Abstract

The scattering phase function and the degree of linear polarization for small crystals oriented randomly in space have been computed using the geometric ray tracing theory and assuming that the crystals are homogeneous and isotropic. Calculations have been carried out for the main crystal geometries. Detection of halos from crystals other than hexagonal water ice is briefly discussed. The crystal size and shape parameters have also been averaged over some simple distributions in order to examine general light scattering properties of sharp-edged particles. A scalar physical optics correction has been developed for the geometric optics phase functions. Results can be applied to light scattering from regoliths and planetary rings, and possibly also to atmospheric halos. Retroreflecting crystals in the regolith would cause an opposition spike, a phenomenon observed for many bright satellites.

© 1989 Optical Society of America

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References

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  1. H. Jacobowitz, “A Method for Computing the Transfer of Solar Radiation through Clouds of Hexagonal Ice Crystals,” J. Quant. Spectrosc. Radiat. Transfer 11, 691–695 (1971).
    [CrossRef]
  2. P. Wendling, R. Wendling, H. K. Weickmann, “Scattering of Solar Radiation by Hexagonal Ice Cyrstals,” Appl. Opt. 18, 2663–2671 (1979).
    [CrossRef] [PubMed]
  3. K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).
  4. Q. Cai, K. N. Liou, “Polarized Light Scattering by Hexagonal Ice Crystals: Theory,” Appl. Opt. 21, 3569–3580 (1982).
    [CrossRef] [PubMed]
  5. K. N. Liou, Q. Cai, P. W. Barber, S. C. Hill, “Scattering Phase Matrix Comparison for Randomly Hexagonal Cylinders and Spheroids,” Appl. Opt. 22, 1684–1687 (1983).
    [CrossRef] [PubMed]
  6. K. N. Liou, Q. Cai, J. B. Pollack, J. N. Cuzzi, “Light Scattering by Randomly Oriented Cubes and Parallelepipeds,” Appl. Opt. 22, 3001–3008 (1983).
    [CrossRef] [PubMed]
  7. Y. Takano, K. Jayaweera, “Scattering Phase Matrix for Hexagonal Ice Crystals Computed from Ray Optics,” Appl. Opt. 24, 3254–3263 (1985).
    [CrossRef] [PubMed]
  8. L. G. Berry, B. Mason, R. V. Dietrich, Mineralogy (Freeman, San Francisco, 1983).
  9. F. Pattloch, E. Tränkle, “Monte Carlo Simulation and Analysis of Halo Phenomena,” J. Opt. Soc. Am. A 1, 520–526 (1984).
    [CrossRef]
  10. E. Whalley, G. E. McLaurin, “Refraction Halos in the Solar System: I. Halos from Cubic Crystals That May Occur in Atmospheres in the Solar System,” J. Opt. Soc. Am. A 12, 1166–1170 (1984).
    [CrossRef]
  11. J. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of Light by Stochastically Rough Particles,” submitted to Appl. Opt.
  12. T. S. Trowbridge, “Retroreflection from Rough Surfaces,” J. Opt. Soc. Am. 9, 1225–1241 (1978).
    [CrossRef]
  13. K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, E. Bowell, “A Possible Explanation for the Anomalously Sharp Opposition Effects,” Bull. Am. Astron. Soc. 19, 850 (1987).
  14. B. Lyot, “Recherches sur la Polarisation de la Lumieres des Planete et de Quelques Substances Terrestres,” Ann. Obs. Paris, 8, 1 (1929).
  15. B. Zellner, J. Gradie, “Minor Planets and Related Objects. XX: Polarimetric Evidence for the Albedos and Compositions of Asteroids,” Astron. J. 81, 262–280 (1976).
    [CrossRef]
  16. B. Zellner, “Minor Planets and Related Objects. VIII: Deimos.,” Astron. J. 77, 183–185 (1972).
    [CrossRef]
  17. A. Dollfus, “Optical Polarimetry of the Galilean Satellites of Jupiter,” Icarus 25, 416–431 (1975).
    [CrossRef]
  18. Y. Ohman, “A Tentative Explanation of the Negative Polarization in Diffuse Reflection,” Stockholm Obs. Ann. 18, 1–10 (1955).
  19. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  20. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  21. K. Muinonen, “Scattering of Light by Crystals: A Modified Kirchhoff Approximation,” Appl. Opt.28, 0000–0000 (1Aug.1989), to be published same issue.
  22. Y. Kuga, A. Ishimaru, “Retroreflectance from a Dense Distribution of Spherical Particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
    [CrossRef]
  23. L. Tsang, A. Ishimaru, “Backscattering Enhancement of Random Discrete Scatterers,” J. Opt. Soc. Am. A 1, 836–839 (1984).
    [CrossRef]
  24. K. Muinonen, “Electromagnetic Scattering by Two Interacting Dipoles,” in Proceedings, 1989 URSI EM Theory Symposium (1989).

