An explicitly solvable case of the construction of the fundamental cavity mode in the form of a Gaussian beam with a complex astigmatism

An analysis is carried out of the ring cavities whose fundamental modes represent Gaussian beams with complex astigmatism. The analysis is made for the case where the block B or C in the cavity ray matrix ABCD of dimension 4×4 is a symmetric nondegenerate matrix. Explicit formulas are obtained, which permits the expression of beam characteristics directly in terms of the cavity ray matrix without the laborious process of finding its eigenvectors. The results obtained in the study can be used for controlling astigmatism in lasers and laser systems, the calculation of polarization of three-dimensional active media in linear and ring optical cavities of complex configuration, etc.