A Semiclassical Approach to Rainbows

We describe a simple semiclassical theory of rainbow scattering based on real-space light paths. Our approach is rooted in concepts that physics students are familiar with — reflection, refraction, and interference — rather than arcane methods often used in mathematical treatments of rainbows, such as Mie theory and complex angular momentum (CAM). The semiclassical theory successfully explains experimental data on rainbow scattering: the relative peak intensities and locations of supernumerary bows, the intensities of intervening minima, the shape of the primary peak, and the diffracted intensity within Alexander's "dark" band. It also agrees with Mie and CAM calculations and naturally accounts for a mysterious "π/2 phase shift" sometimes invoked in the literature to correctly describe supernumerary bows. The semiclassical theory interpolates between Fermat's principle for ray optics and Feynman's "sum over paths" approach to quantum electrodynamics, and thereby provides a conceptual bridge between classical and quantum theories of light.