BIFURCATION TO STANDING AND TRAVELING WAVES IN LARGE ARRAYS OF COUPLED LASERS

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