Dynamics of the OAM in Stable Generalized Structured Laguerre-Gaussian and Rotated Polygon Beams in the Astigmatic Optical System

We have theoretically and experimentally considered the four-parameter generalized structured beam, which can be either in the initial state of a Laguerre-Gaussian beam or a Hermite-Gaussian beam, or in both states simultaneously. In astigmatic transformations, these states can alternate with orbital angular momentum reaching double the radial number with a single topological charge. The astigmatic transformations of a spiral triangular beam containing a composition of higher-order Laguerre-Gaussian modes were also analyzed. We found that such dramatic transformations are controlled by the astigmatic Gouy phase. Moreover, it was revealed that the invariant of such astigmatic transformations is the sum of the squares of the orbital angular momentum and the cross-intensity moment, representing the total orbital angular momentum equal to the square of the initial OAM.