1、文章编号2097-1842（2023）05-1195-11The Poynting vectors,spin and orbital angular momentums ofuniformly polarized cosh-Pearcey-Gauss beams in the far zoneLIAOSai，CHENGKe*，HUANGHong-wei，YANGCeng-hao，LIANGMeng-ting，SUNWang-xuan（College of Optoelectronic Engineering,Chengdu University of Information Technolog
2、y,Chengdu 610225,China）*Corresponding author，E-mail:Abstract:Weproposecosh-Pearcey-Gaussbeamswithuniformpolarization,whicharemainlymodulatedbyahyperboliccosinefunction(n,)andtheanglesrelatedtouniformpolarization(,).Basedonangularspectrumrepresentationandthestationaryphasemethod,thePoyntingvector,Spi
3、nAngularMomentums(SAM)andOrbitalAngularMomentums(OAMs)inthefarzonearestudied.TheresultsshowthatalargernorinthehyperboliccosinefunctioncanpartitionthelongitudinalPoyntingvectors,SAMsandOAMsintomoremulti-lobedparabolicstructures.Differentpolarizationsdescribedby(,)candistinguishtheirPoynt-ingvectorsan
4、dangularmomentumsbetweentheTEandTMterms,thoughthisdoesnotaffectthepatternsofthewholebeam.Furthermore,theweightoftheleftandrightsidesoflongitudinalPoyntingvectors,SAMsandOAMsinTEandTMtermscanbemodulatedbyleft-handedorright-handedellipticalpolariza-tion,respectively.Theresultsinthispapermaybeusefulfor
5、informationstorageandpolarizationimaging.Key words:cosh-Pearcey-Gaussbeams;spinangularmomentum;orbitalangularmomentum;Poyntingvec-tor均匀偏振 cosh-Pearcey-Gauss 光束的远场坡印廷矢量，自旋与轨道角动量廖赛，程科*，黄宏伟，杨嶒浩，梁梦婷，孙望轩（成都信息工程大学光电工程学院,四川成都610225）摘要：本文提出了均匀偏振 cosh-Pearcey-Gauss光束，其主要由双曲余弦函数(n,)和偏振相关角度(,)所调制。基于矢量角谱法和稳相法，研
6、究了该光束的远场坡印廷矢量、自旋角动量和轨道角动量。研究结果表明：较大的双曲余弦函数值能将远场坡印廷矢量、自旋角动量和轨道角动量分割成多瓣抛物线结构。虽然左旋和右旋椭圆偏振不能影响整个光束结构，但却可通过调节 TE 和 TM 项的左半边和右半边的分布权重，进而分辨出光束的远场坡印廷矢量和角动量分布。本文结果对信息存储与偏振成像技术领域有着潜在的应用价值。收稿日期：2022-11-11；修订日期：2022-12-29基金项目：四川省自然科学基金项目(No.23NSFSC1097)SupportedbyNaturalScienceFoundationofSichuanProvince(No.23N
7、SFSC1097)第16卷第5期中国光学（中英文）Vol.16No.52023 年 9 月ChineseOpticsSept.2023关 键 词：cosh-Pearcey-Gauss 光束；自旋角动量；轨道角动量；坡印廷矢量中图分类号：TN929.1文献标志码：Adoi：10.37188/CO.EN.2022-00221IntroductionPearceyfunctionwithnon-convergenceorin-finiteenergyisthesecondtypeofcatastrophicfunc-tion.Itcanbeexpressedbytheintegralformulao
8、foscillating structure1.In 2012,Ring et al.intro-ducedtheGaussianenvelopetoconfineitsenergy,whereinthePearceybeamsweretheoreticallypro-posed and experimentally demonstrated2.It wasfoundthatthePearceybeamsalsohaveself-healingandauto-focusingpropertiesthatareanalogoustotheAirybeamsinrealspacepropagati
9、on.Half-Pear-cey beams3,Bi-Pearcey beams4,symmetric odd-Pearceybeams5andchirpedPearceybeams6havealso been extensively studied.For example,Denget al.foundthatthetunablecircularPearceybeamswithanannularspiral-zonephasepossessthedis-tinctiveabilitytotrapparticles7.Zhaoet al.showedthatthepropertiesofaut
10、ofocusingperformanceandthe inversion effect could be perfectly inheritedwithoutbeingaffectedbyspatialcoherence8.For optical catastrophe beams,their vectorialstructurescanbeanalyzedbyusingvectorangularspectrumand stationary phase methods.For ex-ample,the intensity profiles of the GeneralizedHumbert-G
