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American Journal of Physics

Analogy between optical rays and nonrelativistic particle trajectories: A comment

Jacques Arnaud

Published Nov 1, 1976• 10 citations
Publisher

Abstract

The identification, medium refractive index n = nonrelativistic particle velocity v, in the analogy of geometrical optics with nonrelativistic particle mechanics is physically correct and useful, but can easily be misinterpreted. The conditions under which the analogy holds are outlined. The significance of Descartes’s mechanical models is also discussed.

DOI
10.1119/1.10586

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The identification, medium refractive index n = nonrelativistic particle velocity v, in the analogy of geometrical optics with nonrelativistic particle mechanics is physically correct and useful, but can easily be misinterpreted. The conditions under which the analogy holds are outlined. The significance of Descartes’s mechanical models is also discussed.
We provide a new unified formalism for the analogy between fiber-optics and mechanics. It is based on the scalar theory of light for an optical fiber with length-independent refractive index, and the mechanics of a particle in a time-independent potential. Illustrative examples such as tunneling and precession are given. We also mention the conceptual analogies that arise when we include (i) polarization and (ii) length or time-dependent variations. This should prove useful both in teaching and as a bridge between research areas.
There is a theorem in geometrical optics due to Helmholtz which states that if a ray of light (1) after any number of refractions and reflections at plane or nearly plane surfaces gives rise (among others) to a ray (2) whose intensity is a certain fraction f12 of the intensity of the ray (1), then on reversing the path of the light an incident ray (2)′ will give rise, among others, to a ray (1)′ whose intensity is a fraction f21 of the ray (2)′, such that The surfaces if not plane must be such that their radius of curvature is large compared with the lateral extent of the ray and this lateral extent must itself be large compared with the wave-lengths in the ray. In view of the many recent studies of reflection and transmission (refraction) of beams of electrons at potential walls, it seems worth while to ask whether there exists an analogy of the optical theorem for beams of electrons (or other particles). It is easy to see that the analogy must exist by using the principle of detailed balancing required by an assembly in statistical equilibrium. But a direct “dynamical” proof is also easy and may prove of sufficient practical interest to put on record here.