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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.
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PUBLISHED ONLINE: 31 JULY 2015 | DOI: 10.1038/NPHYS3404
Structured quantum waves
Jérémie Harris
, Vincenzo Grillo
, Erfan Mafakheri
, Gian Carlo Gazzadi
, Stefano Frabboni
Robert W. Boyd
and Ebrahim Karimi
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.
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
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
, 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
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
: 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
. 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
, who demonstrated
exchanges of angular momentum between circularly polarized light
and a doubly refracting plate.
Optical SAM has also drawn interest on fundamental grounds
In particular, an apparent paradox arises in the interaction between
circularly polarized optical plane waves and massive particles
Because plane waves are non-localized
, 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 eectively paraxial, resulting in an unambiguous
eective beam SAM (ref. 10).
In contrast to SAM, OAM is associated with a b eams transverse
phase profile, rather than its polarization. Optical OAM can
be divided into intrinsic OAM, `
, and extrinsic OAM, `
(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 particles rotation about the beam
axis, whereas its intrinsic OAM causes the particle to rotate
about an axis through its centre
, in a manner resembling the
mechanical eect 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
. 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
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
beams cophasal surfaces will experience one full twist after `
wavelengths of propagation
. 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
. 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
, 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
Department of Physics, University of Ottawa, 25 Templeton St., Ottawa, Ontario K1N 6N5, Canada.
CNR-Istituto Nanoscienze, Centro S3, Via G Campi
213/a, I-41125 Modena, Italy.
Dipartimento di Fisica Informatica e Matematica, Università di Modena e Reggio Emilia, via G Campi 213/a, I-41125 Modena,
Institute of Optics, University of Rochester, Rochester, New York 14627, USA.
© 2015 Macmillan Publishers Limited. All rights reserved
© 2015 Macmillan Publishers Limited. All rights reserved
they carry spin angular momenta s
h about their propagation
direction; can occupy OAM eigenstates, in which they car ry orbital
angular momenta `
h about their propagation direction
; and
can also exist in radial mode eigenstates, characterized by well-
defined radial indices p (for LG (ref. 19), Walsh
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
on a vacuum state
state |0i, such that |k;s, p, `i=a
|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=
|k; s, p, `i,
where c
is a complex constant. Beams whose constituent photons
are described by ‘wavefunctions with controlled coecients c
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 coecients, c
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
. 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
. 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
. Exchanges of
angular momentum between the SAM and OAM spaces have
also been demonstrated in inhomogeneous birefringent materials,
such as photonic
and plasmonic
q-plates, and liquid crystal
. The manipulation of polarization, radial and azimuthal
indices has sparked great interest in quantum and classical optics,
leading to applications in coronagraphy
, superdense coding
and quantum information
. 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
However, important distinctions exist between electron and photon
waves. Whereas electrons possess a rest mass m
, 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
) 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
Min. 0
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
+ P
, where P
h =(
r ),
h/4r (
), is the Pauli vector, and =(·) denotes the
imaginary part of its argument. The momentum density ar ising
from P
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
and P
components in this way becomes apparent when evaluating the
electron angular momentum J=
r. The two momentum
density terms P
and P
respectively produce a coordinate-
dependent OAM L =
(r P
) d
r and a coordinate-independent
spin angular momentum, S =
) d
r. This spin is quantized:
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
. 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
© 2015 Macmillan Publishers Limited. All rights reserved
© 2015 Macmillan Publishers Limited. All rights reserved
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 photons 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 eectively
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
. This occurs
owing to the presence of transverse components of the electron
linear momentum P
. 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 @
(r; t) =
) r
(r; t). When the SFSE is
solved with the ansatz (r; t) = (r) exp (iEt/
h), one obtains
the expression (r
+ k
) (r) = 0, where k
:= 2m
. 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
r) exp(i(k
z +`')) (ref. 36), where J
is an `th
order Bessel function of the first kind, and which have well-
studied photonic analogues
. Bessel modes are diractionless, non-
normalizable,and therefore unphysical solutions to the SFSE and the
optical wave equation. If it is assumed that the mode longitudinal
wavenumber, k
, is much larger than the transverse wavenumber,
, 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
(r,';z):=hr, ',z|p,`i, and have been extensively studied
Electron wavefunctions possessing phase terms exp (i`') and
exp (ik
z), such as Bessel and LG modes, carry OAM
