Abstract

A method is presented for obtaining the specific intensity of linearly polarized optical waves propagated in discrete random media. This method is an extension of our previous analysis of circular polarization, applied to linear polarization. Using a combination of small-angle and diffusion solutions in vector form, solutions for the specific intensity are obtained for large particles over a wide range of optical depths. Linearly polarized waves for normal incidence require an analysis of azimuth-dependent terms, and those components contributing to the major scattering process are retained. Copolarized and cross-polarized incoherent intensities are obtained within the framework of a 4 × 4 matrix. A comparison with numerical solutions obtained by the method of extended spherical harmonics is made to demonstrate the validity of the present theory. The ratio of small-angle scattering intensity to total scattering intensity in the forward direction is also represented as a function of optical depths.

© 1989 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  2. J. E. Hansens, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [Crossref]
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  4. H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, New York, 1980).
  5. I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
    [Crossref]
  6. L. Tsang, A. Ishimaru, “Radiative wave and cyclical transfer equations for dense nontenuous media,” J. Opt. Soc. Am. A 2, 2187–2193 (1985).
    [Crossref]
  7. W. G. Tam, A. Zardecki, “Off-axis propagation of a laser beam in low visibility weather conditions,” Appl. Opt. 19, 2822–2827 (1980).
    [Crossref] [PubMed]
  8. S. Ito, “On the theory of pulse wave propagation in media of discrete random scatterers,” Radio Sci. 15, 893–901 (1980).
    [Crossref]
  9. K. Furutsu, “Diffusion equation derived from space–time transport equation,”J. Opt. Soc. Am. 70, 360–366 (1980).
    [Crossref]
  10. S. Ito, K. Furutsu, “Theory of light pulse propagation through thick clouds,”J. Opt. Soc. Am. 70, 366–374 (1980).
    [Crossref]
  11. S. Ito, T. Oguchi, “An approximate method for solving the vector radiative transfer equation in discrete random media,” Radio Sci. 22, 873–879 (1987).
    [Crossref]
  12. T. Oguchi, “Effect of incoherent scattering on attenuation and cross polarization of millimeter wave due to rain: preliminary calculations at 34.8 and 82 GHz for spherical raindrops,” Annu. Telecommun. 35, 380–389 (1980).
  13. A. Ishimaru, R. L.-T. Cheung, “Multiple scattering effects on wave propagation due to rain,” Annu. Telecommun. 35, 373–379 (1980).
  14. R. L.-T. Cheung, A. Ishimaru, “Transmission, backscattering, and depolarization of waves in randomly distributed spherical particles,” Appl. Opt. 21, 3792–3798 (1982).
    [Crossref] [PubMed]
  15. T. Oguchi, “Effect of incoherent scattering on attenuation and depolarization of millimeter and optical waves due to hydrometeors,” Radio Sci. 21, 717–730 (1986).
    [Crossref]
  16. See Ref. 11, App.
  17. R. Burridge, “Spherically symmetric differential equations, the rotation group, and tensor spherical functions,” Proc. Cambridge Philos. Soc. 65, 157–175 (1969).
    [Crossref]
  18. T. Oguchi, “Effect of incoherent scattering on attenuation and cross-polarization of millimeter wave due to rain: preliminary calculations at 34.8 and 82 GHz for spherical raindrops,”J. Radio Res. Lab. 27, 1–51 (1980).
  19. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 865.
  20. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

1987 (1)

S. Ito, T. Oguchi, “An approximate method for solving the vector radiative transfer equation in discrete random media,” Radio Sci. 22, 873–879 (1987).
[Crossref]

1986 (1)

T. Oguchi, “Effect of incoherent scattering on attenuation and depolarization of millimeter and optical waves due to hydrometeors,” Radio Sci. 21, 717–730 (1986).
[Crossref]

1985 (1)

1982 (1)

1980 (7)

T. Oguchi, “Effect of incoherent scattering on attenuation and cross-polarization of millimeter wave due to rain: preliminary calculations at 34.8 and 82 GHz for spherical raindrops,”J. Radio Res. Lab. 27, 1–51 (1980).

