Theory and applications of free-electron vortex states
K. Y. Bliokh, I. P. Ivanov, G. Guzzinati, L. Clark, R. Van Boxem, A. Béché, R. Juchtmans, M. A. Alonso, P. Schattschneider, F. Nori, J. Verbeeck
TTheory and applications of free-electron vortex states
K. Y. Bliokh a,b , I. P. Ivanov c , G. Guzzinati d , L. Clark d,e , R. Van Boxem d , A. B´ech´e d , R. Juchtmans d ,M. A. Alonso f , P. Schattschneider g , F. Nori a,h , J. Verbeeck d a CEMS, RIKEN, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan b Nonlinear Physics Centre, PSPE, The Australian National University, Canberra ACT 0200, Australia c CFTP, Instituto Superior T´ecnico, Universidade de Lisboa, Lisbon, Portugal d EMAT, University of Antwerp, Groenenborgerlaan 171, 2020, Antwerp, Belgium e School of Physics and Astronomy, Monash University, VIC, 3800, Australia f The Institute of Optics, University of Rochester, Rochester NY 14627, USA g TU Wien, University Service Centre for Electron Microscopy, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria h Physics Department, University of Michigan, Ann Arbor, MI 48109-1040, USA
Abstract
Both classical and quantum waves can form vortices : with helical phase fronts and azimuthal currentdensities. These features determine the intrinsic orbital angular momentum carried by localized vortexstates. In the past 25 years, optical vortex beams have become an inherent part of modern optics, withmany remarkable achievements and applications. In the past decade, it has been realized and demonstratedthat such vortex beams or wavepackets can also appear in free electron waves , in particular, in electronmicroscopy. Interest in free-electron vortex states quickly spread over different areas of physics: from basicaspects of quantum mechanics, via applications for fine probing of matter (including individual atoms), tohigh-energy particle collision and radiation processes. Here we provide a comprehensive review of theoreticaland experimental studies in this emerging field of research. We describe the main properties of electron vortexstates, experimental achievements and possible applications within transmission electron microscopy, as wellas the possible role of vortex electrons in relativistic and high-energy processes. We aim to provide a balanceddescription 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 electronmicroscopy and beyond, in any situation where free electrons occur.
Contents1 Introduction 3
Preprint submitted to Physics Reports March 22, 2017 a r X i v : . [ qu a n t - ph ] M a r .4.2 Diffraction grating with a fork dislocation . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.3 Magnetic monopole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Vortex electrons in electric and magnetic fields. Basic aspects. . . . . . . . . . . . . . . . . . 182.6 Longitudinal magnetic field. Landau states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Unusual rotational dynamics in a magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . 242.8 Spin-orbit interaction phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.9 Electron-electron interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Acknowledgements 76Appendix A. Conventions and notations 77 CIS. Even a vortex is a vortex in something. You can’t have a whirlpool withoutwater; and you can’t have a vortex without gas, or molecules or atoms or ions orelectrons or something, not nothing.THE HE-ANCIENT. No: the vortex is not the water nor the gas nor the atoms:it is a power over these things.
George Bernard Shaw “Back to Methuselah”
1. Introduction
Electrons are elementary quantum particles which exhibit wave-particle duality inherent to all quantumobjects [1]. While their particle properties are known from classical electrodynamics (where electrons areconsidered as point charged particles experiencing Lorentz force and Coulomb interaction), the wave featuresof electrons are described in quantum mechanics by the Schr¨odinger equation [2, 3]. Depending on theproblem, either particle or wave picture could be more suitable. For example, confined electron states inatomic orbitals are clearly wave entities.The wave-particle duality of free electrons manifests itself naturally in electron microscopy [1, 4, 5].Indeed, individual electrons are usually well separated from each other, and as a single electron hits thedetector, it appears as a single bright point on the detector. At the same time, the signal accumulatedfrom many electrons clearly exhibits interference patterns characteristic of waves: such as, e.g., two-slitinterference [1, 6, 7], Fig. 1. Therefore, the description of electron evolution in microscopes sometimes relieson classical equations of motion with the Lorentz force, and sometimes requires the use of the Schr¨odingerwave equation.In many cases it is sufficient to assume that the electron’s wave nature reveals itself in the plane-wave-like phase acquired upon the electron propagation. To consider localized electrons, one usually impliessemiclassical Gaussian-like wavepackets with spatial dimensions much larger than the de Broglie wavelength.The centroids of such wavepackets follow classical trajectories (according to the Ehrenfest theorem [3, 8]),while their phase fronts can be locally approximated by a plane wave with the wave vector correspondingto the mean (expectation) value of the electron momentum.
Figure 1: Wave-particle duality of electrons. Single electrons, arriving one by one at the detector, build up the interferencepattern in a two-slit interference experiment in a transmission electron microscope [1].
Plane waves are very basic wave entities, while generic wave fields can exhibit features drastically differentfrom planar phase fronts propagating in the normal direction. Wave fields which are essentially differentfrom plane waves (or smooth Gaussian wavepackets) are often called structured waves.Structured waves naturally appear in problems with external potentials, where plane waves are notsolutions of the wave (Schr¨odinger) equation. Examples include: atomic orbitals, modes of quantum dots or3esonators, Landau states in a magnetic field, surface waves, to name a few. However, even free-space wavesare generically structured. Of course, any free-space wave field is a superposition of multiple plane wavesseen in the momentum (Fourier) representation. But interference of these plane-wave components can leadto rather non-trivial properties of the resulting wave field. This is because most of the important physicalcharacteristics – intensity, current, momentum, etc. – are described by quadratic forms of the wave function,so that the superposition principle is applicable to wave fields, but not to their physical properties.The interference of two plane waves can already be considered as a structured wave field. However, themost interesting and generic forms appear starting from three -wave intereference [9]. Namely, wave fieldsconsisting of three or more interfering plane waves generically contain phase singularities , i.e., dislocationsof phase fronts or vortices [10, 11, 12]. Such singularities appear in the points of destructive interference, r = r s , where the amplitude of the wave function vanishes, | ψ ( r s ) | = 0, while its phase Arg ψ ( r s ) isindeterminate. A vanishing amplitude of a complex field means two real conditions (vanishing of its realand imaginary parts), so that phase singularities generically appear as points in 2D plane or lines in 3Dspace. Most importantly, the phase of the wave function is well-defined around singular points/lines, andgenerically it has a nonzero increment for a countour enclosing the singularity: (cid:72) ∇ Arg ψ ( r ) · d r = 2 π(cid:96) .Here (cid:96) = 0 , ± , ± , ... is an integer winding number (to provide continuity of the phase modulo 2 π ), whichis called the “topological charge” of the vortex. The typical behaviour of the wave function near the phasesingularity is ψ ∝ | r − r s | | (cid:96) | exp( i(cid:96)ϕ s ), where ϕ s is the azimuthal angle around the r = r s point. Such waveforms are called vortices because the probability current density j ∝ Im ( ψ ∗ ∇ ψ ) = | ψ | ∇ Arg ψ swirls aroundphase singularities. For example, Figure 2 shows multiple vortices in a 2D interference field obtained as asuperposition of randomly-directed plane waves. In the 3D case, vortex lines are dislocation lines for phasefronts (i.e., surfaces of constant phase) [10, 11, 12] (see Fig. 3 below). Figure 2: Vortices (phase singularities) in a random 2D interference field [12]. The complex wave function ψ ( r ), r = ( x, y ) isobtained as a superposition of 100 plane waves with randomly-oriented wave vectors k = ( k x , k y ), fixed k , and random phases[12]. (a) The grayscale plot displays the absolute value of the wave function, | ψ ( r ) | . Streamlines of the probability currentdensity, j ∝ ∇ Arg ψ , are shown in orange. (b) Color-coded phase of the wave function, Arg ψ ( r ). (c) Combined representationof the complex ψ ( r ), where the brigtness is proportional to the amplitude, while the color indicates the phase [13]. Black andwhite dots in (b) mark vortices with (cid:96) = 1 and (cid:96) = −
1, respectively. The probability current in (a) forms whirlpools aroundthese points. The typical distance between vortices in such interference patterns is of the order of the wavelength 2 π/k . Since vortices are generic wave forms, they appeared in many early studies of various types of waves. Inoptical fields, an example of a vortex was described in 1950 for the total-internal-reflection of a light beam[14], and a famous textbook [15] reproduces detailed figures from 1952 [16] with multiple optical vorticesin a plane wave diffracted by a half-plane. For quantum matter waves, wave functions with vortices wereknown from the early days of quantum mechanics. Indeed, spherical harmonics, atomic orbitals with orbitalangular momentum [2, 3], and eigenmodes of the Schr¨odinger equation in a magnetic field [17] all containthe exp( i(cid:96)ϕ ) vortex factors. Furthermore, the seminal Dirac paper about magnetic monopoles [18] analysesthe phase singularities in a wave function, and vortex eigenmodes appear in the related Aharonov–Bohmproblem [19].Despite these multiple predecessors, the first systematic study of phase singularities was performed in4974 by Nye and Berry [10] in the context of ultrasonic pulses. Almost simultaneously, Hirschfelder et al. [20, 21] analysed vortices in quantum wave functions. The seminal work by Nye and Berry gave birth tothe field of singular optics , with thousands of studies in the past decades [22, 11, 12]. Vortices were shownto be very important in the analysis of structured wave fields. They form a “singular skeleton”, on whichthe phase and intensity structure hangs [23, 12]. In particular, random wave fields, which are ubiquitous innature, are pierced by numerous vortices [23, 24] (see Fig. 2) and even vortex knots (in the 3D case) [25, 26].In this manner, vortices provide unique information about wave fields, both statistical and as “fingerprints”of individual realizations.
The swirling current around phase singularities suggests that vortices should possess angular-momentum properties. Indeed, assuming cylindrical or spherical coordinates with the azimuthal angle ϕ , vortex wave-fucntions ψ ∝ exp( i(cid:96)ϕ ) are eigenmodes of the z -component of the quantum-mechanical orbital angular mo-mentum (OAM) operator, ˆ L z = − i (cid:126) ∂/∂ϕ , with the eigenvalues (cid:126) (cid:96) [2, 3]. In random wavefields, Fig. 2, thenumbers of positive and negative vortices are approximately equal to each other, and the net OAM approx-imately vanishes. Moreover, only vortices with topological charges (cid:96) = ± | (cid:96) | . Although the OAM eigenmodes with vortices have been known for many years in textbooks on quantummechanics [2, 3], only in 1992 Allen et al. [27] realized that such wave modes can be generated as free-spaceoptical beams. Indeed, the free-space solutions of the wave equation in cylindrical coordinates ( r, ϕ, z ),which propagate along the z -axis have a typical form ψ ( r ) ∝ f ( r ) exp( ik z z + i(cid:96)ϕ ), where f ( r ) is the radialdistribution (which can also slowly change with z for diffracting beams), k z is the longitudinal wave number,and (cid:96) is the azimuthal quantum number. At (cid:96) = 0, such solutions describe usual Gaussian-like wave beams,while higher-order modes with (cid:96) (cid:54) = 0 are the so-called vortex beams , shown in Fig. 3. Such beams haveisolated vortices of topological charge (cid:96) on their axes, helical phase fronts, and spiralling currents. Mostimportantly, being eigenmodes of ˆ L z , vortex beams carry a well-defined OAM (cid:126) (cid:96) per particle (photon in thecase of optics) along their axes: (cid:104) L z (cid:105) = (cid:126) (cid:96) .The presence of a vortex and well-defined OAM dramatically modifies both geometrical and dynamical properties of the wave. Therefore, the description and generation of optical vortex beams in the beginningof the 1990s [28, 27, 29, 30] caused enormous interest and initiated the rapidly-developing field of opticalangular momentum . Since then, optical vortex beams have been intensively studied and have found numer-ous applications in diverse areas, including: optical manipulations of small particles or atoms [31, 32, 33],quantum information and communications [34, 35, 36, 37, 38], quantum entanglement [39, 40], radio com-munications [41, 42], astronomy and astrophysics [43, 44, 45, 46, 47], optical solitons [48, 49, 50], andHall effects [51, 52, 53, 54]. In the past two decades, five books [55, 56, 57, 58, 59] and many reviews[60, 61, 62, 63, 64, 65, 66] about optical vortex beams and OAM were published.Several important physical points have to be made about the OAM of vortex wave states: • The z -directed OAM carried by vortex beams is intrinsic [67, 65], i.e., independent of the choice ofcoordinates. This is in sharp contrast to the extrinsic mechanical OAM of classical point particles, L = r × p (where r and p are the particle coordinates and momentum, respectively), which dependson the choice of the coordinate origin. • Moreover, the mean (expectation) value of the OAM in vortex beams is aligned with the mean mo-mentum: (cid:104) L (cid:105) = (cid:96) (cid:104) p (cid:105) / (cid:104) p (cid:105) . This is also in contrast to point-particle OAM, which is orthogonal to themomentum at every instant of time: L ⊥ p . • The intrinsic OAM and spiraling current density do not contradict the rectilinear propagation of eitherplane waves or classical particles in free space. Indeed, the centroid of a vortex state follows a rectilinear trajectory in free-space (e.g., lies on the axes of vortex beams). Also, vortex beams are superpositions5 igure 3: Vortex beams are cylindrical solutions of the wave equation. They propagate along the z -axis and carry intrinsiclongitudinal orbital angular momentum (OAM) (cid:104) L z (cid:105) = (cid:126) (cid:96) (assuming paraxial approximation) per particle. The 3D schematicsshow the phase fronts (cyan) and probability-current streamlines (orange) for beams with different vortex charges (cid:96) . The 2Dplots show the corresponding transverse wave function distributions ψ ( x, y ) at z = 0; the phase-amplitude representation issimilar to Fig. 2(c). The radial profiles correspond to the Bessel modes analyzed in detail in Sections 2.2 and 2.3 below. of plane waves [see Figs. 5(a) and 6(a) below], but the probability-current streamlines, i.e., Bohmiantrajectories of the particles [68, 8, 69], can be curvilinear in free space [70, 71]. • Vortex states carrying intrinsic OAM is not a collective effect, but a phenomenon that persists on the single-particle level [34]. In other words, these are forms of the single-particle wave function.
Until recently, the majority of studies on phase singularities and free-space vortex beams dealt withoptical fields and other classical waves. At the same time, the universal character of wave equations suggeststhat fundamental results of singular optics and optical angular momentum should be equally applicable to quantum , in particular electron, waves [72]. Moreover, the concept of the OAM in vortex beams essentiallyrelies on the quantum-mechanical operator ˆ L z . Nonetheless, until recently there were only a few studies offree-space quantum wave functions with vortices [20, 21, 73, 74].In 2007 Bliokh et al. [75] suggested that free electrons can be in vortex-beam (or vortex-wavepacket)states carrying intrinsic OAM. They also discussed basic interactions of such vortex electrons with externalfields and possible ways they could be generated. In 2010, free-electron vortex beams were indeed producedin transmission electron microscopes (TEMs) by Uchida and Tonomura [76] and Verbeeck et al. [77]. Oneyear later, McMorran et al. [78] demonstrated the generation of electron vortex beams with OAM up to6 (cid:96) | = 100. This is in enormous contrast with the spin angular momentum (SAM) of electrons, which isrestricted to 1/2 (in units of (cid:126) ). These studies initiated a new research area investigating free-electronvortex states, or, in a wider context, structured quantum waves [79]. Electron vortex beams are currentlyattracting considerable interest, with potential applications in various fields, such as electron microscopy,fundamentals of quantum mechanics, and high-energy physics. The present paper aims to provide the firstcomprehensive review of this emerging area of research.Thus, free-space vortex beams carrying OAM, based on the quantum-mechanical picture of angular mo-mentum, were developed in classical optics, and recently returned to their quantum roots. While similaritiesbetween optical and electron waves are obvious, it is important to mention basic distinctions between op-tical and electron vortices. Apart from the huge difference in wavelengths, electrons, unlike photons, are charged particles, and therefore can directly interact with each other as well as with external electric andmagnetic fields . Moreover, the presence of the OAM means the presence of a vortex-induced magnetic mo-ment carried by vortex electrons. Furthermore, electrons can interact with electromagnetic waves (photons),as well as radiate photons via the Vavilov–Cherenkov or other effects. Vortex electrons can also partici-pate in particle collisions in the context of high-energy physics. All these phenomena enrich the physicsof structured electron waves, as compared to their optical counterparts. At the same time, some features,naturally present in optical waves, are practically absent in eletron optics. First of all, free-electron sourcesin electron microscopy generate unpolarized particles, which are described by the scalar wave function. Thisis in sharp contrast to optics, where the use of spin (polarization) degrees of freedom is ubiquitous both inregular and singular/OAM optics [80, 81, 66]. In addition, electron beams in TEM are highly-paraxial , whilemodern nano-optics often deals with non-paraxial (tighly focused or scattered) fields with wavelength-scaleinhomogenities [82]. The wave nature of electrons is naturally exploited in transmission electron microscopy and holography[1, 4, 5, 83, 84]. Electron microscopes can vastly outperform optical microscopes in terms of spatial resolutionbecause of the extremely small wavelength (of the order of picometers) obtained in accelerated electron waves.This explains the tremendous success of TEMs in exploring the atomic structure of matter.On the one hand, in conventional TEM imaging and holography, a nearly-plane electron wave is producedto interact with a thin sample. The local interaction of the electron wave with microscopic electromagneticpotentials of the specimen leads to deformations of planar phase fronts and produces a structured transmittedwave. This wave contains information about the atomic structure and electromagnetic properties of thesample. Naturally, transmitted waves contain a multitude of vortices, and the well-developed methods ofsingular optics [11, 12] could provide a new insight and a toolbox for electron microscopy [74, 85, 86].On the other hand, recent progress in the deliberate creation of electron vortex beams [76, 77, 78] allowsone to employ incident structured electrons and make use of the new
OAM degrees of freedom .Actually, free-space vortex electron states by themselves offer unique opportunities of studying fundamen-tal quantum-mechanical phenomena. In particular, interactions with external magnetic fields and structuresbring about a number of fundamental effects involving vortices [87, 88, 89, 90]. Recent TEM experimentsfor the first time demonstrated free-electron
Landau states (previously hidden in condensed-matter systems)and their fine internal dynamics [91], as well as the interaction of electron waves with approximate magneticmonopoles [92] (previously only considered theoretically).Most importantly, incident vortex electrons interacting with the specimen in a TEM can unveil newinformation about the sample or increase the resolution of the microscope [93]. In particular, recent exper-iments with electron vortex beams demonstrated their role in chiral energy-loss spectroscopy and magneticdichroism [77, 94, 95, 96, 97, 98, 99, 100]. This is in sharp contrast to optics, where probing magnetic dichro-ism and chirality involve only polarization (spin) degrees of freedom of light and are mostly insensitive tovortices [101, 102, 103, 104]. Moreover, focusing free-electron vortices makes them comparable in size andparameters with orbitals in atoms [100]. This opens a way for magnetic mapping with atomic resolution [105, 106, 107]. 7 .6. High-energy perspective: scattering and radiation
Vortex electrons can also contribute to the study of fundamental interaction phenomena besides electronmicroscopy. Two directions which can be pursued with current technology are: (i) the interaction of vortexelectrons with intense laser fields [108, 109] and (ii) radiation processes with vortex electrons (e.g., theVavilov–Cherenkov and transition radiations), which were predicted to depart from their usual expressions[110, 111, 112, 113]. For instance, vortex electrons (carrying large OAM and magnetic moment) can revealthe magnetic-moment contribution to the transition radiation, which has never been observed before.Even more exciting is the possibility to bring vortex states in quantum-particle collisions [114, 115]. Inall collider-like settings realized thus far, the colliding particles (electrons, positrons, or hadrons) behaveas semiclassical Gaussian-like wavepackets, and their scattering processes can be safely calculated in termsof plane waves. Collisions of vortex particles involve a completely new degree of freedom: the intrinsicOAM. Therefore, in addition to the kinematical distributions and polarization dependences, one can studydependences of the scattering cross-sections on the OAM of the incoming particles. This possibility isparticularly tantalizing in hadronic physics, in the context of the proton spin puzzle [116]. Experimentaldata show that a significant part of the proton spin comes from the orbital angular momentum of the quarksand gluons, but its exact contribution, as well as the whole issue of the SAM–OAM separation, remainsunder hot debate [117, 118, 119]. Vortex electrons can serve as a new OAM-sensitive probe in this problem.Recent theoretical investigations brought several examples of quantities which have been inaccessible sofar, but which could be revealed in vortex-particle collisions [120, 121, 122, 123, 124]. In particular, byscattering vortex electrons on a counterpropagating particle and observing the interference fringes in theirjoint angular distribution, one can directly probe its OAM state [121]. This phenomenon can be employedto probe the proton spin structure. On the experimental side, major challenges still need to be overcome,such as the acceleration of vortex electrons to higher energies and focusing them as tightly as possible ontothe protons. The generation and acceleration of vortex protons is another future milestone to be achievedexperimentally.
The present paper aims to provide the first comprehensive review of the theory and applications offree-electron vortex states carrying OAM. The paper is organized as follows. This Introduction providesa pedagogical and historical overview of wave vortices and vortex OAM in classical and quantum waves.The mostly-theoretical Section 2 describes basic physical properties of free-electron vortex states: their wavefunctions, currents, AM properties, magnetic moment, evolution in external electric and magnetic fields, etc.Section 3 reviews the most important TEM experiments and proposals involving electron vortices: produc-tion methods, OAM measurements, elastic and inelastic interaction with matter, and transfer of mechanicalangular momentum. Several applications and future prospects are also discussed. Readers interested in themostly-experimental TEM part can skip Section 2 and read Section 3 right after the Introduction. Sec-tion 4 describes the main problems involving vortex electrons in high-energy physics: relativistic effects andcollisions of vortex particles. Section 5 explains peculiarities of the radiation processes (Vavilov–Cherenkovand transition radiation) with vortex electrons. Finally, a brief outlook of future perspectives concludes thereview.For the reader’s convenience, in the Appendix A (Tables 2 and 3) we summarize the main abbreviations,conventions, and notations used in this paper. Also the following conventions are used in figures through-out the review. Intensity (i.e., probability-density) distributions are mostly shown using 2D grayscale (ormonochrome) plots, in arbitrary units, with brighter areas corresponding to higher intensities [as, e.g., inFigs. 1 and 2(a)]. The phase distributions are shown using rainbow colors, as in Fig. 2(b). Similarly, rainbow-colored wave vectors [e.g., Fig. 5(a)] indicate mutual phases of plane waves in the Fourier spectrum of thefield. We also use combined intensity-phase (brightness-color) representations of complex wave functions[13], as in Figs. 2(c) and 3. Note also that in most theoretical figures the propagation z -axis is horizon-tal for the sake of convenience, while it is vertical in schemes related to electron-microscopy experiments,corresponding to the actual TEM setup. 8 . Basic properties of electron vortex states As other quantum particles, electrons share both wave and particle properties. We start with the simplestnon-relativistic description of a scalar electron (i.e., without spin) in free space (i.e., without external fields),which is based on the Schr¨odinger wave equation [2, 3]: i (cid:126) ∂ψ∂t + (cid:126) m e ∇ ψ = 0 . (2.1)Here ψ ( r , t ) is the wave function, and m e is the electron mass. Most of the analysis below can be applied toany quantum particle described by the Schr¨odinger equation.The wave properties of the electron reveal themselves in the plane wave solution of the Schr¨odingerequation: ψ = a exp (cid:2) i (cid:126) − ( p · r − E t ) (cid:3) , (2.2)where a is the constant amplitude, p is the wave momentum, and E is the electron energy, which satisfy E = p / m e . Plane waves (2.2) have well-defined momentum, but they are delocalized in space. Therefore,plane-wave solutions (2.2) cannot be normalized and cannot correspond to physical particles.To describe localized electron states, one has to consider superpositions of multiple plane waves (2.2)with different momenta p , which produce an uncertainty in the momentum, δ p , and the corresponding finiteuncertainty in the electron coordinate [2, 3]. To model localized electrons, usually Gaussian distributions inboth momentum and coordinate spaces are implied, which satisfy the Heisenberg uncertainty principle. Letthe electron move along the z -axis with the mean momentum (cid:104) p (cid:105) = p = p ¯ z (hereafter, the overbar standsfor the unit vector in the corresponding direction), and its mean coordinate at t = 0 is (cid:104) r (0) (cid:105) = r = 0.Assuming the azimuthal symmetry of the electron’s state about the z -axis, we can write the Gaussianamplitude envelope of the wave function, a ( r , t ), as a ( r , ∝ exp (cid:18) − r ⊥ w − z l (cid:19) . (2.3)Here r ⊥ = ( x, y ) are the transverse coordinates, while w and l are the width and length of the distribution.If the electron is well-localized in momentum space, i.e., has small momentum uncertainy | δ p | (cid:28) p , thenthe spatial dimensions of its probability density distribution are large as compared with the de Brogliewavelength: w, l (cid:29) (cid:126) /p . Under these conditions, the phase of the electron wave function approximatelyfollows the plane-wave form (2.2) with p = p and the distribution (2.3) propagates as a wavepacket withvelocity v = p /m e , i.e., one can substitute z → ( z − vt ) in Eq. (2.3) at t (cid:54) = 0.The above consideration is valid only in the leading-order approximation in | δ p | /p and neglects thediffraction and dispersion effects. These include a slow spread of both the transverse and longitudinaldimensions of the wavepacket, i.e., variations of w and l during the electron motion, as well as deformationsof the phase front as compared to the plane wave [3, 8].Note that a small uncertainty in the transverse momentum components, δ p ⊥ , represent variations inthe direction of the momentum, while the longitudinal uncertainty δ p (cid:107) (cid:39) δp ¯ z represents variations in the absolute value of the momentum, which is related to the energy uncertainty: δE (cid:39) p δp/m e . Correspondingly,the w and l dimensions of the wavepackets are linked to these uncertainties in the direction of propagationand energy: w ∼ (cid:126) / | δ p ⊥ | and l ∼ (cid:126) /δp . The latter relation can be written in terms of the temporal duration τ of the wavepacket, τ = l/v , and energy uncertainty: τ ∼ (cid:126) /δE .In many problems, only the transverse localization of the electron is important. Then, one can considerstates delocalized in the longitudinal dimension, l = ∞ , and hence monoenergetic : δp = δE = 0. Such statesare called wave beams [125]. It should be noticed, however, that physical beams of electrons (e.g., in electronmicroscopes) consist of many wavepackets propagating one after another and having some finite energy9 igure 4: Schematic pictures of a plane wave, a wavepacket, and a wave beam in both real- and momentum-space representa-tions. The real-space probability density distributions are schematically shown in yellow, while the phase fronts are shown incyan, and azimuthal symmetry about the propagation z -axis is implied. Assuming the paraxial approximation p z (cid:39) p , p ⊥ (cid:28) p ,the characteristic dimensions of the real- and momentum-space distributions satisfy the uncertainty relations l ∼ (cid:126) /δp and w ∼ (cid:126) /δp ⊥ (see explanations in the text). uncertainty δE . Still, dealing with monoenergetic beam solutions significantly simplifies the analysis ofthe problem and allows one to describe most of the phenomena related to the transverse structure of theelectron wave function. Therefore, in most cases below, we will consider monoenergetic electron beams in the paraxial approximation (i.e., | δ p ⊥ | (cid:28) p ), analyzing the effects related to the energy uncertainty separately.We also note that the localization or delocalization of electron states is directly related to the continuousor discrete spectrum of the corresponding quantum parameters (in the case of a complete orthogonal setof modes). The plane waves (2.2) are delocalized in all three dimensions, and therefore are only describedby three components of p with continuous spectra. Wavepackets are localized in three dimensions and,correspondingly, can be characterized by three discrete quantum numbers. Gaussian wavepackets (2.3)correspond to the lowest-level state; higher-order states can be described, e.g., by Hermite-Gaussian modes[125]. In turn, wave beams (or spherical modes [2, 3]) are localized with respect to two dimensions and aredescribed by two discrete quantum numbers related to the transverse modal structure of the beam [125, 57]. The solutions of the Schr¨odinger equation (2.1) can be decomposed via a complete set of orthogonalfree-space modes. There are different sets of such modes, and the convenience of using one or another setis determined by symmetries and other conditions in each particular problem. Furthermore, solving theSchr¨odinger equation in different representations and coordinates naturally leads to different modes. Forexample, planes waves (2.2) represent a complete set of orthogonal modes convenient in the momentumrepresentation. These modes are delocalized and non-normalizable. In the coordinate representation, us-ing Cartesian coordinates, one can obtain
Hermite–Gaussian modes with respect to the three dimensions. This is a very accurate description for electrons in a TEM, except possibly for the source crossover region where electron-electron interaction can become non-negligble, especially at very high (pulsed) beam currents. spherical modes , widely used in atomic physics [3]. These modesare suitable for localized electrons in atoms rather than for electrons freely moving in the longitudinal z -direction in electron microscopes. Combining the z -direction of the electron motion with the isotropy of thefree-space problem with respect to the transverse ( x, y )-coordinates naturally results in the choice of cylindri-cal coordinates ( r, ϕ, z ) [75]. The cylindrical solutions that we describe below allow a convenient analyticaldescription and offer a good approximation to the electron states produced in electron microscopes. We seek monoenergetic beam eigenmodes of the Schr¨odinger equation (2.1), which correspond to theelectron propagating along the z -axis. After substitution ψ ( r , t ) → ψ ( r )exp (cid:0) i (cid:126) − E t (cid:1) , Eq. (2.1) in cylindricalcoordinates takes the form: − (cid:126) m e (cid:20) r ∂∂r (cid:18) r ∂∂r (cid:19) + 1 r ∂ ∂ϕ + ∂ ∂z (cid:21) ψ = E ψ. (2.4)The axially-symmetric solutions of Eq. (2.4) are [87]: ψ B(cid:96) ∝ J | (cid:96) | ( κr ) exp[ i ( (cid:96)ϕ + k z z )] , (2.5)where J (cid:96) is the Bessel function of the first kind, (cid:96) = 0 , ± , ± , ... is an integer number (azimuthal quantumnumber), k z = p z / (cid:126) is the longitudinal wave number, and κ = p ⊥ / (cid:126) is the transverse (radial) wave number.Solutions (2.5) satisfy Eq. (2.4) provided that the following dispersion relation is fulfilled: E = (cid:126) m e k = (cid:126) m e (cid:0) k z + κ (cid:1) , (2.6)where k = p/ (cid:126) .The cylindrically-symmetric modes (2.5) and (2.6) are called Bessel beams [126, 127, 128, 87, 129]. Theyhave a cylindrical probability density distribution, independent of z , i.e., without diffraction. Most impor-tantly, the azimuthal quantum number (cid:96) (also called the topological charge or vortex charge) determines the vortex phase structure in Bessel beams and their orbital angular momentum (OAM) properties [55, 60, 75],which are discussed in Section 2.3 below. The zero-order ( (cid:96) = 0) beam has no vortex and maximal proba-bility density on the axis, i.e., at r = 0. The higher-order ( (cid:96) (cid:54) = 0) modes are characterized by the quantumvortex exp( i(cid:96)ϕ ), spiral phase structure, azimuthal probability current, and the probability density vanishingon the axis: ψ B(cid:96) (cid:12)(cid:12) r =0 = 0. Figure 5 shows the transverse probability density and current distributions inBessel beams (2.5) and (2.6).The Bessel beams represent the simplest theoretical example of vortex beams. Despite the probabilitydensity of Bessel modes decaying as | ψ B(cid:96) | ∼ /r when r → ∞ , these solutions are not properly localized in thetransverse dimensions. Indeed, the integral (cid:82) ∞ (cid:12)(cid:12) ψ B(cid:96) (cid:12)(cid:12) r dr diverges, and the function cannot be normalizedwith respect to the transverse dimensions. The delocalized nature of Bessel beams is reflected in theabsence of diffraction and a single transverse quantum number (cid:96) (instead of two transverse quantum indicesin the properly-localized modes).In terms of the plane-wave spectrum, the Bessel beam (2.5) and (2.6) represents a superposition of planewaves with conically -distributed momenta: p (cid:107) = p = p z ¯ z and | p ⊥ | ≡ p ⊥ = (cid:126) κ , which can be characterizedby the polar angle θ , sin θ = κ/k , Fig. 5(a). This corresponds to the Fourier spectrum:˜ ψ B(cid:96) ( k ⊥ ) ∝ δ ( k ⊥ − κ ) exp( i(cid:96)φ ) , ψ B(cid:96) ( r ) = (cid:90) ˜ ψ B(cid:96) ( k ⊥ ) e i k · r d k ⊥ , (2.7) This means infinite number of particles or energy per unit z -length in the Bessel beams. Therefore, the exact solutions(2.5) cannot be generated in practice, but a good approximation to this solution can be produced in experiments for finiteradial apertures r < r max and propagation distances | z | < z max [127, 128, 129]. igure 5: Bessel-beam solutions (2.5) and (2.6) of the Schr¨odinger equation. (a) The momentum spectrum consists of conically-distributed wave vectors with fixed p ⊥ = (cid:126) κ and p z = (cid:126) k z , i.e., forms a circle. The mean electron momentum is (cid:104) p (cid:105) = (cid:126) k z ¯ z .The mutual phases (colour-coded) of the plane waves in the spectrum increase by 2 π(cid:96) around the circle. This forms a vortex oftopological charge (cid:96) ( (cid:96) = 2 here) and determines the intrinsic orbital angular momentum (OAM) of the electron: (cid:104) L (cid:105) = (cid:126) (cid:96) ¯ z ,Eq. (2.17). (b) Transverse probability density ρ (grayscale plots) and current j (circular arrows) distributions in the Besselbeams with different values of (cid:96) , Eq. (2.13). Here, the radii and thicknesses of the current circles correspond to the positionsand values of the maxima (normalized in each panel independently) of the quantity rj ϕ that determines the contribution tothe OAM, Eq. (2.16). where k = p / (cid:126) is a wave vector with transverse components k ⊥ and azimuthal angle φ in k -space, and δ is the Dirac delta-function. The delocalization of the Bessel modes and absence of diffraction is a directconsequence of the fact that the wave vectors are distributed only azimuthally, while the radial transversecomponent k ⊥ is fixed. To construct vortex beams properly localized (square-integrable) in the transverse dimensions, one canuse at least two alternative ways. First, considering superpositions of multiple Bessel beams with thesame fixed energy E but different wave numbers k z and κ (i.e., introducing some uncertainty δκ in theradial momentum component), results in a general integral form of such modes [130]. However, to dealwith analytical solutions, here we follow the second, simplified way. Namely, we make use of the paraxialapproximation: p ⊥ (cid:28) p and p (cid:107) (cid:39) p ¯ z ( k z (cid:39) k ). In the first-order approximation in p ⊥ /p = k ⊥ /k (cid:28)
1, theSchr¨odinger equation (2.1) or (2.4) can be simplified using the substitution ∂ /∂z (cid:39) − k + 2 ik ∂/∂z . Indoing so, it takes the form of the so-called paraxial wave equation , widely used in optics [125]:2 ik ∂∂z + (cid:20) r ∂∂r (cid:18) r ∂∂r (cid:19) + 1 r ∂ ∂ϕ (cid:21) ψ = 0 . (2.8)Interestingly, this equation has the form of a Schr¨odinger-like equation with the time-like coordinate z andtwo space-like transverse coordinates ( r, ϕ ).The solutions of equation (2.8) in cylindrical coordinates are the Laguerre–Gaussian (LG) beams [125,27, 60, 55, 87]: ψ LG(cid:96),n ∝ (cid:18) rw ( z ) (cid:19) | (cid:96) | L | (cid:96) | n (cid:18) r w ( z ) (cid:19) exp (cid:18) − r w ( z ) + ik r R ( z ) (cid:19) e i ( (cid:96)ϕ + kz ) e − i (2 n + | (cid:96) | +1) ζ ( z ) , (2.9)12here L | (cid:96) | n are the generalized Laguerre polynomials, n = 0 , , , ... is the radial quantum number, w ( z ) = w (cid:112) z /z R is the beam width, which slowly varies with z due to diffraction, R ( z ) = z (cid:0) z R /z (cid:1) isthe radius of curvature of the wavefronts, and ζ ( z ) = arctan( z/z R ). Here, the characteristic transverse andlongitudinal scales of the beam are the waist w (the width in the focal plane z = 0) and the Rayleighdiffraction length z R [125]: w (cid:29) π/k, z R = kw / (cid:29) w . (2.10)The last exponential factor in Eq. (2.9) describes the Gouy phase [131, 125, 132, 133]; it yields an additionalphase difference Φ G = (2 n + | (cid:96) | + 1) π (2.11)upon the beam propagation through its focal point from z/z R (cid:28) − z/z R (cid:29)
1. The Gouy phase isclosely related to the transverse confinement of the modes [133, 134]. The dispersion relation for the LGbeams is simply E = (cid:126) k / m e [cf. Eq (2.6)], while the small transverse wave-vector components are takeninto account in the z -dependent diffraction terms.The Laguerre–Gaussian beams (2.9)–(2.11) are also vortex beams, characterized by the azimuthal quan-tum number (cid:96) and factor exp( i(cid:96)ϕ ). However, in contrast to Bessel beams (2.5) and (2.6), they are properlylocalized and normalizable in the two transverse dimensions. This is because the Fourier spectrum of LGbeams is smoothly distributed over different radial wave-vector components k ⊥ , Fig. 6(a) [cf. Eq. (2.7) andFig. 5(a)]. This radial uncertainty of the momentum is related to the beam waist as δp ⊥ ∼ (cid:126) /w . Thequantum number n corresponds to the radial localization of the LG modes and determines the number ofradial maxima in their probability density distributions (see Fig. 6).Figure 6(b) shows the transverse spatial distributions of the probability densities and currents in the LGbeams with different values of quantum numbers ( (cid:96), n ). The zero-order mode ψ LG , is the standard Gaussianbeam , which can be regarded as an infinitely-long Gaussian wavepacket [cf. Eqs. (2.2) and (2.3)] with l → ∞ and δE = δp = 0. Gaussian beams or wavepackets are often implied in quantum models of freeelectrons, because they do not contain any intrinsic structures and degrees of freedom. In contrast to that,higher-order modes with ( (cid:96), n ) (cid:54) = (0 ,
0) exhibit a variety of structures related to the internal spatial degreesof freedom of localized electrons. In general, LG beams with different ( (cid:96), n ) or Bessel beams with different (cid:96) , constitute a complete set of orthogonal monoenergetic modes for the free-space Schr¨odinger equation(the LG beams being restricted by the paraxial approximation). Therefore, any free-electron state can berepresented as a superposition of these modes. Vortex beams are the most convenient modes when onedeals with monoenergetic electrons with a well-defined propagation direction, and some sort of azimuthal(cylindrical) symmetry in the problem. Importantly, the latter symmetry naturally involves the angularmomentum properties with respect to the propagation direction.
