Generalized Gaussian beam solutions of paraxial optics and their connection to a hidden symmetry

It is shown that the Hermite–Gauss- and Laguerre–Gauss-type solutions of paraxial optics whose corresponding Hermite and Laguerre polynomials have complex arguments are closely related to a hidden symmetry in the parabolic equation. These solutions are generated from the fundamental Gaussian beam solution by applying the powers of the infinitesimal operators of this symmetry group. The Fourier spectrum of these solutions is obtained from the Fourier spectrum of the fundamental Gaussian beam solution in a similar manner. The Gaussian beam solutions containing Hermite polynomials with complex arguments that were derived by Siegman [ J. Opt. Soc. Am.63, 1093 ( 1973)] represent a limiting case of the more-general solutions considered here. The generalized Gaussian beam solutions show a sharp mode picture with exact zeros in the field distribution only in the focal or waist plane. Unconventional applications of the parabolic approximation in optics including focus wave modes are demonstrated.