Dual backgrounds and their stability during frequency comb and second harmonic generation in χ⁽²⁾ microresonators

Light states relevant to the $\chi^{(2)}$ comb generation in optical microresonators possess typically dual backgrounds for coupled first-harmonic (FH) and second-harmonic (SH) envelopes. Stability and/or instability of these is crucial realization stable combs also an efficient SH generation. We explore properties their cases FH pumping resonator. In contrast parameteric oscillation, controlled by a 4th-degree characteristic equation increment. Coefficients this depend not only on wavenumbers perturbations, pump power, dispersion parameters, but FH-SH group velocity difference (temporal walk-off). Our results include characterization regions conditions stability FH- SH-pumping different spectral ranges.