electron vortices - beams with orbital angular momentum · the physical consequences and potential...
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Electron Vortices - Beams with Orbital Angular Momentum
S. M. Lloyd, M. Babiker, G. Thirunavukkarasu and J. Yuan
Department of Physics,University of York,Heslington, York, YO10 5DD,UK⇤
(Dated: 3rd May 2017)
The recent prediction and subsequent creation of electron vortex beams in a number oflaboratories occurred after almost 20 years had elapsed since the recognition of the phys-ical significance and potential for applications of the orbital angular momentum carriedby optical vortex beams. A rapid growth in interest in electron vortex beams followed,with swift theoretical and experimental developments. Much of the rapid progress canbe attributed in part to the clear similarities between electron optics and photonicsarising from the functional equivalence between the Helmholtz equations governing thefree space propagation of optical beams and the time-independent Schrodinger equationgoverning freely propagating electron vortex beams. There are, however, key di↵erencesin the properties of the two kinds of vortex beams. This review is concerned primar-ily with the electron type, with specific emphasis on the distinguishing vortex features:notably the spin, electric charge, current and magnetic moment, the spatial distribu-tion as well as the associated electric and magnetic fields. The physical consequencesand potential applications of such properties are pointed out and analysed, includingnanoparticle manipulation and the mechanisms of orbital angular momentum transferin the electron vortex interaction with matter.
CONTENTS
I. Introduction 1
II. Quantum mechanics of electron vortex beams 3A. Phase properties of vortex beams 3B. Vortex beam solutions of the Schrodinger equation 4
1. Laguerre-Gaussian beams 52. Bessel beams 63. Bandwidth-limited vortex beams 7
C. Mechanical and electromagnetic properties of theelectron vortex beam 91. Inertial mechanical properties 92. Electromagnetic mechanical properties 10
D. Intrinsic spin-orbit interaction (SOI) 12
III. Dynamics of the electron vortex in external field 14A. Parallel propagation 14B. Transverse propagation 15C. Rotational dynamics of vortex beams 16D. Extrinsic spin-orbit interaction 17E. Electron vortex in the presence of laser fields 18
IV. Generation of electron vortex beams 19A. Phase plate technology 19B. Holographic di↵ractive optics 20
1. Binarised amplitude mask 202. Binary phase mask 233. Blazed phase mask 234. Choice of reference waves 24
C. Electron optics methods 251. Spin to orbital angular momentum conversion 252. Magnetic monopole field 263. Vortex lattices 26
D. Hybrid method 26
⇤ [email protected], [email protected]
1. Lens aberrations 262. Electron vortex mode converter 27
V. Vortex beam analysis 27A. Interferometry 27
1. Electron holography 272. Knife-edge and triangle aperture di↵ractive
interferometry 283. Di↵raction 28
B. Mode conversion analysis 28C. Image rotation 29
1. Gouy rotation 292. Zeeman rotation 29
D. Vortex-vortex interactions and collisions 29E. Factors a↵ecting the size of the vortex beam 30
VI. Interaction with matter 31A. Chiral-specific spectroscopy 31
1. Matrix elements for OAM transfer 322. The e↵ect of o↵-axis vortex beam excitation 323. Plasmon spectroscopy 34
B. Propagation in crystalline materials 35C. Mechanical transfer of orbital angular momentum 36D. Polarization radiation 37
VII. Applications, challenges and conclusions 38Recent papers 39
Acknowledgments 39
List of Symbols and Abbreviations 39
References 41
I. INTRODUCTION
Electron vortex beams are a new member of an ex-panding class of experimentally realisable freely propa-
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gating vortex states having well-defined orbital angularmomentum about their propagation axis; the prototypi-cal example of which is the much studied optical vortexbeam. The term vortex beam refers to a beam of particles- electrons, photons or otherwise - that is freely propagat-ing and possesses a wavefront with quantised topologicalstructure arising from a singularity in phase taking theform eil� with � being the azimuthal angle about thebeam axis and l an integer quantum number also knownas the topological charge (or winding number). The topo-logical structure of the wavefront was first described byNye and Berry (1974) as a screw-type dislocation in thewavetrains in analogy with crystal defects.
Over the last two decades optical vortices have beena subject of much interest, after the publication of theseminal work of Allen et al. (1992) in which the quan-tised orbital angular momentum of a Laguerre-Gaussianlaser mode was examined (the earlier discussion of opticalvortices in laser modes by Coullet et al. (1989) did notemphasise the quantisation of the orbital angular mo-mentum about the propagation axis). Since then, op-tical vortices have been intensively studied leading tomany diverse applications (Allen et al., 2003, 1999; An-drews and Babiker, 2012), including optical tweezers andspanners for various applications (Dholakia et al., 2002;Grier, 2003; He et al., 1995; Ladavac and Grier, 2004):micromanipulation (Galajda and Ormos, 2001); classi-cal and quantum communications (Yao and Padgett,2011); phase contrast imaging in microscopy (Baranekand Bouchal, 2013; Furhapter et al., 2005; Zuchner et al.,2011); as well as further proposed applications in quan-tum information and metrology (Molina-Terriza et al.,2007; Yao and Padgett, 2011) and astronomy (Lee et al.,2006; Tamburini et al., 2011; Thide et al., 2007). The dis-cussion of photonic spin and orbital angular momentumin various situations, and the similarities and di↵erencesbetween the two types of angular momentum have led tonew ways of thinking about, and examining orbital an-gular momentum in this context. The spin and orbitalangular momentum can not be clearly separated in gen-eral, i.e without the imposition of the paraxial approx-imation (Barnett and Allen, 1994; O’Neil et al., 2002;Van Enk et al., 1994), which leads to the possibility ofthe entanglement of the two degrees of freedom (Khouryand Milman, 2011; Mair et al., 2001). More subtle quan-tum e↵ects due to the interaction of optical vortices withatoms and molecules involve internal atomic transitionsat near resonance with the beam frequency. Here too, op-tical forces and torques are at play (Allen et al., 1996a;Andersen et al., 2006; Babiker et al., 1994; Lembessiset al., 2011; Surzhykov et al., 2015), leading to the trap-ping and manipulation of individual atoms in certain re-gions of the beam profile, with promising applicationsin the new field of atomtronics (Andersen et al., 2006;Lembessis and Babiker, 2013; Pepino et al., 2010; Sea-man et al., 2007), as well as the proposed generation of
atom vortex beams (Hayrapetyan et al., 2013; Lembessiset al., 2014). A related recent advance in matter vortexbeams is the realisation of neutron vortex beams in thelaboratory (Clark et al., 2015).Although the basic concepts in terms of beam forma-
tion of electron vortices essentially stem from those en-countered in the optical vortex case, the electron vortexis distinguished by additional properties, most notablythe electric charge and half-integer spin. They are thusfermion vortex states characterised by a scalar field in theform of the Schrodinger wavefunction for non-relativisticelectrons and Dirac spinors for the ultra-relativistic elec-tron beams, while optical vortex beams are bosonic statesdescribed by vector fields. Furthermore, there are sub-stantial di↵erences in scale. Currently, electron vorticescreated in a medium-voltage (100-300 kV) electron mi-croscope have de Broglie wavelengths of the order of pi-cometers while optical vortices in the visible range havewavelengths of the order of several hundreds of nanome-ters. Electron vortex beams can thus probe much smallerfeatures than is possible for the optical vortex beams, andas such the range of applications of electron vortices ispredicted to be substantially di↵erent from the existingscope of optical vortex beams.The earliest work on particle vortex beams is
due to Bialynicki-Birula and Bialynicka-Birula (2001);Bialynicki-Birula et al. (2000, 2001). The current re-search activity specifically in electron vortex beams wasstimulated by work due to Bliokh et al. (2007), shortlyfollowed by the experimental realisation in several labo-ratories (McMorran et al., 2011; Uchida and Tonomura,2010; Verbeeck et al., 2010). It has now been establishedthat electron vortices can be created inside electron mi-croscopes and there exist a number of techniques for vor-tex beam creation, including computer generated holo-graphic masks applied in similar ways to those routinelyadopted in the creation of optical vortex beams (Heck-enberg et al., 1992a,b). This review aims to describe therecent developments in the expanding field of electronvortex physics, and highlight significant areas of poten-tial applications. Specifically, electron vortex beams arehoped to lead to novel applications in microscopical anal-ysis, where the orbital angular momentum of the beamis expected to provide new information about the crys-tallographic, electronic and magnetic composition of thesample. Chiral-dependent electron di↵raction has beendetected (Juchtmans et al., 2015, 2016) as well as thedemonstration of magnetic-dependent electron energy-loss spectroscopy (EELS) (Verbeeck et al., 2010), andit is predicted that the high resolution achievable in theelectron microscope will lead to the ability to map mag-netic information at atomic or near-atomic resolution.Additionally, the inherent phase structure of the vor-tex is considered ideal for applications in high resolu-tion phase contrast imaging, as required for biologicalspecimens with low absorption contrast (Jesacher et al.,
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2005). Applications of electron vortex beams are, how-ever, not restricted to di↵raction, spectroscopy and imag-ing - the orbital angular momentum of the beam may alsobe used for the manipulation of nanoparticles (Gnanavelet al., 2012; Verbeeck et al., 2013), leading to electronspanners analogous to the widely used optical tweezersand spanners. Electron vortex states are also relevant inthe context of quantum information and, in particular,the electron vortex may potentially be used to impartangular momentum into vortices in Bose-Einstein con-densates (Fetter, 2001). The orbital angular momentumand magnetic properties of the electron vortex may alsofind potential uses in spintronic applications, either in thecharacterisation of spintronic devices, or in contexts em-ploying spin-polarised current injection, through spin-to-orbital angular momentum conversion processes (Karimiet al., 2012).
Our aim in writing this review has been to strive toprovide a report on the current state of the new subjectof electron vortex beams and their interactions. We haveendeavoured to survey much of the relevant literature,but any omissions of specific references would certainlybe inadvertent and we would apologise for not havingcome across them. For more focused prospectives, thereaders can consult two recently published papers on thesubject (Harris et al., 2015; Verbeeck et al., 2014).
The outline of this review is as follows: section II in-troduces the quantum mechanics governing the propaga-tion of electron vortex beams, namely the wave equationdiscussed in the non-relativistic and relativistic regimes.The mechanical and electromagnetic properties arisingfrom the vortex mass and electric charge are then con-sidered, along with the role of the vortex fields in thespin-orbit interaction within the beam. Section III cov-ers the dynamics of the electron vortex in external fields.Section IV discusses the various methods for the realisa-tion of electron vortex beams in the laboratory that havehitherto been considered, drawing comparison with thecreation of optical vortex beams wherever such an anal-ogy can be identified. Section V deals with the methodsone can use to analyse the various properties associatedwith vortex beams. The interaction of electron vortexbeams with matter is covered in VI. The prospects ofusing electron vortex beams to determine chirality andother magnetic information are discussed in terms of boththeoretical and experimental considerations, with con-cluding remarks about the field given in section VII.
II. QUANTUM MECHANICS OF ELECTRON VORTEXBEAMS
Freely propagating vortex states having the requiredeil� phase factor may be written as solutions to theSchrodinger, Klein-Gordon and Dirac equations, (Bliokhet al., 2007, 2011; Bliokh and Nori, 2012b; van Boxem
et al., 2013; Karlovets, 2012; Schattschneider and Ver-beeck, 2011) yielding non-relativistic, relativistic andspinor electron vortex beams, respectively. The spatialdistributions of these vortex solutions may take variousforms in the relativistic, non-relativistic and paraxial lim-its and each state is characterised by the distinguishingfeature of a vortex, namely the node on the propaga-tion axis. Such states have been mostly described ei-ther by the Bessel functions, prototypes of non-di↵ractingvortex beams (McGloin and Dholakia, 2005), or by theLaguerre-Gaussian (LG) functions which are well knownin optics (Allen et al., 1992), with LG representing abeam with a well defined waist at the focal plane. Sinceboth the Bessel and Laguerre-Gaussian sets of functionsform complete orthonormal basis sets, any beam, vortexor otherwise, may be described in terms of these vor-tex states. In the non-paraxial and relativistic limits ofthe Dirac equation, the spin of the electron also plays arole, and the particular distribution is modified by a spin-orbit interaction intrinsic to the beam (see section II.D).Note that there is an interesting alternative approachdescribing electron vortices as a natural consequence ofthe skyrmion model of a fermion (Bandyopadhyay et al.,2014, 2016, 2017; Chowdhury et al., 2015), but this willnot be discussed any further in this review.The remainder of this section introduces the specific
properties of the electron vortex beam mainly in thenon-relativistic and paraxial limits, focusing on solutionsto the Schrodinger equation. This will not only enabledirect comparison with the commonly applied paraxialsolutions in optics, a comparison facilitated by the func-tional equivalence between the Schrodinger and the scalarHelmholtz equations, but also illustrates the most impor-tant properties of electron vortex beams and serves as abasis for understanding more complex vortex beams.
A. Phase properties of vortex beams
The phase structure of a vortex wave is topologicallydi↵erent to that of a plane wave. In contrast to a planewave, with phase-fronts that are normal to the propa-gation direction, the phase front of the vortex wave de-scribes a helix about the axis of propagation (Nye andBerry, 1974) such that the phase is dependent on the an-gular position about the axis. This topological structurewas first described by Nye and Berry (Nye and Berry,1974) as screw-type dislocations in wave trains, in anal-ogy with crystal defects. The topological charge l (alsocalled the winding number) quantises this winding suchthat there are l twisted wavefronts about the beams axis,or equivalently a phase change of 2⇡l during a full rota-tion about the axis, as shown in Fig. 1. The phase fac-tor of eil� that gives rise to this helical phase structureis a characteristic feature of orbital angular momentum(c.f. the similar phase factor in the azimuthal components
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of the orbital angular momentum-containing hydrogenicwavefunctions). The functions characterising a vortexbeam propagating with a well defined axis (taken alongthe z direction) have a general form which can be conve-niently written in cylindrical coordinates r(⇢,�, z):
l
(r, t) = u(⇢, z)eil�eikz
ze�i!t, (1)
with u(⇢, z) a suitable mode function such as theLaguerre-Gaussian functions (section B1), which arecharacterised by the azimuthal index l and the radialindex p, or the Bessel functions of the first kind (sectionB2), which are charecterised by just the azimuthal indexl. The helical phase structure of the vortex beam leads tothe phase at the core of the beam as indeterminate sinceit is connected to all possible phases of the wave. Thiscentral phase singularity is not physically viable, and iscompensated by the requirement that all functions mustvanish on axis (at the location of the singularity), givingthe beam a cross-sectional distribution in the form of aring, or concentric rings. This has led to the nickname of‘doughnut’ beams for a particular class of vortex beams- the Laguerre-Gaussian vortices with non-zero windingnumber |l|, but zero radial index (p = 0), each being abright ring surrounding a central dark core. The require-ment that all vortex functions must vanish on axis is,however, not su�cient to describe a vortex beam - theremust be some topological di↵erence between a region ofthe beam containing the vortex, and a region that doesnot (Nye and Berry, 1974). For the present purposes,the topology of the vortex describes the connectednessof the phase fronts. The phase front of the vortex istopologically distinct from that of a plane wave, as onecannot be transformed into the other through continuousdeformations. Similarly, the l = 1 phase front cannot bedeformed into the l = 2 or any other l phase front, andthis is the reason why the winding number l has alsobeen termed the topological charge, characterising the‘strength’ of the vortex. As has been pointed out ear-lier, the phase front of any vortex is characterised by thefactor, eil�, leading to a phase singularity along the prop-agation axis. In order to appreciate the significance of thetopological charge we shall assume that the function given in Eq. 1 represents a wavefunction of a vortex beamstate of a particle of mass m. It is easy to show that aclosed loop integration of the probability current densityj(r) = ~
2mi
{ ⇤(r, t)r (r, t)� (r, t)r ⇤(r, t)} along apath C encircling the axis gives a quantised value propor-tional to the topological charge of the beam (Bialynicki-Birula et al., 2000).
I
C
j(r) · ds = 2⇡~m
l (2)
where ds is a line displacement vector tangental to thepath C.
For the vortex beam given in the form Eq. 1 we have
(b) Phase change in thetransverse plane for l = 1
(c) Spiral phasefront for l = 3
(d) Phase change in thetransverse plane for l = 3
FIG. 1: (Color online) The phase character of l = 1 andl = 3 vortex beams. (a) shows the single helix phase frontof an l = 1 vortex around the beam axis. (b) shows the cor-responding continuous phase ramp in one of the transverseplane perpendicular to the beam axis. The total phase changeon one rotation is exactly 2⇡. For the l = 3 vortex there arethree helical surfaces of constant phase, each move aroundthe axis as shown in (c). This leads to a total phase changeon one rotation of exactly 6⇡. We used wrapped phase repre-sentation, with all phase mapped to values between 0 and 2⇡(represented between blue to red in the 2D phase maps in (b)and (d))), so there are three ’artificial’ phase jumps from 0 to2⇡ in the phase-wrap representation as shown in (d). In bothcases, the only real phase discontinuity of significance is onbeam axis.
the normalised probability current density
j(r) =~m
✓
l
⇢�+ k
z
z
◆
. (3)
Integrating this about a loop enclosing the z axis gives2⇡~m
l, while any other closed path gives zero, showing thetopological distinction between a region of space contain-ing the vortex and one that does not. Thus, on circlingthe z-axis an additional phase of 2⇡l is acquired.
B. Vortex beam solutions of the Schrodinger equation
The wavefunction (r, t) describing an electron vortexbeam is a solution of the Schrodinger equation, namely
H (r, t) = E (r, t) (4)
5
where E is the energy eigenvalue. In free space, theHamiltonian H is given by the kinetic energy of the elec-tron beam only:
H =p2
2m(5)
where p is the linear momentum operator. Equation 4can be re-arranged to look like the Helmholtz equationfor monochromatic light:
r2 (r, t) + k2 (r, t) = 0, (6)
where k is the wave vector of the electron wavefunctionand is given by
k2 =2mE~2 (7)
This equivalence is the basis for treating freely propagat-ing electron and light on the same footing, at least at thescalar field level. Indeed the fields of electron microscopy,ion beam physics and accelerator physics started on thisbasis. The same is true for the electron vortex beamresearch.
In general, the vortex solution of the Schrodinger equa-tion requires the complex wavefunctions to be identicallyzero to cope with the phase indeterminacy at the vor-tex core (Bialynicki-Birula et al., 2000). This means thatboth real and imaginary parts of the wavefunction shouldbe zero separately. Each condition defines a surface andthe vortex core can be considered as the intersection ofthe two surfaces, resulting in a line of vortex cores. In thefollowing, we first consider two simple solutions in whichthe vortex core forms a straight line along the z-axis. Wethen introduce a specific and a general solution, which ismore useful in the context of the practical electron vor-tex beams which are usually generated under bandwidth-limited conditions. In general, the vortex lines can becurved, closed and knotted (O’Holleran et al., 2008), butthey can be regarded as a superposition of the simplerstraight vortex lines introduced below.
1. Laguerre-Gaussian beams
Optical vortex beams are most commonly discussedin terms of Laguerre-Gaussian modes, as these are agood approximation to the vortex modes created fromHermite-Gaussian laser modes (Lax et al., 1975; Pad-gett, 1996). The Laguerre-Gaussian vortex beam statearises as a solution of the paraxial approximation of theHelmholtz equation for light or Schrodinger equation forelectrons in free space:
✓
r2? + 2ik
z
@
@z
◆
= 0; (8)
where is a component of the vector field and the sub-script ? in r? indicates di↵erentiation only with respect
R(z)
z=0
zR
w(z)
w0 2w0
FIG. 2: (Color online) Schematic representation of the Gaus-sian profile showing the characteristic parameters, namely thewidth w(z), with w0 the width at the narrowest part of thebeam; z
R
the Rayleigh range and R(z) the in-plane radius ofcurvature at axial position z
to in-plane (transverse) coordinates. This equation de-scribes a component of the relevant field propagating inthe z direction with axial wavevector of magnitude k
z
.The variations along the axis are considered so small thatthe second axial derivative may be neglected (Kogelnikand Li, 1966; Lax et al., 1975). The solutions of Eq. 8represent a vortex for which the magnitude of the trans-verse momentum ~k? is much smaller than the axial mo-mentum ~k
z
(overall k2? + k2z
= k2). A suitable vortexsolution to Eq. 8 is the Laguerre-Gaussian form, whichhas a Gaussian envelope modified radially by a Laguerrepolynomial, with appropriate phase factors. We have,written in cylindrical polar coordinates r = (⇢,�, z),
LGp,l
(r, t) =C
lp
zR
p
z2R
+ z2
p2⇢
w(z)
!
l
Ll
p
✓
2⇢2
w2(z)
◆
⇥ eikz
ze�i!teil�
⇥ e
⇢� ⇢
2
w
2(z)+ ik
z
⇢
2z
2(z2R
+z
2)�i(2p+|l|+1) tan�1
⇣z
z
R
⌘�
(9)
where Ll
p
(x) is the generalised Laguerre polynomial, withazimuthal index l and radial index p � 0, and nor-malisation factor C
lp
=p
2|l|+1p!/[⇡(|l|+ p)!]. The z-dependence of the Gaussian envelope is depicted in Fig. 2,with the characteristic parameters of width w(z) andRayleigh range z
R
given by
w(z) = w0
s
1�✓
2z
kz
w20
◆2
; (10)
zR
=kz
w20
2. (11)
where w0 = w(0) is the beam radius at focus. The ra-dial profile of the beam varies with the indices p and l.The azimuthal index l is responsible for the beam orbitalangular momentum l~ per electron and may take any in-teger value, either positive or negative; whereas the radial
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index p � 0 specifies the number of intensity maxima, i.e.the number of rings in the radial intensity distribution,such that the beam has p + 1 maxima (for l = 0, thebeam has a central spot, and p additional rings). Thetransverse distributions of the Laguerre-Gaussian beamsare displayed in Fig. 3 for various sets of l and p. As canbe seen, the modes with |l| > 0 have a central minimum,and Eq. 9 has the appropriate eil� phase factor, indicat-ing that the Laguerre-Gaussian modes are endowed withthe required feature of vorticity.
