Abstract

An analytic solution to the problem of determining photon direction after successive scatterings in an infinite, homogeneous, isotropic medium, where each scattering event is in accordance with a two-term Henyey-Greenstein phase function, is presented and compared against Monte Carlo simulation results. The photon direction is described by a probability density function of the dot product of the initial direction and the direction after multiple scattering events, and it is found that such a probability density function can be represented as a weighted series of one-term Henyey-Greenstein phase functions.

© 2008 Optical Society of America

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References

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  1. L.G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  2. C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002).
    [CrossRef] [PubMed]
  3. L. Wang and H. Wu, Biomedical Optics: Principles and Imaging (John Wiley and Sons, 2007), Ch. 3.
  4. N. Pfeiffer and G.H. Chapman, “Monte Carlo Simulations of the Growth and Decay of Quasi-Ballistic Photon Fractions with Depth in an Isotropic Medium,” Proc. SPIE 5695, 136–147 (2005).
    [CrossRef]
  5. I. Turcu and R. Bratfalean, “Narrowly peaked forward light scattering on particulate media I. Assessment of the multiple scattering contributions to the effective phase function,” J. Opt. A: Pure Appl. Opt.10, (2008).
    [CrossRef]
  6. W. E. Vargas and G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys.: Condens. Matter 9, 9083–9096 (1997).
    [CrossRef]
  7. H. C. van de Hulst, Multiple Light Scattering, Vol 1 (Academic, 1980).
  8. T. Binzoni1, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey-Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
    [CrossRef]
  9. S. L. Jacques, C. A. Alter, and S. A. Prahl, “Angular Dependence of HeNe laser Light Scattering by Human Dermis,” Laser Life Sci. 1, 309–333 (1987).
  10. V. I. Haltrin , “Two-term Henyey-Greenstein light scattering phase function for seawater,” in IGARSS ’99: Proceeding of the International Geoscience and Remote Sensing Symposium, T. I. Stein , ed. (IEEE, 1999), pp. 1423–1425.
  11. S. A. Prahl, Light Transport in Tissue, App. A1, PhD Thesis, (University of Texas at Austin, 1988).
  12. R. A. Leathers and N. J. McCormick, “Ocean inherent optical property estimation from irradiances,” Appl. Opt. 36, 8685–8698 (1997).
    [CrossRef]

2006 (1)

T. Binzoni1, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey-Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
[CrossRef]

2005 (1)

N. Pfeiffer and G.H. Chapman, “Monte Carlo Simulations of the Growth and Decay of Quasi-Ballistic Photon Fractions with Depth in an Isotropic Medium,” Proc. SPIE 5695, 136–147 (2005).
[CrossRef]

2002 (1)

1997 (2)

W. E. Vargas and G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys.: Condens. Matter 9, 9083–9096 (1997).
[CrossRef]

R. A. Leathers and N. J. McCormick, “Ocean inherent optical property estimation from irradiances,” Appl. Opt. 36, 8685–8698 (1997).
[CrossRef]

1987 (1)

S. L. Jacques, C. A. Alter, and S. A. Prahl, “Angular Dependence of HeNe laser Light Scattering by Human Dermis,” Laser Life Sci. 1, 309–333 (1987).

1941 (1)

L.G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Alter, C. A.

S. L. Jacques, C. A. Alter, and S. A. Prahl, “Angular Dependence of HeNe laser Light Scattering by Human Dermis,” Laser Life Sci. 1, 309–333 (1987).

Binzoni1, T.

T. Binzoni1, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey-Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
[CrossRef]

Boss, E.

Bratfalean, R.

I. Turcu and R. Bratfalean, “Narrowly peaked forward light scattering on particulate media I. Assessment of the multiple scattering contributions to the effective phase function,” J. Opt. A: Pure Appl. Opt.10, (2008).
[CrossRef]

Chapman, G.H.

N. Pfeiffer and G.H. Chapman, “Monte Carlo Simulations of the Growth and Decay of Quasi-Ballistic Photon Fractions with Depth in an Isotropic Medium,” Proc. SPIE 5695, 136–147 (2005).
[CrossRef]

Delpy, D. T.

