Catastrophes in Reconnections of Optical Vortex Rings
A small change in parameters leads to a small change in the outcomes, in most cases. For instance, an optical vortex ring, created in the superposition of two laser beams, can slightly shrink if the phase changes minutely. However, dynamical systems can also experience significant changes due to small perturbations in parameters; the vortex rings can reconnect, nucleate or collapse into a point. Parameters that cause such striking outcomes are at the focal point of catastrophe theory - a branch of bifurcation theory that studies degenerate critical points of a potential function. We present theoretical study of topological configurations of optical vortex rings in elliptic Gaussian beams, which change dramatically at specific beam parameters. We identify two types of catastrophes, cusp and butterfly, near which very small changes of elliptic deformations lead to complex topological transformations of vortex rings.