Abstract

We experimentally examine the primary rainbow created by the illumination of a coated cylinder. We present a simple technique for varying the coating thickness over a wide range of values, and we see evidence for two different scattering regimes. In one, where the coating thickness is large, twin rainbows are produced. In the second, where the coating is thin enough to act as a thin film, a single rainbow is produced whose intensity varies periodically as the coating thickness varies. We find good agreement with previous theoretical predictions.

© 2001 Optical Society of America

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References

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  1. W. Mobius, “Zur Theorie des Regenbogens und ihrer experimentallen Prufung,” Abh. Math.- Phys. Kl. Saechs. Ges. Wiss. 30, 105–254 (1907–1909).
  2. W. Mobius, “Zur Theorie des Regenbogens und ihrer experimentallen Prufung,” Ann. Phys. (Leipzig) 33, 1493–1558 (1910).
    [CrossRef]
  3. A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?,” J. Opt. Soc. Am. 73, 1626–1628 (1983).
    [CrossRef]
  4. D. Marcuse, “Light scattering from elliptical fibers,” Appl. Opt. 13, 1903–1905 (1974).
    [CrossRef] [PubMed]
  5. P. L. Marston, “Rainbow phenomena and the detection of nonsphericity in drops,” Appl. Opt. 19, 680–685 (1980).
    [CrossRef] [PubMed]
  6. P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984).
    [CrossRef]
  7. P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
    [CrossRef] [PubMed]
  8. J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
    [CrossRef]
  9. J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London, Ser. A 438, 397–417 (1992).
    [CrossRef]
  10. K. Sassen, “Optical backscattering from near-spherical water, ice, and mixed phase drops,” Appl. Opt. 16, 1332–1341 (1977).
    [CrossRef] [PubMed]
  11. G. Kaduchak, P. L. Marston, “Hyperbolic umbilic and E6 diffraction catastrophes associated with the secondary rainbow of oblate water drops: observations with laser illumination,” Appl. Opt. 33, 4697–4701 (1994).
    [CrossRef] [PubMed]
  12. G. Kaduchak, P. L. Marston, H. J. Simpson, “E6 diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination,” Appl. Opt. 33, 4691–4696 (1994).
    [CrossRef] [PubMed]
  13. 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]
  14. 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]
  15. J. van Beeck, “Rainbow phenomena: development of a laser-based, non-intrusive technique for measuring droplet size, temperature and velocity,” Ph.D. dissertation (Technische Universiteit Eindhoven, Eindhoven, The Netherlands, 1997).
  16. P. Massoli, F. Beretta, A. D’Alessio, M. Lazzaro, “Temperature and size of single transparent droplets by light scattering in the forward and rainbow regions,” Appl. Opt. 32, 3295–3301 (1993).
    [CrossRef] [PubMed]
  17. C. W. Chan, W. K. Lee, “Measurement of a liquid refractive index by using high-order rainbows,” J. Opt. Soc. Am. B 13, 532–535 (1996).
    [CrossRef]
  18. X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, G. Gréhan, “Characterization of initial disturbances in a liquid jet by rainbow sizing,” Appl. Opt. 37, 8498–8503 (1998).
    [CrossRef]
  19. H. Hattori, H. Yamanaka, H. Kurniawan, S. Yokoi, K. Kagawa, “Using minimum deviation of a secondary rainbow and its application to water analysis in a high-precision, refractive-index comparator for liquids,” Appl. Opt. 36, 5552–5556 (1997).
    [CrossRef] [PubMed]
  20. H. Hattori, H. Kakui, H. Kurniawan, K. Kagawa, “Liquid refractometry by the rainbow method,” Appl. Opt. 37, 4123–4129 (1998).
    [CrossRef]
  21. H. Hattori, “Simulation study on refractometry by the rainbow method,” Appl. Opt. 38, 4037–4046 (1999).
    [CrossRef]
  22. N. Roth, K. Anders, A. Frohn, “Refractive-index measurements for the correction of particle sizing methods,” Appl. Opt. 30, 4960–4965 (1991).
    [CrossRef] [PubMed]
  23. N. Savage, “Laser imaging brings sprays into focus,” Laser Focus World 33, 93–97 (1998).
  24. A. Tokay, K. V. Beard, “A field study of raindrop oscillations. I. Observation of size spectra and evaluation of oscillation causes,” J. Appl. Meteorol. 35, 1671–1687 (1996).
    [CrossRef]
  25. 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]
  26. C. Saekang, P. L. Chu, “Backscattering of light from optical fibers with arbitrary refractive index distributions: uniform approximation approach,” J. Opt. Soc. Am. 68, 1298–1305 (1978).
    [CrossRef]
  27. P. Massoli, “Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets,” Appl. Opt. 37, 3227–3235 (1998).
    [CrossRef]
  28. L. Kai, P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
    [CrossRef] [PubMed]
  29. L. Kai, P. Massoli, A. D’Alessio, “Some far-field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
    [CrossRef]
  30. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36, 5188–5198 (1997).
    [CrossRef] [PubMed]
  31. A. A. Kokhanovsky, T. Y. Nakajima, “The dependence of phase functions of large transparent particles on their refractive indexes and shape,” J. Phys. D 31, 1329–1335 (1998).
    [CrossRef]
  32. G. Gouesbet, G. Grehan, “Generalized Lorenz-Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47, 821–837 (2000).
  33. J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677–4690 (1994).
    [CrossRef] [PubMed]
  34. D. Marcuse, H. M. Presby, “Optical fiber coating concentricity: measurement and analysis,” Appl. Opt. 16, 2383–2390 (1977).
    [CrossRef] [PubMed]
  35. H. M. Presby, D. Marcuse, “Refractive index and diameter determinations of step index glass fibers,” Appl. Opt. 13, 2882–2885 (1974).
    [CrossRef] [PubMed]
  36. M. Minnaert, Light and Color in the Outdoors (Springer-Verlag, New York, 1993), pp. 189–198.
  37. R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, UK, 1980), pp. 1–11.
  38. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44(5), 421–433 (1976).
    [CrossRef]
  39. P. H. Ng, M. Y. Tse, W. K. Lee, “Observation of high-order rainbows formed by a pendant drop,” J. Opt. Soc. Am. A 15, 2782–2787 (1998).
    [CrossRef]
  40. A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  41. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 478.
  42. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 240–246.
  43. 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]
  44. 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]

