Shaped Antennas

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698 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 31,NO. 6 , J U N E 1989

A Method for Producing a Shaped ContourRadiation Pattern Using a Single ShapedReflector and a Single FeedAbstract-Eliminating the corporate feed network in shaped contour

beam antennas will reduce the expense, weight, and RF loss of theantenna system. One way of producing B shaped contour beam withoutusing a feed network is to use a single shaped reflector with a single feedelement. For a prescribed contour beam and feed, an optimizationmethod for designing the reflector shape i s given. As a design example, ashaped reflector is designed to produce a continental United States(CONUS) coverage beam. T he RF performance of the shaped reflector isthen verified by physical optics.

~ Parabolic Reltector/

\ ,eam Forming NetworkI. INTRODUCTION

N MANY APPLICATIONS, spacecraft antennas requireI adiation patterns to be shaped such that the pattern contourfits the shape of the desired coverage region. A commonexample is to design the constant gain levels of the pattern tofollow the projected boundary of the continental United States(CONUS) as viewed from synchronous orbit. The shapedcontour radiation pattern reduces wasted transmitted power byminimizing the illumination of unwanted areas such as theocean in this example.

The two most popular designs for producing shaped contourradiation patterns are the array-fed parabolic reflector [13, [ 2 ]and the direct radiating planar array [3] (see Figs. l(a ) andl e ) , respectively). If the particular application calls for asingle fixed shaped beam (no electronic scanning is needed),then both approaches generally employ passive beamformingnetworks to properly weight the array elements. There are,however, several disadvantages to passive beamforming net-works. One drawback is that a major part of the cost of theantenna is in the construction and tuning of the beamformingnetwork; construction and tuning become more difficult as thefrequency is increased. Another disadvantage is the RF lossassociated with passive beamforming networks. In general,the network loss also becomes larger with increased fre-quency,

It is, however, possible to generate a shaped contourradiation pattern without using a beamforming network. This

Manuscript received November 24, 1987; revised June 30 , 1988. This workwas supported by NASA Grant NAG 3419.A . R . Cherrette was with the Department of Electrical and ComputerEngineering, University of Illinois, Urbana, IL . He is now with Space andCommunication Group, Hughes Aircraft Company, Mail Station W3 19 ,Building S-12, 1950 East Imperial Highway, El Segundo, CA 90245.S . W . Le e is with the Department of Electrical and Computer Engineering,University of Illinois, Urbana, IL 61801.R. Acosta is with NASA-Lewis Research Center, Cleveland, OH 44135.IEEE Log Number 8927256.

(a )Planar Amay

Lti\ Beam Form ing Network(b ), Single Shaped Reflector

\- ingle Feed(C )array. (c) Single shaped reflector with a single feed.Fig. 1. (a) Parabolic reflector with an array feed. (b)Direct radiating planar

is accomplished by using a single feed element and a singleshaped reflector (see Fig. l(c)) . In this case, the far-fieldradiation pattern is configured to the desired shape by properlyshaping the reflector surface. Using this design, a relativelyinexpensive, low-loss shaped beam antenna can be producedwith no electrical tuning other than alignment of the feedelement and reflector.

The subject of single reflector shaping for arbitrarily shapedcontour beams has been an active research area since thebeginning of the 1970s. One of the earliest methods consistsof a wavefront synthesis technique [4]. In this method, thewavefront of the far-field radiation pattern is assumed to becomposed of two parts. The inner part is a spherical wave

OO18-926X/89/0600-0698$01 OO O 1989 IEEE


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CHERRETTE et al. : PRODUCING SHAPED CONTOUR RADIATION PATTERN 69 9limited in angular extent by the approximate shape of thedesired coverage region. The outer part is a ruled surface withthe contour of the boundary of the inner part as its directrix.The wavefront of the feed element is assumed to be spherical.The reflector surface is then completely determined from theincident and reflected wavefronts by applying the principles ofgeometric optics [ 5 ] , [8], [9]. This method yields good resultsprovided design parameters (such as the thickness of the outerpart of the wavefront) are optimized. The design parameteroptimization is necessary because there are usually differencesbetween the desired radiation pattern and the actual radiationpattern calculated from the generated reflector surface. Thisdifference exemplifies the difficulty of handling the geometricoptics caustic associated with the outer part of the reflectedwavefront.

