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Electric field Monte Carlo simulation of polarized light propagation in turbid media

Min Xu

Open Access Open Access

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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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.

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Fig. 2.
Fig. 2. (a) Speckle pattern formed by the angular-resolved backscattering light. (b) Normalized speckle Ix /〈Ix 〉 follows a negative exponential distribution.

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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.

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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.

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Equations (17)

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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 ℜ [ 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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