Abstract

Simple analytical relations for reflection and transmission matrices of plane-parallel layers of random media with discrete particles are presented. They can be used for rapid estimation of intensity and polarization characteristics of reflected and transmitted light beams under arbitrary illumination of a layer. The accuracy of the analytical formulas obtained increases with the optical thickness τ of a layer. Thus equations are applicable only at large values of τ (τ>5). Another limitation is due to the probability of photon absorption β, which should be rather low (β<0.050.1, depending on the optical characteristics in question).

© 2001 Optical Society of America

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References

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  1. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. 1 and 2.
  2. W. M. F. Wauben, “Multiple scattering of polarized radiation in planetary atmospheres,” Ph.D. thesis (Free University of Amsterdam, Enschede, The Netherlands, 1992).
  3. H. C. van de Hulst, “Radiative transfer in thick atmospheres with an arbitrary scattering function,” Bull. Astron. Inst. Neth. 20, 77–86 (1968).
  4. H. Domke, “Linear Fredholm integral equations for radiative transfer problems in finite plane-parallel media. II. Embedding in a semi-infinite medium,” Astron. Nachr. 299, 95–102 (1978).
    [CrossRef]
  5. W. A. de Rooij, “Reflection and transmission of polarized light by planetary atmospheres,” Ph.D. thesis (Free University of Amsterdam, Utrecht, The Netherlands, 1985).
  6. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in a Scattering Medium (Springer-Verlag, Berlin, 1991).
  7. A. A. Kokhanovsky, Light Scattering Media Optics (Wiley-Praxis, Chichester, UK, 1999).
  8. N. Umow, “Chromatische Depolarisation durch Lichtzerstreuung,” Phyz. Z. 6, 674–676 (1905).
  9. B. Hapke, Theory of Reflectance and Emittance Spectroscopy (Cambridge U. Press, Cambridge, UK, 1993).

1978 (1)

H. Domke, “Linear Fredholm integral equations for radiative transfer problems in finite plane-parallel media. II. Embedding in a semi-infinite medium,” Astron. Nachr. 299, 95–102 (1978).
[CrossRef]

1968 (1)

H. C. van de Hulst, “Radiative transfer in thick atmospheres with an arbitrary scattering function,” Bull. Astron. Inst. Neth. 20, 77–86 (1968).

1905 (1)

N. Umow, “Chromatische Depolarisation durch Lichtzerstreuung,” Phyz. Z. 6, 674–676 (1905).

de Rooij, W. A.

W. A. de Rooij, “Reflection and transmission of polarized light by planetary atmospheres,” Ph.D. thesis (Free University of Amsterdam, Utrecht, The Netherlands, 1985).

Domke, H.

H. Domke, “Linear Fredholm integral equations for radiative transfer problems in finite plane-parallel media. II. Embedding in a semi-infinite medium,” Astron. Nachr. 299, 95–102 (1978).
[CrossRef]

Hapke, B.

B. Hapke, Theory of Reflectance and Emittance Spectroscopy (Cambridge U. Press, Cambridge, UK, 1993).

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in a Scattering Medium (Springer-Verlag, Berlin, 1991).

Katsev, I. L.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in a Scattering Medium (Springer-Verlag, Berlin, 1991).

Kokhanovsky, A. A.

A. A. Kokhanovsky, Light Scattering Media Optics (Wiley-Praxis, Chichester, UK, 1999).

Umow, N.

N. Umow, “Chromatische Depolarisation durch Lichtzerstreuung,” Phyz. Z. 6, 674–676 (1905).

van de Hulst, H. C.

H. C. van de Hulst, “Radiative transfer in thick atmospheres with an arbitrary scattering function,” Bull. Astron. Inst. Neth. 20, 77–86 (1968).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. 1 and 2.

Wauben, W. M. F.

W. M. F. Wauben, “Multiple scattering of polarized radiation in planetary atmospheres,” Ph.D. thesis (Free University of Amsterdam, Enschede, The Netherlands, 1992).

Zege, E. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in a Scattering Medium (Springer-Verlag, Berlin, 1991).

Astron. Nachr. (1)

H. Domke, “Linear Fredholm integral equations for radiative transfer problems in finite plane-parallel media. II. Embedding in a semi-infinite medium,” Astron. Nachr. 299, 95–102 (1978).
[CrossRef]

Bull. Astron. Inst. Neth. (1)

H. C. van de Hulst, “Radiative transfer in thick atmospheres with an arbitrary scattering function,” Bull. Astron. Inst. Neth. 20, 77–86 (1968).

Phyz. Z. (1)

N. Umow, “Chromatische Depolarisation durch Lichtzerstreuung,” Phyz. Z. 6, 674–676 (1905).

Other (6)

B. Hapke, Theory of Reflectance and Emittance Spectroscopy (Cambridge U. Press, Cambridge, UK, 1993).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. 1 and 2.

