Oliker Department of Mathematics and Computer Science Emory University Atlanta Georgia email olikermathcsemoryedu Abstract In this paper the problem of synthesis of o64256set shaped single re64258ector antenna is considered This problem has to be so ID: 27250
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ARigorousMethodforSynthesisofO setShapedRe\rectorAntennasVladimirI.OlikerDepartmentofMathematicsandComputerScience,EmoryUniversity,Atlanta,Georgiae-mail:oliker@mathcs.emory.eduAbstractInthispapertheproblemofsynthesisofo setshapedsinglere\rectorantennaisconsidered.Thisproblemhastobesolvedwhenare\rectorantennasystemisrequiredtocontrolthe eldamplitudeand/orphaseonthefar- eldorontheoutputapertureinthenear- eld.Achievinghigheciencyisaveryimportantobjectiveofthedesignandshapedre\rectorantennasareusedforthatpurpose.Theequationsoftheproblemarestronglynonlinearpartialdi erentialequationswhichcannotbeanalyzedbystandardtechniques.Thoughtheproblemhasbeenthesubjectofstudybymanyauthorsforover40years,upuntilrecently,therewerenorigoroustheoreticalresultsresolvingcompletelythequestionsofexistenceandunique-ness.Withfewexceptions,authorshaveattackedtheproblemwithheuristicnumericalprocedures,and,dependingonthespeci cformulation,obtaineddi erentresultsnotalwaysinagreementwitheachother.Inthispaperanewmethodforsolvingthesinglere\rectorproblemispresented.Thenewmethodallowsacompleteandmathematicallyrigorousinvestigationofthisproblem.Furthermore,theproposedmethodlendsitselftoanumericalimplementationandwepresenthereseveralexamples.Keywords:re\rectorantenna,synthesis,nonlinearpartialdi erentialequations,numericsSubjectClassi cation:AMS(MOS):78A50,65N21;CR:G1.8. Author'snote:ThispaperwassubmittedforpublicationtotheJournalofComputationalMethodsinSciencesandEngineering(JCMSE)onAugust12,2001andacceptedDecember12,2001.However,duetoreasonsrelatedtopublishingissuesandcompletelybeyondanycontroloftheauthorthepaperhasneverbeenpublished.ItwastransmittedtoComputedLettersonMarch17,2006.1 1IntroductionRe\rectingpropertiesofquadricsurfaces,suchasellipsoids,paraboloidsandhyperboloidsarewellknownandhavebeenwidelyusedtobuildlenses,mirrors,antennas,andmanyotherre-\rectingandrefractingdevices.However,inmanyengineeringproblemsofre\rector/refractordesignandanalysisthesimpleshapesofquadricsurfacesareoftennotsucienttosatisfytherequirementsofhigheciencyandsigni cantlymorecomplexsystemsofsurfaceshavetobedesignedandmanufactured.Ageneralproblemthatarisesinthetheoryofsynthesisofre\rectorantennasisthatofshapingare\rectingsurfacesothatitwouldredistributeagivengeometricaloptics(GO)feedpowerpatternintoaprespeci edamplitudeaperturedistribution[2],[3],[1],[4],[5]Restrictingthedesigntoaxiallysymmetricsurfacesorcom-binationsofellipsoids,paraboloidsandhyperboloidswillproduceinmanycasesasystemwithblockageandloweciency.Variousversionsofthisproblemarisealsointhetheoryofsynthesisofmirrorsinoptics[9],[8],[10],[11],[12],heattransfer[6],[7],andotherareas.Becauseofimportantpracticalapplications,thisproblemhasbeenstudiedquiteex-tensively.Intheaxiallysymmetriccasewiththesourceontheaxistheproblemreducestosolvinganordinarydi erentialequationwhichcanberigorouslyinvestigatedandsolvednumericallywithanygivenprecision[5].Obviously,anaxiallysymmetricdesignistoore-strictive.Forexample,blockageofthere\rectorbythefeedcannotbeavoidedinsuchdesign.Thus,inmanycircumstancesanon-axiallysymmetricando setre\rectorantennasaredesired.However,inthiscasetheequationsdescribingtheproblemarehighlynonlin-earpartialdi erentialequationswhichuntilrecentlycouldnotbeanalyzedrigorouslybyavailablemathematicalmethods.Duringthelastfourdecadesessentiallythreedi erentapproacheshavebeenusedforformulatingthegeneralproblemanalyticallyandsolvingitnumerically.Inthe rstapproachtheproblemisformulatedasasystemof rstorderpartialdi erentialequationsandamethodresemblingthemethodofcharacteristicsisappliedtosolvethesystemnumerically;see[2,1],andotherreferencesthere.Inthesecondapproachthesameproblemisformulatedasaboundaryvalueproblemforasecondorderpartialdi erentialequationofMonge-Amperetypeforacertaincomplex-valuedfunction[4].Thenalinearizationprocedureisusedtoconstructasolutionclosetosomeaprioriselectedsolution.Inbothapproachesarigorousmathematicalanalysisoftheresultingequationsislackingandconsequentlythevalidityofthenumericsisneverfullyestablished.Thisledtoacontroversyregardingexistenceanduniquenessofsolutionsthatuntilnowhasnotbeenresolved[3,2,1].Athirdapproachbasedonarigorousmathematicalanalysisofasecondorderreal-valuedMonge-Ampereequationwasappliedin[13]toproveexistenceanduniquenessinthespecialcasewhenthesolutionisrequiredtobe\close"toanaxially-symmetricone.2 