Abstract

The propagation of light through turbid media is of fundamental interest in a number of areas of optical science including atmospheric and oceanographic science, astrophysics and medicine amongst many others. The angular distribution of photons after a single scattering event is determined by the scattering phase function of the material the light is passing through. However, in many instances photons experience multiple scattering events and there is currently no equivalent function to describe the resulting angular distribution of photons. Here we present simple analytic formulas that describe the angular distribution of photons after multiple scattering events, based only on knowledge of the single scattering albedo and the single scattering phase function.

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References

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  2. H. C. van de Hulst, Multiple Light Scattering, Vol. 1 (Academic Press, 1980).
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  4. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge Univ. Press, 2006).
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    [CrossRef]
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    [CrossRef]
  14. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters. (Academic, 1994).
  15. T. J. Petzold, T. J. SIO Ref. 72–78, Scripps Institute of Oceanography (U. California, 1972).
  16. G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
    [CrossRef]
  17. J. L. Forand and G. R. Fournier, “Particle distributions and index of refraction estimation for Canadian waters,” Proc. SPIE 3761, 34–44 (1999).
    [CrossRef]
  18. I. Turcu and M. Kirillin, “Quasi-ballistic light scattering – analytical models versus Monte Carlo simulations,” J. Phys.: Conf. Ser. 182, 012035 (2009).
    [CrossRef]

2009 (1)

I. Turcu and M. Kirillin, “Quasi-ballistic light scattering – analytical models versus Monte Carlo simulations,” J. Phys.: Conf. Ser. 182, 012035 (2009).
[CrossRef]

2008 (2)

2007 (1)

M. Sydor, “Statistical treatment of remote sensing reflectance from coastal ocean water: proportionality of reflectance from multiple scattering to source function b/a,” J. Coast. Res. 235, 1183–1192 (2007).
[CrossRef]

2004 (1)

2000 (1)

1999 (2)

P. Flatau, J. Piskozub, and J. R. V. Zaneveld, “Asymptotic light field in the presence of a bubble-layer,” Opt. Express 5(5), 120–124 (1999).
[CrossRef] [PubMed]

J. L. Forand and G. R. Fournier, “Particle distributions and index of refraction estimation for Canadian waters,” Proc. SPIE 3761, 34–44 (1999).
[CrossRef]

1994 (1)

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
[CrossRef]

1941 (1)

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

Brown, I.

Chapman, G. H.

Flatau, P.

Forand, J. L.

J. L. Forand and G. R. Fournier, “Particle distributions and index of refraction estimation for Canadian waters,” Proc. SPIE 3761, 34–44 (1999).
[CrossRef]

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
[CrossRef]

Fournier, G. R.

J. L. Forand and G. R. Fournier, “Particle distributions and index of refraction estimation for Canadian waters,” Proc. SPIE 3761, 34–44 (1999).
[CrossRef]

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
[CrossRef]

Greenstein, J. L.

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

Henyey, L. C.

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

Kirillin, M.

I. Turcu and M. Kirillin, “Quasi-ballistic light scattering – analytical models versus Monte Carlo simulations,” J. Phys.: Conf. Ser. 182, 012035 (2009).
[CrossRef]

McKee, D.

Otremba, Z.

Pfeiffer, N.

Piskozub, J.

Robinson, I. S.

Schwarz, J. N.

Sydor, M.

M. Sydor, “Statistical treatment of remote sensing reflectance from coastal ocean water: proportionality of reflectance from multiple scattering to source function b/a,” J. Coast. Res. 235, 1183–1192 (2007).
[CrossRef]

Turcu, I.

I. Turcu and M. Kirillin, “Quasi-ballistic light scattering – analytical models versus Monte Carlo simulations,” J. Phys.: Conf. Ser. 182, 012035 (2009).
[CrossRef]

Weeks, A. R.

Zaneveld, J. R. V.

