Semiclassical theory of frequency combs generated by parametric modulation of optical microresonators

An optical microresonator, which parameters are periodically modulated in time, can generate optical frequency comb (OFC) spectral resonances equally spaced by the modulation frequency. Significant recent progress in realization of OFC generators based on the modulation of microresonator parameters boosted interest to their further experimental development and theoretical understanding of underlying phenomena. However, most of theoretical approaches developed to date were based on the lumped parameter models which unable to evaluate, analyse, and optimize the effect of spatial distribution of modulation inside microresonators. Here we develop the multi-quantum semiclassical theory of parametrically excited OFCs which solves these problems. As an application, we compare OFCs which are resonantly or adiabatically excited in a racetrack microresonator (RTM) and a Surface Nanoscale Axial Photonics (SNAP) bottle microresonator (SBM). The principal difference between these two types of microresonators consists in much slower propagation speed of whispering gallery modes along the SBM axis compared to the speed of modes propagating along the RTM waveguide axis. We show that, due to this difference, similar OFCs can be generated by an SBM with a much smaller size compared to that of the RTM. Based on the developed theory, we analytically express the OFC spectrum of microresonators through the spatial distribution of modulated parameters and optimize this distribution to arrive at the strongest OFCs generated with minimum power consumption.