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In this third paper of a series describing direction dependent correctionsfor polarimetric radio imaging, we present the the A-to-Z solver methodology tomodel the full Jones antenna aperture illumination pattern (AIP) with Zernikepolynomials. In order to achieve thermal noise limited imaging with modernradio interferometers, it is necessary to correct for the instrumental effectsof the antenna primary beam (PB) as a function of time, frequency, andpolarization. The wideband AW projection algorithm enables those correctionsprovided an accurate model of the AIP is available. We present the A-to-Zsolver as a more versatile algorithm for the modeling of the AIP. It employsthe orthonormal circular Zernike polynomial basis to model the measured fullJones AIP. These full Jones models are then used to reconstruct the fullMueller AIP repsonse of an antenna, in principle accounting for all theoff-axis leakage effects of the primary beam. The A-to-Z solver is generalenough to accomodate any interferometer for which holographic measurementsexist, we have successfully modelled the AIP of VLA, MeerKAT and ALMA as ademonstration of its versatility. We show that our models capture the PBmorphology to high accuracy within the first 1-2 sidelobes, and show theviability of full Mueller gridding and deconvolution for any telescope givenhigh quality holographic measurements.

Direction Dependent Corrections in Polarimetric Radio Imaging III: A-to-Z Solver -- Modeling the full Jones antenna aperture illumination pattern