We consider the propagation of a tilted high order paraxial vortex-beam through a homogeneous anisotropic medium of a uniaxial crystal. We found that the initially circularly polarized beam bearing the lorder optical vortex splits into ordinary and extraordinary beams with a complex vortex structure. After a series of dislocation reactions the vortices gather together at the axis of the partial beam with the initial circular polarization shaping the l-order optical vortex. However, only l-1 vortices gather together on the axis of the partial beam with the orthogonal circular polarization. One optical vortex is shifted along the direction perpendicular to the inclination plane of the beam. Such a vortex displacement induces the transverse shift of the partial beam. In fact, we deal with the beam quadrefringence in a uniaxial, homogeneous anisotropic medium. The first two beams is a result of the splitting of the initial tilted beam into the ordinary and extraordinary ones. The appearance of the second two beams is caused by the transverse shift of one of the circularly polarized components in the initially circularly polarized vortex-beam or both circularly polarized components in the initially linearly polarized beam .We consider this effect both in terms of the conservation law of the angular momentum flux along the crystal optical axis and on the base of the solutions to the paraxial wave equation for the initially circularly and linearly polarized beams. We revealed that the transverse shift of the crystal traveling beam depends on neither a magnitude nor a sign of the vortex topological charge being defined only by a handedness of the initial circular polarization and a sign of the inclination angle of the beam. Also we analyze the deformation of the cross-section of the shifted vortex-beams and its evolution as the beam propagates along the crystal.

The splitting of a single optical vortex into four separate ones in a singular beam is theoretically and experimentally described for the propagation of obliquely incident light in a uniaxial crystal. We also find the condition under which the generated vortices in each of the four individual beams propagate independently without changing their structure and have different locations in all beams for any crystal lengths.

We have considered the paraxial vector erf-Gaussian beams with field distribution in the form of the error function that are shaped by the cone of plane waves with a fractional step of the azimuthal phase distribution modulated by the Gaussian envelope. We have revealed that the initial distributions of the transverse electric and transverse magnetic fields have the form far from standard ones but at the far diffraction field the field distributions recover nearly the symmetric form. We have also revealed that the half-charged vortices in one of the field components can propagates up to the Rayleigh length without essential structural transformations but then splits into an asymmetric net of singly charged vortices.

We implement a novel experimental technique for generating mono- and polychromatic optical beams with on-axis single vortex by manipulating polarization singularities of light in birefringent crystals. We demonstrate that, in contrast to the well-known optical quadrupoles generated by beams propagating along the optical axis of a uniaxial crystal, the beam bearing isolated single-charge on-axis vortex can be generated if the incident beam is tilted with respect to the optical axis at a certain angle.

We describe how the propagation of light through uniaxial crystals can be used as a versatile tool towards the spatial engineering of polarization and phase, thereby providing an all-optical technique for vectorial and scalar singular beam shaping in optics. Besides the prominent role played by the linear birefringence, the influence of circular birefringence (the optical activity) is discussed as well and both the monochromatic and polychromatic singular beam shaping strategies are addressed. Under cylindrically symmetric light-matter interaction, the radially, azimuthally, and spirally polarized eigen-modes for the light field are revealed to be of a fundamental interest to describe the physical mechanisms at work when dealing with scalar and vectorial optical singularities. In addition, we also report on nontrivial effects arising from cylindrical symmetry breaking, e.g. tilting the incident beam with respect to the crystal optical axis.