply adding to it the exact same PTD correction asis done for the conventional solution based on a numericalevaluation of the PO integral. In particular, the PTD edgediffraction correction can be added to the PO solution as a

0018–926X/01$10.00 © 2001 IEEE

CHOU et al.: NOVEL GAUSSIAN BEAM METHOD FOR THE RAPID ANALYSIS OF LARGE REFLECTOR ANTENNAS 881

line integral contribution around the reflector edge [4]–[10],[27]. Such a PTD correction provides an improvement tothe PO based analysis primarily in regions of lower fieldlevels. In the case of reflector antennas the PTD thus providessome improvement in the wide-angle sidelobe region. As analternative to the PTD-based correction to PO, one can firstemploy the present PO-based GB solution in the region ofthe main beam and first few sidelobes, and then switch to theuniform geometrical theory of diffraction (UTD) [10]–[13] raysolution in the region of the intermediate and far-out sidelobesof reflector antennas. One notes that the UTD ray solutionfails in regions of focused or nearly focused fields such aswithin the main beam regions of parabolic reflectors with afeed located at or near the reflector focus. Thus, the commonapproach for avoiding this difficulty of the UTD ray approachin focused regions of the radiated fields is to utilize the POapproximation in those regions where UTD fails; however,because the PO integral is evaluated numerically over the largereflector surface, it thus becomes highly inefficient as indicatedearlier. The UTD approach, on the other hand, requires nointegration on the reflector, and hence it is very efficient inits regions of validity. It is seen that the present GB approachretains all the advantages of the UTD ray method, and it hasthe important added advantage that it also remains valid inregions of focused or nearly focused fields where UTD fails.In fact, the present PO-based GB method may be viewed asa UTD for GBs which is analogous to UTD for rays exceptthat the rays are now replaced here by GBs (where the GBs,unlike rays, remain valid in regions of focused fields). APTD correction to this PO-based GB method, or alternativelyswitching from this GB to a UTD analysis outside the mainbeam region, can be accomplished in a straightforward fashionto improve the PO-based GB method as mentioned above;however, these have not been included in the present work.

The use of GBs as EM field basis functions has been studiedand demonstrated by several authors in the past, notably in[14]–[18] to represent the fields of aperture antennas. TheGabor expansion was used therein to represent the knownaperture field with Gaussian functions as basis elements inthat expansion. The field of each of these Gaussian elementsevolved from the aperture to the external region thereby directlyproviding the near and far fields of the aperture in closedform. The GB expansion employed here has some similaritiesto that in [14]–[16] but at the same time is also significantlydifferent from it in the manner in which the GBs are chosenas will be discussed below. The aperture is discretized in[14]–[16] using a doubly periodic array of lattice points onwhich the Gaussian basis elements are centered. The fieldsexternal to the aperture consist of a set of GBs (where each GBis described within the paraxial region of the fields radiated byeach Gaussian basis element). The GBs are launched radiallyoutward from each of the periodic array of points that arechosen to discretize the aperture as in Fig. 1. Such a fieldrepresentation constitutes a GB expansion in “phase-space.”When the period and of the lattice is specified in the

and directions, then the Gabor expansion automaticallyprovides information on the launching angle and the waistsize of the rotationally symmetric GBs that emanate from

Fig. 1. GB expansion for the fields radiated from a periodic array of latticepoints in the aperture defined overjx j � a=2; jy j � b=2; z = 0. The GBsare launched radially out from each lattice point in the periodic array. The latticepoints may typically correspond to the phase center of each subaperture (of sizeD � D ).

each lattice point. While there are an infinite number ofGBs launched from each of the finite number of latticepoints in [14]–[16], only the finite number of propagatingGBs are significant outside the close neighborhood of theaperture; all the remaining GBs are evanescent and contributenegligibly except for points very close to the aperture. Inpractice, only a few evanescent GBs are typically needed inaddition to the finite number of propagating GBs to accuratelysynthesize the original field distribution in the aperture. Thecoefficients of the GB expansion; i.e., the initial complexamplitudes of the GBs at launch, are found in [14]–[16] usingthe biorthogonality properties of the Gabor expansion. In [16],a Gabor-based GB expansion was also employed for analyzingthe transmission of the fields of a planar aperture throughplane and curved dielectric layers. However, previous workin using the Gabor representation was primarily restrictedto two-dimensional (2-D) geometries in which the aperturewas one-dimensional (1-D) (or linear) in extent; also, thediffraction of GBs was not included in those applications.An analysis of the reflector radiation pattern using a singleGB to model the feed radiation was developed in [17] forthe three-dimensional (3-D) problem, but the reflector edgeswere illuminated negligibly in that case also, and so thediffraction of the GB by the edges was excluded in [17].An analysis of GB diffraction by a straight edge has beenpresented previously in [19]–[22]. It is noted that the workpresented here is based on [1] which allows one to treatthe substantially more general and practical problem of GBreflection and diffraction by a general curved surface whichis truncated in a rather general but locally planar curved edge.

