Abstract

To model bidirectional measurements, a numerical method for computing the incoherent light scattered by a diffusing medium is presented. The results are expressed as a function of the incident and the observer angles (θi, ϕi) and (θf, ϕf), in contrast to the N-flux method, which gives no information about the azimuthal distribution. To solve the multiple-scattering equations, an auxiliary function, expanded on the spherical harmonics, is introduced in the diffusion equation. A set of integral equations on the coefficients are obtained that are well suited for the numerical resolution. The boundary conditions are included in the linear operator of the integral equation, so that each boundary condition is associated with a specific equation. As an illustration, the method is applied to numerical simulations of maps of the light scattered by a thick refractive diffusing layer of refractive index n=1.5, for two directive phase functions and for several incident collimated-beam angles.

© 2002 Optical Society of America

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References

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  1. M. Elias, L. Simonot, M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001).
    [CrossRef]
  2. P. Kubelka, F. Munk, “Ein Beitrag zur Optik des Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).
  3. B. Maheu, J. N. Letouzan, G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 23, 3353–3362 (1984).
    [CrossRef]
  4. P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
    [CrossRef] [PubMed]
  5. J. J. Joshi, D. B. Vaidya, H. S. Shah, “Application of multi-flux theory based on Mie scattering to the problem of modeling the optical characteristics of colored pigmented paint films,” Color Res. Appl. 26, (2001).
    [CrossRef]
  6. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  7. 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]
  8. C. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  9. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]

2001

M. Elias, L. Simonot, M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001).
[CrossRef]

J. J. Joshi, D. B. Vaidya, H. S. Shah, “Application of multi-flux theory based on Mie scattering to the problem of modeling the optical characteristics of colored pigmented paint films,” Color Res. Appl. 26, (2001).
[CrossRef]

1988

1984

1980

1971

1931

P. Kubelka, F. Munk, “Ein Beitrag zur Optik des Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Bohren, C.

C. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Chandrasekhar, S.

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

Elias, M.

M. Elias, L. Simonot, M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001).
[CrossRef]

Gouesbet, G.

Huffman, D. R.

C. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jayaweera, K.

Joshi, J. J.

J. J. Joshi, D. B. Vaidya, H. S. Shah, “Application of multi-flux theory based on Mie scattering to the problem of modeling the optical characteristics of colored pigmented paint films,” Color Res. Appl. 26, (2001).
[CrossRef]

Kubelka, P.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik des Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Letouzan, J. N.

Maheu, B.

Menu, M.

M. Elias, L. Simonot, M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001).
[CrossRef]

Mudgett, P. S.

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik des Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Richards, L. W.

Shah, H. S.

J. J. Joshi, D. B. Vaidya, H. S. Shah, “Application of multi-flux theory based on Mie scattering to the problem of modeling the optical characteristics of colored pigmented paint films,” Color Res. Appl. 26, (2001).
[CrossRef]

Simonot, L.

M. Elias, L. Simonot, M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001).
[CrossRef]

Stamnes, K.

Tsay, S. Chee

Vaidya, D. B.

J. J. Joshi, D. B. Vaidya, H. S. Shah, “Application of multi-flux theory based on Mie scattering to the problem of modeling the optical characteristics of colored pigmented paint films,” Color Res. Appl. 26, (2001).
[CrossRef]

Wiscombe, W.

Wiscombe, W. J.

Appl. Opt.

Color Res. Appl.

J. J. Joshi, D. B. Vaidya, H. S. Shah, “Application of multi-flux theory based on Mie scattering to the problem of modeling the optical characteristics of colored pigmented paint films,” Color Res. Appl. 26, (2001).
[CrossRef]

Opt. Commun.

M. Elias, L. Simonot, M. Menu, “Bidirectional reflectance of a diffuse background covered by a partly absorbing layer,” Opt. Commun. 191, 1–7 (2001).
[CrossRef]

Z. Tech. Phys.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik des Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Other

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

C. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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Figures (6)

Fig. 1
Fig. 1

Geometry of the problem. Incident collimated light, with ui direction, enters a layer with refractive index n and absorption coefficient α, containing diffusing particles with absorption and scattering coefficients K and S. Collimated and diffuse light propagate in the medium. The diffuse and collimated light is collected in the direction uf.

