Analytic solutions for Gaussian gain profile resonators

The problem of transversally gain-limited resonators is studied applying the Fresnel-Huygens integral operator including gain. For simplicity, only symmetric resonators with a Gaussian gain profile confined in a centered thin plate are considered, but the results are valid for more general cases. The solutions obtained are Hermite-Gaussian and Laguerre-Gaussian functions with complex argument and the method yields also the eigenvalues. The analysis of the eigenfunctions leads to the result that the beam-waist is not located in general at the center of the resonator, and could be outside the resonator in some cases.