Thepurposeofthispaperistodescribeanewapproachtothesynthesisproblemforsinglere\rectorantennas.Incontrastwiththeabovementioned rsttwoapproachesthenewapproachisbasedonarigorousmathematicaltheoryanditsvaliditydoesnotdependonspeci cnumericalexamples.Italsodoesnothavethelimitationspresentinthethirdapproach.Theapproachpresentedherewas rstdevelopedin[14]forthefar- eldcase,andthen,forthenear- eldcase,in[15,16].Asomewhatdi erentbutclose\inspirit"approachtosynthesisofdualre\rectorsystemswithacollimatedsourceisdescribedin[17].Note.(AddedinproofMay10,2006).Analternativeapproachtotheproblemofdesigningopticalandantennasystemsconsistingofoneortwore\rectingsurfaceswhichtransformtheenergyofthesourceintoaprescribedenergydistributiononagiventargetsetwasgivenbyT.GlimmandV.Olikerin[22,23].Thisapproachisbasedoncalculusofvariations,inparticular,ontheMonge-Kantorovichtransporttheory.In[14,15,16]theemphasiswasonthemathematicalaspectsoftheproblemsandtheresultsarenotreadilyapplicabletoproblemsofpracticalinterests.Thepresentpaperemphasizesthegeometricsideofthenewapproachand,inadditiontoprovidingarelativelyelementaryexpositionofit,containsalsoanimportantimprovementandanalysisleadingtopracticalcriteriapermittingdesignswithnoblockage.Thepresentedapproachallowstonotonlyresolverigorouslythequestionsofexistenceanduniquenessofsolutionsbutitalsolendsitselfnaturallytonumericalproceduresforconstructingnumericalsolutionswithanyuser-speci edaccuracy.Suchaprocedurehasbeenimplementedbyusintoanumericalcodeandweusedittodevelopo setshapedre\rectorantennastransformingagivenGOfeedpowerpatternintoaprespeci edamplitudedistributionacrossagiven\rataperture.Inessence,thenewapproachisverygeometricandquiteelementary.Whilethemathematicsbehinditissomewhatinvolvedandusesrelativelyadvancedconceptsofweaksolutionstononlinearpartialdi erentialequations,themainideasaretransparentandcanbeexplainedbysimpleandfamiliarmeans.Theorganizationofthispaperisasfollows.Insection2wegiveadetailedstatementoftheproblem,insections3and4wedescribethenewapproachtoitssolution,in5weprovideconditionswhichguaranteethatself-blockageandblockagebythefeedareavoided.Finally,insection6wepresentnumericalexamples.3 2StatementoftheProblemLetx;y;zbeaCartesiancoordinatesystemwithoriginO.WedenotealsobyOanon-isotropicpointsourceofenergyplacedattheoriginandletI(m)denotethefeedpowerpatternasafunctionofdirectionm.Asusual,thegeometricalopticsapproximationisusedtosetupthesynthesisproblem[5,11,4,1]. x y z T O m n R r(m) S D u t r = |r(m)| v = r + tu Figure1:FormulationoftheproblemItisconvenienttoconsiderthedirectionmasapointonaunitsphereScenteredatOasshownontheFig.1.Inthissetting,mistheincidencedirectionpassingthroughtheinputaperturewhichispicturedasaregiononthesphereS.WewillassumethatisaclosedsetonSinthesenseofsettheory.Inordertoderivetherequiredrelationsweconsiderare\rectorsurfaceRthatinterceptstheincidentraysandre\rectsthemaccordingtoSnell'slaw.ThesurfaceRisrepresentedbyitspolarradius(m)withmvaryinginregion.Invectorformitisgivenbyr(m)=(m)m.TherayofdirectionmisincidentuponthesurfaceRatthepointr(m)andisre\rectedindirectionugivenaccordingtoSnell'slawasu=m 2(mn)n;(1)wherenistheunitnormaltosurfaceR.ItisassumedherethatRissmoothandprojectsradiallyontoinaone-to-onefashion.IntheproblemconsideredherewearealsogiveninadvancesomeregionTinspacewhichthere\rectedraysmustreachandproducethereacertainpowerpattern.Forsimplicity,4 theenergydeliveredbyraysreachingTdirectlyfromOisexcludedfromouranalysis.Obviously,thisenergycanbeeasilyaccountedforinthe nalanalysis.InthispaperweassumethatTisa\ratregiononsomeplane.Thisassumptionismadeonlytosimplifythepresentation;itcanbesigni cantlyrelaxed[15].ItisalsoassumedthatTisaboundedandclosedset.WedenotebyL(v)afunctionde nedonTwhichrepresentstherequiredoutputpowerpattern.Finally,ifwedenotebyt(m)thedistancefromthesurfaceRtraveledbythere\rectedrayofdirectionu(m)toreachT,thenwehaveamap (m):!TfromtheinputaperturetotheregionT,thatis,v= (m).Thismapisgivenby (m)=r(m)+t(m)u(m):(2)Next,werelatetheinputpowerpatternI(m)onandtheoutputpowerpatternL(v)onT.LetJ( )betheJacobianofthemap .Theenergyconservationlawalongin