Global continua of solutions to the Lugiato-Lefever model for frequency combs obtained by two-mode pumping

Abstract

We consider Kerr frequency combs in a dual-pumped microresonator as time-periodic and spatially $2\pi$-periodic traveling wave solutions of a variant of the Lugiato-Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation given by $\mathrm{i}{a}_{\tau }=(\zeta -\mathrm{i})a-d{a}_{xx}-|a{|}^{2}a+\mathrm{i}{f}_0+\mathrm{i}{f}_1{\mathrm{e}}^{\mathrm{i}(k_{1}x-{\nu }_1\tau )}$. The main new feature of the problem is the specific form of the source term $f_0+f_1{\mathrm{e}}^{\mathrm{i}(k_{1}x-{\nu }_1\tau )}$ which describes the simultaneous pumping of two different modes with mode indices $k_0=0$ and $k_1\in \mathbb{N}$. We prove existence and uniqueness theorems for these traveling waves based on a-priori bounds and fixed point theorems. Moreover, by using the implicit function theorem and bifurcation theory, we show how non-degenerate solutions from the $1$-mode case, i.e. $f_1=0$, can be continued into the range $f_1\ne 0$. Our analytical findings apply both for anomalous ($d>0$) and normal ($d<0$) dispersion, and they are illustrated by numerical simulations.

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