Rainbow scattering by a coated sphere

We examine the behavior of the first-order rainbow for a coated sphere by using both ray theory and Aden-Kerker wave theory as the radius of the core a(12) and the thickness of the coating δ are varied. As the ratio δ/a(12) increases from 10(-4) to 0.33, we find three classes of rainbow phenomena that cannot occur for a homogeneous-sphere rainbow. For δ/a(12) ≲ 10(-3), the rainbow intensity is an oscillatory function of the coating thickness, for δ/a(12) ≈ 10(-2), the first-order rainbow breaks into a pair of twin rainbows, and for δ/a(12) ≈ 0.33, various rainbow-extinction transitions occur. Each of these effects is analyzed, and their physical interpretations are given. A Debye series decomposition of coated-sphere partial-wave scattering amplitudes is also performed and aids in the analysis.