A later method of reflector shaping replaces the wavefrontsynthesis with a more rigorous aperture phase synthesis [6]. Inthis method, the aperture phase distribution is optimized (for afixed aperture amplitude distribution) to shape the far-fieldpattern. The reflector surface is then determined from theincident and reflected wavefronts in a manner similar to thatgiven by Katagi [4]. Aperture phase synthesis eliminates thecaustic problem and allows more pattern control (i .e. , sidelobesuppression) than wavefront synthesis. However, one problemwith aperture phase synthesis is that the aperture amplitudedistribution and the aperture phase distribution cannot beindependently specified (as in the case of dual shapedreflectors) [7]. The reflector shape that produces the optimizedaperture phase distribution will not produce the apertureamplitude distribution assumed in the phase optimizationprocedure. Jorgensen in [ 6 ] uses a fixed Gaussian amplitudedistribution and assumes the aperture phase has the majorcontribution to beam shaping. This is not a bad approximation,but it begins to break down for more complicated radiationpattern contours by causing the radiation pattern calculatedfrom the synthesized reflector surface to be different from thatfor the desired radiation pattern.

To improve on the previous methods, we must find areflector design procedure that produces an aperture amplitudeand phase distribution pair that satisfies two conditions:

1) the aperture amplitude and phase distributions produce2 ) the aperture amplitude and phase distributions can bethe desired beam shape;

generated by a single shaped reflector.In this paper a method will be presented that ensures that thesynthesized aperture amplitude and phase distributions satisfythe above two conditions.

The synthesis method is an iterative technique that consistsof three basic procedures. In the first section of this paper, theiterative technique is outlined. In the next three sections, eachof the three basic procedures is described in detail. The lastpart of this paper applies the iterative technique to the designof a CONUS beam.

11. REFLECTORHAPINGECHNIQUEThe reflector shaping technique consists of what can be

conveniently thought of as the repetition of three basic

Aperture Plane, et lector J

\Element inAperture Plane\%/ FeedFig. 2. Geometry of the reflector shaping problem.

procedures. Consider the reflector-feed near-field apertureplane diagram shown in Fig. 2 . The first procedure is tooptimize the phase distribution in the near-field aperture planefor a fixed near-field amplitude distribution. This procedure issimilar to the method of [6] except that the aperture is dividedinto an array of square elements where each element isassumed to have uniform amplitude and phase. A least meansquare phase optimization routine is then applied to find thebest phase distribution for the array for a given amplitudedistribution. The second procedure is to calculate the coordi-nates of the reflector surface that create the given phasedistribution. As mentioned previously, the reflector surfacecan be completely determined from the phase distribution byapplying the principles of geometric optics. The third proce-dure is to calculate the aperture amplitude distribution fromthe shaped reflector surface and feed element pattern. The newaperture amplitude distribution will be used in the nextiteration of the phase optimization procedure.

Fig. 3 depicts the iterative technique for determining thereflector shape. Since there must be some starting point in anyiterative technique, a perfect offset parabolic reflector of anarbitrary projected aperture is chosen. This reflector producesa certain amplitude distribution that depends on the feedpattern and a uniform phase distribution. The first procedure,phase optimization, is then performed. The result is theparabolic reflector amplitude distribution combined with anoptimized phase distribution that yields a far-field radiationpattern with the desired shape. Next, the second procedure isapplied and the reflector surface that produces the optimizedphase distribution is determined. Once the reflector surface isknown, the third procedure is performed and the new apertureamplitude distribution is obtained. From the aperture ampli-tude and phase information, the far-field radiation pattern ofthe shaped reflector can be calculated by Fourier transforma-tion. If the far-field pattern meets the specifications, then theiterative process is ended; if not, then the new apertureamplitude distribution is used in the next iteration of the phaseoptimization procedure. The iterative process is then repeateduntil the beam shape meets the specifications, or the beamcontour settles to some particular shape. The convergence of


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CHERRETTE et al.: PRODUCING SHAPED CONTOUR RADIATION PATTERN 701

Fig. 4. Three-dimensional illustration of the cost functionf as a function oftwo phase variables PI and p 2 .

M number of far-field directions where gain isto be optimized.

Minimizing rl, is the criterion for the optimization procedure.Note that when rl, is minimized, the mean square differencebetween the gain desired and the gain achieved is minimized.For phase optimization, the set of phases 0; that minimize rl, issought.

The set of Pi that minimizes rl, can be found by applying themethod of steepest descent. The method takes the followingform:

where Pi(k) = P i on the kthL -

teration of the algorithmBy properly choosing p, rl, will be minimized since thegradient (the last term on the right of (6))gives the direction inP-space that results in the greatest increase in I)(see Fig. 4).Amethod for choosing p that works fairly well in practice is tofirst set p equal to the inverse of the magnitude of the gradientin (6), then reduce p by an additional 0.9 times every time thecost function increases.

