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Abstract
Vortex beams carrying orbital angular momentum have been produced
recently with electron microscopy by interfering an incident electron
beam with a grid containing dislocations. Here, we present an analytical
derivation of vortex wave functions in reciprocal and real space. We
outline their mathematical and physical properties and describe the con-
ditions under which vortex beams can be used in scanning transmission
microscopy to measure magnetic properties of materials at the atomic
scale.
..............................................................................................................................................................................................
Keywords
electron vortex, orbital angular momentum, STEM, EELS, dichroism,
ion vortex
..............................................................................................................................................................................................
Received
26 May 2011, accepted 22 August 2011; online 23 September 2011
..............................................................................................................................................................................................
Physical: Letter
Vortex beams for atomic resolution dichroism
Juan C. Idrobo1,2,* and Stephen J. Pennycook1,2
1Materials Science and Technology Division, Oak Ridge National Laboratory, PO Box 2008, Oak Ridge, TN
37831, USA and2Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA
*To whom correspondence should be addressed. E-mail: jidrobo@gmail.com
It is now possible, using state-of-the-art electron
microscopes, to obtain direct chemical identifi-
cation of individual atoms based on image intensity
[1] together with spectroscopic fine-structure infor-
mation [2]. However, it has not so far been feasible
to measure magnetic properties of individual atoms
or single atomic columns. One way to achieve
atomic-scale magnetic information within an elec-
tron microscope would be to polarize the electron
beam such that each electron carries a specific
orbital angular momentum (OAM).
Vortex photon beams carrying an OAM have been
known (and widely used) for almost 20 years in
optical physics [3,4]. Recently, it has been shown
that vortex beams with desired OAM can also be
produced in a transmission electron microscope
(TEM) [5,6,7].Specifically, by interfering an incident
electron beam with a grid containing dislocations,
Verbeeck et al. [6] showed that electron vortex
beams can be generated to measure magnetic circu-
lar dichroism with a spatial resolution of 250nm
using electron-energy loss spectroscopy (EELS) in
a TEM. Their work also implies that by using a
scanning (S)TEM (see Fig. 1), it may be possible to
achieve the goal of atomically resolved measure-
ments of magnetic properties.
In this letter, we derive an analytical description
in reciprocal and real space of vortex beams carry-
ing an OAM by evaluating the diffracted pattern
formed by a plane wave that has passed through a
grid containing dislocations. We find that a vortex
beam can be expressed by generalized hypergeo-
metric functions. It has an O-ring shape and con-
tains a phase singularity in real and reciprocal
space. Our calculations show that electron vortex
probes,producedwith
aberration-corrected electron microscopes, achieve
a spatial resolution in imaging and spectroscopy
that is ?10 times worse than that for electron
probes without OAM. We find that an alternative
route to achieve atomic resolution is to either accel-
erate the electrons to 2MV (or above) or to use ion
beams (He+) rather than electrons.
An analytical expression of an electron (or ion)
vortex wave function carrying an OAM can be
obtained by calculating the diffraction pattern when
a plane wave goes through a grid containing fork
dislocations. This is equivalent to obtaining the
currentstate-of-the-art
........................................................................................................................................................................................................................................................
Journal of Electron Microscopy 60(5): 295–300 (2011)
doi: 10.1093/jmicro/dfr069
........................................................................................................................................................................................................................................................
Published by Oxford University Press [on behalf of Japanese Society of Microscopy]. All rights reserved.
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Fourier transform of the mathematical function that
describes such a grid. Mathematically, an ideal grid
with l fork dislocations can be defined as cos(lφ−
arcosφ)1, where a is the grid spacing and (r,φ) are
cylindrical coordinates in real space. Thus, the
diffraction pattern is obtained by evaluating the
following integral:
1
2
ð
ðeilf?iarcosfþik?rþ e?ilfþiarcosfþik?rÞdr;
ð1Þ
where k is the vector position in reciprocal space.
Note that Eq. (1) contains two terms. Each term
can be interpreted as a single transmitted wave
function carrying an OAM of +lh?, and the center
of propagation in reciprocal space which is located
at kx=±1/a, ky=0. The diffraction pattern when a
plane wave is transmitted through a real physical
(binary) grid l dislocations follows Bragg’s law and
is composed of a diffracted beam (without OAM)
centered at k=0 plus a set of n pairs of diffracted
beams centered at kx=±n1/a, ky= 0 with an OAM
equal to +nlh?, where n is just the diffraction order
number. The phase and amplitude of each pair of
diffracted electron (ion) beams of order n from the
binary grid can be described by those of an ideal
grid (1), where l is replaced by nl, multiplied by the
function1
general vortex wave function Λl(k) carrying an
OAM of lh?as
ð
The integral in Eq. (2) for a circular grid of radius R
is
2sincðnp=2Þ [4]. Therefore, we can write a
LlðkÞ ¼
eilf?iarcosfþik?rdr:
ð2Þ
LlðkÞ ¼ eilu1
k2
ðkR
0
tJlðtÞdt;
ð3Þ
where Jlis the Bessel function of the first kind with
order l and θ the angular coordinate in reciprocal
space (see the appendix for details). Eq. (3) defines
a vortex wave function in reciprocal space.
