4ZJr0jr 18CHAPTER3RADIATINGSYSTEMSINFREESPACEintothreedi erentregionsnearstaticzonedrintermediateinductionzonedrfarradiationzonedrForthenearzonetheexponentialintheintegraloft ID: 147840
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Chapter3Radiatingsystemsinfreespacewavesarealwaysgeneratedbytemporalchangesofchargeandcurrentdistributions.Thischapterdealswiththeprimary eldsofsuchsources,i.e.therearenoboundariesb 4ZJ(r0;!)jr 18CHAPTER3.RADIATINGSYSTEMSINFREESPACEintothreedi erentregions:near(static)zone:d r intermediate(induction)zone:d rfar(radiation)zone:d rForthenearzonetheexponentialintheintegralofthevectorpotentialcanbereplacedbyunity.Consequently,apartfromtheharmonictimedependence,thespatialbehaviourofthevectorpotentialisidenticaltothestaticcase,andcanbeexpandedintoaseriesofsphericalharmonics:A(r)=Xl;mClmYlm(; )rl+1(3.2)Inthefarzonekr 1,sotheexponentialtermintheintegraloscillatesrapidly.Thedistancejr r0jcanbereplacedbyjr r0jr nr0,wheren=r=r.ThevectorpotentialisnowA(r)=0eikr4rZJ(r0;!)e iknr0d3r0(3.3)Ifthesourcedimensionsaresmallcomparedtothewavelengththenitisreasonabletoexpandthisexpressionintoapowerserieswithrespecttok:A(r)=0eikr4r1Xm=0( ik)mm!ZJ(r0)(nr0)md3r0(3.4)SotheleadingbehaviourisAeikr=r.Thisisasphericalwavewithanangulardependentcoecient.Itisanexercisetoshowthatthe eldsaretransversetorandfallo as1=r,correspondingtoradiation elds.Theintermediateregionisdicult,sincetheapproximationsmadeabovearenotpossible.Leavingmathematicaldetailsasanexercise,wegivetheresulteikjr r0j4jr r0j=ik1Xl=0jl(kr)h(1)(kr�)lXm= lYlm(0; 0)Ylm(; )(3.5)wherejlandh(1)laresphericalBesselandHankelfunctions,respectively,andr=min(r;r0);r�=max(r;r0).Thentheexpansionofthevectorpotential,validforallroutsidethesource,isA(r)=i0kXl;mh(1)l(kr)Ylm(; )ZJ(r0)jl(kr0)Ylm(0; 0)d3r0(3.6)Themathematicalusefulnessofthisexpansionistheexplicitseparationoftheobservationandthesourcepointsinsphericalcoordinates. 3.1.MULTIPOLEEXPANSIONOFTHEVECTORPOTENTIAL193.1.1Electricdipole eldConsiderthe0thtermofEq.3.4:A(r)=0eikr4rZJ(r0)d3r0(3.7)RememberingthecontinuityequationrJ=i!andtakingintoaccountthatJisnon-zeroonlyina nitevolume,weobtainbyintegrationbypartsA(r)= i!0eikr4rZr(r)d3r(3.8)Usingthede nitionoftheelectricdipolemomentfromelectrostatics(p=Rr(r)d3r),thiscanbewrittenasA(r)= i!0eikr4rp(3.9)AninspectionofEq.3.6showsthatthisistheexactformofthe rsttermeverywhereoutsidethesource,notonlyinthefarregion.Anexerciseistocalculatethe eldsB(r)=k2(np)0ceikr4r(1 1ikr)(3.10)E(r)=k2(np)neikr40r+(3n(np) p)(1r3 ikr2)eikr40wheren=r=r.Atthestaticlimit,themagnetic eldvanishesandtheelec-tric eldtakesitsfamiliarstaticdipoleform.Inthefarzone,theradiation eldsareB(r)=k2(np)0ceikr4rE(r)=cBn(3.11)The eldsaretransversetotheradiusvectorfromthesourcetotheobser-vationpoint,andtheyarealsotransversetoeachother,andjEj=cjBj.TheradiatedpoweriscalculatedusingthePoyntingvector.Themean-ingfulquantityisthetime-average,whichfortheharmonictime-dependenceleadstothepowerpersolidangledPradd\n=120Re(r2nEB)=0c3k4322jnpj2(3.12)ThetotalpowerisPrad=0c3k412jpj2(3.13) 20CHAPTER3.RADIATINGSYSTEMSINFREESPACE3.1.2Magneticdipole