EXACT STATIONARY SOLUTION OF A FOKKER-PLANCK EQUATION FOR MULTIMODE LASER ACTION INCLUDING PHASE LOCKING

We first derive the exact stationary solution of a Fokker-Planck equation where the complex drift coefficients are nonlinear functions of the variables, provided the drift and diffusion coefficients fulfill certain conditions. Then we apply the solution to 1. normal multimode action where no phase locking occurs at all. 2. phase locking in a laser with many axial modes having a narrow frequency spacing. The atomic line which supports laser action may be homogeneously or inhomogeneously broadened. In case 1 the modes may be completely arbitrary, i.e. for instance running or standing waves. In case 2 we assume axial modes described by running waves. The treatment is valid in a region not too far below and not too far above laser threshold where the atomic variables adiabatically follow the motion of the lightfield variables. The drift coefficients are taken from the multimode Langevin equations of Haken and Sauermann. The diffusion coefficients are taken from a paper of Arzt et al. The only essential assumption is that the diffusion coefficients may be considered constant over the frequency range where modes participate in the laser process. If our results are specialized to single mode action we obtain Risken's solution. © 1969 Springer-Verlag.