11、aussian beams were obtained by theabovemethodsanditwasnotedthattheside-lobesdecreasewithadecreaseinthetopologicalchargeofthehelicalaxicon9.Jiaet al.foundthattopologicalcharge,truncationparametersandellipticityangleshave a great influence on the far-field vectorialstructures of a Laguerre-Gaussian be
12、am diffractedbyacircularaperture10.Furthermore,ourpreviousworkalsoshowedthatthesymmetryanddirectionof the far-field vectorial structures of a Pearcey-Gaussvortexbeamcanbemodulatedbytopologic-alchargesandnoncanonicalstrength11.Ontheoth-erhand,SpinAngularMomentum(SAM)isassoci-atedwithpolarization.Inas
13、calarfield,SAMcanbedescribedbyitsThirdStokesparametersandPoin-car sphere12.However,for the vector field,thevectorial SAM and Orbital Angular Momentum(OAM)shouldbeconsidered.Inaddition,theintro-ductionof the cosh function will bring more de-greesoffreedomforthePearceybeams.Thispaperisfocusedontheinfl
14、uenceoftheinitial polarization state and cosh function on thePoyntingvector,SAMandOAMofcosh-Pearcey-Gaussbeams(cPeG)inthefarzone.OurproposedbeamswithdifferentpolarizationscanhavevaryingweightsofOAMdensitiesintheTEandTMterms.The results obtained are different from those ofPearcey-Gaussvortexbeamsandp
15、rovidepotentialapplicationsininformationstorageandpolarizationimaging13-15.2Uniformlypolarizedcosh-Pearcey-Gauss(cPeG)beamsAssume the uniformly polarized cPeG beampropagatesinfreespacewithanelectricpermittiv-ity of and a magnetic permeability of.In theCartesiancoordinatesystem,theelectricfieldofthec
16、PeGbeamswithuniformpolarizationatthesourceplanez=0isexpressedby2,16-22(Ex(x,y,0)Ey(x,y,0)=(cosexp(i)sin)PeG(x,y)=x,ycoshn(),（1）PeG(x,y)whereistheazimuthalangleinpolarization,de-notesaphaseretardationanglebetweenthex-andy-componentsoftheelectricfield,andandnarethemodulationparameterandtheorderofthehy
17、perbol-ic cosine function,respectively.The backgroundfunctionofthecPeGbeamsinEq.(1)isgivenbyPeG(x,y)=expx2+y2w20Pe(xx0,yy0),（2）1196中国光学（中英文）第16卷wherew0isthebeamwaistwidthoftheGaussianterm,andPe(x/x0,y/y0)isthePearceyfunctionwithx0andy0ofscalinglengthsinthex-andy-axes,re-spectively.Thevectorangularsp
18、ectrumsofA(p,q)ofcPeGbeamsaregivenby18-21(Ax(p,q)Ay(p,q)=12x(Ex(x,y,0)Ey(x,y,0)expik(px+qy)dxdy,（3）where the wave number k=2/is related to thewavelength.SubstitutingEqs.(1)(2)intoEq.(3),onecanobtaintheanalyticalexpressionsofthevec-torangularspectraas(Ax(p,q)Ay(p,q)=(cosexp(i)sin)nt=0nh=0(nt)(nh)4nw2
19、0(1+iw20/4y20)14expw20(2+2)/4Pew202ix0(1+iw20/4y20)1/4,w20(y0/2x20)2iy0(1+iw20/4y20)1/2,（4）with=kpi(2tn),=kqi(2hn),(nt)=n!t!(nt)!,(nh)=n!h!(nh)!.（5）FromEqs.(4)(5),onecanfindthattheangu-larspectrumiscomposedofaPearceyfunctionandaGaussiantermwithandninthecoshpart.Fig.1(coloronline)showstheinitialinten
20、sityandangu-larspectrumofcPeGbeamsforn=1,2and4.FromFig.1,itcanbeseenthatthecentralzoneintheini-tialintensitydarkenedandthemainlobethatorigin-ates from the Pearcey factor and gradually fadeswiththeincreasingn.TheamplitudesintheangularspectraexhibitaparabolicstructureduetothePear-ceyfactorwithalower-o
