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Magnetic vector electron tomography (VET) is a promising technique that enables better understanding of micro- and nano-magnetic phenomena through the reconstruction of 3D magnetic fields at high spatial resolution. Here we introduce WRAP (Wavelet Regularised A Program), a reconstruction algorithm for magnetic VET that directly reconstructs the magnetic vector potential A using a compressed sensing framework which regularises for sparsity in the wavelet domain. We demonstrate that using WRAP leads to a significant increase in the fidelity of the 3D reconstruction and is especially robust when dealing with very limited data; using datasets simulated with realistic noise, we compare WRAP to a conventional reconstruction algorithm and find an improvement of ca. 60% when averaged over several performance metrics. Moreover, we further validate WRAP's performance on experimental electron holography data, revealing the detailed magnetism of vortex states in a CuCo nanowire. We believe WRAP represents a major step forward in the development of magnetic VET as a tool for probing magnetism at the nanoscale., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2023 The Author(s). Published by Elsevier B.V. All rights reserved.)

Fundamental solid state physics phenomena often occur at very low temperatures, requiring liquid helium cooling in experimental studies. Transmission electron microscopy is a well-established characterization method, which allows probing crucial materials properties down to nanometre and even atomic resolution. Due to the limited space in the object plane, however, suitable liquid-helium cooling is very challenging. To overcome this limitation, resolving power was sacrificed in our Dresden in-situ (S)TEM special, resulting in more than 60 mm usable experimental space in all directions with the specimen in the centre. With the installation of a continuous-flow liquid-helium cryostat, any temperature between 6.5 K and 400 K can be set precisely and kept for days. The information limit of the Dresden in-situ (S)TEM special is about 5 nm. It is shown that the resolution of the Dresden in-situ (S)TEM special is currently not limited by aberrations, but by external instabilities., (Copyright © 2019 Elsevier B.V. All rights reserved.)

Off-axis electron holography is a well-established transmission electron microscopy technique, typically employed to investigate electric and magnetic fields in and around nanoscale materials, which modify the phase of the reconstructed electron wave function. Here, we elaborate on a detailed analysis of the two characteristic intensity terms that are completing the electron hologram, the conventional image intensity and the interference fringe intensity. We show how both are related to elastic and inelastic scattering absorption at the sample and how they may be separated to analyze the chemical composition of the sample. Since scattering absorption is aperture dependent, a quantitative determination of the corresponding attenuation coefficients (reciprocal mean free path lengths) requires the use of holographic image modi with well-defined objective aperture stops in the back-focal plane of the objective lens. The proposed method extends quantitative electron holography to a correlated three-in-one characterization of electric and magnetic fields, Z-contrast and dielectric losses in materials., (Copyright © 2016 Elsevier B.V. All rights reserved.)

The recently derived general transfer theory for off-axis electron holography provides a new approach for reconstructing the electron wave beyond the conventional sideband information limit. Limited ensemble coherence of the electron beam between object and reference area leads to an attenuation of spatial frequencies of the object exit wave in the presence of aberrations of the objective lens. Concerted tilts of the reference wave under the condition of an invariant object exit wave are proposed to diminish the aberration impact on spatial frequencies even beyond the sideband information limit allowing its transfer with maximum possible contrast. In addition to the theoretical considerations outlined in detail, an experimental proof-of-principle is presented. A fully controlled tilt of the reference wave, however, remains as a promising task for the future. The use of a hologram series with varying reference wave tilt is considered for linearly synthesizing an effective aperture for the transfer into the sideband with broader bandwidth compared to conventional off-axis electron holography allowing us to correct the incoherent aberrations in transmission electron microscopy. Furthermore, tilting a reference wave with respect to a plane wave is expected to be an alternative way for measuring the coherent and incoherent aberrations of a transmission electron microscope. The capability of tilting the reference wave is expected to be beneficial for improving the signal-to-noise ratio in dark-field off-axis electron holography as well., (Copyright © 2015 Elsevier B.V. All rights reserved.)

Current developments in TEM such as high-resolution imaging at low acceleration voltages and large fields of view, the ever larger capabilities of hardware aberration correction and the systematic shaping of electron beams require accurate descriptions of TEM imaging in terms of wave optics. Since full quantum mechanic solutions have not yet been established for, e.g., the theory of aberrations, we are exploring semiclassical image formation in the TEM from the perspective of quantum mechanical phase space, here. Firstly, we use two well-known semiclassical approximations, Miller's semiclassical algebra and the frozen Gaussian method, for describing the wave optical generalization of arbitrary geometric aberrations, including nonisoplanatic and slope aberrations. Secondly, we demonstrate that the Wigner function representation of phase space is well suited to also describe incoherent aberrations as well as the ramifications of partial coherence due to the emission process at the electron source. We identify a close relationship between classical phase space and Wigner function distortions due to aberrations as well as classical brightness and quantum mechanical purity., (Copyright © 2014 Elsevier B.V. All rights reserved.)

