Abstract

Geometric optics rainbows and ice-crystal halos that include some effects of a Rayleigh-scattering atmosphere and a cloud of finite optical thickness are simulated in color by the use of a Monte Carlo approach.

© 1994 Optical Society of America

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References

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  1. C. B. Boyer, The Rainbow From Myth to Mathematics (Princeton U. Press, Princeton, N.J., 1987).
  2. R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970).
  3. M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, New York, 1954).
  4. R. G. Greenler, A. J. Malimann, “Circumscribed halos,” Science 176, 128–131 (1972).
    [CrossRef] [PubMed]
  5. R. G. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, Cambridge, 1980).
  6. F. Pattloch, E. Tränkle, “Monte Carlo simulation and analysis of halo phenomena,” J. Opt. Soc. Am. A1, 520–526 (1984).
    [CrossRef]
  7. W. Tape, “Alternate explanations of Lowitz arcs,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 62–65.
  8. E. Tränkle, R. G. Greenler, “Multiple scattering effects in halo phenomena,” J. Opt. Soc. Am. A4, 591–599 (1987).
    [CrossRef]
  9. E. Tränkle, “Computer simulation of rainbows, glory and corona phenomena including multiple scattering,” in Digest of Topical Meeting on Meteorological Optics (Optical Society of America, Washington, D.C., 1968) pp. 70–73.
  10. S. D. Gedzelman, “Visibility of halos and rainbows,” Appl. Opt. 19, 3068–3074, (1980).
    [CrossRef] [PubMed]
  11. S. D. Gedzelman, “Rainbow brightness,” Appl. Opt. 21, 3032–3037 (1982).
    [CrossRef] [PubMed]
  12. For a copy of the programs rainbow.bas and halo.bas either send a 3.5″ diskette in a self-addressed stamped envelop or obtain the file over Internet, using file transfer protocol (ftp). The steps areftp cunim.sci.ccny.cuny.edu 〈Crt〉cd pub〈Crt〉cd optics 〈Crt〉mget 〈Crt〉quit 〈Crt〉
  13. R. L. Lee, “What are ‘All the colors of the rainbow’?” Appl. Opt. 30, 3401–3407, 3545 (1991).
    [CrossRef] [PubMed]
  14. R. J. Kubesh, “Computer display of chromaticity coordinates with the rainbow as an example,” Am. J. Phys. 60, 919–923 (1992).
    [CrossRef]
  15. J. N. Kidder, “Colors of the spectrum: do you wonder where the yellow went?” Appl. Opt. 33, 4727–4732 (1994).
    [CrossRef] [PubMed]
  16. D. K. Lynch, S. D. Gedzelman, A. B. Fraser, “Subsuns, Bottlinger's rings and elliptical halos,” Appl. Opt. 33, 4580–4589 (1994).
    [CrossRef] [PubMed]
  17. C. F. Bottlinger, “An interesting phenomenon seen during a balloon trip,” Meteorol. Z. 25, 74 (1910).
  18. K. Stuchtey, “Untersonnen and Lichtsaulen an Sonne und Mond,” Ann. Phys. 59, 33–55 (1919).
    [CrossRef]
  19. G. H. Liljequist, “Halo-phenomena and ice crystals,” in Norwegian–British–Swedish Antarctic Expedition, 1949–1952, Scientific Results, (Norsk Polarinstitutt, Oslo, 1956), Vol. 2, Part 2A. This has been confirmed by W. Tape and G. P. Können, who note that spectacular displays at the South Pole are highly ephemeral and seem to occur when ice crystals form rapidly in localized pockets of air with elevated humidity.
  20. W. Tape, “South Pole halo display, 2 January 1990,” Appl. Opt. 30 (24), cover photograph (1991).
  21. G. P. Können, S. H. Muller, J. Tinbergen, “Halo polarization profiles and the interfacial angles of ice crystals,” Appl. Opt 33, 4569–4579 (1994).
    [CrossRef] [PubMed]

1994 (3)

1992 (1)

R. J. Kubesh, “Computer display of chromaticity coordinates with the rainbow as an example,” Am. J. Phys. 60, 919–923 (1992).
[CrossRef]

1991 (2)

R. L. Lee, “What are ‘All the colors of the rainbow’?” Appl. Opt. 30, 3401–3407, 3545 (1991).
[CrossRef] [PubMed]

W. Tape, “South Pole halo display, 2 January 1990,” Appl. Opt. 30 (24), cover photograph (1991).

1987 (1)

