Electron Vortices - Beams with Orbital Angular Momentum

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Lloyd, S. M., Babiker, M. orcid.org/0000-0003-0659-5247, Thirunavukkarasu, G.
orcid.org/0000-0002-8978-5304 et al. (1 more author) (2017) Electron vortices : Beams
with orbital angular momentum. Reviews of Modern Physics. 035004. ISSN 0034-6861
https://doi.org/10.1103/RevModPhys.89.035004
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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)
Therecentpredictionandsubsequentcreationofelectronvortexbeamsinanumberof
laboratoriesoccurredafteralmost20yearshadelapsedsincetherecognitionofthephys-
icalsignificanceandpotentialforapplicationsoftheorbitalangularmomentumcarried
byopticalvortexbeams. Arapidgrowthininterestinelectronvortexbeamsfollowed,
with swift theoretical and experimental developments. Much of the rapid progress can
be attributed in part to the clear similarities between electron optics and photonics
arisingfromthefunctionalequivalencebetweentheHelmholtzequationsgoverningthe
freespacepropagationofopticalbeamsandthetime-independentSchro¨dingerequation
governingfreelypropagatingelectronvortexbeams. Thereare,however,keydifferences
in the properties of the two kinds of vortex beams. This review is concerned primar-
ilywiththeelectrontype,withspecificemphasisonthedistinguishingvortexfeatures:
notably the spin, electric charge, current and magnetic moment, the spatial distribu-
tion as well as the associated electric and magnetic fields. The physical consequences
and potential applications of such properties are pointed out and analysed, including
nanoparticle manipulation and the mechanisms of orbital angular momentum transfer
intheelectronvortexinteractionwithmatter.
CONTENTS 1. Lensaberrations 26
2. Electronvortexmodeconverter 27
I. Introduction 1
V. Vortexbeamanalysis 27
II. Quantummechanicsofelectronvortexbeams 3 A. Interferometry 27
A. Phasepropertiesofvortexbeams 3 1. Electronholography 27
B. VortexbeamsolutionsoftheSchro¨dingerequation 4 2. Knife-edgeandtriangleaperturediffractive
1. Laguerre-Gaussianbeams 5 interferometry 28
2. Besselbeams 6 3. Diffraction 28
3. Bandwidth-limitedvortexbeams 7 B. Modeconversionanalysis 28
C. Mechanicalandelectromagneticpropertiesofthe C. Imagerotation 29
electronvortexbeam 9 1. Gouyrotation 29
1. Inertialmechanicalproperties 9 2. Zeemanrotation 29
2. Electromagneticmechanicalproperties 10 D. Vortex-vortexinteractionsandcollisions 29
D. Intrinsicspin-orbitinteraction(SOI) 12 E. Factorsaffectingthesizeofthevortexbeam 30
III. Dynamicsoftheelectronvortexinexternalfield 14 VI. Interactionwithmatter 31
A. Parallelpropagation 14 A. Chiral-specificspectroscopy 31
B. Transversepropagation 15 1. MatrixelementsforOAMtransfer 32
C. Rotationaldynamicsofvortexbeams 16 2. Theeffectofoff-axisvortexbeamexcitation 32
D. Extrinsicspin-orbitinteraction 17 3. Plasmonspectroscopy 34
E. Electronvortexinthepresenceoflaserfields 18 B. Propagationincrystallinematerials 35
C. Mechanicaltransferoforbitalangularmomentum 36
IV. Generationofelectronvortexbeams 19 D. Polarizationradiation 37
A. Phaseplatetechnology 19
B. Holographicdiffractiveoptics 20 VII. Applications,challengesandconclusions 38
1. Binarisedamplitudemask 20 Recentpapers 39
2. Binaryphasemask 23
3. Blazedphasemask 23 Acknowledgments 39
4. Choiceofreferencewaves 24
C. Electronopticsmethods 25 ListofSymbolsandAbbreviations 39
1. Spintoorbitalangularmomentumconversion 25
2. Magneticmonopolefield 26 References 41
3. Vortexlattices 26
D. Hybridmethod 26
I. INTRODUCTION
Electron vortex beams are a new member of an ex-
⇤ [email protected],[email protected] panding class of experimentally realisable freely propa-
2
gating vortex states having well-defined orbital angular atomvortexbeams(Hayrapetyanet al.,2013;Lembessis
momentum about their propagation axis; the prototypi- et al., 2014). A related recent advance in matter vortex
cal example of which is the much studied optical vortex beams is the realisation of neutron vortex beams in the
beam. Thetermvortexbeamreferstoabeamofparticles laboratory (Clark et al., 2015).