Tomography commonly requires a linear relation between the measured signal and the underlying specimen property; for Electron Holographic Tomography this is given by the Phase Grating Approximation (PGA). While largely valid at medium resolution, discrepancies arise at high resolution imaging conditions. We set out to investigate the artefacts that are produced if the reconstruction still assumes the PGA even with an atomic resolution tilt series. To forego experimental difficulties the holographic tilt series was simulated. The reconstructed electric potential clearly shows peaks at the positions of the atoms. These peaks have characterisitic deformations, which can be traced back to the defocus a particular atom has in the holograms of the tilt series. Exchanging an atom for one of a different atomic number results in a significant change in the reconstructed potential that is well contained within the atom's peak., (Copyright © 2014 Elsevier B.V. All rights reserved.)

Here, we study the effect of dynamic scattering on the projected geometric phase and strain maps reconstructed using dark-field electron holography (DFEH) for non-uniformly strained crystals. The investigated structure consists of a {SiGe/Si} superlattice grown on a (001)-Si substrate. The three-dimensional strain field within the thin TEM lamella is modelled by the finite element method. The observed projected strain is simulated in two ways by multiplying the strain at each depth in the crystal by a weighting function determined from a recently developed analytical two-beam dynamical theory, and by simply taking the average value. We demonstrate that the experimental results need to be understood in terms of the dynamical theory and good agreement is found between the experimental and simulated results. Discrepancies do remain for certain cases and are likely to be from an imprecision in the actual two-beam diffraction conditions, notably the deviation parameter, and points to limitations in the 2-beam approximation. Finally, a route towards a 3D reconstruction of strain fields is proposed., (Copyright © 2014 Elsevier B.V. All rights reserved.)

The reduced density matrix completely describes the quantum state of an electron scattered by an object in transmission electron microscopy. However, the detection process restricts access to the diagonal elements only. The off-diagonal elements, determining the coherence of the scattered electron, may be obtained from electron holography. In order to extract the influence of the object from the off-diagonals, however, a rigorous consideration of the electron microscope influences like aberrations of the objective lens and the Möllenstedt biprism in the presence of partial coherence is required. Here, we derive a holographic transfer theory based on the generalization of the transmission cross-coefficient including all known holographic phenomena. We furthermore apply a particular simplification of the theory to the experimental analysis of aloof beam electrons scattered by plane silicon surfaces., (Copyright © 2014 Elsevier B.V. All rights reserved.)

Off-axis electron holography provides access to the phase of the elastically scattered wave in a transmission electron microscope at scales ranging from several hundreds of nanometres down to 0.1nm. In many cases the reconstructed phase shift is directly proportional to projected electric and magnetic potentials rendering electron holography a useful and established characterisation method for materials science. However, quantitative interpretation of experimental phase shifts requires quantitative knowledge about the noise, which has been previously established for some limiting cases only. Here, we present a general noise transfer formalism for off-axis electron holography allowing to compute the covariance (noise) of reconstructed amplitude and phase from characteristic detector functions and general properties of the reconstruction process. Experimentally, we verify the presented noise transfer formulas for two different cameras with and without objects within the errors given by the experimental noise determination., (Copyright © 2014 Elsevier B.V. All rights reserved.)

Dark-field electron holography maps strain in crystal lattices into reconstructed phases over large fields of view. Here we investigate the details of the lattice strain-reconstructed phase relationship by applying dynamic scattering theory both analytically and numerically. We develop efficient analytic linear projection rules for 3D strain fields, facilitating a straight-forward calculation of reconstructed phases from 3D strained materials. They are used in the following to quantify the influence of various experimental parameters like strain magnitude, specimen thickness, excitation error and surface relaxation., (© 2013 Elsevier B.V. All rights reserved.)