Appl. Opt. (1)

Astrophys. J. (1)

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

J. Coast. Res. (1)

M. Sydor, “Statistical treatment of remote sensing reflectance from coastal ocean water: proportionality of reflectance from multiple scattering to source function b/a,” J. Coast. Res. 235, 1183–1192 (2007).
[CrossRef]

J. Phys.: Conf. Ser. (1)

I. Turcu and M. Kirillin, “Quasi-ballistic light scattering – analytical models versus Monte Carlo simulations,” J. Phys.: Conf. Ser. 182, 012035 (2009).
[CrossRef]

Opt. Express (4)

Proc. SPIE (2)

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
[CrossRef]

J. L. Forand and G. R. Fournier, “Particle distributions and index of refraction estimation for Canadian waters,” Proc. SPIE 3761, 34–44 (1999).
[CrossRef]

Other (8)

G. Marsaglia, The Diehard Battery of Tests of Randomness (1995), http://www.stat.fsu.edu/pub/diehard/ .

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters. (Academic, 1994).

T. J. Petzold, T. J. SIO Ref. 72–78, Scripps Institute of Oceanography (U. California, 1972).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

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

H. C. van de Hulst, Multiple Light Scattering, Vol. 2 (Academic Press, 1980).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge Univ. Press, 2006).

A. A. Kokhanovsky, Cloud Optics (Springer, 2006).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of scattering angles used in Eq. (2) for calculating multiple scattering trajectories.

Fig. 2
Fig. 2

nth order scattering phase functions for (a) and (b) HG (g1 = 0.9) and (c) and (d) FF (bbp / bp = 0.018) phase functions up to 50th order scattering obtained from Monte Carlo simulations (lines). Symbols in (a) and (b) are calculated using Eq. (3) with gn = g1 n from Pfeiffer and Chapman [7]. FF phase functions are not defined on the asymmetry parameter, so it is not possible to use the Pfeiffer and Chapman relationship in this manner for these functions.

Fig. 3
Fig. 3

Asymmetry parameters for nth order scattering phase functions, gn , plotted against scattering order for 3 different FF scattering phase functions. Best-fit regression lines confirm that gn = g1 n holds for FF phase functions.

Fig. 4
Fig. 4

Asymmetry parameters for effective multiple scattering phase functions for three FF single scattering phase functions calculated from Monte Carlo simulations (symbols) and Eq. (5) (lines) over a wide range of single scattering albedos.

Fig. 5
Fig. 5

nth order scattering phase functions calculated with 0.1° increments in θ using Eq. (6) (symbols - only every 2° shown for clarity) and from Monte Carlo simulations (solid lines) for a HG single scattering phase function with g1 = 0.924 (a and b), and for a FF single scattering phase function with bbp / bp = 0.018 (c and d).

Fig. 6
Fig. 6

Effective multiple scattering phase functions for a FF single scattering phase function (bbp / bp = 0.018) from Monte Carlo simulations (lines) and calculated using Eq. (7) (symbols) with nth order phase functions derived from Monte Carlo simulations. Increasing single scattering albedo results in more scattering at wide angles for a given single scattering phase function. The single scattering phase function (solid line, no symbols, n = 1) is shown for reference.

Equations (8)

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g 1 = 2 π 0 π β ˜ ( θ ) sin θ cos θ d θ
cos θ n = cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos ψ
β ˜ ( θ ) = 1 4 π 1 g 2 ( 1 + g 2 2 g cos θ ) 3 / 2
β ˜ ( θ ) = 1 4 π ( 1 δ ) 2 δ ν ( [ ν ( 1 δ ) ( 1 δ ν ) ] + 4 u 2 [ δ ( 1 δ ν ) ν ( 1 δ ) ] )
ν = 3 μ 2 , δ = u 2 3 ( n r 1 ) 2 , u = 2 sin ( θ / 2 )
g m s = i = 1 n g i ω i i = 1 n ω i = g 1 ( 1 ω ) ( 1 g 1 ω )
β ˜ n ( θ n ) = 4 π θ 1 , θ 2 , ψ β ˜ n 1 ( θ 1 ) sin ( θ 1 ) Δ θ 1 β ˜ 1 ( θ 2 ) sin ( θ 2 ) Δ θ 2 θ 1 , θ 2 , ψ sin ( θ 1 ) Δ θ 1 sin ( θ 2 ) Δ θ 2
β ˜ m s ( θ ) = β ˜ 1 ( θ ) + ω β ˜ 2 ( θ ) + ω 2 β ˜ 3 ( θ ) + + ω n 1 β ˜ n ( θ ) 1 + ω + ω 2 + + ω n 1

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