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PROGRESS ARTICLE
PUBLISHED ONLINE: 31 JULY 2015 | DOI: 10.1038/NPHYS3404
Structured quantum waves
Jérémie Harris
1
, Vincenzo Grillo
2
, Erfan Mafakheri
3
, Gian Carlo Gazzadi
2
, Stefano Frabboni
2,3
,
Robert W. Boyd
1,4
and Ebrahim Karimi
1
*
The study of structured optical waves has enhanced our understanding of light and numerous experimental methods now
enable the control of the angular momentum and radial distributions. Recently, these wavestructuring techniques have
been successfully applied to the generation and shaping of electron beams, leading to promising practical and fundamental
advances. Here, we discuss recent progress in the emerging field of electron beam shaping, and explore the unique attributes
that distinguish electron beams from their photonic analogues.
T
hroughout its orbit around our solar system, a comet’s
tail invariably points away from the sun. As early as the
seventeenth century, this observation led Johannes Kepler
to conjecture that sunlight might carry linear momentum. Only
in 1905 did John Henry Poynting develop the first theory
describing the momentum density of ele ctromagnetic waves
1
.
Almost simultaneously, Albert Einstein proposed that light mig ht
be comprised of quantized packets of energy. Shortly thereafter,
these energy packets, now known as photons
2
, were understood
to carry quantized momenta p =
¯
hk as well, where
¯
h and k
denote the reduced Planck constant and photon wavevector. Today,
optical linear momentum is a well-understood phenomenon that
explains countless physical observations, from radiation pressure
to Compton scattering. The study of optical linear momentum in
media remains active, and this work has recently culminated in a
solution to the long-standing Abraham–Minkowski dilemma
3
.
Less commonly recognized is the fact that light can carry angular
momentum in addition to its linear momentum. This follows from
the definition of angular momentum, j = r ⇥ p, where r is the
radial vector. The optical angular momentum carried by a paraxial
optical beam has two sources
4
: spin angular momentum (SAM),
s, and orbital angular momentum (OAM), `, so that j = s + `.
Optical SAM is directly associated with the circular polarization
of light, and represents a form of intrinsic angular momentum,
as its magnitude is independent of the position about which it
is measured. SAM is therefore intimately lin ked to the vectorial
(polarization) structure of a beam
5
. A beam of circularly polarized
light necessarily carries SAM, and on interacting with a small
particle will cause it to rotate about its centre. The mechanical
properties of SAM were first explored by Beth
6
, who demonstrated
exchanges of angular momentum between circularly polarized light
and a doubly refracting plate.
Optical SAM has also drawn interest on fundamental grounds
7
.
In particular, an apparent paradox arises in the interaction between
circularly polarized optical plane waves and massive particles
8
.
Because plane waves are non-localized
9
, they represent non-
paraxial beams, and therefore their angular momenta c annot be
unambiguously separated into SAM and OAM components. Despite
their ambiguous SAM content, plane waves do impart a well-
defined SAM onto particles with which they interact. This pec uliar
circumstance can be explained by noting that the only port ion of the
plane wave that is relevant to the interaction is that which overlaps
with the absorbing particle. For the purpose of the interaction,
the plane wave can therefore be considered to be a localized and
truncated beam, for which the shape and size matches that of the
particle, s o that it is effectively paraxial, resulting in an unambiguous
effective beam SAM (ref. 10).
In contrast to SAM, OAM is associated with a b eam’s transverse
phase profile, rather than its polarization. Optical OAM can
be divided into intrinsic OAM, `
int
, and extrinsic OAM, `
ext
(ref. 11). Whereas intrinsic OAM is constant for a particular beam,
extrinsic OAM varies with the axis about which it is measured.
