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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~E ei~k00~xintheairgap,and~Etei~k(~x ~d)onthetransmittedside.(Wehaveremovedanunimportantphasefromthetransmittedsidebyshifting~xbythevector~dpointingfromtheincidenttothetransmittedsideoftheairgap).dkk’k’0k0kinnn=1(air)rIfiistheincidentangle,thentheanglerfromthenormalintheairgapisgivenbySnell'slaw,nsini=sinr,andthetransmittedangleisalsoi(becauseitisthesamedielectric).Weseethatcosr=p1 sin2r=p1 n2sin2iandthatcosrispurelyimaginaryintheeventthatiisgreaterthanthecriticalanglefortotalinternalreection.ToobtainEtandErintermsofEi,wemaymatchtheparallelcomponentsof~Easwellastheparallelcomponentsof~H.Weconsidertwocases.For~Eperpendiculartotheplaneofincidence,thematchingbecomes rstinterfacesecondinterfaceEk:Ei+Er=E++E ;E+ei +E e i =EtHk:n(Ei Er)cosi=(E+ E )cosr;(E+ei E e i )cosr=nEtcosiwherewehaveintroducedthephase =~k0~d=k0dcosr=!dcosrcThematchingconditionsatthe rstinterfacemaybewrittenasE+=12Ei(1+ )+12Er(1 )E =12Ei(1 )+12Er(1+ )(1)wherewehavede ned =ncosicosr=ncosip1 n2sin2iSimilarly,thematchingconditionsatthesecondinterfaceyieldE+=12e i Et(1+ )E =12ei Et(1 )(2)Equating(1)and(2)allowsustosolvefortheratiosEtEi=4 (1+ )2e i (1 )2ei =2 2 cos i(1+ 2)sin ErEi=(1 2)(ei e i )(1+ )2e i (1 )2ei =i(1 2)sin 2 cos i(1+ 2)sin (3)where =ncosip1 n2sin2i; =!dcosrc=!dp1 n2sin2icSolongasiisbelowthecriticalangle,both and arereal.Inthiscase,thetransmissionandreectioncoecientsareT= EtEi 2=4 24 2cos2 +(1+ 2)2sin2 =4 24 2+(1 2)2sin2 R= ErEi 2=(1 2)2sin2 4 2cos2 +(1+ 2)2sin2 =(1 2)2sin2 4 2+(1 2)2sin2 (4)NotethatT+R=1,asexpected.However,thisexhibitsaclassicinterferencebehavior,whereToscillatesbetween(2 =(1+ 2))2and1asthenumberofwavelengthsinthegapvary.For~Eparalleltotheplaneofincidence,we ndinsteadthematchingconditions rstinterfacesecondinterfaceEk:(Ei Er)cosi=(E+ E )cosr;(E+ei E e i )cosr=EtcosiHk:n(Ei+Er)=(E++E );E+ei E e i =nEtTheseequationshavethesamestructureastheperpendicularcase,butwiththeindexofrefractionenteringsomewhatdi erently.We ndthematchingconditionsn 1E+=12Ei(1+ )+12Er(1 )n 1E =12Ei(1 )+12Er(1+ )andn 1E+=12e i Et(1+ )n 1E =12ei Et(1 )wherethistime =cosincosr=cosinp1 n2sin2iTheseexpressionsaresimilarto(1)and(2)above,exceptwiththereplacementE!n 1Eand ! .Hencethetransmissionandreectioncoecientsaregivenbyexpressionsidenticalto(4),exceptwiththereplacement ! .b)forigreaterthanthecriticalanglefortotalinternalreection,sketchtheratiooftransmittedpowertoincidentpowerasafunctionofdmeasuredinunitsofwavelengthinthegap.Tobeconcrete,considerthecasefor~Eperpendiculartotheplaneofincidence.Sinceiisgreaterthanthecriticalangle,both and willbepurelyimaginary.Whatevervaluestheyare,de ne =i; =iThentheratiosEt=EiandEr=Eiin(3)becomeEtEi=2i2icosh+(1 2)sinhErEi= (1+2)sinh2icosh+(1 2)sinhsothatT= EtEi 2=4242+(1+2)2sinh2R= ErEi 2=(1+2)2sinh242+(1+2)2sinh2where= ncosipn2sin2i 1;=!dpn2sin2i 1cInthiscase,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.Amediumofdielectricconstantandconductivitymaybedescribedbyane ectivedielectricconstant=+i!Sincethemediumisnonpermeable,wehave=0.Asaresult,fornormalincidence,theratioofthereectedtotheincidentelectric eldisErEi=1 n1+nwheren=r0=r0+i0!TheamplitudeAandphase'ofthereectedwaveisde nedbyAei'=ErEi=1 p=0+i=0!1+p=0+i=0!Thisexpressionimplicitlyde nesAand'.Tobemoreexplicit,wedecomposethecomplexindexofrefractionintoamagnitudeandphasen=pei =2where=s02+0!2;tan =!(5)TheamplitudeofthereectedwaveisthenA=s(1 n)(1 n)(1+n)(1+n)=s1+jnj2 2n1+jnj2+2n=s1+ 2pcos( =2)1++2pcos( =2)(6)whilethephaseis'=arg1 n1+n=arg(1 n)(1+n)(1+n)(1+n)=tan 1 2=n1 jnj2=tan 1 2psin( =2)1 (7)Notethatsomecaremustbetakenwhenextractingthephase.Inparticular,for'=tan 1(y=x),wemustensurethattheangle'liesintheproperquadrantde nedbythepoint(x;y).Thisiswhywechoosetokeeptheminussigninthenumer