Figure 2 shows the calculated intensity, phase and
amplitude profile of a vortex wave Λ1(k). Λ1(k) pre-
sents an O-ring shape and a phase singularity at k=
0, as can be observed in Fig. 2. The k values where
Λ1(k)=0 (labeled as siin Fig. 2) depend entirely on
Fig. 1. Schematic of how electron vortex beams can be produced in the (S)TEM. In a TEM, a grid with fork dislocations is placed after the
sample and then an aperture is used to select a single electron vortex beam with the desired OAM [6]. In the STEM, the grid with the
dislocation and the beam-selecting aperture (highlighted inside the gray rectangle) are placed before the sample. Extra condenser lenses are
also required to magnify the electron vortex before the probe-forming objective lens and aperture [8]. The resulting electron probe can be
used as in any other STEM but with the benefit that since it carries an effective OAM, magnetic properties of the sample could also be
studied directly.
1For instance, see p. 158 in Ref. [4]. The definition of an ideal grid
is given in the context that the transmitted wave function consists
only of two vortex beams with opposite OAM as is shown in Eq. (1)
and discussed in the main text.
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the radius R, as expected from the diffraction
theory. Larger values of R result in smaller values of
siand vice versa. This means, from a calculation
perspective, that the magnification of an electron
vortex at the objective aperture plane can be con-
trolled by choosing an appropriate value of R.
The analytical expression for a single electron
(ion) vortex wave in real space is obtained by
applying an inverse Fourier transform to Λl (k).
Therefore, a vortex carrying an OAM lh? can be
expressed in real space as
LlðrÞ ¼
ð
e?ik?r?ixðkÞLlðkÞdk:
ð4Þ
In Eq. (4), we have added the term iχ (k) in the
exponential in order to describe the change of
phase that an electron (ion) undergoes due to
imperfections (aberrations) of the objective lenses.
χ (k) is known as the aberration function [9].
Eq. (4) contains all the physics for a general
description of a vortex in real space. However, for
the investigation purpose of this study, which is to
find how small an electron (ion) vortex can be
made in real space, we set χ (k) to zero without
loss of generality in our calculations. The last
approximation is allowed because of the develop-
ment in the last few years of aberration-correction
optics [10]. When aberrations can be neglected
up to values k≤kmax, an electron vortex in real
space is2
LlðrÞ ¼ eilf
ðkmax
0
JlðkrÞ
k
ðkR
0
tJlðtÞdtdk:
ð5Þ
With the definition of vortex beams in real space
given by Eq. (5), we can proceed to investigate the
electron optical conditions at which the electron
vortexes reach their smallest physical size. This can
be done by calculating the intensity profile |Λl(r)|2.
For all our calculations, we define kmax=α/λ, where
α is the half angle of the objective aperture (?30
mrad in state-of-the-art aberration-corrected elec-
tron microscopes) and λ the wavelength of the
electron.
First, we examine how an electron probe changes
in size depending on the number of diffracted rings
in Λl (k) that are allowed within the objective
probe-forming aperture (in other words, the relative
magnification between the aperture in the diffrac-
tion plane and the objective aperture). Figure 3a
shows the electron vortex probe intensity with an
OAM of h?as a function of the number of diffracted
rings, si, allowed in the objective aperture. The
probe was calculated for the case of an electron
microscope operating at 100kV. We find that the
best optical condition to obtain the smallest elec-
tron probe is when only the first diffracted ring (s1)
is allowed through the objective aperture. Choosing
a value smaller than s1for Λl (k) results only in
effectively decreasing the total current of the elec-
tron probe without reducing significantly the elec-
tron confinement radius Rc(defined as the position
where the maximum intensity occurs) or the full-
width at half-maximum on the Gaussian-like profile
node ΔR. This is to be expected, since the spread of
an electron vortex in reciprocal space necessarily
implies a localization of the electron probe in real
space. We find the same behavior for electron
vortexes with higher OAM.
We next calculate how the increase in the OAM
of the electron vortexes affects their confinement in
real space. We find that under optimum optical
Fig. 2. (Color online) Calculated intensity, phase and amplitude
profile of a vortex wave Λl(k) carrying an OAM, h
has been renormalized by choosing the appropriate value of R, such
that the first occurrence when Λ1(k) =0 and k≠0, s1, is equal to
1. Using the renormalization, the values for the next four zeros, s2,
s3, s4and s5, are 1.374, 2.052, 2.469 and 3.112, respectively. Note
that under the above definition s0=0, which occurs when k= 0.
?i.e. (l=1). Λ1(k)
2In order to derive Λl(r) from Eq. (4), we have used the same
approach outlined in the appendix for Λl (k), in particular, the
integral with respect to φ. In the definition of Λl(r), we have left out
phase shifts proportional to l and other integration constants.