In the present application of the GB method, a procedure dif-ferent from the Gabor expansion is utilized to analyze the fieldsradiated by a 3-D offset parabolic reflector that is illuminatedby a realistic feed antenna as mentioned earlier. Furthermore,the feed illumination is such that the effects of diffraction bythe reflector edge are important, as is true in most applications,and hence cannot be neglected. It is noted that a Gabor expan-sion could also have been utilized in the present GB approachto represent the fields incident on the reflector from the given


feed antenna. An advantage of the Gabor type expansion is thatit guarantees completenessa priori and essentially removes anyarbitrariness in the choice of initial beam parameters once thelattice spacing is chosen. However, the Gabor expansion givesrise to nonidentical GBs with nonuniform angular or interbeamspacing at launch thus providing less control on the choice ofinitial beam parameters; also, the coefficients of the GB ex-pansion based on the Gabor representation generally requirea relatively complicated numerical computation of biorthogo-nality integrals. Therefore, a somewhat different approach isemployed here as it is very convenient for engineering appli-cations. Specifically, the set of GBs that are launched fromeach of the lattice points of Fig. 1 in the present approach aresuch that they have the same interbeam spacing,(definedlater) and, except for the different launching directions and po-larization, all of the GBs are otherwise identical and rotation-ally symmetric. Relatively simple rules are developed for thechoice of initial GB parameters in the present approach whichalso offer more control of these parameters. The computationof the GB coefficients (i.e., initial GB launching amplitudes)in the GB expansion are found through a relatively straightfor-ward and efficient point-matching approach, or a least squaresapproach. It is important to note that in the present approach, theinitial launching amplitudes of the GBs in the expansion for thefeed radiation are chosen to match the far zone fields of the feedantenna which illuminates the reflector, rather than to match thefeed antenna aperture field distribution. Consequently only theusual propagating GBs need to be included in the present ex-pansion and no evanescent GBs are required.

It is noted that, for most realistic single feed reflector antennaapplications, only one lattice point [ at in Fig. 1,where is the phase center of the feed in the feed coordinates( )] is sufficient. For feed arrays, it may be necessaryto have more than one lattice point if the reflector is not trulyin the far zone of the entire array. For offset parabolic reflec-tors whose projected apertures range from aboutto indiameter (where free space wavelength), it is seen in thepresent applications that the numberof GBs required is ap-proximately 45 with (corresponding to a single latticepoint at the phase center, of the feed antenna).

The format of this paper is as follows. In Section II, the gen-eral procedure for expanding the fields radiated by the feed an-tenna in terms of GBs is developed. The analysis of the near- andfar-zone radiation pattern of fairly general offset reflector an-tennas is presented next in Section III specifically using the rel-atively simple closed-form expressions developed earlier in [1]for the reflection and diffraction of an arbitrary GB by a curvededge in a curved surface. Several numerical results based on thisGB method are presented in Section IV for analyzing parabolic,ellipsoidal and even shaped reflectors; the computation of theseresults are shown to be extremely fast and accurate. In partic-ular, it is noted that the present approach can provide the nearand far fields of relatively large sized reflectors in a matter ofonly a few seconds on a workstation, whereas, the conventionalnumerical integration of the PO integral over the reflector sur-face requires several hours on the same workstation to furnishthe same results. An time dependence for the fields is as-sumed and suppressed.

Fig. 2. Coordinate system(x ; y ; z ) whose originO is the phasereference for the feed. Also, coordinates(x ; y ; z ) fixed in themth GB launched from thenth lattice point in thez = 0 feed plane.

II. GB EXPANSION FOR THEFEED ANTENNA RADIATION

Let the feed antenna, which illuminates the reflector, radiatethe electromagnetic fields . One can express the fieldsin the far zone of the feed antenna as

(1a)

(1b)

where the radiation pattern functions and in any direc-tion (with as in Fig. 2) are known. Hereis the freespace wavenumber and is the free space impedance .

One may represent in (1a) in terms of a GB expan-sion, i.e., in terms of a sequence of GBs, by

(2)

where propagating rotationally symmetric GBs are launchedfrom each of the lattice points in the feed plane which is the

plane as depicted previously in Fig. 1 such that the an-gular interbeam spacing [see (6)] between any pair of adja-cent GBs is a constant. A typicalth lattice point at where

is shown in Fig. 2. The direction is per-pendicular to the feed plane and it constitutes the feed boresightdirection. Comments will be added later to deal with generalrules for selecting the values of and . The magnetic fieldassociated with each rotationally symmetric GB basis function


in (2) is given by

(3)

where are local coordinates which arefixed in the th propagating GB launched from theth latticepoint with being the axial direction of that GB. Theadditional phase term in (2) simply references the phaseof launched from to the feed coordinate origin at thecentral point . In this work, it is of interest to evaluatein (2) to represent [on the left hand side of (2)] in thefar zone of the feed; hence, it suffices to choose the polarizationvector by

(4)

The above value of is valid within the paraxial region of theGB in (3); this is certainly true in the far zone of the feed. Fur-thermore, in this far-zone representation, the axis of everythGB beam emanating from all the lattice points will always beparallel to each other, so that the polarization in (4) is identicalfor every th GB launched from each of the lattice points.

The coefficients of expansions in (2) can be found in arelatively simple manner, since the feed pattern is known, usingthe point matching technique so that (2) can be reduced to a setof linear simultaneous equations that can be solved using stan-dard matrix inversion. For reflector feed antenna applications,the total coefficients are relatively small in numberso that the size of the matrix to be inverted is not large, therebymaking it very efficient to find . Alternatively, the canbe found via a least squares approach.

