Nonlinear solutions for χ(2) frequency combs in optical microresonators

Abstract

Experimental and theoretical studies of nonlinear frequency combs in ${\ensuremath{\chi}}^{(3)}$ optical microresonators attracted tremendous research interest during the last decade and resulted in prototypes of soliton-based steadily working devices. Realization of similar combs owing to ${\ensuremath{\chi}}^{(2)}$ optical nonlinearity promises new breakthroughs and is a big scientific challenge. We analyze the main obstacles for realization of the ${\ensuremath{\chi}}^{(2)}$ frequency combs in high-$Q$ microresonators and propose two families of steady-state nonlinear solutions, including soliton and periodic solutions, for such combs. Despite periodicity of light fields inside microresonators, the nonlinear solutions can be topologically different and relevant to periodic and antiperiodic boundary conditions. The antiperiodic states are expected to be the most favorable for the comb generation. The found particular solutions exist owing to a large difference in the group velocities between the first and second harmonics, typical of ${\ensuremath{\chi}}^{(2)}$ microresonators, and to the presence of the pump. They have no zero-pump counterparts relevant to conservative solitons. The stability issue for the found comb solutions remains open and requires further numerical analysis.