Abstract

This paper deals with an efficient and accurate simulation algorithm to solve the vector Boltzmann equation for polarized light transport in scattering media. The approach is based on a stencil method, which was previously developed for unpolarized light scattering and proved to be much more efficient (speedup factors of up to 10 were reported) than the classical Monte Carlo while being equally accurate. To validate what we believe to be the new stencil method, a substrate composed of spherical non-absorbing particles embedded in a non-absorbing medium was considered. The corresponding single scattering Mueller matrix, which is required to model scattering of polarized light, was determined based on the Lorenz–Mie theory. From simulations of a reflected polarized laser beam, the Mueller matrix of the substrate was computed and compared with an established reference. The agreement is excellent, and it could be demonstrated that a significant speedup of the simulations is achieved due to the stencil approach compared with the classical Monte Carlo.

© 2010 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2009 (1)

2007 (1)

P. Jenny, S. Mourad, T. Stamm, M. Vöge, and K. Simon, “Computing light statistics in heterogeneous media based on a mass weighted probability density function method,” J. Opt. Soc. Am. A. 24, 2206–2219 (2007).
[CrossRef]

2006 (2)

P. Jenny, M. Vöge, S. Mourad, and T. Stamm, “Modeling light scattering in paper for halftone print,” in Proceedings of the CGIV (IS&T, 2006), pp. 443–447.

R. Lawless, Y. Xie, P. Yang, G. W. Kattawar, and I. Laszlo, “Polarization and effective Mueller matrix for multiple scattering of light by nonspherical ice crystals,” Opt. Express 14, 6381–6393 (2006).
[CrossRef] [PubMed]

2004 (2)

2003 (1)

2002 (1)

X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: a Monte Carlo study,” J. Biomed. Opt. 7, 279–290 (2002).
[CrossRef] [PubMed]

2001 (1)

1999 (1)

1996 (1)

S. K. Gayen and R. R. Alfano, “Emerging optical biomedical imaging techniques,” Opt. Photonics News 7, 16–22 (1996).
[CrossRef]

1991 (1)

1983 (1)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

1979 (1)

W. J. Wiscombe, “Mie scattering calculations: advances in technique and fast, vector-speed computer codes,” Tech. Note NCAR/TN-140+STR (National Center for Atmospheric Research, 1979); ftp://climate.gsfc.nasa.gov/pub/wiscombe/Single_Scatt/Homogen_Sphere/Exact_Mie/NCARMieReport.pdf.

1957 (1)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

Alfano, R. R.

S. K. Gayen and R. R. Alfano, “Emerging optical biomedical imaging techniques,” Opt. Photonics News 7, 16–22 (1996).
[CrossRef]

Baum, B. A.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Cameron, B. D.

Coté, G. L.

Drevillon, B.

Du, H.

Gayen, S. K.

S. K. Gayen and R. R. Alfano, “Emerging optical biomedical imaging techniques,” Opt. Photonics News 7, 16–22 (1996).
[CrossRef]

Hostetler, C. A.

Hu, Y. X.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Jenny, P.

M. Šormaz, T. Stamm, S. Mourad, and P. Jenny, “Stochastic modeling of light scattering with fluorescence using a Monte Carlo-based multiscale approach,” J. Opt. Soc. Am. A 26, 1403–1413 (2009).
[CrossRef]

P. Jenny, S. Mourad, T. Stamm, M. Vöge, and K. Simon, “Computing light statistics in heterogeneous media based on a mass weighted probability density function method,” J. Opt. Soc. Am. A. 24, 2206–2219 (2007).
[CrossRef]

P. Jenny, M. Vöge, S. Mourad, and T. Stamm, “Modeling light scattering in paper for halftone print,” in Proceedings of the CGIV (IS&T, 2006), pp. 443–447.

Kaplan, B.

Kattawar, G. W.

Laszlo, I.

Lawless, R.

Ledanois, G.