The gradient can be easily calculated since rl, is an explicitfunction of P i . The result is

i = I

FO,O,t, =Fig. 5 . Geometry for the surface calculation.

Note that the gradient is also a function of the particular set ofA ; , i = 1, N used in the procedure.By starting with initial phase and amplitude distributions,

&(O) and Ai(0) = 1 , N , a new phase distribution, P,(k) i =1, N , ( A i ( 0 ) = 1, N remain unchanged) is obtained thatminimizes the mean square difference between the gaindesired and the gain achieved at the various far-field direc-tions. By choosing the desired gains and their correspondingdirections to cover the area of interest (CONUS, for example),a shaped contour radiation pattern can be produced. Note thatsidelobes can be suppressed in selected directions by choosingthe gain desired to be small for those directions.

The choice of the least mean square optimization criterion( 5 ) , (6) is arbitrary. The beam shaping method could alsoemploy a minimax algorithm as long as phase is the onlyoptimized variable.

IV. CALCULATIONF TH E REFLECTORURFACEOnce the optimized aperture phase distribution is known, a

reflector surface can be completely determined from thisdistribution by applying the principles of geometric optics.Consider the geometry of Fig. 5 . If one reflection point A onthe reflector surface is assumed to be known, then the lengthof what will be referred to as the reference path d s known andgiven by

d = IFAI + IZA,I (8 )where F denotes the feed position at (0, 0,f and A , denotes apoint in the aperture plane.

Note that the incident ray F A produces the reflected rayAA,, and the total length along this ray path is given by thereference path length. For any other point P on the reflectorsurface, the length of the ray path d is given by~ = ( F P I + I ? P , ( = [ X ~ + ~ ~ + ( ~ - ~ ) ~ I ~

+ [(x- A 2+ ( Y - d 2+ ( z- ,)21 (9)where Podenotes a point in the aperture plane ( x u , , , z,) andP denotes a point on the reflector (x, y , z ) .

In this case the incident ray FP produces the reflected ray?Po. If it is assumed that the phase in the aperture plane iscontinuous and that the phases at points A , and P, in the


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702 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 37 . N O. 6. JUNE 1989aperture plane are given by e(&) and O(Pa), espectively,then the following relation holds

- 1k = - [ O ( Pa ) - e ( A , ) ] + d (10)

where27rh

Equation (10) relates the length difference between the tworay paths to the difference in their corresponding phase valuesin the aperture plane. The right-hand side of (10) is known andthe left-hand side contains the unknowns x, y , and z, thecoordinates of a point on the reflector surface which remainsto be determined. Tw o more independent equations are neededto solve for the three unknowns. These two equations areobtained from the expression for the line passing through thepoints P and P a . n the geometric optics limit, the ray PP, isnormal to the equiphase surface and parallel to the wave vectork at the point P a . More precisely

k = - .

where

FFig. 6 . Incident and reflected ray tubes.

where

Therefore, the two equations for the line passing through PP,are given by

as

(12)X -X a - - Z amx m,

mXm2

U = - .

At the beginning of this derivation two variables were assumedto be known, d , O(A,).These variables are chosen arbitrarily

where

wherefis the focal length of the initial parabolic reflector thatis used to begin the iterative surface shaping technique and z= zais the equation of the aperture plane. Note also that thesolutions of ( lo ), (12), and (13) automatically satisfy Snellslaw for smooth phase distributions [8].

It should be mentioned that if there are discontinuities in theaperture phase distribution, the corresponding reflector sur-face points will not satisfy Snells law. Since, for every pointin the aperture plane, a reflection point on the reflector surfaceis determined, a normal for every reflector point can bedetermined from neighboring reflection points. In this way,

(13)

(14a)

(14b)(14c) Snells law can be tested. It will be shown that discontinuities

in the aperture phase will produce discontinuities in thereflector surface.

