HOMOCLINIC CHAOS FOR RAY OPTICS IN A FIBER: 200 YEARS AFTER LAGRANGE

Abstract

Publisher Summary In the geometrical optics approximation, stable and unstable manifolds of periodic orbits, invariant tori, and hyperbolic invariant manifolds are shown to exist and produce trapping of bundles of light rays near the axis of a translation-invariant, axisymmetric optical fiber, whose squared refractive index is a parabolic function of squared radius. Periodic symmetry-breaking perturbations in the refractive index are shown to destroy this ray trapping and to produce homoclinic tangles, through which nearby trapped rays escape and untrapped rays become temporarily trapped. This chapter discusses axisymmetric, translation-invariant media. Axisymmetric, translation-invariant media, in which the index of refraction is a function of the radius alone, are of considerable theoretical interest. Axisymmetry implies an additional constant of motion and reduction of the Hamiltonian system for light rays to phase plane analysis. The resulting ray paths describe, in a certain sense, perfect optical instruments. The chapter further reviews the effects of perturbations of the refractive index.