Rainbows from ellipsoidal water drops

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 E 6 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 √⅔ 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.