Axially invariant pupil filters

We establish general conditions that complex and real valued pupil filters must satisfy to produce an identical axial response. We show several conditions that permit the generation of families of filters with identical axial response but with different transverse response. In particular we show results for supergaussian rings that depend on three parameters. Two of the parameters can determine the shape of the axial response achieving arbitrarily high focal depth. For a given axial response the third parameter controls the transverse response, permitting the attainment of super-resolution up to a limit. Loss of light throughput and the sidelobe's increase are also studied. The axial behaviour 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.