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We present an experimental visualization of the Terrell effect, an optical phenomenon predicted in 1959 by Roger Penrose and James Terrell, which reveals that the Lorentz contraction of a moving object is not visible in a snapshot photograph. Using fs-laser pulses and a gated intensified camera that allows gating times as short as 300 ps, we achieve a virtual reduction of the speed of light to less than 2 m/s, enabling the visualisation of relativistically moving objects in real time. By capturing light reflected from deliberately Lorentz-contracted objects, our setup effectively reconstructs their visual appearance. This didactic visualization not only commemorates the centennial of Anton Lampa's seminal 1924 paper on relativistic length contraction but also provides the first experimental evidence of the Terrell effect in a laboratory setup. Our results comprise detailed relativistic illustrations, simulations and photographic snapshots of a sphere and a cube, which are animated to velocities close to the speed of light, revealing the apparent rotation effect and the distortion predicted by relativistic theory., Comment: 6 pages, 8 figures

Structured waves are ubiquitous for all areas of wave physics, both classical and quantum, where the wavefields are inhomogeneous and cannot be approximated by a single plane wave. Even the interference of two plane waves, or a single inhomogeneous (evanescent) wave, provides a number of nontrivial phenomena and additional functionalities as compared to a single plane wave. Complex wavefields with inhomogeneities in the amplitude, phase, and polarization, including topological structures and singularities, underpin modern nanooptics and photonics, yet they are equally important, e.g., for quantum matter waves, acoustics, water waves, etc. Structured waves are crucial in optical and electron microscopy, wave propagation and scattering, imaging, communications, quantum optics, topological and non-Hermitian wave systems, quantum condensed-matter systems, optomechanics, plasmonics and metamaterials, optical and acoustic manipulation, and so forth. This Roadmap is written collectively by prominent researchers and aims to survey the role of structured waves in various areas of wave physics. Providing background, current research, and anticipating future developments, it will be of interest to a wide cross-disciplinary audience., Comment: 110 pages, many figures

The topological charge $m$ of vortex electrons spans an infinite-dimensional Hilbert space. Selecting a two-dimensional subspace spanned by $m=\pm 1$, a beam electron in a transmission electron microscope (TEM) can be considered as a quantum bit (qubit) freely propagating in the column. A combination of electron optical quadrupole lenses can serve as a universal device to manipulate such qubits at the experimenter's discretion. We set up a TEM probe forming lens system as a quantum gate and demonstrate its action numerically and experimentally. High-end TEMs with aberration correctors are a promising platform for such experiments, opening the way to study quantum logic gates in the electron microscope.

In light optics, beams with orbital angular momentum (OAM) can be produced by employing a properly-tuned two-cylinder-lens arrangement, also called $\pi$/2 mode converter. It is not possible to convey this concept directly to the beam in an electron microscope due to the non-existence of cylinder lenses in commercial transmission electron microscope (TEM). A viable work-around are readily-available electron optical elements in the form of quadrupole lenses. In a proof-of-principle experiment in 2012, it has been shown that a single quadrupole in combination with a Hilbert phase plate produces a spatially-confined, transient vortex mode. Here, an analogue to an optical $\pi$/2 mode converter is realized by repurposing a CEOS DCOR probe corrector in an aberration corrected TEM in a way that it resembles a dual cylinder lens using two quadrupoles. In order to verify the presence of OAM in the output beam, a fork dislocation grating is used as an OAM analyser. The possibility to use magnetic quadrupole fields instead of, e.g., prefabricated fork dislocation gratings to produce electron beams carrying OAM enhances the beam brightness by almost an order of magnitude and delivers switchable high-mode purity vortex beams without unwanted side-bands.

We propose to use the topological charge instead of the spin variable to span a two-dimensional Hilbert space for beam electrons in a transmission electron microscope (TEM). In this basis, an electron can be considered as a qbit freely floating in vacuum. We show how a combination of magnetic quadrupoles with a magnetic drift tube can serve as a universal device to manipulate such qbits at the experimenter's discretion. High-end TEMs with aberration correctors, high beam coherence and utmost stability are a promising platform for such experiments, allowing the construction of quantum logic gates for single beam electrons in a microscope.