E. Tränkle, R. G. Greenler, “Multiple scattering effects in halo phenomena,” J. Opt. Soc. Am. A4, 591–599 (1987).
[CrossRef]

1984 (1)

F. Pattloch, E. Tränkle, “Monte Carlo simulation and analysis of halo phenomena,” J. Opt. Soc. Am. A1, 520–526 (1984).
[CrossRef]

1982 (1)

1980 (1)

1972 (1)

R. G. Greenler, A. J. Malimann, “Circumscribed halos,” Science 176, 128–131 (1972).
[CrossRef] [PubMed]

1919 (1)

K. Stuchtey, “Untersonnen and Lichtsaulen an Sonne und Mond,” Ann. Phys. 59, 33–55 (1919).
[CrossRef]

1910 (1)

C. F. Bottlinger, “An interesting phenomenon seen during a balloon trip,” Meteorol. Z. 25, 74 (1910).

Bottlinger, C. F.

C. F. Bottlinger, “An interesting phenomenon seen during a balloon trip,” Meteorol. Z. 25, 74 (1910).

Boyer, C. B.

C. B. Boyer, The Rainbow From Myth to Mathematics (Princeton U. Press, Princeton, N.J., 1987).

Fraser, A. B.

Gedzelman, S. D.

Greenler, R. G.

E. Tränkle, R. G. Greenler, “Multiple scattering effects in halo phenomena,” J. Opt. Soc. Am. A4, 591–599 (1987).
[CrossRef]

R. G. Greenler, A. J. Malimann, “Circumscribed halos,” Science 176, 128–131 (1972).
[CrossRef] [PubMed]

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

Kidder, J. N.

Können, G. P.

G. P. Können, S. H. Muller, J. Tinbergen, “Halo polarization profiles and the interfacial angles of ice crystals,” Appl. Opt 33, 4569–4579 (1994).
[CrossRef] [PubMed]

Kubesh, R. J.

R. J. Kubesh, “Computer display of chromaticity coordinates with the rainbow as an example,” Am. J. Phys. 60, 919–923 (1992).
[CrossRef]

Lee, R. L.

R. L. Lee, “What are ‘All the colors of the rainbow’?” Appl. Opt. 30, 3401–3407, 3545 (1991).
[CrossRef] [PubMed]

Liljequist, G. H.

G. H. Liljequist, “Halo-phenomena and ice crystals,” in Norwegian–British–Swedish Antarctic Expedition, 1949–1952, Scientific Results, (Norsk Polarinstitutt, Oslo, 1956), Vol. 2, Part 2A. This has been confirmed by W. Tape and G. P. Können, who note that spectacular displays at the South Pole are highly ephemeral and seem to occur when ice crystals form rapidly in localized pockets of air with elevated humidity.

Lynch, D. K.

Malimann, A. J.

R. G. Greenler, A. J. Malimann, “Circumscribed halos,” Science 176, 128–131 (1972).
[CrossRef] [PubMed]

Minnaert, M.

M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, New York, 1954).

Muller, S. H.

G. P. Können, S. H. Muller, J. Tinbergen, “Halo polarization profiles and the interfacial angles of ice crystals,” Appl. Opt 33, 4569–4579 (1994).
[CrossRef] [PubMed]

Pattloch, F.

F. Pattloch, E. Tränkle, “Monte Carlo simulation and analysis of halo phenomena,” J. Opt. Soc. Am. A1, 520–526 (1984).
[CrossRef]

Stuchtey, K.

K. Stuchtey, “Untersonnen and Lichtsaulen an Sonne und Mond,” Ann. Phys. 59, 33–55 (1919).
[CrossRef]

Tape, W.

W. Tape, “South Pole halo display, 2 January 1990,” Appl. Opt. 30 (24), cover photograph (1991).

W. Tape, “Alternate explanations of Lowitz arcs,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 62–65.

Tinbergen, J.

G. P. Können, S. H. Muller, J. Tinbergen, “Halo polarization profiles and the interfacial angles of ice crystals,” Appl. Opt 33, 4569–4579 (1994).
[CrossRef] [PubMed]

Tränkle, E.

E. Tränkle, R. G. Greenler, “Multiple scattering effects in halo phenomena,” J. Opt. Soc. Am. A4, 591–599 (1987).
[CrossRef]

F. Pattloch, E. Tränkle, “Monte Carlo simulation and analysis of halo phenomena,” J. Opt. Soc. Am. A1, 520–526 (1984).
[CrossRef]

E. Tränkle, “Computer simulation of rainbows, glory and corona phenomena including multiple scattering,” in Digest of Topical Meeting on Meteorological Optics (Optical Society of America, Washington, D.C., 1968) pp. 70–73.

