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 of 1/ √ 3 for the TPA parameter r is exact. In a bifurcation study of the LLE [37], r = 1/ √ 3 was also found as a threshold value for the TPA parameter preventing bifurcations from the steady-state solution a(t , τ ) = a 0 . Further, r < 1/ √ 3 is also a necessary condition for bistability of the continuous-wave (cw) intracavity power A = |a 0 | 2 as a function of the detuning ζ [18].…”