Abstract

General formulations are given for the multiple scattering of a polarized wave incident upon a slab of randomly distributed spherical particles. The radiative transfer equation with Stokes vectors is decomposed into Fourier components, and they are shown for linearly and circularly polarized incident wave. For linear polarization, the copolarized and cross-polarized incoherent intensities show sinusoidal variations with the azimuthal angle. The degree of polarization is also calculated for various directions and optical thickness. The calculations are made for optical waves at 5, 10, and 15 μm in fog and compared with the first-order scattering calculations.

© 1982 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, New York, 1978).
  2. A. Ishimaru, “Multiple Scattering Effects on Optical Propagation in Turbulence and Particles,” in AGARD Conf. Proc. 300 (Optical Communication, Technical Editing and Reproduction, Hanford House, London, 1981), pp. 4-1–4-12.
  3. A. Ishimaru, Opt. Eng. 20, 63 (1981).
    [CrossRef]
  4. H. C. van de Hulst, Multiple Light Scattering, Vol. 2 (Academic, New York, 1980).
  5. S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, 1950).
  6. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  7. R. L.-T. Cheung, “Millimeter and Optical Waves in Rain and Fog,” Ph.D. Dissertation, Electrical Engineering Department, U. Washington, Seattle, 98195 (1981).
  8. A. Ishimaru, R. L.-T. Cheung, Ann. Telecommun. 35, 373 (1980a).
  9. A. Ishimaru, R. L.-T. Cheung, Radio Sci. 15, 507 (1980b).
    [CrossRef]
  10. T. Oguchi, Radio Sci. 16, 691 (1981).
    [CrossRef]
  11. Z. Sekera, J. Opt. Soc. Am. 56, 1732 (1966).
    [CrossRef]
  12. A. I. Carswell, “Laboratory Measurements of Light Propagation in Turbid Media,” in AGARD Conf. Proc. 300 (Optical Communication, Technical Editing and Reproduction, Hanford House, London, 1981), pp. 17-1–17-9.
  13. A. Ishimaru, R. T. Woo, J. A. Armstrong, D. Cheng, to appear in Radio Science (special issue on NASA Propagation Studies).
  14. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  15. M. Kerker, The Scattering of Light (Academic, New York, 1969).
  16. G. C. Moordian, M. Geller, L. B. Stotts, D. H. Stephens, R. A. Krautwald, Appl. Opt. 18, 429 (1979).
    [CrossRef]

1981 (2)

A. Ishimaru, Opt. Eng. 20, 63 (1981).
[CrossRef]

T. Oguchi, Radio Sci. 16, 691 (1981).
[CrossRef]

1980 (2)

A. Ishimaru, R. L.-T. Cheung, Ann. Telecommun. 35, 373 (1980a).

A. Ishimaru, R. L.-T. Cheung, Radio Sci. 15, 507 (1980b).
[CrossRef]

1979 (1)

1966 (1)

Armstrong, J. A.

A. Ishimaru, R. T. Woo, J. A. Armstrong, D. Cheng, to appear in Radio Science (special issue on NASA Propagation Studies).

Carswell, A. I.

A. I. Carswell, “Laboratory Measurements of Light Propagation in Turbid Media,” in AGARD Conf. Proc. 300 (Optical Communication, Technical Editing and Reproduction, Hanford House, London, 1981), pp. 17-1–17-9.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, 1950).

Cheng, D.

A. Ishimaru, R. T. Woo, J. A. Armstrong, D. Cheng, to appear in Radio Science (special issue on NASA Propagation Studies).

Cheung, R. L.-T.

A. Ishimaru, R. L.-T. Cheung, Radio Sci. 15, 507 (1980b).
[CrossRef]

A. Ishimaru, R. L.-T. Cheung, Ann. Telecommun. 35, 373 (1980a).

R. L.-T. Cheung, “Millimeter and Optical Waves in Rain and Fog,” Ph.D. Dissertation, Electrical Engineering Department, U. Washington, Seattle, 98195 (1981).

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

Geller, M.

Ishimaru, A.

A. Ishimaru, Opt. Eng. 20, 63 (1981).
[CrossRef]

A. Ishimaru, R. L.-T. Cheung, Radio Sci. 15, 507 (1980b).
[CrossRef]

A. Ishimaru, R. L.-T. Cheung, Ann. Telecommun. 35, 373 (1980a).

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, New York, 1978).

A. Ishimaru, “Multiple Scattering Effects on Optical Propagation in Turbulence and Particles,” in AGARD Conf. Proc. 300 (Optical Communication, Technical Editing and Reproduction, Hanford House, London, 1981), pp. 4-1–4-12.

