  Kerr-effect-induced changes of the polarization state of light are well known
in pulsed laser systems. An example is nonlinear polarization rotation, which
is critical to the operation of many types of mode-locked lasers. Here, we
demonstrate that the Kerr effect in a high-finesse Fabry-P\'erot resonator can
be utilized to control the polarization of a continuous wave laser. It is shown
that a linearly-polarized input field is converted into a left- or
right-circularly-polarized field, controlled via the optical power. The
observations are explained by Kerr-nonlinearity induced symmetry breaking,
which splits the resonance frequencies of degenerate modes with opposite
polarization handedness in an otherwise symmetric resonator. The all-optical
polarization control is demonstrated at threshold powers down to 7 mW. The
physical principle of such Kerr effect-based polarization controllers is
generic to high-Q Kerr-nonlinear resonators and could also be implemented in
photonic integrated circuits. Beyond polarization control, the spontaneous
symmetry breaking of polarization states could be used for polarization filters
or highly sensitive polarization sensors when operated close to the
symmetry-breaking point.
