Three-dimensional nondiffracting Weber beams

Controlling the shape of higher-dimensional nondiffracting beams is one of the important research topics in current beam propagation theory and engineering practice. This article investigates the three-dimensional Helmholtz equation and derives its exact beam solution, which incorporates Weber functions and a single control parameter. Building upon the nondiffracting beam solution obtained, we analyze the excited states of Weber beams for different values of the control parameter, including the fundamental state, the first excited state, and even- and odd-order Weber beams. Our findings reveal that the fundamental state of Weber beams exhibits a pancake-like shape, while the first excited state forms a tube-like shape. Odd-order beams display toroidal shapes, whereas even-order beams combine toroidal and ellipsoidal shapes. Typically, the toroidal structures exhibit vortex-type energy distributions, with higher intensities appearing at the edges of tubes, while the ellipsoidal structures display Gaussian-type energy distributions, with higher energies concentrated at the center of pancake-like regions. The method proposed in this study for constructing higher-dimensional exact solutions of the Helmholtz equation using a novel coordinate transformation can be extended to other higher-dimensional models.