Classification of Codimension-One Bifurcations in a Symmetric Laser System

Abstract

We consider a class symmetric laser system, particularly rings of n identical semiconductor lasers coupled bidirectionally. The model is described using the Lang-Kobayashi rate equations where the finite propagation time of the light from one laser to another is reflected by a constant delay time parameter in the laser optical fields. Due to the network structure, the resulting system of delay differential equations has symmetry group $\mathbb {D}_n\times \textbf {S}^1$. We follow the discussions in Buono and Collera (SIAM J Appl Dyn Syst 14:1868–1898, 2015) for the general case, then give the important example where n = 4 to explicitly illustrate the general method which is a classification of codimension-one bifurcations into regular and symmetry-breaking. This particular example complements the recent works on rings with unidirectional coupling (Domogo and Collera, AIP Conf Proc 1787:080002, 2016), and on laser networks with all-to-all coupling (Collera, Mathematical and computational approaches in advancing modern science and engineering. Springer, Cham, 2016). Numerical continuation using DDE-Biftool are carried out to corroborate our classification results.