Orbital angular momentum of superpositions of optical vortices after passing through a sector diaphragm

In optical communications, it is desirable to know some quantities describing a light field, that are conserved on propagation or resistant to some distortions. Typically, optical vortex beams are characterized by their orbital angular momentum (OAM) and/or topological charge (TC). Here, we study what happens with the OAM of a superposition of two or several optical vortices (with different TCs) when it is distorted by a hard-edge sector aperture. We discover several cases when such perturbation does not violate the OAM of the whole superposition. The first case is when the incident beam consists of two vortices of the same power. The second case is when the aperture half-angle equals an integer number of $\pi$ divided by the difference between the topological charges. For more than two incident beams, this angle equals an integer number of $\pi$ divided by the greatest common divisor of all possible differences between the topological charges. For two incident vortex beams with real-valued radial envelopes of the complex amplitudes, the OAM is also conserved when there is a $±\pi/2$ phase delay between the beams. When two beams with the same power pass through a binary radial grating, their total OAM is also conserved.