Abstract

Previous observations of oblate drops of water illuminated perpendicular to their axis of symmetry exhibit catastrophe patterns near the primary-rainbow scattering angle [see, e.g., Appl. Opt. 30, 3468 (1991). The present research concerns observation of diffraction catastrophes near the secondary-rainbow scattering angle under similar experimental conditions. Illumination with laser light exhibits similar caustic structures in the secondary rainbow including the hyperbolic umbilic focal section and the E6 or symbolic umbilic focal section. The range of drop aspect ratios explored includes aspect ratios as small as those found for freely falling drops in air as well as highly flattened drops. The new features of the secondary rainbow occur for highly flattened drops.

© 1994 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 13.
  2. W. J. Humphreys, Physics of the Air, 3rd ed. (Dover, New York, 1964), pp. 475–495.
  3. P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
    [CrossRef] [PubMed]
  4. P. L. Marston, E. H. Trinh, “Hyperbolic umbilic catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984).
    [CrossRef]
  5. 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]
  6. H. J. Simpson, P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468–3473 (1991).
    [CrossRef] [PubMed]
  7. P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” in Physical Acoustics, R. N. Thurston, A. D. Pierce, eds. (Academic, Boston, Mass., 1992), Vol. 21, pp. 1–234.
  8. C. E. Dean, P. L. Marston, “Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spherical drops,” Appl. Opt. 30, 3443–3451 (1991).
    [CrossRef] [PubMed]
  9. J. F. Nye, “Rainbow scattering form spherical drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
    [CrossRef]
  10. J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London Ser. A 438, 397–417 (1992).
    [CrossRef]
  11. P. L. Marston, S. E. LoPorto-Arione, G. L. Pullen, “Quadrapole projection of the radiation pressure on a compressible sphere,” J. Acoust. Soc. Am. 69, 1499–1501 (1981);Erratum 71, 511 (1982);Y. Tian, R. G. Holt, R. E. Apfel, “Deformation and location of an acoustically levitated liquid drop,” J. Acoust. Soc. Am. 93, 3096–3104 (1993).
    [CrossRef] [PubMed]
  12. P. L. Marston, G. Kaduchak, “Secondary and higher-order generalized rainbows and unfolded glories of oblate drops: analysis and laboratory observations,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 12–15.
  13. P. L. Marston, G. Kaduchak, “Generalized rainbows and unfolded glories of oblate drops: organization for multiple internal reflections and extension of cusps into Alexander's dark band,” Appl. Opt. 33, 4702–4713 (1994).
    [CrossRef] [PubMed]
  14. M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).

1994 (2)

1992 (1)

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

1991 (2)

1985 (1)

1984 (2)

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

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

1981 (1)

P. L. Marston, S. E. LoPorto-Arione, G. L. Pullen, “Quadrapole projection of the radiation pressure on a compressible sphere,” J. Acoust. Soc. Am. 69, 1499–1501 (1981);Erratum 71, 511 (1982);Y. Tian, R. G. Holt, R. E. Apfel, “Deformation and location of an acoustically levitated liquid drop,” J. Acoust. Soc. Am. 93, 3096–3104 (1993).
[CrossRef] [PubMed]

1980 (1)

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).

Berry, M. V.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).

Dean, C. E.

Humphreys, W. J.

W. J. Humphreys, Physics of the Air, 3rd ed. (Dover, New York, 1964), pp. 475–495.

Kaduchak, G.

LoPorto-Arione, S. E.

P. L. Marston, S. E. LoPorto-Arione, G. L. Pullen, “Quadrapole projection of the radiation pressure on a compressible sphere,” J. Acoust. Soc. Am. 69, 1499–1501 (1981);Erratum 71, 511 (1982);Y. Tian, R. G. Holt, R. E. Apfel, “Deformation and location of an acoustically levitated liquid drop,” J. Acoust. Soc. Am. 93, 3096–3104 (1993).
[CrossRef] [PubMed]

Marston, P. L.

