Abstract

High-order rainbows, up to the 200th order, formed by a pendant water drop and a 50-mW laser beam, have been observed. Some of the characteristics of the high-order rainbows, including angular intensity distributions and angular positions, are reported. Rainbow intensity as a function of order number is also presented. Rainbows beyond the 32nd order have been observed for the first time to our knowledge. Experimental and theoretical results are in reasonable agreement.

© 1998 Optical Society of America

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References

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  1. H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
    [CrossRef]
  2. K. Sassen, “Angular scattering and rainbow formation in pendant drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).
    [CrossRef]
  3. J. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976); “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237(7), 138–144 (1977).
    [CrossRef]
  4. J. A. Lock and J. R. Woodruff, “Non-Debye enhancements in the Mie scattering of light from a single water droplet,” Appl. Opt. 28, 523–529 (1989).
    [CrossRef] [PubMed]
  5. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
    [CrossRef]
  6. P. L. Marston, “Rainbow phenomena and the detection of nonsphericity in drops,” Appl. Opt. 19, 680–685 (1980).
    [CrossRef] [PubMed]
  7. J. P. A. J. van Beeck and 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]
  8. C. W. Chan and W. K. Lee, “Measurement of liquid refractive index using high-order rainbows,” J. Opt. Soc. Am. B 13, 532–535 (1996).
    [CrossRef]
  9. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
    [CrossRef]
  10. R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
    [CrossRef] [PubMed]
  11. H. C. van de Hulst and R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991).
    [CrossRef] [PubMed]
  12. J. A. Lock, “Theory of the observations made of high-order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
    [CrossRef] [PubMed]
  13. For each rainbow between the 18th and the 21st orders, two supernumerary arcs were visible. The supernumerary arcs of the 18th-order rainbow do not show up in the picture because the 5th- and the 18th-order rainbows were attentuated by ~100 times to avoid saturation of the CCD camera.
  14. With our experimental configuration, the 1st-order rainbow cannot be formed, in the sense that there is only a bright patch without a principal peak or supernumerary arc features near the position of the 1st-order rainbow; (p−1)= 1 refers to this bright patch. The (p−1)+13n empirical rule is an accident of the choice of droplet size and the refractive index of water between the red and blue regions (values of roughly 1.332–1.338). By simple geometric optics calculation, one finds that 13n will be replaced by 12n and 14n for refractive-index values of ~1.413 and 1.279, respectively.
  15. We found the peak intensity by integrating the gray-scale values of all pixels within a small portion (~1° of the scattering angle) of the peak region of the rainbow image.
  16. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  17. W. J. Humphreys, Physics of The Air (Dover, New York, 1964).
  18. G(12) is not observable because it is roughly located at the forward scattering direction and is thus masked by the intense scattered light there.

1996

1993

1991

1989

1987

1980

1979

1977

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
[CrossRef]

Chan, C. W.

Lee, W. K.

Lock, J. A.

Marston, P. L.

Nussenzveig, H. M.

Riethmuller, M. L.

Sassen, K.

van Beeck, J. P. A. J.

van de Hulst, H. C.

Wang, R. T.

Woodruff, J. R.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Sci. Am.

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
[CrossRef]

Other

J. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976); “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237(7), 138–144 (1977).
[CrossRef]

For each rainbow between the 18th and the 21st orders, two supernumerary arcs were visible. The supernumerary arcs of the 18th-order rainbow do not show up in the picture because the 5th- and the 18th-order rainbows were attentuated by ~100 times to avoid saturation of the CCD camera.

With our experimental configuration, the 1st-order rainbow cannot be formed, in the sense that there is only a bright patch without a principal peak or supernumerary arc features near the position of the 1st-order rainbow; (p−1)= 1 refers to this bright patch. The (p−1)+13n empirical rule is an accident of the choice of droplet size and the refractive index of water between the red and blue regions (values of roughly 1.332–1.338). By simple geometric optics calculation, one finds that 13n will be replaced by 12n and 14n for refractive-index values of ~1.413 and 1.279, respectively.

We found the peak intensity by integrating the gray-scale values of all pixels within a small portion (~1° of the scattering angle) of the peak region of the rainbow image.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

W. J. Humphreys, Physics of The Air (Dover, New York, 1964).

G(12) is not observable because it is roughly located at the forward scattering direction and is thus masked by the intense scattered light there.

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Figures (6)

Fig. 1
Fig. 1

Experimental setup. (a) Top view. The open circle represents the circumference of the pendant drop in the equatorial plane. (b) Side view.

Fig. 2
Fig. 2

Group G(5) rainbows. (a) Streaks from bottom to top, n=07; exposure time, 200 ms. (b) Streaks from bottom to top, n=615; exposure time, 500 ms.

Fig. 3
Fig. 3

Angular intensity distributions of the G(5) VHOR’s. Curves from top to bottom: (a) n=916, (b) n=1316. The two smooth curves are the theoretical fits for n=9 and n=10. The scattering angles modulo 360° are shown along the horizontal axis.

Fig. 4
Fig. 4

Relative principal peak intensity as a function of order number. Solid curve, theoretical result. The experimental values of the 27th-, 28th-, 18th-, 19th-, 22nd-, and 37th-order rainbows of G(1), G(2), G(5), G(6), G(9), and G(11), respectively, were adjusted such that they coincide with the theoretical values. (The intensity of the 18th-order rainbow is assumed to be unity.) The values within the same group were then scaled by the same factor. The experimental uncertainty is roughly the same as the size of the data points.

Fig. 5
Fig. 5

Theoretical vertical spreads of the rainbows. Curves from bottom to top; n=02. Screen at 500 mm from the droplet. The straight lines between data points are guides for the eye.

Fig. 6
Fig. 6

Part of the group G(5) rainbows formed by (a) a He–Ne laser beam irradiating a distilled-water drop (refractive index, ∼1.3320) and (b) a 532-nm laser beam irradiating an ethylene glycol aqueous solution (refractive index, ∼1.3374). In both (a) and (b) the curves from bottom to top are n=04. In (b) the 5th-order rainbow was attenuated by ∼300 times with neutral-density filters.

Tables (1)

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Table 1 Angular Positions and Angular Widths of the Principal Maximaa

Equations (2)

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d=2r cos ϕ.
1d0110-2 cos ϕR1 1d01p-1.

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