When an OAM-carry ing beam interacts with a small particle,
its extrinsic OAM induces the particle’s rotation about the beam
axis, whereas its intrinsic OAM causes the particle to rotate
about an axis through its centre
12
, in a manner resembling the
mechanical effect of SAM (refs 13,14). An OAM-carrying beam
is described by a ‘wavefunction’ containing a corkscrew (helical)
phase exp(i`'), where ` is an integer, and ' is the azimuthal
angle in the plane transverse to the beam axis
15
. Such beams
therefore have twisted spiral phase f ronts (associated with so-
called optical vortices), whose cophasal surfaces form `-helices
during propagation, and therefore possess non-zero transverse
wavevectors. Beams possessing well-defined OAMs are known as
‘twisted beams’. Any OAM-carrying beam contains at least one
phase singularity, a point of undefined phase in the plane transverse
to the beam propagation axis
16
.
The scales associated with features of the optical field tied
to OAM and SAM are fairly comparable. For SAM, the electric
field of a circularly polarized optical beam rotates once after
propagating by one wavelength. Similarly, an OAM-carrying
beam’s cophasal surfaces will experience one full twist after `
wavelengths of propagation
17
. Despite their similarities, spin and
orbital angular momenta represent independent degrees of freedom
for paraxial optical fields in vacuum or isotropic media. Still another
independent degree of freedom required to fully characterize the
fields’ transverse phase and intensity distributions is the mode radial
parameter
18
. The radial index p ass ociated with Laguerre–Gauss
(LG) modes, which are solutions to the paraxial wave equation, was
recently shown to be quantized in the single-photon regime
19
, and
should not be confused with the beam waist or related parameters.
Indeed the SAM, OAM and radial parameters all represent
quantum indices that can be assigned to individual photons.
Specifically, single photons can exist in SAM eigenstates, in which
1
Department of Physics, University of Ottawa, 25 Templeton St., Ottawa, Ontario K1N 6N5, Canada.
2
CNR-Istituto Nanoscienze, Centro S3, Via G Campi
213/a, I-41125 Modena, Italy.
3
Dipartimento di Fisica Informatica e Matematica, Università di Modena e Reggio Emilia, via G Campi 213/a, I-41125 Modena,
Italy.
4
Institute of Optics, University of Rochester, Rochester, New York 14627, USA.
*
e-mail: ekarimi@uottawa.ca
NATURE PHYSICS | VOL 11 | AUGUST 2015 | www.nature.com/naturephysics 629
© 2015 Macmillan Publishers Limited. All rights reserved
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PROGRESS ARTICLE
NATURE PHYSICS DOI: 10.1038/NPHYS3404
they carry spin angular momenta s
z
=s
¯
h about their propagation
direction; can occupy OAM eigenstates, in which they car ry orbital
angular momenta `
z
=`
¯
h about their propagation direction
20
; and
can also exist in radial mode eigenstates, characterized by well-
defined radial indices p (for LG (ref. 19), Walsh
21,22
or related
modes), where s =±1 and ` = 0, ±1, ±2, ... and p = 0, 1, 2, ...
are the spin, azimuthal and radial indices. To unambiguously
describe the quantum state of a single photon, one therefore
must provide information about each of these parameters, in
addition to the photon wavevector k. Consequently, a single
photon possessing well-defined indices s, `, p and a well-defined
wavevector is an excitation of the electromagnetic field, produced
by the action of a creation operator a
†
k;s,p,`
on a vacuum state
state |0i, such that |k;s, p, `i=a
†
k;s,p,`
|0i (ref. 23). More generally,
an arbitrary single-photon nonseparable state may be described
by a superposition of s, `, p and k eigenstates, in which case
it may be ascribed a ‘wavefunction’ |9i=
P
k;s,p,`
c
p,`
k,s
|k; s, p, `i,
where c
p,`
k,s
is a complex constant. Beams whose constituent photons
are described by ‘wavefunctions’ with controlled coefficients c
p,`
k,s
are said to be structured. Innumerable examples of such beams
exist, including, for example, a category of modes known as
radial vector beams, which are characterized by only two non-
zero coefficients, c
`=1
s=1
=c
`=1
s=1
=1/
p
2, whose radial indices p and
wavevectors are identical. Radial vector beams possess radially
oriented linear polarizations in the plane transverse to their
propagation direction, and have found applications in near-field
microscopy and lithography
24
. An example of a structured beam is
shown in Fig. 1.
