Shear-induced bifurcations and chaos in models of three coupled lasers

Abstract

We study nonlinear dynamics in a linear array of three coupled laser oscillators with rotational $\mathbb{S}^1$ and reflectional $\mathbb{Z}_2$ symmetry. The focus is on a coupled-laser model with dependence on three parameters: laser coupling strength, $\kappa$, laser frequency detuning, $\Delta$, and degree of coupling between the amplitude and phase of the laser, $\alpha$, also known as shear or nonisochronicity. Numerical bifurcation analysis is used in conjunction with Lyapunov exponent calculations to study the different aspects of the system dynamics. First, the shape and extent of regions with stable phase locking in the $(\kappa,\Delta)$ plane change drastically with $\alpha$. We explain these changes in terms of codimension-two and -three bifurcations of (relative) equilibria. Furthermore, we identify locking-unlocking transitions due to global homoclinic and heteroclinic bifurcations and the associated infinite cascades of local bifurcations. Second, vast regions of deterministic chaos emerge in the $(\kappa,\Delta)$ plane for nonzero $\alpha$. We give an intuitive explanation of this effect in terms of $\alpha$-induced stretch-and-fold action that creates horseshoes and discuss chaotic attractors with different topologies. Similar analysis of a more accurate composite-cavity mode model reveals good agreement with the coupled-laser model on the level of local and global bifurcations as well as chaotic dynamics, provided that coupling between lasers is not too strong. The results give new insight into modeling approaches and methodologies for studying nonlinear behavior of laser arrays.