(a) LG00 (b) LG01 (c) LG02
(d) LG10 (e) LG111 (f ) LG12
(g) LG20 (h) LG21 (i) LG22
FIG. 3: (Color online) Intensity distribution patterns for theLG
pl
modes, shown in the z = 0 plane. Intensity is given by| LG
l
|2. The concentric ring structure of the orbital angularmomentum carrying modes is clear, with p + 1 rings. Colourscale shows the intensity variation within individual modes(not the relative intensity variation across all modes). TheLaguerre-Gaussian modes having l < 0 have the same intensitydistributions as shown above, however the phase (not shown)has the opposite sign.
In addition to the phase factor relating to the orbitalangular momentum the Laguerre-Gaussian beam also hasa Gouy phase factor:
exp
�i(2p+ |l|+ 1) arctan
✓
z
zR
◆�
(12)
which is associated with the focusing of the beam at thewaist plane(Feng andWinful, 2001; Petersen et al., 2014).As a result a convergent Gaussian beam experiences a
phase change of ⇡
2 as it passes through the focal planefrom�1 to +1, whereas the phase shift of the Laguerre-Gaussian beam on focusing is given as
�(2p+ |l|+ 1)⇡
2(13)
The Gouy phase shift arises due to the spatial confine-ment of the beam, leading to momentum components inthe transverse direction that contribute to the dynamicphase of the beam (Feng and Winful, 2001; Petersenet al., 2014). Near the focal plane, the rate of changeof the transverse momentum of the Laguerre-Gaussianbeam is larger than that of the fundamental Gaussianbeam due to the more complex radial profile. The mag-nitude of the Gouy phase change thus depends on theradial and azimuthal mode indices.
2. Bessel beams
The Bessel-type electron vortex function takes theform
B
l
(r, t) = Nl
Jl
(k?⇢)eil�eikz
ze�i!t (14)
where Jl
(k?⇢) is the Bessel function of the first kind,of order l, where, as above, l is the topological charge,or winding number. The wavenumbers k
z
and k? arethe axial and transverse wave vector components suchthat |k| =
p
k2z
+ k2?. Nl
is a suitable normalisation fac-tor, determined by the specific boundary conditions ofthe beam. Except for l = 0, all other Bessel functionssatisfy J
l
(0) = 0, so that they are suitable for describ-ing a vortex beam. Their spatial distribution functionsare functions of the radial coordinate ⇢ only, so that incontrast to Laguerre-Gaussian beams, freely propagatingBessel beams are non-di↵ractive (McGloin and Dholakia,2005), and the use of the full Helmholtz or Schrodingerequations coincides with the paraxial limit in this case.The Bessel beams are indeed the simplest type of vor-tex beam and so provide an ideal theoretical platform todetermine the general characteristics of vortex beams.The oscillatory nature of Bessel functions gives the
Bessel beam a cross-section of concentric rings, decreas-ing in brightness away from the axis. This concentricring structure is shown in Fig. 4. However, unlike theLaguerre-Gaussian function, which decays exponentiallywith radial position, the Bessel function is infinite in ex-tent, so that in principle the beam contains an infinitenumber of rings. Each ring of the Bessel beam carriesthe same power in the case of an optical vortex beam(McGloin and Dholakia, 2005) while for electron vortexbeams the relevant property is the current, which im-plies infinite power being carried by the beam, whichis of course physically unrealistic. What is meant bya physical Bessel-type beam is a beam that has am-plitude modulation similar to a Bessel function, over a
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(a) l = 0 (b) l = 1 (c) l = 2
FIG. 4: (Color online) Plots of the intensity | Bl
(r)|2 of aBessel beam with (a) l = 0, (b) l = 1, and (c) l = 2. TheBessel modes with l < 0 have the same intensity distributionsas shown above, however the phase (not shown) has the oppo-site sign.
finite radius, and whose core components behave non-di↵ractively (such that the central maximum or minimumpersists with very little spreading) over a reasonable, butfinite, propagation length (McGloin and Dholakia, 2005).These are achievable by several methods in optics in-cluding axicon lenses, annular apertures and holograms(Durnin et al., 1987; McGloin and Dholakia, 2005), andhave also been generated in electron optics using kino-forms (Grillo et al., 2014a). A kinoform is wavefrontreconstruction device (for a reference see (Jordan et al.,1970))
In the momentum representation k0(k0?,�0, k0
z
) theBessel beam has the form
l
(k0) =i�l
2⇡
eil�0
k?�(k
z
� k0z
)�(k? � k0?); (15)
which is interpreted as a superposition of plane waves ofvarying k? such that k =
p
k2? + k2z
for each wave. Fora given k
z
the possible k? lie on a ring on the surfaceof constant k, so that there is a cone of plane waves ofvarying k that constitute the Bessel beam (Bliokh et al.,2011; McGloin and Dholakia, 2005), with the phase ofeach given by eil'. This is the principle by which axiconlenses produce the rings of a Bessel-type beam (Hermanand Wiggins, 1991). The conical propagation leads toanother interesting property of the Bessel beam, namelythat the original spatial distribution is reconstructed af-ter propagation past an obstruction (MacDonald et al.,1996; McGloin and Dholakia, 2005), as has been demon-strated for electron vortex Bessel beams (Grillo et al.,2014b).
3. Bandwidth-limited vortex beams
In electron vortex beam research, the limited trans-verse spatial coherence of practical electron sourcesmeans that finite radius vortex modes defined using acircular aperture (or pupil) function is more appropriatein real situations. The simplest bandwidth limited vor-tex beam is generated by the Fraunhofer di↵raction of a
FIG. 5: (Color online) The Fourier transform of the Besselbeam results in a set of waves of fixed k
z
and varying kx
andky
, such that k? =p
k2x
+ k2y
. The vortex Bessel beam illus-
trated here has a phase factor eil', so that the phase changesby 2⇡l on rotation about the k
z
axis. This is illustrated forl = 1. The relationship between k
z
and k? fixes the cone an-gle ✓.
plane wave by a spiral phase plate with the transmissionfunction:
(⇢0,�0, z) = eil�0eikz
z (16)
through an aperture of finite radius Rmax
. We have usedthe convention for the momentum/Fourier space repre-sentation of the variables as in Eq. 15, since the trunca-tion is in practice taking place in the aperture plane of aconvergent electron lens as shown in Fig. 6.The di↵racted beam intensity is related to the Fourier
transform of the transmitted wave, which can be writtenas (Kotlyar et al., 2006, 2007; Lubk et al., 2013a)
(⇢,�, z) = eil�2⇡il2�l
⇢
l
R
2+l
max
(2+l)�(1+l) x
1F2(1 +l
2 ; 2 +l
2 ; 1 + l;� 14⇢R
2max
)(17)
wherep
Fq
(a; b; c; z) is the generalized hypergeometricfunction. This is a limiting case of Hypergeometric-Gaussian beams (Karimi et al., 2007) due to di↵ractionof apertured spiral phase masks by a Gaussian beam.To represent the waveform of the arbitrary beam
in similar bandwidth-limited situations, an orthonor-mal basis set characterised by orbital angular mo-mentum has recently been reported for vortex beams(Thirunavukkarasu et al., 2017), including both the az-imuthal and radial quantum numbers l and p, respec-tively. It is based on describing the normal modes of thetransverse wavefront confined to a finite radius at thepupil or aperture plane by an orthonormal set of trun-cated Bessel functions, much like the solutions of the
8
allowed normal modes of surface vibrations on a drumsurface:
TBB
p,l
(⇢0,�0, z) = Np,l
eikz
zeil�0Jl
(kpl
? ⇢0) for ⇢0 R
max
(18)
where the radial and azimuthal indices are p and l re-spectively following the convention used in the case ofLG modes. We again used the cylindrical coordinates(⇢0,�0, z) to describe the location in the aperture planeand R
max
is the radius of the circular aperture. The mag-nitude of the transverse wavevector kpl
? takes the discrete
values �
p,l
R
max
, with �p,l
the (p+ 1)th zero of the lth orderBessel function J
l
, and Rmax
is the radius of the aper-ture. The truncated Bessel functions, whose amplitudesare for ⇢ � R
max
, together with the azimuthal phasefactor form a complete two-dimensional basis set of theOAM modes at the aperture plane. The Fourier trans-form of these truncated Bessel beams forms a conjugateset of quantum basis set which we termed Fourier Trans-formed Truncated Bessel Beams (FT-TBB):
FT�TBB
p,l
(k⇢
,�, z) = il�p,l
J 0l
(�p,l
)eil�Jl
(k⇢
Rmax
)
kpl
?2 � k2
⇢
(19)
where kq
is the transverse wavevector of the di↵ractedbeams. At the focal plane of a lens of power 1/f , the
corresponding radial displacement (⇢) is given by fk
q
k
z
(⇠ fk
q
k0). The corresponding wavefunction in the focal
plane coordinate (⇢,�) becomes:
FT�TBB
p,l
(⇢,�, z) = il�p,l
f
k0J 0l
(�p,l
)eil�Jl
(k⇢
Rmax
)
⇢2p
l
� ⇢2
(20)
where ⇢p,l
is the radius of the most prominent doughnut
ring and is given by �
p,l
f
R
max
k0.
The inset in Fig. 6 shows the schematic phase distribu-tion of some of the low order basis wavefunctions of thetruncated Bessel beams (TBB). Also shown is the conju-gate relationship between the TBB and and the FT-TBB.
The amplitude and phase of the first three p-modes ofthe l = 1 FT-TBB subset are shown in Fig.7 respectively.These results show that the higher order radial modesare distinguished by p+1 bright rings, reminiscent of thecorresponding LG modes (Allen et al., 1992). However,the similarity does not extend to the additional faint ringstructures that can be seen in the amplitude distributionof the FT-TBB1,3 mode. These small ringed structuresare caused by the ringing e↵ect of the sharply definedaperture. Another noticeable feature is that the largestamplitude occurs when k
⇢
approaches kpl
? , in which casethe wavefunction locally becomes a sinc function of theradial coordinate.
FIG. 6: (Color online) The intensity and phase distributionof the transverse wavefunctions of FT-TBB at z = 0. Imageafter Thirunavukkarasu et al. (2017) and are plotted for thesame relative scale.
FIG. 7: (Color online) Computer-simulated fine structure ofthe di↵raction of Truncated Bessel Beams (FT-TBB): Firstrow for intensity and the second row for the correspondingphase distribution. From left to right, for the FT-TBB vortexbeams with l = 1 but with di↵erent radial modes (p = 0, 1 and2). Images adapted after Thirunavukkarasu et al. (2017)
This can also be understood by regarding the originalmask as the product of the unobstructed Bessel beamand a top-hat mask function. The transverse structuresof the vortex beam at the focal plane can be consideredas the convolution of the Fourier transform of the Besselfunction (whose transverse structure is a ring with a ra-dius controlled by the radial size of the first dark zone inthe mask) and that of the top-hat mask (which is the wellknown Airy pattern with side band ring structures). Thisis consistent with the mathematical form of the Fouriertransform of the normal modes in the aperture plane. Asp increases, the size of the first node ring shrinks andthe Bessel ring at the focal plane increases in size. Thisexplains the size changes seen in Fig.7 (first row) for dif-ferent values of p. The convolution of the Bessel ringswith the Airy pattern functions results in side bands, but
9
they preserve the circular symmetry of the main Besselpeaks. The details of the experimental realization of theFT-TBB beams can be found in ref. (Thirunavukkarasuet al., 2017)
The FT-TBB set of beams is one of the bandwidth-limited vortex beams whose spatial frequency is deter-mined by the aperture size. As this FT-TBB set of modesforms a complete orthonormal set, it can be used to de-scribe any such bandwidth-limited vortex beam. For ex-ample, the simplest and most investigated bandwidth-limited electron vortex beam as shown in Eq.17 can beexpanded in terms of linear combinations of the FT-TBBset (Schattschneider and Verbeeck, 2011).
Both LG and Bessel beams are unbound solutions andare often discussed in theoretical developments becauseof their mathematical simplicity. On the other hand,bandwidth-limited beams are required for the precise de-scription of electron vortex beams produced in real situ-ations.
C. Mechanical and electromagnetic properties of theelectron vortex beam
The global mechanical properties of electron vortexmodes stem from the two basic properties of electrons,namely that the finite electron mass leads to inertialposition-dependent mass flux with which are associatedglobal inertial linear and angular momenta of the elec-tron vortex beam while the finite electronic charge leadsto position-dependent electromagnetic fields which arefurther sources of global linear and angular momenta.It is instructive to derive these global properties of theelectron vortex with reference to the Bessel type, formathematical convienence. Here we outline the treat-ment by Lloyd et al. (2013) who were first to show thatthe mechanical and electromagnetic properties of elec-tron vortices emerge directly from the quantum mechan-ical wavefunction of the vortex mode. Concentrating onthe Bessel-type vortex beam for which the wavefunctionis given in Eq.(14) and writing ! = E/~ we have
(r, t) = Nl
Jl
(k?⇢)eikzzeil�e�iEt/~, (21)
The vortex beam is assumed to extend along the axis overa length D which is much larger than the beam width.The normalisation factor N
l
follows straightforwardly inthe form
Nl
=
k2?
2⇡DI(1)l
!
12
, (22)
where I(1)l
is the first moment integral of the Bessel func-tion defined by
I(1)l
=
Z 1
0
|Jl
(x)|2xdx. (23)
1. Inertial mechanical properties
The inertial mechanical properties are associated withthe finite electron mass and these can now be derived asfollows. The vortex wavefunction (r, t) gives rise to alocal mass density ⇢m(r, t) and a mass current densityjm(r, t) which are as follows
⇢m(r, t) = m ⇤(r, t) (r, t); (24)
jm(r, t) =~2i
{ ⇤(r, t)r (r, t)� (r, t)r ⇤(r, t)} .
(25)
These emerge on substituting for (r, t) in the form
⇢m(r, t) = m|Nl
|2|Jl
(k?⇢)|2; (26)
jm(r, t) = ~|Nl
|2✓
l
⇢���+ k
z
z
◆
|Jl
(k?⇢)|2. (27)
where ��� and z form, with ⇢⇢⇢, the standard unit vector setfor cylindrical coordinates. The unit vector z is along thebeam axis.The evaluation of the global inertial linear momentum
of the vortex follows from the realisation that the (local)mass current density (j
m
) is the same as the (local) linearmomentum density (P
m
, i.e. (local) linear momentumper unit volume). The (global) inertial linear momen-tum vector of the Bessel electron vortex beam (Pm) thenfollows by volume integration. We have
Pm =
Z
Pm
(r, t)dV =
Z
jm
(r, t)dV. (28)
We find
Pm =~|Nl
|2D⇥Z 1
0
Z 2⇡
0
d�
⇢
kz
z+l
⇢���
�
|Jl
(k?⇢)|2⇢d⇢.
(29)
It is easy to see that the azimuthal component in theintegrand ofPm when integrated over the volume leads toa zero value because of a vanishing angular integral. Bycontrast the z-component leads to a finite result. Directintegration of the z-component in Eq. 29 gives
Pm = 2⇡~kz
DI(1)l
|Nl
|2z= ~k
z
z. (30)
where we have made use of Eq. 22. The resultPm = ~kz
zis the inertial linear momentum of the Bessel electronvortex beam. Note that the inertial linear momentumis axial, involving only the axial component k
z
of the
10
wavevector component. There are no in-plane compo-nents, neither as azimuthal, nor radial, and there is nodependence on k?. Note also that the azimuthal linearmomentum density is non-zero, but, as we have shownabove, its volume integral vanishes. This is consistentwith the cylindrical symmetry of the vortex beam.
The (local) inertial orbital angular momentum density(Lm) is defined as the moment of the (local) inertial lin-ear momentum density. We have
Lm = r⇥Pm
(r, t)
= ~|Nl
|2 (⇢⇢⇢⇢+ zz)⇥⇢
l
⇢���+ k
z
z
�
|Jl
(k?⇢)|2.
(31)
Integration of this over the volume leads us to the(global) inertial orbital angular momentum vector (Lm).We find
Lm =
Z
LmdV,
= ~|Nl
|2Z
D/2
�D/2
Z 2⇡
0
Z 1
0
⇢
lz� ⇢kz
���� l
⇢z⇢⇢⇢
�
⇥|Jl
(k?⇢)|2⇢d⇢d�dz. (32)
It is easy to verify that the angular and radial integralslead to vanishing results and only the axial componentsurvives. Using Eq. 22 the inertial angular momentumcan be written in the form
Lm = 2⇡l~DI(1)l
|Nl
|2z = ~lz. (33)
Equation 33 shows that, in general, the electron vortexbeam carries only an axial inertial orbital angular mo-mentum equal to ~l and that, as is the case with theinertial linear momentum of the vortex, there are notransverse components. The components of the inertialangular momentum L
x
and Ly
are both zero as well asthe inertial linear momentum components P
x
and Py
. Itturns out that this feature is not a preserve of electronvortex beams alone and holds for all pure vortex beamsincluding the Bessel- and Laguerre-Gaussian-optical vor-tex beams.
2. Electromagnetic mechanical properties
We have so far concentrated on the inertial mechan-ical properties of the electron vortex i.e. those due to
the finite electron mass and such a theory applies to anyelectrically neutral particle vortex. However, an electronvortex also carries electric and magnetic fields E(r) andB(r) by virtue of the finite electric charge and magneticmoment. These fields have been evaluated in Lloyd et al.(2012b) for a Bessel electron vortex beam generated un-der a condition that would be found in an electron mi-croscope. The outlines are as follows.Direct use of Maxwell’s equations with the charge and
current distributions of the electron vortex beam consid-ered as sources enables the the evaluation of the specificelectric and magnetic fields of the electron vortex. TheBessel electron vortex defined in Eq. 14 possesses cylin-drically symmetric charge and current density distribu-tions, each varying only as a function of the radial coordi-nate ⇢, so that the electric and magnetic fields also havesuch a cylindrical symmetry. Respectively, the chargeand current densities for the electron Bessel beam arefound to be
⇢e
(⇢) =� e|Nl
|2J2l
(k?⇢); (34)
je
(⇢) =� e~|Nl
|2m
e
Jl
(k?⇢)
✓
l
⇢���+ k
z
z
◆
. (35)
As with a linear charge and current source, the electronvortex beam possesses an electric field in the radial direc-tion and a magnetic field in the azimuthal direction. Inaddition, due to the helical (solenoid-like) nature of thecharged current density distribution, the Bessel vortexbeam is characterised by an axial magnetic field com-ponent (Lloyd et al., 2012b). The electric field for theBessel beam of order l is found to be
E(⇢) =� ⇢⇢⇢e|N
l
|22"0
⇥
⇢�
J2l
(k?⇢)� Jl�1(k?⇢)Jl+1(k?⇢)
�
. (36)
This is valid for any l, including the non-vortex Besselbeam of l = 0. Similarly, the azimuthal component ofthe magnetic field of the same electron vortex beam isfound to take the form
B�
(⇢) =eµ0~kz
|Nl
|22m
e
⇥
⇢�
J2l
(k?⇢)� Jl�1(k?⇢)Jl+1(k?⇢)
�
. (37)
Finally, the axial component of the magnetic field is givenby the following expression
Bz
(⇢) = eµ0~|N
l
|22m
e
1�4�l⇢2l2F3
⇥
{l, l + 12}; {l + 1, l + 1, 2l + 1};�⇢2
⇤
l2[�(l)]2
!
(38)
wherep
Fq
[{a1 . . . ap}; {b1 . . . bq}; z] is the generalised hy- pergeometric function, and �(x) the gamma function.
11
This general form reduces to a series of products of Besselfunctions for particular values of l (Abramowitz and Ste-gun, 1972). The above expressions Eq. 36, Eq. 37 andEq. 38 are valid for the Bessel beam of infinite radialextent. In order to estimate the field strengths for vor-tices such as would typically be generated in an electronmicroscope, the wavefunction Eq. 14 may be truncatedafter a finite number of rings, i.e.
t
l
(r) = l
(r)⇥(⇢� ⇢l,n
) (39)
where ⇥(x) is the Heaviside step function, and ⇢l,n
= ↵
ln
k?is the radius corresponding to the nth zero of the Besselfunction of order l, such that J
l
(↵ln
) = 0. This wave-function may then be applied to generate the charge andcurrent densities of the truncated beam, (⇢
e
)tl
and jtl
,which are the sources for the electric and magnetic fieldsassociated with the truncated vortex beam. We now con-sider a typical electron vortex beam created in a typicalelectron microscope. We choose a truncated Bessel beamwith a single ring and the following parameters, which aretypical for contemporary electron microscopes
Beam energy: E = 200 keV;
Beam current: I = 1nA;
Axial wavevector: kz
= 2.29104⇥ 1012 m�1;
Radial wavevector: k? = 2.29104⇥ 1010 m�1,
the normalisation factor Nl
can be expressed in terms ofthe beam current I, which is a measurable quantity. Wehave
N2l
=Ik2?m
2⇡e~kz
R
↵1l
0J2l
(x)xdx. (40)
Figures 8 and 9 show the spatial distributions of the fieldcomponents for the truncated Bessel vortex beams hav-ing l = 1 and l = 10. The field strengths are seen tobe rather small for the parameters chosen. However,the field strengths scale linearly with the electric current(I), so that, in principle, electron vortices with largerfield strengths could be produced experimentally. Thez-component is particularly interesting, as it arises dueto the vortex nature of the beam, and is highly localisedin a region of the order of an A. We also note that al-though the field strengths are particularly small, theirgradients are rather large within the core region and theaxial magnetic field gradient is of the order of hundredsof Tm�1. This suggests that the electron vortex canimpart a large force on a magnetic particle. Thus theelectron vortex beam could potentially find applicationsin investigation of quantum mechanical phase e↵ects dueto localised magnetic systems and has implications forthe possibility of observing the Aharonov-Bohm e↵ect(Aharonov and Bohm, 1959) at small scales.
