Lanczos Pseudospectral Propagation Method for Init

arXiv:physics/0410271v1[physics.comp-ph]28Oct2004LanczosPseudospectralPropagationMethodforInitial-ValueProblemsinElectrodynamicsofPassiveMediaAndreiG.Borisova,1andSergeiV.Shabanovb,2aLaboratoiredesCollisionsAtomiqueetMol´eculaires,UMRCNRS-Universit´eParis-Sud8625,91405OrsayCedex,FrancebDepartmentofMathematics,UniversityofFlorida,Gainesville,FL32611,USAAbstractMaxwell’sequationsforelectrodynamicsofdispersiveandabsorptive(passive)me-diaarewrittenintheformoftheSchr¨odingerequationwithanon-HermitianHamil-tonian.TheLanczostime-propagationschemeismodifiedtoincludenon-HermitianHamiltoniansandused,incombinationwiththeFourierpseudospectralmethod,tosolvetheinitial-valueproblem.Thetime-domainalgorithmdevelopedisshowntobeunconditionallystable.Variabletimestepsand/orvariablecomputationalcostspertimestepwitherrorcontrolarepossible.Thealgorithmisappliedtostudytransmis-sionandreflectionpropertiesofioniccrystalgratingswithcylindricgeometryintheinfra-redrange.1email:[e-mail]2email:shabanov@phys.ufl.edu11.Introduction.Thereisademandforfasttime-domainsolversoftheMaxwell’sequa-tionstomodeldynamicsofbroadbandelectromagneticpulsesindispersiveandabsorptivemedia.Thisisdrivenbyvariousapplicationsthatincludephotonicdevices,communications,andradartechnology,amongstmanyothers.Efficiency,accuracy,andstabilityarethekeycriteriaofchoosingaconcretealgorithmforspecificapplications.Inthepastdecades,pseu-dospectralmethodsofsolvingtheinitialvalueproblemfordifferentialequationshavebeenunderintensivestudy[1].Becauseoftheirhighefficiencyandaccuracy,theyhavereplacedfinitedifferencingapproachesinmanytraditionalapplicationsaswellasscientificsimula-tions,e.g.,inquantumchemistry[2].Unconditionallystablepseudospectralalgorithmsareparticularlyattractivefornumericalsimulations.Inthepresentpaperwedevelopanunconditionallystabletime-domainalgorithmforsolv-ingtheinitialvalueproblemforMaxwell’sequationsindispersiveandabsorptive(passive)mediawithsharpinterfaces(discontinuitiesofmediumparameters).Itisatime-steppingalgorithmthatisbasedontheHamiltonianformalismforelectrodynamicsofpassivecontin-uousmedia,theLanczospropagationscheme[3,4],andtheFourierpseudospectralmethod[5].Apartfromtheunconditionalstability,thealgorithmhasadynamicalcontrolofaccu-racy,whichallowsonetoautomaticallyoptimizecomputationalcostswitherrorcontrolateachtimestep.Weapplythealgorithmtothescatteringofbroadbandelectromagneticpulsesongrat-ings,thephotonicdevicesthatcurrentlyattractlotsofattentionbecauseoftheirtransmis-sionandreflectionproperties[6].Asforthepassivemedium,wechooseanioniccrystalmaterial.Fromthenumericalpointofview,themodelofthedielectricpermeabilityofsuchamaterialisratherrepresentativeandusedinthevastnumberofapplications.Fromthephysicalpointofview,theinteresttogratingsandphotoniccrystalsmadeofthiskindofmaterialisduetwotypesofeffectsininteractionwithelectromagneticradiation:Thestructuralandpolaritonicones[7,8].Weshowthatintheinfraredrangethereflectionandtransmissionpropertiesofioniccrystalgratingschangesignificantlyinnarrowfrequencyrangesduetostructuralandpolaritonicresonances.Structuralresonancesareassociatedwiththeexistenceoftrapped(quasistationary)electromagneticmodessupportedbythegratinggeometry(guidedwaveresonances)[9].Polaritonicresonancesareassociatedwithdispersivepropertiesofthematerial.Suchresonancesappearwhentheincidentradiationcancausepolaritonicexcitationsinthemedium.Fromthemacroscopicpointofview,thisoccursintheanomalousdispersionregionofthedielectricconstant.2.Basicequations.Maxwell’sequationsinpassivemediacanbewrittenintheformoftheSchr¨odingerequationinwhichthewavefunctionisamultidimensionalcolumn,composedofelectromagneticfieldcomponentsandthemediumpolarization,andtheHamiltonianis,ingeneral,non-Hermitianwhenattenuationispresent.Theinitial-valueproblem(thetimeevolutionofanelectromagneticpulse)isthensolvedbyfindingthefundamentalsolution(theevolutionoperatorkernel)fortheSchr¨odingerequation.Herethisideaisappliedtotheioniccrystalmaterialwhosedielectricpropertiesatthefrequencyωaredescribedbythedielectricconstantε(ω)=ε∞+(ε0−ε∞)ω2Tω2T−ω2−iηω,(1)2whereε∞,0areconstants,ωTistheresonantfrequency,andηistheattenuation.Inparticular,transmissionandreflectionpropertiesoftheperiodicgratingstructureofcircularparallelcylindersmadeofsuchamaterialarestudied.LetPbeadispersivepartofthetotalpolarizationvectorofthemedium.ThenD=ε∞E+P,whereDandEaretheelectricinductionandfield,respectively.ByusingtheFouriertransform,itisstraightforwardtodeducethatPsatisfiesthesecond-orderdifferentialequation¨P+η˙P+ω2TP=ε∞ω2pE,(2)wheretheoverdotdenotesthetimederivative,ω2p=(ε0−ε∞)ω2T/ε∞ifε0−ε∞ispositive,otherwise,ω2p→−ω2pin(2).Equation(2)mustbesolvedwiththezeroinitialconditions,P=˙P=0att=0.DefineasetofauxiliaryfieldsQ1,2byP=√ε∞ωpQ1/ωTand˙Q1=ωTQ2.Fornon-magneticmedia(μ=1),theMaxwell’sequationsand(2)canbewrittenastheSchr¨odingerequation,i˙ψ(t)=Hψ(t),inwhichthewavefunctionandtheHamiltonianaredefinedbyψ=ε1/2∞EBQ1Q2,H=0icε−1/2∞∇×0−iωp−ic∇×ε−1/2∞000000iωTiωp0−iωT−iη,(3)wherecisthespeedoflightinthevacuum,andBisthemagn