Patterns, fishing and nonlinear optics.

Abstract

Motivated by a conversation with my brother, a deep sea fisherman off the east coast of Scotland, I review the concepts which unify the topic of pattern formation in nonequilibrium systems. As a specific example of a pattern-forming system, I go on to examine pattern formation in nonlinear optics and I discuss two nonlinear optical systems in considerable detail. The first, counterpropagating laser beams in a nonlinear Kerr medium, results in the prediction and numerical observation of hexagonal patterns in a self-focusing medium, and of square patterns in a self-defocusing medium. Furthermore, a novel Hopf bifurcation is observed which destabilises the hexagons and an explanation in terms of a coupled-amplitude model is given. The other system, namely the mean-field model of propagation in a nonlinear cavity, also gives rise to hexagonal patterns in a self-focusing medium. By extending this model to include the vector nature of the electric field, polarisation patterns are predicted and observed for a self-defocusing medium. Roll patterns dominate close to threshold, while farther from threshold labyrinthine patterns are found. By driving the system very hard, a transition to a regime consisting of polarisation domains connected by fronts is also observed. Finally, numerical algorithms appropriate for solving the model equations are discussed and an alternative algorithm is presented which may be of use in pattern-forming systems in general.