Abstract

Light scattering properties of spheroidal particles oriented randomly with their long axes in the horizontal are studied. A computational scheme has been developed to calculate the scattering matrices as well as the extinction and scattering cross sections and asymmetry factors in a manner consistent with our previous treatment of 3-D orientations. The single scattering properties strongly depend on the elevation angle of incident light. The dependence of the extinction and scattering cross sections is particularly prominent, while the dependence of the single scattering albedo and asymmetry factor is rather small but still significant. For a given elevation angle, the scattered intensity is a function of not only the scattering angle but also the azimuth angle of emergence. Implications of the anisotropic scattering by horizontally oriented nonspherical particles with respect to the elevation angle are also discussed.

© 1983 Optical Society of America

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References

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  1. M. Kerker, The Scattering of Light (Academic, New York, 1969), pp. 595–613 and 104–127.
  2. K. Sassen, J. Meteorol. Soc. Jpn. 58, 422 (1980).
  3. C. M. R. Piatt, J. Appl. Meteorol. 17, 482 (1978).
    [CrossRef]
  4. K.-N. Liou, J. Atmos. Sci. 29, 524 (1972).
    [CrossRef]
  5. G. L. Stephens, J. Atmos. Sci. 37, 435 (1980).
    [CrossRef]
  6. C. M. R. Platt, D. W. Reynolds, N. L. Abshire, Mon. Weather Rev. 108, 195 (1980).
    [CrossRef]
  7. K. Sassen, K.-N. Liou, J. Atmos. Sci. 36, 838 (1979).
    [CrossRef]
  8. R. F. Coleman, K.-N. Liou, J. Atmos. Sci. 38, 1260 (1981).
    [CrossRef]
  9. Q. Cai, K.-N. Liou, Appl. Opt. 21, 3569 (1982).
    [CrossRef] [PubMed]
  10. Y. Takano, S. Asano, submitted to J. Meteorol. Soc. Jpn.00, 000 (198x).
  11. S. Asano, M. Sato, Appl. Opt. 19, 962 (1980).
    [CrossRef] [PubMed]
  12. In Liou’s scheme, the integration range is limited to 0 ≤ β ≤ π/2 for symmetrical scattering bodies. This may, in general, cause erroneous results in computation of the scattering matrix.
  13. S. Asano, G. Yamamoto, Appl. Opt. 14, 29 (1975).
    [PubMed]
  14. S. Asano, Appl. Opt. 18, 712 (1979).
    [CrossRef] [PubMed]
  15. A. J. Gibson, L. Thomas, S. K. Bhattacharyya, J. Atmos. Terr. Phys. 39, 657 (1977).
    [CrossRef]
  16. P. Chýlek, J. Opt. Soc. Am. 67, 175 (1977).
    [CrossRef]
  17. The mean shadow area of horizontally oriented finite-sized particles such as circular and hexagonal columns and plates may take maximum values at intermediate elevation angles. For those particles, the elevation angle dependence of the scattering and extinction cross sections may be rather complicated in comparison with the case of spheroidal particles.
  18. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 103–113 and 179–183.
  19. G. L. Stephens, J. Atmos. Sci. 37, 2095 (1980).
    [CrossRef]
  20. S. Asano, submitted to J. Meteorol. Soc. Jpn.

1982 (1)

1981 (1)

R. F. Coleman, K.-N. Liou, J. Atmos. Sci. 38, 1260 (1981).
[CrossRef]

1980 (5)

G. L. Stephens, J. Atmos. Sci. 37, 2095 (1980).
[CrossRef]

K. Sassen, J. Meteorol. Soc. Jpn. 58, 422 (1980).

G. L. Stephens, J. Atmos. Sci. 37, 435 (1980).
[CrossRef]

C. M. R. Platt, D. W. Reynolds, N. L. Abshire, Mon. Weather Rev. 108, 195 (1980).
[CrossRef]

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

1979 (2)

