Abstract

Over a dozen rainbows have been observed in a single water droplet. They appear as glare spots on the water droplet which take on coloration at the appropriate rainbow angles. The appearance of rainbows as colored glare spots in this situation is understood in terms of the caustics created in the vicinity of the droplet by the refracting light rays. The angular positions of the glare spots are understood in terms of the Fourier transform of the geometric scattering amplitude. The rainbow glare spots are also found to appear numerically in the Fourier transform of the Mie scattered fields. An additional glare spot produced by rays at grazing incidence and not attributable to geometric optics also appears numerically in the Fourier transformed Mie fields.

© 1987 Optical Society of America

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  1. Accounts of the observation of third-order rainbows are given in C. B. Boyer, The Rainbow, from Myth to Mathematics (Thomas Yoseloff, New York, 1959), p. 271;a recent account is D. E. Pedgley, “A Tertiary Rainbow,” Weather 41, 401 (1986).
  2. R. A. R. Tricker, Introduction to Meterological Optics (American-Elsevier, New York, 1970), p. 57.
  3. R. Greenler, Rainbows, Halos, and Glories (Cambridge, U. P., London, 1980), pp. 6 and 7.
  4. C. W. Querfeld, “Mie Atmospheric Optics,” J. Opt. Soc. Am. 55, 105 (1965).
    [CrossRef]
  5. J. V. Dave, “Scattering of Visible Light by Large Water Spheres,” Appl. Opt. 8, 155 (1969).
    [CrossRef] [PubMed]
  6. K.-N. Liou, J. E. Hansen, “Intensity and Polarization for Single Scattering by Polydisperse Spheres: a Comparison of Ray Optics and Mie Theory,” J. Atmos. Sci. 28, 995 (1971).
    [CrossRef]
  7. In regard to the observability of high-order rainbows in the atmosphere, Ref. 4 predicts that the blue component of the fifth-order rainbow should lie in Alexander's band for scattering from spherical water droplets and should possibly be visible.
  8. The observability of high-order rainbows in nonspherical droplets is examined in K. Sassen, “Angular Scattering and Rainbow Formation in Pendant Drops,” J. Opt. Soc. Am. 69, 1083 (1979).
    [CrossRef]
  9. J. Walker, “Multiple Rainbows from Single Drops of Water and Other Liquids,” Am. J. Phys. 44, 421 (1976).
    [CrossRef]
  10. J. Walker, “How to Create and Observe a Dozen Rainbows in a Single Drop of Water,” Sci. Am. 237, 138 (July1977).
    [CrossRef]
  11. J. Walker, “Mysteries of Rainbows, Notably their Rare Supernumerary Arcs,” Sci. Am. 242, 174 (June1980).
    [CrossRef]
  12. Photographs of the rainbow glare spots are given in Ref. 2, pp. 44–46 and in Ref. 10. Many diagrams of the rainbow glare spots are also given in Refs. 9 and 10.
  13. Drawings of these caustics appear many places in the literature. Among them are H. M. Nussenzveig, “The Theory of the Rainbow,” Sci. Am. 236, 116 (Apr.1977);Ref. 1, Figs. 61–62; and W. J. Humphreys, Physics of the Air (Dover, New York, 1964), Fig. 174.
    [CrossRef]
  14. W. J. Humphreys, Physics of the Air (Dover, New York, 1964), pp. 484–492.
  15. Sassen8 conjectures that the visibility of high-order rainbows is further enhanced for nonspherical droplets.
  16. A. C. Holland, J. S. Draper, “Analytical and Experimental Investigation of Light Scattering from Polydispersions of Mie Particles,” Appl. Opt. 6, 511 (1967).
    [CrossRef] [PubMed]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 10.
  18. D. D. Cooke, M. Kerker, “Response Calculations for Light-Scattering Aerosol Particle Counters,” Appl. Opt. 14, 734 (1975).
    [CrossRef] [PubMed]
  19. S. I. Rubinow, “Scattering from a Penetrable Sphere at Short Wavelengths,” Ann. Phys. NY 14, 305 (1961).
    [CrossRef]
  20. H. M. Nussenzvieg, “High-Frequency Scattering by a Transparent Sphere I. Direct Reflection and Transmission,” J. Math. Phys. 10, 82 (1969).
    [CrossRef]
  21. H. M. Nussenzvieg, “High-Frequency Scattering by a Transparent Sphere II. Theory of the Rainbow and the Glory,” J. Math. Phys. 10, 125 (1969).
    [CrossRef]
  22. V. Khare, H. M. Nussenzvieg, “Theory of the Rainbow,” Phys. Rev. Lett. 33, 976 (1974).
    [CrossRef]
  23. H. M. Nussenzveig, “Complex Angular Momentum Theory of the Rainbow and the Glory,” J. Opt. Soc. Am. 69, 1068 (1979).
    [CrossRef]
  24. The Mie series was summed via the method described in W. J. Wiscombe, “Improved Mie Scattering Algorithms,” Appl. Opt. 19, 1505 (1980). Since the spatial frequencies for an FFT of these data were too widely spaced, the Fourier transform was performed directly via numerical integration of Eq. (1).
    [CrossRef] [PubMed]
  25. D. Ludwig, “Diffraction by a Circular Cavity,” J. Math. Phys. 11, 1617 (1970).
    [CrossRef]

