Stability analysis of the mode-locking dynamics in a laser cavity with a passive polarizer

A low-dimensional model is constructed via a variational formulation that characterizes the mode-locking dynamics in a laser cavity with a passive polarizer. The theoretical model accounts explicitly for the effects of the passive polarizer with a Jones matrix. In combination with the nonlinear interaction of the orthogonally polarized electromagnetic fields, the evolution of the mode-locked state reduces to the nonlinear interaction of the amplitude, width, and phase chirp. This model allows for an explicit analytic prediction of the steady-state mode-locked state (fixed point) and its corresponding stability. The stability analysis requires a center manifold reduction, which reveals that the solution decays to the mode-locked state on a timescale dependent on the gain bandwidth and the net cavity gain. Quantitative and qualitative agreement is achieved between the full governing model and the low-dimensional model, thus providing for an excellent design tool for characterizing and optimizing mode-locking performance.