T. Binzoni1, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey-Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
[CrossRef]

Gandjbakhche, A. H.

T. Binzoni1, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey-Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
[CrossRef]

Greenstein, J. L.

L.G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Haltrin, V. I.

V. I. Haltrin , “Two-term Henyey-Greenstein light scattering phase function for seawater,” in IGARSS ’99: Proceeding of the International Geoscience and Remote Sensing Symposium, T. I. Stein , ed. (IEEE, 1999), pp. 1423–1425.

Henyey, L.G.

L.G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Jacques, S. L.

S. L. Jacques, C. A. Alter, and S. A. Prahl, “Angular Dependence of HeNe laser Light Scattering by Human Dermis,” Laser Life Sci. 1, 309–333 (1987).

Leathers, R. A.

Leung, T. S.

T. Binzoni1, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey-Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
[CrossRef]

McCormick, N. J.

Mobley, C. D.

Niklasson, G. A.

W. E. Vargas and G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys.: Condens. Matter 9, 9083–9096 (1997).
[CrossRef]

Pfeiffer, N.

N. Pfeiffer and G.H. Chapman, “Monte Carlo Simulations of the Growth and Decay of Quasi-Ballistic Photon Fractions with Depth in an Isotropic Medium,” Proc. SPIE 5695, 136–147 (2005).
[CrossRef]

Prahl, S. A.

S. L. Jacques, C. A. Alter, and S. A. Prahl, “Angular Dependence of HeNe laser Light Scattering by Human Dermis,” Laser Life Sci. 1, 309–333 (1987).

S. A. Prahl, Light Transport in Tissue, App. A1, PhD Thesis, (University of Texas at Austin, 1988).

Rüfenacht, D.

T. Binzoni1, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey-Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
[CrossRef]

Sundman, L. K.

Turcu, I.

I. Turcu and R. Bratfalean, “Narrowly peaked forward light scattering on particulate media I. Assessment of the multiple scattering contributions to the effective phase function,” J. Opt. A: Pure Appl. Opt.10, (2008).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering, Vol 1 (Academic, 1980).

Vargas, W. E.

W. E. Vargas and G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys.: Condens. Matter 9, 9083–9096 (1997).
[CrossRef]

Wang, L.

L. Wang and H. Wu, Biomedical Optics: Principles and Imaging (John Wiley and Sons, 2007), Ch. 3.

Wu, H.

L. Wang and H. Wu, Biomedical Optics: Principles and Imaging (John Wiley and Sons, 2007), Ch. 3.

Appl. Opt. (2)

Astrophys. J. (1)

L.G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

J. Phys.: Condens. Matter (1)

W. E. Vargas and G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys.: Condens. Matter 9, 9083–9096 (1997).
[CrossRef]

Laser Life Sci. (1)

S. L. Jacques, C. A. Alter, and S. A. Prahl, “Angular Dependence of HeNe laser Light Scattering by Human Dermis,” Laser Life Sci. 1, 309–333 (1987).

Phys. Med. Biol. (1)

T. Binzoni1, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey-Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
[CrossRef]

Proc. SPIE (1)

N. Pfeiffer and G.H. Chapman, “Monte Carlo Simulations of the Growth and Decay of Quasi-Ballistic Photon Fractions with Depth in an Isotropic Medium,” Proc. SPIE 5695, 136–147 (2005).
[CrossRef]

Other (5)

I. Turcu and R. Bratfalean, “Narrowly peaked forward light scattering on particulate media I. Assessment of the multiple scattering contributions to the effective phase function,” J. Opt. A: Pure Appl. Opt.10, (2008).
[CrossRef]

L. Wang and H. Wu, Biomedical Optics: Principles and Imaging (John Wiley and Sons, 2007), Ch. 3.

H. C. van de Hulst, Multiple Light Scattering, Vol 1 (Academic, 1980).

V. I. Haltrin , “Two-term Henyey-Greenstein light scattering phase function for seawater,” in IGARSS ’99: Proceeding of the International Geoscience and Remote Sensing Symposium, T. I. Stein , ed. (IEEE, 1999), pp. 1423–1425.

