  In the field of structured light, the study of optical vortices and their
vectorial extension--vectorial vortex beams--has garnered substantial interest
due to their unique phase and polarisation properties, which make them
appealing for many potential applications. Combining the advantages of vortex
beams and anisotropic materials, new possibilities for electromagnetic field
tailoring can be achieved in nonlinear optics, quantum and topological
photonics. These applications call for a comprehensive modelling framework that
accounts for properties of both anisotropic materials and vector vortex beams.
In this paper, we describe a semi-analytical model that extends the vectorial
diffraction theory to focused vortex beams propagating through a uniaxial slab,
considering the cases of scalar and vectorial vortexes in the common framework
of a Laguerre-Gaussian modes. The model aims to provide a comprehensive
description of the methodology, enabling the implementation of complex beams
transmission through, reflection from and propagation in uniaxial anisotropic
materials for specific applications. We apply the developed approach to
propagation of high-order vortex beams in uniaxial materials with various
dispersion characteristics> elliptic, hyperbolic and epsilon-near-zero regimes.
We show how variations of the medium anisotropy modify the beam structure due
to the vectorial nature of their interaction, which results from the different
permittivities of the medium for transverse and longitudinal field components.
The applicability of the approach can be extended to artificially structured
media if they can described by effective medium parameters. The developed
formalism will be useful for modelling of interaction of complex beams with
uniaxial materials, allowing a common framework for a large variety of
situations, which can also be extended beyond the electromagnetic waves.
