A numerical technique has been developed for analyzing the transverse modes of waveguide lasers with external mirrors. Propagation outside the guide is computed with the Fresnel-Kirchhoff diffraction integral and within the guide by decomposing the fields into the characteristic modes of the guide structure. The transverse modes of the entire waveguide-mirror system fall into a number of distinct classes: TE0m, TM0m, EH1m, EH2m, etc. For each class of modes, the, corresponding guide modes form a complete and orthogonal set and may be used as basis vectors to describe those modes. This reduces the mode analysis of the waveguide resonator to the diagonalization of a small (5 × 5 or 10 × 10) complex matrix. Guide losses, coupling losses, and mode shapes will be discussed for a number of interesting cases, with the Fresnel number of the waveguide ranging from 0.1 to 1.0 and with various values of mirror curvature and position. It will be shown that some vales of resonator parameters are particularly advantageous for achieving single mode operation.
Richard L. Abrams and Arthur N. Chester, "Resonator Theory for Hollow Waveguide Lasers," Appl. Opt. 13, 2117-2125 (1974)