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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References

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  1. S. Chandrasekhar, Radiative Transfer, (Oxford Clarendon Press; 1950).
  2. S. Chandrasekhar and D. Elbert, �??The Illumination and polarization of the sunlight sky on Rayleigh scattering,�?? Trans. Am. Phil. Soc. 44, (1956).
  3. K. F. Evans and G. L. Stephens, �??A new polarized atmospheric radiative transfer model,�?? J. Quant. Spectrosc. Radiat Transfer. 46, 413-423, (1991).
    [CrossRef]
  4. S. A. Prahl, M. J. C. van Gemert, A. J. Welch, �??Determining the optical properties of turbid media by using the adding-doubling method,�?? Appl. Opt. 32, 559-568, (1993).
    [CrossRef] [PubMed]
  5. G. W. Kattawar and G. N. Plass �??Radiance and polarization of multiple scattered light from haze and clouds,�?? Appl. Opt. 7, 1519-1527, (1967).
    [CrossRef]
  6. G. W. Kattawar and G. N. Plass, �??Degree and direction of polarization of multiple scattered light. 1: Homogeneous cloud layers,�?? Appl. Opt. 11, 2851-2865, (1972).
    [CrossRef] [PubMed]
  7. G. W. Kattawar and C. N. Adams �??Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,�?? Limnol. Oceanogr. 34, 1453-1472, (1989).
    [CrossRef]
  8. P. Bruscaglione, G. Zaccanti, W. Qingnong �??Transmission of a pulsed polarized light beam through thick turbid media: numerical results,�?? Appl. Opt. 32, 6142-6150, (1993).
    [CrossRef]
  9. S. Bianchi, A. Ferrara, C. Giovannardi, �??Monte Carlo simulations of dusty spiral galaxies: extinction and polarization properties,�?? American Astronomical Society 465, 137-144, (1996).
  10. S. Bianchi Estinzione e polarizzazione della radiazione nelle galassie a spirale, Tesi di Laura (in Italian), 1994.
  11. A. Ambirajan and D.C. Look, �??A backward Monte Carlo study of the multiple scattering of a polarized laser beam,�?? J. Quant. Spectrosc. Radiat. Transfer 58, 171-192, (1997).
    [CrossRef]
  12. A. S. Martinez and R. Maynard �??Polarization Statistics in Multiple Scattering of light: a Monte Carlo approach,�?? in Localization and Propagation of classical waves in random and periodic structures (Plenum Publishing Corporation New York, 1993).
  13. A. S. Martinez Statistique de polarization et effet Faraday en diffusion multiple de la lumiere Ph.D. Thesis (in French and English), 1984.
  14. S. Bartel and A. Hielsher, �??Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,�?? Appl. Opt. 39, 1580-1588, (2000).
    [CrossRef]
  15. B. D. Cameron, M. J. Rakovic, M. Mehrubeoglu, G. Kattawar, S. Rastegar, L.-H. Wang, G. L. Cote, �??Measurement and calculation of the two-dimensional backscattering Mueller matrix of a turbid medium,�?? Opt. Lett. 23, 485-487, (1998).
    [CrossRef]
  16. Daniel Côté, I. Alex Vitkin �??Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations,�?? Opt. Express 13, 148-163, (2005).
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  17. 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]
  18. M. J. Rakovic, G. W. Kattawar, M. Mehrubeoglu, B. D. Cameron, L.-H. Wang, S. Rastegar, G. L. Cote, �??Light backscattering polarization patterns from turbid media: theory and experiment,�?? Appl. Opt. 38, 3399-3408, (1999).
    [CrossRef]
  19. H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, �??Monte Carlo and Multicomponent Approximation Methods for Vector Radiative Transfer by use of Effective Mueller Matrix Calculations,�?? Appl. Opt. 40, 400�??412, (2001).
    [CrossRef]
  20. B. Kaplan, G. Ledanois, and B. Villon, �??Mueller Matrix of Dense Polystyrene Latex Sphere Suspensions: Measurements and Monte Carlo Simulation,�?? Appl. Opt. 40, 2769�??2777, (2001).
    [CrossRef]
  21. I. Lux and L.Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Prss, Boca Raton, Fla., 1991).
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    [CrossRef]
  23. <a href= "http://omlc.ogi.edu/software/polarization/">http://omlc.ogi.edu/software/polarization/</a>
  24. N. Metropolis and S. Ulam, �??The Monte Carlo method,�?? J. Am. Stat. Assoc. 44, 335-341, (1949).
    [PubMed]
  25. S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, �??A Monte Carlo model of light propagation in tissue,�?? in Dosimetry of Laser Radiation in Medicine and Biology, G. Mueller and D. Sliney, eds., SPIE IS 5, 102-111, (1989).
  26. L. H Wang, S. L. Jacques, L-Q Zheng, �??MCML �?? Monte Carlo modeling of photon transport in multilayered tissues,�?? Computer Methods and Programs in Biomedicine 47, 131-146, (1995).
    [CrossRef] [PubMed]
  27. A. N. Witt, �??Multiple scattering in reflection nebulae I: A Monte Carlo approach,�?? Astophys. J. Suppl. Ser. 35, 1-6, (1977).
    [CrossRef]
  28. J. J. Craig, Introduction to robotics. Mechanics and controls, (Addison-Weseley Publishing Company, 1986).
  29. D. Benoit, D. Clary �??Quaternion formulation of diffusion quantum Monte Carlo for the rotation of rigid molecules in clusters,�?? J. Chem. Phys. 113, 5193-5202, (2000).
    [CrossRef]
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    [CrossRef]
  31. J. B. Kuipers, Quaternions and rotation sequences. A primer with applications to orbits, aerospace, and virtual reality (Princeton University Press, 1998)
    [PubMed]
  32. C. Bohren and D. R. Huffman, Absorption and scattering of light by small particles, (Wiley Science Paperback Series, 1998).
    [CrossRef]
  33. <a href= "http://omlc.ogi.edu/software/mie/index.html">http://omlc.ogi.edu/software/mie/index.html</a>
  34. K. I. Hopcraft, P. C. Y. Chang, J. G. Walker, E. Jakeman, �??Properties of Polarized light-beam multiply scattered by Rayleigh medium,�?? in Light Scattering from Microstructure F. Moreno and F. Gonzales eds., Vol. 534, Lecture Notes in Physics, (Springer-Verlag, 2000), pp. 135-158.
    [CrossRef]
  35. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, the art of Scientific Computing, (Cambridge University Press, 1992).
  36. J. W. Hovenier, �??Symmetry Relationships for Scattering of Polarized Light in a Slab of Randomly Oriented Particles,�?? Journal of the Atmospheric Sciences 26, 488-499, (1968).
    [CrossRef]
  37. K. Shoemake, �??Animating rotation with quaternion curves,�?? Computer Graphics 19, 245-254, (1985).
    [CrossRef]
  38. J.C. Ramella-Roman, S.A.Prahl, S.L. Jacques, are preparing a manuscript to be called �??Three Monte Carlo programs of polarized light transport into scattering media: part II�??.

