Fine-structure mapping of structured Laguerre-Gaussian beam states on the orbital Poincaré sphere

Elements of a 4D symplectic matrix P of second-order intensity moments are theoretically found and experimentally measured for paraxial structured Laguerre-Gaussian (sLG) beams with two control parameters—the amplitude e and phaseq. It is shown that only three out of ten elements of the matrix P are independent and directly measurable in the experiment. The remaining elements are determined through them. This made it possible to calculate the orbital Stokes parameters (S1, S2, S3), with their the sum of squares forming the invariant S of a first-order optical system. In terms of the elements of the matrix P, the orbital Stokes parameters are parameterized by the parameter q of the sLG beam. The invariant S can be treated as the radius of a 2D sphere (orbital Poincaré sphere) in the Cartesian coordinates (S1<, S2, S3). These coordinates map the sLG beam states onto the sphere, where the mapped trajectories acquire a complete spatial form inherent in two parametric sLG beam. The variation of the amplitude parameter controls the area covered by the trajectory, its shape and positions of the trajectory self-crossing points. It should be noted that the area of the sphere covered by the trajectory on the sphere with an adiabatic change in the beam parameter is directly related to the geometric Berry phase. Thus, with propagation and variations of the phase parameter, the sLG beam can acquire a controlled geometric phase, which can be considered as an additional degree of freedom.