Despite their usual independence in vacuum and isotropic
media, SAM, OAM and radial indices can be coupled within
media in general, and even in vacuum under tight focusing
conditions
25
. For example, a Gaussian beam with radial index p =0
and flat phase front ` =0 can be converted into a superposition
of modes with non-zero p values, each carrying one unit of
OAM, by passing through a spiral phase plate
26
. Exchanges of
angular momentum between the SAM and OAM spaces have
also been demonstrated in inhomogeneous birefringent materials,
such as photonic
27
and plasmonic
28
q-plates, and liquid crystal
droplets
29
. The manipulation of polarization, radial and azimuthal
indices has sparked great interest in quantum and classical optics,
leading to applications in coronagraphy
30
, superdense coding
31
and quantum information
32
. Only recently, however, have optical
structuring techniques been applied to the generation and shaping
of electron beams.
Electron waves
Since the existence of matter waves was first proposed by de Broglie
in 1924, numerous experiments have demonstrated the remarkable
correspondence bet ween the wavelike behaviours of matter and
light. Electron waves, in particular, have drawn attention in fields
from electron microscopy to nanofabrication. The similarities
between electron and light waves, now verified the oretically and
in countless experiments, suggest that photons’ quantum indices
can be applied just as well in specifying the states of elec trons
33
.
However, important distinctions exist between electron and photon
waves. Whereas electrons possess a rest mass m
e
, a charge e, and
obey fermionic statistics, photons are massless, neutral bosons. A
related point of contrast is that the non-relativistic (relativistic)
wavelike behaviours of electrons and light are prescribed by the
Pauli–Schrödinger (Dirac
34
) and Maxwell equations.
Free electrons carry linear momentum given by the de Broglie
wave relation as
¯
hk =2⇡
¯
h/, where is the electron wavelength.
Electron wavelengths can be significantly shorter than even those
associated with X-rays. This has motivated the de velopment
of electron microscopy techniques that exploit short electron
wavelengths to achieve sub-ångström resolutions unattainable by
2π
Min. 0
0
Max.
a
b
Min.
Max.
2π
Figure 1 | An example of structured light. a, Composite image showing
transverse distributions of polarization and intensity associated with a
structured photon beam constructed from an equal superposition of the
single-photon states |k;s =1, p =2, ` =1i and |k;s =1, p =1,` =3i,
where |k;s,p,`i represents a Laguerre–Gauss (LG) mode propagating
along a direction indicated by the wavevector k, with spin, radial and
azimuthal parameters respectively denoted s, p and `. Arrows on the
polarization ellipses shown in the figure indicate polarization handedness.
b, Schematic illustrating the transverse intensity, phase and polarization
distributions associated with the two component single-photon states that
give rise to the pattern shown in a. Together, the intensity and phase
distributions carry all information conveyed by the ` and p transverse
indices, and these are therefore combined in the same ket, with the spin
index s specified separately.
standard optical microscopes. More generally in the non-relativistic
limit, the linear momentum density of the electron wavefunction
(r, t) is given by P = P
0
+ P
s
, where P
0
=
¯
h =(
⇤
r ),
P
s
=
¯
h/4r ⇥(
⇤
), is the Pauli vector, and =(·) denotes the
imaginary part of its argument. The momentum density ar ising
from P
s
is always oriented azimuthally, and so makes no net
contribution to the linear momentum of the beam. The reason
for separating the linear momentum density into P
0
and P
s
components in this way becomes apparent when evaluating the
electron angular momentum J=
R
(r⇥P)d
3
r. The two momentum
density terms P
0
and P
s
respectively produce a coordinate-
dependent OAM L =
R
(r ⇥P
0
) d
3
r and a coordinate-independent
spin angular momentum, S =
R
(r⇥P
s
) d
3
r. This spin is quantized:
S=
¯
h /2 (ref. 35). From this discussion, it can be seen that electron
spin is inextricably linked to the linear momentum distribution in
the transverse plane, and hence can act as an important structural
parameter for electron wavefunctions
7
. Major electron structural
features tied to spin and orbital angular momentum are illustrated
in Fig. 2.
