Geometrical optics and geodesics in thin layers

Tomasz Radożycki
Phys. Rev. A 98, 063802 – Published 3 December 2018

Abstract

The propagation of a light ray in a 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 nonlinear 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.

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  • Received 3 September 2018

DOI:https://doi.org/10.1103/PhysRevA.98.063802

©2018 American Physical Society

Physics Subject Headings (PhySH)

Atomic, Molecular & Optical

Authors & Affiliations

Tomasz Radożycki*

  • Faculty of Mathematics and Natural Sciences, College of Sciences, Cardinal Stefan Wyszyński University, Wóycickiego 1/3, 01-938 Warsaw, Poland

  • *t.radozycki@uksw.edu.pl

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Issue

Vol. 98, Iss. 6 — December 2018

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