A. Ishimaru, R. T. Woo, J. A. Armstrong, D. Cheng, to appear in Radio Science (special issue on NASA Propagation Studies).

Kerker, M.

M. Kerker, The Scattering of Light (Academic, New York, 1969).

Krautwald, R. A.

Moordian, G. C.

Oguchi, T.

T. Oguchi, Radio Sci. 16, 691 (1981).
[CrossRef]

Sekera, Z.

Stephens, D. H.

Stotts, L. B.

van de Hulst, H. C.

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

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Woo, R. T.

A. Ishimaru, R. T. Woo, J. A. Armstrong, D. Cheng, to appear in Radio Science (special issue on NASA Propagation Studies).

Ann. Telecommun. (1)

A. Ishimaru, R. L.-T. Cheung, Ann. Telecommun. 35, 373 (1980a).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

A. Ishimaru, Opt. Eng. 20, 63 (1981).
[CrossRef]

Radio Sci. (2)

A. Ishimaru, R. L.-T. Cheung, Radio Sci. 15, 507 (1980b).
[CrossRef]

T. Oguchi, Radio Sci. 16, 691 (1981).
[CrossRef]

Other (10)

A. I. Carswell, “Laboratory Measurements of Light Propagation in Turbid Media,” in AGARD Conf. Proc. 300 (Optical Communication, Technical Editing and Reproduction, Hanford House, London, 1981), pp. 17-1–17-9.

A. Ishimaru, R. T. Woo, J. A. Armstrong, D. Cheng, to appear in Radio Science (special issue on NASA Propagation Studies).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

M. Kerker, The Scattering of Light (Academic, New York, 1969).

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

S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, 1950).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

R. L.-T. Cheung, “Millimeter and Optical Waves in Rain and Fog,” Ph.D. Dissertation, Electrical Engineering Department, U. Washington, Seattle, 98195 (1981).

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, New York, 1978).

A. Ishimaru, “Multiple Scattering Effects on Optical Propagation in Turbulence and Particles,” in AGARD Conf. Proc. 300 (Optical Communication, Technical Editing and Reproduction, Hanford House, London, 1981), pp. 4-1–4-12.

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Figures (9)

Fig. 1
Fig. 1

Geometry showing the wave ( E 1 , E 2 ) incident on a particle in the direction (μ′,ϕ′) and the scattered wave (E1,E2) in the direction (μ,ϕ).

Fig. 2
Fig. 2

Transmitted incoherent specific intensities Ix and Iy (solid curves) at the optical depths of τ0 = 1 and τ0 = 5.0 at θ = 6.7° and λ = 10 μm. The dashed curves Ixs and Iys are the first-order scattering solution.

Fig. 3
Fig. 3

Transmitted inchoherent specific intensities under the same conditions as Fig. 2 except θ = 33°.

Fig. 4
Fig. 4

Baskscattered incoherent specific intensities at θ = 147°.

Fig. 5
Fig. 5

Backscattered incoherent specific intensities at θ = 173.3°.

Fig. 6
Fig. 6

Degree of polarization at τ0 = 1 and τ0 = 5 for different θ.

Fig. 7
Fig. 7

Transmitted incoherent specific intensities Ix and Iy at λ = 5, 10, and 15 μm and θ = 6.7°, ϕ) = 0.

Fig. 8
Fig. 8

Backscattered incoherent specific intensities Ix and Iy, at λ = 5, 10, and 15 μm and θ = 173.3°, ϕ = 0.

Fig. 9
Fig. 9

Degree of polarization as a function of τ0.

Equations (43)

