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.

Hybrid organic-inorganic perovskites have high potential as materials for solar energy applications, but their microscopic properties are still not well understood. Atomic-resolution scanning transmission electron microscopy has provided invaluable insights for many crystalline solar cell materials, and we used this method to successfully image formamidinium lead triiodide [CH(NH2)2PbI3] thin films with a low dose of electron irradiation. Such images reveal a highly ordered atomic arrangement of sharp grain boundaries and coherent perovskite/PbI2 interfaces, with a striking absence of long-range disorder in the crystal. We found that beam-induced degradation of the perovskite leads to an initial loss of formamidinium [CH(NH2)2+] ions, leaving behind a partially unoccupied perovskite lattice, which explains the unusual regenerative properties of these materials. We further observed aligned point defects and climb-dissociated dislocations. Our findings thus provide an atomic-level understanding of technologically important lead halide perovskites.

A model for a new electron-vortex beam production method is proposed and experimentally demonstrated. The technique calls on the controlled manipulation of the degrees of freedom of the lens aberrations to achieve a helical phase front. These degrees of freedom are accessible by using the corrector lenses of a transmission electron microscope. The vortex beam is produced through a particular alignment of these lenses into a specifically designed astigmatic state and applying an annular aperture in the condenser plane. Experimental results are found to be in good agreement with simulations.

The recent demonstration of electron vortex beams has opened up the new possibility of studying orbital angular momentum (OAM) in the interaction between electron beams and matter. To this aim, methods to analyze the OAM of an electron beam are fundamentally important and a necessary next step. We demonstrate the measurement of electron beam OAM through a variety of techniques. The use of forked holographic masks, diffraction from geometric apertures, diffraction from a knifeedge and the application of an astigmatic lens are all experimentally demonstrated. The viability and limitations of each are discussed with supporting numerical simulations.

We investigate topological aspects of sub-nm electron vortex beams upon elastic propagation through atomic scattering potentials. Two main aspects can be distinguished: (i) Significantly reduced delocalization compared to a similar non-vortex beam if the beam centers on an atomic column and (ii) site symmetry dependent splitting of higher-order vortex beams. Furthermore, the results provide insight into the complex vortex line fabric within the elastically scattered wave containing characteristic vortex loops predominantly attached to atomic columns and characteristic twists of vortex lines around atomic columns.