From consideration of the physical mechanism of the matching of gaussian modes by means of a single thin lens, a straightforward method is developed for determining the position and the focal length of this lens, when the position and the size of the beam waists are given. The analytic results, although based on a quadratic equation, are unambiguous. An approximate solution is given for the case when the ratio of the beam waists approaches unity. The position of the matching lens may be determined graphically.
G. E. Francois, F. M. Librecht, and J. J. Engelen, "Mode Matching with a Single Thin Lens," Appl. Opt. 10, 1157-1159 (1971)