Dispersion of Homogeneous and Inhomogeneous Waves

DispersionofHomogeneousandInhomogeneousWavesintheYeeFinite-DifferenceTime-DomainGridJohnB.Schneider(correspondingauthor)SchoolofElectricalEngineeringandComputerScienceWashingtonStateUniversityP.O.Box642752Pullman,[e-mail]:5093353818Phone:[phone]RobertJ.KruhlakNonlinearOpticsLaboratoryDepartmentofPhysicsWashingtonStateUniversityP.O.Box642814Pullman,[e-mail]1AbstractThenumericaldispersionrelationgoverningthepropagationofhomogeneousplanewavesinafinite-differencetime-domain(FDTD)gridiswellknown.However,homogeneousplanewaves,bythemselves,donotformacompletebasissetcapableofrepresentingallvalidfielddistributions.Acompletebasissetisobtainedbyincludinginhomogeneouswaveswhere,inthephysicalworld,constantphaseplanesmustbeorthogonaltoconstantamplitudeplanesforlosslessmedia.InthispaperwepresentadispersionanalysisforbothhomogeneousandinhomogeneousplanewavesintheYeeFDTDgrid.Weshowthat,ingeneral,theconstantamplitudeandconstantphaseplanesofinhomogeneousplanewavesarenotorthogonal,buttheyapproachorthogonalityforfinediscretization.Thedispersionanalysisalsoshowsthat,forverycoarselyresolvedfields,homogeneouswaveswillexperienceexponentialdecayastheypropagateandtheymaypropagatefasterthanthespeedoflight.Boundsareestablishedforthespeedofpropagationwithinthegrid,aswellasthehighestfrequencyandtheshortestwavelengththatcanbecoupledintothegrid.AnalysisisrestrictedtotheclassicYeealgorithmbutasimilarapproachcanbeusedtoanalyzeothertime-domainfinite-differencemethods.Indexterms:FDTDmethods1IntroductionThesecond-orderYeefinite-differencetime-domain(FDTD)technique[1]isarguablythemostrobustandsuccessfulnumericaltechniqueavailabletodaytosolveproblemsinelectromagneticwavepropagation.Thetechniquehasbeenthesubjectofthreebooks[2,3,4]aswellasnearly3,000journalandconferencepapers[5,6].DespitethevastattentiontheFDTDmethodhasreceived,thedispersionrelationforthecompleteplane-wavebasissethasneverbeenderived.Rules-of-thumbhavebeendevelopedforsuitablediscretizations(e.g.,[7]).However,theseruleshavebeenbasedsolelyonconsiderationofhomogeneous(propagating)wavesandmaynotbeusefulwheninhomogeneous(evanescent)fieldsplayasignificantroleinagivenproblem.TaflovehaspreviouslyderivedthedispersionrelationforhomogeneouswavesintheFDTDgrid[8](seealso[3]).Theequationhederivediscorrect,buttheassumptionthatsomecoarselyresolvedfieldshaveaphasevelocityofzeroisnot.AsTaflovereported,thephasevelocityde-creasesasthediscretizationbecomesmorecoarse.However,aswasshownin[9],athreshold2eventuallyisreachedbeyondwhichafurtherincreaseinthecoarsenessresultsinawavenumberthatiscomplex.Thephasevelocityforthesewaveswithcomplexwavenumbersactuallyincreaseswithgridcoarsenessandatnopointisitzero.Infact,certainspectralcomponents,whichwerefertoassuperluminal,haveaphasevelocitygreaterthanthespeedoflight.One-dimensional(orgrid-aligned)superluminalFDTDpropagationwasexploredin[9].TheanalysisdemonstratingsuperluminalbehaviorintheFDTDgridwillberestatedintheSec.2andresultswillbegivenforobliquelypropagatingwaves.Insection3wepresentthedispersionrelationforinhomogeneousplanewaves(thefamiliarhomogeneousdispersionrelationisaspecialcaseofthemoregeneralinhomogeneousone).WeshowthattheplanesofconstantphaseandconstantamplitudearenotnecessarilyorthogonalinalosslessFDTDgrid.WealsodemonstratehowthedispersionrelationcanbeusedtodeterminetheattenuationconstantassociatedwithtotalinternalreflectioninFDTDsimulations.Thisanalysisshowshowthedispersionrelationcanbeusedtoquantifytheerrorinevanescentfields.Forthecaseoftotalinternalreflection,itisfoundthattheerrorintheattenuationconstantintherarermediumisstronglygovernedbythediscretizationinthedensermedium.WerestricttheanalysistotheoriginalYeealgorithmsinceitisstilloneofthemostpopularandrobusttime-domainfinite-differencetechniques.Althoughsimulationsofreal(physicalworld)wavephenomenaarenotdonenearthediscretizationlimitsupportedbytheYeegrid,innearlyallFDTDsimulationstherewillbesomeenergypresentinthiscoarsely-resolvedportionofthespectrum.OuranalysisclearlydefinesthebehavioroftheYeegridatandnearthediscretizationlimitforbothhomogeneousandinhomogeneouswavepropagation.AlthoughtheYeealgorithmisthefocushere,asimilaranalysis,i.e.,onethatdoesnotrestrictconsiderationtorealwavenumbers,canbeappliedequallywelltoothertechniques.2HomogeneousWavesInthissectionwereexaminethedispersionrelationforhomogeneousplanewavesintheYeegrid.StartingfromthefamiliarFDTDdispersionrelation,itisshownthatwaveswithcomplexwavenumbersaresupportedbythegridforcoarsediscretizations.Thesecomplexwavesattenuate3astheypropagateandcan,dependingonthegridresolution,propagatefasterorslowerthanthespeedoflightinthephysicalworld.Weestablishboundsonthegreatestpropagationspeedwithinthegrid,aswellasonthehighestfrequencyandtheshortestwavelengththatcanbecoupledintothegrid.Thespecialcasesofpropagationalongthegriddiagonalandpropagationalongoneofthegridaxesareusedtoestablishthelimitingbehaviorofthegrid.Unliketheanalysispresentedin[9]whichonlyconsideredone-dimensional(grid-aligned)propagation,norestrictionsarep