  These supplementary materials detail some calculations and some experimental
results related to the propagation of light in the photonic billiard. We first
justify why we focus only on the rays that are transmitted through the
cylinders and justify the geometrical optics approximation. Then we explain the
dynamical properties of ray propagation, demonstrate that asymptotically the
Lyapunov exponent grows as $\lambda$ $\sim$ ln t$\star$ where t$\star$ = T /R
is the photonic crystal period T divided by the cylinder radius R. Finally we
close these supplementary materials by presenting some experimental results
demonstrating the exponential sensitivity to the initial conditions.
