  Spontaneous symmetry breaking is an important concept in many areas of
physics. A fundamentally simple symmetry breaking mechanism in electrodynamics
occurs between counter-propagating electromagnetic waves in ring resonators,
mediated by the Kerr nonlinearity. The interaction of counter-propagating light
in bi-directionally pumped microresonators finds application in the realisation
of optical non-reciprocity (for optical diodes), studies of PT-symmetric
systems, and the generation of counter-propagating solitons. Here, we present
comprehensive analytical and dynamical models for the nonlinear
Kerr-interaction of counter-propagating light in a dielectric ring resonator.
In particular, we study discontinuous behaviour in the onset of spontaneous
symmetry breaking, indicating divergent sensitivity to small external
perturbations. These results can be applied to realise, for example, highly
sensitive near-field or rotation sensors. We then generalise to a
time-dependent model, which predicts new types of dynamical behaviour,
including oscillatory regimes that could enable Kerr-nonlinearity-driven
all-optical oscillators. The physics of our model can be applied to other
systems featuring Kerr-type interaction between two distinct modes, such as for
light of opposite circular polarisation in nonlinear resonators, which are
commonly described by coupled Lugiato-Lefever equations.