In general the vortex electric field is radial and is afunction of the radial coordinate ⇢ only. It can be written
FIG. 8: (Color online) The electric fields of the Bessel beamsof finite radial extent, for (a) l = 1 and (b) l = 10 respectively.Adapted from Lloyd et al. (2012b)
succinctly as:
E(⇢) = ⇢E⇢
(⇢), (41)
while the vortex magnetic field has two orthogonal com-ponents, one axial and another azimuthal
B(⇢) = zBz
(⇢) + ���B�
(⇢). (42)
The (local) electromagnetic linear momentum densityemerges straightforwardly as follows
Pem =✏0E⇥B,
=✏0n
zE⇢
B�
� ���E⇢
Bz
o
. (43)
and the corresponding (globle) electromagnetic linearmomentum of the vortex beam follows by volume inte-gration. Evaluating the z-integral yields
Pem = ✏0D
Z 1
0
⇢d⇢
Z 2⇡
0
d�n
zE⇢
B�
� ���E⇢
Bz
o
.
(44)
Since the fields are functions only of ⇢ (Lloyd et al.,2012b), once again we note that the � component givesa vanishing integral and we are left only with the axialcomponent. We have
Pem = z2⇡✏0D
Z 1
0
E⇢
B�
⇢d⇢. (45)
12
FIG. 9: (Color online) The magnetic fields of the Besselbeam of finite radial extent, for (a) l = 1 and (b) 10 respec-tively. Note that the z-components of the magnetic fields aretwo orders of magnitude smaller than the �-component. Wehave assumed that the vortex beam is su�ciently long, as fora current carrying solenoid, allowing us to ignore beam endinge↵ects. Adapted from Lloyd et al. (2012b)
The radial integral can be done numerically using expres-sions for the fields as functions of ⇢.
Next we evaluate the electromagnetic angular momen-tum contributions as the integrals of the moment of theelectromagnetic linear momentum density. We have
Lem =
Z
dV r⇥Pem,
= ✏0
Z
dVn
⇢⇢⇢zE⇢
Bz
� ���⇢E⇢
B�
� z⇢E⇢
Bz
o
,
(46)
and it is easy to verify that we have a vanishing � integral.The integral of the ⇢ component also vanishes and onlyan axial component remains. We then have
Lem = �z2⇡✏0D
Z 1
0
E⇢
Bz
⇢2d⇢. (47)
Equations 45 and 47 are two further contributions to themechanical properties of the electron vortex due to thevortex electromagnetic nature to be added to Eqs. 30and 33 arising from the inertial properties.
Lloyd et al. (2013) considered orders of magnitude thatcould arise in a feasible experimental arrangement as-
suming electron vortices created inside a 1 nA electronmicroscope of energy 200 keV and transverse wavevectorcomponent k? = 0.01k
z
. Estimates for the electromag-netic linear and orbital angular momenta in this scenarioare found to be as follows
Pem ⇡ 10�34kg m s�1 (48)
Lem ⇡ 10�48J s (49)
These are both extremely small compared to the inertialcounterparts. The ratios are as follows
Pem
Pm⇡ 10�12; (50)
Lem
Lm⇡ 10�14, (51)
so for practical purposes the electromagnetic linear andorbital angular momenta in such an electron microscopevortex beam are very small. However, in other contextsthe electromagnetic contributions could be non-negligibleas for example when the vortex beam is created in a linearaccelerator.Lloyd et al. (2013) also speculated on the orders of
magnitude when electron vortex beams are used to rotatenano-particles. They focused on the e↵ects of the vortexbeam on a nano-particle in the form of a small cylinderof radius R and length d = R whose axis is assumed tocoincide with that of the vortex beam. A laser beam actsto first levitate the nanoparticle against gravity as wellas the axial force of the vortex beam so that we only haverotational dynamics. For orientation as to orders of mag-nitude we assume that a minimum angular momentum of~ is transferred to the nanoparticle. We can estimate avalue for the angular frequency of a nanoparticle of fusedsilica of mass density of approximately 2.2⇥ 103kg m�3
and radius of 10�8 m and we find using ~ = I⌦ is
⌦ ⇡ 87.6Hz (52)
where I is the moment of inertia of the particle aboutits axis. This is much higher than the angular frequencyreported by Gnanavel et al. (2012) and Verbeeck et al.(2013) for a gold nanoparticle on a support. The ex-periments seem to indicate that the rotation is dampeddue to friction between the nanoparticle and the support.The set-up described in Lloyd et al. (2013) in which thenanoparticle is optically levitated would eliminate the ef-fects of friction due to a support. Nanoparticles of fusedsilica would be easier to rotate as a controlled optical lev-itation of a metallic nano-particle would be more di�cultto achieve.
D. Intrinsic spin-orbit interaction (SOI)
The spin-orbit interaction arising in the non-relativistic limit is well understood for electrons with or-bital angular momentum bound within atomic orbitals; a
13
similar phenomenon can be described for a vortex propa-gating in a radially inhomogeneous, but axially invariantfield (Leary et al., 2009, 2008; Lloyd et al., 2012b). Thesource of this coupling is well-known in the case of anexternal field, however in the case of relativistic electronvortices, an intrinsic spin-orbit interaction is also shownto arise (Bliokh et al., 2011). The origin of the interactionis di↵erent in the two cases - the extrinsic, non-relativisticcoupling compared to the intrinsic coupling within therelativistic beams - however the coupling mechanisms aresimilar, and may be viewed as the electron travelling inan e↵ective magnetic field, or alternatively as a mani-festation of geometric phase (Bliokh et al., 2011; Learyet al., 2008). The geometrical origins of the spin-orbitcoupling have also been invoked to explain the origin ofspin-orbit interaction in optical vortices, which of courseare una↵ected by a magnetic field (Allen et al., 1996b;Berard and Mohrbach, 2006; Bliokh, 2006; Leary et al.,2009). In this section the e↵ects and basis of the spin-orbit interaction in electron vortex beams are discussed.
In the relativistic and non-paraxial limits, the electronvortex beam exhibits an intrinsic spin-orbit interaction(Bliokh et al., 2011), in which the orbital angular mo-mentum depends upon the spin polarisation of the beam.Setting c = 1, the relativistic and non-paraxial electronvortex eigenstates are found to be (Bliokh et al., 2011)
l
=eikz
z�i!t
p2
"
p
1 + m
E
w�
z
cos ✓weil�J
l
(k?⇢)
+ i
0
B
B
@
00
�� sin ✓0
1
C
C
A
ei(l�1)�Jl�1(k?⇢)
+ i
0
B
B
@
000
↵ sin ✓
1
C
C
A
ei(l+1)�Jl+1(k?⇢)
#
.
(53)
where w, the two component spinor characterizing theelectron polarization in the rest frame with E= m, isgiven by:
w =
✓
↵�
◆
, (54)
with ↵ and � are projections of the spin polarizationstate in the basic eigenstates of S
z
in the electron restframe and |↵|2 + |�|2 = 1. =
p
1� m
E and the angle✓ describing the angle of the cone of Bessel plane waves,as shown in Fig. 5.
It can be seen that two ‘extra’ modes arise in the rela-tivistic electron vortex solutions, making the relativisticelectron vortex a mixed state of l and l± 1 Bessel modesfor spin s = ±1/2. These modes represent contributionsof the small components of the Dirac spinor, and vanish
in both the non-relativistic and paraxial limits. In thenon-relativistic limit ! 0, and in the paraxial limitwe have sin ✓ ! 0 so that the pure l mode is recovered.Such solutions demonstrate that spin and orbital angularmomentum are not always separately well-defined exceptwithin the paraxial limit, with the relativistic electronvortex solutions showing similarities to the non-paraxialoptical vortex solutions which also demonstrate an in-trinsic spin-orbit interaction (Barnett and Allen, 1994;Bliokh et al., 2010; Jauregui, 2004). Such an SOI is de-scribed as intrinsic as it does not require the influenceof an external field or propagation medium; the non-paraxial or relativistic solutions are eigenmodes of thetotal angular momentum operator, but not separately ofthe spin and orbital angular momentum operators (Bar-nett and Allen, 1994; Bliokh et al., 2010, 2011).
The origin of the intrinsic spin-orbit interaction is ge-ometric in nature, arising from the momentum depen-dence of the modified spin and orbital angular momen-tum operators required to maintain the invariance of thetotal orbital angular momentum J. This leads to a Berrygauge field k⇥S
k
2 , with corresponding curvature kk
3 havinga monopole structure (Bliokh et al., 2010, 2011). Thismonopole curvature leads to the accumulation of Berryphase about the momentum spectrum of the Bessel beam(see Fig. 5), shifting the relative phase of the plane wavesand modifying the orbital angular momentum of thebeam. The solid angle of the curvature field subtendedby the Bessel beam spectrum determines the magnitudeof the coupling; a larger range of k? - i.e. a larger coneopening angle ✓ - increases the Berry phase shift acrossthe spectrum. Thus, in the paraxial approximation thespin-orbit interaction disappears, and spin and orbitalangular momentum are fully separable. The geometri-cal arguments discussed here are also applicable to theextrinsic spin-orbit interactions mentioned above, withthe gauge and curvature in such cases originating from agross orbital trajectory of the beam, such as in the pho-tonic spin and orbital Hall e↵ects (Bliokh et al., 2008;Bliokh, 2006) and the motion of electrons in an inhomo-geneous e↵ective magnetic field (Berard and Mohrbach,2006; Karimi et al., 2012; Leary et al., 2008) or the prop-agation of photons in an inhomogeneous medium (Berardand Mohrbach, 2006; Leary et al., 2009).
Bliokh and Nori (2012b) extended the treatment of theelectron vortex to polychromatic beams and showed thatsuch beam can carry intrinsic OAM at an arbitrary angleto the mean momentum of the beam.
14
III. DYNAMICS OF THE ELECTRON VORTEX INEXTERNAL FIELD
Electrons subject to an external magnetic field obey aSchrodinger equation in the form
H =(p� eA)2
2m . (55)
where pkin = p�eA is the kinetic linear momentum op-erator with A the magnetic vector potential. For exam-ple, for a magnetic field B in the axial, i.e. z-, direction,the magnetic vector potential is azimuthal in direction:
A =B⇢
2���. (56)
Solutions of the Schrodinger equation in the presenceof several di↵erent field configurations have been investi-gated (Bliokh et al., 2012; van Boxem et al., 2013; Gal-latin and McMorran, 2012; Greenshields et al., 2012,2014; Schattschneider et al., 2017; Velasco-Martınezet al., 2016). The Landau configuration involves a con-stant magnetic field in a fixed direction (Bliokh and Nori,2012a; Gallatin and McMorran, 2012; Greenshields et al.,2012; Landau and Lifshitz, 1977), whilst the Aharanov-Bohm configuration can involve a single line of flux(Bliokh and Nori, 2012a). The vortex propagation direc-tion may be transverse (Gallatin and McMorran, 2012),parallel (Bliokh et al., 2012; Greenshields et al., 2012) tothe direction of the field or in an arbitrary orientation(Greenshields et al., 2014), and the interaction betweenthe magnetic moment with the external field leads to in-teresting dynamics, as we now explain.
A. Parallel propagation
For the case when the electron vortex is propagatingin the same direction as the magnetic field (B = B
z
z)the non-relativistic Hamiltonian of the system takes theform (Greenshields et al., 2012)
H =p2z
2m+
p2?2m
+1
2m!2
L
⇢2 + !L
(Lz
z+ gs
S) (57)
where Lz
is the axial component of the orbital angularmomentum operator, S the spin vector operator, g
s
thegyromagnetic ratio and !
L
= eB
z
2m the Larmor frequency.Here the subscripts z and? stand for axial and transversevector components respectively.
In the Aharonov-Bohm configuration, with a single lineof flux, the solutions to Eq. 55 take the form of Besselbeams, with current density winding around the flux line(Bliokh and Nori, 2012a). Similarly, the coaxially prop-agating eigenstates of Eq. 55 for a vector potential asin Eq. 56 have the form of non-di↵racting Laguerre-Gaussian modes, with a fixed ’magnetic’ width (w
B
) andmagnetic Rayleigh range (z
B
) given by the strength of
the field (Bliokh and Nori, 2012a; Greenshields et al.,2012).
wB
= 2
s
~|eB| ; z
B
= 2
p2Em|eB| . (58)
The presence of the field alters the probability currentdensity (Bliokh and Nori, 2012a); j ! 1
m
h |p� eA| i,so that the z-component of the observable kinetic OAMmay be written as
hLz
i = m s ⇢j�
dV
h | i (59)
The Bessel beam eigenstates of the single flux line takethe form
AB = NJ|l�↵|(k?)ei(l�+k
z
z), (60)
which are clearly still eigenstates of Lz
, with eigenvalue~l, but the observable OAM is now di↵erent. Bliokh etal. (Bliokh and Nori, 2012a) introduced a parameter ↵ =e'/2⇡~ where ' is the magnetic flux. The expectationvalue of the observable kinetic OAM is then given by:
hLz
i = ~(l � ↵); (61)
while the charge density distribution is also altered - sothat Bessel beams with l = ±1 do not have symmetriccharge distributions in the presence of an external field.This is due to a Zeeman interaction between the magneticmoment of the beam and the field. Here, though, due tothe infinitesimal localisation of the flux line along a nodeof the Bessel beam, there is no energy or phase shiftwithin the beam.If the vector potential A relates to a constant mag-
netic field B, the beam then undergoes phase and energyshifts. In this case, the eigenfunctions of Eq. 55 take theform of non-di↵racting Laguerre-Gaussian modes, beingdescribed radially by the Laguerre-Gaussian functions,but having no width variation (Bliokh and Nori, 2012a;Greenshields et al., 2012):
L =Nl,p
1
wB
p2⇢
wB
!|l|
L|l|p
✓
2⇢2
w2B
◆
⇥
e�⇢
2/w
2Beil�eikl,p
z (62)
with wB
given by Eq. 58, and the longitudinal (axial)momentum now depends on the OAM and radial quan-tum numbers. Since the beam propagates parallel to thefield, there is no transverse deflection, and the gross tra-jectory is unchanged, however the circulation within thevector potential alters the beam phase as well as the ob-servable orbital angular momentum, depending on therelative direction of the circulation of the field and the
15
electron vortex. The energy eigenvalues take the form
E =~2k2
z
2m� ~!
L
l + ~|!L
| (2p+ |l|+ 1)
=~2k2
z
2m+ ~|!
L
| (2n+ 1) ; (63)
i.e. the dynamics is that of free motion with superposedquantised Landau levels of index n
L
, such that
nL
= p+|l|2(1 + sgn(Bl)) . (64)
The second term in the first line of Eq.63 has the formof a Zeeman interaction between the field and the or-bital angular momentum, while the third has the sameform as the Gouy phase term in the di↵racting Laguerre-Gaussian beam, relating to the transverse confinement ofthe mode (Feng and Winful, 2001). The Landau levelsdescribed by Eq. 64 are due to the combined e↵ects of theZeeman and Gouy phase shifts (Bliokh and Nori, 2012a;Greenshields et al., 2012), where the Zeeman energy shiftmay be written in terms of the observable orbital angularmomentum
E =~2k2
z
2m� !
L
hLz
i ; (65)
with hLz
i incorporating the Gouy phase shift
hLz
i = ~✓
l + sgn(B) h2⇢2
w2B
i◆
= ~ [l + sgn(B) (2p+ |l|+ 1)] . (66)
The Gouy and Zeeman energy shifts lead to additionalphase accumulation on the propagation of the vortexmode, described by the phase factor of e�i�(z/z
B
), with
� = [l + sgn(B) (2p+ |l|+ 1)] . (67)
Unlike the Bessel beam eigenfunctions of the singleflux line, the charge density distribution of the LandauLaguerre-Gaussian beams are not a↵ected by the externalfield. On the other hand, the current density is altered ina surprising way, as shown in Fig. 10. If the beam orbitalangular momentum vector is parallel to the field directionthen the current density flow is in the same direction asthat of the free-space Laguerre-Gaussian mode, and theobservable OAM is enhanced. For those modes with l 0the observable orbital angular momentum turns out tobe independent of the orbital angular momentum of themode, and is always greater than zero (Bliokh and Nori,2012a; Greenshields et al., 2012), so that there is now cir-culation in the current density of those modes with l = 0.Additionally, the current density of the modes with l < 0changes direction at the radius of maximum intensity,
⇢m
= wB
q
|l|2 , due to competition between the negative
intrinsic vortex current, which dominates at ⇢ < ⇢m
, and
FIG. 10: (Color online) The e↵ect of the magnetic fieldon the circulating current of the non-di↵racting Laguerre-Gaussian vortex modes, for a magnetic field directed in thepositive z-direction, such that the vector potential has thesame sense of circulation as the l > 1 vortex modes. The localcharacter of the azimuthal current is indicated with arrows -it can be seen that the presence of the field induces circula-tion, even in the modes for which l = 0. For those modes withl > 0 the observable OAM (listed at the top of each figure)is increased from that of the bare mode, while those beamswith l 0 have a fixed observable OAM, determined by thefield strength and the radial quantum number p. For thosebeams with l < 0 the azimuthal current changes direction atthe point of maximum intensity. The top (bottom) panel is forthe non-di↵racting LG vortex modes with radial index p = 0(p = 1). The scale bar length is 2 w
B
. Image adapted fromBliokh et al. (2012).
the positive circulation induced by the external potential(Bliokh and Nori, 2012a).Greenshields et al. (2014) explored the conceptual is-
sue of conservation of orbital angular momentum for anelectron beam in a uniform colinear magnetic field. Theyshowed that the electric field associated with an electronbeam with an extended probability distribution, such asthat discussed by Lloyd et al. (2012b) for electron Besselbeam, when coupled to the external magnetic field, con-tribute an angular momentum which precisely ensuresthe conservation of the canonical orbital angular momen-tum of the electron beam in a magnetic field.
B. Transverse propagation
Substituting Eq.56 into Eq.55 and choosing Laguerre-Gaussian solutions propagating perpendicular to themagnetic field reveals interesting dynamics relating tothe orientation of the OAM vector projected onto theelectron trajectory - or the helicity of the beam (Gallatinand McMorran, 2012)
⌃ = L · p/|p| (68)
In such a field, an electron wavepacket with orbital an-gular momentum follows the gross circular trajectory ex-pected from classical electrodynamics, orbiting around
16
FIG. 11: (Color online) Phase and intensity of the electronwavepacket around the circular cyclotron trajectory. Thewavepacket can be seen to rotate at !
L
= !c
/2, such thatafter one rotation the OAM vector points opposite to the orig-inal direction. The length and width of the wavepacket alsooscillates on rotation, with the cyclotron frequency. Note thatthe scale of the plots of <( 1) is roughly five orders of mag-nitude smaller than that of the | 1|2 plots, so that the phasecan be seen. Image from Gallatin and McMorran (2012).
the z-axis at the cyclotron frequency !c
= e|B|/m, withthe expected quantised Landau levels. The spin helicityis conserved, however the orbital angular momentum vec-tor is found to precess with the Larmor frequency (equalto half the cyclotron frequency), !
L
= !c
/2. Thus, ontraversing a cyclotron orbit, the resulting orbital angularmomentum vector now points in the direction oppositeto the initial orientation, as shown in Fig. 11 (Gallatinand McMorran, 2012).
In addition to the rotation of the angular momentumvector, the physical extent of the wavepacket is also foundto be oscillatory when the propagating states are not ex-act eigenstates of the Hamiltonian Eq.55 (Gallatin andMcMorran, 2012). Competition between di↵ractive ef-fects arising from the propagation of the beam and focus-ing e↵ects arising due to the confinement of the harmonicpotential of the gross circular motion cause the lengthand width of the wavepacket to oscillate, as can be seenin Fig. 11. These e↵ects can be balanced to avoid such os-cillations for wavepackets having the characteristic widthand the Rayleigh range determined by the strength of thefield (Bliokh and Nori, 2012a; Gallatin and McMorran,2012; Greenshields et al., 2012):
The rotation of the orbital angular momentum helic-ity has important implications for electron vortex beamssubject to transverse external fields. In particular, thetransverse field components of the magnetic electron
lenses may cause some reorientation of the orbital angu-lar momentum vector as the beam spirals about the focalpoint. In particular this e↵ect will have consequences forthe orientation of the beam orbital angular momentumwhen the sample is inside the lens field, as the change inorientation will not be reversed by the transverse fieldsof the opposite direction, at the exit of the lens. On theother hand, these transverse components are small, sothat the dominant e↵ect is expected to come from thevortex propagating coaxially with the vector potential.
