Abstract

The classical ray optics approximation has been applied to compute the angular scattering of light by finite-sized hexagonal ice crystals in the form of columns and plates. The results are presented at a wavelength of 0.55 μm for a random orientation of the crystals either in space or in a plane. The results are also compared to those of ice spheres. For the first time the angular light scattering of platelike crystals and a quantitative description of the 46° halo are given. Contrary to earlier studies, it is shown that both plate- and columnlike ice crystals show a strong backscattering. In agreement with previous studies, ice spheres are found to scatter—when compared to ice prisms—less energy at angles near 90°. With regard to the effect of orientation on the light scattering, it is shown that columnlike crystals randomly oriented in a plane behave rather like spherical particles and not like columns randomly oriented in space.

© 1979 Optical Society of America

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References

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  1. K. N. Liou, J. Atmos. Sci. 29, 524 (1972).
    [CrossRef]
  2. K. Sassen, K. N. Liou, S. Hunter, “Scientific Report I” (U. of Utah, Department of Meteorology, Salt Lake City, 1977).
  3. H. Jacobowitz, J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
    [CrossRef]
  4. K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964).
  6. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  7. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  8. P. V. Hobbs, Ice Physics (Clarendon Press, Oxford, 1974).
  9. R. C. Weast, Ed., Handbook of Chemistry and Physics (CRC Press, Cleveland, 1966–67).
  10. A. Ono, J. Atmos. Sci. 26, 138 (1969).
    [CrossRef]
  11. P. Huffman, W. R. Thursby, J. Atmos. Sci. 26, 1073 (1969).
    [CrossRef]
  12. J. R. Hodkinson, I. Greensleaves, J. Opt. Soc. Am. 53, 577 (1963).
    [CrossRef]
  13. N. K. Nikiforova, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol Sci. 8, 243 (1977).
    [CrossRef]

1977 (1)

N. K. Nikiforova, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol Sci. 8, 243 (1977).
[CrossRef]

1972 (1)

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

1971 (2)

H. Jacobowitz, J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
[CrossRef]

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

1969 (2)

A. Ono, J. Atmos. Sci. 26, 138 (1969).
[CrossRef]

P. Huffman, W. R. Thursby, J. Atmos. Sci. 26, 1073 (1969).
[CrossRef]

1963 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964).

Greensleaves, I.

Hansen, J. E.

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

Hobbs, P. V.

P. V. Hobbs, Ice Physics (Clarendon Press, Oxford, 1974).

Hodkinson, J. R.

Huffman, P.

P. Huffman, W. R. Thursby, J. Atmos. Sci. 26, 1073 (1969).
[CrossRef]

Hunter, S.

K. Sassen, K. N. Liou, S. Hunter, “Scientific Report I” (U. of Utah, Department of Meteorology, Salt Lake City, 1977).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Jacobowitz, H.

H. Jacobowitz, J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
[CrossRef]

Liou, K. N.

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

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

K. Sassen, K. N. Liou, S. Hunter, “Scientific Report I” (U. of Utah, Department of Meteorology, Salt Lake City, 1977).

Nikiforova, N. K.

N. K. Nikiforova, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol Sci. 8, 243 (1977).
[CrossRef]

Ono, A.

A. Ono, J. Atmos. Sci. 26, 138 (1969).
[CrossRef]

Pavlova, L. N.

N. K. Nikiforova, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol Sci. 8, 243 (1977).
[CrossRef]

Petrushin, A. G.

N. K. Nikiforova, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol Sci. 8, 243 (1977).
[CrossRef]

Sassen, K.

K. Sassen, K. N. Liou, S. Hunter, “Scientific Report I” (U. of Utah, Department of Meteorology, Salt Lake City, 1977).

Snykov, V. P.

N. K. Nikiforova, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol Sci. 8, 243 (1977).
[CrossRef]

Thursby, W. R.

P. Huffman, W. R. Thursby, J. Atmos. Sci. 26, 1073 (1969).
[CrossRef]

Van de Hulst, H. C.

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

Volkovitsky, O. A.

N. K. Nikiforova, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol Sci. 8, 243 (1977).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964).

J. Aerosol Sci. (1)

N. K. Nikiforova, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol Sci. 8, 243 (1977).
[CrossRef]

J. Atmos. Sci. (4)

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

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

A. Ono, J. Atmos. Sci. 26, 138 (1969).
[CrossRef]

P. Huffman, W. R. Thursby, J. Atmos. Sci. 26, 1073 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

H. Jacobowitz, J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
[CrossRef]

Other (6)

K. Sassen, K. N. Liou, S. Hunter, “Scientific Report I” (U. of Utah, Department of Meteorology, Salt Lake City, 1977).

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964).

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

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

P. V. Hobbs, Ice Physics (Clarendon Press, Oxford, 1974).

R. C. Weast, Ed., Handbook of Chemistry and Physics (CRC Press, Cleveland, 1966–67).

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Figures (11)

Fig. 1
Fig. 1

Scattering geometry.