h` and
linear momenta
per electron, oriented along their propagation
direction. Such wavefunctions have linear momentum densities
(r) =
h(`/r ' + k
(r), and probability c urrent densities
(r) = P
, where ' and z denote the azimuthal and axial
unit vectors, and
(r) :=| (r)|
. Because elec trons carry charge,
the presence of an azimuthal probability current produces an
eective loop of charge current about the propagation axis
. 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
, where µ
) is the electron Bohr magneton, s1/2
is the electron spin index and g
'2.002 is the electron g -factor
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
dier 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
wavefunctions energy; as k
or p increase, so does the transverse
energy carrie d by the electron wavefunction
. The radial index-
dependence of the electron transverse energy is an important
consideration in electron/magnetic field interactions
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 eorts 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
If the optical thickness of a material can be increased by ` de
Broglie wavelengths over one full rotation about the phase plates
rt(r,') ·d' =`, the plate will imprint a helical phase
© 2015 Macmillan Publishers Limited. All rights reserved
© 2015 Macmillan Publishers Limited. All rights reserved
ab e
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
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
. c, SEM image of an ultrathin needle
(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
(on the micrometre scale). e, Schematic
of the structured electron beams emerging from an electron phase hologram. The spreading of various diraction orders can be observed to occur rapidly
on propagation, and the distinct transverse profiles associated with five dierent diracted 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
. This investigation followed earlier work, which in 2009 also
proposed the use of such holograms for structured electron beam
. 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
(r, ') produced from the interference of reference
(r, ') and desired target
(r, ') wavefunctions, such that
(r, ') =|
(r, ') +
(r, ')|
, and by designing a mask with
thickness func tion t(r, ') = t
(r, ')], where 2[·] denotes
the Heaviside function, and t
is the optical thickness of the
. When an electron beam is made incident on such a
hologram, a series of diracted beams are produced, each carrying
a dierent 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 eciencies and low absorption losses
. This technique was
shown to produce high-quality electron Bessel
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
. In a striking demonstration of the ee ctiveness of phase
holograms, one study reported the successful generation of electron
Airy beams
, 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
, achieving
generation eciencies of 32%. Another demonstrated twisting of
electron beams using a simulated magnetic monopole constructed
from a thin, nanoscale magnetic needle
. This latter strategy
could benefit from an unusually high conversion eciency. 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
, ‘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
. 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
. When this beam is
phase flattened, a spin-polarized Gaussian electron wavefunction is
recovered, with theoretical eciencies and degrees of polarization
of up to 50% and 97.5%, respectively. Various electron structuring
strategies are illustrated in Fig. 3.
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 dierent magnetic field
© 2015 Macmillan Publishers Limited. All rights reserved
© 2015 Macmillan Publishers Limited. All rights reserved
2, 1
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
r)expi(`' 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 beams
transverse mode. The eect 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
Electric and magnetic field eec 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
. 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
. As this intrinsic OAM transfer is likely to be
significantly less ecient in the case of photon/atom interactions,
structured electron beams therefore represent a possible avenue for
the exploration of truly novel physical eects.
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
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
. This suggests that structured electron beams could in
principle be used to simulate the eects 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 eects. 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 eects.
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
Received 8 April 2015; accepted 15 June 2015;
published online 31 July 2015
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(CERC) Program.
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© 2015 Macmillan Publishers Limited. All rights reserved
... Less familiar vortex states of free electrons have attracted much attention in recent years, reviews on the theory and applications of such states can be found in Refs. [29][30][31]. Of particular interest in the context of rescattering are papers developing the theory of collisions of vortex electrons with atoms [32][33][34][35][36] and molecules [37]; see also a review article [38] and references therein. ...
... OAM represents the rotational property of microwave photons. Furthermore, OAM can be divided into intrinsic OAM and extrinsic OAM [28][29][30] . Specifically, the intrinsic OAM reflects the internal rotation of the wave packet structure, while the extrinsic OAM depends on the spatial rotation of EM waves and varies with the change in the referred coordinate, e.g., the extrinsic OAM is expressed as L ext ¼ hri hPi 28,30 , where r and P denote the vectors of the position and the linear momentum, respectively. ...
