D Joannopoulos Department of Physics and Center for Material Science and Engineering Massachusetts Institute of Technology Cambridge Massachusetts 02139 Received July 25 2002 revised manuscript received October 3 2002 accepted October 21 2002 We pre ID: 23396
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Temporalcoupled-modetheoryfortheFanoresonanceinopticalresonatorsShanhuiFanandWonjooSuhDepartmentofElectricalEngineering,StanfordUniversity,Stanford,California94305J.D.JoannopoulosDepartmentofPhysicsandCenterforMaterialScienceandEngineering,MassachusettsInstituteofTechnology,Cambridge,Massachusetts02139ReceivedJuly25,2002;revisedmanuscriptreceivedOctober3,2002;acceptedOctober21,2002WepresentatheoryoftheFanoresonanceforopticalresonators,basedonatemporalcoupled-modeformal-ism.Thistheoryisapplicabletothegeneralschemeofasingleopticalresonancecoupledwithmultipleinput dtSjv021 tDa1~^ku*!us1 fromports1to,respectively,withthecouplingcon-(ForcompactnessofpresentationweadoptDirac'sbracketnotationtodescribevectorsthatareindexedtothelabelsoftheports.)Theresonantmode,onceexcited,coupleswiththeoutgoingwavesattheportswiththecouplingconstantsInadditiontotheresonance-assistedcouplingbetweentheports,theincomingandoutgoingwavesintheportscanalsocouplethroughadirectpathway,asdescribedbyascatteringmatrix.Thepresenceofthedirectpath-wayisanessentialaspectoftheFanoeffect.Hencetheheremustbetakentobeanarbitraryscatteringmatrix,i.e.,anyunitaryandsymmetricmatrix.Equations(1)and(2)representageneralizationofthestandardtemporalcoupled-modetheory,inwhichisadiagonalmatrix.Ourtheorythusassumesthesamere-gimeofvalidityasthestandardtemporal-coupled-modetheory;i.e.,thisapproachisstrictlyvalidonlywhenthewidthoftheresonanceisfarsmallerthantheresonancefrequency.IthasbeenshowninRef.17thatinthisre-gimethecouplingconstantscanbetakentobefrequencyindependentandthatthefrequencyshiftduetotheexpo-nentialdecayofthemodetotheportsisasecond-ordereffectandcanbeincorporatedintothetheorythrougharenormalizationofThecoef®cients,andarenotindependent;rather,theyarerelatedbyenergy-conservationandtime-reversalsymmetryconstraints.Belowwewillexploittheconse-quenceoftheseconstraintstodevelopaminimumsetofparametersthatcompletelycharacterizethesystem.First,forexternallyincidentexcitationsatafre-,wecanwritethescatteringmatrixforthesystemdescribedbyEqs.(1)and(2)as .(3)Sincethescatteringmatrixhastobesymmetricbecauseoftime-reversalsymmetry,wehave.(4)(Thusthecoef®cientsarenotindependentandmustsatisfy,etc).Also,withincoming-waveamplitudes,theamplitudeoftheresonantmodeis .(5)Insteadofconsideringthecaseinwhichtheresonatorisexcitedbyexternallyincidentwaves,letusnowconsideranalternativesituationinwhichtheexternalin-cidentwaveisabsent,i.e.,0,andat0thereisa®niteamplitudeoftheresonance.At0,thereso-nantmodedecaysexponentiallyintothetwoports,as dt2 ,(6)whichrequiresthat.(7)Now,letusperformatime-reversaltransformationfortheexponentialdecayprocessasdescribedbyEq.(6).Thetime-reversedcaseisrepresentedbyfeedingthereso-natorwithexponentiallygrowingwavesatacomplexfre-(1/),withamplitudesat0equal.Suchexcitationscausearesonanceamplitude0togrowexponentiallyintime.UsingEq.(5)atthecomplexfrequency(1/),wehave 2/t5~^kud&a!* andtherefore.(8)CombiningEqs.(4),(7),and(8),weareledtoanimpor-tantconclusion:.(9)Thetime-reversedexcitationalsohastosatisfytheconditionthatnooutgoingwaveshalloccuruponsuchexcitations;i.e.,,(10) Fig.1.Schematicofanopticalresonatorsystemcoupledwithmultipleports.Thearrowsindicatetheincomingandoutgoingwaves.Thedashedlinesarereferenceplanesforthewaveam-plitudesintheports.570J.Opt.Soc.Am.A/Vol.20,No.3/March2003etal. Thusthecouplingconstantshavetosatisfyafurther.