Zernike expansion of pupil filters: optimization of the signal concentration factor

Amplitude pupil filters for optimizing the signal concentration factor for a point spread function of given transverse and/or axial widths are derived. The pupil is expanded in a basis of Zernike polynomials. It is shown that the pupil that maximizes the signal concentration factor for a given transverse gain has a quadratically varying amplitude profile, as was shown in a previous paper, while the pupil that maximizes the signal concentration factor for a given axial gain has a quartic amplitude profile.