外文资料--Monte Carlo Simulations of Spatial Patterns of the Degree of (1).PDF

MonteCarloSimulationsofSpatialPatternsoftheDegreeofPolarisationforBackscatteringLightsinTurbidMediawithBirefringenceYinqiFengandYaoqinLiuOptoMechatronicEquipmentTechnologyBeijingAreaMajorLaboratoryBeijingInstituteofPetrochemicaltechnologyBeijing102617,P.R.CZhengChangDepartmentofMechanicalEngineeringBeijingInstituteofPetrochemicaltechnologyBeijing102617,P.R.CAbstractInOpticalCoherenceTomography(OCT)technology,ascontrastmechanismandacharacterizationoftheopticalpolarizationpropertiesofabiologicaltissue，itisveryimportantofpolarizationinwhichbirefringenceisoneoftherelativephenomena.WeanalyzedthepropagationofpolarizedlightinalinearlybirefringentturbidmediumandsimulatedthedegreeofpolarizationofdiffuselybackscatteredlightsusingtheStocksMuellerformalismandMonteCarloalgorithm.Statisticaldistributionsofthechangesofpolarizationintheturbidmediumwithdifferentbirefringentparametersaredemonstrated.Keywords-Polarizedlight,linearlybirefringent,MonteCarloSimulation,StocksMuellerformalism.I.INTRODUCTIONTherehasbeenanincreasinginterestinthepropagationinrandomlyscatteringmediaofpolarizedlightinthefieldsofopticalimagingandspectroscopyinmedicaldiagnostictechniquesforcancerandforothertissuepathologies,especiallyinOpticalCoherenceTomography(OCT)technologyinwhichpolarizationisascontrastmechanismandacharacterizationoftheopticalpolarizationpropertiesofabiologicaltissue.Theoreticalanalysesandexperimentalstudieshavebeencarriedoutsimultaneously.Forexample,Hielscher,et.al.1usedaStokesvector/Muellermatrixapproachtopolarizedlightscatteringinordertoachievefullexperimentalcharacterizationoftheopticaltheopticalpropertiesofasampleunderinvestigation.TheygeneralizedtheconceptofaneffectiveMuellermatrixandmeasuredthetwo-dimensionalMuellermatrixofbackscatteredlight.Rakovicet.al.2,3andBartelet.al.4developedMonteCarloalgorithmstostudythebackscatteredintensitypatternsandcomparedtheirsimulationresultswithexperimentaldata.Rakovicet.alestimatedthecontributionofeachscatteredphotonbyanescapefunctionfromaparticularscatteringlocation,andBartelet.al.tracedthepolarizationstateofindividualphotonsuntiltheyareeitherabsorbedorleavethemedium.Recently,MonteCarlotechniquestodescribethepropagationofpolarizedlightinbothisotropicturbidmediumandlinearlybirefringentturbidmediaarepresented5-8.However,comparingWangsresults7ofthesymmetricalrelationsinthe16MuellermatrixelementsforisotropicturbidmediawiththatoftheexperimentalresultsfromHielsheret.al.1,itisnoticedthatthereexistsaninconsistency.9Hielschersanalyticalandnumericalmodelsthatdescribepolarizedlightpropagationinisotropicturbidmediaareextendedtomultiplyscatteringturbidmediawithlinearbirefringenceinthispaper.ThedetaileddescriptionsofpolarizedlightbackscatteredfromabirefringentturbidmediumarebasedonthebasicconceptsoftheStokes-Muellerformalism10andMonteCarloalgorithmpreviouslydevelopedbyWangandJacques11.Wesimulatepolarization-dependentphotonpropagationthroughmultiplyscatteringturbidmediawithlinearbirefringenceandshowthatoursimulationstracethepolarizationstatesofindividualphotonsuntiltheyareeitherabsorbedorleaveabirefringentmediumanddeterminetheeffectivebackscatteredMullermatrixelementswhichhavedifferentsymmetricalrelationshipsandshapeswiththatofWangsresults6,7.II.ANALYTICALANDNUMERICALMODELA.Stokes-MuellerFormulationThebasicStokes-MuellerformalismandMonteCarlotracingmethodhavebeendescribedearlier2,4,10.Withinthisformalism,thepolarizationstateoflightcanbecompletelydeterminedbyfourparameterswhichmakeuptheStokesvectorS:+=*lrrllrrlrrllrrllEEEEiEEEEEEEEEEEEVUQIS,(1)whereErandElaretwoorthogonalelectricalfieldcomponentsinaplaneperpendiculartothepropagationdirection.The“*”indicatesthecomplexconjugate,andthebracketsrepresentensembleaverage(ortimeaverageofergodic,stationaryprocesses).NotethatthecomponentsQandThisworkissupportedbya2007grant(4072009)fromGeneralProgramofBeijingNaturalScienceFundandthegrantsfromthereturnedscholarsandstudentsfundingprogramsofMinistryofEducationofChinaandBeijingMinistryofPersonnel.978-1-4244-4713-8/10/$25.002010IEEEUoftheStokesvectordependonthechoiceofhorizontalandverticaldirections.IfthebasisvectorsleKandreKarerotatedthroughananglelookingagainstthedirectionoflightpropagation,thetransformationfromStokesparameters(I,Q,U,V)toStokesparameters(I,Q,U,V)relativetotherotatedaxesleKandreKis=VUQIVUQIRVUQI10000)cos(2)sin(200)sin(2)cos(200001)(2)whereR()isastandard44rotationmatrix.Itisobviousfrom(2)thattheStokesparametersI,Q2+U2,andVareinvariantundertherotationofreferencedirections.Inaddition,theStokesparametersarenotallindependent:2222VUQI+.(3)Equalityholdsfor100percentpolarizedlight.Theinequalityin(3)leadsnaturallytothenotationofthedegreeofpolarization(DOP)IVUQ/222+=,(4)thedegreeoflinearpolarization(DOLP)IUQ/221+=,(5)andthedegreeofcircularpolarization(DOCP)IV/2=.(6)Stokescomponent*rrllEEEEI+=isatotalintensityinanobviousinterpretation.WhentheirradianceIisnormalizedtounity,StokesVectorscanbenotedas1,0,0,0,1,1,0,0,1,0,1,0,and1,0,0,1,fornaturallight,horizontallinearlypolarizedlight,45degreelinearlypolarizedlight,andright-handcircularpolarizedlight,respectively.WhenapolarizationstateofascatteredlightisexpressedthroughStokesVectorS,itcanberelatedtothepolarizationstateofanincidentlightSbytheMuellermatrixMofanopticalelementoramaterial.Asthescatteringisinvolved,thisMuellermatrixisalsoknownasthescatteringmatrix:=VUQImmmmVUQImmmmmmmmmmmm44434241343332312423222114131211.(7)Thatis,Muellermatrixdescribestherelationbetween“incident”and“transmitted”or“reflected”Stokesvectors.ThemeasurementofMuellermatrixcompletelycharacterizestheopticalelementormaterialintermsofitsopticalproperties.Forexample,theMuellermatrixforanideallinearretarderis7+=cossinC-sinS0sinCcosS)cos-(1CS0sinS-)cos-(1CScosC00001),(2222222222222222CSM(8)whereC2=cos(2),S2=sin(2),isananglebetweenthehorizontalcomponentoftheincidentelectricfield(directioninleK)andoneoftheaxesofthelinearretarder(1eK)(Fig.1),andtheretardanceisthephasedifferenceintroducedbythelinearretarder.Figure1.e1ande2specifytheaxesofanidealinearretarderB.MonteCarloCalculationsByemployingaboveStokes-Muellerformalismofpolarizedlight,aMonteCarlomethodtobeusedtocalculatetheeffectivebackscatteredMuellermatrixM=mijforabirefringentturbidmediumdefinedby(7)whichconnectsanincidentStokesVectorSwithanoutgoingvectorSofbackscatteredlightfromthebirefringentturbidmediumwillbeconsideredinthefollowing.Atthebeginning,aphotonissupposedtoinjectorthogonallyintoabirefringentturbidmediumattheorigin,whichcorrespondstoacollimatedarbitrarilynarrowbeamofphotons.Thepositionanddirectionofthephotonisspecifiedintheglobalsystem(i,j,k)bytheCartesiancoordinates(x,y,z),whichinitializedto(0,0,0)anddirectionalcosines(ux,uy,uz),whichinitializedto(0,0,1).Thedirectionoftheslowaxisofthebirefringentmediumisalongthex-axis,whichistheorientationwithhigherrefractiveindex.LetS0=I0,Q0,U0,V0TbetheStokesvectorthatcorrespondstotheirradianceofincidentlightwithrespecttothex-zplane,wherethesuperscriptTdenotestranspose.Oncestepsizesithatiscalculatedbasedonthesamplingoftheprobabilitydistributionforphotonsfreepathisspecified,thephotonisreadytobemovedinthebirefringentturbidmedium.Wewilltracealargenumberofrandomlychosenphotontrajectoriesthatallstartattheoriginal(0,0,0).Eachtrajectoryisdefinedbythecollectionofpoints(xi,yi,zi),i=1,2,n,wherethescatteringtakesplace.Tokeepthetrackofaphotonspolarizationstatesasitundergoesmultiplescatteringeventsinabirefringentturbidmedium,itisassignedthefour-componentStokesVectorSiandalocalcoordinatesystem(eri,eli,e3i)inwhichSiisdefined(Fig.2).Theprimecorrespondstoalocalreferenceplaneinalocalcoordinatesystem,whichisdifferentfromthex-zreferenceplaneintheglobesystem(i,j,k).FollowingthesimilartracingmethodofBartelandHielscher4,thelocalcoordinatesystemischosensothate3ipointsthedirectionofpropagationofthephoton,anderi,andeliareorientedparallelandperpendiculartothelocalreferenceplane,respectively.Foreachscatteringevent,thesamplingdeflectionangleandazimuthalanglefromuniformlygeneratedrandomnumbersisperformedwiththerejectionmethod4accordingtothefollowingprobability-densityfunction5,whichisafunctionoftheincidentStocksvectorS0=I0,Q0,U0,V0T:000)2sin()2cos()()(),(IUQba+=,(9)Onceiandiarechosen,bothalocalcoordinatesystem(eri,eli,e3i)andStokesVectorSiofthephotonpacketisupdateduponeachscatteringeventinthebirefringentturbidmedium.Figure2.LocalcoordinatesystemsofaphotonpriortoandafterascatteringIntheFig.2,thelocalcoordinatesystem(eri,eli,e3i)priortoasinglescatteringeventisrotatedaboute3iwithangleitoobtainedatransitioncoordinatesystem(eri,eli,e3i).Duetothescatteringevent,thesystem(eri,eli,e3i)isrotatedagainabouteriwiththescatteringangleitoobtainedanothertransitioncoordinatesystem(eri,eli,e3i).Thenasastepsizesithatthephotontakesisspecified,thesystem(eri,eli,e3i)ismadeaparallelmovealongthepropagationdirectione3itoobtainanewlocalcoordinatesystem(eri+1,eli+1,e3i+1).ThephotonpositionspecifiedineachlocalcoordinatesystembytheCartesiancoordinates(eri,eli,e3i)canbeexpressedbyavectoriiililiririeaeaeaa33KKKK+=.(10)Thisvectorgivenlocallyisupdatedas=+iiiliriiiiiiiiiiiiilirsaaaaaa00cossin-0cossincoscossinsinsin-sincos-cos3i1311,(11)By(9),thenewdirectionofthephotonpacketintheglobalsystem(i,j,k)isfoundtobeupdatedas=100cossin-0cossincoscossinsinsin-sincos-cos1-01111i22222iiiiiiiiiiiiiiiiiiiiiiiiiiiiuz-uzuy-uzuzuy-uz-uxux-uzuzux-uzuyuzuyux(12)Ontheotherhand,StokesvectorSigiveninalocalcoordinatesystem(eri,eli,e3i)isfirstprojectedintoalocalscatteringplanebytherotationalMuellermatrixR(-i)definedin(2).Atthispoint,theStokesvector)(iiSRisgivenwithrespecttothesystem(eri,eli,e3i).Itshouldbenotedthattherotationangleinthesenseofthedirectionoflightpropagationis-ibecausethedirectionofrotationiscounter-clockwisebytheusualdefinitionofR()in(2).AswesimplyconsiderthecaseofMiescattering,asinglescatteringprocesswithscatteringangleicanbedescribedbyasingle-scatteringMuellermatrix)(iM,whichtakesarelativelysimpleform10,12=)d()e(00)e(-)d(0000)a()b(00)b()(a)(iiiiiiiiiM,(13)thentheresultingStokesvector)()(iiiSRMisgivenwithrespecttothesystem(eri,eli,e3i).InabirefringentmediumbirefringenteffectonaphotonpacketcanbedescribedbytheMuellermatrixofretarder),(iiM(equation8)instepsizesi,StokesvectorSiinthelocalcoordinatesystemisthenupdatedas)()(),(1iiiiiiSRMMS=+,(14)whichisgivenwithrespecttothenewlocalsystem(eri+1,eli+1,e3i+1)with1)(0=MandR(-0)=1fori=0.iintheMuellermatrixofaretarder),(iiMisanazimuthalanglebetweenthebirefringentslowaxis(alongthex-axisatthemoment)andthehorizontalcomponentofelectricalfieldinlocalcoordinate(alongelioreli+1),anditisupdatedas:21cosiiiiuzuzux=(15)whichinitializedto1.Iftheangleofthephotonpacketistooclosetonormalofthemediumsurface,thenthefollowingformulasshouldbeusediiicoscoscos=.(16)AndtheretardanceiintheMuellermatrix),(iiMcanbeobtainedbyiisn2=(17)wheresiisthestepsize,isthewavelengthofthelightinvacuum,andnisthedifferencebetweenthemaximumandminimumrefractiveindicesintheplaneperpendiculartothepropagationorientationofthephotonpacket,whichcanbeshownas6,7fifisfsnnnnnn+=2/122)sin()cos(18)nsandnfaretherefractiveindicesofthebirefringentmedium(thelinearbirefringenceisns-nf)alongtheslowaxisandthefastaxis,respectively,andiistheanglebetweentheslowaxisofthemedium(alongthex-axisintheglobesystem(i,j,k)andthepropagationorientationofthephotonpacket(alonge3iore3i+1),whichcanbeupdatedasiiux=cos.(19)Oncethephotonhasbeenscattered,someattenuationofthephotonweightduetoscatteringbytheinteractionsitemustbecalculated.Atthestartpoint,itisassociatedw0=1.Ifthephotonpacketscurrentweightiswi,thenthephotonweighthastobeupdatedbyitsiww=+1(20)wheresisthescatteringcoefficient.AndwigivesacontributiontotheStokesvectorSiofaphotonpacketafterithasbeenscattereditimesinthefollowingway:)()(),(1iiiiiiiSRMMwS=+(21)Actually,tsisthesingle-scatteringalbedoandexpressestheremainingenergyafterthephotonpackethasbeenscattered.TheresultantStokesVectorSnisgiveninalocalreferenceplaneintherandomlyorientedcoordinatesystem(ern,eln,e3n)afteralargenumbernofscatteringevents.AStokesvectorofanexitingphotonatthedetectormustbedeterminedintermsofx-zreferenceplaneintheglobalsystem(i,j,k).Bykeepingtrackofthelocalreferenceplanesandusingtheextendeddefinitionofthereferenceplanesrotation9,thesamepolarizationstatedescribedbySncanalsobedescribedby)()()(11121nniinnnSRSRRRS=(22)whichcorrespondstothebackscatteredlightwithrespecttothex-zreferenceplaneintheglobalsystem(i,j,k).AspointedbyBartelandHielscher4,atotaloffourrunswiththefollowingincidentStocksvectors,T40T30T20T101)0,0,1(0)1,0,1(0)0,1,1(0)0,0,1(SSSS(23)issufficienttodetermineallelementsofabackscatteringMuellerMatrixofaturbidmedium.InpresentMonteCarlosimulationoflightscatteringandpropagationinabirefringentturbidmedium,weareabletoaccessallStokescomponentsdirectlyratherthanonlyintensities,thenitisstraightforwardtodeterminetheactualbackscatteredMuellermatrixM=mijwithauxiliarymatrixsij:()()TTTTTTTTssssssSSSSMmsssssSSSSMmssss