Let f and g be analytic functions on the open unit disk ${\mathbb D}$ such that $|f|=|g|$ on a set A. We give an alternative proof of the result of Perez that there exists c in the unit circle ${\mathbb T}$ such that $f=cg$ when A is the union of two lines in ${\mathbb D}$ intersecting at an angle that is an irrational multiple of $\pi $, and from this, deduce a sequential generalization of the result. Similarly, the same conclusion is valid when f and g are in the Nevanlinna class and A is the union of the unit circle and an interior circle, tangential or not. We also provide sequential versions of this result and analyze the case $A=r{\mathbb T}$. Finally, we examine the most general situation when there is equality on two distinct circles in the disk, proving a result or counterexample for each possible configuration.

The surface deformation of the main reflector in a large radio telescope is closely related to its working efficiency, which is important for some astronomical science studies. Here, we present a deep learning-based surface deformation recovery framework using non-interferometric intensity measurements as input. The recurrent convolutional neural network (RCNN) is developed to establish the inverse mapping relationship between the surface deformation of the main reflector and the intensity images at the aperture plane and at a near-field plane. Meanwhile, a physical forward propagation model is adopted to generate a large amount of data for pre-training in a computationally efficient manner. Then, the inverse mapping relationship is adjusted and improved by transfer learning using experimental data, which achieves a 15-fold reduction in the number of training image sets required, which is helpful to facilitate the practical application of deep learning in this field. In addition, the RCNN model can be trained as a denoiser, and it is robust to the axial positioning error of the measuring points. It is also promising to extend this method to the study of adaptive optics.

The spatial distribution of beams with orbital angular momentum in the far field is known to be extremely sensitive to angular aberrations, such as astigmatism, coma and trefoil. This poses a challenge for conventional beam optimization strategies when a homogeneous ring intensity is required for an application. We developed a novel approach for estimating the Zernike coefficients of low-order angular aberrations in the near field based solely on the analysis of the ring deformations in the far field. A fast, iterative reconstruction of the focal ring recreates the deformations and provides insight into the wavefront deformations in the near field without relying on conventional phase retrieval approaches. The output of our algorithm can be used to optimize the focal ring, as demonstrated experimentally at the 100 TW beamline at the Extreme Light Infrastructure - Nuclear Physics facility.

This article is the second within a three-part series on Fourier ptychography, which is a computational microscopy technique for high-resolution, large field-of-view imaging. While the first article laid out the basics of Fourier ptychography, this second part sheds light on its algorithmic ingredients. We present a non-technical discussion of phase retrieval, which allows for the synthesis of high-resolution images from a sequence of low-resolution raw data. Fourier ptychographic phase retrieval can be carried out on standard, widefield microscopy platforms with the simple addition of a low-cost LED array, thus offering a convenient alternative to other phase-sensitive techniques that require more elaborate hardware such as differential interference contrast and digital holography.

A reformulated implementation of single-sideband ptychography enables analysis and display of live detector data streams in 4D scanning transmission electron microscopy (STEM) using the LiberTEM open-source platform. This is combined with live first moment and further virtual STEM detector analysis. Processing of both real experimental and simulated data shows the characteristics of this method when data are processed progressively, as opposed to the usual offline processing of a complete data set. In particular, the single-sideband method is compared with other techniques such as the enhanced ptychographic engine in order to ascertain its capability for structural imaging at increased specimen thickness. Qualitatively interpretable live results are obtained also if the sample is moved, or magnification is changed during the analysis. This allows live optimization of instrument as well as specimen parameters during the analysis. The methodology is especially expected to improve contrast- and dose-efficient in situ imaging of weakly scattering specimens, where fast live feedback during the experiment is required.

Recent work has revived interest in the scattering matrix formulation of electron scattering in transmission electron microscopy as a stepping stone toward atomic-resolution structure determination in the presence of multiple scattering. We discuss ways of visualizing the scattering matrix that make its properties clear. Through a simulation-based case study incorporating shot noise, we shown how regularizing on this continuity enables the scattering matrix to be reconstructed from 4D scanning transmission electron microscopy (STEM) measurements from a single defocus value. Intriguingly, for crystalline samples, this process also yields the sample thickness to nanometer accuracy with no a priori knowledge about the sample structure. The reconstruction quality is gauged by using the reconstructed scattering matrix to simulate STEM images at defocus values different from that of the data from which it was reconstructed.

