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Eigen-Polarization beams in uniaxial crystals

We present a full set of the solution of the parabolic wave equation in the form of mode beams with the eigen polarizations. The theoretical analysis showed that the paraxial fundamental Gaussian beam can transform into a singular beam bearing the double charged optical vortex: the left-hand circularly polarized beam converts into the right-hand polarized beam with a positive topological charge whereas the right-hand one generates the left-hand polarized beam with a negative charged vortex. Moreover, the device can transform high-order singular beams in such a way that the negative topological charge of the vortex in the right-hand polarized beam raises by two units but the positive vortex charge in the same beam diminishes by two units.