1980 (2)

1979 (2)

1977 (2)

J. Walker, “How to Create and Observe a Dozen Rainbows in a Single Drop of Water,” Sci. Am. 237, 138 (July1977).
[CrossRef]

Drawings of these caustics appear many places in the literature. Among them are H. M. Nussenzveig, “The Theory of the Rainbow,” Sci. Am. 236, 116 (Apr.1977);Ref. 1, Figs. 61–62; and W. J. Humphreys, Physics of the Air (Dover, New York, 1964), Fig. 174.
[CrossRef]

1976 (1)

J. Walker, “Multiple Rainbows from Single Drops of Water and Other Liquids,” Am. J. Phys. 44, 421 (1976).
[CrossRef]

1975 (1)

1974 (1)

V. Khare, H. M. Nussenzvieg, “Theory of the Rainbow,” Phys. Rev. Lett. 33, 976 (1974).
[CrossRef]

1971 (1)

K.-N. Liou, J. E. Hansen, “Intensity and Polarization for Single Scattering by Polydisperse Spheres: a Comparison of Ray Optics and Mie Theory,” J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

1970 (1)

D. Ludwig, “Diffraction by a Circular Cavity,” J. Math. Phys. 11, 1617 (1970).
[CrossRef]

1969 (3)

H. M. Nussenzvieg, “High-Frequency Scattering by a Transparent Sphere I. Direct Reflection and Transmission,” J. Math. Phys. 10, 82 (1969).
[CrossRef]

H. M. Nussenzvieg, “High-Frequency Scattering by a Transparent Sphere II. Theory of the Rainbow and the Glory,” J. Math. Phys. 10, 125 (1969).
[CrossRef]

J. V. Dave, “Scattering of Visible Light by Large Water Spheres,” Appl. Opt. 8, 155 (1969).
[CrossRef] [PubMed]

1967 (1)

1965 (1)

1961 (1)

S. I. Rubinow, “Scattering from a Penetrable Sphere at Short Wavelengths,” Ann. Phys. NY 14, 305 (1961).
[CrossRef]

Boyer, C. B.

Accounts of the observation of third-order rainbows are given in C. B. Boyer, The Rainbow, from Myth to Mathematics (Thomas Yoseloff, New York, 1959), p. 271;a recent account is D. E. Pedgley, “A Tertiary Rainbow,” Weather 41, 401 (1986).

Cooke, D. D.

Dave, J. V.

Draper, J. S.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 10.

Greenler, R.

R. Greenler, Rainbows, Halos, and Glories (Cambridge, U. P., London, 1980), pp. 6 and 7.

Hansen, J. E.

K.-N. Liou, J. E. Hansen, “Intensity and Polarization for Single Scattering by Polydisperse Spheres: a Comparison of Ray Optics and Mie Theory,” J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

Holland, A. C.

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964), pp. 484–492.

Kerker, M.

Khare, V.

V. Khare, H. M. Nussenzvieg, “Theory of the Rainbow,” Phys. Rev. Lett. 33, 976 (1974).
[CrossRef]

Liou, K.-N.

K.-N. Liou, J. E. Hansen, “Intensity and Polarization for Single Scattering by Polydisperse Spheres: a Comparison of Ray Optics and Mie Theory,” J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

Ludwig, D.

D. Ludwig, “Diffraction by a Circular Cavity,” J. Math. Phys. 11, 1617 (1970).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, “Complex Angular Momentum Theory of the Rainbow and the Glory,” J. Opt. Soc. Am. 69, 1068 (1979).
[CrossRef]

Drawings of these caustics appear many places in the literature. Among them are H. M. Nussenzveig, “The Theory of the Rainbow,” Sci. Am. 236, 116 (Apr.1977);Ref. 1, Figs. 61–62; and W. J. Humphreys, Physics of the Air (Dover, New York, 1964), Fig. 174.
[CrossRef]

Nussenzvieg, H. M.

