Generalized fractional one-dimensional Hermite-Gaussian beams

We introduce the generalized fractional Hermite-Gaussian (GFHG) beams, a type of structured light based on the fractionalization of Hermite functions. By incorporating a noninteger free parameter, these beams constitute a bridge between Hermite-Gaussian beams of integer order but exhibit an asymmetric light field distribution in their transverse profile. We show that the fractionalization can be seen as the result of a convolution between a Gaussian beam and a power law function. Remarkably, and in contrast to traditional Hermite-Gaussian beams of integer order, we find that to ensure light localization, there is a critical threshold value between the fractional Hermite functions and the corresponding Gaussian apodization. We demonstrate their numerical and theoretical propagation in free space and gradient-index (GRIN) media. Finally, we highlight a possible connection between these beams and nonlocal nonlinear media.