S. A. Prahl, Light Transport in Tissue, App. A1, PhD Thesis, (University of Texas at Austin, 1988).

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Figures (4)

Fig. 1.
Fig. 1.

Probability distribution of scattering cosine for OTHG phase function, anisotropy factor g=0.9, scattering order n=1..4, symbols are Monte Carlo results, lines are phg, n (cos θ).

Fig. 2.
Fig. 2.

Probability distribution of scattering cosine for OTHG phase function, anisotropy factor g=0.9, scattering order n=10,20,..,50, symbols are Monte Carlo results, lines are phg, n (cos θ).

Fig. 3.
Fig. 3.

Probability distribution of scattering cosine for TTHG phase function, α=0.9, g α =0.9, β=0.1, g β =0.0, scattering order n=1..4,10, symbols are Monte Carlo results, lines are phg2, n (cos θ).

Fig. 4.
Fig. 4.

Probability distribution of scattering cosine for TTHG phase function, α=0.96, g α =0.9, β=0.04, g β =-0.26, scattering order n=1..4,10, symbols are Monte Carlo results, lines are phg2, n (cos θ).

Equations (18)

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p h g ( Ω 0 · Ω , g ) = p h g ( cos θ , g ) = 1 2 [ 1 g 2 ( 1 + g 2 2 g cos θ ) 3 2 ]
p hg ( cos θ , g ) = Σ l = 0 ( l + 1 2 ) g l p l ( cos θ )
p hg 2 ( cos θ , α , g α , β , g β ) = α p hg ( cos θ , g α ) + β p hg ( cos θ , g β )
p hg 2 ( cos θ , α , g α , β , g β ) = Σ l = 0 ( l + 1 2 ) ( α g α l + β g β l ) P l ( cos θ )
p hg , n ( cos θ , g ) = Σ l = 0 ( l + 1 2 ) ( g l ) n P l ( cos θ ) = p hg ( cos θ , g n )
p hg 2 , n ( cos θ , α , g α , β , g β ) = Σ l = 0 ( l + 1 2 ) ( α g α l + β g β l ) n P l ( cos θ )
p hg 2 , 2 ( cos θ , α , g α , β , g β ) = α 2 p hg , α α ( cos θ , g α ) + α β p hg , α β ( cos θ , g α β )
+ β α p hg , β α ( cos θ , g β α ) + β 2 p hg , β β ( cos θ , g β )
p hg 2 , 0 ( cos θ , α , g α , β , g β ) = p hg ( cos θ , 1 )
p hg 2 , 1 ( cos θ , α , g α , β , g β ) = α p hg ( cos θ , g α ) + β p hg ( cos θ , g β )
p hg 2 , 2 ( cos θ , α , g α , β , g β ) = α 2 p hg ( cos θ , g α 2 ) + 2 α β p hg ( cos θ , g α g β )
+ β 2 p hg ( cos θ , g β 2 )
p hg 2 , 3 ( cos θ , α , g α , β , g β ) = α 3 p hg ( cos θ , g α 3 ) + 3 α 2 β p hg ( cos θ , g α 2 g β )
+ 3 α β 2 p hg ( cos θ , g α g β 2 ) + β 3 p hg ( cos θ , g β 3 )
p hg 2 , n ( cos θ , α , g α , β , g β ) = Σ i = 0 n ( n i ) α ( n i ) β i p hg ( cos θ , g α ( n i ) g β i )
cos θ n = Σ i = 0 n ( n i ) α ( n i ) β i g α ( n i ) g β i
cos θ ( g ) = { 1 2 g [ 1 + g 2 ( 1 g 2 1 g + 2 g ξ ) 2 ] if g 0 2 ξ 1 if g 0
cos θ ( α , g α , g β ) = { cos θ ( g α ) if χ < α cos θ ( g β ) otherwise

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