  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.
