Soliton interaction and bound state formation in coupled Kerr resonators

TL;DR

Solitary waves in two weakly coupled Kerr microresonators driven by independent injections are analyzed to understand soliton interactions. The authors derive coupled Lugiato–Lefever equations, perform weak-coupling asymptotics, and validate with numerical simulations to map soliton clusters. They identify three cluster types with fixed separations $ξ_-$ and show how pump phase difference $δ$ and cavity-length mismatch $V$ control binding, motion, and stability, predicting an Arnold tongue structure and bifurcations. The results offer actionable strategies for controlling soliton ensembles in integrated photonics, enabling programmable binding, routing, and synchronization of solitons.

Abstract

Soliton dynamics in coupled Kerr microcavities is an important aspect of frequency comb technologies, with applications in optical communication and precision metrology. We investigate a minimal system consisting of two nearly identical coupled Kerr microresonators, each operating in the soliton regime and driven by a separate coherent beam, and analyze the mechanisms that govern their soliton interactions. In the weak-coupling regime, the system supports multiple soliton clusters characterized by distinct soliton separations and stability. Numerical simulations indicate that asymmetric perturbations can alter soliton separations or destroy these states, while the imposed pump phase difference plays a key role in cluster selection. Together, these findings highlight previously unexplored regimes of dissipative soliton organization and suggest new strategies for controlling soliton ensembles in integrated photonic platforms.

Figures (13)

• Figure 1: a) Quantities $C_+$ (dotted line) and $C_-$ (solid line) as functions of the variable $\xi$. b) A magnified view of the neighborhood of zero from panel (a). Parameter values: $\theta = 3.5$, $\eta_1=\eta_2 = 1.9$, $\delta = 0$, $b = 1$, $L = 20.0$.
• Figure 2: Amplitude of the first and second solitons, $|A_1|$ and $|A_2|$, corresponding to stable states obtained via numerical integration of the coupled LLEs. (a) Bound state of moving solitons at $\delta = 2.64$, $V = 0$, $\eta_1=\eta_2=1.9$, and $L=20.0$. (b) Soliton anti-bunching at $\delta = 0$, $V = 0$, $\eta_1=\eta_2=1.9$, and $L=12.0$. (c) Same as (a) but with different injection rates $\eta_1=2.0$ and $\eta_2=1.8$. (d) Soliton bunching at $\delta = 0$, $V = 0.2$, and $L=20.0$.Other parameters: $\theta = 3.5$, $\eta_1=\eta_2 = 1.9$, $b = 1$.
• Figure 3: a) Total intensity ($|A_1|^2 + |A_2|^2$) of CW solutions as a function of the coupling coefficient $\kappa$. Symmetric solutions ($|A_1|^2 = |A_2|^2$) are shown in blue, and asymmetric solutions ($|A_1|^2 \neq |A_2|^2$) in red. Solid lines represent stable states; dashed lines indicate unstable ones. The points labeled 'P' and 'SN' mark the pitchfork and saddle-node bifurcations, respectively. 'M' denotes the onset of modulational instability. b) Real parts of the dominant eigenvalues from the linear stability spectrum of a modulationally unstable asymmetric state at $\kappa = 1.5$. c) Same as in (b), but for a modulationally unstable symmetric state at $\kappa = 3$. Parameters: $\theta = 3.5$, $\eta_1=\eta_2 = 1.9$, $V = 0$, $\delta = 0$, and $b = 1$.
• Figure 4: a) Total intensity ($|A_1|^2 + |A_2|^2$) of CW solutions as a function of the coupling coefficient $\kappa$ for the case of normal dispersion. Parameters are the same as in Fig. \ref{['fig:uniform']}(a), except with $b = -1$.
• Figure 5: Shaded area indicates the parameter region where bunching soliton clusters are stable. Parameters: $\theta = 3.5$, $\eta_1=\eta_2 = 1.9$, $V = 0$, $b = 1$.