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We have derived the corresponding equations and found their solutions bothfor nonparaxial and paraxial beams. The paraxial solutions we have presented inthe form of the generalized Hermite-Gaussian beams propagating perpendicular tothe optical axis of a uniaxial crystal. We have also constructed thegeneralized Laguerre-Gaussian beams at the z=0 plane and analyzed theirevolution in a homogeneous isotropic medium. Comparing it with the evolution ofthe standard Laguerre-Gaussian beams with and in the crystal we have revealedthat the additional elliptic deformation of the extraordinary beam results intopological reactions that essentially distorts field structure for the accountof different rotation rates of the vortex row originated from the centereddegenerate optical vortex and the conoscopic pattern. We have predictedconversion of the vortex topological charge at the beam axis similar to that inastigmatic lenses and analyzed the radical differences with this process. Wehave revealed the synchronic oscillations of the spin angular momentum and thesign of the vortex topological charge at the beam axis.

Non-canonical propagation of high-order elliptic vortex-beams in a uniaxial crystal