Light chaotic dynamics and ray engineering transformed from curved to flat space

Two-dimensional curved surfaces have attracted increasing attention recently, as an analog model of General Relativity with reduced dimensionality. Here, inspired by the concept of transformation optics, we demonstrate that a curved surface is fundamentally equivalent to a table billiard with nonuniform distribution of refractive index. We rigorously prove these two systems share the same dynamics in terms of both light rays and waves. 

From our study on chaotic dynamics of light on a typical family of surfaces of revolution, we find that the degree of chaos is fully controlled by the single curvature-related parameter of the curved surface. This statement is verified by exploring in the transformed billiard the dependence with this geometric parameter of the Poincaré surface of section, the Lyapunov exponent and the statistics of eigenmodes and eigenfrequency spectrum. We reveal that curvature provides a degree of freedom in chaos engineering, as well as potentialities to design nonuniform billiards and cavities.

Following the flow of transformation, we propose to engineer the trajectories of light rays by inheriting exotic geodesics on curved surfaces. We show that chaos can be turned off, and all light rays can further be rectified to be periodic. More interestingly, we illustrate spiraling trajectories with extremely long “life time”, transformed from photon sphere of Schwarzschild blackhole. An implementation is suggested for real-time control of chaos and rectification of light rays.