Measurement of the orbital Stokes parameters in structured astigmatic beams in critical planes of the first-order optical system

In this work, a method for measuring the orbital Stokes parameters of an inhomogeneous astigmatic beam in an optical system of the first order is developed and experimentally implemented. Unlike the method of measuring orbital parameters in homogeneous beams, our technique employs properties of critical planes that arise in a space between cylindrical and spherical lenses, as well as the observation plane. Theoretically and experimentally, it is revealed that in the critical planes, not only is there equality of beam radii in the x and y directions, but also the Gouy phase difference in the x and y components is equal to π/2 and 2πn, n=0, 1,2, ... in the first and second critical planes, respectively. Additionally, in the critical planes, the reciprocity principle between the transverse intensity moment Wxy and OAM is fulfilled, as well as the correspondence principle between all beam parameters in the second critical plane and the front plane of the cylindrical lens. These properties of the structured beam make it possible to find all three orbital Stokes parameters by measuring three Wxx, Wyy, and Wxy intensity moments with a 4% measurement error, as well as to map the beam states onto the orbital Poincare sphere.