eldThenexttermofthemultipoleexpansionisA(r)=0eikr4r(1r ik)ZJ(r0)nr0d3r0(3.14)whichisvalideverywhereoutsidethesourceregion.Itisusefultoseparatetheintegrandintosymmetricandantisymmetricpartswithrespecttor0andJ:(nr0)J=12[(nr0)J+(nJ)r0]+12(r0J)n(3.15)AnexerciseistoshowthatA(r)=i0keikr4r(1 1ikr)nm 0ck2eikr8r(1 1ikr)Zr0(nr0)(r0)d3r0(3.16)wheremisthemagneticdipolemomentofthecurrentsystem:m=12ZrJ(r)d3r(3.17)The rsttermofthevectorpotentialhasthesameformasthemagnetic eldoftheelectricdipole eldintheprevioussubsection.Sothe eldsareobtainedfromthe eldsoftheprevioussubsectionleadingtoE(r)= k2(nm)0ceikr4r(1 1ikr)B(r)=k2(nm)neikr40cr+(3n(nm) m)(1r3 ikr2)eikr40cr(3.18)Thesecondtermofthevectorpotentialismorecomplicated.Sincetheintegralinvolvessecondmomentsofthechargedensity,thistermcorre-spondstoanelectricquadrupole.Wewillnotstudyitfurther,andwealsoneglectallhighermultipolesforwhichthepresenttechniqueiste-dious.Agenerallymorepowerfulmethoddealswithvectormultipole elds(Sect.3.3).3.2ExamplesofradiatingsystemsTheprevioussectionwassomewhatabstractinconsideringthemultipoleexpansionofthevectorpotentialandrelated elds.Nowwepresentsomesimpleconcreteradiatingsystems. 3.2.EXAMPLESOFRADIATINGSYSTEMS21qÐqxyzL/2ÐL/2Figure3.1:Simpledipoleantenna.3.2.1RadiatingdipoleantennaConsideranelectricdipoleconsistingoftwosmallspheresinthezaxisatpointsz=L=2(Fig.3.1).Theyareconnectedbyawirewhosecapacitanceisnegligible.Ifthechargeoftheuppersphereisq(t)thenthechargeoftheloweroneis q(t).ConservationofchargeyieldsthecurrentdensityJ(r;t)=I(t)(x)(y)(L=2 z)(z+L=2)ez(3.19)whereI=+dq=dt.ThevectorpotentialhasonlythezcomponentAz(r;t)=04ZL=2 L=2I(t jr z0ezj=c)jr z0ezjdz0(3.20)Beforecontinuing,thereadershouldthinkqualitatively,whichcomponentsthe eldhas.Althoughthisisdynamicsystem,impressionsfromelectro-andmagnetostaticsarequiteusefulinthiscase.Atalargedistances(r L)wecanapproximatejr z0ezj=(r2 2z0ezr+z02)1=2r z0cos(3.21)whereistheanglebetweentheradiusvectorroftheobservationpointandthezaxis.Inthedenominatorofthevectorpotential,z0coscanbeignoredatlargedistances.Intheretardationtermitisnegligibleifz0cos=cissmallcomparedtothetimescaleofthecurrent,forexample,totheperiodTofaharmonicallyvaryingcurrent.Sincez0cosL=2,wecanomitz0cos=conlyifL=2 cT=(3.22)IfthisisthecasethenatlargedistancesAz(r;t)=04LrI(t r=c)(3.23) 22CHAPTER3.RADIATINGSYSTEMSINFREESPACEThesimplestwaytodeterminethescalarpotentialistoapplytheLorenzgaugerA+1c2@'@t=0(3.24)yielding@'@t= L40@@z1rI(t r=c)(3.25)=L40zr3I(t r=c)+zr2cI0(t r=c)whereI0referstodi erentiationofIwithrespecttot r=c.OntheotherhandI=+q0,so'(r;t)=L40zr2q(t r=c)r+I(t r=c)c(3.26)Consideraharmonicallyoscillatingdipoleq(t r=c)=q0cos!(t r=c)I(t r=c)=I0sin!(t r=c)= !q0sin!(t r=c)(3.27)InsphericalcoordinatesAr=04I0Lrcossin!(t r=c)A= 04I0Lrsinsin!(t r=c)(3.28)A =0Themagnetic eldB=rAhasnowonlya componentB =04I0Lrsin!ccos!(t r=c)+1rsin!(t r=c)(3.29)Thecomponentsoftheelectric eldE= @A=@t r'areEr=2I0Lcos40sin!(t r=c)r2c cos!(t r=c)!r3E= I0Lsin401!r3 !rc2cos!(t r=c) 1r2csin!(t r=c)E =0(3.30)ItiseasytoshowthatatlargedistancesE=cBer.TheradialcomponentofthePoyntingvectorexpressesthepowerradi-atedperaunitsolidangle:dPd\n=R2EB =0=120c(I0L!4c)2sin2cos2!