WALOP-South: a four camera one shot imaging polarimeter for the PASIPHAE survey. Paper III -- PSF modelling
The two WALOP instruments, built for the PASIPHAE survey, will measure the linear polarization of large numbers of stars in the Galactic polar regions in the SDSS-$r$ band. They are designed with a wide field-of-view, enabling measurement of the Stokes parameters $I$, $q$, and $u$ for multiple stars simultaneously within a $35^\prime \times 35^\prime$ region of sky for WALOP-South and $30^\prime \times 30^\prime$ region for WALOP-North. In this paper, we present a polar shapelet-based PSF photometry framework for well-sampled stellar point sources applicable to WALOP-type wide-field polarimeters. Polar shapelets are a set of orthogonal basis functions, constructed from Gauss-Hermite or Gauss-Laguerre polynomials, that are well-suited to modelling localized PSF in a compact and efficient way. We developed an efficient PSF modelling method that uses polar shapelets as basis functions to reconstruct the spatial variation of the PSF shape across the CCD using Zemax-simulated images of one of the WALOP instruments, and show that a limited number of shapelet coefficients are sufficient to capture this variation consistently across different CCD locations. To simulate realistic star images, we introduce random sub-pixel shifts in the star centroids in the Zemax-simulated images, and account for this using a two-step iterative method that alternately estimates the PSF model and the sub-pixel centroid shift. Applying PSF photometry to the target faint stars, we demonstrate that the photometric accuracy of approximately 0.15% is achievable, and that the reconstructed PSF model can be incorporated into the photometry across different seeing conditions, meeting the polarimetric science requirements of the PASIPHAE survey.