We consider Kerr frequency combs in a dual-pumped microresonator as time-periodic and spatially $$2\pi $$ 2 π -periodic traveling wave solutions of a variant of the Lugiato–Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation given by $$\textrm{i}a_\tau =(\zeta -\textrm{i})a-d a_{x x}-|a|^2a+\textrm{i}f_0+\textrm{i}f_1\textrm{e}^{\textrm{i}(k_1 x-\nu _1 \tau )}$$ i a τ = ( ζ - i ) a - d a xx - | a | 2 a + i f 0 + i f 1 e i ( k 1 x - ν 1 τ ) . The main new feature of the problem is the specific form of the source term $$f_0+f_1\textrm{e}^{\textrm{i}(k_1 x-\nu _1 \tau )}$$ f 0 + f 1 e i ( k 1 x - ν 1 τ ) which describes the simultaneous pumping of two different modes with mode indices $$k_0=0$$ k 0 = 0 and $$k_1\in \mathbb {N}$$ k 1 ∈ 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., $$f_1=0$$ f 1 = 0 , can be continued into the range $$f_1\not =0$$ f 1 ≠ 0 . Our analytical findings apply both for anomalous ($$d>0$$ d > 0 ) and normal ($$d<0$$ d < 0 ) dispersion, and they are illustrated by numerical simulations.
Kerr frequency combs generated in high-Q microresonators offer an immense potential in many applications, and predicting and quantifying their behavior, performance and stability is key to systematic device design. Based on an extension of the Lugiato-Lefever equation we investigate in this paper the perspectives of changing the pump scheme from the well-understood monochromatic pump to a dual-tone configuration simultaneously pumping two modes. For the case of anomalous dispersion we give a detailed study of the optimal choices of detuning offsets and division of total pump power between the two modes in order to optimize single-soliton comb states with respect to performance metrics like power conversion efficiency and bandwidth. Our approach allows also to quantify the performance metrics of the optimal single-soliton comb states and determine their trends over a wide range of technically relevant parameters.
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