Fractional orbital angular momentum of a Gaussian beam with an embedded off-axis optical vortex

We theoretically and experimentally study asymmetric Gaussian optical vortices generated via diffracting a Gaussian beam by an off-axis spiral phase plate or amplitude "fork" hologram. It is shown that depending on the magnitude of the off-axis shift the laser beam can acquire a crescent-shaped cross-section, which rotates on propagation. We obtain an analytical expression for the orbital angular momentum (OAM) of such a beam, which appears to be fractional. We show that with increasing off-axis shift, the OAM decreases parabolically. The experimental results are in qualitative agreement with theory and numerical simulation.