Intensity features of the general model vortex higher‑order cosh‑Gaussian beam in free space and gradient-index media

In the framework of the generalized Huygens–Fresnel diffraction integral and paraxial approximation theory, the propagation features of the general model vortex higher-order cosh-Gaussian beam (GMvHchGB) in a free space (FS) and a gradient-index medium (GIM) are studied. The analytical formulas of the GMvHchGB propagating through the considered media are derived. According to the derived formulas, some numerical examples of the normalized intensity distributions and corresponding phase structure of the GMvHchGB in both FS and GIM are performed under different parameter conditions during propagation setups. The obtained results show that the beam with higher decentered parameters, the GMvHchGB propagating in FS case cannot maintain its initial shape during the propagation process, whereas the beam propagation in the GIM case will keep the original shape unchanged with a repeated self-focusing along propagation distances. Finally, the numerical results of both intensity and phase distributions demonstrated that the beam order N significantly influences the vortex property of the beam, which enhances the clear importance of the current work.