Tricker, R. A. R.

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

Am. J. Phys. (1)

R. J. Kubesh, “Computer display of chromaticity coordinates with the rainbow as an example,” Am. J. Phys. 60, 919–923 (1992).
[CrossRef]

Ann. Phys. (1)

K. Stuchtey, “Untersonnen and Lichtsaulen an Sonne und Mond,” Ann. Phys. 59, 33–55 (1919).
[CrossRef]

Appl. Opt (1)

G. P. Können, S. H. Muller, J. Tinbergen, “Halo polarization profiles and the interfacial angles of ice crystals,” Appl. Opt 33, 4569–4579 (1994).
[CrossRef] [PubMed]

Appl. Opt. (6)

J. Opt. Soc. Am. (2)

F. Pattloch, E. Tränkle, “Monte Carlo simulation and analysis of halo phenomena,” J. Opt. Soc. Am. A1, 520–526 (1984).
[CrossRef]

E. Tränkle, R. G. Greenler, “Multiple scattering effects in halo phenomena,” J. Opt. Soc. Am. A4, 591–599 (1987).
[CrossRef]

Meteorol. Z. (1)

C. F. Bottlinger, “An interesting phenomenon seen during a balloon trip,” Meteorol. Z. 25, 74 (1910).

Science (1)

R. G. Greenler, A. J. Malimann, “Circumscribed halos,” Science 176, 128–131 (1972).
[CrossRef] [PubMed]

Other (8)

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

C. B. Boyer, The Rainbow From Myth to Mathematics (Princeton U. Press, Princeton, N.J., 1987).

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

M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, New York, 1954).

E. Tränkle, “Computer simulation of rainbows, glory and corona phenomena including multiple scattering,” in Digest of Topical Meeting on Meteorological Optics (Optical Society of America, Washington, D.C., 1968) pp. 70–73.

For a copy of the programs rainbow.bas and halo.bas either send a 3.5″ diskette in a self-addressed stamped envelop or obtain the file over Internet, using file transfer protocol (ftp). The steps areftp cunim.sci.ccny.cuny.edu 〈Crt〉cd pub〈Crt〉cd optics 〈Crt〉mget 〈Crt〉quit 〈Crt〉

W. Tape, “Alternate explanations of Lowitz arcs,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 62–65.

G. H. Liljequist, “Halo-phenomena and ice crystals,” in Norwegian–British–Swedish Antarctic Expedition, 1949–1952, Scientific Results, (Norsk Polarinstitutt, Oslo, 1956), Vol. 2, Part 2A. This has been confirmed by W. Tape and G. P. Können, who note that spectacular displays at the South Pole are highly ephemeral and seem to occur when ice crystals form rapidly in localized pockets of air with elevated humidity.

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Figures (2)

Fig. 1
Fig. 1

(a) 22° and 46° halos produced by a perfect cloud of 10,000 randomly falling crystals with aspect ratio a = 0.5 in a vacuum with solar zenith angle Z = 65°. (b) Same as Fig. 1(a), but for a Rayleigh atmosphere with turbidity β = 2 and cloud optical depth τ = 1 located at pressure pc = 300 mbars. Note here how the brightness of the bottom of the halo is severely reduced, (c) Same as Fig. 1(b), but for β = 1, τ = 0.1, and pc = 950 mbars. In this case, the top of the 22° halo is faint because of the small optical thickness of the cloud.

Fig. 2
Fig. 2

Halo complex for Z = 67° produced by a perfect cloud in a vacuum with 30,000 pencil crystals with a = 1/3 falling in Parry orientation.

Equations (8)

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p RAY = 1 exp { [ 0.008 λ 4 sec ( Z ) ] } ,
n rain = 1.3601 0.044 λ ( μ m ) ,
P 1 = exp [ k λ 4 β sec ( Z ) p c ] ,
P 2 = 1 exp [ τ sec ( Z ) ] .
P 3 = { sec ( ϕ H ) sec ( Z ) sec ( ϕ H ) { exp [ τ sec ( ϕ H ) ] exp [ τ sec ( Z ) ] } ϕ H Z τ sec ( Z ) exp [ τ sec ( Z ) ] ϕ H = Z .
P 4 = exp [ k β sec ( ϕ H ) λ 4 ( 10 5 p c ) ] .
P 3 = 1 exp { τ [ sec ( Z ) + sec ( ϕ H ) ] } sec ( Z ) + sec ( ϕ H )
n ice = 1.3203 0.0333 λ ( μ m ) ,

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