Threshold of complexity and Arnold tongues in Kerr-ring microresonators

We show that the threshold condition for two pump photons to convert into a pair of the sideband ones in Kerr microresonators with high-quality factors breaks the pump laser parameter space into a sequence of narrow in frequency and broad in power Arnold tongues. Instability tongues become a dominant feature in resonators with the finesse dispersion parameter close to and above one. As pump power is increased, the tongues expand and cross by forming a line of cusps, i.e., the threshold of complexity, where more sideband pairs become unstable. We elaborate theory for the tongues and threshold of complexity, and report the synchronisation and frequency-domain symmetry breaking effects inside the tongues.

Figure 1

(a),(b) Arnold tongues and interplay of the FWM, $\mathcal{W}={\mathcal{W}}_{\mu F}$, and complexity, $\mathcal{W}={\mathcal{W}}_{\mu C}$, thresholds for the quasi-continuous ($|{\mathcal{F}}_d|=0.005, Q={10}^9$) and sparse ($|{\mathcal{F}}_d|=0.5, Q={10}^{11}$) residual spectra; $|D_2|/2\pi =1\mathrm{kHz}, \eta =0.5$, residual finesse ${\mathcal{F}}_d=D_2/\kappa =D_{2}Q/{\omega }_0$. The color scheme shows the number of the simultaneously unstable $\pm \mu$ sideband pairs. Lines dropping below $\mathcal{W}={\mathcal{W}}_{\mu F}$ and reaching $\mathcal{W}=0$ show a pair of the synchronization tongues. (c),(d) Resonances in the residual spectrum for small (c) and large (d) $|{\mathcal{F}}_d|$. Plots show Lorentzian lines, $1/(1+4{\mathrm{\Delta }}_{\mu }^2/{\kappa }^2)$, for $\mu$ starting from 0. (e) Distribution of the dynamical regimes across the $\left|\mu \right|=35$ instability tongue and its neighbourhood for ${\mathcal{F}}_d=0.5$. Black color corresponds to the repetition rate locking to $\left|\mu \right|D_1$, see Fig. 2. Yellow marks the unlocked regimes with the $|d_{\mu }|=\mathrm{const}$ symmetry breaking, see Fig. 2. Grey is the weak chaos with the symmetry breaking, see Fig. 2. Blue is the multimode complexity with dense continuous spectra, see Fig. 2. The intra-resonator cw power at the tongue minima $g_{35}^{\left(2\right)}/\gamma =\kappa /2\gamma =0.1$ W (FWM threshold), and $g_{35}^{\left(1\right)}/\gamma =g_{36}^{\left(2\right)}/\gamma \simeq {\alpha }_{35}\kappa /2\gamma =1.75$ W at the threshold of complexity.

Figure 2

(a) Repetition rate locking [$d_{\mu }=0$, black area in Fig. 1]; (b) Frequency-domain symmetry breaking [$d_{\mu }=\text{const}$, yellow area in Fig. 1]; (c) Few-mode chaos [chaotic variations of $d_{\mu }$, grey area in Fig. 1]. (d) Multimode complexity [blue area in Fig. 1]. Top two rows show the $\mu =\pm 35$ sideband RF spectra $|S_{\mu }{\left(\mathrm{\Delta }\right)|}^2$ [37]. The third row shows the mode powers at $t=16\mathrm{ms}$. The fourth row is the space-time dynamics.