Spherical Gauss-Laguerre beam propagation in 4D space-time

In this paper, what we believe to be a novel class of beams, which are referred to as the spherical Gauss-Laguerre beams, are proposed. The beams propagate stably in the anomalous dispersive media, within which the second order derivative with respect to t could be combined with the two-dimensional (2D) Laplacian operator in the transverse direction and forms a three-dimensional (3D) Laplacian operator, which describes the beam propagation in the z direction within the four-dimensional (4D) x-y-z-t space-time. The wave equation is solved by the variable separation method and the analytical expression for the spherical Gauss-Laguerre beams is derived. The beams have a 3D Gaussian field distribution with a variable beam waist with respect to the propagation distance. Unlike any 2D spatial vortex beams, the 3D beams could possess either the spatial vortex or the spatiotemporal optical vortex (STOV) by choosing the vortex plane in the 3D x-y-t space-time. The derived spherical Gauss-Laguerre beam expression in the 4D space-time is verified by the numerical simulations with excellent agreement.