Conservation theorems for the orbital angular momentum of a superposition of shifted optical vortices

We proved two conservation theorems for the orbital angular momentum (OAM) of a superposition of identical optical vortices with an arbitrary radially symmetric shape, integer topological charge n , and arbitrary off-axis shift. The normalized OAM of such a superposition for any real weights equals the OAM of each individual beam contributing to the superposition. If the beam centers are located on a straight line passing through the origin, then, even if the superposition weights are complex, the normalized OAM of the whole superposition equals that of each contributing beam. These theorems allow of generating vortex laser beams with different (not necessarily radially symmetric) intensity distribution, but with the same OAM. The results of numerical simulation are given for superpositions of Bessel beams, Hankel-Bessel, Bessel-Gaussian and Laguerre-Gaussian beams.