Several strategies in phase retrieval are unified by an iterative "difference map" constructed from a pair of elementary projections and three real parameters. For the standard application in optics, where the two projections implement Fourier modulus and object support constraints, respectively, the difference map reproduces the "hybrid" form of Fienup's input-output map when a particular choice is made for two of the parameters. The geometric construction of the difference map illuminates the distinction between its fixed points and the recovered object, as well as the mechanism whereby the form of stagnation encountered by alternating projection schemes is avoided. When support constraints are replaced by object histogram or atomicity constraints, the difference map lends itself to crystallographic phase retrieval. Numerical experiments with synthetic data suggest that structures with hundreds of atoms can be solved.
Aberration-corrected optics have made electron microscopy at atomic resolution a widespread and often essential tool for characterizing nanoscale structures. Image resolution has traditionally been improved by increasing the numerical aperture of the lens (α) and the beam energy, with the state-of-the-art at 300 kiloelectronvolts just entering the deep sub-ångström (that is, less than 0.5 ångström) regime. Two-dimensional (2D) materials are imaged at lower beam energies to avoid displacement damage from large momenta transfers, limiting spatial resolution to about 1 ångström. Here, by combining an electron microscope pixel-array detector with the dynamic range necessary to record the complete distribution of transmitted electrons and full-field ptychography to recover phase information from the full phase space, we increase the spatial resolution well beyond the traditional numerical-aperture-limited resolution. At a beam energy of 80 kiloelectronvolts, our ptychographic reconstruction improves the image contrast of single-atom defects in MoS substantially, reaching an information limit close to 5α, which corresponds to an Abbe diffraction-limited resolution of 0.39 ångström, at the electron dose and imaging conditions for which conventional imaging methods reach only 0.98 ångström.
We have used the method of x-ray diffraction microscopy to image the complex-valued exit wave of an intact and unstained yeast cell. The images of the freeze-dried cell, obtained by using 750-eV x-rays from different angular orientations, portray several of the cell's major internal components to 30-nm resolution. The good agreement among the independently recovered structures demonstrates the accuracy of the imaging technique. To obtain the best possible reconstructions, we have implemented procedures for handling noisy and incomplete diffraction data, and we propose a method for determining the reconstructed resolution. This work represents a previously uncharacterized application of x-ray diffraction microscopy to a specimen of this complexity and provides confidence in the feasibility of the ultimate goal of imaging biological specimens at 10-nm resolution in three dimensions. coherent x-ray diffraction imaging ͉ x-ray microscopy X -ray diffraction microscopy is a recently developed method in which only the coherent diffraction pattern of the sample is measured. It provides a path to high resolution without the limitations imposed by an x-ray optical system. The idea to image a noncrystalline object by phasing and inverting its diffraction pattern goes back to a suggestion by Sayre (1, 2) and was first demonstrated with x-rays by Miao et al. (3). In this article, we report the imaging of the complex-valued exit wavefront (both phase and magnitude) of a whole freeze-dried and unstained yeast cell. The images, at 30-nm resolution from multiple angular orientations of the cell, required an exposure of approximately one minute each using 750-eV x-rays (1 eV ϭ 1.602 ϫ 10 Ϫ19 J). This demonstration paves the way for the application of 3D x-ray diffraction microscopy (XDM) (4, 5) to frozen-hydrated samples in the future.High-resolution 3D images of biological samples are currently made by at least three methods: zone-plate x-ray microscopy (6-9), transmission electron microscopy (10, 11), and x-ray crystallography. All three have particular strengths and limitations. Both water-window (7-9) and multi-keV (12) zone-plate microscopes are currently limited to Ϸ60-nm 3D resolution by details of zone-plate resolution, depth of field, and operation. On the other hand, high-resolution transmission electron microscopes, although capable of extraordinary resolution, are limited by multiple electron scattering to specimens thinner than 0.5-1 m (10, 13). The third method, x-ray crystallography, traditionally yields the highest resolution structures and is the structural technique of choice, but it is limited to specimens that can be crystallized. In summary, the traditional structural techniques do not provide a capability for 3D imaging of an intact eukaryotic cell with resolution around 10 nm, and it is toward this end that our present efforts are directed.Since its introduction, XDM has been demonstrated with metal test objects in two dimensions (3, 14) and three dimensions (4) and with stained biological specimens (15) an...
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