It may be slightly surprising to find that the electron spin
and photon SAM spaces have the same dimensionality, given that
the electron is a fermion, and the photon a boson. This can
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NATURE PHYSICS DOI: 10.1038/NPHYS3404
PROGRESS ARTICLE
Spin-up
a
b
c
Spin-down
Figure 2 | Electron structural features associated with angular
momentum. a, Schematic displaying the spin-dependent transverse linear
momentum distributions respectively associated with spin-up (s=1/2) and
spin-down (s=1/2) electron beams. The electron spin gives rise to a
vanishing net transverse linear momentum. b, Cophasal surfaces
associated with an electron wavepacket carrying OAM ` =1. Intensity is
indicated by colour, with yellow (red) denoting the regions of highest
(lowest) transverse probability. c, Classical (Bohmian) trajectories of
individual electrons propagating in a Laguerre–Gauss mode characterized
by an azimuthal index ` =1. Colours are used to indicate the phases
associated with each electron trajectory.
be explained with an appeal to special relativity, which suggests
that a photon’s longitudinal polarization must exper ience infinite
length contraction, be cause the photon travels at the speed of light.
Therefore, despite being a spin-1 particle, the photon can effectively
access only two (transverse) degrees of freedom, producing the
observed correspondence between light and electron waves, in
this regard.
When bound to atomic nuclei, electrons possess OAM in
addition to their spin. Perhaps more surprisingly, however, even free
electrons can be made to carry OAM on propagation
33
. This occurs
owing to the presence of transverse components of the electron
linear momentum P
0
. As with OAM-carry ing photon beams,
such electron wavefunctions possess spiral phase structures, their
wavefunctions containing phase terms exp (i`'). The similarities
between the descriptions of electron wavefunctions and photon
beams arise from the spinless free particle Schrödinger equation
(SFSE), i
¯
h @
t
(r; t) =
¯
h
2
/(2m
e
) r
2
(r; t). When the SFSE is
solved with the ansatz (r; t) = (r) exp (iEt/
¯
h), one obtains
the expression (r
2
+ k
2
e
) (r) = 0, where k
2
e
:= 2m
e
E/
¯
h
2
. This
formula is formally identical to the optical wave equation, so
that the transverse electron wavefunctions obtained from it will
match the transverse modes of the electric field associated with
photon beams. When solved in cylindrical coordinates r, ', z,
this form of the SFSE g ives rise to a family of Bessel beam
solutions, for which the time-indep endent wavefunctions are given
by (r) /J
`
(k
?
r) exp(i(k
k
z +`')) (ref. 36), where J
`
is an `th
order Bessel function of the first kind, and which have well-
studied photonic analogues
37
. Bessel modes are diffractionless, non-
normalizable,and therefore unphysical solutions to the SFSE and the
optical wave equation. If it is assumed that the mode longitudinal
wavenumber, k
k
, is much larger than the transverse wavenumber,
k
?
=
p
k
2
e
k
2
k
, the SFSE reduces to a form analogous to the optical
paraxial wave equation. The resulting expression possesses solutions
in the form of LG modes, introduced earlier for photon beams.
Such modes have transverse wavefunctions |p, `i at any given axial
position z, which in the position representation are expressed as
LG
p,`
(r,';z):=hr, ',z|p,`i, and have been extensively studied
38
.
