Abstract: 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 of 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.

Abstract: 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.

Abstract: By focusing electrons on probes with a diameter of 50 pm, aberration-corrected scanning transmission electron microscopy (STEM) is currently crossing the border to probing subatomic details. A major challenge is the measurement of atomic electric fields using differential phase contrast (DPC) microscopy, traditionally exploiting the concept of a field- induced shift of diffraction patterns. Here we present a simplified quantum theoretical interpretation of DPC. This enables us to calculate the momentum transferred to the STEM probe from diffracted intensities recorded on a pixel array instead of conventional segmented bright- field detectors. The methodical development yielding atomic electric field, charge and electron density is performed using simulations for binary GaN as an ideal model system. We then present a detailed experimental study of SrTiO3 yielding atomic electric fields, validated by comprehensive simulations. With this interpretation and upgraded instrumentation, STEM is capable of quantifying atomic electric fields and high-contrast imaging of light atoms.

Abstract: 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 nondiffracting 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. Interestingly, the nonrotating superpositions reproduce stable multivortex configurations that appear in rotating superfluids. Our results open an avenue for the direct electron-microscopy observation of fundamental properties of free quantum-electron states in magnetic fields.

Abstract: The coupling of angstrom-sized electron probes with spin-polarized electronic transitions shows that the inelastically scattered probe electron is in a mixed state containing electron vortices with nonzero 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.

Abstract: Electron vortex beams were only recently discovered and their potential as a probe for magnetism in materials was shown. Here we demonstrate a method to produce electron vortex beams with a diameter of less than 1.2 Å. 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.

Abstract: As electron energy-loss spectroscopy (EELS) and x-ray absorption spectroscopy (XAS) probe the same transitions from coreshell states to unoccupied states above the Fermi energy, it should always be possible to apply the two techniques to the same physical phenomena, such as magnetic dichroism, and obtain the same information. Indeed, the similarity in the expression of the electron and x-ray cross-sections had been already exploited to prove the equivalence of x-ray magnetic linear dichroism and anisotropy in EELS, by noting that the polarization vector of a photon plays the same role as the momentum transfer in electron scattering. Recently, the same was proven true for x-ray magnetic circular dichroism (XMCD) by establishing a new TEM technique called EMCD (electron energy-loss magnetic chiral dichroism) (Schattschneider P et al 2006 Nature 441 4868), which makes use of special electron scattering conditions to force the absorption of a circularly polarized virtual photon. The intrinsic advantage of EMCD over XMCD is the high spatial resolution of electron microscopes, which are readily available. Among the particular obstacles in EMCD that do not exist for synchrotron radiation, is the notoriously low signal and the very particular scattering conditions necessary to observe a chiral dichroic signal. In spite of that, impressive progress has been made in recent years. The signal strength could be considerably increased, and some innovations such as using a convergent beam have been introduced. EMCD has evolved into several techniques, which make full use of the versatility of the TEM and energy filtering, spectroscopy or STEM conditions (Rubino S 2007 Magnetic circular dichroism in the transmission electron microscope PhD Thesis Vienna University of Technology, Vienna, Austria).

Abstract: Vortex beams (also known as beams with a phase singularity) consist of spiralling wavefronts that give rise to angular momentum around the propagation direction. Vortex photon beams are widely used in applications such as optical tweezers to manipulate micrometre-sized particles and in micro-motors to provide angular momentum1, 2, improving channel capacity in optical3 and radio-wave4 information transfer, astrophysics5 and so on6. Very recently, an experimental realization of vortex beams formed of electrons was demonstrated7. Here we describe the creation of vortex electron beams, making use of a versatile holographic reconstruction technique in a transmission electron microscope. This technique is a reproducible method of creating vortex electron beams in a conventional electron microscope. We demonstrate how they may be used in electron energy-loss spectroscopy to detect the magnetic state of materials and describe their properties. Our results show that electron vortex beams hold promise for new applications, in particular for analysing and manipulating nanomaterials, and can be easily produced.

Abstract: According to theory, x-ray magnetic circular dichroism in a synchrotron is equivalent to energy loss magnetic chiral dichroism (EMCD) in a transmission electron microscope (TEM). After a synopsis of the development of EMCD, the theoretical background is reviewed and recent results are presented, focusing on the study of magnetic nanoparticles for ferrofluids and Heusler alloys for spintronic devices. Simulated maps of the dichroic strength as a function of atom position in the crystal allow evaluating the influence of specimen thickness and sample tilt on the experimental EMCD signal. Finally, the possibility of direct observation of chiral electronic transitions with atomic resolution in a TEM is discussed.

Abstract: Considering the rapid technical development of transmission electron microscopes, we investigate the possibility to map electronic transitions in real space on the atomic scale. To this purpose, we analyse the information carried by the scatterer's initial and final state wave functions and the role of the different atomic transition channels for the inelastic scattering cross section. It is shown that the change in the magnetic quantum number in the transition can be mapped. Two experimental set-ups are proposed, one blocking half the diffraction plane, the other one using a cylinder lens for imaging. Both methods break the conventional circular symmetry in the electron microscope making it possible to detect the handedness of electronic transitions as an asymmetry in the image intensity. This finding is of important for atomic resolution energy-loss magnetic chiral dichroism (EMCD), allowing to obtain the magnetic moments of single atoms.

Abstract: We present energy filtered holography experiments on a thin foil of Al. By propagating the reduced density matrix of the probe electron through the microscope, we quantitatively predict the fringe contrast as a function of energy loss. Fringe contrast simulations include the effect of Fresnel fringes created at the edges of the defocused biprism, the effect of partial coherence in combination with inelastic scattering, and the effect of a finite energy distribution of the incoming beam. (c) 2007 Elsevier B.V. All rights reserved.

Abstract: A theoretical framework is described to understand the results of plasmon holography experiments leading to insight in the meaning of the experimental results and pointing out directions for future experiments. The framework is based on the formalism of mutual intensity to describe how coherence is transferred through an optical system. For the inelastic interaction with the object, an expression for the volume. plasmon excitations in a free electron gas is used as a model for the behaviour of aluminium. The formalism leads to a clear graphical intuitive tool for under-standing the experiments. It becomes evident that the measured coherence is solely related to the angular distribution of the plasmon scattering in the case of bulk plasmons. After describing the framework, the special case of coherence outside a spherical particle is treated and the seemingly controversial idea of a plasmon with a limited coherence length obtained front experiments is clarified. (C) 2004 Elsevier B.V. All rights reserved.