E&M 七 3 4 6 8

Physics505Fall2007HomeworkAssignment#11|SolutionsTextbookproblems:Ch.7:7.3,7.4,7.6,7.87.3Twoplanesemi-inniteslabsofthesameuniform,isotropic,nonpermeable,losslessdielectricwithindexofrefractionnareparallelandseparatedbyanairgap(n=1)ofwidthd.Aplaneelectromagneticwaveoffrequency!isindicentonthegapfromoneoftheslabswithangleofindicencei.Forlinearpolarizationbothparalleltoandperpendiculartotheplaneofincidence,a)calculatetheratioofpowertransmittedintothesecondslabtotheincidentpowerandtheratioofreectedtoincidentpower;Weintroduce(complex)electriceldvectorsoftheform~Eiei~k
~xand~Erei~k0
~xontheincidentside,~E+ei~k0
~xand~Eei~k00
~xintheairgap,and~Etei~k
(~x~d)onthetransmittedside.(Wehaveremovedanunimportantphasefromthetransmittedsidebyshifting~xbythevector~dpointingfromtheincidenttothetransmittedsideoftheairgap).dkk’k’0k0kinnn=1(air)rIfiistheincidentangle,thentheanglerfromthenormalintheairgapisgivenbySnell'slaw,nsini=sinr,andthetransmittedangleisalsoi(becauseitisthesamedielectric).Weseethatcosr=p1sin2r=p1n2sin2iandthatcosrispurelyimaginaryintheeventthatiisgreaterthanthecriticalanglefortotalinternalreection.ToobtainEtandErintermsofEi,wemaymatchtheparallelcomponentsof~Easwellastheparallelcomponentsof~H.Weconsidertwocases.For~Eperpendiculartotheplaneofincidence,thematchingbecomesrstinterfacesecondinterfaceEk:Ei+Er=E++E;E+ei+Eei=EtHk:n(EiEr)cosi=(E+E)cosr;(E+eiEei)cosr=nEtcosiwherewehaveintroducedthephase=~k0
~d=k0dcosr=!dcosrcThematchingconditionsattherstinterfacemaybewrittenasE+=12Ei(1+)+12Er(1)E=12Ei(1)+12Er(1+)(1)wherewehavedened=ncosicosr=ncosip1n2sin2iSimilarly,thematchingconditionsatthesecondinterfaceyieldE+=12eiEt(1+)E=12eiEt(1)(2)Equating(1)and(2)allowsustosolvefortheratiosEtEi=4(1+)2ei(1)2ei=22cosi(1+2)sinErEi=(12)(eiei)(1+)2ei(1)2ei=i(12)sin2cosi(1+2)sin(3)where=ncosip1n2sin2i;=!dcosrc=!dp1n2sin2icSolongasiisbelowthecriticalangle,bothandarereal.Inthiscase,thetransmissionandreectioncoecientsareT=EtEi2=4242cos2+(1+2)2sin2=4242+(12)2sin2R=ErEi2=(12)2sin242cos2+(1+2)2sin2=(12)2sin242+(12)2sin2(4)NotethatT+R=1,asexpected.However,thisexhibitsaclassicinterferencebehavior,whereToscillatesbetween(2=(1+2))2and1asthenumberofwavelengthsinthegapvary.For~Eparalleltotheplaneofincidence,wendinsteadthematchingconditionsrstinterfacesecondinterfaceEk:(EiEr)cosi=(E+E)cosr;(E+eiEei)cosr=EtcosiHk:n(Ei+Er)=(E++E);E+eiEei=nEtTheseequationshavethesamestructureastheperpendicularcase,butwiththeindexofrefractionenteringsomewhatdierently.Wendthematchingconditionsn1E+=12Ei(1+)+12Er(1)n1E=12Ei(1)+12Er(1+)andn1E+=12eiEt(1+)n1E=12eiEt(1)wherethistime=cosincosr=cosinp1n2sin2iTheseexpressionsaresimilarto(1)and(2)above,exceptwiththereplacementE!n1Eand!.Hencethetransmissionandreectioncoecientsaregivenbyexpressionsidenticalto(4),exceptwiththereplacement!.b)forigreaterthanthecriticalanglefortotalinternalreection,sketchtheratiooftransmittedpowertoincidentpowerasafunctionofdmeasuredinunitsofwavelengthinthegap.Tobeconcrete,considerthecasefor~Eperpendiculartotheplaneofincidence.Sinceiisgreaterthanthecriticalangle,bothandwillbepurelyimaginary.Whatevervaluestheyare,dene=i;=iThentheratiosEt=EiandEr=Eiin(3)becomeEtEi=2i2icosh+(12)sinhErEi=(1+2)sinh2icosh+(12)sinhsothatT=EtEi2=4242+(1+2)2sinh2R=ErEi2=(1+2)2sinh242+(1+2)2sinh2where=ncosipn2sin2i1;=!dpn2sin2i1cInthiscase,thereisnooscillatorybehaviorinthetransmittedpower,butonlyexponentialsuppressionastheairgapiswidened.ItiseasytoseethatT!1whend!0(correspondingto!0)andthatTfallsexponentiallyto0whend!1(whichisthesameas!1).Forn=1:5(approximatelytheindexofrefractionofglass),thecriticalanglefortotalinternalreectionisi042.AsketchofTasafunctionofdlookslike0.511.520.20.40.60.81Td/(2π/k0)=1.5ni=75i=60i=457.4Aplane-polarizedelectromagneticwaveoffrequency!infreespaceisincidentnor-mallyontheatsurfaceofanonpermeablemediumofconductivityanddielectricconstant.a)Calculatetheamplitudeandphaseofthereectedwaverelativetotheincidentwaveforarbitraryand.Amediumofdielectricconstantandconductivitymaybedescribedbyaneectivedielectricconstant=+i!Sincethemediumisnonpermeable,wehave=0.Asaresult,fornormalincidence,theratioofthereectedtotheincidentelectriceldisErEi=1n1+nwheren=r0=r0+i0!TheamplitudeAandphase'ofthereectedwaveisdenedbyAei'=ErEi=1p=0+i=0!1+p=0+i=0!ThisexpressionimplicitlydenesAand'.Tobemoreexplicit,wedecomposethecomplexindexofrefractionintoamagnitudeandphasen=pei=2where=s02+0!2;tan=!(5)TheamplitudeofthereectedwaveisthenA=s(1n)(1n)(1+n)(1+n)=s1+jnj22n1+jnj2+2n=s1+2pcos(=2)1++2pcos(=2)(6)whilethephaseis'=arg1n1+n=arg(1n)(1+n)(1+n)(1+n)=tan12=n1jnj2=tan12psin(=2)1(7)Notethatsomecaremustbetakenwhenextractingthephase.Inparticular,for'=tan1(y=x),wemustensurethattheangle'liesintheproperquadrantdenedbythepoint(x;y).Thisiswhywechoosetokeeptheminussigninthenumer