Abstract

In this paper, we have modified the geometric ray tracing theory for the scattering of light by hexagonal cylinders to cubes and parallelepipeds. Effects of the real and imaginary parts of the refractive index and aspect ratio of the particle on the scattering phase function and the degree of linear polarization are investigated. Causes of the physical features in the scattering polarization patterns are identified in terms of the scattering contribution due to geometric reflections and refractions. The single-scattering phase function and polarization data presented in this paper should be of some use for the interpretation of observed scattering and polarization data from planetary atmospheres and for the physical understanding of the transfer of radiation in an atmosphere containing nonspherical particles.

© 1983 Optical Society of America

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References

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  1. S. Asano, M. Sato, Appl. Opt. 19, 962 (1980).
    [CrossRef] [PubMed]
  2. P. W. Barber, C. Yeh, Appl. Opt. 14, 2864 (1975).
    [CrossRef] [PubMed]
  3. J. B. Pollack, J. N. Cuzzi, J. Atmos. Sci. 37, 868 (1980).
    [CrossRef]
  4. Q. Cai, K. N. Liou, Appl. Opt. 21, 3569 (1982).
    [CrossRef] [PubMed]
  5. K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980), p. 146.
  6. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 43.
  7. V. Vonk, Nature London 162, 330 (1948).
    [CrossRef]
  8. R. H. Zerull, R. H. Giese, in Planets, Stars and Nebulae Studied with Photopolarimetry, T. Gehrels, Ed. (U. Arizona Press, Tucson, 1974), p. 901.
  9. R. J. Perry, A. J. Hunt, D. R. Huffman, Appl. Opt. 17, 2700 (1978).
    [CrossRef] [PubMed]
  10. I. Kirmaci, G. Ward, Appl. Opt. 18, 3328 (1979).
    [CrossRef] [PubMed]
  11. K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
    [CrossRef]
  12. K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 852 (1979).
    [CrossRef]

1982 (1)

1980 (2)

J. B. Pollack, J. N. Cuzzi, J. Atmos. Sci. 37, 868 (1980).
[CrossRef]

S. Asano, M. Sato, Appl. Opt. 19, 962 (1980).
[CrossRef] [PubMed]

1979 (2)

I. Kirmaci, G. Ward, Appl. Opt. 18, 3328 (1979).
[CrossRef] [PubMed]

K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 852 (1979).
[CrossRef]

1978 (1)

1975 (1)

1971 (1)

K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

1948 (1)

V. Vonk, Nature London 162, 330 (1948).
[CrossRef]

Asano, S.

Barber, P. W.

Cai, Q.

Cuzzi, J. N.

J. B. Pollack, J. N. Cuzzi, J. Atmos. Sci. 37, 868 (1980).
[CrossRef]

Giese, R. H.

R. H. Zerull, R. H. Giese, in Planets, Stars and Nebulae Studied with Photopolarimetry, T. Gehrels, Ed. (U. Arizona Press, Tucson, 1974), p. 901.

Hansen, J. E.

K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

Huffman, D. R.

Hunt, A. J.

Kirmaci, I.

Liou, K. N.

Q. Cai, K. N. Liou, Appl. Opt. 21, 3569 (1982).
[CrossRef] [PubMed]

K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 852 (1979).
[CrossRef]

K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980), p. 146.

Perry, R. J.

Pollack, J. B.

J. B. Pollack, J. N. Cuzzi, J. Atmos. Sci. 37, 868 (1980).
[CrossRef]

Sassen, K.

K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 852 (1979).
[CrossRef]

Sato, M.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 43.

Vonk, V.

V. Vonk, Nature London 162, 330 (1948).
[CrossRef]

Ward, G.

Yeh, C.

Zerull, R. H.

R. H. Zerull, R. H. Giese, in Planets, Stars and Nebulae Studied with Photopolarimetry, T. Gehrels, Ed. (U. Arizona Press, Tucson, 1974), p. 901.

Appl. Opt. (5)

J. Atmos. Sci. (3)

J. B. Pollack, J. N. Cuzzi, J. Atmos. Sci. 37, 868 (1980).
[CrossRef]

K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 852 (1979).
[CrossRef]

Nature London (1)

V. Vonk, Nature London 162, 330 (1948).
[CrossRef]

Other (3)

R. H. Zerull, R. H. Giese, in Planets, Stars and Nebulae Studied with Photopolarimetry, T. Gehrels, Ed. (U. Arizona Press, Tucson, 1974), p. 901.