V. Khare, H. M. Nussenzvieg, “Theory of the Rainbow,” Phys. Rev. Lett. 33, 976 (1974).
[CrossRef]

H. M. Nussenzvieg, “High-Frequency Scattering by a Transparent Sphere II. Theory of the Rainbow and the Glory,” J. Math. Phys. 10, 125 (1969).
[CrossRef]

H. M. Nussenzvieg, “High-Frequency Scattering by a Transparent Sphere I. Direct Reflection and Transmission,” J. Math. Phys. 10, 82 (1969).
[CrossRef]

Querfeld, C. W.

Rubinow, S. I.

S. I. Rubinow, “Scattering from a Penetrable Sphere at Short Wavelengths,” Ann. Phys. NY 14, 305 (1961).
[CrossRef]

Sassen, K.

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meterological Optics (American-Elsevier, New York, 1970), p. 57.

Walker, J.

J. Walker, “Mysteries of Rainbows, Notably their Rare Supernumerary Arcs,” Sci. Am. 242, 174 (June1980).
[CrossRef]

J. Walker, “How to Create and Observe a Dozen Rainbows in a Single Drop of Water,” Sci. Am. 237, 138 (July1977).
[CrossRef]

J. Walker, “Multiple Rainbows from Single Drops of Water and Other Liquids,” Am. J. Phys. 44, 421 (1976).
[CrossRef]

Wiscombe, W. J.

Am. J. Phys. (1)

J. Walker, “Multiple Rainbows from Single Drops of Water and Other Liquids,” Am. J. Phys. 44, 421 (1976).
[CrossRef]

Ann. Phys. NY (1)

S. I. Rubinow, “Scattering from a Penetrable Sphere at Short Wavelengths,” Ann. Phys. NY 14, 305 (1961).
[CrossRef]

Appl. Opt. (4)

J. Atmos. Sci. (1)

K.-N. Liou, J. E. Hansen, “Intensity and Polarization for Single Scattering by Polydisperse Spheres: a Comparison of Ray Optics and Mie Theory,” J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

J. Math. Phys. (3)

D. Ludwig, “Diffraction by a Circular Cavity,” J. Math. Phys. 11, 1617 (1970).
[CrossRef]

H. M. Nussenzvieg, “High-Frequency Scattering by a Transparent Sphere I. Direct Reflection and Transmission,” J. Math. Phys. 10, 82 (1969).
[CrossRef]

H. M. Nussenzvieg, “High-Frequency Scattering by a Transparent Sphere II. Theory of the Rainbow and the Glory,” J. Math. Phys. 10, 125 (1969).
[CrossRef]

J. Opt. Soc. Am. (3)

Phys. Rev. Lett. (1)

V. Khare, H. M. Nussenzvieg, “Theory of the Rainbow,” Phys. Rev. Lett. 33, 976 (1974).
[CrossRef]

Sci. Am. (3)

J. Walker, “How to Create and Observe a Dozen Rainbows in a Single Drop of Water,” Sci. Am. 237, 138 (July1977).
[CrossRef]

J. Walker, “Mysteries of Rainbows, Notably their Rare Supernumerary Arcs,” Sci. Am. 242, 174 (June1980).
[CrossRef]

Drawings of these caustics appear many places in the literature. Among them are H. M. Nussenzveig, “The Theory of the Rainbow,” Sci. Am. 236, 116 (Apr.1977);Ref. 1, Figs. 61–62; and W. J. Humphreys, Physics of the Air (Dover, New York, 1964), Fig. 174.
[CrossRef]

Other (8)

W. J. Humphreys, Physics of the Air (Dover, New York, 1964), pp. 484–492.

Sassen8 conjectures that the visibility of high-order rainbows is further enhanced for nonspherical droplets.

Accounts of the observation of third-order rainbows are given in C. B. Boyer, The Rainbow, from Myth to Mathematics (Thomas Yoseloff, New York, 1959), p. 271;a recent account is D. E. Pedgley, “A Tertiary Rainbow,” Weather 41, 401 (1986).

R. A. R. Tricker, Introduction to Meterological Optics (American-Elsevier, New York, 1970), p. 57.

R. Greenler, Rainbows, Halos, and Glories (Cambridge, U. P., London, 1980), pp. 6 and 7.

Photographs of the rainbow glare spots are given in Ref. 2, pp. 44–46 and in Ref. 10. Many diagrams of the rainbow glare spots are also given in Refs. 9 and 10.

In regard to the observability of high-order rainbows in the atmosphere, Ref. 4 predicts that the blue component of the fifth-order rainbow should lie in Alexander's band for scattering from spherical water droplets and should possibly be visible.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 10.

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