W. G. Tam, A. Zardecki, “Off-axis propagation of a laser beam in low visibility weather conditions,” Appl. Opt. 19, 2822–2827 (1980).
[Crossref] [PubMed]

S. Ito, “On the theory of pulse wave propagation in media of discrete random scatterers,” Radio Sci. 15, 893–901 (1980).
[Crossref]

K. Furutsu, “Diffusion equation derived from space–time transport equation,”J. Opt. Soc. Am. 70, 360–366 (1980).
[Crossref]

S. Ito, K. Furutsu, “Theory of light pulse propagation through thick clouds,”J. Opt. Soc. Am. 70, 366–374 (1980).
[Crossref]

T. Oguchi, “Effect of incoherent scattering on attenuation and cross polarization of millimeter wave due to rain: preliminary calculations at 34.8 and 82 GHz for spherical raindrops,” Annu. Telecommun. 35, 380–389 (1980).

A. Ishimaru, R. L.-T. Cheung, “Multiple scattering effects on wave propagation due to rain,” Annu. Telecommun. 35, 373–379 (1980).

1974 (1)

J. E. Hansens, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

1969 (1)

R. Burridge, “Spherically symmetric differential equations, the rotation group, and tensor spherical functions,” Proc. Cambridge Philos. Soc. 65, 157–175 (1969).
[Crossref]

1959 (1)

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[Crossref]

Burridge, R.

R. Burridge, “Spherically symmetric differential equations, the rotation group, and tensor spherical functions,” Proc. Cambridge Philos. Soc. 65, 157–175 (1969).
[Crossref]

Chandrasekhar, S.

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

Cheung, R. L.-T.

R. L.-T. Cheung, A. Ishimaru, “Transmission, backscattering, and depolarization of waves in randomly distributed spherical particles,” Appl. Opt. 21, 3792–3798 (1982).
[Crossref] [PubMed]

A. Ishimaru, R. L.-T. Cheung, “Multiple scattering effects on wave propagation due to rain,” Annu. Telecommun. 35, 373–379 (1980).

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 865.

Furutsu, K.

Hansens, J. E.

J. E. Hansens, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

Ishimaru, A.

Ito, S.

S. Ito, T. Oguchi, “An approximate method for solving the vector radiative transfer equation in discrete random media,” Radio Sci. 22, 873–879 (1987).
[Crossref]

S. Ito, “On the theory of pulse wave propagation in media of discrete random scatterers,” Radio Sci. 15, 893–901 (1980).
[Crossref]

S. Ito, K. Furutsu, “Theory of light pulse propagation through thick clouds,”J. Opt. Soc. Am. 70, 366–374 (1980).
[Crossref]

Kušcer, I.

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[Crossref]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 865.

Oguchi, T.

S. Ito, T. Oguchi, “An approximate method for solving the vector radiative transfer equation in discrete random media,” Radio Sci. 22, 873–879 (1987).
[Crossref]

T. Oguchi, “Effect of incoherent scattering on attenuation and depolarization of millimeter and optical waves due to hydrometeors,” Radio Sci. 21, 717–730 (1986).
[Crossref]

T. Oguchi, “Effect of incoherent scattering on attenuation and cross-polarization of millimeter wave due to rain: preliminary calculations at 34.8 and 82 GHz for spherical raindrops,”J. Radio Res. Lab. 27, 1–51 (1980).

T. Oguchi, “Effect of incoherent scattering on attenuation and cross polarization of millimeter wave due to rain: preliminary calculations at 34.8 and 82 GHz for spherical raindrops,” Annu. Telecommun. 35, 380–389 (1980).

Ribaric, M.

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[Crossref]

Tam, W. G.

Travis, L. D.

J. E. Hansens, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

Tsang, L.

Van de Hulst, H. C.

H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, New York, 1980).

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

Zardecki, A.

Annu. Telecommun. (2)

T. Oguchi, “Effect of incoherent scattering on attenuation and cross polarization of millimeter wave due to rain: preliminary calculations at 34.8 and 82 GHz for spherical raindrops,” Annu. Telecommun. 35, 380–389 (1980).

A. Ishimaru, R. L.-T. Cheung, “Multiple scattering effects on wave propagation due to rain,” Annu. Telecommun. 35, 373–379 (1980).

