Abstract

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

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References

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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.”

1992 (1)

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.

1991 (2)

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

R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
[CrossRef] [PubMed]

1987 (1)

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).

1986 (1)

1983 (1)

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).

1979 (4)

For a notable exception, see K. Sassen, “Angular scattering and rainbow formation in pendant drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).

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).

S. D. Mobbs, “Theory of the rainbow,” J. Opt. Soc. Am. 69, 1089–1092 (1979).
[CrossRef]

G. P. Können, J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979).
[CrossRef] [PubMed]

1908 (1)

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).

Airy, G. B.

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!

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 112–113, 477–481.

Boyer, C. B.

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.

de Boer, J. H.

Fraser, A. B.

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).

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.

Gorraiz, J.

Horvath, H.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 112–113, 477–481.

Hulst, H. C. van de

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981; reprint of 1957 Wiley edition), p. 247.

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964; reprint of 1940 McGraw-Hill edition), pp. 491–494.

Können, G. P.

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).

G. P. Können, J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979).
[CrossRef] [PubMed]

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.

Kubesh, R. J.

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.

Lee, R. L.

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

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.

Livingston, W.

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.

Lynch, D. K.

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.

McCartney, E. J.

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).

Mie, G.

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).

Mobbs, S. D.

Nussenzveig, H. M.

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).

H. M. Nussenzveig, “The theory of the rainbow,” in Atmospheric Phenomena (Freeman, San Francisco, 1980), pp. 60–71.

Raimann, G.

Sassen, K.

For a notable exception, see K. Sassen, “Angular scattering and rainbow formation in pendant drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982), pp. 158–164.

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970), pp. 179–181.

van de Hulst, H. C.

Wang, R. T.

Wyszecki, G.

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982), pp. 158–164.

Am. J. Phys. (1)

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.

Ann. Phys. (1)

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).

Appl. Opt. (4)

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

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).

J. Opt. Soc. Am. (2)

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).

For a notable exception, see K. Sassen, “Angular scattering and rainbow formation in pendant drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).

J. Opt. Soc. Am. (1)

Weatherwise (1)

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).

Other (29)

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.

See Ref. 9, p. 247.

Ref. 6, p. 70.

Similar Airy underestimates of intensities in Alexander’s dark band are evident in Ref. 7, p. 1073 (Fig. 3).

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982), pp. 158–164.

Ref. 26, pp. 306–309.

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.

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.

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).

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).

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.

Consistent with my definition of the natural rainbow, naturalistic here means “as seen in naturally occurring polydisperse bows.”

All chromaticity and luminance differences are calculated with the real-number data that underlie Figs. 14–17.

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.

Können and de Boer clearly show this phase relationship (Ref. 11, p. 1964).

Ref. 6, p. 70.

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.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981; reprint of 1957 Wiley edition), p. 247.

H. M. Nussenzveig, “The theory of the rainbow,” in Atmospheric Phenomena (Freeman, San Francisco, 1980), pp. 60–71.

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!

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.

Ref. 2, p. 313.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964; reprint of 1940 McGraw-Hill edition), pp. 491–494.

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970), pp. 179–181.

Ref. 11, p. 1963.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 112–113, 477–481.

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.

Ref. 17, pp. 300–304.

Exceptions include bows seen from mountains, hills, and airplanes in flight.

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Figures (30)

Fig. 1
Fig. 1

Comparison of Mie and Airy intensity distribution functions at size parameter x = 628.3 for wavelength λ = 0.5 μm and a drop radius of r = 50 μm. In Figs. 1 and 2, C rms is the root-mean-square distance between the two theories’ intensities averaged over deviation angle θ = 137°–145°.

Fig. 2
Fig. 2

Comparison of Mie and Airy intensity distribution functions at x ∼ 1885 for λ = 0.5 μm and r = 150 μm.

Fig. 3
Fig. 3

Relative solar spectral radiance at a Sun elevation of ∼45°, derived from measurements at University Park, Pa., on 5 October 1987. This is the spectral illuminant assumed in calculating Figs. 425 and 27–30, and its CIE 1976 uniform chromaticity scale coordinates are u′ = 0.1986 and v′ = 0.4713.

Fig. 4
Fig. 4

Normalized Mie and Airy theory primary luminances as functions of deviation angle θ for a single 500-μm-radius raindrop (both polarizations). The two theories’ intensity distribution functions are convolved with Fig. 3’s illuminant to produce spectrally integrated luminances. These luminances are plotted in arbitrary units that are also used in Figs. 57 and 1821.

Fig. 5
Fig. 5

Normalized Mie and Airy theory primary luminances as functions of deviation angle θ for a single 150-μm-radius drizzle drop (both polarizations).

Fig. 6
Fig. 6

Normalized Mie and Airy theory primary luminances as functions of deviation angle θ for a single 50-μm-radius cloud drop (both polarizations).

Fig. 7
Fig. 7

Normalized Mie and Airy theory primary luminances as a function of deviation angle θ for a single 10-μm-radius cloud drop (both polarizations).

Fig. 8
Fig. 8

Portion of the CIE 1976 UCS diagram, showing the chromaticity of Fig. 3’s illuminant (+) and separate u′(θ), v′(θ) chromaticity curves for Airy and Mie theory primary cloudbows at 10-μm radius (both polarizations). The two curves’ mean colorimetric separation Δ u ,   v ¯ = 0.0080, compared with a mean MacAdam JND of 0.004478 in the u′, v′ region spanned by the Mie and the Airy primaries.