(t R=c)(3.31) 3.2.EXAMPLESOFRADIATINGSYSTEMS23Integrationoverasphericalsurfaceyieldsthetotalradiatedpower:P=ISnda=10R2Z0EB 2sind(3.32)WhenR!1,onlytheradiation eldsproportionalto1=rcontribute,soPrad=ISnda=(I0L)260!2c3cos2!(t R=c)(3.33)Thisistheinstantaneousradiatedpower.IntegrationovertheperiodT=2=!providestheaveragepower,whichisgenerallyamorerelevantquan-tity:hPavei=L2!260c3I202=23r00L2I202(3.34)ThisisanalogoustotheaveragepowerRI20=2dissipatedinanACcircuitwhoseresistanceisRandcurrentI0cos!t.Soitismeaningfultode netheradiationresistanceRr=23r00L2789L2\n(3.35)Amagneticdipoleisanalysedinthesameway.Itcanbemodelledasacircularloopcarryingatime-harmoniccurrentI0cos!t.Thedipolevectorisperpendiculartotheplaneoftheloop.Becausethecurrenthasonlythe component,theonlynon-zerocomponentofthevectorpotentialisA (r;t)=0I0a4Z20cos!(t jr r0j=c)jr r0jcos d (3.36)whereaistheradiusoftheloop.Thedipoleapproximationpresumesthatr aand!a c.Therestofthecalculationisleftasanexercise.Adi erencetoanelectricdipoleisthattheelectric eldisnowtangentialtoanysphericalsurfacecenteredatthecurrentloop.3.2.2HalfwavelengthantennaTheresultobtainedintheprevioussectiondoesnotyieldthecorrectradia-tionpowerofatrueradioantenna,becauseanantennaisusuallynotshortcomparedtothewavelength,andthecurrentistypicallyfedintothecentre,nottotheends.Consideranantennawhoselengthisexactlyequaltoahalfwavelength.Thisisarealisticexample,sinceforexample,thewavelengthofa100MHzwaveis3m.Theantennacanbeformallyconstructedofin nitesimaldipoles 24CHAPTER3.RADIATINGSYSTEMSINFREESPACEconsideredintheprevioussection.Assumethattheantennaisinthez-axisin( =4;+=4)andthatitscurrentisI(z0;t)=I0sin!tcos2z0(3.37)Thisiszeroatbothends.Theelementatz0producesaradiationelectric eldwhose-componentisdE=I0sin40Rc2!cos!(t R=c)cos2z0dz0(3.38)HereRisthedistancefromdz0totheobservationpoint,andtermsoftheorderof1=R2areignored.The componentofthemagnetic eldisdB =04I0!Rcsincos!(t R=c)cos2z0dz0(3.39)Togetthetotal eldsEandB ,wemustcalculatetheintegralK=Z=2 =21Rcos!(t R=c)cosudu(3.40)whereu=2z0=.Again,R=r z0cosandatlargerwecanreplace1=Rby1=r.Thecosinetermrequiresmorecare:K1rZ=2 =2cos[!(t r=c)+ucos]cosudu(3.41)ThisisequaltoK=1rRe(ei!(t r=c)Z=2 =2eiucoscosudu)yieldingK=2rcos!(t r=c)cos[(=2)cos]sin2(3.42)Substitutingthisgivesthe eldsE=I020rccos!(t r=c)cos[(=2)cos]sin(3.43)B =0I02rcos!(t r=c)cos[(=2)cos]sin(3.44)Anexerciseistocalculatethetime-averagedradiatedpower:hPi=14r00I20Z0cos2[(=2)cos]sin2sind(3.45)Theintegralisapproximately1,219,sotheradiatingpowerofahalf-wave-lengthantennaishPi73\nI202(3.46) 3.2.EXAMPLESOFRADIATINGSYSTEMS253.2.3Centre-fedlinearantennaThethirdexampleisathinlinearantennaoflengthdexcitedbyacoaxialcableacrossasmallgapatitsmidpoint.ThecurrentdensityisJ(r)=Isin(kd=2 kjzj)(jzj d=2)(x)(y)ez(3.47)andthetime-dependenceisagainharmonic.Thevectorpotentialinthefarzone(kr 1)isaccordingtoEq.3.3A(r)=0Ieikr4rZd=2 d=2sin(kd=2 kjzj)e