Abstract

A Monte Carlo method based on tracing the multiply scattered electric field is presented to simulate the propagation of polarized light in turbid media. Multiple scattering of light comprises a series of updates of the parallel and perpendicular components of the complex electric field with respect to the scattering plane by the amplitude scattering matrix and rotations of the local coordinate system spanned by the unit vectors in the directions of the parallel and perpendicular electric field components and the propagation direction of light. The backscattering speckle pattern and the backscattering Mueller matrix of an aqueous suspension of polystyrene spheres in a slab geometry are computed using this Electric Field Monte Carlo (EMC) method. An efficient algorithm computing the Mueller matrix in the pure backscattering direction is detailed in the paper.

© 2004 Optical Society of America

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References

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Appl. Opt. (7)

Inverse Problems (1)

S. R. Arridge, �??Optical tomography in medical imaging,�?? Inverse Problems 15, R41�??R93 (1999).
[CrossRef]

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 (2)

J. Quant. Spectrosc. Radiat. Transfer (1)

K. F. Evans and G. L. Stephens, �??A new polarized atmospheric radiative transfer model,�?? J. Quant. Spectrosc. Radiat. Transfer 46, 413�??423 (1991).
[CrossRef]

J. Res. Natl. Bur. Stand. (1)

J. von Neumann, �??Various techniques used in connection with random digits,�?? J. Res. Natl. Bur. Stand. 5, 36�??38 (1951).

Laser speckle and related phenomena (1)

J. W. Goodman, �??Statistical properties of laser speckle patterns,�?? in Laser speckle and related phenomena, J. C. Dainty, ed., pp. 9�??75 (Springer-Verlag, Berlin, 1975).
[CrossRef]

Opt. Exp. (1)

I. Berezhnyy and A. Dogariu, �??Time-resolved Mueller matrix imaging polarimetry,�?? Opt. Exp. 12(19), 4635�??4649 (2004).
[CrossRef]

Opt. Lett. (1)

Opt. Photon. News (1)

S. K. Gayen and R. R. Alfano, �??Emerging optical biomedical imaging techniques,�?? Opt. Photon. News 7 (3), 17�??22 (1996).

Ph.D. thesis (1)

J. C. Ramella-Roman, �??Imaging skin pathologies with polarized light: empirical and theoretical studies,�?? Ph.D. thesis, OGI School of Science & Engineering at Oregon Health & Science University (2004).

Phys. Rev. (1)

D. S. Saxon, �??Tensor Scattering Matrix for the Electromagnetic Field,�?? Phys. Rev. 100(6), 1771�??1775 (1955).
[CrossRef]

Phys. Rev. Lett. (3)

P.-E. Wolf and G. Maret, �??Weak Localization and Coherent Backscattering of Photons in Disordered Media,�?? Phys. Rev. Lett. 55(24), 2696�??2699 (1985).
[CrossRef]

M. P. V. Albada and A. Lagendijk, �??Observation of Weak Localization of Light in a Random Medium,�?? Phys. Rev. Lett. 55(24), 2692�??2695 (1985).
[CrossRef]

E. Akkermans, P. E. Wolf, and R. Maynard, �??Coherent backscattering of light by disordered media: analysis of the peak line shape,�?? Phys. Rev. Lett. 56(14), 1471�??1474 (1986).
[CrossRef]

Phys. Today (1)

A. Yodh and B. Chance, �??Spectroscopy and imaging with diffusing light,�?? Phys. Today 48 3, 38�??40 (1995).

SIAM J. Sci. Comput. (1)

A. D. Kim and M. Moscoso, �??Chebyshev Spectral methods for radiative transfer,�?? SIAM J. Sci. Comput. 23, 2074�??2094 (2002).
[CrossRef]

TOPS Adv. in Opt. Imag. and Phot. Migr. (1)

G. W. Kattawar, M. J. Rakovi�?, and B. D. Cameron, �??Laser backscattering polarization patterns from turbid media: theory and experiments,�?? in Advances in Optical Imaging and Photon Migration, J. G. Fujimoto and M. S. Patterson, eds., vol. 21 of OSA TOPS, pp. 105�??110 (1998).

