swings and roundabouts optical poincaré spheres for polarization and gaussian beams 2017 m. r. dennis资料.pdf

该【swings and roundabouts optical poincaré spheres for polarization and gaussian beams 2017 m. r. dennis资料 】是由【李十儿】上传分享，文档一共【16】页，该文档可以免费在线阅读，需要了解更多关于【swings and roundabouts optical poincaré spheres for polarization and gaussian beams 2017 m. r. dennis资料 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..Downloadedfromhttp://rsta./onJanuary11,2017Swingsandroundabouts:opticalPoincaré:DennisMR,,UniversityofBristol,Swingsandroundabouts:opticalPoincaréBristolBS81TL,,UniversityofRochester,Rochester,:,USAhttp://dx./.0441MRD,0000-0003-1147-1804Accepted:6September2016TheconnectionbetweenPoincaréspheresforpolarizationandGaussianbeamsisexplored,Onecontributionof14toathemeissuefocusingontheinterpretationofellipticpolarization‘Opticalorbitalangularmomentum’.intermsoftheisotropictwo-dimensionalharmonicoscillatorinHamiltonianmechanics,itscanonicalSubjectAreas:,quantumphysics,mathematicalleadstotheinterpretationofstructuredGaussianmodes,theHermite?Gaussian,Laguerre?GaussianphysicsandgeneralizedHermite?Laguerre?GaussianmodesaseigenfunctionsofoperatorscorrespondingtotheKeywords:classicalconstantsofmotionofthetwo-dimensionalharmonicoscillator,Hermite–Gaussian,oscillator,whichacquireanextrasigni?canceasLaguerre–?OpticalAuthorforcorrespondence:orbitalangularmomentum?.-mail:mark.******@,notleastbecausebothinvolvethemathematicaltheoryofwavesincludingideassuchasFouriertransforms,cavitymodes,,waveopticalphenomenaareoftenpresentedinquantummechanicalterms,sometimesasspecialcasesoftheirquantummechanicalcounterparts,suchastheinterpretationofthebandwidththeorembyHeisenberg?suncertaintyrelations[1],functionsinHilbertspacesrepresenting2017TheAuthor(s).:..Downloadedfromhttp://rsta./onJanuary11,2017(a)(b)LG102.........................................................éspheresforellipticpolarizationandofGaussianbeamsofmodeorder1.(a)-handedsenseinthenorthernhemisphere,left-handedinthesouthernhemisphere,.(b),andLGvortex375modes(ofpositiveornegativesign):20150441haveasinglevortexonaxiswithanellipticalcore.(Onlineversionincolour.)quantumstatesandoptical?elds,andthefree-spaceparaxialequationasSchr?dinger?sequation,?eldquantization,,wetaketheopportunitytoexploreanothersuchanalogy:theconnectionbetweenpolarization?statesofellipticpolarization,asparametrizedbyStokesparametersandthePoincarésphere?andtheanalogousrepresentationsofhigh-orderGaussianlasermodes,especiallythecelebratedHermite?Gaussian(HG)andLaguerre?Gaussian(LG)ésphereparametrizationoflinear,circularandellipticstatesofpolarization,andHG,LGaswellasthelessfamiliargeneralizedHermite?Laguerre?Gaussian(GG)modes[2?4],hasbeenmuchexploredoverthelast25years,followingtheimportantobservationofthe?equivalence?ofthePoincarésphereforpolarizationandGaussianmodesofmodeorderunitybyPadgett&Courtial[5],asshownin?,frominterpretingpolarizationintermsoftheHamiltonianmechanicsofanisotropictwo-dimensionaloscillator,andGaussianmodesfromitscanonicalquantizationintermsofoperators[6,7].Thenotionofangularmomentum,inthesenseofboththespinangularmomentumencapsulatedbythethirdStokesparameterS3andtheorbitalangularmomentumoperatorofwhichtheLGmodesareeigenfunctions,,especiallyin[2?4,8?10],althoughwedrawstrongermathematicalanalogiesbetweentheclassicalHamiltonianstructureofthePoincarésphereforpolarization,?swingsandroundabouts?natureofharmonicoscillatororbitsunderlieseverything,especiallytheangular-momentum-carryingnatureofcircularorbitsandLGmodes[11].Ourexpositionwillbepedagogical,withtheaimofessibletonewentrantstothe?eld,,butpotentiallymisleading,tothinkofopticalangularmomentumofstructuredGaussianbeamsdirectlyintermsofthree-(asconsideredhere),thereisonlyonepossibledirectionofspinororbitalangularmomentum,namelythepropagationdirection,:..Downloadedfromhttp://rsta./onJanuary11,2017(a)((b)c)(d)3.........................................................,anduponpropagation.(a)IntensityofHG21inthefocalplane.(b)AstheHG21beampropagates,,whicharealwaysatthepositionsofthezerosofthescaledHermitepolynomials.(c)IntensityofLG11inthefocalplane.