We show here that the product of two Laguerre-Gaussian (LG) beams, i.e. double LG beams (dLG), can be represented as finite superposition of conventional LG beams with certain coeffi-cients that are expressed via zero-argument Jacobi polynomials. This allows obtaining an explicit expression for the complex amplitude of the dLG beams in the Fresnel diffraction zone. Generally, such beams do not retain their structure, changing shape upon free-space propagation. However, if both LG beams are of the same order, we obtain a special case of a "squared" LG beam, which is Fourier-invariant. Another special case of the dLG beams is obtained when the azimuthal indices of the Laguerre polynomials are equal to n – m and n + m. For such a beam, an explicit expression is obtained for the complex amplitude in the Fourier plane. We show that if the lower indices of the constituent LG beams are the same, such a double LG beam is also Fourier-invariant. Similar to conventional LG beams, the product of LG beams can be used for optical data transmission, since they are characterized by azimuthal orthogonality and carry an orbital angular momentum equal to the topological charge.

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 transformed 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.

We calculate the topological charge (TC) of a coherent axial superposition of different-"color" Laguerre-Gaussian (LG) beams, each having a different wavelength and TC. It is found that the TC of such a superposition is equal to the TC of the longer-wavelength constituent LG beam regardless of the weight coefficient of this beam in the superposition and its corresponding TC. It is interesting that the instantaneous TC of such a superposition is conserved and the (time-averaged) intensity distribution of the "colored" optical vortex changes its light "gamut": whereas in the near field with increasing radius, colors of the light rings (rainbow) change according to their TC in the superposition from the smaller TC to the larger one, upon free-space propagation (to the far field), with increasing radius, the ring colors in the rainbow get arranged in the reverse order from the larger TC to the smaller one. It is also shown that choosing the wavelengths (blue, green, and red)in a special way in a three-color superposition of single-ringed LG beams allows obtaining a time-averaged white light ring at a certain distance.