Other (8)

A. Ishimaru, Wave propagation and scattering in random media, I and II (Academic, New York, 1978).

S. Chandrasekhar, Radiative transfer (Dover, New York, 1960).

EMC is available at <a href="http://www.sci.ccny.cuny.edu/~minxu">http://www.sci.ccny.cuny.edu/~minxu</a>.

H. C. van de Hulst, Light scattering by small particles (Dover, New York, 1981).

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 1983).

R. Y. Rubinstein, Simulation and the Monte Carlo method (John Wiley & Sons, 1981).
[CrossRef]

W. H. Press, S. A. Teukolsky,W. T. Vetterling, and B. P. Flannery, Numerical recipes in C (Cambridge University press, 1996).

I. Lux and L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).

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Figures (4)

Fig. 1.
Fig. 1.

A photon moving along s is scattered to s′ with a scattering angle θ and an azimuthal angle ϕ inside a local coordinate system spanned by orthonormal bases (m,n, s). e 1,2 and e′ 1,2 are the unit vectors parallel and perpendicular to the current scattering plane spanned by s and s′ prior to and after scattering. The local coordinate system (m,n, s) is rotated to (m′,n′, s′) after scattering.

Fig. 2.
Fig. 2.

(a) Speckle pattern formed by the angular-resolved backscattering light. (b) Normalized speckle Ix /〈Ix 〉 follows a negative exponential distribution.

Fig. 3.
Fig. 3.

Backscattering Mueller matrix for the slab. All 4×4 matrix element are displayed as a two-dimensional image of the surface, 20ls ×20ls in size, with the laser being incident in the center. The displayed Mueller matrix has been normalized by the maximum light intensity of the (1,1) element. The parameters of the slab is given in the text.

Fig. 4.
Fig. 4.

Reduced backscattering Mueller matrix for the slab. All 4×4 elements of the reduced Mueller matrix is displayed as a one-dimensional curve versus the distance ρ/ls from the origin. The reduced backscattering Mueller matrix is 2×2 block diagonal.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

s = m sin θ cos ϕ + n sin θ sin ϕ + s cos θ
e 1 = m cos ϕ + n sin ϕ
e 2 = m sin ϕ + n cos ϕ
e 1 = μ m cos ϕ + μ n sin ϕ sin θ s ,
( m ' n ' s ' ) = A ( m n s )
A = ( cos θ cos ϕ cos θ sin ϕ sin θ sin ϕ cos ϕ 0 sin θ cos ϕ sin θ sin ϕ cos θ ) ,
( E 1 E 2 ) = B ( E 1 E 2 )
B = [ F ( θ , ϕ ) ] 1 2 ( S 2 cos ϕ S 2 sin ϕ S 1 sin ϕ S 1 cos ϕ ) .
F ( θ , ϕ ) = ( S 2 2 cos 2 ϕ + S 1 2 sin 2 ϕ ) E 1 2 + ( S 2 2 sin 2 ϕ + S 1 2 cos 2 ϕ ) E 2 2
+ 2 ( S 2 2 S 1 2 ) cos ϕ sin ϕ [ E 1 ( E 2 ) * ]
p ( θ ) = 0 2 π p ( θ , ϕ ) d ϕ = S 2 2 + S 1 2 Q sca x 2
M bs ( ρ , ϕ ) = R ( ϕ ) M bs ( ρ , ϕ = 0 ) R ( ϕ )
R ( ϕ ) = ( 1 0 0 0 0 cos 2 ϕ sin 2 ϕ 0 0 sin 2 ϕ cos 2 ϕ 0 0 0 0 1 ) .
I o = M 0 bs ( ρ ) I i
I i ( I i ) T = R ( ϕ ) I i I i T R ( ϕ ) = R ( ϕ ) ( 1 0 0 0 0 1 2 0 0 0 0 1 4 0 0 0 0 1 4 ) R ( ϕ ) .
D = [ I i ( I i ) T ] 1 = ( 1 0 0 0 0 3 cos 4 ϕ sin 4 ϕ 0 0 sin 4 ϕ 3 + cos 4 ϕ 0 0 0 0 4 ) ,
M 0 bs ( ρ ) = I ' o ( I ' i ) T D

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