Modal structure of two-dimensional nondiffracting Weber beams

The control of the shape and spatial distributions of light beams is one of the hottest topics in current optical research. To this end, we construct an exact solution of the two-dimensional (2D) Helmholtz equation, using coordinate transformation and variable separation methods. The analytical solution of the new Weber beam is expressed in terms of Weber functions, which feature a nonnegative integer mode parameter that controls the structure of the beam. Since the problem is 2D, the solution involves the products of two Weber functions with different coordinates, each governed by its own mode parameter. By analyzing Weber functions for different values of the parameter, we obtain intensity distributions and mode patterns of zero-, even-, and odd-order Weber beams, demonstrating their propagation characteristics. This study reveals the modal structure of nondiffracting Weber beams, provides a novel method for controlling other nondiffracting beams, and promotes the development of beam modulation technologies.