Abstract

Interference of the 11th- and higher-order rainbows formed by a pendant water drop was studied by use of laser beams of wavelengths 632.8 and 532 nm. Time variations of the corresponding glare-point intensity, due to vibration of the droplet, are in agreement with results of a simple model calculation. Our results are compared with the “non-Debye enhancement” of the 11th-order rainbow reported by Lock and Woodruff.

© 2003 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 Guassian 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. Clearly, the light wave need not be plane wave; any light beam of diameter larger than the size of the pendent drop yields similar glare points. There are glare points that are not located in the equatorial plane, which do not concern us here.
  13. K. Sassen, “Infrared (10.6-μm) radiation induced evaporation of large water drops,” J. Opt. Soc. Am. 71, 887–891 (1981).
    [CrossRef]
  14. 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).
  15. For wavelengths 400, 488, and 700 nm, with refractive-index values in water17 1.34355, 1.3374, and 1.33047, the deviations of the 24th-order rainbow ray from the 11th-order rainbow ray, calculated by use of 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.
  16. Here 270° is equivalent to the 90° in Ref. 4.
  17. H. Eisenberg, “Equation of the refractive index of water,” J. Chem. Phys. 43, 3887–3892 (1965).
    [CrossRef]
  18. These values are calculated by use of the Airy approximation. Geometric optics yields values 268.05° and 271.2°, respectively.
  19. Vertical spreads of the rainbows depend on the distance of the laser focal point from the droplet, laser beam divergence, and size of the droplet.
  20. Vibrations can cause complicated changes in shape of the equatorial cross section of the pendant drop. However, our experimental result can be basically explained by an up-down oscillation of the pendant drop.
  21. M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley, New York, 1986).
  22. O. A. Basaran and D. W. DePaoli, “Nonlinear oscillations of pendant drops,” Phys. Fluids 6, 2923–2943 (1994), Fig. 7.
    [CrossRef]

1998

1996

1994

O. A. Basaran and D. W. DePaoli, “Nonlinear oscillations of pendant drops,” Phys. Fluids 6, 2923–2943 (1994), Fig. 7.
[CrossRef]

1993

1991

1989

1988

1987

1981

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]

Basaran, O. A.

O. A. Basaran and D. W. DePaoli, “Nonlinear oscillations of pendant drops,” Phys. Fluids 6, 2923–2943 (1994), Fig. 7.
[CrossRef]

Chan, C. W.

DePaoli, D. W.

O. A. Basaran and D. W. DePaoli, “Nonlinear oscillations of pendant drops,” Phys. Fluids 6, 2923–2943 (1994), Fig. 7.
[CrossRef]

Eisenberg, H.

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

Lee, W. K.

Lock, J. A.

Ng, P. H.

Sassen, K.

Tse, M. Y.

van de Hulst, H. C.

Walker, J.

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

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

Wang, R. T.

Woodruff, J. R.

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

Phys. Fluids

O. A. Basaran and D. W. DePaoli, “Nonlinear oscillations of pendant drops,” Phys. Fluids 6, 2923–2943 (1994), Fig. 7.
[CrossRef]

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

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).

For wavelengths 400, 488, and 700 nm, with refractive-index values in water17 1.34355, 1.3374, and 1.33047, the deviations of the 24th-order rainbow ray from the 11th-order rainbow ray, calculated by use of 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.

Here 270° is equivalent to the 90° in Ref. 4.

Clearly, the light wave need not be plane wave; any light beam of diameter larger than the size of the pendent drop yields similar glare points. There are glare points that are not located in the equatorial plane, which do not concern us here.

These values are calculated by use of the Airy approximation. Geometric optics yields values 268.05° and 271.2°, respectively.

Vertical spreads of the rainbows depend on the distance of the laser focal point from the droplet, laser beam divergence, and size of the droplet.

Vibrations can cause complicated changes in shape of the equatorial cross section of the pendant drop. However, our experimental result can be basically explained by an up-down oscillation of the pendant drop.

M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley, New York, 1986).

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Figures (4)

Fig. 1
Fig. 1

Experimental setup. L1, L2, focusing lenses; PMT, photomultiplier.

Fig. 2
Fig. 2

Interference signal obtained by use of (a) 632.8-nm wavelength, laser power ∼7 mW, and (b) 532-nm wavelength, laser power ∼40 mW. The signal is normalized to facilitate comparison with results shown in Fig. 3.

Fig. 3
Fig. 3

Interference signal obtained by the model calculation, (a) μ=1.332 and (b) μ=1.3347. See text for details.

Fig. 4
Fig. 4

Interference signal after the optical table was tapped once. Note that t=0.014 s and t=0.084 s correspond to two consecutive times at which the droplet was instantaneously at rest. The time difference thus corresponds to a half-period of the oscillation.

Equations (9)

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ϕ11-ϕ24=ϕ24-ϕ37=ϕ37-ϕ50==ϕ,
ϕ11-ϕ37=ϕ24-ϕ50=ϕ37-ϕ63==2ϕ,
ϕ11-ϕ50=ϕ24-ϕ63=ϕ37-ϕ76==3ϕ,
,
ϕ13×2πλ×2a sin3×360°2×13.
I=rNir+rsNsNiriscos(ϕr-ϕs),
r, s=11, 24, 37,,
I=r=0N-1Ircos rϕ,
ϕ=A sin 2πft,

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