The Lugiato-Lefever equation with nonlinear damping caused by two photon absorption

Abstract

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.

Analysis of Kerr comb generation in silicon microresonators under the influence of two-photon absorption and fast free-carrier dynamics

“…Note that the upper limit for r of 1 3 is exact and is in agreement with results also obtained from a bifurcation study of the LLE including TPA [32]. For the values of r and eff τ σ ′ ′ specified by Eq.…”