  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 SNAP (Surface
Nanoscale Axial Photonics) 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.
