  Thus, the results of our studies lie in developing and implementing the basic principles of digital sorting the Laguerre-Gauss modes by radial numbers both for a non-degenerate and a degenerate state of a vortex beam subject to perturbations in the form of a hard-edged aperture of variable radius. The digital sorting of LG beams by the orthogonal basis involves the use of higher-order intensity moments, and subsequent scanning of the modulated beam images at the focal plane of a spherical lens. As a result, we obtain a system of linear equations for the squared mode amplitudes and the cross amplitudes of the perturbed beam. The solution of the equations allows one to determine the amplitudes of each LG mode and restore both the real mode array and the combined beam as a whole. First, we developed a digital sorting algorithm, and then two types of vortex beams were experimentally studied on its basis: a single LG beam and a composition of single LG beams with the same topological charges(azimuthal numbers) and different radial numbers . The beam was perturbed by means of a circular hard-edged aperture with different radii R. As a result of the perturbation, a set of secondary LG modes with different radial numbers k is appeared that is characterized by an amplitude spectrum . The spectrum obtained makes it possible to restore both the real array of LG modes and the perturbed beam itself with a degree of correlation not lower than. As a measure of uncertainty induced by the perturbation we measured the informational entropy (Shannon's entropy) 
Digital sorting perturbed laguerre-gaussian beams by radial numbers via   high order intensity moments