Abstract

Anisotropy in the polarization of the backscattered light from a polarized laser beam incident upon a scattering medium has been observed experimentally. When the beam is viewed through an oriented polarizer, characteristic patterns in the backscattered light are observed. We present here a simple explanation of these patterns, using the theory of incoherent scattering of light by spheres. It appears that the major contribution to the observed patterns comes from the double scattering of light.

© 1998 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. M. Dogariu, T. Asakura, “Polarization-dependent backscattering patterns from weakly scattering media,” J. Opt. (Paris) 24, 271–278 (1993).
    [CrossRef]
  4. M. Dogariu, T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. 35, 2234–2239 (1996).
    [CrossRef]
  5. P. L. Marston, “Uniform Mie-theoretic analysis of polarized and cross-polarized optical glories,” J. Opt. Soc. Am. 73, 1816–1818 (1983).
    [CrossRef]
  6. D. Eliyahu, M. Rosenbluh, I. Freund, “Angular intensity and polarization dependence of diffuse transmission through random media,” J. Opt. Soc. Am. A 10, 477–491 (1993), and references therein.
    [CrossRef]
  7. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  8. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).
  9. B. D. Cameron, M. J. Raković, M. Mehrubeoglu, G. W. Kattawar, S. Rastegar, L. V. Wang, G. Coté, “Measurement and calculation of the two-dimensional backscattering Mueller matrix of a turbid medium,” Opt. Lett. 23, 485–487 (1998).
    [CrossRef]

1998 (1)

1996 (1)

M. Dogariu, T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. 35, 2234–2239 (1996).
[CrossRef]

1993 (2)

1985 (1)

1983 (1)

1980 (1)

Asakura, T.

M. Dogariu, T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. 35, 2234–2239 (1996).
[CrossRef]

M. Dogariu, T. Asakura, “Polarization-dependent backscattering patterns from weakly scattering media,” J. Opt. (Paris) 24, 271–278 (1993).
[CrossRef]

Cameron, B. D.

Carswell, A. I.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

Coté, G.

Dogariu, M.

M. Dogariu, T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. 35, 2234–2239 (1996).
[CrossRef]

M. Dogariu, T. Asakura, “Polarization-dependent backscattering patterns from weakly scattering media,” J. Opt. (Paris) 24, 271–278 (1993).
[CrossRef]

Eliyahu, D.

Freund, I.

Kattawar, G. W.

Marston, P. L.

Mehrubeoglu, M.

Pal, S. R.

Rakovic, M. J.

Rastegar, S.

Rosenbluh, M.

van de Hulst, H. C.

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

Wang, L. V.

Appl. Opt. (2)

J. Opt. (Paris) (1)

M. Dogariu, T. Asakura, “Polarization-dependent backscattering patterns from weakly scattering media,” J. Opt. (Paris) 24, 271–278 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

M. Dogariu, T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. 35, 2234–2239 (1996).
[CrossRef]

Opt. Lett. (1)

Other (2)

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

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

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Figures (2)

Fig. 1
Fig. 1

Geometry of double scattering.

Fig. 2
Fig. 2

Radiance of the light backscattered from water suspensions of polysterene spheres of diameter (a)–(c) d 0 = 6.8 μm and (d)–(f) d 0 = 0.085 μm. Initial polarization was along the x axis. (a), (d) Total radiance distributions; (b), (e) light viewed through the polarizer oriented along the y (crossed patterns) axis; (c), (f) light viewed through the polarizer oriented along the x axis (parallel pattern).

Equations (32)