ikzcosdzez(3.48)resultinginA(r)=0Ieikr2krcos(kd2cos) cos(kd2)sin2ez(3.49)Anexerciseistocalculatethetime-averagedradiationpowerperunitsolidangle:dPd\n=I2820cjcos(kd2cos) cos(kd2)sinj2(3.50)3.2.4RadiationduetoamovingchargedparticleTheelectromagnetic eldduetoamovingchargedparticleisfamiliarfromthecourseofelectrodynamics.Wegiveashortoverviewoftheresultshere.Thechargeandcurrentdensitiesofachargedparticleare(r;t)=q(r rq(t))(3.51)J(r;t)=q_rq(t)(r rq(t))(3.52)AstraightforwardwayistosolvetheinhomogeneouswaveequationsofthevectorandscalarpotentialsintheLorenzgaugeusingthemethodofGreen'sfunctions.Thepotentialsare'(r;t)=q401(1 n )Rret=q401R R ret(3.53)A(r;t)=q40c (1 n )Rret=q40c R R ret(3.54)whereR=r rq,n=R=Rand =v=c.Thesubscriptretreferstotheevaluationoftheexpressionsattheretardedtimet0de nedbyt0+jr rq(t0)j=c=t(3.55) 26CHAPTER3.RADIATINGSYSTEMSINFREESPACETheobservermeasuresthe eldsatrattimet.Atediousdi erentiationyieldstheelectric eldE(r;t)=q40"(1 2)(R R )+R((R R )_ )=c(R R)3#ret(3.56)andthemagnetic eldB(r;t)=1cRRretE(r;t)(3.57)Theelectric eldisbasicallyasumofaCoulomb eldanda eldduetoaccelerationoftheparticle.Theformerisofnofurtherinteresthere,sinceitsPoynting\ruxvanishesatin nity.Thelattertermistheradiation eldvanishingas1=R.SothecorrespondingPoynting\ruxremains niteatin nity.Inotherwords,radiationcarriesthe eldenergy(andmomentumandangularmomentum)awayfromtheparticle.Theradiation eldsofaslowlymovingparticle( 1)areErad(r;t)=q40c2n(n_v)=R(3.58)Brad(r;t)=1cRRE=q40c3_vn=R(3.59)Poynting'svectorisS=10EradBrad=q21620c3jR_vj2R5R(3.60)Thisbehavesas1=R2,sothePoynting\ruxdoesnotvanish.Theradiationpowerperunitsolidangled\nisdPd\n=q2_v21620c3sin2(3.61)whereistheanglebetween_vandn.Integrationyieldsthetotalpower(Larmor'sformula)P=q2_v260c3(3.62)Forparticlesmovingathighvelocities,thedi erencebetweentandt0issigni cant.Theradiatedenergyduringtheintervalt1=t0+R(t01)=c:::t2=t02+R(t02)=cisW=Zt2t1[Sn]retdt=Zt0t01Sndtdt0dt0(3.63) 3.2.EXAMPLESOFRADIATINGSYSTEMS27Itismeaningfultode nethepowerradiatedperunitareaintermsofthecharge'sowntime:Sndt=dt0=Sn(1 n ).ThepowerradiatedperunitsolidangleisthenobtainedinastraightforwardmannerusingPoynting'svectorresultingindP(t0)d\n=q21620c n((n )_ ) 2(1 n )5(3.64)Tointerpretthis,wemayimaginethattheparticleisacceleratedonlyforashortintervalduringwhichthevelocityandaccelerationvectorsremainnearlyconstant.IfwearefarawaysothatnandRdonotpracticallychangeduringtheaccelerationintervalthenthisformulagivestheangulardistributionoftheradiatedpower.If !1thee ectofthedenominatorofdP=d\nincreasesandtheradiatedenergy\ruxconcentratesmoreparalleltothevelocity.ThetotalradiatedpowerisP=q260c\r6(_ 2 ( _ )2)(3.65)Thisresultcanbeobtainedintwoways,byadirectintegration,orbyusingtherelativisticformulation.However,weignorethedetailedcalculationhere.RadiationduetoasystemofmovingchargedparticlesConsiderasetofslowlymovingchargesv cwhichareassumedtobefarawayfromtheobservationpoint.Moreexplicitly,allchargesarewithinavolumeV1forthetimewhenthewavereachestheobserver.Further,thescalesofV1areassumedsmallcomparedtothewavelengthandtothedistancetotheobserver.LettheoriginbeinsideV1,denotecoordinatesofthechargesbyr0,oftheobservationpointbyr,andde neR=r r0.NowR=jr