We now describe the main observable characteristics of electron vortex beams. First, the probabilitydensity and probability current density in quantum electron states are determined by [2, 3]: ρ = (cid:12)(cid:12) ψ (cid:12)(cid:12) , j = 1 m e ( ψ | ˆ p | ψ ) = (cid:126) m e Im ( ψ ∗ ∇ ψ ) . (2.12)Here, ˆ p = − i (cid:126) ∇ is the canonical momentum operator, and we use the notation ( ψ | ... | ψ ) ≡ Re (cid:0) ψ † ...ψ (cid:1) forthe local expectation value of an operator.Substituting the wave function (2.5) into Eq. (2.12), we obtain the probability density and current inthe Bessel beams (see Fig. 5): ρ B | (cid:96) | ( r ) ∝ (cid:12)(cid:12) J | (cid:96) | ( κr ) (cid:12)(cid:12) , j B(cid:96) ( r, ϕ ) = (cid:126) m e (cid:18) (cid:96)r ¯ ϕ + k z ¯ z (cid:19) ρ B | (cid:96) | ( r ) , (2.13)where ¯ ϕ is the unit vector of the azimuthal coordinate. The (cid:96) -dependent azimuthal component of theprobability current (2.13), together with its longitudinal component, result in a spiraling current , Fig. 3.13 igure 6: Same as in Fig. 5 but for the Laguerre–Gaussian (LG) beams (2.9)–(2.11) and (2.14). The Bessel and LG beamshave similar azimuthal and OAM properties. However, the LG beams are also characterized by the momentum spectrum with radial distribution of the wave vectors (a). This provides the radial confinement of the electron, which is characterized bythe additional radial quantum number n . In this manner, ( n + 1) is the number of radial maxima (rings) in the intensitydistribution (b). This is a common feature of all vortex beams [27, 60, 55, 70, 75, 87].For LG beams (2.9), the probability current density also has a radial component related to the diffraction.The probability density and azimuthal (cid:96) -dependent current component in the LG beams are (see Fig. 6): ρ LG | (cid:96) | ,n ( r, z ) ∝ (cid:18) r w ( z ) (cid:19) | (cid:96) | (cid:12)(cid:12)(cid:12)(cid:12) L | (cid:96) | n (cid:18) r w ( z ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) − r w ( z ) (cid:19) , j LG(cid:96),n ϕ ( r, z ) = (cid:96) (cid:126) m e r ρ LG | (cid:96) | ,n ( r, z ) . (2.14)The azimuthal probability current in vortex beams is directly related to the z -directed orbital angularmomentum of such states. The electron OAM can be defined either as the expectation value of the OAMoperator or via the circulation of the probability current. Normalizing this per electron, we have: (cid:104) L (cid:105) = (cid:104) ψ | ˆ L | ψ (cid:105)(cid:104) ψ | ψ (cid:105) = m e (cid:82) r × j d r (cid:82) ρ d r , (2.15)where ˆ L = r × ˆ p is the canonical OAM operator [2, 3], and the inner product involves the volume integral (cid:82) ... d r . The definition (2.15) is suitable for wavepackets localized in three dimensions. For 2D-localizedbeams one should use integrals over the two transverse dimensions: (cid:82) ... d r → (cid:82) ... d r ⊥ . This means thatfor wave beams we deal with linear densities per unit z -length and normalize quantities per electron per unit z -length [60, 55]. In this manner, the longitudinal component of the OAM in a beam becomes in cylindrical14oordinates: (cid:104) L z (cid:105) = (cid:104) ψ | ˆ L z | ψ (cid:105)(cid:104) ψ | ψ (cid:105) = m e (cid:82) rj ϕ d r ⊥ (cid:82) ρ d r ⊥ , (2.16)where ˆ L z = i (cid:126) ∂/∂φ .For any vortex beam with ψ (cid:96) ∝ exp( i(cid:96)ϕ ) and j ϕ = (cid:96) ( (cid:126) /m e r ) ρ (including the Bessel and LG beams),Eq (2.15) results in [75, 87]: (cid:104) L z (cid:105) = (cid:126) (cid:96). (2.17)Thus, an electron in a vortex-beam state carries a well-defined, longitudinal OAM, which is determinedby the azimuthal quantum number (cid:96) . Furthermore, vortices ψ (cid:96) are eigenmodes of the OAM operator ˆ L z :ˆ L z ψ (cid:96) = (cid:96) ψ (cid:96) . Notably, the OAM (2.17) is intrinsic , i.e., independent of the choice of the coordinate origin[67, 65]. Although the radius vector r is present in the canonical OAM operator ˆ L and in the local OAMdensity under the integral in Eq. (2.15), it disappears in the final expectation value (cid:104) L z (cid:105) .Note that the extrinsic OAM can be calculated as [65]: (cid:104) L ext (cid:105) = (cid:104) r (cid:105) × (cid:104) p (cid:105) . (2.18)Here (cid:104) r (cid:105) = (cid:104) ψ | r | ψ (cid:105)(cid:104) ψ | ψ (cid:105) = (cid:82) r ρ d r (cid:82) ρ d r (2.19)is the electron centroid, whereas (cid:104) p (cid:105) = (cid:104) ψ | ˆ p | ψ (cid:105)(cid:104) ψ | ψ (cid:105) = m e (cid:82) j d r (cid:82) ρ d r (2.20)is the expectation value of the electron momentum. For cylindrical vortex beams, (cid:104) p (cid:105) (cid:107) ¯ z and (cid:104) r ⊥ (cid:105) = 0, sothat the longitudinal component of the extrinsic OAM (2.18) vanishes : (cid:104) L ext z (cid:105) = 0.The longitudinal intrinsic OAM of free electrons is a remarkable and somewhat counterintuitive quantumproperty. Based on classical-mechanics intuition, one can expect that the angular momentum is producedby a rotational (i.e., curvilinear) motion. However, free-space electrons, in the absence of any externalforces, must propagate along straight rectilinear trajectories. This apparent contradiction is removed if wecarefully distinguish local and integral properties. For classical point particles, which cannot have any internalstructure, there are no intrinsic properties. In contrast, quantum (wave) beams or wavepackets inevitablyhave finite sizes and inhomogeneous distributions of local densities. In particular, the local probabilitycurrent density with the azimuthal component [see Eqs. (2.13) and (2.14)] implies spiraling streamlines, i.e.,spiral Bohmian trajectories r Bohm ( t ) of electrons [68, 8], Fig. 7. At the same time, the quantum-classicalcorrespondence (Ehrenfest theorem) requires only the trajectory of the electron averaged position (centroid) to be rectilinear [3, 8]. In agreement with this, (cid:104) r ⊥ (cid:105) = 0, and the electron centroid coincides with therectilinear beam axis. In the general case, the centroid (2.19) always follows a rectilinear trajectory forany localized quantum state of free-space electrons. Thus, internal spiraling streamlines of the probabilitycurrent density generate the intrinsic OAM in free-electron vortex states, while electron’s center of gravityalways follows a rectilinear trajectory. Note also that vortex beams are superpositions of multiple planewaves , and therefore are solutions of free-space wave (Schr¨odinger) equation. In terms of geometrical optics,these plane waves determine a family of rectilinear rays which are tangent to the rotationally-symmetric(e.g., cylindrical) surface, associated with the maximum probability density in the vortex beam [70, 78].However, the superposition principle is valid for the wave functions but not for the probability currents,which are quadratic forms. Therefore, streamlines of the probability current of a superposition of planewaves are curvilinear in the generic case [8, 70]. Figure 7 shows an example of rays, current streamlines,and centroid trajectory in a vortex Bessel beam.Notably, non-relativistic scalar electrons in a vortex-beam state somewhat resemble “massless particleswith spin (cid:96) ”. Indeed, the spin angular momentum (SAM) of massless relativistic particles is aligned withtheir momentum, so that helicity is a well-defined quantum number. Vortex electrons carry similar OAMwith well-defined longitudinal component, i.e., “orbital helicity”. However, in contrast to the real SAM of15 igure 7: Geometrical-optics rays, streamlines of the probability current, and centroid trajectory in a Bessel beam with (cid:96) = 2(see Figs. 3 and 5). The rays form a two-parameter family of straight lines which are tangent to a cylindrical surface (here weshow rays touching the cylinder at a given z and different azimuthal angles ϕ ) [70]. The directions and color-coded phases of therays correspond to the wave vectors k in the beam spectrum, Fig. 5(a). In contrast, the streamlines of the probability-current j ,Eqs. (2.12) and (2.13), (Bohmian trajectories) are curvilinear [68, 8]. For Bessel beams, these are spirals lying on the cylindricalsurface [70]. The azimuthal component of this current generates the OAM of the beam, Eq. (2.15)–(2.17). Finally, the centroid (cid:104) r (cid:105) , Eq. (2.19), obviously coincides with the beam axis and corresponds to the rectilinear motion of the classical electron. the electron, which is limited by (cid:126) /
2, the intrinsic OAM can take on arbitrarily large values of (cid:126) (cid:96) . As such,the OAM of electron vortex states can have important consequences in the dynamics of electrons and theirinteractions with external fields, atoms, and other particles.
After introducing the vortex-beam solutions of the Schr¨odinger equation, it is important to discuss thebasic ways of how such states can be generated with accelerated electrons in electron microscopes. Here weonly briefly describe the main concepts, while a more detailed description of experimental techniques willbe given in Section 3.2. Using analogies and differences of electron optics as compared to light optics, threeways of generating electron vortex beams were put forward in the original theoretical work [75].
The first method is a straightforward analogy of spiral phase plates used for photons in different frequencyranges [30, 135, 136]. When free electrons move through a solid-state plate, they acquire an additionalphase ∆Φ as compared with free-space propagation [5, 1]. This phase is proportional to the plate thickness d : ∆Φ = ξd , and is analogous to the phase delay of an optical wave propagating through a dielectricplate. Therefore, a plate with spiral thickness varying with the azimuthal angle, d = ζϕ , will create thecorresponding spiral phase in the transmitted wave: ∆Φ = ξζϕ . Thus, if the incident wave was a planewave, the transmitted wave will carry a vortex exp( iξζϕ ) with topological charge (cid:96) = ξζ , see Fig. 8(a).This idea was used in the first experiment [76] demonstrating the production of a free-electron vortex in aTEM, by employing a spiral-thickness-like region in a stack of graphite flakes. Since the phase change atthe step was not an integer times 2 π in that experiment, the output electron wave possessed a non-integervortex [137, 138], which can be regarded as a superposition of several vortex states with different quantumnumbers (cid:96) [139]. Later, experiments with accurate spiral phase plates producing electron vortex beams withinteger OAM were reported [140, 141] (see Section 3.2.1 below). The second way of generating electron vortex states also represents a TEM adoption of the analogousoptical method. The vortex structure in a wave field represents a screw dislocation of the phase front[10, 11, 12]. Considering the diffraction of a basic Gaussian-like beam on a diffraction grating with an edgedislocation (“fork”), the edge dislocation in the grating produces screw dislocations in the diffracted beams[28, 29], see Fig. 8(b). If the dislocation in the grating is of order (cid:96) , then the N th order of diffraction16ransforms the incident Gaussian-like beam ( (cid:96) = 0) into a vortex beam with ( (cid:96) = N (cid:96) ). This methodwas first used for the efficient generation of high-quality electron vortex beams with integer (cid:96) in [77] (forthe (cid:96) = 1 grating dislocation). Soon after, this method was extended up to (cid:96) = 25 in [78]. Notably,this experiment demonstrated electron vortex beams with the topological charge up to (cid:96) = 100 (in the N = 4 diffraction order). Thus, this technique allows the generation of quantum electron states with anintrinsic OAM of hundreds and even thousands of (cid:126) [142], which is impossible with spin angular momentum.For details and the state-of-the-art holographic techniques for the production of electron vortex beams seeSection 3.2.2. Finally, the third fundamental method of generating electron vortices has no straightforward optical coun-terpart. Namely, it exploits the interaction of electrons with external magnetic fields and vector-potentials.Indeed, in contrast to photons, electrons are charged particles, and this opens a route to interesting in-teractions of electron vortices with magnetic fields and structures (various examples of these are describedbelow). Quantum phenomena of electron-field interactions appear in the electron phase and involve thevector-potential A ( r ). A famous example is the Aharonov–Bohm effect [19, 1], intimately related to theso-called Dirac phase [18] Φ D = e (cid:126) c (cid:90) C A ( r ) · d r (2.21)for an electron moving along a contour C in the presence of the vector-potential. Hereafter, e = −| e | is theelectron charge and c is the speed of light.Importantly, the vector-potential of a magnetic flux line (an infinitely thin solenoid) has the form of a vortex : A ( r ) = ( (cid:126) cα m /er ) ¯ ϕ , where α m is the dimensionless magnetic-flux strength ( α m = 1 correspondingto two magnetic-flux quanta) [19]. This hints that the Dirac phase (2.21) from a magnetic flux line canproduce a vortex phase exp( i(cid:96)ϕ ) with the quantum number (cid:96) = α m . To produce such vortex, one has toconsider a transition of an electron without vortex ( (cid:96) = 0) in the region without magnetic flux ( α m = 0)to the region with the flux α m (cid:54) = 0. Notably, the end of the flux line represents nothing but a magneticmonopole of strength α m [18, 143]. Thus, scattering of an electron wave by a magnetic monopole generatesan electron vortex of strength (cid:96) = α m [5, 75], Fig. 8(c). Recently, this was demonstrated experimentallyby using a thin magnetic needle with a sharp end, approximating a magnetic flux line with a monopole[92, 144].Generation of an electron vortex by a magnetic monopole can be understood in terms of angular-momentum conservation. For simplicity, let us consider a classical point electron moving in a magnetic-monopole field. Although it might seem that the monopole is a spherically-symmetric object, the usualangular momentum of the electron, L = r × p , is not conserved. Indeed, the Lorentz force from themonopole is not central and it originates from the non-symmetric vector-potential. However, there existsanother integral of motion, the generalized angular momentum [143]: L (cid:48) = L − (cid:126) α m r . (2.22)Here ¯ r is the unit radius vector, and we assume that the monopole is located at the origin. The z -componentof L (cid:48) must be conserved in the electron scattering by the monopole. When the electron comes from z → −∞ and the scattered electron is observed at z → + ∞ , the radius-vector ¯ r changes from − ¯ z to +¯ z , so that theelectron OAM must change as L out z = L in z + (cid:126) α m .For classical electrons, this additional OAM can be explained by the Lorentz force from the monopole.The monopole magnetic field can be written as B = ( (cid:126) cα m / e ) r / | r | . In the eikonal approximation, apoint-like electron approximately follows a straight-line trajectory passing at the radial distance r from the Importantly, although in theoretical considerations it is convenient to consider the magnetic flux line (also called
Diracstring ) aligned with the propagation z -axis, observable quantities are independent of its orientation and involve only themonopole charge α m . igure 8: Schematics of basic methods for the generation of electron vortex beams: (a) a spiral phase plate with the 2 π(cid:96) phase-shift increment around its center (here (cid:96) = 1); (b) a diffraction grating (hologram) with a fork-like edge dislocation oforder (cid:96) (here (cid:96) = 1); and (c) a magnetic monopole of dimensionless charge α m (here α m = 2). See also explanations in thetext. monopole. Then, the Lorentz force from the monopole deflects the electron, so that it gains a transverse(azimuthal) momentum p ϕ = (cid:126) α m /r . As a result, the electron acquires the OAM L z = p ϕ r = (cid:126) α m .This shift of the electron’s angular momentum in the presence of a magnetic flux appears in both classical-particle and quantum-wave considerations [145, 87], provided we consider the kinetic rather than canonicalOAM (see Section 2.6 below). Electrons are charged particles which interact with electromagnetic fields. The classical equations ofmotion of a point electron in an external electric and magnetic fields are [146]:˙ p = e E + ec ˙ r × B , ˙ r = p m e . (2.23)Here the overdot stands for the time derivative, r and p are the coordinates and momentum of the electron,while E and B are the electric and magnetic fields.Quantum wavepacket or beam states of electrons have finite dimensions and therefore can possess in-ternal properties, in addition to the electric charge e . Finite-size electron states are characterized by thedistributions of the charge density ρ e = eρ and electric current density j e = e j . Most importantly, thecoiling current density in vortex electron states acts as a solenoid and generates a magnetic moment . Themagnetic moment of a localized electron state can be defined as [146]: M = 12 c (cid:82) r × j e d r (cid:82) ρ d r = e m e c (cid:104) L (cid:105) . (2.24)In particular, the longitudinal z -directed magnetic moment of electron vortex beams (per unit z -length) infree space equals [75, 147]: M z = e (cid:126) m e c (cid:96) ≡ − µ B (cid:96), (2.25)where µ B = | e | (cid:126) / (2 m e c ) is the Bohr magneton. 18hus, vortex electrons carry a longitudinal magnetic moment (2.25) proportional to the quantized OAMand anti-parallel to it. Note that this magnetic moment corresponds to the gyromagnetic ratio with g -factor g = 1, while g = 2 for the magnetic moment generated by the spin [3, 148]. The presence of the magneticmoment should modify the equations of motion (2.23).We first consider the interaction of the magnetic moment or intrinsic OAM with an external electric field E (we set B = 0 here); this can result in a spin-orbit-type interaction. In fact, since the intrinsicangular momentum has an orbital origin in our case, this should rather be called orbit-orbit interaction between the intrinsic OAM (vortex) and extrinsic OAM (trajectory) (see [51, 52, 53] for such effects inoptical vortex beams). This interaction couples the intrinsic OAM (cid:104) L (cid:105) and extrinsic trajectory parameters (cid:104) r (cid:105) and (cid:104) p (cid:105) in the equations of motion. To describe such semiclassical evolution, we consider localized (butsufficiently large) paraxial electron wavepacket and assume that the intrinsic OAM maintains its form duringthe wavepacket propagation, (cid:104) L (cid:105) = (cid:126) (cid:96) (cid:104) p (cid:105) /p ( |(cid:104) p (cid:105)| (cid:39) p ). In this case, the “orbit-orbit interaction” becomesequivalent to that of massless spinning particles with spin (cid:96) in an external scalar potential [149]. Using theBerry-connection formalism [150], the semiclassical equations of motion take the form [75]: (cid:104) ˙ p (cid:105) = e E , (cid:104) ˙ r (cid:105) = (cid:104) p (cid:105) m e + (cid:126) (cid:96) (cid:104) ˙ p (cid:105) × (cid:104) p (cid:105) p , (2.26) (cid:104) ˙ L (cid:105) = − [ e E × (cid:104) p (cid:105) ] × (cid:104) L (cid:105) p . (2.27)The last term in Eq. (2.26) and equation (2.27) describe the mutual influence of the intrinsic OAM and thetrajectory. It can be readily shown that these equations are consistent with the assumed form (cid:104) L (cid:105) = (cid:126) (cid:96) (cid:104) p (cid:105) /p ,i.e., the “orbital helicity” (cid:104) L (cid:105) · (cid:104) p (cid:105) /p = const is an integral of motion of Eqs. (2.26) and (2.27). Equation(2.27) is an analogue of the Bargman–Michel–Telegdi equation [151, 152, 153] for the precession of theintrinsic angular momentum in an external electric field. In turn, the last term in Eq. (2.27) describesthe OAM-dependent ( (cid:96) -dependent) transverse transport of the electron, Fig. 9(a). This is an analogue ofthe intrinsic spin Hall effect, known in condensed-matter physics [154, 150], high-energy physics [149, 155],and optics (for photons) [156, 66]. Here it should rather be called orbital Hall effect . The typical valueof the transverse (cid:96) -dependent shift of the electron trajectory is (cid:126) /p , i.e., the de Broglie wavelength of theelectron [75]. Therefore, this effect is extremely small, and practically unobservable, for free electrons inelectron microscopes. Note, however, that an analogous orbital Hall effect has been successfully measuredfor optical vortex beams interacting with dielectric inhomogeneities [53], because subwavelength accuracyis quite achievable in modern optics. Furthermore, similar spin Hall effect of electrons in condensed-mattersystems result in the observable accumulation of opposite spin polarizations on the opposite edges of thesample with an applied electric field [157]. Thus, the orbital Hall effect for vortex electrons, Eq. (2.26),could still play a role, e.g., in condensed-matter phenomena [158, 159].Let us now consider the interaction of the intrinsic OAM with an external magnetic field B (we set E = 0for simplicity). One could expect that the interaction between the magnetic moment of the electron (2.24)and the external magnetic field is described by a Zeeman-like energy ∆ E = − M · B [75]. However, expression(2.17) for the intrinsic OAM and the corresponding Eq. (2.25) for the magnetic moment are derived using the free-space vortex-beam solutions, i.e., without any external fields. In the presence of external fields one hasto find a solution of the corresponding Schr¨odinger equation, and it will contain a self-consistent distributionof charges, currents, and fields, including their interactions [87]. As we show in the next Section 2.6, thisdrastically modifies the values of the electron OAM and its magnetic moment in the presence of a magneticfield.Moreover, in the presence of a magnetic field, the dynamical evolution of the electron is described bythe kinetic momentum (cid:104) (cid:112) (cid:105) and OAM (cid:104) (cid:76) (cid:105) . These quantities differ from their canonical counterparts, (cid:104) p (cid:105) and OAM (cid:104) L (cid:105) , by the vector-potential contribution [87] (such that kinetic and canonical quantities coincidein the absence of the vector-potential). Below we formally introduce kinetic characteristics, and here onlymake one important point. Namely, generically, an electron in a magnetic field cannot keep its intrinsicOAM parallel to its momentum, i.e., (cid:104) (cid:76) (cid:105) (cid:107) (cid:104) (cid:112) (cid:105) , which was assumed in all solutions considered above [75].19 igure 9: (a) The orbital Hall effect of vortex electron states in an electric field E [75], Eqs. (2.26). Here, (cid:104) p (cid:105) is the initial meanmomentum of the electron. Bended by the electric field, the trajectories of the centroids of the vortex beams or wavepacketsexperience transverse (cid:96) -dependent shifts of the order of the de Broglie wavelength. (b) Cyclotron motion of a vortex electronin a uniform magnetic field B . The mean kinetic momentum (cid:104) (cid:112) (cid:105) precesses about the magnetic field with the cyclotron angularfrequency Ω c , Eq. (2.28). At the same time, the kinetic OAM (cid:104) (cid:76) (cid:105) of a vortex electron state precesses with the Larmorfrequency Ω L = Ω c /
2, Eq. (2.29). Therefore, the initially-alligned momentum and OAM change their mutual direction duringthis evolution [75, 89] (see also Fig. 14 below).