-electrons,photonsorotherwise-thatisfreelypropagat- Although the basic concepts in terms of beam forma-
ing and possesses a wavefront with quantised topological tion of electron vortices essentially stem from those en-
structure arising from a singularity in phase taking the countered in the optical vortex case, the electron vortex
form eilφ with � being the azimuthal angle about the is distinguished by additional properties, most notably
beam axis and l an integer quantum number also known the electric charge and half-integer spin. They are thus
asthetopologicalcharge(orwindingnumber). Thetopo- fermionvortexstatescharacterisedbyascalarfieldinthe
logical structure of the wavefront was first described by formof theSchr¨odinger wavefunction fornon-relativistic
Nye and Berry (1974) as a screw-type dislocation in the electrons and Dirac spinors for the ultra-relativistic elec-
wavetrains in analogy with crystal defects. tronbeams,whileopticalvortexbeamsarebosonicstates
Over the last two decades optical vortices have been described by vector fields. Furthermore, there are sub-
a subject of much interest, after the publication of the stantial differences in scale. Currently, electron vortices
seminal work of Allen et al. (1992) in which the quan- created in a medium-voltage (100-300 kV) electron mi-
tised orbital angular momentum of a Laguerre-Gaussian croscope have de Broglie wavelengths of the order of pi-
lasermodewasexamined(theearlierdiscussionofoptical cometers while optical vortices in the visible range have
vortices in laser modes by Coullet et al. (1989) did not wavelengths of the order of several hundreds of nanome-
emphasise the quantisation of the orbital angular mo- ters. Electronvortexbeamscanthusprobemuchsmaller
mentum about the propagation axis). Since then, op- featuresthanispossiblefortheopticalvortexbeams,and
tical vortices have been intensively studied leading to as such the range of applications of electron vortices is
many diverse applications (Allen et al., 2003, 1999; An- predicted to be substantially different from the existing
drewsandBabiker,2012),includingopticaltweezersand scope of optical vortex beams.
spanners for various applications (Dholakia et al., 2002; The earliest work on particle vortex beams is
Grier, 2003; He et al., 1995; Ladavac and Grier, 2004): due to Bialynicki-Birula and Bialynicka-Birula (2001);
micromanipulation (Galajda and Ormos, 2001); classi- Bialynicki-Birula et al. (2000, 2001). The current re-
cal and quantum communications (Yao and Padgett, search activity specifically in electron vortex beams was
2011); phase contrast imaging in microscopy (Baranek stimulated by work due to Bliokh et al. (2007), shortly
andBouchal,2013;Fu¨rhapteretal.,2005;Zu¨chneretal., followed by the experimental realisation in several labo-
2011); as well as further proposed applications in quan- ratories (McMorran et al., 2011; Uchida and Tonomura,
tum information and metrology (Molina-Terriza et al., 2010;Verbeecket al.,2010). Ithasnowbeenestablished
2007;YaoandPadgett,2011)andastronomy(Leeet al., that electron vortices can be created inside electron mi-
2006;Tamburinietal.,2011;Thid´eetal.,2007). Thedis- croscopesandthereexistanumberoftechniquesforvor-
cussion of photonic spin and orbital angular momentum tex beam creation, including computer generated holo-
in various situations, and the similarities and differences graphic masks applied in similar ways to those routinely
betweenthetwotypesofangularmomentumhaveledto adopted in the creation of optical vortex beams (Heck-
new ways of thinking about, and examining orbital an- enberg et al., 1992a,b). This review aims to describe the
gular momentum in this context. The spin and orbital recent developments in the expanding field of electron
angular momentum can not be clearly separated in gen- vortex physics, and highlight significant areas of poten-
eral, i.e without the imposition of the paraxial approx- tial applications. Specifically, electron vortex beams are
imation (Barnett and Allen, 1994; O’Neil et al., 2002; hopedtoleadtonovelapplicationsinmicroscopicalanal-
Van Enk et al., 1994), which leads to the possibility of ysis, where the orbital angular momentum of the beam
the entanglement of the two degrees of freedom (Khoury is expected to provide new information about the crys-
andMilman,2011;Mairet al.,2001). Moresubtlequan- tallographic, electronic and magnetic composition of the
tumeffectsduetotheinteractionofopticalvorticeswith sample. Chiral-dependent electron diffraction has been
atoms and molecules involve internal atomic transitions detected (Juchtmans et al., 2015, 2016) as well as the
atnearresonancewiththebeamfrequency. Heretoo,op- demonstration of magnetic-dependent electron energy-
tical forces and torques are at play (Allen et al., 1996a; loss spectroscopy (EELS) (Verbeeck et al., 2010), and
Andersen et al., 2006; Babiker et al., 1994; Lembessis it is predicted that the high resolution achievable in the
et al., 2011; Surzhykov et al., 2015), leading to the trap- electron microscope will lead to the ability to map mag-
ping and manipulation of individual atoms in certain re- netic information at atomic or near-atomic resolution.