S. Asano, Appl. Opt. 18, 712 (1979).
[CrossRef] [PubMed]

K. Sassen, K.-N. Liou, J. Atmos. Sci. 36, 838 (1979).
[CrossRef]

1978 (1)

C. M. R. Piatt, J. Appl. Meteorol. 17, 482 (1978).
[CrossRef]

1977 (2)

A. J. Gibson, L. Thomas, S. K. Bhattacharyya, J. Atmos. Terr. Phys. 39, 657 (1977).
[CrossRef]

P. Chýlek, J. Opt. Soc. Am. 67, 175 (1977).
[CrossRef]

1975 (1)

1972 (1)

K.-N. Liou, J. Atmos. Sci. 29, 524 (1972).
[CrossRef]

Abshire, N. L.

C. M. R. Platt, D. W. Reynolds, N. L. Abshire, Mon. Weather Rev. 108, 195 (1980).
[CrossRef]

Asano, S.

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

S. Asano, Appl. Opt. 18, 712 (1979).
[CrossRef] [PubMed]

S. Asano, G. Yamamoto, Appl. Opt. 14, 29 (1975).
[PubMed]

S. Asano, submitted to J. Meteorol. Soc. Jpn.

Y. Takano, S. Asano, submitted to J. Meteorol. Soc. Jpn.00, 000 (198x).

Bhattacharyya, S. K.

A. J. Gibson, L. Thomas, S. K. Bhattacharyya, J. Atmos. Terr. Phys. 39, 657 (1977).
[CrossRef]

Cai, Q.

Chýlek, P.

Coleman, R. F.

R. F. Coleman, K.-N. Liou, J. Atmos. Sci. 38, 1260 (1981).
[CrossRef]

Gibson, A. J.

A. J. Gibson, L. Thomas, S. K. Bhattacharyya, J. Atmos. Terr. Phys. 39, 657 (1977).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light (Academic, New York, 1969), pp. 595–613 and 104–127.

Liou, K.-N.

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

R. F. Coleman, K.-N. Liou, J. Atmos. Sci. 38, 1260 (1981).
[CrossRef]

K. Sassen, K.-N. Liou, J. Atmos. Sci. 36, 838 (1979).
[CrossRef]

K.-N. Liou, J. Atmos. Sci. 29, 524 (1972).
[CrossRef]

Piatt, C. M. R.

C. M. R. Piatt, J. Appl. Meteorol. 17, 482 (1978).
[CrossRef]

Platt, C. M. R.

C. M. R. Platt, D. W. Reynolds, N. L. Abshire, Mon. Weather Rev. 108, 195 (1980).
[CrossRef]

Reynolds, D. W.

C. M. R. Platt, D. W. Reynolds, N. L. Abshire, Mon. Weather Rev. 108, 195 (1980).
[CrossRef]

Sassen, K.

K. Sassen, J. Meteorol. Soc. Jpn. 58, 422 (1980).

K. Sassen, K.-N. Liou, J. Atmos. Sci. 36, 838 (1979).
[CrossRef]

Sato, M.

Stephens, G. L.

G. L. Stephens, J. Atmos. Sci. 37, 2095 (1980).
[CrossRef]

G. L. Stephens, J. Atmos. Sci. 37, 435 (1980).
[CrossRef]

Takano, Y.

Y. Takano, S. Asano, submitted to J. Meteorol. Soc. Jpn.00, 000 (198x).

Thomas, L.

A. J. Gibson, L. Thomas, S. K. Bhattacharyya, J. Atmos. Terr. Phys. 39, 657 (1977).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 103–113 and 179–183.

Yamamoto, G.

