Spontaneous pattern formation in the interaction of two counter-propagative waves in a high-aspect-ratio single-longitudinal-mode laser

A single-longitudinal-mode laser in which the transverse plane is much longer than wider is investigated. This system is described by the Maxwell–Bloch equations, and has traveling-wave solutions. By means of the reductive perturbation method, evolution equations for the envelope of the traveling waves are derived near the laser threshold. Two nonlinear equations for the amplitude of the two traveling waves propagating along the x transverse coordinate in opposite directions are obtained, and solved numerically. Several stable patterns, periodic in space and possibly in time are found, starting from initial conditions with different spatial profiles. The stability of the patterns depends on the orientation with respect to the transverse x -coordinate.