Electron wavefunctions possessing phase terms exp (i`') and
exp (ik
k
z), such as Bessel and LG modes, carry OAM
¯
h` and
linear momenta
¯
hk
k
per electron, oriented along their propagation
direction. Such wavefunctions have linear momentum densities
P
`
(r) =
¯
h(`/r ' + k
k
z)⇢
`
(r), and probability c urrent densities
j
`
(r) = P
`
/m
e
, where ' and z denote the azimuthal and axial
unit vectors, and ⇢
`
(r) :=| (r)|
2
. Because elec trons carry charge,
the presence of an azimuthal probability current produces an
effective loop of charge current about the propagation axis
39
. This
supplements the charge current already present due to the intrinsic
SAM of the electron wavefunction. These spin and OAM-induced
charge c urrents respectively produce magnetic moments sg
e
µ
B
and
`µ
B
, where µ
B
=e
¯
h/(2m
e
) is the electron Bohr magneton, s=±1/2
is the electron spin index and g
e
'2.002 is the electron g -factor
34
.
These magnetic moments allow interactions between the electron
wavefunction and external magnetic fields, which cannot occur for
analogous photon beams. Just as remarkably, the OAM c arried by
twisted electron be ams has even been shown to give ris e to electric
and magnetic fields associated with t hese beams themselves, which
differ significantly from the fields associated with electron beams
carrying no OAM (ref. 40).
As with photon beams, an additional radial parameter is
required to fully specify the transverse field distributions of electron
wavefunctions. This parameter can be discrete or continuous,
depending on the electron mode. Bessel modes possess continuous
radial parameters k
?
, whereas LG modes possess discretized
radial indices p. In either case, the radial parameter dictates t he
quantization condition for the transverse component of the electron
wavefunction’s energy; as k
?
or p increase, so does the transverse
energy carrie d by the electron wavefunction
41,42
. The radial index-
dependence of the electron transverse energy is an important
consideration in electron/magnetic field interactions
43
and plays a
crucial role in electron/electron interactions as well.
Together, t he radial, azimuthal and spin indices confer spat ial
structure on an electron wavefunction, and, along with linear mo-
mentum, unambiguously specify the states of individual electrons.
Recently, great progress has been made in efforts to produce electron
beams with tailored structure, by controlling these parameters.
Experimental generation of structured electron waves
Structured electron beam generation was first reported by Uchida
and Tonomura in 2010 (ref. 44), using spiral phase plates
consisting of spontaneously stacked graphite thin f ilms to impart
OAM onto incident electron beams. Spiral phase plates are
produced by inducing azimuthally and uniformly increasing ‘optical’
thicknesses t(r, ') around the axis of ‘optically’ dense materials
45
.
If the optical thickness of a material can be increased by ` de
Broglie wavelengths over one full rotation about the phase plate’s
axis,
H
rt(r,') ·d' =`, the plate will imprint a helical phase
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© 2015 Macmillan Publishers Limited. All rights reserved
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PROGRESS ARTICLE
NATURE PHYSICS DOI: 10.1038/NPHYS3404
ab e
dc
z
2nd order
1st order
0th order
−1st order
−2nd order
Figure 3 | Electron structuring techniques. a, Scanning electron microscope (SEM) image of a spontaneously stacked graphite sheet (on the nanometre
scale), used as a spiral phase plate
44
to impart OAM on an incident electron beam. b, SEM image of an amplitude hologram (on the micrometre scale)
applied to the generation of twisted electron beams
46
. c, SEM image of an ultrathin needle
55
(on the micrometre scale) to simulate a magnetic monopole,
capable of imparting OAM on incident electron plane waves. d, SEM image of the first electron phase hologram
49
(on the micrometre scale). e, Schematic
of the structured electron beams emerging from an electron phase hologram. The spreading of various di�raction orders can be observed to occur rapidly
on propagation, and the distinct transverse profiles associated with five di�erent di�racted orders are shown in the bottom inset.
exp (i`') on any incident electron wavefunction. Although the
stacked graphite films used to structure the azimuthal phases of
electron wavefunctions were found to produce structured beams
of reasonable quality, the approach taken in this work could not
readily be generalized to produce beams with |`|6=1, as it depended
on the is olat ion of spontaneously stacked graphite films, which
could not be produced at will or in customized configurations by
an exper imenter.
