Optical Billiards

Abstract: This paper investigates the dynamics of optical billiards, a generalization of classic billiards, where light rays travel within a refractive medium and reflect elastically at the boundary. Inspired by studies of acoustic modes in rapidly rotating stars, optical billiards are approached using Riemannian geometry, treating the refractive index as a metric. The discrete dynamics is explored through geodesics, curvature properties, highlighting similarities and differences with classical billiards. Examples include geodesic circular billiards and refractive media with zero Gaussian curvature, revealing integrability conditions and singular behaviors. The study establishes foundational results for optical billiards, emphasizing their dependence on boundary convexity and refractive index.