Conical Refraction with Flat-Topped Vortex Hollow Beam through Biaxial Crystals

This paper presents a new theoretical method to determine conical refraction (CR) with a light beam by examining the flat-topped vortex hollow beam in a biaxial crystal. The analytical formula for the considered CR is derived in a closed form. Some graphical representations are plotted to show how the source beam parameters N and M, the control parameter ρ0 of CR between the cylinder radius R0 and the waist radius ω, and the propagation distance z, affect the transverse intensity pattern behind the biaxial crystal. Results show that the circularly polarized flat-topped vortex hollow beams entering the crystal will transform the classical double-ring pattern into a multi-ring one consisting of bright rings. The evolution properties of CR with vortex beams upon propagation is also discussed and compared with the reference beam. Through numerical simulations, we established a good agreement between the integrated expressions method and our analytical formula. The present study generalizes the cases of the circularly polarized Gaussian and the flat-topped Gaussian transformed by the CR in a biaxial crystal.