For the present application, it is important to note that oneis concerned here with the development of a GB expansion forrepresenting the radiation pattern of a feed antenna in whichthe feed is always relatively small in size as compared to thereflector which it illuminates so that it is essentially in the farzone of the feed; hence, for this application, and even for a feedarray which can be used to illuminate a reflector under similarconditions, it suffices to have just a single lattice point (i.e.,

only) at in Fig. 2 from where rotationally symmetricGBs are launched. Thus, there are a total of only

GBs in this case. The case where a large feedarray needs to be decomposed into subarrays is the topic of aseparate paper which also addresses the problem of reflectorfeed array synthesis in contoured beam applications [23]. Beforedetermining the number of GBs, one can define a cone of halfangle whose apex is at . This cone half angle ismeasured from the cone axis which is also the feed axis (orfeed boresight); it is chosen to adequately cover the solid angle

Fig. 3. Cone of half-angle� which encloses the reflector with a feed atO .The reflector has an offset as shown by the distanced. The projected reflectordiameter isD, and the reflector focal length isF .

subtended by the reflector edge from the point as shownin Fig. 3. It is important to note that the feed pattern needs tobe represented in terms of the GB expansion only within thiscone of half angle to calculate the fields scattered from thereflector surface; thus it is not necessary to find a GB expansionfor the feed radiation outside this cone. It is convenient next tointroduce the mapping

(5)

It is clear from (5) that and that

with . Since is undefined asit is convenient to let in this case. The angular inter-

beam separation is chosen to be a constant between all the GBs.In particular, this condition can be satisfied if the interbeam an-gular spacings and in the space satisfy thefollowing:

(6)

where is a constant. The axial direction of each of theGBs launched from the single point is shown by dots inthe spherically angular space of Fig. 4. The beamsin Fig. 4 can be regrouped into a set of beams in the

space. Thus, , and for each th GBthere is a corresponding pair of numbers .

According to Fig. 4, one allows to vary from to alongthe axis where

(7)

(8)

and denotes the integer portion of the argument. For agiven , the number of GBs in the direction is

(9)


Fig. 4. Dots indicate the directions of theM GBs in the space(� ; � ). This(� ; � ) space covers the region on the reflector surface and its neighborhoodwhich lies within the cone of half angle� of Fig. 3.

The index in the direction is chosen such that ifis an evennumber then

(10)

otherwise

(11)

Using (8) and (10) in yields

(12)

Up to this point, it is assumed that is known so that istherefore also known for a given . With this information,one can solve for the coefficients in (2) for the

case of interest using the point matching techniqueas discussed earlier.

For the case, one is referred to Fig. 1 which indicateshow GBs are launched from lattice points. As indicatedearlier, one may need if the reflector is not almost in thefar zone of a feed array. In such situations, usually suf-fices. Points of the rectangular lattice may correspond to thephase centers of subapertures into which the original aper-ture ; is discretized. This situation can betreated in exactly the same fashion as done for the single lat-tice ( ) case, because the GBs launched from eachthlattice point in Fig. 1 are required to representonly the radiation field which is generated by each of the corre-sponding th subaperture containing only theth lattice point.Furthermore, one can define a for each th lattice point

in the same way as defined in Fig. 4 for the case,except that in Fig. 3 must now be replaced by of Fig. 2.The procedure to find the amplitudes of the GBs launchedfrom the th lattice is then exactly analogous to that done abovefor the case.

It now remains to present some rules for determiningandthe parameter in (3) which indicates the size of the GB waist.Here, is the same for each of the identical, rotationally sym-metric GBs at launch. It was determined through numerical ex-perimentation in [6], [27] that

(13)

with (where free space wavelength). This resultwas obtained in two dimensions (2-D) by first assuming thatthe radiation pattern function (in this case,and ) is slowlyvarying with respect to . It was also assumed for the 2-Dcase that for a given GB in the far field, only the two nearest oradjacent GBs overlap significantly in the vicinity of the axis ofthe given beam. However, the condition in (13) is found, againvia numerical experimentation, to be also sufficient for guaran-teeing an adequate sampling of the feed pattern for the three-di-mensional (3-D) case of interest in this work. That this condi-tion in (13) should work well for 3-D is also due to the choiceof almost identical interbeam angular spacing as illustrated inFig. 4 where it is seen that the GBs along diametrically oppo-site sides of any hexagonal lattice resemble three overlappingGBs of the 2-D case mentioned earlier. The choice of the rela-tionship between and defined in (13) was found to giveexcellent reproduction of the radiation pattern when used in theGB expansion.

