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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

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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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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

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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

References

1. Loudon, R. & Baxter, C. Contributions of John Henry Poynting to the

understanding of radiation pressure. Proc. R. Soc. A 468, 1825–1838 (2012).

2. Einstein, A. Über einen die erzeugung und verwandlung des lichtes

betreﬀenden heuristischen gesichtspunkt. Ann. Phys. 322, 132–148 (1905).

3. Barnett, S. M. Resolution of the Abraham–Minkowski dilemma. Phys. Rev. Lett.

104, 070401 (2010).

4. Darwin, C. Notes on the theory of radiation. Proc. R. Soc. Lond. A 136,

36–52 (1932).

5. Poynting, J. The wave motion of a revolving shaft, and a suggestion as to the

angular momentum in a beam of circularly polarised light. Proc. R. Soc. Lond.

A 82, 560–567 (1909).

6. Beth, R. A. Mechanical detection and measurement of the angular momentum

of light. Phys. Rev. 50, 115–125 (1936).

7. Ohanian, H. C. What is spin? Am. J. Phys. 54, 500–505 (1986).

8. Khrapko, R. Question# 79. does plane wave not carry a spin? Am. J. Phys. 69,

405–405 (2001).

9. Humblet, J. Sur le moment d’impulsion d’une onde elec tromagnetique. Physica

10, 585–603 (1943).

10. Allen, L. & Padgett, M. Response to question# 79. do es a plane wave carry spin

angular momentum? Am. J. Phys. 70, 567–568 (2002).

11. O’Neil, A., MacVicar, I., Allen, L. & Padgett, M. Intrinsic and extrinsic nature

of the orbital angular momentum of a light beam. Phys. Rev. Lett. 88,

053601 (2002).

12. He, H., Friese, M., Heckenberg, N. & Rubinsztein-Dunlop, H. Direct

observation of transfer of angular momentum to absorptive particles from a

laser beam with a phase singularity. Phys. Rev. Lett. 75, 826–829 (1995).

13. Simpson, N., Dholakia, K., Allen, L. & Padgett, M. Mechanical equivalence of

spin and orbital angular momentum of light: An optical spanner. Opt. Lett. 22,

52–54 (1997).

14. Padgett, M. & Bowman, R. Tweezers with a twist. Nature Photon. 5,

343–348 (2011).

15. Allen, L., Beijersbergen, M. W., Spreeuw, R. & Woerdman, J. Orbital angular

momentum of lig ht and the transformation of Laguerre–Gaussian laser modes.

Phys. Rev. A 45, 8185–8189 (1992).

16. Berry, M. V. Paraxial beams of spinning light. Proc. SPIE 3487,

http://dx.doi.org/cbmvrn (1998).

17. Paterson, L. et al. Controlled rotation of optically trapped microscopic

particles. Science 292, 912–914 (2001).

18. Karimi, E. & Santamato, E. Radial coherent and intelligent states of paraxi al

wave equation. Opt. Lett. 37, 2484–2486 (2012).

19. Karimi, E. et al. Exploring the quantum nature of the radial degree of

freedom of a photon via Hong-Ou-Mandel interference. Phys. Rev. A 89,

013829 (2014).

20. Mair, A., Vaziri, A., Weihs, G. & Zeilinger, A. Entanglement of the orbital

angular momentum states of photons. Nature 412, 313–316 (2001).

21. Walsh, J. L. A closed set of normal orthogonal functions. Am. J. Math. 45,

5–24 (1923).

22. Salakhutdinov, V., Eliel, E. & Löﬄer, W. Full-field quantum correlations of

spatially entangled photons. Phys. Rev. Lett. 108, 173604 (2012).

23. Romero, L. D., Andrews, D. & Babiker, M. A quantum electrodynamics

framework for the nonlinear optics of twisted beams. J. Opt. B 4,

S66–S72 (2002).

24. Dorn, R., Quabis, S. & Leuchs, G. Sharper focus for a radially polarized light

beam. Phys. Rev. Lett. 91, 233901 (2003).

