Propagation of the first order annular Bessel Gaussian beams in a uniaxial crystal along the optical axis governed by the Pockels effect.

We investigate the first order annular Bessel Gaussian beams propagating in a strontium barium niobate (SBN) crystal governed by the Pockels effect. The left-hand circularly polarized incident waves propagating along the optical axis in the crystal give rise to the right-hand circularly polarized vortex fields with a topological charge of 2. If an external dc field is applied along the optical axis of the SBN crystal, the anisotropy strength can be controlled rapidly, and the rotational invariance around the optical axis is still maintained. In this case, the normalized intensity in the focal region, the power exchange between left-hand and right-hand components, and the exchange between the spin and orbit contributions of the angular momentum flux can be manipulated. We also consider the case where the incident waves are linearly polarized along the x axis. The cylindrical symmetry of two Cartesian components is broken during propagation due to the anisotropy, and the intensity distribution as well as the polarization state of the beams can be regulated electrically.