In this work, we theoretically and experimentally demonstrate a method for measuring the orbital Stokes parameters of structured Laguerre–Gaussian beams in the critical planes formed in a first-order optical system consisting of a cylindrical and a spherical lens. The positions of the critical planes are determined by the condition of equality of the beam radii in the x- and y-directions at the Rayleigh length {z₀} = 2{f₂₋}. These planes are characterized by specific values of the Gouy phase difference { ₗₘ}: in the first critical plane it equals ₗₘ^{ (1) } = /2, while in the second it takes discrete values ₗₘ^{ (2) } = 2 n, \, \, n = 0, 1, 2, ~. By employing the reciprocity effect between the transverse intensity moment {Wₗₘ} and the orbital angular momentum { ₙ}, related through { ₙ} = 4Wxy, we carried out the measurement of the third orbital Stokes parameter. Owing to the self-healing effect in this system, it was shown that a structured Laguerre–Gaussian beam is reconstructed in the second critical plane, while its astigmatic analogue is formed in the first. For an astigmatic structured Laguerre–Gaussian beam, the situation is reversed: reconstruction occurs in the first critical plane, and its astigmatic analogue appears in the second. This behavior ensures the applicability of the method to both structured Laguerre–Gaussian beams and their astigmatic modifications. The experimental results demonstrate good agreement with numerical modeling with a measurement error not exceeding four percent. It is also shown that structured Laguerre–Gaussian beams can be mapped onto the orbital Poincaré sphere, which opens up prospects for their practical use.
Khalilov et al. (2025) studied this question.