Spatiotemporal Airy beam modulated by elegant Laguerre and elegant Gaussian function in free space

Airy elegant Gaussian beam is the airy beam which is modulated by the elegant Gaussian. According to the (2+1)D paraxial wave equation describing the wave propagation of the electric field, Airy elegant Gaussian solution to the paraxial wave equation in spatial field are accessed. The solution of the Airy-Elegant-Largeurre-Gaussian(AiELG) beam, which is the spatiotemporal Airy elegant Gaussian beam modulated by the elegant Larguerre function, can be accessed by solving the (3+1)D Schrödinger equation in the free space. We found that their intensity profiles are determined by three parameters: the azimuthal mode number, the radial mode number and the modulation depth. It is worth mentioning that the radial mode number of AiELG wave packet will change in propagation.