Fig. 2
Fig. 2

Method for finding crystal face hit by the refracted ray.

Fig. 3
Fig. 3

Energy per unit scattering angle reflected and refracted from hexagonal ice columns as function of scattering angle. The radiation is incident perpendicularly to the c axis of the columns.

Fig. 4
Fig. 4

Contributions from selected rays to scattering peaks of hexagonal ice prisms due to refraction and internal reflections (incident radiation perpendicular to c axis, T = total reflection).

Fig. 5
Fig. 5

Energy per unit scattering angle reflected and refracted from hexagonal ice columns for different column lengths as function of scattering angle. The columns are randomly oriented in space.

Fig. 6
Fig. 6

Angular scattering due to diffraction, reflection, and refraction for hexagonal ice columns randomly oriented in space as function of scattering angle and column length c. The diffraction peak is calculated for a Gaussian distribution of the column radius a. The mode radius is am = 30 μm and σa = 2.55.

Fig. 7
Fig. 7

Angular scattering for hexagonal ice columns randomly oriented in space and for columns randomly oriented in a plane (incident radiation normal to c axis of columns) as function of scattering angle. (Diffraction peak for Gaussian distribution according to variation of column radius a: am = 30 μm, σa = 2.55.)

Fig. 8
Fig. 8

Angular scattering functions for hexagonal ice columns and for optically equivalent ice spheres. The columns are randomly oriented in space. The results for ice sphere are obtained from Mie theory for a Gaussian distribution of the radii of the spheres (rm = 61.7 μm, σr = 5.24). For the diffraction peak we varied the column radii according to the same kind of distribution (am = 30 μm, σa = 2.55).

Fig. 9
Fig. 9

Angular scattering for hexagonal ice plates randomly oriented in space as function of scattering angle and crystal size. (Diffraction peak for Gaussian distribution according to variation of plate radius a: am = 25 μm, σa = 2.12; am = 200 μm, σa = 17.0; am = 500 μm, σa = 42.5.)

Fig. 10
Fig. 10

Angular scattering functions for hexagonal ice columns, hexagonal ice plates, and ice spheres of the same volume. The columns and plates are assumed to be randomly oriented in space. The results for ice spheres are obtained from Mie theory for a Gaussian distribution of the radii of the spheres (rm = 51.2 μm, σr = 4.35). For the diffraction peak we used the same distribution and varied the column and the plate radii according to: am (plate) = 100 μm, σ a P = 8.49; am (column) = 30 μm, σ a c = 2.55.

Fig. 11
Fig. 11

Comparison of measured and calculated angular scattering functions for hexagonal ice columns and plates randomly oriented in space. The scattered intensities are normalized to unity at the scattering angle of 10°. The incident light is unpolarized with a wavelength of 0.63 μm. (Diffraction peak for Gaussian distribution of column and plate radii according to: am (plate) = 20 μm, σ a P = 5.1; am (column) = 15 μm, σ a c = 4.25.) The calculations are performed for a refractive index for ice of n = 1.308.

Equations (13)

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1 4 π Ω G ( θ ) d Ω = 1.
G 1 , 2 = G 1 , 2 F + p = 0 N G 1 , 2 P
G = ( G 1 + G 2 ) / 2.
r 1 = - sin ( θ i - θ t ) sin ( θ i + θ t ) ,             r 2 = tan ( θ i - θ t ) tan ( θ i + θ t ) .
cos θ = sin ϑ sin ϑ 0 cos ( φ 0 - φ ) + cos ϑ 0 cos ϑ .
φ 0 = π 2 × R N ,             cos ϑ 0 = 1 - R N ,
φ 0 = π 2 × R N ,             cos φ 0 = 0.
A 1 / i = 1 N A i then to ( A 1 + A 2 ) / i = 1 N A i
( A 1 + A 2 + A j ) / i = 1 N A i             ( j N ) ,
G 1 F c = G 2 F c = 2 π 4 k 2 × c × a 0 π sin 2 ϑ 0 [ 0 2 π sin 2 ( X cos Φ ) ( X cos Φ ) 2 × sin 2 ( Y sin Φ ) ( Y sin Φ ) 2 d Φ ] d ϑ 0 ,
G 1 F p = G 2 F p = ( k a ) 2 π 3 0 π 0 2 π ( cos ϑ 0 + cos ζ ) 2 | J 1 ( k a ξ ) k a ξ | 2 d Φ d ϑ 0 ,
ξ = [ sin 2 θ + cos 2 ϑ 0 ( cos 2 θ - cos 2 Φ sin 2 θ - 2 cos θ + 1 ) - 2 sin ϑ 0 cos ϑ 0 cos Φ sin θ ( cos θ - 1 ) ] 1 / 2 .
Ω P 11 ( θ ) d Ω = 4 π .

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