nitesimalraytubescanbeexpressedas[18]L( (m))jJ( (m))j=I(m):(3)Now,there\rectorproblemisformulatedasthefollowinginverseproblem.SupposewearegiveninadvancetheregiononSanda\ratregionTonsomeplane,anon-negativefunctionI(m)de nedonandanon-negativefunctionL(v)de nedonT.Theproblemistodetermineare\rectingsurfaceRsuchthatthemap ,de nedby(2),mapsontoTandthecondition(3)issatis edforallmintheinteriorof.Anecessaryconditionthatmustbesatis edbythefunctionsIandLfollowsimme-diatelyfrom(3).Namely,integrating(3)andusingtheformulaforchangingthevariableunderintegralweobtainZDI(m)d(m)=ZDL(\r(m))jJ(\r(m))jd(m)=ZTL(v)d(v);whered(m)istheareaelementonthesphereSandd(v)istheareaelementonT.ItfollowsthatifL(v)isagivenoutputpowerpatternontheoutputapertureTthenthefollowing\balance"equationisnecessaryforsolvabilityofthere\rectorproblem:ZDI(m)d(m)=ZTL(v)d(v):(4)Fromthepointofviewofdi erentialequationstheequations(1),(2)and(3)arenotconvenientforstudyingthere\rectorproblembecausethepositionvectoroftheunknownre\rectorRenterstheseequationsinaverycomplicatedway.Di erentwaysforrewritingtheseequationshavebeensuggested[2,3,4,5,11,19,20].Someofthemaremoreconvenientthanotherbut,exceptfortheaxiallysymmetriccase,theresultingequationsarealwayshighlynonlinearandcomplicated,and,consequently,diculttoinvestigate.Fortunately,inourapproachwedonotneedtheseequationsinexplicitform.5 3TheNewApproach3.1AnObservationandtheMainStrategyInsomesensetheapproachdescribedbelowresemblestheclassicalapproachinwhichonewouldliketousetheknownre\rectingpropertiesofquadricstobuildre\rectors.Consider rstthespecialcasewhentheoutputapertureTconsistsofonlyonepointv.TherequiredoutputpowerisinterpretedinthiscaseastheDirac functionwithapositive\mass"Lvconcentratedatthepointv.Sincetherighthandsideinthebalanceequation(4)isthetotalenergyrequiredatv,thisequationinthiscaseassumestheformZDI(m)d(m)=Lv:(5)Inthefollowingourconstructionswillinvolveellipsoids.Toavoidconfusion,letusnotethatthroughoutthepaperbyanellipsoidwealwaysmeanthesurfaceofthesolid.LetE(v)beanellipsoidofrevolutionwiththeaxisofrevolutionpassingthroughthepointsOandv,withoneofthefociatOandtheotheratv.Itspolarradiusisgivenby(m)=d 1 m^v;(6)wheredisthefocalparameterofE(v),theeccentricity,and^v=v jvj:HereandthroughoutthepaperweidentifythepointvwiththevectororiginatingatOandterminatingatv.SincethedistancejOvj(=jvj)betweenthefociisspeci ed, xingapositived xesonesuchellipsoid.TheeccentricityofE(v)canthenbedeterminedfromtheknownformulajvj=2d 1 2;(7)whichcanberesolvedas=vuut 1+d2 jvj2 d jvj:(8)Thewellknowngeometricpropertiesofsuchanellipsoidimplythatanyrayofdirec-tionmoriginatingatOisre\rectedbyE(v)andreachesvwhichisacausticpointofthisGO\row.ThetotalpowerofthesourceZDI(m)d(m)6 iscarriedtothefocusvbytheGOrayspassingthroughtheapertureandtransformedbyE(v).By(5)ZDI(m)d(m)=Lv:Thus,inthisspecialcasethepieceoftheellipsoidE(v)thatprojectsradiallyontoisasolutionofthere\rectorproblem.Uniqueness.ItisclearthatanyellipsoidofrevolutionwiththesamefociOandvisalsoasolutioninthiscase.However,asitfollowsfrom(6)and(8),therelationbetweenpolarradiiofanytwosuchsolutionsisnotlinear.Inparticular,thetwosolutionsarenotrelatedbyahomothetyrelativetoO.Thus,weseethatinthiscasethere\rectorproblemhasin nitelymanygeometricallydistinctsolutions.3.2Re\rectorsDe nedbyaFiniteNumberofEllipsoidsConsidernowamoregeneralcasewhenthesetTconsistsofa nitenumberofpointsandtheoutputpowerpatternrequiredonTisacollectionofmassesconcentratedatthesepoints.Inthiscase,ourplanistoconstructthere\rectorsurfacefrompiecesofconfocalellipsoidsofrevolutionwiththecommonfocusatOandaxeswithdirectionsde nedbythepointsinT.Morespeci cally,letTbegivenasa nitesetofpoints,T=fv1;:::;vNg,andtherequiredoutputpowerpatternonTisgivenasasumLT=NXi=1Li(vi);whereLiarepositivenumbersequaltotheoutputpowersrequiredatpointsvi.ThefunctionLTshouldbeunderstoodhereinthedistributionalsense,thatis,whenappliedtoaunitprobeatthepointvi,itproducesthepowerLi.Similarly,therighthandsideofthebalanceequation(4)shouldbeunderstoodinthiscaseasthetotalpowerQ=PNi=1LiandweassumethatthenumbersLiaresuchthatthefollowingbalanceequationissatis ed:ZDI(m)d(m)=Q:(9)Ultimately,there\rectorsurfaceRinthiscasewillbebuiltofpiecesofconfocalellipsoidsofrevolutionE(vi)withthecommonfocusatOandaxesofrevolutionalongOvi;i=1;:::;N.However,theconstructionofsuchare\rectorinthiscaseisabitmoreinvolvedthanbeforeandweneedtointroduceprecisede nitions.TosimplifythenotationweputEiE(vi).ThepolarradiusofEiisgivenbyi(m)=di 