1987 (1)

K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, E. Bowell, “A Possible Explanation for the Anomalously Sharp Opposition Effects,” Bull. Am. Astron. Soc. 19, 850 (1987).

1985 (1)

1984 (4)

1983 (2)

1982 (1)

1979 (1)

1978 (1)

T. S. Trowbridge, “Retroreflection from Rough Surfaces,” J. Opt. Soc. Am. 9, 1225–1241 (1978).
[CrossRef]

1976 (1)

B. Zellner, J. Gradie, “Minor Planets and Related Objects. XX: Polarimetric Evidence for the Albedos and Compositions of Asteroids,” Astron. J. 81, 262–280 (1976).
[CrossRef]

1975 (1)

A. Dollfus, “Optical Polarimetry of the Galilean Satellites of Jupiter,” Icarus 25, 416–431 (1975).
[CrossRef]

1972 (1)

B. Zellner, “Minor Planets and Related Objects. VIII: Deimos.,” Astron. J. 77, 183–185 (1972).
[CrossRef]

1971 (1)

H. Jacobowitz, “A Method for Computing the Transfer of Solar Radiation through Clouds of Hexagonal Ice Crystals,” J. Quant. Spectrosc. Radiat. Transfer 11, 691–695 (1971).
[CrossRef]

1955 (1)

Y. Ohman, “A Tentative Explanation of the Negative Polarization in Diffuse Reflection,” Stockholm Obs. Ann. 18, 1–10 (1955).

1929 (1)

B. Lyot, “Recherches sur la Polarisation de la Lumieres des Planete et de Quelques Substances Terrestres,” Ann. Obs. Paris, 8, 1 (1929).

Barber, P. W.

Berry, L. G.

L. G. Berry, B. Mason, R. V. Dietrich, Mineralogy (Freeman, San Francisco, 1983).

Bowell, E.

K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, E. Bowell, “A Possible Explanation for the Anomalously Sharp Opposition Effects,” Bull. Am. Astron. Soc. 19, 850 (1987).

Cai, Q.

Cuzzi, J. N.

Dietrich, R. V.

L. G. Berry, B. Mason, R. V. Dietrich, Mineralogy (Freeman, San Francisco, 1983).

Dollfus, A.

A. Dollfus, “Optical Polarimetry of the Galilean Satellites of Jupiter,” Icarus 25, 416–431 (1975).
[CrossRef]

Gradie, J.

B. Zellner, J. Gradie, “Minor Planets and Related Objects. XX: Polarimetric Evidence for the Albedos and Compositions of Asteroids,” Astron. J. 81, 262–280 (1976).
[CrossRef]

Hill, S. C.

Irvine, W. M.

J. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of Light by Stochastically Rough Particles,” submitted to Appl. Opt.

Ishimaru, A.

Jackson, J. D.

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

Jacobowitz, H.

