The propagation dynamics of the Airy Gaussian vortex beams in uniaxial crystals orthogonal to the optical axis has been investigated analytically and numerically. The propagation expression of the beams has been obtained. The propagation features of the Airy Gaussian vortex beams are shown with changes of the distribution factor and the ratio of the extraordinary refractive index to the ordinary refractive index. The correlations between the ratio and the maximum intensity value during the propagation, and its appearing distance have been investigated.

The propagation of right-hand circularly polarized Airy-Gaussian beams (RHCPAiGBs) through slabs of right-handed materials (RHMs) and left-handed materials (LHMs) is investigated analytically and numerically with the transfer matrix method. An approximate analytical expression for the RHCPAiGBs passing through a paraxial ABCD optical system is derived on the basis of the Huygens diffraction integral formula. The intensity and the phase distributions of the RHCPAiGBs through RHMs and LHMs are demonstrated. The influence of the parameter χ₀ on the propagation of RHCPAiGBs through RHM and LHM slabs is investigated. The RHCPAiGBs possess transverse-momentum currents, which shows that the physics underlying this intriguing accelerating effect is that of the combined contributions of the transverse spin and transverse orbital currents. Additionally, we go a step further to explore the radiation force including the gradient force and scattering force of the RHCPAiGBs.

Propagation of Airy-Gaussian vortex (AiGV) beams through the gradient-index medium is investigated analytically and numerically with the transfer matrix method. Deriving the analytic expression of the AiGV beams based on the Huygens diffraction integral formula, we obtain the propagate path, intensity and phase distributions, and the Poynting vector of the first- and second-order AiGV beams, which propagate through the paraxial ABCD system. The ballistic trajectory is no longer conventional parabolic but trigonometric shapes in the gradient-index medium. Especially, the AiGV beams represent the singular behavior at the propagation path and the light intensity distribution. The phase distribution and the Poynting vector exhibit in reverse when the AiGV beams through the singularity. As the order increases, the main lobe of the AiGV beams is gradually overlapped by the vortex core. Further, the sidelobe weakens when the AiGV beams propagate nearly to the singularity. Additionally, the figure of the Poynting vector of the AiGV beams proves the direction of energy flow corresponding to the intensity distribution. The vortex of the second-order AiGV beams is larger, and the propagation velocity is faster than that of the first order.

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