1 i(m^vi);(10)7 wheredi�0andi,0i1,are,respectively,thefocalparameterandeccentricityofEi,whilemisanarbitrarypointonthesphereS.ThesolidbodyboundedbyEiisdenotedbyBi.Next,weconsidertheintersectionofthebodiesBiandputB=N\i=1Bi:SinceeachofBiisaconvexbody,thesetBisalsoconvex.ThefocalparametersoftheellipsoidsEicanbeselectedsothatthepointsviliestrictlyinsideBforalli.Inordertoestablishthiswe rstnotethatby(10)wehaveforallmonSi(m)&#x-1.2;虔di=2:(11)Theinequalityin(11)isstrictbecause0i1.Theinequality(11)meansthataballofradiusdi=2withthecenteratO tsstrictlyinsideEi.BecauseofthesymmetryofEiaballofthesameradiusbutwiththecenteratthesecondfocusviisalsocontainedinsideEi.Let!(T)denotethediameterofthesetT,thatis,themaximaldistancebetweenanytwopointsinT.Ifwechoosedi2!(12)foralli=1;:::;Nthaneachoftheellipsoidsde nedbydiwillcontainthesetTstrictlyinside.Therefore,thesolidBand,ofcourse,thesurfaceboundingitwillalsocontainsTstrictlyinside.Becausetherequiredre\rectorwillbeconstructedasapartoftheboundaryofthesolidB,theabovepropertywillbeimportantforestablishingthatnopartofthere\rectorconstitutesablockagetore\rectedrays(priortoreachingthepointsvioftheoutputaperture).Nowwecanstatetheprecisede nitionofare\rector.De nition1TheconvexsurfaceboundingthesolidBiscalledare\rectorde nedbytheellipsoidsE1;;EN.Obviously,anyfamilyofellipsoidsEi,i=1;:::;N,de nesare\rectorredirectingtheraysfromthesourceOtotheirrespectivefocivi.However,theamountofenergydeliveredateachofthevidependsonhowmuchofthetotalenergyfromOis\intercepted"bythecorrespondingellipsoidpriortootherellipsoidsinthefamily.Thus,themainproblemistoselecttheellipsoidsEisothatforeachi=1;:::;NtheenergyarrivingatviisequaltoLi.Aswewillseeinthefollowingsections,thisisaccomplishedbysolvingasystemofequationsfortherequiredfocalparameters.8 E 1 E 2 O v 1 v 2 A C B 1 B 2 Figure2:Thegeometryofthere\rectorin2D3.2.1ASpecial2DCaseLetusexplainthemainideaforconstructingthere\rectorinthesimpleplanarcaseshownonFig.2.Inthis gureE1andE2aretwoellipseswithacommonfocusOandTconsistsoftwopointsv1andv2.ThelatterarealsothesecondfociofE1andE2,respectively.Itfollowsfromde nition1thatthere\rectorinthiscaseistheclosedarcAB1CB2A.Denotethisre\rectorbyR.NotethateveryrayemanatingfromOandre\rectingo Rpasseseitherthroughv1orv2.TheinputapertureistheangleB1OB2.ItisconvenienttoassumethattheinputapertureisextendedtotheentireunitcirclecenteredatO.InordertoaccountforsuchapertureextensionweassumethatoutsideoftheactualinputapertureB1OB2thefeedpowerdensityiszero.Then,onlytheraysintheangleB1OB2areofinterest.Ifthesumofpowersrequiredatv1andv2isequaltothetotalinputpower(thatis,theequation(9)appliedtothiscaseissatis ed),thenthecontributionofeachoftheellipsestothepoweratthecorrespondingfocusdependsonhowmuchofthetotalinputpowerisinterceptedandredirectedbythecorrespondingellipse.This,ontheotherhand,canbecontrolledbytheappropriatechoiceofthefocalparameters.Forexample,ifthefocalparameterofE2issolargethatE1isstrictlyinsidethesetboundedbyE2thenalltheraysfromOareinterceptedbyE1.Inthiscase,thereisnopowercontributiontov2andallpowergoestov1.Similarly,ifthefocalparameterofE2issucientlysmallthenE2willinterceptalltheraysfromOandallpowerwillbedeliveredtov2.Intheformercase,keepingthefocalparameterofE1 xedanddecreasingthefocalparameterofE2wecanachieveanydesireddistributionofthetotalpowerbetweenthepointsv1andv2.Oncethedesireddistributionisachieved,thepartofthere\rectoroutsidethearcB1AB2isdeleted.Suchdeletiondoesnota ecttheattainedenergydistribution,sincetheinputpowerdensityfordirectionsoutsidetheangleB1OB2iszero.Then,the9 remainingpartofthere\rectorcontainedintheangleB1OB2givestherequiredre\rector.3.2.2ConstructingRe\rectorsWithEllipsoidsWenowusethesameideatobuildare\rectorin3Dincasewhenthenumberofellipsoidsde ningthere\rectormaybearbitrary(but nite).Wewillseelaterthatthisistheimportantcaseinpractice.Inordertodescribetherequiredconstructionweneedtointroducetworelatednotions.Firstofall,itisconvenienttoassumethatthefeedpowerpatternI(m)isde nedovertheentiresphereSbysettingI(m)0outside.ThenwecanworkwiththeentiresphereSastheinputaperture.Theresultingre\rectorwillbeaclosedsurface.Oncethisre\rectorisconstructedwewilldeletetheunnecessarypartandobtaintherequiredre\rector.LetT=fv1;:::;vNgandletEi;i=1;:::;N,beafamilyofellipsoidsde ningthere\rectorRasinde nition1.LetEjbeoneoftheellipsoidsofthefamily.Byde