Mehrubeoglu, M.

Mourad, S.

M. Šormaz, T. Stamm, S. Mourad, and P. Jenny, “Stochastic modeling of light scattering with fluorescence using a Monte Carlo-based multiscale approach,” J. Opt. Soc. Am. A 26, 1403–1413 (2009).
[CrossRef]

P. Jenny, S. Mourad, T. Stamm, M. Vöge, and K. Simon, “Computing light statistics in heterogeneous media based on a mass weighted probability density function method,” J. Opt. Soc. Am. A. 24, 2206–2219 (2007).
[CrossRef]

P. Jenny, M. Vöge, S. Mourad, and T. Stamm, “Modeling light scattering in paper for halftone print,” in Proceedings of the CGIV (IS&T, 2006), pp. 443–447.

Rakovic, M. J.

Rastegar, S.

Simon, K.

P. Jenny, S. Mourad, T. Stamm, M. Vöge, and K. Simon, “Computing light statistics in heterogeneous media based on a mass weighted probability density function method,” J. Opt. Soc. Am. A. 24, 2206–2219 (2007).
[CrossRef]

Šormaz, M.

Stamm, T.

M. Šormaz, T. Stamm, S. Mourad, and P. Jenny, “Stochastic modeling of light scattering with fluorescence using a Monte Carlo-based multiscale approach,” J. Opt. Soc. Am. A 26, 1403–1413 (2009).
[CrossRef]

P. Jenny, S. Mourad, T. Stamm, M. Vöge, and K. Simon, “Computing light statistics in heterogeneous media based on a mass weighted probability density function method,” J. Opt. Soc. Am. A. 24, 2206–2219 (2007).
[CrossRef]

P. Jenny, M. Vöge, S. Mourad, and T. Stamm, “Modeling light scattering in paper for halftone print,” in Proceedings of the CGIV (IS&T, 2006), pp. 443–447.

van de Hulst, H. C.

Vöge, M.

P. Jenny, S. Mourad, T. Stamm, M. Vöge, and K. Simon, “Computing light statistics in heterogeneous media based on a mass weighted probability density function method,” J. Opt. Soc. Am. A. 24, 2206–2219 (2007).
[CrossRef]

P. Jenny, M. Vöge, S. Mourad, and T. Stamm, “Modeling light scattering in paper for halftone print,” in Proceedings of the CGIV (IS&T, 2006), pp. 443–447.

Wang, L. V.

Wang, R. T.

Wang, X.

X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: a Monte Carlo study,” J. Biomed. Opt. 7, 279–290 (2002).
[CrossRef] [PubMed]

Wei, H.

Winker, D. M.

Wiscombe, W. J.

W. J. Wiscombe, “Mie scattering calculations: advances in technique and fast, vector-speed computer codes,” Tech. Note NCAR/TN-140+STR (National Center for Atmospheric Research, 1979); ftp://climate.gsfc.nasa.gov/pub/wiscombe/Single_Scatt/Homogen_Sphere/Exact_Mie/NCARMieReport.pdf.

Xie, Y.

Xu, M.

Yang, P.

Appl. Opt. (5)

J. Biomed. Opt. (1)

X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: a Monte Carlo study,” J. Biomed. Opt. 7, 279–290 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. A. (1)

P. Jenny, S. Mourad, T. Stamm, M. Vöge, and K. Simon, “Computing light statistics in heterogeneous media based on a mass weighted probability density function method,” J. Opt. Soc. Am. A. 24, 2206–2219 (2007).
[CrossRef]

Opt. Express (2)

Opt. Photonics News (1)

S. K. Gayen and R. R. Alfano, “Emerging optical biomedical imaging techniques,” Opt. Photonics News 7, 16–22 (1996).
[CrossRef]

Other (4)

P. Jenny, M. Vöge, S. Mourad, and T. Stamm, “Modeling light scattering in paper for halftone print,” in Proceedings of the CGIV (IS&T, 2006), pp. 443–447.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

W. J. Wiscombe, “Mie scattering calculations: advances in technique and fast, vector-speed computer codes,” Tech. Note NCAR/TN-140+STR (National Center for Atmospheric Research, 1979); ftp://climate.gsfc.nasa.gov/pub/wiscombe/Single_Scatt/Homogen_Sphere/Exact_Mie/NCARMieReport.pdf.

