Polarized light propagation through scattering media: time ...light.6–10 Polarization propagation in biological tissues is a complicated process that is fundamental to tissue optics

Journal of Biomedical Optics 8(4), 608–617 (October 2003)

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Polarized light propagation through scattering media:time-resolved Monte Carlo simulations andexperiments

Xueding WangLihong V. WangTexas A&M UniversityDepartment of Biomedical EngineeringOptical Imaging LaboratoryCollege Station, Texas 77843-3120E-mail: [email protected]

Chia-Wei SunChih-Chung YangNational Taiwan UniversityGraduate Institute of Electro-Optical EngineeringDepartment of Electrical EngineeringTaipei, Taiwan, Republic of China

Abstract. A study of polarized light transmitted through randomlyscattering media of a polystyrene-microsphere solution is described.Temporal profiles of the Stokes vectors and the degree of polarizationare measured experimentally and calculated theoretically based on aMonte Carlo technique. The experimental results match the theoreti-cal results well, which demonstrates that the time-resolved MonteCarlo technique is a powerful tool that can contribute to the under-standing of polarization propagation in biological tissue. Analysisbased on the Stokes-Mueller formalism and the Mie theory shows thatthe first scattering event determines the major spatial patterns of thetransmitted Stokes vectors. When an area detected at the output sur-face of a turbid medium is circularly symmetrical about the incidentbeam, the temporal profile of the transmitted light is independent ofthe incident polarization state. A linear relationship between the av-erage order of the scatters and the light propagation time can be usedto explain the exponential decay of the degree of polarization of trans-mitted light. © 2003 Society of Photo-Optical Instrumentation Engineers.[DOI: 10.1117/1.1606462]

Keywords: polarized light; scattering; ultrafast optics; Monte Carlo; light propaga-tion; time-resolved.

Paper 02070 received Oct. 14, 2002; revised manuscript received Apr. 23, 2003;accepted for publication May 1, 2003.

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1 IntroductionRecently there has been increased interest in the propagatiof polarized light in randomly scattering media, such as bio-logical tissues, because of its potential applications, particularly in biomedical imaging and diagnosis. Optical technologyoffers significant advantages for imaging human tissues because it employs nonionizing radiation and provides a highcontrast between early cancers and the background normtissues. However, the random scattering of light in biologicatissues deteriorates the imaging resolution, which presents thmain challenge associated with optical imaging. Ultrafast optics can be used to enhance optical imaging and diagnosiWith this technique, a short light pulse is applied to the bio-logical tissues, and then the transmitted ballistic and quasballistic photons that carry more information about biologicaltissue properties are extracted through various gating techniques. Since multiply scattered photons usually have longepath lengths than weakly scattered photons, they can be ecluded by time gating.1–5 Polarization gating can be used asan alternative method for extracting the ballistic and quasiballistic components because weakly scattered light preserveits original polarization better than multiply scatteredlight.6–10 Polarization propagation in biological tissues is acomplicated process that is fundamental to tissue opticsMany parameters, such as size, shape, refractive index, cocentration of the scattering particles, and incident polarizationstate play important roles in the scattering of light.11,12

608 Journal of Biomedical Optics d October 2003 d Vol. 8 No. 4

d From: http://biomedicaloptics.spiedigitallibrary.org/ on 07/12/2016 Terms

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In order to improve the quality of optical imaging usinultrafast light sources, more study is necessary to fully undstand the evolution of the polarization state in scattering mdia, as well as the time- and polarization-dependent distrition of light transmitted through the media. The Monte Ca~MC! technique offers a flexible and accurate approachthese problems, since it can trace detailed information inlarization propagation inside and outside a scattering medwith complex geometries and can score multiple physiquantities simultaneously.13,14

By employing a streak camera and an ultrafast lasource, in this study we analyze the temporal profiles ofStokes components of light transmitted through scatteringdia made of polystyrene-microsphere solutions, where thecident light pulses are linearly polarized and circularly polized, respectively. A time-resolved MC technique based onStokes-Mueller formalism is developed to analyze polarition propagation through these scattering media. The polartion distributions after a single scattering and after multipscatterings are compared and discussed. The MC-simulStokes vectors of transmitted light are compared withmeasurements obtained in experiments. We study and expsome of the properties of the polarization-dependent timresolved transmitted light that can be observed from theperimental and MC results. Polystyrene-microsphere solutiwith various particle diameters and scattering concentrati

1083-3668/2003/$15.00 © 2003 SPIE

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Polarized light propagation . . .

