Paraxial matrix description of astigmatic and cylindrical mirror resonators with twisted axes for laser spectroscopy

A complete matrix description of ray-optic propagation in an astigmatic multipass cell is presented, with expressions for the coupling of coordinates. A pair of transforms puts the 4 X 4 paraxial system matrix into block-diagonal form, allowing a solution using Sylvester's theorem. A variation on the Jones matrix calculus is employed wherein the ray coordinates on both resonator mirrors are simultaneously represented as a single state of the system. The formulations are applicable to resonators with any degree of astigmatism and axial twist. Examples are presented of beam paths and the boundary shapes of beam spots. The shape of the pattern boundaries, as a function of the coordinate coupling coefficient, influences the practical availability of patterns.