(d)AstheLG11beampropagatesfromitsfocalplane,itspreadsmaintainingtheintensitypattern,,suchastheonerepresentedbythegreysurface,,thisisanypairoforthogonalellipticpolarizationstates(,orright-andleft-handedcircularpolarizations),andthese375:20150441basisstatesareparametrizedbythePoincarésphere[12].ThissphereisanalogoustotheBlochsphereforquantumspin1/2because,forlightbeamswitha?xeddirectionofpropagation,theelectric?eldmustbetransverse,plexsuperpositionofleftandrightcircularpolarizations,orequivalently,(-dimensionalJonesvectors)areantipodalonthePoincarésphere,aswewilldiscussin§,thiswillbedescribedusingbasesofGaussianlasermodes?especiallytheHGandLGbasissets?(z=0),soafundamentalGaussianbeamhasamplitude(2/π)1/2w?1exp(?[x2+y2]/w2),wherew0representsthewaistwidthofthebeam[13],00andisnormalized(itssquare,integratedovertheplane,givesunity).ThisGaussianhasthesamefunctionalformasthegroundstateofatwo-dimensionalquantumharmonicoscillator,whichisjusti?edphysically[8,13?15]intermsofthecurvedmirrorsinthelasercavityhavingtheeffectontheparaxiallypropagatingwave,inthefocalplane,,breakingthecavity?spureaxialsymmetry;theHGmodes(TEMmodes)arehigher-ordermodesofsuchcavities,givenby√√1x2+y22x2yHGmn(x,y)=√m+n?1exp?2HmHn,()w02πm!n!w0w0w0whereHm,HndenoteHermitepolynomials[16],andm,narenon-negativeintegers0,1,2,...(thefundamentalGaussianbeingthecasem=n=0).HGmodesarecharacterizedbyanodal?grid?asseenin?gure2a,b,pletelyfactorizedintotwofunctions,onedependingonx(indexedbym)andoneony(indexedbyn),andN=m+,thereareN+1modes,forwhich(m,n)=(N,0),(N?1,1),...,(0,N).ThesetofHGmnmodesisorthonormal(withrespecttointegrationovertheplanewithuniformweight),andonpropagationthemodesmaintainthesameintensitypattern,eventothefar?eld:theFouriertransformofanHGbeamisfunctionallythesameas().Onpropagation,themodesacquirean(N+1)-dependentGouyphasefactor;theintensitypatternofsuperpositionsofmodeswithdifferentNchangesonpropagation,[11],whichareexpressedinplanepolarcoordinatesR,φinthewaistplaneas21+||p!1R2||||2R2LGp(R,φ)=1+||exp?2Rexp(iφ)Lp2,()π(p+||!)w0w0w0:..Downloadedfromhttp://rsta./onJanuary11,2017whoseradialdependenceisdeterminedbytheassociatedLaguerrepolynomialL||[16],p4p=0,1,2,...andisapositiveornegativeinteger0,±1,±2,....LGmodesfactorizeintoanR-dependentfunctiontimesexp(iφ);thislatterisaneigenfunctionoftheorbitalangular.........................................................?i?φ=?i(x?y?y?x)witheigenvalue.urasmodesoflasercavitieswithmirrorswithnon-negligiblesphericalaberration;duetoresidualastigmatismorlocalizedcavitydefects,thereisacouplingofbothsignsofangularmomentum,sotheresultingurastherealandimaginarypartsof(),andsodonotcarryasenseofright-orleft-,havethesamefunctionalformastheirFouriertransformandmaintaintheirintensitypatternonpropagation,asshownfortheexampleLG11in?gure2c,,=||+2p:foreachNthereareagainN+1modeswhere(,p)=(?N,0),(?N+2,1),...,(N?2,1),(+N,0).EachLGmodewithmodeorderNcanbeexpressedasasuperpositionoftheN+1HGmodesofthesamemodeorder,andviceversa[2];&Courtial[5]observedthatanysuperpositionofGaussianbeamsofmodeorderN=1375canberepresentedonasphere,analogoustothePoincarésphereofpolarization,asrepresented:20150441in?gure1b:thecircularmodesLG±1,uratthepoles,andlinearmodesHG10,HG01withanyuraroundtheequator;intermediate?elliptic?,,wewillmakemuchofthetwo-dimensionalisotropicharmonicoscillator,,,inthiswork,theclassicalpictureisthatofelliptic?rays?parametrizedbythePoincarésphere,heGaussianbeams(aseigenfunctionsofcertainnaturaloperators).Althoughthelanguage(andindeedthetwo-dimensionalharmonicoscillatorsystem)issuggestiveofphotonsandquantumoptics,allourclassical,,wewillconsidertheStokesparametersandPoincarésphere,discussinghistoricalapproachesandtheformulationintermsoftheHamiltonianmechanicsofatwo-,HGandGGmodesareconsideredin§3astheeigenstatesofoperatorsnaturallyarisingfromthecanonicalquantizationoftheoscillator,andtheconnectionisstrengthenedin§4,inwhichasemiclassicalpicturerelatingthe?classical?polarizationsphereandthe?quantum?,therectilinear?swinging?oflinearpolarization(andHGmodes)contrastswiththe?roundabout?motionofcircularpolarization(andLGmodes).:Stokes,Poincaré,GibbsandHamiltoninctdescriptionofthePoincarésphere,in[17,par157,],Thetwopoles...correspondtothecircularvibrations,andthevariouspointsofthe?rstmeridian[?s3greatcircle][-handed]inthenorthernhemisphere,andleft[-handed][onthesphere];thelociofpoints[ofthesameaxisalignment]...arethe[great]circlespassingthroughthetwopoles,that1MilesPadgettposedtooneofus(.)ataRankPrizemeetingin2003,towhichthispapershouldbeconsideredthe(belated)answer.:..Downloadedfromhttp://rsta./onJanuary11,2017istosay,;the5locusofthepointscorrespondingtoagivenshapeisalineoflatitude..........................................................-dimensionalJonesvectorE=(Ex,Ey);inER