Abstract

We both theoretically and experimentally examine the behavior of the first- and the second-order rainbows produced by a normally illuminated glass rod, which has a nearly elliptical cross section, as it is rotated about its major axis. We decompose the measured rainbow angle, taken as a function of the rod’s rotation angle, into a Fourier series and find that the rod’s refractive index, average ellipticity, and deviation from ellipticity are encoded primarily in the m = 0, 2, 3 Fourier coefficients, respectively. We determine these parameters for our glass rod and, where possible, compare them with independent measurements. We find that the average ellipticity of the rod agrees well with direct measurements, but that the rod’s diameter inferred from the spacing of the supernumeraries of the first-order rainbow is significantly larger than that obtained by direct measurement. We also determine the conditions under which the deviation of falling water droplets from an oblate spheroidal shape permits the first few supernumeraries of the second-order rainbow to be observed in a rain shower.

© 1998 Optical Society of America

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References

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  1. C. B. Boyer, The Rainbow From Myth to Mathematics (Princeton U. Press, Princeton, N.J., 1987), p. 309.
  2. W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentallen Prüfung,” Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. 30, 105–254 (1907–1909).
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    [CrossRef]
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  12. J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, New York, 1987), pp. 129–180.
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  14. H. R. Pruppacher, R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain droplets,” J. Atmos. Sci. 28, 86–94 (1971).
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  15. A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?” J. Opt. Soc. Am. 73, 1626–1628 (1983).
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  19. C. L. Adler, J. A. Lock, B. R. Stone, C. J. Garcia, “High-order interior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A 14, 1305–1315 (1997).
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  24. J. A. Lock, C. L. Adler, B. R. Stone, P. D. Zajak, “Amplification of high-order rainbows of a cylinder with an elliptical cross section,” Appl. Opt. 37, 1527–1533 (1998).
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  25. M. Minnaert, Light and Colour in the Open Air (Dover, New York, 1954), p. 173.
  26. R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, Cambridge, 1989), p. 20.
  27. A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
    [CrossRef]
  28. H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, The Netherlands, 1978), pp. 23–25.
  29. A. B. Fraser, “Chasing rainbows,” Weatherwise 36, 280–289 (1983).
    [CrossRef]

1998 (3)

1997 (4)

1995 (1)

G. Gouesbet, “Scattering of a first order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
[CrossRef]

1994 (1)

J. A. Lock, T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[CrossRef]

1992 (1)

1988 (1)

1987 (1)

1983 (2)

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]

1974 (2)

1971 (1)

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

1910 (1)

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentallen Prüfung,” Ann. Phys. 33, 1493–1558 (1910).
[CrossRef]

Adler, C. L.

Barton, J. P.

Bowman, J. J.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, New York, 1987), pp. 129–180.

Boyer, C. B.

C. B. Boyer, The Rainbow From Myth to Mathematics (Princeton U. Press, Princeton, N.J., 1987), p. 309.

Feshbach, H.

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 1407–1423.

Fraser, A. B.

Garcia, C. J.

Gedzelman, S. D.

Gouesbet, G.

G. Gouesbet, “Scattering of a first order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
[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.

R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, Cambridge, 1989), p. 20.

Hovenac, E. A.

Klett, J. D.

H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, The Netherlands, 1978), pp. 23–25.

Können, G. P.

Lock, J. A.

Marcuse, D.

Marston, P. L.

McCollum, T. A.

J. A. Lock, T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[CrossRef]

Minnaert, M.

M. Minnaert, Light and Colour in the Open Air (Dover, New York, 1954), p. 173.

Möbius, W.

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentallen Prüfung,” Ann. Phys. 33, 1493–1558 (1910).
[CrossRef]

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentallen Prüfung,” Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. 30, 105–254 (1907–1909).

Morse, P.

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 1407–1423.

Mount, C. M.

Pitter, R. L.

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

Presby, H. M.

Pruppacher, H. R.

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

H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, The Netherlands, 1978), pp. 23–25.

Sassen, K.

Senior, T. B. A.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, New York, 1987), pp. 129–180.

Stone, B. R.

Thiessen, D. B.

Uslenghi, P. L. E.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, New York, 1987), pp. 129–180.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 297–328.

Zajak, P. D.

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

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentallen Prüfung,” Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. 30, 105–254 (1907–1909).

