  In this work, an analytic expression is derived for the optical field
generated by an off-axis Gaussian beam diffracted by a Discrete
Vortex-Producing Lens. With this system, a multifocal arrangement of asymmetric
optical vortices is obtained whose topological charge values change with the
position along the optical axis. This scheme allows obtaining a principal
asymmetric vortex corresponding with the topological charge value of the phase
mask and secondary ones with different charges. With the analytical expression,
the effects induced by the discretization and misalignment on the irradiance
and phase of each vortex can be simultaneously studied. A signal-to-noise ratio
expression is derived to verify if the noise of the multifocal system might
affect importantly the optical vortices of interest. Finally, we proposed new
metrics to measure the off-axis displacement using the phase of the vortex. We
concluded that noise is not a problem, then a Discrete Vortex-Producing Lens
can be used as a continuous Spiral Phase Plate with the bonus that secondary
optical vortices can be used in the same way as principal optical vortices.
