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MonteCarloSimulationsofSpatialPatternsoftheDegreeofPolarisationforBackscatteringLightsinTurbidMediawithBirefringenceYinqiFengandYaoqinLiuOpto–MechatronicEquipmentTechnologyBeijingAreaMajorLaboratoryBeijingInstituteofPetrochemicaltechnologyBeijing102617,P.R.Chinafengyinqibipt.edu.cnZhengChangDepartmentofMechanicalEngineeringBeijingInstituteofPetrochemicaltechnologyBeijing102617,P.R.Chinachangzhengbipt.edu.cnAbstractInOpticalCoherenceTomographyOCTtechnology,ascontrastmechanismandacharacterizationoftheopticalpolarizationpropertiesofabiologicaltissue,itisveryimportantofpolarizationinwhichbirefringenceisoneoftherelativephenomena.WeanalyzedthepropagationofpolarizedlightinalinearlybirefringentturbidmediumandsimulatedthedegreeofpolarizationofdiffuselybackscatteredlightsusingtheStocksMuellerformalismandMonteCarloalgorithm.Statisticaldistributionsofthechangesofpolarizationintheturbidmediumwithdifferentbirefringentparametersaredemonstrated.KeywordsPolarizedlight,linearlybirefringent,MonteCarloSimulation,StocksMuellerformalism.I.INTRODUCTIONTherehasbeenanincreasinginterestinthepropagationinrandomlyscatteringmediaofpolarizedlightinthefieldsofopticalimagingandspectroscopyinmedicaldiagnostictechniquesforcancerandforothertissuepathologies,especiallyinOpticalCoherenceTomographyOCTtechnologyinwhichpolarizationisascontrastmechanismandacharacterizationoftheopticalpolarizationpropertiesofabiologicaltissue.Theoreticalanalysesandexperimentalstudieshavebeencarriedoutsimultaneously.Forexample,Hielscher,et.al.1usedaStokesvector/Muellermatrixapproachtopolarizedlightscatteringinordertoachievefullexperimentalcharacterizationoftheopticaltheopticalpropertiesofasampleunderinvestigation.TheygeneralizedtheconceptofaneffectiveMuellermatrixandmeasuredthetwodimensionalMuellermatrixofbackscatteredlight.Rakovicet.al.2,3andBartelet.al.4developedMonteCarloalgorithmstostudythebackscatteredintensitypatternsandcomparedtheirsimulationresultswithexperimentaldata.Rakovicet.alestimatedthecontributionofeachscatteredphotonbyanescapefunctionfromaparticularscatteringlocation,andBartelet.al.tracedthepolarizationstateofindividualphotonsuntiltheyareeitherabsorbedorleavethemedium.Recently,MonteCarlotechniquestodescribethepropagationofpolarizedlightinbothisotropicturbidmediumandlinearlybirefringentturbidmediaarepresented58.However,comparingWangsresults7ofthesymmetricalrelationsinthe16MuellermatrixelementsforisotropicturbidmediawiththatoftheexperimentalresultsfromHielsheret.al.1,itisnoticedthatthereexistsaninconsistency.9Hielschersanalyticalandnumericalmodelsthatdescribepolarizedlightpropagationinisotropicturbidmediaareextendedtomultiplyscatteringturbidmediawithlinearbirefringenceinthispaper.ThedetaileddescriptionsofpolarizedlightbackscatteredfromabirefringentturbidmediumarebasedonthebasicconceptsoftheStokesMuellerformalism10andMonteCarloalgorithmpreviouslydevelopedbyWangandJacques11.WesimulatepolarizationdependentphotonpropagationthroughmultiplyscatteringturbidmediawithlinearbirefringenceandshowthatoursimulationstracethepolarizationstatesofindividualphotonsuntiltheyareeitherabsorbedorleaveabirefringentmediumanddeterminetheeffectivebackscatteredMullermatrixelementswhichhavedifferentsymmetricalrelationshipsandshapeswiththatofWangsresults6,7.II.ANALYTICALANDNUMERICALMODELA.StokesMuellerFormulationThebasicStokesMuellerformalismandMonteCarlotracingmethodhavebeendescribedearlier2,4,10.Withinthisformalism,thepolarizationstateoflightcanbecompletelydeterminedbyfourparameterswhichmakeuptheStokesvectorS⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎣⎡−⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡lrrllrrlrrllrrllEEEEiEEEEEEEEEEEEVUQIS,1whereErandElaretwoorthogonalelectricalfieldcomponentsinaplaneperpendiculartothepropagationdirection.Theindicatesthecomplexconjugate,andthebracketsrepresentensembleaverageortimeaverageofergodic,stationaryprocesses.NotethatthecomponentsQandThisworkissupportedbya2007grant4072009fromGeneralProgramofBeijingNaturalScienceFundandthegrantsfromthereturnedscholarsandstudentsfundingprogramsofMinistryofEducationofChinaandBeijingMinistryofPersonnel.9781424447138/10/25.00©2010IEEEUoftheStokesvectordependonthechoiceofhorizontalandverticaldirections.IfthebasisvectorsleKandreKarerotatedthroughanangleφlookingagainstthedirectionoflightpropagation,thetransformationfromStokesparametersI,Q,U,VtoStokesparametersI,Q,U,VrelativetotherotatedaxesleKandreKis⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡VUQIVUQIRVUQI10000cos2sin200sin2cos200001φφφφφ2whereRφisastandard44rotationmatrix.Itisobviousfrom2thattheStokesparametersI,Q2U2,andVareinvariantundertherotationofreferencedirections.Inaddition,theStokesparametersarenotallindependent2222VUQI≥.3Equalityholdsfor100percentpolarizedlight.Theinequalityin3leadsnaturallytothenotationofthedegreeofpolarizationDOPIVUQ/222Φ,4thedegreeoflinearpolarizationDOLPIUQ/221Φ,5andthedegreeofcircularpolarizationDOCPIV/2Φ.6StokescomponentrrllEEEEIisatotalintensityinanobviousinterpretation.WhentheirradianceIisnormalizedtounity,StokesVectorscanbenotedas1,0,0,0,1,1,0,0,1,0,1,0,and1,0,0,1,fornaturallight,horizontallinearlypolarizedlight,45degreelinearlypolarizedlight,andrighthandcircularpolarizedlight,respectively.WhenapolarizationstateofascatteredlightisexpressedthroughStokesVectorS,itcanberelatedtothepolarizationstateofanincidentlightSbytheMuellermatrixMofanopticalelementoramaterial.Asthescatteringisinvolved,thisMuellermatrixisalsoknownasthescatteringmatrix⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛VUQImmmmVUQImmmmmmmmmmmm44434241343332312423222114131211.7Thatis,MuellermatrixdescribestherelationbetweenincidentandtransmittedorreflectedStokesvectors.ThemeasurementofMuellermatrixcompletelycharacterizestheopticalelementormaterialintermsofitsopticalproperties.Forexample,theMuellermatrixforanideallinearretarderis7⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡δδδδδδδδδβδcossinCsinS0sinCcosScos1CS0sinScos1CScosC00001,2222222222222222CSM8whereC2cos2β,S2sin2β,βisananglebetweenthehorizontalcomponentoftheincidentelectricfielddirectioninleKandoneoftheaxesofthelinearretarder1eKFig.1,andtheretardanceδisthephasedifferenceintroducedbythelinearretarder.Figure1.e1ande2specifytheaxesofanidealinearretarderB.MonteCarloCalculationsByemployingaboveStokesMuellerformalismofpolarizedlight,aMonteCarlomethodtobeusedtocalculatetheeffectivebackscatteredMuellermatrixM{mij}forabirefringentturbidmediumdefinedby7whichconnectsanincidentStokesVectorSwithanoutgoingvectorSofbackscatteredlightfromthebirefringentturbidmediumwillbeconsideredinthefollowing.Atthebeginning,aphotonissupposedtoinjectorthogonallyintoabirefringentturbidmediumattheorigin,whichcorrespondstoacollimatedarbitrarilynarrowbeamofphotons.Thepositionanddirectionofthephotonisspecifiedintheglobalsystemi,j,kbytheCartesiancoordinatesx,y,z,whichinitializedto0,0,0anddirectionalcosinesux,uy,uz,whichinitializedto0,0,1.Thedirectionoftheslowaxisofthebirefringentmediumisalongthexaxis,whichistheorientationwithhigherrefractiveindex.LetS0I0,Q0,U0,V0TbetheStokesvectorthatcorrespondstotheirradianceofincidentlightwithrespecttothexzplane,wherethesuperscriptTdenotestranspose.Oncestepsizesithatiscalculatedbasedonthesamplingoftheprobabilitydistributionforphotonsfreepathisspecified,thephotonisreadytobemovedinthebirefringentturbidmedium.Wewilltracealargenumberofrandomlychosenphotontrajectoriesthatallstartattheoriginal0,0,0.Eachtrajectoryisdefinedbythecollectionofpointsxi,yi,zi,i1,2,n,wherethescatteringtakesplace.Tokeepthetrackofaphotonspolarizationstatesasitundergoesmultiplescatteringeventsinabirefringentturbidmedium,itisassignedthefourcomponentStokesVectorSiandalocalcoordinatesystemeri,eli,e3iinwhichSiisdefinedFig.2.Theprimecorrespondstoalocalreferenceplaneinalocalcoordinatesystem,whichisdifferentfromthexzreferenceplaneintheglobesystemi,j,k.FollowingthesimilartracingmethodofBartelandHielscher4,thelocalcoordinatesystemischosensothate3ipointsthedirectionofpropagationofthephoton,anderi,andeliareorientedparallelandperpendiculartothelocalreferenceplane,respectively.Foreachscatteringevent,thesamplingdeflectionangleθandazimuthalangleφfromuniformlygeneratedrandomnumbersisperformedwiththerejectionmethod4accordingtothefollowingprobabilitydensityfunction5,whichisafunctionoftheincidentStocksvectorS0I0,Q0,U0,V0T0002sin2cos,IUQbaφφθθφθρ,9Onceθiandφiarechosen,bothalocalcoordinatesystemeri,eli,e3iandStokesVectorSiofthephotonpacketisupdateduponeachscatteringeventinthebirefringentturbidmedium.Figure2.LocalcoordinatesystemsofaphotonpriortoandafterascatteringIntheFig.2,thelocalcoordinatesystemeri,eli,e3ipriortoasinglescatteringeventisrotatedaboute3iwithangleφitoobtainedatransitioncoordinatesystemeri,eli,e3i.Duetothescatteringevent,thesystemeri,eli,e3iisrotatedagainabouteriwiththescatteringangleθitoobtainedanothertransitioncoordinatesystemeri,eli,e3i.Thenasastepsizesithatthephotontakesisspecified,thesystemeri,eli,e3iismadeaparallelmovealongthepropagationdirectione3itoobtainanewlocalcoordinatesystemeri1,eli1,e3i1.ThephotonpositionspecifiedineachlocalcoordinatesystembytheCartesiancoordinateseri,eli,e3icanbeexpressedbyavectoriiililiririeaeaeaa33KKKK.10Thisvectorgivenlocallyisupdatedas⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛iiiliriiiiiiiiiiiiilirsaaaaaa00cossin0cossincoscossinsinsinsincoscos3i1311θθφθφθφφθφθφ,11By9,thenewdirectionofthephotonpacketintheglobalsystemi,j,kisfoundtobeupdatedas⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎟⎟⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛100cossin0cossincoscossinsinsinsincoscos101111i22222iiiiiiiiiiiiiiiiiiiiiiiiiiiiuzuzuyuzuzuyuzuxuxuzuzuxuzuyuzuyuxθθφθφθφφθφθφ12Ontheotherhand,StokesvectorSigiveninalocalcoordinatesystemeri,eli,e3iisfirstprojectedintoalocalscatteringplanebytherotationalMuellermatrixRφidefinedin2.Atthispoint,theStokesvectoriiSRφ−isgivenwithrespecttothesystemeri,eli,e3i.ItshouldbenotedthattherotationangleinthesenseofthedirectionoflightpropagationisφibecausethedirectionofrotationiscounterclockwisebytheusualdefinitionofRφin2.AswesimplyconsiderthecaseofMiescattering,asinglescatteringprocesswithscatteringangleθicanbedescribedbyasinglescatteringMuellermatrixiMθ,whichtakesarelativelysimpleform10,12⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛de00ed0000ab00baiiiiiiiiiMθθθθθθθθθ,13thentheresultingStokesvectoriiiSRMφθisgivenwithrespecttothesystemeri,eli,e3i.InabirefringentmediumbirefringenteffectonaphotonpacketcanbedescribedbytheMuellermatrixofretarder,iiMβδequation8instepsizesi,StokesvectorSiinthelocalcoordinatesystemisthenupdatedas,1iiiiiiSRMMSφθβδ−,14whichisgivenwithrespecttothenewlocalsystemeri1,eli1,e3i1with10θMandRφ01fori0.