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Figures (10)

Fig. 1
Fig. 1

Sketch of a single scattering event.

Fig. 2
Fig. 2

Illustration of two stencils. Shown are the positions of equally probable realizations after the times (a) Δ t = l s and (b) Δ t = 5 l s .

Fig. 3
Fig. 3

Aligning stencil with propagation direction and major axis of polarization ellipse.

Fig. 4
Fig. 4

Coefficients of backscattering Mueller matrix M b s calculated with the stencil method.

Fig. 5
Fig. 5

Same as Fig. 4, but enlarged around center.

Fig. 6
Fig. 6

Quantitative comparison between backscattering Mueller matrices calculated with stencil and Monte Carlo methods. The grayscale values represent the normalized difference between the two methods.

Fig. 7
Fig. 7

Coefficients of backscattering Mueller matrix M b s calculated with the stencil method. Inclination of incident laser beam is 45°.

Fig. 8
Fig. 8

Same as Fig. 7, but enlarged around center.

Fig. 9
Fig. 9

Quantitative comparison between backscattering Mueller matrices calculated with stencil and Monte Carlo methods. Inclination of incident laser beam is 45°. The grayscale values represent the normalized difference between the two methods.

Fig. 10
Fig. 10

Computational cost of Monte Carlo and stencil methods as a function of mean free path lengths.

Tables (4)

Tables Icon

Table 1 Test Cases to Validate Implementation of Lorenz–Mie Theory a

Tables Icon

Table 2 Extinction and Scattering Efficiencies Computed with Lorenz–Mie Theory

Tables Icon

Table 3 Scattering Amplitude Functions S 1 ( 0 ) = S 2 ( 0 ) Computed with Lorenz–Mie Theory

Tables Icon

Table 4 Scattering Amplitude Functions S 1 ( π ) = S 2 ( π ) Computed with Lorenz–Mie Theory

Equations (43)

