  The topological phase acquired by vector vortex optical beams is
investigated. Under local unitary operations on their polarization and
transverse degrees of freedom, the vector vortices can only acquire discrete
geometric phase values, 0 or $\pi\,$, associated with closed paths belonging to
different homotopy classes on the SO(3) manifold. These discrete values are
demonstrated through interferometric measurements and the spin-orbit mode
separability is associated to the visibility of the interference patterns. The
local unitary operations performed on the vector vortices involved both
polarization and transverse mode transformations with birefringent wave plates
and astigmatic mode converters. The experimental results agree with our
theoretical simulations and generalize our previous results obtained with
polarization transformations only.

Quantum Physics