In optics, mode conversion is an elegant way to switch between Hermite Gaussian and Laguerre Gaussian beam profiles and thereby impart orbital angular momentum onto the beam and to create vortices. In optics such vortex beams can be produced in a setup consisting of two identical cylinder lenses. In electron optics, quadrupole lenses can be used for the same purpose. Here we investigate generalized asymmetric designs of a quadrupole mode converter that may be realized within the constraints of existing electron microscopes and can steer the development of dedicated vortex generators for high brilliance electron vortex probes of atomic scale., Comment: 9 pages 10 Figures

Treating Coulomb scattering of two free electrons in a stationary approach, we explore the momentum and spin entanglement created by the interaction. We show that a particular discretisation provides an estimate of the von Neumann entropy of the one-electron reduced density matrix from the experimentally accessible Shannon entropy. For spinless distinguishable electrons the entropy is sizeable at low energies, indicating strong momentum entanglement, and drops to almost zero at energies of the order of 10 keV when the azimutal degree of freedom is integrated out, i.e. practically no entanglement and almost pure one-electron states. If spin is taken into account, the entropy for electrons with antiparallel spins should be larger than in the parallel-spin case, since it embodies both momentum and spin entanglement. Surprisingly, this difference, as well as the deviation from the spin-less case, is extremely small for the complete scattering state. Strong spin entanglement can however be obtained by post-selecting states at scattering angle pi/2.

We discuss the feasibility of detecting spin polarized electronic transitions with a vortex filter. This approach does not rely on the principal condition of the standard energy loss magnetic chiral dichroism (EMCD) technique, the precise alignment of the crystal, and thus paves the way for the application of EMCD to new classes of materials and problems. The dichroic signal strength in the L$_{2,3}$-edge of ferromagnetic cobalt is estimated on theoretical grounds. It is shown that magnetic dichroism can, in principle, be detected. However, as an experimental test shows, count rates are currently too low under standard conditions.

Standard electron optics predicts Larmor image rotation in the magnetic lens field of a TEM. Introducing the possibility to produce electron vortex beams with quantized orbital angular momentum brought up the question of their rotational dynamics in the presence of a magnetic field. Recently, it has been shown that electron vortex beams can be prepared as free electron Landau states showing peculiar rotational dynamics, including no and cyclotron (double-Larmor) rotation. Additionally very fast Gouy rotation of electron vortex beams has been observed. In this work a model is developed which reveals that the rotational dynamics of electron vortices are a combination of slow Larmor and fast Gouy rotations and that the Landau states naturally occur in the transition region in between the two regimes. This more general picture is confirmed by experimental data showing an extended set of peculiar rotations, including no, cyclotron, Larmor and rapid Gouy rotations all present in one single convergent electron vortex beam.

Both classical and quantum waves can form vortices: with helical phase fronts and azimuthal current densities. These features determine the intrinsic orbital angular momentum carried by localized vortex states. In the past 25 years, optical vortex beams have become an inherent part of modern optics, with many remarkable achievements and applications. In the past decade, it has been realized and demonstrated that such vortex beams or wavepackets can also appear in free electron waves, in particular, in electron microscopy. Interest in free-electron vortex states quickly spread over different areas of physics: from basic aspects of quantum mechanics, via applications for fine probing of matter (including individual atoms), to high-energy particle collision and radiation processes. Here we provide a comprehensive review of theoretical and experimental studies in this emerging field of research. We describe the main properties of electron vortex states, experimental achievements and possible applications within transmission electron microscopy, as well as the possible role of vortex electrons in relativistic and high-energy processes. We aim to provide a balanced description including a pedagogical introduction, solid theoretical basis, and a wide range of practical details. Special attention is paid to translate theoretical insights into suggestions for future experiments, in electron microscopy and beyond, in any situation where free electrons occur., Comment: 87 pages, 34 figures

Transmission electron microscopy has been a promising candidate for mapping atomic orbitals for a long time. Here, we explore its capabilities by a first principles approach. For the example of defected graphene, exhibiting either an isolated vacancy or a substitutional nitrogen atom, we show that three different kinds of images are to be expected, depending on the orbital character. To judge the feasibility of visualizing orbitals in a real microscope, the effect of the optics aberrations is simulated. We demonstrate that, by making use of energy-filtering, it should indeed be possible to map atomic orbitals in a state-of-the-art transmission electron microscope.

