Abstract

The secondary rainbow scattering angle for spheroidal drops of water is virtually independent of aspect ratio for most visible wavelengths. For most solar heights the residual aspect-ratio dependence shifts the bow toward a smaller deviation angle if the drop size increases. These two facts explain why the supernumeraries of the secondary rainbow are never seen in rain showers. At high solar elevations the flattening of drops results in a shift of the secondary rainbow toward a larger deviation angle. It is shown that this shift is still large enough to cause the formation of the first supernumerary in red light. This red supernumerary of the secondary rainbow may be observable by eye in natural showers if a red filter is used to remove the obscuring contribution of shorter wavelengths to the light of the rainbow. For indices of refraction far from that of water, a strong aspect-ratio dependence of the secondary rainbow angle is shown to be present. Some possible implications of this for the formation of a hyperbolic umbilic diffraction catastrophe in the secondary rainbow pattern are indicated.

© 1987 Optical Society of America

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References

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  1. A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?”J. Opt. Soc. Am. 73, 1626–1629 (1983).
    [CrossRef]
  2. M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, New York, 1954).
  3. W. Möbius, “Zur Theorie des Regenboges und ihren experimentellen Prüfung,” Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. 30, 108–254 (1907).
  4. W. Möbius, “Zur Theorie des Regenboges und ihren experimentellen Prüfung,” Ann. Phys. 33, 1493–1558 (1910). (This shortened version of Ref. 3 does not contain the derivation of this formula.)
    [CrossRef]
  5. F. Volz, “Der Regenbogen,” in Handbuch der Geophysik VIII, F. Linke, F. Möller, eds. (Bornträger, Berlin, 1961), pp. 977–982.
  6. H. R. Pruppacher, R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,”J. Atmos. Sci. 28, 86–94 (1971).
    [CrossRef]
  7. H. R. Prupacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, The Netherlands, 1978).
    [CrossRef]
  8. A. W. Green, “An approximation for the shapes of large raindrops,”J. Appl. Meteorol. 14, 1578–1583 (1975).
    [CrossRef]
  9. P. L. Marston, “Rainbow phenomena and the detection of nonsphericity in drops,” Appl. Opt. 19, 680–685 (1980).
    [CrossRef] [PubMed]
  10. P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
    [CrossRef]
  11. G. P. Können, J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979).
    [CrossRef] [PubMed]
  12. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 205, 245–246.
  13. A. P. M. Baede, “Charge transfer between neutrals at hyperthermal energies,” in Molecular Scattering: Physical and Chemical Applications, Vol. XXX of Advances in Chemical Physics, K. P. Lawley, ed. (Wiley, New York, 1975), p. 474.
  14. A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,”J. Atmos. Sci. 29, 211–212 (1972).
    [CrossRef]
  15. R. G. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, 1980), Fig. 1–13, p. 19.
  16. G. P. Können, Polarized Light in Nature (Cambridge U. Press, Cambridge, 1985), Plate 29, p. 55.

1984 (1)

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
[CrossRef]

1983 (1)

1980 (1)

1979 (1)

1975 (1)

A. W. Green, “An approximation for the shapes of large raindrops,”J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

1972 (1)

A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,”J. Atmos. Sci. 29, 211–212 (1972).
[CrossRef]

1971 (1)

H. R. Pruppacher, R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,”J. Atmos. Sci. 28, 86–94 (1971).
[CrossRef]

1910 (1)

W. Möbius, “Zur Theorie des Regenboges und ihren experimentellen Prüfung,” Ann. Phys. 33, 1493–1558 (1910). (This shortened version of Ref. 3 does not contain the derivation of this formula.)
[CrossRef]

1907 (1)

W. Möbius, “Zur Theorie des Regenboges und ihren experimentellen Prüfung,” Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. 30, 108–254 (1907).

Baede, A. P. M.

A. P. M. Baede, “Charge transfer between neutrals at hyperthermal energies,” in Molecular Scattering: Physical and Chemical Applications, Vol. XXX of Advances in Chemical Physics, K. P. Lawley, ed. (Wiley, New York, 1975), p. 474.

de Boer, J. H.

Fraser, A. B.

A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?”J. Opt. Soc. Am. 73, 1626–1629 (1983).
[CrossRef]

A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,”J. Atmos. Sci. 29, 211–212 (1972).
[CrossRef]

Green, A. W.