C. Rotational dynamics of vortex beams
By rotation dynamics, we mean the rotation of thetransverse structure of the vortex beams as a function ofthe distance along the beam axis. As the beams are trav-elling waves, this can also be considered as time-evolutionof the wavefronts. This is to be distinguished from thetime-dependent changes in the transverse structure of thevortex beams at a given plane perpendicular to the beamaxis. Such time-dependent changes are due to the coher-ent interference of electron waves with di↵erent energies,a process which is more challenging to investigate exper-imentally.The phase change due to the influence of a constant
magnetic field parallel to the beam axis may be observedin the form of rotations of asymmetric superposition ofvortex states (Bliokh and Nori, 2012a; Greenshields et al.,2012; Guzzinati et al., 2013). This e↵ect is very similarto the Faraday e↵ect in optics causing rotation of spinpolarisation (Faraday, 1936; Nienhuis et al., 1992). Theorientation of linearly polarised optical beams rotateson propagation through a magnetic field, due to circu-lar birefringence acting oppositely on the two spin com-ponents. This e↵ect is not observed for optical orbitalangular momentum, although a ‘mechanical Faraday ef-fect’ can be induced with the rotation of the mediumthough which the optical vortex superposition propagates(Franke-Arnold et al., 2011). In electron beams such anorbital Faraday e↵ect arises due to the Zeeman interac-tion between the field and the vortex magnetic moment,with apparent rotation due to the di↵erence in the ac-tion of the field on the phase of the vortices counter- andco-propagating with the field (Bliokh and Nori, 2012a;Greenshields et al., 2012). Whilst observable in the elec-tron microscope as an image rotation, it is quite distinctfrom the Lorentz force rotation that is well known inelectron lens systems, with the Faraday rotation occur-ring even when there is no transverse motion of the beamaxis with respect to the field.Although the presence of the field causes a change in
the phase as described above, the rotation e↵ect is ob-servable only in the superposition of vortex states, due tothe rotational symmetry of the single vortex mode. Thesuperposition may have a zero or an non-zero net angu-
17
lar momentum, to be referred to as ‘balanced’ or ‘un-balanced’ superposition, respectively. In each case therotation is independent of l, depending instead on theenergy of the beam and the strength of the field. For thebalanced superposition given by
b
= LG
l,p
+ LG
�l,p
0 (69)
the intensity distribution is that of a ‘petal’ pattern with2l lobes. As discussed above, the change in phase onpropagation is given as �z/z
B
. For superposition withfixed energy, the change in phase due to the presence ofthe field will necessarily be di↵erent for each vortex com-ponent, with the deviation from the kinetic phase factorgiven as k = k
z
⌥ sgn(B)�/zB
for the ±l modes respec-tively. The Gouy phase shift a↵ects both components inthe same way, whereas the Zeeman term leads to an l de-pendent phase of ⌥ sgn(B)lz/z
B
for the ±l components.The phase di↵erence between the two beams is observ-able as a rotation of the interference pattern by an angle(Bliokh and Nori, 2012a; Greenshields et al., 2012)
��l,�l
= sgn(B)z/zB
. (70)
This is independent of l, varying with the strength of thefield and the energy of the beam through the characteris-tic length z
B
= ~kz
/m!L
, so the characteristic frequencyof the image rotation is the Larmor frequency.
Unbalanced superposition of two or more vortex statesare those for which the net angular momentum of thebeam is non-zero. For example:
ub
= LG
0,p + LG
l,p
0 . (71)
Such a superposition has an intensity profile charac-terised by l o↵-axis vortices. In the case of the unbal-anced superposition under the influence of the magneticfield, existence of image rotations depends on the relativedirection of the field and the beam propagation (Bliokhand Nori, 2012a). For the case when the beam is propa-gating along the field direction with a net positive OAM,i.e. l sgn(B) > 0 the mode rotates on propagation, withthe rotation angle given by
��0,l = 2 sgn(B)z/zB
, (72)
which is, once again, independent of the value of the or-bital angular momentum. On the other hand, for thosesituations having l sgn(B) < 0 there is no image rota-tion at all, due to the additional Gouy phase terms can-celling with the Zeeman phase acquired from the counter-circulating field (Bliokh and Nori, 2012a).
D. Extrinsic spin-orbit interaction
The extrinsic spin-orbit interaction (SOI) for electronvortex beams arises from the magnetic moment of the
electron interacting with an e↵ective magnetic field dueto its motion within an inhomogeneous external poten-tial. In order to facilitate SOI in the vortex beam,the external field must be radially inhomogeneous andaxially invariant. In the non-relativistic limit, the ap-propriate SOI may be derived by performing a Foldy-Wouthuysen transformation of the Dirac equation in thepresence of fields (Bjorken and Drell, 1964). The Foldy-Wouthuysen transformation is a unitary transformationU = eiS(t), where S(t) is an odd, self-adjoint opera-tor. The Foldy-Wouthuysen transformation diagonalisesthe Dirac Hamiltonian such that the particle and anti-particle solutions are not mixed, allowing the small com-ponents of the spinor wavefunctions to be systemati-cally incorporated and their residual e↵ects ultimatelyneglected (Foldy and Wouthuysen, 1950). The transfor-mation takes the form
HFW = eiS(t)
✓
H� i~ @@t
◆
e�iS(t) (73)
where H is the Dirac Hamiltonian. Applying the trans-formation, and expanding in powers of (mc2)�1 yieldsa series expansion of the Dirac Hamiltonian for particlesolutions. The first few terms give the Pauli equation,the non-relativistic Schrodinger equation with relativis-tic corrections, including the spin-orbit interaction term
HFW =mc2 + e�+p2
2m� p4
8m3c2
� e~2m
��� ·B� e~28m2c2
r ·E
� e~4m2c2
��� · (E⇥ p� p⇥E) . (74)
where � is the Coulomb potential. The last term is rel-evant for the spin-orbit interaction - since we deal withexternal electrostatic field we have r⇥E = 0; addition-ally E = �r�, so that the SOI may be written as (Learyet al., 2008; Lloyd et al., 2012b)
HSO = � e~4m2c2
✓
1
⇢
@�
@⇢
◆
��� · (⇢⇢⇢⇥ p)
= � e
2m2c2
✓
1
⇢
@�
@⇢
◆
S · L. (75)
It can be seen that, in contrast to the atomic SOI, onlythe z-components of the spin and orbital angular mo-menta are relevant. This SOI Hamiltonian may now beapplied perturbatively to the non-relativistic vortex so-lutions, to find the energy shift for the parallel or anti-parallel spin and orbital angular momenta:
�(l,s) = �~2ls h | ⇠ | i (76)
where
⇠ =e
2m2c2
✓
1
⇢
@�
@⇢
◆
. (77)
18
For a positive Coulomb potential, with electric fieldpointing radially outwards, the parallel (anti-parallel)states shift upward (downward) in energy (Leary et al.,2008; Lloyd et al., 2012b). This causes a splitting of theparallel and anti-parallel states, which can be observed inthe rotation of superposition of parallel and anti-parallelstates having the same s but opposite l. The interfer-ence between such states gives rise to a characteristic‘petal’-like interference pattern. If the parallel and anti-parallel states have slightly di↵erent beam energies thenthe petal-pattern will rotate as a function of time. On theother hand, if the energy is kept fixed then the two stateswill have di↵erent axial momenta, and the pattern willrotate as a function of position (Leary et al., 2008). Thisallows for the possibility of observation of the relativesplitting between the parallel and anti-parallel angularmomentum vortex states.
Consider again the electron vortex of section III. Theextrinsic spin-orbit coupling results may be used to findthe approximate energy splitting of an electron in a vor-tex beam, travelling within the mean field generated bythe beam current. For an electron vortex with l = 1,the electric field is of the order of a few hundred Vm�1,giving a splitting of E of the following magnitude:
�l=1 ⇡ 3⇥ 10�13 eV (78)
This is very small - too small for direct measurementwithin the electron microscope, however indirect mea-surement might be a possibility, such as those involvingspin-flip processes of a spin-polarised beam.
In the optical vortex case, the role of the external fieldis played by the refractive index of an inhomogeneous,anisotropic medium (Marrucci et al., 2006). The cou-pling between the spin and orbital angular momentumof light has been exploited in the generation of polarisedoptical vortices, using specially structured liquid crystalcells, known as q-plates (Marrucci et al., 2006; Marrucci,2013). Particular choices of the liquid crystal structureallow for the complete conversion of spin angular mo-mentum into orbital angular momentum, controlling thesign of the resulting OAM through the polarisation ofthe input mode. A similar conversion process for elec-tron vortex beams, involving q-plates, involving a spe-cially structured magnetic field configuration, has alsobeen proposed (Grillo et al., 2013; Karimi et al., 2012).
E. Electron vortex in the presence of laser fields
The Schrodinger equation of a quasi-relativistic elec-tron vortex beam in the presence of electromagnetic fieldhas been set out in Lloyd et al. (2012a). For relativisticelectron beams, a second order Dirac equation incorpo-rating a transverse electromagnetic field A is given by
(c = 1):
(p� eA)2 �m2 � ie
2Fµ⌫
�µ⌫
�
= 0 (79)
where pµ = (i@t
,�i5) is the electron four momen-tum operator, F
µ⌫
= @µ
A⌫
� @⌫
Aµ
is the electromag-netic field tensor. �µ⌫ is the spin tensor defined by2�µ⌫ = �µ�⌫ � �⌫�µ, with �µ the 4x4 Dirac matrices.Bialynicki-Birula and Radozycki (2006) have shown thatclassical (cyclotron) and quantum (Landau) orbits of acharged particle in a constant magnetic field can be con-trolled by electromagnetic waves with embedded vortexlines.One area of interest is in the interaction of electron vor-
tex beam in the presence of strong field generated by anultrashort light pulse. Hayrapetyan et al. (Hayrapetyanet al., 2014) examined the case of a head-on collision of arelativistic electron vortex beam and a short laser pulseand have shown that the orbital angular momentum com-ponents of the laser field couple to the total angular mo-mentum of the electrons, causing the centre of the beamto be shifted with respect to the centre of the field-freeelectron vortex beam. Theoretical estimates suggest thata shift of 0.02 nm may be induced in a 300 kV electronvortex beams using a moderately strong laser pulse, soexperimental observation is challenging but feasible.Ivanov (2012b) considered the case for electron-photon
interactions and demonstrated the entanglement arisingfrom conservation of the sum of the helicity shown inEq.(80) below for particle-particle collisions also appliesin the case of electron-photon interactions. There arealso plans to use inverse Compton scattering of laser lightby electron vortex beams to generate structured X-raybeams (Seipt et al., 2014)In addition to the dynamics of the interactions of freely
propagating vortices and anti-vortices, collisions betweenelectrons and other particles carrying orbital angular mo-mentum may also be considered (Ivanov and Serbo, 2011;Ivanov, 2012b; Jentschura and Serbo, 2011; Seipt et al.,2014), with implications for generating high energy vor-tices of various species. In such two-particle scatteringsituations, there are several possible outcomes - the par-ticles may be scattered to a range of final states, in-cluding plane-plane or vortex-plane wave states (Ivanovand Serbo, 2011; Jentschura and Serbo, 2011), or vortex-vortex entangled states (Ivanov, 2012b). In high energyelectron-particle scattering, due to the scattering geome-try, such collision processes are suitably described by con-sideration of the orbital helicity of each particle, i.e. theprojection of the orbital angular momentum onto theparticles momentum. When both particles are allowedto scatter into a final vortex state from an initial plane-vortex collision it is found that there are two degrees ofentanglement between the two states - the transverse mo-mentum k? and the orbital helicity, ⌃. It is found that
19
the sum of the helicities of the two final states is approx-imately the same as the helicity of the incident vortexstate, i.e.
⌃f,1 + ⌃f,2 ⇡ ⌃
i
, (80)
the larger helicity tending to accompany the larger trans-verse momentum, which may fall within a range deter-mined by the transverse momentum of the incident vor-tex (Ivanov, 2012b). This result may be applied to gener-ate vortex-entangled particles or di↵erent species by col-lision, or additionally opens up the possibility of applyingvortex states to high energy particle physics through col-liding high energy vortex electrons with protons or otherparticles.
IV. GENERATION OF ELECTRON VORTEX BEAMS
Due to the similarities between the wave equations forelectron and optical vortices, it has in many cases beenrelatively straightforward to adapt successful ideas fromoptical vortex research to similar ends in electron vortexapplications. For the particulars of optical vortex gener-ation, we refer the reader to the recent reviews (Molina-Terriza et al., 2007; Yao and Padgett, 2011). Broadly, themethods by which electron vortex beams may be gener-ated may be classified into three main categories involv-ing phase plates, di↵ractive optics and electron optics;with those based on phase plates and di↵ractive opticsbeing analogues of optical vortex technologies, and elec-tron optics methods forming an entirely new area. Ini-tial success in electron vortex beam production involvedmaterials-based di↵ractive elements and phase plates,however the electron optics approach appears promisingin terms of versatility and overcoming the e�ciency lim-itations of the holographic plates (Yuan, 2014). In addi-tion, photoemission has been considered to be a possiblesource of electron vortex beams (Takahashi and Nagaosa,2015).
A. Phase plate technology
The concept of the phase plate for an electron beamis not new (Nagayama, 2011), with early examples suchas the Zernike phase plate designed for enhancing phasecontrast of biological materials (Kanaya et al., 1958). Inthe absence of any external fields, the e↵ective refractiveindex for an electron travelling within a solid is given by(Reimer and Kohl, 2008)
ne↵ = 1� eU
EE0 + E2E0 + E (81)
where E0 is the electron rest mass energy, E the kineticenergy of the incident electron, U the material specificmean inner potential. For example, ne↵ is approximately
1.00082 in silicon nitride for 100 keV electrons. The rel-ative phase shift for a material of thickness �t is thengive by
�� = 2⇡(ne↵ � 1)�t
�(82)
For example, to achieve a relative phase delay of ⇡ fora 200 keV beam, which has a wavelength of 2.5 pm, athickness di↵erence in a silicon nitride film of 42 nm isrequired (Shiloh et al., 2014) (the precise thickness mayvary (Bhattacharyya et al., 2006; Grillo et al., 2014a;Harvey et al., 2014), depending on many experimentalfactors such as the crystallinity of the film, the surfacecoating both intended and incidental such as carbon con-tamination).Spiral phase plates are well known in optical vortex
beam generation (Sueda et al., 2004; Turnbull, 1996; Yaoand Padgett, 2011), and consist of a thin film plate of arefractive material whose thickness changes continuouslyabout the axis. The helical shaped thickness profiles ofthe plates impart angular momenta to a transmittinglaser beam. Phase plates may be produced for millimetrewavelengths down to optical wavelengths (Sueda et al.,2004; Turnbull, 1996; Yao and Padgett, 2011).The first experimental demonstration of an electron
vortex beam by Uchida and Tonomura (2010) involvedthe use of a stepped spiral phase plate constructed ofstacked graphite flakes. Their stepped phase plate wasmade of spontaneously stacked flakes of graphite, lead-ing to a discrete - rather than a continuous change ofits thickness profile. The edges of the steps cause ex-tra phase jumps, to appear at di↵erent points in thebeam cross-section - in addition to the discrete 2⇡ phasechange of the desired vortex structure. This was ob-served in Uchida and Tonomura’s experimental resultsvia interference patterns and in-plane phase profile. Thetransmitted beam thus did not demonstrate the requiredcharacteristics of a pure vortex state with integer orbitalangular momentum, but was nevertheless the first ex-perimental demonstration of a freely propagating mixedvortex state. However, the particular arrangement of thegraphite flakes cannot be properly controlled and theylead to phase defects. In addition, being made of car-bon, under the influence of the high energy electron beamthe flakes are subject to damage and deformation, andthe phase plate loses its integrity. As such, the steppedgraphite phase plate is not suitable for long-term, repro-ducible vortex generation.The production of continuous spiral phase plates has
recently been attempted using focused ion beam etchingof a silicon nitride membrane (Shiloh et al., 2014). How-ever, the resulting doughnut-shaped beam profile has anopening, indicating a non-integer vortex beam has beenproduced (Berry, 2004), probably because of the di�-culty of achieving precise refractive index and wavelengthmatching requirements. It has been estimated that 1 nm
20
thickness error results in 2 � 3% unwanted variation inphase. On the other hand, a similar phase variation canbe obtained by a change of 1 kV in the electron energy,indicating that there is some room for adjustment af-ter the phase plate has been made. Thus the challengeof fabricating the spiral phase plate for electron vortexbeam is to produce a thickness profile to the requiredsmoothness and precision. Another inherent problem ofusing matter-based electron-optical elements is the un-avoidable scattering of the transmitted electrons otherthan those arising from mean inner potential. This cancause undesired e↵ects such as additional phase shift orloss of coherence.
B. Holographic di↵ractive optics
Holographic reconstruction is a well known techniquein both optical and electron microscopy (Gabor, 1948;Saleh and Teich, 1991; Tonomura, 1987). It has been usedto increase image resolution by reconstructing the im-age from the interference pattern between the di↵ractedand non-di↵racted components of the incident wave. Thesame principles of holography may be used to reconstructan image or a wavefunction. The method involves passingan input wave though a computer generated hologram(CGH) generated via the interference between a refer-ence wave and the required output mode. Most of theholographic di↵ractive optical masks produced to dateare constructed out of thin membrane films of materi-als such as silicon nitride. This can either be of varyingthicknesses for phase grating structures or can be of auniform thickness with additional heavy metal coating,forming a binary amplitude grating structure. There hasalso been a proposal to use the Kapitza-Dirac e↵ect toproduce electron vortex beams by the formation of op-tical dislocated gratings (Handali et al., 2015). When apulsed laser source is used this method enables pulsedelectron vortex beams to be produced for use in time-resolved studies.
The holograms employed in vortex optics are con-structed from the interference pattern between the vortexmode and a non-vortex reference wave:
I2holo
(r) =| vortex
+ ref
|2 (83)
where the choice is often for the reference wave to be atilted plane wave, although other waves such as sphericalreference waves (Saitoh et al., 2012; Verbeeck et al., 2012)can also be used.
The holographic masks for vortex beam generation areversatile and are relatively easy to produce; however,they also have their drawbacks. Most of the masks re-ported to date are based on binary amplitude modula-tion holography (Clark et al., 2012; McMorran et al.,2011; Verbeeck et al., 2010), and as such produce very
low intensity in the desired di↵raction orders, of approx-imately 6% of the incident beam intensity. This is be-cause most of the beam intensity incident on the mask isblocked, although improvements have recently been re-ported with the use of phase modulation holograms (seesection IV.B.3). Furthermore, since multiple di↵ractionmodes are produced in the binary holographic mask, theisolation of a mode of interest for specific applications canbe tricky (Idrobo and Pennycook, 2011; Krivanek et al.,2014; Pohl et al., 2016) and this feature limits the practi-cal use of holographic masks as a vortex beam generator.
1. Binarised amplitude mask
Direct interference of a vortex beam with a plane ref-erence wave produces a characteristic pattern of fringeswith smoothly varying intensity. The reproduction ofthis continuous variation, either in terms of varying filmthickness or some other properties of the materials, is as,or even more, technically demanding as the fabrication ofa smoothly varying spiral phase plate. A more practicalalternative is to replace the continuous intensity variationby a discrete variation, such as the selective total removalof material in order to mask the beam, such that emptyspaces correspond to fringe maxima, and material regionsof beam blocking thickness correspond to minima of theinterference pattern. Such a mask is known as a binaryamplitude hologram, and is well known in optics (Lee,1979). For electron beams, binary holographic masksmay be relatively simply fabricated by focused ion beam(FIB) etching or electron lithography. Using such a bi-narised CGH mask, Verbeeck et al. (2010) produced thefirst pure electron vortex beam. The principles behind
FIG. 12: The typical forked mask pattern produced by bi-narisation of the interference pattern between a Bessel vortexwave and a plane reference wave for (a) l = 1 beam, and (b)l = 3 beam.
generating a suitable binarised transmission grating canbe demonstrated using the case of a simple vortex modetravelling in the z direction (neglecting any normalisation
21
factors, see section II.B.3) given by
vortex
(r) ⌘ l
(r) = eil�eikz
z (84)
The hologram pattern is generated by the interference ofthis mode with a reference plane wave travelling at anangle such that k
x
is the component of the plane wavemomentum orthogonal to the z direction:
ref
(r) ⌘ p
(r) = eikx
xeikz
z (85)
Any component of the plane wave momentum in the zdirection does not contribute to the interference pattern.The interference is constructed by evaluating the super-position of the two waves at z = 0:
Iholo
=1
4|
l
+ p
|2, (86)
which is particular to the beam of interest. The prefactoris chosen to produce a pattern which oscillates between0 and 1. For the phase vortex, the characteristic patternis an edge dislocation, with l + 1 edges - also known asa fork dislocation. The interference pattern may then bebinarised by clipping the pattern, for example
Ibiholo(⇢,�) =
(
1 Iholo
� 0.5, ⇢ Rmax
0 Iholo
< 0.5, ⇢ > Rmax
(87)
for a maximum aperture radius Rmax. A similar pro-cess has been followed to produce holographic masks us-ing Bessel or Laguerre-Gaussian vortex functions (Clarket al., 2012). Despite the very di↵erent radial struc-tures of the Bessel, Laguerre-Gaussian or the simple vor-tex beam of Eq. 84, the resulting binarised holographicmasks are very similar (Clark et al., 2012), indicating thebinarisation primary picks out the strong intensity vari-ation due to the phase variation and ’irons out’ the moresubtle amplitude variation of the di↵erent types of vortexmodes.
