耦合模理论-coupled-mode-theory

1Lecture6:Coupled-modetheoryModeexpansionSingle-waveguidemodecouplingMultiple-waveguidemodecouplingTwo-modecouplingCodirectionalcouplingContradirectionalcouplingPhasematchingReferences:ThislecturefollowsthematerialsfromPhotonicDevices,Jia-MingLiu,Chapter4.2Coupled-modetheoryCoupled-modetheorydealswiththecouplingofspatialmodesofdifferentspatialdistributionsordifferentpolarizations,orboth.ThenormalmodefieldsspatialdependenceinalosslesswaveguideatasinglefrequencycanbegivenasThetwoMaxwell’sequationsbecome)(exp),()(ziyxErE)(exp),()(ziyxHrHHiE0EiHThenormalmodeswithfieldsgivenbyE(r)andH(r)arecharacteristicsolutionsofMaxwell’sequations.Modeexpansion34ModeexpansionThenormalmodesareorthogonalandcanbenormalizedtohavetheorthonormalityrelation.Theyformabasisforlinearexpansionofanyopticalfieldatagivenfrequencyinthewaveguide:whereEandHarenormalizedmodefieldssatisfyingtheorthonormalityrelation,andthesummationsumsoveralldiscreteindicesoftheguidedmodes(andintegratesoverallcontinuousindicesoftheradiationandevanescentmodes).Inanidealwaveguidewherethesemodesaredefined,thenormalmodesdonotcouple.Then,theexpansioncoefficientsAareconstantsthatareindependentofx,yandz.)(exp),(ˆ)(ziyxEArE)(exp),(ˆ)(ziyxHArH5PowerandorthogonalityThenormalmodesareorthogonalandcanbenormalizedtohavetheorthonormalityrelation.Exceptforevanescentfields,theenergyofthefieldsinawaveguideflowsonlyinthelongitudinaldirection.Theintensityofawaveguidemodeisgivenbywhichisafunctionofxandy.Thepower,P,ofthemodeisobtainedbyintegratingI(x,y)overtheentiretransversecrosssectionofthewaveguide.zHEHEzSSIˆ)(ˆ)(***6PowerandorthogonalityForTEandTMmodes,thepowerobtainedbyintegratingI(x,y)canbegivenasInalosslessisotropicwaveguide,themodefieldshavethefollowingorthogonalityrelation:dxdyEPTE202dxdyHyxPTM2),(12PdxdyzHEHEˆ**7PowerandorthogonalityThemodefieldscanbenormalizedtohavethefollowingorthonormalityrelation:wheretheplussignisforforward-propagatingmodeswhiletheminussignisforbackward-propagatingmodes.TheelectricandmagneticfieldpatternsofaparticularmodearerepresentedbythenormalizedmodefielddistributionsE(x,y)andH(x,y).HereistheKroneckerdeltafunctionfordiscretemodes.Foranonplanarwaveguide,=mnand=m’n’,andmm’nn’Foraplanarwaveguide,=m,=m’,andmm’dxdyzHEHEˆˆˆˆˆ**8PowerandorthogonalityForTEmodes,theorthonormalityrelationcanbegivenasForTMmodes,Theorthogonalityrelationortheorthonormalityrelationindicatethatpowercannotbetransferredbetweendifferentmodesinalinear,losslesswaveguide.Foranisotropicorlossywaveguides,theorthogonalityconditionsformodesofsuchwaveguideshaveotherforms.dxdyEE*0ˆˆ2dxdyHHyx*ˆˆ),(129ModeexpansionWhenthereisaspatiallydependentperturbationtoawaveguide,themodesdefinedbytheunperturbedidealwaveguidearenolongerexactnormalmodesoftheperturbedwaveguide.Theycannowbecoupledbytheperturbationastheypropagatealongthewaveguide.Ifthefieldsarestillexpandedintermsofthenormalmodesoftheunperturbedwaveguide,theexpansioncoefficientsarenolongerconstantsofpropagationbutvarywithzasthefieldspropagatedownthewaveguide:)exp(),(ˆ)()(ziyxEzArE)exp(),(ˆ)()(ziyxHzArHSingle-waveguidemodecoupling1011Single-waveguidemodecouplingConsiderthecouplingbetweennormalmodesinasinglewaveguidethatissubjecttosomeperturbation.ThespatiallydependentperturbationtothewaveguidecanberepresentedbyaperturbingpolarizationP(r)atfrequency.ThefollowingMaxwell’sequationsareusedThefieldsintheperturbedwaveguidearegovernedbythesetwoequationswithP0.ThenormalmodefieldsoftheunperturbedwaveguidealsosatisfythesetwoequationswithP=0.HiE0PiEiH12Single-waveguidemodecouplingUsingthetwoMaxwell’sequations,wehaveIfwetake(E1,H1)tobethoseoftheperturbedmodefieldsand(E2,H2)tobethenormalmodefields,wehaveP1=PandP2=0.Substitutingtheseintotheaboveequationandintegratingbothsidesoftheresultantequationoverthecrosssectionofthewaveguide,wehave)()(1*2*211**212PEPEiHEHEPdxdyEeidxdyzHEHEezAdzdzizi***)(ˆˆ)ˆˆˆˆ()(13Single-waveguidemodecouplingByapplyingtheorthonormalityrelation,wefindfromabovethefollowingcoupled-modeequation:wheretheplussignisusedwhen0andmodeisaforward-propagatingmode,andtheminussignisusedwhen0andmodeisabackward-propagatingmode.Theresultcanbeusedformodecouplingcausedbyanykindofspatiallydependentperturbationonthecharacteristicsofthewaveguide.E.g.Pcanbeaperturbingpolarizationduetotheeffectsofnonlinearopticalinteractionsonthefieldsatfrequencyinthewaveguide.PdxdyEeidzdAzi*ˆ14Single-waveguidemodecouplingForthecasewheretheperturbationcanberepresentedbyachangeinlinearpolarizationasWehaveisthecouplingcoefficientbetweenmodeandmode.zieEAEPˆ)(zieAidzdA)(dxdyEEˆˆ*15Single-waveguidemodecouplingThisresultcanalsobeextendedtoanisotropicwaveguidesbyconsideringPtobeapolari