gions of the beam profile, with promising applications Additionally, the inherent phase structure of the vor-
in the new field of atomtronics (Andersen et al., 2006; tex is considered ideal for applications in high resolu-
Lembessis and Babiker, 2013; Pepino et al., 2010; Sea- tion phase contrast imaging, as required for biological
man et al., 2007), as well as the proposed generation of specimens with low absorption contrast (Jesacher et al.,
3
2005). Applications of electron vortex beams are, how- et al., 2013; Karlovets, 2012; Schattschneider and Ver-
ever,notrestrictedtodiffraction,spectroscopyandimag- beeck, 2011) yielding non-relativistic, relativistic and
ing-theorbitalangularmomentumofthebeammayalso spinor electron vortex beams, respectively. The spatial
be used for the manipulation of nanoparticles (Gnanavel distributions of these vortex solutions may take various
et al., 2012; Verbeeck et al., 2013), leading to electron formsintherelativistic,non-relativisticandparaxiallim-
spanners analogous to the widely used optical tweezers its and each state is characterised by the distinguishing
and spanners. Electron vortex states are also relevant in feature of a vortex, namely the node on the propaga-
the context of quantum information and, in particular, tion axis. Such states have been mostly described ei-
the electron vortex may potentially be used to impart therbytheBesselfunctions,prototypesofnon-diffracting
angular momentum into vortices in Bose-Einstein con- vortex beams (McGloin and Dholakia, 2005), or by the
densates (Fetter, 2001). The orbital angular momentum Laguerre-Gaussian (LG) functions which are well known
and magnetic properties of the electron vortex may also in optics (Allen et al., 1992), with LG representing a
findpotentialusesinspintronicapplications,eitherinthe beam with a well defined waist at the focal plane. Since
characterisation of spintronic devices, or in contexts em- both the Bessel and Laguerre-Gaussian sets of functions
ployingspin-polarisedcurrentinjection,throughspin-to- form complete orthonormal basis sets, any beam, vortex
orbital angular momentum conversion processes (Karimi or otherwise, may be described in terms of these vor-
et al., 2012). tex states. In the non-paraxial and relativistic limits of
Our aim in writing this review has been to strive to the Dirac equation, the spin of the electron also plays a
provide a report on the current state of the new subject role,andtheparticulardistributionismodifiedbyaspin-
ofelectronvortexbeamsandtheirinteractions. Wehave orbit interaction intrinsic to the beam (see section II.D).
endeavoured to survey much of the relevant literature, Note that there is an interesting alternative approach
but any omissions of specific references would certainly describing electron vortices as a natural consequence of
be inadvertent and we would apologise for not having the skyrmion model of a fermion (Bandyopadhyay et al.,
come across them. For more focused prospectives, the 2014, 2016, 2017; Chowdhury et al., 2015), but this will
readerscanconsulttworecentlypublishedpapersonthe not be discussed any further in this review.
subject (Harris et al., 2015; Verbeeck et al., 2014). The remainder of this section introduces the specific
The outline of this review is as follows: section II in- properties of the electron vortex beam mainly in the
troducesthequantummechanicsgoverningthepropaga- non-relativisticandparaxiallimits,focusingonsolutions
tion of electron vortex beams, namely the wave equation to the Schr¨odinger equation. This will not only enable
discussed in the non-relativistic and relativistic regimes. direct comparison with the commonly applied paraxial
The mechanical and electromagnetic properties arising solutions in optics, a comparison facilitated by the func-
from the vortex mass and electric charge are then con- tionalequivalencebetweentheSchr¨odingerandthescalar
sidered, along with the role of the vortex fields in the Helmholtzequations,butalsoillustratesthemostimpor-
spin-orbit interaction within the beam. Section III cov- tant properties of electron vortex beams and serves as a
ersthedynamicsoftheelectronvortexinexternalfields. basis for understanding more complex vortex beams.
Section IV discusses the various methods for the realisa-
tionofelectronvortexbeamsinthelaboratorythathave
hitherto been considered, drawing comparison with the A. Phase properties of vortex beams
creation of optical vortex beams wherever such an anal-
ogy can be identified. Section V deals with the methods The phase structure of a vortex wave is topologically
one can use to analyse the various properties associated different to that of a plane wave. In contrast to a plane
with vortex beams. The interaction of electron vortex wave, with phase-fronts that are normal to the propa-
beams with matter is covered in VI. The prospects of gation direction, the phase front of the vortex wave de-
using electron vortex beams to determine chirality and scribes a helix about the axis of propagation (Nye and
othermagneticinformationarediscussedintermsofboth Berry,1974)suchthatthephaseisdependentonthean-
theoretical and experimental considerations, with con- gular position about the axis. This topological structure
cluding remarks about the field given in section VII. was 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 (also
II. QUANTUM MECHANICS OF ELECTRON VORTEX called the winding number) quantises this winding such
BEAMS thatthereareltwistedwavefrontsaboutthebeamsaxis,
or equivalently a phase change of 2⇡l during a full rota-
Freely propagating vortex states having the required tion about the axis, as shown in Fig. 1. The phase fac-
