Abstract

The optics of rainbows and fogbows is investigated theoretically for monodisperse drops using Mie theory. Included in the calculations are a realistic solar illumination spectrum and the finite size of the sun. Drop sizes range from 3 to 300 μm (3800 > X > 38). Results are presented on the location, width, contrast, polarization, and color of both primary and secondary rainbows. Particular attention is given to rainbows formed in small drops (fogbows).

© 1991 Optical Society of America

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References

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  1. The best set of quantitative observations of natural fogbows seems to be that from Ben Nevis published in 1887 by Omond in the Proceedings of the Royal Society of Edinburgh and analyzed by J. C. McConnell, “The theory of fog-bows,” Philos. Mag. 29, 453–461 (1890).
  2. G. H. Liljequist, “Halo phenomena and ice crystals,” Norwegian-British-Swedish Antarctic Expedition, 1949–1952, Scientific Results, Vol. 2, Part 2, Special Studies (1956); F. Palmer, “Unusual rainbows,” Am. J. Phys. 13, 203–204 (1945); R. A. Brown “Occurrence of supernumerary fogbows at subfreezing temperatures,” Mon. Weather Rev. 94, 47–48 (1966); W. C. Livingston, “The cloud contrast bow as seen from high flying aircraft,” Weather 34, 16 (1979).
    [CrossRef]
  3. A. de Ulloa, “Relazion historica del viage á la America meridional,” Part 1, Section 2, 592–593 (1748).
  4. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957; reprinted by Dover, New York, 1981).
  5. C. W. Querfeld, “Mie atmospheric optics,” J. Opt. Soc. Am. 55, 105–106 (1965).
    [CrossRef]
  6. J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [CrossRef]
  7. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).
  8. W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
    [CrossRef]
  9. J. E. MacDonald, “The shape and aerodynamics of large raindrops,” J. Meteorol. 11, 478–494 (1954); A. W. Green, “An approximation for the shape of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
    [CrossRef]
  10. K. V. Beard, H. T. Ochs, R. J. Kubesh, “Natural oscillations of small raindrops,” Nature (London) 342, 408–410 (1989).
    [CrossRef]
  11. F. X. Kneizys, E. P. Shettle, L. W. Abreu, G. P. Anderson, J. H. Chetwynd, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users guide to lowtran7,” AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Bedford, Mass., 1988).
  12. F. E. Volz, “Some aspects of the optics of rainbows and the physics of rain,” in Physics of Precipitation (American Geophysica Union, Washington, D.C., (1960), pp. 280–286A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,” J. Atmos. Sci. 29, 211–222 (1972); “Why can the supernumerary bows be seen in a rain shower?” J. Opt. Soc. Am. 73, 1626–1628 (1983).
    [CrossRef]
  13. J. A. Lock, “Observability of atmospheric glories and supernumerary rainbows,” J. Opt. Soc. Am. A 6, 1924–1930 (1989).
    [CrossRef]
  14. In the geometric limit, the angle of minimum deviation represents a hard limit like an opaque edge. We know from wave theory that such an edge results in diffraction that puts light into the shadow, or in this case into smaller deviation angles. The spacing of the diffraction ripples is small for large drops. For large drops there will be many diffraction ripples sitting on top of the geometric rainbow. Because they are faint, they are not generally noticed. However, with decreasing drop size, the diffraction ripples increase in visibility (width and contrast) until a point is reached where the first-order diffraction ripple (=zeroth supernumerary) matches the contribution from the classical rainbow. For smaller drops yet diffraction dominates and the rainbow, by now faint and pale, results almost exclusively from diffraction.
  15. There seems to be little in the way of reliable measurements of the natural rainbow’s width in the literature. Most discussions refer to the width being ~2°, apparently based on the difference between the angles of minimum deviation for red and blue light [e.g., R. A. R. Tricker, Meteorological Optics (American Elsevier, New York, 1970), who suggests a width of 1.7.]. However, when the solar distribution of energy and finite width to the sun’s diameter are included, the bow widens considerably.

1989 (2)

K. V. Beard, H. T. Ochs, R. J. Kubesh, “Natural oscillations of small raindrops,” Nature (London) 342, 408–410 (1989).
[CrossRef]

J. A. Lock, “Observability of atmospheric glories and supernumerary rainbows,” J. Opt. Soc. Am. A 6, 1924–1930 (1989).
[CrossRef]

1974 (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

1968 (1)

W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
[CrossRef]

1965 (1)

1956 (1)

