Eigenfrequencies of a composite ring resonator

An inhomogeneous ring optical resonator of a special type is considered, which contains two identical dielectric plates of finite thickness separated by an arbitrary distance. The refraction coefficient of these plates is significantly higher than that of the medium filling the rest of the resonator. This system can be treated as a ring resonator formed by two linear resonators coupled through the plates confining them. The classical spectral problem for such a resonator is solved in the plane-wave approximation. It is shown that in the case of a comparatively low reflectance from the plates, it is possible to obtain analytically a physically acceptable description of the spectrum of eigenfrequencies and modes. The method for solving the spectral problem is proposed in which the analytic approach is combined with the numerical experiment. It is shown that the resonator spectrum is simple and is formed by a sequence of doublets. The modes corresponding to these doublets are real and orthogonal. Conditions are found under which the splitting of eigenfrequencies in doublets disappears.