eilφ phase factor may be written as solutions to the tor of eilφ that gives rise to this helical phase structure
Schr¨odinger, Klein-Gordon and Dirac equations, (Bliokh is a characteristic feature of orbital angular momentum
et al., 2007, 2011; Bliokh and Nori, 2012b; van Boxem (c.f.thesimilarphasefactorintheazimuthalcomponents
4
of the orbital angular momentum-containing hydrogenic
wavefunctions). The functions characterising a vortex
beam propagating with a well defined axis (taken along
the z direction) have a general form which can be conve-
niently written in cylindrical coordinates r(⇢,�,z):
(r,t)=u(⇢,z)eilφeikzze�iωt, (1)
l
with u(⇢,z) a suitable mode function such as the
Laguerre-Gaussian functions (section B1), which are
(b) Phasechangeinthe
characterised by the azimuthal index l and the radial
transverseplanefor l =1
index p, or the Bessel functions of the first kind (section
B2), which are charecterised by just the azimuthal index
l. Thehelicalphasestructureofthevortexbeamleadsto
the phase at the core of the beam as indeterminate since
it is connected to all possible phases of the wave. This
central phase singularity is not physically viable, and is
compensated by the requirement that all functions must
vanish on axis (at the location of the singularity), giving
the beam a cross-sectional distribution in the form of a
ring,orconcentricrings. Thishasledtothenicknameof
‘doughnut’ beams for a particular class of vortex beams (c) Spiral phase (d) Phasechangeinthe
front for l = 3 transverseplanefor l =3
- the Laguerre-Gaussian vortices with non-zero winding
number |l|, but zero radial index (p = 0), each being a
FIG. 1: (Color online) The phase character of l = 1 and
brightringsurroundingacentraldarkcore. Therequire- l = 3 vortex beams. (a) shows the single helix phase front
ment that all vortex functions must vanish on axis is, of an l = 1 vortex around the beam axis. (b) shows the cor-
however, not sufficient to describe a vortex beam - there responding continuous phase ramp in one of the transverse
must be some topological difference between a region of plane perpendicular to the beam axis. The total phase change
on one rotation is exactly 2π. For the l = 3 vortex there are
the beam containing the vortex, and a region that does
three helical surfaces of constant phase, each move around
not (Nye and Berry, 1974). For the present purposes,
the axis as shown in (c). This leads to a total phase change
the topology of the vortex describes the connectedness
on one rotation of exactly 6π. We used wrapped phase repre-
of the phase fronts. The phase front of the vortex is sentation, with all phase mapped to values between 0 and 2π
topologically distinct from that of a plane wave, as one (represented between blue to red in the 2D phase maps in (b)
cannotbetransformedintotheotherthroughcontinuous and (d))), so there are three ’artificial’ phase jumps from 0 to
deformations. Similarly, the l =1 phase front cannot be 2π in the phase-wrap representation as shown in (d). In both
cases, the only real phase discontinuity of significance is on
deformed into the l = 2 or any other l phase front, and
beam axis.
this is the reason why the winding number l has also
been termed the topological charge, characterising the
‘strength’ of the vortex. As has been pointed out ear-
the normalised probability current density
lier, the phase front of any vortex is characterised by the
factor,eilφ,leadingtoaphasesingularityalongtheprop-
~ l
agationaxis. Inordertoappreciatethesignificanceofthe j(r)= φˆ+k zˆ . (3)
m ⇢ z
topological charge we shall assume that the function ✓ ◆
giveninEq. 1representsawavefunctionofavortexbeam
Integrating this about a loop enclosing the z axis gives
state of a particle of mass m. It is easy to show that a 2π~l, whileanyotherclosedpathgiveszero, showingthe
closed loop integration of the probability current density m
topologicaldistinctionbetweenaregionofspacecontain-
j(r) = ~ { ⇤(r,t) (r,t) (r,t) ⇤(r,t)} along a
2mi r � r ing the vortex and one that does not. Thus, on circling
pathC encirclingtheaxisgivesaquantisedvaluepropor-
the z-axis an additional phase of 2⇡l is acquired.
tional to the topological charge of the beam (Bialynicki-
Birula et al., 2000).
2⇡~ B. Vortex beam solutions of the Schro¨dinger equation
j(r)·ds= l (2)
m
IC Thewavefunction (r,t)describinganelectronvortex
where ds is a line displacement vector tangental to the beam is a solution of the Schr¨odinger equation, namely
path C.
For the vortex beam given in the form Eq. 1 we have H (r,t)=E (r,t) (4)
5
where E is the energy eigenvalue. In free space, the
w(z)
Hamiltonian H is given by the kinetic energy of the elec-
z=0
tron beam only:
p2 R(z) w0 2w0
H= (5)
2m
where p is the linear momentum operator. Equation 4
can be re-arranged to look like the Helmholtz equation
for monochromatic light: z
R
2 (r,t)+k2 (r,t)=0, (6)
r
FIG. 2: (Color online) Schematic representation of the Gaus-
where k is the wave vector of the electron wavefunction sian profile showing the characteristic parameters, namely the
and is given by width w(z), with w0 the width at the narrowest part of the
beam; zR the Rayleigh range and R(z) the in-plane radius of
2mE
k2 = (7) curvature at axial position z
~2
Thisequivalenceisthebasisfortreatingfreelypropagat-
ingelectronandlightonthesamefooting,atleastatthe
to in-plane (transverse) coordinates. This equation de-
scalarfieldlevel. Indeedthefieldsofelectronmicroscopy,
scribes a component of the relevant field propagating in
ion beam physics and accelerator physics started on this
the z direction with axial wavevector of magnitude k .