A. W. Green, “An approximation for the shapes of large raindrops,”J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

Greenler, R. G.

R. G. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, 1980), Fig. 1–13, p. 19.

Klett, J. D.

H. R. Prupacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, The Netherlands, 1978).
[CrossRef]

Können, G. P.

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

G. P. Können, Polarized Light in Nature (Cambridge U. Press, Cambridge, 1985), Plate 29, p. 55.

Marston, P. L.

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
[CrossRef]

P. L. Marston, “Rainbow phenomena and the detection of nonsphericity in drops,” Appl. Opt. 19, 680–685 (1980).
[CrossRef] [PubMed]

Minnaert, M.

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

Möbius, W.

W. Möbius, “Zur Theorie des Regenboges und ihren experimentellen Prüfung,” Ann. Phys. 33, 1493–1558 (1910). (This shortened version of Ref. 3 does not contain the derivation of this formula.)
[CrossRef]

W. Möbius, “Zur Theorie des Regenboges und ihren experimentellen Prüfung,” Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. 30, 108–254 (1907).

Pitter, R. L.

H. R. Pruppacher, R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,”J. Atmos. Sci. 28, 86–94 (1971).
[CrossRef]

Prupacher, H. R.

H. R. Prupacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, The Netherlands, 1978).
[CrossRef]

Pruppacher, H. R.

H. R. Pruppacher, R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,”J. Atmos. Sci. 28, 86–94 (1971).
[CrossRef]

Trinh, E. H.

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 205, 245–246.

Volz, F.

F. Volz, “Der Regenbogen,” in Handbuch der Geophysik VIII, F. Linke, F. Möller, eds. (Bornträger, Berlin, 1961), pp. 977–982.

Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. (1)

W. Möbius, “Zur Theorie des Regenboges und ihren experimentellen Prüfung,” Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. 30, 108–254 (1907).

Ann. Phys. (1)

W. Möbius, “Zur Theorie des Regenboges und ihren experimentellen Prüfung,” Ann. Phys. 33, 1493–1558 (1910). (This shortened version of Ref. 3 does not contain the derivation of this formula.)
[CrossRef]

Appl. Opt. (2)

J. Appl. Meteorol. (1)

A. W. Green, “An approximation for the shapes of large raindrops,”J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

J. Atmos. Sci. (2)

A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,”J. Atmos. Sci. 29, 211–212 (1972).
[CrossRef]

H. R. Pruppacher, R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,”J. Atmos. Sci. 28, 86–94 (1971).
[CrossRef]

J. Opt. Soc. Am. (1)

Nature (1)

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
[CrossRef]

Other (7)

H. R. Prupacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, The Netherlands, 1978).
[CrossRef]

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

F. Volz, “Der Regenbogen,” in Handbuch der Geophysik VIII, F. Linke, F. Möller, eds. (Bornträger, Berlin, 1961), pp. 977–982.

R. G. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, 1980), Fig. 1–13, p. 19.

G. P. Können, Polarized Light in Nature (Cambridge U. Press, Cambridge, 1985), Plate 29, p. 55.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 205, 245–246.

A. P. M. Baede, “Charge transfer between neutrals at hyperthermal energies,” in Molecular Scattering: Physical and Chemical Applications, Vol. XXX of Advances in Chemical Physics, K. P. Lawley, ed. (Wiley, New York, 1975), p. 474.

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Figures (5)

Fig. 1
Fig. 1

Path of the ray of minimum deviation for the secondary rainbow. For a spherical drop (solid lines), the angle of refraction β of the Descartes ray is very close to 45°. Therefore, seen from the center of the drop, the difference in position angle of two subsequent hits at the wall is about 90°. If the drop becomes spheroidal (dashed lines), the path of the Descartes ray changes. However, because of this difference of 90° in position angle, the changes in light path at hits (1) and (2) are almost entirely compensated for by the subsequent changes at hits (3) and (4), so that the aspect-ratio dependence of the direction of the emerging ray remains very small. This explains why supernumeraries of the secondary rainbow in showers are extremely rare.

