Reversible Self-Replication of Spatiotemporal Kerr Cavity Patterns

We uncover a novel and robust phenomenon that causes the gradual self-replication of spatiotemporal Kerr cavity patterns in cylindrical microresonators. These patterns are inherently synchronized multifrequency combs. Under proper conditions, the axially localized nature of the patterns leads to a fundamental drift instability that induces transitions among patterns with a different number of rows. Self-replications, thus, result in the stepwise addition or removal of individual combs along the cylinder’s axis. Transitions occur in a fully reversible and, consequently, deterministic way. The phenomenon puts forward a novel paradigm for Kerr frequency comb formation and reveals important insights into the physics of multidimensional nonlinear patterns.

Figure 1

(a) Sketch of the driven microcylinder. (b),(c) Snaking diagrams of pattern families $P_{\mathrm{odd}}$ and $P_{\mathrm{even}}$ on the $(\delta ,U)$ plane for ${\sigma }_z=15$ and ${\sigma }_z=55$, respectively. Thick traces denote stability. Dots in (c) correspond to the profiles $|\psi (x,z)|$, shown in insets (i)–(viii). Panels’ areas span over $x\in [-16,16)$ (full circumference, $2\pi R$), $z\in [-20,20]$. Here, $h_0=\xi (x)=1$. Scaling: $\mathrm{\Delta }\delta =0.1\iff 15.6\mathrm{MHz}$, ${\sigma }_z=10\iff 4.9\mathrm{mm}$, $\mathrm{\Delta }z=40\iff 1.96\mathrm{cm}$, and $h=1\iff 46\mathrm{mW}$/mm.

Figure 2

Existence and stability charts for (a) $P_{\mathrm{odd}}$ and (b) $P_{\mathrm{even}}$ on the $(\delta ,{\sigma }_z)$ plane [$h_0=1$, $\xi (x)=1$]. Patterns are stable (unstable) within the black (colored) areas. Domains for extended hexagonal pattern (${\sigma }_z\to \infty$) are also shown. The red contour encloses the MI area. (c) Drift eigenvalue vs ${\sigma }_z$ at $\delta =0.84$ for the patterns in (a) and (b). Areas in (c) [color matched with (a) and (b)] mark the drift bands for $P_{\mathrm{odd}}$ (purple) and $P_{\mathrm{even}}$ (gray). Scaling: $\mathrm{\Delta }\delta =0.1\iff 15.6\mathrm{MHz}$ and ${\sigma }_z=10\iff 4.9\mathrm{mm}$.

Figure 3

Pattern self-replication and self-erasure under the two-dimensional localization of the pump, $h_0\approx 9$, ${\sigma }_x=2$, and varying ${\sigma }_z(t)$ with $\delta =0.84$. (a),(b) Evolution of ${\sigma }_z$ and center of mass. Dots in (a) and (b) mark the entrance into the drift-instability regions for $P_{\mathrm{odd}}$ (purple) and $P_{\mathrm{even}}$ (gray). The inset in (a) shows the drift bands for $\xi =1$ [cf. Fig. 2]. The dashed green rectangle enlarges the $P_2\to P_3$ transition from (b), while distributions $|\psi (x,z){|}^2$ correspond to labels (i)–(iii). Axes of panels (i)–(iii): $x\in [-16,16)$ (full circumference), $z\in [-12,12]$. Scaling: ${\sigma }_z=10\iff 4.9\mathrm{mm}$, $⟨z⟩=1\iff 490\mu \mathrm{m}$, $t=1\iff {10}^3$ round-trips, $\tau \approx 6.4\mathrm{ps}$, total propagated time $\sim 2.9\mathrm{ms}$, $\mathrm{\Delta }z=24\iff 1.2\mathrm{cm}$, ${\sigma }_x=2\iff \pi R/8$, and $h_0=9\iff 3$ W/mm.

Figure 4

Robust multicomb spectra for (a) $P_1$, (b) $P_3$, and (c) $P_8$ (${\sigma }_z$ in labels) for the same parameters as Fig. 3. (d)–(f) Spectra at the selected $z$ positions (white arrows) in (a)–(c), respectively. All power levels are in the range $[-150,0]\mathrm{dB}$. The background field ${\psi }_0(x,z)$ was subtracted prior to Fourier transforms. Scaling: $\mathrm{\Delta }{\kappa }_x=1\iff 1/40\mu {\mathrm{m}}^{-1}$. At ${\lambda }_p=1.55\mu \mathrm{m}$, $\mathrm{\Delta }{\kappa }_x=48\iff {\lambda }_0\in [1.35,1.82]\mu \mathrm{m}$. $\mathrm{\Delta }z=40\iff 1.96\mathrm{cm}$.