r0jr rr0r(3.66)andtheretardedscalarpotentialis'(r;t)=140ZV1(r0;t R=c)RdV0140ZV1(r0;t r=c+rr0=cr)r (rr0)=rdV0(3.67)Useofthebinomialseries(r rr0=r) 1=r 1+r 2(rr0=r)+:::(3.68) 28CHAPTER3.RADIATINGSYSTEMSINFREESPACEandtheTaylorexpansionr0;t rc+rr0cr=r0;t rc+rr0cr@@t r0;t r=c+:::(3.69)gives'(r;t)=140rZV1r0;t rcdV0+140r3rZV1r0r0;t rcdV0+140r2crddtZV1r0r0;t rcdV0+:::Thisisthefamiliarmultipoleexpansion:'(r;t)=140Qr+rp(t r=c)r3+r_p(t r=c)cr2(3.70)TheretardedvectorpotentialisA(r;t)=04ZV1J(r0;t r=c+rr0=cr)r (rr0)=rdV0=04rZV1Jr0;t rcdV0+:::(3.71)Itisanexercisetoshowthatfora nitevolumeVRVJdV=dp=dt,soA(r;t)=04r_p(t r=c)(3.72)Nextweonlyconsiderradiation eldswiththespatialdependencer 1.Because_pisafunctionof(t r=c),then@_p=@r= (1=c)p,andE(r;t)= 04rp(t r=c)+140rp(t r=c)c2r3r(3.73)Anexerciseistoshowthatthemagneticradiation eldisB(r;t)=rA(r;t)= 04cr2rp(3.74)andthatE(r;t)= crrB(r;t)(3.75)Allderivativesareevaluatedattheretardedtimet r=c,whichcanbeassumedidenticaltoallparticlesinV1.Theradiation eldisatransverseelectromagneticwave,whosePoyntingvectorS=cB20rr=11620c3r5(rp)2r(3.76) 3.2.EXAMPLESOFRADIATINGSYSTEMS29Settingz-axisparalleltopyieldsS=p2sin21620c3r2rr(3.77)Themaximumintensityisobtainedintheperpendiculardirectiontop.TodeterminetheradiatedpowerPrad,considerasphericalsurfaceatalargedistance:Prad=I@VSnda=160p2c3(3.78)Thisresultshowsthatasetofchargedparticlesradiatesiftherelateddipolemomenthas"acceleration".Itmayalsohappenthatthedipolemomentisindependentoftime,althoughparticleshaveacceleration.Insuchacase,itwouldbenecessarytoconsiderhighertermsinthemultipoleexpansion.3.2.6AuroralkilometricradiationAuroralkilometricradiation(AKR)isemittedbyelectronsintheauroralaccelerationregionatheights2000-20000km1.ThefrequencyisequaltotheLarmorfrequencyofelectrons(30-600kHz)anditswavelengthisafewkilometres.Themaximumradiationpowerduringmagneticstormsisabout1GW,whichisabout1%ofthepowerofparticleprecipitationintotheionosphere.ThesolarradiationpowerincidentattheEarthisinturnabout108GW.ThisisofthesameorderofmagnitudeastheinfraredradiationemittedbytheEarth.SoAKRisnotsigni cantfromtheenergyviewpoint.However,thenumberofphotonsperunittimecanbelargerforAKRthanforinfraredradiation.Thisisremarkable,sincethephoton\ruxde-termineshowdistantobjectsitispossibletoobserve,assumingthatthedetectortechnologyissophisticated.ItmightbepossibletoobserveAKRemitteduptoadistanceofabout100lightyears(numericalargumentationisanexercise).Inotherwords,ifthereareanyearth-likeexoplanetswithinthedistanceof100lytheycouldbeobservedduetotheirAKRemission.Thereareafewproblemsinsuchanobservation.First,thedetectorshouldbeinspace,becauseAKRisdissipatedintheionosphereduetoitslowfrequency.Itisevidentlypossibletoconstructsuchaspaceinstrumentwiththepresenttechnology.AnotherproblemisthattheSunisamorein-tenseradiationsourceatAKRfrequenciesthantheEarth,atleastmostofthetime.Concerningexoplanets,theirhoststarscoulddisturbtheobserva-tionofplanetaryAKR.Asolutionisinterferometry:signalmustbereceivedattwositessimultaneously.Anexerciseistoshowthatthismeasurementcouldbepossibletoperforminoursolarsystem.Thethirddicultymaybescintillationduetointerplanetaryplasmas.Finally,thereadershould1AKRsectionanditsexercisesarebasedonideaspresentedbyPekkaJanhunen(FMI). 