Optimal point spread function design for 3D imaging.

Optical imaging of single nanoscale objects such as a quantum dot, metallic nanoparticle, or a single molecule provides a powerful window into a variety of biological or material systems, and the physical problem of extracting maximum information from single emitters is an important goal. One application is single-particle tracking (SPT [1]), which relies upon extracting the spatial trajectory of a single moving molecular label, quantum dot, or metallic nanoparticle from a series of images. For example, a single mRNA particle can be localized and followed in a living cell in real-time [2]. Another application of single-molecule localization is “super-resolution” (SR) microscopy, [3–5] which works by ensuring that only a sparse subset of labels on an extended object (e.g. a cellular structure) are emitting in each imaging frame. One localizes the single emitters just as in SPT; the multitude of localizations are then reconstructed into a single, high-resolution image. This enables the spatial resolving power of SR microscopy to surpass the classical diffraction resolution limit by 5- to 10-fold. Historically, single-particle localization was used for 2D imaging, namely, inferring the x,y coordinates of each emitter, e.g. by centroid-fitting of by fitting to a 2D Gaussian [6]. However, the third spatial dimension, z, or the depth of an emitter, can also be inferred from its measured 2D image. This can be done by considering how the shape of the microscope’s point spread function (PSF) varies with emitter position. The PSF of a microscope is the image that is detected when observing a point source. For a standard microscope, to a good approximation, the PSF in focus (i.e. z=0) resembles a circular Airy pattern, and its shape is invariant to lateral shifts (x,y) of the emitter – however it will change upon defocus (z). Unfortunately, the standard PSF spreads out (defocuses) quickly with z which limits the range over which z can be determined. Importantly, to obtain much more useful 3D position information, the PSF of the microscope can be altered – for example by pupil (Fourier) plane processing [7,8]. Phase modulating the electromagnetic field in the Fourier plane is a low-loss method to encode z-information in the shape of the image on the camera. Examples of this include astigmatic PSFs [9,10], double-helix (DH-PSF) microscopy [11–13] or segmented phase ramps [14]. The precision to which a single emitter can be localized depends on several factors. These include the emitter’s brightness (detected photon flux), background fluorescence, detector pixel size, and detection noise [15,16]. Another key factor is the shape of the PSF itself. For example, in astigmatism-based 3D imaging, the PSF is altered to have an elliptical shape, and the z position of the emitter can be determined by the relative widths of the PSF along the two principal axes [9,10]. The double-helix PSF [12,13] is composed of two spots, with the angle between a line connecting them and the camera axis encoding the z position of the emitter. Among existing PSFs for 3D imaging, the double-helix PSF has been shown to allow a larger depth of field than astigmatism (~2–3 μm vs ~0.5–0.7 μm) [17], and a recently suggested PSF based on accelerating beams [18] demonstrates high, uniform precision over a 3 μm range. The purpose of this paper is to fundamentally improve upon these previous schemes. Here, we address the problem of finding a feasible and optimally informative PSF. Namely, we ask the question – given an imaging scenario with certain characteristics (e.g. magnification, noise level, pixel size, emitter signal) – what is the pupil plane pattern that would yield maximal physical information about the 3D position of an emitter, and what is the resulting optimal PSF? In other words – since localization precision depends on the PSF of the system – can we design the system to have a PSF that would yield the best possible precision in determining x, y, and z, compared to any other PSF? We regard such a PSF as optimally informative. A powerful measure of the effectiveness of a PSF for encoding an emitter’s position is based on Fisher information [17,19,20], a concept from statistical information theory. Fisher information is a mathematical measure of the sensitivity of an observable quantity (the PSF) to changes in its underlying parameters (emitter position). Using the Fisher information function, one may compute the Cramer-Rao lower bound (CRLB), which is the theoretical best-case x,y,z precision that can be attained (with any unbiased estimator) given a PSF and a noise model. With the right estimator, the best-case localization precision represented by the CRLB can be approached in practice [21–23]. Traditionally, the CRLB has been used as an analysis tool, i.e. to evaluate the performance of an existing PSF design, which is often conceived using physical intuition and reasonable requirements (e.g. a significant change of the PSF over the z-range of interest, and concentration of emitted light into small spots). The CRLB has also been used to fine-tune an existing PSF [24]. To find the optimal pupil plane pattern, and thereby the optimal PSF, we propose a new approach to PSF design –we treat the PSF as a free design parameter of the imaging system, and generate PSFs with optimal photon-efficient 3D position encoding, with no prior constraints on the shape of the PSF. This is achieved by CRLB optimization – that is, we directly solve the mathematical optimization problem of minimizing the CRLB (and hence improving the precision bound) of the system, and use the resulting PSF. Such a PSF will provide optimal precision by definition. Physically reasonable requirements are accounted for by using realistic imaging and noise models, including pixelation, photon shot-noise Poisson statistics, and background fluorescence. This enables us to demonstrate, for typical experimental conditions and without scanning, the highest theoretical precision to date over a 3 μm axial range, as well as <50 nm experimental precision for an unprecedented ~5 μm range.