nition,thesolidB(vj)mustcontainthere\rectorR.Weneedtodistinguishthecaseswhenthere\rectorRislyingstrictlyinsideB(vj)andwhensuchinclusionisnotstrict,thatis,whenEjandRhavecommonpoints.De nition2WesaythatEjissupportingtoRifthesetCj=EjTRisnotempty.ThesetCjiscalledthe\contact"setofEjwithR.Thisde nitionisillustratedinthe2DcaseonFig.3.Inthis gure,allthreeellipsesE3aresupporting.However,ifthefocalradiusd3ofE3isincreasedtheellipseE3ceasestobesupportingandC3becomesempty.Weneedtoallowforthatbecausewhenweconstructare\rectorwedonotknowaprioriifaparticularellipsoidfromagivenfamilyconstitutesapartofthere\rectorornot.Nowwede neasetthatdeterminesthecontributionofeachellipsoidtotheoutputpowerpattern.De nition3LetRbeare\rectorasaboveandE(v)anellipsoidfromthefamilyde ningR.DenotebyV(v)theradialprojectionofC(v)onSbyraysfromO.Thissetiscalledthevisibilitysetofv.ThisnotionisillustratedonFig.4.PutG(R;v)=ZV(v)I(m)d(m):10 E 1 E 2 O v 1 v 2 A C B 1 B 2 E 3 v 3 Figure3:Supportingellipses E 1 E 2 v 1 v 2 V(v ) 2 V(v ) 1 S Figure4:VisibilitysetsEvidently,thevisibilitysetV(v)consistsofdirectionsmofallraysemittedfromOwhicharere\rectedbyRandreachv.Therefore,thequantityG(R;v)givesthetotalpowerdeliveredbyre\rectorRtothepointv.Note,thatbecauseofthewaywerede nedI(m)overS,onlythecontributiontothisintegralfromthesetTV(v)playsarole.Also,notethatifC(v)isemptythenV(v)isempty,andG(R;v)=0.Now,there\rectorproblemwiththeoutputapertureconsistingofa nitenumberofpointscanbeformulatedasfollows.LetT=fv1;:::;vNgandL1;:::;LNarepositivenumberssuchthatthebalancecondition(9)issatis ed.Theproblemistodetermineare\rectorRde nedbyellipsoidsEiwithfociatOandvi;i=1;2;:::;N;suchthatG(R;vi)=Liforeachi=1;2;:::;N:(13)SinceeachoftheellipsoidsEiisuniquelyde nedbyitsfocalparameterdi,thedeterminationofthere\rectorRmeansdeterminingNfocalparametersd1;:::;dNsothatthecorrespondingre\rectorsatis es(13).11 Thesystem(13)isanonlinearsystemwithrespecttothevariablesd1;:::;dNandinthenextsectionwedescribeaprocedureforsolvingit.3.2.3TheAlgorithmforSolvingSystem(13)Thesystem(13)issolvedbyaniterativeprocedurestartingwithaninitialre\rectorR0.There\rectorR0isconstructedfromthedata.We rstdescribethisconstruction.LetRbesomere\rectorde nedbyellipsoidsEi;i=1;:::;N,withoneofthefociatOandtheotheratv1;:::;vN,respectively.WewanttoshowthatthefocalparametersofEicanbeselectedsothatforallminandalli�1wehave1(m)i(m):(14)ThisinequalityimpliesthatalltheenergyfromthesourceOisredirectedbythere\rectortothepointv1.Welet,asbefore,M=maxijvijand!=diameterofT.Naturally,itisalwaysassumedthatthesetTisatapositivedistancefromO,andthusminijvij�0.Letalso\ri=maxDm^viand\r=maxi\ri.Also,itisassumedthattheraysfromthesourcepassingthroughtheinputapertureareseparatedfromtheraysfromthesourcedirectedtowardspointsontheoutputaperture,thatis,\r1:(15)Putd1= M;(16)where isapositivenumberwhichisadesignparameterwhosespeci cchoicewillbediscussedinsection3.2.4.Then,using(7),weobtain21 1 21=jv1j d11 and1p 1+ 2 :Itfollowsfrom(10)thatforthepolarradiusoftheellipsoidE1overtheinputaperturewehavetheestimate1(m)=d1 1 1m^v1 M 1 (p 1+ 2 )\r1;(17)thatis,theinequality(14)issatis ed.12 Put =2 1 (p 1+ 2 )\r1anddi= M:Thenforpointsmindomainweobtain,using(10)and(17),i(m)=di 1 im^vidi 2= M 2= M 1 (p 1+ 2 )\r11(m):(18)LetR0denotethere\rectorconstructedwithellipsoidsE1;:::;EN.Becauseof(18)theraysfromOthroughareallintercepted rstbyE1.Usingthenotationintroducedearlier,andtheassumption(9),wecanexpressthisfactasG(R0;v1)=ZSI(m)d(m)=Q;G(R0;vi)=0;fori�1:Theconstructionoftheinitialre\rectorisnowcomplete.Wenowbeginmodifyingthere\rectorR0.InthisprocessonlythefocalparametersofellipsoidsEi;i�1;willbechanging,whileE1willremain xed.WebeginwithE2.Decreasecontinuouslythefocalparameterd2andconsiderthere\rectorsde nedbythesameellipsoidsasbeforeexceptforE2whichisreplacedbytheellipsoidwithreducedfocalparameterd2.The\intermediate"re\rectorsaredenotedbyR0int.Theparameterd2isdecreaseduntiltheequalityG(R0int;v2)=L2isachieved.ThismusthappenbecauseG(R0int;v2)changescontinuouslywithd2onthesegment[0;Q].ThecontinuityofG(R0int;v2)followsfromthefactthatitdependscontinuouslyonthevisibilitysetV(v2)andthelatterincreasescontinuouslyasd2decreases.Moreprecisely,ifintheinitialpositiontheellipsoidE2issupportingthenV(v2)willincreasewhend2isdecreased.IfE2isnotsupportingattheinitialstagethenV(v2)isemptyuntilE2becomessupporting(whend2becomessucientlysmall).Inaddition,ifd2isnearzerothentheellipsoidE2isinterceptingalltheraysfromOthroughandthecorrespondingvisibilitysetV(v2)=S.ThenG(R0int;V(v2))=ZSI(m)d(m)=Q:Thelatterisimpossible,sincebyconstructiond2wasdecreasingonlywhileG(R0int;v2)L2.LetusshowthattheequalityG(R0int;v2)=L2isattainedwhiled2�d1(1 \r)=2:(19)ThisestimateisrequiredinordertoshowthattheellipsoidE2duringtheabovemodi cation(andthesubsequentonestobedescribedbelow)doesnotdegenerateintoastraightlinesegmentjoiningOandv2.ToestablishtherequiredpropertyassumethatforsomeintermediatepositionofE2wehaved2=d1(1 \r)=2.