30CHAPTER3.RADIATINGSYSTEMSINFREESPACEuseher/hisimaginationtothinkofwhatwecouldinferfromanobservationofAKRfromanEarth-likeexoplanet.3.3Vectormultipole eldsThesphericalharmonicexpansionisusefulinelectro-andmagnetostaticproblemshavingsomesymmetrywithrespecttotheoriginofthecoordi-natesystem.Fortime-varying eldsthescalarsphericalharmonicexpansioncanbegeneralisedtoavectorform.Thisisalsomorepowerfulthantheap-proachbasedontheexpansionofthevectorpotential(Sect.3.1).Thevectormultipoleapproachisusefulinboundaryvalueproblemswithasphericalsymmetryandinstudiesofradiationfromalocalisedsource.3.3.1BasicsphericalwavesolutionsWestartwiththescalarwaveequationr2 (r;t) 1c2@2 (r;t)@t2=0(3.79)FouriertransformwithrespecttotimeleadstotheHelmholtzequationr2 (r;!)+k2 (r;!)=0(3.80)Separationinsphericalcoordinatesleadstothefamiliarexpansion (r;!)=Xl;mfl(r)Ylm(; )(3.81)whereYlmaresphericalharmonics.Thereadermay(should)showthatthegeneralsolutioncanbewrittenas (r;!)=Xl;m(A(1)mh(1)l(kr)+A(2)lmh(2)l(kr))Ylm(; )(3.82)Hereh(1;2)listhesphericalHankelfunctionde nedbyh(1;2)l(x)=(2x)1=2(Jl+1=2(x)iNl+1=2(x))=jl(x)inl(x)(3.83)whereJandNaretheBesselandNeumannfunctions,andjandnarethesphericalBesselandNeumannfunctions,respectively.Theirexplicitexpressionsareeasilycomputedfromjl(x)=( x)l(1xddx)lsinxxnl(x)= ( x)l(1xddx)lcosxx(3.84) 3.3.VECTORMULTIPOLEFIELDS31Forotherusefulformulas,seeJacksonorArfkenandWeber,forexample.Nextweconsidermorecloselythesphericalharmonics,whichsatisfytheequation (1sin@@(sin@@)+1sin2@2@ 2)Ylm=l(l+1)Ylm(3.85)Itisconvenienttode neanoperatorL=1irr(3.86)Readersfamiliarwithquantummechanicsmayrecognizethatthisisba-sicallytheorbitalangularmomentumoperator.ItscomponentscanbewrittenasL+=Lx+iLy=ei (@@+icot@@ )L =Lx iLy=e i ( @@+icot@@ )Lz= i@@ (3.87)TheoperatorLa ectsonlyangularvariablesandisindependentofr.AstraightforwardcalculationshowsthatLL=L2=L2+L2y+L2zistheoperatoronthelefthandsideofEq.3.85:L2Ylm=l(l+1)Ylm(3.88)Theproofofthefollowingresultsisanexercise:rL=0L+Ylm=q(l m)(l+m+1)Yl;m+1L Ylm=q(l+m)(l m+1)Yl;m 1LzYlm=mYlmL2L=LL2LL=iLLjr2=r2Ljr2=1r@2@r2r L2r2irL=rr2 r(1+r@@r)LF=ir(rF)L(rF)=ir2(rF) ir@@r(r2rF)(3.89) 32CHAPTER3.RADIATINGSYSTEMSINFREESPACE3.3.2Multipoleexpansionoftheelectromagnetic eldsNextweconsidertheelectromagnetic eldinasource-freeregion.Assumingtime-harmonic elds(e i!t)anddenotingk=!=c,themagnetic eldsati estheHelmholtzequationandthedivergence-freecondition,andtheelectric eldisobtainedfromthemagnetic eld:(r2+k2)B=0rB=0E=ickrB(3.90)Alternatively,(r2+k2)E=0rE=0B= ikcrE(3.91)Thegoalistorepresentthe eldsasmultipoleexpansionswithexplicitlyseparatedradialandangulardependences.Onepossibilityisto rstnotethatforanywell-behavedvectorfunctionF,r2(rF)=r(r2F)+2rF(3.92)Consequently,outsideofthesourceregionthescalarfunctionsrBandrEsatisfytheHelmholtzequation(r2+k2)(rB)=0;(r2+k2)(rE)=0(3.93)ThegeneralsolutionforthesescalarfunctionsisgivenbyEq.3.82.Nowwede neamagneticmultipole