General theorem on the divergence of vortex beams

The propagation and divergence properties of beams carrying orbital angular momentum (OAM) play a crucial role in many applications. Here we present a general study on the divergence of optical beams with OAM. We show that the mean absolute value of the OAM imposes a lower bound on the value of the beam divergence. We discuss our results for two different definitions of the divergence, the so-called rms or encircled energy. The bound on the rms divergence can be expressed as a generalized uncertainty principle, with applications in long-range communication, microscopy, and two-dimensional quantum systems.