It therefore remains to determine. Firstly, one notes that thehalf width of each of the rotationally symmetric GBs launchedfrom the feed plane has a spot area at any axialdistance from the launch point, where for the special GBwhose axis is along (see Fig. 3) the is given by

(14)

A solution for in terms of is obtained from (14) as

(15)

where it is generally more advantageous to pick the smaller ofthe two values in (15) to ensure a smoother GB representationof the feed pattern with fewer beams as is evident from (13).The result in (15) provides a real valuedonly if

(16)

From (14) and (16), it follows that the spot size areais givenby

(17)


It was also indicated in [1, Appendix B] that a simpleclosed-form approximation is available to accurately locate thecomplex point of diffraction associated with the diffraction ofan incident GB by the edge of the reflector which it illuminates;this simple closed-form result is valid provided(with defined in [1], and also later in Section III of thispaper) which leads to

(18)

In (18), is the diameter of the circular boundary which cir-cumscribes the projected aperture projected in anyconstantplane by the reflector as for example in Fig. 3. Clearlymusttherefore satisfy both the above conditions, namely the ones in(17) and (18), simultaneously. Therefore, combining (17) and(18) yields

or

(19)

Consequently, one may pick a convenient value ofwhich sat-isfies (19); this value of in turn provides a choice for the valueof via (15) and hence also for via (13) thereby specifyingeach GB completely at launch. For more complex feed patterns,one can always decrease from its value in (13) while keeping

the same as in (15); this allows the use of more GBs that maybecome necessary in this case to more accurately reproduce thecomplexity in the pattern. In general, the conditions in (13) and(15) are sufficient for most types of simple feeds.

III. SCATTERING OF EACHGB IN THE EXPANSION FOR THE

FEED RADIATION WHICH ILLUMINATES THE REFLECTOR

SURFACE

Once the far-zone magnetic field radiated from the feed an-tenna is expanded in terms of a set of GBs as shown in Section II,then each GB which is launched from the feed plane in this ex-pansion now illuminates the reflector from where it is scattered.It is noted that the reflector may not be fully in the far zone of thefeed [as in the case where in (2)] even though the GBexpansion is obtained from a knowledge of the far zone feedradiation pattern, and the GB expansion thus obtained remainsvalid even within the near zone (but excluding the extreme nearzone of the feed). The net field scattered by the reflector whenit is illuminated by the feed antenna is thus represented as a su-perposition of the scattered fields produced by each GB whenthey become incident on the reflector. The explicit results for thescattering of an arbitrary GB by a smooth, electrically large re-flecting surface containing an edge were developed in [1]; theseresults which were obtained via an asymptotic evaluation of thePO integral for the scattered field are summarized below forconvenience.

In [1], it was assumed that the surface of the original reflectorcould be approximated by a local paraboloid at the point

(a)

(b)

Fig. 5. (a) The projected area of the parabolic surfaceS on thez = 0 planeand (b) its mapping to the(� ; ) coordinate system. The quantities� ; �and� are explicitly shown.

where the th incident GB axis intersects the reflector surface(or its mathematical extension past the edge). The local parab-oloid at is given by

(20)

in which is normal to the original surface at , andare the original principal surface directions at ,

respectively, as in Fig. 5. Also, is a unit vector normal to theplane containing the edge with the principal radii of curvatures

, given by

(21)

where denotes the coordinate system for thelocal parabolic surface at . It is assumed that the incidentmagnetic field of the th rotationally symmetric GB incidentat is given by [1]

(22)


where is the position vector in theincident GB coordinate system which is also centered atand is the incident magnetic field at , i.e., at

. The , , and are the elements ofthe GB complex curvature (or phase) matrix at .In particular, (22) reduces at to the following:

(23)

with

(24a)

and

(24b)

It is clear that appearing in (23) and (24a) represents the dis-tance associated with the th GB. The transformationmatrix between the coordinates fixed in the incident GB and thelocal surface coordinate system both of which are centered at

is

(25)

Clearly, the transformation matrix in (22) is known. Thescattered field which is produced when a GB is incidenton the local parabolic surface can be written asymptotically forlarge as

(26)

where represents the reflected field contribution to thescattered field which actually would be the only contribution ifthe edge were absent. Also, represents the edge diffractedfield contribution, and is a transition function that is ob-tained via uniform asymptotics, which provides a modificationto the reflected component of the field resulting from the pres-ence of the edge. Indeed, when the edge is present, only a partof the incident GB is reflected from the surface, and the rest isdiffracted by the edge; thus, the effect of the surface reflectionis modified by the edge and the extent of this modification de-pends on the proximity of to the edge as measured by theparameter of the transition function . It is noted that

denotes either the scattered electric field or the scatteredmagnetic field according to the function which isdefined below. The individual field components of are sum-marized in detail next.

A. Reflection Contribution

The reflected field is given by [1]

(27)

where

for

for .

(28)

and

(29)

In (23), denotes the distance between the observation pointand the complex stationary or reflection point on the complexextension of the surface at in the coordinatesystem at , and denotes its complex direction pointing to-ward the observer at also in the system.It is noted that the quantities and aregiven in closed form as

(30a)

(30b)

(30c)

and

(31)

in which the coefficients and are

(32)


(33)

(34)

(35)

(36)

(37)

B. Edge Diffraction Contribution

The edge diffracted field is given by [1]

(38)

where

(39)

(40)

(41)

and

(42)

The coordinates above pertain to a complex saddlepoint associated with edge diffraction. It is convenient to employpolar coordinates in which to express ; in particular,let

(43a)

(43b)

At this saddle point, the values of and are denotedby and , respectively, or

(44a)

(44b)

Fig. 6. Near-field pattern at 5 GHz of an offset parabolic reflector illuminatedby a Huygens source located at the focus where(D = 40 ; F = 36 ; d =0 ; z = 40 ). Forty-eight GBs are used.

where as in Fig. 6(a) and (b) with

(45)

(46)

(47)

and

(48)

(49)

The derivatives and are explicitly given in [1] by

(50)

and

(51)

It is noted that the saddle point which corresponds toin an appropriate polar coordinate system [1] is the

solution of

(52)

where are given by [1]

(53)

(54)


(55)

(56)

(57)

with and as

(58)

(59)

(60)

(61)

The approximate but accurate closed-form solution of (52) isshown in [1], and it will not be repeated here.

