Orbital angular momentum of an elliptically symmetric laser beam after passing an elliptical spiral phase plate

We obtain a simple closed expression for the normalized orbital angular momentum (OAM) (OAM per unit power) of an arbitrary paraxial light beam of an elliptical cross-section, diffracted by an elliptical spiral phase plate (SPP), rotated by an arbitrary angle around the optical axis. Note that the ellipticities of the beam and of the SPP can be different. It is shown that when an elliptical beam illuminates an elliptical SPP, the normalized OAM of the output beam is maximal / minimal if the ellipses of the beam cross-section and the SPP are parallel / orthogonal. The results can be used in optical trapping, e.g. for continuously changing the OAM transferred to a particle by rotating the SPP around the optical axis.