  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.
