Generalized rainbows and unfolded glories of oblate drops: organization for multiple internal reflections and extension of cusps into Alexander's dark band.

Oblate drops of water can produce caustics where, unlike a simple Airy caustic, more than two rays merge. We extend previous treatments of generalized primary rainbows based on catastrophe optics [Opt. Lett. 10, 588 (1985); Proc. R. Soc. (London) A 438, 397 (1992)] to rays having (p - 1) = 2 to 5 internal reflections. The analysis is for a horizontally illuminated ellipsoid with a vertical symmetry axis. Aspect ratios causing a vanishing of the vertical curvature at the equator for the outgoing wave front are found from generalized ray tracing. In response to infinitesimal deformation, the axial caustic of real glory rays unfolds producing cusps. Laboratory observations with laser illumination demonstrate that cusps resulting from rays with five internal reflections extend into Alexander's dark band when the drop's aspect ratio is near 1.08. The evolution of this p = 6 scattering pattern as cusps meet the quinary rainbow is suggestive of an E(6) catastrophe. For ellipsoids of varying aspect ratio and refractive index N, there is an organizing singularity associated with an exceptionally flat outgoing wave front from spheres with N = p.