The binary masks for an l = 1 and an l = 3 Besselbeam are displayed in Fig. 12, showing the correspond-ing l + 1 edge dislocations. Such a mask pattern is thenembedded into something opaque to the radiation of in-terest - a printed film (Heckenberg et al., 1992a,b) or aspatial light modulator (Yao and Padgett, 2011) for opti-cal beams, or, in the case of electron beams, a FIB etchedmetal or silicon nitride film (McMorran et al., 2011; Ver-beeck et al., 2010) - and placed into the path of the in-cident beam. Di↵raction of the beam through the maskproduces the desired vortex beams.
The far-field di↵raction pattern resulting from thetransmission of a plane wave through a CGH mask isgiven by the Fourier transform of the mask. This pro-duces a non-di↵racted zero-order beam, along with a se-ries of vortex beams and their complex conjugates. Fromordinary di↵raction grating theory, we know that even
(a)
(b)
(c)
FIG. 13: (Color online) The far field di↵raction pattern andphase distribution of the binary CGH amplitude mask to pro-duce (l = 1, p = 1) FT-TBB beams shown in Fig. 7. (a) showsthe di↵racted beams, with several di↵raction orders present;high intensity is indicated by white, zero by black. The phaseof the beams is shown in (b), the opposite phase of the twosets of sidebands can be seen, with the nth order beams dis-playing a phase change of 2⇡n. The rainbow scale indicatesphase change from 0 (red) to 2⇡ (purple). (c) shows the in-tensity of the various vortex beams. The central minima of allthe di↵racting beams are clear in the intensity patterns.
harmonics will be absent if the widths of the masked grat-ing elements is half of their spacing (Born et al., 1997).When this condition is only approximately satisfied, acommon occurrence for many binary amplitude masks,the odd order di↵raction beams are usually more intensethan the even order beams, an example for the di↵ractionof a binary amplitude CGH mask is shown in Fig. 13.
A characteristic feature of the binarisation process isto introduce higher di↵raction orders, generated from thestep edge approximation to the sinusoidal transmissionvariation expected from the straightforward superposi-tion of the vortex and the reference waves. This allowsfor the generation of higher order vortex beams thanthose encoding the original interference pattern, due tothe inclusion of the high order harmonics of the originalsuperposition in the binary holographic mask. Takingadvantage of this mechanism, very high orders of orbitalangular momentum have been demonstrated in electronbeams - using a CGH mask designed for l = 25 beams,vortices having l = 100 have been demonstrated (McMor-ran et al., 2011). Naturally, these higher order di↵racted
22
beams are significantly less intense than the first orderbeams. Recently, it has been shown that an electron vor-tex beam of winding number l = 200 can be generatedfrom the first order di↵raction peak (Grillo et al., 2015).
The CGH mask itself is not chiral, and so, unlike aphase plate, cannot impart orbital angular momentum tothe transmitted beam by directly modulating the phaseof the wavefront. Instead, the mask decomposes the in-put plane wave into a set of left- and right-handed vor-tices, so that the vanishing total orbital angular momen-tum of the incident wave is conserved (and so the maskmay also act as a mode analyser (Saitoh et al., 2012)).The various di↵raction orders propagate from the maskat some angle �
s
to the optical axis of the incident beam,so that in the far-field di↵erent vortex beams are angu-larly separated. The magnitude of the transverse wave-vector of the reference wave, k
x
, relative to the longi-tudinal wave-vector, k
z
, determines the angles at whichthe di↵racted beams exit the hologram, such that a largekx
increases the angular separation between the di↵erentdi↵racted orders (Heckenberg et al., 1992a; McMorranet al., 2011). The angle of the first order di↵racted beam,for a Bessel vortex beam, is
�s
=�
d=
kx
kz
(88)
for grating separation d, which is related inversely to kx
.The nth order di↵racted beam emerges at an angle n�
s
while the zero-order beam propagates along the originaldirection of the incident wave. A particular di↵ractionorder of interest may be realigned to this optical axis byilluminating the hologram with a beam with transversemomentum nk
x
(Schattschneider et al., 2012b). Thetransverse momentum of the vortex beam itself k? is de-termined by the size of the mask aperture; for the Besselbeam we have
k? =�1lRmax
(89)
with �1l ⇡ 3.81 the first zero of the Bessel function Jl
(x).A similar relationship applies for other vortex beams. Forexample, in the case of the Laguerre-Gaussian modes,�1l is replaced by the relevant radius of the Laguerre-Gaussian mode at z = 0.
In a binary hologram, the phase information is encodedas a lateral shift of the interference fringes, suggestingthat the phase structure of the vortex beam can be re-covered whatever the shape of the fringe (Clark et al.,2012).
CGH masks are much more versatile and controllablethan spiral phase plates as described in section IV.A.Each di↵raction order is a pure phase vortex of strengthnl as each is a unique vortex ‘harmonic’ of the incidentvortex beam with orbital angular momentum of integerorder l~. The CGH masks may be constructed out of ma-terials that are resistant to beam damage, and will have
a longer useful lifetime than a spiral phase plate con-structed of graphite thin films - in addition the results aredirectly reproducible, and in principle any order of orbitalangular momentum may be specified, as demonstrated bythe production of electron vortex beams with high valuesof orbital angular momentum (McMorran et al., 2011).On the other hand, it should be noted that the binaryCGH mask itself will block much of the incoming beam,so that only ⇠ 50% of the incident intensity is trans-mitted. Approximately 25% of the incident intensity ischanneled into the zero order beam, with the higher orderbeam decreasing in intensity. The first order di↵ractedbeams share ⇠ 12% of the incident intensity, which is notsu�cient for many experimental applications.
Forked apertures as described above have been usedto generate electron vortices in transmission electron mi-croscopes (TEM) (McMorran et al., 2011; Schattschnei-der et al., 2012b; Verbeeck et al., 2011a, 2010), and arenow a standard technique in experimental electron vor-tex physics. The first proof-of-principle demonstrationinvolved a 5 µm diameter CGH mask cut from plat-inum foil, with a single fork dislocation generating leftand right handed vortex beams. The second instance ofthis holographic vortex generation involved silicon nitridefilms milled with very high resolution features (McMor-ran et al., 2011); the high resolution milling allowed forthe cutting of a narrowly spaced grating-like linear pat-terns, so that the di↵racting beams produced had a largeangular separation. The high resolution milling also en-abled the fine features of holographic masks for generat-ing vortex beams of higher order to be accurately repro-duced. A forked mask encoding a vortex beam with atopological charge l = 25 was demonstrated. For struc-tural stability of this fine featured CGH mask, the edgedislocation is not reproduced within the central region.However, leaving a solid block at the very centre of themask did not seem to significantly impair the function ofthe mask, and vortices with clear central dark cores wereproduced with the fourth order di↵racted beam carry-ing 100~ orbital angular momentum (McMorran et al.,2011), demonstrating the versatility of the uses of binaryCGH masks over spiral phase plates.
The phase structure and vorticity of the resultingbeams were confirmed by observation of the forked inter-ference fringes (McMorran et al., 2011; Verbeeck et al.,2010), and by the persistence of the axial dark core ofthe vortex on propagation and di↵raction (McMorranet al., 2011). A beam that simply has an annular profilewill spread radially both inwards and outwards, obliter-ating the central zero intensity away from the focal point,whereas a beam with a phase singularity must preservethis singularity, as the orbital angular momentum mustbe conserved.
23
FIG. 14: (Color online) Comparison of the theory and thesimulation of the non-zero order approximate Bessel type elec-tron vortex beam. The topological order of the beams are de-noted by n, which is identical to the l defined in this review.Image adapted from Fig. 4 of Grillo et al. (2014b).
2. Binary phase mask
The binarised CGH masks discussed above rely on theamplitude modulation of the transmitted wave in orderto separate the orbital angular momentum components.Di↵raction can also occur through a phase modulatingCGH grating (Grillo et al., 2014b; Harvey et al., 2014)and o↵ers an alternative method for the generation ofvortex beams.
As in the case of the phase plate technology, a phaseCGH mask can be produced by imprinting the desiredphase change into a silicon nitride film with the thicknessvariation given by:
��(⇢,�) = CU�t(⇢,�) (90)
where C is a coe�cient determined by the energy of theelectron beam and U is the mean inner potential as de-fined in the context of Eq.81. The advantages of thephase grating technique are the increased control overthe radial structure of the vortex beam and that it couldpotentially enable higher intensities to be transmittedinto desired orders.
As a demonstration of the control of the radial struc-ture of the vortex beams, Grillo et al. (2014b) were ableto demonstrate approximate non-di↵racting and self-healing electron Bessel beams over a range of 0.4 m (com-pared with 0.16 m for a conventional electron beam withthe same spot size). This was done by encoding the fol-
lowing phase modulation into a phase mask:
��l
(⇢,�) = ��0sgn
cos
✓
k?⇢+ l�+2⇡
d⇢ cos�
◆�
(91)
where��0 and d are the maximum size of the phase mod-ulation and the wavelength of the reference plane waveused respectively and other symbols have the meaning de-fined in Eq.14 . Fig. 14 shows two kinds of non-zero orderbeams representing approximate vortex Bessel beams.The multiple ring structure seen is a characteristic of theelectron probability distribution of the Bessel beam. Thenon-di↵racting nature of the Bessel beam has been ob-served both in the optical (McGloin and Dholakia, 2005)and now also in the electron Bessel beams (Grillo et al.,2014b).The non-di↵ractive nature of the Electron Bessel
beams might be particularly useful in electron tomogra-phy (Midgley and Dunin-Borkowski, 2009), where it canenable di↵erent planes within a material to be imagedwith the same resolution, without the need to correct forthe focus.As almost all the electrons are transmitted through the
phase holographic mask, the e�ciency of vortex beamgeneration is expected to be higher than that of the am-plitude holographic mask. For an ideal binary phaseholographic mask, the power of di↵racted electrons atthe first order of di↵raction can reach up to approxi-mately 40% of the incident beam (Magnusson and Gay-lord, 1978), which is about 4 times higher than that gen-erated by the amplitude holograms. However, the pat-tern generated in the FIB turns into an approximate si-nusoidal form, and thus the e�ciency of the generatedbeam is approximately 17% smaller than an ideal binaryprofile (40%) with the same size of the thickness variation(Grillo et al., 2014b; Harvey et al., 2014).
3. Blazed phase mask
The increase in e�ciency is an important goal for elec-tron vortex beam generation (Yuan, 2014), since signif-icant improvements in vortex beam brightness are re-quired for the various applications currently being pur-sued. Using a blazed grating approach, Grillo et al.demonstrated increased transmission of 25% intensityinto the l = 1 vortex mode. As before, the specific pat-tern to be milled is the characteristic forked pattern -however with the introduction of blazing to the gratingpattern, a higher intensity may be projected into the de-sired mode. As can be seen in Fig. 15 the mask fabricatedby Grillo et al. (2014a) was a good approximation to aperfectly blazed grating, and as such the results show asignificant intensity decrease in the zero-order l = 0 vor-tex, with substantial increase in the first order beam with
24
FIG. 15: (Color online) The blazed phase grating as demon-strated in (Grillo et al., 2014a). (a) shows an SEM image ofthe mask, with electron-energy-loss measured thickness profilein (b). (c) shows typical heights of the grating maxima andminima, with an ideal blazed profile in (d) for comparison.Image adapted from Grillo et al. (2014a).
l = 1; the intensity distribution of the di↵raction ordersis now strongly asymmetric.
Nevertheless, one fundamental issue with this phasehologram approach is the inevitable energy spread thatwill arise within the beam, due to the inelastic scatteringof electrons as they pass through the mask. For certainapplications this will not be a problem as the inelasti-cally scattered components of the electron beam can beremoved by energy filtering. However, for applicationssuch as spectroscopy, this may result in unacceptableloss of intensity and the presence of inelastically scat-tered components. In such instances, the use of a purelyphase shifting device - such as those using electron opticsalone - will be required.
It is worth noting that the theoretical maximum ef-ficiency for an ideal blazed phase holograms is 100% inoptics. As no absorption and scattering are involved,this value refers both to absolute e�ciency and relativetransmitted e�ciency in optics (Grillo et al., 2014a). Inthe context of refractive-material-based blazed gratingsfor electron vortex beam generation, the absolute gen-eration e�ciency will be limited by the inevitable largeangle elastic scattering and incoherence due to inelasticscattering processes. As a result, the absolute e�ciencyis much lower than the relative transmission e�ciencyfor the electron vortex case. At the time of writing ofthis review, the best transmission e�ciency reported byGrillo et al (Grillo et al., 2016) is 37% out of a theoret-ical limit of 38% for their partially blazed phase mask.On the other hand, McMorran’s group (McMorran et al.,2017) reported a much larger value of 70% for the abso-lute transmission e�ciency for their blazed mask. Clearlymore work is needed to investigate the practical limits tothe generation e�ciencies using optimized blazed phasemasks.
4. Choice of reference waves
The holographic mask production techniques outlinedabove may also be applied with di↵erent reference waves.Another common choice in optics is a wave with a spher-ical wavefront (Heckenberg et al., 1992a; Kotlyar et al.,2006), sharing an axis with the desired mode. This alsoproduces a characteristic interference pattern, a spiralwith l arms, as shown for the l = 1 and l = 3 vortices inFig. 16, alongside the corresponding binarised mask. Theaction of these holographic masks on an incident planewave is very similar to that described above for the forkedmask, however instead of the beams being separated byan angle, they are separated along the propagation di-rection. For the l = 0 non-vortex beam, the use of aspherical reference wave of curvature 1/f results in thefamiliar Fresnel zone plate, as shown in Fig. 16a andFig. 16d. The vortex and zero-order modes transmittedthrough the spiral hologram produced using this refer-ence spherical wave focuses at di↵erent points separatedby a distance f (Heckenberg et al., 1992b; Verbeeck et al.,2012) - when the beam as a whole is properly focused byan electron lens, the zero-order beam will be in the focalplane of the electron lens, while the first order di↵ractedbeams are focused at a distance f in front and behind thefocal plane of the electron lens. For electron microscopy,this has the advantage that over- or under-focusing thebeam enables the di↵erent vortices to be brought into fo-cus onto the focal plane, where they may then be utilisedwith minimum involvement from the other orders in thebeam (Verbeeck et al., 2012). Unlike the forked masks,these under- or over-focused vortex beams should be use-ful for scanning electron microscopy; although the addi-tional vortex modes will lead to a background contribu-tion, reducing the signal to noise ratio of the vortex modein focus.
The use of a spiral holographic mask has been demon-strated for electron vortices of various topological charges(Saitoh et al., 2012; Verbeeck et al., 2012). As with thehigh order forked mask of Ref. (McMorran et al., 2011),the stability of the mask structure requires a reinforced(Saitoh et al., 2012) centre or supporting struts (Verbeecket al., 2012); it was found in both simulations and exper-iments that the supporting struts did not significantlyimpair the integrity of the vortices produced (Verbeecket al., 2012). However, one issue with the application ofa spiral mask is that the coaxial presence of the di↵erentdi↵raction orders leads to a relatively large backgroundsignal, causing the intensity of the centre of the vortex tobe increased from zero (Verbeeck et al., 2012). In orderto reduce this overlapping e↵ect as much as possible, thefocal length of the corresponding Fresnel zone plate mustbe very long. A long focal length f means the arms ofthe spiral would decrease in separation rapidly towardsthe edge of the aperture. This requires very fine featuresin the holographic mask, a requirement similar to the
25
FIG. 16: Spiral interference patterns and masks of vorticesinterfering with spherical waves. (a) The in-plane intensitypattern for a non-vortex beam interfering with an outwardpropagating spherical wave; binarisation of this intensity pat-tern, forming a Fresnel zone plate, is shown in (d). (b) and(e) show the continuous and binarised interference patternsrespectively for an l = 1 vortex beam interfering with thespherical wave. (c) and (f) show the same for the l = 3 vortexbeam.
case involving a large kx
giving a high di↵raction angle�s
and decreasing the grating separation Eq. 89 in thedislocated grating mask. Additionally a highly coherentbeam with a large convergence angle is required, stretch-ing the limits of current microscope and FIB technology.
C. Electron optics methods
Material based electron optics is not flexible, requir-ing a specific mask for a single type of beam generation,and is also highly ine�cient, as noted above. In gen-eral, modern electron microscopy avoids the use of suchelements, instead relying almost exclusively on carefullycontrolled electrostatic and magnetic interactions to con-trol the trajectory of the electron beams. With the ex-ception of apertures, the shaping of the electron beamis in general accomplished by purely phase-transformingdevices.
Modern aberration-corrected electron microscopy al-ready has many multiple lenses in addition to the stan-dard round lens (Rose, 2008). These existing multiplelens systems have been exploited to generate a struc-tured phase shift to covert a non-vortex beam into abeam containing electron vortices, with the promise ofhigh e�ciency (Clark et al., 2013). In addition, the lensaberration itself has also been used by Petersen et al.,(Petersen et al., 2013) to show that electron di↵ractioncatastrophes can be created in this way, containing arraysof intensity zeros threading vortex cores.
1. Spin to orbital angular momentum conversion
One possibility is the generation of electron vorticesfrom spin-polarised electron beams using q-filters, inanalogy with the q-plates of optics. So called ‘q-plates’have been available in optics since 2006 and have foundapplications in quantum information - these devices arebased on patterned liquid crystal (LC) filters which al-low conversion of spin polarised optical beams to oppo-sitely spin-polarised vortex beams (Marrucci et al., 2012,2006). The action of the q-plate is based on the bire-fringent action of the LC which is also inhomogeneouslypatterned, such that spin and orbital angular momentaare exchanged between the beam and the LC. The LCthickness and patterning may be tuned in such a wayas to facilitate the transfer of the spin and angular mo-mentum of the beam, such that changes of spin angularmomentum (SAM) by 2 are accompanied by a change inOAM of �2q, where q refers to the order of the singulardefect of the patterning of LC optic axis. Other, morecomplex LC patterning may also be utilised to producevector beams with specific polarisation profiles (Cardanoet al., 2012).In an electron spin-to-orbital angular momentum con-
version device the conversion e↵ect is produced by theaction of a spatially varying magnetic field on the beamover a specifically matched length, inducing a spatiallydependent geometric (Berry) phase (Karimi et al., 2012).Such a device takes the form of a spatially inhomoge-neous Wien filter, with the trajectory altering action ofthe magnetic field balanced by the presence of an orthog-onal electric field with the same spatial inhomogeneity.As with the LC filter, the multipolar fields of the electronq-filters also contain a topological defect (a field zero) atthe centre, the order of which determines the value ofq. Quadrupole fields give a q value of �1, for hexapolefields q = �2, and so forth. The filter is most e↵ective forannular beams, so that it is more e�cient to add or sub-tract q units of orbital angular momentum from a vortexbeam (for example, for discrimination between di↵erentvortex modes) rather than create a vortex beam from aGaussian or other non-vortex beam.On transmission through the q-filter a precession of the
electron spin is induced by the magnetic field. This isaccompanied by a position-dependent geometric (Berry)phase shift, which is the source of the orbital angularmomentum of the resulting beam (Karimi et al., 2012).The two spin components of the electron beams are op-positely a↵ected, so that the resulting beam is in a mixedstates having orbital angular momentum l ± q and spinangular momentum ⌥1/2 (Schattschneider et al., 2017).Since high brightness spin-polarised electron beams arenot currently available, the q-filter is not viable as amethod of generating useful electron vortices - however,conversely the q-filter may be used with l = ±1 vor-tex beam and appropriate apertures and OAM sorters to
26
obtain non-vortex spin-polarised electron probes for var-ious applications (Grillo et al., 2013; Karimi et al., 2012,2014).
2. Magnetic monopole field
The field distribution of a magnetic monopole wouldprovide the ideal phase shift to convert a plane wave intoa vortex beam (Wu and Yang, 1976). Such a monopolefield has been considered for use in electron microscopyas early as 1992 (Kruit and Lenc, 1992). In the absenceof such monopoles, the edge fields of a suitable thin wiremagnetised along its long axis have been shown to in-duce a phase ramp around the beam that approaches2⇡ (Beche et al., 2013; Blackburn and Loudon, 2014),opening the possibility of producing high-intensity vor-tex beams, in addition to improvements in phase con-trast imaging. This method is potentially versatile, sincethe phase change of the beam depends on the magneticflux contained within the needle, so that higher and non-integer vortex states may also be created (Blackburn andLoudon, 2014). When such a needle is placed in themiddle of the aperture for the vortex forming lens, over90% incident beam can be converted (Beche et al., 2016).This high e�ciency is a very useful property for manyapplications. This method also has the distinct featureto be independent of the energy of the electron beam,making it a potential avenue for producing ultrashortpulsed electron vortex beams. The main challenges inthe use of monopole-like fields to create vortices will bethe control of the fields themselves - the field geometryand strength are significantly a↵ected by the thickness,width and magnetisation of the needle. Slight artifactsof the fabrication process may substantially a↵ect theshape of the fields, as discussed in detail in (Blackburnand Loudon, 2014), resulting in some mixing of di↵erentvortex states (Beche et al., 2016).
