  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.