25. Zhao, Y., Edgar, J. S., Jeﬀries, G. D., McGloin, D. & Chiu, D. T. Spin-to-orbital

angular momentum conversion in a strongly focused optical beam. Phys. Rev.

Lett. 99, 073901 (2007).

NATURE PHYSICS | VOL 11 | AUGUST 2015 | www.nature.com/naturephysics 633

© 2015 Macmillan Publishers Limited. All rights reserved

© 2015 Macmillan Publishers Limited. All rights reserved

PROGRESS ARTICLE

NATURE PHYSICS DOI: 10.1038/NPHYS3404

26. Karimi, E., Zito, G., Piccirillo, B., Marrucci, L. & Santamato, E.

Hypergeometric-Gaussian modes. Opt. Lett. 32, 3053–3055 (2007).

27. Marrucci, L., Manzo, C. & Paparo, D. Optical spin-to-orbital angular

momentum conversion in inhomogeneous anisotropic media. Phys. Rev. Lett.

96, 163905 (2006).

28. Karimi, E. et al. G enerat ing optical orbital angular momentum at visible

wavelengths using a plasmonic metasurface. Light 3, e167 (2014).

29. Brasselet, E., Murazawa, N., Misawa, H. & Juodkazis, S. Optical vortices from

liquid crystal droplets. Phys. Rev. Lett. 103, 103903 (2009).

30. Foo, G. et al. Optical vortex coronagraph. Opt. Lett. 30, 3308–3310 (2005).

31. Barreiro, J. T., Wei, T.-C. & Kwiat, P. G. Beating the channel capacity limit for

linear photonic superdense coding. Nature Phys. 4, 282–286 (2008).

32. Molina-Terriza, G., Torres, J. P. & Torner, L. Twisted photons. Nature Phys. 3,

305–310 (2007).

33. Bliokh, K. Y., Bliokh, Y. P., Savel’ev, S. & Nori, F. Semiclassical dynamics of

electron wave packet states with phase vortices. Phys. Rev. Lett. 99,

190404 (2007).

34. Bliokh, K. Y., Dennis, M. R. & Nori, F. Relativistic electron vortex beams:

Angular momentum and spin–orbit interaction. Phys. Rev. Lett. 107,

174802 (2011).

35. Pauli, W. Über den Zusammenhang des Abschlusses der Elektronengruppen

im Atom mit der Komplexstruktur der Spektren. Z. Phys. A 31,

765–783 (1925).

36. Grillo, V. et al. Generation of nondiﬀracting ele ctron Bessel beams. Phys. Rev. X

4, 011013 (2014).

37. Durnin, J. Exact solutions for nondiﬀracting beams. i. The scalar theory. J. Opt.

Soc. Am. 4, 651–654 (1987).

38. Siegman, A. E. Lasers (Mill Valle y, 1986).

39. Bliokh, K. Y., Schattschneider, P., Verbeeck, J. & Nori, F. Electron vortex beams

in a magnetic field: A new twist on Landau levels and Aharonov–Bohm states.

Phys. Rev. X 2, 041011 (2012).

40. Lloyd, S. M., Babiker, M., Yuan, J. & Kerr-Edwards, C. Electromagnetic vortex

fields, spin, and spin–orbit interactions in electron vortices. Phys. Rev. Lett.

109, 254801 (2012).

41. Greenshields, C., Stamps, R. L. & Franke-Arnold, S. Vacuum Faraday eﬀect for

electrons. New J. Phys. 14, 103040 (2012).

42. Guzzinati, G., Schattschneider, P., Bliokh, K. Y., Nori, F. & Verbeeck, J.

Observation of the Larmor and Gouy rotations with electron vortex beams.

Phys. Rev. Lett. 110, 093601 (2013).

43. Schattschneider, P. et al. Imaging the dynamics of free-electron Landau states.

Nature Commun. 5, 4586 (2014).

44. Uchida, M. & Tonomura, A. Generation of electron beams carrying orbital

angular momentum. Nature 464, 737–739 (2010).

