In this work, we investigate the trajectories of Laguerre–Gaussian beam states on the Poincaré sphere under general astigmatic transformation. It is shown that rotating the cylindrical lens shifts the trajectory into a plane perpendicular to the sphere’s axis, resulting in a precessional motion that depends on the rotate angle and the beam’s orbital angular momentum. The results are useful for analyzing and controlling the phase properties of light in optical systems.
Bretsko et al. (Mon,) studied this question.