(Sinceintheinitialpositiond22d1andd2ischanging13 continuously,itisclearthatif(19)isnottruethend2=d1(1 \r)=2mustholdforsomecon gurationofellipsoidsforminganintermediatere\rector.)Thenforpointsminwehave2(m)=d1(1 \r) 2(1 2m^v2)d1(1 \r) 2(1 2\r2)d1(1 \r) 2(1 \r2)d1 2:Ontheotherhand,itfollowsfrom(10)that1(m)d1 2:Therefore,forpointsmin2(m)1(m):ThenwemusthaveG(R0int;v2)=ZSI(m)d(m)=Q�L2;whichisincontradictionwiththewayE2wasmodi ed.Theestimate(19)isestablished.We xthevalued2forwhichG(R0int;v2)=L2andthecorrespondingellipsoidE2.IfN=2thenwearedone;otherwise,thesameprocedureisrepeatedforeachoftheremainingellipsoidsfori�2(itisassumedthatN2;thecasewhenN=1wasdescribedinsection3.1).Theresultingre\rectorwedenotebyR1.ItisclearthatiftheellipsoidEiissupportingandthefocalparameterdiisdecreasingthentheenergyG(R0int;vi)isincreasingwhileG(R0int;vj)arenon-increasingforallj=i.Thatis,theintermediatere\rectorsR0intandR1satisfytheinequalitiesG(R1;v1)L1;G(R1;vi)Li;fori�1;andNXi=1G(R1;vi)=NXi=1Li=ZSI(m)d(m):IfG(R1;vi)=Liforeachi=1;2;:::;N;theprocessterminates.Otherwise,itcontinuesbyresettingR0=R1andrepeatingtheabovesteps.Asaresult,asequenceofre\rectorsRk;k=0;1;:::;withfocalparametersdk1;:::;dkNisconstructed.Foreachi�1dkidk+1i:(20)Intheterminologyofpartialdi erentialequationsthere\rectorsRkaresupersolutionsofthesystem(13).Theestimate(19)holdsforanydide ningellipsoidEkifori�1andanyre\rectorRk.Themonotonicity(20)ofthefocalparametersimpliesthatthesequenceofre\rectorsRkconvergestosomere\rectorRandbecauseoftheestimate(19),appliedtoeachoftheelementsofthesequencesdki,noneoftheellipsoidsEkidegeneratesintoastraightlinesegment.Ifwedenotebyd1;:::;dNthefocalparametersoftheellipsoidsformingRthenbyconstructionwehavedidki;foralli1andallk=0;1;2;:::(21)14 Letusshowthatthefocalparametersd1;:::;dNofthere\rectorRsatisfythesystem(13).Supposethatforsomej�1wehaveG(R;vj)Lj:Thenwecandecreasedjbyasucientlysmallamountandconstructanewre\rectorR0forwhichG(R0;v1)L1;G(R0;vi)Li;fori&#x-272;&#x.280;1:Forthisre\rectorwehaved0jdjwhichisimpossiblebecauseof(21).Itcanbeshown(see[16])thatforagiventolerancetheaboveprocessterminatesina nitenumberofstepswiththedeterminedre\rectorRsatisfying(13)withinthespeci edtolerance.There\rectordeterminedbytheabovealgorithmisaclosedconvexsurface.However,itsportionoutsideofthepartthatprojectsradiallyontotransmitsnoenergy(sinceI(m)0whenm2Sn).Deletingthepartofthere\rectorsurfaceoverSnweobtainthe nalsolution(withinauserspeci edtolerance).Bylettingthetolerancetendto0weobtainasequenceofre\rectorsconvergingtoare\rectorsatisfyingthesystem(13)exactly.3.2.4UniquenessofSolutiontoSystem(13)Asthenoteattheendofsection3.1indicateswecannotexpectthatthedeterminedsolutionisunique.Thesameobservationappliestore\rectorsconstructedbysolvingthesystem(13).Indeed,intheconstructionofthere\rectordescribedinprecedingsection,the rstellipsoidwas xedonlyuptothechoiceoftheparameter andthereforeanotherre\rectorsatisfying(13)canbeconstructedbysimplychoosingdi erentlythisparameter.However,theuniquenessstillholdsinthefollowingsense.LetRandR0betwosolutionsofthesystem(13).LetRbedeterminedbyellipsoidswithfocalparametersd1;d2;:::;dNandR0byd01;d02;:::;d0N.Supposethatthetwore\rectorshaveacommonellipsoid.Thenthetwore\rectorscoincide.Thisisprovedin[15].Thus,inparticular, xingtheparameter xesuniquelyasolution.4DistributedOutputPowerPatternsLetnowTbearegiononsomeplaneandL(v);v2T;afunctionde ningtherequiredoutputpowerpatternonT.Assumethatthebalanceequation(4)issatis ed.There\rector15 approximatingtherequiredre\rectorwithinanyuserspeci edtoleranceisconstructedinthiscaseviaaprocedurewhichreducesthisproblemtotheproblemofconstructingare\rectorde nedbya nitenumberofellipsoids.Thisprocedureisasfollows.PartitiontheregionTintosmallsubregionsTi;i=1;2;:::;NsothatT=SNi=1Ti.PutLi=ZTiL(v)d(v):PickapointviineachTiandsolvethesystem(13)withthepointsv1;:::;vNandpowersL1;:::;LNatthecorrespondingpoints.Denotetheresultingre\rectorbyRN.ItcanbeshownthatasthediametersofthesetsTishrinktozerothecorrespondingsequenceofre\rectorsfRNgconvergestoare\rectorRsolvingthere\rectorproblemwiththepowerpatternLspeci