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Orbital angular momentum of a laser beam behind an off-axis spiral phase plate.

Here 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 an off-axis spiral phase plate (SPP). The formula shows that the OAM is zero if the incident light is bounded by a diaphragm, and the SPP center is outside it. Another consequence of the obtained formula is that if the incident light is limited by an annular diaphragm then the full angular momentum is the same, regardless of the position of the SPP center inside the shaded circle of the annular diaphragm. This means that it is more appropriate to illuminate the SPP by beams with an annular intensity distribution because, in this case, the inaccurate alignment of the SPP center and the center of the annular intensity distribution do not affect the total OAM of the beam. In addition, we obtain a formula for the OAM density of such a light beam in the initial plane. The experimentally measured OAM is in good agreement with that calculated by the obtained formula.