American Astronomical Society (1)

S. Bianchi, A. Ferrara, C. Giovannardi, �??Monte Carlo simulations of dusty spiral galaxies: extinction and polarization properties,�?? American Astronomical Society 465, 137-144, (1996).

Appl. Opt. (8)

P. Bruscaglione, G. Zaccanti, W. Qingnong �??Transmission of a pulsed polarized light beam through thick turbid media: numerical results,�?? Appl. Opt. 32, 6142-6150, (1993).
[CrossRef]

S. A. Prahl, M. J. C. van Gemert, A. J. Welch, �??Determining the optical properties of turbid media by using the adding-doubling method,�?? Appl. Opt. 32, 559-568, (1993).
[CrossRef] [PubMed]

S. Bartel and A. Hielsher, �??Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,�?? Appl. Opt. 39, 1580-1588, (2000).
[CrossRef]

M. J. Rakovic, G. W. Kattawar, M. Mehrubeoglu, B. D. Cameron, L.-H. Wang, S. Rastegar, G. L. Cote, �??Light backscattering polarization patterns from turbid media: theory and experiment,�?? Appl. Opt. 38, 3399-3408, (1999).
[CrossRef]

G. W. Kattawar and G. N. Plass �??Radiance and polarization of multiple scattered light from haze and clouds,�?? Appl. Opt. 7, 1519-1527, (1967).
[CrossRef]

G. W. Kattawar and G. N. Plass, �??Degree and direction of polarization of multiple scattered light. 1: Homogeneous cloud layers,�?? Appl. Opt. 11, 2851-2865, (1972).
[CrossRef] [PubMed]

H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, �??Monte Carlo and Multicomponent Approximation Methods for Vector Radiative Transfer by use of Effective Mueller Matrix Calculations,�?? Appl. Opt. 40, 400�??412, (2001).
[CrossRef]

B. Kaplan, G. Ledanois, and B. Villon, �??Mueller Matrix of Dense Polystyrene Latex Sphere Suspensions: Measurements and Monte Carlo Simulation,�?? Appl. Opt. 40, 2769�??2777, (2001).
[CrossRef]

