Light scattered by an oblate drop of water has been observed to produce cusp caustics in the general vicinity of the rainbow region [ Marston P. L. Trinh E. H., 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 − 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.
© 1991 Optical Society of America
Original Manuscript: September 13, 1990
Published: August 20, 1991
Cleon E. Dean and Philip L. Marston, "Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops," Appl. Opt. 30, 3443-3451 (1991)