Abstract

Oblate drops of water illuminated perpendicular to their symmetry axis exhibit catastrophe patterns near the primary-rainbow scattering angle. Previous patterns include the hyperbolic umbilic focal section and separate lips events [see, e.g., H. J. Simpson and P. L. Marston, Appl. Opt. 30, 3468 (1991)]. The present observations concern a much higher-order singularity analyzed by J. F. Nye [Proc. R. Soc. London Ser. A 438, 397 (1992)], the E6 or symbolic umbilic, in the scattering by levitated drops with monochromatic and collimated white-light illumination. Photographs show the colors produced when the drop is illuminated by white light. The E6 occurs when the Gaussian curvature of the scattered wave front vanishes in both principal directions, resulting in a high degree of directional focusing. This type of focusing, although only slightly explored, is applicable to the more general problem of scattering from penetrable spheroids.

© 1994 Optical Society of America

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References

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  1. P. L. Marston, E. H. Trinh, “Hyperbolic umbilic catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984);P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
    [CrossRef] [PubMed]
  2. J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532(1984).
    [CrossRef]
  3. 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]
  4. H. J. Simpson, “The lips event for light back scattered from levitated water drops,” M. S. degree project rep. (Washington State University, Pullman, Wash., 1988);see also P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989).
    [CrossRef]
  5. P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” in Physical AcousticsR. N. Thurston, A. D. Pierce, eds. (Academic, Boston, Mass., 1992), Vol. 21, pp. 1–234.
  6. J. F. Nye, “Optical caustics from liquid drops under gravity: observations of the parabolic and symbolic umbilics,” Phil. Trans. R. Soc. London Ser. A 92, 25–44 (1979).
  7. M. V. Berry, “Waves and Thom's theorem,” Adv. Phys. 25, 1–26(1976).
    [CrossRef]
  8. M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 259–343 (1980).
  9. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 13.
  10. G. P. Konnen, J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979).
    [CrossRef] [PubMed]
  11. J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London Ser. A 438, 397–417 (1992).
    [CrossRef]
  12. J. Callahan, “Caustics of the harmonic double cusp near an E6 point,” Proc. R. Soc. London Ser. A 382, 319–334 (1982).
    [CrossRef]
  13. J. W. Bruce, P. J. Giblin, Curves and Singularities, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 294–295.
  14. E. H. Trinh, “Compact acoustic levitation device for studies in fluid dynamics and material science in the laboratory and microgravity,” Rev. Sci. Instrum. 56, 2059–2065 (1985).
    [CrossRef]
  15. 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]
  16. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  17. W. J. Humphreys, Physics of the Air, 3rd ed. (Dover, New York, 1964), pp. 489–493.

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)

E. H. Trinh, “Compact acoustic levitation device for studies in fluid dynamics and material science in the laboratory and microgravity,” Rev. Sci. Instrum. 56, 2059–2065 (1985).
[CrossRef]

1984 (2)

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984);P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
[CrossRef] [PubMed]

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

1982 (1)

J. Callahan, “Caustics of the harmonic double cusp near an E6 point,” Proc. R. Soc. London Ser. A 382, 319–334 (1982).
[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, 259–343 (1980).

1979 (2)

G. P. Konnen, J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979).
[CrossRef] [PubMed]

J. F. Nye, “Optical caustics from liquid drops under gravity: observations of the parabolic and symbolic umbilics,” Phil. Trans. R. Soc. London Ser. A 92, 25–44 (1979).

1976 (2)

M. V. Berry, “Waves and Thom's theorem,” Adv. Phys. 25, 1–26(1976).
[CrossRef]

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

Berry, M. V.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 259–343 (1980).

M. V. Berry, “Waves and Thom's theorem,” Adv. Phys. 25, 1–26(1976).
[CrossRef]

Bruce, J. W.

J. W. Bruce, P. J. Giblin, Curves and Singularities, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 294–295.

Callahan, J.

J. Callahan, “Caustics of the harmonic double cusp near an E6 point,” Proc. R. Soc. London Ser. A 382, 319–334 (1982).
[CrossRef]

de Boer, J. H.

Giblin, P. J.

J. W. Bruce, P. J. Giblin, Curves and Singularities, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 294–295.

Humphreys, W. J.

W. J. Humphreys, Physics of the Air, 3rd ed. (Dover, New York, 1964), pp. 489–493.

Konnen, G. P.

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.

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]

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984);P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
[CrossRef] [PubMed]

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, “Geometrical and catastrophe optics methods in scattering,” in Physical AcousticsR. 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 from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532(1984).
[CrossRef]

J. F. Nye, “Optical caustics from liquid drops under gravity: observations of the parabolic and symbolic umbilics,” Phil. Trans. R. Soc. London Ser. A 92, 25–44 (1979).

