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We analyze the quantum limit of sensitivity in four-dimensional scanning transmission electron microscopy (4D-STEM), which has emerged as a favored technique for imaging the structure of a wide variety of materials, including biological and other radiation-sensitive materials. 4D-STEM is an indirect (computational) imaging technique, which uses a scanning beam, and records the scattering distribution in momentum (diffraction) space for each beam position. We find that, in measuring a sample's electrostatic potential, the quantum Fisher information from 4D-STEM can match that from real-space phase-contrast imaging. Near-optimum quantum Fisher information is achieved using a delocalized speckled probe. However, owing to the detection in the diffraction plane, 4D-STEM ultimately enables only about half of the quantum limit, whereas Zernike phase-contrast imaging enables the quantum limit for all spatial frequencies admitted by the optical system. On the other hand, 4D-STEM can yield information on spatial frequencies well beyond those accessible by phase-contrast TEM. Our conclusions extend to analogous imaging modalities using coherent scalar visible light and x-rays.

We recently reported [1,2] measurements of the charge density fluctuations in the strange metal cuprate Bi$_{2.1}$Sr$_{1.9}$Ca$_{1.0}$Cu$_{2.0}$O$_{8+x}$ using both reflection M-EELS and transmission EELS with $\leq$10 meV energy resolution. We observed the well-known 1 eV plasmon in this material for momentum $q\lesssim$ 0.12 r.l.u., but found that it does not persist to large $q$. For $q\gtrsim0.12$ r.l.u., we observe a frequency-independent continuum, similar to that observed in early Raman scattering experiments [3,4], that correlates highly with the strange metal phase [2]. In his Comment (arXiv:2103.10268), Joerg Fink claims we do not see the plasmon, and that our results are inconsistent with optics, RIXS, and the author's own transmission EELS measurements with $\sim$100 meV resolution from the early 1990's [5,6]. The author claims we have made a trigonometry error and are measuring a larger momentum than we think. The author asserts that the two-particle excitations of cuprate strange metals are accurately described by weakly interacting band theory in RPA with corrections for conduction band carrier lifetimes and Umklapp effects. Here, we show that the author's Comment is in contradiction with known information from the literature. At $q\lesssim0.12$ r.l.u. we see the same 1 eV plasmon as other techniques. Moreover we compute our momentum correctly, adjusting the sample and detector angles during an energy scan to keep $q$ fixed. The only discrepancy is between our data and the results of Ref. [5] for $q\gtrsim0.12$ r.l.u. where, because of the coarse resolution used, the data had to be corrected for interference from the elastic line. A reexamination of these corrections in early transmission EELS measurements would likely shed light on this discrepancy., Comment: 5 pages, 3 figures. Response to arXiv:2103.10268. Wording updated in selected locations

This article is written to serve as an introduction and survey of imaging with synthetic aperture radar (SAR). The reader will benefit from having some familiarity with harmonic analysis, electromagnetic radiation, and inverse problems. After an overview of the SAR problem and some main concepts, the SAR imaging problem is contextualized in terms of classical harmonic analysis. Within this context, we consider partial Fourier sums of off-centered Fourier data and correspondingly the convolutional kernels resulting from conventional SAR image formation techniques. Following this, we revisit imaging of random complex signals from frequency data as in SAR, providing simpler derivations of some previous results and extending these ideas to the continuous setting. These concepts are tied in with the derived convolutional kernels, and it is deduced how good an image approximation is when it is obtained from only a small band of high frequency Fourier coefficients. Finally, regularization methods are presented to improve the quality of SAR images. Corresponding MATLAB software is made available for reproducibility of most figures and to facilitate further exploration of the methods presented here.

Electrons with kinetic energies of the order 100 keV are capable of exciting atomic vibrational states from a distance of microns. Despite such a large interaction distance, our detailed calculations show that the scattering physics permits a high-energy electron beam to locate vibrational excitations with atomic-scale spatial resolution. Pursuits to realize this capability experimentally could potentially benefit numerous fields across the physical sciences.

A detailed study has been carried out on $\gamma^\prime$ (AlAg$_2$) precipitates in Al-Ag and Al-Ag-Cu alloys to reconcile the conflicting reports on chemical ordering and stacking faults in this phase. High angle annular dark field scanning transmission electron microscopy and convergent beam electron diffraction show no indication of chemical ordering on alternate basal planes of $\gamma^\prime$ precipitates in alloys aged at 473 K for 2-23 h. Precipitates were visible as Ag-rich regions with 1-13 fcc$\rightarrow$hcp stacking faults, corresponding to $\gamma^\prime$ platelets with thicknesses ranging from 0.69-6.44 nm. There were no systematically absent thicknesses. Growth ledges with a riser height equal to the $c$-lattice parameter (0.46 nm) were directly observed for the first time. Genuine stacking faults within the precipitates were extremely rare and only observed in thicker precipitates. In precipitates with 1-3 stacking faults there was also substantial Ag in the surrounding fcc layers of the matrix, indicating that Ag strongly segregated to the broad, planar precipitate-matrix interfaces. This segregation is responsible for previous reports of stacking faults in $\gamma^\prime$ precipitates. The results indicate that the early stages of $\gamma^\prime$ precipitate growth are interfacially controlled., Comment: Author accepted manuscript, 19 pages, 10 figures

Atomic-scale mapping of the chemical elements in materials is now possible using aberration-corrected electron microscopes but delocalization and multiple scattering can confound image interpretation. Here we report atomic-resolution measurements with the elastic and inelastic signals acquired on an absolute scale. By including dynamical scattering in both the elastic and inelastic channels we obtain quantitative agreement between theory and experiment. Our results enable a close scrutiny of the inelastic scattering physics and demonstrate the possibility of element-specific atom counting.