Am. J. Phys. (1)

J. A. Lock, T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[CrossRef]

Ann. Phys. (1)

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentallen Prüfung,” Ann. Phys. 33, 1493–1558 (1910).
[CrossRef]

Appl. Opt. (6)

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. (1)

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

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (6)

Part. Part. Syst. Charact. (1)

G. Gouesbet, “Scattering of a first order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
[CrossRef]

Weatherwise (1)

A. B. Fraser, “Chasing rainbows,” Weatherwise 36, 280–289 (1983).
[CrossRef]

Other (7)

H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, The Netherlands, 1978), pp. 23–25.

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 1407–1423.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, New York, 1987), pp. 129–180.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 297–328.

C. B. Boyer, The Rainbow From Myth to Mathematics (Princeton U. Press, Princeton, N.J., 1987), p. 309.

M. Minnaert, Light and Colour in the Open Air (Dover, New York, 1954), p. 173.

R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, Cambridge, 1989), p. 20.

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Figures (9)

Fig. 1
Fig. 1

(a) Cylinder has a cross section consisting of two half-ellipses denoted by the index j = 1, 2 joined smoothly at the points R and L. The length of the common semimajor axis of the two half-ellipses is a, and the length of their differing semiminor axes are b 1 and b 2. The x′ y′ z′ coordinate system is attached to the cylinder. (b) The cylinder is rotated by an angle ξ about the z′ axis. Incident light rays propagate in the -y direction of a fixed laboratory coordinate system.

Fig. 2
Fig. 2

Angles γ p of the normal to the surface and the angles δ p of the interior rays for the 0 ≤ p ≤ 2 interactions of a light ray with the surface.

Fig. 3
Fig. 3

Deviation of the first-order rainbow angle θ2 R (ξ) of a two-half-ellipse cross-section cylinder from the Descartes first-order rainbow angle θ2 D as a function of the rotation angle ξ for a refractive index n = 1.474, average ellipticity ∊ave = 0.060, and ellipticity difference Δ∊ = 0 (curve a), Δ∊ = 0.06 (curve b), Δ∊ = 0.12 (curve c), and Δ∊ = 0.18 (curve d). The range of Δ∊ here is much larger than in Table 3 because the dependence of the rainbow angle on Δ∊ is weak. The value of ∊ave is also different than that in Table 3.

Fig. 4
Fig. 4

Deviation of the second-order rainbow angle θ3 R (ξ) for a two-half-ellipse cross-section cylinder from the Descartes second-order rainbow angle θ3 D as a function of the rotation angle ξ for a refractive index n = 1.474, average ellipticity ∊ave = -0.037, and ellipticity difference Δ∊ = 0 (curve a), Δ∊ = 0.01 (curve b), Δ∊ = 0.02 (curve c), Δ∊ = 0.03 (curve d), and Δ∊ = 0.04 (curve e). These parameters are identical to those of Table 4.

Fig. 5
Fig. 5

Experimental first-order rainbow deviation angle as a function of the rotation angle ξ and the theoretical fit of the two-half-ellipse cross-section model with n = 1.474, ∊ave = -0.037, and Δ∊ = 0.026.

Fig. 6
Fig. 6

Experimental second-order rainbow deviation angle as a function of the rotation angle ξ and the theoretical fit of the two-half-ellipse cross-section model with n = 1.474, ∊ave = -0.037, and Δ∊ = 0.026.

Fig. 7
Fig. 7

Experimental light intensity I in the vicinity of the first-order rainbow as a function of scattering angle Δθ. The peak of the principal rainbow maximum corresponds to Δθ = 0°.

Fig. 8
Fig. 8

Negative of the average ellipticity -∊ave and the ellipticity difference Δ∊ as functions of the equal-volume-sphere radius a 0 of raindrops falling at terminal velocity and derived from the parameterization of Ref. 14. The filled circles are the analytical approximation to -∊ave of Ref. 27, and the open circles are the linearized approximation of Ref. 16.

Fig. 9
Fig. 9

Deviation angle of the second-order rainbow θ3 R with respect to the Descartes second-order rainbow angle θ3 D as a function of the equal-volume-sphere radius a 0 of raindrops falling at terminal velocity for solar elevation angles of (a) 10°, (b) 20°, (c) 40°. In each graph, the lowest curve is the principal Airy maximum, the middle graph is the first supernumerary maximum, and the highest curve is the second supernumerary maximum.

