Hyperbolic geometrical optics: Hyperbolic glass

We study the geometrical optics generated by a refractive index of the form $n(x,y)=1/y$ $(y>0)$, where $y$ is the coordinate of the vertical axis in an orthogonal reference frame in $\R^2$. We thus obtain what we call "hyperbolic geometrical optics" since the ray trajectories are geodesics in the Poincar\'e-Lobachevsky half--plane $\HH^2$. Then we prove that the constant phase surface are horocycles and obtain the \emph{horocyclic waves}, which are closely related to the classical Poisson kernel and are the analogs of the Euclidean plane waves. By studying the transport equation in the Beltrami pseudosphere, we prove(i) the conservation of the flow in the entire strip $0<y\leqslant 1$ in $\HH^2$, which is the limited region of physical interest where the ray trajectories lie; (ii) the nonuniform distribution of the density of trajectories: the rays are indeed focused toward the horizontal x axis, which is the boundary of $\HH^2$. Finally the process of ray focusing and defocusing is analyzed in detail by means of the sine--Gordon equation.