Abstract

Propagation of polarized light through a scattering medium has been studied with a Monte Carlo code to obtain polarized backscattered images. Studies of these backscattered patterns obtained with polarized illumination can be used as a technique to characterize the medium anisotropy factor g. First we present the different steps of the Monte Carlo simulation that describe polarized light propagation in a turbid medium. Monte Carlo is a good tool to simulate the backscattered polarized light but is time-consuming. Therefore, we consider two ways to decrease the computation time. The first way deals with angle sampling of the light direction. The second takes advantage of backscattered image symmetry to divide the simulation time by a factor of 4. By combining these two techniques we significantly decrease the code computation time.

© 2003 Optical Society of America

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References

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  1. R. Bolt, J. ten Bosch, “On determination of optical parameters for turbid materials,” Waves Random Media 4, 233–242 (1994).
    [CrossRef]
  2. L. Gobin, L. Blanchot, H. Saint-Jalmes, “Integrating the digitized backscattered image to measure absorption and reduced-scattering coefficients in vivo,” Appl. Opt. 38, 4217–4227 (1999).
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  3. A. H. Hielscher, J. R. Mourant, I. J. Bigio, “Influence of particle size and concentration on the diffuse backscattering of polarized light from tissue phantoms and biological cell suspensions,” Appl. Opt. 36, 125–135 (1997).
    [CrossRef] [PubMed]
  4. M. J. Raković, G. W. Kattawar, M. Mehrübeoğlu, B. D. Cameron, L. V. Wang, S. Rastegar, G. L. Coté, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. 38, 3399–3408 (1999).
    [CrossRef]
  5. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge University, New York, 1986).
  6. S. Bartel, A. H. Hielscher, “Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. 39, 1580–1588 (2000).
    [CrossRef]
  7. L. V. Wang, S. L. Jacques, L. Zheng, “Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
    [CrossRef] [PubMed]
  8. L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  9. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  10. G. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
    [CrossRef]
  11. C. Brosseau, Fundamentals of Polarized Light: a Statistical Optics Approach (Wiley, New York, 1998).
  12. A. S. Martinez, R. Maynard, “Faraday effect and multiple scattering of light,” Phys. Rev. B 50, 3714–3732 (1994).
    [CrossRef]
  13. F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
    [CrossRef]

2000 (1)

1999 (2)

1997 (1)

1995 (1)

L. V. Wang, S. L. Jacques, L. Zheng, “Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

1994 (2)

A. S. Martinez, R. Maynard, “Faraday effect and multiple scattering of light,” Phys. Rev. B 50, 3714–3732 (1994).
[CrossRef]

R. Bolt, J. ten Bosch, “On determination of optical parameters for turbid materials,” Waves Random Media 4, 233–242 (1994).
[CrossRef]

1989 (1)

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

1941 (1)

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Bartel, S.

Bigio, I. J.

Blanchot, L.

Bohren, G. F.

G. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
[CrossRef]

Bolt, R.

R. Bolt, J. ten Bosch, “On determination of optical parameters for turbid materials,” Waves Random Media 4, 233–242 (1994).
[CrossRef]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: a Statistical Optics Approach (Wiley, New York, 1998).

Cameron, B. D.

Coté, G. L.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge University, New York, 1986).

Gobin, L.

Greenstein, J. L.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Henyey, L. G.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hielscher, A. H.

Huffman, D. R.

G. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
[CrossRef]

Jacques, S. L.

L. V. Wang, S. L. Jacques, L. Zheng, “Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Kattawar, G. W.

MacKintosh, F. C.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Martinez, A. S.

A. S. Martinez, R. Maynard, “Faraday effect and multiple scattering of light,” Phys. Rev. B 50, 3714–3732 (1994).
[CrossRef]

Maynard, R.

A. S. Martinez, R. Maynard, “Faraday effect and multiple scattering of light,” Phys. Rev. B 50, 3714–3732 (1994).
[CrossRef]

Mehrübeoglu, M.

Mourant, J. R.

Pine, D. J.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge University, New York, 1986).

Rakovic, M. J.

Rastegar, S.

Saint-Jalmes, H.

ten Bosch, J.

R. Bolt, J. ten Bosch, “On determination of optical parameters for turbid materials,” Waves Random Media 4, 233–242 (1994).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge University, New York, 1986).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge University, New York, 1986).

Wang, L. V.

