Measurements of Pancharatnam–Berry phase in mode transformations on hybrid-order Poincaré sphere

We report direct measurements of the Pancharatnam-Berry (PB) phase in mode transformations on a hybrid-order Poincaré sphere. This geometric phase arises when the vector vortex states undergo a cyclic transformation over a closed circuit on a hybrid-order Poincaré sphere. The measured PB phase is proportional to the solid angle of the closed circuit, as well as to the variation of the total angular momenta between north and south poles. More importantly, a zero PB phase has been demonstrated, despite the vector vortex states taken through a closed circuit on the hybrid-order Poincaré sphere. This interesting phenomenon can be explained as being due to the zero Berry curvature.