eldoforder(l;m)byrB(M)lm=l(l+1)kcgl(kr)Ylm(; )rE(M)lm=0(3.94)wheregl(kr)=A(1)h(1)l(kr)+A(2)lh(2)l(kr)(3.95)Themagneticandelectric eldsarerelatedbykcrB=1ir(rE)=1i(rr)E=LE(3.96)Theelectric eldofthemagneticmultipolemustthensatisfytheequationLE(M)lm(r;; )=l(l+1)gl(kr)Ylm(; )(3.97) 3.3.VECTORMULTIPOLEFIELDS33andrE(M)lm=0.BecausetheoperatorLactsonlyontheangularvariables,theradialdependenceofE(M)lmisgivenbygl(kr).Asshownintheprevioussubsection,thee ectofLonYlmistoraiseorlowerm,butnottomodifylatall.RememberingthatL2Ylm=l(l+1)Ylm,weseerathereasilythattheelectric eldmustbeE(M)lm(r;; )=gl(kr)LYlm(; )(3.98)SinceB(M)lm= ikcrE(M)lm(3.99)the eldsofamagneticmultipoleoforder(l;m)arenowspeci edapartfromthecoecientsA(1;2)l.Inthesameway,wede neanelectricmultipoleoforder(l;m)byrE(E)lm= l(l+1)ckfl(kr)Ylm(; )rB(E)lm=0(3.100)Theelectricmultipole eldsareB(E)lm(r;; )=fl(kr)LYlm(; )E(E)lm=ickrB(E)lm(3.101)Theradialfunctionfl(kr)hasasimilarexpressiontogl(kr).WearenowreadytowritethegeneralsolutionoftheMaxwellequa-tionsinthesource-freeregion.Formoreconvenientnotations,wede neanormalizedformofthevectorsphericalharmonic:Xlm(; )=1pl(l+1)LYlm(; )(3.102)Ithasthefollowingorthogonalityproperties:ZX0m0Xlmd\n=ll0mm0ZXl0m0(rXlm)d\n=0(3.103)Consequently,the eldsareB=XlmaE(l;m)fl(kr)Xlm ikcaM(l;m)r(gl(kr)Xlm)E=XlmickaE(l;m)r(fl(kr)Xlm)+aM(l;m)gl(kr)Xlm(3.104) 34CHAPTER3.RADIATINGSYSTEMSINFREESPACETheunknowncoecientsareobtainedfromaM(l;m)gl(kr)=kpl(l+1)ZYlmrBd\naE(l;m)fl(kr)= kpl(l+1)ZYlmrEd\n(3.105)Thus,weonlyhavetoknowrBandrEattwodi erentradiir1andr2inthesource-freeregionforacompletesolution,includingtherelativemagnitudesofA(1;2)l.3.3.3Nearandfarzonemultipole eldsTostudythenearzone elds(kr 1),wenotethattheradialdependenceisgivenbyh(1;2)l(kr)=jl(kr)inl(kr).Withasmallkr,thesphericalHankelfunctionbehaveslike(kr) (l+1),andthemagnetic eldforanelectricmultipole(l;m)isB(E)lm klLYlmrl+1(3.106)wherethespeci cproportionalitycoecientischosenforconvenience.Theelectric eldisE(E)lm=ickrB(E)lm ilrL(Ylmrl+1)(3.107)Usinganoperatoridentity,theelectric eldtakestheformE(E)lm (rr2 r(1+r@@r))Ylmlrl+1(3.108)The rsttermvanishes,sinceYlm=rl+1isasolutionoftheLaplaceequation.ItfollowsthatE(E)lm r(Ylmrl+1)(3.109)Fromthebasiccourseofelectrodynamicswerememberthatthisisexactlytheelectrostaticmultipole eld.Themagneticmultipole eldsaretreatedinthesamewayjustbyinterchangingE(E)! B(M);B(E)!E(M).Forthefarzone(kr 1),weinvestigateoutgoingwavesfromalocal-izedsource,correspondingtotheradialdependencegivenbyh(1)(kr).Itsasymptoticbehaviourimpliesthatthemagnetic eldofanelectricmultipoleisB(E)lm( i)l+1eikrkrLYlm(3.110)andtheelectric eldisE(E)lm( i)lck2(r(eikrr)LYlm+eikrrrLYlm)(3.111) 3.3.VECTORMULTIPOLEFIELDS35Keepingonlytheleadingtermsandusingthesameoperatoridentityasforthenearzonecalculation,thisbecomesE(E)lm ( i)l+1ceikrkr(nLYlm 1k(rr2 r)Ylm)(3.112)wheren=r=r.Thesecondtermisclearlysomeangularfunctiontimes1=r,soitisignorable.Notsurprisingly,theradiationzone eldshaveanexactrelationshipE(E)lm=cB(E)lmn(3.113)Again,themagneticmultipole eldsareobtainedbyasimilarinterchangelikewiththenearzonecase.3.3.4EnergyofmultipoleradiationWestarttheenergyconsiderationbystudyingonlyalinearsuperpositionofelectricmultipoles(l;m)withavaryingmbuta