We developed a new alterative technique of the digital sorting of Laguerre–Gaussian beams (LG) by radial numbers resorting to algebra of the high-order intensity moments. The term “digital mode sorting” involves sorting the main mode characteristics (in the form of a mode spectrum) by the computer cells. If necessary, the spatial mode spectrum can be reproduced, for example, by means of a spatial light modulator. In the experiment, we investigated both a single LG mode and a composition of LG modes with the same topological charge but different radial numbers subjected to perturbations via a hard-edged circular aperture. The LG beams sorting was accomplished by monitoring the amplitude spectrum of the triggered secondary LG modes then recovering the sorted modes and the perturbed beam as a whole. We have revealed degenerate states of the perturbed LG beam composition when the one $k$th mode in the amplitude spectrum can be related to a set of LG modes with the same radial numbers. In order to decrypt and to sort beams in such a degenerate state, it is necessary to know several keys, the number of which is equal to the number of LG modes in the initial wave composition. We were also able to analyze and to sort such degenerate mode states. For monitoring the measure of uncertainty arising in the perturbed beam, we measured informational entropy (Shannon entropy).
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