Abstract

The angular location of the recently discovered cusp pattern in the far-field scattering from an oblate spheroid is calculated as a function of the aspect ratio D/H. The calculation assumes the diameter Dλ and is limited to illumination perpendicular to the short axis of the spheroid. It agrees with observations for water drops in the range 1.22 < D/H < 1.37 with D ~ 1 mm.

© 1985 Optical Society of America

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References

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  1. P. L. Marston, E. H. Trinh, Nature (London) 312, 529 (1984).
    [CrossRef]
  2. J. F. Nye, Nature (London) 312, 531 (1984).
    [CrossRef]
  3. H. M. Nussenzveig, J. Math Phys. N.Y. 10, 125 (1969).
    [CrossRef]
  4. G. P. Können, J. H. deBoer, Appl. Opt. 18, 1961 (1979).
    [CrossRef] [PubMed]
  5. M. V. Berry, C. Upstill, Prog. Opt. 18, 257 (1980).
    [CrossRef]
  6. R. Thom, Stabilité Structurelle et Morphogénèse (Benjamin, Reading, Mass., 1972).
  7. M. V. Berry, J. F. Nye, F. J. Wright, Philos. Trans. R. Soc. London Ser. A 291, 453 (1979).
    [CrossRef]
  8. V. I. Arnold, Singularity Theory (Cambridge U. Press, Cambridge, 1981).
  9. T. Pearcey, Philos. Mag. 37, 311 (1946).
  10. M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958).
  11. The two-dimensional diffraction integral reduces to Pearcey’s integral with an integrand exp[is4 + is2 (θ − θ3)a2 + is ζa1] and an integration variable s = a0∊; the aj are functions of D, H, λ, and μ.
  12. S. Asano, G. Yamamoto, Appl. Opt. 14, 29 (1975).
    [PubMed]

1984 (2)

P. L. Marston, E. H. Trinh, Nature (London) 312, 529 (1984).
[CrossRef]

J. F. Nye, Nature (London) 312, 531 (1984).
[CrossRef]

1980 (1)

M. V. Berry, C. Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

1979 (2)

M. V. Berry, J. F. Nye, F. J. Wright, Philos. Trans. R. Soc. London Ser. A 291, 453 (1979).
[CrossRef]

G. P. Können, J. H. deBoer, Appl. Opt. 18, 1961 (1979).
[CrossRef] [PubMed]

1975 (1)

1969 (1)

H. M. Nussenzveig, J. Math Phys. N.Y. 10, 125 (1969).
[CrossRef]

1946 (1)

T. Pearcey, Philos. Mag. 37, 311 (1946).

Arnold, V. I.

V. I. Arnold, Singularity Theory (Cambridge U. Press, Cambridge, 1981).

Asano, S.

Berry, M. V.

M. V. Berry, C. Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

M. V. Berry, J. F. Nye, F. J. Wright, Philos. Trans. R. Soc. London Ser. A 291, 453 (1979).
[CrossRef]

deBoer, J. H.

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958).

Können, G. P.

Marston, P. L.

P. L. Marston, E. H. Trinh, Nature (London) 312, 529 (1984).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, J. Math Phys. N.Y. 10, 125 (1969).
[CrossRef]

Nye, J. F.

J. F. Nye, Nature (London) 312, 531 (1984).
[CrossRef]

M. V. Berry, J. F. Nye, F. J. Wright, Philos. Trans. R. Soc. London Ser. A 291, 453 (1979).
[CrossRef]

Pearcey, T.

T. Pearcey, Philos. Mag. 37, 311 (1946).

Thom, R.

R. Thom, Stabilité Structurelle et Morphogénèse (Benjamin, Reading, Mass., 1972).

Trinh, E. H.

P. L. Marston, E. H. Trinh, Nature (London) 312, 529 (1984).
[CrossRef]

Upstill, C.

M. V. Berry, C. Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

Wright, F. J.

M. V. Berry, J. F. Nye, F. J. Wright, Philos. Trans. R. Soc. London Ser. A 291, 453 (1979).
[CrossRef]

Yamamoto, G.

Appl. Opt. (2)

J. Math Phys. N.Y. (1)

H. M. Nussenzveig, J. Math Phys. N.Y. 10, 125 (1969).
[CrossRef]

Nature (London) (2)

P. L. Marston, E. H. Trinh, Nature (London) 312, 529 (1984).
[CrossRef]

J. F. Nye, Nature (London) 312, 531 (1984).
[CrossRef]

Philos. Mag. (1)

T. Pearcey, Philos. Mag. 37, 311 (1946).

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

M. V. Berry, J. F. Nye, F. J. Wright, Philos. Trans. R. Soc. London Ser. A 291, 453 (1979).
[CrossRef]

Prog. Opt. (1)

M. V. Berry, C. Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

Other (4)

R. Thom, Stabilité Structurelle et Morphogénèse (Benjamin, Reading, Mass., 1972).

V. I. Arnold, Singularity Theory (Cambridge U. Press, Cambridge, 1981).

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958).

The two-dimensional diffraction integral reduces to Pearcey’s integral with an integrand exp[is4 + is2 (θ − θ3)a2 + is ζa1] and an integration variable s = a0∊; the aj are functions of D, H, λ, and μ.

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Figures (5)

Fig. 1
Fig. 1

Rays through a spheroidal drop in the horizontal equatorial plane.

Fig. 2
Fig. 2

Photographs of the scattering into the rainbow region. The horizontal scattering angle θ increases from left to right. The ratio D/H of the drops increases from (a) to (c).

Fig. 3
Fig. 3

(a) Locations of once-reflected (twice-refracted) rays as they leave a drop. (b) Partitioning of a scattering pattern with D/Hq4.

Fig. 4
Fig. 4

The dependence of D/H on the cusp location θ3. The prediction is the curve, and the ● are measurements. ▲ is the measured q4 from Ref. 1.

Fig. 5
Fig. 5

Drops having D/H of (a) 1.21 and (b) 1.14. The camera’s aperture is at θ ≃ 143°, which is close to θ3 for (a) but is well inside the two-ray region for (b). Ray 1 is visible near the left edge of each drop; however, in (a) skew rays merging with it form a bright arc. The bright patch near each center is not related to the laser illumination. It is from the diffuse source used to highlight the profiles.

Equations (1)

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q = μ [ 2 w ( w - w ) ] - 1 / 2 ,             w = ( μ 2 - S ) 1 / 2 ,

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