Possible realization of directional optical waveguides in photonic crystals with broken time reversal symmetry

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1、PossibleRealizationofDirectionalOpticalWaveguidesinPhotonicCrystalswithBrokenTime-ReversalSymmetryF.D.M.HaldaneandS.RaghuDepartmentofPhysics,PrincetonUniversity,PrincetonNJ08544-0708(Dated:August30,2007)WeshowhowinprincipletoconstructanalogsofquantumHalledgestatesin“photoniccrystals”madewithnon-

2、reciprocal(Faraday-effect)media.Theseform“one-waywaveguides”thatallowelectromagneticenergytoflowinonedirectiononly.PACSnumbers:42.70.Qs,03.65.VfInthisletter,wedescribeanoveleffectinvolvinganneutralbosonswhichdonotinteractinlinearmedia;fur-interfacebetweentwomagneto-opticphotoniccrystalsthermore,pho

3、tonicbandscanbedescribedclassically,in(periodic“metamaterials”thattransmitelectromagnetictermsofMaxwell’sequations.However,theintegerQHEwaves)whichcantheoreticallyactasa“one-waywaveg-caninprincipleoccurwithoutanyuniformmagneticfluxuide”,i.e.,achannelalongwhichelectromagneticen-density(justwithbro

4、kentime-reversalsymmetry)ashasergycanpropagateinonlyasingledirection,withnoexplicitlyshownbyoneofusinagraphene-likemodelpossibilityofbeingback-scatteredatbendsorimper-ofnon-interactingBlochelectrons[6];thusLandau-levelfections.Theunidirectionalphotonicmodesconfinedquantizationisnotanessentialrequ

5、irementforthequan-tosuchinterfacesarethedirectanalogsofthe“chi-tumHalleffect.raledge-states”ofelectronsinthequantumHalleffectWehavetranscribedthekeyfeaturesoftheelectronic(QHE)[1,2].ThekeyenablingingredientisthepresencemodelofRef.[6]tothephotoniccontext.Theedge-statesof“non-reciprocal”(Faraday-effe

6、ct)mediathatbreaksareapropertyofaone-particleeigenstateproblemsim-time-reversalsymmetryinthemetamaterial.ilartotheMaxwellnormal-modeproblem,soarerepli-Justasintheelectroniccase,everytwodimensionalcatedinthephotonicsproblem.(TheQHEitselfhasnophotonicbandischaracterizedbyatopologicalinvariantphoto

7、nicanalog,asitfollowsfromthePauliprincipleofknownastheChernnumber[5],anintegerthatvanishesfillingallone-particlestatesbelowtheFermilevel.)identicallyunlesstime-reversalsymmetryisbroken.IfTheMaxwellnormal-modeprobleminloss-fre