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Fig. 1 (a) The schematic diagram of the experimental setup, where P is a polarizer, HW is a zero-order half-wave plate, QW is a zero-orderquarter-wave plate, and A is a linear analyzer. (b) The response of the detection system to the laser pulse.

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have been studied. Here we present the results for 380-npolystyrene-microsphere solutions with scatterer concentrations of 0.133 and 0.066%.

2 Experimental SetupThe experimental setup is shown in Fig. 1. A titanium:sap-phire ~Ti:sapphire! laser is used to provide 800-nm lightpulses with a FWHM of 100 fs. The laser beam is split intotwo branches, one for triggering the streak camera and thother for propagating through the scattering media. An average power of 200 mW is applied to the input surface of themedia. The incident polarization state is controlled by a po-larizer, a zero-order half-wave plate, and a zero-order quartewave plate. Meanwhile, the detected polarization state is controlled by a zero-order quarter-wave plate and a lineaanalyzer. The light transmitted from the scattering media isdirected to the streak camera with a fiber bundle. The diameter of the receiving area is 1 mm, and the receiving angle othe fiber bundle is about 30 deg. The temporal resolution othe operational mode of the streak camera is 4.74 ps. Becauof the dispersion of the optical signal in the fiber bundle andthe limited resolution of the streak camera, the response of thdetection system to the laser pulse has a FWHM of 20 ps, ashown by the normalized profile in Fig. 1~b!.

The scattering media are composed of polystyrenemicrosphere solutions contained in a transparent plastic cubcuvette, with a size of53532 cm, whereL, the thickness ofthe cuvette along the laser axis, is 2 cm. The polystyrenemicrosphere solutions occupy the cuvette to about 80% of thvolume. The absorption coefficient of the solutions is lowenough to be regarded as zero. The polystyrene microspherhave a diameter2a of 380 nm613 nm.The refractive indicesof the background medium,nb , and the polystyrene micro-spheres,ns , are 1.329 and 1.565, respectively, at a wave-length ~l! in vacuo of 800 nm. The speed of light,c, in thescattering media is 0.0226 cm/ps. The polystyrenemicrosphere solution is diluted to different concentrations toachieve different scattering mean-free-paths~SMFP!, l s . Theincident laser beam enters the medium from the center of thinput surface. The fiber bundle and the incident beam araligned with each other so that the area detected on the outpsurface of the medium has a circular symmetry about the axiof the incident light.


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3 Monte Carlo AnalysisThe geometry of multiple scattering events in a polystyremicrosphere solution is shown in Fig. 2~a!. The MC simula-tion that describes the multiple scattering events of lightturbid media is based on radiative theory, which assumesscattering events are independent and ignore coherent effA narrow pencil beam propagates downward along the z-ainto a plane-parallel slab of scattering medium with a thicness ofL. The incident point is~0, 0, 0! in the laboratorycoordinate system~x, y, z!. Photon packets are scattered in tmedium by microspheres before exiting the upper or lowsurface of the medium. At each scattering event, the scatteangles of the photon packets are statistically selected accing to Mie theory.15 During propagation, the polarization evolutions of photon packets are traced through the StokMueller formalism.16,17

To begin, we analyze the first scattering event thatphoton packets encounter after they enter the input surfaca scattering medium. The scattering geometry is shown in2~b!, whereQP@0,p# is the polar angle between the propgation orientations before and after the scattering, andfP@0,2p) is the azimuthal angle between the reference plaand the scattering plane. OnceQ andf are known, the propa-gation orientation after the scattering is determined. TStokes vector of light after the first scattering event canexpressed as

S1~f,Q!5R~2f!M~Q!R~f!S0. ~1!

