How the astigmatic Gouy phase controls invariants of structured beams

The properties of Fourier-invariants of a generalized structured Laguerre-Gaussian beam in an optical system with two spherical lenses and a cylindrical lens between them are theoretically and experimentally investigated. The first spherical lens forms the characteristic geometry of a generalized structured Laguerre-Gaussian beam at the input of the cylindrical lens. The second spherical lens performs the Fourier transform of an astigmatic beam. A photodetector is placed in the focal plane of a spherical lens and is shifted along the beam axis together with it. The analysis of the orbital Stokes parameters is carried out by the methods of second-order intensity moment matrices. It is found that the orbital Stokes parameters, including the orbital angular momentum, do not change during free propagation of structured astigmatic beams of any order for any control parameters and are their invariants. It is shown that even a slight violation of the Fourier-invariance conditions leads to the destruction of Fourier-invariants. The processes of the formation and destruction of invariants are controlled by the astigmatic Gouy phase.