Geometrical optics and geodesics in thin layers

TL;DR

This paper investigates how light rays propagate in thin curved layers, showing that in very thin films the paths follow geodesics on the surface, with finite thickness causing deviations and potential chaotic behavior.

Contribution

It derives the equations governing light trajectories in thin layers, linking them to geodesics on curved surfaces, and analyzes the effects of finite thickness on these paths.

Findings

Light rays in very thin layers follow geodesics on the surface.

Finite thickness causes deviations from geodesic paths.

Trajectory equations exhibit chaotic features with sensitivity to initial conditions.

Abstract

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.