Fig. 2
Fig. 2

Intensity of the secondary rainbow as a function of scattering angle θ, integrated over a Marshall–Palmer drop-size distribution and over the solar disk. The curve for C = 0° is valid for the rainbow base if the solar elevation h is low and for its top if h ≃ 30°; C = 1° can be realized at the rainbow top if the solar elevation is 40° or more. For the definition of C, see Eq. (14). The interval of θ has been extended over 180°, so that the scattering angle equals the deviation angle. The Descartes secondary rainbow angle for spheres, denoted by θr(sphere),is about 231°. The intensity of the light caused by external reflection at the drops is also indicated.

Fig. 3
Fig. 3

Intensity of the primary rainbow as a function of scattering angle θ near the base of the bow at low solar elevation and at its top, integrated over the same drop-size distribution as the one applied in Fig. 2 and over the solar disk. Same intensity units as for Fig. 2. C is defined by Eq. (5). The Descartes primary rainbow angle for spheres, denoted by θr(sphere), is about 1380.

Fig. 4
Fig. 4

Visibility diagram for the first supernumerary of the secondary rainbow in showers. If Vis < 0.06, supernumerary formation is impossible or highly unlikely; if Vis ≥ 0.06, there is a chance for such formation. Vis = 0.15 corresponds to the situation in Fig. 2 for C = 1°. The line for C = 0° represents the situation at which the rainbow intensity distribution is independent of flattening of drops. The most favorable conditions for supernumerary formation occur at high solar elevations and long wavelengths. If the solar elevation exceeds 51°, the rainbow top is below the horizon.

Fig. 5
Fig. 5

Supernumerary of the secondary rainbow in an artificially made water spray, apparently consisting of a rather monodisperse drop-size distribution. To bring out the supernumerary, the Ektachrome original has been copied in black and white through a red filter. This procedure improves the visibility of the feature significantly. In the original, parts of the second supernumerary of the secondary rainbow are discernible too. (Photography by W. C. Livingston.)

Tables (1)

Tables Icon

Table 1 Relationship between the Quantity C, Which Determines the Strength of the Dependence of the Rainbow Angle on Drop Distortion, and the Visibility Parameter Vis of the First Supernumerary of the Secondary Rainbow

Equations (20)

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ρ = ( a - b ) / ( a + b ) .
ρ 0.050 r 2
Δ θ r θ r ( spheroid ) - θ r ( sphere ) = 180 ° π 16 ρ sin β cos 3 β cos ( 2 h - 42 ° ) ,
ρ = ρ ( 1 - sin 2 ζ cos 2 h ) , tan h = tan h / cos ζ .
Δ θ r = 13 ° r 2 cos ( 2 h - 42 ° ) C r 2 .
Δ θ r = - 180 ° π 64 ρ sin β cos 3 β cos 2 β cos ( 2 h - 51 ° ) .
cos 2 β = 5 n 2 - 9 4 n 2 .
δ β β - 45 ° = 180 ° π 5 4 9 / 5 ( 9 / 5 - n ) .
C 0 = 32 20 δ β = 86 ° ( 1.342 - n ) .
Δ θ r = C 0 r 2 cos ( 2 h - 51 ° ) C 1 r 2 .
Δ θ r ( second order ) = 180 ° π ρ 2 [ - 32 3 cos 2 ( 2 h - 51 ° ) + 16 sin ( 4 h - 102 ° ) ] - 20 C 0 ρ 2 [ 10 + 5 sin ( 4 h - 102 ° ) - 28 3 cos 2 ( 2 h - 51 ° ) ] .
Δ θ r = C 1 r 2 + C 2 r 4 ,
C 1 = C 0 cos ( 2 h - 51 ° ) , C 2 = - 1.56 ° cos 2 ( 2 h - 51 ° ) + 2.34 ° sin ( 4 h - 102 ° ) - C 0 [ ½ + ¼ sin ( 4 h - 102 ° ) - 7 / 15 cos 2 ( 2 h - 51 ° ) ] .
Δ θ r = C r 2 ,
C = C 1 + 0.5 C 2 .
I ( r , θ ) r 7 / 3 Ai 2 { - 3.1 ( 500 λ ) 2 / 3 r 2 / 3 [ θ - θ r ( sphere ) - C r 2 ] } r 7 / 3 Ai 2 [ f ( r , θ ) ] .
4 C r s 2 = θ - θ r ( sphere ) .
θ 1 , 2 ( λ , h ) = 0.71 ° C 1 / 4 ( λ , h ) ( λ 640 ) 1 / 2 .
d N / d r exp ( - 6 r ) .
V i s = I sup - I min I sup + I min ,

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