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Applied Optics

Applied Optics


  • Vol. 37, Iss. 9 — Mar. 20, 1998
  • pp: 1506–1519

Mie theory, Airy theory, and the natural rainbow

Raymond L. Lee, Jr.  »View Author Affiliations

Applied Optics, Vol. 37, Issue 9, pp. 1506-1519 (1998)

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Compared with Mie scattering theory, Airy rainbow theory clearly miscalculates some monochromatic details of scattering by small water drops. Yet when monodisperse Airy theory is measured by perceptual (rather than purely physical) standards such as chromaticity and luminance contrast, it differs very little from Mie theory. Considering only the angular positions of luminance extrema, Airy theory’s errors are largest for small droplets such as those that dominate cloudbows and fogbows. However, integrating over a realistic drop-size distribution for these bows eliminates most perceptible color and luminance differences between the two theories.

© 1998 Optical Society of America

OCIS Codes
(010.1290) Atmospheric and oceanic optics : Atmospheric optics
(090.5640) Holography : Rainbow holography
(260.5430) Physical optics : Polarization
(290.4020) Scattering : Mie theory

Original Manuscript: May 14, 1997
Revised Manuscript: September 12, 1997
Published: March 20, 1998

Raymond L. Lee, "Mie theory, Airy theory, and the natural rainbow," Appl. Opt. 37, 1506-1519 (1998)

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  1. G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc.6, 379–403 (1838). Airy read his paper before the Society in May 1836 and March 1838. Remarkably (at least from the biased standpoint of atmospheric optics), Airy makes no mention of this signal accomplishment in his Autobiography (Cambridge U. Press, Cambridge, 1896). Instead, Airy’s memorable events from 1836–1838 include his improved filing system for Greenwich Observatory’s astronomical papers!
  2. C. B. Boyer, The Rainbow: From Myth to Mathematics (Princeton U. Press, Princeton, N.J., 1987; reprint of 1959 Thomas Yoseloff edition), pp. 304–310.
  3. Ref. 2, p. 313.
  4. For a notable exception, see K. Sassen, “Angular scattering and rainbow formation in pendant drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).
  5. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908). An English translation is available as G. Mie, “Contributions to the optics of turbid media, particularly of colloidal metal solutions,” Royal Aircraft Establishment library translation 1873. (Her Majesty’s Stationery Office, London, 1976).
  6. H. M. Nussenzveig, “The theory of the rainbow,” in Atmospheric Phenomena (Freeman, San Francisco, 1980), pp. 60–71.
  7. Reference 6’s arguments appear in greater detail in H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
  8. Perpendicular (⊥) and parallel (|) directions here are measured with respect to the scattering plane defined by Sun, water drop, and observer. This plane’s orientation changes around the rainbow arc.
  9. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981; reprint of 1957 Wiley edition), p. 247.
  10. S. D. Mobbs, “Theory of the rainbow,” J. Opt. Soc. Am. 69, 1089–1092 (1979). [CrossRef]
  11. G. P. Können, J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979). [CrossRef] [PubMed]
  12. Ref. 6, p. 70.
  13. R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991). [CrossRef] [PubMed]
  14. W. J. Humphreys, Physics of the Air (Dover, New York, 1964; reprint of 1940 McGraw-Hill edition), pp. 491–494.
  15. R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970), pp. 179–181.
  16. Ref. 11, p. 1963.
  17. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 112–113, 477–481.
  18. Ref. 7, p. 1073 (Fig. 3). Note that Fig. 1’s Mie curve includes small-scale structure due to external reflections, whereas Nussenzveig’s figure does not.
  19. Ref. 17, pp. 300–304.
  20. Exceptions include bows seen from mountains, hills, and airplanes in flight.
  21. The effects of this smoothing on rainbow luminance are also shown in A. B. Fraser, “Chasing rainbows: numerous supernumeraries are super,” Weatherwise 36, 280–289 (1983).
  22. Rather than combine color and luminance differences in a single color-difference measure, I show them separately here. Although rainbow observers cannot make this separation, it does let me address more readily the issues raised in Refs. 6, 7, 9, and 13. Optically speaking, cloudbows and fogbows differ very little, so the terms can be used interchangeably.
  23. See Ref. 9, p. 247.
  24. J. Gorraiz, H. Horvath, G. Raimann, “Influence of small color differences on the contrast threshold: its application to atmospheric visibility,” Appl. Opt. 25, 2537–2545 (1986). [CrossRef] [PubMed]
  25. Similar Airy underestimates of intensities in Alexander’s dark band are evident in Ref. 7, p. 1073 (Fig. 3).
  26. G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982), pp. 158–164.
  27. Ref. 26, pp. 306–309.
  28. D. K. Lynch, W. Livingston, Color and Light in Nature (Cambridge U. Press, Cambridge, 1995), p. 119 (Fig. 4.10A). Also see R. A. Anthes, J. J. Cahir, A. B. Fraser, H. A. Panofsky, The Atmosphere, 3rd ed. (Merrill, Columbus, Ohio, 1981), Plate 19b, opposite p. 468.
  29. Eliminating deviation angles outside the primary where the Mie and the Airy 150-μm chromaticities diverge noticeably (θ < 137.8°) reduces Δu′, v′¯ to only 0.008901. That still exceeds the 50-μm cloud drop’s Δu′, v′¯ of 0.00571.
  30. Reference 26, pp. 138–139. Similar techniques are used in R. J. Kubesh, “Computer display of chromaticity coordinates with the rainbow as an example,” Am. J. Phys. 60, 919–923 (1992). Figures 14–17 each contain more than 219,000 pixels.
  31. All chromaticity and luminance differences are calculated with the real-number data that underlie Figs. 14–17.
  32. In a personal communication, G. P. Können (Royal Netherlands Meteorological Institute, De Bilt, The Netherlands) kindly extended Ref. 11’s mathematics to include both polarizations of the Airy secondary.
  33. Können and de Boer clearly show this phase relationship (Ref. 11, p. 1964).
  34. Ref. 6, p. 70.
  35. R. L. Lee, “What are ‘all the colors of the rainbow’?,” Appl. Opt. 30, 3401–3407, 3545 (1991).
  36. In fact, secondary supernumeraries are seen only rarely in nature. See G. P. Können, “Appearance of supernumeraries of the secondary rainbow in rain showers,” J. Opt. Soc. Am. A4, 810–816 (1987).
  37. E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles (Wiley, New York, 1976), pp. 163, 170 (Figs. 3.19 and 3.22).
  38. Although Nussenzveig notes in passing that monodisperse ripples will be averaged out “over a range of size parameters,” he does not dwell on the point (Ref. 7, p. 1079).
  39. R. L. Lee, A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science (Penn State Press, University Park, Pa., to be published), Figs. 8-22 and 8-23.
  40. Consistent with my definition of the natural rainbow, naturalistic here means “as seen in naturally occurring polydisperse bows.”

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