2000 (1)

G. Gouesbet, G. Grehan, “Generalized Lorenz-Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47, 821–837 (2000).

1999 (2)

1998 (7)

1997 (3)

1996 (3)

1995 (1)

1994 (5)

1993 (1)

1992 (1)

J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London, Ser. A 438, 397–417 (1992).
[CrossRef]

1991 (1)

1985 (1)

1984 (2)

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984).
[CrossRef]

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[CrossRef]

1983 (1)

1980 (1)

1978 (1)

1977 (2)

1976 (1)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44(5), 421–433 (1976).
[CrossRef]

1974 (2)

1951 (1)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

1910 (1)

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

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 478.

Aden, A. L.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Adler, C. L.

Anders, K.

Bauckhage, K.

Beard, K. V.

A. Tokay, K. V. Beard, “A field study of raindrop oscillations. I. Observation of size spectra and evaluation of oscillation causes,” J. Appl. Meteorol. 35, 1671–1687 (1996).
[CrossRef]

Beretta, F.

Chan, C. W.

Chu, P. L.

Corbin, F.

D’Alessio, A.

L. Kai, P. Massoli, A. D’Alessio, “Some far-field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
[CrossRef]

P. Massoli, F. Beretta, A. D’Alessio, M. Lazzaro, “Temperature and size of single transparent droplets by light scattering in the forward and rainbow regions,” Appl. Opt. 32, 3295–3301 (1993).
[CrossRef] [PubMed]

Fraser, A. B.

Frohn, A.

Garcia, C. J.

Gouesbet, G.

