Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops

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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 E. H. Trinh , Nature London312, 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 − Uc)3 = − d∞V2, where U and V are horizontal and vertical scattering angles and Uc is the cusp point direction. The far-field opening rate d∞ is calculated for the transverse cusp. It is shown that d∞ 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 wavefront and hence to d∞. Measurements of d∞. for scattering laser light from acoustically levitated drops appear to support the calculation. As q goes to q4 ≈ 1.31, the critical value for generation of a hyperbolic–umbilic focal section, the predicted d∞ goes to infinity. The nature of the divergence was numerically investigated as was the rate at which d∞ vanishes as q approaches critical values for lips and transition events.