Fig. 2: C-point polarization singularity and associated co-polarized singular phase in the eigen parameter space(δ, χ).

A Distribution of the polarization ellipses of \({|}\Psi \rangle\) in the eigen parameter space. The evolution of the input state creates a C-point polarization singularity at \(\delta=\pi,\chi=0\), containing the antipodal LHCP state at the singular point. Three different closed loops outside (solid circle), crossing (dashed circle), and encircling (dotted circle) the singular C-point are considered. B The corresponding trajectories of \({|}\Psi \rangle\), along with the co-polarized phase distribution are displayed on the Poincaré sphere. The yellow star represents the position on the Poincaré sphere where the singularity appears. C–F The decomposition of the \({|}\Psi \rangle\) into the co-and cross-polarized channels is performed by the respective projection operators, representing amplitude and phase distribution in the eigen-parameter space. Notably, at the C-point, the co-polarized phase becomes singular and circulation of this topologically protected phase enables complete 2π wavefront modulation in a spin-preserved manner.