  We present a rigorous statistical-mechanics theory of nonlinear many mode
laser systems. An important example is the passively mode-locked laser that
promotes pulse operation when a saturable absorber is placed in the cavity. It
was shown by Gordon and Fischer [1] that pulse formation is a first-order phase
transition of spontaneous ordering of modes in an effective "thermodynamic"
system, in which intracavity noise level is the effective temperature. In this
paper we present a rigorous solution of a model of passive mode locking. We
show that the thermodynamics depends on a single parameter, and calculate
exactly the mode-locking point. We find the phase diagram and calculate
statistical quantities, including the dependence of the intracavity power on
the gain saturation function, and finite size corrections near the transition
point. We show that the thermodynamics is independent of the gain saturation
mechanism and that it is correctly reproduced by a mean field calculation. The
outcome is a new solvable statistical mechanics system with an unstable
self-interaction accompanied by a natural global power constraint, and an exact
description of an important many mode laser system.