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d I ( x , s , λ , t ) d s = γ t I ( x , s , λ , t ) + γ s 4 π 4 π M ( s , s ) I ( x , s , λ , t ) d ω ,
I = E l E l + E r E r = a l 2 + a r 2 ,
Q = E l E l E r E r = a l 2 a r 2 ,
U = E l E r + E l E r = 2 a l a r   cos   δ ,
V = i ( E l E r E l E r ) = 2 a l a r   sin   δ ,
( E l new E r new ) = [ F ( θ , ϕ ) ] 1 / 2 ( S 2   cos   ϕ S 2   sin   ϕ S 1   sin   ϕ S 1   cos   ϕ ) ( E l old E r old ) ,
| E l old | 2 + | E r old | 2 = | E l new | 2 + | E r new | 2 .
F ( θ , ϕ ) = { ( | S 2 | 2 cos 2 ϕ + | S 1 | 2 sin 2 ϕ ) | E l old | 2 + ( | S 2 | 2 sin 2 ϕ + | S 1 | 2 cos 2 ϕ ) | E r old | 2 + 2 ( | S 2 | 2 | S 1 | 2 ) cos   ϕ   sin   ϕ   Re ( E l old E r , old ) } / ( | E l old | 2 + | E r old | 2 ) .
Q = I   cos ( 2 χ ) cos ( 2 ψ ) ,
U = I   cos ( 2 χ ) sin ( 2 ψ ) ,
V = I   sin ( 2 χ ) .
δ = ± π / 2 ,
a l = 1 + cos ( 2 χ init ( j ) ) 2 ,
a r = 1 a l 2 ,
M b s ( ρ , ϕ ) = I o I i T I i I i T ,
I i I i T = ( 1 0 0 0 0 1 / 2 0 0 0 0 1 / 4 0 0 0 0 1 / 4 ) ,
S 1 ( θ ) = n = 1 2 n + 1 n ( n + 1 ) { a n π n ( cos   θ ) + b n τ n ( cos   θ ) } ,
S 2 ( θ ) = n = 1 2 n + 1 n ( n + 1 ) { b n π n ( cos   θ ) + a n τ n ( cos   θ ) } .
n max = x + 4 x 1 / 3 + 2
π n ( cos   θ ) = d P n ( cos   θ ) d   cos   θ ,
τ n ( cos   θ ) = cos   θ π n ( cos   θ ) ( 1 cos 2 θ ) d π n ( cos   θ ) d   cos   θ .
π n = 2 n 1 n 1 cos   θ π n 1 n n 1 π n 2 ,
τ n = n   cos   θ π n ( n + 1 ) π n 1 ,
π 0 = 0 ,     π 1 = 1 ,     π 2 = 3   cos   θ ,     τ 0 = 0 ,     τ 1 = cos   θ ,
τ 2 = 3   cos ( 2 θ ) .
a n = m 2 j n ( m x ) [ x j n ( x ) ] j n ( x ) [ m x j n ( m x ) ] m 2 j n ( m x ) [ x h n ( 1 ) ( x ) ] h n ( 1 ) ( x ) [ m x j n ( m x ) ] ,
b n = j n ( m x ) [ x j n ( x ) ] j n ( x ) [ m x j n ( m x ) ] j n ( m x ) [ x h n ( 1 ) ( x ) ] h n ( 1 ) ( x ) [ m x j n ( m x ) ] ,
a n = [ D n ( m x ) / m + n / x ] x j n ( x ) x j n 1 ( x ) [ D n ( m x ) / m + n / x ] x h n ( 1 ) ( x ) x h n 1 ( 1 ) ( x ) ,
b n = [ D n ( m x ) m + n / x ] x j n ( x ) x j n 1 ( x ) [ D n ( m x ) m + n / x ] x h n ( 1 ) ( x ) x h n 1 ( 1 ) ( x ) .
h n ( 1 ) ( z ) = j n ( z ) + i y n ( z ) .
f n 1 ( z ) + f n + 1 ( z ) = 2 n + 1 z f n ( z )
j 0 ( z ) = sin ( z ) / z ,     j 1 ( z ) = sin ( z ) / z 2 cos ( z ) / z ,
y 0 ( z ) = cos ( z ) / z ,     y 1 ( z ) = cos ( z ) / z 2 sin ( z ) / z .
D n ( z ) = [ m x j n ( m x ) ] m x j n ( m x ) ,
D n 1 ( z ) = n z 1 D n ( z ) + n / z ,
D n start ( z ) = 0 ,     n start = max ( n max , abs ( z ) ) + 16.
p ( θ , ϕ ) = P 11 ( θ ) + P 12 ( θ ) [ Q   cos ( 2 ϕ ) + U   sin ( 2 ϕ ) ] / I ,
P 11 ( θ ) = 2 ( | S 2 | 2 + | S 1 | 2 ) / ( Q sca x 2 ) ,
P 12 ( θ ) = 2 ( | S 2 | 2 | S 1 | 2 ) / ( Q sca x 2 ) .
1 2 0 π P 11 ( θ ) sin ( θ ) d θ = 1.
1 2 0 θ P 11 ( θ ) sin ( θ ) d θ = ξ .
p ( ϕ θ ) = 1 + P 12 ( θ ) P 11 ( θ ) [ Q   cos ( 2 ϕ ) + U   sin ( 2 ϕ ) ] I ,
0 ϕ p ( ϕ θ ) d ϕ = ξ .

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