Anatomy of Phase Locking in Hyperparametric Oscillations Based on Kerr Nonlinearity

We investigate the dynamical origin of synchronization and phase locking of hyperparametric oscillations in Kerr-nonlinear media. These oscillations occur in the presence of parametric gain and, although arising from modulational instability of random vacuum fluctuations with arbitrary phases, lead to phase-locked states in the form of pulse trains. Using few-mode approximations of the Lugiato-Lefever equation (LLE), we find that the pumped mode injection-locks to the driving laser pump following the Adler equation. Based on experimentally motivated assumptions, we derive analytical expressions, which reveal the essence of phase locking in frequency combs and confirm them through numerical integration of the LLE. Clear understanding of the phenomenon of phase locking in optical microresonators can lead to devising novel techniques for achieving phase-locked states or improving the coherence properties of frequency combs. Our results are mathematically generic and apply to other systems described by an externally driven damped nonlinear Schödinger equation.