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Orbital angular momentum (OAM), which was first discovered in the optical field, represents a new dimension of electromagnetic waves. However, the detection of OAM microwave photons, i.e., vortex microwave photons, at room temperature is difficult due to their low energy. Here we report a prototype of a vortex microwave photon detection device based on vortex electrons. Our OAM detection device efficiently distinguishes the intrinsic OAM in the microwave band, which is helpful for exploring new physical dimensions. In addition, the detection device can be enhanced with a vortex electron sorting device designed with electron holograms so that OAM microwave photon demultiplexing can be achieved. Finally, the OAM detection device has high practicability; i.e., not only it can be used at room temperature, but also it is much smaller than a particle accelerator system. To illustrate the significance of this method, we demonstrate an on-off keying transmission system based on our OAM detection device.
... Moreover, we showed how the aberrations exhibiting local minima or maxima in the interior of the circle defining the Zernike polynomials are those which can more significantly affect the topology of the singular skeleton. These considerations can be useful not only for com-munication purposes but also in devising setups to generate knotted fields in more delicate scenarios, e.g. in nanophotonics experiments [10] or in setups for other kinds of structured quantum waves [33,34], e.g. electrons and neutrons [35]. ...
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Optical knots and links, consisting of trajectories of phase or polarisation singularities, are intriguing nontrivial three-dimensional topologies. They are theoretically predicted and experimentally observed in paraxial and non-paraxial regimes, as well as in random and speckle fields. Framed and nested knots can be employed in security protocols for secret key sharing, quantum money, and topological quantum computation. The topological nature of optical knots suggests that environmental disturbances should not alter their topology; therefore, they may be utilised as a resilient vector of information. Hitherto, the robustness of these nontrivial topologies under typical disturbances encountered in optical experiments has not been investigated. Here, we provide the experimental analysis of the effect of optical phase aberrations on optical knots and links. We demonstrate that Hopf links, trefoil and cinquefoil knots exhibit remarkable robustness under misalignment and phase aberrations. The observed knots are obliterated for high aberration strengths and defining apertures close to the characteristic optical beam size. Our observations recommend employing these photonics topological structures in both classical and quantum information processing in noisy channels where optical modes are strongly affected and not applicable.
... Because of their mutual interactions they exhibit a collective dynamics known as spin waves (magnons) [8]. This collective spin state can support orbital angular momentum (OAM) similar to all other waves [9][10][11][12][13], and in principle the magnetic substance not only carries intrinsic spin angular momentum (SAM) but also OAM [14]. While the typical EdH effect relies only on the creation of polarized SAM, the excitation of OAM modes can also result in a mechanical rotation. ...
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Magnons, collective spin excitations in magnetic crystals, have attracted much interest due to their ability to couple strongly to microwaves and other quantum systems. In compact magnetic crystals, we show that there are magnonic modes that can support orbital angular momentum and that these modes can be driven by linearly polarized microwave fields. Because of conservation of angular momentum, exciting such magnon modes induces a mechanical torque on the crystal. We study a levitated magnetic crystal, a yttrium iron garnet (YIG) microsphere, where such orbital angular momentum magnon modes are driven by microwaves held in a microwave high-Q microwave cavity. We find that the YIG sphere experiences a mechanical torque and can be spun up to ultralarge angular speeds exceeding 10 GHz.
... The arrays can be laid out on the square centimeter areas typical of usual neutron scattering targets. This suggests possibilities for the direct integration of other structured wave techniques, such as the generation of Airy and Bessel beams (35)(36)(37), into neutron sciences. Furthermore, we discuss the applications toward characterization of materials, helical neutron interactions, and spin-orbit correlations. ...
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Methods of preparation and analysis of structured waves of light, electrons, and atoms have been advancing rapidly. Despite the proven power of neutrons for material characterization and studies of fundamental physics, neutron science has not been able to fully integrate these techniques because of small transverse coherence lengths, the relatively poor resolution of spatial detectors, and low fluence rates. Here, we demonstrate methods that are practical with the existing technologies and show the experimental achievement of neutron helical wavefronts that carry well-defined orbital angular momentum values. We discuss possible applications and extensions to spin-orbit correlations and material characterization techniques.
