"Analytical Continuation'' of Flattened Gaussian Beams

A purely analytical extension of the flattened Gaussian beams [Opt. Commun. $\mathbf{\text{107,}}$ 335 (1994)] to any values of the beam order, is here proposed. Due to it, the paraxial propagation problem of axially symmetric, coherent flat-top beams through arbitrary $ABCD$ optical systems can definitely be solved in closed form via a particular bivariate confluent hypergeometric function.