edonT.Furthermore,foragiventoleranceonecanconstructare\rectorapproximatingRwithinthistoleranceina nitenumberofsteps.Themathematicshereissomewhatinvolvedandwereferthereaderforfurtherdetailstothepapers[15,16].5AvoidingBlockage5.1AvoidingSelf-blockageLetusshownowthatbyappropriatechoiceoftheparameter in(16)there\rectorcanbedesignedsothatself-blockagecanbecompletelyavoided.Byself-blockagewemeanhereasituationwhenapartoftheoutputapertureTisblockedbysomepartsofthere\rectorsurfacefromtheraysre\rectedo otherpartsofthere\rector.Again,fromthepointofviewofnumericsandapplications,thecasewhenthere\rectorisconstructedwitha nitenumberofellipsoidsistheonethatneedstobeconsideredthoughthedescribedresultisalsovalidingeneral.Ithasbeenexplainedattheendofsection3.2thatwithourconstructionofare\rectortheself-blockagewillnothappenifforallellipsoidsformingthere\rectorthefocalparameterssatisfytheinequality(12).Therefore,itfollowsfrom(19)thatif ischosensothat 4! M(1 \r)(22)then,settingd1= M,weareguaranteedthatallellipsoidsintheconstructedre\rectorwillcontainTinsideandnoself-blockagewilloccur.16 5.2AvoidingBlockagebytheFeedAcommonprobleminaxiallysymmetricre\rectorsisthatiftheradiationsourceispositionedontheaxisthenitbecomesasecondaryscattererandthatleadstoenergylosses.Fromtheconstructiondescribedinsection3.2.3itfollowsthatthefollowingconditionguaranteesthatnoblockageofre\rectedrayswilloccur.LetCDdenoteaconegeneratedbyraysfromOthroughthepointsintheinputaperture.Letl1;:::;lNbetheraysfromOpassingthroughv1;:::;vN,respectively.Sinceourre\rectorsareconstructedofellipsoids,itisclearthatforare\rectedraytopassthroughOandvitheinputapertureshouldincludeadirectionm= ^vi.Suchpossibilityisexcludedifwerequirethatnotworays,onefromCDandonefromthesetl1;:::;lN,formtogetheracompletestraightline.Whenthetargetisaspeci edregionnotlimitedtoa nitenumberofpointsthisconditionshouldbeformulatedasfollows.LetCTbeaconegeneratedbyraysfromOthroughthepointsintheoutputapertureT.Inordertoavoidblockageofre\rectedraysbythesourceO,thepositionsoftheinputapertureandoutputapertureTshouldbesuchthatnotworays,onefromCDandonefromCT,formtogetheracompletestraightline.Inpractice,ofcourse,oneshouldalsotakeintoaccountthesizeofthefeed.6ExamplesInthissection,wepresentfourexamplescalculatednumericallywiththealgorithmde-scribedinsection3.2.3.Thecodeusedforcomputingsolutionspresentedinthispaperisasigni cantlyimprovedversionoftheoriginalcodeusedforcalculatingtheresultsin[16].EXAMPLE1.Inthe rstexample,shownschematicallyonFig.5,therewereused25nodesuniformlydistributedovertheoutputapertureT.Thesolutionwasconstructedwith25ellipsoidswithonefocusatthesourceOandsecondfociatthenodesonT.Inthisexample,thediameterofT!(T)=p 2m,distancefromthesourcetothecentralnodeonTish=200m,themaximumamongalldistancesfromthesourcetonodesonTis200:005m,andtheparameter\r :4938639.Usingtheinequality(22),itcanbedeterminedthatself-blockagewillbeavoidedifthefocalparameterofthe rstellipsoidd14! 1 \r3:786727.Theactualcomputationswerecarriedoutwithd1=3:8.Thisgivesfortheparameter :019.Since3:799712!=2p 2,thecondition(12)issatis ed.Infact,thisinequalityshowsthatinthiscasewecouldhavetakend1justslightlybiggerthan2p 2(seeexample2below).Becausetheoutputenergypatternwasrequiredtobeuniformandthevariationofthe17 T OUTPUT INPUT APERTURE D,5ENERGY FEED q,f) = 10exp(-3q), q - ANGULAR DISTANCEFROM CENTRAL RAY FEED REFLECTOR OUTPUT ENERGY PATTERN L(v) IS REQUIRED TO BE UNIFORM L(v) h h - DISTANCE IN METERS FROM SOURCE TO THE PLANE OF OUTPUT APERTURE A O B AOB = 135LE 2) o o o Figure5:Geometryofthesinglere\rectorinexample1.feedenergydensityisnotverysigni cantalloftheresultingellipsoidshadfocalparameterscloseto3:8withthemaximum=3:80618andminimum=3:79972.Ittook348iterationstoreachasolutionforwhichmaxjcomputedoutputdensitydistribution requireduniformdistributionj0:01:Thewallclockrun-timewas19minutesona300MHZSiliconGraphicscomputerwithoneprocessor.Thedesignedre\rectorhasthefollowingdimensions:thedistancefromthesourcetothere\rectoralongthecentralrayis2:25mandthediameterofthere\rectoris1:27m.Itisimportanttonotethatintheaboveinequalitywearecomparingoutputdensitydistributionwiththedensitydistributionforthetruesolutionandtheboundontherighthandsideisuser-speci