Weitz, D. A.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Zheng, L.

L. V. Wang, S. L. Jacques, L. Zheng, “Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Zhu, J. X.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Appl. Opt. (4)

Astrophys. J. (1)

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Comput. Methods Programs Biomed. (1)

L. V. Wang, S. L. Jacques, L. Zheng, “Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Phys. Rev. B (2)

A. S. Martinez, R. Maynard, “Faraday effect and multiple scattering of light,” Phys. Rev. B 50, 3714–3732 (1994).
[CrossRef]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Waves Random Media (1)

R. Bolt, J. ten Bosch, “On determination of optical parameters for turbid materials,” Waves Random Media 4, 233–242 (1994).
[CrossRef]

Other (4)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge University, New York, 1986).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

G. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
[CrossRef]

C. Brosseau, Fundamentals of Polarized Light: a Statistical Optics Approach (Wiley, New York, 1998).

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Figures (10)

Fig. 1
Fig. 1

Local frame rotations that show the new direction of the photon.

Fig. 2
Fig. 2

Local frame rotation to the detector normal direction.

Fig. 3
Fig. 3

Final local frame rotation to the detector frame.

Fig. 4
Fig. 4

Mie distribution S(θ, ϕ) and the constant comparison function (straight line) drawn as a function of θ with ϕ = 0.

Fig. 5
Fig. 5

Comparison function: m 11(θ) + |m 12(θ)| (thick curve) compared with Mie distribution m 11(θ) + m 12(θ)(Q o cos 2ϕ + U o sin 2ϕ)/I o with Q o /I o = -1, U o /I o = 0, and ϕ = 0 as a function of θ.

Fig. 6
Fig. 6

Simulation time by use of the Henyey-Greenstein function for sampling θ and a uniform sampling for ϕ divided by time by use of the rejection method with m 11(θ) + |m 12(θ)| as the comparison function.

Fig. 7
Fig. 7

Symmetry of the Stokes elements Q and U for a (1100) incident Stokes vector. The white and dark areas represent positive and negative intensities, respectively. The parallel axis is the horizontal axis.

Fig. 8
Fig. 8

Symmetry of the Stokes element Q for a (1100) incident Stokes vector. The dark and white areas represent negative and positive intensities, respectively.

Fig. 9
Fig. 9

Backscattered Stokes vector with a (1000) incident Stokes vector (the parallel axis is the horizontal axis). From left to right, I, Q, U, and V are, respectively, normalized by the maximum of I. The medium parameters are μ s = 15.5 cm-1, μ a = 0.01 cm-1, and g = 0.80. The thickness is 2 cm, the matrix size is 60 × 60, the pixel size is 670 μm, and the gray-level scale ranges from -0.01 to 0.01.

Fig. 10
Fig. 10

Simulation time ratio of the rejection method for different anisotropy factors between the constant comparison function m 11(0) and m 11(θ) + |m 12(θ)|.

Equations (18)

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S=IQUV=EE*+EE*EE*-EE*EE*+EE*iEE*-EE*
Ss=MθRϕSi,
Rϕ=10000cos 2ϕsin 2ϕ00-sin 2ϕcos 2ϕ00001,
Mθ=m11θm12θ00m12θm11θ0000m33θm34θ00-m34θm33θ,
I=m11θI+m12θQ cos2ϕ+U sin2ϕ,
refθi, θt=a+ba-b00a-ba+b0000c0000c
a= 12|tanθi-θt|2|tanθi+θt|2, b= 12|sinθi-θt|2|sinθi+θt|2, c= sinθi-θttanθi-θtsinθi+θttanθi+θt
transθi, θt=a+ba-b00a-ba+b0000c0000c
a= 12|2 sin θi cos θt|2|sinθi+θtcosθi-θt|2, b= 12|2 sin θt cos θi|2|sinθi+θt|2, c= 4 sin2 θtcos2 θisin2θi+θtcosθi-θt.
Sr=refθi, θtRϕSi,
St=transθi, θtRϕSi.
rayθ=1+cos2 θ2cos2 θ-1200cos2 θ-121+cos2 θ20000cos θ0000cos θ.
So=wRϕorayθS1.
Sθ, ϕ=m11θ+m12θQo cos2ϕ+Uo sin2ϕ/Io
fθ=m11θ+|m12θ|
Fθ=0θ fθsinθdθ.
θ=F-1I1,
I3=random×fθo.

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