Nonlinear dynamics and Kerr frequency comb formation in lattices of coupled microresonators

ARTICLE https://doi.org/10.1038/s42005-023-01438-z OPEN Nonlinear dynamics and Kerr frequency comb formation in lattices of coupled microresonators 1234567890():,; Aleksandr Tusnin 1,2 ✉, Alexey Tikan 1,2 ✉, Kenichi Komagata 1,3 & Tobias J. Kippenberg 1,2 ✉ Recently, substantial progress has been made in the understanding of microresonators frequency combs based on dissipative Kerr solitons (DKSs). However, most of the studies have focused on the single-resonator level. Coupled resonator systems can open new avenues in dispersion engineering and exhibit unconventional four-wave mixing (FWM) pathways. However, these systems still lack theoretical treatment. Here, starting from general considerations for the N-(spatial) dimensional case, we derive a model for a one-dimensional lattice of microresonators having the form of the two-dimensional Lugiato-Lefever equation (LLE) with a complex dispersion surface. Two fundamentally different dynamical regimes can be identified in this system: elliptic and hyperbolic. Considering both regimes, we investigate Turing patterns, regularized wave collapse, and 2D (i.e., spatio-temporal) DKSs. Extending the system to the Su-Schrieffer-Heeger model, we show that the edge-state dynamics can be approximated by the conventional LLE and demonstrate the edge-bulk interactions initiated by the edge-state DKS. 1 Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland. 2 Center for Quantum Science and Engineering, EPFL, Lausanne, Switzerland. 3Present address: Laboratoire Temps-Fréquence, Avenue de Bellevaux 51, 2000 Neuchâtel, Switzerland. ✉email: aleksandr.tusnin@epfl.ch; alexey.tikan@epfl.ch; tobias.kippenberg@epfl.ch COMMUNICATIONS PHYSICS | (2023)6:317 | https://doi.org/10.1038/s42005-023-01438-z | www.nature.com/commsphys 1 ARTICLE O COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-023-01438-z ver the past decade, it has been shown that continuous wave-driven Kerr nonlinear resonators host a variety of coherent dissipative structures1,2. In the anomalous dispersion regime, they give rise to dissipative Kerr solitons (DKS)3, while in the normal dispersion regime, platicons4,5, or interlocked switching waves, have been generated. These coherent dissipative structures give rise to a wide range of nonlinear dynamical phenomena, ranging from breathers6,7 and soliton switching8 to chaotic behavior9. Mathematically, in leading order, the dynamics can be described by the 1D driven-dissipative nonlinear Schrödinger equation (NLSE)10 known as the Lugiato-Lefever equation (LLE)11,12, and extension thereof, e.g., to include multi-mode dynamics or the Raman nonlinearity13. In this framework, a variety of nonlinear phenomena have been observed4,5,14–18. On the application side, in particular, the DKS formation process has been utilized and has enabled photonic integrated microresonator-based optical frequency comb generation with applications ranging from coherent communications19 and neuromorphic computing20 to atomic clocks21. Yet to date, almost all experimental and theoretical works on ‘dissipative structures’ in optically driven Kerr nonlinear resonators (be it fiber22,23 or microresonator based) have focused on the single resonator case, and only recently extended to the dimer case24–26. The recent advances in ultra-low loss nonlinear integrated platforms, particularly silicon nitride27,28, have dramatically reduced the threshold for optical parametric oscillations and concomitant dissipative structure generation — at and below the milliwatt level. This indicates that large-scale arrays of coupled Kerr nonlinear resonators that combine spatial and synthetic frequency dimensions29 are within experimental reach — yet their nonlinear dynamics under continuous-wave driving remain largely unexplored, both theoretically and experimentally. Such systems are expected to exhibit rich nonlinear dynamics and novel 2D dissipative structures that have combined spatial and temporal dimensions. Even the simple case of a photonic dimer has demonstrated a variety of emergent nonlinear dynamics24,25 and phenomena such as soliton hopping and recurrent dispersive waves. 1D and 2D lattices are particularly attractive as they allow significantly more complex dispersion landscapes — opening ways to engineer dispersion beyond the traditional approaches. Therefore, chains of resonators are expected to provide a pathway to octave-spanning DKS30, which is an enduring outstanding challenge in the field. Such spectra are required for the selfreferencing of micro-combs31. Lattices of nonlinear resonators also allow studying topological systems (e.g., the Su-SchriefferHeeger (SSH) model or honeycomb lattices32,33), which over the past decade have been extensively studied in the linear regime in photonics. Nonlinear effects, and spatial solitons in particular, have already been observed and studied in arrays of coupled optical waveguides34–36. Crucially these nonlinear effects however did not include parametric frequency conversion (i.e., parametric oscillations), which is the underlying physical principle for soliton microcombs. However, a first analysis of DKS formation was carried out in37, where the authors studied Kerr nonlinear version of the photonic 2D Haldane model made of coupled multi-mode optical microresonators with anomalous dispersion that are coupled via link resonators. In this manuscript, we analyze dissipative structures in optically driven coupled chains of nonlinear resonators in the absence and presence of the edge modes. First, we provide a general description of nonlinear dynamics in an arbitrary N-dimensional lattice of resonators and derive an effective (N+1)D coupledmode equation (Fig. 1a, b). Next, we study the 1D chain of equally coupled resonators (see Fig. 1a), providing a leading order model in the form of the 2D continuous-discrete LLE. This model demonstrates a net difference in comparison with its lower 2 dimensional counterpart. In particular, we observe fundamentally distinct nonlinear regimes attributed to the local dispersion profile: elliptic and hyperbolic. Investigating Turing patterns and chaotic states in this system, we observe a drastic difference between the considered regimes that emerge due to the four-wave mixing (FWM) pathways structure mimicking the gain lobes profiles. Performing dynamical simulations, we show the effect of regularized wave collapses and the emergence of solitons that are inherently two-dimensional spatiotemporal mode-locked structures. Furthermore, we study a dimerized chain of coupled resonators described by the simplest topological model, the SSH model (Fig. 1c). We focus on the aspect of DKS generation in the edge state that is localized in the middle of the photonic bandgap. Specifically, we show that edge-state solitons, in the absence of interactions with the bulk, can be approximated by the conventional single-resonator DKS (Fig. 1d). However, (...truncated)