# Theory and applications of free-electron vortex states

## TL;DR

This paper reviews the development, properties, and applications of free-electron vortex states, highlighting their significance in quantum mechanics, microscopy, and high-energy physics, and suggesting directions for future research.

## Contribution

It provides a comprehensive overview of theoretical and experimental advances in free-electron vortex states, including new insights and potential applications across various physics domains.

## Key findings

- Electron vortex states carry intrinsic orbital angular momentum.
- Vortex electron beams have been experimentally realized and utilized in microscopy.
- Potential applications include probing matter at atomic scales and high-energy particle processes.

## 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.