Tilted Poincaré sphere geodesics.

We provide the first, to the best of our knowledge, experimental demonstration of a geometric phase generated in association with closed Poincaré sphere trajectories comprising geodesic arcs that do not start, end, or necessarily even include, the north and south poles that represent pure Laguerre-Gaussian modes. Arbitrarily tilted (elliptical) single vortex states are prepared with a spatial light modulator, and Poincaré sphere circuits are driven by beam transit through a series of π-converters and Dove prisms.