Abstract
We investigate the two-dimensional normalized linear Schrödinger equation describing the propagation of diffraction-free beams in free space using the traditional variable separation method. We discover that each beam component satisfies the standard Weber differential equation. From this fact, we construct exact solutions utilizing two parabolic cylinder functions, which are denoted as the two-dimensional (2D) parabolic cylinder beams with diffraction-free characteristics that can be described by two mode numbers. By using numerical simulation and choosing appropriate mode numbers, we exhibit intensity profiles of the 2D parabolic cylinder beams, which display a matrix structure. It is demonstrated that the beams can be completely controlled by the choice of two mode numbers.
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