Analytical description of optical vortex beams generated by discretized vortex producing lenses

Discretized vortex-producing lenses programmed on low performance spatial light modulators have been used for the generation of optical vortices. However, the description of these vortices has been supported only by numerical simulations. In this work, a general analytical treatment (any topological charge - any discretization levels) for the propagation of a Gaussian beam through a discretized vortex-producing lens is presented. The resulting field could be expressed as a sum of Kummer beams with different amplitudes and topological charges focalized at different planes, whose characteristics of formation can be modified by tuning the parameters of the setup. Likewise, vortex lines are analyzed to understand the mechanism of formation of observed topological charges, which appear in specific planes. Conservation of the topological charge is demonstrated. Theoretical predictions are supported by experiments.