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

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

J. Radio Res. Lab. (1)

T. Oguchi, “Effect of incoherent scattering on attenuation and cross-polarization of millimeter wave due to rain: preliminary calculations at 34.8 and 82 GHz for spherical raindrops,”J. Radio Res. Lab. 27, 1–51 (1980).

Opt. Acta (1)

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[Crossref]

Proc. Cambridge Philos. Soc. (1)

R. Burridge, “Spherically symmetric differential equations, the rotation group, and tensor spherical functions,” Proc. Cambridge Philos. Soc. 65, 157–175 (1969).
[Crossref]

Radio Sci. (3)

T. Oguchi, “Effect of incoherent scattering on attenuation and depolarization of millimeter and optical waves due to hydrometeors,” Radio Sci. 21, 717–730 (1986).
[Crossref]

S. Ito, T. Oguchi, “An approximate method for solving the vector radiative transfer equation in discrete random media,” Radio Sci. 22, 873–879 (1987).
[Crossref]

S. Ito, “On the theory of pulse wave propagation in media of discrete random scatterers,” Radio Sci. 15, 893–901 (1980).
[Crossref]

Space Sci. Rev. (1)

J. E. Hansens, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

Other (6)

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

H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, New York, 1980).

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

See Ref. 11, App.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 865.

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

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Figures (5)

Fig. 1
Fig. 1

Geometry of the problem and coordinate system defining the Stokes vector.

Fig. 2
Fig. 2

Geometrical relation between a local scattering plane OP1P2 and meridian planes for incident and scattered waves.

Fig. 3
Fig. 3

Horizontally and vertically polarized incoherent intensities, Ih and Iv, in the forward direction versus optical depths. A horizontally polarized wave is incident upon a semi-infinite medium of fog with radius a = 1.0 μm. Solid curves show approximate solutions, and broken curves show numerical solutions obtained by the extended spherical-harmonics method.

Fig. 4
Fig. 4

Same as Fig. 3, except that the radius is a = 2.0 μm.

Fig. 5
Fig. 5

Ratio of small-angle scattering to total scattering intensity of horizontally polarized waves in the forward direction versus optical depths.

Equations (58)