S0 and S1 are Stokes vectors of light before and after tscattering. R~f! is the rotation matrix connecting the twStokes vectors that describe the same polarization staterespect to the reference plane and the scattering plane, retively. The reference plane coincides with the scattering plafter a counterclockwise rotation by an anglef around theorientation of the light propagation. R~f! has the form of

R~f!5F 1 0 0 0

0 cos~2f! sin~2f! 0

0 2sin~2f! cos~2f! 0

0 0 0 1

G . ~2!

Journal of Biomedical Optics d October 2003 d Vol. 8 No. 4 609


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Fig. 2 (a) The geometry of multiple scattering events in a slab of turbid medium. (b) The geometry of the first scattering event during lightpropagation.

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M~Q! is the scattering matrix in the Mie regime, which hasthe form of

M~Q!5F a~Q! b~Q! 0 0

b~Q! a~Q! 0 0

0 0 d~Q! 2e~Q!

0 0 e~Q! d~Q!

G , ~3!

where the simulations ofa(Q), b(Q), d(Q), ande(Q) havebeen shown in Ref. 15. The probability density function~PDF! of polar angleQ, which determines the distribution ofQ in @0, p#, satisfies the normalization requirement13:

2pE0

p

a~Q!sin~Q!dQ51, ~4!

which shows that the intensity distribution, as a function ofQafter a single scattering event, is independent of the polarization state before the scattering. Once theQ is determined, thePDF of the polar anglef as a function of the incident StokesvectorS05@S0 ,S1 ,S2 ,S3#T can be expressed as

rQ~f!5H 11b~Q!

a~Q!

@S1 cos~2f!1S2 sin~2f!#

S0J Y 2p.

~5!

When the incident light is linearly polarized with the orienta-tion of polarization along the x-axis(S05H5@1;1;0;0#T),the Stokes vector after the single scattering can be expressas



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5F S0

S1

S2

S3

G5F a~Q!1b~Q!cos~2f!

b~Q!cos~2f!1a~Q!cos2~2f!1d~Q!sin2~2f!

b~Q!sin~2f!1@a~Q!2d~Q!#sin~2f!cos~2f!

2e~Q!sin~2f!

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When the incident light is right circularly polarized(S05R5@1;0;0;1#T), the Stokes vector after the single scatterican be expressed as

S1~f,Q!5F S0

S1

S2

S3

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b~Q!cos~2f!1e~Q!sin~2f!

b~Q!sin~2f!2e~Q!cos~2f!

d~Q!

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~7!

For an 800-nm light scattered by 380-nm-diameter postyrene microspheres, the size parameter,ka52pnba/l, is2.0 and the light scattering is in the Mie regime. The thredimensional distributions of the Stokes vectorsS1 for the Hand R incident polarization states are shown in the left cumns in Fig. 3 and Fig. 4, respectively. The intensity distbutions ofS0 are normalized at the peak point where the poangleQ50. Patterns ofS1 /S0 , S2 /S0 and S3 /S0 show thecomparative intensity distribution of the Stokes componewith respect toS0 .

For both the H and R incident polarization states, all of tlight energy propagates forward(0<Q,45 deg) after asingle scattering event. The integral ofS0 along the azimuthalanglef leads to the intensity distribution as a function of th



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Fig. 3 (Left column) Three-dimensional distributions of the Stokes components of light after a single scattering event and (right column) the spatialpatterns of the Stokes components of light transmitted through a slab of turbid medium, where the incident light is linearly polarized with theorientation of polarization along the x-axis; S1 , S2 , and S3 are normalized with respect to the intensity distribution S0 .


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Fig. 4 (Left column) Three-dimensional distributions of the Stokes components of light after a single scattering event and (right column) the spatialpatterns of the Stokes components of light transmitted through a slab of turbid medium, where the incident light is right circularly polarized; S1 , S2 ,and S3 are normalized with respect to the intensity distribution S0 .

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polar angleQ, which is constant for any incident polarizationstate. For the H incident state, the Stokes componentsS0 andS1 depend on the azimuthal anglef from 0 to 2p, while forthe R incident state, the Stokes componentsS0 and S3 areindependent off. The Stokes componentsS2 and S3 for H,andS1 andS2 for R, have cloverleaflike patterns projected on



the x-y-plane, as shown in Fig. 3 and Fig. 4. The two positleaves ~bright leaves! and the two negative leaves~darkleaves! in each pattern are symmetrically distributed arouthe center point.

