Complex ABCD transformations for optical ring cavities with losses and gain

Complex ABCD field transformations are investigated for inhomogeneous optical ring cavities with losses and gain. It is shown that the sets of eigenfunctions, corresponding to counterpropagating waves, are really biorthogonal: the functions in each of these sets are really orthogonal relative to one another, and have a complex weighting factor independent of the mode number. Bidirectional and unidirectional stability conditions are formulated for such cavities. These conditions are qualitatively different from those for loss-free cavities. A simple algorithm is proposed for the evaluation of the ABCD matrix for a medium with an arbitrary longitudinal inhomogeneity along the beam.