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.

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]

H. J. Simpson, “The lips event for light back scattered from levitated water drops,” M. S. degree project rep. (Washington State University, Pullman, Wash., 1988);see also P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989).
[CrossRef]

Trinh, E. H.

E. H. Trinh, “Compact acoustic levitation device for studies in fluid dynamics and material science in the laboratory and microgravity,” Rev. Sci. Instrum. 56, 2059–2065 (1985).
[CrossRef]

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984);P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
[CrossRef] [PubMed]

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 259–343 (1980).

van de Hulst, H. C.

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

Walker, J. D.

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

Adv. Phys. (1)

M. V. Berry, “Waves and Thom's theorem,” Adv. Phys. 25, 1–26(1976).
[CrossRef]

Am. J. Phys. (1)

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

Appl. Opt. (2)

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)

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984);P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
[CrossRef] [PubMed]

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

Phil. Trans. R. Soc. London Ser. A (1)

J. F. Nye, “Optical caustics from liquid drops under gravity: observations of the parabolic and symbolic umbilics,” Phil. Trans. R. Soc. London Ser. A 92, 25–44 (1979).

Proc. R. Soc. London Ser. A (2)

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

J. Callahan, “Caustics of the harmonic double cusp near an E6 point,” Proc. R. Soc. London Ser. A 382, 319–334 (1982).
[CrossRef]

Prog. Opt. (1)

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 259–343 (1980).

Rev. Sci. Instrum. (1)

E. H. Trinh, “Compact acoustic levitation device for studies in fluid dynamics and material science in the laboratory and microgravity,” Rev. Sci. Instrum. 56, 2059–2065 (1985).
[CrossRef]

Other (5)

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. 489–493.

J. W. Bruce, P. J. Giblin, Curves and Singularities, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 294–295.

H. J. Simpson, “The lips event for light back scattered from levitated water drops,” M. S. degree project rep. (Washington State University, Pullman, Wash., 1988);see also P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989).
[CrossRef]

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

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Figures (4)

Fig. 1
Fig. 1

Ray diagram for production of the primary rainbow scattered from a spherical drop or, for the case of the experiment, the equatorial plane of a spheroidal drop. Light is incident from the left, where it is refracted at the outer surface of the drop, reflected within, and refracted back into the atmosphere. Rays 1 and 2 hit the drop with different angles of incidence i but have the same scattering angle θ. The Descartes ray is the limit of two rays that share the same angle of incidence, forming the directional fold caustic. The horizontal curvature of the outgoing virtual wave front vanishes along this ray. The special focusing observed for oblate drops is associated with a vanishing vertical curvature.

Fig. 2
Fig. 2

Acoustic levitation assembly with an approximate height of 8 in. (20.32 cm) utilized in the experiments. Two piezoelectric disks in contact with a beryllium copper sheet are driven with a sinusoidal voltage that excites vibrations in the aluminum mechanical transformer. The vibrations excite an acoustic standing wave in the cavity below the reflector. A water drop is inserted and levitated just below a velocity antinode. The drop (not shown to scale) is oblate. Illumination is incident perpendicular to the drop's symmetry axis. The weight of the horn assembly is supported by the beryllium copper plate.

Fig. 3
Fig. 3

Diagram for fitting the vertical curvature of an oblate drop, y is the symmetry axis, and the curvature at k is determined by measuring x′ and y′ from negatives of the drop profile.

Fig. 4
Fig. 4

Photographic sequence of caustics observed in the levitation experiments under monochromatic illumination. The photographs span a horizontal angular field of approximately 21°. Backscattering is to the right of the photographs out of the field of view. The photographs are shown in order of increasing aspect ratio D/H, in which the drop diameters range from 1.41 to 1.84 mm from a–i. a–e Exhibit a curvature reversal of the outer fold line, d–f Bracket the E6 event, which is approximately represented by e. In g and h two hyperbolic umbilics separate but do not themselves unfold until i, in which an aperture restriction within the drop causes them to disappear (although a slight aperture restriction from the camera is evident in several of the photographs). Aspect ratios corrected as in Eq. (2) for the photographs in a, c, e, g, and i are 1.49, 1.55, 1.61, 1.68, and 1.74 respectively. (The aspect ratio is given for every other photograph, as the interval between consecutive photographs is of the order of the measurement uncertainties.)

Equations (3)

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x 2 ( D / 2 ) 2 + y 2 ( H / 2 ) 2 = 1
D H = [ ( D x ) x ] 1 / 2 y .
D H = [ 9 n 2 8 ( n 2 1 ) ] 1 / 2 = 1.605 ( n < 2 ) .

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