We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then $f=g$ up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of $\pi $ . We also prove that if f and g are functions in the Nevanlinna class, and if $|f|=|g|$ on the unit circle and on a circle inside the unit disc, then $f=g$ up to the multiplication of a unimodular constant.

Phase retrieval is necessary for propagation-based phase-contrast imaging (PB-PCI). Arhatari established a model for predicting the impact of the sample-to-detector distance and the system noise on the phase retrieval performance. We have extended Arhatari's model to account for the parameters of excessive source size, finite detector resolution, and geometrical magnification for more practical cases. However, there exist interaction effects among these parameters resulting in difficulty of predicting the phase retrieval performance. In this study, we found that optimizing the trade-off among these parameters for phase retrieval is consistent with the improvement of edge enhancement to noise ratio (EE/N) in the “forward problem” of the PB-PCI. Hence, we engaged in establishing a relationship between EE/N and phase retrieval performance in terms of the “forward problem” and “inverse problem” of the PB-PCI, respectively. Our results showed that, at fixed detector resolution, phase retrieval from the phase-contrast projections at the same EE/N level resulted in the consistent phase retrieval performance. Therefore, the performance of phase retrieval can be predicted based on the EE/N level and be quantitatively optimized by increasing EE/N.

A three-wavelength coherent-modulation-imaging (CMI) technique is proposed to simultaneously measure the fundamental, second and third harmonics of a laser driver in one snapshot. Laser beams at three wavelengths (1053 nm, 526.5 nm and 351 nm) were simultaneously incident on a random phase plate to generate hybrid diffraction patterns, and a modified CMI algorithm was adopted to reconstruct the complex amplitude of each wavelength from one diffraction intensity frame. The validity of this proposed technique was verified using both numerical simulation and experimental analyses. Compared to commonly used measurement methods, this proposed method has several advantages, including a compact structure, convenient operation and high accuracy.

Computational methods and a program to obtain crystal structures that have the perfectly identical diffraction patterns, i.e. structure factors with the same absolute values and the same lattice symmetry are discussed. This is directly related to the uniqueness of solutions in crystal structure determination of single-crystal/powder-crystal samples from diffraction data. In order to solve the problem, it is necessary to solve a system of quadratic equations. The framework of positive-semidefinite programming is used herein to solve the system efficiently.

We present a deterministic approach to the ptychographic retrieval of the wave at the exit surface of a specimen of condensed matter illuminated by X-rays. The method is based on the solution of an overdetermined set of linear equations, and is robust to measurement noise. The set of linear equations is efficiently solved using the conjugate gradient least-squares method implemented using fast Fourier transforms. The method is demonstrated using a data set obtained from a gold–chromium nanostructured test object. It is shown that the transmission function retrieved by this linear method is quantitatively comparable with established methods of ptychography, with a large decrease in computational time, and is thus a good candidate for real-time reconstruction.

In low-energy electron microscopy (LEEM) we commonly encounter images which, beside amplitude contrast, also show signatures of phase contrast. The images are usually interpreted by following the evolution of the contrast during the experiment, and assigning gray levels to morphological changes. Through reconstruction of the exit wave, two aspects of LEEM can be addressed: (1) the resolution can be improved by exploiting the full information limit of the microscope and (2) electron phase shifts which contribute to the image contrast can be extracted. In this article, linear exit wave reconstruction from a through-focal series of LEEM images is demonstrated. As a model system we utilize a heteromolecular monolayer consisting of the organic molecules 3,4,9,10-perylene tetracarboxylic dianhydride and Cu-II-Phthalocyanine, adsorbed on a Ag(111) surface.

Knowing the (geometric) covariogram of a convex body is equivalent to knowing, for each direction u, the distribution of the lengths of the chords of that body which are parallel to u. We prove that the covariogram determines convex polygons, among all convex bodies, up to translation and reflection. This gives a partial answer to a problem posed by Matheron.