(11)Hencethecouplingconstantsingeneralcannotbearbi-trarybutareinsteadrelatedtothescatteringmatrixofthedirectprocess.TocheckthatEqs.(9)and(11)indeedproduceaself-consistenttemporalcoupled-modetheory,weneedtoen-surethatthescatteringmatrix,asde®nedbyEq.(3),isunitary.Forthispurpose,wenotethat ~v2v0!21~1/t!21Cud&*^du 2j~v2v0!1~1/t!1udu*C1 TakingadvantageofEq.(11)anditscomplexconjugate,wecanindeedprovetheunitarypropertyofthematrix ~v2v0!21t212udu 2j~v2v0!1~1/t!12udu .(14)Equations(1)(14)areapplicabletothegeneralprob-lemofasingleopticalmodecoupledwithmultipleinputandoutputports.Below,wewillapplythegeneralfor-malismtotwo-portstructures.Inthiscase,itcanbeshownthatoncethemagnitudesofthecouplingconstantsare®xed,thephasesofthecouplingconstantscanbedeterminedfromthescatteringmatrixofthedi-rectprocess.Thetheorycanbefurthersimpli®ed,how-ever,whenweconsiderstructureswithmirrorsymmetry.Forthesestructures,ifweplacethereferenceplanessymmetricallyoneachsideofthestructurewithrespecttothemirrorplane,thescatteringmatrixhastobesuchthatthetwodiagonalelementsareequal.Thuswehave.Thescatteringmatrixforthedirecttransportprocessalsoacquiresaspecialformrjtjtr,(15)r,t,andarerealconstantswithUsingEqs.(7)and(11),wecandetermine,andconsequentlythescatteringmatrixfortheoverallsys-temasrjtjtr Herethesigncorrespondstothecasewherethereso-nantmodeiseven(odd)withrespecttothemirrorplane,inwhichcase.FromEq.(16),theinten-sityre¯ectioncoef®cientistherefore AsymmetricLorentzianlineshapeisreproducedonlywheneitheriszero.Inallothercases,thesystemexhibitsaFanoasymmetriclineshape.Thusourtheorydirectlypredictstheline-shapefunctionoftheFanophe-3.NUMERICALVALIDATIONOFTHETHEORYThetheoreticalderivationaboveshouldbeapplicabletoanysingle-modeopticalresonatorsystem.Tocheckthevalidityofthetheory,wecomparethetheoreticalpredic-tionsto®rst-principlessimulationsofonetypeofopticalresonance:theguidedresonanceinaphotoniccrystalslab.Forde®niteness,weconsideracrystalconsistingofasquarelatticeofairholes,eachwitharadiusof0.2isthelatticeconstant,introducedintoadielectricslabwithadielectricconstantof12andathicknessof.We®ndthatthetwolowest-frequencyresonantstatesatoccurat0.37and0.39(c/),wherecisthespeedoflightinvacuum.Weshallnowfocusontheseresonancesandinvestigatetheirlineshapes.Using®nite-differencetime-domainsimulations,wecalculatethetransmissionspectraoflightthatisnormallyincidentontheslab[Fig.2(a)]. Fig.2.(a)Photoniccrystalstructureconsistingofasquarelat-ticeofairholesofradius0.2inadielectricslabwithdielectricconstant12andathicknessof0.5.Thearrowindicatesthedirectionoftheincidentlight.(b)Theintensitytransmissionspectrumthroughsuchastructure.Thecirclesaretheresultsfromthe®nite-differencetime-domainsimulations.ThesolidcurveisdeterminedfromanalytictheoryasrepresentedbyEq.(16).(c)Thesameplotasin(b),exceptthatthefrequencyrangeisnowrestrictedto[0.36(c/a),0.42(c/a)]toexhibitfurtherdetailsoftheresonancelineshape.etal.Vol.20,No.3/March2003/J.Opt.Soc.Am.A571 Thesimulatedtransmissionspectrum,shownascirclesinFig.2(b),consistsofFanoresonantlineshapessuperim-posedonasmoothFabryPerotbackground.Tocom-parethesimulationswiththeory,wedeterminefromthesimulationsthefrequencyandthewidthoftheresonancebystudyingtheexponentialtemporaldecayofthereso-nanceamplitudeaftertheexcitation.Thescatteringma-forthedirecttransmissionprocessisestablishedby®ttingthebackgroundinthesimulatedspectrumtothetransmissioncoef®cientsthroughauniformslabwiththesamethicknessandwithaneffectivedielectricconstant.Usingtheseparameters,wethencalculatethetheoreticalspectrumusingEq.