Abstract

Two methods for solving the radiative transfer equation are compared with the aim of computing the angular distribution of the light scattered by a heterogeneous scattering medium composed of a single flat layer or a multilayer. The first method [auxiliary function method (AFM)], recently developed, uses an auxiliary function and leads to an exact solution; the second [discrete-ordinate method (DOM)] is based on the channel concept and needs an angular discretization. The comparison is applied to two different media presenting two typical and extreme scattering behaviors: Rayleigh and Mie scattering with smooth or very anisotropic phase functions, respectively. A very good agreement between the predictions of the two methods is observed in both cases. The larger the number of channels used in the DOM, the better the agreement. The principal advantages and limitations of each method are also listed.

© 2003 Optical Society of America

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References

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  1. M. Elias, G. Elias, “New and fast calculation for incoherent multiple scattering,” J. Opt. Soc. Am. A 19, 894–901 (2002).
    [CrossRef]
  2. A. da Silva, C. Andraud, J. Lafait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. (to be published).
  3. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  4. J.-P. Briton, B. Maheu, G. Grehan, G. Gouesbet, “Monte Carlo simulation of multiple cross section in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
    [CrossRef]
  5. K. Stamnes, S. Chee Tsay, W. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layer media,” Appl. Opt. 27, 2502–2510 (1988).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. W. J. Wiscombe, “Improve Mie scattering algorithm,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]

2002

1992

J.-P. Briton, B. Maheu, G. Grehan, G. Gouesbet, “Monte Carlo simulation of multiple cross section in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

1988

1980

1971

Andraud, C.

A. da Silva, C. Andraud, J. Lafait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. (to be published).

Barrera, R. G.

A. da Silva, C. Andraud, J. Lafait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. (to be published).

Briton, J.-P.

J.-P. Briton, B. Maheu, G. Grehan, G. Gouesbet, “Monte Carlo simulation of multiple cross section in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Chee Tsay, S.

da Silva, A.

A. da Silva, C. Andraud, J. Lafait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. (to be published).

Elias, G.

Elias, M.

Gouesbet, G.

J.-P. Briton, B. Maheu, G. Grehan, G. Gouesbet, “Monte Carlo simulation of multiple cross section in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

Grehan, G.

J.-P. Briton, B. Maheu, G. Grehan, G. Gouesbet, “Monte Carlo simulation of multiple cross section in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

Jayaweera, K.

Lafait, J.

A. da Silva, C. Andraud, J. Lafait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. (to be published).

Maheu, B.

J.-P. Briton, B. Maheu, G. Grehan, G. Gouesbet, “Monte Carlo simulation of multiple cross section in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

Mudgett, P. S.

Richards, L. W.

Robin, T.

A. da Silva, C. Andraud, J. Lafait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. (to be published).

Stamnes, K.

Wiscombe, W.

Wiscombe, W. J.

Appl. Opt.

J. Opt. Soc. Am. A

Part. Part. Syst. Charact.

J.-P. Briton, B. Maheu, G. Grehan, G. Gouesbet, “Monte Carlo simulation of multiple cross section in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

Other

A. da Silva, C. Andraud, J. Lafait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. (to be published).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

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Figures (9)

Fig. 1
Fig. 1

Geometry and notation.

Fig. 2
Fig. 2

Total flux, reflectance, and transmittance vectors used in a multilayered medium.

Fig. 3
Fig. 3

Phase functions p(θ) of bead type 1 (quasi-isotropic) and bead type 2 (very anisotropic).

Fig. 4
Fig. 4

Collimated flux transmitted (T) and reflected (R) by (a) medium 1 and (b) medium 2 as a function of the incidence angle θi.

Fig. 5
Fig. 5

Total scattered flux transmitted (T) and reflected (R) by (a) medium 1 and (b) medium 2 as a function of the incidence angle θi: AFM (solid curves), DOM (dots).

Fig. 6
Fig. 6

Directional scattered flux transmitted (T) and reflected (R) by (a) medium 1 and (b) medium 2 in logarithmic scale for normal incidence as a function of the observation angle θs: AFM (solid curves), DOM (circles; N=20 for medium 1 and N=80 for medium 2), DOM (triangles; N=60 for medium 1 and N=240 for medium 2).

Fig. 7
Fig. 7

Directional scattered flux transmitted (T) and reflected (R) by both media as a function of the azimuthal observation angle ϕs for an incidence angle θi=30°. Medium 1 (N=60): (a) R, (b) T, medium 2 (N=240): (c) R, (d) T. Different angles of observation are presented: θs=30° (cross for DOM), 45° (triangle for DOM), and 60° (circle for DOM). The AFM is represented by solid curves everywhere.

Fig. 8
Fig. 8

Angular distribution of scattered transmitted flux for medium 2, issued from the AFM, for θi=45°, ϕi=0°: (a) single scattering, (b) higher-order scattering, (c) total scattering.