Indeed, as follows from Eqs. (2.23), semiclassical electrons approximately follow cyclotron trajectories, andthe mean momentum evolves (at least, in the classical limit (cid:126) →
0) as (cid:104) ˙ (cid:112) (cid:105) = em e c (cid:104) (cid:112) (cid:105) × B . (2.28)Thus, the momentum precesses around the magnetic-field direction with the cyclotron frequency Ω c = eB/ ( m e c ). At the same time, the evolution of the intrinsic OAM (cid:104) (cid:76) (cid:105) in the magnetic field is describedby the Zeeman energy term and the corresponding Larmor precession equation (Bargman–Michel–Telegdiequation with g = 1 factor in a magnetic field) [3]: (cid:104) ˙ (cid:76) (cid:105) = e m e c (cid:104) (cid:76) (cid:105) × B . (2.29)This means that the electron OAM precesses about the magnetic-field direction with the Larmor frequencyΩ L = eB/ (2 m e c ). Since the Larmor and cyclotron frequencies differ by a factor of two, Ω c = 2 Ω L , themomentum (cid:104) (cid:112) (cid:105) and OAM (cid:104) (cid:76) (cid:105) cannot be parallel in such evolution, with the exception of the case (cid:104) (cid:76) (cid:105) (cid:107)(cid:104) (cid:112) (cid:105) (cid:107) B , Fig. 9(b). This means that the initial free-space form of the electron vortex states, (cid:104) (cid:76) (cid:105) = (cid:126) (cid:96) (cid:104) (cid:112) (cid:105) /p ,cannot survive in a magnetic field with a non-zero transverse component [75, 89].The two-frequency evolution of Eqs. (2.28) and (2.29) results in very interesting dynamics of vortexelectrons in a magnetic field which is considered in detail below. Note that the evolution of the SAM of theelectron does not face such a problem. Because of the g = 2 factor for spin, the frequency of its precessionbecomes twice the Larmor frequency, i.e., the cyclotron one [151, 152, 153, 3]. Therefore, in contrast to theOAM, the SAM precession is synchronized with the momentum evolution, and the helicity (projection ofthe spin onto the momentum direction) is conserved. We now provide a self-consistent quantum treatment of electron vortex modes in a magnetic field B .The free-space electron Hamiltonian underlying the Schr¨odinger equation (2.1) is modified in a magneticfield as: ˆ H = ˆ p m e → ˆ (cid:112) m e = 12 m e (cid:16) ˆ p − ec A (cid:17) , (2.30)20here ˆ p = − i (cid:126) ∇ is the canonical momentum operator, ˆ (cid:112) = ˆ p − ec A is the kinetic (or covariant) momentumshifted by the vector-potential A ( r ) generating magnetic field B = ∇ × A .The presence of the coordinate-dependent solenoidal vector-potential considerably complicates the Schr¨o-dinger equation, and it allows a simple analytical solution only in some cases, such as the following case ofa uniform and constant magnetic field B . Choosing the z -axis to be directed along the field, B = B ¯ z , theproblem acquires the cylindrical symmetry natural for vortex-beam solutions. Moreover, in this geometry,the vector-potential can be chosen to have only an azimuthal component, i.e., to form a vector-potentialvortex : A = Br ϕ . (2.31)The corresponding stationary Schr¨odinger equation (2.4) with a uniform magnetic field in cylindrical coor-dinates becomes: − (cid:126) m e (cid:34) r ∂∂r (cid:18) r ∂∂r (cid:19) + 1 r (cid:18) ∂∂ϕ + iσ r w m (cid:19) + ∂ ∂z (cid:35) ψ = E ψ. (2.32)Here w m = 2 (cid:112) (cid:126) c/ | eB | = (cid:112) (cid:126) /m e | Ω L | is the magnetic length parameter, and σ = sgn( B ) = ± L (rather than the cyclotron frequencyΩ c = 2 Ω L ) is the fundamental frequency in the quantum-mechanical problem [160, 87]. This is related toLarmor’s theorem, the conservation of angular momentum, and this will be clearly seen below from thequantum picture of the electron evolution.The solutions of Eq. (2.32) are known as Landau states [2, 3, 17, 161], and they have the form of non-diffracting LG beams (see Fig. 10) [87, 88]: ψ L(cid:96),n ∝ (cid:18) rw m (cid:19) | (cid:96) | L | (cid:96) | n (cid:18) r w m (cid:19) exp (cid:18) − r w m (cid:19) exp [ i ( (cid:96)ϕ + k z z )] , (2.33)where the wave number k z must obey the dispersion relation considered below, Eq. (2.35). The Landaustates (2.33) are identical to the LG beams (2.9) with the beam waist w = w m at z = 0.We also introduce a longitudinal scale z m = v/ | Ω L | determined by the Larmor frequency and the electronvelocity v = (cid:112) E/m e . The transverse magnetic length w m and longitudinal Larmor length z m representcounterparts of the beam waist and Rayleigh length of the free-space LG beams (2.9) but here they areuniquely determined by the electron energy and magnetic field strength: w m = 2 √ (cid:126) c (cid:112) | eB | , z m = 2 c √ Em e | eB | , i.e., z m = (cid:115) E (cid:126) | Ω L | w m . (2.34)The fact that eigenmodes (2.33) in the magnetic field are non-diffracting and transversely confined (i.e.,possess a discrete radial quantum number n ) reflects the boundedness of classical electron orbits in a magneticfield. While the diffracting LG beams (2.9) represent approximate paraxial solutions of the Schr¨odinger equa-tion, Landau LG modes (2.33) yield exact solutions of the problem with magnetic field. In doing so, thewave numbers satisfy the following dispersion relation [87]: E = (cid:126) k z m e − (cid:126) Ω L (cid:96) + (cid:126) | Ω L | (2 n + | (cid:96) | + 1) ≡ E (cid:107) + E Z + E G (cid:124) (cid:123)(cid:122) (cid:125) E ⊥ . (2.35) Note that here we choose B = B z rather than B = | B | , so that quantities B , Ω L , and Ω c can be either positive or negativedepending on the direction of the magnetic field, σ = sgn( B ). In optics, non-diffracting LG modes entirely analogous to Eq. (2.33) appear in parabolic-index optical fibers [162]. This isrelated to the fact that the Schr¨odinger equation in a uniform magnetic field can be mapped onto a two-dimensional quantum-oscillator problem [3] igure 10: Same as in Figs. 5(b) and 6(b) but for the Landau states (2.33)–(2.42) with different values of ( (cid:96), n ) in a uniformmagnetic field B = B ¯ z [here σ = sgn( B ) = +1]. Although the probability density distributions of the Landau modes is entirelysimilar to those of the LG beams, Fig. 6(b), their current densities and angular-momentum properties differ significantly. Inparticular, the kinetic OAM of Landau states, (cid:104) (cid:76) z (cid:105) , cannot be zero or negative even for (cid:96) <
0, and its minimum value is (cid:126) . Thisis because of the always-positive (for σ = +1) vector-potential contribution to the azimuthal probability current, Eqs. (2.38)and (2.39). Here E (cid:107) = (cid:126) k z / m e is the energy of the free longitudinal motion, while the quantized transverse-motionenergy in Eq. (2.35) can be written as E ⊥ = (cid:126) | Ω L | (2 N L + 1) , N L = n + 12 | (cid:96) | [1 + sgn( σ(cid:96) )] = 0 , , , ... . (2.36)Thus, Eq. (2.36) describes the structure of quantized Landau energy levels [2, 3, 17, 161, 87, 88]. Equa-tion (2.35) shows that Landau energies consist of two terms [87]: E ⊥ = E Z + E G . The first one, E Z = − (cid:126) Ω L (cid:96) = − M z B , represents the Zeeman energy of the free-space magnetic moment (2.25) in a magneticfield B . The second term E G = (cid:126) | Ω L | (2 n + | (cid:96) | + 1) can be associated with the Gouy phase (2.11) ofthe diffractive LG modes. (Recall that the Gouy-phase term is related to the transverse kinetic energyof spatially-confined modes [132, 133], which shifts the propagation constants and eigenfrequencies of thewaveguide and resonator modes [125].) As we show below, the Zeeman and Gouy-phase contributions areseparately observable and lead to a remarkable behavior of the electron probability density distributions ina magnetic field.Obviously, the transverse probability-density distributions of Landau modes (2.33) are entirely analogousto those of the LG modes (2.9) [Fig. 10, cf. Eq. (2.14) and Fig. 6]: ρ L | (cid:96) | ,n ( r ) ∝ (cid:18) r w m (cid:19) | (cid:96) | (cid:12)(cid:12)(cid:12)(cid:12) L | (cid:96) | n (cid:18) r w m (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) − r w m (cid:19) . (2.37)However, their probability-current and AM properties differ significantly from the free-space solutions. Thisis because the definitions of the gauge-invariant probability current density and of the kinetic momentum/AMare essentially modified by the presence of the vector-potential. Namely, the probability current density is22ow determined as the local expectation value of the kinetic (covariant) momentum operator (2.30) [2, 3]: j = 1 m e ( ψ | ˆ (cid:112) | ψ ) = (cid:126) m e Im ( ψ ∗ ∇ ψ ) − em e c A ρ. (2.38)This means that the vector potential A produces an additional probability current in quantum electronstates. For Landau states (2.33) the current (2.38) yields: j L(cid:96),n ( r, ϕ ) = (cid:126) m e (cid:20) r (cid:18) (cid:96) + σ r w m (cid:19) ¯ ϕ + k z ¯ z (cid:21) ρ L | (cid:96) | ,n ( r ) . (2.39)Here, the σ -dependent term is the vector-potential contribution. It is worth noticing that for the counter-circulating vortex exp( i(cid:96)ϕ ) and vector-potential A ϕ , (cid:96)σ <
0, the azimuthal current in (2.39) changes its signat r = r | (cid:96) | ≡ | (cid:96) | w m / √
2, i.e., around the first radial maximum of the LG mode. For r < r | (cid:96) | the current fromthe vortex exp( i(cid:96)ϕ ) prevails, whereas for r > r | (cid:96) | the contribution from the vector-potential A ϕ becomesdominant (see Fig. 10).Taking into account the vortex-like form of the vector-potential (2.31) and its appearance in the azimuthalprobability current (2.39), it should also contribute to the OAM of the electron. In fact, one can define two OAM quantities in the presence of a magnetic field. The first one is the canonical
OAM, which is determinedby the canonical OAM operator ˆ L = r × ˆ p . Its longitudinal component ˆ L z = − i (cid:126) ∂/∂ϕ acts only on thevortex phase factor exp( i(cid:96)ϕ ) in Landau modes (2.33). Hence, similar to free-space vortex beams, Landaustates are eigenmodes of the canonical OAM operator and have the same expectation value of the canonicalOAM as in Eq. (2.17) [87]: ˆ L z ψ L(cid:96),n = (cid:96) ψ L(cid:96),n , (cid:104) L z (cid:105) = (cid:68) ψ (cid:12)(cid:12)(cid:12) ˆ L z (cid:12)(cid:12)(cid:12) ψ (cid:69) (cid:104) ψ | ψ (cid:105) = (cid:126) (cid:96). (2.40)The second OAM of the electron in a magnetic field is the kinetic OAM determined by the kinetic momentumoperator: ˆ (cid:76) = r × ˆ (cid:112) or the probability current density: (cid:104) (cid:76) (cid:105) = (cid:68) ψ (cid:12)(cid:12)(cid:12) ˆ (cid:76) (cid:12)(cid:12)(cid:12) ψ (cid:69) (cid:104) ψ | ψ (cid:105) = m e (cid:82) r × j d r (cid:82) ρ d r . (2.41)It is kinetic OAM (2.41) that describes the mechanical action of the electron OAM and observable rotationaldynamics in electron states.Substituting characteristics of Landau modes, Eqs. (2.33), (2.37), and (2.39), into Eq. (2.41) (with thebeam substitution d r → d r ⊥ ) and using Eq. (2.36), we arrive at [87] (cid:104) (cid:76) z (cid:105) = (cid:126) [ (cid:96) + σ (2 n + | (cid:96) | + 1)] = (cid:126) σ (2 N L + 1) . (2.42)Equation (2.42) reveals nontrivial properties of the electron OAM in a magnetic field, Fig. 10. First, itshows that the sign of the kinetic OAM is solely determined by the direction of the magnetic field, σ ,and is independent of the vortex charge (cid:96) . This is because after the integration (2.41) the vector-potentialcontribution to the azimuthal current always exceeds the vortex one. Note also that for parallel OAMand magnetic field, σ(cid:96) >
0, the canonical OAM (cid:126) (cid:96) is enhanced (in absolute value) by the magnetic-fieldcontribution: (cid:10) (cid:76) ↑↑ z (cid:11) = (cid:126) [2 (cid:96) + σ (2 n + 1)]. At the same time, in the opposite case of anti-parallel OAM andmagnetic field, σ(cid:96) <
0, the kinetic OAM takes the form (cid:10) (cid:76) ↑↓ z (cid:11) = (cid:126) σ (2 n + 1), i.e., becomes independent ofthe vortex charge (cid:96) . This is caused by the partial cancellation of the counter-circulating azimuthal currentsproduced by the vortex exp( i(cid:96)ϕ ) and by the magnetic vector-potential A ϕ . Second, the value of (cid:104) (cid:76) z (cid:105) isindependent of the magnitude of the magnetic field, | B | . This is because the radius of the beam changes The vector-potential contribution to the kinetic OAM is sometimes called “diamagnetic angular momentum” [163]. w m ∝ / (cid:112) | B | , Eq. (2.33), the angular velocity Ω L ∝ | B | , whereas the mechanical OAM behaves as (cid:76) z ∝ Ω L w m . Third, in contrast to the classical electron motion in a magnetic field, which can have zeroOAM, Eq. (2.42) shows that there is a minimal kinetic OAM of quantum Landau states: |(cid:104) (cid:76) z (cid:105)| min = (cid:126) .Importantly, modified definitions of the probability current density (2.38) and kinetic OAM (2.41) alsoaffect the value of the electron magnetic moment in the presence of a magnetic field. Indeed, using thedefinition (2.24) with the modified current density (2.38), we obtain [87]: M = e c (cid:82) r × j d r (cid:82) ρ d r = e m e c (cid:104) (cid:76) (cid:105) . (2.43)Thus, the magnetic moment of the electron in a magnetic field is determined by the kinetic OAM. Since (cid:104) (cid:76) (cid:105) is always aligned with B and e <
0, the magnetic moment (2.43) is anti-parallel to the magnetic field.This determines the diamagnetic response of free scalar electrons in a magnetic field [161, 164].The magnetic moment of the Landau states, M z , shares all the unusual properties of the kinetic OAM(2.42) and differs strongly from the magnetic moment of free-space vortex electrons, Eq. (2.25). Using themagnetic moment (2.43), the transverse energy of the electron, Eqs. (2.35) and (2.36), can be written as asingle Zeeman term: E ⊥ = − M z B. (2.44)It now includes both the “pure” Zeeman term from the coupling of the free-space magnetic moment (2.25)with the field as well as the Gouy-phase term. Notably, the latter term can be considered as a nonlinear (with respect to the field) effect of the interaction of the vector-potential current − ( e/m e c ) A ρ (“diamagneticangular momentum”) with the magnetic field B [87].Most peculiarities of the electron Landau states in a magnetic field are contained in their dispersionrelation (2.35) and (2.36), depending on quantum numbers (cid:96) and σ , and the corresponding OAM values(2.42). These quantities bring about rather nontrivial rotational dynamics when various superpositions ofLandau modes propagate in a magnetic field [87, 88, 89, 165, 91, 166, 163, 167]. Below we show someexamples of peculiar rotations of electron vortex states in a magnetic field. In terms of the cylindrically-symmetric Landau modes, the rotational dynamics of asymmetric electronstates in a magnetic field appear from the interference of different modes (2.33)–(2.35) acquiring differentphases during propagation. Assuming that all the modes have the same fixed energy E and that they areparaxial, i.e., E ⊥ (cid:28) E , w m (cid:28) z m , one can write the longitudinal wave number as k z (cid:39) k + ∆ k z , where (cid:126) k = √ Em e and ∆ k z = − [ σ(cid:96) + (2 n + | (cid:96) | + 1)] z − m . (2.45)Thus, the Larmor length z m , Eq. (2.34), determines the characteristic longitudinal scale of the beam evo-lution. During the propagation along the z -axis, the correction (2.45) to the wave number k yields anadditional phase Φ LZG = ∆ k z z. (2.46)This phase depends on both vortex and magnetic-field properties. We call it Landau–Zeeman–Gouy phase[87] because of its intimate relation to the Landau levels, Zeeman coupling [the σ(cid:96) -term in (2.45)], and Gouyphase [the (2 n + | (cid:96) | + 1)-term in (2.45)]. The interplay between the Zeeman and Gouy terms results in richdynamics of various Landau-mode superpositions.We first consider the simplest superposition of two Landau modes (2.33) with equal amplitudes, thesame radial index n , and opposite vortex charges ± (cid:96) : ψ = ψ L − (cid:96),n + ψ L(cid:96),n . Such superposition carries nonet canonical OAM, (cid:104) L z (cid:105) = 0, and its transverse probability density distribution represents a flower-likepattern with 2 | (cid:96) | petals: | ψ | ∝ cos ( (cid:96)ϕ ) (see Fig. 11). The phases (2.46) of the two interfering modes ψ L ± (cid:96),n differ only in their Zeeman terms : ∆Φ
LZG = ∓ (cid:96)σz/z m . Combining these terms with the azimuthal vortexdependencies as exp ( ± i(cid:96)ϕ ) → exp [ ± i(cid:96) ( ϕ − σz/z m )], one can see that this results in the rotation of the24 igure 11: (a) Superpositions of the Landau states (2.33)–(2.35) ψ L(cid:96),n + ψ L − (cid:96),n (with n = 0 and (cid:96) = 1 , Larmor rotation (2.47) during the propagation in the magnetic field B [87, 88]. The transverse probability-density (grayscale)and probability-current (orange arrows) distributions are shown for different propagation distances z . The net canonical OAMof such superpositions vanishes, (cid:104) L (cid:105) = 0, and their centroids (shown by yellow spheres for the (cid:96) = 1 superposition) obeyrectilinear trajectories parallel to the magnetic field. (b) Experimental demonstration of this Larmor rotation in a TEM [165].The holographic aperture (top) produces superpositions of the LG modes with ( (cid:96), n ) = ( ± ,
0) in the first diffraction orders(bottom). Changing magnification in the imaging magnetic lens (i.e., the effective magnetic field B ) from 41 · × (yellow) to55 · × (white) produces a rotation of the image by the angle ∆ ϕ = 106 ◦ corresponding to the Larmor rotation (2.47). Thisis an example of image rotation which is well known in electron microscopy [4]. interference pattern by the angle [87, 88] (Fig. 11)∆ ϕ (0) = σ zz m . (2.47)Since z/z m = | Ω L | z/v , the rotation (2.47) is characterized by the Larmor frequency Ω L . Such rotation of asuperposition of two opposite- (cid:96) vortex modes in a magnetic field was recently observed in [165]. In fact, theLarmor rotation of images in a magnetic field is well known in transmission electron microscopy [4]. Theabove theory provides a convenient quantum-mechanical description of this effect. Indeed, any superposition(image) carrying no net angular momentum and consisting of pairs of opposite- (cid:96) modes will undergo thesame Larmor rotation (2.47).As another example, we now consider a superposition of two Landau modes (2.33) with the same radialindex n , and vortex charges 0 and (cid:96) : ψ = ψ L ,n + a ψ L(cid:96),n , where a is some constant amplitude [87]. Such asuperposition has a nonzero net canonical OAM (cid:104) L z (cid:105) ∝ (cid:96) , and is characterized by a pattern of | (cid:96) | off-axisvortices (Fig. 12). Landau modes with different | (cid:96) | involve the Gouy term in the difference of phases (2.45)and (2.46). Namely, the ψ L(cid:96),n mode acquires an additional phase ∆Φ
LZG = − ( (cid:96)σ + | (cid:96) | ) z/z m as comparedwith the ψ L ,n mode. From here, it follows that the superposition ψ with parallel OAM and magnetic field, (cid:96)σ >
0, exhibits a rotation of the interference pattern by the angle∆ ϕ ↑↑ = 2 σ zz m . (2.48)In contrast to this, the superposition with anti-parallel OAM and magnetic field, (cid:96)σ <
0, shows no rotationat all: ∆ ϕ ↑↓ = 0 . (2.49)Equation (2.48) describes the rotation of the image with the double-Larmor (i.e., cyclotron) frequency igure 12: (a) Same as in Fig. 11(a) but for superpositions ψ L ,n + aψ L(cid:96),n (with a = 2, n = 0, and (cid:96) = 1 , (cid:104) L (cid:105) parallel to the magnetic field B . These superpositions undergo cyclotron (double-Larmor)rotation (2.48) during the propagation [87]. The superposition with (cid:96) = 1 has an off-axis centroid, which follows a classicalcyclotron trajectory, in agreement with Eqs. (2.50). (b) Analogous superpositions with (cid:96) = − , − anti-parallel to the magnetic field experience zero rotation , Eq. (2.49) [87]. This is due to the cancellation of the vortex andvector-potential contributions to the azimuthal probability current (2.38). Ω c = 2Ω L . Note also that non-rotating superpositions, Eq. (2.49), consist of modes with the kinetic OAMvalues independent of the vortex charge: (cid:10) (cid:76) ↑↓ z (cid:11) = (cid:126) σ (2 n + 1). For n = 0, these correspond to the lowestLandau energy level. Figure 12 shows examples of the cyclotron and zero rotations described by Eqs. (2.48)and (2.49).Equations (2.47)–(2.49) represent an intriguing result. Namely, the rotational dynamics of quantumelectron states with OAM in a magnetic fields is characterized by three frequencies: (i) Larmor, (ii) cyclotron(double-Larmor), and (iii) zero frequency [87]. This is in sharp contrast to the classical electron evolution(2.28), which is described by a single cyclotron rotation [146]. In spite of such difference, the quantumevolution (2.47)–(2.49) is fully consistent with the classical evolution (2.28). Indeed, according to theEhrenfest theorem, the expectation values of the electron coordinates and momentum must obey classicalequations of motion [3, 8]. Importantly, one should take the expectation values of the kinetic quantumquantities, which correspond to classical trajectories. Using the kinetic momentum, Eq. (2.30), the equationsof motion (2.23) and (2.28) become: (cid:104) ˙ (cid:112) (cid:105) = em e c (cid:104) (cid:112) (cid:105) × B , (cid:104) ˙r (cid:105) = (cid:104) (cid:112) (cid:105) m e . (2.50)Explicit calculations for the superpositions of Landau states considered above show that for Larmor-rotatingand non-rotating states, Eqs. (2.47) and (2.49), the mean kinetic momentum is always aligned with themagnetic field: (cid:104) (cid:112) (cid:105) (cid:107) B (cid:107) ¯ z , i.e., (cid:104) (cid:112) ⊥ (cid:105) = 0. Therefore, the centroid (cid:104) r ⊥ (cid:105) of such states lies on a rectilinear trajectory parallel to the magnetic field. In contrast to this, states rotating with the cyclotron angularvelocity, Eq. (2.48), can have a non-zero transverse mean momentum, (cid:104) (cid:112) ⊥ (cid:105) (cid:54) = 0, and their centroids (cid:104) r ⊥ (cid:105) trace classical cyclotron orbits along the beam propagation, see Figs. 11 and 12.The nontrivial rotational dynamics of quantum electron states is closely related to the summation of the igure 13: Experimental observation [91] of the three-frequency rotational dynamics of electron vortex states in a magneticfield, Eqs. (2.47)–(2.49) and (2.51). (a) Schematics of the TEM experimental setup. A holographic fork mask generates a rowof vortex beams with different azimuthal indices (cid:96) = ..., − , − , , , , ... . These beams are focused by a magnetic lens andare studied in the region of maximal quasi-uniform magnetic field. The focal plane is shifted few Rayleigh ranges below theobservation plane z = 0 to reduce the Gouy-phase rotation. A knife-edge stop is placed at z k <
0, where it blocks half of eachof the beams (to break the azimuthal symmetry of the probability density distributions). The spatial rotational dynamics ofthe cut beams propagating to the observation plane was observed by varying the position z k of the knife edge (see details in[91]). (b) A quantitative analysis of the (cid:96) -dependent beam rotations. The azimuthal orientations of the cut modes ∆ ϕ withrespect to the extrapolated reference azimuth ϕ = ϕ | z k =0 are plotted versus z k . Three lines correspond to the zero, Larmorand cyclotron rotations, Eq. (2.51). vortex and vector-potential contributions to the probability current (2.38) and (2.39). For parallel OAM andmagnetic field, the two contributions produce azimuthal currents of the same sign, which result in the double-Larmor (cyclotron) rotation. For anti-parallel OAM and magnetic field, the two azimuthal contributionscancel each other, which produces a non-rotating state [87].Furthermore, the above “three-frequency dynamics” immediately follows from the expression (2.39) forthe probability current in Landau states. Indeed, one can define the local value of the electron angularfrequency as Ω( r ) = v ϕ ( r ) /r = j φ ( r ) / ( rρ ( r )). Calculating the expectation value of this quantity, we obtain[91]: (cid:104) Ω (cid:105) = (cid:82) ∞ Ω( r ) ρ ( r ) r dr (cid:82) ∞ ρ ( r ) r dr = (cid:96)σ < L for (cid:96) = 02Ω L for (cid:96)σ > . (2.51)These expressions can be regarded as internal angular velocities of electrons in pure Landau states. Notably,they correspond exactly to expressions (2.47)–(2.49) for the rotations of Landau-mode superpositions withsimilar OAM properties (i.e., zero, parallel, and anti-parallel OAM with respect to the magnetic field).Recenty, all three kinds of rotations (2.51) in Landau modes were observed experimentally [91]. In thatexperiment, the cylindrically-symmetric probability density distribution of Landau modes was broken by asharp aperture stop, which cut half of the beam, and then the rotational evolution of such half-beams wastraced, Fig. 13. Truncated modes can be considered as superpositions of multiple pure Landau modes, andthis explains the exact correspondence between the internal dynamics (2.51) and superposition rotations(2.47)–(2.49).Other remarkable aspects of the electron vortex beams dynamics in a magnetic field were considered in27ecent works [89, 166, 163, 167]. In particular, the radial dynamics of vortex mode superpositions and theangular momentum conservation was analysed in [166]. Afterwards, Ref. [163] investigated the evolutionof vortex beams shifted and tilted with respect to the z -axis. Of course, such shifted/tilted beams canalso be considered as superpositions of multiple pure Landau modes. However, in this case it is instructiveto separate the internal vortex properties and external dynamics of the vortex/beam centroid. Notably,the shifted/tilted Landau vortex mode preserves its shape with respect to the centroid, while the centroidfollows a classical cyclotron trajectory, Eqs. (2.50). This allows separating not only canonical (vortex) andvector-potential contributions to the OAM, but also its intrinsic and extrinsic parts.Using the electron centroid (2.19), the intrinsic and extrinsic parts of the kinetic OAM are [cf. Eq. (2.18)][65]: (cid:10) (cid:76) ext (cid:11) = (cid:104) r (cid:105) × (cid:104) (cid:112) (cid:105) , (cid:10) (cid:76) int (cid:11) = (cid:104) (cid:76) (cid:105) − (cid:10) (cid:76) ext (cid:11) . (2.52)Recall that (cid:104) (cid:76) (cid:105) = (cid:104) r × (cid:112) (cid:105) , while (cid:104) (cid:112) (cid:105) is the expectation value of the kinetic momentum, defined similarlyto Eq. (2.20) but with the operator ˆ (cid:112) . It follows from Eqs. (2.52) that the intrinsic OAM does not changeits value under transverse spatial translations, while the extrinsic OAM is transformed according to the“parallel-axis theorem” of classical mechanics: r → r + r ⊥ : (cid:10) (cid:76) int (cid:11) → (cid:10) (cid:76) int (cid:11) , (cid:10) (cid:76) ext (cid:11) → (cid:10) (cid:76) ext (cid:11) + r ⊥ × (cid:104) (cid:112) (cid:105) . (2.53)For example, the superposition ψ L , + a ψ L , of the Landau modes, shown in Fig. 12(a), has non-zero trans-verse momentum (cid:104) (cid:112) ⊥ (cid:105) , shifted off-axis centroid (cid:104) r ⊥ (cid:105) , and, hence, non-zero longitudinal component of theextrinsic OAM (2.52): (cid:104) (cid:76) ext z (cid:105) = ( (cid:104) r ⊥ (cid:105) × (cid:104) (cid:112) ⊥ (cid:105) ) z . We note that the intrinsic-extrinsic separation (2.52) andproperties (2.53) are generic and independent of the presence of a magnetic field.So far we considered only vortex beams propagating along the magnetic field or slightly tilted withrespect to it. As the opposite limiting case, one can consider an electron vortex in the orthogonal magneticfield. Such problem was analyzed in detail [89] for paraxial electron wavepackets with vortices. Thisyielded a remarkable example of the intrinsic (vortex) and extrinsic (centroid) evolution associated with theLarmor and cyclotron rotations, see Fig. 14. Let the uniform magnetic field be still aligned with the z -axis, B = B ¯ z , whereas the vortex evolution occurs in the transverse ( x, y )-plane. For instance, let the vortexwavepacket be oriented along the x –axis, with some initial momentum along this axis, (cid:104) (cid:112) ( t = 0) (cid:105) ≡ (cid:112) (cid:107) ¯ x .Then, the wavepacket undergoes rotational evolution (in time) in the ( x, y )-plane. Namely, in agreementwith the Ehrenfest theorem, the centroid of the wavepacket follows the cyclotron orbit, Eq. (2.50). At thesame time, the orientation of the wavepacket, together with the vortex core and associated intrinsic OAM (cid:104) (cid:76) int (cid:105) , experiences the Larmor precession (2.29) with half the cyclotron frequency. In doing so, the π -anglerotation of the wavepacket orientation (during the 2 π rotation of its centroid) brings its probability densitydistribution back to the original distribution, but now with the intrinsic OAM pointing in the oppositedirection, Fig. 14.This example also provides a nice illustration of different types of the electron OAM: canonical, kinetic,intrinsic, and extrinsic. Let the wavepacket centroid lie in the z = 0 plane: (cid:104) z (cid:105) = 0. First, since thevector-potential does not have a z -component, A z = 0, the in-plane OAM has a purely canonical origin (the“diamagnetic angular momentum” has only the z -component): (cid:104) (cid:76) ⊥ (cid:105) = (cid:104) L ⊥ (cid:105) (here we keep the ⊥ subscriptto denote the ( x, y )-plane). Second, the cyclotron motion of the centroid implies that (cid:104) (cid:112) z (cid:105) = (cid:104) p z (cid:105) = 0. Itfollows from here that the in-plane OAM also has purely intrinsic origin: (cid:104) (cid:76) ⊥ (cid:105) = (cid:104) (cid:76) int (cid:105) and (cid:104) (cid:76) ext ⊥ (cid:105) = 0,Eqs. (2.52). At the same time, the cyclotron motion of the electron centroid produces the z -directed extrinsic OAM: (cid:104) (cid:76) ext (cid:105) = (cid:104) r ⊥ (cid:105) × (cid:104) (cid:112) ⊥ (cid:105) = σr (cid:112) ¯ z , where r = |(cid:104) r ⊥ (cid:105)| is the radius of the cyclotron orbit, (cid:112) = |(cid:104) (cid:112) ⊥ (cid:105)| isthe absolute value of the kinetic momentum of the electron, and σ = sgn( B ) is the direction of the magneticfield, Fig. 14. This extrinsic OAM has both canonical and “diamagnetic” (vector-potential) contributionsbecause (cid:104) (cid:112) ⊥ (cid:105) (cid:54) = (cid:104) p ⊥ (cid:105) . Until now we considered electrons described by a scalar wave function ψ . However, real electrons arefermions, i.e., vector particles with intrinsic spin degrees of freedom. Since spin produces intrinsic angular28 igure 14: Temporal evolution of an electron wavepacket with a vortex ( (cid:96) = 1 here) in an orthogonal magnetic field B [89].The centroid of the wavepacket follows the classical cyclotron orbit in agreement with Eqs. (2.50). Accordingly, the meanmomentum (cid:104) (cid:112) (cid:105) also undergoes the cyclotron precession (2.28) with period T c = 2 π/ | Ω c | . At the same time, the intrinsic OAM (cid:104) (cid:76) int (cid:105) due to the vortex experiences the Larmor (half-cyclotron) precession (2.29), cf. Fig. 9(b). Because of this, the vortexorientation rotates by an angle π during the 2 π cyclotron rotation of the electron, and the initially parallel (cid:104) (cid:112) (cid:105) and (cid:104) (cid:76) int (cid:105) become anti-parallel after one period T c . The cyclotron motion of the electron centroid also produces an extrinsic OAM (cid:104) (cid:76) ext (cid:105) ,Eq. (2.52) parallel to the magnetic field (see explanations in the text). momentum, it is interesting to consider its interplay with the OAM due to vortices. Here we only brieflydescribe spin properties of electrons, because most of electron-microscopy systems use unpolarized electronbeams, i.e., essentially the scalar electrons considered above.Spin is a fundamental relativistic property, and its self-consistent description requires the use of the Diracequation rather than the Schr¨odinger equation (2.1) [148, 168]. Proper consideration of vortex solutions ofthe Dirac equation will be given in Section 4, and here we only list the main results following from theproper relativistic description of spin degrees of freedom of the electron [147]. First, the Dirac electron wavefunction Ψ has four components (it is a bi-spinor), and the spin operator is a 4 × ˆS = (cid:126) (cid:18) ˆ σ ˆ σ (cid:19) . (2.54)Here ˆ σ is the vector of 2 × pc (cid:28) E (cid:39) m e c , only the uppertwo components of the wave function, Ψ + , play a role (the other two components describe positron states). Unike previous sections using nonrelativistic kinetic energy, in this section, we imply relativistic energy E , including therest-mass contribution.
