Dynamical properties of lasers coupled face to face

We derive a reduced model to describe two identical lasers coupled face to face. Two limits are introduced in the Maxwell-Bloch equations: adiabatic elimination of the material polarization and large distance between the two lasers. The resulting model describes coupled homogeneously broadened lasers, including semiconductor lasers. It consists of two coupled delay differential equations with delayed linear cross-coupling and an instantaneous self-coupling nonlinearity. The study is analytical and numerical. We focus on the properties of steady and periodic amplitudes of the electric fields. In steady state, there are symmetric, antisymmetric, and asymmetric solutions with respect to a permutation of the two fields. A similar classification holds for the periodic states. The stability of these solutions is determined partly analytically and partly numerically. A homoclinic point is associated with the asymmetric periodic solutions.