Since the phase distribution in the aperture plane is known, thepartial derivatives of the phase with respect to x and y can beapproximated by difference expressions. The only unknownsin(12)and(13)arex,y,and~.Therefore, 10),(12),and(13)can then be solved for the three unknowns (x ,y , z ) yielding apoint on the reflector surface. The solution of the threeequations is given by

Q2(xa, O ) -x: -Y : - 3+ 2zaf - f *Z = +za2(0xo+ YY a+ 2 a - f ) -2Q(xa, Y , ) ( o ~ y 2 + 1)

(154

V . CALCULATIONF THE APERTURELANE MPLITUDEDISTRIBUTIONAfter the surface has been determined at a discrete set of

points, the aperture amplitude distribution can be calculated.All points on the reflector surface can be tested to see if Snellslaw is satisfied. For those points that satisfy Snells law, theamplitude of the exit ray can be calculated in the apertureplane by the projected area ratio of a ray tube (see Fig. 6). It is


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CHERRETTE et al. : PRODUCING SHAPED CONTOUR RADIATION PATTERN 703

, . . . .&'2 -

Fig. 7. Cross sections of the ray tubes at the reflector and in the aperture

well known [5], [9] that

0 -plane.

-2

+ + + + * + * ++ + + * + + * * * * + * * + ++ * + + + + + + + + + ++ + I + +* + + + *+ +**:++ * * + + + + + +*++* * + *+ +-

where-412

-4 -2Far-field points over w hich the gain is optimized (desired gain

(17)Fig. 8. >28.0 dBi for all points).

IErI(E aarealarea,?

magnitude of the reflected field at themagnitude of the reflected field in theaperture planeprojected area of the ray tube at the reflec-tor surfaceprojected area of the ray tube in the apertureplane.

A small ray tube can be formed by selecting three rays in_closeproximity. Consider Fig. 6; the three incident rays are FP (Z,J ) ,FP(Z + 1, J ) and Ff(Z, J + 1). The correspondingreflected rays are P(Z, J)P,(Z, J ) ,P(Z + 1, J)P,(Z + 1,J ) nd P(Z, J + l)&Z, J + 1). As the aperture points andthe reflection points are brought closer together, (17) can beused to approximate the aperture field E , at point Pa(Z, J ) .The approximation becomes exact as the areas go to zero. The

reflector surface Nearfield Aperture

_ _ ( kk 4 eedF - 2 5 0 h R = 1 2 5 h

projected areas are given by (see Fig. 7)12area, = - h(side 1)k; f i

whereh = (side 2)Jl -(cos CY)^

(side 3),?- side l),? side 2)22 (side l)(side 2)os CY=

side 1= IP(Z, J) -P( Z+ 1, J) Iside 2 = JP(Z,J)-P(Z, J + 1)(

side 3 = IP(Z+ 1, J ) - P ( Z , J + 1)1& = normalized incident ray directionfi =normal to the surface at P(Z, J )

1area2= - (cell)lr2 2cell= area of square patch in apertureCr= normalized reflected ray direction

2= normal to aperture plane.

Z A = 1 4 0 A H = 1 5 5 A

Feed Pattern E - Plane cosH Plane c o s ' oq = 11 25

Fig. 9. Geometry of the initial parabolic reflector.The field quantity JErIn (17) is known from the givenprimary feed pattern; therefore, the amplitude distribution inthe aperture can be calculated from (17).

VI. DESIGNF A CONUS BEAMThe iterative technique outlined in the previous sections was

used to design a CONUS shaped contour radiation pattern. Tobegin the design procedure, 73 directions in the far field werechosen to cover the projected CONUS map as viewed from aparticular synchronous orbit position as shown in Fig. 8. Next,an initial offset parabolic reflector and feed element wereselected for the starting point of the iterative process. Thisreflector and feed are shown in Fig. 9, and the far-field patternis shown in Fig. 10. The amplitude and phase of the referencepolarized electric field component were then calculated overthe aperture plane on a 47 by 47 point square grid. Theresulting 2209 square patches (elements) in the aperture planewere each 0.543 X on a side. The normalized referencepolarized electric field component due to each patch was19)


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CHERRETTE er al. PRODUCING SHAPED CONTOUR RADIATION PATTERN 70 5

Fig. 16 . Aperture phase distribution after the fourth iteration,

Fig. 17 . A plot of the difference (in z-values) between the shaped reflectorsurface and the initial parabolic reflector surface. The shaped su rface datawas interpolated on to a uniform x - y grid from the randomly spaced dataobtained f rom the numerical results. The peak-to-peak variation is about0.5 A .