G. H. Liljequist, “Halo phenomena and ice crystals,” Norwegian-British-Swedish Antarctic Expedition, 1949–1952, Scientific Results, Vol. 2, Part 2, Special Studies (1956); F. Palmer, “Unusual rainbows,” Am. J. Phys. 13, 203–204 (1945); R. A. Brown “Occurrence of supernumerary fogbows at subfreezing temperatures,” Mon. Weather Rev. 94, 47–48 (1966); W. C. Livingston, “The cloud contrast bow as seen from high flying aircraft,” Weather 34, 16 (1979).
[CrossRef]

1954 (1)

J. E. MacDonald, “The shape and aerodynamics of large raindrops,” J. Meteorol. 11, 478–494 (1954); A. W. Green, “An approximation for the shape of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

1890 (1)

The best set of quantitative observations of natural fogbows seems to be that from Ben Nevis published in 1887 by Omond in the Proceedings of the Royal Society of Edinburgh and analyzed by J. C. McConnell, “The theory of fog-bows,” Philos. Mag. 29, 453–461 (1890).

Abreu, L. W.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, G. P. Anderson, J. H. Chetwynd, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users guide to lowtran7,” AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Bedford, Mass., 1988).

Anderson, G. P.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, G. P. Anderson, J. H. Chetwynd, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users guide to lowtran7,” AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Bedford, Mass., 1988).

Beard, K. V.

K. V. Beard, H. T. Ochs, R. J. Kubesh, “Natural oscillations of small raindrops,” Nature (London) 342, 408–410 (1989).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Chetwynd, J. H.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, G. P. Anderson, J. H. Chetwynd, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users guide to lowtran7,” AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Bedford, Mass., 1988).

Clough, S. A.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, G. P. Anderson, J. H. Chetwynd, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users guide to lowtran7,” AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Bedford, Mass., 1988).

de Ulloa, A.

A. de Ulloa, “Relazion historica del viage á la America meridional,” Part 1, Section 2, 592–593 (1748).

Gallery, W. O.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, G. P. Anderson, J. H. Chetwynd, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users guide to lowtran7,” AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Bedford, Mass., 1988).

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Irvine, W. M.

W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
[CrossRef]

Kneizys, F. X.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, G. P. Anderson, J. H. Chetwynd, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users guide to lowtran7,” AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Bedford, Mass., 1988).

Kubesh, R. J.

K. V. Beard, H. T. Ochs, R. J. Kubesh, “Natural oscillations of small raindrops,” Nature (London) 342, 408–410 (1989).
[CrossRef]

Liljequist, G. H.

G. H. Liljequist, “Halo phenomena and ice crystals,” Norwegian-British-Swedish Antarctic Expedition, 1949–1952, Scientific Results, Vol. 2, Part 2, Special Studies (1956); F. Palmer, “Unusual rainbows,” Am. J. Phys. 13, 203–204 (1945); R. A. Brown “Occurrence of supernumerary fogbows at subfreezing temperatures,” Mon. Weather Rev. 94, 47–48 (1966); W. C. Livingston, “The cloud contrast bow as seen from high flying aircraft,” Weather 34, 16 (1979).
[CrossRef]

Lock, J. A.

MacDonald, J. E.

J. E. MacDonald, “The shape and aerodynamics of large raindrops,” J. Meteorol. 11, 478–494 (1954); A. W. Green, “An approximation for the shape of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

McConnell, J. C.

The best set of quantitative observations of natural fogbows seems to be that from Ben Nevis published in 1887 by Omond in the Proceedings of the Royal Society of Edinburgh and analyzed by J. C. McConnell, “The theory of fog-bows,” Philos. Mag. 29, 453–461 (1890).

Ochs, H. T.

K. V. Beard, H. T. Ochs, R. J. Kubesh, “Natural oscillations of small raindrops,” Nature (London) 342, 408–410 (1989).
[CrossRef]

Pollack, J. B.

W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
[CrossRef]

Querfeld, C. W.

Selby, J. E. A.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, G. P. Anderson, J. H. Chetwynd, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users guide to lowtran7,” AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Bedford, Mass., 1988).

Shettle, E. P.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, G. P. Anderson, J. H. Chetwynd, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users guide to lowtran7,” AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Bedford, Mass., 1988).

Travis, L. D.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Tricker, R. A. R.

There seems to be little in the way of reliable measurements of the natural rainbow’s width in the literature. Most discussions refer to the width being ~2°, apparently based on the difference between the angles of minimum deviation for red and blue light [e.g., R. A. R. Tricker, Meteorological Optics (American Elsevier, New York, 1970), who suggests a width of 1.7.]. However, when the solar distribution of energy and finite width to the sun’s diameter are included, the bow widens considerably.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957; reprinted by Dover, New York, 1981).

Volz, F. E.