Using the Stokes-Mueller formalism, we trace the evotion of the polarization state of each photon packet throu


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the successive multiplication of the matrices M~Q! and R~f!.The detailed method for tracing the photon propagation wabased on an idea in Ref. 18. The Stokes vector,Sn

f s , of atransmitted photon packet that has been scatteredn times byspherical scattering particles in a turbid medium can be brieflyexpressed as

Snf s~x,y;ms ,ma!5@ms /~ma1ms!#

n3R~fL!M~Qn!

3R~fn!...M~Q1!R~f1!S0, ~8!

wherems , ma are the scattering coefficient and the absorptioncoefficient, respectively, and~x,y! is a detection point on thelower surface of the turbid medium in the laboratory coordi-nate.@ms /(ma1ms)#n denotes the energy remaining after thephoton packet has been scatteredn times.R(f i) andM(Q i)( i 51,2,...,n) are the rotation matrix and scattering matrix forthe i ’ th scattering event.R(fL) transforms the Stokes of lightin the local coordinate system back to the Stokes in the laboratory coordinate system before the photon packet is receiveby the detector. In the MC simulation, the receiving area andthe receiving angle are the same as those in the experimen

The spatial patterns of the transmitted Stokes componenfor the H and R incident polarization states are shown in theright columns in Fig. 3 and Fig. 4, respectively, where thethicknessL of the 380-nm polystyrene-microsphere solutionsis 1 cm, the anisotropic factorg is 0.65, the scatterer concen-tration is 0.066%, the scattering coefficientms is 4.61 cm21,and the absorption coefficientma is set to be 0. The SMFPand the factorL/ l s are simulated to be 0.22 cm and 4.61,respectively. Each pattern in the x-y plane shows a0.630.6 cmarea on the output surface of the scattering mediumTo make the spatial patterns ofS0 clearer, we show the deci-bel values of the intensities. Other Stokes components,S1 ,S2 , and S3 , for each position are normalized withS0 . Thepatterns of transmitted Stokes components show differenpreferential orientations, which are mainly determined by in-cident polarization states. According to Eqs.~5! to ~7!, theintegral values of the intensityS0 in any circular area centeredat the axis of the incident light are the same for the H and Rincident polarization states, although the intensity distribu-tions of S0 for the H and R states show different preferentialorientations.

The remaining degree of polarization~DOP! and the pat-terns of the Stokes vectors of transmitted light come from thelowly scattered photon packets. Considering a scattering medium with a greater factor ofL/ l s , the ratio of lowly scatteredlight in the total transmitted light is lower. Therefore, the re-maining DOP and the spatial patterns of the Stokes vectorare weaker. However, the shapes of the patterns of the Stokcomponents remain the same for scattering media with different factors ofL/ l s . Our MC-simulated polarization patternsof transmitted light are similar to the experimental and analytical results obtained by previous researchers.19,20,16,13

We can see in Fig. 3 and Fig. 4 that projections of the 3-Dpatterns after single scattering events on the x-y-plane lead2-D polarization distributions that match well with the corre-sponding 2-D patterns of the transmitted Stokes componentThis demonstrates that the key features of 2-D spatially distributed polarization of light, transmitted through a scatteringmedium containing spherical particles, are determined by th


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first scattering event that the light encounters, while the baground level in the polarization pattern is affected by the mtiply scattering events.

In time-resolved measurements of transmitted Stokes cponents, the incident beam and the fiber bundle for detecare aligned with each other. The detected Stokes componcome from the integral of the transmitted photon packetsthe detection area((x21y2)1/2,0.5 mm) centered at the~0,0! point in the x-y-plane. We expect that the signals ofS2 andS3 for the H incident polarization state, andS1 andS2 for theR incident polarization state will be zero, because in the pterns of these Stokes components, the positive leavesnegative leaves are distributed symmetrically in respect tocenter. Therefore, for the H and R incident polarization stathe DOP of transmitted light comes entirely from Stokes coponentsS1 and S3 , respectively. When the detection areanot centered at the~0, 0! point, these Stokes components,S2 ,S3 for the H andS1 , S2 for the R, may not be zero and thevalues are highly dependent on the detection area on theplane.