Fig. 9
Fig. 9

Angular distribution of scattered reflected flux for medium 2, issued from the AFM, for the same configuration as that in Fig. 8.

Tables (1)

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Table 1 Characteristics of the Two Media

Equations (37)

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dw±(u, τ)dτ=w±(u, τ)|μ|±q4π w±(u, τ)|μ|p(u, u)dΩ±q4πj Wj±(τ)|μj|p(u, uj),
f(u, τ)= w(u, τ)|μ|p(u, u)dΩ,
S(u, τ)=j Wj(τ)|μj|p(u, uj),
p(u, u)=l=0νmaxalPl(cos ψ),
Pl(cos ψ)=m=0lηm (l-m)!(l+m)!Plm(μ)Plm(μ)×cos[m(ϕ-ϕ)],
p(u, u)=m=0νmaxηmpm(μ, μ)cos[m(ϕ-ϕ)],
pm(μ, μ)=l=mνmaxal (l-m)!(l+m)!Plm(μ)Plm(μ).
w(μ, τ)=m=0νmaxwm(μ, τ) cos mϕ,
dw±m(μ, τ)dτ=w±m(μ, τ)|μ|±q4πfm(μ, τ)±q4πSm(μ, τ),
dwidτ=-wiμi+q2j=1N wj|μj|pijωj,
dwdτ=Sw.
wi=α=1NAiαCα exp(λατ)withi[1, N].
fa=T12fin(0)+R21fc,fout(0)=T21fc+R12fin(0),
fafc=β12fin(0)fout(0).
fout(h)fin(h)=β23fbfd.
fbfd=Pfafc,
fout(h)fin(h)=β23Pβ12fin(0)fout(0).
Wair-(0)=R0W0T0[1+(1-2R0)exp(-2h/μ0)]
Wair+(h)=W0T0 exp[-(h/μ0)].
w+(μ, ϕ, τ)=w+(μ, ϕ, 0)exp(-τ/μ)+q4π0τt(μ, ϕ, s)exp[-(τ-s)/μ]ds,
w-(-μ, ϕ, τ)=w-(-μ, ϕ, h)exp[(τ-h)/μ]+q4πτht(-μ, ϕ, s)exp[(τ-s)/μ]ds.
t(μ, ϕ, τ)=W0μ0l,mtlm(τ)Plm(μ)cos(mϕ),
w+(μ, ϕ, τ)=W0μ0l,malm(μ, τ)Plm(μ)cos(mϕ),
w-(-μ, ϕ, τ)=W0μ0l,mblm(μ, τ)Plm(μ)cos(mϕ),
alm(μ, τ)=alm(μ, 0)exp(-τ/μ)+q4π0τtlm(s)×exp[-(τ-s)/μ]ds,
blm(μ, τ)=blm(μ, h)exp[(τ-h)/μ]+(-1)l+m q4πτhtlm(s)×exp[-(τ-s)/μ]ds.
flm(τ)=l1=mνmax0hHm(l, l1, τ, s)tl1m(s)ds+πηm(l-m)!(l+m)!pll1=mνmaxLm(l, l1, τ),
Hm(l, l1, τ, s)=q4πηmα(l, m)plKm(l, l1, τ-s)for0<s<τ,
Hm(l, l1, τ, s)=q4π(-1)l+l1ηmα(l, m)×plKm(l, l1, τ-s)forτ<s<h,
Km(l, l1, x)=0l exp(-x/μ)μPlm(μ)Pl1m(μ)dμ.
Lm(l, l1, τ)=01{al1m(μ, 0)exp(-τ/μ)+(-1)l+mbl1m(μ, h)×exp[(τ-h)/μ]}Plm(μ)Pl1m(μ)dμ/μ.
w+(μ, ϕ, 0)=R(μ)w-(-μ, ϕ, 0)atτ=0,
w-(-μ, ϕ, 0)=R(μ)w+(μ, ϕ, 0)atτ=h,
flm(τ)=l1=mνmax0hHm(l, l1, τ, s)+q4ηm (l-m)!(l+m)!p1Qm(l, l1, τ, s)×[fl1m(s)+Sl1m(s)]l1mds,
Qm(l, l1, τ, s)=(-1)l1+mAm(l, l1, τ+s)+(-1)l1+mAm(l, l1, 2h-τ-s)+Bm(l, l1, 2h+τ-s)+(-1)l+l1Bm(l, l1, 2h-τ+s),
Am(l, l1, x)=μ=01 R(μ)Plm(μ)Pl1m(μ)exp-(x/μ)dμ1-R2(μ)exp-(2h/μ)μ,
Bm(l, l1, x)=μ=01 R2(μ)Plm(μ)Pl1m(μ)exp-(x/μ)dμ1-R2(μ)exp-(2h/μ)μ.

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