Propagation properties of Gaussian vortex beams through the gradient-index medium

Based on the extended Huygens–Fresnel principle, the analytical expression for Gaussian vortex beams propagating through a gradient-index medium is obtained. The propagation trajectory, the intensity distributions, and the phase distributions of Gaussian vortex beams propagating in the gradient-index medium are demonstrated, and the effects of gradient-index parameter β and topological charge m on these propagation properties are analysed in detail. It is found that the propagation trajectory and intensity distributions change on periodically with the increasing of propagation distance, and the periodicities are determined by the gradient-index parameters. As the increasing of the gradient-index parameters, the periodic variations on the propagation trajectory and intensity distributions are slowed down. Interestingly, the phase always circulates about the central null in the medium, but the rotation direction will change. The phase distributions also change on periodically in the medium, but the periodicities are different when the topological charge is odd or even. Furthermore, increasing the topological charge will increase the size of the beam spot and the circular dark hollow in the medium.