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J LP = [ I Q U V ] = [ E l E l * + E r E r * E l E l * - E r E r * 2 Re E l E r * 2 Im E l E r * ] ,
d J LP ( r , Ω ) d ρ = - σ t J LP ( r , Ω ) + Ψ LP ( Ω , Ω ) J LP ( r , Ω ) d Ω ,
Ψ LP = Ψ 1 + Ψ 2 .
d j 1 ( r , Ω ) d ρ = - σ t j 1 ( r , Ω ) + Ψ 1 ( Ω , Ω ) j 1 ( r , Ω ) d Ω .
d j 2 ( r , Ω ) d ρ = - σ t j 2 ( r , Ω ) + Ψ LP ( Ω , Ω ) j 2 ( r , Ω ) d Ω + W ( r , Ω ) ,
W ( r , Ω ) - ( Ψ LP - Ψ 1 ) j 1 ( r , Ω ) d Ω .
Ψ LP ( Ω , Ω ) = R L ( - π + β ) S LP ( Ω , Ω ) R L ( α ) ,
R L ( χ ) = [ 1 0 0 0 0 cos 2 χ sin 2 χ 0 0 - sin 2 χ cos 2 χ 0 0 0 0 1 ] ,
S 1 ( Ω , Ω ) = ( σ s / 2 π ) [ J 1 ( k a Ω - Ω ) / Ω - Ω ] 2 ,
cos [ 2 ( α + β ) ] cos [ 2 ( ϕ - ϕ ) ] , sin [ 2 ( α + β ) ] sin [ 2 ( ϕ - ϕ ) ] ,
Ψ 1 ( Ω , Ω ) = R L ( ϕ - ϕ ) S 1 ( Ω , Ω ) ,
j 1 ( z , Ω ) = R L ( ϕ - ϕ 0 ) j f ( z , Ω ) ,
d j f ( r , Ω ) d ρ = - σ t j f ( r , Ω ) + S 1 ( Ω , Ω ) j f ( r , Ω ) d Ω ,
j 2 ( z , Ω ) = s j 2 s ( z , θ ) exp [ i s ( ϕ - ϕ 0 ) ] ,
Ψ LP ( Ω , Ω ) = s Ψ LP s ( θ , θ ) exp [ i s ( ϕ - ϕ ) ] ,
Ψ LP s ( θ , θ ) = l P L ( s , l ; θ ) Q L ( l ) P L ( s , l ; θ ) ,
P L ( s , l ; θ ) = [ P s 0 l 0 0 0 0 P s 2 l ( + ) - i P s 2 l ( - ) 0 0 i p s 2 l ( - ) p s 2 l ( + ) 0 0 0 0 P s 0 ] ,
P s 2 l ( ± ) = ( 1 / 2 ) [ P s 2 l ( cos θ ) ± P s - 2 l ( cos θ ) ] ,
Ψ 1 s ( θ , θ ) = l P L ( s , l ; θ ) S f ( l ) P L ( s , l ; θ ) ,
cos θ d j 2 s ( z , θ ) d z = - σ t j 2 s ( z , θ ) + 2 π Ψ LP s ( θ , θ ) j 2 s ( z , θ ) sin θ d θ + W s ( z , θ ) ,
W s ( z , θ ) = 2 π [ Ψ LP s ( θ , θ ) - Ψ 1 s ( θ , θ ) ] j 1 s ( z , θ ) sin θ d θ , j 1 s ( z , θ ) = 1 2 [ 1 1 0 0 ] j f ( z , θ )             for s = 0 = 1 2 [ 0 1 ± i 0 ] j f ( z , θ )             for s = ± 2.
j 2 - s = ( j 2 s ) *
[ real imaginary imaginary real ]
j 2 - s = j 2 s for the first and second elements = - j 2 s for the third and fourth elements ,
j 2 s ( z , θ ) = l P L ( s , l ; θ ) j s l ( z ) .
j 2 s ( z , θ ) = l = s s + 1 P L ( s , l ; θ ) j s l ( z ) ,
j 2 0 ( z , θ ) = j 00 ( z ) + cos θ j 01 ( z ) .
I 2 0 ( z ) = [ j 00 ( z ) j 01 ( z ) ] ,
d d z I 2 0 ( z ) = M 2 0 I 2 0 ( z ) + F 2 0 ( z ) ,
M 2 0 = - [ 0 σ t ( 1 - q 1 ) 3 σ t ( 1 - q 0 ) 0 ] , F 2 0 ( z ) = 3 σ t 4 π [ q 1 - 1 2 q 0 - 1 2 ] exp ( - σ t z / 2 ) ,
q 0 = 4 π Q w + ( 0 ) / σ t = W 0 , q 1 = 4 π Q w + ( 1 ) / σ t = μ W 0 .
d 2 j 00 ( z ) d z 2 - 3 σ t 2 A 0 j 00 ( z ) = - 3 4 π σ t 2 B 0 exp ( - σ t z / 2 ) ,
A 0 = ( 1 - q 0 ) ( 1 - q 1 ) , B 0 = ( 1 - q 1 ) [ q 0 - ( 1 / 2 ) ] + ( 1 / 2 ) [ q 1 - ( 1 / 2 ) ] .
