Self-organization in Kerr-cavity-soliton formation in parametric frequency combs

We show that self-organization and synchronization underlie Kerr-cavity-soliton formation in parametric frequency combs. By reducing the Lugiato-Lefever equation to a set of phase equations, we find that self-organization arises from a two-stage process via pump-degenerate and pump-nondegenerate four-wave mixing. The reduced phase equations are akin to the Kuramoto model of coupled oscillators and intuitively explain the origin of the pump phase offset, predict antisymmetrization of the intracavity field before phase synchronization, and clarify the role of chaos in Kerr-cavity-soliton formation in parametric combs.

Figure 1

(a) Synchronization of oscillators with distinct natural frequencies to a phase-locked state with a single frequency. A large fraction of the oscillator phases lock to $\psi$, the average phase. (b) Synchronization transition of the coherence $R$. (c) Broadband frequency comb generation via continuous-wave pumped four-wave mixing in silicon nitride microrings. (d) Mode locking of the cavity modes results in equidistant frequency spacings.

Figure 2

Scheme for (a) pump-degenerate and (b) pump-nondegenerate four-wave mixing. The amplitude of the central pump mode is $\rho$ times stronger than the nonpump modes. (c) Energy diagrams for the PD-FWM, PND-FWM, and nondegenerate (ND)-FWM processes.

Figure 3

(a),(b) Three stages of evolution of the intracavity field predicted by (a) the KPE and (b) the LLE. Solid gray curves represent the initial random phase profile. Dotted red curves show the phase evolution after 370 (KPE) or 308 (LLE) round trips; both phase profiles depict the antisymmetrization of the spectral phase due to the pump degenerate FWM processes. Black curves show the final phase profile after 3394 round trips (both models). The phases have become synchronized by the pump nondegenerate terms. Note the slight offset of the pump phase from the phases of the other cavity modes. (c),(d) The final spectral phase profiles of the KPE and LLE systems deviate from a pure linear profile, including the pump phase offsets. (Insets) Temporal pulse shapes of the KPE and LLE.

Figure 4

Coherence $R$ (red-dotted) and normalized average phase difference ${\theta }_o$ (green) for the (a) Kerr phase equations (KPE) and the (b) Lugiato-Lefever equation (LLE). Both systems show an abrupt transition to a stable phase-synchronized state. Coherence $R$ (red-dotted) and phase symmetry $R_{\mathrm{asym}}$ (blue) for the (c) KPE and (d) LLE.

Figure 5

Time evolution of the coherence for the KPE for three values of $N_o=140$, 170, and 200. The equations are only stable for $185>N_o>155$, which is consistent with the soliton condition of the LLE.