Depending on the length of the propagation paths,transmitted photon packets are recorded in different unitsthe time space. Since the incident photons are launched sitaneously, we convolute the temporal profiles of the transmted Stokes vectors with the response of the detection systo the laser pulse to match the experimental measuremeThe time-resolved transmitted Stokes vector has the form

S~ t !5@S0~ t !;S1~ t !;S2~ t !;S3~ t !#T. ~9!

The time-resolved DOP is obtained from

DOP~ t !5AS1

2~ t !1S22~ t !1S3

2~ t !

S0~ t !. ~10!

4 Time-Resolved Results and DiscussionIn the following results, the 0 point on the time axis depicthe time when the light is received by the detector aftertransmits the cuvette with water. Therefore the time whenincident light pulse enters the scattering medium is2L/c,which equals288.6 ps when the thicknessL of the slab is 2cm. The temporal profiles of the Stokes components of tramitted light have been normalized with respect to the mamum value ofS0 , which makes the comparison between tMC simulations and experimental measurements possible

For a 380-nm polystyrene-microsphere solution withconcentration of 0.133%, the scattering coefficientms and theanisotropic factorg are simulated to be 9.22 cm21 and 0.65,respectively. The SMFP,l s , and the factorL/ l s are 0.11 cmand 18.44. The time-resolved Stokes componentsS0 and S1and the DOP of the transmitted light when the incident lighH state polarized are shown in Figs. 5~a!, 5~b!, and 5~c!, re-spectively. The FWHM ofS0 is 208 ps. The maximum intensity of S1 is 0.43. The time-resolved Stokes componentsS0andS3 and the DOP of the transmitted light when the incidelight is R state polarized are shown in Figs. 5~d!, 5~e!, and5~f!, respectively. The FWHM ofS0 is 206 ps. The maximumintensity of S3 is 0.58. For both the linearly and circularlincident polarization states, the experimental results~scatteredcircles!, including the Stokes components and the DOP, ma



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Fig. 5 Time-resolved Stokes components S0 (a) and S1 (b) and degree of polarization (c) of transmitted light where the incident light is linearlypolarized. Time-resolved Stokes components S0 (d) and S1 (e) and degree of polarization (f) of transmitted light where the incident light is circularlypolarized. The scattering medium is a 380-nm polystyrene-microsphere solution with a scatterer concentration of 0.133%. The circles depict theexperimental measurements and the solid lines show the MC simulations. The dashed lines in (a) and (d) show the simulation results based on thediffusion model. The order of scatters as functions of the propagation time is also shown as dashed lines in (c) and (f) for linearly and circularlyincident polarization states, respectively.

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well with the results from the MC simulations~solid lines!. InFig. 5, the Stokes componentsS2 , S3 for the H incident stateandS1 , S2 for the R incident state are not presented becausthey show very weak values compared with the intensities oS0 and can be regarded as zero. This phenomenon can bexplained by considering the symmetrical patterns of thesStokes components and the circular symmetrical distributionof the detection area around the axis of the incident lasebeam, which was discussed in Sec. 3.



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It can be observed that for linearly and circularly incidepolarization states, the time-resolved profiles ofS0 of trans-mitted light are similar, which shows that the temporal distbution of the light transmittance is independent of the incidpolarization state. According to our detection system,S0 is theintegral of the light intensity within the detection area that iscircular area centered at the~0, 0! point on the x-y-plane. Asshown in Sec. 3, although the spatial patterns ofS0 are differ-ent for various incident polarization states, the total light e


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ergies that enter the detection area are the same. We know ththe trajectories of the photon packets in the turbid medium ardetermined by both the PDF of polar angleQ and the PDF ofazimuthal anglef in successive single scattering events.However, by considering the circular symmetrical distributionof the detection area around the incident light axis, we candeduce that the detected temporal profile ofS0 is independentof the PDF off but totally determined by the PDF ofQ insuccessive single scattering events. Moreover, as we have dcussed, the intensity distribution as a function of the polarangleQ after a single scattering is independent of the polar-ization state. Therefore, with the detection geometry in theexperiment, the detected temporal profiles ofS0 of the trans-mitted light are the same for various incident polarizationstates.