Fig. 18 . The shaped reflector surface

2 . 0IolU-z 0.004W_1

- 2 . 0-4 . 0 -2:o 0.0 2:o 4:

AZIMUTH (degrees)Fig. 19 . Cross-polarization pattern for the shaped reflector after the fourthiteration.

discontinuities were detected by testing to see if Snells lawwas satisfied. At the discontinuity points (edge points) thereflected fields were neglected when calculating the far-fieldpattern. The electrical performance of the shaped reflector waschecked by applying physical optics. Fig. 20 shows thephysical optics far-field radiation pattern for the interpolated

-4 .0 -2.0 0.0 2 . 0 4. 0AZIMUTH (degrees)

Fig. 20 . Physical optics far-field pattern for the shaped reflector. Thepattern was calculated using physical optics and the interpolated reflecto rsurface after the fourth iteration.

shaped reflector surface of Fig. 18 . Note there is very littledifference between the physical optics pattern and the geomet-ric optics pattern of Fig. 14.

Vl7. CONCLUSIONA method has been presented for synthesizing a single

reflecting surface that accurately produces a shaped contourradiation pattern when the radiation contour shape and feedpattern are specified. The method is quite general in that it willtake into account discontinuities in the reflecting surface.Results were generated for a CONUS beam design, and verygood beam shaping resulted. The electrical performance of theshaped reflector surface was also verified by physical optics.The major problems with this method are the surfacediscontinuities that can be generated, and the lack of control inedge shape. These are due to the discrete phase optimizationprocedure used. Recent work seems to show that thesedifficulties can be overcome by fitting a two-dimensionalglobal polynomial to each newly synthesized aperture phasedistribution. The polynomial insures smoothness and allowsinterpolation of the phase at points that were discarded.Another approach being studied is to optimize the coefficientsof a set of smooth functions (i.e., polynomials) that describethe phase, instead of optimizating the phase values directly. Inany case, the iterative technique outlined in this paper (seeFig. 3) can be used to arrive at an optimized shaped reflectorsurface.

REFERENCESA . W. Rudge, K. Milne, A. D. Olver. and P. Knight, The Handbookof Antenna Design, vol. 1. London: Peter Peregrinus, 1982, sec.3 . 5 .W. V. T. Rusch, The current state of the reflector antenna art,IEEE Trans. Antennas Propagat. , vol. AP-32, pp . 313-329, 1984.A . R . Cherrette and D. C . D. Chang, Phased array contour beamshaping by phase optimization, in Antennas Propagat. Symp. Inl.Symp. Dig. , Vancouver, BC, 1985, pp . 475-478.T. Katagi and Y . Takeichi, Shaped-beam horn-reflector antennas,IEEE Trans. Antennas Propagat. , vol. AP-23, pp . 752-763, 1975.P. H . Pathak, Techniques for high frequency problems, in AntennaHandbook, Y. T. Lo and S . W. Lee , MS . New York: V an NostrandReinhold, 1988, ch . A7.R. Jorgensen, Coverage shaping of contoured-beam antennas byaperture field synthesis, Proc. Inst. Elec. Eng., vol. 127, pp . 201-208, 1980.B. S . Westcott, Shaped Reflector Aitenna Design. Letchworth,England: Res. Studies Press, 1983, pp. 79-82.


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706 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 37, NO . 6, J UNE 1989R. E. Collin and F. I Zucker, Anlenna Theory, part 2 . New York:McGr aw-Hill, 1969, pp. 21-23, 33-34.S . W . Lee, Electromagnetic reflectlon from a conduction surface:geometrical optics solution, ZEEE Trans. Antennas Propagat. , vol.

apphed electromagnetlc theory, reflector antennas, adaptive antennas, andbeamforming techniques.

AP-23, pp. 184-191, 1975.Shung-Wu Le e (S83-M66-SM73-F81), for a biography please see page205 of the February 1989 issue of this TRANSACTIONS.

Alan R. Cherrette (M85) received the B . S . and

the area of reflectorcurrently at Hughes A

M.S. degrees from Michigan State University, EastLansing , in 1982 and 1983, respectively, and thePh.D. degree from the University of Illinois,Urbana- Champ aign, in 19 88, all in electrical engi-neering.From 1983 to 1985 he was with Hughes AircraftCompany, El Segundo, CA , where he was involvedin the design and development of spacecraft anten-nas. Fro m 1985 to 198 8 he was a Hughes fellow atthe University of Illinois, where his research w as inantennas and geometric theory of diffraction. He is.ircraft Company, where his research interests include

Roberto J. Acosta (M87) received the B.S.E.E.degree. from the University of Puerto R ico and theM.S.E.E. degree from the University of Toledo,OH , in 1977 and 198 0, respectively. He is currentlyworking toward the Ph.D. degree at the Universityof Akron, OH .Since 1977 he has been on the staff of NASALewis Research Center, Cleveland, OH. His re-search interest is in the area of communicationsatellite antennas.Mr. Acosta is a member of Phi Kappa Phi.

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