Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder. I. Theory.

A sequence of rainbows is produced in light scattering by a particle of high symmetry in the short-wavelength limit, and a supernumerary interference pattern occurs to one side of each rainbow. Using both a ray-tracing procedure and the Debye-series decomposition of first-order perturbation wave theory, I examine the spacing of the supernumerary maxima and minima as a function of the cylinder rotation angle when an elliptical-cross-section cylinder is normally illuminated by a plane wave. I find that the supernumerary spacing depends sensitively on the cylinder-cross-section shape, and the spacing varies sinusoidally as a function of the cylinder rotation angle for small cylinder ellipticity. I also find that relatively large uncertainties in the supernumerary spacing affect the rainbow angle only minimally.