The spin and orbital angular momentum of photons can perturb during propagation in few mode optical fiber, and hence, in the receiver, perturbations must be adjusted. For polarization adjustment, polarization controllers have been developed previously. In this paper, we show that in the presence of external effects such as tension and tortion, the entanglement between the different degrees of freedom of a photon does not change. A device for simultaneous adjustment of polarization and spatial distribution in few mode fiber is proposed. In addition to modification, this device can also be used to produce different modes in fiber.
We report the use of an electrostatic micro-electromechanical systems-based device to produce high quality electron vortex beams with more than 1000 quanta of orbital angular momentum (OAM). Diffraction and off-axis electron holography experiments are used to show that the diameter of the vortex in the diffraction plane increases linearly with OAM, thereby allowing the angular momentum content of the vortex to be calibrated. The realization of electron vortex beams with even larger values of OAM is currently limited by the breakdown voltage of the device. Potential solutions to overcome this problem are discussed.
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This work presents a general framework for quantum interference between processes that can involve different fundamental particles or quasi-particles. This framework shows that shaping input wavefunctions is a versatile and powerful tool for producing and controlling quantum interference between distinguishable pathways, beyond previously explored quantum interference between indistinguishable pathways. Two examples of quantum interference enabled by shaping in interactions between free electrons, bound electrons, and photons are presented: i) the vanishing of the zero-loss peak by destructive quantum interference when a shaped electron wavepacket couples to light, under conditions where the electron's zero-loss peak otherwise dominates; ii) quantum interference between free electron and atomic (bound electron) spontaneous emission processes, which can be significant even when the free electron and atom are far apart, breaking the common notion that a free electron and an atom must be close by to significantly affect each other's processes. Conclusions show that emerging quantum wave-shaping techniques unlock the door to greater versatility in light-matter interactions and other quantum processes in general.
The electron vortex beam (EVB)-carrying quantized orbital angular momentum (OAM) plays an essential role in a series of fundamental research. However, the radius of the transverse intensity profile of a doughnut-shaped EVB strongly depends on the topological charge of the OAM, impeding its wide applications in electron microscopy. Inspired by the perfect vortex in optics, herein, we demonstrate a perfect electron vortex beam (PEVB), which completely unlocks the constraint between the beam size and the beam's OAM. We design nanoscale holograms to generate PEVBs carrying different quanta of OAM but exhibiting almost the same beam size. Furthermore, we show that the beam size of the PEVB can be readily controlled by only modifying the design parameters of the hologram. The generation of PEVB with a customized beam size independent of the OAM can promote various in situ applications of free electrons carrying OAM in electron microscopy.
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div class="title">Holographic Generation of Highly Twisted Electron Beams - Volume 21 Issue S3 - Vincenzo Grillo, Gian Carlo Gazzadi, Erfan Mafakheri, Stefano Frabboni, Ebrahim Karimi, Robert W Boyd
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Free electrons can possess an intrinsic orbital angular momentum, similar to those in an electron cloud, upon free-space propagation. The wavefront corresponding to the electron's wavefunction forms a helical structure with a number of twists given by the \emph{angular speed}. Beams with a high number of twists are of particular interest because they carry a high magnetic moment about the propagation axis. Among several different techniques, electron holography seems to be a promising approach to shape a \emph{conventional} electron beam into a helical form with large values of angular momentum. Here, we propose and manufacture a nano-fabricated phase hologram for generating a beam of this kind with an orbital angular momentum up to 200$\hbar$. Based on a novel technique the value of orbital angular momentum of the generated beam are measured, then compared with simulations. Our work, apart from the technological achievements, may lead to a way of generating electron beams with a high quanta of magnetic moment along the propagation direction, and thus may be used in the study of the magnetic properties of materials and for manipulating nano-particles.
