Complex argument Hermite–Gaussian and Laguerre–Gaussian beams

Hermite–Gaussian and Laguerre–Gaussian beams with complex arguments of the type introduced by Siegman [ J. Opt. Soc. Am.63, 1093 ( 1973)] are shown to arise naturally in correction terms of a perturbation expansion whose leading term is the fundamental paraxial Gaussian beam. Additionally, they can all be expressed as derivatives of the fundamental Gaussian beam and as paraxial limits of multipole complex-source point solutions of the reduced-wave equation.