A numerical procedure that uses an explicit finite difference method to solve the wave equation is described. This technique results in a propagation algorithm that can accurately propagate an arbitrary electric field through a uniform medium or a medium that is nonuniform, transversely flowing, saturable, and contains index inhomogeneities. By using the propagation algorithm to propagate an arbitrary field back and forth between two resonator mirrors, the three-dimensional transverse mode and the output beam characteristics for a laser resonator can be determined. The advantage of the finite difference method is that unlike integral techniques the computational accuracy and efficiency improve as the resonator Fresnel number increases. The computational techniques are explained, and results for several specific empty cavity confocal unstable resonators are presented and compared to results obtained using an established calculation technique. The application of the finite difference method to inhomogeneous laser media is described, and computational results for an existing CO2 gas dynamic laser are presented and compared to measured data. The medium kinetics and shock wave models used in the calculations are described.
D. B. Rensch, "Three-Dimensional Unstable Resonator Calculations with Laser Medium," Appl. Opt. 13, 2546-2561 (1974)