Electronic states are responsible for most material properties, including chemical bonds, electrical and thermal conductivity, as well as optical and magnetic properties. Experimentally, however, they remain mostly elusive. Here, we report the real-space mapping of selected transitions between p and d states on the {\AA}ngstr\"om scale in bulk rutile (TiO2) using electron energy-loss spectrometry (EELS), revealing information on individual bonds between atoms. On the one hand, this enables the experimental verification of theoretical predictions about electronic states. On the other hand, it paves the way for directly investigating electronic states under conditions that are at the limit of the current capabilities of numerical simulations such as, e.g., the electronic states at defects, interfaces, and quantum dots.

Energy-loss magnetic chiral dichroism (EMCD) allows for the quantification of magnetic properties of materials at the nanometer scale. It is shown that with the support of simulations that help to identify the optimal conditions for a successful experiment and upon implementing measurement routines that effectively reduce the noise floor, EMCD measurements can be pushed towards quantitative magnetic measurements even on individual nanoparticles. With this approach, the ratio of orbital to spin magnetic moments for the Fe atoms in a single L$1_0$ ordered FePt nanoparticle is determined to be ${m_l}/{m_s} = 0.08 \pm 0.02$. This finding is in good quantitative agreement with the results of XMCD ensemble measurements., Comment: 35 pages, 10 figures

The theoretical possibility to use an electron microscope as a spin polarizer is studied. It turns out that a Bessel beam passing a standard magnetic objective lens is intrinsically spin polarized. In the limit of infinitely small detectors on axis, the spin polarisation tends to 100 % . Increasing the detector size, the polarisation decreases rapidly, dropping below 10^-4 for standard settings of medium voltage microscopes. For extremely low voltages, the figure of merit increases by two orders of magnitude, approaching that of existing Mott detectors. Our findings may lead to new desings of spin filters, an attractive option in view of its inherent combination with the electron microscope, especially at low voltage.

Landau levels and states of electrons in a magnetic field are fundamental quantum entities underlying the quantum-Hall and related effects in condensed matter physics. However, the real-space properties and observation of Landau wave functions remain elusive. Here we report the first real-space observation of Landau states and the internal rotational dynamics of free electrons. States with different quantum numbers are produced using nanometer-size electron vortex beams, with a radius chosen to match the waist of the Landau states, in a quasi-uniform magnetic field. Scanning the beams along the propagation direction, we reconstruct the rotational dynamics of the Landau wave functions with angular frequency ~100 GHz. We observe that Landau modes with different azimuthal quantum numbers belong to three classes, which are characterized by rotations with zero, Larmor, and cyclotron frequencies, respectively. This is in sharp contrast to the uniform cyclotron rotation of classical electrons, and in perfect agreement with recent theoretical predictions., Comment: 10 pages, 4 figures

Electron vortex beams were only recently discovered and their potential as a probe for magnetism in materials was shown. Here we demonstrate a new method to produce electron vortex beams with a diameter of less than 1.2 \AA. This unique way to prepare free electrons to a state resembling atomic orbitals is fascinating from a fundamental physics point of view and opens the road for magnetic mapping with atomic resolution in an electron microscope.

We discuss the feasibility of detecting magnetic transitions with focused electron vortex probes, suggested by selection rules for the magnetic quantum number. We theoretically estimate the dichroic signal strength in the L$_{2,3}$ edge of the ferromagnetic d metals. It is shown that under realistic conditions, the dichroic signal is undetectable for nanoparticles larger than ~1 nm. This is confirmed by a key experiment with nanometer sized vortices.