z
basis. The same is true for the electron vortex beam
Thevariationsalongtheaxisareconsideredsosmallthat
research.
the second axial derivative may be neglected (Kogelnik
Ingeneral,thevortexsolutionoftheSchr¨odingerequa- and Li, 1966; Lax et al., 1975). The solutions of Eq. 8
tionrequiresthecomplexwavefunctionstobeidentically
represent a vortex for which the magnitude of the trans-
zteexrocotroec(oBpiaelywniitchkit-hBeiruplhaaeset ailn.,d2et0e0r0m).inTahcyisamtetahnestvhoart- vmeernsetummom~kent(uomve~rakl?l kis2m+uckh2s=makl2le)r. tAhasnutithaebalexivaolrmteox-
bothrealandimaginarypartsofthewavefunctionshould z ? z
solution to Eq. 8 is the Laguerre-Gaussian form, which
be zero separately. Each condition defines a surface and
has a Gaussian envelope modified radially by a Laguerre
the vortex core can be considered as the intersection of
polynomial, with appropriate phase factors. We have,
thetwosurfaces,resultinginalineofvortexcores. Inthe
written in cylindrical polar coordinates r=(⇢,�,z),
following, we first consider two simple solutions in which
thevortexcoreformsastraightlinealongthez-axis. We l
C z p2⇢ 2⇢2
then introduce a specific and a general solution, which is LG(r,t)= lp R Ll
more useful in the context of the practical electron vor- p,l zR2 +z2 w(z)! p✓w2(z)◆
texbeamswhichareusuallygeneratedunderbandwidth- peikzze�iωteilφ
⇥
limited conditions. In general, the vortex lines can be
curved, closedandknotted(O’Holleranet al.,2008), but e⇢�w2ρ2(z)+2(izkR2zρ+2zz2)�i(2p+|l|+1)tan�1⇣zzR⌘�
they can be regarded as a superposition of the simpler ⇥ (9)
straight vortex lines introduced below.
whereLl(x)isthegeneralisedLaguerrepolynomial,with
p
azimuthal index l and radial index p 0, and nor-
1. Laguerre-Gaussian beams malisation factor C = 2|l|+1p!/[⇡(|l|�+p)!]. The z-
lp
dependenceoftheGaussianenvelopeisdepictedinFig.2,
Optical vortex beams are most commonly discussed p
with the characteristic parameters of width w(z) and
in terms of Laguerre-Gaussian modes, as these are a
Rayleigh range z given by
R
good approximation to the vortex modes created from
Hermite-Gaussian laser modes (Lax et al., 1975; Pad- 2
2z
gaeritste,s1a9s96a).solTuhtieonLaogfutehrerep-aGraauxsiasliaanppvroorxteimxabteioanmosfttahtee w(z)=w0s1�✓kzw02◆ ; (10)
k w2
Helmholtz equation for light or Schr¨odinger equation for z = z 0. (11)
R
electrons in free space: 2
where w = w(0) is the beam radius at focus. The ra-
@ 0
r2?+2ikz@z =0; (8) dial profile of the beam varies with the indices p and l.
✓ ◆ Theazimuthalindexl isresponsibleforthebeamorbital
where is a component of the vector field and the sub- angularmomentuml~perelectronandmaytakeanyin-
script in ? indicatesdifferentiationonlywithrespect tegervalue,eitherpositiveornegative;whereastheradial
? r
6
indexp 0specifiesthenumberofintensitymaxima,i.e. phase change of π as it passes through the focal plane
� 2
the number of rings in the radial intensity distribution, from to+ ,whereasthephaseshiftoftheLaguerre-
�1 1
such that the beam has p+1 maxima (for l = 0, the Gaussian beam on focusing is given as
beam has a central spot, and p additional rings). The
⇡
transverse distributions of the Laguerre-Gaussian beams (2p+|l|+1) (13)
� 2
are displayed in Fig. 3 for various sets of l and p. As can
be seen, the modes with |l|>0 have a central minimum, The Gouy phase shift arises due to the spatial confine-
and Eq. 9 has the appropriate eilφ phase factor, indicat- ment of the beam, leading to momentum components in
ing that the Laguerre-Gaussian modes are endowed with the transverse direction that contribute to the dynamic
the required feature of vorticity. phase of the beam (Feng and Winful, 2001; Petersen
et al., 2014). Near the focal plane, the rate of change
of the transverse momentum of the Laguerre-Gaussian
beam is larger than that of the fundamental Gaussian
beam due to the more complex radial profile. The mag-
nitude of the Gouy phase change thus depends on the
radial and azimuthal mode indices.