xedl.Thentheoutgoing eldsareBl=XmaE(l;m)Xlmh(1)(kr)El=ickrBl(3.114)Thetime-averagedenergydensityfortime-harmonic eldsisu=04(EE+c2BB)(3.115)IntheradiationzoneE=cBn,sotheenergydUinasphericalshell[r;r+dr]isdU=20c2drk2Xm;m0a(l;m0)aE(l;m)Z(4)Xlm0Xlmd\n(3.116)wheretheasymptoticexpressionofthesphericalHankelfunctionisapplied.Usingtheorthogonalityintegralofvectorsphericalharmonics,thisyieldsdUdr=20c2k2XmjaE(l;m)j2(3.117)ForageneralsuperpositionofelectricandmagneticmultipolestheresultisdUdr=20c2k2Xl;m(jaE(l;m)j2+jaM(l;m)j2)(3.118) 36CHAPTER3.RADIATINGSYSTEMSINFREESPACENextwediscussthepowerofmultipoleradiation.Atthelimitkr 1the eldsareB=eikrkrXlm( i)l+1(aE(l;m)Xlm+aM(l;m)nXlm)E=cBn(3.119)TheaveragepowerradiatedperunitsolidangleiscalculatedfromPoynting'svectoryieldingdPd\n=c20k2jXlm( i)l+1(aE(l;m)Xlmn+aM(l;m)Xlm)j2(3.120)Theelectricandmagneticmultipolesofagiven(l;m)havethesameangulardependence,butperpendicularpolarizations.Sothemultipoleordercanbedeterminedbymeasuringtheangulardistributionofradiatedpower.Todistinguishbetweentheelectricandmagneticnatureoftheradiatingsource,polarizationmustbedetected.TheangularpowerdistributionofasinglemultipoleisdP(l;m)d\n=c20k2ja(l;m)Xlmj2(3.121)3.3.5MultipolemomentsThenexttaskistorelatethemultipoleexpressionsdirectlytothesourceterms.Weassumethatthechargedensity(r)e i!tandcurrentdensityJ(r)e i!tareknown.Wecouldalsoconsidertheintrinsicmagnetizationasasourceterm,butweneglectithereforbrevity(seeJacksonforacompletediscussion).Again,westartwiththeMaxwellequationsrE==0rB=0rE ikcB=0rB+(ik=c)E=0J(3.122)De ningE0=E+iJ=(!0)andusingtheequationofthecurrentcontinuity,weobtainrE0=0rB=0rE0 ikcB=irJ=(!0)rB+(ik=c)E0=0(3.123) 3.3.VECTORMULTIPOLEFIELDS37Ifwehadincludedmagnetization,itscurlwouldappearintherightsideofthelastequation.NotethatE0=Eoutsideofsources.TheinhomogeneousHelmholtzwaveequationsfollowfromthecurlequa-tions:(r2+k2)B= 0rJ(r2+k2)E0= ir(rJ)=(0ck)(3.124)Weusethesametrickasearlierandtakethescalarproductwithvectorr,andusethevectoridentityr(rF)=(rr)F= iLF.Thisyieldsscalarequations(r2+k2)rB= i0LJ(r2+k2)rE0=L(rJ)=(0ck)(3.125)TheseequationscanbesolvedusingthemethodofGreen'sfunctions(seethecourseofelectrodynamics).Sinceweareinterestedinoutgoingwaves,thesolutionisrB(r)=i04Zeikjr r0jjr r0jL0J(r0)d3r0rE0(r)= 140ckZeikjr r0jjr r0jL0r0J(r0)d3r0(3.126)ThemultipolecoecientsaE;aMareobtainedfromaM(l;m)gl(kr)=kpl(l+1)ZYlmrBd\naE(l;m)fl(kr)= kpl(l+1)ZYlmrE0d\n(3.127)asderivedinSect.3.3.2.Theradialdependencemustbefl(kr)=gl(kr)=h(1)(kr),sincethiscorrespondstooutgoingradiation.Next,weneedEq.3.5:eikjr r0j4jr r0j=ik1Xl=0jl(kr)h(1)(kr�)lXm= lYlm(0; 0)Ylm(; )(3.128)wherenowr=r0;r�=r,becausewestudytheregionoutsideofthesource.ItfollowsthatZd\nYlm(; )eikjr r0j4jr r0j=4ikh(1)(kr)jl(kr0)Ylm(0; 0)(3.129) 38CHAPTER3.RADIATINGSYSTEMSINFREESPACEConsequently,themultipolecoecientsareaM(l;m)= 0k2pl(l+1)Zjl(kr)Ylm(; )LJ(r)d3r(3.130)aE(l;m)=ik0cpl(l+1)Zjl(kr)Ylm(; )LrJ(r)d3rAcentre-fedlinearantenna(Sect.3.2.3)providesanillustratingconcreteexercise.