Stability analysis of the self-phase-locked divide-by-2 optical parametric oscillator

The properties of all-optical phase-coherent frequency division by 2, based on a self-phase-locked continuous-wave (cw) optical parametric oscillator (OPO), are investigated theoretically. The coupled field equations of an OPO with intracavity quarter-wave plate are solved analytically in steady-state, yielding a condition for self-phase-locked operation. In the self-phase-locked state, two different values for the pump power at threshold are obtained. By using a linear stability analysis, it is proven that only the lower threshold value is stable, whereas the higher threshold value is unstable. The analytical investigations of the steady-state field values further reveal a twofold symmetry in phase space. The theoretical consideration is completed by a numerical analysis based on the integration of the envelopes of the three OPO fields, which allows for studying the temporal evolution of different initial values. The numerical investigation of the OPO subharmonic phases shows that the two-phase eigenstates are equivalent with respect to experimental parameters and are assumed by the self-phase-locked OPO in dependence of the initial phases of the subharmonic fields, dividing phase space into two symmetric basins of attraction