H. Jacobowitz, “A Method for Computing the Transfer of Solar Radiation through Clouds of Hexagonal Ice Crystals,” J. Quant. Spectrosc. Radiat. Transfer 11, 691–695 (1971).
[CrossRef]

Jayaweera, K.

Karttunen, H.

K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, E. Bowell, “A Possible Explanation for the Anomalously Sharp Opposition Effects,” Bull. Am. Astron. Soc. 19, 850 (1987).

Kuga, Y.

Liou, K. N.

Lumme, K.

K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, E. Bowell, “A Possible Explanation for the Anomalously Sharp Opposition Effects,” Bull. Am. Astron. Soc. 19, 850 (1987).

J. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of Light by Stochastically Rough Particles,” submitted to Appl. Opt.

Lyot, B.

B. Lyot, “Recherches sur la Polarisation de la Lumieres des Planete et de Quelques Substances Terrestres,” Ann. Obs. Paris, 8, 1 (1929).

Mason, B.

L. G. Berry, B. Mason, R. V. Dietrich, Mineralogy (Freeman, San Francisco, 1983).

McLaurin, G. E.

E. Whalley, G. E. McLaurin, “Refraction Halos in the Solar System: I. Halos from Cubic Crystals That May Occur in Atmospheres in the Solar System,” J. Opt. Soc. Am. A 12, 1166–1170 (1984).
[CrossRef]

Muinonen, K.

K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, E. Bowell, “A Possible Explanation for the Anomalously Sharp Opposition Effects,” Bull. Am. Astron. Soc. 19, 850 (1987).

J. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of Light by Stochastically Rough Particles,” submitted to Appl. Opt.

K. Muinonen, “Scattering of Light by Crystals: A Modified Kirchhoff Approximation,” Appl. Opt.28, 0000–0000 (1Aug.1989), to be published same issue.

K. Muinonen, “Electromagnetic Scattering by Two Interacting Dipoles,” in Proceedings, 1989 URSI EM Theory Symposium (1989).

Ohman, Y.

Y. Ohman, “A Tentative Explanation of the Negative Polarization in Diffuse Reflection,” Stockholm Obs. Ann. 18, 1–10 (1955).

Pattloch, F.

Peltoniemi, J.

K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, E. Bowell, “A Possible Explanation for the Anomalously Sharp Opposition Effects,” Bull. Am. Astron. Soc. 19, 850 (1987).

J. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of Light by Stochastically Rough Particles,” submitted to Appl. Opt.

Pollack, J. B.

Takano, Y.

Tränkle, E.

Trowbridge, T. S.

T. S. Trowbridge, “Retroreflection from Rough Surfaces,” J. Opt. Soc. Am. 9, 1225–1241 (1978).
[CrossRef]

Tsang, L.

van de Hulst, H. C.

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

Weickmann, H. K.

Wendling, P.

Wendling, R.

Whalley, E.

E. Whalley, G. E. McLaurin, “Refraction Halos in the Solar System: I. Halos from Cubic Crystals That May Occur in Atmospheres in the Solar System,” J. Opt. Soc. Am. A 12, 1166–1170 (1984).
[CrossRef]

Zellner, B.

B. Zellner, J. Gradie, “Minor Planets and Related Objects. XX: Polarimetric Evidence for the Albedos and Compositions of Asteroids,” Astron. J. 81, 262–280 (1976).
[CrossRef]

B. Zellner, “Minor Planets and Related Objects. VIII: Deimos.,” Astron. J. 77, 183–185 (1972).
[CrossRef]

Ann. Obs. Paris (1)

B. Lyot, “Recherches sur la Polarisation de la Lumieres des Planete et de Quelques Substances Terrestres,” Ann. Obs. Paris, 8, 1 (1929).

