Propagation of focused scalar and vector vortex beams in anisotropic media: A semianalytical approach

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 polarization properties, which make them appealing for many potential applications. Combining the advantages of vortex beams and anisotropic materials, unique possibilities for electromagnetic field tailoring and manipulation can be achieved in nonlinear optics, quantum and topological photonics. These applications call for a comprehensive modeling framework that accounts for properties of both anisotropic materials and vector vortex beams. In this paper, we describe a semianalytical model that extends the vectorial diffraction theory to the case of focused vortex beams propagating through a uniaxial slab, considering both the cases of scalar and vectorial vortices in the common framework of a Laguerre-Gaussian mode basis. The model aims to provide a comprehensive description of the methodology, enabling the implementation of complex beam transmission through, reflection from, and propagation in uniaxial anisotropic materials for specific applications. As a demonstration of its versatility, we apply the developed approach to describe propagation of high-order vortex beams in uniaxial materials with various dispersion characteristics, exploring the 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 be described by effective medium parameters. The developed formalism will be useful for modeling 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.