It was claimed (Lloyd et al., PRL 108 (2012) 074802) that energy loss magnetic chiral dichroism (EMCD) with electron vortex beams is feasible, and has even advantages over the standard setup with Bragg diffracted waves. In this Comment, we show that Lloyd et al. ignored an important constraint on the proposed selection rule for the transfer of angular momentum in the interaction, namely that it is only valid for an atom located in the very center of the vortex. As an experimental consequence, the EMCD signal will only be strong for extremely small nanoparticles of 1 to 2 nm diameter., Comment: Submitted to Physical Review Letters 11 July 2012. Accepted for publication 3 April 2013. "Copyright (2013) by the American Physical Society." http://prl.aps.org/

We demonstrate how the mixed dynamic form factor (MDFF) can be interpreted as a quadratic form. This makes it possible to use matrix diagonalization methods to reduce the number of terms that need to be taken into account when calculating the inelastic scattering of electrons in a crystal. It also leads in a natural way to a new basis that helps elucidate the underlying physics. The new method is applied to several cases to show its versatility. In particular, predictions are made for directly imaging atomic orbitals in crystals.

Electron vortex beams carrying intrinsic orbital angular momentum (OAM) are produced in electron microscopes where they are controlled and focused using magnetic lenses. We observe various rotational phenomena arising from the interaction between the OAM and magnetic lenses. First, the Zeeman coupling, proportional to the OAM and magnetic field strength, produces an OAM-independent Larmor rotation of a mode superposition inside the lens. Second, hen passing through the focal plane, the electron beam acquires an additional Gouy phase dependent on the absolute value of the OAM. This brings about the Gouy rotation of the superposition image proportional to the sign of the OAM. A combination of the Larmor and Gouy effects can result in the addition (or subtraction) of rotations, depending on the OAM sign. This behaviour is unique to electron vortex beams and has no optical counterpart, as Larmor rotation occurs only for charged particles. Our experimental results are in agreement with recent theoretical predictions., Comment: 5 pages, 5 figures

A mode converter for electron vortex beams is described. Numerical simulations, confirmed by experiment, show that the converter transforms a vortex beam with topological charge $m=\pm 1$ into beams closely resembling Hermite-Gaussian HG$_{10}$ and HG$_{01}$ modes. The converter can be used as a mode discriminator or filter for electron vortex beams. Combining the converter with a phase plate turns a plane wave into modes with topological charge $m=\pm 1$. This combination serves as a generator of electron vortex beams of high brilliance.

We examine the propagation of the recently-discovered electron vortex beams in a longitudinal magnetic field. We consider both the Aharonov-Bohm configuration with a single flux line and the Landau case of a uniform magnetic field. While stationary Aharonov-Bohm modes represent Bessel beams with flux- and vortex-dependent probability distributions, stationary Landau states manifest themselves as non-diffracting Laguerre-Gaussian beams. Furthermore, the Landau-state beams possess field- and vortex-dependent phases: (i) the Zeeman phase from coupling the quantized angular momentum to the magnetic field and (ii) the Gouy phase, known from optical Laguerre-Gaussian beams. Remarkably, together these phases determine the structure of Landau energy levels. This unified Zeeman-Landau-Gouy phase manifests itself in a nontrivial evolution of images formed by various superpositions of modes. We demonstrate that, depending on the chosen superposition, the image can rotate in a magnetic field with either (i) Larmor, (ii) cyclotron (double-Larmor), or (iii) zero frequency. At the same time, its centroid always follows the classical cyclotron trajectory, in agreement with the Ehrenfest theorem. Remarkably, the non-rotating superpositions reproduce stable multi-vortex configurations that appear in rotating superfluids. Our results open up an avenue for the direct electron-microscopy observation of fundamental properties of free quantum electron states in magnetic fields., Comment: 21 pages, 10 figures, 1 table, to appear in Phys. Rev. X

In this work, the transition matrix elements for inelastic electron--electron scattering are investigated. The angular part is given by spherical harmonics. For the weighted radial wave function overlap, analytic expressions are derived in the Slater-type and the hydrogen-like orbital models. These expressions are shown to be composed of a finite sum of polynomials and elementary trigonometric functions. Hence, they are easy to use, require little computation time, and are significantly more accurate than commonly used approximations.

The propagation of electron beams carrying angular momentum in crystals is studied using a multislice approach for the model system Fe. It is found that the vortex beam is distorted strongly due to elastic scattering. Consequently, the expectation value of the angular momentum as well as the local vortex components change with the initial position of the vortex and the propagation depth, making numerical simulations indispensable when analyzing experiments.