A second seminal study was carried out shortly thereafter,
demonstrating the conferral of OAM to electron beams by using
amplitude holograms constructed from 100-nm-thick platinum
foil
46
. This investigation followed earlier work, which in 2009 also
proposed the use of such holograms for structured electron beam
generation
47
. This technique benefits from greater versatility, and
allows for the generation of electron beams with high OAM content.
Amplitude holograms are constructed by simulating the intensity
pattern I
int
(r, ') produced from the interference of reference
ref
(r, ') and desired target
t
(r, ') wavefunctions, such that
I
int
(r, ') =|
ref
(r, ') +
t
(r, ')|
2
, and by designing a mask with
thickness func tion t(r, ') = t
0
2[I
int
(r, ')], where 2[·] denotes
the Heaviside function, and t
0
is the optical thickness of the
mask
48
. When an electron beam is made incident on such a
hologram, a series of diffracted beams are produced, each carrying
a different OAM.
For t hree years, the holographic generation of structured electron
beams was only possible using amplitude holograms. In 2014,
however, a new class of holographic mask, already widely used in
photon optics and known as t he phase hologram, was reported to
achieve electron beam shaping by directly imprinting controllable
phases onto electron wavefunctions, resulting in unprecedented
high efficiencies and low absorption losses
49,50
. This technique was
shown to produce high-quality electron Bessel
36
and LG beams, with
various azimuthal parameters `.
Following this early work, a number of studies explored
applications of holographic masks to the generation of electron
beams with high OAM, some achieving values up to 200
¯
h per
electron
51,52
. In a striking demonstration of the effe ctiveness of phase
holograms, one study reported the successful generation of electron
Airy beams
53
, and observed their unusual transverse ‘acceleration’
and ‘self-healing’ properties.
In addition to the spiral phase plate and holographic techniques,
a number of alternative approaches also allow for elect ron beam
shaping. One study reported the generation of electron beams
with helical phase fronts by manipulating ab errations associated
with the corrector lenses of an electron microscope
54
, achieving
generation efficiencies of 32%. Another demonstrated twisting of
electron beams using a simulated magnetic monopole constructed
from a thin, nanoscale magnetic needle
55
. This latter strategy
could benefit from an unusually high conversion efficiency. An
additional advantage distinguishing this magnetic needle method
from other electron beam shaping techniques is its independence
from the acceleration voltage applied to incident electrons. A
variety of distinct strategies therefore exist to generate electrons
with structured azimuthal phases. In contrast, little work has been
done, as yet, on structuring the radial p arameters of electron
beams. Electron spin has also received relatively little attention in
this regard, despite also being an important structural parameter.
This is largely due to the challenge of generating spin-polarized
free electron (SPFE) beams; indeed, t he generation of SPFEs was
thought to be disallowed by Pauli, whose view was later echoed
by Bohr in his statement that
56
, ‘it is impossible to observe the
spin of the electron, separated fully from its orbital momentum,
by means of experiments based on the concept of classical particle
trajectories.’ Recently, however, a strategy has been proposed to
produce SPFEs by passing an unpolarized elec tron beam through
a magnetic phase grating
57
. Another approach has been proposed to
address this challenge, employing complementary and topologically
charged ele ctric and magnetic fields to couple electron spin and
orbital angular momentum degrees of freedom, producing a spin-
polarized, OAM-carrying electron beam
58
. When this beam is
phase flattened, a spin-polarized Gaussian electron wavefunction is
recovered, with theoretical efficiencies and degrees of polarization
of up to 50% and 97.5%, respectively. Various electron structuring
strategies are illustrated in Fig. 3.