j 00 ( z ) = 2 3 σ t ( 1 - q 1 ) - 1 × { d j 00 ( z ) d z - 3 4 π σ t [ q 1 - ( 1 / 2 ) ] exp ( - σ t z / 2 ) } .
2 3 T d j 22 ( z ) d z + 5 7 C 1 d j 23 ( z ) d z = - A 2 j 22 ( z ) + 5 2 S 2 ( z ) , C 1 d j 22 ( z ) d z + 1 3 T d j 23 ( z ) d z = - A 3 j 23 ( z ) + 7 2 S 3 ( z ) ,
A 2 = σ t E - ( 4 π / 5 ) Q L ( 2 ) , A 3 = σ t E - ( 4 π / 7 ) Q L ( 3 ) ,
T = [ 0 0 0 0 0 0 - i 0 0 i 0 0 0 0 0 0 ] , C 1 = ( 1 / 3 ) [ 3 / 5 0 0 0 0 1 0 0 0 0 1 0 0 0 0 3 / 5 ] ,
S 2 ( z ) = 2 π d θ sin θ P L ( 2 , 2 ; θ ) × d θ sin θ [ Ψ LP 2 ( θ , θ ) - Ψ 1 2 ( θ , θ ) ] j 1 2 ( z , θ ) ( 1 / 5 ) [ Q L ( 2 ) - S f ( 2 ) ] [ 0 1 i 0 ] exp ( - σ t z / 2 ) , S 3 ( z ) = ( 1 / 7 ) [ Q L ( 3 ) - S f ( 3 ) ] [ 0 1 i 0 ] exp ( - σ t z / 2 ) .
Q L ( l ) = Q L 1 ( l ) + Q L 2 ( l ) ,
Q L - 1 = [ Q w + 0 Q v + Q v - 0 Q w - ] , Q L 2 = [ 0 Q v r 0 0 Q v r 0 0 0 0 0 0 - i Q v i 0 0 i Q v i 0 ] .
Q v r , i Q v i Q w + , Q w - .
d d z I 4 2 ( z ) = M 4 2 I 4 2 ( z ) + F 4 2 ,
I 4 2 ( z ) = [ j ˜ 22 ( z ) j ˜ 23 ( z ) ]
M 4 2 = ( 7 / 3 ) [ - T ˜ A ˜ 2 ( 5 / 7 ) A ˜ 3 A ˜ 2 - 2 T ˜ A ˜ 3 ] , F 4 2 = ( 35 / 6 ) [ T ˜ A ˜ 2 ( 0 ) - S ˜ 3 ( 0 ) - S ˜ 2 ( 0 ) + ( 14 / 5 ) T ˜ S ˜ 3 ( 0 ) ] exp ( - σ t z / 2 ) .
0 π / 2 P ˜ L ( s , l ; θ ) j ˜ 2 s ( z = 0 , θ ) cos θ sin θ d θ = 0 ,             l = 2.
I 4 2 p ( z ) = - ( σ t 2 + M 4 2 ) - 1 F 4 2 ,
I 4 2 c ( z ) = n = 1 2 c n β n exp ( λ n z ) ,
I 4 2 ( z ) = I 4 2 p ( z ) + I 4 2 c ( z ) .
j ˜ 2 2 ( z , θ ) = [ Q 2 U 2 ] = P ˜ L ( 2 , 2 ; θ ) I ˜ a ( z ) + P ˜ L ( 2 , 3 ; θ ) I ˜ b ( z ) ,
Ψ CP s ( θ , θ ) = l P ( s , l ; θ ) Q CP ( l ) P ( s , l ; θ ) ,
Q CP ( l ) = [ Q v + + Q v + - Q v - + Q v - - Q v + - Q w + + Q w - - Q v - + Q v - + Q w - - Q w + + Q v + - Q v - - Q v - + Q v + - Q v + + ] ,
T = [ 0 1 1 0 1 0 0 1 i 0 0 - i 0 - 1 1 0 ] ,
Q L ( l ) = T Q CP T - 1 = [ Q w + Q v r 0 0 Q v r Q v + 0 0 0 0 Q v - - i Q v i 0 0 i Q v i Q w - ] ,
Q w ± = Q w + + ± Q w - - , Q v ± = Q v + + ± Q v - - , Q v i = Q v + - - Q v - + , Q v r = Q v + - + Q v - + ,
a n = b n 1 / 2
S f = [ S w ( l ) 0 S v ( l ) S v ( l ) 0 S w ( l ) ] ,
S w ( l ) = ( 1 / 4 k 2 ) n = 1 n a n = 1 n a u ( n + n - l + δ ) C n a C n a W n n ( l ) , S v ( l ) = ( 1 / 4 k 2 ) n = 1 n a n = 1 n a u ( n + n - l + δ ) C n a C n a V n n ( l ) ,
C n a = ( 2 n + 1 ) / n ( n + 1 ) , W n n ( l ) = K ( n , n ) ( 1 , - 1 ; l , 0 ) 2 , V n n ( l ) = K ( n , n ) ( 1 , 1 ; l , 2 ) 2 , K ( n , n ) = n ( n + 1 ) n ( n + 1 ) ,

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