Bifurcation to standing and traveling waves in large arrays of coupled lasers

Abstract

We consider the equations for N coupled class-B lasers in a ring geometry. We investigate the first bifurcation to time-periodic solutions in the limit of (i) small damping, (ii) small coupling, and (iii) large N. We identify the relevant scaling between these three quantities from the linear stability analysis and derive a nonlinear partial differential equation for the slow time and slow space evolution of the time-periodic traveling-wave modes. The equation is similar but not identical to the Ginzburg-Landau equation derived in the areas of fluid or chemical instabilities. It contains an additional term and the boundary conditions are different if N is even or odd. We next determine solutions of this equation. If N is even, the first bifurcation corresponds to a time-periodic standing wave and its amplitude is identical to the amplitude previously obtained for N even but arbitrary [Li and Erneux, Phys. Rev. A 46, 4252 (1992)]. If N is odd, the bifurcation diagram is quite different. We find two primary bifurcations to traveling-wave solutions and one secondary bifurcation to a standing-wave solution. Our analytical results are in agreement with a detailed numerical study of the original laser equations.