Vectorial phase retrieval in super-resolution polarization microscopy

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). Yet, this shape can be significantly affected by aberrations and other imperfections in the imaging system, leading to an 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 by a set of PSF measurements at varying focal planes and for various polarization projections. This PZ-stack of PSFs, which contains diversity in both phase and polarization projection, 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 modeling and characterization is made freely available.