Spontaneous symmetry breaking and trapping of temporal Kerr cavity solitons by pulsed or amplitude-modulated driving fields

We report on a systematic study of temporal Kerr cavity soliton dynamics in the presence of pulsed or amplitude-modulated driving fields. In stark contrast to the more extensively studied case of phase modulations, we find that Kerr cavity solitons are not always attracted to maxima or minima of driving field amplitude inhomogeneities. Instead, we find that the solitons are attracted to temporal positions associated with specific driving field values that depend only on the cavity detuning. We describe our findings in light of a spontaneous symmetry breaking instability that physically ensues from a competition between coherent driving and nonlinear propagation effects. In addition to identifying a previously unfamiliar type of Kerr cavity soliton behavior, our results provide valuable insights into practical cavity configurations employing pulsed or amplitude-modulated driving fields.

Figure 1

(a) and (b) Steady-state intracavity field solutions (red curves) for peak driving amplitudes (a) $S_0=1.9$ and (b) $S_0=2.3$. Gray dashed curves show the corresponding Gaussian driving field profiles. (c) and (d) Dynamical intracavity field evolutions corresponding to (a) and (b), respectively. The initial soliton position ${\tau }_0=5$. The dashed vertical magenta line highlights the position of the maximum driving field amplitude ($\tau =0$). Note the different $x$ axes in (c) and (d).

Figure 2

(a) Driving field amplitude at the steady-state CS position $[S_{\mathrm{t}}=S\left({\tau }_{\mathrm{CS}}\right)]$ as a function of the peak driving amplitude $S_0$. (b)–(d) Red curves show steady-state intracavity field profiles for three different driving amplitudes: (b) $S_0=1.9$, (c) $S_0=2.1$, and (d) $S_0=2.5$. Gray dashed curves show the corresponding Gaussian driving field profiles $S\left(\tau \right)$. The dash-dotted horizontal blue line indicates the critical driving value $S_{\mathrm{c}}=1.98$. A detuning $\mathrm{\Delta }=4$ was used in all calculations.

Figure 3

Steady-state field profiles (red curves) for a cosinusoidal driving field with modulation frequencies (a) $\omega =0.05$, (b) $\omega =0.1$, and (c) $\omega =0.2$. The CSs were initially excited at (a) ${\tau }_0=-1$, (b) ${\tau }_0=1$, and (c) ${\tau }_0=-1$. Gray dashed curves show the corresponding driving field profiles, while the dash-dotted horizontal blue curve highlights $S_{\mathrm{c}}=1.98$. All calculations use $S_0=2.3$ and $\mathrm{\Delta }=4$. Note the different fast time axes in (a)–(c).

Figure 4

Critical driving field values $S_{\mathrm{c}}$ as a function of detuning. Red solid circles correspond to values extracted from numerical simulations, while the solid black curve highlights the minimum driving amplitude needed for CS existence: $S_{\min }={(8\mathrm{\Delta }/{\pi }^2)}^{1/2}$. The dashed black curve highlights the up-switching point $S_{\uparrow }={[2/27({\mathrm{\Delta }}^3+9\mathrm{\Delta }+\sqrt{{\mathrm{\Delta }}^2-3})]}^{1/2}$, above which the homogeneous response of the LLE is monostable. Cavity solitons can exist between the dashed and solid curves.

Figure 5

Cavity soliton symmetry breaking bifurcation curve for a Gaussian driving field with ${\tau }_{\mathrm{G}}=20$ and $\mathrm{\Delta }=4$. Blue and red curves show the steady-state positions of the CS solutions as a function of the maximum driving amplitude for symmetric and asymmetric states, respectively, with the dashed part being unstable. The dash-dotted vertical line indicates the critical driving value $S_{\mathrm{c}}$.

Figure 6

Drift coefficient $a(S_{\mathrm{H}},\mathrm{\Delta })$ calculated from the neutral mode of a steady-state CS solution for a range of homogeneous driving strengths $S_{\mathrm{H}}$ and cavity detunings $\mathrm{\Delta }$. The solid black curves correspond to $S_{\min }$ and $S_{\uparrow }$ as defined in the caption of Fig. 4; CSs only exist between these two curves. The black dashed curve shows the critical driving field values $S_{\mathrm{c}}$ obtained from direct numerical simulations of the LLE with a Gaussian driving field (the same data are shown in Fig. 4). The inset shows the curve $a(S_{\mathrm{H}},\mathrm{\Delta }=4)$.

Figure 7

Simulation results from an Ikeda-like cavity map. (a) Steady-state intracavity field profile (red curve) for a Gaussian driving field (gray dashed curve). (b) Evolution of the CS's center of mass over four cavity transits.