Three-dimensional spatiotemporal nondiffracting parabolic cylinder beams

In this paper, the propagation characteristics of three-dimensional spatiotemporal nondiffracting parabolic cylinder beams in free space are studied. From the (3+1)-dimensional paraxial wave equation, we obtain an exact solution by the method of separating variables. The solution is constructed using parabolic cylinder functions, described by the three optical mode numbers. The spatiotemporal intensity distribution of the parabolic cylinder beam at five times Rayleigh range with different mode parameters is presented analytically and simulated numerically. Our results indicate that the intensity distribution can be well controlled by adjusting the values of the three mode numbers. They are of great significance for improved understanding of the formation mechanism of parabolic cylinders and other nondiffracting beams.