(a) LG00 (b) LG01 (c) LG02 2. Bessel beams
The Bessel-type electron vortex function takes the
form
lB(r,t)=NlJl(k?⇢)eilφeikzze�iωt (14)
where Jl(k?⇢) is the Bessel function of the first kind,
of order l, where, as above, l is the topological charge,
(d) LG10 (e) LG111 (f) LG12 or winding number. The wavenumbers kz and k? are
the axial and transverse wave vector components such
that |k|= k2+k2. N is a suitable normalisation fac-
z ? l
tor, determined by the specific boundary conditions of
p
the beam. Except for l = 0, all other Bessel functions
satisfy J (0) = 0, so that they are suitable for describ-
l
ing a vortex beam. Their spatial distribution functions
are functions of the radial coordinate ⇢ only, so that in
contrasttoLaguerre-Gaussianbeams,freelypropagating
(g) LG20 (h) LG21 (i) LG22 Besselbeamsarenon-diffractive(McGloinandDholakia,
FIG. 3: (Color online) Intensity distribution patterns for the 2005), and the use of the full Helmholtz or Schr¨odinger
LG modes, shown in the z = 0 plane. Intensity is given by equations coincides with the paraxial limit in this case.
pl
|ψLG|2. The concentric ring structure of the orbital angular The Bessel beams are indeed the simplest type of vor-
l
momentum carrying modes is clear, with p+1 rings. Colour tex beam and so provide an ideal theoretical platform to
scale shows the intensity variation within individual modes
determine the general characteristics of vortex beams.
(not the relative intensity variation across all modes). The
The oscillatory nature of Bessel functions gives the
Laguerre-Gaussian modes having l<0 have the same intensity
Bessel beam a cross-section of concentric rings, decreas-
distributions as shown above, however the phase (not shown)
has the opposite sign. ing in brightness away from the axis. This concentric
ring structure is shown in Fig. 4. However, unlike the
Laguerre-Gaussian function, which decays exponentially
with radial position, the Bessel function is infinite in ex-
In addition to the phase factor relating to the orbital
tent, so that in principle the beam contains an infinite
angularmomentumtheLaguerre-Gaussianbeamalsohas
number of rings. Each ring of the Bessel beam carries
a Gouy phase factor:
the same power in the case of an optical vortex beam
z (McGloin and Dholakia, 2005) while for electron vortex
exp i(2p+|l|+1)arctan (12)
� z beams the relevant property is the current, which im-
 ✓ R◆�
plies infinite power being carried by the beam, which
which is associated with the focusing of the beam at the is of course physically unrealistic. What is meant by
waistplane(FengandWinful,2001;Petersenetal.,2014). a physical Bessel-type beam is a beam that has am-
As a result a convergent Gaussian beam experiences a plitude modulation similar to a Bessel function, over a
7
(a) l=0 (b) l=1 (c) l=2
FIG. 4: (Color online) Plots of the intensity |ψB(r)|2 of a
l
Bessel beam with (a) l = 0, (b) l = 1, and (c) l = 2. The
Bessel modes with l < 0 have the same intensity distributions
as shown above, however the phase (not shown) has the oppo-
site sign.
finite radius, and whose core components behave non-
diffractively(suchthatthecentralmaximumorminimum
FIG. 5: (Color online) The Fourier transform of the Bessel
persistswithverylittlespreading)overareasonable,but
finite,propagationlength(McGloinandDholakia,2005). kbye,amsucrhestuhltastikn?a=setpokfx2w+avkey2s.oTfhfiexevdorktzexanBdesvsaerlybineagmkxilalunsd-
These are achievable by several methods in optics in- trated here has a phase factor eilϕ, so that the phase changes
cluding axicon lenses, annular apertures and holograms by 2πl on rotation about the kz axis. This is illustrated for
(Durnin et al., 1987; McGloin and Dholakia, 2005), and l = 1. The relationship between kz and k? fixes the cone an-
have also been generated in electron optics using kino- gle θ.
forms (Grillo et al., 2014a). A kinoform is wavefront
reconstruction device (for a reference see (Jordan et al.,
1970))
In the momentum representation k0(k0 ,�0,k0) the plane wave by a spiral phase plate with the transmission
? z function:
Bessel beam has the form
i�leilφ0 (⇢0,�0,z)=eilφ0eikzz (16)
˜l(k0)= 2⇡ k? �(kz �kz0)�(k?�k?0 ); (15) throughanapertureoffiniteradiusR . Wehaveused
max
which is interpreted as a superposition of plane waves of the convention for the momentum/Fourier space repre-
varying k? such that k = k?2 +kz2 for each wave. For sentation of the variables as in Eq. 15, since the trunca-
a given kz the possible k?plie on a ring on the surface tion is in practice taking place in the aperture plane of a
of constant k, so that there is a cone of plane waves of convergent electron lens as shown in Fig. 6.
