Abstract

We measured the supernumerary spacing parameter of the first- and second-order rainbows of two glass rods, each having an approximately elliptical cross section, as a function of the rod’s rotation angle. We attribute large fluctuations in the supernumerary spacing parameter to small local inhomogeneities in the rod’s refractive index. The low-pass filtered first-order rainbow experimental data agree with the prediction of ray-tracing–wave-front modeling to within a few percent, and the second-order rainbow data exhibit additional effects that are due to rod nonellipticity.

© 2001 Optical Society of America

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References

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  1. W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentallen Prufung,” Abh. Math.-Phys. Kl. Saechs. Ges. Wiss. 30, 105–254 (1907–1909).
  2. W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentallen Prufung,” Ann. Phys. (Leipzig) 33, 1493–1558 (1910).
    [CrossRef]
  3. G. P. Können, “Appearance of supernumeraries of the secondary rainbow in rain showers,” J. Opt. Soc. Am. A 4, 810–816 (1987).
    [CrossRef]
  4. A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?” J. Opt. Soc. Am. 73, 1626–1628 (1983), color plate 1.
    [CrossRef]
  5. A. B. Fraser, “Chasing rainbows,” Weatherwise 36, 280–287 (1983).
    [CrossRef]
  6. J. A. Lock, “Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder. I. Theory,” Appl. Opt. 39, 5040–5051 (2000).
    [CrossRef]
  7. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sect. 13.23, pp. 243–246.
  8. R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
    [CrossRef] [PubMed]
  9. J. P. A. J. van Beeck, “Rainbow phenomena: development of a laser-based, non-intrusive technique for measuring droplet size, temperature and velocity,” Ph.D. dissertation (Eindhoven Technische Universiteit, Eindhoven, The Netherlands, 1997), p. 78.
  10. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington D.C., 1964), Sect. 10.4, pp. 446–447, 478.
  11. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  12. M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
    [CrossRef]
  13. C. L. Adler, J. A. Lock, B. R. Stone, “Rainbow scattering by a cylinder with a nearly elliptical cross section,” Appl. Opt. 37, 1540–1550 (1998).
    [CrossRef]
  14. 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).
    [CrossRef]
  15. J. A. Lock, C. L. Adler, “Debye-series analysis of the first-order rainbow produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A 14, 1316–1328 (1997).
    [CrossRef]
  16. 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).
    [CrossRef]
  17. V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
    [CrossRef]
  18. Ferguson’s Cut Glass Originals, 4292 Pearl Road, Cleveland, Ohio 44109.
  19. J. A. Lock, C. L. Adler, E. A. Hovenac, “Exterior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder: semiclassical scattering theory analysis,” J. Opt. Soc. Am. A 17, 1846–1856 (2000).
    [CrossRef]
  20. N. Roth, K. Anders, A. Frohn, “Refractive-index measurements for the correction of particle sizing methods,” Appl. Opt. 30, 4960–4965 (1991).
    [CrossRef] [PubMed]
  21. J. P. A. J. van Beeck, M. L. Riethmuller, “Nonintrusive measurements of temperature and size of single falling raindrops,” Appl. Opt. 34, 1633–1639 (1995).
    [CrossRef] [PubMed]
  22. P. L. Marston, “Rainbow phenomena and the detection of nonsphericity in drops,” Appl. Opt. 19, 680–685 (1980).
    [CrossRef] [PubMed]
  23. J. P. A. J. van Beeck, M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
    [CrossRef] [PubMed]
  24. H. Lohner, P. Lehmann, K. Bauckhage, “Detection based on rainbow refractometry of droplet sphericity in liquid–liquid systems,” Appl. Opt. 38, 1127–1132 (1999).
    [CrossRef]

2000 (2)

1999 (1)

1998 (2)

1997 (2)

1996 (1)

1995 (1)

1991 (2)

1987 (1)

1983 (2)

1980 (1)

1976 (2)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

1974 (1)

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

1910 (1)

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

Adler, C. L.

Anders, K.

