Dynamical transverse laser patterns. I. Theory

We consider a cylindrically symmetrical laser with spherical mirrors and describe the dynamics in terms of the competition among different Gauss-Laguerre modes of the cavity. In this paper we focus on the case in which the mode competition leads the laser to a dynamical state that, according to the values of the control parameters, can be periodic or quasiperiodic. The linear stability analysis of the single-mode stationary solutions, in which the laser oscillates with the fundamental ${\mathrm{TEM}}_{00}$ or the ${\mathrm{TEM}}_{01}^{\mathrm{*}}$ mode, provides an initial guideline in our search for the various spatiotemporal patterns which emerge. We consider cases in which the gain line activates one, two, or three frequency-degenerate families of modes. The motion of optical vortices, from the simple rotation to creation and annihilation in pairs is analyzed, together with the correlated movement of the peaks of the intensity distribution in the traverse plane. We study also the patterns which appear when the cylindrical symmetry of the system is broken. The parameters of our calculations correspond closely to those which characterize ${\mathrm{Na}}_2$ lasers, ${\mathrm{CO}}_2$ lasers, and Nd-doped yttrium aluminum garnet lasers.