The coupling between Angstrom-sized electron probes and spin polarised electronic transitions shows that the inelastically scattered probe is in a mixed state containing electron vortices with non-zero orbital angular momentum. These electrons create an asymmetric intensity distribution in energy filtered diffraction patterns, giving access to maps of the magnetic moments with atomic resolution. A feasibility experiment shows evidence of the predicted effect. Potential applications are column-by-column maps of magnetic ordering, and the creation of Angstrom-sized free electrons with orbital angular momentum by inelastic scattering in a thin ferromagnetic foil.

The holographic mask technique is used to create freely moving electrons with quantized angular momentum. With electron optical elements they can be focused to vortices with diameters below the nanometer range. The understanding of these vortex beams is important for many applications. Here we present a theory of focused free electron vortices. The agreement with experimental data is excellent. As an immediate application, fundamental experimental parameters like spherical aberration and partial coherence are determined., Comment: 4 pages, 5 figures

We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of such tilings are of types p3, p31m, p4, p4g, and p6. There are no isohedral tilings with symmetry groups p3m1, p4m, or p6m that have polyominoes or polyiamonds as fundamental domains. We display the algorithms' output and give enumeration tables for small values of n. This expands on our earlier works (Fukuda et al 2006, 2008).

Magnetic Circular Dichroism (MCD) is a standard technique for the study of magnetic properties of materials in synchrotron beamlines. We present here a new scattering geometry in the Transmission Electron Microscope through which MCD can be observed with unprecedented spatial resolution. A convergent electron beam is used to scan a multilayer Fe/Au sample and record energy loss spectra. Differences in the spectra induced by the magnetic moments of the Fe atoms can be resolved with a resolution of 2 nm. This is a breakthrough achievement when compared both to the previous EMCD resolution (200 nm) or the best XMCD experiments (approx. 20 nm), with an improvement of two and one order of magnitude, respectively., Comment: 11 pages, 3 figures, submitted to Ultramicroscopy

We present a derivation of the orbital and spin sum rules for magnetic circular dichroic spectra measured by electron energy loss spectroscopy in a transmission electron microscope. These sum rules are obtained from the differential cross section calculated for symmetric positions in the diffraction pattern. Orbital and spin magnetic moments are expressed explicitly in terms of experimental spectra and dynamical diffraction coefficients. We estimate the ratio of spin to orbital magnetic moments and discuss first experimental results for the Fe L_{2,3} edge., Comment: 11 pages, 2 figures

Recently it was demonstrated (Schattschneider et al., Nature 441 (2006), 486), that an analogue of the X-ray magnetic circular dichroism (XMCD) experiment can be performed with the transmission electron microscope (TEM). The new phenomenon has been named energy-loss magnetic chiral dichroism (EMCD). In this work we present a detailed ab initio study of the chiral dichroism in the Fe, Co and Ni transition elements. We discuss the methods used for the simulations together with the validity and accuracy of the treatment, which can, in principle, apply to any given crystalline specimen. The dependence of the dichroic signal on the sample thickness, accuracy of the detector position and the size of convergence and collection angles is calculated., Comment: 9 pages, 6 figures, submitted to Physical Review B

We describe a new experimental setup for the detection of magnetic circular dichroism with fast electrons (EMCD). As compared to earlier findings the signal is an order of magnitude higher, while the probed area could be significantly reduced, allowing a spatial resolution of the order of 30 nm. A simplified analysis of the experimental results is based on the decomposition of the Mixed Dynamic Form Factor S(q,q',E) into a real part related to the scalar product and an imaginary part related to the vector product of the scattering vectors q and q'. Following the recent detection of chiral electronic transitions in the electron microscope the present experiment is a crucial demonstration of the potential of EMCD for nanoscale investigations., Comment: 12 pages, 6 figures, submitted to Ultramicroscopy

This document describes a two-year project designed to prepare, desk-top publish, and class-test supplemental materials geared specifically toward a slower-paced, one-year Calculus I course that integrates the review of precalculus topics as they are needed. This course replaces the traditional two-semester Precalculus-Calculus I sequence for underprepared students. This report includes an executive summary involving project overview, statement of purpose, background and origins, project description, evaluation/project results and conclusions. A 1992 draft of the text: "A Companion To Calculus"; an evaluation report of the two-year project; announcements and program materials from the dissemination conference, and two papers related to this project are contained in appendices. (ASK)