Exact solution of the nonlinear laser passive mode locking transition

We present the first statistical mechanics study of a passively mode locked laser which includes all the main physical processes, saturable absorption, Kerr nonlinearity, parabolic gain filtering and group velocity dispersion, assuming the soliton condition. We achieve an exact solution in the thermodynamic limit, where the ratio of the cavity length to the pulse width, the duty cycle, tends to infinity. The thermodynamics depends on a single dimensionless parameter $\gamma$, the ratio of the correlation length to the pulse width. The phase diagram consists of one ordered, mode-locked phase and one disordered, continuous wave phase, separated by a first order phase transition at $\gamma=9$. The model belongs to a new class of solvable statistical mechanics models with a non-trivial phase diagram. The results are obtained with a fully controlled transfer matrix calculation, showing rigorously that passive mode locking is a thermodynamic phase transition.