45. Beijersbergen, M., Coerwinkel, R., Kristensen, M. & Woerdman, J.

Helical-wavefront laser beams produced with a spiral phaseplate. Opt.

Commun. 112, 321–327 (1994).

46. Verbeeck, J., Tian, H. & Schattschneider, P. Production and application of

electron vortex beams. Nature 467, 301–304 (2010).

47. McMoran, B. Electron Diﬀraction and Interferometry Using Nanostructure

PhD thesis, Univ. Arizona (2009).

48. Bazhenov, V. Y., Vasnetsov, M. & Soskin, M. Laser beams with screw

dislocations in their wavefronts. JETP Lett. 52, 429–431 (1990).

49. Grillo, V. et al. Highly eﬃcient electron vortex beams generated by

nanofabricated phase holograms. Appl. Phys. Lett. 104, 043109 (2014).

50. Shiloh, R., Lereah, Y., Lilach, Y. & Arie, A. Sculpturing the electron wave

function using nanoscale phase masks. Ultramicroscopy 144, 26–31 (2014).

51. McMorran, B. J. et al. Electron vortex beams with high quanta of orbital

angular momentum. Science 331, 192–195 (2011).

52. Grillo, V. et al. Holographic generation of highly twisted electron beams. Phys.

Rev. Lett. 114, 034801 (2015).

53. Voloch-Bloch, N., Lereah, Y., Lilach, Y., Gover, A. & Arie, A. Generation of

electron Airy beams. Nature 494, 331–335 (2013).

54. Clark, L. et al. Exploiting lens aberrations to create electron-vortex beams.

Phys. Rev. Lett. 111, 064801 (2013).

55. Béché, A., Van B oxem, R., Van Tendeloo, G. & Verbeeck, J. Magnetic monopole

field exposed by electrons. Nature Phys. 10, 26–29 (2014).

56. Garraway, B. & Stenholm, S. Does a flying electron spin? Contemp. Phys. 43,

147–160 (2002).

57. McGregor, S., Bach, R. & Batelaan, H. Transverse quantum Stern-Gerlach

magnets for electrons. New J. Phys. 13, 065018 (2011).

58. Karimi, E., Marrucci, L., Grillo, V. & Santamato, E. Spin-to-orbital angular

momentum conversion and spin-polarization filtering in electron beams. Phys.

Rev. Lett. 108, 044801 (2012).

59. Hayrapetyan, A. G. et al. Interaction of relativistic electron-vortex beams with

few-cycle laser pulses. Phys. Rev. Lett. 112, 134801 (2014).

60. Lloyd, S., Babiker, M. & Yuan, J. Quantized orbital angular momentum transfer

and magnetic dichroism in the interaction of electron vortices with matter.

Phys. Rev. Lett. 108, 074802 (2012).

61. Yuan, J., Lloyd, S. M. & Babiker, M. Chiral-specific electron-vortex-beam

spectroscopy. Phys. Rev. A 88, 031801(R) (2013).

62. Schattschneider, P., Löﬄer, S., Stöger-Pollach, M. & Verbeeck, J. Is magnetic

chiral dichroism feasible with electron vortices? Ultramicroscopy 136,

81–85 (2014).

63. Ivanov, I. P. & Karlovets, D. V. Polarization radiation of vortex elect rons with

large orbital angular momentum. Phys. Rev. A 88, 043840 (2013).

64. Kaminer, I., Nemirovsky, J., Re chtsman, M., Bekenstein, R. & Segev, M.

Self-accelerating Dirac particles and prolonging the lifetime of relativistic

fermions. Nature Phys. 11, 261–267 (2015).

65. Verbeeck, J. et al. Atomic scale electron vortices for nanoresearch. Appl. Phys.

Lett. 99, 203109 (2011).

66. Lloyd, S., Babiker, M. & Yuan, J. Mechanical properties of electron vortices.

Phys. Rev. A 88, 031802 (2013).

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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