Symmetry-Breaking Bifurcations in Laser Systems with All-to-All Coupling
We consider a system of n semiconductor lasers with all-to-all coupling that is described using the Lang-Kobayashi rate equations. The lasers are coupled through their optical fields with delay arising from the finite propagation time of the light from one laser to another. As a consequence of the coupling structure, the resulting system of delay differential equations is equivariant under the symmetry group S n ×S1. Since symmetry gives rise to eigenvalues of higher multiplicity, implementing a numerical bifurcation analysis to our laser system is not straightforward. Our results include the use of the equivariance property of the laser system to find symmetric solutions, and to correctly locate steady-state and Hopf bifurcations. Additionally, this method identifies symmetry-breaking bifurcations where new branches of solutions emerge.
This work was funded by the UP System Emerging Interdisciplinary Research Program (OVPAA-EIDR-C03-011). The author also acknowledged the support of the UP Baguio through RLCs during the A.Y. 2014–2015.
- 3.Engelborghs, K., Luzyanina, T., Samaey, G.: DDE-BIFTOOL v. 2.00: A MATLAB package for bifurcation analysis of delay differential equations. Technical report TW-330. Department of Computer Science, K.U. Leuven, Leuven (2001)Google Scholar
- 8.Gray, G.R., Ryan, A.T., Agrawal, G.P., Gage, E.C.: Optical-feedback-induced chaos and its control in semiconductor lasers. In: SPIE’s 1993 International Symposium on Optics, Imaging, and Instrumentation, San Diego, pp. 45–57. International Society for Optics and Photonics (1993)Google Scholar
- 13.Verduyn Lunel, S.M., Krauskopf, B.: The mathematics of delay equations with an application to the Lang-Kobayashi equation. In: Krauskopf, B., Lenstra, D. (eds.) Fundamental Issues of Nonlinear Laser Dynamics. AIP Conference Proceedings, vol. 548. American Institute of Physics, New York (2000)Google Scholar