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1、PHYSICALREVIEWE80,0567022009Spectralelementmethodforbandstructuresofthree-dimensionalanisotropicphotoniccrystalsMaLuoandQingHuoLiuDepartmentofElectricalandComputerEngineering,DukeUniversity,Durham,NorthCarolina27708,USAReceived1July2009;published9November200
2、9AspectralelementmethodSEMisintroducedforaccuratecalculationofbandstructuresofthree-dimensionalanisotropicphotoniccrystals.ThemethodisbasedontheÞnite-elementframeworkwithcurvi-linearhexahedralelements.Gauss-Lobatto-Legendrepolynomialsareusedtoconstructthebas
3、isfunctions.Inordertosuppressspuriousmodes,mixed-ordervectorbasisfunctionsareemployedandtheBlochperiodicboundaryconditionisimposedintothebasisfunctionswithtangentialcomponentsattheboundarybymulti-plyingaBlochphasefactor.TheÞeldsandcoordinatesinthecurvilinearhex
4、ahedralelementsaremappedtothereferencedomainbycovariantmapping,whichpreservesthecontinuityoftangentialcomponentsoftheÞeld.NumericalresultsshowthattheSEMhasexponentialconvergenceforbothsquare-latticeandtriangular-latticephotoniccrystals.Thesamplingdensityassmall
5、as3.4pointsperwavelengthcanachieveaccuracyashighas99.9%.ThebandstructuresofseveralmodiÞedwoodpilephotoniccrystalsarecalculatedbyusingtheSEM.DOI:10.1103/PhysRevE.80.056702PACSnumbers:02.70.Hm,42.70.Qs,42.25.Fx,52.40.DbI.INTRODUCTIONisaspecialkindofhigh-orderFE
6、M,whichusesGauss-Lobatto-LegendreGLLpolynomialstoconstructthebasisInordertoinvestigatethephysicalpropertiesofphotonicfunctions.Inourimplementation,analyticmappingbetweencrystalsPCs1Ð3andengineertheapplicationofthesethereferencedomainandrealspaceisusedtoin
7、creasethenovelartiÞcialmaterials,ahighaccuracyandhighefÞciencyaccuracy.ThesolutionsconvergeexponentiallyataverysolverforbandstructuresofPCsisnecessary.Thebandhighspeed.TheexponentialconvergencespeedremainsthestructurescontainmuchinformationaboutthePCssuchassame
8、evenfortriangular-latticePCsandanisotropicmaterialtheexistenceofabandgap,densityofstates,andpropaga-PCs.Inthispaper,theSEMisextendedtosolvebandstruc-tionpropertiesgivenbygro