29n this case, the spin is described by Pauli matrices: ˆs = (cid:126) ˆ σ . (2.55)The canonical spin operators (2.54) and (2.55) seem to be completely independent of the spatial (orbital)degrees of freedom. However, the vector and spatial degrees of freedom are essentially coupled in the Diracequation (where differential operators are multiplied by matrix operators), and, hence, in its spinor solutionsΨ( r , t ). Using a plane-wave solution of the Dirac equation, Ψ p ( r , t ), with a well-defined momentum p andenergy E = (cid:112) p c + m e c , the expectation value of the spin operator (2.54) becomes [148, 168, 147, 169]: S = Ψ † p ˆS Ψ p Ψ † p Ψ p = m e c E s + (cid:18) − m e c E (cid:19) p ( p · s ) p . (2.56)Here s is the expectation value of the non-relativistic spin (2.55); it can be regarded as spin in the electronrest frame. Equation (2.56) clearly indicates a coupling between spin and momentum properties of the Diracelectron, i.e., the spin-obit interaction (SOI). While the non-relativistic spin s can have arbitrary direction,independently of the electron momentum, the relativistic spin has a p -dependent correction. In the ultra-relativistic (or massless) limit m e c /E →
0, the spin becomes “enslaved” by the momentum direction: S (cid:107) p .The relativistic SOI manifests itself even in the free-space solutions of the Dirac equation. For example,one can construct vortex Bessel-beam solutions of the Dirac equation [147, 108], i.e., vector analogues ofthe scalar Eq. (2.5) and Fig. 5. We choose the non-relativistic spin to be parallel or anti-parallel to thepropagation z -axis, s = s ¯z , s = ± /
2, whereas the spatial vortex properties are characterized by thevortex charge (cid:96) as well as the radial and longitudinal momentum components p ⊥ = (cid:126) κ and p z = (cid:126) k z .Calculating the expectation values of the spin and orbital AM (defined with the suitable spatial integrationfor wavepackets or beams), we obtain [147]: (cid:104) L (cid:105) = (cid:126) ( (cid:96) + Λ s ) ¯z , (cid:104) S (cid:105) = (cid:126) ( s − Λ s ) ¯z , (2.57)where Λ = (cid:18) − m e c E (cid:19) (cid:16) κk (cid:17) (2.58)is the dimensionless parameter, which determines the strength of the SOI effects.Equations (2.57) demonstrate that the SAM and OAM of relativistic vortex solutions are inevitablycoupled with each other, and part of the non-relativistic SAM is converted to the OAM. This phenomenon iscalled spin-to-orbital AM conversion , and it is well known in non-paraxial (e.g., focused or scattered) opticalfields [170, 171, 172, 173, 174, 66]. This effect manifests itself in the spin-dependent spatial distributions ofthe probability density and current in the beams. In particular, the probability density distribution in theDirac Bessel beams becomes (cf. Eq. (2.13)) [147]: ρ B(cid:96),s ( r ) ∝ (cid:18) − Λ2 (cid:19) (cid:12)(cid:12) J | (cid:96) | ( κr ) (cid:12)(cid:12) + Λ2 (cid:12)(cid:12) J | (cid:96) +2 s | ( κr ) (cid:12)(cid:12) . (2.59)Thus, the probability density distributions differ for the beams with the same (cid:96) and opposite s , as shownin Fig. 15. Namely, the beams with (cid:96)σ > (cid:96)σ < | (cid:96) | = 1 vortex and anti-parallel spin s = − (cid:96)/ r = 0, Fig. 15(a) (this effect is known and observed in optics [175, 176, 174]). TheSOI properties of relativistic vortex electrons, described in [147], were recently confirmed in [177, 178]. Since photons are massless particles, relativistic SOI phenomena are inherent in optics [66]. For optical couterparts ofEqs. (2.57)–(2.59) see [173]. igure 15: Transverse spin-dependent probability-density and probability-current distributions in Bessel-beam states of rela-tivistic Dirac electrons [147] [cf., scalar non-relativistic Bessel beams in Fig. 5(b)]. These distributions are shown for (cid:96) = 1 (a)and (cid:96) = 3 (b), for the opposite spin states: s = ± / − ” signs), see Eqs. (2.57)–(2.59). The param-eters are: p = (cid:126) k = 2 . m e c and κ = 0 . k , i.e., Λ (cid:39) .
3. One can see that the probability-density distributions (2.59) differfor the beams with parallel and anti-parallel SAM and OAM, i.e., (cid:96)σ > (cid:96)σ <
0. This signals the spin-orbit interaction(SOI) in free-space relativistic vortex electrons. In particular, states with | (cid:96) | = 1 and (cid:96)σ < r = 0. Equation (2.58) shows that this SOI effect has two independent sources of smallness: it becomes small(i) in the non-relativistic case ( E − m e c ) (cid:28) E and (ii) in the paraxial case κ (cid:28) k . In modern TEMs,electrons are often accelerated to relativistic energies, so that the first factor in (2.58) is not necessarilysmall. However, electron beams are always highly paraxial, and the strongest focusing currently possible inTEMs is lower than κ/k ∼ − , i.e., Λ ∼ − . Therefore, the relativistic SOI effects are very difficult toobserve in TEM experiments. Even the most noticeable effect of non-zero probability density in the centerof a vortex mode with s = − (cid:96)/
2, Fig. 15(a), in practice becomes non-observable [179] due to decoherenceeffects related to the source-size broadening, which blurs the vortex core even in the non-relativistic scalarcase [180] (see details in Section 3.2.6).Dealing with the Dirac equation also allows rigorous calculations of the magnetic moment of relativis-tic vortex elecrons. For the Dirac–Bessel electron beams carrying SAM and OAM (2.57), this yields [cf.Eqs. (2.24) and (2.25)] [147]: M = e c E [ (cid:104) L (cid:105) + 2 (cid:104) S (cid:105) ] . (2.60)This reflects the well known fact that the orbital and spin AM contribute to the magnetic moment with the g = 1 and g = 2 factors, respectively [3]. Note that the magnetic moment (2.60) corresponds to free-space Dirac–Bessel solutions. In the presence of a magnetic field, one has to solve the relativistic Landau problemand find relativistic counterparts of Eqs. (2.42) and (2.43) with the kinetic OAM [168, 181].Importantly, intrinsic relativistic spin-orbit interactions can be strongly enhanced in artificial structures.In optics, various anisotropic structures enable efficient manipulations of spin (polarization) degrees offreedom and couple these to the orbital properties of light [81, 182, 183, 66]. Recently, there was a veryinteresting proposal to use similar inhomogeneous magnetic structures (in particular, Wien filters) for thespin-to-orbital angular momentum conversion and spin filtering of electron beams [90]. This can create anew platform for exploring electron SOI phenomena (which are so far mostly restricted to solid-state andcondensed-matter electrons).
One of the crucial differences between electrons and photons is that electrons are charged particles,and, therefore, can interact with each other even in free space. Since such interaction is of electromagneticnature, the self-consistent description of the electron dynamics should involve the Maxwell equations forelectromagnetic fields. One can describe the collective behaviour of electrons in a beam using a plasma model,which typically couples the classical equations of motion of electrons and Maxwell equations. However, inour case of coherent electron waves, we need to use the Schr¨odinger wave equation (2.1) (we again assumenon-relativistic scalar electrons) instead of the classical equations of motion.31he most straightforward way to describe the “electrons + fields” system is to consider the electriccharge and current distributions associated with the quantum probability distributions (2.12): ρ e = e | ψ | , j e = e (cid:126) m e Im ( ψ ∗ ∇ ψ ) . (2.61)These electric charge and current densities are sources of electromagnetic fields according to Maxwell equa-tions [146]: ∇ · E = 4 πρ e , ∇ × E = − c ∂ B ∂t , ∇ · B = 0 , ∇ × B = − c ∂ E ∂t + 4 πc j e . (2.62)In turn, electric and magnetic fields can be expressed via the electromagnetic potentials: B = ∇ × A and E = −∇ V − c − ∂ A /∂t , which enter the Schr¨odinger equation: i (cid:126) ∂ψ∂t − (cid:20) m e (cid:16) − i (cid:126) ∇ − ec A (cid:17) + eV (cid:21) ψ = 0 . (2.63)Equations (2.61)–(2.63) represent a complete self-consistent set for the electron wave function ψ and elec-tromagnetic potentials A and V . In particular, considering stationary solutions and neglecting magneticinteractions (which have additional relativistic smallness), the interaction is described via the scalar poten-tial V , which fulfils the Poisson equation following from Eqs. (2.61) and (2.62): ∆ V = − πe | ψ | . Coupledto the Schr¨odinger equation (2.63) (with A = 0), it describes the Coulomb interaction between electrons.As a result, the Schr¨odinger equation becomes effectively nonlinear . This may potentially lead to electronvortex solitons and other interesting nonlinear phenomena [50].Some examples of the electron-electron interactions in vortex beams were considered in [184]. Moreover,the simplest consequences of Coulomb repulsion (“the space-charge effect”), such as additional defocussingof electron beams, are known in electron microscopy [185, 186]. However, it should be noticed that the modelof coupled Schr¨odinger–Maxwell equations (2.61)–(2.63) has a significant drawback. Namely, it describesa non-zero interaction even for a single electron in a wavepacket or beam state described by the wavefunction ψ . But an electron cannot interact with itself, and only the interaction with other electrons makessense [187]. Thus, a proper description of the electron-electron interactions in a beam should involve somepair characteristics and quantum multi-body methods. In particular, the interaction must depend on theaverage distance between individual electrons in the beam, transverse and longitudinal sizes of individualelectron wavepackets, etc. Such accurate description of interacting electrons and fields remains a challenge.Recently, this problem was analysed using a Hartree–Fock approach in [188], where non-diffracting vortex-beam solutions with balanced electron-electron interaction were found.32 . Vortex beams in electron microscopy Electron microscopes are popular instruments, used to characterize materials on the micro, nano andatomic scales. A typical electron microscope is designed to impinge a beam of accelerated electrons onto asample to produce an image resulting from the interaction of the electrons with the material. Many types ofinteractions can be exploited, and the electrons can either be made to interact with the surface of materialsin scanning electron microscopy (SEM) or to interact with the internal structure of a layer of the material,thin enough to allow electrons to pass through, in transmission electron microscopes (TEMs) [4, 83, 84].At present, the majority, if not all, of the research on electron vortex beams uses the TEM setup. Inhindsight, this is partially a coincidence of suitable instruments being available, but if one realizes that amodern TEM is constructed to offer highly-coherent electron beams to obtain information about materials,one can see the close connection with laser optics — a standard platform to explore optical vortex beams.Indeed, a schematic of a TEM (Fig. 16) shows that it can provide a fixed optical bench for electron optics ,with a large number of adjustable magnetic lenses, the ability to insert custom-designed apertures, andchoice of what is placed in the sample stage.Transmission electron microscopes are typically used in one of two main operating modes. The conven-tional
TEM technique illuminates the sample with a broad, planar wavefront and either the image or far-fielddiffraction pattern is recorded, where all parts of the image are recorded simultaneously. Alternatively, theelectron beam can be focused onto a small spot in the sample plane. Varying the position of such an electron“ probe ” generates the image of the sample formed in a raster scanning fashion. This technique is known as scanning -TEM (STEM). It enables one to gather additional information about the sample on a point-wisebasis, e.g., by spectroscopic techniques [83].In TEM and STEM, typical electron acceleration (kinetic) energies are E ∼ λ = 2 π/k ∼ aberration-limited due to the severe spherical aberrations that areintrinsic to cylindrically-symmetric magnetic lenses [189, 190]. These spherical aberrations can be correctedwith non-cylindrically-symmetric lenses, but at the expense of a vast increase in complexity, cost, andstability requirements. These aberration correctors can be placed in the illumination and/or imaging systemof the microscope. Nevertheless, severe higher-order aberrations remain, and the best resolution obtainedin an electron microscope to date is around 40–50 pm, still much larger than the wavelength [191, 192, 193].Despite these limitations, electron microscopy offers one of the highest resolution imaging and spectroscopytechniques, with great flexibility to study various properties of a wide range of materials.Below we will discuss some of the ways in which electron optics of commercially available TEMs can beused to produce, measure and study electron vortex beams and their interaction with matter. We will reviewthe generation of the electron vortex beams, the measurements of the OAM carried by such beams, andpeculiarities of their interaction with materials. We will also focus on emerging applications and examinethe potential of vortex electrons to provide novel characterization methods in TEM and beyond. Since the first experimental demonstration of electron vortex beams in 2010 [76, 77], a veritable zooof methods for the production of electron beams with OAM has been developed. Some of these methodshave achieved greater popularity, while others remain more exotic. Here we comprehensively review thesemethods, compare their efficiencies, OAM-mode purity, as well as experimental advantages and drawbacks.We will also discuss which methods are most appropriate for certain categories of experiment or prospectiveapplications.Several important points should be made about electron beams in TEMs. First, electron beams are alwayshighly paraxial . Even strongly-focused STEM probes are characterized by convergence angles k ⊥ /k (cid:39) θ ∼ − –10 − . Therefore, the paraxial approximation k z (cid:39) k is always justified in TEM.33 igure 16: Simplified schematic of a transmission electron microscope (TEM), indicating the essential planes and aperturepositions that will be referred to when describing various experiments. The microscope can broadly be divided into an illumination stage and an imaging stage , each comprising several lenses and apertures, with a subsequent section containingthe detectors and cameras used for the collection of data. Second, TEM electrons can achieve kinetic energies E comparable with their rest-mass energy m e c .Therefore, weak relativistic effects can become noticable. At the same time, the TEM electrons are un-polarized , and no observable spin effects occur. In practice, the analysis of Section 2, based on the non-relativistic Schr¨odinger equation and classical equations of motion, remains valid at TEM energies. Theonly noticable effects are corrections to the electron wavelength (momentum) from the relativistic disper-sion E = (cid:112) m e c + p c − m e c and the relativistic modification of the electron mass m e → γm e , where γ = 1 / (cid:112) − v /c is the Lorentz factor.Third, due to instrumental factors, the wave beams produced in TEMs are not exactly Bessel or Gaussianbeams described in Section 2.2. Instead, their transverse Fourier spectrum is characterised by a uniformintensity for all wave vectors below a certain cutoff frequency κ max (corresponding to a circular aperture):˜ ψ ( k ⊥ ) ∝ Θ( κ max − k ⊥ ), where Θ is the Heaviside step function. The real-space distribution ψ ( r ⊥ ) for thespectrum (3.1) is the well-known Airy disc [15]. Electron vortex beams are produced from the incoming34eams (3.1), and, therefore, are characterized by a similarly abrupt Fourier spectrum with an additionalazimuthal phase: ˜ ψ ( k ⊥ ) ∝ Θ( κ max − k ⊥ ) exp( i(cid:96)φ ) . (3.1)Such circular-aperture beams look similar to the Laguerre–Gaussian beams (Fig. 6) but they also exhibitadditional rings (radial maxima) similar to those seen in Bessel beams (Fig. 5), although with smallerquickly-decaying amplitudes (see the far-field patterns in Figs. 8 and 18). One can design special hologramsto generate beams approximating any desired radial distrubution, e.g., Bessel beams [129], but the naturaland most efficient TEM beams have a radial spectrum corresponding to the uniformly-illuminated circularaperture.We finally note that, similarly to optical systems, the Fourier transform corresponds to the transitionfrom near-field to the far-field zone (e.g., in the diffraction from an aperture). Therefore, both the real-space [ ψ ( r )] and Fourier [ ˜ ψ ( k )] distributions mentioned above can appear as real-space distributions ψ ( r )at different planes of the TEMs. When electrons travel through a thin material in the TEM, they experience a phase shift depending onthe average electrostatic potential inside the material, which changes the electron momentum as comparedto travelling in vacuum [84]. This global phase shift is only a zero-order effect, describing the interactionof the electron with the space-averaged potential ¯ V produced by the atoms making up the material. Theactual microscopic electrostatic potential V ( r ) causes additional scattering effects, leading, e.g., to atomic-resolution images and a variety of dynamical Bragg scattering effects which are commonly exploited inthe TEM to gain information about the sample. In the following, we neglect the microscopic-field effects,assuming that the TEM has been setup such that either the high-frequency atomic-field information isdiscarded (e.g., by using a restrictive objective aperture which acts as a low-pass filter) or averaged out byundersampling and/or averaging in the detection plane. In this case, the phase shift caused by a homogeneousmaterial can be written as [84]:∆Φ( r ⊥ ) = C E (cid:90) d ( r ⊥ )0 V ( r ) dz ≡ C E ¯ V ( r ⊥ ) d ( r ⊥ ) , (3.2)where d is the thickness of the material (which can vary with the transverse coordinates r ⊥ ), ¯ V is themean inner potential (MIP) (which is assumed to be r ⊥ -independent for homogeneous materials), and C E = k e ( E + E ) /E ( E + 2 E ) is the interaction constant. Here E = m e c is the electron rest-mass energy,and E is the acceleration energy of the microscope. For example, C E = 0 . − nm − for E = 200 keV.For many materials, the MIP ¯ V is in the range of 5–30 V [194]. Then, for E = 200 keV, an amorphous SiO layer of d (cid:39)
85 nm causes a phase shift (3.2) ∆Φ = 2 π .Similar phase shifts are produced in optics using transparent materials with refractive index n (cid:54) = 1. Thisallows one to shape the phase front of the transmitted wave with plates of space-varying thickness d ( r ⊥ ). Inparticular, photonic vortex beams were generated in optics, millimeter waves, and X-rays using spiral phaseplates with azimuthally-varying thickness d ∝ ϕ corresponding to the phase shift ∆Φ = 2 π(cid:96)ϕ [30, 135, 136].The same method can be employed for electrons, Fig. 8(a).Historically, the first deliberately-made electron vortex beam was demonstrated by exploiting the for-tuitous helical stacking of three graphite flakes [76]. Even though this setup is only a rudimentary ap-proximation of a spiralling thickness profile, it was sufficient to demonstrate the presence of typical vortexcharacteristics. Progressing from this first demonstration, several researchers made attempts to perfect thesetup in order to obtain high-purity electron vortex states [140, 141]. In order to avoid directional scatteringdue to the Bragg diffraction, amorphous materials are preferred. Also preferrable are low-density materials The actual spectrum of electron vortex beams can behave differently and vanish in a small area around the center k ⊥ = 0,but it is well approximated by a uniform amplitude in most of the circle k ⊥ < κ max . igure 17: Spiral phase plates for electron optics [see also Fig. 8(a)]. These devices exploit the internal potential of a material,which alters the momentum of the electron travelling inside the material, thereby imprinting a phase shift (3.2). (a) Experi-mental TEM image of a spiral phase plate, produced by the focused-electron-beam-induced deposition (FEBID) of SiO2 [141].The black central spot is made of platinum and acts as a beam stopper (green arrow). (b) Atomic force microscopy heightprofile of the plate revealing its spiral shape. Since the material is uniform, the phase shift is proportional to the thickness.(c) A through-focus series of the transverse intensity distributions for the electron vortex beam produced by the spiral phaseplate (i.e., the far-field pattern of the plate) [141]. A dark core caused by the destructive interference in the beam center doesnot vanish upon focusing or defocusing: a typical signature of a vortex beam. with weak MIP, because the higher thickness needed to obtain a 2 π phase shift requires less precise thick-ness control, and the discrete atomic nature or surface roughness of the material are less significant over thehigher volume.Focused ion beam (FIB) milling was used in [140] to create an accurate spiral phase plate by millingaway material from a commercially available SiN grid. In addition to making high-purity vortex beams, thetechnique allows the fabrication of many different types of phase plates. Instead of milling material away,one can also use additive manufacturing offered by ion- or electron-beam-induced deposition of amorphousmaterials. In Ref. [141] the focused-electron-beam-induced deposition (FEBID) of SiO on an ultrathinsubstrate of SiN was used to create another version of the spiral phase plate, shown in Fig. 17(a,b). In orderto remove a part of the beam that could pass through the hole in the center of the spiral, a small platinumbeam stopper was deposited via FEBID. Since SiO is an insulator, the phase plate had to be coated witha thin layer of carbon to prevent unwanted charging effects when used in the microscope. The phase platewas then introduced in the sample plane of a TEM (operating in Lorentz mode). Figure 17(c) showsa through-focus series of the focused electron beam (probe) in the far-field plane of the spiral phase plate.The characteristic destructive-interference area at the vortex-beam core is clearly visible and does not vanishupon focusing.Even though the above spiral phase plates are highly versatile (the thickness profile can in principleencode for any desired phase profile), the technique has several significant drawbacks. Namely, charging andcontamination need to be carefully avoided and the phase plate has to be designed for a specific accelerationvoltage ( C E depends on the acceleration energy E and electron wave number k ). The thickness of the phaseplate also has to be carefully calibrated, since the MIP is not exactly known due to the less-than-perfectquality of the deposited or milled material. In practice, this means that the fabrication procedure has to go This technique requires a very high lateral coherence length, see Section 3.2.6. A potentially significant advantage of the phase-plate method is the fact that no magnetic materials are used, and the plate can work well even in the veryhigh magnetic fields inside electron-optical lenses.
The complexity of precise height control to create a spiral phase plate, as well as the limitation of usingonly one acceleration voltage, can be lifted by employing holographic gratings , as in Fig. 8(b). Such gratingswith dislocations, generating different vortex beams in different diffraction orders, were first introduced inoptics [28, 29].To describe their holographic reconstructive action, let us start from the desired target wave (a vortexbeam with topological charge (cid:96) in our case), which can be written as: ψ target ( r, ϕ, z ) = a ( r, z ) exp( i(cid:96) ϕ + ikz ) , (3.3)where a ( r, z ) is the amplitude (assumed to be real-valued, for simplicity), and k is the wave number of theparaxial beam. Consider the interference of this target wave with a reference wave , which can be chosen as,e.g., a plane-like wave slightly tilted in the x -direction: ψ ref ( r, ϕ, z ) = a ( r, z ) exp( ik x x + ikz ) , (3.4)where k x (cid:28) k , and for the sake of simplicity we assume the same amplitude a ( r, z ) as in the target wave.The interference of the two waves (3.3) and (3.4) leads to the following intensity pattern in the referenceplane z = 0: ρ int ( r, ϕ ) = | ψ target ( r, ϕ,
0) + ψ ref ( r, ϕ, | = a ( r,
0) [1 + cos( (cid:96) ϕ − k x r cos ϕ )] . (3.5)This equation describes a fork-like interference pattern with an edge dislocation of the order (cid:96) and gratingperiod x g = 2 π/k x , as shown in Fig. 18(a).Now consider an aperture with a transmission function T ( r, ϕ ) mimicking the interference pattern (3.5)(in amplitude, phase, or both). Illuminating it with the untilted reference plane wave ψ ∝ exp ( ikz ) allowsthe reconstruction of the tilted target vortex wave. Indeed, assuming an amplitude-modulating aperturewith T ( r, ϕ ) ∝ ρ int ( r, ϕ ), the transmitted wave becomes: ψ reconstruct ( r, ϕ, z ) = T ( r, ϕ ) exp ( ikz ) ∝ a ( r,
0) exp ( ikz ) [exp( i(cid:96) ϕ − ik x x ) + 2 + exp( − i(cid:96) ϕ + ik x x )] . (3.6)Thus, the electron wave after interaction with such an aperture is a superposition of: (i) the target wave,but now with the tilt, (ii) the untilted reference wave, and (iii) the complex conjugate of the target wave,tilted in the opposite direction, see Fig. 8(b). Moving away from the z = 0 plane containing the apertureseparates the three terms due to their tilts. The beams are typically observed in the far field ( z = ∞ ) of theaperture plane (essentially, the Fourier transform of the aperture-plane waves), where the three componentsare well separated as long as their Fourier components contain spatial frequencies below k x / ψ reconstruct ( k ⊥ ) ∝ ˜ a ( k ⊥ , ∗ [ ˜ ψ target ( k ⊥ + k x ¯x ) + 2˜ a ( k ⊥ ,
0) + ˜ ψ ∗ target ( k ⊥ − k x ¯x )] . (3.7)where ˜ ψ ( k ⊥ ) and ˜ a ( k ⊥ ,
0) are the Fourier transform of ψ ( r ⊥ ,
0) and a ( r ⊥ , ∗ ” denotes the convo-lution of functions. This is, however, not a physical limit but a technological one, as the effect could be counteracted with a set of limitingapertures to erase the high-frequency information, and demagnifying lenses to reduce the size of this purified vortex. igure 18: Schematics of the holographic reconstruction concept, Eqs. (3.3)–(3.8), and generation of electron vortex beamsusing binarized fork holograms. (a) The interference of the target wave (vortex beam with (cid:96) = 1 here) with a reference wave(tilted plane wave) yields a characteristic interference pattern with an edge dislocation (fork) of order (cid:96) . For practical reasons,this pattern is usually binarized, Eq. (3.8). (b) The resulting pattern is then milled in a thin film that is placed as a holographicaperture in a TEM. When illuminated with a coherent electron beam, this aperture produces an array of vortex beams of order (cid:96) = N(cid:96) in different diffraction orders N = 0 , ± , ± , ... . For non-binarized holograms only the first orders N = 0 , ± A problem with the above reconstruction method is that currently it is near-impossible to make a smoothsinusoidally-modulating hologram for electrons. An often-used approximation is to binarize the sinusoidalpattern (3.5) to either block ( T = 0) or transmit ( T = 1) the electrons. Mathematically, such binarizedtransmission function can be written as: T ( r, ϕ ) = 12 Θ( r max − r ) { (cid:96) ϕ − k x r cos ϕ )] } , (3.8)where we assumed that a ( r,
0) = Θ( r max − r ), corresponding to the circular aperture and Eq. (3.1), andthe resulting binarized hologram is shown in Fig. 18(a). This binarization leads to higher-order diffractionharmonics with increasing vortex charge (cid:96) = N (cid:96) and tilt N k x , where N = ± , ± , ± , ... [195, 196], seeFig. 18(b). The intensity of the N th-order diffraction spot scales as 1 /N . In some experiments using thistechnique, also faint even-order harmonics are present due to imperfections in the binary aperture [78, 96].Note also that the above considerations on the intensity of the diffracted spots are true as long as the barsand slits in the grating have the same widths. The general case is similar to the well known problem ofmultiple-slit diffraction, and the relative intensities of different diffraction orders are determined by the ratiobetween the width of bars and slits [15].To achieve high values of the OAM ( (cid:126) (cid:96) per one electron) carried by electron vortex beams, one should(i) use the grating with high-order dislocation | (cid:96) | (cid:29) | N | >
1. This provided the first demonstration of high-OAM electron vortex beams up to (cid:96) = 100 [78],38hile currently similar methods reached (cid:96) = 1000 [197, 142]. Such high-order vortex electrons carry hugemagnetic moments, Eqs. (2.24) and (2.25), exceeding the usual spin magnetic moment (Bohr magneton) byseveral orders of magnitude, which can be important in experiments involving magnetic interactions.A typical binary, amplitude modulating, holographic grating is shown in Fig. 19(a). The aperture hasa typical diameter ranging between 10 µ m (for optical-bench-like experiments) to 30 µ m (used to generateSTEM probes), and is made out of a 0.5–1 µ m thick sputtered gold film deposited on a commercial 200 nmthick Si N support film. The period of the bars, limited by the resolution reachable by the focused ionbeam milling (typically of the order of the gold-film thickness), is about x g = 500 nm . Note that this periodand the wavelength λ = 2 pm correspond to a very small diffraction angle θ d = k x /k = 0 . ) and the sample allowone to vary the magnification and separation of the generated beams in the sample plane. Typically, inthe STEM mode, the convergence angle of the probe is calibrated for a given radius of the aperture r max as θ max = r max /L , where L is the equivalent camera length of the lens system in a given configuration.Using a forked grating aperture of the same diameter produces probes which are separated by the distance Lθ d . Then, taking typical parameters r max = 15 µ m and θ max = 20 mrad, yields L = 0 .
