Periodic Orbits in a Simple Ray-Splitting System

We study ray dynamics in a square billiard that allows mode conversion and is parametrised by $\kappa$, the ratio of the velocities of the two modes. At $\kappa \to 1^{+}$, conversion occurs at every reflection and periodic orbits proliferate exponentially. As $\kappa$ 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 $\kappa >\sqrt{2}$ and show numerically that exponential proliferation persists around $\kappa \simeq \sqrt{2}$ but as $\kappa$ increases, a crossover to sub-exponential behaviour occurs for short periods. We discuss these results in the light of conservation laws.