Appl. Opt. (5)

Astron. J. (2)

B. Zellner, J. Gradie, “Minor Planets and Related Objects. XX: Polarimetric Evidence for the Albedos and Compositions of Asteroids,” Astron. J. 81, 262–280 (1976).
[CrossRef]

B. Zellner, “Minor Planets and Related Objects. VIII: Deimos.,” Astron. J. 77, 183–185 (1972).
[CrossRef]

Bull. Am. Astron. Soc. (1)

K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, E. Bowell, “A Possible Explanation for the Anomalously Sharp Opposition Effects,” Bull. Am. Astron. Soc. 19, 850 (1987).

Icarus (1)

A. Dollfus, “Optical Polarimetry of the Galilean Satellites of Jupiter,” Icarus 25, 416–431 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

T. S. Trowbridge, “Retroreflection from Rough Surfaces,” J. Opt. Soc. Am. 9, 1225–1241 (1978).
[CrossRef]

J. Opt. Soc. Am. A (4)

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

H. Jacobowitz, “A Method for Computing the Transfer of Solar Radiation through Clouds of Hexagonal Ice Crystals,” J. Quant. Spectrosc. Radiat. Transfer 11, 691–695 (1971).
[CrossRef]

Stockholm Obs. Ann. (1)

Y. Ohman, “A Tentative Explanation of the Negative Polarization in Diffuse Reflection,” Stockholm Obs. Ann. 18, 1–10 (1955).

Other (7)

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

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

K. Muinonen, “Scattering of Light by Crystals: A Modified Kirchhoff Approximation,” Appl. Opt.28, 0000–0000 (1Aug.1989), to be published same issue.

J. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of Light by Stochastically Rough Particles,” submitted to Appl. Opt.

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

L. G. Berry, B. Mason, R. V. Dietrich, Mineralogy (Freeman, San Francisco, 1983).

K. Muinonen, “Electromagnetic Scattering by Two Interacting Dipoles,” in Proceedings, 1989 URSI EM Theory Symposium (1989).

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Figures (7)

Fig. 1
Fig. 1

Parallelepiped (a) and hexagonal crystals (b).

Fig. 2
Fig. 2

Scattering phase function and degree of linear polarization for ice plate, column and cube. For crystal parameters, peaks and halos see text and Table I. Notice the negative polarization in the backward scattering domain.

Fig. 3
Fig. 3

Phase function and polarization for orthoclase and anorthite crystals. The backscattering peak and the negative polarization characterize the backward scattering of the monoclinic crystal.

Fig. 4
Fig. 4

Phase function and polarization for enstatite and forsterite crystals. The shape of the enstatite crystal differs only slightly from orthorhombic, which causes the turbulent behavior in the polarization curve.

Fig. 5
Fig. 5

Phase function and polarization for augite crystal. The phase function is normalized to 4π ω ˜0 instead of 4π. Increasing absorption decreases the level of the phase function, increases polarization, and destroys the halos.

Fig. 6
Fig. 6

Phase function and polarization for distributions of sharp-edged ice particles with m = 1.31. The smooth phase function and the small polarization in the last case closely resemble the scattering by stochastically rough particles.11

Fig. 7
Fig. 7

Phase function and polarization for distributions of sharp-edged silicate particles with m = 1.55. The backscattering peak vanishes and there results a slight increase in the phase function towards backward direction.

Tables (3)

Tables Icon

Table I Refractive indices, m, axial ratios c:2a for hexagonal crystals and a:b:c for parallelepiped ones, Interfacial angles α,β,γ in degrees, angles for specular features in degrees, phase function values at the forward and backward directions, polarization minimums Pmin and maximums Pmax, backscattering peak parameters B, C and T defined in Sec. III and asymmetry factors g for crystals and gs for equal-refractive-index spheres for the nonabsorbing cases

Tables Icon

Table II Refractive indices m, absorption factors nka, single scattering albedos ω ˜0, phase function values at the forward and backward directions, polarization minimums Pmin and maximums Pmax and asymmetry factors g for the triclinic augite crystal (see Table I)