Appl. Opt. (4)

J. Appl. Meteorol. (1)

C. M. R. Piatt, J. Appl. Meteorol. 17, 482 (1978).
[CrossRef]

J. Atmos. Sci. (5)

K.-N. Liou, J. Atmos. Sci. 29, 524 (1972).
[CrossRef]

G. L. Stephens, J. Atmos. Sci. 37, 435 (1980).
[CrossRef]

K. Sassen, K.-N. Liou, J. Atmos. Sci. 36, 838 (1979).
[CrossRef]

R. F. Coleman, K.-N. Liou, J. Atmos. Sci. 38, 1260 (1981).
[CrossRef]

G. L. Stephens, J. Atmos. Sci. 37, 2095 (1980).
[CrossRef]

J. Atmos. Terr. Phys. (1)

A. J. Gibson, L. Thomas, S. K. Bhattacharyya, J. Atmos. Terr. Phys. 39, 657 (1977).
[CrossRef]

J. Meteorol. Soc. Jpn. (1)

K. Sassen, J. Meteorol. Soc. Jpn. 58, 422 (1980).

J. Opt. Soc. Am. (1)

Mon. Weather Rev. (1)

C. M. R. Platt, D. W. Reynolds, N. L. Abshire, Mon. Weather Rev. 108, 195 (1980).
[CrossRef]

Other (6)

Y. Takano, S. Asano, submitted to J. Meteorol. Soc. Jpn.00, 000 (198x).

In Liou’s scheme, the integration range is limited to 0 ≤ β ≤ π/2 for symmetrical scattering bodies. This may, in general, cause erroneous results in computation of the scattering matrix.

The mean shadow area of horizontally oriented finite-sized particles such as circular and hexagonal columns and plates may take maximum values at intermediate elevation angles. For those particles, the elevation angle dependence of the scattering and extinction cross sections may be rather complicated in comparison with the case of spheroidal particles.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 103–113 and 179–183.

M. Kerker, The Scattering of Light (Academic, New York, 1969), pp. 595–613 and 104–127.

S. Asano, submitted to J. Meteorol. Soc. Jpn.

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Figures (11)

Fig. 1
Fig. 1

Geometry of the scattering description for a prolate spheroid in a horizontal plane. Orientation of the spheroid is specified by the incidence angle ζ and azimuth angle χ in the XYZ coordinate system, where the incident wave vector (i)K is in the polar axis OZ, and the X axis is in the plane containing the incident direction and the zenith and/or nadir. Particle orientation is also specified by the elevation angle ɛ and the orientation angle β in the horizontal plane. The meanings of other symbols are the same as those in Asano and Sato.11

Fig. 2
Fig. 2

Variations of the polar angles (ζ,χ) and the angles θ, ϕ, and γ in Fig. 1, as a function of the orientation angle β, at the observation point (Ө,Φ) = (30°,45°) for elevation angle ɛ = 45°.

Fig. 3
Fig. 3

Angular distribution of the scattered intensity P(Ө,Φ) for unpolarized incident light, as a function of the scattering angle Ө, in three scattering planes, Φ = 0°, 90°, and 180°, for horizontally oriented prolate spheroids with m ˜ = 1.310, a/b = 3, and α = 24 for elevation angle ɛ = 45°.

Fig. 4
Fig. 4

Angular distribution of the degree of linear polarization p(Ө,Φ) for single scattering of unpolarized light for the same case as in Fig. 3.

Fig. 5
Fig. 5

Normalized phase functions averaged over azimuth angles Φ, as a function of the scattering angle Ө, for horizontally oriented prolate spheroids with m ˜ = 1.310, a/b = 3, and α = 24 for the incidence of unpolarized light at elevation angles ɛ = 5°, 45°, and 90°. The phase functions for the same spheroids in 3-D random orientation and for spheres of the same surface area (the mean size parameter of 12.5) are also shown, respectively, by thick and thin solid lines.

Fig. 6
Fig. 6

Angular distribution of the azimuthally averaged degree of linear polarization for single scattering of unpolarized incident light for the same case as in Fig. 5.

Fig. 7
Fig. 7

Normalized phase functions averaged over azimuth angles Φ, as a function of the scattering angle Ө, for horizontally oriented absorbing oblate spheroids with m ˜ = 1.290 + 0.0945 i, a/b = 5, and α = 24 for the incidence of unpolarized light at elevation angles ɛ = 0°, 45°, and 90°. The phase function for the same spheroids oriented randomly in 3-D space is shown by a thick solid line.

