Symmetry-Breaking Bifurcations in Rings of Delay-Coupled Semiconductor Lasers

Abstract

We consider symmetric rings of delay-coupled lasers modeled using the Lang--Kobayashi (LK) rate equations with unidirectional and bidirectional coupling. Because of phase symmetry the networks have symmetry groups $\mathbb{Z}_n\times\mathbf{S}^1$ (unidirectional) and $\mathbf{D}_n\times\mathbf{S}^1$ (bidirectional). Our first main result is a characterization of isotropy subgroups of those actions from which we determine sufficient conditions for the existence of basic classes of compound laser modes (CLMs) valid for all ring sizes. Case studies of the $n=3$ and $n=8$ coupled LK equations are presented, including branches of CLMs obtained via DDE-Biftool. Using the block diagonalization of the linearization coming from the isotypic decomposition we classify the symmetry-breaking type of steady-state and Hopf bifurcation points at fully symmetric CLMs and in the case $n=3$ obtain explicit location of bifurcation points. We complement the study using DDE-Biftool to perform branch continuation at steady-stat...