Dissipative Kerr soliton frequency combs generated in high-Q microresonators may unlock novel perspectives in a variety of applications and crucially rely on quantitative models for systematic device design. Here, we present a global bifurcation study of the Lugiato-Lefever equation which describes Kerr comb formation. Our study allows systematic investigation of stationary comb states over a wide range of technically relevant parameters. Quantifying key performance parameters of bright and dark-soliton combs, our findings may serve as a design guideline for Kerr comb generators.
Kerr frequency comb generation relies on dedicated waveguide platforms that are optimized toward ultralow loss while offering comparatively limited functionality restricted to passive building blocks. In contrast to that, the silicon-photonic platform offers a highly developed portfolio of high-performance devices, but suffers from strong two-photon absorption (TPA) and subsequent free-carrier absorption (FCA) at near-infrared telecommunication wavelengths, thereby rendering Kerr comb generation a challenge. Here we present a model to investigate the impact of TPA and FCA on Kerr comb formation. Our model combines a modified version of the Lugiato-Lefever equation with a refined relation to precisely describe the fast space and time dependence of the free-carrier concentration along the circumference of the microresonator. Using this refined model, we derive conditions for modulation instability, in particular for necessary pump powers depending on TPA parameters and free-carrier lifetimes. We validate our analytical predictions by time integration and study the impact of fast free-carrier dynamics on Kerr comb formation. We find that silicon microresonators may be suitable for Kerr comb generation in the NIR, provided that the dwell time of the TPA-generated free carriers in the waveguide core is reduced by a reverse-biased p-i-n-junction and that the pump parameters are chosen appropriately.
In this paper we investigate the effect of nonlinear damping on the Lugiato-Lefever equation|a| 2 a + if on the torus or the real line. For the case of the torus it is shown that for small nonlinear damping κ > 0 stationary spatially periodic solutions exist on branches that bifurcate from constant solutions whereas all nonconstant solutions disappear when the damping parameter κ exceeds a critical value. These results apply both for normal (d < 0) and anomalous (d > 0) dispersion. For the case of the real line we show by the Implicit Function Theorem that for small nonlinear damping κ > 0 and large detuning ζ 1 and large forcing f 1 strongly localized, bright solitary stationary solutions exists in the case of anomalous dispersion d > 0. These results are achieved by using techniques from bifurcation and continuation theory and by proving a convergence result for solutions of the time-dependent Lugiato-Lefever equation.
In this paper we investigate the effect of nonlinear damping on the Lugiato-Lefever equation|a| 2 a + if on the torus or the real line. For the case of the torus it is shown that for small nonlinear damping κ > 0 stationary spatially periodic solutions exist on branches that bifurcate from constant solutions whereas all nonconstant solutions disappear when the damping parameter κ exceeds a critical value. These results apply both for normal (d < 0) and anomalous (d > 0) dispersion. For the case of the real line we show by the Implicit Function Theorem that for small nonlinear damping κ > 0 and large detuning ζ 1 and large forcing f 1 strongly localized, bright solitary stationary solutions exists in the case of anomalous dispersion d > 0. These results are achieved by using techniques from bifurcation and continuation theory and by proving a convergence result for solutions of the time-dependent Lugiato-Lefever equation.
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