75 mm and probesseparated by Lθ d = 75 nm. This is a factor of r max /x g larger than the diffraction limit of the probes. Alarger separation can be achieved by altering the condenser lenses to obtain a higher effective L but thisat the same time sacrifices the size of the diffraction-limited probe by lowering the convergence angle. Ingeneral, holographic apertures allow one to produce electron vortex beams, focused on the sample, rangingin size from atomic scale to hundreds of nanometers.An example of a resulting far-field pattern for the hologram with (cid:96) = 1 is presented in Fig. 19(a).The central beam is surrounded by two vortex beams with (cid:96) = ± (cid:96) = ± (cid:96) = 9 dislocation[198]. It generates (cid:96) = ± any structured modes . Forexample, Fig. 19(b) shows a holographic aperture and the corresponding far-field intensity distributions,which correspond to the superposition of (cid:96) = 3 and (cid:96) = − spherical wave as a reference wave [201, 199]. In this case,the corresponding hologram takes the form of a (spiral) zone plate, and the different diffracted orders areseparated along the z -axis , i.e., can be brought into focus one at a time by varying the defocus ∆ z , Fig. 19(d).This approach has been applied for the production of vortex beams for STEM imaging, where only onevortex beam is in focus on the sample while the other beams generate an out-of-focus background [199].An obvious drawback of this technique is that all out-of-focus diffraction orders still remain present at alltimes. Therefore, an OAM-dependent interaction with the target can only be studied if its spatial scale issmall enough to be neglected for the out-of-focus waves. Such focus-dependent vortex beams can be usedfor studying crystals, where the structure of the lattice can be probed with a single diffraction order, if theprobe can be made small enough to resolve atomic distances.So far, we considered only holograms, which modulate the intensity of the incident wave. Hologramsfabricated with electron-transparent materials, such as carbon or silicon nitride, modulate the phase of theelectron beam, similar to phase plates in Section 3.2.1. This approach has an advantage that the wholeincoming beam is transmitted through the aperture, doubling the intensity with respect to the binary The aperture can also be placed in other aperture planes like, e.g., the sample plane or the selected-area plane, as long asthe lateral coherence length of the incoming electron beam extends over the whole diameter of the aperture in that plane. igure 19: Examples of vortex beams and other structured modes generated using holographic reconstructions. The SEMmicrographs of the holograms are shown alongside the beams they produce in TEM. The 2 µ m white scalebars and 10 nmyellow scalebars are shown in the hologram and far-field beam images, respectively. (a) A fork hologram with (cid:96) = 1. The (cid:96) = 0 beam is visible in the center, with the first sidebands with (cid:96) = ±
1, and the higher-order beams with (cid:96) = ± (cid:96) = 9, which generates (cid:96) = ± (cid:96) = 3 and (cid:96) = − (cid:96) = 1) with a spherical referencewave [199]. The generated probes with (cid:96) = 1 and (cid:96) = − along the propagation z -axis , i.e., appear at certainplanes with defocusing ∆ z > z < amplitude grating approach. Such holograms also allow one to minimize or even cancel the intensity in theundesired reference part of the wave, thereby obtaining a higher fraction of the diffracted intensity in thedesired first-order beams [202]. In addition, since this method uses thinner films and relies on structureswith a depth of only several (or tens of) nanometers, the lateral resolution of the milling can be muchhigher, allowing finer and more sophisticated masks. Lastly, using a phase hologram with a blazed profile,one can enhance the fraction of the intensity directed in a single first-order diffracted beam (e.g., N = 1)while suppressing the opposite ( N = −
1) beam [129, 202]. The experimentally-demonstrated efficiency ofsuch blazed phase holograms is up to 40% of the incoming beam ending up in the desired vortex state.Phase holograms also have practical limitations. Carbon has a tendency to migrate under the beam,compromising the quality of the hologram and altering the phase shift, while the Si N films tend to chargedue to the emission of secondary electrons. A gold or chromium coating with a thickness of few nanometerscan be used to reduce the charging, at the cost of a lower diffraction efficiency. Furthermore, since thephase shift depends on the electron’s energy, the performance of a phase-modulated mask depends on the40cceleration voltage used, which is not the case for binary amplitude holograms (with the exception thatthe positions of different diffraction orders depend on the electron wavelength).As an alternative approach to holographic reconstruction – holograms producing on-axis structuredelectron beams – have also been demonstrated [140]. Such holograms are calculated from the target wavethrough an iterative Fourier-transform algorithm and are imprinted on a transparent film to obtain a purephase mask. While this approach allows one to sculpt the intensity distribution of the beam with a highflexibility, its efficiency is low. Due to the impossibility of producing subwavelength structures for fastelectrons at TEM energies, most of the wave intensity still forms an on-axis central spot, which is about 400times more intense than the generated target wave. Speckling and multiple diffraction orders also occur inthe pattern. An alternative method to produce electron vortex beams relies on earlier experiments in optics using thelinear relation between the so-called Hermite–Gaussian (HG, which in the low-order HG mode resemblesp-orbitals) and Laguerre–Gaussian (LG) beams [27, 203]. Indeed a proper linear combination of two HGbeams can create an LG beam and vice versa. Starting from a single-mode laser that produces a HG beam,it is possible to create a vortex LG mode using a mode converter based on two cylindrical lenses [203].A similar approach can be applied to electrons [204]. It starts with a phase plate which changes thephase of the electron wave by π in a half-plane [e.g., ∆Φ = π Θ( x )], generating a HG–like mode in theelectron microscope. Then, exploiting the tuneability of astigmatism through quadrupolar lenses availablein any TEM, one can introduce cylindrical distortions to electron optics to make it analogous to the opticalsetup with cylindrical lenses. The conversion of electron vortex beams into HG–like beams and vice versawas demonstrated in [204].The advantage of this method is that almost all the electrons of the beam end up in the desired mode,and the sign of the OAM can be easily reversed using only tuneable lenses, which are part of any standardelectron microscope. The drawback of the technique is that it still uses a half-plane phase plate which suffersfrom charging and contamination and needs to be carefully tuned in thickness for each acceleration voltage.Recently an alternative setup producing HG-like beams was demonstrated using the Aharanov–Bohmeffect around a magnetized needle (an analogue of the magnetic-flux line) that divides an aperture into twohalf-planes [205]. When carefully tuning the magnetic flux in the needle to an odd number of flux quanta, aphase difference between the two half planes becomes exactly π , independently of the acceleration voltage.This setup has an additional benefit: charging and contamination are strongly reduced, while only a smallpart of the beam is absorbed when hitting the needle. Combining this superior setup to generate HG beamsin combination with two astigmatic lenses could be an attractive way to produce vortex electrons of highpurity and intensity. In visible-light optics, spatial light modulators allow flexible manipulations of both the phase and am-plitude of the wave. Unsurprisingly, this tool has become indispensable for many tasks involving structuredwaves, such as optical vortex beams [59, 206, 33, 207]. An electron analogue of spatial light modulatorswould be an ideal tool to deal with structured electron waves, but, unfortunately, it does not exist yet [208].In recent years, hardware aberration correctors have been developed [209, 210, 211], to counteract spher-ical aberrations which are intrinsic to cylindrically-symmetric magnetic lenses [190]. Aberration correctorsconsist of a sequence of adjustable multipolar lenses connected by transfer doublets. By modifying therelative strength of these magnetic multipoles, the electron wavepacket can be deformed, adjusting thelower-order Seidel aberrations of the lens system. The aberration corrector is typically adjusted such thatthe electron beam impinging on the sample approximates either a planar wavefront (for the TEM work), ora spherical wave focusing to a small probe (for the STEM imaging). Negative spherical aberration can alsobe used, to enhance contrast when imaging some challenging specimens [212].Recently, it was shown that a more radical adjustment of the aberrations can be used to directly createelectron vortex beams [213]. The vortex electron phase should exhibit the linear azimuthal depedenceΦ( r, ϕ ) (cid:39) (cid:96)ϕ . However, the control software of the aberration corrector is designed to measure the wavefront41f the beam in terms of the Seidel aberrations [214], none of which depends linearly on ϕ . Nonetheless, theΦ (cid:39) (cid:96)ϕ dependence can be neatly expanded into a sawtooth Fourier series, with a period of 2 π . Such anexpansion reveals that the aberrations of the image shift, along with the n -fold orders of astigmatism, canbe manipulated into a spiral phase structure, but only for a limited radial range.The radial range of the beam can be easily limited by inserting an annular aperture into the column. Indoing so, the angular size of the aperture should be tuned to the desired mode purity (a broader annulusprovides a less ideal vortex phase structure, but higher intensity) and to the obtainable aberrations in themicroscope (the required astigmatism values vary with the aperture size). In the FEI Titan microscopeused in [213], an annular aperture provided an angular selection from 5 . . (cid:96) = 1 state, and almost 50% of the initial beam intensity transmitted.The advantage of this method is its flexibility: the opposite-order vortex beam can be obtained byswitching between aberration settings, as long as hysteresis and drift in the corrector are not too significant.The drawbacks are: the annular aperture severely limiting the total beam current, the non-atomic size ofthe vortex probe, and the less-than-perfect mode purity. Close relations between magnetic monopoles and phase singularities in electron waves were first em-phasized in the pioneering Dirac paper [18]. Later, this led to the suggestion that an electron plane waveinteracting with a magnetic monopole can be converted into a vortex state [215, 5, 75]. At first glance, thismight seem to be a fantasy, given that no magnetic monopole has ever been observed in nature. However,recently it was shown that an approximate monopole, manufactured ad-hoc, works very well in practice[92, 144].The experiments [92, 144] employed a ferromagnetic needle , magnetized along the needle’s length in asingle-domain magnetic state, and with a thickness chosen so that the magnetic flux through the sectionequals an integer number α m of double magnetic-flux quanta. Then, the magnetic field around a tip of theneedle approximates the monopole field, while the magnetic flux passing through the needle plays the roleof the so-called Dirac string (ideally, an infinitely-thin magnetic-flux line) [18]. Placing such an approximatemonopole in the center of the electron beam produces a transmitted vortex beam with topological charge (cid:96) = α m , see Section 2.4 and Fig. 8(c). Making use of an extra round aperture ensures that incoming electronsinteract with the region close to the chosen tip of the needle, leaving the other tip of the needle (the opposite − α m monopole) outside the electron beam, Fig. 20(a).This provides a highly efficient method for the production of electron vortex beams, with several impor-tant advantages. First, the phase profile is independent of the acceleration voltage. Second, the transmit-tance is near 100% as the only part that absorbs electrons is the needle, whose “shadow” is small comparedto the area of the aperture. Third, the purity of the generated vortices can be very high, as long as thelength of the needle is much larger than the aperture diameter. This assures that the magnetic field aroundthe tip approximates well the monopole one. Finally, the magnetization of the needle can in principle bereversed, allowing the dynamical control of the sign of the topological charge (cid:96) .The results of the most recent magnetic-monopole experiment [144] are presented in Fig. 20. A 60 nmthick pure nickel (Ni) film was deposited on a silicon-nitride (Si N ) film and covered by a 1 µ m sputteredgold layer. A focused ion beam was used to cut a very elongated bar out of this ferromagnetic film. Then,this needle was extracted from its original film and deposited on another gold-plated Si N grid, with one ofits ends positioned over the centre of a precut 20 µ m round aperture, Fig. 20(a). The needle was fabricatedwith a very high aspect ratio (300 nm wide by 50 µ m long) in order to increase the shape anisotropy, whichforces the needle to be in a single-domain magnetic state, with the magnetization along the needle axis. Thecross-section of the needle, in combination with the saturation field of the material, was chosen to create amagnetic flux equalling two flux quanta, which corresponds to (cid:96) = α m = 1 and a 2 π phase shift betweenelectrons passing on the two opposite sides of the needle.In practice, finding the exact width needed for the needle cross-section requires iteratively measuringthe phase in a TEM with electron holography [see Fig. 20(b)] and going back to the FIB. The reason forthis difficulty is likely caused by the FIB, where a fraction of the Ga ions used for the milling ends up42 igure 20: Generation of electron vortex beams using the magnetic monopole-like field of a magnetized-needle tip [144], seealso Fig 8(c). (a) SEM view of a thin magnetic rod placed over a round aperture. (b) Color-coded phase shift around the tipof the needle measured and reconstructed through electron holography. (c) A through-focus series of the far-field images of theaperture illuminated with a plane electron wave. The dark spot in the center indicates the vortex character of the transmittedbeam, and the radial “shadow” from the magnetic needle is also seen. implanted in the Ni film altering its magnetic properties and the thickness of the films used, which limitsthe milling resolution. The lithographic production of highly controlled Ni needles seems very attractive forfuture exploration.Once this artificial “magnetic monopole” was fabricated, it was placed in the sample plane of the TEM(operating in Lorentz mode to avoid the presence of the strong magnetic field in the lenses), and a seriesof through-focus images was recorded in the far-field plane (i.e., in the diffraction mode). The results areshown in Fig. 20(c). One can easily recognize the typical doughnut-like profile of the transmitted vortexbeam with a dark central region of the destructive interference. The electron hologram acquired at thetip of the magnetic needle, Fig. 20(b), also provides the measured phase shift. In this particular case, aninterpolation of the phase over the whole aperture showed the expectation value of the OAM per electron tobe (cid:104) L z (cid:105) (cid:39) . (cid:126) . Placing this aperture in the condenser aperture plane allows the production of high-qualityatomic-size focused vortex beams for STEM experiments [144]. The application of the methods described above imposes several practical constraints to be kept in mind.Electron guns emit electrons from a small but finite area of a sharp tip. Electrons emanating from differentpositions are typically mutually-incoherent and the emitting tip behaves as an extended incoherent source [15, 216]. Suppose that each individual electron can be considered as a perfectly coherent wavepacket. Ifthe goal is to create the smallest possible electron vortex beam, one uses the setup employed to generatethe STEM electron probe, where the sample plane is conjugated with the source plane. This means thatthe ideal vortex probe formed by a single emitted electron will be superimposed incoherently with a largenumber of such vortices, each slightly displaced with respect to the other. This causes a blurring of thevortex intensity profile, leading to the appearance of a finite intensity in the dark r = 0 core of the vortex[180].This effect can be reduced by demagnifying the source with the condenser lenses either by changing the43xperimental “spot size” (lens settings) and/or by varying the strength of the “gun lens”, depending on theexact technical characteristics of the microscope. In any case, this demagnification will always require atrade off in intensity unless a brighter electron source can be chosen. For this reason, the field emission gunsare currently preferred, but work is underway to develop superior options [217]. Current probe-correctedmicroscopes tend to have a source-size broadening of the order of the smallest probe achievable on thatinstrument. Indeed, as far as the probe size is concerned, the source-size broadening does not considerablydeteriorate the spatial resolution (for STEM imaging), as long as it remains smaller than the diffraction limitimposed by the aberrations. However, for vortex beams, the same effect destroys the vortex by blurring thedark core, and the source-size broadening becomes a limiting factor. For larger-diameter electron vortices,the source-size broadening becomes insignificant and near-perfect coherence can be achieved.If the aperture is placed in either the sample plane or the selected-area plane, the situation is differentand the most important factor becomes how paraxial the incoming beam can be made. Indeed the spatialcoherence in the aperture plane needs to extend over the size of the aperture, which is inversely proportionalto the angular spread of the beam. In the sample plane, this requires extremely small angular spread or verysmall aperture sizes. Typically, the low-magnification modes in TEMs can reach spatial coherence coveringa 50 µ m aperture. This allows one to test the apertures in the sample plane, which provides higher flexibilityand speed in terms of transferring the aperture in and out of the vacuum.Next, the material of the apertures is important for the electron energy loss spectroscopy (EELS).Indeed, fast electrons excite transitions in the infrared to ultraviolet range in the material, experiencing thecorresponding energy loss. These losses can be significant for trajectories of electrons close to the edge ofthe aperture (interacting with the metallic film) or travelling through the light phase-plate material. Thisgenerates a background spectrum exclusively due to the aperture (not the sample), with spectral featuresdepending on the material chosen and mixing with the actual EELS spectrum of the sample. This effect canbe particularly noticable for the MIP-based phase plates and fork binary holograms, while the magnetic-needle setup is preferred in this respect as only a small amount of material is present in the aperture. Charging is another factor which complicates the design of phase apertures. Secondary-electron emissiontypically causes the aperture to charge positively, if there is no way for an electric current to flow fromthe aperture to ground, replenishing the lost electrons. Metallic materials are preferred but are oftencrystalline, which leads to unwanted Bragg diffraction, and are generally heavier, which leads to higherscattering. Amorphous insulating materials can be used, but coating with a thin layer of metal can solveor reduce this issue. Different work functions can also make a difference in the secondary-electron emissionand, hence, should also be considered.To compare different methods of the generation of electron vortex beams, we summarize their mainfeatures in Table 1. Undoubtedly, other methods for electron vortex production can be envisioned, and thislist is likely to expand in the future.
In many cases, in addition to the production of electron vortex beams, one also needs to measure theOAM (cid:104) L z (cid:105) carried by the beam. It is important to remember that the result of such measurements depend,in the generic case, on the position of the z -axis with respect to the electron beam. So far, we mostlyassumed cylindrical symmetry of the electron vortex beams, but real-life conditions often deviate from thissymmetry, e.g., due to the occurence of optical aberrations or scattering by non-cylindrically-symmetricobjects. In this case, the direct proportionality between the topological charge (cid:96) and OAM breaks down[221], meaning that measuring the topological charge is not generically equivalent to measuring the OAM[222].At the quantum-mechanical level, the z -component of the OAM of a paraxial wave can be calculated byintegrating the OAM density, proportional to rj ϕ , Eqs. (2.15) and (2.16). Therefore, an ideal method shouldbe able to measure this quantity independently of the radial component of the current, j r , and independentlyof the shape of the wave. As we will see, all the methods developed so far fall short of this requirement.Note that wave propagation, whether free or operated by lenses, is intimately linked to the Fourier transform44 ethod Efficiencytheory/exp. Advantages Disadvantages Papers Holograms(Binary) ∼
10% / ∼
10% Versatile, straightforward,high quality EVBs Inefficient, multiplediffracted beams [77, 78, 202],[218, 219],[199, 100,200]Holograms(Phase) 100% / ∼
40% Efficient, versatile,high quality EVBs Charging, multiplediffracted beams, aperturesare specific to a given valueof kinetic energy [218, 202][140, 129]MIP phaseplate 100% / ∼
55% Efficient, single outputbeam Charging, difficult fabrica-tion, aperture is specific toa given kinetic energy [140, 141]Magneticphase plate >
95% / ∼
92% Efficient, single outputbeam, independent ofkinetic energy Difficult fabrication, can-not be used inside a strongmagnetic field such as theobjective back focal plane [92, 220, 144]Aberrationtuning ∼
50% / ∼
32% Single output beam,more efficient thanbinary holograms Limited by aberrationcorrection technology [213, 208]Astigmaticmodeconversion 100% / N/A Efficient, single outputbeam, tuneable in realtime Charging, apertures arespecific to a given value ofkinetic energy [204, 205]
Table 1: Summary of different methods for the generation of electron vortex beams. Presented are: theoretically expected andexperimentally achieved efficiencies (defined as the fraction of the impinging intensity to end up in the desired vortex state),the main advantages and drawbacks, and relevant bibliographic references. and, as such, allows the efficient mapping of the transverse momentum components p ⊥ into coordinates r ⊥ .This simple relation lies as the heart of diffractive techniques, that reveal the linear momentum spectrumof the outgoing wave in the diffraction pattern, as used, e.g., in TEM studies of reciprocal crystal lattices.If a similar process could be devised for the longitudinal OAM, it would facilitate studies of a variety ofphenomena.An efficient far-field sorting of different OAM states was suggested in optics [223]. By employing atransverse-coordinate transformation r ⊥ → r (cid:48)⊥ that converts every r = const circle into a straight line (e.g., y (cid:48) = const), it is in principle possible to transform the 2 π(cid:96) azimuthal phase into a linear phase ramp, therebycausing a shift in the far-field intensity proportional to (cid:96) . Mathematically this can be done using a conformalmap mimicking the transition from Cartesian to log-polar coordinates: x (cid:48) = − a arctan (cid:16) yx (cid:17) , y (cid:48) = − a ln (cid:32) (cid:112) x + y b (cid:33) , (3.9)where a and b are constant parameters. Such conformal mapping can be realized via ad-hoc phase plates,and efficient OAM sorting was demonstrated for optical beams [223, 224]. However, the limitations of thephase-plate technology for electrons hampered the adoption of this method in the TEM. Only recently45he first demonstration of this method has been realised using holographic gratings (with all the ensuinglimitations in terms of efficiency) [225]. There were also proposals to employ more efficient electrostaticoptical components, but these have yet to be realised experimentally [226].Several other approaches to OAM measurement have been developed in optics, with sensitivity up to thesingle-photon level [34, 227]. However, these methods cannot be directly adopted by electron microscopydue to the limited flexibility of electron-optical elements (e.g., because of the absence of beam splitters forelectrons). Therefore, the techniques that have so far been demonstrated with electrons offer a much lowerlevel of generality and/or detection efficiency than those available for photons [228, 225]. The first measurement of the topological charge has been demonstrated by projecting a vortex beamon a fork-grating hologram [230, 228], Fig. 21(a,b). The far-field pattern generated by the hologram inthis condition is similar to the one obtained for an impinging plane wave (Fig. 18) and is formed by aone-dimensional array of vortex beams. The relative intensities of the diffracted beams are, to a goodapproximation, unaltered [196, 230], while their topological charges (cid:96) (cid:48) are now given by (cid:96) (cid:48) = N (cid:96) + (cid:96) where (cid:96) is the dislocation order of the fork and (cid:96) is the topological charge of the impinging beam. In particular, adiffracted beam that satisfies N (cid:96) = − (cid:96) will be a non-vortex (cid:96) (cid:48) = 0 mode with the maximum intensity in thecenter. This allows the determination of the topological charge of the impinging beam via the dislocationcharge of the fork hologram, (cid:96) , and the diffraction order N of the beam without the typical doughnut-like intensity distribution Fig. 21(b). A sufficiently small pinhole centered on a diffracted spot can readilydiscriminate between the vortex and non-vortex modes and detect the (cid:96) (cid:48) = 0 component with the maximalintensity in the center. This method has been suggested as possible basis for automated measurements of thetopological charge of the impinging beam using a set of appropriately-placed pinholes. Such measurements,however, would be highly inefficient as most of the beam intensity would be discarded due to the limitedtransparency of the mask, division of the intensity into several diffraction orders, and the subsequent limitedtransmittance of the pinhole [228].The topological charge of a vortex beam can also be revealed by diffraction from geometric apertures .The common feature of these methods is that the diffraction pattern given by the geometrical aperture isaltered by the unique phase profile of the vortex beam in a way that reveals the topological charge. Forexample, a triangular aperture generates a triangular lattice of spots, with the side of the triangle having | (cid:96) | + 1 spots and the orientation of the triangle depending on sgn( (cid:96) ) via the Gouy phase [228], see Fig. 21(c).While such geometric apertures have not been widely used to measure the OAM of electron vortex beams,the diffraction patterns reveal a variety of remarkable physical phenomena [231].A particular case of geometric aperture is a knife-edge aperture that blocks half (or a finite segment) ofthe vortex beam at its waist, Fig. 21(d). This produces a C-shaped beam, which rotates, upon propagationto the far field, as an effect of the Gouy phase, in the direction determined by sgn( (cid:96) ) [232, 233, 165].Importantly, this effect can be regarded as a direct manifestation of the azimuthal probability current j ϕ in vortex beams. Indeed, the spiraling current density in a cylindrically-symmetric vortex beam produceszero transverse momentum: (cid:104) p ⊥ (cid:105) = 0, Eq. (2.20). Blocking part of the beam breaks this symmetry, and theresulting C-shaped beam acquires non-zero transverse momentum (cid:104) p ⊥ (cid:105) (cid:54) = 0, which leads, upon propagation,to the transverse shift of the diffracted-beam centroid (cid:104) r ⊥ (cid:105) [173], see Fig. 21(d). Variations of this methodhave been employed to confirm the vortex character of doughnut-shaped beams [92] and to investigaterotational vortex-beam dynamics in a magnetic field [91] (Fig. 13).A more complete method for the analysis of the OAM spectrum is offered by a multi-pinhole interfer-ometer (MPI) [229]. This is an interferometer comprising a set of small holes; for the OAM measurementthe most interesting configuration is n pinholes evenly distributed around a circle, Fig. 21(f). When a waveis projected on such an interferometer, the far-field intensity distribution is determined by relative phasesbetween the pinholes, which allows one to retrieve the value of (cid:96) . The MPI, however, is more than justanother type of geometrical aperture, as it can be used to obtain a quantitative OAM decomposition of thebeam, even for mixed OAM states. Namely, recording the diffraction pattern allows one to determine the46 igure 21: Diffraction-based methods for measurements of the topological charges of electron vortex beams [228, 229]. (a)Schematic diagram of the main idea. The vortex beam impinges on a specially-shaped aperture (or a phase plate) producingthe far-field diffraction pattern which reveals the value of (cid:96) . In (b–f) the patterns produced by different elements are shown,with the yellow and blue far-field images corresponding to the (cid:96) > (cid:96) < (cid:96) + 1. The knife-edge aperture (d) causes atransverse deformation and shift of the beam intensity that depends on the sign and value of the OAM. The astigmatic phaseplate (e) also causes a multi-lobed pattern where the number of lobes is proportional to (cid:96) + 1. The multi-pihole interferometer(f) produces patterns which are related to the topological charge of the original wave function. Furthermore, the autocorrelationfunction of these patterns allows one to obtain an approximate OAM spectrum of the incoming wave [229]. autocorrelation function of the interferometer by employing the Wiener–Khinchin theorem: (cid:65) ( ψ ) = (cid:70) − (cid:16) | ˜ ψ | (cid:17) , (3.10)where (cid:65) denotes the autocorrelation, and (cid:70) − indicates the inverse Fourier transform. This means that theautocorrelation function of the MPI-transmitted wave ψ can be obtained by the inverse Fourier transfor-mation of the far-field diffraction patterns | ˜ ψ | shown in Fig. 21(f). Since (cid:65) ( ψ ) exhibits peaks at positions47orresponding to the distances between similar objects, it will show peaks at displacements from the centreequivalent to the displacement vector between different pinholes (in addition to the central peak which isalways present). Furthermore, each peak will be characterized by the phase equal to the phase differencebetween the two contributing pinholes. Once the phase differences are obtained from the autocorrelationfunction, it is possible to perform the OAM-harmonic decomposition to obtain the OAM spectrum of theoriginal beam [229]. While such quantitative analysis is a major step forward, the MPI method suffers from afew limitations. First, the pinholes need to be small enough to consider the phase as approximately constantinside each pinhole. Second, they should be distant enough so that the peaks in the autocorrelation functiondo not overlap. Third, since we are sampling the wave at only n positions, the MPI cannot distinguishbetween the vortex mode of the orders (cid:96) and (cid:96) + n , as they yield identical phase differences in the n pinholepositions. The topological charge can also be measured by directly manipulating the wave phase, as in the conformal-map method [223]. For electrons, a much simpler approach was demonstrated using mode conversion withan astigmatic plate [204, 228]. When a quadratic phase plate is applied to a vortex mode by tuning thequadrupolar electron stigmators, the doughnut-like intensity profile of the beam is split into a number oflinearly-arranged intensity lobes as shown in Fig. 21(e), where the number of lobes is equal to | (cid:96) | + 1. Fur-thermore, the orientation of the pattern with respect to the phase plate (at the ± π/ (cid:96) )[234, 235].Due to the ubiquitous presence of electron stigmators, this method is particularly convenient to employwithin the TEM: it requires the manual adjustment of only one freely tunable parameter. Therefore, thistechnique can be an ideal way to confirm the vortex-beam order during the preparation of a more complexexperimental setup (which can be then readjusted to the astigmatism-free condition). Recently, a method has been proposed [236] to measure the OAM of a vortex electron, by exploiting itsmagnetic moment, Eqs. (2.24) and (2.25). Namely, when a vortex electron carrying magnetic moment passesthrough a hollow metallic cylinder, it induces eddy currents in the cylinder without changing the electronvortex state. Simulations performed for a 20 µ m long and 1 µ m wide platinum cylinder, and an electronwith a kinetic energy E = 100 keV, showed the eddy currents to be of the order of several picoamperes:an amount which would be measurable with current technology. The same simulations have estimated theenergy loss of the electron to be extremely small, ∼ − eV, supporting the nondestructive nature ofthis approach [236]. However, since the current pulse in the cylinder contains the same amount of energy,this extremely low value also means that it would be extremely difficult, if not impossible, to measure it inpractical experimental conditions. Beside producing and detecting electron vortices, one might wonder if we can also use spiral phase platesas an imaging filter by changing the topological charge of the transfer function of the microscope. This can,in principle, be obtained by mounting a spiral phase plate in the back focal plane of the objective lens ofthe microscope. In practice, this could be a true spiral phase plate or an alternative means to add a vortexphase factor to this plane, changing the conventional transfer function of the microscope as:exp[ i ∆Φ( k ⊥ ) ] → exp[ i ∆Φ( k ⊥ ) + i(cid:96)φ ] , (3.11)where ∆Φ( k ⊥ ) is the aberration function describing the phase behaviour of the objective lens, (cid:96) is thetopological charge of the spiral phase plate, and φ is the azimuthal angle in the k ⊥ -plane. Doing so resultsin a redistribution of the image intensities (i.e., probability densities): ρ ( r ⊥ ) → ρ (cid:96) ( r ⊥ ) . (3.12)48n real experimental conditions, spiral phase plates do not conserve the total current. Indeed, the centerof the plate, where the phase (and the thickness) is indeterminate, is usually blocked by putting a smallamount of an opaque material, which effectively removes the lowest frequency components from the filteredimage, see Fig. 17(a).Applying such a spiral phase filter results in an image where the intensity in each point is directlyproportional to the weight of the (cid:96) th OAM component measured with respect to the same point [237].Recording a series of such images ρ (cid:96) with (cid:96) covering a range of values provides a point-by-point OAMspectrum of the conventional image wave ψ . This can, in principle, allow the reconstruction of the wholecomplex wave field ψ (otherwise unknown, as we typically record only ρ = | ψ | ). However, in practicethis would require some form of a programmable phase plate which could easily change its topological chargeto allow a rapid recording of the image series. Such plates currently do not exist in TEM [208].Lacking such an (cid:96) -varying phase plate one, could resort to only measuring the image intensities with twoopposite topological charges: e.g., (cid:96) = ±
