Finsler Spinoptics

FinslerSpinopticsC.DUVALCentredePhysiqueTheorique,CNRS,Luminy,Case907F-13288MarseilleCedex9(France)yMarch142008AbstractTheobjectiveofthisarticleistobuildupageneraltheoryofgeometricalopticsforspinninglightraysinaninhomogeneousandanisotropicmediummodeledonaFinslermanifold.TheprerequisitesoflocalFinslergeometryarereviewedtogetherwiththemainpropertiesoftheCartanconnectionusedinthiswork.Then,theprinciplesofFinslerianspinopticsareformulatedonthegroundsofpreviousworkonRiemannianspinoptics,andrelyingonthegenericcoadjointorbitsoftheEuclideangroup.Anewpresymplecticstructureontheindicatrix-bundleisintroduced,whichgivesrisetoafoliationthatsignicantlydepartsfromthatgeneratedbythegeodesicspray,andleadstoaspecicanomalousvelocity,duetothecouplingofspinandtheCartancurvature,andrelatedtotheopticalHalleect.Keywords:Presymplecticmanifolds,Geometricalspinoptics,FinslerstructuresMathematicsSubjectClassication2000:78A05,70G45,58B20Preprint:CPT-P29-2007mailto:[e-mail]6207duCNRSassocieeauxUniversitesd'Aix-MarseilleIetIIetUniversiteduSudToulon-Var;LaboratoirealiealaFRUMAM-FR2291arXiv:0707.0200v3[math-ph]15Apr2008Contents1Introduction22Finslerstructures:acompendium72.1Finslermetrics..............................72.1.1Anoverview............................72.1.2Introducingspecialorthonormalframes............82.1.3Thenon-linearconnection....................92.2Finslerconnections............................112.2.1TheChernconnection......................112.2.2TheCartanconnection.....................133GeometricalopticsinFinslerspaces143.1Finslergeodesics.............................143.2Geometricalopticsinanisotropicmedia................163.2.1TheFermatPrinciple......................163.2.2Finsleroptics...........................183.2.3Theexampleofbirefringentsolidcrystals...........194GeometricalspinopticsinFinslerspaces214.1SpinopticsandtheEuclideangroup...................224.1.1Coloredlightrays........................234.1.2ThespinningandcoloredEuclideancoadjointorbits.....244.2SpinopticsinFinsler-Cartanspaces...................264.2.1MinimalcouplingtotheCartanconnection..........264.2.2TheFinsler-Cartanspintensor.................284.2.3LawsofgeometricalspinopticsinFinsler-Cartanspaces...305Conclusionandoutlook351IntroductionGeometryandopticshavemaintainedalastingrelationshipsinceEuclid'sOpticswherelightrayswererstinterpretedasorientedstraightlinesinspace(or,putinmodernterms,asoriented,nonparametrized,geodesicsofEuclideanspace).Onecan,withal,tracebacktheoriginofthecalculusofvariationstoFermat'sPrincipleofleastopticalpath.Thisprinciplehasservedasthebasisofgeometricalopticsininhomogeneous,isotropic,mediaandprovedafundamentalmathematicaltoolinthedesignofoptical(andelectronic)devicessuchasmirrors,lenses,etc.,andintheunderstandingofcaustics,andopticalaberrations.2AlthoughMaxwell'stheoryofwaveopticshasunquestionablyclearlysupersededgeometricalopticsasabonadetheoryoflight,theseminalworkofFermathasopenedthewaytowidebranchesofmathematics,physics,andmechanics,namely,tothecalculusofvariationsinthelarge,modernclassical(andquantum)eldtheory,LagrangianandHamiltonianorpresymplecticmechanics.ItshouldbestressedthatFermat'sPrinciplehas,inessence,acloserelationshiptomodernFinslergeometry,asitrestsonaspecic\LagrangianF(x;y)=n(x)qijyiyj(1.1)wherey=(yi)standsforthe\velocityoflight,andn(x)0forthevalueofthe(smooth)refractiveindexofthemedium,atthe\locationxinEuclideanspace.(NotethatEinstein'ssummationconventionistacitlyunderstoodthroughoutthisarticle.)Asamatteroffact,thefunction(1.1)isaFinslermetric,namelyapositivefunction,homogeneousofdegreeoneinthevelocity,smoothwherevery6=0,andsuchthattheHessiangij(x;y)=12F2
yiyjispositive-denite(seeSection2.1).Al-though,thisisaveryspecialcaseofFinslermetric|itactuallydenesaconformal-lyatRiemannianmetrictensor,viz.,gij(x)=n2(x)ij|,thisfactisworthnotingforfurthergeneralization.ThegeodesicsoftheFermat-Finslermetric(1.1)areafairlygoodmathematicalmodelforlightraysinrefractive,inhomogeneous,andnondispersivemedia|providedpolarizationoflightisignored!IthasquiterecentlybeenenvisagedtoconsiderageneralFinslermetricF(x;y)todescribeanisotropyofopticalmedia,astheFinslermetrictensor,gij(x;y),de-pends,ingeneralnontrivially,onthedirectionofthevelocity,y,or\elementdesupportinthesenseofCartan[18].Thisenablesonetoaccountforthefactthat[3,28]Inananisotropicmedium,thespeedoflightdependsonitsdirection,andtheunitsurfaceisnolongerasphere.(Finsler,1969)TheFermatPrinciplehasalsobeenreformulatedinthepresymplecticframeworkin[16],andgeneralizedin[17]tothecontextofanisotropicmedia.Bytheway,3theregularityconditionimposed,inthelatterreference,amountstodemandingaFinslerstructure.OnethustakesforgrantedthatorientedFinslergeodesicsmaydescribelightraysinanisotropicmedia.AFinslerianversionoftheFermatPrinciplenowstatesthatthesecond-orderdierentialequationsgoverningthepropagationoflightstemfromthegeodesicsprayofa(three-dimensional)Finslerspace,(M;F),givenbytheReebvectoreldofthecontact1-form$=!3;(1.2)where!3=Fyidxiis,he