(16)andplotitasasolidcurveinFig.2(b)and2(c).Thereisexcellentagreementbetweentheoryandsimulations.4.FINALREMARKSInconcluding,wenotethatforstructuresinwhichtheresonancesaresuf®cientlyclosetoeachother,ageneraltheoryincorporatingmultipleresonancesisneeded.Suchatheorywillbedevelopedinfutureresearch.Inaddition,whilethescattering-matrixapproachhasbeenusedintheanalysisofgratingsandinthegeneralcaseofarbitraryscatters,andmanyaspectsoftheFanoreso-nancecanbeobtainedinastructure-independentfashionbyusingthesymmetrypropertiesofthescatteringmatrixalone(aspreviouslyreportedinstudiesofphase-coherenttransportinmesoscopicsemiconductors),ourtheorydoescontainadditionaldynamicinformationabouttheresonanceamplitude.Withthisinformation,temporalcoupled-modetheorycanbereadilyappliedinsituationswithmorethanoneresonantmodeandwithnonlin-Ðsituationsinwhichastraightforwardapplica-tionofscattering-matrixformalismalonewouldhavebeenmoredif®cult.Thuswebelievethatthetheorypre-sentedhereshouldbeusefulforsynthesizingresponsefunctionsin®lterandsensorapplications.ThesimulationswereperformedthroughtheNationalScienceFoundation'sNationalProgramforAdvancedComputationalInfrastructures.ShanhuiFanacknowl-edgesthesupportofa3Muntenuredfacultyaward.CorrespondingauthorShanhuiFancanbereachedbye-mailatshanhui@stanford.edu.1.R.W.Wood,``Ontheremarkablecaseofunevendistributionofalightinadiffractivedgratingspectrum,''Philos.Mag.402(1902).2.U.Fano,``Thetheoryofanomalousdiffractiongratingsandofquasi-stationarywavesonmetallicsurfaces(Sommer-feld'swaves),''J.Opt.Soc.Am.,213222(1941).3.A.HesselandA.A.Oliner,``AnewtheoryofWood'sanoma-liesonopticalgratings,''Appl.Opt.,12751297(1965).4.D.Maystre,``Generalstudyofgratinganomaliesfromelec-tromagneticsurfacemodes,''inElectromagneticSurface,A.D.Boardman,ed.(Wiley,Chichester,UK,1982).5.H.L.Bertoni,L.-H.S.Cheo,andT.Tamir,``Frequency-selectivere¯ectionandtransmissionbyaperiodicdielectriclayer,''IEEETrans.AntennasPropag.,7883(1989).6.S.S.Wang,R.Magnuson,J.S.Bagby,andM.G.Moharam,``Guided-moderesonanceinplanardielectriclayerdiffrac-tiongratings,''J.Opt.Soc.Am.A,14701474(1990).7.R.MagnusonandS.Wang,``Newprinciplesforoptical®l-ters,''Appl.Phys.Lett.,10221024(1992).8.M.Neviere,E.Popov,andR.Reinisch,``Electromagneticresonancesinlinearandnonlinearoptics:phenomenologi-calstudyofgratingbehaviorthroughthepolesandzerosofthescatteringoperator,''J.Opt.Soc.Am.A,5139.S.PengandG.M.Morris,``Resonantscatteringfromtwo-dimensionalgratings,''J.Opt.Soc.Am.A,99310.A.Sharon,D.Rosenblatt,A.A.Friesem,H.G.Weber,H.Engel,andR.Steingrueber,``Lightmodulationwithreso-nantgrating-waveguidestructures,''Opt.Lett.,15641566(1996).11.S.M.Norton,T.Erdogan,andG.M.Morris,``Coupled-modetheoryofresonant-grating®lters,''J.Opt.Soc.Am.A639(1997).12.T.TamirandS.Zhang,``Resonantscatteringbymultilay-ereddielectricgratings,''J.Opt.Soc.Am.A,160713.G.Levy-YuristaandA.A.Friesem,``Verynarrowspectral®lterswithmultilayeredgratingwaveguidestructures,''Appl.Phys.Lett.,15961598(2000).14.S.Fan,``Sharpasymmetriclineshapesinside-coupledwaveguide-resonatorsystems,''Appl.Phys.Lett.,910912(2002).15.M.Kanskar,P.Paddon,V.Pacradouni,R.Morin,A.Busch,J.F.Young,S.R.Johnson,J.MacKenzie,andT.Tiedje,``Observationofleakyslabmodesinanair-bridgedsemicon-ductorwaveguidewithatwo-dimensionalphotoniclattice,''Appl.Phys.Lett.,14381440(1997).16.V.N.Astratov,I.S.Culshaw,R.M.Stevenson,D.M.Whit-taker,M.S.Skolnick,T.F.Kraus,andR.M.DeLaRue,``Resonantcouplingofnear-infraredradiationtophotonicbandstructurewaveguides,''J.LightwaveTechnol.2057(1999).17.H.A.Haus,WavesandFieldsinOptoelectronicsHall,EnglewoodCliffs,N.J.,1984).18.S.FanandJ.D.Joannopoulos,``Analysisofguidedreso-nanceinphotoniccrystalslabs,''Phys.Rev.B,art.no.235112(2002).19.P.Vincent,``Singularityexpansionsforcylindersof®niteconductivity,''Appl.Phys.,239248(1978).20.J.U.NockelandA.D.Stone,``Resonancelineshapesinquasi-one-dimensionalscattering,''Phys.Rev.B,1741517432(1994).572J.Opt.Soc.Am.A/Vol.20,No.3/March2003etal.