3. Vortex lattices
Vortices were originally described as screw-type dis-locations in wave fields produced by the interference ofthree or more plane waves (Nye and Berry, 1974). In elec-tron holography (Tonomura, 1987), electrostatic biprismshave been used to split the wavefront of a plane wave intotwo tilted plane waves which are deflected towards eachother to produce standing wave patterns in the overlapregion. By using two electrostatic prisms, we can ob-tain two overlapping standing waves. The crossings ofthe nodal lines in the standing waves are locations ofvortices which form a regular array. Both square and tri-angular arrays of vortices are produced in this manner byadjusting the relative angles of the two sets of standingwaves (Dwyer et al., 2015; Niermann et al., 2014). Such
lattices will have a pattern of vortices and anti-vorticesregularly arranged, with locally varying orbital angularmomentum density. In the square lattice, the resultingwavefront contains an array of dark spots; phase singular-ities of topological strength |l| = 1 with phase circulationalternately left- or right-handed about adjacent cores. Inpractice, due to position dependent phase shifting aber-rations within the electron lenses or that caused by thesample, there is an overall shift in the position of thevortex cores across the array. This e↵ect can be used tomap out the phase shift in the electron-optical systemsor that caused by the sample. Regular vortex arrays mayalso find applications in spectroscopy, so that the analy-sis might be performed simultaneously over a larger re-gion of the sample, while reducing the average distancebetween the vortex lines and the atoms, and thus the o↵-axis contributions (see section VI.A below) (Niermannet al., 2014).A more complex form of interference of waves is that
which causes di↵raction catastrophes and caustics, whichcan also lead to formation of vortex arrays (Nye, 2006;Petersen et al., 2013). A lattice of vortex-antivortex pairshas been shown to surround the caustics formed fromboth astigmatic and coma induced di↵raction catastro-phes in the electron microscope, along with the existenceof additional vortices inside the caustic. The wave fieldsof such caustics present the opportunity to study com-plex vortex behaviors, with twisted vortex trajectoriesand loops being apparent (Petersen et al., 2013).
D. Hybrid method
1. Lens aberrations
Clark et al. used electron optical aberration to gener-ate an azimuthal phase ramp at the back-focal plane toproduce an isolated electron vortex beam (Clark et al.,2013). Comparison of the aberration correction seriesand the Fourier series of the vortex phase shows thata good approximation to the vortex phase ramp can bemade by minimising all aberrations except for the variousorders of astigmatism. Matching the astigmatism to thedesired phase shifts and applying an annular aperture toensure beam passage through the relevant regions of thelens and corrector allow for the appropriate phase rampto be created (Clark et al., 2013). The beam generatedexperimentally using such a set up was demonstrated tobe a good approximation to the l = 1 vortex mode, ex-hibiting the axial minimum and the 0 to 2⇡ phase vari-ation. The modal decomposition showed that only 32%of the transmitted beam was in the l = 1 vortex mode.However, the presence of other vortex modes, and theclear spatial asymmetry of the beam demonstrate thatthe beam produced is not as pure as that created withholographic masks (Clark et al., 2013). On the other
27
hand, the intensity of the vortex generated is approx-imately twice that achieved using the amplitude binaryholographic mask technique; this method is more e�cientfor the generation of the |l| = 1 vortex mode, howeverthe generation of higher order vortices may prove to bemuch more challenging due to the significantly steeperphase ramp required.
2. Electron vortex mode converter
A mode converter for electron beams has been de-scribed (Schattschneider et al., 2012a), acting in an anal-ogous way to laser mode converters in optics (Beijersber-gen et al., 1993). The electron vortex implementationinvolves making use of astigmatic correctors specific tothe electron microscope, as well as the use of a Hilbertphase plate. This is therefore a hybrid technique encom-passing the phase plate and electron optics approachesdiscussed above.
The Laguerre-Gaussian vortex mode may be describedas a linear superposition of two Hermite-Gaussian modeswith a phase di↵erence of ⇡/2. The Hermite-Gaussianmodes do not themselves carry orbital angular momen-tum, however by exploiting the di↵erence in Gouy phasefor astigmatic Hermite-Gaussian modes, such a superpo-sition can be produced, resulting in a Laguerre-Gaussianmode with well defined orbital angular momentum andphase singularity (Beijersbergen et al., 1993). The ex-perimental procedure for electron vortices described bySchattschneider et al. (2012a) relies on a lens with vari-able astigmatism, so that the focal points of the x andy transverse parameters may be set independently. Thismay then be used to generate a Laguerre-Gaussian modefrom a Hermite-Gaussian mode. An approximation toa Hermite-Gaussian mode has been generated using aHilbert phase plate (Danev et al., 2002), which impartsa phase-shift of ⇡ between the two halves of the beam,similar to the phase di↵erence of ⇡ between the two lobesof the Hermite-Gaussian mode. The Hermite-Gaussianmode is then passed through the astigmatic converter,oriented at 45� to the transverse axes of the astigmaticconverter, so that the astigmatism acts on the two x andy components. The foci of the astigmatic lens are set sothat a relative Gouy phase di↵erence are created betweenthe two transverse profiles in the back focal plane to ob-tain a Laguerre-Gaussian beam profile (Schattschneideret al., 2012a).
A proof-of-principle experimental result has beendemonstrated, however although a phase singularity isapparent at the centre of the back focal plane, the re-sulting profile does not have rotational symmetry, andso is not a pure Laguerre-Gaussian mode. The discrep-ancy from the simulated results arises due to defocusand, importantly, strong beam absorption in transmis-sion through the Hilbert phase plate (Schattschneider
et al., 2012a). Nevertheless, the electron vortex modeconverter is an attractive prospect if these e↵ects can beovercome, as it enables the generation of electron vor-tices of high intensity, of up to 90% of the incident planewave intensity, as opposed to ⇡6% using the holographicmasks.
V. VORTEX BEAM ANALYSIS
Any study of the electron vortex beam will inevitablyinvolve some characterisation of its properties. This willalso be useful in, for example, examining transfer of or-bital angular momentum in experiments involving inter-actions with various forms of matter and light. We nowbriefly review the techniques that have been developedfor electron vortex beam analysis.
A. Interferometry
The simplest approach is to examine the intensity pro-file of the beam cross-section. For a given point in thewavefront in a vortex beam, the existence of the conju-gate point with antisymmetry in phase ensures that theircoherent superposition on the beam axis will always leadto destructive interference. This is responsible for thepersistence of the central dark spot in the vortex beam.Thus the observation of the doughnut ring in the beamcross-section is taken to be a signature of a non-zero topo-logical charge. However we also need to know the signand the magnitude of the topological charge and the ra-dial structure. For example, the radial structure of theLaguerre-Gaussian beam is specified by the radial indexp in Eq. 9. As for the identification of the pure OAMstates, some progress has been made, as we summarisebelow.
1. Electron holography
The interference of a pure vortex beam with a planereference wave results in a forked interference fringe, aproperty that is fundamental to the holographic masktechnology introduced in section IV.B. The interferenceexperiment can be performed in an electron microscopeequipped with an electron biprism (Tonomura, 1987). Abiprism is usually a positively charged wire. Two half ofthe wavefront on either sides of the charged wire will ex-perience transverse shear because the negatively chargedelectrons will be attracted towards the positively chargedwire. The resulting overlap of the wavefronts forms holo-graphic fringes. If only a parallel beam falls onto theregions containing the charged wire, linear grating-likeinterference structure will be observed. In traditionalelectron holography, a sample is inserted in the path ofone half of the split wavefront so that the extra phase
28
and amplitude modification can be recorded as the ad-ditional shift of the fringe as well as the modulation ofthe fringe contrast. In vortex beam analysis, the samplemay be replaced by a vortex converter such as a spiralphase plate or an holographic di↵raction mask. The vor-tex phase structure imprint on the interference fringes isa dislocation or a fork structure, such that the number ofthe forks in the interference pattern is equal to the mag-nitude of the topological charge and the orientation of thefork can be related to the sign of the topological chargeof the beam. This electron holographic method was usedby Uchida and Tonomura (2010) to demonstrate for thefirst time the phase structure of an electron vortex beam.
2. Knife-edge and triangle aperture di↵ractive interferometry
Electron holography using the electron biprisim tech-nique is only available in specialised microscopes, so al-ternative methods had to be developed. The simplestholography method without the use of electronic biprismis the knife-edge holography method. This creates a ref-erence waves by utilising the beam-bending e↵ect asso-ciated with the Fresnel-di↵raction of electrons from theedge of an aperture. Verbeeck et al. (2010) used this todemonstrate the production of vortex beams by a holo-graphic mask.
A useful variation of this method is the use of a triangu-lar aperture. The interference of the Fresnel fringes fromthree neighbouring edges of the triangle results in a tri-angular lattice structure which depends strongly on thetopological charge of the vortex beam (Hickmann et al.,2010). It is then a simple matter to count the number ofspots in the resulting interference pattern to determinethe topological charge.
It is to be noted that the above-mentioned methodsonly work in out-of-focus conditions.
3. Di↵raction
The phase structure of the vortex beam can also bestudied by the di↵raction method. A hologram maskcan be used to determine the topological charge of thebeam (Guzzinati et al., 2014; Saitoh et al., 2013). Inthis method, a forked grating structure, formed from theinterference of a plane wave and a vortex wave of topo-logical charge of l
grating
, is used as a beam analyser. Thevortex beam is di↵racted by the grating structure and farfield di↵raction pattern is examined. The di↵raction ofthe dislocated grating adds n ⇤ l
grating
to the topologicalcharge of the resulting di↵racted beams, where n is theorder of the di↵raction and takes the values of 0,±1,±2,etc. The total topological charge of the di↵racted beamsis now given by:
m = nlgrating
+ l (92)
In particular, the di↵racted beam with m = 0 has anintense central spot and so can be easily identified. Thisallows the OAM content of the incident beam to be eas-ily determined using Eq. 92, if gratings with di↵erenttopological charges are employed. Other methods suchas the simple multiple pinhole plates have also been sug-gested, with the mixed results due to possible e↵ects suchas aliasing e↵ects (Clark et al., 2014).In general, the identification of the vortex beams of
mixed orders is di�cult. But Saitoh et al. (Saitoh et al.,2013) suggested that if a pin-hole aperture is also usedon the di↵racted beam, the vortex beam can be sortedinto di↵erent components according to their topologicalcharge.
B. Mode conversion analysis
One of the simplest methods is based on the mode con-version of Hermite-Gaussian modes to generate Laguerre-Gaussian ones (Allen et al., 1992; Courtial and Pad-gett, 1999). Here, the mode conversion is run in re-verse and the vortex beam is decomposed into Hermite-Gaussian beams with characteristic modes. In optics thisis achieved using cylindrical lens while in electron optics(Guzzinati et al., 2014; Shiloh et al., 2015) use is madeof a astigmatism corrector which is readily available in atypical electron microscope. Obviously for higher ordervortex beams, we might have mixed radial modes sharingthe same OAM quantum number. This may result in acomplex superposition of the pattern. Thus the methodis probably more useful for vortex beams of small topo-logical charges. The sign of the topological charge can beread out through the sense of the rotation of the result-ing Hermite-Gaussian pattern because of the l-dependentGouy rotation.The approaches adopted so far can all be traced to orig-
inal methods developed for characterising optical vortexbeams and they work well for single pure OAM states.The challenge occurs when one has to deal with mixedOAM states. Optical approaches such as multipoint in-terferometers (Berkhout and Beijersbergen, 2008), geo-metric transformations by phase manipulation (Berkhoutet al., 2010), and the use of multiple interferometers ina cascade set-up (Leach et al., 2004) are currently dif-ficult to implement in the existing electron microscopeset-up, so a new approach is required, especially for thecharacterisation in the single electron region. For prac-tical reasons, the holographic di↵raction mask and theaperture masking methods are di�cult to implement foratomic size vortex beams, making the astigmatism trans-formation method currently the most realistic method foratomic scale pure vortex beams. For e�cient sorting ofelectron vortex beams in general, electron optical are pre-ferred and a refractive device has recently been proposedby McMorran et al. (2017).
29
C. Image rotation
Because of the azimuthal dependence of a pure vor-tex beam is encoded in the phase factor eil� indepen-dent of z, the cross-sectional image of any pure vor-tex beams is circularly symmetric so any potential ro-tation is not detectable. This method is therefore notsuitable for the characterisation of the rotation of thepure states. On the other hand, the circular symme-try will be broken in a mixed state vortex beam, re-sulting in an asymmetrical cross-sectional beam intensity.This can be achieved by di↵erent methods, ranging fromsimply cutting a pure vortex beam by a knife-edge-likemask to create an asymmetrical cross-sectional distribu-tion (Guzzinati et al., 2014), to the careful preparationof a superposition of pure vortex beams (Greenshieldset al., 2012) or a vortex-endowed C-shaped beam (Mous-ley et al., 2015). The rotation of the transverse imageof the electron vortex beam can be measured as a func-tion of the propagation distance or as a function of themagnetic field strength. In that way, measurements ofthe image rotation along the z-direction can be used tostudy the Zeeman or Gouy e↵ects of the electron vortexbeam (Schachinger et al., 2015).
1. Gouy rotation
The Gouy phase is due to spatial confinement of thevortex beams near focus (Feng and Winful, 2001; Pe-tersen et al., 2014). In the absence of an external mag-netic field or when the field strength is so small that themagnetic e↵ect can be ignored to the first order, the maincontribution to the image rotation is due to the variationof the Gouy phase di↵erence between OAM beams withdi↵erent topological charges. The Gouy phase di↵erencechanges rapidly near the beam waist, so that the Gouyrotation is most easily observed around the focus plane,as shown by Guzzinati et al. (2013).
2. Zeeman rotation
One of the consequence of using magnetic lens to fo-cus electron beams is that all electron beam trajecto-ries with transverse velocity components undergo Lar-mor precession. In electron microscopy, this results inthe well known image rotation phenomenon (Reimer andKohl, 2008). Early microscopists had to take this into ac-count when comparing microscope images taken at di↵er-ent magnifications as the excitation of the magnetic fieldin the lens changes. In modern transmission microscopy,lens design is such that the rotation e↵ect due to di↵erentlenses cancels out, so electron microscopists need not beconcerned with such e↵ect.
The presence of the external magnetic field also causes
complex and interesting changes in the phase of the elec-tron vortex beam as described in section III. However, fornanoscale vortex beams that are being generated insideelectron microscopes where the saturated field strengthis typically in the order of 2T, the e↵ect of the magneticfield would be very weak (Babiker et al., 2015). To see themagnetic field induced rotation, one has to go either tomuch stronger magnetic fields or vortex beams of largetransverse structures. In general, such rotation shouldbe accompanied by changes in the radial direction dueto competition of the beam di↵raction and the confininge↵ects of the magnetic field (Greenshields et al., 2012).In special cases, when the vortex beam has the charac-teristic beam width w
B
given in Eq. 58, one can studythe rotation of the Landau states given by Eq. 62.These image rotations have been experimentally ob-
served in the electron microscope, for both the balancedand unbalanced suppositions (Guzzinati et al., 2013).The experimental situation di↵ers from the theoreticaltreatments outlined above in that the fields are not uni-form, and the beams have a well defined focal plane. Assuch, the action of the Zeeman and Gouy phase shiftsboth contribute, with the Gouy shifts dominating in thevicinity of the focus of the beam, due to the transverseconfinement (Feng and Winful, 2001), and the Zeemanshift is more apparent at large radial distances fromthe beam axis, due to the magnetic fields of the lenses.For the unbalanced superposition, the di↵erence betweenthose states, with the net OAM aligned and anti-alignedwith the direction of the magnetic field, is clearly shownin the net addition of the separated rotations due to theZeeman and Gouy terms (Guzzinati et al., 2013). SuchZeeman-Gouy phase e↵ects may find applications in vor-tex beam analysis, since this o↵ers a method by whichoppositely oriented vortices may be di↵erentiated, by theobservation - or lack - of image rotations in known mag-netic fields. For beams with a Landau state transversemode given by Eq. 62, rotation either in Larmor, cy-clotron (double-Larmor) or zero frequency have been ob-served (Schattschneider et al., 2014b), consistent withthe prediction of Bliokh and Nori (2012a). However, theuse of the knife-edge resulted in the mode broadeningdue to an approximate uncertainty principle for angularposition and angular momentum (Franke-Arnold et al.,2004), so the measurement can not be taken for a vortexbeam in a pure Landau state.
D. Vortex-vortex interactions and collisions
General considerations of vortex-vortex collisions havebeen discussed (Berry and Dennis, 2007, 2012; Bialynicki-Birula et al., 2000, 2001) for the cases where there aretwo or more phase singularities present in the wavefield. In such cases, the behavior of the vortices be-comes somewhat complicated, with possible phenomena
30
including the creation, annihilation and crossing of vortexlines. For the electron vortex, experimental demonstra-tions of wavefields with two (Hasegawa et al., 2013) orseveral (Niermann et al., 2014) phase singularities havebeen achieved in the electron microscope. In the firstcase, a specially prepared holographic mask with twophase defects (edge dislocations) was used to embed twophase singularities into the resulting first order beams(Hasegawa et al., 2013), while the second involved twoorthogonally-acting biprisms arranged in such a manneras to generate a lattice of vortices through wave interfer-ence.
For the case when two vortices are present in the beam,the behaviour and interaction of the two vortices can beexamined as the beam passes through focus. Holographicmasks incorporating two edge defects were produced togenerate two vortices of topological change |l| = 1 slightlydisplaced from one another - the two vortices may eitherbe aligned or anti-aligned, in which case the behaviouras they propagate takes on a di↵erent character. For thecase when the two vortices are aligned, they are found toprecess about each other within the first order di↵ractedbeams, whereas for the case when the vortices are anti-aligned the lines of phase singularity are attracted toeach other, eventually annihilating. Both e↵ects are at-tributed to the change in Gouy phase as the vortices passthrough the focal point (Hasegawa et al., 2013) - for thealigned beams the Gouy phase shift occurs in the samedirection, so that the beam rotates in the same directionat the same rate, whereas the phase shift is opposite forthe anti-aligned vortices, causing them to annihilate.
E. Factors a↵ecting the size of the vortex beam
The possibility of using electron vortex beams to probethe properties of materials with atomic resolution re-quires the generation of atomic scale vortex beams withcross-sections in the Angstrom scale. Sub-Angstrom fo-cused electron probes have been demonstrated and avail-able more than a decade ago (Batson et al., 2002). Ina modern transmission electron microscope, a probe sizeas small as 0.5 A can be achieved using a highly coher-ent source with a large convergence angle and correc-tive optics for the minimising of aberrations in the probeforming lens (Erni et al., 2009). Can we replace the cir-cular aperture with a top-hat like transmission functionwith azimuthal phase gradient to produce electron vor-tex beam, such as the beam defined in Section II.B.3, inthe Angstrom range? To answer this question, we needto first define what is meant by the size of non-vortexbeams as well as that of vortex beams.
A simple estimate of the ring size of the single donutvortex beam can be obtained by considering the quan-tized orbital angular momentum of a circulating particleflux in a circle of radius ⇢
l
. We make use of the standard
expression for the angular momentum L = ⇢⇢⇢l
⇥p and thatthe maximum size of the linear momentum transferredby the di↵raction of electron beams passing through anaperture subtending a half angle ↵ is equal to ↵k0, where~k0 is the momentum of the electrons. The size of thedonut ring is then approximately given by:
⇢l
⇠ l~↵~k0
=l�
2⇡↵(93)
For all the beams passing through such an aperture,the uncertainty principle also implies that the minimumbeam size (�⇢) due to di↵ractive broadening, is given by:
�⇢ ⇠ ~�p
=~
↵~k0⇠ �
2⇡↵(94)
These order-of-magnitude estimates show that both⇢l
and �⇢ are controlled by the convergence angle sub-tended by the lens aperture (↵ = R
max
/f) but for dif-ferent physical reasons. To be more precise, one needsto know the exact electron wavefunctions because thespatial distributions of the electrons are known to be dif-fuse in space and vary for di↵erent types of vortex beamsas reviewed in section II.B. For example, the size of thedonut rings (⇢
l
) in LG beams introduced in section II.B.1scales with
pl (Lembessis and Babiker, 2016), but for
bandwidth limited vortex beams produced by the Fraun-hofer di↵raction of a finite-radius plane waves imping-ing on a spiral phase plate (section II.B.3), the ring sizescales with l (Curtis and Grier, 2003). For the FT-TBBdiscussed in section II.B.3, there is a non-linear depen-dence on l. On the other hand, the di↵raction-limitedwidth of an ideal non-vortex beam is given by the sizeof the Airy disk (�⇢ = 1.22�/↵ (Airy, 1834)). Com-bining the two di↵erent e↵ects an approximate empiricalformula for the overall vortex beam size (⇢
l
) emerges as(Curtis and Grier, 2003):
⇢l
= 2.585�f
⇡Rmax
✓
1 +l
9.80
◆
. (95)
This suggests that the di↵raction limited singly chargedvortex beam size is only about 10% larger than thenon-vortex beam. With a microscope capable of 0.5Anon-vortex beam, a singly charged vortex beam of sub-Anstrom size should be possible.Experimentally, most studied electron vortex beams
can be e↵ectively obtained as some forms of bandwidth-limited beams with wavefunction truncation by a hardaperture at the plane of the focusing lens (Beche et al.,2016; Schattschneider et al., 2012b; Verbeeck et al.,2011a). The scalar di↵raction theory of such a case hasbeen thoroughly investigated in the course of optical vor-tex studies (Kotlyar et al., 2005) and this compares fa-vorably with the theoretical investigations when finitesource size e↵ects or even spherical aberration, is takeninto account, where appropriate (Schattschneider et al.,
31
FIG. 17: Vortex probes at focus of the condenser lens of aJEOL 2200FS aberration corrected tranmission electron mi-croscope operating at 200kV. The central spot is the imageof the non-vortex beam, while the other spots are images ofthe di↵racted vortex beams produced by a forked di↵ractivehologram with a binary pattern similar to that given in Fig.12. Sizes of the 1st and 2nd vortex cores at FWHM are 2.6nm and 3.4 nm, respectively. Due to incoherent e↵ect, the in-tensity dip is only partially visible for the 2nd order vortexbeam.