Tomographic techniques play a crucial role in imaging methods such as transmission electron microscopy (TEM) due to their unique capabilities to reconstruct three-dimensional object information. However, the accuracy of the two standard tomographic reconstruction techniques, the weighted back-projection (W-BP) and the simultaneous iterative reconstruction technique (SIRT) is reduced under common experimental restrictions, such as limited tilt range or noise. We demonstrate that the combination of W-BP and SIRT leads to an improved tomographic reconstruction technique: the weighted SIRT. Convergence, resolution and reconstruction error of the W-SIRT are analyzed by a detailed analytical, numerical, and experimental comparison with established methods. Our reconstruction technique is not restricted to TEM tomography but can be applied to all problems sharing single axis imaging geometry., (© 2013 Elsevier B.V. All rights reserved.)

Electron holography initially was invented by Dennis Gabor for solving the problems raised by the aberrations of electron lenses in Transmission Electron Microscopy. Nowadays, after hardware correction of aberrations allows true atomic resolution of the structure, for comprehensive understanding of solids, determination of electric and magnetic nanofields is the most challenging task. Since fields are phase objects in the TEM, electron holography is the unrivaled method of choice. After more than 40 years of experimental realization and steady improvement, holography is increasingly contributing to these highly sophisticated and essential questions in materials science, as well to the understanding of electron waves and their interaction with matter., (© 2013 Elsevier B.V. All rights reserved.)

A new generalized linear transfer theory describing the signal and noise transfer in image detectors is presented, which can be applied to calculate the pixelwise first and second statistical moment of arbitrary experimental images including correlation between pixels. Similar to the existing notion of a point spread function describing the transfer of the first statistical moment (the average), a noise spread function is introduced to characterize the spatially resolved transfer and generation of noise (second central moment, covariance). It is also shown that previously used noise characteristics like the noise power spectrum and detection quantum efficiency, derived from plainly illuminated images, contain only partial information of the complete noise transfer., (Copyright © 2012 Elsevier B.V. All rights reserved.)

A novel generalized linear transfer theory describing the signal and noise transfer in image detectors has been developed in Part I (Niermann, this issue, [1]) of this paper. Similar to the existing notion of a point spread function (PSF) describing the transfer of the first statistical moment (the average), a noise spread function (NSF) was introduced to characterize the spatially resolved transfer of noise (central second moment, covariance). Following the theoretic results developed in Part I (Niermann, this issue, [1]), a new experimental method based on single spot illumination has been developed and applied to measure 2D point and 4D noise spread functions of CCD cameras used in TEM. A dedicated oversampling method has been used to suppress aliasing in the measured quantities. We analyze the 4D noise spread with respect to electronic and photonic noise contributions., (Copyright © 2012 Elsevier B.V. All rights reserved.)

The standard treatment for the different plane wave components of incoming electrons in transmission electron microscope imaging is an incoherent superposition. However, projectile electrons in transmission electron microscopes are localized in space, and therefore have to be described as coherent wave-packets. Moreover, recent developments towards ultrafast electron microscopy and dynamic transmission electron microscopy require a description using highly localized wave-packets. Here we will extend the standard stationary modeling of the elastic scattering processes in high-resolution microscopy to a fully time-dependent approach, by using the direct solution of the time-dependent Schrödinger equation. We will draw the connection to the detection of coherent wave-packets, giving explicit implications for the reconstructed waves in off-axis electron holography. Additionally the description of incoherent aberrations is extended to incorporate the influence of the biprism accurately, leading to a modified form of the damping of spatial frequencies., (Copyright 2010. Published by Elsevier B.V.)

Electron-holographic tomography (EHT), that is, the combination of off-axis electron holography with electron tomography, was successfully applied for the quantitative 3D mapping of electrostatic potentials at the nanoscale. Here we present the first software package (THOMAS) for semi-automated acquisition of holographic tilt series, a prerequisite for efficient data collection. Using THOMAS, the acquisition time for a holographic tilt series, consisting of object and reference holograms, is reduced by a factor of five on average, compared to the previous, completely manual approaches. Moreover, the existing software packages for retrieving amplitude and phase information from electron holograms have been extended, now including a one-step procedure for holographic tilt series reconstruction. Furthermore, a modified SIRT algorithm (WSIRT) was implemented for the quantitative 3D reconstruction of the electrostatic potential from the aligned phase tilt series. Finally, the application of EHT to a polystyrene latex sphere test-specimen and a pn-doped Ge 'needle'-shaped specimen are presented, illustrating the quantitative character of EHT. For both specimens the mean inner potential (MIP) values were accurately determined from the reconstructed 3D potential. For the Ge specimen, additionally the 'built-in' voltage across the pn junction of 0.5V was obtained., (Copyright 2010 Elsevier B.V. All rights reserved.)