Abstract

We have conducted numerical studies on interference of the nth- and higher-order rainbows (n=8, 9, 10, and 11) formed by a spherical water drop based on Mie scattering theory. Wavelengths studied include 632.8, 532, 514.5, and 488nm. Our results are compared with the “non-Debye enhancement” of the 11th-order rainbow reported by Lock and Woodruff [Appl. Opt. 28, 523 (1989)] .

© 2007 Optical Society of America

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References

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  1. K. Sassen, "Angular scattering and rainbow formation in pendant drops," J. Opt. Soc. Am. 69, 1083-1089 (1979).
    [CrossRef]
  2. J. Walker, "Multiple rainbows from single drops of water and other liquids," Am. J. Phys. 44, 424-433 (1976).
    [CrossRef]
  3. J. Walker, "How to create and observe a dozen rainbows in a single drop of water," Sci. Am. 237, 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, "Cooperative effects among partial waves in Mie scattering," J. Opt. Soc. Am. A 5, 2032-2044 (1988).
    [CrossRef]
  6. 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]
  7. P. H. Ng, M. Y. Tse, and W. K. Lee, "Observation of high-order rainbows formed by a pendant drop," J. Opt. Soc. Am. B 15, 2782-2787 (1998).
    [CrossRef]
  8. 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]
  9. R. T. Wang and H. C. van de Hulst, "Rainbows: Mie computations and the Airy approximation," Appl. Opt. 30, 106-117 (1991).
    [CrossRef] [PubMed]
  10. H. C. van de Hulst and R. T. Wang, "Glare points," Appl. Opt. 30, 4755-4763 (1991).
    [CrossRef] [PubMed]
  11. 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]
  12. According to geometrical optics calculation, the 11th-order rainbow ray repeats its path after 13 internal reflections if the refractive index is 1.33236 (close to that of red light in water). See also H. M. Nussenzveig, "Complex angular momentum theory of the rainbow and the glory," J. Opt. Soc. Am. 69, 1068-1079 (1979).
    [CrossRef]
  13. For wavelengths 400, 488, and 700 nm, with refractive index values in water 1.34355, 1.3374, and 1.33047, the deviations of the 24th-order rainbow ray from the 11th-order rainbow ray, calculated by using geometric optics, are 3×360°+14.10°, 3×360°+2.84°, and 3×360°−2.42°, respectively. As the incident angles of higher-order rainbow forming rays approach fast enough to an asymptotic value, similar results hold for the 37th-, 50th-, ..., order rainbows.
  14. H. Eisenberg, "Equation of the refractive index of water," J. Chem. Phys. 43, 3887-3892 (1965).
    [CrossRef]
  15. Here 270° is equivalent to 90° in .
  16. The program was downloaded in 1994, but currently the web site cannot be found. It was originally in single precision, and we have converted it into double precision. We have checked that results of the program are consistent with another computer program, MiePlot downloaded from http://www.philiplaven. com. Source code of dave.for is available upon request.
  17. P. H. Ng, P. Y. So, C. W. Chan, and W. K. Lee, "Interference of the eleventh- and higher-order rainbows formed by a pendant water drop," J. Opt. Soc. Am. B 20, 2395-2399 (2003).
    [CrossRef]
  18. The enhancement is larger than that with θw=2.0°, since there are more portions of electric field of the 11th- and the 24th-order rainbows are in phase for θw=4.8°.
  19. They used an observation angle of 90°, which is equivalent to 270° in this work as stated in note 15. We chose an observation angle of 268.6°, because this is the position of the main peak of the 11th-order rainbow for m=1.3317 and λ=632.8 nm.

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1991

1989

1988

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1979

1977

J. Walker, "How to create and observe a dozen rainbows in a single drop of water," Sci. Am. 237, 138-144 (1977).
[CrossRef]

1976

J. Walker, "Multiple rainbows from single drops of water and other liquids," Am. J. Phys. 44, 424-433 (1976).
[CrossRef]

1965

H. Eisenberg, "Equation of the refractive index of water," J. Chem. Phys. 43, 3887-3892 (1965).
[CrossRef]

Am. J. Phys.

J. Walker, "Multiple rainbows from single drops of water and other liquids," Am. J. Phys. 44, 424-433 (1976).
[CrossRef]

Appl. Opt.

J. Chem. Phys.

H. Eisenberg, "Equation of the refractive index of water," J. Chem. Phys. 43, 3887-3892 (1965).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Sci. Am.

J. Walker, "How to create and observe a dozen rainbows in a single drop of water," Sci. Am. 237, 138-144 (1977).
[CrossRef]

Other

The enhancement is larger than that with θw=2.0°, since there are more portions of electric field of the 11th- and the 24th-order rainbows are in phase for θw=4.8°.

They used an observation angle of 90°, which is equivalent to 270° in this work as stated in note 15. We chose an observation angle of 268.6°, because this is the position of the main peak of the 11th-order rainbow for m=1.3317 and λ=632.8 nm.

Here 270° is equivalent to 90° in .

The program was downloaded in 1994, but currently the web site cannot be found. It was originally in single precision, and we have converted it into double precision. We have checked that results of the program are consistent with another computer program, MiePlot downloaded from http://www.philiplaven. com. Source code of dave.for is available upon request.

For wavelengths 400, 488, and 700 nm, with refractive index values in water 1.34355, 1.3374, and 1.33047, the deviations of the 24th-order rainbow ray from the 11th-order rainbow ray, calculated by using geometric optics, are 3×360°+14.10°, 3×360°+2.84°, and 3×360°−2.42°, respectively. As the incident angles of higher-order rainbow forming rays approach fast enough to an asymptotic value, similar results hold for the 37th-, 50th-, ..., order rainbows.

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Figures (3)

Fig. 1
Fig. 1

Intensity of glare point of G ( 11 ) versus droplet radius a. The observation angle θ o is 271.4°, and the wavelength λ is 532 nm . The refractive index m is 1.3342. θ w is set to 2.0°. For the labels of the small narrow peaks, see the text.

Fig. 2
Fig. 2

Fourier transform of the electric field of the scattered light wave by a spherical water droplet (incident plane wave) observed at 271.4° with θ w = 2.0 ° . The wavelength is 532 nm , and the refractive index is 1.3342. The solid and dotted curves correspond to a = 1749.983 μ m and a = 1749.998 μ m , respectively. Each curve contains 5000 equally spaced data points.

Fig. 3
Fig. 3

Intensity versus w, with all parameters the same as those in Fig. 2. Acceptance angle values: upper curves, 4.8°, lower curves, 2.0°. Each curve contains 5000 equally spaced data points. The glare point at w 0.72 is due to specular reflection.

Tables (1)

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Table 1 Fourier Transform of the Perpendicularly Polarized Electric Field of Scattered Light Wave by a Spherical Water Droplet a

Equations (1)

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Δ a = λ 26 m cos θ r .

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