2012b). In situations where a small convergent beamis used (Schattschneider et al., 2012b; Verbeeck et al.,2011b), the ring diameter of the vortex beam can be accu-rately measured and the experimentally obtained valuesagree with those emerging from optical di↵raction theory.For example, in a TEM with a round condenser aperturesubtending a convergence angle of 21 mrad, the size ofthe Airy-pattern of the non-vortex beam is 1 A using theRayleigh resolution criterion for the case an electron vor-tex beam with l = 1 and a minimum full width at halfmaximum (FWHM) diameter of 1.2 A. The reason thatit is the FWHM that is used in this case instead of thering size is the broadening e↵ect of the spatial distribu-tion due to the finite source size (incoherent broadening),estimated at 0.7 A. Incoherent broadening e↵ects, whenthese are significant, result in a less visible central dip ofthe vortex beam (for an example, see Fig. 17). Incoher-ent broadening plays an increasingly important role inatomic size electron vortex beam experiments as a size-limiting factor (Lofgren et al., 2016), but its influence canbe taken into account in the same manner as done rou-tinely in describing focused non-vortex beams commonlyused in scanning transmission electron microscopy (Kirk-land, 2010). For an electron vortex beam generated in-side a 300kV scanning transmission electron microscope aprobe size of the order of 0.87 A has been demonstrated(Beche et al., 2016), invalidating the early too pessimisticprediction (Idrobo and Pennycook, 2011). As the currentresolution of electron microscopes is not yet wavelengthlimited, there is still scope to reduce the vortex beam sizefurther.
VI. INTERACTION WITH MATTER
The interaction of an electron vortex beam with sin-gle atoms has been considered (Lloyd et al., 2012a,c; vanBoxem et al., 2015, 2014; Yuan et al., 2013), as well as
Mott scattering (Serbo et al., 2015) when the spin-orbitinteraction is taken into account. The related radiativecapture of a vortex electron by an ion has also been inves-tigated (Matula et al., 2014). We focus on chiral specificinteractions in this section.
A. Chiral-specific spectroscopy
The practical generation of electron vortex beams wasaccompanied by the suggestion that such vortex probesmight initiate, as a first application, a new type of elec-tron energy loss spectroscopy (EELS) involving orbitalangular momentum transfer (McMorran et al., 2011;Uchida and Tonomura, 2010; Verbeeck et al., 2010). Thefirst experiment on EELS using electron vortex beamswas reported by Verbeeck et al. (2010). In this experi-ment, a 50 nm thick Fe film was placed inside the fieldof the objective lens, such that it was magnetically satu-rated. A non-vortex beam was transmitted through theiron film and the transmitted beam then passed througha forked holographic mask at a slight defocus, such thatthe various orbital angular momentum components wereseparated into distinct vortex beams. Comparing theenergy-loss spectra of the two first order transmitted vor-tex beams showed a dichroic e↵ect in the iron L2 and L3
edges, understood to indicate a transfer of orbital angularmomentum between the beam and the internal electronicstates of the iron atoms. The electron vortex energy-loss spectrum corresponds well to similar X-ray magneticcircular dichroism (XMCD) spectra (Carra et al., 1993;Thole et al., 1992), so that the magnetisation of the sam-ple is said to be clearly identified. However, since this2010 result was reported there have been no further ex-perimental reports of an observed magnetic dichroism,and there has been much discussion as to whether elec-tron vortex beams could provide an advantage over ex-isting methods in electron beam chiral dichroism spec-troscopy. Nevertheless, a great deal of theoretical workhas been carried out, uncovering the subtle physics in-volving orbital angular momentum transfer and the bestconditions for its observation (Rusz and Bhowmick, 2013;Schattschneider et al., 2014a; Yuan et al., 2013).Comparing the results of the iron dichroism experi-
ment with the well known XMCD spectra of iron suggeststhat there is a similar transfer of orbital angular momen-tum between the beam electron and the internal atomicstates, in contrast to the case of optical vortices, in whichno orbital angular momentum transfer can arise in dipoletransition (Andrews et al., 2004; Babiker et al., 2002;Jauregui, 2004) (see also (Alexandrescu et al., 2005))nor any were observed (Araoka et al., 2005; Giammancoet al., 2017; Lo✏er et al., 2011). The mechanisms of theatomic-vortex interactions are quite di↵erent in the op-tics and electron cases - in the optics case the interactionHamiltonian arising from the minimal coupling prescrip-
32
tion does not exhibit the required chirality to mediateorbital angular momentum transfer, in contrast to thelong-range Coulomb interaction between the atomic andvortex electrons (Lloyd et al., 2012a,c).
1. Matrix elements for OAM transfer
Writing the interaction Hamiltonian as the sum of theCoulomb interactions between the atomic constituentsand the vortex electron, we have,
Hint = � e2
4⇡✏0
✓
1
|rv
�R+ m
e
M
q| �1
|rv
�R+ m
p
M
q|
◆
(96)
where M = mp
+ me
is the mass of the atom, and therelevant position vectors are shown in Fig. 18. This inter-action Hamiltonian may now be expanded as a multipolarseries and applied as a scattering perturbation to a setof initial and final states of the well-known hydrogenicwavefunctions and vortex wavefunctions to yield the se-lection rules of the interaction (Lloyd et al., 2012a,c). Inthe leading (dipole) order the transition matrix elementis found to reduce to the following
Mfi
=e2
4⇡✏0
⇣
C+1l
�[(L+l),(L0+l
0+1)]�m,m
0�1 +
C�1l
�[(L+l),(L0+l
0�1)]�m,m
0+1 +
C0l
�[(L+l),(L0+l
0)]�m,m
0
⌘
. (97)
where the Cl
’s are complex functions of the internal coor-dinate q to the first order. The selection rules deduciblefrom the matrix element show that in the dipole approx-imation a single unit of orbital angular momentum maybe absorbed (released) by the atomic electron from (to)the combined orbital angular momentum of the electronvortex and atomic centre of mass. The combination ofthe vortex and centre of mass orbital angular momentaallows for the possibility of the rotation of the centre ofmass, provided the atom is not fixed; which lays the foun-dations for the manipulation of larger particles thoughOAM exchange (Gnanavel et al., 2012; Verbeeck et al.,2013). Further analysis of the quadrupole and higher or-der interaction terms demonstrate similar selection rulesin which zero, one or two units of orbital angular momen-tum may be transferred - higher order multipole terms oforder n mediate the transfer of zero or n units of orbitalangular momentum.
2. The e↵ect of o↵-axis vortex beam excitation
In expanding the interaction Hamiltonian about theatomic centre of mass, the above analysis does not fullydemonstrate the complications arising from the extrin-sic nature of the vortex orbital angular momentum. It
FIG. 18: (Color online) The atomic model system interact-ing with a vortex beam whose cylindrical axis is along thez-direction of the laboratory frame of reference. The vectorsre
, rp
,R and q refer, respectively, to the position vectors ofthe atomic electron, the nucleus, the centre of mass, the in-ternal position relative to the centre of mass and the positionvariable of the vortex electron. The corresponding �’s rep-resent the azimuthal angles and these are important for thedescription of the phase factors.
is illuminating to determine the modal expansion of thebeam from the perspective of the atomic nucleus, andcompute the weighted scattering amplitudes to the var-ious possible transfer channels. The interaction Hamil-tonian remains the same as that of Eq. 96, but nowthe initial and final vortex states have to be written asexpansions about the centre of mass frame, rather thanbeing given in the laboratory frame, as before. This isaccomplished by use of the Bessel function addition the-orem (Abramowitz and Stegun, 1972) (see the figure forthe relevant notation)
Jl
(k?) =e�il(�v
��
n
)⇥1X
p=�1Jl�p
(k?⇢n)Jp(k?⇢0v
)ei(l�p)�neip�
0v .
(98)
This results in the original vortex beam of topologicalcharge l being described with respect to a new axis - theorigin of which is common with the the atomic nucleus -in terms of an infinite series of Bessel functions J
p
(k?⇢0v
),with weighting functions given by J
l+p
(k?⇢n). Thus, therelative location of the atom with respect to the axis ofthe incident vortex beam determines the precise modesthat the atomic electron ‘sees’ to interact with. For anatom situated directly on the beam axis, the original p =
33
l mode is the only contribution, since in this case onlythe weighting term J0(k?⇢n) is non-zero (also see theleft panel of Fig.19). However, for an atom displacedfrom the beam axis, the next atom-centred vortex modesp = l ± 1 become significant even at small distances,of the order of a fraction of the radius correspondingto the first Bessel function zero �11/k? ⇡ 0.1 nm forl = 1, i.e. within the first ring of the Bessel beam. It isclear that slight displacement from the beam axis leads tothe contributions of vortex modes with winding numbersdi↵erent from the overall angular momentum quantumnumber of the beam l (also see the right panel of Fig.19).The expanded Bessel function may then be used to definean e↵ective operator
O = h 0f
|Hint| 0i
i (99)
which determines the selection rules when applied to theatomic states, where 0
i(f) refers to the initial (final)wavefunction involving the expanded Bessel function ofEq. 98. Applying the Bessel addition theorem a sec-ond time allows for the expansion in terms of the in-plane dipole moment of the atom, making apparent thespecific multipolar character of the transfer interaction(Yuan et al., 2013).
When the atom is o↵ axis, there are several availablechannels for atomic excitation, both in view of the mul-tipolar nature and the specific Bessel mode of the expan-sion, as is illustrated in Fig. 19. It can be seen that, incontrast to the straightforward on-axis case, for an atomlocalised at an o↵-axis position the change in the orbitalangular momentum of the beam does not necessarily in-dicate a corresponding change in the OAM of the internaldynamics of the atom. At first sight this would seem topresent a large obstacle to the use of vortex beams toprobe chiral information. However, after taking into ac-count the relative intensities of the various interactionchannels, one can show that for the dipole excitationcase, the o↵-axis contributions are an order of magnitudesmaller than the on-axis contribution, with intensity de-caying rapidly further away from the axis, so that theprincipal transitions in any such chiral spectroscopy ex-periment are those having the XMCD-like selection rulesof Eq. 97. Additionally, higher order multipole transi-tions will contribute to the background signal, but theseare also found to be much smaller in magnitude comparedto the dipole contribution, due to the much smaller over-lap of the Bessel functions involved (Yuan et al., 2013).
The o↵-axis contributions to an experimental electron-energy loss spectrum may be further reduced to producean acceptable signal-to-noise ratio, by making use of aconfocal TEM set-up, ensuring that the signal contribu-tions come from those atoms lying on or very close to themicroscope axis, and reducing the non-chiral signal fromatoms displaced from the axis (Schattschneider et al.,
FIG. 19: (Color online) Illustration of the various interac-tion channels available in (a) the case in which the atom issituated on-axis; and (b) the o↵-axis case. In (a), any changein OAM of the atomic electron is immediately apparent as achange in OAM of the transmitted beam, �l = l0 � l. The casein (b) is more complicated due to the expansion modes havingvarious OAM p, which may each transfer any number of unitsof OAM to the atomic electron. The resulting changes �l val-ues may correspond to a variety of combinations of expansionmodes and multipolar transitions, as shown. However, due tothe relative strengths of the expansion modes and multipolaratomic transitions, the dominant interaction channels arisefrom the dipole interaction with the p = l mode. Notice thatp is used here only to indicate the OAM modes involved in ano↵-axis vortex beam, not as a radial index as in the rest of thetext. Image from Yuan et al. (2013).
2014a). As pointed out above, interactions in these re-gions are most likely to involve the p = l modes, and thusthe change in orbital angular momentum of the vortex inthe laboratory frame is indicative of the atomic changein magnetic quantum number. A schematic of such anexperiment is shown in Fig. 20 - a vortex beam is incidenton a sample, with the resulting transmitted beam splitinto the the various orbital angular momentum compo-nents by a vorticity analyser. After the passing throughthe sample, the transmitted beam contains several dif-ferent OAM component l0 from the various interactions,along with the original value l from electrons passingthough unscattered. Analysis of the various OAM com-ponents will allow the determination of the change in theOAM of the atom - it has been shown that a suitable anal-yser may take the form of a forked holographic mask inconjunction with a pinhole (Saitoh et al., 2013). The pin-hole will enable the isolation of specific OAM componentsfor measurement of EELS spectra for that channel, allow-ing detection of only those transmitted electrons havingl0 = 0; for example, those that have su↵ered a loss (gain)of one unit of orbital angular momentum in the interac-tion with an l = +1(�1) incident vortex. The pinholeacts to select only those transitions that have both p = lcontributions and that are scattered to l0 = p0 = 0 states.
34
FIG. 20: (Color online) Schematic of suggested experimentalset up for OAM based spectroscopy using electron vortices. Avortex beam, produced by a holographic mask or other suit-able method (not shown) is incident onto a thin sample in thespecimen plane. After interaction with the sample, the trans-mitted beam is then passed though a forked mask in order toseparate the various vortex components. A pinhole placed inthe di↵raction plane allows isolation of those modes that havean OAM of 0 after passing though the mask - these can thenbe detected to obtain EELS.
Repeating the experiment using a vortex beam of oppo-site OAM, i.e. �l will enable a dichroism spectrum to beobtained.
The advantage of this method over the experimentdemonstrated in (Verbeeck et al., 2010) should be an in-crease in signal to noise ratio, since the forward and re-verse interactions are treated separately (the additionalo↵-axis features and higher-multipole excitations alsocontribute to a background noise with a non-vortex inci-dent beam). In this way, probing specific atomic transi-tions is feasible with varying incident vortices, includinghigher order multipole transitions using incident vorticeswith orbital angular momentum greater than ±~. Onthe other hand, since the transmitted intensities in thesecases are expected to be small, the experimental condi-tions must be optimised so that long collection times maybe utilised. Experimental feasibility study is encourag-ing (Schachinger et al., 2017), especially for amorphousmagnetic materials.
Since they were first reported, it has been suggestedthat electron vortex beams may be combined with theatomic resolution microscope probes to enable chiralspectroscopy with atomic resolution (Idrobo and Pen-nycook, 2011; Lloyd et al., 2012c; Rusz and Bhowmick,2013; Verbeeck et al., 2011a, 2010; Yuan et al., 2013).However, there is currently some debate as to the con-ditions under which such high resolution will be achiev-able, and the limits of application of the vortex beam(Pohl et al., 2015). Specifically, are the sub-nanometrescale vortex beams described in Verbeeck et al. (2011a)suitable for atomic resolution dichroism experiments?It has been argued that the sub-nanometre FWHM ofsuch beams is not su�cient due to the inherent inco-herence in the microscope. On the other hand, it hasalso been argued that it is only in the atomic resolu-tion limit that vortex chiral dichroism experiments willgive any improvement over the intrinsic EMCD e↵ectdue to the crystal structure acting to di↵ract the vor-tex modes (Rusz and Bhowmick, 2013). Simulations ofinelastic scattering of vortex beams through iron crystalsup to 20 nm thick shows that magnetic information isavailable only when the radius of the vortex beam is ofthe order of the atomic radius. In this case, the energyfiltered di↵raction signal shows a magnetic componentof approximately 10% of the background, non-magneticsignal, and is strongly dependent on the position of inci-dence within the unit cell, so displaying atomic resolution(Rusz and Bhowmick, 2013).The inelastic scattering of an electron vortex beam by
atoms has also been further investigated for hydrogen by(van Boxem et al., 2015), by including the explicit radialdistribution functions.
3. Plasmon spectroscopy
In addition to the research into chiral excitation ofcore electron transitions which is relevant to the exci-tation of magnetic sublevels in the inner shell of theatoms. Asenjo-Garcia et al. have calculated the chiralplasmon response (Asenjo-Garcia and Garcıa de Abajo,2014) which is due to the coupling of the charges in thevortex beam with the electric field of the collective mo-tion of the valence electrons.The application of vortex beams to EELS may also
have potential in mapping the magnetic response of ma-terials in the form of magnetic plasmon resonances (Mo-hammadi et al., 2012). Magnetic plasmon resonances innanoparticle arrays are expected to lead to the produc-tion of metamaterials, exhibiting negative permittivityand permeability in the optical range (Podolskiy et al.,2002; Sarychev et al., 2006), such that vortex basedmagnetic plasmon EELS (vortex-EELS) would providean invaluable tool for the characterisation of metamate-rial response. EELS is already well applied in the de-
35
termination of electric plasmon resonances of nanopar-ticles (Bosman et al., 2007; Horl et al., 2013; Nelayahet al., 2007), vortex-EELS should provide a complemen-tary technique, with the additional possibility of gath-ering information on both the electric and magnetic re-sponses of nanoparticles simultaneously in a single exper-iment, since the electron vortex will also induce electronplasmon resonances in addition to magnetic plasmon res-onances.
A theoretical treatment of the magnetic response ofan array of split ring resonators has been demonstrated(Mohammadi et al., 2012), allowing direct comparisonof the electric and magnetic plasmon resonance spectraand spatial distribution. Making use of the duality ofthe electric and magnetic fields allows the relationshipsbetween the induced resonance field and the beam cur-rent - related by Green’s functions - to be recast into aninduced magnetic field, regulated by a magnetic Green’sfunction, and induced by the e↵ective magnetic currentof the vortex beam (Mohammadi et al., 2012). Fromhere, the magnetic EELS spectra can be calculated usingstandard finite-di↵erence time-domain techniques. For asplit-ring resonator the results of such simulations showa strong magnetic response on the inside of the ring, con-trasting with the electric response at the ends of the arms,as shown in Fig. 21. The calculated spatial profiles areconsistent with previous theoretical work on electric andmagnetic resonances of nanoparticles of similar sizes andshapes (Enkrich et al., 2005; Sarychev et al., 2006). Crit-ically, the intensity of the magnetic response is withinan order of magnitude of the electric response, indicat-ing that measurement of magnetic plasmon resonancesshould be experimentally accessible (Mohammadi et al.,2012). Naturally, in an experimental situation, the vor-tex beam will induce electric plasmon resonances at thesame time as the magnetic resonances, which for meta-material development will necessarily be within similarenergy ranges - notably within the visible light spectrum.Energy filtering is therefore insu�cient to fully isolate themagnetic component of the EELS spectrum. For nanos-tructures for which the electric response is well under-stood, separating the magnetic signal may be possible bysubtracting a separately measured non-vortex-EELS sig-nal from the vortex-EELS map, or by filtering the trans-mitted signal by its OAM content (Mohammadi et al.,2012). This latter method requires further investigationinto the role of orbital angular momentum transfer inmagnetic resonances, as it is not immediately apparentthat OAM transfer is a requirement for vortex-inducedmagnetic plasmon resonance.
B. Propagation in crystalline materials
In order that experiments involving vortex beams trav-elling in real materials to be appropriately interpreted, it
FIG. 21: (Color online) Spatial maps of electron energy lossprobability for (a) magnetic resonance at 0.863 eV, with anl = 1 beam, and (b) electric resonance at 0.8 eV with a planewave beam, both at 100 keV beam energy. The two plasmonresonance maps show markedly di↵erent spatial profiles, aswell as the magnetic resonance response being approximatelyan order of magnitude less than the electric response. Imagesfrom Mohammadi et al. (2012)
is necessary that the way such beam propagates throughcrystalline structures is well understood. Any elec-tron probe propagating through a crystal will experi-ence strong elastic scattering from Coulomb interactionwith the atomic nuclei, proportional to the thickness ofthe sample and tilt relative to the beam axis, as well aschanneling along the atomic columns (Reimer and Kohl,2008; Williams and Carter, 2009). Due to this scatter-ing potential, the trajectory of the vortex and the localorbital angular momentum density of the beam will bealtered, since the crystal potential breaks the cylindricalsymmetry of the beam, leading to coherent superpositionof OAM eigenstates that change in composition throughthe crystal (Lo✏er et al., 2012). Because of the exchangeof orbital angular momentum between the lattice andthe beam the local values of orbital angular momentumwithin the crystal may be quite di↵erent from those ofthe original, incident vortex beam (Lo✏er et al., 2012;Lubk et al., 2013a,b); additionally the trajectories of thevortex lines are no longer simple, involving oscillatorymotions, looping and the generation of vortex-anti-vortexpairs (Lubk et al., 2013a,b). It is worth mentioning thatvortex structure can also be produced by dynamical scat-tering of crystals by a non-vortex incident electron beam(Allen et al., 2001) because of multiple scattering (Nyeand Berry, 1974).Multislice simulations of the propagation of vortex
beams through iron (Lo✏er et al., 2012) and strontiumtitanate (Lubk et al., 2013a) crystals have been carriedout to explore the complex dynamics arising from theinteraction. Both the Fe and SrTiO3 materials are rel-evant since Fe is a simple and widely available materialand is of specific interest in understanding the chiral spec-troscopy EELS results reported in (Verbeeck et al., 2010),while the more complex SrTiO3 crystal consists of atomicspecies of varying mass, allowing for more complex dy-
36
namics. In both cases, investigations of the phase andamplitude of the wavefunction within the crystal demon-strate that the resulting exit wave depends strongly notonly on the thickness of the crystal, but also the posi-tion of the incident beam in the unit cell, as well as thetopological charge of the vortex.
