Homoclinic chaos for ray optics in a fiber

Abstract

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 may escape, and untrapped rays may become at least temporarily trapped. Melnikov's technique is used to prove that the perturbations cause the stable and unstable manifolds of the unperturbed periodic orbits, invariant tori, and hyperbolic invariant manifolds to develop transverse intersections and therefore, to form homoclinic tangles. These tangles imply either homoclinic chaos by the Smale horseshoe mechanism and the Poincaré-Birkhoff-Smale theorem, or Arnol'd diffusion. In both situations, lobe dynamics will dominate phase space transport, seen here as a flux of (initially) untrapped rays passing through the trapping region.