Fig. 2
Fig. 2

Total diffused emerging flux [curve (1)] and collimated reflected flux [curve (2)] emerging at z=0, both normalized to the incident flux, as a function of the incident angle θi for q=0.9, h=10, n=1.5 with p(cos γ)=1+cos γ.

Fig. 3
Fig. 3

Angular distribution of the emerging diffused light for q=0.9, h=10, n=1.5, p(cos γ)=1+cos γ, and θi=0°. (a) Single scattering, (b) higher-order scattering, (c) total scattering.

Fig. 4
Fig. 4

Angular distribution of the emerging diffused light for the same configuration as in Fig. 3, for θi=30°.

Fig. 5
Fig. 5

Angular distribution of the emerging diffused light for the same configuration as in Fig. 3, for θi=60°.

Fig. 6
Fig. 6

Angular distribution of the emerging diffused light for the same configuration as in Fig. 5, p(cos γ)=(1+cos γ)3/2.

Equations (65)

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dI1=-(α+K+S)I1dl,
dI2=SI1dlp(cos γ)dΩ2/4π,
cos θdI(u, Z)dZ=-(K+S+α)I(u, Z)+S4πI(u1, Z)p(u, u1)dΩ1+S4πjIj(Z)p(u, uj).
cos θdI(u, z)dz=-I(u, z)+q4πI(u1, z)p(u, u1)dΩ1+q4πjIj(z)p(u, uj).
w(u, z)=|cos θ|I(u, z).
dw+(u, z)dz=-w+(u, z)|cos θ|+q4πw(u1, z)|cos θ1|p(u, u1)dΩ1+q4πjWj(z)|cos θj|p(u, uj)(0<θ<π/2),
dw-(u, z)dz=w-(u, z)|cos θ|-q4πw(u1, z)|cos θ1|p(u, u1)dΩ1-q4πjWj(z)|cos θj|p(u, uj)(π/2<θ<π).
Fi=12πϕ=02πΔθiw(θ, ϕ)sin θdθdϕ.
f(u, z)=w(u1, z)|cos θ1|p(u, u1)dΩ1
S(u, z)=jWj(z)|cos θj|p(u, uj),
w+(u, z)=w+(u, 0)exp(-z/|cos θ|)+q4π0z[f(u, s)+S(u, s)]×exp[-(z-s)/|cos θ|]ds,
w-(u, z)=w-(u, h)exp[(z-h)/|cos θ|]+q4πzh[f(u, s)+S(u, s)]×exp[(z-s)/|cos θ|]ds.
w+(u, 0)=0,
w-(θ, ϕ, h)=R(π-θ)w+(π-θ, ϕ, h).
w+(π-θ, ϕ, h)
=q4π0h[f(π-θ, ϕ, s)+S(π-θ, ϕ, s)]×exp[-(h-s)/|cos θ|]ds.
w+(θ, ϕ, z)=q4π0z[f(θ, ϕ, s)+S(θ, ϕ, s)]×exp[-(z-s)/|cos θ|]ds,
w-(θ, ϕ, z)=q4πexp[(z-h)/|cos θ|]R(π-θ)×0h[f(π-θ, ϕ, s)+S(π-θ, ϕ, s)]×exp[-(h-s)/|cos θ|]ds+q4π×zh[f(θ, ϕ, s)+S(θ, ϕ, s)]×exp[(z-s)/|cos θ|]ds.
W(z)=Winc exp(-z/cos θi),
S(u, z)=Winc exp[-z/cos θi)p(u, ui)/cos θi.
w+(u, z)=q4π0z[f(u, s)+S(u, s)]×exp[-(z-s)/|cos θ|]ds,
w-(u, z)=q4πzh[f(u, s)+S(u, s)]×exp[(z-s)/|cos θ|]ds,