F. E. Volz, “Some aspects of the optics of rainbows and the physics of rain,” in Physics of Precipitation (American Geophysica Union, Washington, D.C., (1960), pp. 280–286A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,” J. Atmos. Sci. 29, 211–222 (1972); “Why can the supernumerary bows be seen in a rain shower?” J. Opt. Soc. Am. 73, 1626–1628 (1983).
[CrossRef]

Icarus (1)

W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
[CrossRef]

J. Meteorol. (1)

J. E. MacDonald, “The shape and aerodynamics of large raindrops,” J. Meteorol. 11, 478–494 (1954); A. W. Green, “An approximation for the shape of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Nature (London) (1)

K. V. Beard, H. T. Ochs, R. J. Kubesh, “Natural oscillations of small raindrops,” Nature (London) 342, 408–410 (1989).
[CrossRef]

Norwegian-British-Swedish Antarctic Expedition, 1949–1952, Scientific Results (1)

G. H. Liljequist, “Halo phenomena and ice crystals,” Norwegian-British-Swedish Antarctic Expedition, 1949–1952, Scientific Results, Vol. 2, Part 2, Special Studies (1956); F. Palmer, “Unusual rainbows,” Am. J. Phys. 13, 203–204 (1945); R. A. Brown “Occurrence of supernumerary fogbows at subfreezing temperatures,” Mon. Weather Rev. 94, 47–48 (1966); W. C. Livingston, “The cloud contrast bow as seen from high flying aircraft,” Weather 34, 16 (1979).
[CrossRef]

Philos. Mag. (1)

The best set of quantitative observations of natural fogbows seems to be that from Ben Nevis published in 1887 by Omond in the Proceedings of the Royal Society of Edinburgh and analyzed by J. C. McConnell, “The theory of fog-bows,” Philos. Mag. 29, 453–461 (1890).

Space Sci. Rev. (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Other (7)

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

A. de Ulloa, “Relazion historica del viage á la America meridional,” Part 1, Section 2, 592–593 (1748).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957; reprinted by Dover, New York, 1981).

In the geometric limit, the angle of minimum deviation represents a hard limit like an opaque edge. We know from wave theory that such an edge results in diffraction that puts light into the shadow, or in this case into smaller deviation angles. The spacing of the diffraction ripples is small for large drops. For large drops there will be many diffraction ripples sitting on top of the geometric rainbow. Because they are faint, they are not generally noticed. However, with decreasing drop size, the diffraction ripples increase in visibility (width and contrast) until a point is reached where the first-order diffraction ripple (=zeroth supernumerary) matches the contribution from the classical rainbow. For smaller drops yet diffraction dominates and the rainbow, by now faint and pale, results almost exclusively from diffraction.

There seems to be little in the way of reliable measurements of the natural rainbow’s width in the literature. Most discussions refer to the width being ~2°, apparently based on the difference between the angles of minimum deviation for red and blue light [e.g., R. A. R. Tricker, Meteorological Optics (American Elsevier, New York, 1970), who suggests a width of 1.7.]. However, when the solar distribution of energy and finite width to the sun’s diameter are included, the bow widens considerably.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, G. P. Anderson, J. H. Chetwynd, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Users guide to lowtran7,” AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Bedford, Mass., 1988).

F. E. Volz, “Some aspects of the optics of rainbows and the physics of rain,” in Physics of Precipitation (American Geophysica Union, Washington, D.C., (1960), pp. 280–286A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,” J. Atmos. Sci. 29, 211–222 (1972); “Why can the supernumerary bows be seen in a rain shower?” J. Opt. Soc. Am. 73, 1626–1628 (1983).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Spectrum of incident sunlight at sea level with a solar altitude of 20°.

Fig. 2
Fig. 2

I(Θ) with drop radius a as a parameter.

Fig. 3
Fig. 3

Logarithm of I(Θ) with drop radius a as a parameter s11fda. plt (PAM).

Fig. 4
Fig. 4

Rainbow properties as a function of drop size

Fig. 5
Fig. 5

Width of primary and secondary rainbows versus drop radius a.

Fig. 6
Fig. 6

Contrast versus drop radius a.

Fig. 7
Fig. 7

Monochromatic bows at a wavelengths of 0.40 and 0.65 μm for a = 100 μm.

Fig. 8
Fig. 8

Color purity parameter ϕ vs drop radius a.

Fig. 9
Fig. 9

Position of supernumeraries.

Fig. 10
Fig. 10

Polarization p vs drop size.

Fig. 11
Fig. 11

Maximum linear polarization.

Tables (2)

Tables Icon

Table I Summary of Rainbow Calculations for the Primary and Secondary Bowsa

Tables Icon

Table II Summary of Rainbow Calculations for the Supernumerary Bowsa

Equations (1)

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I ( θ ) = K S ( θ ) * S 11 ( a , λ , θ ) I ( λ ) d λ ,

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