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]

P. L. Marston, G. Kaduchak, “Generalized rainbows and unfolded glories of oblate drops: organization for multiple internal reflections and extension of cusps into Alexander's dark band,” Appl. Opt. 33, 4702–4713 (1994).
[CrossRef] [PubMed]

H. J. Simpson, P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468–3473 (1991).
[CrossRef] [PubMed]

C. E. Dean, P. L. Marston, “Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spherical drops,” Appl. Opt. 30, 3443–3451 (1991).
[CrossRef] [PubMed]

P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
[CrossRef] [PubMed]

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

P. L. Marston, S. E. LoPorto-Arione, G. L. Pullen, “Quadrapole projection of the radiation pressure on a compressible sphere,” J. Acoust. Soc. Am. 69, 1499–1501 (1981);Erratum 71, 511 (1982);Y. Tian, R. G. Holt, R. E. Apfel, “Deformation and location of an acoustically levitated liquid drop,” J. Acoust. Soc. Am. 93, 3096–3104 (1993).
[CrossRef] [PubMed]

P. L. Marston, G. Kaduchak, “Secondary and higher-order generalized rainbows and unfolded glories of oblate drops: analysis and laboratory observations,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 12–15.

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” in Physical Acoustics, R. N. Thurston, A. D. Pierce, eds. (Academic, Boston, Mass., 1992), Vol. 21, pp. 1–234.

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 form spherical drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[CrossRef]

Pullen, G. L.

P. L. Marston, S. E. LoPorto-Arione, G. L. Pullen, “Quadrapole projection of the radiation pressure on a compressible sphere,” J. Acoust. Soc. Am. 69, 1499–1501 (1981);Erratum 71, 511 (1982);Y. Tian, R. G. Holt, R. E. Apfel, “Deformation and location of an acoustically levitated liquid drop,” J. Acoust. Soc. Am. 93, 3096–3104 (1993).
[CrossRef] [PubMed]

Simpson, H. J.

Trinh, E. H.

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

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 13.

Appl. Opt. (4)

J. Acoust. Soc. Am. (1)

P. L. Marston, S. E. LoPorto-Arione, G. L. Pullen, “Quadrapole projection of the radiation pressure on a compressible sphere,” J. Acoust. Soc. Am. 69, 1499–1501 (1981);Erratum 71, 511 (1982);Y. Tian, R. G. Holt, R. E. Apfel, “Deformation and location of an acoustically levitated liquid drop,” J. Acoust. Soc. Am. 93, 3096–3104 (1993).
[CrossRef] [PubMed]

Nature (London) (2)

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

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

Opt. Lett. (1)

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]

Prog. Opt. (1)

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).

Other (4)

P. L. Marston, G. Kaduchak, “Secondary and higher-order generalized rainbows and unfolded glories of oblate drops: analysis and laboratory observations,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 12–15.

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” in Physical Acoustics, R. N. Thurston, A. D. Pierce, eds. (Academic, Boston, Mass., 1992), Vol. 21, pp. 1–234.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 13.

W. J. Humphreys, Physics of the Air, 3rd ed. (Dover, New York, 1964), pp. 475–495.

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Figures (3)

Fig. 1
Fig. 1

Ray diagram for production of the primary and the secondary rainbows from a spherical drop or from the equatorial plane of an oblate drop. Light is incident from the right. The rays shown are for the extreme deviation angles of the primary and the secondary rainbows. The horizontal curvature of the outgoing virtual wave fronts vanishes along the Descartes ray of the primary and the secondary rainbows.

Fig. 2
Fig. 2

Photographic sequence of caustics observed in the levitation experiments under monochromatic illumination. Backscattering is to the right of the photographs out of the field of view. Drop diameters vary from 1.39 to 1.83 mm. a depicts the secondary (left) and primary (right) bows for a moderately oblate drop spanning an angular field of ∼ 23°. The aspect ratio of the drop is less than 1.31. b is an isolated view of a vertical fold centered near the secondary-rainbow scattering angle, c and d exhibit a cusp catastrophe emanating from larger scattering angles, where the drop in d is slightly flatter than in c. e is a manifestation of the HUFS. The measured aspect ratio of the optically equivalent ellipsoids are b, 1.22; c, 1.36; d, 1.37; e, 1.43. These photographs span an angular field of ∼ 16°.

Fig. 3
Fig. 3

Continuation of Fig. 2, except with greater aspect ratios (1.49 < D/H < 1.57). Drop diameters vary from 1.45 to 1.78 mm. a depicts the secondary rainbow following a curvature reversal before the primary's curvature reversal, b is a manifestation of the secondary E6 focal section (although partially cut off by an aperture restriction), whereas the primary bow is shown on the right, c–e show only the angular region of the secondary bow. They vary from just before a curvature reversal of the outer fold to a near manifestation of the E6 focal section in the secondary rainbow. The measured aspect ratios of the optically equivalent ellipsoid are 1.49, 1.57, and 1.50 in a, b, and c, respectively, whereas the aspect ratios for d and e are close to (but less than) 1.57. Photographs a and c–e span angular fields of ≈ 16°, whereas b spans an angular field of ≈ 15°.

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