f(u, z)=q4πϕ1=02πdϕ1D1+D2[f(u1, s)+S(u1, s)]×exp(-|z-s|/|cos θ1|)|cos θ1|p(u, u1)sin θ1dθ1ds,
D1:0<θ1<π/2and0<s<z,
D2:π/2<θ1<πandz<s<h.
J=Δzexp(-x/|cos θ1|)|cos θ1|dx
n sin θ0=sin θi.
W(z)=WincT(θi)exp(-z/cos θ0),
S(u, z)=WincT(θi)exp(-z/cos θ0)×p(u, u0)/cos θ0.
w+(θ, ϕ, 0)=R1(θ)w-(π-θ, ϕ, 0),
f(θ, ϕ, z)=q4πϕ1=02πdϕ1D1+D2[f(θ1, ϕ1, s)+S(θ1, ϕ1, s)]exp(-|z-s|/|cos θ1|)|cos θ1|×p(cos γ)sin θ1dθ1ds+q4πϕ1=02πθ1=0π/2s=0h[f(π-θ1, ϕ1, s)+S(π-θ1, ϕ1, s)]R1(θ1)×exp[-(z+s)]/cos θ1]cos θ1×p(cos γ)sin θ1dθ1d.
p(cos γ)=n=0pnPn(cos γ),
Pn(cos γ)=m=0nα(n, m)Pnm(cos θ)×Pnm(cos θ1)cos m(ϕ-ϕ1),
p(u, u1)=n=0m=0nα(n, m)pnPnm(cos θ)×Pnm(cos θ1)cos m(ϕ-ϕ1).
S(u, z)=Wiexp(-z/cos θi)cos θin=0m=0nα(n, m)pn×Pnm(cos θ)Pnm(cos θi)cos mϕ.
f(u, z)=Wicos θin=0m=0ngn(m)(z)Pnm(cos θ)cos mϕ
ϕ1=02π cos m1ϕ1 cos m(ϕ1-ϕ)dϕ1=πηmδm,m1 cos mϕ,
gn(m)(z)=q4pnηmn1=mD1+D2[gn1(m)(s)+exp(-s/cos θi)pn1α(n1, m)Pn1m(cos θi)]×exp(-|z-s|/|cos θ1|)|cos θ1|α(n, m)Pn1m(cos θ1)×Pnm(cos θ1)sin θ1dθ1ds.
K(m)(x, n, n1)=α(n, m)θ=0π/2exp(-x/cos θ)cos θ×Pnm(cos θ)Pn1m(cos θ)sin θdθ
gn(m)(z)=n1=ms=0hH(m)(n, n1, z, s)×[gn1(m)(s)+bn1(m)(s)]dsfornm,
bn1(m)(s)=exp(-s/cos θi)pn1α(n1, m)Pn1m(cos θi),
H(m)(n, n1, z, s)
=q4pnηmK(m)(|z-s|, n, n1)
for0<s<z,
H(m)(n, n1, z, s)
=q4pnηm(-1)n+n1K(m)(|z-s|, n, n1)
forz<s<h.
f(u, z)=WincT(θi)cos θ0n=0m=0ngn(m)(z)Pnm(cos θ)cos mϕ,
gn(m)(z)=n1=ms=0hH1(m)(n, n1, z, s)
×[gn1(m)(s)+bn1(m)(s)]dsfornm,
bn1(m)(s)=exp(-s/cos θ0)pn1α(n1, m)Pn1m(cos θ0),
H1(m)(n, n1, z, s)
=H(m)(n, n1, z, s)+q4pnηm(-1)m+n1U(m)(z+s, n, n1),
U(m)(x, n, n1)
=α(n, m)θ=0π/2exp(-x/cos θ)cos θ×R1(θ)Pnm(cos θ)Pn1m(cos θ)sin θdθ.
gn,i(m)=n1=mNj=1QHn,n1,i,j(m)×[gn1,j(m)+bn1,j(m)],mnN,1iQ,
Hn,n1,i,j(m)=zj-Δz/2zj+Δz/2H(m)(n, n1, zi, s)ds.
g(z)=s=0hH(z, s)[g(s)+b(s)]ds,
b(s)=exp(-s/cos θi),
H(z, s)=q2K(|z-s|),
K(x)=θ=0π/2exp(-x/cos θ)cos θsin θdθ.
ws(θs, ϕ)=T(π-θs)w-(θ, ϕ, 0)cos θsn2 cos θ.
X=2 sin(θ/2)cos ϕ,Y=2 sin(θ/2)sin ϕ.
dXdY=12sin θdθdϕ=12dΩ.
p(cos γ)=1+cos γ,

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