By employing the radiative transfer theory based on thediffusion approximation,21 we simulate the temporal profilesof light transmittance through a slab of turbid medium tocompare with the results from the MC simulations and theexperiments. Accurate results can be obtained when the twconditions in Eq.~11! are satisfied:

m0!~12g!ms and L@ l t8 , ~11!

where l t8 is the transport mean-free-path that equals1/@ma

1(12g)ms#. Because of the low absorption of ourpolystyrene-microsphere solutions, the first condition in Eq~11! is true. For the 0.133% polystyrene-microsphere solutionL/ l t8 is about 6.5, while for the 0.066% solution,L/ l t8 is about3.2. Therefore we can expect that the diffusion model describes the temporal transmittance for the 0.133% solutionmore accurately than for the 0.066% solution.

We convolute the time-dependent transmittance based othe radiative transfer theory with the response of the detectiosystem to the laser pulse to simulate theS0 profile measuredin the experiments, which are shown in Figs. 5~a! and 5~d!with the dashed curves normalized at the peak value. Thresults from the diffusion model match the MC simulation andthe experimental results well after 250 ps. At times before 250ps, there is considerable discrepancy between them becauthe photons exiting the scattering medium at an earlier timehave not encountered multiple scattering and the distributioof their propagations is not adequately isotropic to be described by diffusion theory.

The MC-simulated average order of scatters,^N&, as afunction of the light propagation time are shown in Figs. 5~c!and 5~f! as dashed lines. After about 20 ps,^N& shows a goodlinear relationship with the propagation time, which can beexpressed as

^N&5c~ t1L/c!

l s5cS t1

L

c Dms , ~12!

where t1L/c is the time of light propagation between theincident point and the detection point. Whent is less than 20ps, the main part of the detected light comes from the ballisticand quasi-ballistic component that does not encounter adequate scatters to achieve a homogeneous distribution in thscattering medium. Therefore, during the nonlinear transitionperiod from 0 to 20 ps, the order of scatters^N& as a functionof the propagation time cannot be matched by Eq.~12!. When


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t approaches 0,N& will decrease to 0. For the H and R incdent polarization states, the temporal profiles of^N& are simi-lar.

Figures 5~c! and 5~f! show that for both linear and circulaincident polarization states, the DOP of the transmitted liwith respect to the propagation time decays exponentiallyter about 20 ps whenN& is greater than 16. In a previoupublication, Bicout et al.11 presented the exponential relationship between the DOP of light and the factord/ l s for theforward propagation mode, whered is the thickness of theturbid medium. Ambirajan and Look22 found that the degreeof linear polarization of backward-scattered light decreaexponentially with respect to the order of scatters. Becausthe linear relationship between^N& and the propagation timeof the light, the exponential decay of the DOP of the tranmitted light as a function of time can be explained.

For a 380-nm polystyrene-microsphere solution withconcentration of 0.066%, the scattering coefficientms is 4.61cm21, the SMFPl s is 0.22 cm, and the factorL/ l s is 9.22,which is half that of the 0.133% solution. The time-resolvStokes componentsS0 andS1 and the DOP of the transmittelight when the incident light is linearly polarized are shownFigs. 6~a!, 6~b!, and 6~c!, where the FWHM ofS0 is 22 ps, themaximumS1 is 0.97, and the FWHM of the DOP is 62 pThe time-resolved Stokes componentsS0 andS3 and the DOPof the transmitted light when the incident light is circularpolarized are shown in Figs. 6~d!, 6~e!, and 6~f!, where theFWHM of S0 is 20 ps, the maximumS3 is 0.98, and theFWHM of the DOP is 71 ps. The experimental results~scat-tered circles! matched well with the MC simulations~solidlines! for this 0.066% polystyrene–microspheres solution.

The FWHM ofS0 for this 0.066% solution is less than thafor the 0.133% polystyrene-microsphere solution, regardlof the incident polarization state. Because of the lower sctering coefficient of this medium, the percentage of lowscattered light in the total transmittance is higher. Most tramitted light comes from the ballistic and quasi-ballistic component, which has a comparatively shorter duration thhighly scattered light. Therefore the intensity ofS0 in Fig. 6decreases much faster than that for the 0.133% solutMoreover, since the main part ofS0 comes from the ballisticand quasi-ballistic component, the simulation result~dashedlines! based on the diffusion model does not fit with the prfiles measured in experiments.