ed.Ifthisboundisdecreasedthenumberofiterationswillincrease,insomecases,quitesigni cantly.Asnapshotofthecomputergeneratedpictureofthefoundre\rectorisshowninFig.6.Notethattheapexofthere\rectorisdisplacedslightlyinthedownwarddirection.Thisisduetothefactthattheoutputapertureisatananglewiththeinputaperture(=135obetweencentralraysfromthesourcetotheinputapertureandtothetarget;seeFig.5).AstheangleAOBbecomessmallerthisdisplacementbecomesmoresigni cant;seeexample2below.EXAMPLE2.Inthesecondexamplethesettingwassimilartotheexample1,butthefollowingparameterswerechanged: AOB=90oandthefocalparameterofthe rstellipsoidwastaken=1:7.Thediameterofthisre\rectoris1:3m.Thedistancefrom18 Figure6:Aview(fromtheback)ofthere\rectorinexample1.thesourcetothere\rectoralongthecentralrayis1:7m.Therun-timeinthiscasewas13minutes.Asnapshotofthecomputergeneratedpictureofthere\rectorisshowninFig.7.Itclearlyshowsfurtherdisplacementofthesurfaceapexindownwarddirection.EXAMPLE3.Inthethirdexamplealldatawerethesameasinexample1exceptforthedistancehtothetargetaperturewhichwastaken=200;000m.Thisresultedonlyasmallchangeinthechoiceofthefocalparameterofthe rstellipsoidwhichwastaken=3:772.Thediameterofthedeterminedre\rectoris1:26mandthedistancefromthesourcetore\rectoralongthecentralrayis2:23m.Apictureofthisre\rectorisshownonFig.8.Thisexampleshowsthattheproposedtechniquescanbealsousedfordesigningre\rectorswithpre-speci edenergypatternsacrossfar- eldregions.EXAMPLE4.Thepurposeofthisexampleistoillustratethebehaviorofthealgorithmwhenthenumberofellipsoidsintheproblemincreases.Inthisexample,allofthedataremainsthesameasinexample1exceptforthenumberofellipsoidswhichwastakentobe36.Inthiscase,thealgorithmmade613iterations;therun-timewas1hour20minutes.Itsdiameteris1:27mandthedistancefromsourcetore\rectoralongthecentralrayis2:24m.Theexample4andotherexperimentsthatweperformedshowthatthecomputationaltimegrowswiththenumberofellipsoidsusedinthesolution.Whiletestingthecodeonnumerousexamplesweobservedthatthealgorithm ndsquicklyarelativelygoodapproxi-mationandthenitslowsdownastheiteratesapproachthetruesolution.Suchbehavioris19 Figure7:Aviewofthere\rectorinexample2.verytypicalfornumericalschemesdealingwithequationscontainingstrongnonlinearities.In[21]wepresentedastrategyforimprovingsigni cantlytheconvergencepropertiesinasimilarcaseofanonlinearproblem.Currently,weareinvestigatingtheapplicabilityofthisstrategytotheproblemconsideredhere.Itshouldbenoted,however,thatforpracticaldesignapplicationsthecurrentversionofthecodewillmostlikelybesucient.Thefactthatthealgorithmdoesnotrequiretheknowledgeofaninitialapproximationmakesitparticularlyattractive.Ofcourse,thefactthatitissupportedbyarigorousmathematicaljusti cationisofcriticalimportance,sincethismakesthenumericalresultsreliable.7ConclusionInthispaperanewmethodforsynthesisofo set,shaped,singlere\rectorantennasispre-sented.Thesynthesisproblemisconsideredingeometricalopticsapproximation.Incon-trastwithpreviouslyknownGOsynthesismethodsthevalidityofthemethodpresentedhereisestablishedbymathematicallyrigorousarguments.Ourresultscompletelyresolvethequestionsofexistenceanduniquenessofsolutionstotheabovesynthesisproblem.Is-suesregardingself-blockageandblockagebythefeedhavebeeninvestigatedandconditionswhichguaranteeavoidanceofsuchblockagesarepresented.Thisleadstodesignsofre\rectorantennaswithhigheciency.Thenewmethodhasbeenimplementedinacomputercode20 Figure8:Aviewofthere\rectorinexample3.andseveraldesigncasesarepresentedhere.Thecorrespondingnumericalprocedureisaniterativeonewhichstartswithaninitialsolution.Aimportantadvantageofournumericalprocedureisthattheinitialsolutioncanbeexplicitlyandeasilydeterminedfromtheinitialdata.Theproposedmethodanditsnumericalimplementationshouldhaveapplicationsinvariousimportantproblemsofdesignofcontourbeamshapingspacecraftandgroundsystemsinwhichhighgainisrequiredando setgeometrymustbeutilizedtoachievehigheciency.Acknowledgement.TheresearchofVladimirOlikerwaspartiallysponsoredbytheAFOSRundercontractF49620-97-C-0007andbytheEmoryUniversityResearchCom-mittee.References[1]V.Galindo-Israel,W.A.Imbriale,R.Mittra,andK.Shogen,Onthetheoryofthesyn-thesisofo setdual-shapedre\rectors{caseexamples,IEEETransactionsonAntennasandPropagation39,No.5,620-626(1991).21 [2]V.Galindo-Israel,W.A.Imbriale,andR.Mittra,Onthetheoryofthesynthesisofsingleanddualo 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