  We report the use of a phase retrieval procedure based on maximum likelihood
estimation (MLE) to produce an improved, experimentally calibrated model of a
point spread function (PSF) for use in three-dimensional (3D) localization
microscopy experiments. The method estimates a global pupil phase function
(which includes both the PSF and system aberrations) over the full axial range
from a simple calibration scan. The pupil function is used to refine the PSF
model and hence enable superior localizations from experimental data. To
demonstrate the utility of the procedure, we apply it to experimental data
acquired with a microscope employing a tetrapod PSF with a 6 micron axial
range. The phase-retrieved model demonstrates significant improvements in both
accuracy and precision of 3D localizations relative to the model based on
scalar diffraction theory. The localization precision of the phase-retrieved
model is shown to be near the limits imposed by estimation theory, and the
reproducibility of the procedure is characterized and discussed. Code which
performs the phase retrieval algorithm is provided.