W. M. F. Wauben, “Multiple scattering of polarized radiation in planetary atmospheres,” Ph.D. thesis (Free University of Amsterdam, Enschede, The Netherlands, 1992).

W. A. de Rooij, “Reflection and transmission of polarized light by planetary atmospheres,” Ph.D. thesis (Free University of Amsterdam, Utrecht, The Netherlands, 1985).

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in a Scattering Medium (Springer-Verlag, Berlin, 1991).

A. A. Kokhanovsky, Light Scattering Media Optics (Wiley-Praxis, Chichester, UK, 1999).

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Figures (3)

Fig. 1
Fig. 1

Dependence of the degree of light polarization reflected from a water cloud with gamma particle size distribution at a wavelength of 443 nm on the incidence angle at nadir observation at τ=8, 15, and 30 according to numerical radiative transfer calculations (broken curves) and Eqs. (35) and (32) (symbols).

Fig. 2
Fig. 2

Dependence of the degree of light polarization reflected from a water cloud with gamma particle size distribution at a wavelength of 443 nm on the inverse optical thickness at nadir observation and incidence angle equal to 37° according to approximate Eq. (35) (solid curve) and numerical radiative transfer calculations (symbols).

Fig. 3
Fig. 3

Dependence of the degree of light polarization reflected from a water cloud on the probability of photon absorption at nadir observation and incidence angle equal to 30° and 37° according to numerical radiative transfer calculations (symbols) and Eq. (36) (lines) at optical thickness 7. The phase matrix coincides with the phase matrix used for calculations presented in Figs. 1 and 2.

Equations (43)

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St(μ, ϕ)=Tˆ(μ, μ0, ϕ)F,
Sr(μ, ϕ)=Rˆ(μ, μ0, ϕ)F
Rˆ(μ, μ0, ϕ)=Rˆ(μ, μ0, ϕ)-fTˆ(μ, μ0).
Tˆ(μ, μ0)=tK(μ)KT(μ0),
t=m exp(-kτ)1-f2.
f=s exp(-kτ),
m=201dμμ[PT(μ)P(μ)-PT(-μ)P(-μ)],
K(μ)=m-1[P(μ)-2 01dξξRˆ(μ, ξ)P(-ξ)],
s=201dμμKT(μ)P(-μ),
Rˆ(μ, ξ)=12π02πdϕRˆ(μ, ξ, ϕ)
(1-kμ)P(μ)=ω02-11dξZˆ(μ, ξ)P(ξ),
k=[3(1-g)(1-ω0)]1/2,
m=8k3(1-g),
s=1-4kα3(1-g),
Rˆ(μ, μ0, ϕ)=Rˆ0(μ, μ0, ϕ)-4k3(1-g)K¯0(μ)K0T(μ0),
g=120πp(θ)sin θ cos θdθ
α=301dμμ2S(μ),
201dμμS(μ)=1.
m=1-exp(-2y),
s=exp(-y),
y=4k3(1-g),
Rˆ(μ, μ0, ϕ)=Rˆ(μ, μ0, ϕ)-Tˆ(μ, μ0)exp(-y-kτ),
Tˆ(μ, μ0)=tK0(μ)K0T(μ0),
mK(μ)KT(μ0)[1-exp(-2y)]K0(μ)K0T(μ0).
t=1-exp(-2y)exp(kτ)-exp(-2y-kτ)
t=401μdμ01μ0dμ0T11(μ, μ0).
201μdμK01(μ)dμ=1,
t=shysh(x+y),
x=kτ.
R(μ, μ0, ϕ)=R(μ, μ0, ϕ)-T(μ, μ0)exp(-y-kτ),
T(μ, μ0)=tK0(μ)K0(μ0),
Rˆ(μ, μ0, ϕ)=Rˆ0(μ, μ0, ϕ)-Tˆ(μ, μ0),
Tˆ(μ,μ0)=tK0(μ)K0T(μ0),
t=1/[1+34τ(1-g)].
P(μ0)=P(μ0)1-u(μ0)(1-r),
u(μ0)=K01(1)K01(μ0)R11(μ0, 1),P(μ0)=-R21(μ0, 1)R11(μ0, 1),
P(μ0)=P(μ0)1-1.18t,
ψ(a)=Aaμ exp-μaa0,
P(μ0)=-R21(μ0, 1)R11(μ0, 1)-K01(μ)K01(μ0)t exp(-x-y)
P(μ0)=P*(μ0)1-u*(μ0)t exp(-x-y),
u*(μ0)=K01(1)K01(μ0)R11*(μ0, 1).
P(μ0)=c(μ0, τ)P*(μ0),
c(μ0, τ)=11-u*(μ0)t exp(-x-y)

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