Greenler, R.

R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, UK, 1980), pp. 1–11.

Grehan, G.

G. Gouesbet, G. Grehan, “Generalized Lorenz-Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47, 821–837 (2000).

Gréhan, G.

Guo, L. X.

Han, X.

Hattori, H.

Jamison, J. M.

Kaduchak, G.

Kagawa, K.

Kai, L.

L. Kai, P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
[CrossRef] [PubMed]

L. Kai, P. Massoli, A. D’Alessio, “Some far-field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
[CrossRef]

Kakui, H.

Kerker, M.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Kokhanovsky, A. A.

A. A. Kokhanovsky, T. Y. Nakajima, “The dependence of phase functions of large transparent particles on their refractive indexes and shape,” J. Phys. D 31, 1329–1335 (1998).
[CrossRef]

Kurniawan, H.

Lazzaro, M.

Lee, W. K.

P. H. Ng, M. Y. Tse, W. K. Lee, “Observation of high-order rainbows formed by a pendant drop,” J. Opt. Soc. Am. A 15, 2782–2787 (1998).
[CrossRef]

C. W. Chan, W. K. Lee, “Measurement of a liquid refractive index by using high-order rainbows,” J. Opt. Soc. Am. B 13, 532–535 (1996).
[CrossRef]

Lehmann, P.

Lin, C.-Y.

Lock, J. A.

Lohner, H.

Marcuse, D.

Marston, P. L.

Massoli, P.

Minnaert, M.

M. Minnaert, Light and Color in the Outdoors (Springer-Verlag, New York, 1993), pp. 189–198.

Mobius, W.

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

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

Nakajima, T. Y.

A. A. Kokhanovsky, T. Y. Nakajima, “The dependence of phase functions of large transparent particles on their refractive indexes and shape,” J. Phys. D 31, 1329–1335 (1998).
[CrossRef]

Ng, P. H.

P. H. Ng, M. Y. Tse, W. K. Lee, “Observation of high-order rainbows formed by a pendant drop,” J. Opt. Soc. Am. A 15, 2782–2787 (1998).
[CrossRef]

Nye, J. F.

J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London, Ser. A 438, 397–417 (1992).
[CrossRef]

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[CrossRef]

Presby, H. M.

Ren, K. F.

Riethmuller, M. L.

Roth, N.

Saekang, C.

Sassen, K.

Savage, N.

N. Savage, “Laser imaging brings sprays into focus,” Laser Focus World 33, 93–97 (1998).

Simpson, H. J.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 478.

Stone, B. R.

Tokay, A.

A. Tokay, K. V. Beard, “A field study of raindrop oscillations. I. Observation of size spectra and evaluation of oscillation causes,” J. Appl. Meteorol. 35, 1671–1687 (1996).
[CrossRef]

Trinh, E. H.

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984).
[CrossRef]

Tse, M. Y.

P. H. Ng, M. Y. Tse, W. K. Lee, “Observation of high-order rainbows formed by a pendant drop,” J. Opt. Soc. Am. A 15, 2782–2787 (1998).
[CrossRef]

van Beeck, J.

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

van Beeck, J. P. A. J.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 240–246.

Walker, J. D.

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44(5), 421–433 (1976).
[CrossRef]

Wu, Z.

Wu, Z. S.

Yamanaka, H.

Yokoi, S.

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

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

Am. J. Phys. (1)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44(5), 421–433 (1976).
[CrossRef]

Ann. Phys. (Leipzig) (1)

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

Appl. Opt. (21)

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

P. Massoli, F. Beretta, A. D’Alessio, M. Lazzaro, “Temperature and size of single transparent droplets by light scattering in the forward and rainbow regions,” Appl. Opt. 32, 3295–3301 (1993).
[CrossRef] [PubMed]

L. Kai, P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
[CrossRef] [PubMed]

J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677–4690 (1994).
[CrossRef] [PubMed]

G. Kaduchak, P. L. Marston, H. J. Simpson, “E6 diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination,” Appl. Opt. 33, 4691–4696 (1994).
[CrossRef] [PubMed]

