Periodic orbits in a simple ray-splitting system.

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 → 1+, conversion occurs at every reflection and periodic orbits proliferate exponentially. As κ increases beyond √ 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 > √ 2 and show numerically that exponential proliferation persists around κ ≃ √ 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. PACS numbers: 05.45.+b, 03.65.Sq Typeset using REVTEX ∗email:[email protected] †on leave from Theoretical Physics Division, Bhabha Atomic Research Centre, Bombay 400 085 1