Directly measuring the total orbital angular momentum in astigmatic structured beams and the loss of several degrees of freedom

In this article, a model of the astigmatic structured beam obtained by the ABCD matrix method is shown to be simpler and more visual compared to the approach of integral transforms [Volyar AV, Abramochkin EG, Bretsko MV, Khalilov SI, Akimova YE. Control of giant orbital angular momentum bursts of structured Laguerre-Gaussian beams in a medium with general astigmatism. Computer Optics 2024; 48(1): 35-46. DOI: 10.18287/2412-6179-CO-1395]. We study in detail physical mechanisms of shaping super-high orbital angular momentum (OAM) after a cylindrical lens during general astigmatic transformations. We also theoretically substantiate and experimentally confirm a new technique for measuring a total OAM in the structured beam based on a single measurement of the crossed intensity moment as a result of computer processing of the intensity pattern. It is shown that such simplified measurements are based on the degeneracy of the off-diagonal elements W and M of the submatrices, which reduces the number of additional degrees of freedom of the structured beam from ten to seven.