Article

# Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes

Applied Optics (Impact Factor: 1.78). 08/1991; 30(24):3468-73. DOI: 10.1364/AO.30.003468

Source: PubMed

**ABSTRACT**

Oblate drops of water illuminated perpendicular to their symmetry axis generate a hyperbolic-umbilic diffraction catastrophe near the primary rainbow [P. L. Marston and E. H. Trinh, Nature London 312, 529-531 (1984)]. Observations were made of this diffraction catastrophe generated by white-light illumination of acoustically levitated drops of water in air. The observations suggest what generalized rainbows would look like if they were produced in nature when sunlight illuminates large raindrops. Unlike the usual rainbow arc, the transverse cusp of the unfolded catastrophe is not distinctly colored. The hyperbolic-umbilic focal section is distinctly colored as is another diffraction catastrophe generated in the rainbow region when the drop is highly oblate.

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**ABSTRACT:**Light scattered by an oblate drop of water has been observed to produce cusp caustics in the general vicinity of the rainbow region [P. L. Marston and E. H. Trinh, Nature London 312, 529-531 (1984)]. The principal curvatures of the generic local wave front that produces the far-field transverse cusp are examined. This wave front is shown to generate a transverse cusp curve (U - U(c))(3) = - d(infinity)V(2), where U and V are horizontal and vertical scattering angles and U(c) is the cusp point direction. The far-field opening rate d(infinity) is calculated for the transverse cusp. It is shown that d(infinity) has a simple dependence on the parameters of the generic wave front. We define the aspect ratio of the drop q = D/H, where H is the height and D is the equatorial width for the scattering drop. The method of generalized ray tracing is used to relate q to principal curvatures and shape parameters of the outgoing wave front and hence to d(infinity). Measurements of d(infinity) for scattering laser light from acoustically levitated drops appear to support the calculation. As q goes to q(4) approximately 1.31, the critical value for generation of a hyperbolic-umbilic focal section, the predicted d(infinity) goes to infinity. The nature of the divergence was numerically investigated as was the rate at which d(infinity) vanishes as q approaches critical values for lips and transition events. - [Show abstract] [Hide abstract]

**ABSTRACT:**New features of the primary rainbow have been revealed in recent experiments by Marston and co-workers in which light is incident on levitated water drops. The drops are oblate spheroids. This paper provides a treatment of the problem by geometrical optics, using the principles of catastrophe optics. It predicts analytically the axial ratios of the drops necessary to produce certain landmark features of the caustics, such as two hyperbolic umbilic foci on the equatorial plane, two kinds of lips events and two E6 catastrophes. The sequence of caustics produced as the axial ratio of a water drop is changed is shown to be organized by two higher-order singularities which would correspond to drops of refractive index 2 and axial ratios of 1 and surd 2/3 respectively. The unfoldings of these two higher singularities explain qualitatively all the significant events for a water drop. They also lead to the prediction that two previously unsuspected hyperbolic umbilic foci will be formed in the vertical plane of symmetry when the spheroid is prolate with axial ratio 1.625. Rainbow scattering, that is, the formation of caustics, can occur whenever photons or other particles are scattered by atomic nuclei, atoms, molecules or crystals. Ion channelling through crystals, and the scattering of atoms and molecules by crystal surfaces are examples. The approach described here, of locating higher organizing centres, is equally applicable throughout this broader field. - [Show abstract] [Hide abstract]

**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 E(6) 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 E(6) 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.

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