Bauckhage, K.

Berry, M. V.

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

Fraser, A. B.

Frohn, A.

Garcia, C. J.

Hovenac, E. A.

Khare, V.

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

Können, G. P.

Lehmann, P.

Lock, J. A.

Lohner, H.

Marston, P. L.

Möbius, W.

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

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

Nussenzveig, H. M.

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

Riethmuller, M. L.

Roth, N.

Stone, B. R.

van Beeck, J. P. A. J.

van de Hulst, H. C.

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

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sect. 13.23, pp. 243–246.

Walker, J. D.

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Wang, R. T.

Zajak, P. D.

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

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

Adv. Phys. (1)

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

Am. J. Phys. (1)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Ann. Phys. (Leipzig) (1)

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

Appl. Opt. (9)

J. A. Lock, “Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder. I. Theory,” Appl. Opt. 39, 5040–5051 (2000).
[CrossRef]

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

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).
[CrossRef]

C. L. Adler, J. A. Lock, B. R. Stone, “Rainbow scattering by a cylinder with a nearly elliptical cross section,” Appl. Opt. 37, 1540–1550 (1998).
[CrossRef]

N. Roth, K. Anders, A. Frohn, “Refractive-index measurements for the correction of particle sizing methods,” Appl. Opt. 30, 4960–4965 (1991).
[CrossRef] [PubMed]

J. P. A. J. van Beeck, M. L. Riethmuller, “Nonintrusive measurements of temperature and size of single falling raindrops,” Appl. Opt. 34, 1633–1639 (1995).
[CrossRef] [PubMed]

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

J. P. A. J. van Beeck, M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
[CrossRef] [PubMed]

H. Lohner, P. Lehmann, K. Bauckhage, “Detection based on rainbow refractometry of droplet sphericity in liquid–liquid systems,” Appl. Opt. 38, 1127–1132 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Phys. Rev. Lett. (1)

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

Weatherwise (1)

A. B. Fraser, “Chasing rainbows,” Weatherwise 36, 280–287 (1983).
[CrossRef]

Other (4)

J. P. A. J. van Beeck, “Rainbow phenomena: development of a laser-based, non-intrusive technique for measuring droplet size, temperature and velocity,” Ph.D. dissertation (Eindhoven Technische Universiteit, Eindhoven, The Netherlands, 1997), p. 78.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington D.C., 1964), Sect. 10.4, pp. 446–447, 478.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sect. 13.23, pp. 243–246.

Ferguson’s Cut Glass Originals, 4292 Pearl Road, Cleveland, Ohio 44109.

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Figures (6)

Fig. 1
Fig. 1

(a) Deviation of the p = 2 rainbow angle from its average value and (b) supernumerary spacing parameter of a 1.0-mm-radius glass rod with refractive index n = 1.511 and eccentricity ∊ = 0.00155 as a function of the rod’s rotation angle. The solid circles are the experimental data of Ref. 1, and the dashed curves are the predictions of the ray-tracing–wave-front modeling procedure of Ref. 6.

Fig. 2
Fig. 2

Beam of a 3-mW He–Ne laser is attenuated by a polarizing filter, expanded by an afocal telescope, and is incident on a glass rod mounted on a rotation stage. A beam block prevents interference of reflected light from the rod with the p = 2 and p = 3 rainbows. The rainbow pattern is recorded by a CCD camera placed in the scattering near zone.

Fig. 3
Fig. 3

Supernumerary spacing parameter of the p = 2 rainbow of a 8.05-mm-radius glass rod with refractive index n = 1.474 and eccentricity ∊ = -0.037 as a function of the rod’s rotation angle. The solid circles are the experimental data obtained from (a) the first and second supernumerary maxima and (b) the first and third supernumerary maxima. The solid curves are the low-pass filtered experimental data, and the dashed curves are the predictions of the ray-tracing–wave-front modeling procedure of Ref. 6.