Tables Icon

Table III Refractive indices m, statistical parameters defined in Eq. (8), phase function values at the forward and backward directions, polarization minimums Pmin and maximums Pmax and asymmetry factors g for sharp-edged particles and gs for equal-refractive-index spheres for the averaged scattering characteristics (Sec. IV)

Equations (14)

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[ x y z ] = [ cos τ cos λ cos τ sin λ sin τ sin λ cos λ 0 sin τ cos λ sin τ sin λ cos τ ] [ x y z ] ,
I ¯ r = R ¯ ¯ · K ¯ ¯ · A I ¯ i I ¯ t = T ¯ ¯ · K ¯ ¯ · A I ¯ i ,
A = exp ( 2 n k s ) K ¯ ¯ = [ 1 0 0 0 0 cos 2 Ψ sin 2 Ψ 0 0 sin 2 Ψ cos 2 Ψ 0 0 0 0 1 ] ,
R ¯ ¯ = 1 2 [ r l r l * + r r r r * r l r l * r r r r * 0 0 r l r l * r r r r * r l r l * + r r r r * 0 0 0 0 2 R e ( r l r r * ) 2 I m ( r l r r * ) 0 0 2 I m ( r l r r * ) 2 R e ( r l r r * ) ] T ¯ ¯ = n 2 cos θ t 2 n 1 cos θ i · [ t l t l * + t r t r * t l t l * t r t r * t l t l * t r t r * t l t l * + t r t r * 0 0 0 0 × 0 0 0 0 2 R e ( t l t r * ) 2 I m ( t l t r * ) 2 I m ( t l t r * ) 2 R e ( t l t r * ) ] ,
I ¯ s = ω ˜ 0 G 0 4 π r 2 P ¯ ¯ · I ¯ i 4 π d Ω 4 π P 11 = 1 ,
P = P 21 P 11 g = 4 π d Ω 4 π cos θ P 11 .
P 11 ( θ ) = 1 4 π 0 π 0 2 π d τ d λ sin τ ω ˜ ( τ , λ ) G ( τ , λ ) ω ˜ 0 G 0 × 1 2 π 0 2 π d ϕ P 11 ( τ , λ; θ , ϕ ) ,
p ( ξ ) = 1 2 Δ ξ Ө ( ξ ξ 0 + Δ ξ ) Ө ( ξ 0 + Δ ξ ξ ) ,
Ө ( ξ ) = { 1 , ξ 0 0 , ξ < 0 ·
P ˜ 11 ( x , Ω ) = 4 π d Ω ' 4 π D ( x , Ω Ω ) P 11 ( Ω ) ,
D ( x , ϑ ) = x 2 [ 2 J 1 ( x sin ϑ ) x sin ϑ ] 2 cos ϑ Ө ( 90 ° ϑ ) + J 0 ( x ) 2 + J 1 ( x ) 2 4 π d Ω 4 π D ( x , Ω ) = 1 , cos ϑ = e ˆ Ω · e ˆ Ω .
f ˜ ( x , θ ) ( t 4 t 1 ) x cos θ exp ( t 2 x 2 sin 2 θ ) + t 4 x 1 + t 3 x 2 sin 2 θ θ 90 ° ,
t 1 = 0 . 85 , t 2 = 0 . 18 , t 3 = 0 . 51 , t 4 = t 3 π t 2 x t 1 [ 1 exp ( t 2 x 2 ) ] 2 t 2 x arc sin ( t 2 x / 1 + t 2 2 x 2 ) t 3 [ 1 exp ( t 3 x 2 ) ] t 3 .
P ˜ 11 ( x , θ ) = B + C D ( x , 180 ° θ ) + T f ˜ ( x , θ ) , θ 175 ° P ˜ 11 ( x , 180 ° ) = B + C x 2 4 π + T t 1 x

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