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.

We give an exact self-consistent operator description of the spin and orbital angular momenta, position, and spin-orbit interactions of nonparaxial light in free space. Both quantum-operator formalism and classical energy-flow approach are presented. We apply the general theory to symmetric and asymmetric Bessel beams exhibiting spin-and orbital-dependent intensity profiles. The exact wave solutions are clearly interpreted in terms of the Berry phases, quantization of caustics, and Hall effects of light, which can be readily observed experimentally

We explore the behavior of a class of fully correlated optical beams that span the entire surface of the Poincaré sphere. The beams can be constructed from a coaxial superposition of a fundamental Gaussian mode and a spiral-phase Laguerre-Gauss mode having orthogonal polarizations. When the orthogonal polarizations are right and left circular, the coverage extends from one pole of the sphere to the other in such a way that concentric circles on the beam map onto parallels on the Poincaré sphere and radial lines map onto meridians. If the beam waist parameters match, the map is stereographic and the beam propagation corresponds to a rigid rotation about the pole. We present an experimental example of how a symmetrically stressed window can produce these beams and show that the predicted rotation indeed occurs when moving through the beams' focus.

We present a general theory of spin-to-orbital angular momentum (AM) conversion of light in focusing, scattering, and imaging optical systems. Our theory employs universal geometric transformations of non-paraxial optical fields in such systems and allows for direct calculation and comparison of the AM conversion efficiency in different physical settings. Observations of the AM conversions using local intensity distributions and far-field polarimetric measurements are discussed.

A general family of scalar structured Gaussian beams naturally emerges from a consideration of families of rays. These ray families, with the property that their transverse profile is invariant upon propagation (except for a global rescaling), have two parameters, the first giving a position on an ellipse naturally represented by a point on a ray-family analog of the Poincaré sphere (familiar from polarization optics), and the other determining the position of a curve traced out on this Poincaré sphere. This construction naturally accounts for the well-known families of Gaussian beams, including Hermite-Gaussian, Laguerre-Gaussian, and generalized Hermite-Laguerre-Gaussian beams, but is far more general, opening the door for the design of a large variety of propagation-invariant beams. This ray-based description also provides a simple explanation for many aspects of these beams, such as "self-healing" and the Gouy and Pancharatnam-Berry phases. Further, through a conformal mapping between a projection of the Poincaré sphere and the physical space of the transverse plane of a Gaussian beam, the otherwise hidden geometric rules behind the beam's intensity distribution are revealed. While the treatment is based on rays, a simple prescription is given for recovering exact solutions to the paraxial wave equation corresponding to these rays.

The group velocity of 'space-time' wave packets -propagation-invariant pulsed beams endowed with tight spatio-temporal spectral correlations -can take on arbitrary values in free space. Here we investigate theoretically and experimentally the maximum achievable group delay that realistic finite-energy space-time wave packets can achieve with respect to a reference pulse traveling at the speed of light. We find that this delay is determined solely by the spectral uncertainty in the association between the spatial frequencies and wavelengths underlying the wave packet spatiotemporal spectrum -and not by the beam size, bandwidth, or pulse width. We show experimentally that the propagation of space-time wave packets is delimited by a spectral-uncertainty-induced 'pilot envelope' that travels at a group velocity equal to the speed of light in vacuum. Temporal walkoff between the space-time wave packet and the pilot envelope limits the maximum achievable differential group delay to the width of the pilot envelope. Within this pilot envelope the spacetime wave packet can locally travel at an arbitrary group velocity and yet not violate relativistic causality because the leading or trailing edge of superluminal and subluminal space-time wave packets, respectively, are suppressed once they reach the envelope edge. Using pulses of width ∼ 4 ps and a spectral uncertainty of ∼ 20 pm, we measure maximum differential group delays of approximately ±150 ps, which exceed previously reported measurements by at least three orders of magnitude.

Geometric phases are a universal concept that underpins numerous phenomena involving multicomponent wave fields. These polarization-dependent phases are inherent in interference effects, spin-orbit interaction phenomena, and topological properties of vector wave fields. Geometric phases have been thoroughly studied in two-component fields, such as two-level quantum systems or paraxial optical waves. However, their description for fields with three or more components, such as generic nonparaxial optical fields routinely used in modern nano-optics, constitutes a nontrivial problem.Here we describe geometric, dynamical, and total phases calculated along a closed spatial contour in a multi-component complex field, with particular emphasis on 2D (paraxial) and 3D (nonparaxial) optical fields. We present several equivalent approaches: (i) an algebraic formalism, universal for any multi-component field; (ii) a dynamical approach using the Coriolis coupling between the spin angular momentum and reference-frame rotations; and (iii) a geometric representation, which unifies the Pancharatnam-Berry phase for the 2D polarization on the Poincaré sphere and the Majoranasphere representation for the 3D polarized fields. Most importantly, we reveal close connections between geometric phases, angular-momentum properties of the field, and topological properties of polarization singularities in 2D and 3D fields, such as C-points and polarization Möbius strips.

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