Nonlinear periodic orbit solutions and their bifurcation structure at the origin of soliton hopping in coupled microresonators

Microresonator frequency combs, essential for future integrated optical systems, rely on dissipative Kerr solitons generated in a single microresonator to achieve coherent frequency comb generation. Recent advances in the nanofabrication of low-loss integrated nonlinear microresonators have paved the way for the exploration of coupled-resonator systems. These systems provide significant technological advantages, including higher conversion efficiency and the generation of dual dispersive waves. Beyond their practical benefits, coupled-resonator systems also reveal novel emergent nonlinear phenomena, such as soliton hopping, a dynamic process in which solitons periodically transfer between coupled resonators. In this study, we employ a dynamical system approach and the corresponding well-established numerical techniques, extensively developed within the context of hydrodynamics and transitional turbulence, to investigate the bifurcation structure of periodic orbit solutions of the coupled Lugiato-Lefever equations that underlie soliton hopping in photonic dimers and trimers. Our main finding uncovers a fundamental difference in the origin of the hopping process in dimers and trimers. We demonstrate that in dimers, hopping emerges from a branch of stable soliton solutions, whereas in trimers, it originates from an unstable branch. This distinction leads to a significant difference in pump power requirements. We relate the bifurcation structure of the periodic orbits including their stability to the observed dynamics in simulated laser scans mimicking typical experimental investigations. Subcritical Hopf bifurcations of unstable equilibrium branches specifically explain observed hysteresis, the coexistence of multiple attractors at the same parameter values, and the importance of choosing a specific path in parameter space to reliably achieve a desired dynamical regime.