Elliptic critical points in paraxial optical fields

Generic critical points of elliptically polarized, paraxial optical fields include (i) C-points, which are isolated points of circular polarization, (ii) stationary points of the azimuthal angle that measures the orientation of the major axes of the ellipses, and (iii) stationary points of the form factor that measures the ratio of the minor to major axes of the ellipses. These newly defined elliptic stationary points are introduced, and using a mapping of ellipse fields onto a complex Stokes field, topological constraints are formulated that tie them to C-points and to a-lines, which are lines of constant azimuthal orientation, as well as to L-lines, which are lines of linear polarization. Experiments on random ellipse fields are described that verify the most important of these constraints, the sign rule. A mapping of C-points onto phase vortices is made that permits construction of a large variety of C-points with predetermined topological charges, Poincaré-Hopf indices, and three-dimensional trajectories. This mapping also shows that C-points in random ellipse fields should exhibit essentially complete screening of their topological charges. This result is verified by large-scale computer simulations.