A Fourier-invariant squared Laguerre-Gaussian vortex beam

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