Spectral properties of a ring optical resonator with an arbitrary longitudinal inhomogeneity

A ring optical resonator with an arbitrary but continuous change in the permittivity of the filling medium along the resonator axis is considered. It is shown that in the case of a small deviation of the permittivity from its average value, the double degeneracy of eigenfrequencies inherent in a homogeneous resonator is removed and the corresponding modes acquire the properties of standing waves. Simple universal expressions are derived to calculate eigenfrequencies and distribution coefficients in modes. Conditions are found under which splitting in the frequency spectrum of an inhomogeneous resonator is absent. The general results obtained in the paper can be used in numerical experiments.