Non-uniform pupil filters placed on the exit pupil of an optical system produce effects on the point spread function (PSF), both in the transverse plane, generating apodization or superresolution, and axially, varying the depth of focus. Usually amplitude-only or phase-only pupil filters are considered. These pupil filters can be made programmable by using a liquid crystal spatial light modulator (LC-SLM). In these cases the LC-SLM must be configured in order to produce an amplitude-only or phase-only modulation versus the applied voltage. These situations can be obtained through the polarization configuration of the display. In most cases these filters have been treated using paraxial scalar theory, and the filters are axially symmetric. Filters can be constructed of an array of rings, or alternatively can consist of a continuously variable transmittance. Filters can be designed by application of formulae for different design parameters, by optimization, or by exhaustive search. Filters consisting of an array of rings can involve many independent variables. Here we consider two of the simplest examples, those of a filter consisting of only two or three rings. Improved expressions for the performance parameters for transverse and axial gains for complex pupil filters are revised. These expressions can be used to predict the behavior of filters that give a small axial shift in the focal intensity maximum and also predict the changes in gain for different observation planes The axial behavior is also studied for the case of optical systems with high aperture. The introduction of a generalized pupil function allows the determination of the conditions to obtain axially invariant pupil filters.