The Fundamental Solution of the Time-Dependent Sys

TheFundamentalSolutionoftheTime-DependentSystemofCrystalOptics.RobertBurridgeEarthResourcesLaboratoryMassachusettsInstituteofTechnology42CarletonStreet,E34-450Cambridge,MA02142-1324JianliangQianDepartmentofMathematicsUniversityofCalifornia,LosAngeles405HilgardAvenueLosAngeles,CA90095-1555February1,2004AbstractWesetuptheelectromagneticsystemanditsplane-wavesolutionswiththeassociatedslownessandwavesurfaces.WetreattheCauchyinitial-valueproblemfortheelectricvectorandmakeexplicitthequantitiesnecessaryfornumericalevaluation.WeusetheHerglotz-Petrovskiirepresentationasanintegralaroundloopswhich,foreachpositionandtimeformtheintersectionofaplaneinthespaceofslownesseswiththeslownesssurface.Thefield,especiallyitssingularities,isstronglydependentonthevaryinggeometryoftheseloops.Wegivewithoutderivationthestatictermcorrespondingtothemodewithzerowavespeed.Numericalevaluationofthesolutionispresentedgraphicallyfollowedbysomeconcludingremarks.1Introduction1.1GeneralintroductionCrystalopticsissimilarto,butsimplerthan,anisotropicelasticity.Forinstanceitsslownesssurfacehasconicalpoints,incommonwithmanyelasticitysystems,andthereareconicalpointsonthewavesurface.ItalsohasathirdinterestingfeatureassociatedwiththerˆoleofthedivergenceinrelationtoMaxwell’sequations,namelythefactthatonecharacteristicspeediszero(actuallytwocoincidentzeros),sothattheslownesssurfaceisquarticratherthansexticasmightbeexpectedfromthedimensionality-onequadraticsheetoftheslownesssurfaceliesatinfinity.Remarkablythewavesurfaceisanotherquarticsurfaceofthesamealgebraictype,butwithreciprocalparameters.SeeforinstanceBornandWolf(1989)foraveryfullandreadableaccountoftheplane-wavetheoryofthissystemandtheassociatedgeometry.Thesystemofcrystalopticsisofgreatintrinsicandhistoricalinterest,thelatterbecauseHamilton’spredictionin1833ofinternalconicalrefraction,andLloyd’sexperimentalconfirmationcloselythereafter,ledtothewideacceptanceofFresnel’swavetheoryoflight.Theintrinsicinterestislargelycenteredaroundtheremarkablegeometricalpropertiesoftheslownesssurfaceandwavesurface,whicharebothofatypeknownasFresnel’swavesurface(Salmon,1915).WeillustratenumericallytheanalyticexpressionforthefundamentalsolutionofthesystemintermsofrealloopintegralsaccordingtotheHerglotz-Petrovskiiformula,whichmayalsobeappliedreadilytootherconstant-coefficienthyperbolicsystems.Petrowskii(1945)expressedthesolutionintermsofnon-realcyclesincomplexspace.Atiyah,Bott,andGarding(1970,1973)placedPetrovskii’sworkonamodernbasis,andDeHoopandSmit(1995)recentlyelaboratedthisinathree-dimensionalelastodynamicsetting.ButfollowingJohn(1955)andGelfandandShilov(1964)wewillstaywiththerepresentationintermsofrealintegrals.Burridge(1967)usedittoobtainthegeometricalarrivals(seebelow),andthesingularitydue1totheconicalpointsoftheslownesssurfaceatfieldpointsintheinterioroftheconeofinternalconicalrefractionforcubicelasticmedia.Butthatworklackednumericalillustrationsandthetreatmentoftheconicalpointwasnotuniformneartheconicalsurfaceitself.Althoughwestilldonotgivetheuniformtime-dependentasymptoticanalysisforthisregion,wedopresentnumericalsolutionsclosetoandonthis‘coneofinternalconicalrefraction’.Thegeometricalarrivalsmentionedabovearesingularitiesinthefieldassociatedwithslownessesξwhichare‘stationarypoints’wheretheplaneξ·x=ttouchestheslownesssurfaceandatwhichtheslownesssurfacehasfinitenon-zeroGaussiancurvature,andsuchwavearrivalsaregovernedbythesimplestformofgeometricalraytheory.ForinstanceMovskinetal.(1993)havederivedtheGreen’sfunctioninthefrequencydomainanddiscussedvariousimportantdirectionsandconesofdirectionsinrelationtothefield,namelyinthedirectionsofgeneratorsoftheconeofinternalconicalrefraction,andinthedirectionsofthebiradials,i.e.thedirectionsoftheconicalpointsonthewavesurface,andtheyobtainasymptoticapproximationstothefieldatlargedistancesintheneighborhoodsofthesedirections.Inthispaperwestudythesecond-ordervectorequationforEobtainedbyeliminatingtheotherdependentvariablesfromMaxwell’sequationsandtheconstitutivelawsofcrystaloptics.Thisequationislikethesecond-orderelastodynamicequationforparticledisplacementandmaybeobtainedfromthatofisotropicinfinitesimalelasticitybysettingtheLam´econstantλ=−2,andµ=1,sothatλ+2µ=0,andthedensityρ=σ(seebelow).SeeEvery(1981)theeffectsofcurvatureoftheslownesssurfacenearcrystalsymmetryaxesincubiccrystalacoustics,andShuvalovandEvery(1996)formoregeneralsymmetries.1.2OutlineofthispaperInSection2wesetuptheelectromagneticsystemanditsplane-wavesolutionswiththeassociatedgeometricalentitiessuchastheslownesssurface,andthewavesurface,andweshowtheirremarkablytightlyknitrelationshiptotheenergyellipsoidanditsparameterizationbyellipticcoordinates.InSection3wesetupandsolvetheCauchyinitial-valueproblemforEandmakeexplicitsomequantitieswithaviewtonumericalevaluation.InSection4wefollowtheHerglotz-Petrovskiiprocedureoftransformingtheintegralrepresentationtoanintegralaroundloopswhich,foreachx,t,formtheintersectionoftheplaneξ·x=twiththeslownesssurface.Asx,tvarythegeometryoftheseloopsvaries,andthefield,especiallyitssingularities,arestronglyd