A priori bounds and global bifurcation results for frequency combs modeled by the Lugiato-Lefever equation

In nonlinear optics $2\pi$-periodic solutions $a\in C^2([0,2\pi];\mathbb{C})$ of the stationary Lugiato-Lefever equation $-d a"= ({\rm i} -\zeta)a +|a|^2a-{\rm i} f$ serve as a model for frequency combs, which are optical signals consisting of a superposition of modes with equally spaced frequencies. We prove that nontrivial frequency combs can only be observed for special ranges of values of the forcing and detuning parameters $f$ and $\zeta$, as it has been previously documented in experiments and numerical simulations. E.g., if the detuning parameter $\zeta$ is too large then nontrivial frequency combs do not exist, cf. Theorem 2. Additionally, we show that for large ranges of parameter values nontrivial frequency combs may be found on continua which bifurcate from curves of trivial frequency combs. Our results rely on the proof of a priori bounds for the stationary Lugiato-Lefever equation as well as a detailed rigorous bifurcation analysis based on the bifurcation theorems of Crandall-Rabinowitz and Rabinowitz. We use the software packages AUTO and MATLAB to illustrate our results by numerical computations of bifurcation diagrams and of selected solutions

Cascaded half-harmonic generation of femtosecond frequency combs in mid-IR

For the growing demand of frequency combs in mid-infrared (mid-IR), known as the "molecular fingerprint" region of the spectrum [1], down conversion of near-IR frequency combs through half- harmonic generation offers numerous benefits including high conversion efficiency and intrinsic phase and frequency locking to the near-IR pump [2]. Hence cascaded half-harmonic generation promises a simple path towards extending the wavelength coverage of stable frequency combs. Here, we report a two-octave down-conversion of a frequency comb around 1 {\mu}m through cascaded half-harmonic generation with ~64% efficiency in the first stage, and ~18% in the second stage. We obtain broadband intrinsically-frequency-locked frequency combs with ~50-fs pulses at ~2 {\mu}m and ~110-fs pulses at ~4 {\mu}m. These results indicate the effectiveness of half-harmonic generation as a universal tool for efficient phase- and frequency-locked down-conversion, which can be beneficial for numerous applications requiring long-wavelength coherent sources

Dual pumped microresonator frequency combs

A study is made of the nonlinear dynamics of dual pumped microresonator Kerr frequency combs described by a driven and damped nonlinear Schr\"odinger equation, with an additional degree of freedom in the form of the modulation frequency. A truncated four wave model is derived for the pump modes and the dominant sideband pair which is found to be able to describe much of the essential dynamical behaviour of the full equation. The stability of stationary states within the four wave model is investigated and numerical simulations are made to demonstrate that a large range of solutions, including cavity solitons, are possible beyond previously considered low intensity patterns.Comment: 7 pages, 9 figures, submitted to Phys. Rev.