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Three Monte Carlo programs of polarized light transport into scattering media: part I

Jessica C. Ramella-Roman, Scott A. Prahl, and Steve L. Jacques

Open Access Open Access

Abstract

Propagation of light into scattering media is a complex problem that can be modeled using statistical methods such as Monte Carlo. Few Monte Carlo programs have so far included the information regarding the status of polarization of light before and after a scattering event. Different approaches have been followed and limited numerical values have been made available to the general public. In this paper, three different ways to build a Monte Carlo program for light propagation with polarization are given. Different groups have used the first two methods; the third method is original. Comparison in between Monte Carlo runs and Adding Doubling program yielded less than 1 % error.

©2005 Optical Society of America

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Figures (12)
Fig. 1.
Fig. 1. Flow chart of Polarized Monte Carlo program, the white cells are used in Standard Monte Carlo programs the gray cells are specific of Polarized Light Monte Carlo programs

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Fig. 2.
Fig. 2. Meridian planes geometry. The photon’s direction of propagation before and after scattering is I 1 , and I 2 respectively. The plane COA defines the meridian plane before and after scattering.

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Fig. 3.
Fig. 3. Left Fig. shows the Mie phase function for incident linear polarized light and the right Fig. is the phase function for incident circular polarized light.

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Fig. 4.
Fig. 4. Visual description of the rotations necessary to transfer the reference frame from one meridian plane to the next. Initially the electrical field E is defined with respect to the meridian plane COA.

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Fig. 5.
Fig. 5. For a scattering event to occur the Stokes must be referenced with respect to the scattering plane BOA. The electrical field E (blue lines) is rotated so that E ‖ is parallel to BOA.

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Fig. 6.
Fig. 6. After a scattering event the Stokes vector is defined respect to the plane BOA.

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Fig. 7.
Fig. 7. Second reference plane rotation. The electric field is now rotated so that E‖ is in the meridian plane COB.

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Fig. 8.
Fig. 8. The triplet w , v , u is rotated of an angle β. The rotation is about the axis u , left image, the rotated w and v vectors are shown on the right. u remains unchanged.

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Fig. 9.
Fig. 9. Second rotation on the w , v , u triplet. The rotation is of an angle α about the vector v as shown on the left. The rotated vector are shown on the right. v remains unchanged.

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Fig. 10.
Fig. 10. Rotation about an axis u of an angle ε. This rotation will bring v parallel to the Z-axis and the plane w0u in a meridian plane.

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Fig. 11.
Fig. 11. Effect of a rotation about an axis u of an angle ε on the w and v vectors. v is parallel to the Z-axis and the plane wOu is in a meridian plane.

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Fig. 12.
Fig. 12. Rotation of the photon reference frame about the Z-axis. All three vectors u , v , w are affected by this rotation. Thus the final Stokes vector is:

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Tables (2)

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Table 1. Reflectance mode, comparison between Evans adding-doubling code and the meridian plane Monte Carlo program. The results do not include the final rotation for a single detector.

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Table 2. Transmission mode, comparison between Evans adding doubling code and the meridian plane Monte Carlo program. The results are not corrected for a single detector.

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Equations (40)

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q = ( q o , q 1 , q 2 , q 3 )
q = q 0 + Γ = q 0 + i q 1 + j q 2 + k q 3
L ( T ) = q * T q
Δ s = − ln ( ζ ) μ t
x ′ = x + u x Δ s
y ′ = y + u y Δ s
z ′ = z + u z Δ s
albedo = μ s μ s + μ a
( I Q U V ) out = [ I · W Q · W U · W V · W ]
P ( α , β ) = s 11 ( α ) I o + s 12 ( α ) [ Q o cos ( 2 β ) + U o sin ( 2 β ) ]
M ( α ) = [ s 11 ( α ) s 12 ( α ) 0 0 s 12 ( α ) s 11 ( α ) 0 0 0 0 s 33 ( α ) s 34 ( α ) 0 0 − s 34 ( α ) s 33 ( α ) ]
s 11 = 1 2 ( ∣ S 2 ∣ 2 + ∣ S 1 ∣ 2 )
s 12 = 1 2 ( ∣ S 2 ∣ 2 − ∣ S 1 ∣ 2 )
s 33 = 1 2 ( S 2 * S 1 + S 2 S 1 * )
s 34 = − i 2 ( S 1 S 2 * − S 2 S 1 * )
P ( α ) = s 11 ( α )
R ( β ) = [ 1 0 0 0 0 cos ( 2 β ) sin ( 2 β ) 0 0 − sin ( 2 β ) cos ( 2 β ) 0 0 0 0 1 ]
u ̂ x = sin ( α ) cos ( β )
u ̂ y = sin ( α ) cos ( 2 β )
u ̂ z = cos ( α ) u z ∣ u z ∣
u ̂ x = 1 1 − u z 2 sin ( α ) [ u x u y cos ( β ) − u y sin ( β ) ] + u x cos ( α )
u ̂ y = 1 1 − u z 2 sin ( α ) [ u x u z cos ( β ) − u x sin ( β ) ] + u y cos ( α )
u ̂ z = 1 − u z 2 sin ( α ) cos ( β ) [ u y u z cos ( β ) − u x sin ( β ) ] + u z cos ( α )
cos γ = − u z + u ̂ z cos α ± ( 1 − cos 2 α ) ( 1 − u ̂ z 2 )
S new = R ( − γ ) M ( α ) R ( β ) S
p ″ = p · cos ( σ ) + sin ( σ ) · ( k × p ) + [ 1 − cos ( σ ) ] · ( k · p ) · k
R euler ( k , σ ) = [ k x k x v + c k y k x v − k z s k z k x v + k y s k x k y v + k z s k y k y v + c k y k z v − k x s k x k z v − k y s k y k z v + k x s k z k z v + c ]
S new = M ( α ) R ( β ) S
q β = β + u = β + i u x + j u y + k u z
q β 1 = cos ( β 2 )
q β 2 = u x sin ( β 2 )
q β 3 = u y sin ( β 2 )
q β 4 = u z sin ( β 2 )
q α = α + v = α + i v x + j v y + k v z
φ = tan − 1 ( u y u x )
φ = − tan − 1 ( u y u x )
w = v × u
ε = 0 when v z = 0 and u z = 0
ε = tan − 1 ( v z − w z ) in all other cases
S final = R ( φ ) R ( ε ) S
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