We demonstrate single-molecule fluorescence imaging beyond the optical diffraction limit in 3 dimensions with a wide-field microscope that exhibits a double-helix point spread function (DH-PSF). The DH-PSF design features high and uniform Fisher information and has 2 dominant lobes in the image plane whose angular orientation rotates with the axial (z) position of the emitter. Single fluorescent molecules in a thick polymer sample are localized in single 500-ms acquisitions with 10-to 20-nm precision over a large depth of field (2 m) by finding the center of the 2 DH-PSF lobes. By using a photoactivatable fluorophore, repeated imaging of sparse subsets with a DH-PSF microscope provides superresolution imaging of high concentrations of molecules in all 3 dimensions. The combination of optical PSF design and digital postprocessing with photoactivatable fluorophores opens up avenues for improving 3D imaging resolution beyond the Rayleigh diffraction limit.microscopy ͉ photoactivation ͉ superresolution ͉ computational imaging ͉ PSF engineering F luorescence microscopy is ubiquitous in biological studies because light can noninvasively probe the interior of a cell with high signal-to-background and remarkable label specificity. Unfortunately, optical diffraction limits the transverse (x-y) resolution of a conventional fluorescence microscope to approximately /(2NA), where is the optical wavelength and NA is the numerical aperture of the objective lens (1). This limitation requires that point sources need to be Ͼ Ϸ200 nm apart in the visible wavelength region to be distinguished with modern high-quality fluorescence microscopes. Diffraction causes the image of a single-point emitter to appear as a blob (i.e., the point-spread function or PSF) with a width given by the diffraction limit. However, if the shape of the PSF is measured, then the center position of the blob can be determined with a far greater precision (termed superlocalization) that scales approximately as the diffraction limit divided by the square root of the number of photons collected, a fact noted as early as Heisenberg in the context of electron localization with photons (2) and later extended to point objects (3, 4) and single-molecule emitters (5-8). Because single-molecule emitters are only a few nanometers in size, they represent particularly useful point sources for imaging, and superlocalization of single molecules at room temperature has been pushed to the 1-nm regime (9) in transverse (2-dimensional) imaging. In the third (z) dimension, diffraction also limits resolution to Ϸ2n /NA 2 with n the index of refraction, corresponding to a depth of field of Ϸ500 nm in the visible wavelength region with modern microscopes. Improvements in 3D localization beyond this limit are also possible by using astigmatism (10, 11), defocusing (12), or simultaneous multiplane viewing (13).Until recently, superlocalization of individual molecules was unable to provide true resolution beyond the diffraction limit (superresolution) because the concentration of emi...

We demonstrate three-dimensional tracking of fluorescent microparticles, with a computational optical system whose point spread function (PSF) has been engineered to have two twisting lobes along the optical axis, generating a three-dimensional (3D) double-helix (DH) PSF. An information theoretical comparison in photon limited systems shows that the DH-PSF delivers higher Fisher information for 3D localization than the standard PSF. Hence, DH-PSF systems provide better position estimation accuracy. Experiments demonstrate average position estimation accuracies under 14nm and 37nm in the transverse and axial dimensions respectively. The system determines the 3D position of multiple particles with a single image and tracks them over time while providing their velocities.

Rotating point spread functions (PSFs) present invariant features that continuously rotate with defocus and are important in diverse applications such as computational imaging and atom/particle trapping. However, their transfer function efficiency is typically very low. We generate highly efficient rotating PSFs by tailoring the range of invariant rotation to the specific application. The PSF design involves an optimization procedure that applies constraints in the Gauss-Laguerre modal plane, the spatial domain, and the Fourier domain. We observed over thirty times improvement in transfer function efficiency. Experiments with a phase-only spatial light modulator demonstrate the potential of high-efficiency rotating PSFs.

Photo-activation localization microscopy is a far-field superresolution imaging technique based on the localization of single molecules with subdiffraction limit precision. Known under acronyms such as PALM (photo-activated localization microscopy) or STORM (stochastic optical reconstruction microscopy), these techniques achieve superresolution by allowing only a sparse, random set of molecules to emit light at any given time and subsequently localizing each molecule with great precision. Recently, such techniques have been extended to three dimensions, opening up unprecedented possibilities to explore the structure and function of cells. Interestingly, proper engineering of the three-dimensional (3D) point spread function (PSF) through additional optics has been demonstrated to theoretically improve 3D position estimation and ultimately resolution. In this paper, an optimal 3D single-molecule localization estimator is presented in a general framework for noisy, aberrated and/or engineered PSF imaging. To find the position of each molecule, a phase-retrieval enabled maximum-likelihood estimator is implemented. This estimator is shown to be efficient, meaning it reaches the fundamental Cramer-Rao lower bound of x, y, and z localization precision. Experimental application of the phase-retrieval enabled maximum-likelihood estimator using a particular engineered PSF microscope demonstrates unmatched low-photon-count 3D wide-field single-molecule localization performance.computational imaging | nanoscopy | point spread function engineering | pupil encoding | inverse problems

Accurate estimation of the three-dimensional (3D) position of particles is critical in applications like biological imaging, atom/particle-trapping, and nanomanufacturing. While it is well-known that localization accuracy better than the Rayleigh resolution limit is possible, it was recently shown that, for photon-limited cases, 3D point spread functions (PSFs) can be shaped to increase accuracies over a 3D volume [Pavani and Piestun, Opt. Express 16, 22048 (2008)]. Here, we show that in the detector-limited regime, the gain in accuracy occurs in all three dimensions throughout the axial range of interest. The PSF is shaped as a double helix, resulting in a system with fundamentally better 3D localization accuracies than standard PSF systems, capable of achieving single-image subnanometer accuracies.