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\title{Nonlinear phenomena with whispering gallery modes}

\author{\vspace*{-3mm} Boris Sturman}
\affiliation{Institute for Automation and Electrometry of Russian Academy of Sciences,
Novosibirsk 630090, Russia}

\maketitle

Whispering gallery modes (WGMs), which are strongly localized near the surface of a
resonator owing to the total internal reflection, are known since the time of Lord
Rayleigh. During the last decade, a lot of research interest has been attracted to
optical WGM micro-resonators made of different transparent solid-state materials --
glasses and crystals (both $\chi^{2}$ and $\chi^{3}$). The sizes range from tens of
$\mu$m to several~mm, the shape is typically axisymmetric (spherical or not), and
different fabrication techniques can be used. The problem of coupling light in and out
of the resonator is reliably solved: Different couplers (prism, fiber, etc.) can
transfer up to $100\%$ of pump light to individual WGMs.

The main figures of merits for WGMs are the quality factor $Q$ and the modal
cross-section $\sigma$. The values of $Q$ reach nowadays $10^{11}$ bringing the line
width to a sub-MHz range, and the values of $\sigma$ can be as small as $\sim
10^2\,\mu$m$^2$. Correspondingly, the light intensity in WGM, $I \propto PQ/\sigma$ can
be huge even for a very small pump power $P$. Thus, very weak coherent continuous-wave
light sources can initiate strong nonlinear effects. The latter can be very specific.
Furthermore, a weak material nonlinearity can be easily compensated by high quality
factors of the interacting waves. The window of transparency becomes a crucial issue.

The back side of the ultra-strong intensity enhancement is discreteness of the frequency
spectrum. It causes problems for the phase matching, so that additional tuning means
should be used.

Among the nonlinear effects in WGM resonators are generation of frequency combs and
transition to chaos, excitation of mechanical vibrations and acoustical WGMs, and also
modified optical parametric oscillation and second-harmonic generation. My goal is to
overview these and some other nonlinear phenomena.




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