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P 1 = P 0 exp - β T | z | ,
d P 2 = β s d z M Θ R ϕ P 1 d ω 1 ,
R ϕ = 1 0 0 0 0 cos   2 ϕ - sin   2 ϕ 0 0 sin   2 ϕ cos   2 ϕ 0 0 0 0 1
d P 3 = d P 2 exp - β T r .
d P 4 = β s d r   exp - β T | z | M π - Θ d P 3 d ω 2 ,
d I bs = R ϕ d P 4 d s 3 d ω 2 = R ϕ d P 4 d s 2 d ω 2 = R ϕ d P 4 d s 2 d r d ω 2 d z = R ϕ d P 4 d ω 1 d ω 2 d r d z r 2 = β s 2 exp - β T | z | + | z | + r R ϕ M π - Θ × M Θ R ϕ P 0 d z d z r 2 .
I bs ρ ,   ϕ = β s 2 - 0 - 0 d z d z r 2 exp - β T | z | + | z | + r × R ϕ M π - Θ M Θ R ϕ P 0 ,
r = ρ 2 + z - z 2 1 / 2 ,     tan   Θ = ρ z - z .
I bs ρ ,   ϕ = β s 2 M bs ρ ,   ϕ ;   β T P 0 = β s 2 R ϕ L ρ β T R ϕ P 0 ,
L ρ s = 1 2 ρ s 0 π / 2 d Θ   exp - ρ s cot Θ / 2 M Θ M π - Θ + M π - Θ M Θ ,     ρ s = ρ β T .
M Θ = a Θ b Θ 0 0 b Θ a Θ 0 0 0 0 d Θ - e Θ 0 0 e Θ d Θ , 2 π   0 π   a Θ sin   Θ d Θ = 1 .
a Θ = 3 16 π 1 + cos 2   Θ ,     b Θ = 3 16 π - 1 + cos 2   Θ , d Θ = 3 8 π cos   Θ ,     e Θ = 0 .
M ij bs ρ s ,   ϕ = 1 2 ρ s 0 π / 2 d Θ   exp - ρ s cot Θ / 2 F ij Θ ,   ϕ .
I ρ s ,   ϕ = β s 2 M 11 bs ρ s ,   ϕ + M 12 bs ρ s ,   ϕ .
I c ρ s ,   ϕ = β s 2 2 M 11 bs ρ s ,   ϕ + M 12 bs ρ s ,   ϕ - M 21 bs ρ s ,   ϕ - M 22 bs ρ s ,   ϕ ;
I p ρ s ,   ϕ = β s 2 2 M 11 bs ρ s ,   ϕ + M 12 bs ρ s ,   ϕ + M 21 bs ρ s ,   ϕ + M 22 bs ρ s ,   ϕ .
I ρ s ,   ϕ = β s 2 f 0 ρ s + f 2 ρ s cos   2 ϕ ,
I c ρ s ,   ϕ = β s 2 f 0 ρ s + f 4 ρ s 4 1 - cos   4 ϕ ,
I p ρ s ,   ϕ = β s 2 3 f 0 ρ s - f 4 ρ s 4 + f 2 ρ s cos   2 ϕ + f 0 ρ s + f 4 ρ s 4 cos   4 ϕ ,
f 0 ρ s = 1 ρ s 0 π / 2 exp - ρ s cot Θ / 2 a Θ a π - Θ + b Θ b π - Θ d Θ ,
f 2 ρ s = 1 ρ s 0 π / 2 exp - ρ s cot Θ / 2 a Θ b π - Θ + b Θ a π - Θ d Θ ,
f 4 ρ s = 1 ρ s 0 π / 2 exp - ρ s cot Θ / 2 d Θ d π - Θ - e Θ e π - Θ d Θ .
f 0 Ray ρ s = 9 128 π 2 1 ρ s 0 π / 2 exp - ρ s cot Θ / 2 × 1 + cos 4   Θ d Θ ,
f 2 Ray ρ s = - 9 128 π 2 1 ρ s 0 π / 2 exp - ρ s cot Θ / 2 × 1 - cos 4   Θ d Θ ,
f 4 Ray ρ s = - 9 64 π 2 1 ρ s 0 π / 2 exp - ρ s cot Θ / 2 cos 2   Θ d Θ .
I bs ρ ,   ϕ = β s 2 R ϕ L ρ R ϕ P 0 , L ρ = - 0 d z   - 0 d z r 2 exp - β T | z | + r + | z | × M π - Θ M Θ ,
r = ρ 2 + z - z 2 1 / 2 = ρ sin   Θ , tan   Θ = ρ z - z ,     z = z - ρ tan   Θ .
L ρ = 1 ρ - 0 d z   0 π - arctan   ρ / | z | d Θ   exp - β T - 2 z + ρ × cot Θ / 2 M π - Θ M Θ .
L ρ = 1 ρ 0 π / 2 d Θ   - 0 d z   exp - β T - 2 z + ρ × cot Θ / 2 M π - Θ M Θ + 1 ρ π / 2 π d Θ   - ρ   cot   Θ d z   exp - β T - 2 z + ρ × cot Θ / 2 M π - Θ M Θ .
L ρ = 1 ρ 0 π / 2 d Θ   - 0 d z   exp - β T - 2 z + ρ × cot Θ / 2 M π - Θ M Θ + 1 ρ π / 2 π d Θ   - 0 d z   exp - β T - 2 z + ρ × tan Θ / 2 M π - Θ M Θ .
L ρ = 1 2 ρ β T 0 π / 2 d Θ   exp - β T ρ   cot Θ / 2 × M π - Θ M Θ + M Θ M π - Θ ,
F 11 = 2 a Θ a π - Θ + b Θ b π - Θ , F 12 = 2 b Θ a π - Θ + a Θ b π - Θ cos   2 ϕ , F 13 = - 2 b Θ a π - Θ + a Θ b π - Θ sin   2 ϕ , F 14 = 0 , F 21 = F 12 , F 22 = a Θ a π - Θ + b Θ b π - Θ - d Θ d π - Θ + e Θ e π - Θ + a Θ a π - Θ + b Θ b π - Θ + d Θ d π - Θ - e Θ e π - Θ cos   4 ϕ , F 23 = - a Θ a π - Θ + b Θ b π - Θ + d Θ d π - Θ - e Θ e π - Θ sin   4 ϕ , F 24 = 2 d Θ e π - Θ + e Θ d π - Θ sin   2 ϕ , F 31 = - F 13 , F 32 = - F 23 , F 33 = - a Θ a π - Θ - b Θ b π - Θ + d Θ d π - Θ - e Θ e π - Θ + a Θ a π - Θ + b Θ b π - Θ + d Θ d π - Θ - e Θ e π - Θ cos   4 ϕ , F 34 = - 2 d Θ e π - Θ + e Θ d π - Θ cos   2 ϕ , F 41 = 0 , F 42 = F 24 , F 43 = - F 34 , F 44 = 2 d Θ d π - Θ - e Θ e π - Θ .

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