21、rdern=1inFig.1(d).Theseparabolicspectracanbepartitionedintomorecom-plexstructuresbyincreasingtheorderofthecoshpartsnasshowninFigs.1(e)(f).Ahigher-ordern(n=4)leadstotheappearanceofadarkzoneofan-gularspectruminthecenterasshowninFig.1(f),andthemaximalvaluesarelocatedonthesidelobeoftheparabola.Onecanfin
22、dthattheangularspec-trumcanbemodifiedwithahigher-orderhyperbol-iccosinefunction,i.e.andn,anditsfar-zonevec-(,)=(/4,/8)(,)=(0,0)(,)=(/4,3/4)n=4n=2n=1Angular spectrumInitial intensity1.00.500.50.50.51.01.01.0y/mmx/mm1.01.01.00.50.50.80.60.40.201.00.80.60.40.2000.51.01.00.500.51.010 10 51001.01.50.5000
23、.50.51.01.01.01.00.50.51.00.80.60.40.20000.51.01.00.50(d)(a)(b)(c)(f)(e)0510kq/mm1kq/mm10.51.01.00.500.51.01.00.500.51.010 105510501.00.500.51.01.00.500510kq/mm1kq/mm110105510500510kq/mm1kq/mm1Fig.1TheinitialintensitiesandangularspectraofthecPeGbeamswithdifferentinitialpolarizationstatesforn=1,2and4
24、.(a),(d):=/4,=/8;(b),(e):=0,=0;(c),(f):=/4,=3/4.Thephasedistributionandpolarizationstatesarede-scribedinthetopformlefttorightinFigs.1(a)(c).Blue:left-handedellipticalpolarization;Red:right-handedelliptic-alpolarization;Black:linearpolarization.Theparametersarex0=y0=100mand=1.4第5期LIAOSai,et al.:ThePo
25、yntingvectors,spinandorbitalangularmomentumsof.1197torialstructuresinSAMandOAMisworthyoffur-ther analysis.The theoretical design of the cPeGbeamsbyencodingtheamplitudeandphaseoftheirFourierspectrumsontoaspatiallightmodulator,asshowninFig.2(coloronline),wheretheuniformlylinearly,circularlyorelliptica
26、llypolarizedlightcanbeproducedbywaveplates.cosh-Pearcey-GaussBeamsWave plateBSPolarizerLaserffSLMFig.2ThetheoreticaldesignschemeofthecPeGbeamswithuniformpolarization.BS:BeamSplitter;SLM:SpatialLightModulator3Far-zonevectorialstructureofcPeGbeamsAnarbitraryvectorialelectromagneticfieldcanbeseparated
27、into two orthogonal terms of Trans-verseElectric(TE)andTransverseMagnetic(TM)fields,anditcanbeasthesumofTEandTMtermsanddefinedasE=ETE+ETMandH=HTE+HTMwhereE and H denote the whole electric and magneticfields,respectively10,18,20.Thefar-zonebehavioroftheelectromagneticfield,i.e.EandHcanbefur-theranaly
28、zedbyETE,ETM,HTEandHTM,respect-ively.Theseelectricandmagneticfieldscanbeob-tainedbyvectorangularspectraandthestationaryphasemethodinthefarzone18-23.OurattentionisdirectedtothedependenceofthePoyntingvector,spinandorbitalangularmomentumdensitiesofthecPeGbeamsinuniformpolarization(,).Further-more,thein
29、fluencesoftheparametersandtheorderofthehyperboliccosinefunctionandnonthevectorialstructuresofETE,ETM,HTEandHTMinthefarzonearealsoinvestigatedindetail.Thecalcula-tionparametersare=633nm,x0=y0=100m(un-lessotherwisestated),w0=1mmandz0=ky02.3.1PoyntingvectorThe magnitude and direction of energy flowduri
30、ngelectromagneticwavepropagationcanbede-scribed by the Poynting vector.The longitudinalcomponentofthePoyntingvectorisproportionaltothe light intensity,and its transversal componentshowsthemagnitudeanddirectionofthecomplexpower flow in a fixed plane.The transversal andlongitudinal components of the s