C. Uniform Transition Function

The transition function is given by [1]

erf (62)

where denotes the Error function erfand the parameter depends on

the complex distance between the complex saddle point forreflection and the complex saddle point for edge diffraction[see (63) at the bottom of the page], which is given in [1]. Thechoice for the branch of the square root is based on a Taylorexpansion around the complex saddle point for reflection, i.e.,

sign Re sign (64)

where

(65)

and denote a set of appropriate polar coordinatevalues [1] corresponding to . Thus,

(66)

Fig. 7. Near-field pattern at 5 GHz of an offset parabolic reflector illuminatedby a Huygens source located at the focus where(D = 40 ; F = 36 ; d =0 ; z = 100 ). Forty-eight GBs are used.

with . In (65) and (66), is given by [1]

(67)

However, it issuggestedthat forsmalltheTaylorexpansionforof (63), which is essentially shown in (64), may be employed

directly to avoid any numerical error in the transition region.

IV. NUMERICAL RESULTS AND DISCUSSION

In this section, some numerical results are presented for theGB-based analysis of the fields radiated from an offset parabolicreflector illuminated by a single feed or feed array, as well asfrom an offset ellipsoidal reflector, and from a shaped reflectordesigned with a single feed to produce a contoured beam for thecontinentalUnitedStates(CONUS).Theaccuracyandefficiencyof thepresentGBapproach isalsodiscussedbycomparing itwitha reference solution obtained via a conventional direct numericalevaluation of the PO integral over the surface of the reflector.

The first case of interest here is the investigation of the nearfield pattern of an offset parabolic reflector. The geometry forthis case is shown in Fig. 3 where the diameterof the pro-jected aperture is given by with , and thefocal length . Also is the location of the outputplane, and the near-field patterns are in the plane with

. The feed antenna is a simple Huygens source located atthe focus, and the frequency is 5 GHz. The Huygens source istilted so that its boresight (or axis) makes an anglein theplane given by , and its electric current moment lies inthe plane. Figs. 6 and 7 show the near field patterns of

(63)


(a)

(b)

(c)

Fig. 8. Co-polar and cross-polar far-field pattern at 10 GHz of an offsetparabolic reflector used in Fig. 6 when it is illuminated by a Huygens sourcelocated at the focus. 32 GBs are used. (a)� = 0 plane (co-polar). (b)� = 90 plane (co-polar). (c)� = 0 plane (cross-polar). Note that there isno cross-polar field in the� = 90 plane.

this reflector at two different output planes. In particular, Fig. 6shows the pattern in the output plane located at whichis relatively very close to the reflector, whereas Fig. 7 shows thepattern in the output plane located at which is still inthe near zone but much farther from the reflector. Both Figs. 6and 7 show very good agreement with the reference solution. Itis noted that the present GB approach is based on the asymptotic

(a)

(b)

Fig. 9. Far-field pattern at 10 GHz of an offset parabolic reflector used in Fig. 6when it is illuminated by a defocussed Huygens source located at(0:; 0:; 30 ).Thirty-two GBs are used. (a)� = 0 plane. (b)� = 90 plane.

Fig. 10. CPU time required on a silicon graphics workstation for the referencenumerical PO integration-based far field computation compared with that basedon GB expansion. The time shown is for computing 1802 pattern points for thereflector used in Figs. 6–9. Note that the vertical scale is logarithmic.

development in [1] that employs a Fresnel approximation withinthe PO radiation integral for the scattered field produced by eachincident GB. It is expected that some errors will occur if thefield point is not in the Fresnel zone over the spot area on the


(a)

(b)

Fig. 11. Far-field pattern at 121.803 GHz for the parabolic reflector utilizedin Figs. 6–9 except that it is illuminated by a9� 9 element rectangular arraywith an interelement spacing of a half wavelength. The center of the array isat the focus of the parabolic reflector. (a)� = 0 plane. (b)� = 90 plane.Eighty-nine GBs are used.

reflector which is made by each GB in the feed expansion. Fig. 7shows the improvement in accuracy when all the field points areproperly located in the Fresnel zone at a larger distance

as compared to . Also, in the present application,the spot area of the incident GB on the surface of the reflectoris assumed to be small compared with the total surface area ofthe offset reflector as discussed in Section II. The total numberof GBs used is 48 in Figs. 6 and 7.

The second case of interest is the computation of the far-fieldpattern for the same reflector as the one used in Figs. 6 and 7.Again, the Huygens source feed is at the focus and its axis istilted at an angle in the plane as before for this reflectorwhose far zone pattern is shown in Fig. 8 at 10 GHz. When thefeed is defocussed, the far-zone pattern changes to that in Fig. 9.It is noted that, being an asymptotic high-frequency solution,the accuracy of the GB based solution increases with frequency.

Fig. 12. Near-field pattern at 11.803 GHz of an offset ellipsoidal reflectorilluminated by a Huygen’s source placed on the axis at a distance of 70 infrom the origin (see Fig. 3). Note that (D = 80 in, d = 0 in, z = 100 in).Approximately 50 GBs are used.

