Periodic Orbits in a Simple Ray-Splitting System

Abstract

We study ray dynamics in a square billiard that allows mode conversion and is parametrised by $κ$, the ratio of the velocities of the two modes. At $κ\rightarrow 1^+$, conversion occurs at every reflection and periodic orbits proliferate exponentially. As $κ$ increases beyond $\sqrt{2}$, the collection of daughter rays explore only three momentum directions and mode conversion is progressively inhibited. We provide an algorithm for determining periodic orbits when $κ> \sqrt{2}$ and show numerically that exponential proliferation persists around $κ\simeq \sqrt{2}$ but as $κ$ increases, a crossover to sub-exponential behaviour occurs for short periods. We discuss these results in the light of conservation laws.