RESONANT QUASI-PERIODIC PATTERNS IN A 3-DIMENSIONAL LASING MEDIUM

Starting from the Maxwell-Bloch equations for a three-dimensional (3D) ring-cavity laser, we analyze stability of the nonlasing state and de monstrate that, at the instability threshold, the wave vectors of the critical perturbations belong to a paraboloid in the 3D space. Then, w e derive a system of nonlinear evolution equations above the threshold . The nonlinearity in these equations is cubic. For certain sets of fo ur spatial modes whose vectors belong to the critical paraboloid, the cubic nonlinearity gives rise to a resonant coupling between them. Thi s is a nontrivial example of a nonlinear dissipative system in which t he cubic terms are resonant. The equations for the four coupled amplit udes have two different solutions that are simultaneously stable: the single-mode one and a solution in which all the amplitudes are equal, while a certain combination of the phases is pi. The latter solution g ives rise to a quasiperiodic pattern in the infinite 3D cavity. We als o consider effects of the boundary conditions and demonstrate that if the cavity's cross section is a trapezium it may support the quasiperi odic four-mode state rather than suppressing it. Using the Lyapunov fu nction, we find that for the ring-laser configuration the four-mode st ate is metastable. However, we demonstrate that for a Fabry-Perot cavi ty, where diffusion washes out the standing-wave grating, this state i s absolutely stable. We also consider a number of more complicated pat terns. We demonstrate that adding a pair of resonant vectors, or any n umber of nonresonant ones, always produces an unstable solution. A set containing several resonant quarters without resonant coupling betwee n them may be stable, but it is less energetically favorable than a si ngle quartet.