Nhng tin b trong Quang hc, Quang ph vÃ ng dng VI ISSN 1859 - 4271

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Advances in Optics, Photonics, Spectroscopy & Applications VI ISSN 1859 - 4271SIMULATION OF MULTI-PORT DEVICES COMBINING CO- ANDCONTRA- DIRECTIONAL COUPLINGR. CoïssonDepartment of Physics, Parma University, I-43100 Parma, ItalyE-mail:roberto.coison@fis.unipr.itIngeneral,<strong>ph</strong>otonicsdealswithinteractionofwaveswitheachotherandwithmaterialstructures(whichalsocanbecomposedofresonantoscilatorsorvarioustypesofpropaga<s<strong>trong</strong>>tin</s<strong>trong</strong>>gwavesorspatialyperiodicstructures).Intheapproximationofsmalparametricefectsorsmalnonlinearity,wavescanbeseenassuperpositionofmodesinterac<s<strong>trong</strong>>tin</s<strong>trong</strong>>gwitheachotherorwiththematerialstructure,andthesecoupled-waveapproachesareknownsincealongtimeago[1-4]FibreopticalcouplersandBraggreflectorsarealreadybasicelementsofopticaldevicesandsystems.Butnewhorizonsopenupwithmicro-andnano-structuredmaterialsanddevices:submicronorderedaraysofdielectric,semiconductorormetalelementsbehaveascoupledoscilators,formingwaveguides,“<strong>ph</strong>otoniccrystals”,plasmonicstructures,metamaterials.Forexampleanarayofcoupled oscilatorscanbehaveashaving powerflow oppositetowavevector,wavescanbeguidedina2-D or3-D arayofelements(dielectric,oscilator,quantumdots,…),etc.Itwouldbetoolongtogiveanexhaustivelistoftheliteratureonthesubject,wejustgivesomeexamplesclosertothepresentdiscusion[5-10]ThesesystemsareusualystudiedwithdetailedmethodslikeFEM,butforexploringnewposibilitiesitisusefultousethecoupled-waveframework,whichalowsrapidtrial-and-erorwithasimpleandflexibletoolascanbegivenbymatrix-orientedprogramminglanguageslikeMatlaborScilab[11],andthiswewanttoilustratewithasimpleexample,combiningco-andcontra-directionalcouplingandshowingacomplexinput-outputbehaviour.Co-directionalwavecoupling,from microwavestovisible,hasawidevarietyofrealizations:waveseitherdirectlyinterac<s<strong>trong</strong>>tin</s<strong>trong</strong>>g(withnottoodiferentwavevectors),orthroughaspatialyperiodicstructure(staticrefractiveindexvariation,waveguidecorugation,acousticwave)havingsuitablewavevectorforeficientcoupling(so-caledLongPeriodGra<s<strong>trong</strong>>tin</s<strong>trong</strong>>gs).Contra-directionalcoupling isaconceptwidely applied in opticalfibredevicesandsemiconductorlasers.Bragggra<s<strong>trong</strong>>tin</s<strong>trong</strong>>gdevicesor“1-D<strong>ph</strong>otoniccrystals”havebeendevelopedformanyapplicationsandwithavarietyofconfigurations:distributedBraggreflectorsforfibrelasercavities,λ/4-shiftedgra<s<strong>trong</strong>>tin</s<strong>trong</strong>>gsforsemiconductorlasercavities,chirpedgra<s<strong>trong</strong>>tin</s<strong>trong</strong>>gsforfibredispersioncompensation,slightlynonlinearstructuresforswitching,AgeneralmodularcomputerprogramusingMatlaborScilabhasbeenwriten[12],forthereflection,transmissionandinternalfieldcalculationsofanyquasi-periodicBraggstructure,includingiterativesolutionsofnonlinearcases.Combiningco-andcontra-directionalinteractionscanprovideacomplexinput-outputbehaviour,withas<strong>trong</strong>dependenceonvariousparameters,whichmakesi<s<strong>trong</strong>>tin</s<strong>trong</strong>>teres<s<strong>trong</strong>>tin</s<strong>trong</strong>>gforthedevelopmentofdevices,andsomeexampleshavebeenproposedintheliterature[5-10].Inthiscase,ingeneral,boundaryconditionsarenotalonthesameside,andthisimpliestheneedfora“partialinversion”ofamatrix:intheequationAx=y,boththevectorsxandycontainbothknownandunknownelements,soasuitablelineartransformationisneededtofindtheunknowns;anoperationthathasbeenimplementedinScilab[13].40
Những tiến bộ <strong>trong</strong> <strong>Quang</strong> học, <strong>Quang</strong> <strong>ph</strong>ổ và Ứng dụng VI ISSN 1859 - 4271Inordertoilustratethecomplexinput-outputofsuchsystemscombiningco-directionalandcontra-directionalcouplingofmodes,asimpleexamplewithtwowaveguidesandfourmodeswilbeshown.Anywaythiscanbeeasilygeneralizedtomorecomplexcases,applyingittonewideasofdevices.Co-directional and contra-directional couplingF(z)B(z)z=021| | | | | | | | | | | | | | | | | | | | | | |Bragg gra<s<strong>trong</strong>>tin</s<strong>trong</strong>>gs + coupling| | | | | | | | | | | | | | | | | | | | | | |4z=L3dFdzdBdz11==dF2αF2+ κ1B1=dzdB−α'B2+κ1'F1dz−α'F2=1αB+ κ1+2κ' B2F22F 1 (0)=1, F 2 (0)=B 1 (L)=B 2 (L)=0Two boundary conditions at z=0, two at z=LUnknowns: F 1 (L), F 2 (L), B 1 (0), B 2 (0)(transmission and reflection coefficients)Thefolowingfigureshowseachofthefouroutputsasfunctionsofdetuningfrom exactBraggcondition,withinpu<s<strong>trong</strong>>tin</s<strong>trong</strong>>onesideofoneofthewaveguides.Thisshowsthataslightchangeinwavelengt<strong>hc</strong>andirecttheligh<s<strong>trong</strong>>tin</s<strong>trong</strong>>toadiferentoutput.Also,fromarelativelywidebandinputonecangettwoorthreenarowbands(andbea<s<strong>trong</strong>>tin</s<strong>trong</strong>>g…).Andthesecondoneshowsthefouroutputswhentheco-directionalcouplingconstantisvaried,asforexampleitisrealizedwithanacousticwave,whoseintensitycanbevaried.Itshowsthatbyvaryingtheacousticintensity,theoutputcanbedirectedtoanyofthe4outputs[14,15].41
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