Orbital angular momentum of an arbitrary axisymmetric light field after passing through an off-axis spiral phase plate

We obtain a simple formula for the relative total orbital angular momentum (OAM) of a paraxial light beam with arbitrary rotationally symmetric complex amplitude passed through a spiral phase plate (SPP) whose center is shifted from the optical axis. The formula shows that the OAM equals zero if the incident beam is bounded by an aperture and the SPP center is outside this aperture. For the incident beam bounded by an annular aperture, there is another interesting consequence of the obtained expression. The total OAM of such a beam is the same regardless of the position of the SPP center within the shaded circle of the aperture. Thus, it would be appropriate to illuminate the SPP by beams with an annular intensity distribution, since in this case an inaccurate alignment of the SPP center and the center of the annular intensity distribution does not affect the total OAM of the beam. We also obtain an expression for the OAM density of such a beam in the initial plane.