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980), p. 146.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 43.

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Figures (7)

Fig. 1
Fig. 1

Geometry of the orientation of a parallelepiped with respect to the incident electric vector of a geometric ray. The incident electric vector is described by the OX′Y′Z′ coordinate, while the orientation of the rectangle is fixed in the OXYZ coordinate.

Fig. 2
Fig. 2

Scattering phase functions for random oriented cubes with sizes of 100 μm using a wavelength of 0.55 μm and a refractive index of 1.31 corresponding to external reflection (p = 0), two refractions (p = 1), and internal reflections (p ≥ 2).

Fig. 3
Fig. 3

Effects of the real refractive index on the scattering phase function for randomly oriented cubes as a function of the scattering angle.

Fig. 4
Fig. 4

Effects of the aspect ratio on the scattering phase function for randomly oriented particles as a function of the scattering angle for refractive indices of 1.31 (a) and 2 (b).

Fig. 5
Fig. 5

(a) Effects of the imaginary refractive index on the scattering phase function for randomly oriented cubes as a function of the scattering angle using a real refractive index of 1.57. (b) Comparisons of the present results with those calculated by Pollack and Cuzzi based on a semi-empirical theory and measured by Zerull and Giese from a microwave analog experiment.

Fig. 6
Fig. 6

Effects of the real mr and imaginary mi refractive index on the degree of linear polarization for randomly oriented cubes as a function of the scattering angle: (a) real part (mi = 0); (b) imaginary part (mr = 1.57).

Fig. 7
Fig. 7

Influence of the particle shape on the linear polarization pattern as a function of the scattering angle. Included for comparisons in the figure are cubes, parallelepipeds, hexagons, and spheres. All the polarization curves are computed from geometric ray tracing programs.

Tables (2)

Tables Icon

Table I Percentage of the Total Scattered Energy for Various Refractive Indices and Sizes. The Internal Reflections up to Four are used in the Calculations

Tables Icon

Table II Single-Scattering Albedos ω 0 and Asymmetry Factors g for Randomly Oriented Cubes with a Real Refractive Index of 1.57

Equations (19)

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x = b / 2 y = a / 2 x = b / 2 y = a / 2 z = h / 2 z = h / 2 }
cos α n = cos n π / 2 cos β n = sin n π / 2 cos γ n = 0 } n = 0 , 1 , 2 , 3 , cos α n = 0 cos β n = 0 cos γ n = cos [ ( n 4 ) π ] } n = 4 , 5 ,
u p = j u 0 λ r B exp ( jkr ) d x d y ,
u p = j u 0 k 2 λ r i = 1 6 ( g i P i C i h i P i D i ) .
[ E l E r ] Z OP = [ A 2 A 3 A 4 A 1 ] [ E x 0 E y 0 ] Z O X ,
A = A f + A s = [ A 2 f A 3 f A 4 f A 1 f ] + [ A 2 s A 3 s A 4 s A 1 s ] ,
[ I Q U V ] = F ( θ , ϕ ) [ I 0 Q 0 U 0 V 0 ] ;
P 11 = 4 π σ s F 11 ,
F 11 = 1 2 k = 1 4 | A k | 2 ,
σ s = 0 2 π 0 π ( E l E l * + E r E r * ) sin θ d θ d ϕ .
4 π P 11 ( Ω ) d Ω / 4 π = 1 .
P 11 ( θ , ϕ ; η , ψ 2 ) = 1 2 π 0 2 π P 11 ( θ , ϕ ; η , ψ 2 , ψ 1 ) d ψ 1 ,
P 11 ( θ ) = 1 4 π 0 2 π 0 π P 11 ( θ , 0 ; η , ψ 2 ) sin η d η d ψ 2 ,
L P ( θ ) = P 12 ( θ ) P 11 ( θ ) ,
ω 0 = σ sca / σ ext , g = 1 2 1 1 P ( cos θ ) cos θ d cos θ ,
ω 0 = 1
ω 0 = 0.926
ω 0 = 0.870
ω 0 = 0.555

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