The phase problem can be considered as one of the cornerstones of quantum mechanics intimately co... more The phase problem can be considered as one of the cornerstones of quantum mechanics intimately connected to the detection process and the uncertainty relation. The latter impose fundamental limits on the manifold phase reconstruction schemes invented to date in particular at small magnitudes of the quantum wave. Here, we show that a rigorous solution of the Transport of Intensity Reconstruction (TIE) scheme in terms of a linear elliptic partial differential equation for the phase provides reconstructions even in the presence of wave zeros if particular boundary conditions (BCs) are given. We furthermore discuss how partial coherence hampers phase reconstruction and show that a modified version of the TIE reconstructs the curl-free current density at arbitrary (in-)coherence. This opens the way for a large variety of new applications in fields as diverse as astrophysics, geophysics, photonics, acoustics, and electron microscopy, where zeros in the respective wave field are a ubiquito...
The reduced dimensionality in two-dimensional materials leads a wealth of unusual properties, whi... more The reduced dimensionality in two-dimensional materials leads a wealth of unusual properties, which are currently explored for both fundamental and applied sciences. In order to study the crystal structure, edge states, the formation of defects and grain boundaries, or the impact of adsorbates, high resolution microscopy techniques are indispensible. Here we report on the development of an electron holography (EH) transmission electron microscopy (TEM) technique, which facilitates high spatial resolution by an automatic correction of geometric aberrations. Distinguished features of EH beyond conventional TEM imaging are the gap-free spatial information signal transfer and higher dose efficiency for certain spatial frequency bands as well as direct access to the projected electrostatic potential of the 2D material. We demonstrate these features at the example of h-BN, at which we measure the electrostatic potential as a function of layer number down to the monolayer limit and obtain ...
The use of phase plates in the back focal plane of a microscope is a well established technique i... more The use of phase plates in the back focal plane of a microscope is a well established technique in optical microscopy to increase the contrast of weakly interacting samples and is gaining interest in electron microscopy as well. In this paper we study the spiral phase plate (SPP), also called helical, vortex, or two-dimensional Hilbert phase plate, that adds an angularly dependent phase of the form e^iℓϕ to the exit wave in Fourier space. In the limit of large collection angles, we analytically calculate that the average of a pair of ℓ=±1 SPP images is directly proportional to the gradient squared of the exit wave, explaining the edge contrast previously seen in optical SPP work. The difference between a clockwise-anticlockwise pair of SPP images and conditions where this difference vanishes and the gradient of the exit wave can be seen from one single SPP image, are discussed. Finally, we demonstrate how with three images, one without and one with each of an ℓ=±1 SPP, may give enou...
An entry from the Inorganic Crystal Structure Database, the world's repository for inorganic ... more An entry from the Inorganic Crystal Structure Database, the world's repository for inorganic crystal structures. The entry contains experimental data from a crystal diffraction study. The deposited dataset for this entry is freely available from the joint CCDC and FIZ Karlsruhe Access Structures service and typically includes 3D coordinates, cell parameters, space group, experimental conditions and quality measures.
2D randommetal networks possess unique electrical and optical properties, such as almost hundred ... more 2D randommetal networks possess unique electrical and optical properties, such as almost hundred percent optical transparencies and low sheet resistance, which are closely related to their disordered structure and may be exploited in various applications. Here we present a detailed experimental and theoretical investigation of their plasmonic properties, revealing Anderson (disorder-driven) localized surface plasmon (LSP) resonances of very large quality factors and spatial localization close to the theoretical maximum. The LSPs typically consist of multiple field hotspots with a welldefined correlation distance and couple to electromagnetic plane waves. Moreover, they disappear above a geometry-dependent threshold at ca. 1.6 eV in the investigated networks, explaining their large transparencies in the optical spectrum.
The uniqueness of holography lies in the fact that, in addition to the amplitude (intensity), it ... more The uniqueness of holography lies in the fact that, in addition to the amplitude (intensity), it provides the phase distribution from the object. The phase represents additional object information, for example about electric and magnetic fields down to an atomic scale, which are invisible in conventional intensity images. Therefore, most applications of holography evaluate mainly the reconstructed phase images.
Magnetic skyrmions are stable topological solitons with complex non-coplanar spin structures. The... more Magnetic skyrmions are stable topological solitons with complex non-coplanar spin structures. Their nanoscopic size and the low electric currents required to control their motion has opened a new field of research, skyrmionics, that aims for the usage of skyrmions as information carriers. Further advances in skyrmionics call for a thorough understanding of their three-dimensional (3D) spin texture, skyrmion–skyrmion interactions and the coupling to surfaces and interfaces, which crucially affect skyrmion stability and mobility. Here, we quantitatively reconstruct the 3D magnetic texture of Bloch skyrmions with sub-10-nanometre resolution using holographic vector-field electron tomography. The reconstructed textures reveal local deviations from a homogeneous Bloch character within the skyrmion tubes, details of the collapse of the skyrmion texture at surfaces and a correlated modulation of the skyrmion tubes in FeGe along their tube axes. Additionally, we confirm the fundamental prin...
Dark-field electron holography maps strain in crystal lattices into reconstructed phases over lar... more 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.
