Facing the phase problem in Coherent Diffractive Imaging via Memetic Algorithms

Abstract

Coherent Diffractive Imaging is a lensless technique that allows imaging of matter at a spatial resolution not limited by lens aberrations. This technique exploits the measured diffraction pattern of a coherent beam scattered by periodic and non-periodic objects to retrieve spatial information. The diffracted intensity, for weak-scattering objects, is proportional to the modulus of the Fourier Transform of the object scattering function. Any phase information, needed to retrieve its scattering function, has to be retrieved by means of suitable algorithms. Here we present a new approach, based on a memetic algorithm, i.e. a hybrid genetic algorithm, to face the phase problem, which exploits the synergy of deterministic and stochastic optimization methods. The new approach has been tested on simulated data and applied to the phasing of transmission electron microscopy coherent electron diffraction data of a SrTiO 3 sample. We have been able to quantitatively retrieve the projected atomic potential, and also image the oxygen columns, which are not directly visible in the relevant high-resolution transmission electron microscopy images. Our approach proves to be a new powerful tool for the study of matter at atomic resolution and opens new perspectives in those applications in which effective phase retrieval is necessary.Full-field and scanning microscopes can be either lens-based or lensless imaging systems. Coherent Diffractive Imaging (CDI) is a lensless technique that permits imaging matter at a spatial resolution not limited by lens aberrations. The seminal idea of CDI was due to David Sayre in 1952 1 but it was only experimentally demonstrated for X-rays in 1999 2 and, more recently, also for electrons, using a Transmission Electron Microscope (TEM), giving rise to the Electron Diffractive Imaging (EDI) [3][4][5] .The goal is to retrieve a qualitative/quantitative image of a scattering function related to a physical property of the scattering object, such as the electron density (X-ray CDI) or the atomic potential (EDI). High Resolution TEM (HRTEM) images of the projected atomic potential are phase-contrast images limited by the high-order aberrations of the objective lens, which distort the phase of the scattered wave function, giving rise to images of the sample, which in general are not immediately interpretable in terms of its atomic structure 6 . Instead, diffraction patterns of scattering objects are not affected by these aberrations. Therefore they contain, in principle, undistorted information on the scattering function at a better spatial resolution with respect to lens-based imaging systems 3-5 . The diffracted intensity, for weak-scattering objects, is proportional only to the modulus of the Fourier Transform (FT) of the scattering function. Any phase information, which is experimentally lost (phase problem 1 ), has to be retrieved by means of suitable algorithms. The lensless image of the scattering function, obtained by means of an inverse FT of the diffraction pattern once that the correct phase ...

Effective Pattern Intensity Artifacts Treatment for Electron Diffractive Imaging

“…The determination of the specimen atomic projected-potential enables us to distinguish between crystal atomic columns with different compositions and, for example in the case of the SrTiO 3 specimen oriented with the (001) plane perpendicular to the direction of the primary electron beam, enables us to distinguish the atomic column containing Sr, or Ti+O, or only O, gathering fundamental information regarding the properties of the specimen. Also in the case of the KEDI experiments, a proper treatment of the spurious intensities in the electron diffraction patterns and a new phasing algorithm enabled us to retrieve, for the first time, the image of the atomic projected potential at a resolution of 65 pm starting from completely random phases [32]. Moreover, X-ray diffraction patterns could also be affected by several artifacts, which could benefit from some of the methods presented here.…”

“…This enabled us to develop and demonstrate [31] new methods to manage the dynamical effects, always present in TEM experiments on standard specimen thicknesses, and to quantitatively derive the atomic projected potentials even in the case of atoms of light elements in the crystal matrix of heavy elements [31]. The diffraction data of a KEDI experiment, free from spurious intensities, were also used to image, for the first time and by using new phasing algorithms, the structure of a specimen starting from random phases [32]. We believe that the methods used here to treat the spurious diffracted intensities are helpful for all the methods based on electron diffraction.…”

Mask-based approach to phasing of single-particle diffraction data. II. Likelihood-based selection criteria

“…The efficiency of iterative phasing methods may be enhanced by a preliminary global search in the configurational space of phase sets (Lunin et al, 1990(Lunin et al, , 2012(Lunin et al, , 2016Truong et al, 2017;Webster & Hilgenfeld, 2001;Colombo et al, 2017). In general, global search methods, which are more robust in examining the configurational space of possible phase sets, are more time-consuming than iteration methods.…”

Statistical and computational intelligence approach to analytic continuation in Quantum Monte Carlo

“…This operator performs a deterministic optimization of the spectra; however, when used alone, it is only able to get to local maxima of the fitness. The combined use of stochastic and deterministic operators yields a so-called memetic algorithm [37][38][39]. • Flattening.…”