The integral equation that describes the mode structure of unstable resonators of rectangular aperture is solved by a new method that makes use of an asymptotic expansion of the integral for large Fresnel number. This expansion is carried out to terms of order 1/F, where F is the Fresnel number, and includes effects of diffraction at the edges of the feedback mirror, as well as a term corresponding to the geometric-optics approximation. Results obtained by this method are in good agreement with those obtained by gaussian integration, even at an effective Fresnel number of unity. The method has the advantage that it involves finding the roots of a polynomial, rather than the eigenvalues of a matrix, and therefore requires less computer time and storage. The results of the present work are at variance with a class of theories based on geometric optics, as regards the higher-loss modes even for very large Fresnel number, although the lowest-loss symmetric mode is correctly given by geometric optics in this limit.
Paul Horwitz, "Asymptotic theory of unstable resonator modes," J. Opt. Soc. Am. 63, 1528-1543 (1973)