Eigenfrequencies of a nonreciprocal nonuniformly filled ring optical resonator

A nonreciprocal ring optical resonator with the simplest type of nonuniformity in the form of two interacting scatterers is considered. A boundary problem of a periodic type allowing one to write the characteristic equation for determining eigenfrequencies is formulated. Analytic expressions for eigenfrequencies and the distribution coefficient of travelling waves in the mode are derived under the assumption of relative smallness of the reflection coefficient and the nonreciprocity parameter. The peculiarities of splitting these frequencies and the structure of the eigenfrequencies (resonator modes) corresponding to them are discussed.