Measuring the topological charge of optical vortices: from the orbital angular momentum of individual photons to the angular uncertainty principle

We discuss several techniques for measuring the topological charge, m, of optical vortices by utilizing their m-fold rotational symmetry. Such a measurement also amounts to a measurement of the orbital angular momentum (OAM) per photon in units of hbar. The first technique involves the rotational Doppler shift, which was recently suggested as a way to measure the OAM of light beams for optical-communications purposes. The second technique involves interfering an optical vortex with the same vortex, but rotated about the vortex axis by a certain angle. This results in constructive interference for some m values, and destructive interference for other m values. This technique can, in principle, be used to measure the OAM of individual photons. The third technique uses an optical ring resonator that rotates the beam during each round trip. Such a resonator can be tuned to be resonant only for optical vortices with a given values of m. We have used such a resonator to demonstrate an angular form of the uncertainty principle.