Fig. 4
Fig. 4

Supernumerary spacing parameter of the p = 2 rainbow of a 2.44-mm-radius glass rod with refractive index n = 1.502 and eccentricity ∊ = 0.0054 as a function of the rod’s rotation angle. The solid circles are the experimental data that we obtained from averaging the results of the first and second, first and third, and first and fourth supernumerary maxima; the solid curve is the low-pass filtered experimental data; and the dashed curve is the prediction of the ray-tracing–wave-front modeling procedure of Ref. 6.

Fig. 5
Fig. 5

Supernumerary spacing parameter of the p = 3 rainbow of a 8.05-mm-radius glass rod with refractive index n = 1.474 and eccentricity ∊ = -0.037 as a function of the rod’s rotation angle. The solid circles are the experimental data that we obtained from averaging the results of the first and second and the first and third supernumerary maxima, the solid curve is the low-pass filtered experimental data, and the dashed curve is the prediction of the ray-tracing–wave-front modeling procedure of Ref. 6.

Fig. 6
Fig. 6

Supernumerary spacing parameter of the p = 3 rainbow of a 2.44-mm-radius glass rod with refractive index n = 1.502 and eccentricity ∊ = 0.0054 as a function of the rod’s rotation angle. The solid circles are the experimental data that we obtained from averaging the results of the first and second and the first and third supernumerary maxima, the solid curve is the low-pass filtered experimental data, and the dashed curve is the prediction of the ray-tracing–wave-front modeling procedure of Ref. 6.

Tables (2)

Tables Icon

Table 1 Magnitude of the Coefficients in the Fourier-Series Decomposition of h 2 (ξ) for the 8.05- and 2.44-mm-Radius Glass Rods a

Tables Icon

Table 2 Magnitude of the Coefficients in the Fourier-Series Decomposition of h 3 (ξ) for the 8.05- and 2.44-mm-Radius Glass Rods a

Equations (30)

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

x=2πa/λ.
Iθ  Ai2-x2/3θ-θpc/hpc1/3,
cosϕic=n2-1/p2-11/2, sinϕtc=1/nsinϕic, θpc=p-1π+2ϕic-2pϕtc,
hpc=p2-12p2-n21/2p2n2-13/2.
N0.02x2/3
=b/a-1,
Iθ, ξ  Ai2-xave2/3θ-θpRξ/hpξ1/3,
θ2Rξ=θ2c-Δθ2R cos2ξ+θ2c+O2,
Δθ2R=8 sinϕtccos3ϕtc
h2ξh2c+Δh2 cos2ξ+Φ2,
Δh219sinϕtc3/4cosϕtc-10/3
Φ2250°n-285°
1.018 793=π/180xave2/3θamaxξ-θ2Rξ/h2ξ1/3,
θ2Rξ=θamaxξ-1.018 793 180/π×h2ξ1/3/xave2/3,
θ2Rξθamaxξ-1.018 793 180/π×h2c1/3/xave2/3.
1.018 793=π/180xave2/3θamaxξ-θ2Rξ/h2ξ1/3, 3.248 198=π/180xave2/3θbmaxξ-θ2Rξ/h2ξ1/3,
θ2Rξ=1.456 980θamaxξ-0.456 980θbmaxξ,
h2ξ=0.479 80610-6xave2θbmaxξ-θamaxξ3,
δθ2Rmeasuredδθab,
δh2measured0.02349δθabxaveh2c2/3.
h2ξ=e0+m=1 em cosmξ+m=1 fm sinmξ,
h3ξ=g0+m=1 gm cosmξ+m=1 jm sinmξ.
dn=nn2-11/2δθpD/2p2-n21/2.
dh=-np2-123p2-2n2-1dn/p2p2-n21/2n2-15/2;
da/a=dh/2h.
dn/n=n2-11/2π/180δθab/24-n21/2,
da/a=n211-2n2π/180δθab/4n2-11/24-n23/2.
dn/n=32n2-12/9n4,
da/a=1611-2n2n2-1/9n24-n2.
da/a/dn/n=n211-2n2/2n2-14-n2.

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