Hard and soft excitation regimes of Kerr frequency combs

We theoretically study the stability conditions and excitation regimes of hyperparametric oscillation and Kerr frequency comb generation in continuously pumped nonlinear optical microresonators possessing an anomalous group velocity dispersion. We show that both hard and soft excitation regimes are possible in the microresonators. Selection between the regimes is achieved via change in the parameters of the pumping light.

Figure 1

Normalized amplitude of the light within the optically pumped mode and relative amplitude of the oscillation sidebands vs pumping frequency. Stability regions corresponding to soft (S) and hard (H) excitation are shown by solid green lines. Unstable solutions are depicted by solid red lines. The amplitude of the field within the optically pumped mode with no sidebands generated ($B\equiv 0$) is shown by the solid blue line and dashed (red) line. The dashed line represents the unstable solution for the amplitude of the pumped mode when the modulation sidebands are absent.

Figure 2

Normalized amplitude of the light within the optically pumped mode and the relative amplitude of the oscillation sidebands vs normalized amplitude of the pumping light. The detuning of the pumping light from the corresponding optical mode is fixed at $\omega ={\omega }_0-4.7{\gamma }_0$ (the mode is pumped at its red wing). The dashed (blue) line represents the undisturbed solution for the light accumulated in the pumped mode, $B\equiv 0$. Red and green solid lines describe unstable and stable-steady state solutions respectively.

Figure 3

Normalized amplitude of the optically pumped mode and relative amplitude of the oscillation sidebands vs pumping frequency found for the same conditions as used in Fig. 1. Frequency combs shown were generated when the parameters of the system corresponding to points H and S were selected.

Figure 4

Normalized amplitude of the light within the optically pumped mode and relative amplitude of the oscillation sidebands vs normalized amplitude of the pumping light. The detuning is (a, b) $\omega ={\omega }_0-15{\gamma }_0$ (hard excitation) and (c, d) $\omega ={\omega }_0+0.92{\gamma }_0$ (soft excitation). The power for the first sideband is taken from the carrier.

Figure 5

Phase diagram for the pump and the first oscillation sideband in the case of soft excitation. The dashed (blue) line corresponds to nonzero initial conditions, and the solid (red) line to zero initial conditions. The system converges to the same steady-state solution with different phases of the oscillation sideband.

Figure 6

Phase diagram for the pump and the first oscillation sideband in the case of hard excitation. The system does not oscillate if the initial field in the pumped mode is 0 (red line). The oscillation is excited if the initial value of the pump is not zero.

Figure 7

Temporal behavior of the normalized optical field within the resonator. Subpicosecond optical pulses are formed. Inset: Optical pulse envelope found from the analytical expression presented in [15].