The paper presents a theoretical study of the eigenmodes of a misaligned ring multi-mirror laser cavity one or several arms of which are filled with an inhomogeneous medium. We start with posing the general problem of calculation of the radiation characteristics for a ring resonator with arbitrary inhomogeneities of the medium and mirrors, in the paraxial approximation. Then the general relations are specified for the lens-like resonator model in which the real and imaginary parts of the refractive index, as well as the phase and amplitude corrections performed by the mirrors, quadratically depend on the transverse coordinates. Explicit expressions are obtained for the eigen frequency and spatial characteristics of the radiation via the coefficients of the Hermite-Gaussian functions describing the complex amplitude distributions at the mirrors. The main results are formulated in terms of linear relations between positions of the resonator mode axis at the mirrors and the misalignment parameters (small shifts and tilts of the mirrors). The explicit results display the coefficients of the above linear relations. They are calculated for the 3-mirror cavity with using a specially developed approach employing peculiar conditions to the coefficients form that follow from very general considerations of two groups: 1) Independence of the resulting frequency shifts on the sequence in which the misalignments are made (this is the base of the energy method for the resonator analysis, which is essentially generalized and improved); 2) Geometrical symmetry of the resonator. The results of calculations can be used in order to control the radiation characteristics in relation to the resonator misalignments, in particular, for analysis of the output radiation stability and sensitivity to small changes of the cavity configuration.
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