An analytical method of matching the mode of an input laser to the lowest-order mode of a passive resonant ring laser gyro is described, as are the steps in determining the location and focal length of cylindrical mode matching lenses. Results were obtained with no mode matching, with a compromise spherical lens, with horizontal mode matching only, and with the proper cylindrical mode matching lenses. Compared with no mode matching, the latter case shows that the amplitude of the lowest-order mode is increased ~2.5 times. In addition, the number and intensity of higher-order modes are reduced to near zero, and the relative intensity of the lowest-order mode to the higher-order mode increased from ~5 to ~60 times greater.
Thomas Dale Baxter, Theodore T. Saito, Gerald L. Shaw, Richard T. Evans, and Raymond A. Motes, "Mode matching for a passive resonant ring laser gyroscope," Appl. Opt. 22, 2487-2491 (1983)