31、eparated TE andTMtermsaregivenby22S,x=12ReE,yH,zE,zH,y,（6）S,y=12ReE,zH,xE,xH,z,（7）S,z=12ReE,xH,yE,yH,x,（8）where=TEorTM,Rerepresentstherealpartandtheasteriskiscomplexconjugation.Theeffectofuniformpolarization(,)onthePoyntingvectorsofthecPeGbeams,includingTE,TMtermsandwholebeam,atz=50z0infreespaceissh
32、owninFig.3,wherethemagnitudeanddirectionofthetransversePoyntingvectorsareshownbyar-rows.OnecanseethatthepatternsinPoyntingvec-tors are parabolic structures due to the Pearceyfactor,andtheenergyflowmovesfromthetoptothesideoftheparabola.Althoughtheparabolicpat-1198中国光学（中英文）第16卷ternsofawholebeamareunaf
33、fectedbyinitialpo-larizations,thesymmetryoflongitudinalPoyntingvectorsintheTEandTMtermscanbemodulatedbyuniformpolarization(i.e.,).TheparabolicpatternsoftheTEandTMtermsincPeGbeamsaresymmetric about horizontally linear polarization(=0)asshowninFigs.3(b)and(e)(coloron-line),andotherpolarizationsregulat
34、eparabolicpat-ternsintheTEandTMterms.Forexample,theleftsideoftheTEtermismoreinfluentialintheleft-handed(LH)ellipticalpolarization(,)=(/4,/8)inFig.3(a)(coloronline),buttherightonedomin-atestheTEtermintheright-handed(RH)ellipticalpolarization(,)=(/4,3/4)in Fig.3(g)(coloronline).Furthermore,thetopologi
35、calchargeintheTEandTMtermswillchangeiftheuniformpolar-izations(,)aremodulated,andthenettopologic-alchargeisalwayszerounderuniformpolarization(,)due to the topological charges of the twotermsbeingopposite.Fig.4(coloronline)showsthePoyntingvectorsofcPeGbeamsinfreespacefordifferentnandatz=50z0,wherethe
36、LHellipticalpolarizationisgivenby(,)=(/4,/8).OnecanfindthattheTE,TMandwholetermspresentmul-tilobedparabolicstructuresinlongitudinalPoyntingvectors,and the number of partitioned lobes in-creaseswithanincreaseinnor.Theintensitiesofthedarkzonesontheparabolicpatternsarenonzero.TC=+1Whole beamTM termTE t
37、erm(,)=(/4,/8)(,)=(0,0)(,)=(/4,3/4)TC=1TC=1TC=+1y/mmx/mm(a)(b)(c)5.05.02.52.5002.52.55.05.05.05.02.52.5002.52.55.05.05.05.02.52.5002.52.55.05.0(d)(e)(f)5.05.02.52.5002.52.55.05.05.05.02.52.5002.52.55.05.05.05.02.52.5002.52.55.05.0(g)(h)(i)5.05.02.52.5002.52.55.05.05.05.02.52.5002.52.55.05.05.05.02.5
38、2.5002.52.55.05.01.00.80.60.40.20Fig.3Normalized longitudinal Poynting vector(backgrounds)and transversal Poynting vector(arrows)of cPeG beamsfordifferentuniformpolarizations(,)atz=50z0.(a),(d),(g):TEterm;(b),(e),(h):TMterm;(c),(f),(i):wholebeam.Theparametersaren=2and=1.4.Theredpointsymbolizestopolo
39、gicalchargel=+1,andthewhitepointde-notes l=1第5期LIAOSai,et al.:ThePoyntingvectors,spinandorbitalangularmomentumsof.1199TE termTM termWhole beamTC=+1TC=1(a)(b)(c)5.05.02.52.5002.52.55.05.05.05.02.52.5002.52.55.05.05.05.02.52.5002.52.55.05.0TC=+1TC=1(d)(e)(f)5.05.02.52.5002.52.55.05.05.05.02.52.5002.52
40、.55.05.05.05.02.52.5002.52.55.05.0y/mmx/mm1.00.80.60.40.20n=4=1n=1=3Fig.4NormalizedlongitudinalPoyntingvector(backgrounds)andtransversePoyntingvector(arrows)ofcPeGbeamsfordifferentnandatz=50z0,wheretheuniformpolarizationis(,)=(/4,/8)3.2SpinangularmomentumAngularmomentumcarriedbytheelectromag-neticfi
