Abstract

By using Taylor expansions, simple expressions are obtained for the deflection of light by ice crystals. With these simplified formulas, the intensity distributions of halos as a function of scattering angle are calculated analytically near the halo angle. It is found that the intensity distributions of halos depend on the number of degrees of freedom of the generating set of crystals. The differences in the purity of the colors of various types of halo are explained subsequently on the basis of their intensity distributions. An analytical description of the shape of the halo or of the halocaustic near the halo angle is obtained also. On the basis of the obtained intensity distributions, the polarization of refraction halos as a function of scattering angle is calculated, in which both contributions (birefringence of ice and polarization by refraction) are taken into account. It is found that the polarization of parhelia and tangent arcs shows a strong maximum near the inner edge of the halo over an angular ange of 0.1°, followed by a similar maximum of reversed polarization at 0.5° from the first one. The 22° halo shows a less strong maximum near its edge over an angular range of 0.5°. Halos at 46° from the sun also show a strong polarization near their inner edges, but the direction of the polarization is perpendicular to the polarization of the 22° halo edges. The possibility for detecting ice crystals on Venus by polarimetry near the halo angle is discussed briefly.

© 1983 Optical Society of America

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References

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  1. G. P. Können, "Polarization of haloes and double refraction," Weather 32, 467–468 (1977).
  2. G. P. Können, Polarized Light in Nature (Cambridge U. Press, Cambridge, to be published); (Dutch ed., Thieme, Zutphen, the Netherlands, 1980).
  3. R. White, "Intensity plots of the parhelia," Q. J. R. Meteorol. Soc. 103, 169–175 (1977).
  4. A. B. Fraser and G. J. Thompson, "Analytic sun pillar model," J. Opt. Soc. Am. 70, 1145–1148 (1980).
  5. R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, 1980).
  6. R. S. McDowell, "The formation of parhelia at higher solar elevations," J. Atmos. Sci. 31, 1876–1884 (1974).
  7. R. White, "Deviation produced by anisotropic prisms," J. Opt. Soc. Am. 70, 281–287 (1980).
  8. G. P. Können and J. H. de Boer, "Polarized rainbow," Appl. Opt. 18, 1961–1965 (1979).
  9. W. J. Humphreys, Physics of the Air (Dover, New York, 1964).
  10. R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970).
  11. W. Tape, "Analytic foundations of halo theory," J. Opt. Soc. Am. 70, 1175–1192 (1980).
  12. W. Tape, "Folds, pleats and halos," Am. Sci. 70, 467–474 (1982).
  13. For the properties of the Dirac delta function and the Heaviside step function see, e.g., A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1965), Vol. I.
  14. P. V. Hobbs, Ice Physics (Clarendon, Oxford, 1974), p. 202.
  15. H. C. van de Hoiust, Scattering of Light by Small Particles (Wiley, New York, 1957), p. 245.
  16. A. B. Fraser, "Why can the supernumerary bows be seen in a rain shower?" J. Opt. Soc. Am. 73, 1626–1628 (1983).
  17. A. B. Fraser, "What size of ice crystals causes the halo?" J. Opt. Soc. Am. 69, 1112–1118 (1979).
  18. J. Veverka, "A polarimetric search for a Venus halo during the 1969 inferior conjunction," Icarus 14, 282–283 (1971).

1983 (1)

1982 (1)

W. Tape, "Folds, pleats and halos," Am. Sci. 70, 467–474 (1982).

1980 (3)

1979 (2)

1977 (2)

G. P. Können, "Polarization of haloes and double refraction," Weather 32, 467–468 (1977).

R. White, "Intensity plots of the parhelia," Q. J. R. Meteorol. Soc. 103, 169–175 (1977).

1974 (1)

R. S. McDowell, "The formation of parhelia at higher solar elevations," J. Atmos. Sci. 31, 1876–1884 (1974).

1971 (1)

J. Veverka, "A polarimetric search for a Venus halo during the 1969 inferior conjunction," Icarus 14, 282–283 (1971).

de Boer, J. H.

Fraser, A. B.

Greenler, R.

R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, 1980).

Hobbs, P. V.

P. V. Hobbs, Ice Physics (Clarendon, Oxford, 1974), p. 202.

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964).

Können, G. P.

G. P. Können and J. H. de Boer, "Polarized rainbow," Appl. Opt. 18, 1961–1965 (1979).

G. P. Können, "Polarization of haloes and double refraction," Weather 32, 467–468 (1977).

G. P. Können, Polarized Light in Nature (Cambridge U. Press, Cambridge, to be published); (Dutch ed., Thieme, Zutphen, the Netherlands, 1980).

McDowell, R. S.

R. S. McDowell, "The formation of parhelia at higher solar elevations," J. Atmos. Sci. 31, 1876–1884 (1974).

Messiah, A.

For the properties of the Dirac delta function and the Heaviside step function see, e.g., A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1965), Vol. I.

Tape, W.

W. Tape, "Folds, pleats and halos," Am. Sci. 70, 467–474 (1982).

W. Tape, "Analytic foundations of halo theory," J. Opt. Soc. Am. 70, 1175–1192 (1980).

Thompson, G. J.

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970).

van de Hoiust, H. C.

H. C. van de Hoiust, Scattering of Light by Small Particles (Wiley, New York, 1957), p. 245.

Veverka, J.

J. Veverka, "A polarimetric search for a Venus halo during the 1969 inferior conjunction," Icarus 14, 282–283 (1971).

White, R.

R. White, "Deviation produced by anisotropic prisms," J. Opt. Soc. Am. 70, 281–287 (1980).

R. White, "Intensity plots of the parhelia," Q. J. R. Meteorol. Soc. 103, 169–175 (1977).

Am. Sci. (1)

W. Tape, "Folds, pleats and halos," Am. Sci. 70, 467–474 (1982).

Appl. Opt. (1)

Icarus (1)

J. Veverka, "A polarimetric search for a Venus halo during the 1969 inferior conjunction," Icarus 14, 282–283 (1971).

J. Atmos. Sci. (1)

R. S. McDowell, "The formation of parhelia at higher solar elevations," J. Atmos. Sci. 31, 1876–1884 (1974).

J. Opt. Soc. Am. (5)

Q. J. R. Meteorol. Soc. (1)

R. White, "Intensity plots of the parhelia," Q. J. R. Meteorol. Soc. 103, 169–175 (1977).

Weather (1)

G. P. Können, "Polarization of haloes and double refraction," Weather 32, 467–468 (1977).

Other (7)

G. P. Können, Polarized Light in Nature (Cambridge U. Press, Cambridge, to be published); (Dutch ed., Thieme, Zutphen, the Netherlands, 1980).

R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, 1980).

W. J. Humphreys, Physics of the Air (Dover, New York, 1964).

R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970).

For the properties of the Dirac delta function and the Heaviside step function see, e.g., A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1965), Vol. I.

P. V. Hobbs, Ice Physics (Clarendon, Oxford, 1974), p. 202.

H. C. van de Hoiust, Scattering of Light by Small Particles (Wiley, New York, 1957), p. 245.

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