We study the angular structure of polarization of light transmitted through a nematic liquid crystal (NLC) cell by analyzing the polarization state as a function of the incidence angles and the polarization of the incident wave. The polarization-resolved angular (conoscopic) patterns emerging after the NLC cell illuminated by the convergent light beam are described in terms of the polarization singularities such as $C$ points (points of circular polarization) and $L$ lines (lines of linear polarization). For the homeotropically aligned cell, the Stokes polarimetry technique is used to measure the polarization resolved conoscopic patterns at different values of the ellipticity of the incident light, ${ϵ}_{\mathrm{ell}}^{\left(\mathrm{inc}\right)}$, impinging onto the cell. Using the exact analytical expressions for the transfer matrix we show that variations of the ellipticity, ${ϵ}_{\mathrm{ell}}^{\left(\mathrm{inc}\right)}$, induce transformations of the angular pattern exhibiting the effect of avoided $L$-line crossings and characterized by topological events such as creation and annihilation of the $C$ points. The predictions of the theory are found to be in good agreement with the experimental results.

There exists a substantial body of theory that predicts mutual screening of signed topological singularities (topological charges) in random optical fields (speckle patterns). Such screening appears to be rather mysterious because there are neither energetic nor entropic reasons for its existence. We present the first experimental confirmation of mutual screening by the stationary points of the intensity, the canonical optical scalar field, and of mutual screening by $C$ points in elliptically polarized light, the generic optical vector field. We also elucidate specific aspects of the geometry and topology of these fields that we argue give rise to screening.

The canonical point singularity of elliptically polarized light is an isolated point of circular polarization, a C point. As one recedes from such a point the surrounding polarization figures evolve into ellipses characterized by a major axis of length a, a minor axis of length b, and an azimuthal orientational angle α: at the C point itself, α is singular (undefined) and a and b are degenerate. The profound effects of the singularity in α on the orientation of the ellipses surrounding the C point have been extensively studied both theoretically and experimentally for over two decades. The equally profound effects of the degeneracy of a and b on the evolving shapes of the surrounding ellipses have only been described theoretically. As one recedes from a C point, a and b generate a surface that locally takes the form of a double cone (i.e., a diabolo). Contour lines of constant a and b are the classic conic sections, ellipses or hyperbolas depending on the shape of the diabolo and its orientation relative to the direction of propagation. We present measured contour maps, surfaces, cones, and diabolos of a and b for a random ellipse field (speckle pattern).