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Light beams with a helical phase-front possess orbital angular momentum along their direction of propagation in addition to the spin angular momentum that describes their polarisation. Until recently, it was thought that these two 'rotational' motions of light were largely independent and could not be coupled during light-matter interactions. However, it is now known that interactions with carefully designed complex media can result in spin-to-orbit coupling, where a change of the spin angular momentum will modify the orbital angular momentum and vice versa. In this work, we propose and demonstrate that the birefringence of plasmonic nanostructures can be wielded to transform circularly polarised light into light carrying orbital angular momentum. A device operating at visible wavelengths is designed from a space-variant array of subwavelength plasmonic nano-antennas. Experiment confirms that circularly polarised light transmitted through the device is imbued with orbital angular momentum of +/- 2 (h) over bar (with conversion efficiency of at least 1%). This technology paves the way towards ultrathin orbital angular momentum generators that could be integrated into applications for spectroscopy, nanoscale sensing and classical or quantum communications using integrated photonic devices.
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Landau levels and states of electrons in a magnetic field are fundamental quantum entities underlying the quantum Hall and related effects in condensed matter physics. However, the real-space properties and observation of Landau wave functions remain elusive. Here we report the real-space observation of Landau states and the internal rotational dynamics of free electrons. States with different quantum numbers are produced using nanometre-sized electron vortex beams, with a radius chosen to match the waist of the Landau states, in a quasi-uniform magnetic field. Scanning the beams along the propagation direction, we reconstruct the rotational dynamics of the Landau wave functions with angular frequency ~100 GHz. We observe that Landau modes with different azimuthal quantum numbers belong to three classes, which are characterized by rotations with zero, Larmor and cyclotron frequencies, respectively. This is in sharp contrast to the uniform cyclotron rotation of classical electrons, and in perfect agreement with recent theoretical predictions.
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Electron beams are extensively used in lithography, microscopy, material studies and electronic chip inspection. Today, beams are mainly shaped using magnetic or electric forces, enabling only simple shaping tasks such as focusing or scanning. Recently, binary amplitude gratings achieved complex shapes. These, however, generate multiple diffraction orders, hence the desired shape, appearing only in one order, retains little of the beam energy. Here we demonstrate a method in electron-optics for arbitrarily shaping electron beams into a single desired shape, by precise patterning of a thin-membrane. It is conceptually similar to shaping light beams using refractive or diffractive glass elements such as lenses or holograms - rather than applying electromagnetic forces, the beam is controlled by spatially modulating its wavefront. Our method allows for nearly-maximal energy transference to the designed shape, and may avoid physical damage and charging effects that are the scorn of commonly-used (e.g. Zernike and Hilbert) phase-plates. The experimental demonstrations presented here - on-axis Hermite-Gauss and Laguerre-Gauss (vortex) beams, and computer-generated holograms - are a first example of nearly-arbitrary manipulation of electron beams. Our results herald exciting prospects for microscopic material studies, enables electron lithography with fixed sample and beam and high resolution electronic chip inspection by structured electron illumination.
The Aharonov-Bohm effect predicts that two parts of the electron wavefunction can accumulate a phase difference even when they are confined to a region in space with zero electromagnetic field. Here we show that engineering the wavefunction of electrons, as accelerating shape-invariant solutions of the potential-free Dirac equation, fundamentally acts as a force and the electrons accumulate an Aharonov-Bohm-type phase-which is equivalent to a change in the proper time and is related to the twin-paradox gedanken experiment. This implies that fundamental relativistic effects such as length contraction and time dilation can be engineered by properly tailoring the initial conditions. As an example, we suggest the possibility of extending the lifetime of decaying particles, such as an unstable hydrogen isotope, or altering other decay processes. We find these shape-preserving Dirac wavefunctions to be part of a family of accelerating quantum particles, which includes massive/massless fermions/bosons of any spin..
We study the interaction of relativistic electron-vortex beams (EVBs) with laser light. Exact analytical solutions for this problem are obtained by employing the Dirac-Volkov wave functions to describe the (monoenergetic) distribution of the electrons in vortex beams with well-defined orbital angular momentum. Our new solutions explicitly show that the orbital angular momentum components of the laser field couple to the total angular momentum of the electrons. When the field is switched off, it is shown that the laser-driven EVB coincides with the field-free EVB as reported by Bliokh et al. [Phys. Rev. Lett. 107, 174802 (2011).