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

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.