G. Kaduchak, P. L. Marston, “Hyperbolic umbilic and E6 diffraction catastrophes associated with the secondary rainbow of oblate water drops: observations with laser illumination,” Appl. Opt. 33, 4697–4701 (1994).
[CrossRef] [PubMed]

P. Massoli, “Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets,” Appl. Opt. 37, 3227–3235 (1998).
[CrossRef]

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]

H. Hattori, “Simulation study on refractometry by the rainbow method,” Appl. Opt. 38, 4037–4046 (1999).
[CrossRef]

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]

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]

D. Marcuse, “Light scattering from elliptical fibers,” Appl. Opt. 13, 1903–1905 (1974).
[CrossRef] [PubMed]

H. M. Presby, D. Marcuse, “Refractive index and diameter determinations of step index glass fibers,” Appl. Opt. 13, 2882–2885 (1974).
[CrossRef] [PubMed]

K. Sassen, “Optical backscattering from near-spherical water, ice, and mixed phase drops,” Appl. Opt. 16, 1332–1341 (1977).
[CrossRef] [PubMed]

D. Marcuse, H. M. Presby, “Optical fiber coating concentricity: measurement and analysis,” Appl. Opt. 16, 2383–2390 (1977).
[CrossRef] [PubMed]

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

Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36, 5188–5198 (1997).
[CrossRef] [PubMed]

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]

X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, G. Gréhan, “Characterization of initial disturbances in a liquid jet by rainbow sizing,” Appl. Opt. 37, 8498–8503 (1998).
[CrossRef]

H. Hattori, H. Yamanaka, H. Kurniawan, S. Yokoi, K. Kagawa, “Using minimum deviation of a secondary rainbow and its application to water analysis in a high-precision, refractive-index comparator for liquids,” Appl. Opt. 36, 5552–5556 (1997).
[CrossRef] [PubMed]

H. Hattori, H. Kakui, H. Kurniawan, K. Kagawa, “Liquid refractometry by the rainbow method,” Appl. Opt. 37, 4123–4129 (1998).
[CrossRef]

J. Appl. Meteorol. (1)

A. Tokay, K. V. Beard, “A field study of raindrop oscillations. I. Observation of size spectra and evaluation of oscillation causes,” J. Appl. Meteorol. 35, 1671–1687 (1996).
[CrossRef]

J. Appl. Phys. (1)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

J. Mod. Opt. (1)

G. Gouesbet, G. Grehan, “Generalized Lorenz-Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47, 821–837 (2000).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

J. Phys. D (1)

A. A. Kokhanovsky, T. Y. Nakajima, “The dependence of phase functions of large transparent particles on their refractive indexes and shape,” J. Phys. D 31, 1329–1335 (1998).
[CrossRef]

Laser Focus World (1)

N. Savage, “Laser imaging brings sprays into focus,” Laser Focus World 33, 93–97 (1998).

Nature (London) (2)

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984).
[CrossRef]

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[CrossRef]

Opt. Lett. (1)

Part. Part. Syst. Charact. (1)

L. Kai, P. Massoli, A. D’Alessio, “Some far-field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London, Ser. A 438, 397–417 (1992).
[CrossRef]

Other (5)

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

M. Minnaert, Light and Color in the Outdoors (Springer-Verlag, New York, 1993), pp. 189–198.

R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, UK, 1980), pp. 1–11.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 478.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 240–246.

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Figures (16)

Fig. 1
Fig. 1

Coated particle geometry. The inner radius is a 1 and the coating thickness is δ. The core index of refraction is n 1 and the coating index is n 2. The ray paths creating the α and β rainbows are indicated on the diagram.

Fig. 2
Fig. 2

Theoretical values for the scattering intensity near the primary rainbow for a coated sphere. The calculations are based on the Aden–Kerker extension to Mie theory. Refractive indices: n 1 = 1.5, n 2 = 1.33, a 1 = 2.6 mm. The intensity is shown on a linear scale to make a comparison with Fig. 5(b) easier: (a) δ/a 1 = 2 × 10-3, (b) δ/a 1 = 10-3, (c) δ/a 1 = 3 × 10-3.