Fig. 13. Far-field pattern at 11.803 GHz of an offset ellipsoidal reflectorilluminated by a Huygen’s source placed on the axis at a distance of 70 in fromthe origin (see Fig. 3). Note that (D = 80 in, d = 0 in) for this case, whichis the same as for Fig. 12 except that this figure depicts the far-field pattern.Approximately 50 GBs are used.

For the near-field case in Figs. 6 and 7, a total of 48 GBs is usedas indicated above; in the far zone one can further reduce thenumber of GBs to 32 as in Figs. 8 and 9 and this number of GBsstays the same even when the frequency is doubled. It is notedthat Fig. 8(c) indicates the cross-polar field radiated in the farzone by the reflector. The CPU time for the GB approach is ap-proximately 10 s on a Silicon Graphics workstation for a totalof 1802 field point samples used in each of the plots of Figs. 8and 9; on the other hand, the CPU time for the reference numer-ical PO solution increases rapidly with frequency as shown inFig. 10, and is typically given in terms of hours. The numericalPO integration code used here, to provide the reference solu-tion, employs a surface discretization in which the surface el-ements are square wavelengths in size. It is possibleto improve the efficiency of this numerical PO code somewhat;


Fig. 14. Normalized co-polarization contours based on the GB approach for CONUS coverage by a shaped concave reflector with a(cos � ) feed pattern at 12GHz with l = 18:51. Approximately 200 GBs were used.

Fig. 15. Normalized co-polarization gain contours based on the numerical PO integration approach for the same shaped reflector case as in Fig. 14.

however, even with that improvement, the numerical PO codewill still require significantly much larger computational time,especially for higher frequencies, than the novel GB method ofsolution presented here.

The third case considered here is the computation of the farfield pattern of the same parabolic reflector as the one utilized in

the computations of Figs. 6–9, except that it is now fed by aelement planar square antenna array whose center is at the focusand whose boresight points toward the center of the reflectorat an angle as in the case of Figs. 7–9. Each radiating arrayelement consists of Huygens source identical to that employedfor the calculations in Figs. 6–9. The interelement array spacing


TABLE IVALUES FOR THECOEFFICIENTSC AND D OF THESURFACE MODIFIED

JACOBI POLYNOMIAL -BASED EXPANSION FOR ASHAPED REFLECTOR THAT

PRODUCES ACONUS CONTOUREDBEAM SHOWN IN FIGS. 14AND 15

is half a free space wavelength in this case. The total number ofGBs employed here is 89. Also, in this case, and the89 GBs are launched from the single lattice point at the focus.The far field pattern is computed at 11.803 GHz in theand planes, respectively, for this array fed parabolicreflector case; this pattern is shown in Fig. 11 along with thereference numerical PO solution.

The fourth case considered is the near field pattern at 11.803GHz of an offset ellipsoidal reflector fed by a Huygens sourcewhich is located on the axis (or the axis of symmetry of theparent ellipsoid) at a distance from the coordinateorigin in Fig. 3. The value of , and the offset distance ofFig. 3 are indicated below in Fig. 12 for this case. The nearfield result shown in Fig. 12 is evaluated at in the– (or ) plane and compared with the numerical PO

reference solution. It is noted that the feed tilt angleis chosenso that the feed axis (or boresight) strikes the center of the offsetellipsoidal reflector surface. A total of approximately 50 GBsare used. Fig. 13 shows the far field pattern in the planefor the same reflector as in Fig. 12 and at the same frequency,namely 11.803 GHz.

The final case considered here is the calculation of thefar-zone contoured beam produced by a shaped concave re-flector designed for CONUS coverage. The reflector shape wasobtained from elsewhere [24] and was given as an expansionin terms of a product of the modified Jacobi polynomials

and trigonometric functions. Such a surface representation isbeing commonly used for describing shaped reflectors and itis specifically given in [25]. The coefficients andof this surface modified Jacobi polynomial based expansion,

as in [25], where the indicesand here are distinct from andnot to be confused with the GB expansion of Section II, aresummarized below in Table I. A type feed pattern isassumed where . This shaped reflector was designedin [24] to produce three gain zones over the CONUS coverageregion for compensating rain attenuation exactly as shown in[25, Fig. 4]. Referring to Fig. 3, it is noted that this generalconcave shaped reflector has , , feedat a distance of from the origin. The feed tiltangle for this case. Normalized constant far zonegain contours are shown in Fig. 14 for this CONUS coverage.The GB-based calculation of the contoured beam over anangular (elevation azimuth) grid of points to coverthe CONUS region is shown in Fig. 14. The correspondingnumerical PO-based reference solution for this case is shownin Fig. 15 for comparison. Both Figs. 14 and 15 show nor-malized gain contours for co-polarized pattern contributions;the cross-polarized contributions are also found to agree justas well as the co-polarized ones, but are not shown here forspace limitations [see Fig. 8(c) for the cross-polar radiationfor a different case]. The GB method used approximately 200GBs and took less than 5 minutes of CPU time, whereas thenumerical PO method used to provide reference results forcomparison took about 6 h for the same calculations.

