Three-dimensional localized Airy-Hermite-Gaussian and Airy-Helical-Hermite-Gaussian wave packets in free space.

We have investigated the 3D Airy-Hermite-Gaussian and Airy-Helical-Hermite-Gaussian wave packets by solving the (3+1)D free-space Schrödinger equation in Cartesian coordinates analytically. The complex amplitudes of these wave packets are described by the product of the superposition of two Airy functions, the Hermite-Gaussian functions and the helical Hermite-Gaussian functions. The different 3D spatiotemporal profiles, the photoelastics and vortex rings can be obtained by selecting different values of the following parameters: the mode number, the decay factor and the propagation distance.