  In single molecule orientation localization microscopy, valuable information
about the orientation and longitudinal position of each molecule is often
encoded in the shape of the point spread function (PSF). This shape, though,
can be affected significantly by aberrations and other imperfections in the
imaging system, leading to erroneous estimation of the measured parameters. A
basic solution is to model the aberrations as a scalar mask in the pupil plane
that is characterized through phase retrieval algorithms. However, this
approach is not suitable for cases involving polarization-dependent
aberrations, introduced either through unintentional anisotropy in the elements
or by using birefringent masks for PSF shaping. Here, this problem is addressed
by introducing a fully vectorial model in which the polarization aberrations
are represented via a spatially-dependent Jones matrix, commonly used to
describe polarization-dependent elements. It is then shown that these
aberrations can be characterized from a set of PSF measurements at varying
focal planes and for various polarization projections. This PZ-stack of PSFs,
which contains both phase and polarization projection diversity, is used in a
phase retrieval algorithm based on nonlinear optimization to determine the
aberrations. This methodology is demonstrated with numerical simulations and
experimental measurements. The pyPSFstack software developed for the modeling
and characterization is made freely available.