Astophys. J. Suppl. Ser. (1)

A. N. Witt, �??Multiple scattering in reflection nebulae I: A Monte Carlo approach,�?? Astophys. J. Suppl. Ser. 35, 1-6, (1977).
[CrossRef]

Computer Graphics (1)

K. Shoemake, �??Animating rotation with quaternion curves,�?? Computer Graphics 19, 245-254, (1985).
[CrossRef]

Computer Methods and Programs in Biomed. (1)

L. H Wang, S. L. Jacques, L-Q Zheng, �??MCML �?? Monte Carlo modeling of photon transport in multilayered tissues,�?? Computer Methods and Programs in Biomedicine 47, 131-146, (1995).
[CrossRef] [PubMed]

J. Am. Stat. Assoc. (1)

N. Metropolis and S. Ulam, �??The Monte Carlo method,�?? J. Am. Stat. Assoc. 44, 335-341, (1949).
[PubMed]

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. Chem. Phys. (1)

D. Benoit, D. Clary �??Quaternion formulation of diffusion quantum Monte Carlo for the rotation of rigid molecules in clusters,�?? J. Chem. Phys. 113, 5193-5202, (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

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. Quant. Spectrosc. Radiat. Transfer (1)

A. Ambirajan and D.C. Look, �??A backward Monte Carlo study of the multiple scattering of a polarized laser beam,�?? J. Quant. Spectrosc. Radiat. Transfer 58, 171-192, (1997).
[CrossRef]

Journal of the Atmospheric Sciences (1)

J. W. Hovenier, �??Symmetry Relationships for Scattering of Polarized Light in a Slab of Randomly Oriented Particles,�?? Journal of the Atmospheric Sciences 26, 488-499, (1968).
[CrossRef]

Limnol. Oceanogr. (1)

G. W. Kattawar and C. N. Adams �??Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,�?? Limnol. Oceanogr. 34, 1453-1472, (1989).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (1)

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, �??A Monte Carlo model of light propagation in tissue,�?? in Dosimetry of Laser Radiation in Medicine and Biology, G. Mueller and D. Sliney, eds., SPIE IS 5, 102-111, (1989).

Trans. Am. Phil. Soc. (1)

S. Chandrasekhar and D. Elbert, �??The Illumination and polarization of the sunlight sky on Rayleigh scattering,�?? Trans. Am. Phil. Soc. 44, (1956).

Other (13)

S. Chandrasekhar, Radiative Transfer, (Oxford Clarendon Press; 1950).

A. S. Martinez and R. Maynard �??Polarization Statistics in Multiple Scattering of light: a Monte Carlo approach,�?? in Localization and Propagation of classical waves in random and periodic structures (Plenum Publishing Corporation New York, 1993).

A. S. Martinez Statistique de polarization et effet Faraday en diffusion multiple de la lumiere Ph.D. Thesis (in French and English), 1984.

S. Bianchi Estinzione e polarizzazione della radiazione nelle galassie a spirale, Tesi di Laura (in Italian), 1994.

J. J. Craig, Introduction to robotics. Mechanics and controls, (Addison-Weseley Publishing Company, 1986).

<a href= "http://omlc.ogi.edu/software/polarization/">http://omlc.ogi.edu/software/polarization/</a>

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

J.C. Ramella-Roman, S.A.Prahl, S.L. Jacques, are preparing a manuscript to be called �??Three Monte Carlo programs of polarized light transport into scattering media: part II�??.

J. B. Kuipers, Quaternions and rotation sequences. A primer with applications to orbits, aerospace, and virtual reality (Princeton University Press, 1998)
[PubMed]

C. Bohren and D. R. Huffman, Absorption and scattering of light by small particles, (Wiley Science Paperback Series, 1998).
[CrossRef]

<a href= "http://omlc.ogi.edu/software/mie/index.html">http://omlc.ogi.edu/software/mie/index.html</a>

K. I. Hopcraft, P. C. Y. Chang, J. G. Walker, E. Jakeman, �??Properties of Polarized light-beam multiply scattered by Rayleigh medium,�?? in Light Scattering from Microstructure F. Moreno and F. Gonzales eds., Vol. 534, Lecture Notes in Physics, (Springer-Verlag, 2000), pp. 135-158.
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, the art of Scientific Computing, (Cambridge University Press, 1992).

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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

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.

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.

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.

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.

Fig. 6.
Fig. 6.

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

Fig. 7.
Fig. 7.

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

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.

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.

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.

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.

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:

Tables (2)

Tables Icon

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.

Tables Icon

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.

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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