In this article, a general analytical treatment (any topological charge---any number of discretization levels) for the diffraction of a Gaussian beam through a discretized vortex-producing lens is presented. In the proposal, the field is expressed as a sum of Kummer beams with different amplitudes and topological charges, which are focalized at different planes on the propagation axis. Likewise, it is demonstrated that characteristics of diffracted light can be modified by tuning the parameters of the setup. Vortex lines are analyzed to understand the internal mechanism of measurable topological charges that appear in specific planes, apparently violating topological charge conservation. Conservation of the topological charge is verified and theoretical predictions are supported by experiments.