Fig. 9
Fig. 9

Close-up view of Fig. 8. Note that the u′, v′ scaling is isotropic in this and all subsequent UCS diagrams.

Fig. 10
Fig. 10

Perspective view that combines chromaticities (Fig. 9) and luminances (Fig. 7) for Mie and Airy theory 10-μm-radius primary cloudbows (both polarizations). The combined luminance and chromaticity curves are also shown projected on the u′, v′ chromaticity plane.

Fig. 11
Fig. 11

Chromaticity curves for the 50-μm-radius Airy and Mie theory primaries differ by Δ u ,   v ¯ = 0.00571 (both polarizations).

Fig. 12
Fig. 12

Chromaticity curves for the 150-μm-radius Airy and Mie theory primaries differ by Δ u ,   v ¯ = 0.01037 (both polarizations). Breaks in the Airy curve mark the positions of its primary and first supernumerary.

Fig. 13
Fig. 13

Chromaticity curves for the 500-μm-radius Airy and Mie theory primaries differ by Δ u ,   v ¯ = 0.01280 (both polarizations).

Fig. 14
Fig. 14

Map of Airy theory colors for monodisperse primary rainbows and cloudbows. The map assumes (1) Fig. 3’s solar spectrum as the illuminant, (2) Eq. (2)’s sun-width smoothing filter, (3) both rainbow polarizations, (4) spherical, nonabsorbing water drops. At each drop radius, colors’ luminances are normalized by the maximum luminance for that radius. A colorimetrically calibrated version of this figure can be seen at the www address given in the acknowledgment.

Fig. 15
Fig. 15

Map of Mie theory colors for monodisperse primary rainbows and cloudbows. Figure 14’s assumptions (1)–(4) also hold here. A colorimetrically calibrated version of this figure can be seen at the www address given in the acknowledgment.

Fig. 16
Fig. 16

Map of Airy theory colors for monodisperse secondary rainbows and cloudbows. Figure 14’s assumptions (1)–(4) also hold here. A colorimetrically calibrated version of this figure can be seen at the www address given in the acknowledgment.

Fig. 17
Fig. 17

Map of Mie theory colors for monodisperse secondary rainbows and cloudbows. Figure 14’s assumptions (1)–(4) also hold here. A colorimetrically calibrated version of this figure can be seen at the www address given in the acknowledgment.

Fig. 18
Fig. 18

Normalized Mie and Airy theory secondary luminances as functions of deviation angle θ for a single 500-μm-radius raindrop (both polarizations). The illuminant is Fig. 3’s sunlight spectrum.

Fig. 19
Fig. 19

Normalized Mie and Airy theory secondary luminances as functions of deviation angle θ for a single 150-μm-radius drizzle drop (both polarizations).

Fig. 20
Fig. 20

Normalized Mie and Airy theory secondary luminances as functions of deviation angle θ for a single 50-μm-radius cloud drop (both polarizations).

Fig. 21
Fig. 21

Normalized Mie and Airy theory secondary luminances as functions of deviation angle θ for a single 10-μm-radius cloud drop (both polarizations).

Fig. 22
Fig. 22

Chromaticity curves for the 10-μm-radius Airy and Mie theory secondaries differ by Δ u ,   v ¯ = 0.01585 (both polarizations). The mean MacAdam JND in the u′, v′ region spanned by the Mie and the Airy secondaries =0.004773.

Fig. 23
Fig. 23

Chromaticity curves for the 50-μm-radius Airy and Mie theory secondaries differ by Δ u ,   v ¯ = 0.01944 (both polarizations).

Fig. 24
Fig. 24

Chromaticity curves for the 150-μm-radius Airy and Mie theory secondaries differ by Δ u ,   v ¯ = 0.03324 (both polarizations).

Fig. 25
Fig. 25

Chromaticity curves for the 500-μm-radius Airy and Mie theory secondaries differ by Δ u ,   v ¯ = 0.04019 (both polarizations).

Fig. 26
Fig. 26

Deirmendjian modified gamma distribution for drop sizes typical of stratocumulus or fog, 1–40-μm radii. This polydispersion produces the smoothed Mie and Airy theory data shown in Figs. 2730.

Fig. 27
Fig. 27

Normalized luminances for Mie and Airy primary fogbows as functions of deviation angle θ for Fig. 26’s drop-size distribution (both polarizations). In this and subsequent figures, the illuminant is Fig. 3’s sunlight spectrum.

Fig. 28
Fig. 28

Normalized luminances for Mie and Airy secondary fogbows as functions of deviation angle θ for Fig. 26’s drop-size distribution (both polarizations).

Fig. 29
Fig. 29

Chromaticity curves for the Airy and the Mie primary fogbows differ by Δ u ,   v ¯ = 0.00243 (both polarizations).

Fig. 30
Fig. 30

Chromaticity curves for the Airy and the Mie secondary fogbows differ by Δ u ,   v ¯ = 0.00710 (both polarizations).

Tables (1)

Tables Icon

Table 1 Summary of Chromaticity Distances Δ u ,   v ¯ and mean contrast Crms between Mie and Airy Theory Rainbows and Cloudbowsa

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

C rms = i = 1 N L v , Airy θ i - L v , Mie θ i L v , Mie θ i 2 / N 1 / 2
L S θ = θ - w θ + w 1 - θ - ϑ w 2 1 / 2 L v ϑ d ϑ ,
Δ u ,   v = u θ Mie - u θ Airy 2 + v θ Mie - v θ Airy 2 1 / 2 ,

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