Tables (6)

Tables Icon

Table 1 First Five Even Fourier Coefficients in Degrees of the First-Order Rainbow Deviation Angle θ2 R (ξ) for an Elliptical Cross-Sectional Cylinder with Refractive Index n = 1.474 and Ellipticities ∊ = -0.001, -0.01, and -0.1 as Defined in Eq. (17)a

Tables Icon

Table 2 First Five Even Fourier Coefficients in Degrees of the Second-Order Rainbow Deviation Angle θ3 R (ξ) for an Elliptical Cross-Sectional Cylinder with Refractive Index n = 1.474 and Ellipticities ∊ = -0.001, -0.01, and -0.1 as Defined in Eq. (17)a

Tables Icon

Table 3 First Six Fourier Coefficients in Degrees of the First-Order Rainbow Deviation Angle θ2 R (ξ) for a Two-Half-Ellipse Cross-Sectional Cylinder with Refractive Index n = 1.474, Average Ellipticity ∊ave = -0.037, and Various Values of the Ellipticity Difference Δ∊ as Defined in Eq. (23)

Tables Icon

Table 4 First Six Fourier Coefficients in Degrees of the Second-Order Rainbow Deviation Angle θ3 R (ξ) for a Two-Half-Ellipse Cross-Sectional Cylinder with Refractive Index n = 1.474, Average Ellipticity ∊ave = -0.037, and Various Values of the Ellipticity Difference Δ∊ as Defined in Eq. (23)

Tables Icon

Table 5 First Six Fourier Coefficients in Degrees of the Experimental First-Order Rainbow Deviation Angle and of θ2 R (ξ) for a Cylinder with a Two-Half-Ellipse Cross Section, Refractive Index n = 1.474, Average Ellipticity ∊ave = -0.037, and Ellipticity Difference Δ∊ = 0.026

Tables Icon

Table 6 First Six Fourier Coefficients in Degrees of the Experimental Second-Order Rainbow Angle and of θ3 R (ξ) for a Cylinder with a Two-Half-Ellipse Cross Section, Refractive Index n = 1.474, Average Ellipticity ∊ave = -0.037, and Ellipticity Difference Δ∊ = 0.026

Equations (24)

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x 2 a 2 + y 2 b 1 2 = 1     for   y 0 , x 2 a 2 + y 2 b 2 2 = 1     for   y < 0 ,
y = - β j x ± α j A j 2 - x 2 1 / 2 ,
A j 2 = b j 2 / a 2 sin 2   ξ + cos 2   ξ ,
α j = b j / a / A j 2 ,
β j = b j 2 / a 2 - 1 sin   ξ   cos   ξ / A j 2 ,
tan   γ p = A j 2 - x p 2 x p y p + β j A j 2 .
δ 0 = γ 0 + arcsin cos   γ 0 / n ,
δ 1 = 2 γ 1 - δ 0 ,
δ 2 = 2 γ 2 - δ 1 - π .
tan   δ p = y p + 1 - y p / x p + 1 - x p
θ 2 = 3 π / 2 + arcsin n   sin γ 2 - δ 1 - γ 2 ,
θ 3 = 3 π / 2 + arcsin n   sin γ 3 - δ 2 - γ 3 .
cos   ϕ i D = n 2 - 1 p 2 - 1 1 / 2 ,
sin   ϕ t D = 1 / n sin   ϕ i D ,
θ p D = p - 1 π + 2 ϕ i D - 2 p ϕ t D .
x R = a   cos   ξ ,     x L = - a   cos   ξ , y R = a   sin   ξ ,     y L = - a   sin   ξ .
= b / a - 1 ,
θ 2 R ξ = θ 2 D - 8   sin   ϕ t D cos 3   ϕ t D × cos 2 ξ + θ 2 D + O 2 ,
θ 3 R ξ = θ 3 D + 32   sin   ϕ t D cos 3   ϕ t D × cos   2 ϕ t D cos 2 ξ + θ 3 D + O 2 ,
θ 2 R ξ = E 0 + m = 1   E m cos m ξ + m = 1   F m sin m ξ ,
θ 3 R ξ = G 0 + m = 1   G m cos m ξ + m = 1   H m sin m ξ .
ave = b 1 / a + b 2 / a 2 - 1 = 1 + 2 2
Δ = b 1 / a - b 2 / a = 1 - 2
1 = b 1 / a - 1 ,     2 = b 2 / a - 1 .

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