Fig. 8
Fig. 8

Backscattering linear depolarization ratios as a function of the elevation angle ɛ for horizontally oriented prolate spheroids with m ˜ = 1.310, α = 24, and a/b = 3, 4, and 5. δV and δH are the linear depolarization ratios at the backscattering for the incidence of waves polarized linearly in the directions perpendicular and parallel, respectively, to the scattering plane.

Fig. 9
Fig. 9

Extinction cross sections k 2 C ext ¯ and asymmetry factors cos θ ¯ , as a function of the elevation angle ɛ, for horizontally oriented prolate spheroids with m ˜ = 1.310, a/b = 3, and α = 8, 16, and 24. The extinction cross sections are given in nondimensional form by multiplying by k2, where k is the wave number of incident light. Values for the same spheroids but in 3-D random orientation are given by line segments with 3D.

Fig. 10
Fig. 10

Nondimensional cross sections for extinction k 2 C ext ¯ and scattering k 2 C sca ¯, albedos for single scattering ω ¯, and asymmetry factors cos θ ¯ as a function of the elevation angle ɛ for horizontally oriented prolate spheroids with m ˜ = 1.290 + 0.0945 i, a/b = 3, and α = 10 and 24. Values for the same spheroids but in 3-D random orientation are given by segments with 3D.

Fig. 11
Fig. 11

Nondimensional cross sections for extinction k 2 C ext ¯ and scattering k 2 C sca ¯, albedos for single scattering ω ¯, and asymmetry factors cos θ ¯ as a function of the elevation angle ɛ for horizontally oriented oblate spheroids with m ˜ = 1.290 + 0.0945 i, a/b = 5, and α = 10 and 24. Values for the same spheroids but in 3-D random orientation are given by line segments with 3D.

Equations (10)

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ζ ζ ( , β ) = cos 1 ( cos · cos β ) ,
χ χ ( , β ) = cos 1 ( sin · cos β / sin ζ ) , ( 0 π / 2 , 0 β 2 π ) .
F H ( θ, Φ ; ) = 1 2 π 0 2 π Z [ θ, Φ ; ζ ( , β ) , χ ( , β ) ] d β ,
ζ ( , β + π ) = π ζ ( , β ) , χ ( , β + π ) = π + χ ( , β ) , }
θ ( β + π ) = π ϕ ( β ) , ϕ ( β + π ) = 2 π ϕ ( β ) , γ ( β + π ) = π + γ ( β ) . }
C sca ( ζ , χ ) = 1 2 [ C 1 , sca ( ζ ) + C 2 , sca ( ζ ) ] + 1 2 [ C 2 , sca ( ζ ) C 1 , sca ( ζ ) ] · ( cos 2 χ · Q 0 I 0 sin 2 χ · U 0 I 0 ) ,
C sca ¯ ( ) = 1 π 0 π / 2 { C 1 , sca [ ζ ( , β ) ] + C 2 , sca [ ζ ( , β ) ] } d β .
C ext ¯ ( ) = 1 π 0 π / 2 { C 1 , ext [ ζ ( , β ) ] + C 2 , ext [ ζ ( , β ) ] } d β ,
cos θ ¯ = 1 C sca ¯ ( ) [ 1 π 0 π / 2 cos ζ ( , β ) · ( { cos θ 1 · C 1 , sca [ ζ ( , β ) ] } A + { cos θ 2 · C 2 , sca [ ζ ( , β ) ] } A ) d β + 1 π 0 π / 2 sin ζ ( , β ) · ( { cos θ 1 · C 1 , sca [ ζ ( , β ) ] } B + { cos θ 2 · C 2 , sca [ ζ ( , β ) ] } B ) d β ] .
p ( θ, Φ ) = f ˜ 21 / f ˜ 11 ,

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