βiintheMuellermatrixofaretarder,iiMβδisanazimuthalanglebetweenthebirefringentslowaxisalongthexaxisatthemomentandthehorizontalcomponentofelectricalfieldinlocalcoordinatealongelioreli1,anditisupdatedas21cosiiiiuzuzux−β15whichinitializedto1.Iftheangleofthephotonpacketistooclosetonormalofthemediumsurface,thenthefollowingformulasshouldbeusediiiφθβcoscoscos.16AndtheretardanceδiintheMuellermatrix,iiMβδcanbeobtainedbyλπδiisn2Δ17wheresiisthestepsize,λisthewavelengthofthelightinvacuum,andΔnisthedifferencebetweenthemaximumandminimumrefractiveindicesintheplaneperpendiculartothepropagationorientationofthephotonpacket,whichcanbeshownas6,7fifisfsnnnnnn−Δ−2/122sincosαα18nsandnfaretherefractiveindicesofthebirefringentmediumthelinearbirefringenceisnsnfalongtheslowaxisandthefastaxis,respectively,andαiistheanglebetweentheslowaxisofthemediumalongthexaxisintheglobesystemi,j,kandthepropagationorientationofthephotonpacketalonge3iore3i1,whichcanbeupdatedasiiuxαcos.19Oncethephotonhasbeenscattered,someattenuationofthephotonweightduetoscatteringbytheinteractionsitemustbecalculated.Atthestartpoint,itisassociatedw01.Ifthephotonpacketscurrentweightiswi,thenthephotonweighthastobeupdatedbyitsiwwµµ120whereµsisthescatteringcoefficient.AndwigivesacontributiontotheStokesvectorSiofaphotonpacketafterithasbeenscattereditimesinthefollowingway,1iiiiiiiSRMMwSφθβδ−21Actually,tsµµisthesinglescatteringalbedoandexpressestheremainingenergyafterthephotonpackethasbeenscattered.TheresultantStokesVectorSnisgiveninalocalreferenceplaneintherandomlyorientedcoordinatesystemern,eln,e3nafteralargenumbernofscatteringevents.AStokesvectorofanexitingphotonatthedetectormustbedeterminedintermsofxzreferenceplaneintheglobalsystemi,j,k.Bykeepingtrackofthelocalreferenceplanesandusingtheextendeddefinitionofthereferenceplanesrotation9,thesamepolarizationstatedescribedbySncanalsobedescribedby11121nniinnnSRSRRRS⎟⎠⎞⎜⎝⎛−−⋅⋅⋅−−∑−−φφφφ22whichcorrespondstothebackscatteredlightwithrespecttothexzreferenceplaneintheglobalsystemi,j,k.AspointedbyBartelandHielscher4,atotaloffourrunswiththefollowingincidentStocksvectors,T40T30T20T1010,0,,101,0,,100,1,,100,0,,1≡≡≡≡SSSS23issufficienttodetermineallelementsofabackscatteringMuellerMatrixofaturbidmedium.InpresentMonteCarlosimulationoflightscatteringandpropagationinabirefringentturbidmedium,weareabletoaccessallStokescomponentsdirectlyratherthanonlyintensities,thenitisstraightforwardtodeterminetheactualbackscatteredMuellermatrixM{mij}withauxiliarymatrix{sij}TTTTTTTTssssssSSSSMmsssssSSSSMmsssssSSSSMmsSSMm1444134312421141141040443424141434133312321131131030433323131424132312221121121020423222121413121111041312111,s,,sm,m,m,,s,s,sm,m,m,,s,s,sm,m,m,,,s,s,sm,m,m,−−−−−−−−−−−−−−−−−−24Thematrix{sij}ismadebythecomponentsofthedetectedStokesvectorsjSj14wherethesuperscriptj14indicatesoneoftheincidentStokesvectorin23.TherawdatasijgivesaStokescomponentineachgridelementinatwodimensionalgirdsystemthatcouldbesetupinxandydirection.Aftertracingmultiplephotonpackets,Stokesvectorsatthedetectormaysimplybesummedupinthetwodimensionalgirdsystemtoyieldanaverageanswerofthebirefringentturbidmedium.III.SIMULATIONRESULTSANDDISCUSSIONThestatisticaldistributionofthechangesinpolarizationinabirefringentturbidmediacanbeobservedthroughthespatialpatternsofthedegreeofpolarizationDOPorthedegreeoflinear/circularpolarizationDOLP/DOCPofthebackscatteringlight,whichhavebeendefinedin57.MonteCarloprogramsforsimulationofpolarizedlightinalinearlybirefringentturbidmediumandisotropicturbidmediumarecodedtotakeintoaccountoftheprincipleofaboveStocksMuellerformalismandMonteCarloalgorithmincludingpolarizationeffectandbirefringence.Theslowaxisofthebirefringentmediumisalongthexaxis,yaxisandzaxis,respectively,intheglobalcoordinatesystemi,j,k.Thebirefringencevalueis1.33x103.Therefractiveindicesofthebirefringentmediumalongtheslowaxisandthefastaxisarens1.330andnfns1.33x103,respectively.Fortheisotropicsamplewithnobirefringent
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