1. In this case, the difference and sum of the image intensities havebeen demonstrated to yield [237]: ρ + ρ − ∝ |∇ ⊥ ψ | , ρ − ρ − ∝ ( ∇ ⊥ × j ⊥ ) · ¯ z , (3.13)where j ⊥ is the probability current density for the original wave function ψ in the plane of interest (typi-cally the plane immediately after the sample, containing the so-called exit wave). Remarkably, the secondEq. (3.13) enables one to reconstruct the curl of the transverse current density, which is typically omittedin conventional transport of intensity reconstructions [85]. Furthermore, the first Eq. (3.13) includes thegradient of the phase of the wave function ψ , and, hence, can improving the contrast in highly transparent weak phase objects (where intensity is practically uniform).A practical realization of the vortex filtering setup remains difficult due to the unwanted scattering andcontamination issues with phase plates in the objective back focal plane. The most attractive option mightseem to be placing a magnetic needle (with the monopole-like field acting as spiral phase plate) in the backfocal plane, but this is hampered by the presence of a strong magnetic field in that plane which wouldlikely magnetize the needle in the unwanted up/down direction rather than along the needle. A workablealternative is the use of the reciprocity theorem, turning the electron trajectories upside down (time reversal)and creating a bright-field STEM setup with an incoming vortex probe. This setup works reasonably well,relying on different vortex-probe production schemes, but is highly dose-inefficient, as it requires a verylimited acceptance angle, discarding the majority of the electrons. When electron vortex beams interact with matter, the cylindrical symmetry is typically broken, and thebeam evolves in a highly nontrivial way through the material. In this process, the electron OAM changessignificantly for all but the thinnest sample and can even change its sign for certain thickness [238, 239, 240].This process can be effectively simulated using standard multislice numerical calculations solving the non-relativistic Schr¨odinger equation for a fast electron wavepacket travelling through the potential produced bythe atoms. Extending these simulations to include the incoming vortex states is trivial as long as the sourcecode of the software is open [241, 242, 243].This thickness dependence seems to undermine all hopes for using vortex electron beams to obtain uniqueinformation through scattering by materials except for very thin samples. However, even though the electronOAM varies upon propagation through a crystal, the phase singularity in the vortex centre is far more stable.Indeed, the net topological charge is conserved during the wave evolution, and the phase singularity has thetendency to follow a crystallographic channel of atoms [244]. This leads to the peculiar effect: a focusedvortex beam entering the sample along a column of atoms remains captured in this potential channel muchlonger than a beam with the same opening angle but without the vortex phase structure [244]. Thus,electron vortex beams could potentially be used to suppress the unwanted dechanneling effects that occuras conventional focused beams travel through the sample. Such dechanneling leads to the leaking of thebeam’s intensity to atomic columns neighbouring the initial one, which is detrimental for the localization49 igure 22: The scattering cross-section for electron Bessel beams with an energy of 300 keV, the vortex charges (cid:96) = 0 and (cid:96) = 1, and different values of the transverse momentum κ (expressed via the opening angle θ : sin θ = κ/k ), on a screenedCoulomb potential corresponding to an iron atom [245, 246]. The scattering amplitude is cylindrically symmetric (up to thevortex phase factor), and the plots show its dependence on the polar scattering angle θ . Positions of the maxima approximatelycorrespond to the opening angles θ of the incident Bessel beams. of, e.g., spectroscopic signals, such as EELS or energy dispersive X-ray (EDX) spectroscopy. So far, thisattractive proposition has not yet been demonstrated experimentally, likely due to the effect of source-sizebroadening (see Section 3.2.6).In order to gain further insights in the elastic scattering of a vortex probe on materials, it is usefulto have an analytical description of the elastic-scattering amplitude. In Refs. [245, 247], the Rutherfordscattering amplitude of Bessel beams by a screened Coulomb potential has been analytically derived. Thisshowed that the transverse momentum structure of an electron beam (with or without OAM) can have asignificant impact on elastic scattering, even by a simple Coulomb potential. Figure 22 shows the scatteringamplitudes for beams with (cid:96) = 0 and (cid:96) = 1, and different values of the transverse momentum κ . Thescattering amplitude of a Bessel beam can be seen as the convolution of a ring of tilted plane waves [seeFig. 5(a)] with the plane-wave scattering amplitude by the spherically-symmetric potential. This coherentsuperposition of the plane-wave Rutherford scattering peaks leads to a ring-like scattering intensity structure(depending on the polar scattering angle θ ) even for non-vortex beams ( (cid:96) = 0). For (cid:96) = 1, the intensityvanishes in the forward direction θ = 0, and this dip becomes more visible as the transverse momentum κ increases.To calculate the elastic scattering in the case of an arbitrary potential, produced by a configuration ofatoms in a material, one can adopt the so-called kinematic approximation . Assuming the scattered partof the wave to be much smaller than the incoming part, it considers the potential as a small perturbationand takes into account only single-scattering events. In this approximation, the scattering amplitude for anincoming wave ψ ( r ) scattered on a potential V ( r ) to a plane wave ψ (cid:48) ( r ) = exp( i k (cid:48) · r ) with the wave vector k (cid:48) can be written as [4]: f ( k (cid:48) ) = (cid:104) k (cid:48) | V | ψ (cid:105) ∝ (cid:90) d r e − i k (cid:48) · r V ( r ) ψ ( r ) . (3.14)This scattering amplitude can be presented as a convolution of the Fourier transform (FT) of the potentialand that of the incoming wave: f ( k (cid:48) ) = (cid:70) [ V · ψ ]( − k (cid:48) ) = [ ˜ V ∗ ˜ ψ ]( − k (cid:48) ) . (3.15)For an incoming plane wave , ψ ( r ) = exp( i k · r ), the FT ˜ ψ ( k ) is a delta-function and the diffraction pattern50ill be determined solely by the FT of the potential: ˜ V ( k ). For elastic scattering, the energy of thescattered electron (and, hence, the length of the wave vector) is conserved, k = k (cid:48) , and the diffractionpattern is determined in k -space by the intersection of ˜ V ( k ) and a sphere with radius k , called the Ewaldsphere [83, 84]. This means that the momentum transferred in the scattering process, q = k − k (cid:48) , mustsatisfy ˜ V ( q ) (cid:54) = 0, which is the momentum matching condition. A schematic example for a potential V ( r )periodic along the propagation z -axis is given in Fig. 23(a). The FT of the potential, ˜ V ( k ), corresponds todiscrete planes perpendicular to the z -axis, so that the resulting diffraction pattern consists of rings whichcoincide with the zeroth and higher-order Laue zones in conventional electron diffraction [83, 84].The diffraction pattern can be significantly altered when modifying the probe (incoming wave) ψ ( r ).For instance, in convergent-beam electron diffraction (CBED) the scattering amplitude is convolved with theFT of the probe, ˜ ψ ( k ) ∝ Θ( κ max − k ⊥ ), Eq. (3.1), i.e., a disc with uniform phase and amplitude [83, 84],Fig. 23(b). In turn, the FT of a converging vortex electron beam is a disc with the azimuth-dependentphase: ˜ ψ ( k ⊥ ) ∝ exp( i(cid:96)φ ) Θ( κ max − k ⊥ ) [248], see Fig. 23(c).To show the effect of the vortex-probe diffraction, now consider a 3D crystal model. The FT of thepotential, ˜ V ( k ), can now be modelled by a lattice of delta-functions (not explicitly shown in Fig. 23) givingrise to Bragg spots in the plane-wave diffraction pattern. Since the symmetry of the crystal potential isconstrainted by Friedel’s law, requiring ˜ V ( q ) = ˜ V ∗ ( − q ), the zero-order Laue zone must be centrosymmetric [4]. In other words, scatterings with q and − q momentum transfers always have equal probabilities | f ( k (cid:48) ) | ,Fig. 23(a), which yields centrosymmetric diffraction patterns (even for non-centrosymmetric, e.g., chiral,cystals). In the case of CBED, the Bragg spots become discs because of the convolution with the FTof the probe, ˜ ψ ( k ). However, Friedel’s law is still satisfied, [ ˜ V ∗ ˜ ψ ]( q ) = [ ˜ V ∗ ˜ ψ ] ∗ ( − q ), and opposite- q scatterings are still equiprobable, Fig. 23(b), and the resulting diffraction pattern is always centrosymmetric.In contrast to this, the vortex probe circumvents Friedel’s law, and non-centrosymmetric crystals produce non-centrosymmetric diffraction patterns [249, 250]. This is because for the vortex wave function ˜ ψ ( k ),generically [ ˜ V ∗ ˜ ψ ]( q ) (cid:54) = [ ˜ V ∗ ˜ ψ ] ∗ ( − q ), and opposite- q scatterings can have different probabilities, Fig. 23(c).This opens up the vortex-beam probing of complex crystal symmetries, which are hidden for standardtechniques. An important application of the elastic scattering of vortex beams on crystals is the probing of the chirality of crystalline materials. Chirality is the property of any 3D object that cannot be superimposedwith its mirror image [251]. A chiral object and its mirror image are said to be the two enantiomers ofthe same structure. Chirality is ubiquitous in nature, and it underpins many physical phenomena, such as,e.g., optical activity. The two enantiomers of the same compound often have different physical, chemical orbiological properties. A chiral crystal is characterized by a space group possessing only proper symmetryelements [rotations and screw-axes] and no improper symmetry elements [mirror(-glide) planes and (roto-)inversion centers]. Mirroring the crystal with respect to any plane results in a fundamentally differentcrystal.Determining the chirality of a sample in a TEM is challenging because high-resolution imaging techniquesrecord only a 2D projection of the atomic arrangement. If the sample is mirrored with respect to the ( x, y )plane, the crystal is replaced with its enantiomer, while the projection remains identical. Moreover, as wediscussed above, the exact space group (specifically, chirality) of the crystal cannot be determined using theconventional (plane-wave or CBED) electron diffraction. In recent works [249, 248] electron-vortex probingof crystal chirality was suggested and demonstrated experimentally. A kinematical treatment of an electronvortex beam of order (cid:96) focused on a Q -fold screw axis of the chiral crystal, shows that the N th-order Lauezone is centrosymmetric if and only if [249] (cid:96) − χN = nQ. (3.16)Here, χ = ± n is an integer number. In Ref. [249] thismethod was demonstrated experimentally by focusing electron vortex beam on the three-fold screw axis ofa Mn Sb O crystal. 51 igure 23: Visual representation of Eq. (3.15) for elastic electron scattering k → k (cid:48) , Eqs. (3.14) and (3.15), in a periodic crystalpotential V ( r ). (a) In the plane-wave regime, the scattering is determined by the intersection of the Fourier transform (FT)of the potential, ˜ V ( k ) (here shown as discrete planes for a z -periodic potential), with the Ewald sphere of radius k (shownin gray) [83, 84]. The corresponding zero- and first-order Laue zones are indicated by the small blue circle near the pole ofthe sphere and by the red ring, respectively. According to Friedel’s law, ˜ V ( q ) = ˜ V ∗ ( − q ), and scattering events with opposite q = k − k (cid:48) vectors are equally probable and the diffraction pattern is centrosymmetric. (b,c) For non-plane-wave probes, thediffraction condition is determined by the convolution of ˜ V ( k ) with the FT of the incoming wave, ˜ ψ ( k ). (b) In convergentbeam electron diffraction (CBED) ˜ ψ ( k ) is a disc with a constant phase and Friedel’s law remains valid after the convolution.(c) For a vortex probe, ˜ ψ ( k ) is a disc with an azimuthally dependent phase. Convolving it with ˜ V ( k ) breaks Friedel’s lawand generically results in different probabilities for the q and − q event. In particular, the corresponding non-centrosymmetricdiffraction patterns allow one to probe chirality of crystals [249, 248]. Elastic electron scattering on magnetic fields forms another interesting proposal for experiments. Wehave already seen that a magnetic monopole converts a plane wave into a vortex beam, and actual magnetic52elds around (nano) materials can be interpreted as emanating from “local magnetic monopoles”, inside andon the surface of the material with a density ρ m ∝ ∇ · m , where m is the local magnetization of the material.Even though the actual magnetic field is strictly divergence-free, the scattered electron wave neverthelessexhibits a current density which curls near the “magnetic poles”, in a way similar to how an approximatemagnetic monopole can be used to generate electron vortex beams.This effect can be exploited to directly visualize the “monopole density” in a magnetic material in theTEM by obtaining the curl of the in-plane probability current, ∇ ⊥ × j ⊥ , as detailed in Section 3.4 (i.e., usingthe difference of two TEM images obtained with opposite- (cid:96) spiral phase plates). As mentioned above, thissetup is currently difficult to obtain, but the experiment [252] suggests that the reciprocal setup, with anincoming vortex beam directed on the sample and a bright-field detector accepting electrons scattered alongthe optical axis, behaves as predicted. This effect is highly attractive as it localizes the measured signal atthe virtual monopoles. This leads to a potentially superior signal-to-noise ratio in the measurement as thefields can be derived at a later stage by convolution with the field kernel of a single monopole. Still, furtherexperiments are needed to explore this effect, but this prediction highlights once more why an image filterbased on a tunable spiral phase plate would be highly desirable. The fast electrons in a TEM can also interact with the sample inelastically , leaving both the sample andthe fast electron with energies different from the initial ones. This phenomenon is actively exploited in thetechnique of electron energy loss spectroscopy (EELS), where the kinetic energy distribution of an ensembleof fast electrons is recorded after passing through the sample [253]. Such a spectrum contains informationabout all excitation events in the sample and their respective probability, revealing information such as:which elements are in the sample, what is their concentration, and how are they bound to each other. Allthis information can be combined with the atomic resolution provided by a focused STEM probe, whichexplains the widespread use of this technique.
A simple model, that can illuminate a number of important features of inelastic vortex-electron scatter-ing, is the Bessel-beam scattering on a single-electron atom in the Born approximation. For plane waves,this scenario provides a building block of electron-matter interaction, which is used in EELS. The resultspresented below can be deduced analytically, as shown in [254].We consider the same geometry as for the elastic scattering: a centered z -propagating Bessel beamscatters on a single-electron atom. Using the same mathematical formalism as developed for elastic Bessel-beam scattering [245], one can determine analytical expressions for all the partial scattering amplitudes forthis system. The Bessel beam represents a coherent superposition of plane waves with conically distributedwave vectors k , see Fig. 5(a). Therefore, its scattering into a plane wave with the fixed wave vector k (cid:48) is determined as an integral of the plane-wave scattering amplitude f ( k , k (cid:48) ) over incoming k vectors ormomentum-transfer vectors q = k − k (cid:48) , Fig. 24 (see Section 4.4 below).The resulting scattering amplitudes for various atomic transitions, e.g. 1s → ± , can be written down inan analytic (athough rather cumbersome) form [254, 255]. In particular, these vortex-scattering amplitudesdescribe properties of the angular-momentum exchange with the atom, Fig. 25. Namely, when the scatteringof a vortex electron with topological charge (cid:96) is accompanied by the change of the magnetic quantum numberof the atom (with respect to the z -axis), ∆ m , the scattered wave will correspond to the vortex state with (cid:96) (cid:48) = (cid:96) − ∆ m . The cases (cid:96) = ∆ m and (cid:96) (cid:48) = − ∆ m correspond to the “vortex → plane wave” and “planewave → vortex” scatterings, as shown in Fig. 25. Importantly, the scattering amplitudes for these reciprocalcases are exactly equal to each other, assuming the same parameters in the incoming and outgoing vortexstates. Such “inelastic OAM reciprocity” enables one to chose between the two equivalent probing schemesinvolving vortex states of either incident or scattered electrons.53 igure 24: Schematic of the electron Bessel-beam scattering by an atom into a plane wave. The incident Bessel beam consistsof multiple plane waves with the k -vectors conically-distributed with the polar (convergence) angle θ (shown semitransparenthere). These are scattered into a single plane wave with the wave vector k (cid:48) (polar scattering angle θ ). The correspondingmomentum-transfer vectors are q = k − k (cid:48) . In the dipole approximation, the plane-wave scattering of a fast electron by an isotropic point scatterercan be characterized by the following scattering amplitude [84, 253]: f ( k , k (cid:48) ) ∝ (cid:104) i | q · r | f (cid:105) , (3.17)where q = k − k (cid:48) is the momentum-transfer vector, whereas | i (cid:105) and | f (cid:105) are the initial and final quantumstates of the scatterer. In the q (cid:28) k approximation, the selection rules for the angular-momentum ( l ) andmagnetic ( m ) quantum numbers of the atom read [253]:∆ l = ± , ∆ m = − , , +1 . (3.18)For the 1s →
2p atomic transition (corresponding to the K absorption edge), three separate transitions top x , p y , and p z are possible, involving a momentum transfer q in the corresponding x , y , and z directions.Using projective measurements, one can selectively measure excitations to these states attached to Cartesiancoordinates. This method involves the so-called linear dichroism , (also known as “anisotropic scattering crosssection”) which gives access to the population of the different 2p states in anisotropic materials. A typicalexample can be found in the studies of graphite and graphene, where the in-plane σ ∗ bonds can be selectivelyprobed off-axis, while the excitation to the π ∗ states leaves fast electrons close to the z -axis of the incomingelectrons [256, 257].The above linear dichroism works well for anisotropic materials, but fails to identify the angular-momentum properties of the 2p orbitals. The linear combinations p ± = p x ± i p y describe the orbitalstates with opposite OAM. Conventional EELS, with an incoming plane wave, cannot distinguish betweenexcitations to the p + and p − states, because both scatterings lead to the same ring-like probability distri-bution in the far field. Having access to probabilities of these angular-momentum excitations would behighly desirable as it could provide an important information about magnetic properties of the sample.One way to overcome this limitation is to allow two coherent plane waves to interact with the sample One might think that the two states could be distinguished in energy. But even if the energy splitting is high enough tobe resolved in the EELS spectrum, still one cannot know which energy corresponds to which transition. igure 25: Examples of the inelastic electron scattering accompanied by the angular-momentum exchange with the atom[254, 255]. The OAM difference between the incident and scattered electron waves, (cid:96) − (cid:96) (cid:48) , corresponds to the change in themagnetic quantum number of the atom, ∆ m . The examples shown here correspond to the reciprocal cases of “vortex → planewave” and “plane wave → vortex” scatterings with θ = θ and equal probability amplitudes. simultaneously, so that constructive or destructive interference in the diffraction plane reveals the differencebetween exciting to the p + or p − states [258, 259, 260]. This technique is called energy loss magnetic chiraldichroism (EMCD). It allows one to obtain magnetic information similar to that given by the X-ray magneticchiral dichroism [261], but using EELS and now at a nanometer-scale resolution [262, 107]. The drawbackof this technique is the difficulty in setting up the two plane waves, which need to be operated by a preciselycontrolled diffraction condition of the crystal lattice which also contains atoms. This restricts the use ofthe EMCD to crystalline materials with precisely known thickness and where the correct orientation can bepredicted with precision. On the other hand, by using the specimen crystal itself as a beam splitter for thetwo (or more) coherent plane waves, we have locally and temporally extremely stable site selectivity by theintrinsic phase lock in the periodic lattice [263].The atomic transitions 1s → ± are characterized by changes in the angular momentum of the atomby ∆ m = ±
1. In this case, the plane-wave scattering amplitude (3.17) takes the form of [254]: f ± ( k , k (cid:48) ) ∝ f ( q ⊥ ) exp( − i ∆ m φ q ) , (3.19)where q ⊥ = | q ⊥ | , and φ q is the azimuthal angle of the q vector. Equation (3.19) is equivalent to thetransmission function of a spiral phase plate, which means that the outgoing electron is in the vortex statewith (cid:96) (cid:48) = − ∆ m , Fig. 25(b). The total scattering probability | f | is independent of the m -dependentphase term and therefore does not allow distinguishing between the excitation of the p + and p − states. Butdetecting the OAM phase properties of the outgoing electrons will provide the desired magnetic information.In particular, if the densities of unoccupied states in the p + or p − orbitals are different, there should be apreferential scattering channel to the corresponding (cid:96) (cid:48) = ∓ Thus, retrieving the desired magnetic information about the sample requires an efficient far-field OAMsorter for transmitted electrons. Such an ideal sorter is not currently available (as discussed in Section 3.3),although promising setups have been proposed [226]. The first hints of this effect have been demonstratedon a ferromagnetic iron sample, using a forked holographic grating as an OAM filter before the electron In practice, the 2p →
3d transitions are often investigated for the L , edges of transition metals. This complicates theabove description with the spin-orbit splitting, but essentially the same imbalance remains (albeit less strong than in the caseconsidered above). For a detailed treatment, see [264]. focusedelectron vortex probe with topological charge (cid:96) = ± ± states, Fig. 25(a). In this case, considering electron scattering in the forward direction(i.e., into a z -directed plane wave), the approximate EELS selection rules become [254]:∆ l = ± , ∆ m = (cid:96). (3.20)These selection rules signify vortex-dependent magnetic dichroism for electrons. This is in sharp contrast tooptics, where probing magnetic dichroism and chirality involve only polarization (spin) degrees of freedomof light and are mostly insensitive to vortices [101, 102, 103, 104]. Note that the limited validity of the dipoleapproximation for electron scattering leads to a limited validity of the above selection rules [266, 267, 268].The selection rules (3.20) should in principle give access to the magnetic state of an atom with a resolu-tion defined by the probe size, even down to the single atomic column . The setup was shown to be viable ontheoretical grounds, with an increasing level of details included [264, 265, 269]. At the same time, numeri-cal simulations revealed that elastic electron scattering on the crystal lattice (Bragg diffraction, channelingeffects) is rather detrimental to the vortex character of the incoming probe (it breaks the cylindrical sym-metry, making L z a bad quantum number). This sets an upper limit for the thickness of the crystal used.Moreover, the magnitude of the magnetic signal in the EELS spectrum is expected to be at best ∼
10% ofthe EELS signal in the L , edge of the transition metal. This makes the experiments extremely challengingin view of the low signal-to-noise ratios often encountered in atomic-resolution EELS. Note also that thismethod has the desired sensitivity to the magnetic properties only when the probe is positioned precisely onthe atom columns and is of atomic size. Deviations from these parameters will further reduce the magneticsensitivity and can even switch the sign of the effect when the probe is displaced from the atom column oris too large [269].Despite the tremendous experimental efforts that were put into realizing the OAM-magnetic probingexperiment by several groups [98, 269, 270, 271, 272], the results are still inconclusive. Nonetheless, theinelastic scattering of shaped beams has been succesfully applied to the study of surface plasmons [205], andthis success in a similar case supports the validity of the description outlined above. However, as comparedto the case of surface plasmons, experiments on inelastic scattering by atoms are heavily limited by twofactors. First, the source-size broadening is much more important due to the much smaller scale of thescattering objects (atomic scale rather than 10s of nm). Second, the scattering cross-sections are severalorders of magnitude smaller, resulting in a weaker signal. Limiting the incoherent source-size broadeningwithout losing too much intensity is currently the main challenge (see Section 3.2.6). Making significantprogress on this issue requires the development of brighter electron sources [217].Some groups have proposed and demonstrated other symmetry-breaking mechanisms in the electronprobe to reveal the atomic-scale magnetic signal. In particular, Refs. [273, 271] deliberately added four-foldastigmatism through the probe aberration corrector, while the works [105, 107] dealt with off-axis inelasticscattering as a function of the probe displacement with respect to the atomic column. These experimentsshowed an atomically-resolved magnetic signal albeit at very low signal-to-noise ratios.For the actual application of the above techniques, it is crucial to make a significant improvement of thesignal-to-noise ratio. The most likely approach to this challenge is to deal with the “plane wave → vortex”scattering with the subsequent effective OAM sorting in the far field, prior to the EELS spectrometer. Electron vortex beams have also been considered as a possible tool in the spectroscopic investigation oflocalised surface plasmon resonances (SPRs). SPRs are collective excitation of electrons (in metals) andelectromagnetic fields, which appear due to the confinement of the conduction electron gas in metals andsmall nanoparticles [274]. Due to their peculiar properties, such as strong localized electromagnetic fields andhigh sensitivity to nanometer-scale environmental changes, they offer a unique platform for sub-wavelength56 igure 26: OAM-resolved inelastic scattering cross-section for an electron beam impinging on a chiral cluster of plasmonicnanoparticles. (a) Schematic representation of the proposed experiment. An electron beam impinges on a nanoparticle cluster,and then the inelastically-scattered electron wave is filtered with an OAM analyzer. (b) The energy loss spectra for each OAMcomponent, with differences clearly visible for the ∆ (cid:96) = ± (cid:96) = ± optics, nano-photonics, and optoelectronic devices [274, 275, 276]. EELS plays an important role in thisresearch because it allows the intensity of the optical electric field produced by SPRs to be mapped [277]:Γ( r ⊥ , ω ) ∝ (cid:90) | E z ( r ⊥ , z, ω ) | dz, (3.21)where Γ( r ⊥ , ω ) is the probability of the electron beam focused on the position r ⊥ to lose an energy (cid:126) ω ,whereas E z is the z -component of the electric SPR field, and ω is the SPR frequency. While this is a powerfultool to investigate SPRs, the loss of information caused by recording only the electric-field intensity does notallow one to obtain the charge distribution or the direction of the electric field (see, e.g., [278]). Furthermore,important phenomena, such as circular dichroism of chiral plasmonic structures cannot be investigated atthe nanoscale using electron plane waves. Electron vortex beams have been proposed as a potential tool toaddress these limitations.In Ref. [279] the OAM exchange in the inelastic interaction between fast electron beams and chiral assemblies of metal nanoparticles was studied. Calculations of the OAM- and energy-resolved EELS cross-section showed that there is a significant OAM transfer between the electron beam and the SPRs of thenanoassembly, Fig. 26 [279]. Performing the OAM-resolved EELS on such structures would reveal strong(up to ∼ (cid:96) = ±
1, ∆ (cid:96) = ±
2, etc. This effect strongly depends on the axis with respect to whichthe OAM is measured. This can be used for the local probing of the chirality of the plasmon resonanceson the nanometer scale, including biomolecules [279]. Preliminary experiments have provided encouragingresults, although the interpretation requires further clarification [280].We again note that this is in contrast to optical probing of chiral nanoparticles and molecules [251].Circular dichroism enables optical probing of chiral objects using only polarization (spin or helicity) degreesof freedom of light [281, 282, 283, 284]. At the same time, optical vortices, are insensitive to chiral propertiesof matter and do not produce dichroism [101, 103, 102].The interplay between the (cid:96) -fold azimuthal symmetry of electron vortex beams and the discrete ro-tational symmetry of SPR modes was theoretically investigated in [285]. The interaction between thesetwo symmetries could allow selective probing of desired SPR modes. For instance, the (cid:96) = 1 vortex beam57 igure 27: Angular-monentum transfer from electron vortex beams to nanoparticles [286]. (a) Schematic of the experiment.Vortex beams with different topological charges (cid:96) , generated with a holographic mask, are focused on the sample, so thatonly one of the beams impinges on the selected gold nanoparticle. (b) Experimental images displaying clockwise and counter-clockwise rotations of a nanoparticle (indicated by arrows) in the (cid:96) = − (cid:96) = 1 vortex beams. preferentially excites the dipole resonance, while the (cid:96) = 2 vortex couples more strongly to the quadrupole one. More generally, it was recently shown that, in the inelastic interaction between an electron beam anda plasmonic nanostructure, a signature of the scalar electromagnetic potential potential of SPR, V SPR , isimprinted in the phase of the scattered electron wave [205]. By accepting only electrons scattered along thepropagation z -axis (i.e., the non-vortex mode with (cid:96) (cid:48) = 0), it becomes possible to detect only resonanceswhose potential possesses the same symmetry as the impinging electron beam. This was also tested experi-mentally using a two-lobed beam, reminiscent of an HG Hermite–Gaussian mode. In agreement with thetheory and numerical simulations, the experiments [205] showed that such a beam can selectively detect thedipole excitation mode while suppressing the higher-order excitations.
