  Modal coupling between frequency-degenerate resonances of an optical ring
resonator is a commonly observed phenomenon that results in adverse mode
splitting. Traditionally, this coupling is attributed to Rayleigh scattering of
a propagating electromagnetic wave into its associated degenerate
counter-propagating mode from small perturbations to the dielectric material of
the resonator. We have chosen to reframe the problem of intracavity Rayleigh
scattering by considering the optical ring resonator as an infinitely-long,
one-dimensional photonic crystal (PhC) that possesses a lattice constant equal
to the perimeter of the ring. Through application of Bloch-Floquet theory, we
show that modal coupling between degenerate resonances of a ring can
effectively be described as the formation of photonic frequency bands in the
dispersion relation of the resonator. We additionally demonstrate that the
Bragg planes of the PhC lattice coincide with the phase matching conditions for
constructive interference in the ring. Finally, we show that the magnitude of
frequency splitting of a particular resonance is proportional to its associated
coefficient in the Fourier expansion of the ring's periodic dielectric
function.