In Figs. 6~c! and 6~f!, the temporal profiles of the DOPdecay exponentially for both linearly and circularly incidepolarization states. The DOP of the light transmitted throuthe 0.066% polystyrene-microsphere solution decays mslowly compared with the results for the 0.133% solutioThis is not surprising since light propagating through this mdium with a lower scattering coefficient encounters fewscattering events during the same propagation length. Slarly to the 0.133% solution, theN& of light transmittedthrough this solution also shows a linear relationship wrespect to the propagation time after 25 ps. For this solutthe slope of theN& as a function of time is half that for the0.133% solution because its SMFP is double that for0.133% solution.

For initially polarized light transmitted through the 380-npolystyrene-microsphere solutions, the average order of sters^N& at the time when the DOP decreases from 1 to 0.5



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Fig. 6 Time-resolved Stokes components S0 (a) and S1 (b) and degree of polarization (c) of transmitted light, where the incident light is linearlypolarized. Time-resolved Stokes components S0 (d) and S1 (e) and degree of polarization (f) of transmitted light where the incident light is circularlypolarized. The scattering medium is a 380-nm polystyrene-microsphere solution with a scatterer concentration of 0.066%. The circles depict theexperimental measurements and the solid lines show the MC simulations. The dashed lines in (a) and (d) show the simulation results based on thediffusion model. The order of scatters as functions of the propagation time is also shown as dashed lines in (c) and (f) for linearly and circularlyincident polarization states, respectively.

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more than 10, which is obviously greater than that observefor backscattered light.11,23–24 Compared with the backscat-tered light, the transmitted light in its early stage of propaga-tion encounters mainly forward scattering events with smalscattering angles. Because the DOP attenuation for smalangle scattering events is weaker than that for large-anglscattering events, it follows that the transmitted light keeps itspolarization state better than the backscattered light.



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5 ConclusionA time-resolved MC technique based on the Stokes-Mueformalism is employed in this study of initially polarized utrafast laser pulse propagation through 380-nm polystyremicrosphere solutions with different concentrations. Simlated temporal profiles of the Stokes components and the Dare compared with the measurements obtained in experim


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that employ a streak camera with a 4.74-ps resolution. A saisfactory match between them proves the accuracy of this MCalgorithm. The correlation between the 3-D Stokes components of light after the single scattering event and the spatiadistribution of the Stokes components on the output surface othe turbid medium shows that the polarization patterns otransmitted light are mainly determined by the first scatteringevent that the light encounters during propagation. The spatiadistribution of the transmitted intensity is dependent on thepolarization state of the incident light, which can be forecasthrough the PDF of the azimuth anglef in the first scatteringevent. When the detection area has a circular symmetricadistribution around the axis of the incident light, the temporalprofile of transmittance is independent of the incident polar-ization state but is determined mainly by the PDF of the polaangleQ in successive single scattering events. By consideringthe linear relationship between the average order of scatteand the light propagation time, we can interpret the exponential decay of the DOP of the transmitted light with respect tothe propagation time.

Compared with the diffusion model, simulations based onthe MC technique present a more precise match with the experimental measurements, especially when lowly scatterelight is the dominating component in the total transmittanceIn addition to the propagation of light intensity, the MC simu-lation based on the Stokes-Mueller formalism presents detailed information on the evolution of polarization in turbidmedia. This can help us understand the phenomena observin laser–tissue interactions and can potentially help us improve the quality of optical imaging that utilizes time gatingor polarization gating techniques. Because of the nature of thMC simulation, coherent phenomena cannot be modeledHowever, this simulation method can be applied in the noncoherent regime or in cases where the coherent effects aremoved.

AcknowledgmentsThis project is sponsored in part by National Institutes ofHealth grants R01 CA71980 and R21 RR15368, National Science Foundation grant BES9734491, and Texas Higher Education Coordinating Board grant 000512-0063-2001.

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Polarized light propagation through scattering media: time ...light.6–10 Polarization propagation in biological tissues is a complicated process that is fundamental to tissue optics