Mapping structured Laguerre-Gaussian beam states onto the orbital Poincaré sphere in the form of controllable spatial trajectories

The structured Laguerre–Gaussian (LG) beam is a two-parameter superposition of 2 n + ℓ +1 Hermite–Gaussian modes (where n and ℓ are a radial number and a topological charge of the initial LG beam) whose orbital angular momentum oscillations are controlled by phases and amplitude parameters. But we succeeded in reducing its representation to a simple sum of a standard LG mode and a hybrid Hermite–Laguerre–Gaussian (HLG) beam that is a key point in understanding a hidden geometry of the structured LG (sLG) beams and implementations of its unique prosperities. In assents, the hybrid HLG beam is mapped onto the orbital Poincaré sphere in the form of a plane trajectory along a main meridian of the sphere. However, the most intriguing thing is as follows. First, once we slightly perturb the HLG beam with a single LG mode, the flat trajectory turns into a complex multi-petalled tracery with multiple self-intersections due to cyclic variation of the phase parameter of the sLG beam. Moreover, the shape of the tracery as well as the birth and destruction of the self-intersection points can be controlled with the amplitude parameter. However, it is worth noting that when changing the beam parameters cyclically, the area outlined by the trajectory on the sphere is directly related to the geometric phase acquired by the sLG beam that can be treated as an additional degree of freedom for transmitting big data. In the article, we study the sLG beam properties and its mapping onto the orbital Poincarè sphere in the framework of a symplectic 4×4 matrix formalism while the orbital Stokes parameters are experimentally measured, and we have found good agreement between theory and experiment.