Polarization patterns in the transverse plane generically contain singularities: points of circular polarization (C points), lines of linear polarization (L lines), instantaneous zeros (disclinations) and component zeros. We investigate the geometry of ellipse fields at these singularities, using the Stokes parameters and others to characterize the singular geometry and morphology. Comparison is made with analogous structures on random surfaces, namely umbilic points and parabolic lines. The densities and correlations of the different types of polarization singularities are calculated in random polarization fields, and compared to the statistics of phase singularities and random surfaces.
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