Limits of the correction ring applicability in an apodizer with a circular serrated aperture

A theoretical investigation has been carried out of the evolution of a monochromatic laser beam profile after the beam exit from an apodizer. The optical scheme of the apodizer consists of a circular serrated aperture with a narrow opaque corrective ring and a spatial filter based on the Kepler telescope with a pinhole. Characteristic parameters of the apodizer have been found, which allows one to use the corrective ring for the improvement of the apodized beam profile quality and reduce the beam diffraction distortion at beam propagation over distances corresponding to the Fresnel numbers $NF\approx5\div2$.