41、eld,intandemwithitsmechanicalproperties,iscomprisedofspinandorbitalcomponents.Theintrinsicspinangularmomentumisassociatedwithleft-andright-handedcircularpolarizations,whiletheorbitalcomponentoriginatesfromthespatiallyhelicalphaseofthewavefield.Thetotalangularmo-mentumoftheelectromagneticfieldcanbewr
42、ittenasthesumofthespinandorbitalcomponents13,24-25J=L+S=2iwdrj=x,y,zE,j(r)E,j+2iwdrEE,（9）where=TEorTM,isthepermittivityofthevacu-um,istheangularfrequency,andthefirstandsecond terms on the right side of Eq.(9)denoteSAMandOAM,respectively.Figs.5and6(coloronline)givethedependen-ciesoftheSAMoftheTEterm,
43、TMtermandwholebeamwithuniformpolarization(,),coshpara-metersandtheordernumberninfreespace,re-spectively.For linear polarization,the transversalandlongitudinalSAMsdonotexist.ForLHorRHelliptical polarization,the longitudinal SAMs arelessorlargerthanzero,respectively.Thelongitud-inalSAMintheleftsideofT
44、Etermshasanadvant-ageoverthatontherightside,butforTMterms,theoppositeistrue.ThesuperpositionoftheleftsideintheTEtermandtherightsideintheTMtermres-ults in the multilobed parabolic structures of thewholebeaminSAMdensities,andthepeakvaluesof their SAM densities are located on two sidesratherthanthevert
45、exoftheirparabolas.FromFig.6,itisclearthattheSAMdensitiescanbepartitionedbylargercoshparametersandordernumbersn.Inaddition,thetransverseSAMofawholebeamisde-terminedbytheTMtermowingtothez-componentoftheTEtermnotexisting,andthesevectorialar-rowspresentdifferentdirections.Thearrowsintherightsidepointto
46、theleftinFigs.5(c),butthoseintheleftsidepointtotherightasshowninFig.5(f).MoredetailsontheevolutionoftransversalSAMinwholebeamsareprovidedinSupplementaryVideos1200中国光学（中英文）第16卷1and2(seethearticleonline),whereinonecanfindthatanalternativevariationoftransverseSAMsispresented.Itsdirectionintheleftsideva
47、riesfromrighttoleftforLHpolarizationwhere0/2and=/8asshowninVideo1,whereasitschangedir-ection is opposite for the RH polarization with/2and=/8inVideo2,whichindicatesthatthedirectionoftransversalSAMsiscloselyrelatedtoLHorRHpolarization.(,)=(/4,3/4)(,)=(/4,/8)TE termTM termWhole beam0.51.01.00.50000500
48、050005555050050.51.01.00.500.51.01.00.500.51.01.00.500.51.01.00.500.51.01.00.5055 555 5(a)(b)(c)(d)(e)(f)x/mmx/mmx/mm050555 5x/mmx/mmy/mmy/mmy/mm05y/mmy/mm005055 5x/mmy/mmFig.5NormalizedlongitudinalSAM(3Dand2D)andtransverseSAM(arrows)ofcPeGbeamsatz=50z0.(a)(c):=/4,=/8;(d)(f):=/4,=3/4.Theotherparamet
49、ersarethesameasthoseinFig.3TE termTM termWhole beam0.51.01.00.50000500050005555050050.51.01.00.500.51.01.00.500.51.01.00.500.51.01.00.500.51.01.00.5055 555 5(a)(b)(c)(d)(e)(f)x/mmx/mmx/mm050555 5x/mmx/mmy/mmy/mmy/mm05y/mmy/mm005055 5x/mmy/mmn=4=1n=1=3Fig.6Normalizedlongitudinal(3Dand2D)SAMandtransve
50、rseSAM(arrows)ofcPeGbeams(n=1,4)atz=50z0.(a)(c):n=1,=3;(d)(f):n=4,=1.TheotherparametersarethesameasthoseinFigs.5(=/4,=/8)3.3OrbitalangularmomentumThedependencies of longitudinal and trans-verseOAMsofTE,TMandwholebeamsondiffer-ent polarization in free space are shown in Fig.7(coloronline).Althoughthe