Fig. 3
Fig. 3

Experimental diagram. Laser power is 3 mW and the distance from the cylinder center to the camera is d = 5.6 cm. The laser was polarized parallel to the cylinder axis. The neutral-density (ND) filter had an optical density of 0.3.

Fig. 4
Fig. 4

Diagram of a typical pendant droplet investigated in the experiment. The cylinder diameter is 2a 1 = 5.2 mm. The laser beam is incident a distance roughly 5 mm above the end of the rod.

Fig. 5
Fig. 5

(a) Digitized image of the twin rainbows. δ/a 1 ∼ 10-2. (b) Intensity profile of data from image (a). Angular separation of the rainbows is 0.8°.

Fig. 6
Fig. 6

Angular deviation of each bow from θ0 R as a function of time.

Fig. 7
Fig. 7

Calculated value of the coating thickness as a function of time by use of the data from Fig. 6.

Fig. 8
Fig. 8

Intensity of the main rainbow peak as a function of time (after the rainbows merged). The top scale is an interpolated value for the coating thickness by use of the data from Fig. 7.

Fig. 9
Fig. 9

(a) Digitized image of the rainbow at t = 30 s after the merging. (b) Intensity profile of data from image (a). The intensity profile was made from the area enclosed by the dashed lines in (a).

Fig. 10
Fig. 10

Doubly coated particle geometry. Core index, n 1; coating 1 index, n 2; coating 2 index, n 3. The hypothesized ray paths for the α, β, and γ rainbows are shown.

Fig. 11
Fig. 11

Video image of the triplet rainbows (α, β, and γ) and the three γ′ rainbows. The β rainbow is almost invisible on camera, but its position is indicated.

Fig. 12
Fig. 12

Angular position of the α, β, and γ rainbows as a function of time. The angle is measured from the center of the visual field on the camera.

Fig. 13
Fig. 13

Angular position of the γ′ complex of rainbows as a function of time. The angle is measured from the center of the visual field on the camera. After t = 3 s, the γ2′ and γ3′ rainbows appear to pass through each other, making the identification of either one ambiguous.

Fig. 14
Fig. 14

Ray paths and scattering angles for rainbows that are due to multiple internal reflections inside the coatings. We label these the γ′ paths.

Fig. 15
Fig. 15

Video image of a large vertical section of the twin rainbows from a water-coated glass cylinder. The water coating is thicker at the bottom because of gravitational sagging of the water layer.

Fig. 16
Fig. 16

Computer simulation of the moiré interference pattern shown in Fig. 15. Vertical and horizontal scales are in normalized units.

Equations (16)

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θαR=θ0R+2δa14-n123n22+n12-41/2-4-n12n12-11/2,
θβR=θ0R+2δa124-n123n22+n12-41/2-4-n12n12-11/2.
ΔθαβR=2δa14-n123n22+n12-41/2
EAi-x12/3/h1/3θ-θ0R,
h=p2-12p2p2-n121/2n12-13/2.
δ/a1>0.660h1/3/x12/33n22+n12-44-n121/2
δth=a123n22+n12-44-n121/2θαR-θβR.
M21=24-n121/23n22+n12-41/2,
M31=24-n121/23n32+n12-41/2,
N1=24-n121/2n12-11/2.
θαR=θ0R+δ2aM21-N1+δ3aM31-N1,
θβR=θ0R+δ2a2M21-N1+δ3aM31-N1,
θγR=θ0R+δ2a2M21-N1+δ3a2M31-N1.
θγR=θ0R+δ2a3M21-N1+δ3a3M31-N1.
Iz=cαAi-z2+cβAi-z-Δzαβ2+2cαcβAi-zAi-z-Δzαβcosn2x1h11/3sin ϕR2 Δzαβ,
sin ϕR2=4-n123n221/2.

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