Intertwining Intensity Moments and Orbital Angular Momenta via Astigmatic Transforms of Structured Beams

We have shown theoretically and experimentally that for defining the orbital angular momentum (OAM) of structured Laguerre-Gaussian (sLG) beams with a broken symmetry, it is sufficient to measure the only element of the symplectic intensity moment matrix. It looks like an intertwining of two elements belonging to two submatrices of the general symplectic matrix of intensity moments under the condition when the amplitude parameter of the sLG beam is equal to unity. But once slightly to deviate the amplitude parameter from this value, as the intertwining at once unravels. Besides, we revealed that by measuring such a matrix element of the sLG beam in free space enables us to exactly predict the OAM behavior of the sLG beam in a first-order optical system with an astigmatic element, without any measuring in the astigmatic system itself.