  The propagation of a light ray in thin layer (film) within geometrical optics
is considered. It is assumed that the ray is captured inside the layer due to
reflecting walls or total internal reflection (in the case of a dielectric
layer). It has been found that for a very thin film (the length scale is
imposed by the curvature of the surface at a given point) the equations
describing the trajectory of the light beam are reduced to the equation of a
geodesic on the limiting curved surface. There have also been found corrections
to the trajectory equation resulting from the finite thickness of the film.
Numerical calculations performed for a couple of exemplary curved layers (cone,
sphere, torus and catenoid) confirm that for thin layers the light ray which is
repeatedly reflected, propagates along the curve close to the geodesic but as
the layer thickness increases, these trajectories move away from each other.
Because the trajectory equations are complicated non-linear differential
equations, their solutions show some chaotic features. Small changes in the
initial conditions result in remarkably different trajectories. These chaotic
properties become less significant the thinner the layer under consideration.