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I 1 = E 1 E 1 * ; I 2 = E 2 E 2 * ; U = 2 Re E 1 E 2 * ; V = 2 Im E 1 E 2 * ;
μ d d τ [ I ] = - [ I ] + [ S ] [ I ] d ω + [ I i ] ,
ρ σ t = 0 σ t ( D ) N ( D ) d D ,
[ S ] = 1 ( ρ σ t ) [ ρ f 11 2 ρ f 12 2 ρ Re ( f 11 f 12 * ) - ρ Im ( f 11 f 12 * ) ρ f 21 2 ρ f 22 2 ρ Re ( f 21 f 22 * ) - ρ Im ( f 21 f 22 * ) ρ 2 Re ( f 11 f 21 * ) ρ 2 Re ( f 12 f 22 * ) ρ Re ( f 11 f 22 * + f 12 f 21 * ) - ρ Im ( f 11 f 22 * - f 12 f 21 * ) ρ 2 Im ( f 11 f 21 * ) ρ 2 Im ( f 12 f 22 * ) ρ Im ( f 11 f 22 * + f 12 f 21 * ) ρ Re ( f 11 f 22 * - f 12 f 21 * ) ] .
[ E 1 E 2 ] = exp ( i k R ) R [ f 11 f 12 f 21 f 22 ] [ E 1 E 2 ] .
f 11 = ( l , l ) T 1 + ( r , r ) T 2 , f 12 = - ( r , l ) T 1 + ( l , r ) T 2 , f 21 = - ( l , r ) T 1 + ( r , l ) T 2 , f 22 = ( r , r ) T 1 + ( l , l ) T 2 , ( l , l ) = [ ( 1 - μ 2 ) ( 1 - μ 2 ) ] 1 / 2 + μ μ cos ( ϕ - ϕ ) , ( l , r ) = - μ sin ( ϕ - ϕ ) , ( r , l ) = μ sin ( ϕ - ϕ ) , ( r , r ) = cos ( ϕ - ϕ ) , T 1 ( χ ) = A r r ( χ ) - χ A l l ( χ ) 1 - χ 2 , T 2 ( χ ) = A l l ( χ ) - χ A r r ( χ ) 1 - χ 2 ,
χ = cos = [ ( 1 - μ 2 ) ( 1 - μ 2 ) ] 1 / 2 cos ( ϕ - ϕ ) + μ μ , μ = cos θ , μ = cos θ .
A l l = i K S 2 *             and             A r r = i K S 1 * .
[ I r i ] = I 0 [ 1 0 0 0 ] exp ( - τ ) δ ( μ - 1 ) δ ( ϕ ) .
[ I i ] = [ S ] [ I r i ] d ω = I 0 [ F ] exp ( - τ ) , [ F ] = [ F 1 ( μ , ϕ ) F 2 ( μ , ϕ ) F 3 ( μ , ϕ ) F 4 ( μ , ϕ ) ] ,
F 1 = ρ f 11 2 ρ σ t , F 2 = ρ f 21 2 ρ σ t , F 3 = ρ [ 2 Re ( f 11 f 21 * ) ] ρ σ t , F 4 = ρ [ 2 Im ( f 11 f 21 * ) ] ρ σ t ,
f 11 = f 11 ( μ , ϕ , μ = 1 i , ϕ = 0 ) , f 21 = f 21 ( μ , ϕ , μ = 1 i , ϕ = 0 ) .
[ I i ] = I 0 { [ F ] 0 + [ F ] a cos 2 ϕ + [ F ] b sin 2 ϕ } exp ( - τ ) ,
[ F ] 0 = ½ [ F 01 F 02 0 0 ] , [ F ] a = ½ [ F 01 - F 02 0 0 ] , [ F ] b = [ 0 0 - Re ( F 03 ) - Im ( F 03 ) ] , F 01 = ρ A l l ( μ ) 2 ρ σ t ,             F 02 = ρ A r r ( μ ) 2 ρ σ t ,             F 03 = ρ [ A l l ( μ ) A r r * ( μ ) ] ρ σ t .
[ I ] = m = 0 [ I ] m a cos m ϕ + m = 1 [ I ] m b sin m ϕ .
[ S ] = 1 2 π [ S ] 0 a + 1 π m = 1 [ S ] m a cos m ( ϕ - ϕ ) + 1 π m = 1 [ S ] m b sin m ( ϕ - ϕ ) .
[ S ] = [ [ S 1 ] [ S 2 ] [ S 3 ] [ S 4 ] ] .
[ S ] m a = [ [ S 1 ] m a 0 0 [ S 4 ] m a ] ,             [ S ] m b = [ 0 [ S 2 ] m a [ S 3 ] m b 0 ] ,
[ S 1 ] m a = 0 2 π [ S 1 ] cos m ( ϕ - ϕ ) d ( ϕ - ϕ ) , [ S 2 ] m b = 0 2 π [ S 2 ] sin m ( ϕ - ϕ ) d ( ϕ - ϕ ) , [ S 3 ] m b = 0 2 π [ S 3 ] sin m ( ϕ - ϕ ) d ( ϕ - ϕ ) , [ S 4 ] m a = 0 2 π [ S 4 ] cos m ( ϕ - ϕ ) d ( ϕ - ϕ ) .
μ d d τ [ I ] 0 = - [ I ] 0 + - 1 1 [ L ] 0 [ I ] 0 d μ + [ F ] 0 exp ( - τ ) ,