The OAM exchange between electron waves and the sample in inelastic scattering can also be revealedby observing the sample state. Indeed, in the extreme case when the electron is absorbed by the sample,the total angular momentum of the sample is increased by the OAM carried by the electron. This was usedin optics for the detection of mechanical properties of optical OAM beams, which produced the rotation ofabsorptive particles [287, 32, 221, 288, 65]. Similar angular-momentum transfer from electron vortex beamsto small particles can become noticable when the current of the electron beam is high, while the moment ofinertia of the particle is low. In the absence of friction, the particle would start to rotate and the rotationalvelocity would keep increasing as more and more electrons are absorbed. In reality, any particle is in contactwith a substrate (typically a support film), and significant friction exists between the support film andthe particle, causing at best a low rotational speed that is dominated by a balance between the frictionand the flux of the incoming OAM quanta. Recent experiments have demonstrated this effect [286, 289],as shown in Fig. 27. A quantitative interpretation is complicated due to many factors: unknown frictionbetween particle and support; strong dependence of the effect on the orientation of crystalline particles;simultaneous effects of phonon, plasmon, core-loss and elastic scatterings; and rather unpredictable effect ofcarbon contamination and beam damage effects occurring at the high electron currents [286, 290]. Therefore,it is very desirable to preform such experiments on particles in a liquid medium in an environmental sampleholder or using diamagnetically-levitated particles. This might offer direct mechanical measurements ofthe angular momentum carried by electron vortex states. Charging due to secondary-electron generation will likely make this difficult even though we have demonstrated diamagneticsample levitation inside an SEM chamber. . High-energy processes with vortex electrons High-energy processes, such as collisions of electrons, require that the electrons be treated in a fullyrelativistic manner. We start this section with a reminder of this formalism and with a review of the exactvortex solutions of the free Dirac equation, which were constructed in [147, 108, 291]. These expressionshave different forms but are equivalent to each other. Depending on the specific problem, one of these formsmay be more appropriate than the others. Below we will describe these solutions in detail, with the aim toprovide a convenient reference for future calculations of vortex electron scattering processes. Throughoutthis section, we will use the relativistic units (cid:126) = c = 1.Before going into details, we mention that exact solutions of the Dirac equation in cylindrical coordinates,either free or in the presence of external fields or potentials, have been known since decades. In particular,the first edition of the monograph [292] published back in 1982 [293] already contained free solutions ofthe Dirac equation exhibiting vortex-like azimuthal dependences. Various formal aspects of such solutions,with different boundary conditions, were studied later by several authors, see, e.g., [294, 295, 296, 297, 298].However it was only in [147] and later publications that the exact vortex solutions of the Dirac equationwere written in a way convenient both for theoretical exploration of its angular-momentum “anatomy” andfor usage in scattering processes. In this section we expose these recent developments.We start with the scalar Bessel-beam solutions (2.5) in the Fourier-integral form and with the restoredexp( − iEt ) factor: ψ k z κ(cid:96) ( r , t ) = e − iEt (cid:90) d k ⊥ (2 π ) ˜ ψ κ(cid:96) ( k ⊥ ) e i k · r , (4.1)where the Fourier amplitude corresponds to the conical distribution of wave vectors, Eq. (2.7) and Fig. 5(a):˜ ψ κ(cid:96) ( k ⊥ ) = ( − i ) (cid:96) δ ( k ⊥ − κ ) κ e i(cid:96)φ . (4.2)Here ( k ⊥ , φ, k z ) are the cylidnrical coordinates in k -space, and the subscripts explicitly indicate all continuous-and dicrete-spectrum parameters of the solutions. The wave functions (4.1) and (4.2) correspond to acomplete orthogonal set of solutions of the scalar wave equation with a definite energy E = (cid:112) k + m e ,longitudinal momentum p z = k z , and z -component of the OAM L z = (cid:96) . These solutions are normalized as (cid:90) d r ψ ∗ k z κ(cid:96) ( r , t ) ψ k (cid:48) z κ (cid:48) (cid:96) (cid:48) ( r , t ) = 1 κ δ ( κ − κ (cid:48) ) δ ( k z − k (cid:48) z ) δ (cid:96)(cid:96) (cid:48) , (4.3)where δ ab is the Kronecker delta.A relativistic electron with spin is described by the multi-component (bi-spinor) wave function Ψ( r , t ).In this case, the Bessel states can be introduced as a straigthforward generalization of scalar Bessel modes:Ψ k z κ(cid:96)s ( r , t ) = e − iEt (cid:90) d k ⊥ (2 π ) ˜ ψ κ(cid:96) ( k ⊥ ) u k s e i k · r . (4.4)This equation differs from Eq. (4.1) only by the presence of the bispinor u k s which corresponds to theplane-wave solution with momentum k and in the spin state which we generically denote by s . Note thatEq. (4.4) includes the assumption that the Fourier amplitude ˜ ψ κ(cid:96) is the same as in (4.2) and, in particular,that it does not depend on the spin state of the electron. There are two crucial aspects in which Eq. (4.4) differs from the scalar case. First, the quantity (cid:96) can nolonger be interpreted as the z -component of the electron OAM. In fact, the solution (4.4) is not an eigenstate This assumption is not mandatory. One can build vortex electron solutions as superpositions of plane-wave electrons withcontinuously varying k ⊥ accompanied with the varying spin state, i.e., with inhomogeneous polarization [90]. We will notdiscuss such solutions, apart from the mere possibility of introducing a spin- and azimuthal-angle-dependent phase factor, seethe discussion after Eq. (4.11).
59f the OAM operator ˆ L z because even the individual plane-wave components in the superposition do notpossess a well-defined OAM [see Eq. (2.57)]. This is a manifestation of the fact that the OAM and spinoperators, ˆ L z and ˆ S z , do not commute with the Dirac Hamiltonian [148, 168], which leads to the intrinsicspin-orbit interaction [147] (see Section 2.8). The solution (4.4) has a well-defined z -component of the total angular momentum, though, ˆ J z = ˆ L z + ˆ S z , and we will describe below its relation with the parameter (cid:96) .Second, we have the freedom to choose the basis for describing the polarization state s . One possibilityexploited in [147] is to define polarization states with respect to the same z -axis . Another possibility is touse the helicity basis [108, 291], which is especially convenient for scattering processes involving high-energyelectrons.Let us first take the former option and consider the plane-wave bispinor u k s , whose polarization state isdefined with respect to the fixed z -axis. The bispinor has the following form [148]: u k s = (cid:18) (cid:112) E + W s (cid:112) E − ( ˆ σ · ¯ k ) W s (cid:19) , (4.5)where ˆ σ is the vector of 2 × s = ˆ σ / E ± = E ± m e , ¯ k = k /k is the unit vector in the k -direction. The basis spinors W s areeigenvectors of the non-relativistic spin ˆ s z with the eigenvalues s = ± / W +1 / = (cid:18) (cid:19) , W − / = (cid:18) (cid:19) . (4.6)It is instructive to express ˆ σ · ¯ k using spherical coordinates ( k, φ, θ ) in k -space:ˆ σ · ¯ k = ˆ σ + sin θe − iφ + ˆ σ − sin θe iφ + ˆ σ z cos θ, (4.7)where σ ± = ( σ x ± iσ y ) / k z κ(cid:96), + ( r , t ) = e i ( k z z − Et ) π (cid:112) E + (cid:112) E − cos θ e i(cid:96)ϕ J (cid:96) ( κr ) + i θ (cid:112) E − e i ( (cid:96) +1) ϕ J (cid:96) +1 ( κr ⊥ ) , Ψ k z κ(cid:96), − ( r , t ) = e i ( k z z − Et ) π (cid:112) E + − (cid:112) E − cos θ e i(cid:96)ϕ J (cid:96) ( κr ) − i θ (cid:112) E − e i ( (cid:96) − ϕ J (cid:96) − ( κr ) , (4.8)where r = | r ⊥ | , as before, and sin θ = κ/k fixes the polar angle of the conical Bessel plane-wave spectrum,Fig. 5(a). The solutions (4.8) represent Bessel beams for Dirac electrons, i.e., the electron counterparts ofoptical vector Bessel beams [299, 173]. The probability density and current distributions for these
Dirac–Bessel beams with different (cid:96) and s are shown in Fig. 15. The presence of two terms with distinct (cid:96) forany spin state s = ± / L z = − i∂/∂ϕ . These are not eigenmodes of the z -component of the full relativistic (4 ×
4) SAMoperator (2.54) ˆ S = diag( ˆ σ , ˆ σ ) either. Instead, vector Bessel beams (4.8) are eigenmodes of the total angularmomentum operator ˆ J z = ˆ L z + ˆ S z with the eigenvalues J z = (cid:96) + s . As described in Section 2.8, one observesthe intrinsic spin-orbit interaction (SOI) in the free Bessel electron beam, whose strength is determined bythe dimensionless parameter √ Λ = sin θ (cid:112) E − /E , Eq. (2.58). This interaction becomes weak, Λ (cid:28)
1, bothin the non-relativistic limit E − (cid:28) E and in the paraxial approximation θ (cid:28)
1; the vortex quantum number (cid:96) corresponds to the approximate OAM L z (cid:39) (cid:96) in either of these limits. For non-paraxial relativistic electronvortex beams, the SOI is significant and leads to spin-dependent probability densities (see Fig. 15) [147].We now explore the second option, choosing the helicity basis for the electron spin states, which is more60onvenient for high-energy electrons. Consider again the plane-wave spinor (4.5) but now use, instead of W s , the eigenstates of the helicity operator ˆ χ = ˆ s · ¯ k = ˆ σ · ¯ k / χ W ( χ ) = χ W ( χ ) , χ = ± / . (4.9)These spinors can be explicitly written as W (+1 / ( k ) = (cid:18) cos θ sin θ e iφ (cid:19) , W ( − / ( k ) = (cid:18) − sin θ e − iφ cos θ (cid:19) . (4.10)Then, we can replace ˆ σ · ¯ k → χ inside the bispinor (4.5) to arrive at u k χ = (cid:18) (cid:112) E + W ( χ ) χ (cid:112) E − W ( χ ) (cid:19) . (4.11)The φ -dependence is still present inside the spinors W ( χ ) , and it is different for its upper and lower compo-nents, displaying once again that the Dirac electron is not in a fixed-OAM state. Next, we obtain electronBessel beams with a given helicity by substituting the bi-spinors (4.11) into Eq. (4.4) with the scalar Fouriercomponents ˜ ψ κ(cid:96) , Eq. (4.2) [108]. Akin to the Bessel beams with a well-defined non-relativistic spin com-ponent s z = s , Eqs. (4.8), these solutions are eigenmodes of the total angular momentum operator ˆ J z with the eigenvalues J z = (cid:96) + χ , involving the helicity χ instead of s . Thus, alternatively, one can usethe helicity-dependent Fourier components ˜ ψ κ ( J z − χ ) , where J z denotes the total angular momentum of theBessel electron irrespective of its helicity χ [291]. Note that in the paraxial approximation θ (cid:28)
1, the SOIeffects become negligible and the s -based and χ -based solutions approximately coincide with each other.Finally, the link between the fixed-spin basis and helicity basis can be established with the aid of Wigner’s D -function [2]: W ( χ ) ( k ) = (cid:88) s = ± / D / sχ ( φ, θ, − φ ) W s , (4.12)where W ( χ ) and W s are given by (4.10) and (4.6), respectively. The explicit form of the Wigner D -functionfor the spin-1 / D / sχ ( φ, θ, − φ ) = e − isφ d / sχ ( θ ) e iχφ , d / sχ ( θ ) = δ sχ cos θ − s δ s, − χ sin θ . (4.13)With this notation, the bispinor (4.11) takes yet another form: u k χ = (cid:88) s = ± / e i ( χ − s ) φ d / sχ ( θ ) U s ( E, χ ) , U s ( E, χ ) = (cid:18) (cid:112) E + W s χ (cid:112) E − W s (cid:19) . (4.14)Unlike expressions (4.5) and (4.11), the bispinor U s itself is now free from the φ -dependence, which reap-pears only in the Wigner D -function. This representation is convenient for calculating high-energy electronscattering processes, as demonstrated in [291]. Passing to the coordinate representation (4.4), we arrive at:Ψ k z κ(cid:96)χ ( r , t ) = e i ( k z z − Et ) π (cid:88) s = ± / i ( χ − s ) e i ( (cid:96) + χ − s ) ϕ d / sχ ( θ ) J (cid:96) + χ − s ( κr ) U s ( E, χ ) , (4.15)which is the helicity-basis counterpart of Eqs. (4.8). The exact vortex solutions of the Dirac equation can also be found for an electron moving in the fieldof an electromagnetic (EM) wave . These solutions can be named
Volkov–Bessel beams as they extend the61ell-known Volkov solutions [300, 148] to the Bessel vortex electron. These solutions were constructed andinvestigated first in [108] and later in [109, 301], and they offer insights into modifications of the vortexelectron properties in a strong laser field. In this subsection, we denote the electron momentum by p ,reserving the letter k for the wave vector of the electromagnetic wave.The Volkov–Bessel solutions are constructed in the same way as the usual Bessel states. One uses thebasis of plane-wave Volkov solutions with four-momentum p µ to combine them as in (4.4) with the sameFourier amplitudes ˜ ψ κ(cid:96) ( p ⊥ ). The Dirac equation in the field of an EM wave reads [148]:[ γ µ (ˆ p µ − eA µ ) − m e ]Ψ = 0 , (4.16)where γ µ are the Dirac matrices, ˆ p µ = ( i∂ t , − i ∇ ) is the electron canonical four-momentum operator, and A µ is the electromagnetic four-potential. The EM wave is described by the wave four-vector (cid:107) µ = ( ω, (cid:107) ),satisfying (cid:107) µ (cid:107) µ = 0, and the four-potential A µ , in the traditionally-chosen Lorentz gauge, satisfies (cid:107) µ A µ = 0.The EM plane wave depends on the coordinates via the single phase variable ξ ≡ (cid:107) µ r µ = ω t − (cid:107) · r . TheDirac equation (4.16) can be recast in the form of a second-order equation with a simpler spinorial structure: (cid:2) ˆ (cid:112) µ ˆ (cid:112) µ − m e − ieF µν σ µν (cid:3) Ψ = 0 . (4.17)Here, ˆ (cid:112) µ = ˆ p µ − eA µ is the kinetic electron four-momentum, F µν = ∂ µ A ν − ∂ ν A µ is the EM field tensor,and σ µν = ( γ µ γ ν − γ ν γ µ ) / p µ = ( E, p ) satisfying p µ p µ = m e is [300, 148]: Ψ p ( r ) ∝ (cid:20) e p µ k µ ) ( γ µ k µ )( γ µ A µ ) (cid:21) u p,s exp( i (cid:83) ) , (4.18)where u p,s is the plane-wave Dirac bi-spinor (with any spin state “ s ”), and (cid:83) is the action: (cid:83) = − p µ r µ − e ( p µ (cid:107) µ ) (cid:90) dξ ( p µ A µ ) + e p µ (cid:107) µ ) (cid:90) dξ ( A µ A µ ) . (4.19)In order to construct the Volkov–Bessel solution, one must specify the relative kinematics of the EM wave andthe reference axis used to define the vortex electron. The simplest convention is to take counter-propagating EM wave and electron vortex beam, Fig. 28(a). Assuming propagation of the vortex electron along the z -axis, the wave four-vector of the EM wave is (cid:107) µ = ω (1 , , , − φ can be performedexactly.An elegant way to evaluate this integral is to group the first two terms in (4.19). Then, the resultingintegral becomes exactly as for the free Bessel electron, with the replacement of the transverse coordinates: r ⊥ → R ⊥ = r ⊥ + e ( p µ (cid:107) µ ) (cid:90) dξ A ⊥ . (4.20)One then obtains a sum of two Bessel functions [as in Eqs. (4.8) or (4.15], depending on the chosen spinstate), but with the substitutions κr → κR ⊥ in the Bessel fucntions and ϕ → ϕ R in the vortex phase factors(here ϕ R is the azimuthal coordinate of R ⊥ ). The last factor in Eq. (4.19) represents, for a monochromaticplane wave, an integral over a constant, generating an extra phase ∝ ( A µ A µ ) ξ . Just as for the usual Volkovsolution, it amounts to the replacement of the energy and the longitudinal momentum by the quasi -energyand quasi -momentum.One can now calculate the dynamical properties (e.g., the expectation values of the spin and OAM)of the Volkov–Bessel solutions [108]. Since the effective radial coordinate R ⊥ explicitly depends on time,one can consider both instantaneous and effective time-averaged quantities (marked with the overbar here).For the time-averaged quantities, the z -propagating Volkov plane-wave state with a well-defined helicity χ igure 28: (a) Schematics of the Volkov–Bessel problem: a Bessel-beam solution for the Dirac electron and the field of anelectromagnetic (EM) plane wave. Here the electron vortex beam and the EM wave counterpropagate with the mean electronmomentum (cid:104) p (cid:105) and the EM wave vector (cid:107) . The EM wave is linearly y -polarized. (b) Temporal variations of the transverseprobability-density distribution in a paraxial Volkov–Bessel beam shown in the panel (a) [109]. Here, the electron parametersare: azimuthal vortex index (cid:96) = 3, energy E = 300 keV, and opening angle θ = 0 .
02 rad. Instead of a plane EM wave, here afive-cycle laser pulse was used with the central frequency ω c = 10 Hz and electric-field amplitude E = 10 V/cm. The timerange of 6.5 fs approximately corresponds to the duration of the pulse. displays a reduced spin (cid:104) ¯ S z (cid:105) : (cid:104) ¯ S z (cid:105) = χ − e A ⊥ ω/ E ( p µ (cid:107) µ )1 + e A ⊥ ω/ E ( p µ (cid:107) µ ) . (4.21)This effective depolarization is a purely kinematic effect caused by the spin precession in the external EMfield, and its strength is governed by the dimensionless parameter η = e A ⊥ m e . (4.22)In a strong laser field, η ∼ > (cid:104) ¯ S z (cid:105) is substantially reduced and can even change sign with respect tothe helicity χ . For the Volkov–Bessel electron beams, the same effect is responsible for the shift of thetime-averaged total angular momentum (cid:104) ¯ J z (cid:105) , but the effect is always less than one, even for large | (cid:96) | . Forultrarelativistic electrons, the exact reversal point is shifted to higher η . However in such strong fields, oneneeds to complement the picture with inelastic scattering processes.An alternative approach to the azimuthal integrals for the Volkov–Bessel solutions was advocated in[109, 301]. Considering an EM wave linearly polarized along the y -axis , A µ = (0 , , A, e if sin φ = + ∞ (cid:88) n = −∞ J n ( f ) e inφ , f = eκ ( (cid:107) µ p µ ) (cid:90) dξ A ( ξ ) . (4.23)The Volkov–Bessel solution then takes the form of an infinite sum of the Bessel modes with vortex number63 (cid:96) + n ) weighted with i n J n ( f ). Although this representation is not very convenient for an infinite EM planewave, it can be used to investigate the response of the Bessel electron to a strong few-cycle laser pulse[109]. In this case, only a few terms in the above summation are important. The numerical analysis of [109]demonstrated transverse y -oscillations of the Bessel-beam probability-density distribution as the laser pulsepasses through the electron. Figure 28(b) illustrates such temporal variations of the transverse probabilitydensity distribution in a paraxial Volkov–Bessel beam. Nuclear and particle physics is another area where vortex electrons (and in general, vortex states ofparticles) can emerge as a novel tool for the experimental exploration of fundamental interactions. Invirtually all situations involving collisions of particles, be it a fixed-target or a collider-like setting, are welldescribed with plane waves. The fact that real colliding particles are wavepackets is inessential; exceptionsexist [302] but are extremely rare. Vortex states bring in a new degree of freedom, the OAM, which can beexploited in collisions. The instrumentation which would allow one to prepare, accelerate to high energies,and collide vortex electrons, protons and other particles, does not exist yet, but first exploratory studies ofaccelerating twisted electrons to multi-MeV energies are underway in JLab [303]. Anticipating that dedicatedexperimental efforts will eventually make such experiments possible, it is timely to ask what opportunitiesthis new instrument can offer for nuclear and high-energy physics.This question leads us to the problem of the theoretical description of the scattering processes of high-energy vortex particles. Below we will overview the general kinematic novelties which arise in collisions ofvortex states, and mention particular processes investigated so far.Several schemes for collisions of vortex states are possible, as shown in Fig. 29. First, a vortex state (V) can scatter, either elastically or inelastically, on a fixed scattering center, and the final state is usuallyassumed to be a plane wave (PW), Fig. 29(a). The prototypical problem here is the (screened) Rutherfordscattering of vortex electrons on atoms, either elastic [245, 291] or inelastic [96], as well as radiative captureof vortex electrons by atoms [304, 305]. In these examples, a non-relativistic Schr¨odinger-equation treatmentfor incoming spinless (scalar) vortex wave is sufficient to grasp the essential details, but at larger energies,the full relativistic treatment is needed.Second, one can consider the scattering process in the collider-like kinematics, when two incoming par-ticles scatter into a certain final state. For such processes, the quantum-field-theoretic treatment is moreappropriate. Different vortex vs. plane waves settings have been considered: V+PW → PW+PW [115, 306],V+V → PW+PW [115, 307, 120, 308, 121, 122, 124, 123], V+PW → V+PW [114, 309, 115, 310], and evenV+PW → V+V [311], Figs. 29(b)–(e). The fully-vortex scattering setting V+V → V+V can be viewed asa particular case of the general formalism of collision of arbitrarily shaped wavepackets [123]. Each col-lision setting has its own challenges and novelties, both in the theoretical description and in the possibleexperimental realization.
In the fixed-target scattering, V → PW [Fig. 29(a)], the scattering center can absorb any momentumtransfer. Since the initial state is a coherent superposition of many plane waves, individual plane wavescattering amplitudes with different momentum transfers interfere in the total amplitude (for the generalwavepacket scattering on atoms, we refer to the recent pedagogical exposition [312]). For example, for theBessel beam with parameters κ and (cid:96) , we can write the V → PW scattering amplitude as f ( κ, (cid:96) ; k (cid:48) ) = (cid:90) d k ⊥ (2 π ) ˜ ψ κ(cid:96) ( k ⊥ ) f ( k ; k (cid:48) ) , (4.24)where f ( k ; k (cid:48) ) denotes the usual plane wave scattering amplitude for the initial and final momenta k and k (cid:48) .In the specific case of screened Rutherford scattering, the plane-wave scattering amplitude is azimuthallysymmetric, and one observes that f ( κ, (cid:96) ; k (cid:48) ) ∝ exp( i(cid:96)φ (cid:48) ), with φ (cid:48) being the azimuthal angle of the final64 igure 29: Schematic diagrams of scattering processes involving vortex particles. (a) Fixed-target scattering of vortex (V) intoa plane wave (PW). (b) Single-vortex scattering (collision) V+PW → PW+PW. (c) Vortex-vortex scattering into plane waves:V+V → PW+PW. (d,e) Vortex-into-vortex scattering V+PW to V+PW and V+PW to V+V. momentum k (cid:48) , as a natural consequence of the OAM conservation [245]. The polar angle distributiondisplays characteristic dependences on κ and (cid:96) [245, 291].In Eq. (4.24), the axis used to define the vortex state passes exactly through the scattering center. Thecase when the reference axis is displaced from the scattering center by the impact parameter b ⊥ can betreated either by including the extra factor exp( − i b ⊥ · k ⊥ ) in the integral (4.24) [304, 291] or by makinguse of the Bessel addition theorem to rewrite this state in terms of aligned vortices. The latter method wasused in the elastic [245] and inelastic [96] vortex-electron-atom scattering, as well as in the analysis of atomexcitation by off-axis vortex photons [104].In the case of a single scattering center, an experimental control of this transverse shift, at least withinthe transverse wavelength 1 /κ , will be a challenge to overcome. Alternatively, one can consider an amor-phous macroscopic fixed target, where a single vortex beam is scattered by many centers with uniformlydistributed values of b ⊥ . This setting is easy to realize experimentally, but averaging over b ⊥ smears outthe distributions and erases interesting signals. For example, the properties of photons emitted in radiativecapture of vortex electrons [304] and of electrons elastic scattering [291] by atoms are insensitive to the elec-tron vorticity (cid:96) and depend only on κ . Fortunately, this uniform averaging does not apply to crystals andin particular to chiral crystals; in fact, elastic scattering of vortex electron beams can be used to investigatechirality in crystalline materials, see Section 3.5.1. 65 .5. Single-vortex scattering The single-vortex scattering, V+PW → PW+PW [Fig. 29(b)], is essentially identical to plane-wavescattering [115, 306]. To describe it, we start with plane-wave two-particle scattering with four-momenta k µ , = ( E , , k , ) for the incoming particles and k µ (cid:48) , = ( E (cid:48) , , k (cid:48) , ) for the outgoing particles. The plane-wave S -matrix element can be written as S P W = i (2 π ) δ (4) ( k µ + k µ − k µ (cid:48) − k µ (cid:48) ) (cid:77) (cid:0) k µ , k µ ; k µ (cid:48) , k µ (cid:48) (cid:1)(cid:112) E E E (cid:48) E (cid:48) . (4.25)The invariant amplitude (cid:77) is calculated according to the standard Feynman rules. Transition to the vortexstate is done by integrating (4.25) over the plane-wave components of the initial vortex state [114]. Forexample, for the pure Bessel state, similarly to Eqs. (4.1) and (4.2), we have: S V = (cid:90) d k ⊥ (2 π ) ˜ ψ κ(cid:96) ( k ⊥ ) S P W . (4.26)This simple expression exhibits an important effect. In contrast to the fixed-center scattering, now oneintegrates not only the scattering amplitude (cid:77) but also the kinematical δ -function in Eq. (4.25). The delta-function eliminates this integration and effectively removes the coherence of plane-wave components insidea vortex state. As the result, the single-vortex cross-section is represented as the azimutally averaged planewave cross-section, dσ V = (cid:82) ( dφ k / π ) dσ P W ( k ), and is (cid:96) -independent [115]. The situation can become lesstrivial if the vortex state has an inhomogeneous polarization state with a polarization singularity [12], or ifa superposition of two (cid:96) -states is used (see examples in Section 2.7). In the latter case, this scattering canbe used, for example, to produce X-ray beams with accurately structured intensity distributions [306]. The double-vortex scattering V+V → PW+PW [Fig. 29(c)] opens up novel physics opportunities, as itallows one to measure quantities which are not observable in the usual plane-wave collisions. Consider thisprocess with two Bessel vortex states with parameters κ , (cid:96) and κ , (cid:96) defined with respect to the sameaxis. Similarly to Eq. (4.26), we now have: S V V = (cid:90) d k ⊥ (2 π ) d k ⊥ (2 π ) ˜ ψ κ (cid:96) ( k ⊥ ) ˜ ψ κ , − (cid:96) ( k ⊥ ) S P W (cid:0) k µ , k µ ; k µ (cid:48) , k µ (cid:48) (cid:1) . (4.27)The standard procedure, based on Fermi’s golden rule, to calculate the cross-section is to square the S -matrix element, regularize the squares of the delta-functions with a finite volume and finite observationtime, normalize appropriately the initial and final states, divide the probability by the total observationtime and the relative flux, and finally integrate the result over final phase space. For the vortex-vortexscattering, this calculation follows the same route but the expressions differ significantly from the standardcase [114]. The finite-volume normalization rules for the Bessel vortex states are different [309, 115, 108].Separation of the event rate into the flux and cross-section becomes ambiguous, which is a generic featureof the wavepacket scattering formalism, see the classic description in [302] and the recent developmentin [123]. One needs to define in a reasonable way the generalized flux and generalized cross-section. Inthe works [309, 115, 108], slightly different expressions were proposed, but in the paraxial approximation,which is sufficient for all practical purposes, and with a smoothly behaving invariant amplitude (cid:77) , all theseexpressions coincide. Below, we will omit the word “generalized” when describing the cross-sections.The most salient feature of the vortex-vortex collision is that the final state kinematics acquires a newdegree of freedom with respect to the plane-wave collision. In the all-plane-wave two-particle scattering,the total momentum is well-defined, K = k + k and is conserved during the process. As a result, thefinal momenta k (cid:48) and k (cid:48) are maximally correlated: if k (cid:48) is fixed, k (cid:48) has no freedom left, as it must beequal to k (cid:48) = K − k (cid:48) . In the wavepacket collision, with a certain distributions over the initial momenta66 igure 30: Two kinematical configurations in the transverse plane contributing to the integral (4.29) for the final state withtransverse momentum K ⊥ . The two invariant plane-wave amplitudes (cid:77) a and (cid:77) b differ from each other and interfere in (4.30). k and k , this correlation is relaxed. The final momenta k (cid:48) and k (cid:48) represent now independent, althoughpartially correlated, degrees of freedom. As a result, the cross-section is now differential in both k (cid:48) and k (cid:48) ,or alternatively, differential in k (cid:48) and K . For generic normalized wavepackets, it can be represented as dσ = dσ R ( K ) d K , (4.28)where dσ is the usual plane-wave cross-section taken together with its appropriate final phase space, and R ( K ) is a certain function usually peaked at the sum of the average momenta of the two initial wavepackets[302, 120]. In the plane-wave limit, R ( K ) → δ (3) ( k + k − K ), and dσ is recovered.For pure Bessel beams, the non-trivial kinematical correlations concern only the transverse momenta: dσ/d K ⊥ ∝ | F ⊥ | , where F ⊥ = (cid:90) d k ⊥ (2 π ) d k ⊥ (2 π ) ˜ ψ κ ,(cid:96) ( k ⊥ ) ˜ ψ κ , − (cid:96) ( k ⊥ ) δ (2) ( k ⊥ + k ⊥ − K ⊥ ) (cid:77) (cid:0) k µ , k µ ; k µ (cid:48) , k µ (cid:48) (cid:1) . (4.29)In this expression, the value of K is fixed by choice of the two final plane waves. Since this integral containsfour delta-functions and four integrations, it can be evaluated exactly. It is non-zero only when κ , κ , and | K ⊥ | satisfy the triangle inequalities | κ − κ | ≤ | K ⊥ | ≤ κ + κ , and in this case it gets contributions fromexactly two plane-wave configurations (a) and (b) shown in Fig. 30.These two configurations are just reflections of each other with respect to the direction of K ⊥ . Denotingthe plane wave scattering amplitude evaluated at these two initial kinematics as (cid:77) a and (cid:77) b , we obtain[120]: F ⊥ ∝ (cid:77) a exp( i(cid:96) δ + i(cid:96) δ ) + (cid:77) b exp( − i(cid:96) δ − i(cid:96) δ ) , (4.30)where δ and δ are the inner angles of the ( κ , κ , | K ⊥ | ) triangle, see Fig. 30. The net result is that thecross-section contains an additional term proportional to the interference between two different plane waveamplitudes with equal final but different initial momenta: dσ ∝ | (cid:77) a | + | (cid:77) b | + 2 Re [ (cid:77) a (cid:77) ∗ b exp(2 i(cid:96) δ + 2 i(cid:96) δ )] . (4.31)A more accurate analysis [120] with normalized wavepackets of Bessel states is necessary to regularizethe end-point singularities, and it shows that this interference term can be extracted via the azimuthalasymmetry of the cross-section.The expression (4.31) was the starting point in demonstrating [120, 308, 121] that the scattering of twovortex states allows one to probe the overall phase Φ (cid:77) of the scattering amplitude (cid:77) = | (cid:77) | exp( i Φ (cid:77) ).In the usual plane-wave collision, the cross-section dσ ∝ | (cid:77) | is completely insensitive to the phase Φ (cid:77) and its variation with kinematical parameters. In vortex-vortex scattering, the two interfering plane-waveamplitudes (cid:77) a and (cid:77) b correspond to different initial and the same final momenta, which implies different67 igure 31: Differential cross-section, in arbitrary units, as a function of the total momentum K = k + k , for fixed k (cid:48) , forpurely real Born-level elastic electron scattering amplitude (left), and for the momentum-transfer-dependent Coulomb phase(4.32) (right) [122]. The parameters used here are: E = 2 . k (cid:48) = 500 keV with k (cid:48) = k (cid:48) ¯ x , J z = 1 / J z = 13 /
2, and κ and κ are Gaussian-distributed around the values of 200 and 100 keV with the widths 10 and 5 keV, respectively. Theresults are averaged over different helicities of the incoming particles. In the right-hand panel, the fine-structure constant α isartificially set to 10 to enhance the visibility of the up-down asymmetry. momentum transfers. The phase Φ (cid:77) can depend on this momentum transfer. For example, in the elasticscattering of charged particles the amplitude acquires the well-known Coulomb phase, which, for largeenergies and small scattering angles θ , can be written asΦ (cid:77) ( θ ) = Φ (cid:77) + 2 α log(1 /θ ) , (4.32)where α is the fine-structure constant, and Φ (cid:77) is an angle-independent quantity which, strictly speaking,requires infrared regularization and can be sensitive to the details of the process. Thus, the interferenceterm in (4.31) is proportional to cos[2 (cid:96) δ + 2 (cid:96) δ + Φ (cid:77) ( θ a ) − Φ (cid:77) ( θ b )] and it is sensitive to the dependenceΦ (cid:77) ( θ ). This quantity can be extracted from the azimuthal asymmetry of the differential cross-section [120].In Refs. [121, 122], this idea was investigated in detail with the example of the moderately relativisticelastic electron-electron scattering using the Dirac electrons described by bispinors (4.11) and their vortexcombinations. A typical K ⊥ -distribution of the cross-section is shown in Fig. 31. The left and right plotscorrespond, respectively, to the purely real Born-level scattering amplitude and to the amplitude with theCoulomb phase Φ (cid:77) ( θ ) taken into account, with the value of α artificially set to 10 in (4.32) for the purposeof illustration. One can see the interference fringes arising from the interference between the two kinematicalconfigurations of Fig. 30. The left plot is symmetric with regard to the horizontal line (i.e., the directionof k (cid:48) ⊥ ) because the cross-section contains no terms proportional to sin( φ (cid:48) − φ K ) (with φ (cid:48) and φ K beingthe azimuthal angles of k (cid:48) and K ), while the right plot shows a distorted pattern. This distortion canbe quantified in terms of up-down asymmetry [121, 122]. It is this asymmetry that is proportional to thephase difference between the two contributions and can lead to a direct measurement of Φ (cid:77) ( θ ). For therealistic α ≈ / − –10 − , which may be experimentallyaccessible with sufficient statistics.In the works [124, 123], the above suggestion was considered as a particular case of a more general setting.In order to probe the overall phase of the scattering amplitude, one needs to collide states with manifestlybroken azimuthal symmetry, which necessarily goes beyond the plane-wave approximation. Developingfurther the Wigner-function-based theoretical formalism for collision of arbitrary wavepackets, Refs. [124,123] showed that the cross-section contains a new phase-sensitive term proportional to the mean value ofthe following “effective impact parameter”: b eff = b ⊥ − ∂ Φ ( k ⊥ ) ∂ k ⊥ + ∂ Φ ( k ⊥ ) ∂ k ⊥ . (4.33)68ere, b ⊥ = r ⊥ − r ⊥ is the usual impact parameter, i.e., the transverse separation of the centers of thetwo colliding wavepackets, while Φ i ( k i ⊥ ) describe their additional phases beyond the center-of-mass motion: ψ i ( k i ) ∝ exp[ − i r i · k i + i Φ i ( k i ⊥ )]. This term can be extracted from the data via the asymmetry definedas the relative difference between the cross-sections with b eff and − b eff . The average value of this operatorcan be non-zero either for an off-center collision of wavepackets ( b ⊥ (cid:54) = 0) or for the head-on collision ofwavepackets with non-trivial phase fronts, such as vortex electron beams. In fact, with the definition ofasymmetry adopted in [124, 123], the phase singularity needs to be shifted away from the collision axis inorder to produce a non-zero asymmetry. This development opens up several complementary ways to probethe phase of the scattering amplitude, which await experimental verification.When experiments with vortex protons and other hadrons become possible, the above method can beapplied to hadronic processes. It will then offer additional information on hadronic interactions whichcannot be accessed in conventional experiments. One example is the small-angle elastic pp scattering withmomentum transfer of the order of 0 . Pomeron , an emergent strongly-interacting object whose theoretical description is still debated [313]. Withvortex proton scattering, one can measure the dependence of the phase of the full amplitude in the momentumtransfer. In this way, one gets a new observable against which the Pomeron models can be tested [308,124, 123]. So far, the Pomeron phase can be accessed only in the very narrow t -region via the strong-Coulomb interference, and, in addition, it also relies on the good knowledge of the Coulomb phase. Withvortex protons, one should be able to probe this phase over the entire t region, including the diffractiondip region where a strong variation of the phase is expected in some models. Another example is theintermediate-energy hadroproduction reactions such as γp → K + Λ, which involve hadronic resonances inseveral competing partial waves [314]. Although the relative phases between these contributions can beaccessed, disentangling them would become easier if the information on the overall phase were available.