Outlook
Much theoretical work has been done with a view to applying
the unique properties of electron beams to various measure-
ment schemes, and to t he study of fundamental physics. For
example, a thorough treatment of the interaction between elec-
tron beams and magnetic fields was reported in 2012 (ref. 39),
showing how the radi al intensity and charge current distribu-
tions of LG and Bessel electron beams are altered by their
propagation through regions containing different magnetic field
632 NATURE PHYSICS | VOL 11 | AUGUST 2015 | www.nature.com/naturephysics
© 2015 Macmillan Publishers Limited. All rights reserved
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NATURE PHYSICS DOI: 10.1038/NPHYS3404
PROGRESS ARTICLE
B
z
⏐2, 1〉
Max.
ab
Max.
Min.
Min.
⏐1, −3〉
Figure 4 | Structured electron beams and their interaction with uniform
magnetic fields. a, Transverse intensity and charge current distributions
associated with electron states |p =2, ` =1i and |p =1, ` =3i in the
absence of any external magnetic fields, where |p,`i denotes an LG mode
with radial and azimuthal parameters p and `. The size and orientation of
the arrows show the respective magnitude and direction of the local charge
current density; the probability density is indicated by colour. b, Intensity
and charge current distributions associated with the electron
wavefunctions shown in a, in the presence of a uniform external magnetic
field. The beams shown in a and b are plotted so as to possess identical
beam waists for ease of comparison.
configurations. It was found that electron Bessel beams t ake the
form
Bessel
/J
|`↵|
(k
?
r)expi(`' k
k
z) w hen propagating along a
line of magnetic flux proportional to the flux parameter ↵, so that
changes in its magnitude lead to changes in the order of the beam’s
transverse mode. The effect of external magnetic fields on specific
Laguerre–Gauss modes are illustrated in Fig. 4. Additional theoret-
ical studies have since explored the possibility of imparting OAM
onto electron beams by means of photon–electron interactions
59
.
Electric and magnetic field effec ts provide a promising avenue for
the shaping of electron beams. Electron beams’ magnetic field sen-
sitivity might also be exploited for magnetic field sensing in mate-
rials science
60
. Further still, recent work has proposed the potential
application of structured electron beams to the selective excitation
of atomic states, via a transfer of intrinsic electron OAM to electrons
in individual atoms
61,62
. As this intrinsic OAM transfer is likely to be
significantly less efficient in the case of photon/atom interactions,
structured electron beams therefore represent a possible avenue for
the exploration of truly novel physical effects.
Structured ele ctron beams have also be en considered as tools in
the study of polarization radiation (PR) and related phenomena.
PR arises from the movement of electrons within spatially
inhomogeneous media, which induces a polarization current
density in the materials. The magnitude of t he PR produced by an
electron is proportional to its angular momentum. Observation of
PR has proven elusive, however, owing to the small magnitude of the
electron intrinsic spin. This limitation could be overcome by using
structured electron waves with high OAM values
63
.
Further, a recent the oretical study exploring the exotic features
of certain structured fermion beams in the relativistic regime
demonstrated that such beams could behave as if under the
influence of ‘virtual forces’, even in the total absence of external
potentials
64
. This suggests that structured electron beams could in
principle be used to simulate the effects of potentials and forces on
electrons and other fermions. Remarkably, this study also revealed
that certain tailored structures imparted on fermion beams can
cause these be ams’ constituent particles to experience controllable
time dilation and length contraction effects. This observation
indicates a host of experimental possibilities, including applications
in which decaying particles might have their lifetimes extended to a
controllable degree using thes e relativistic effects.
Advances in beam structuring have already paved the way to
major developments in many areas. As a new and rapidly advancing
field, electron beam shaping holds a wealth of p otential for the study
of hitherto inaccessible physical phenomena, and the development
of novel and exciting applications in ele ctron microscopy and
related areas
65,66
.
Received 8 April 2015; accepted 15 June 2015;
published online 31 July 2015
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Acknowledgements
J.H., R.W.B. and E.K. acknowledge the support of the Canada E xcellence Research Chairs
(CERC) Program.
Additional information
Reprints and permissions information is available online at www.nature.com/reprints.
Correspondence should be addressed to E.K.
Competing financial interests
The authors declare no competing financial interests.
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