Propagation of a phase flipped Gaussian beam through a paraxial optical ABCD system

Using the Huygens-Fresnel diffraction integral, we obtain a closed-form analytic expression for the amplitude distribution of a phase flipped Gaussian (PFG) beam passing through a paraxial optical ABCD system. The resulting expression is related in a simple way to the corresponding expression for a Gaussian beam and contains error functions of complex argument. Both ideal and non-ideal PFGs are considered. Numerical results are presented for the special case of free-space propagation. Intensity patterns of the propagating PFG are explained by means of the power series and asymptotic expansions of the error function.