Hopping induced inversions and Pancharatnam excursions of C-points

In this Letter, we show the acquisition of the Pancharatnam phase by a C-point singularity when it is subjected to discrete cyclic polarization transformations. The changes in state of polarizations (SOPs) are mapped onto a Poincaré sphere as geodesical closed trajectories. The Pancharatnam phase acquired by a C-point is equal to the solid angle subtended by the closed trajectories at the center of the Poincaré sphere. We show this by considering index hopping induced inversions of C-points. For example, a lemon from the North Pole of a Poincaré sphere is first converted into a star whose location can be traced to the South Pole of the Poincaré sphere and retrieved back as a lemon at the North Pole to complete a closed geodesical trajectory on the Poincaré sphere. Depending on the trajectory, it is shown that the lemons (stars) acquire different amounts of the Pancharatnam phase, attributable to the amount of rotation in the SOP pattern of the lemons (stars).