[ I ] 0 = [ I 01 I 02 ] ,             [ L ] 0 = [ S 1 ] 0 a ,             [ F ] 0 = ½ [ F 01 F 02 ] .
[ I ] = [ I ] 0 a + [ I ] 2 a cos 2 ϕ + [ I ] 2 b sin 2 ϕ ,
[ I ] 0 a = [ I 01 I 02 0 0 ] ,             [ I ] 2 a = [ I 1 a I 2 a 0 0 ] ,             [ I ] 2 b = [ 0 0 U b V b ] .
μ d d τ [ I ] 2 = - [ I ] 2 = + - 1 1 [ L ] [ I ] 2 d μ + [ F ] 2 exp ( - τ ) ,
[ I ] 2 = [ I 1 a I 2 a U b V b ] ,             [ F ] 2 = [ ½ F 01 - ½ F 02 - Re ( F 03 ) - Im ( F 03 ) ] ,             [ L ] = [ [ S 1 ] 2 a [ S 2 ] 2 b - [ S 3 ] 2 b [ S 4 ] 2 a ] .
[ I ] 0 = [ I ] 2 = 0 for 0 μ 1 at τ = 0 , [ I ] 0 = [ I ] 2 = 0 for 0 μ - 1 at τ = τ 0 .
[ I ] = [ I 1 I 2 U V ] = [ I 01 I 02 0 0 ] + [ I 1 a cos 2 ϕ I 2 a cos 2 ϕ U b sin 2 ϕ V b sin 2 ϕ ] .
[ I x I y U x y V x y ] = [ E x E x * E y E y * 2 Re ( E x E y * ) 2 Im ( E x E y * ) ] = [ R ] [ I 1 I 2 U V ] ,
[ R ] = [ cos 2 θ cos 2 θ sin 2 ϕ - ½ sin 2 ϕ cos θ 0 cos 2 θ cos 2 θ cos 2 ϕ ½ sin 2 ϕ cos θ 0 cos 2 θ cos 2 θ - sin2 ϕ cos 2 ϕ cos θ 0 0 0 0 cos θ ] .
m = [ ( I 1 - I 2 ) 2 + U 2 + V 2 ] 1 / 2 ( I 1 + I 2 ) .
[ I 1 I 2 U V ] = I 0 [ F 1 F 2 F 3 F 4 ] × [ exp ( - τ 0 ) - exp ( - τ 0 μ ) 1 - μ ] ,             1 > μ > 0 ,
F 1 = ρ A l l ( μ ) 2 cos 2 ϕ ρ σ t ,             F 2 = ρ A r r ( μ ) 2 sin 2 ϕ ρ σ t , F 3 = - ρ Re [ A l l ( μ ) A r r * ( μ ) ] sin 2 ϕ ρ σ t , F 4 = - ρ Im [ A l l ( μ ) A r r * ( μ ) ] sin 2 ϕ ρ σ t ,
[ I 1 I 2 U V ] = I 0 [ F 1 F 2 F 3 F 4 ]     [ 1 - exp ( - τ 0 + τ 0 μ ) 1 - μ ] .
I x = ρ cos θ cos 2 A l l ( μ ) + sin 2 ϕ A r r ( μ ) 2 ρ σ t I 0 [ F s ] , I y = ρ sin 2 ϕ cos 2 ϕ cos θ A l l ( μ ) - A r r ( μ ) 2 ρ σ t I 0 [ F s ] ,
[ F s ] = [ exp ( - τ 0 ) - exp ( - τ 0 μ ) 1 - μ ] for transmission ( 1 > μ > 0 ) = [ 1 - exp ( - τ 0 + τ 0 μ ) 1 - μ ] for backscattering ( 0 > μ > - 1 ) .
[ I r i ] = I 0 [ ½ ½ 0 1 ] exp ( - τ ) δ ( μ - 1 ) δ ( ϕ ) ,
[ I i ] = I 0 [ F ] c exp ( - τ ) , [ F ] c = [ ½ F 01 ½ F 02 ± Im ( F 03 ) ± Re ( F 03 ) ] ,
[ S ] = [ [ S 1 ] 0 a 0 0 [ S 4 ] 0 a ] .
μ d d τ [ I ] c = - [ I ] c + - 1 1 [ S 1 ] 0 a [ I ] c d μ + [ F ] 0 exp ( - τ ) , μ d d τ [ U ] c = - [ U ] c + - 1 1 [ S 4 ] 0 a [ U ] c d μ + [ F ] μ exp ( - τ ) ,
[ I ] c = [ I 1 I 2 ] ,             [ U ] c = [ U V ] ,             [ F ] u = [ ± Im ( F 03 ) Re ( F 03 ) ] ,
E + = 1 2 ( E 1 - i E 2 ) ,             E - = 1 2 ( E 1 + i E 2 ) .
[ I CP ] = [ I c I + I - I c * ] ,             I c = ½ ( I 1 - I 2 - i U ) , I 1 = ½ ( I 1 + I 2 - V ) , I 2 = ½ ( I 1 + I 2 + V ) .
[ I 1 I 2 ] = I 0 [ F ] 0 [ F s ] ,

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