A Fourier-invariant squared Laguerre-Gaussian vortex beam

It is shown that a squared Laguerre-Gaussian (LG) vortex beam is Fourier-invariant and retains its structure at the focus of a spherical lens. In the Fresnel diffraction zone, such a beam is trans-formed into superposition of conventional LG beams, the number of which is equal to the number of rings in the squared LG beam. If there is only one ring, then the beam is structurally stable. A more general beam, which is a “product” of two LG beams, is also considered. Such a beam will be Fourier-invariant if the number of rings in two LG beams in the “product” is the same. The considered beams complement the well-known family of LG beams, which are intensively studied as they remain stable during their propagation in turbulent media.