Optimal point spread function design for 3D imaging

Abstract

To extract from an image of a single nanoscale object maximum physical information about its position, we propose and demonstrate a framework for pupil-plane modulation for 3D imaging applications requiring precise localization, including single-particle tracking and superresolution microscopy. The method is based on maximizing the information content of the system, by formulating and solving the appropriate optimization problem--finding the pupil-plane phase pattern that would yield a point spread function (PSF) with optimal Fisher information properties. We use our method to generate and experimentally demonstrate two example PSFs: one optimized for 3D localization precision over a 3 μm depth of field, and another with an unprecedented 5 μm depth of field, both designed to perform under physically common conditions of high background signals.

FIG 1

(Color online): Experimental pupil plane modulation setup with a phase mask in the Fourier plane of a 4f optical processing unit.

FIG 2

(Color online): (a) Saddle-Point (SP) PSF, optimized for high background, low signal 3D precision over a 3 μm depth range. (b) Calculated PSF for various z positions (stated). Scale bar: 1 μm (in sample space). (c) Calculated CRLB of the SP PSF for x, y, z vs. astigmatic (Astigm.) and DH PSF as a function of z. 3500 detected signal photons per frame and β = 50 were considered. Astigmatic axes are at 45⁰ relative to camera axes, hence the similar x and y behavior.

FIG 3

(Color online): (a) Experimental realizations of the Saddle-Point (SP) PSF for various z positions (stated). Scale bar: 1 μm (in sample space). Images were re-scaled to min/max intensity on an individual basis. (b) Experimental measurements of statistical localization precisions as the standard deviations σ_x, σ_y, σ_z of localization outcomes from 500 camera frames. Bars show standard deviations derived from n = 4 independent experiments. (c) Experimental histograms of localization outcomes along the x, y and z dimensions, recorded at z = −1 μm (dotted lines in (b), data from 4 measurements). Extracted σ_x, σ_y, σ_z are stated.