varying k that constitute the Bessel beam (Bliokh et al., The diffracted beam intensity is related to the Fourier
2011; McGloin and Dholakia, 2005), with the phase of transformof the transmittedwave, which canbe written
each given by eilϕ. This is the principle by which axicon as (Kotlyar et al., 2006, 2007; Lubk et al., 2013a)
lenses produce the rings of a Bessel-type beam (Herman ˜(⇢,�,z)= eilφ2πil2�lρlRm2+alxx
and Wiggins, 1991). The conical propagation leads to (2+l)Γ(1+l)
another interesting property of the Bessel beam, namely F (1+ l;2+ l;1+l; 1⇢R2 )(17)
that the original spatial distribution is reconstructed af- 1 2 2 2 �4 max
ter propagation past an obstruction (MacDonald et al., where pFq(a;b;c;z) is the generalized hypergeometric
function. This is a limiting case of Hypergeometric-
1996; McGloin and Dholakia, 2005), as has been demon-
strated for electron vortex Bessel beams (Grillo et al., Gaussian beams (Karimi et al., 2007) due to diffraction
of apertured spiral phase masks by a Gaussian beam.
2014b).
To represent the waveform of the arbitrary beam
in similar bandwidth-limited situations, an orthonor-
3. Bandwidth-limited vortex beams mal basis set characterised by orbital angular mo-
mentum has recently been reported for vortex beams
In electron vortex beam research, the limited trans- (Thirunavukkarasu et al., 2017), including both the az-
verse spatial coherence of practical electron sources imuthal and radial quantum numbers l and p, respec-
means that finite radius vortex modes defined using a tively. It is based on describing the normal modes of the
circular aperture (or pupil) function is more appropriate transverse wavefront confined to a finite radius at the
in real situations. The simplest bandwidth limited vor- pupil or aperture plane by an orthonormal set of trun-
tex beam is generated by the Fraunhofer diffraction of a cated Bessel functions, much like the solutions of the
8
allowed normal modes of surface vibrations on a drum
surface:
TBB(⇢0,�0,z)=N eikzzeilφ0J (kpl⇢0)for⇢0 R
p,l p,l l ?  max
(18)
where the radial and azimuthal indices are p and l re-
spectively following the convention used in the case of
LG modes. We again used the cylindrical coordinates
(⇢0,�0,z) to describe the location in the aperture plane
andR istheradiusofthecircularaperture. Themag-
max
nitudeofthetransversewavevectorkpl takesthediscrete
?
values λp,l , with � the (p+1)th zero of the lth order
Rmax p,l
Bessel function J , and R is the radius of the aper-
l max
ture. The truncated Bessel functions, whose amplitudes
FIG. 6: (Color online) The intensity and phase distribution
are for ⇢ Rmax, together with the azimuthal phase of the transverse wavefunctions of FT-TBB at z = 0. Image
�
factor form a complete two-dimensional basis set of the after Thirunavukkarasu et al. (2017) and are plotted for the
OAM modes at the aperture plane. The Fourier trans- same relative scale.
form of these truncated Bessel beams forms a conjugate
setofquantumbasissetwhichwetermedFourierTrans-
formed Truncated Bessel Beams (FT-TBB):
J (k R )
FT�TBB(k ,�,z)=il� J0(� )eilφ l ρ max
p,l ρ p,l l p,l kpl2 k2
? � ρ
(19)
where k is the transverse wavevector of the diffracted
q
beams. At the focal plane of a lens of power 1/f, the
corresponding radial displacement (⇢) is given by fkq
kz
( fkq). The corresponding wavefunction in the focal
⇠ k0
plane coordinate (⇢,�) becomes:
FIG. 7: (Color online) Computer-simulated fine structure of
f J (k R )
FT�TBB(⇢,�,z)=il� J0(� )eilφ l ρ max the diffraction of Truncated Bessel Beams (FT-TBB): First
p,l p,lk l p,l ⇢2 ⇢2
0 pl � row for intensity and the second row for the corresponding
(20) phase distribution. From left to right, for the FT-TBB vortex
beams with l=1 but with different radial modes (p=0,1 and
where ⇢ is the radius of the most prominent doughnut 2). Images adapted after Thirunavukkarasu et al. (2017)
p,l
ring and is given by λp,lf .
Rmaxk0
TheinsetinFig. 6showstheschematicphasedistribu-
tion of some of the low order basis wavefunctions of the
truncatedBesselbeams(TBB).Alsoshownistheconju- This can also be understood by regarding the original
gaterelationshipbetweentheTBBandandtheFT-TBB. mask as the product of the unobstructed Bessel beam
and a top-hat mask function. The transverse structures
The amplitude and phase of the first three p-modes of of the vortex beam at the focal plane can be considered
thel=1FT-TBBsubsetareshowninFig.7respectively. as the convolution of the Fourier transform of the Bessel
These results show that the higher order radial modes function (whose transverse structure is a ring with a ra-
aredistinguishedbyp+1brightrings,reminiscentofthe dius controlled by the radial size of the first dark zone in
corresponding LG modes (Allen et al., 1992). However, themask)andthatofthetop-hatmask(whichisthewell
thesimilaritydoesnotextendtotheadditionalfaintring knownAirypatternwithsidebandringstructures). This
structuresthatcanbeseenintheamplitudedistribution is consistent with the mathematical form of the Fourier
of the FT-TBB mode. These small ringed structures transformofthenormalmodesintheapertureplane. As
1,3
are caused by the ringing effect of the sharply defined p increases, the size of the first node ring shrinks and
aperture. Another noticeable feature is that the largest the Bessel ring at the focal plane increases in size. This
amplitudeoccurswhenk approacheskpl , inwhichcase explains the size changes seen in Fig.7 (first row) for dif-
ρ ?