It is clear that the present GB method for analyzing reflectorantennas is accurate, very fast, physically appealing, and versa-tile. From the result of Fig. 14, it is obvious that the GB methodshould be extremely fast for also synthesizing shaped reflec-tors in contoured beam applications. The application of this GBmethod to shaped reflector synthesis is currently under study. Inaddition, the extension of this GB method for an efficient anal-ysis/synthesis of dual reflectors is also currently under study.

REFERENCES

[1] H.-T. Chou and P.H. Pathak, “Uniform asymptotic solution for the EMreflection and diffraction of an arbitrary Gaussian beam by a smoothsurface with an edge,”J. Radio Sci., vol. 32, no. 4, pp. 1319–1336,July–Aug. 1997.

[2] P. Y. Ufimstev, “Method of edge waves in the physical theory of diffrac-tion,” Translation prepared by the U.S. Air Force Foreign TechnologyDivision, Wright-Patterson AFB, OH, released for public distributionSept. 7, 1971.

[3] D. I. Butorin, N. A. Martynov, and P. Ya. Ufimstev, “Asymptotic Expres-sions for the Elementary Edge Waves,,”Soviet J. Comm. Tech. Electron.,vol. 33, no. 9, pp. 17–26, 1988.

[4] P. H. Pathak, “High frequency techniques for antenna analysis,”Proc.IEEE, vol. 80, pp. 44–65, Jan. 1992.

[5] K. M. Mitzner, “Incremental length diffraction coefficients,” AircraftDiv., Northrop Corp., Tech. Rep. AFAL-TR-73-296, April 1974.

[6] A. Michaeli, “Elimination of infinities in equivalent edge currents—PartI,” IEEE Trans. Antennas Propagat., vol. AP-34, pp. 912–918, July1986.

[7] E. F. Knott, “The relationship between Mitzners ILDC and Michaelisequivalent currents,”IEEE Trans. Antennas Propagat., vol. AP-33, pp.112–114, Jan. 1985.

[8] E. F. Knott and T. B. A. Senior, “Comparison of three high-frequencydiffraction techniques,”Proc. IEEE, vol. 62, pp. 1468–1478, Nov. 1974.


[9] R. A. Shore and A. D. Yaghijian, “Incremental diffraction coefficientsfor planar surfaces,”IEEE Trans. Antennas Propagat., vol. 36, pp.55–70, Jan. 1988.

[10] P. H. Pathak, “Techniques for high frequency problems,” inAntennaHandbook, Theory Application and Design, Y. T. Lo and S. W. Lee,Eds. New York: Van Nostrand Reinhold, 1988.

[11] R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory ofdiffraction for an edge in a perfectly conducting surface,”Proc. IEEE,vol. 62, pp. 1448–1461, Nov. 1974.

[12] S. W. Lee and G. A. Deschamps, “A uniform asymptotic theory of EMdiffraction by a curved wedge,”IEEE Trans. Antennas Propagat., vol.AP-24, pp. 25–34, Jan. 1976.

[13] D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe,Introduc-tion to the Uniform Geometrical Theory of Diffraction. Norwood, MA:Artech House, 1990.

[14] P. D. Einziger, S. Raz, and M. Shapira, “Gabor representation and aper-ture theory,”J. Opt. Soc. Amer., vol. 3, pp. 508–522, Apr. 1986.

[15] J. Maciel and L. B. Felsen, “Systematic study of fields due to extendedaperture by Gaussian beam discretization,”IEEE Trans. Antennas Prop-agat., vol. 37, pp. 884–892, July 1989.

[16] , “Gaussian beam analysis of propagation from an extended planeaperture distribution through plane and curved dielectric layers, Parts Iand II,” IEEE Trans. Antennas Propagat., vol. 38, pp. 1607–1624, Oct.1990.

[17] F. J. V. Hasselmann and L. B. Felsen, “Asymptotic analysis of parabolicreflector antennas,”IEEE Trans. Antennas Propagat., vol. AP-30, pp.677–685, 1982.

[18] R. J. Burkholder and P. H. Pathak, “Analysis of EM penetration into andscattering by electrically large open waveguide cavities using Gaussianbeam shooting,”Proc. IEEE, vol. 79, pp. 1401–1412, Oct. 1991.

[19] E. Heyman and R. Ianconescu, “Pulsed beam diffraction by a perfectlyconducting wedge: Local scattering models,”IEEE Trans. AntennasPropagat., vol. 34, pp. 519–528, 1995.

[20] G. A. Suedan and E. V. Jull, “Two-dimensional beam diffraction by ahalf-plane and wide slit,”IEEE Trans. Antennas Propagat., vol. 35, pp.1077–1083, 1987.

[21] , “Beam diffraction by planar and parabolic reflector,”IEEE Trans.Antennas Propagat., vol. 39, pp. 521–527, 1991.

[22] H. T. Anastassiu and P. H. Pathak, “High-frequency analysis of Gaussianbeam scattering by a two-dimensional parabolic contour of finite width,”Radio Science, vol. 30, no. 3, pp. 493–503, May–June 1995.

[23] H.-T. Chou, P. H. Pathak, and R. J. Burkholder, “Feed array synthesis forreflector antennas in contoured beam applications via an efficient andnovel Gaussian beam technique,” Radio Science, submitted for publica-tion.

