Stokes singularity relations.

Polarization singularities in paraxial vector optical fields are analyzed in terms of the phase singularities of complex Stokes scalar fields. Six independent relationships are obtained that connect the topological charges of these singularities on special closed contours with the charges of singularities that are enclosed by these contours. These relationships, which have been confirmed by experimental data and computer simulations, imply topological polarization correlations of an infinite range.

Elliptic critical points: C-points, a-lines, and the sign rule.

The critical points of generic paraxial ellipse fields consist of singular points of circular polarization, called C -points, and azimuthal stationary points, i.e., maxima, minima, and saddle points. We define these stationary points here and review their properties. The sign rule for ellipse fields requires that the sign of the singularity indices I(C)=+/-1/2 of the C -points on non-self-intersecting lines of constant azimuthal ellipse orientation (modulo pi/2), i.e., a -lines, alternate along the line. We verify this rule experimentally, using a newly developed interferometric technique to measure C -points and a -lines in an elliptically polarized random optical field.