Optical Billiards

TL;DR

This paper explores the dynamics of optical billiards, where light rays reflect within a medium modeled by Riemannian geometry, revealing new integrability conditions and behaviors distinct from classical billiards.

Contribution

It introduces a geometric framework for optical billiards, analyzing geodesic trajectories and curvature effects, and establishes foundational results linking boundary shape and refractive index.

Findings

Geodesic circular billiards exhibit integrable behavior.

Zero Gaussian curvature media show unique singular behaviors.

Boundary convexity influences optical billiard dynamics.

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.