The phase problem can be considered as one of the cornerstones of quantum mechanics intimately co... more The phase problem can be considered as one of the cornerstones of quantum mechanics intimately connected to the detection process and the uncertainty relation. The latter impose fundamental limits on the manifold phase reconstruction schemes invented to date in particular at small magnitudes of the quantum wave. Here, we show that a rigorous solution of the Transport of Intensity Reconstruction (TIE) scheme in terms of a linear elliptic partial differential equation for the phase provides reconstructions even in the presence of wave zeros if particular boundary conditions (BCs) are given. We furthermore discuss how partial coherence hampers phase reconstruction and show that a modified version of the TIE reconstructs the curl-free current density at arbitrary (in-)coherence. This opens the way for a large variety of new applications in fields as diverse as astrophysics, geophysics, photonics, acoustics, and electron microscopy, where zeros in the respective wave field are a ubiquito...
The reduced dimensionality in two-dimensional materials leads a wealth of unusual properties, whi... more The reduced dimensionality in two-dimensional materials leads a wealth of unusual properties, which are currently explored for both fundamental and applied sciences. In order to study the crystal structure, edge states, the formation of defects and grain boundaries, or the impact of adsorbates, high resolution microscopy techniques are indispensible. Here we report on the development of an electron holography (EH) transmission electron microscopy (TEM) technique, which facilitates high spatial resolution by an automatic correction of geometric aberrations. Distinguished features of EH beyond conventional TEM imaging are the gap-free spatial information signal transfer and higher dose efficiency for certain spatial frequency bands as well as direct access to the projected electrostatic potential of the 2D material. We demonstrate these features at the example of h-BN, at which we measure the electrostatic potential as a function of layer number down to the monolayer limit and obtain ...
The use of phase plates in the back focal plane of a microscope is a well established technique i... more The use of phase plates in the back focal plane of a microscope is a well established technique in optical microscopy to increase the contrast of weakly interacting samples and is gaining interest in electron microscopy as well. In this paper we study the spiral phase plate (SPP), also called helical, vortex, or two-dimensional Hilbert phase plate, that adds an angularly dependent phase of the form e^iℓϕ to the exit wave in Fourier space. In the limit of large collection angles, we analytically calculate that the average of a pair of ℓ=±1 SPP images is directly proportional to the gradient squared of the exit wave, explaining the edge contrast previously seen in optical SPP work. The difference between a clockwise-anticlockwise pair of SPP images and conditions where this difference vanishes and the gradient of the exit wave can be seen from one single SPP image, are discussed. Finally, we demonstrate how with three images, one without and one with each of an ℓ=±1 SPP, may give enou...
An entry from the Inorganic Crystal Structure Database, the world's repository for inorganic ... more An entry from the Inorganic Crystal Structure Database, the world's repository for inorganic crystal structures. The entry contains experimental data from a crystal diffraction study. The deposited dataset for this entry is freely available from the joint CCDC and FIZ Karlsruhe Access Structures service and typically includes 3D coordinates, cell parameters, space group, experimental conditions and quality measures.
2D randommetal networks possess unique electrical and optical properties, such as almost hundred ... more 2D randommetal networks possess unique electrical and optical properties, such as almost hundred percent optical transparencies and low sheet resistance, which are closely related to their disordered structure and may be exploited in various applications. Here we present a detailed experimental and theoretical investigation of their plasmonic properties, revealing Anderson (disorder-driven) localized surface plasmon (LSP) resonances of very large quality factors and spatial localization close to the theoretical maximum. The LSPs typically consist of multiple field hotspots with a welldefined correlation distance and couple to electromagnetic plane waves. Moreover, they disappear above a geometry-dependent threshold at ca. 1.6 eV in the investigated networks, explaining their large transparencies in the optical spectrum.
The uniqueness of holography lies in the fact that, in addition to the amplitude (intensity), it ... more The uniqueness of holography lies in the fact that, in addition to the amplitude (intensity), it provides the phase distribution from the object. The phase represents additional object information, for example about electric and magnetic fields down to an atomic scale, which are invisible in conventional intensity images. Therefore, most applications of holography evaluate mainly the reconstructed phase images.
Magnetic skyrmions are stable topological solitons with complex non-coplanar spin structures. The... more Magnetic skyrmions are stable topological solitons with complex non-coplanar spin structures. Their nanoscopic size and the low electric currents required to control their motion has opened a new field of research, skyrmionics, that aims for the usage of skyrmions as information carriers. Further advances in skyrmionics call for a thorough understanding of their three-dimensional (3D) spin texture, skyrmion–skyrmion interactions and the coupling to surfaces and interfaces, which crucially affect skyrmion stability and mobility. Here, we quantitatively reconstruct the 3D magnetic texture of Bloch skyrmions with sub-10-nanometre resolution using holographic vector-field electron tomography. The reconstructed textures reveal local deviations from a homogeneous Bloch character within the skyrmion tubes, details of the collapse of the skyrmion texture at surfaces and a correlated modulation of the skyrmion tubes in FeGe along their tube axes. Additionally, we confirm the fundamental prin...
Dark-field electron holography maps strain in crystal lattices into reconstructed phases over lar... more 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.
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