Eigenfrequencies and eigenmodes of a ring optical resonator with thin dielectric plates

A ring optical resonator containing arbitrarily placed dielectric plates is considered. The resonator inhomogeneity introduced by the plates is insignificant: the thickness of plates and the excess of their refractive index over the average refractive index of the resonator obey the condition of smallness. It is shown that the eigenfrequency spectrum of this resonator is simple and represents an equidistant sequence of weakly split frequency doublets. Splitting in each doublet is found. Modes in the form of perturbed standing waves are quantitatively described. The formalism of the shift matrices along the trajectories of the differential equation, which makes it possible to obtain the result in the simplest way, is used to solve the above spectral problem.