For an incident vortex beam with l = 1 the ex-pectation value of orbital angular momentum withinthe iron crystal is found to oscillate with propagation,and may take values significantly di↵erent to the orig-inal, including non-integer values and even reversingin sign (Lo✏er et al., 2012). For the SrTiO3 crys-tal, the vortex is found to channel strongly along theatomic columns, and is protected from delocalisationcompared to a non-vortex beam, remaining in an ap-proximate angular momentum eigenstate within a certainradius, beyond which the wavefunction exhibits Rankine-like vortex behaviour (Lubk et al., 2013a; Swartzlan-der and Hernandez-Aranda, 2007). Additionally, vortex-anti-vortex loops are spontaneously generated about thesurrounding atomic columns. They manifest themselvesas vortex-anti-vortex pairs in the x-y plane as the vortexforming the loop propagates in the z-direction and thenturns back on itself (Lubk et al., 2013a). When the vor-tex beam is incident o↵ centre on an atomic column thecore of the vortex is found to circulate around the atomiccolumn, while the centre of mass line remains stationary.
For higher order vortex beams with l > 1 interestingdynamics arise that are dependent on the the specificsymmetry considerations about the point of incidence.The l > 1 vortex beam splits into a number of vortexbeams of various orders, the specific order and arrange-ment of which depend on the specific symmetry. Asa result, the local values of orbital angular momentumhave a complicated dependence on the strength of theincident vortex state and the specific symmetry of thematerial as well as the propagation length and positionwithin the unit cell. This is related to the extrinsic na-ture of the orbital angular momentum as discussed above(see section VI.A.2), and indicates that atoms at di↵er-ent positions within the sample will be subject to modeswith vastly di↵erent orbital angular momenta. This in-troduces complications in, for example, electron energyloss spectroscopy, as the atoms within the sample inter-act with vortices of various strengths. For relatively thicksamples then, particular care must be taken in analy-sis requiring direct observation of phase and intensitycontrast; however filtering the separated scattered vor-tex states will go some way to ameliorating the phasecomplications.
It has been shown that an electron vortex beam canpropagate through atomic columns to a considerable dis-tance by coupling to the 2p columnar orbital with thesame angular momentum about the propagation axis.This shows that the divergence of the electron vortexbeam can be counteracted by interacting with an atomic
column (Xin and Zheng, 2012). The interaction of anelectron vortex beam with an atomic column has alsobeen studied by Xie et al. (2014).One application of electron vortex beams to crystalline
materials is in the determination of the chirality of enan-tiomorphic crystals through di↵raction pattern analysis(Juchtmans et al., 2015, 2016). The other predicted ap-plication of the electron vortex beam is to make use ofthe interplay between the elastic and inelastic scatteringto determine the magnetic dichroism spectroscopy (Ruszand Bhowmick, 2013; Rusz et al., 2014). Here, as dis-cussed earlier, the small size of the electron vortex beamis important in highlighting the advantage of using elec-tron vortex beams over conventional non-vortex beams.
C. Mechanical transfer of orbital angular momentum
The electron vortex beam carries both linear momen-tum and orbital angular momentum of ~k
z
and ~l re-spectively, each along the axial direction. As for op-tical vortices, the total linear and angular momentumof the electron vortex beam have components which arenon-zero only in the axial direction, while both the lin-ear and angular momentum density vectors (defined inEq. 27 and 31) have additional components in the ra-dial and azimuthal directions (Lloyd et al., 2013; Speiritsand Barnett, 2013). These local densities contribute tothe di↵ractive e↵ects within the beam, and the azimuthalmomentum density provides the requisite angular motioncontributing to the total angular momentum; howevercomponents of the total orbital angular momentum donot exist neither radially nor azimuthally.For the electron vortex, in addition to the mechanical
momentum due to the electron mass current, the elec-tric and magnetic fields may also contribute to the totalbeam momentum and angular momentum. As with me-chanical momenta, the contributions to the total linearand angular momentum vectors are found to exist onlyin the axial directions, despite the field momentum den-sities having additional radial and azimuthal components(Lloyd et al., 2013). The contributions from the electricand magnetic fields of the electron vortex beam are small;for the typical beam generated within an electron micro-scope, as described above, the linear and angular momen-tum contributions due to the electromagnetic fields areapproximately 10�12 and 10�14 that of the mechanicalmomenta respectively.As in the case of optical vortices being used to trap
and rotate objects from atoms to particles of micron size(Andersen et al., 2006; Barreiro and Tabosa, 2003; Emileet al., 2014; Franke-Arnold et al., 2008; He et al., 1995;Ru↵ner and Grier, 2012), the influence of an electron vor-tex beam has been shown to induce rotation in nanoparti-cles (Gnanavel et al., 2012; Verbeeck et al., 2013). Usingthe forked holographic mask technique to generate elec-
37
tron vortex beams within a JEOL 2200FS double aberra-tion corrected scanning transmission electron microscopeoperated at 200 keV, gold nanoparticles on carbon sup-port were observed to rotate under the influence of thevortex beams (Gnanavel et al., 2012). The nanometrescale vortex beams produced are shown in the focal planein Fig. 17, having FWHM of 2.6 nm and 3.4 nm respec-tively for the first and second order beams. No centralnodes are observed in the first order beams due to par-tial coherence e↵ects. In order to minimise these e↵ects,the experiment was performed at a slight defocus suchthat the beam profile fully covers the 5 nm diameter goldnanoparticle.
The e↵ects of the vortex beam on the nanoparticle wereobserved using a video capture, with a rate of 0.83 framesper second. Initially, the structural changes, translationand rotation of the particle are minimal; however it wasfound that after approximately 5 minutes of illuminationsignificant damage had occurred to the carbon substratewith the particle essentially detached, and having alsoundergone some structural damage. At this point, theparticle is relatively free of the van der Waals interac-tion and e↵ects due viscous trapping potentials, and ro-tation is observed to occur at an average rate of 3.75� perminute, significantly faster than previous reports involv-ing beams with no orbital angular momentum. Selectedframes indicating the nanoparticle rotation are shown inFig. 22. Although the precise mechanism of the rotationis rather complicated, the existence of the azimuthal com-ponent of the linear momentum density is necessary to ef-fect rotation about the beam axis. This rotation is clearlyshown to occur due to the vortex nature of the beam bythe relatively high rotation rate, and the change of thedirection of rotation when the particle is illuminated witha similar beam carrying an opposite OAM (Gnanavelet al., 2012). A similar rotation of gold nanoparticleshas been observed on silicon nitride support (Verbeecket al., 2013).
It has been proposed that the angular momentumtransfer between the beam and the particle leading torotation arises due to the breaking of the cylindrical sym-metry of the beam (Verbeeck et al., 2013) by the parti-cle. The exact mechanism of orbital angular momentumtransfer between the beam and the nanoparticle dependson a number of variables, notably the relative size of thenanoparticle and the beam, as well as the material prop-erties of the particle, which a↵ect the beam scatteringdynamics within the crystal potential, in addition to theexperimental parameters (Verbeeck et al., 2013). Fur-thermore, the requirement that the sample support bedamaged before any rotation is observed suggests thatfriction between the nanoparticle and the support is thelimiting factor in this case. Indeed, after prolonged il-lumination under the beam, the nanoparticle eventuallybecomes coated with carbon from the support and ceasesto rotate.
FIG. 22: (Color online) Four snapshots of Au nanoparticlesrotated by 2nd order vortex beams selected from a video at1.2 sec. intervals. The centre dark core surrounded by thebright ring of the 1st order vortex beams is partially visible atthe bottom right corner. The angles of lattice fringes noted infigure correspond to 99.5, 99.0, 87.0 and 84.5 degrees respec-tively. Insets show the corresponding FFT.
This suggests that the electron vortex beam may be-come a useful tool in the investigation of friction at thenanoscale, which is still not well understood (Mo et al.,2009). Experiments involving the rotation of variousspecies of nanoparticle on a range of supports may thusbe considered useful in the characterisation of nanoscalefriction. Additionally, a friction-free control environmentcould be provided by rotation of particles whilst theyare levitated by an optical beam (Lloyd et al., 2013).Similar experiments may also be considered to exploreviscous forces, for example by using nanoparticles sus-pended in liquids in a liquid-cell sample holder. Theelectron vortex provides a method by which particles maybe moved transverse to a surface, so that the friction be-tween various surfaces and particles may be investigateddirectly; this transverse motion may also find applica-tion in nano-manipulation for various uses (Falvo andSuperfine, 2000), including molecular biophysics applica-tions (Balzer et al., 2013; Bormuth et al., 2009).
D. Polarization radiation
The magnetic moments associated with the orbital an-gular momentum of the electron vortex beam can be ar-bitrarily large, in principle, as the axial orbital angularmomentum can be very large. The polarisation radia-tion associated with the passing of a fast moving mag-netic dipole moments can display some interesting e↵ectsnot seen before (Ivanov and Karlovets, 2013a,b; Konkovet al., 2014), such as the circular polarisation of the emit-ting radiation. The challenge, however is to to be ableto produce coherent vortex beams of high topologicalcharges, not in the form of a distribution of a numberof vortices of lower order topological charges (Freund,1999; Ricci et al., 2012).
38
VII. APPLICATIONS, CHALLENGES ANDCONCLUSIONS
We have outlined the various methods currently usedfor the realisation of electron vortices in the laboratory.We have also emphasised the quantum nature of electronvortices as freely propagating de-Broglie vortex waves en-dowed with the property of orbital angular momentumabout their propagation axis, which also coincides withthe vortex core. The intrinsic properties of electron vor-tices have been pointed out, specifically their mass andcharge distributions and how these determine their mo-mentum and orbital angular momentum contents, as wellas their spin and its coupling to the orbital angular mo-mentum. This is particularly illuminating in terms of re-visiting some of the basics concepts involved. Progress inthe study of their interaction with matter has been sum-marised, especially in connection with magnetic systemsand the issue of transfer of orbital angular momentum tothe internal dynamics of atomic systems. The possibilityof using electron vortices to rotate nanoparticles and cur-rent experimental work on this has also been described.
The study of electron vortex beams and their inter-actions does clearly benefit from the highly advancedstate of electron optics used to produce high resolutionelectron microscopy, electron spectroscopy and electronbeam lithography. This sets electron beam technologyapart from all other matter beam technologies, with theneutron, ion and atom beams being the distant competi-tors and it is clear currently that it is much easier togenerate electron vortex beams than other matter vortexbeams. The downside of the existing electron optics tech-nology is that they are bulky and, unlike optical systems,not easily reconfigurable because of the connected vac-uum system essential for the free passage of the electronbeams. This means that most of the existing vortex beamresearch has to be conducted within the existing elec-tron microscopes designed with the science of advancedmaterials in mind and time-shared with real-world ap-plications as well. Nevertheless, where existing researchfacilities could be employed with little modification rapidprogress has been made. In particular, this is evident inthe development of electron vortex beam technology inelectron microscopy and our understanding of its char-acteristics and there is some proof-of-principle demon-strations of its unique capabilities. Much more researchis required to understand the intrinsic nature of someof its characteristics as well as the need for developingoptimised set-ups for vortex beam experiments so thatpractical applications can be properly tested. Amongthe optimised set-ups to be realised would be an electronvortex beam source similar to that available for opticalvortex beams (Cai et al., 2012). This would be a use-ful alternative to the various beam conversion schemesthat have been discovered so far. There are some en-couraging experimental evidence for that (Schmidt et al.,
2014). The electron repulsion that would be expected in ahigh brightness (many-electron) beam needs to be takeninto account. A feasibility study on shape-preservingmany electron vortex beam design has been conductedby Mutzafi et al. (2014) and the result is promising, in-dicating that the generation of a high current electronvortex beam is possible.It is probably too early to apply vortex beams rou-
tinely to study the physics of materials because althoughthe existing electron beam set-ups are optimised for elec-tron microscopy, a number of electron vortex beam ex-periments have already led to applications to be realisedin principle. This fact allows us to contemplate on whatpossible developments in the future might be.As pointed out at the outset, more applications of
electron vortex beams are expected to follow, in whichthe orbital angular momentum of the beam is expectedto provide new information about crystallographic, elec-tronic and magnetic properties of the sample. Magnetic-dependent EELS has already been demonstrated, basedon the principles of electron vortex beams, and it is pre-dicted that the high resolution achievable in the elec-tron microscope will lead to the ability to map mag-netic information at atomic or near-atomic resolution.The determination of magnetic structure at the nanoscalehas always placed a significant demand on electron mi-croscopy. The linear and chiral dichroic spectroscopies(Schattschneider et al., 2006; Yuan and Menon, 1997),based on non-vortex beams have been developed, withthe linear dichroism being useful in the study of spin-orientation in antiferromagnetic materials and the chiraldichroism for spin orientation in ferromagnetic materi-als. Electron vortex spectroscopy o↵ers an alternativemethod to access chiral-dependent electronic excitations(Yuan et al., 2013).The orbital angular momentum and magnetic proper-
ties of the electron vortex beams may also find potentialuses in spintronic applications, either in the character-isation of spintronic devices, or in contexts employingspin-polarised current injection, through spin-to-orbitalangular momentum conversion processes (Karimi et al.,2012). The reverse process of spin-filter using spin-orbitcoupling in electron Bessel beams has been investigatedtheoretically (Schattschneider et al., 2017).Additionally, the inherent phase structure of the vor-
tex is considered ideal for applications in high resolutionphase contrast imaging, as required for biological speci-mens with low absorption contrast (Jesacher et al., 2005)or in revealing local orbital angular momentum density(Juchtmans and Verbeeck, 2016).Applications of electron vortex beams are, however,
not restricted to spectroscopy and imaging - the orbitalangular momentum of the beam may also be used forthe manipulation of nanoparticles (Gnanavel et al., 2012;Verbeeck et al., 2013), leading to electron spanners analo-gous to the widely used optical spanners. Electron vortex
39
states are also relevant in the context of quantum infor-mation processing and, in particular, low-energy electronvortex beams may potentially be used to impart angularmomentum on Bose-Einstein condensates (Fetter, 2001).
It has also been predicted that the large magnetic mo-ments associated with electron vortex beams of higherorbital angular momentum can be used to produce novelpolarisation radiation when passing through materials(Ivanov and Karlovets, 2013a,b). This has led to a driveto produce electron beams with very large topologicalcharges (Grillo et al., 2015).
Both the kinematic and dynamical di↵raction of elec-tron vortex beams by chiral crystals have been shown tobe sensitive to the helicity of the vortex beam, suggest-ing a useful characterising tool for the study of chiralcrystals.
The study of an electron vortex beam with strong laserfields has also opened up the possibility of acceleratingnon-relativistic twisted electrons using focused electro-magnetic fields (Karlovets, 2012) or beam steering usingthe high electric fields due to an ultrashort pulsed beam(Bandyopadhyay et al., 2015; Hayrapetyan et al., 2014).Research into laser-electron beam interaction also allowscoherent optical vortex beams to be produced (Hemsinget al., 2013). The inverse photoemission process involv-ing an electron vortex beam is another interesting prob-lem for further investigation (Matula et al., 2014; Zaytsevet al., 2017).
In particle physics experiments, the use of vortexbeams instead of approximate plane waves would permita direct measurement to be made as to how the over-all phase of the plane wave scattering amplitude changeswith the scattering angle (Ivanov et al., 2016; Ivanov,2012b; Karlovets, 2016). This may be important formany high energy experiments in hadron physics. It istherefore highly desirable to introduce electron vortexbeams into high energy particle accelerators.
Vortex beam-beam interactions also open up the pos-sibility of the creation of two vortex-entangled beams,with implications for quantum information processing(Ivanov, 2012a,b).
In conclusion, our review of the area of electron vortexbeams has indicated that much progress has already beenmade in a relatively short period of time, particularlyover the last few years. The emphasis so far has been pri-marily on the fundamental aspects of the electron vortexbeams. However, there are still many unexplored topics,ranging from novel vortex beams (Wang and Li, 2011),to image contrast enhancement and to exploring applica-tions of vortex beams in quantum information processing,just to mention a few. The range and complexity of thephenomena involving electron vortex beams point to abrighter future for further developments in this field.
Recent papers
As a sign of growing interest in the vortex beamphysics, interesting development is still continuing dur-ing the refereeing process of this review. In particular,we highlight the recent proposal to perform a measure-ment of the OAM content of an electron vortex beam(Larocque et al., 2016) using the microscopic version ofthe magnetic braking experiment (Donoso, 2009; Donosoet al., 2011; Saslow, 1992) or by weak measurement (Qiuet al., 2016), the possible detection of magnetic contrastin magnetic crystal using Zeeman e↵ect (Edstrom et al.,2016a,b) as well as the recent debate about the exis-tence of vortex structure in relativistic electron beams(Barnett, 2017; Bialynicki-Birula and Bialynicka-Birula,2017).
ACKNOWLEDGMENTS
The authors wish to thank the UK Engineering andPhysical Sciences Research Council (EPSRC) for finan-cially supporting our own research in this area (GrantNo. EP/J022098/1).
LIST OF SYMBOLS AND ABBREVIATIONS
A vector potential of externally applied axialmagnetic field
↵ convergence angle of a focused electronbeam
↵ a parameter characterizing the magneticcorrection to the e↵ective vortex beamtopological charge
D e↵ective length of the vortex beam�� rotation angle�l,s
spin-orbit energy�� phase change�t film thickness of phase masks for electron
beamsE energy eigenvalue of the electron vortex
beamE and B electric and magnetic vector fields associ-
ated with the electron vortex beamFµ⌫
electromagnetic field tensor
p
Fq
generalised hypergeometric functiongs
gyromagnetic ratio�(x) gamma functionH HamiltonianI beam electric currentIn nth moment integral of the Bessel functionJl
(x) Bessel function of the first kind of order lj(r, t) probability current densityjm
(r, t) mass current density, equivalent to inertialinear momentum density P
m
40
J total angular momentum of the electronvortex beam
k wavevector of the electron wave.
p
1� m
E
kz
axial wavenumber of the electron vortexbeam.
k? transverse wavenumber of the electron vor-tex beam.
l winding number; also referred to as topo-logical charge or azimuthal index
Iholo
hologram intensityLl
p
(x) generalised Laguerre polynomial with az-imuthal index l and radial index p
Lm
inertial angular momentum density of theelectron vortex beam
Lm
inertial angular momentum of the electronvortex beam
Lem
electromagnetic angular momentum den-sity of the electron vortex beam
Lem
global electromagnetic angular momentumof the electron vortex beam
Lz
axial component of the orbital angular mo-mentum operator
�p,l
pth zeros of Bessel function of order l�p,l
wavelengthM
fi
matrix element of interaction HamiltonianH
int
between quantum states i and fn(r, t) probability density (= ⇤ )nL
quantised Landau level indexne↵ e↵ective refractive indexn order of di↵raction orderN
l
normalization constant for Bessel beamwavefunction of order l
!L
Larmor frequency!c
cyclotron frequency! angular frequency of the vortex beam⌦ angular frequency of nanoparticlesPm
inertia linear momentum density of theelectron vortex beam
Pm
global inertia linear momentum of the elec-tron vortex beam
Pem
linear momentum density associated withthe fields of the electron vortex beam
Pem
global electromagnetic linear momentumof the electron vortex beam
p radial indexp (canonical) linear momentum vector or
canonical linear momentum operator inquantum formalism
pkin (kinetic) linear momentum vector operator(= p� eA)
p electron four momentum operator� azimuthal variable in cylindrical polar co-
ordinates
��� unit vector in the azimuthal direction
' magnetic flux� electrostatic potential (⇢,�, z) electron vortex wavefunction in cylindrical
coordinates LG
p,l
wave-function of Laguerre-Gaussian elec-tron vortex beam of winding number l andradial index p
B
l
wavefunction of Bessel electron vortexbeam of winding number l
LG
B
wavefunction of electron vortex wavefunc-tion in a constant magnetic field B
AB wavefunction of electron vortex wavefunc-tion threading a single magnetic flux
tB
p,l
wavefunction of truncated Bessel beam ofwinding number l and radial index p
four-component relativistic wavefunctionof an electron vortex beam
q topological order of defects in liquid crystal⇢ in-plane radial variable in cylindrical polar
coordinates⇢m
(r, t) mass density (= mn(r, t))⇢e
(r, t) charge density (= en(r, t))⇢⇢⇢ unit vector in the in-plane radial directionr(⇢,�, z) position vector in cylindrical coordinates⇢m
radius of maximum intensity of the vortexbeam
Rmax
radius of the aperture/maskR(z) curvature of the wavefront of a gaussian
beamS spin half angular momentum vector oper-
atorS(t) an odd, self-adjoint operator used in Foldy
and Wouthuysen transforms spin quantum number⌃ helicity of the beam�i
ith component of Pauli matrix�µ⌫
Spin tensor✓ angle of the cone of Bessel plane waves⇥(x) Heaviside step functionu(⇢, z) transverse mode functionw(z) beam radius at coordinate z from focus
planew0 beam radius at focus;w
B
beam width in the presence of magneticfield
w two component spinor characterizing theelectron polarization in the rest frame withE = m.
⇠ spin-orbit coupling constantz z- variable in cylindrical polar coordinatesz unit vector in the z- directionzR
Rayleigh range of the focused beamzB
magnetic Rayleigh range
41
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