The vortex-into-vortex scattering process V+PW → V+PW [Fig. 29(d)] brings new challenges. Thecalculation of the strictly forward or backward scattering does not pose any difficulty [114, 309], as one canuse the same reference axis to describe the initial and final vortex states. The orbital angular momentum isnaturally transferred from the initial to the final vortex state: (cid:96) (cid:48) = (cid:96) . For off-forward scattering, the situationis more complicated. While [114] argued that one still has (cid:96) (cid:48) (cid:39) (cid:96) , the analysis of [115] showed that the entire (cid:96) (cid:48) -region from −∞ to + ∞ contributes to the cross-section. In [310], the origin of the discrepancy was tracedback to the usage of non-normalizable pure Bessel beams. If one uses normalizable vortex wavepackets and,in addition, if one chooses its own reference axis for each vortex state, then the controversy is resolved. Thisresult stresses the usage of the orbital helicity [75] (i.e., the OAM component along the propagation axis) asthe physically-relevant quantity rather than the OAM defined with respect to a fixed axis.Additional difficulties arise if one views the process V+PW → V+PW or V+PW → V+V [Fig. 29(d,e)]to produce high-energy vortex states [114, 309, 311]. Formally, the two outgoing waves are momentum-entangled , and there is no pre-existent way to label one particle as a vortex state and the other as a planewave [311]. Only after one particle is projected on an approximate plane wave, and is measured by thedetector with a momentum uncertainty less than κ , the other particle emerges in a vortex state. Whetherthis projection can be performed on an event-by-event basis with the existing technology remains unclear.Further on, the simultaneous energy and momentum conservation in the off-forward V+PW → V+PWscattering implies that the outgoing vortex state is not monochromatic . Indeed, different plane wave com-ponents of the final vortex state correspond to different energies not only of the vortex state itself, but alsoof the recoil plane wave. Thus, the coherence among the plane wave components required to form a vortexstate is lost, or at best is hidden [115, 306].As far as specific high-energy processes are concerned, the only example considered in detail was theinverse Compton backscattering [114, 309, 306]. Here, optical photons scatter almost backward off high-energy electrons and take a sizable portion of the electron energy. This process is well known and isroutinely used, for example, at the SPring-8 and HIgammaS facilities [315] to produce GeV-range photonsfor subsequent hadronic photoproduction experiments. This process was calculated for vortex initial and69nal photons [114, 309], while the electrons, both initial and final, were assumed to be plane waves. In thestrictly forward scattering, the final photon is upconverted into the GeV energy range while retaining itsOAM. For slightly off-forward kinematics and with a due care mentioned above, the final orbital helicity isalso close to the initial one [310]. Another application of this process was considered in [306]. Here, theplane wave photons scatter off energetic vortex electrons and turn into a flux of energetic photons withstructured intensity distribution in the transverse plane. Preparing the initial vortex electrons in custom-tailored superpositions of different values of OAM, one can accurately shape the transverse distribution ofthe final X-ray pulse.
5. Radiation by vortex electrons
Electrons can radiate. They emit electromagnetic (EM) radiation via bremsstrahlung when the electrontrajectory is deflected by external fields or through polarization radiation (an umbrella term including theVavilov–Cherenkov radiation, diffraction radiation, transition radiation, etc.), when moving in or near apolarizable medium. One can ask whether the radiation from vortex electrons differs in any aspect fromthe plane-wave case, and if so, what additional information it encodes. For vortex electrons, two types ofEM radiation have been investigated theoretically so far: the
Vavilov–Cherenkov radiation [110, 111] and transition radiation [112, 316, 113]. In both cases, the vortex nature of the electron leads to several distinctfeatures of the radiation it emits. These features, if experimentally detected, should provide additionalinsight into the radiation process, and could serve as a complementary and convenient diagnostic tool formeasuring the parameters of vortex electrons.
The Vavilov–Cherenkov radiation [317, 318, 319] from a vortex electron was investigated in [110, 111].Both works treated the problem in the full quantum-electrodynamical approach and focused on the spectral, d Γ /dω , and spectral-angular, d Γ /dωd Ω γ , distributions of the photon emission rate Γ ( ω is the photonfrequency and Ω γ is the solid angle spanned by the photon wave-vector directions), as well as on thepolarization properties.As a short reminder, within the standard quantum treatment of the Vavilov–Cherenkov radiation froma plane-wave electron, the emission process is described as a “decay” of the initial electron with the four-momentum momentum p µ into the final electron p µ (cid:48) and the in-medium photon with momentum (cid:126) (cid:107) µ [320].Note that in this section we restore the constants (cid:126) and c . In what follows, E stands for the initial relativisticenergy of the electron, β = v/c = pc/E is its dimensionless velocity, while (cid:126) ω is the energy of the emittedphoton. This radiation has the following spectral-angular distribution [320]: d Γ dωd Ω γ = α π (cid:20) β sin θ Ch + ( (cid:126) ω ) βE ( (cid:110) − (cid:21) δ (cos θ (cid:107) p − cos θ Ch ) , (5.1)where (cid:110) ( ω ) is the frequency-dependent refraction index of the medium, θ (cid:107) p is the angle between the emittedphoton and the initial electron, and θ Ch is the Cherenkov cone opening angle given bycos θ Ch = 1 β (cid:110) + (cid:126) ω E n − β (cid:110) . (5.2)Clearly, the emission angle satisfies 0 ≤ θ Ch ≤ ◦ , and the requirement that cos θ Ch ≤ (cid:126) ω ≤ (cid:126) ω cutoff = 2 E ( β (cid:110) − / ( (cid:110) − igure 32: Schematics of the Vavilov–Cherenkov radiation by a vortex electron [110, 111]. Here, θ is the opening angle of thevortex electron, θ Ch is the opening angle of the Cherenkov cone for the plane-wave case, the gray ring represents the annularregion of directions where the photons can be emitted. to the optical/UV region would strongly suppress the intensity of radiation and therefore make the directobservation of these cut-off effects extremely challenging [111].Equation (5.1) is written for an unpolarized initial electron and after summation over final electron andphoton polarizations. Actually, the radiated Cherenkov photons are almost 100% linearly polarized, withthe polarization lying in the scattering plane. If the initial electron is polarized, then the Cherenkov lightacquires a non-zero degree of circular polarization [111].If the initial electron is in a vortex state, the angular distribution of the Vavilov–Cherenkov radiationchanges [110, 111]. A pure Bessel beam is a superposition of plane-wave electrons with a conical distributionof momenta p , which is characterized by the polar angle θ (sin θ = (cid:126) κ/p ), Fig. 5(a). All these plane-wavecomponents radiate, but if we do not detect the final electron and only study the photon angular distribution,this radiation adds up incoherently . Indeed, in each plane-wave radiation subprocess the four-momentumconservation dictates p µ = p µ (cid:48) + (cid:126) (cid:107) µ . If one measures the intensity of radiation in a certain direction withoutdetecting the final electron, one fixes (cid:107) µ and integrates over all p µ (cid:48) . But then different Fourier componentsof the initial vortex electron carry different values of p µ and, due to the fixed (cid:107) µ , correspond to different finalmomenta p µ (cid:48) . Since the interference requires that all the final-state particles remain exactly in the samestate, no interference is possible in this kinematics. Thus, averaging the plane-wave intensity (5.1) overthe appropriate initial electron momentum p µ , while keeping (cid:107) µ fixed, we will obtain the spectral-angulardistribution for the vortex electron.In doing so, the most salient feature is the angular distribution. It has a ring shape shown in Fig. 32and spans over the photon’s polar angles θ (cid:107) with respect to the mean electron propagation direction (the z -axis): | θ Ch − θ | ≤ θ (cid:107) ≤ θ Ch + θ . (5.3)For a Bessel electron with a definite value of the z -component of the total angular momentum J z , the angulardistribution summed over the final helicities is azimuthally symmetric and grows towards the boundaries ofthe ring. This leads to two remarkable phenomena which cannot be produced with plane-wave electrons.First, if θ Ch + θ > π/
2, then some photons are emitted in the backward hemisphere with respect to theoverall propagation direction of the vortex electrons [110]. In order to observe this effect, one would needto achieve large opening angles of the vortex state or take a medium with a very large refractive index.71 igure 33: The spectral-angular distribution of the photon emission rate Γ as a function of the emitted-photon spherical angles θ (cid:107) and φ (cid:107) (mapped onto the ( x, y ) plane of the far-field detector) for an electron in a superposition of two vortex states with( J z ) − ( J z ) = 3 and equal amplitudes [111]. The radii of the annular region of the radiation are determined by Eq. (5.3). Second, if the two opening angles match, θ Ch = θ , the Cherenkov radiation strongly peaks in the forward direction, see the middle panel of Fig. 33. For perfect Bessel beams with a fixed value of the momentumpolar angle θ p = θ , the intensity of the Vavilov-Cherenkov radiation will grow near the forward direction as1 /θ (cid:107) . For realistic vortex electrons, θ p is not fixed but is distributed over an angular region with width δθ ,Fig 6(a), this bright emission will smear over a spot of comparable angular size δθ . To observe this forwardemission, one would need, first, vortex electrons with a sufficient transverse coherence length and exhibitingseveral radial intensity rings (originating from the Bessel distribution with θ ) to guarantee δθ /θ (cid:28) θ = 20 mrad, which implies 1 − cos θ (cid:39) · − , one must adjust the electron velocity tothe emission threshold velocity with the accuracy of 10 − .After integration over all photon emission angles, the spectral distribution d Γ /dω for the vortex electronsand, generically, for any monochromatic wavepacket, coincides with the plane-wave spectral density. This is aconsequence of the incoherent summation of the radiation from different plane-wave components. Althoughthe initial electron is a coherent superposition of different plane waves, once we integrate over the final-electron parameters, this coherence is lost. In this respect, attributing a special role to the coherence as theorigin of the spectral-angular features as done in [110] is unjustified.The polarization properties of the Cherenkov light also change when one switches from the plane-waveto the vortex electron. The geometry of the problem shows that, for a pure Bessel state, the radiationintensity into any given direction inside the ring is an incoherent sum of two plane-wave intensities withdifferent initial electron kinematics. As a result, the emitted light remains linearly polarized but its degreeof polarization changes. In particular, the photons can become linearly polarized in the direction orthogonalto the scattering plane [111].Finally, if instead of a single vortex state, the electron is taken in a superposition of two vortex stateswith total angular momenta ( J z ) and ( J z ) , then the electron probability-density distribution becomesazimuthally-inhomogeneous, see Figs. 11 and 12. The multi-petal structure of the Vavilov–Cherenkov ra-diation from such structured electrons mimics the electron density profile in the focal plane, as shown inFig. 33. Thus, Cherenkov radiation emerges as a convenient macroscopic diagnostic tool for such electrons. The transition radiation occurs when an electron crosses the boundary separating two media with differ-ent permittivity or permeability [321, 322]. The simplest example is the electron impinging from the vacuum72nto a conductive plane. In all cases studied experimentally, this radiation is associated with the particle’s electric charge . However, classical electrodynamics predicts that magnetic moments and higher-order mul-tipoles can also radiate [321, 322]. This contribution to radiation has never been detected for any kind ofpolarization radiation due to its weakness.Vortex electrons with large OAM (cid:96) can make the observation of this contribution possible. Indeed, as wediscussed in Sections 2.5 and 2.8, paraxial free-space vortex electrons possess longitudinal magnetic moment[75, 147] M = ( ec/ E ) ( (cid:104) L (cid:105) + 2 (cid:104) S (cid:105) ), Eqs. (2.24), (2.25), and (2.60). Assuming longitudinal polarization,i.e., spin component s = ± /
2, the absolute value of the magnetic moment is M = γ − | (cid:96) + 2 s | µ B , where µ B is the Bohr magneton and γ is the Lorentz factor. Therefore, large values of (cid:96) would strongly enhancethe magnetic-moment contribution to the transition radiation and, via interference with the electric-chargecontribution, it can lead to an observable signal.The essence of this idea can be explained in the following way [323, 113, 112, 316]. Within classicalelectrodynamics, an electron with velocity v can be viewed as a pointlike source equipped with the electriccharge e and magnetic moment M . Accordingly, it is described with the electric and magnetic currentdensities j e = e v δ ( r − v t ) and j m = γ − c ∇ × [ M δ ( r − v t )]. The curl leads to an extra factor iω/c inthe Fourier components of the radiation field. As a result, the relative strength of the magnetic momenttransition radiation always bears the following small factor [322]: (cid:15) = M ωγ e c . (5.4)The radiation energy contains this factor squared. For optical/UV photons and for moderately relativisticplane-wave electrons ( (cid:96) = 0), (cid:15) = (cid:126) ω/E ∼ − ; for slower electrons, the radiation is much weaker. Anadditional difficulty arises from the fact that the quantum corrections bear the same suppressing factor (cid:126) ω/E . A calculation which keeps the electron’s magnetic moment contribution but neglects quantum effectsis, strictly speaking, inconsistent, and a full quantum treatment is needed. Thus, for non-vortex electronscarrying magnetic moment from the spin, the magnetic moment contribution is: (i) suppressed by manyorders of magnitude with respect to the usual charge radiation, which makes it undetectable, and (ii) is notcleanly calculable within classical electrodynamics. However, for vortex electrons with large OAM, | (cid:96) | (cid:29) d (cid:87) = d (cid:87) e + d (cid:87) m + d (cid:87) em . The quantity of interest is the interference term, whichis linear in the small parameter (cid:15) . There are two subtleties, which make the extraction of this interferenceintricate. First, the curl in the magnetic current leads to an extra i factor in the EM field radiated bythe magnetic moment, and as the result, the interference vanishes in the case of a transparent medium oran ideal conductor. Fortunately, for a realistic medium with complex permittivity, the interference termbecomes non-zero. Second, since M is a pseudovector, the interference term contains the triple scalarproduct ¯ (cid:107) · ( M × ¯ z ), where ¯ (cid:107) is the direction of the emitted photon, and ¯ z is the normal to the vacuum-metalinterface, Fig. 34(a). It vanishes for the normal incidence, as well as for oblique incidence, after the fullsolid angle integration. It can be observed only at oblique incidence and only in the differential distribution.Refs. [112, 316] proposed to extract the interference term via the left-right asymmetry: I LR = (cid:87) L − (cid:87) R (cid:87) L + (cid:87) R , (cid:87) L,R = (cid:90) d Ω γL,R d (cid:87) d (cid:126) ωd Ω γ , (5.5)where d Ω γL and d Ω γR indicate the two hemispheres lying to the left and to the right of the incidence plane,i.e., characterized by the angles ˜ ϕ (cid:107) > ϕ (cid:107) <
0, as shown in Fig. 34(a).Numerical calculations [112, 316] show that 300 keV vortex electrons with (cid:96) = 1000 and impacting on analuminium plate at large incidence angles produce a left-right asymmetry at the percent level, and shouldbe well observable for a 1 nA electron current. For lower values of (cid:96) , the asymmetry is proportionally small,73 igure 34: (a) Schematics of the backward transition radiation produced by a vortex electron, carrying electric charge e andmagnetic moment M (cid:107) p , at oblique incidence on a metal surface z = 0 [112, 316, 113]. The direction of the electron momentum p in the ( x, z ) plane of incidence is determined by the polar angle θ p , while the emitted out-of-plane photon is characterizedby the wave vector (cid:107) with the angles ˜ θ (cid:107) and ˜ ϕ (cid:107) , as shown in the scheme. (b) The interference between the magnetic-momentand charge contributions leads to the left-right ( ˜ ϕ (cid:107) → − ˜ ϕ (cid:107) ) asymmetry of the transition radiation intensity [112]. The ˜ ϕ (cid:107) -dependence of the spectral-angular density of the transition radiation energy (cid:87) . The parameters are: E = 300 keV, (cid:126) ω = 5eV, θ p = 70 ◦ , and the observation angles are centered around ˜ θ (cid:107) = − ◦ . The curves corresponds to (cid:96) = 0 (solid black curve),1000 (dashed red curve) and 10000 (blue dotted curve). but also seems to be within reach. The experimental detection of this asymmetry would provide the firstever direct demonstration that magnetic moments radiate.In the works [323, 113], an alternative route to detecting the interference term d (cid:87) em was explored. There,the degree of circular polarization in the emitted light was the quantity of interest. Transition radiation of apoint electric charge is linearly polarized [322, 321]. The interference term induces an elliptical polarizationin the out-of-plane photons, quantified by another pseudoscalar quantity, the photon’s helicity or the Stocksparameter (cid:83) [80]. At generic angles, the value of (cid:83) is very small, being suppressed by the same factor (cid:15) .However, near particular directions, where the electric-charge radiation vanishes, the value of (cid:83) can be verylarge. Thus, measuring (cid:83) in the direction of the intensity minimum will also unveil the transition radiationemitted by the vortex electron’s magnetic moment. The calculations reported in [323, 113] show that forOAM (cid:96) = 100, the value of (cid:83) can be as large as 70%. However this large elliptic polarization appears onlyfor directions range ∼ . Concluding remarks Ten years have passed since the prediction [75] of free-electron vortex states carrying intrinsic orbitalangular momentum (OAM) and their first generation [76, 77, 78] in transmission electron microscopes(TEMs) few years later. In this paper, we have reviewed the main theoretical and experimental achievementsin investigations of vortex electrons during the first decade of this rapidly developing field.First, we have provided a pedagogical introduction and a solid theoretical basis for the researchers startingtheir work in this emerging field. In particular, we have introduced the main concepts of phase singularities,angular momentum, and vortex wave beams/packets as applied to electron waves. We have also consideredthe main interaction phenomena involving vortex electrons, including: their nontrivial behaviour in externalelectromagnetic fields, spin-orbit interactions and other relativistic effects, a variety of elastic and inelasticscattering processes, radiation processes, etc.Second, we have described the main features and peculiarities of TEM experiments with electron vortexbeams. In particular, we provided a detailed analysis of various methods of their creation, a wealth ofpractical details, as well as of various ways of the OAM measurements in electron beams. Importantly, wehave described numerous vortex-induced phenomena, which appear in the interactions of electron vortexbeams with various kinds of samples in TEMs. This is the most promising direction for applications ofvortex beams in electron microscopy, especially for the atomic-resolution mapping of magnetic and chiralproperties .Third, we discussed possible novel phenomena involving vortex electrons outside of the TEM context. Themost exciting opportunities arise for higher energies, where interactions with strong laser fields, quantumparticle collisions, and radiation phenomena can reveal new features depending on the OAM degrees offreedom. The generation of vortex states of high-energy electrons or other quantum particles is a futuremilestone to be achieved experimentally.
Vortex electrons can be considered as only one example in a much wider context of structured statesof quantum particles in free space . Indeed, on the one hand, one can consider various structured modes,such as Hermite–Gaussian-like beams [204, 205] and Airy beams [219, 324, 208, 124]. On the other hand,similar vortex or non-vortex states can be explored for other quantum particles, including neutrons (alreadydemonstrated experimentally) [325], atoms [326, 327] (demonstrated in confined BEC systems [328]), ions[329], and even macroscopic objects such as fullerene molecules [330, 331]. Prospects on why these newstates of quantum matter waves would be useful and how this could be implemented experimentally arecurrently actively discussed.In view of the substantial body of work that was already presented here, one could wonder if thereremains much more to explore with respect to electron vortices in TEMs. It is our firm belief that theseinitial experiments only scratch the surface. For example, it became clear that a vortex detector that wouldmeasure and sort the electron OAM modes would be highly beneficial for electron energy loss spectroscopy(EELS) and magnetic chiral dichroism experiments, but the methods we presented are still a long way fromsuch a versatile instrument. Also, in terms of signal-to-noise ratio and the source-size broadening effect,important steps need to be taken in order to bring atomic resolution mapping of magnetic states closer toreality. New technology breakthroughs in direct electron detectors and electron gun design are, however,slowly providing this progress. There is also a clear potential for using elastic scattering effects in obtainingmagnetic and chirality information. Even though the theory seems well established, there are still manyunexplored areas with potential for applications and further research.In the domain of high energies, we are at the beginning of a long journey. The theoretical formalism fordescribing high-energy collisions with vortex electrons has been developed and applied to a few basic QEDscattering processes. Calculations demonstrate that vortex electrons will give access to quantities which aredifficult or impossible to measure in the usual collision settings. One can now apply this formalism to various75nelastic processes such as bremsstrahlung by vortex electrons, production of hadrons by vortex photonscolliding with protons, and eventually the deep inelastic scattering of ultrarelativistic vortex electrons onhadrons and nuclei, with the aim to access, in a radically different way, the nucleon dynamics inside nucleiand the spin and orbital angular momentum contributions to the proton’s spin. One can also investigatewhat new venues in hadronic and nuclear physics will open up if protons, nuclei, and other particles canalso be experimentally prepared in vortex states.The experimental facilities in high-energy physics face big challenges to deal with vortex electrons: oneneeds to create and manipulate ultra-relativistic vortex electrons, transfer the vortex-electron know-how fromelectron microscopes to collider-like setting, and achieve an even stronger focusing. Therefore, dedicatedexperimental efforts are needed, and they will become worth investing when theorists prove that there willbe a clear scientific payoff.It should be noticed that bringing the spin degrees of freedom into play would considerably enrich physicalphenomena involving vortex electron states. This is already well explored in optics [66], and spin-polarizedelectron sources are used in high-energy domain [332]. At the same time, electron microscopy only startsdeveloping this direction [333, 334, 90].In the meanwhile, optical vortex beams and OAM states of photons, out of which vortex electrons wereborn, still form a very active research field. In general, higher energies require more expensive scientificinstruments. Therefore, optics has an important advantage that optical technology is far more likely toappear in applications. In comparison, electron beam technologies are more likely to be limited to expensivescientific instruments, which typically take much longer to develop and affect the world around us.Thus, electron vortex beams are still in an early stage of development, and many opportunities for futureresearch are open. We hope that we provided a solid basis for the researchers venturing into this excitingdirection, and that this review can help them to see the bigger picture and to avoid pitfalls that might occuralong the way.
Acknowledgements
We acknowledge discussions with Mark R. Dennis and Andrei Afanasev. This work was supported by theRIKEN Interdisciplinary Theoretical Science Research Group (iTHES) Project, the Multi-University Re-search Initiative (MURI) Center for Dynamic Magneto-Optics via the Air Force Office of Scientific Research(AFOSR) (Grant No. FA9550-14-1-0040), Grant-in-Aid for Scientific Research (A), Core Research for Evolu-tionary Science and Technology (CREST), the John Templeton Foundation, the Australian Research Coun-cil, the Portuguese Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) (contract IF/00989/2014/CP1214/CT0004under the IF2014 Program), contracts UID/FIS/00777/2013 and CERN/FIS-NUC/0010/2015 (partiallyfunded through POCTI, COMPETE, QREN, and the European Union), Austrian Science Fund Grant No.I543-N20, the European Research Council under the 7th Framework Program (FP7) (ERC Starting GrantNo. 278510 VORTEX), and FWO PhD Fellowship grants (Aspirant Fonds Wetenschappelijk Onderzoek-Vlaanderen). 76 ppendix A. Conventions and notationsNotation DescriptionAbbreviations:
OAM orbital angular momentumSAM spin angular momentumSOI spin-orbit interactionLG Laguerre–GaussianTEM transmission electron microscopeSTEM scanning transmission electron microscopeMIP mean internal potentialEELS electron energy loss spectroscopyEMCD energy loss magnetic chiral dichroismFT Fourier transformEM electromagnetic
Fundamental constants: (cid:126)
Planck’s constant c speed of light e = −| e | electron’s charge m e electron’s mass α fine-structure constant Units:
Gaussian units are used throughout this review. In addition, the (cid:126) = c = 1 units are used in Section 4. Conventions:r , p , L , etc. 3D vectors r ⊥ , p ⊥ , etc. 2D vectors in the plane orthogonal to the main directionˆ p , ˆ L , etc. quantum-mechanical operators of the corresponding quantities (cid:104) r (cid:105) , (cid:104) p (cid:105) , (cid:104) L (cid:105) , etc. expectation (mean) values of the corresponding operators/quantities(normalized per one electron)¯ x , ¯ y , ¯ z , ¯ ϕ , etc. unit vectors of the corresponding coordinates˜ ψ ( k ), ˜ V ( k ), or (cid:70) ( ψ ), (cid:70) ( V ) Fourier transforms of the corresponding functions ψ ( r ), V ( r ), etc.˜ ψ ( k ) ∗ ˜ V ( k ) convolution of functions r µ , k µ , etc. four-vectors in Minkowski spacetime( k µ r µ ) scalar product of four-vectors Special functions: J (cid:96) Bessel functions of the first kind L (cid:96)n Laguerre–Gaussian polinomialsΘ Heaviside step function δ ( δ ab ) Dirac delta function (Kronecker delta) Table 2: Abbreviations, conventions, and general notations used in this review. otation Descriptionr radius-vector( r, ϕ, z ) cylindrical coordinates (note that r = r ⊥ (cid:54) = | r | ) p momentum (canonical) k wave vector( k ⊥ , φ, k z ) cylindrical coordinates in the wave-vector space( θ, φ, k ) spherical coordinates in the wave-vector space κ fixed radial component of the wave vector in Bessel beams θ fixed polar angle in Bessel beamsˆ H Hamiltonian ψ scalar wave functionΨ multi-component wave functionΦ phase of the wave function E energy (either kinetic or full-relativistic, depending on the problem) ρ probability density or intensity ( ρ e : electric charge density) j probability current ( j e : electric current) L (canonical) orbital angular momentum S spin angular momentum ( s : non-relativistic spin angular momentum) J total angular momentum M magnetic moment (cid:112) kinetic momentum in the presence of a vector-potential (cid:76) kinetic orbital angular momentum in the presence of a vector-potential v velocityΩ angular velocity (angular frequency) of electron’s circular motion(Ω L : Larmor frequency, Ω c : cyclotron frequency) (cid:96) vortex topological charge (azimuthal quantum number) (cid:96) order of the fork-like dislocation in holograms generating vortex beams w beam width ( w : beam waist, w m : transverse magnetic length) z L longitudinal magnetic (Larmor) length A magnetic vector-potential V scalar electric potential (voltage) B magnetic field strength E electric field strength σ sign (direction) of the magnetic field α m dimensionless magnetic flux or magnetic-monopole charge g g -factor of electron’s angular momentum in a magnetic field q = k − k (cid:48) mometum-transfer parameter in the k → k (cid:48) scattering l, m angular-momentum and magnetic quantum numbers for atomic orbitals γ Lorentz factorΛ dimensionless spin-orbit interaction parameter for the Dirac electron χ helicity of Dirac electrons ω and (cid:107) frequency and wave vector of electromagnetic waves (photons) in Sections 4.2 and 5 Table 3: The main physical quantities and their notations used in this review. eferencesReferences [1] A. 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