Representation of shifted vortex beams of arbitrary order as a combination of nonshifted vortices

In this paper, we investigate the focusing of beams with a displaced vortex of arbitrary order. For these purposes, we use the following mathematical model: the beam is represented as a combination of beams having a nonshifted vortex. It is shown theoretically that an optical vortex of an arbitrary integer order m displaced within an axisymmetric beam is equivalent to the finite sum of nonshifted vortices of orders from 0 to m inclusive. If the order of the displaced vortex beam is non-integer, then the sum is replaced by an infinite series. Numerical simulation was carried out under sharp focusing conditions in the Debye approximation. The obtained pictures of the focused displaced vortex beams, regardless of the order of the vortex and the magnitude of the displacement, have a qualitatively identical form - the shape of the Crescent. To obtain focal pictures of another type, the illuminating beam must contain an optical vortex of the opposite sign. The obtained results are relevant for multichannel communication systems based on the separation of laser beams carrying orbital angular momentum.