Point spread function (PSF) engineering has extended far-field localization microscopy into three dimensions by encoding the axial position of each emitter into the shape of its image on the detector. By fitting the observed PSF to a model function, one can extract position information with sub-diffraction precision. However, in practice this procedure is often complicated by optical aberrations present in the imaging system, which distort the shape of the observed PSF relative to the model function. The mismatch between the model and observed PSFs can limit the accuracy and precision achieved by the localization procedure.
Here, we present a simple method to experimentally improve the model PSF by phase retrieval of the pupil function of the imaging system using a set of images of an isolated emitter at different displacements from the focal plane. The pupil function is estimated by adding a phase term consisting of a combination of Zernike modes to the theoretical electric field at the back focal plane of the microscope. The amplitudes of the Zernike modes are determined by maximizing the likelihood function over all pixels in the experimental data set. Importantly, since all data is taken with the phase mask in place, we account for any aberrations it introduces. Using the resulting pupil function, we generate a model PSF which is significantly improved over the theoretical model in both the accuracy and precision of experimental emitter localizations. We also provide a MATLAB package which performs the entire fitting procedure, from phase retrieval to single-emitter localization.
Petar N. Petrov, Yoav Shechtman, and William E. Moerner, "Localization beyond the diffraction limit (Conference Presentation)," Proc. SPIE 10071, Single Molecule Spectroscopy and Superresolution Imaging X, 1007108 (Presented at SPIE BiOS: January 28, 2017; Published: 24 April 2017); https://doi.org/10.1117/12.2251307.5380600068001.
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