the wavefunction locally becomes a sinc function of the ferent values of p. The convolution of the Bessel rings
radial coordinate. withtheAirypatternfunctionsresultsinsidebands,but
9
they preserve the circular symmetry of the main Bessel 1. Inertial mechanical properties
peaks. The details of the experimental realization of the
FT-TBB beams can be found in ref. (Thirunavukkarasu The inertial mechanical properties are associated with
et al., 2017) the finite electron mass and these can now be derived as
The FT-TBB set of beams is one of the bandwidth- follows. The vortex wavefunction (r,t) gives rise to a
limited vortex beams whose spatial frequency is deter- local mass density ⇢ (r,t) and a mass current density
m
minedbytheaperturesize. AsthisFT-TBBsetofmodes j (r,t) which are as follows
m
forms a complete orthonormal set, it can be used to de-
scribe any such bandwidth-limited vortex beam. For ex- ⇢m(r,t)=m ⇤(r,t) (r,t); (24)
ample, the simplest and most investigated bandwidth-
limited electron vortex beam as shown in Eq.17 can be
~
expandedintermsoflinearcombinationsoftheFT-TBB j (r,t)= { ⇤(r,t) (r,t) (r,t) ⇤(r,t)}.
m
set (Schattschneider and Verbeeck, 2011). 2i r � r
BothLGandBesselbeamsareunboundsolutionsand (25)
are often discussed in theoretical developments because
These emerge on substituting for (r,t) in the form
of their mathematical simplicity. On the other hand,
bandwidth-limitedbeamsarerequiredfortheprecisede-
scription of electron vortex beams produced in real situ- ⇢m(r,t)=m|Nl|2|Jl(k?⇢)|2; (26)
ations.
l
jm(r,t)=~|Nl|2 ⇢���ˆˆˆ+kzˆz |Jl(k?⇢)|2. (27)
C. Mechanical and electromagnetic properties of the ✓ ◆
electron vortex beam
where���ˆˆˆandˆz form, with⇢⇢⇢ˆˆˆ, the standard unit vector set
forcylindricalcoordinates. Theunitvectorˆzisalongthe
The global mechanical properties of electron vortex
beam axis.
modes stem from the two basic properties of electrons,
namely that the finite electron mass leads to inertial The evaluation of the global inertial linear momentum
position-dependent mass flux with which are associated of the vortex follows from the realisation that the (local)
global inertial linear and angular momenta of the elec- masscurrentdensity(jm)isthesameasthe(local)linear
tron vortex beam while the finite electronic charge leads momentum density (Pm, i.e. (local) linear momentum
to position-dependent electromagnetic fields which are per unit volume). The (global) inertial linear momen-
further sources of global linear and angular momenta. tumvectoroftheBesselelectronvortexbeam(Pm)then
It is instructive to derive these global properties of the follows by volume integration. We have
electron vortex with reference to the Bessel type, for
mathematical convienence. Here we outline the treat- P = P (r,t)dV = j (r,t)dV. (28)
m m m
ment by Lloyd et al. (2013) who were first to show that Z Z
the mechanical and electromagnetic properties of elec-
We find
tron vortices emerge directly from the quantum mechan-
ical wavefunction of the vortex mode. Concentrating on P =~|N |2D
m l
⇥
the Bessel-type vortex beam for which the wavefunction 1 2π l
is given in Eq.(14) and writing ! =E/~ we have d� kzˆz+ ⇢���ˆˆˆ |Jl(k?⇢)|2⇢d⇢.
(r,t)=NlJl(k?⇢)eikzzeilφe�iEt/~, (21) Z0 Z0 ⇢ � (29)
Thevortexbeamisassumedtoextendalongtheaxisover It is easy to see that the azimuthal component in the
a length D which is much larger than the beam width. integrandofP whenintegratedoverthevolumeleadsto
m
The normalisation factor N follows straightforwardly in a zero value because of a vanishing angular integral. By
l
the form contrast the z-component leads to a finite result. Direct
integration of the z-component in Eq. 29 gives
1
k2 2
?
Nl = 2⇡DI(1)! , (22) Pm =2⇡~kzDIl(1)|Nl|2ˆz
l
=~k ˆz. (30)
z
whereI(1) isthefirstmomentintegraloftheBesselfunc-
l
tion defined by wherewehavemadeuseofEq. 22. TheresultPm =~kzˆz
is the inertial linear momentum of the Bessel electron
1
I(1) = |J (x)|2xdx. (23) vortex beam. Note that the inertial linear momentum
l l
Z0 is axial, involving only the axial component kz of the