[24] W. Theunissen, private communication, July 1998.[25] D.-W. Duan and Y. Rahmat-Samii, “A generalized diffraction synthesis

technique for high performance reflector antennas,”IEEE Trans. An-tennas Propagat., vol. AP-43, Jan, 1995.

[26] P. Y. Ufimstev, “Method Drayevykh Voin V Fizicheskoy TheoriiDifraktsi,” Izd-Vo Sov. Radio, pp. 1–243, 1962. Russian version of [2].

[27] A. Michaeli, “Elimination of infinities in equivalent edge currents—PartII,” IEEE Trans. Antennas Propagat., vol. AP-34, pp. 1034–1037, Aug.1986.

Hsi-Tseng Chouwas born in ChangHua, Taiwan, in 1966. He received the B.S.degree from National Taiwan University in 1988 and the M.S. and Ph. D. degreesfrom The Ohio State University, Columbus, in 1993 and 1996, respectively, allin electrical engineering.

He joined the Department of Electrical Engineering, Yuan Ze University,Taiwan, in August 1998 and is currently an Assistant Professor there. He workedwith China Ryoden Co., Ltd. in 1990 for one year and then joined the Electro-Science Laboratory at The Ohio State University as a Graduate Research As-sociate from 1991 to 1996. After receiving his Ph.D. degree in 1996, he con-tinued his work at the ElectroScience Laboratory as a Post-Doctoral Researchertill 1998. His current research interests include antenna design, electromagneticscattering, development of asymptotic high-frequency solutions in the formatof the uniform geometrical theory of diffraction (UTD), novel Gaussian beamtechniques for the application of reflector antenna analysis/synthesis, and hy-brid MoM-UTD type solutions for periodic structures.

Dr. Chou received a Young Scientist Award from the International Union ofRadio Science(URSI) and a best paper award from the ElectroScience Labora-tory both in 1999, and has been invited to present papers at many internationaltechnical conferences. Dr. Chou is a member of the IEEE Antennas and Propa-gation Society and an elected member of US Commission B of URSI.

Prabhakar H. Pathak (M’76–SM’81–F’86) received the B.Sc. (hons.) degreein physics from St. Xaviers College, Univ. of Mumbai, India, in 1962, the B.S.degree in electrical engineering from the Louisiana State University, BatonRouge, and the M.S. and Ph.D. degrees in electrical engineering from TheOhio State University, Columbus, in 1970 and 1973, respectively.

He was an Instructor in the Dept. of Electrical Engineering at the Univ. of Mis-sissippi during the academic year 1965–66, before coming to The Ohio StateUniversity. Since 1973 he has been at Ohio State University ElectroScienceLab., Dept. of Electrical Eng., where he is currently a Professor. Dr. Pathak hascontributed to the development of the uniform geometrical theory of diffrac-tion (UTD) and some of its extensions, all of which are being widely used in theanalysis of antenna and scattering problems associated with complex structures,such as aircraft, spacecraft and ships. Some of this research has also been in-volved with an analysis of the problems of diffraction by discontinuities in thegeometrical as well as in the electrical properties of a surface; the latter cate-gory includes surface wave structures. In addition, he has been involved withthe development of efficient hybrid methods for analysis of microstrip typeantennas, and more recently for dealing with complex radiating structures in-cluding open-ended cavities. Currently, his work continues to be in the areasof asymptotic high-frequency methods and the analytical inversion of the so-lutions obtained therefrom into the time domain to arrive at a time dependentprogressing wave picture for transient radiation and scattering, and also in thedevelopment of novel and fast methods of analysis/synthesis of electromagneticradiation by electrically large general reflector antenna systems and large-sizedfinite arrays. His research interests are broadly in the areas of electromagnetictheory, mathematical methods, antennas, and scattering. He has participated inseveral invited lectures and short courses on the uniform geometrical theory ofdiffraction and other high-frequency methods, both in the U.S. and abroad. Hehas authored and co-authored chapters on the subject of high-frequency diffrac-tion for five books.

Dr. Pathak is a member of Sigma Xi and a member of the U.S. Commission Bof URSI. He was named an IEEE Antennas and Propgation Society (AP-S) dis-tinguished lecturer for a three-year term beginning in 1991. Currently he servesas the chair of the IEEE AP-S Distinguished Lecturer Program. He received the1996 Schelkunoff Best Paper Award from IEEE AP-S, and also the George Sin-clair Award in 1996. He is the recipient of the IEEE Third Millennium Medalwhich was awarded in July 2000 by the AP-S.

Robert J. Burkholder (S’85–M’89–SM’97) received the B.S., M.S., and Ph.D.degrees in electrical engineering from The Ohio State University, Columbus, in1984, 1985, and 1989, respectively.

From 1989 to the present, he has been with The Ohio State University Electro-Science Laboratory, Department of Electrical Engineering, where he currentlyhas the titles of Research Scientist and Adjunct Associate Professor. His re-search specialties are high-frequency asymptotic techniques and their hybridcombination with numerical techniques for solving large-scale electromagneticradiation and scattering problems. He has contributed extensively to the EManalysis of large cavities, such as jet inlets/exhausts, and is currently workingon the more general problem of EM radiation, propagation, and scattering inrealistically complex environments.

Dr. Burkholder is a full member of URSI Commission B, and a member ofthe Applied Computational Electromagnetics Society (ACES).

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