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An optical microresonator, which parameters are periodically modulated intime, can generate optical frequency comb (OFC) spectral resonances equallyspaced by the modulation frequency. Significant recent progress in realizationof OFC generators based on the modulation of microresonator parameters boostedinterest to their further experimental development and theoreticalunderstanding of underlying phenomena. However, most of theoretical approachesdeveloped to date were based on the lumped parameter models which unable toevaluate, analyse, and optimize the effect of spatial distribution ofmodulation inside microresonators. Here we develop the multi-quantumsemiclassical theory of parametrically excited OFCs which solves theseproblems. As an application, we compare OFCs which are resonantly oradiabatically excited in a racetrack microresonator (RTM) and a SNAP (SurfaceNanoscale Axial Photonics) bottle microresonator (SBM). The principaldifference between these two types of microresonators consists in much slowerpropagation speed of whispering gallery modes along the SBM axis compared tothe speed of modes propagating along the RTM waveguide axis. We show that, dueto this difference, similar OFCs can be generated by an SBM with a much smallersize compared to that of the RTM. Based on the developed theory, weanalytically express the OFC spectrum of microresonators through the spatialdistribution of modulated parameters and optimize this distribution to arriveat the strongest OFCs generated with minimum power consumption.

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