J.C. Idrobo and S.J. Pennycook
Vortex beams for atomic resolution dichroism
297
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conditions (defined earlier), ΔR for electron vor-
texes with different OAMs increases slightly with
the OAM (Fig. 3b). We also find that their confine-
ment radius Rc increases linearly and in quasi-
discrete values with the OAM. The last result is due
to the quantization of angular momentum and can
be understood in classical terms by the expression
that relates angular momentum L, with the linear
momentum p and the radius of rotation r of a par-
ticle, i.e. L =r×p. Since the linear (transferred)
momentum is the same for all the electron vortexes
(i.e. the electron vortexes have the same kinetic
energy and are formed with the same objective
aperture), the orbit in which the electrons rotate
necessarily needs to increase to satisfy the conser-
vation of angular momentum. The smallest probe
for an electron vortex at 100kV (OAM=h?) has an
Rc=0.370nm and a ΔR= 0.404nm. If images or EEL
spectra were going to be taken using that electron
probe, then the structural and spectroscopy infor-
mation obtained would have a spatial resolution d
of 1.14nm, where d=2Rc+ΔR. To put the last
result into perspective, under the same optical
conditions, an electron probe without the OAM is
about 10 times smaller, which allows the acquisition
of images and spectra with atomic resolution.
Increasing the acceleration voltage to 300kV pro-
duces an electron vortex probe with an Rc=0.200
nm and a ΔR=0.215nm, resulting in d= 0.615nm
(see Fig. 4a). Increasing α to 40mrad gives Rc=
0.150nm, ΔR=0.162nm and d= 0.462nm. Even in
the unrealistic case that α could be increased up to
60mrad, we find that electron vortex beams would
not be small enough to provide imaging and spec-
troscopy at the atomic scale3. If the electrons are
accelerated to 2MV, then the vortex probe shrinks
to values Rc=0.050nm, ΔR=0.055nm and d=0.155
nm(Fig. 4a), achieving
resolution.
Finally, we calculate the probe size for an STEM
employing a beam of helium ions. We find that
in a proof of principle He+microscope operating
thegoal ofatomic
Fig. 3. (Color online) Intensity profiles of electron vortex probes
carrying an OAM. (a) Electron vortex probe intensity with an OAM
of h
Fig. 2) that are allowed within the objective aperture. (b) Under
optimal optical conditions, electron vortexes with different OAMs
have very similar ΔR. At the same time, their confinement radius, Rc,
increases linearly and in quasi-discrete values with the OAM.
?as a function of the number of diffracted rings (sishown in
Fig. 4. (Color online) Intensity profiles of electron and helium
vortex probes carrying the OAM. (a) Electron vortex probes
calculated for electron microscopes operating at 100 kV, 300 kV and
2 MV. The inset shows a two-dimensional density plot of the
electron vortex at 100kV (the scale bar is 1nm). (b) He vortex
probes for a microscope operating at 40kV compared with the
300kV electron probe shown in (a). The horizontal axis in (b) is in
a logarithmic scale.
3When α =60mrad, we obtain that Rc=0.100nm, ΔR= 0.107nm and
d= 0.307nm.
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without aberration-corrected lenses (40kV, α= 0.1
mrad) [11], the probe size has an Rc=2.030nm,
ΔR= 2.237nm and d=6.297nm (Fig. 4b). In an
aberration-corrected He microscope allowing a
value α =4mrad, the He vortex probe is similar to
that of a 2MV electron vortex probe (i.e. Rc= 0.050
nm, ΔR=0.057nm and d=0.157nm).
In conclusion, we derive an analytical description
in reciprocal and real space of vortex beams carry-
ing the OAM. We find that in current aberration-
corrected STEM, vortex probes can be confined to
a diameter of 0.6nm, which allows for the mapping
of magnetic properties at the sub-nanometer level.
But in order to achieve atomic resolution, our cal-
culations indicate that either electrons have to be
accelerated to 2MV (or above) or the microscope
needs to use ion beams (He+) instead of electrons.
Funding
This work was supported by the National Science
Fundation (grant number DMR-0938330, J.C.I.).
Oak Ridge National Laboratory’s SHaRE User
Facility (J.C.I.), which is sponsored by the Office of
Basic Energy Sciences, U.S. Department of Energy
and the Office of Basic Energy Sciences, Materials
Sciencesand Engineering
Department of Energy (S.J.P.).
Division, U.S.
Acknowledgements
We thank O.L. Krivanek, A.R. Lupini and M. Prange for useful
discussions.
Appendix
The integral shown in Eq. (2) can be evaluated by
expanding k · r as kr(cosφcosθ+sinφsinθ), where
φ and θ are the angular coordinates in real and reci-
procal space, respectively. For simplification pur-
poses, it is also convenient to define ξ=r(−a+k
cosθ) and η=krsinθ. The terms in the exponential
iξcosφ and iηsinφ then can be replaced by using
the definition of Bessel functions through a Laurent
series to obtain the following expression:
LlðkÞ¼
X
1
n;m¼?1
in
ðR
0
rJnðjÞJmðhÞdr
ð2p
0
eiðlþnþmÞfdf:
ðA.1Þ
The integral with respect to φ vanishes for all
values of n and m that do not satisfy the condition
l= −(n+m). Thus, the only remaining terms after
integrating with respect to φ are
2p
X
X
1
n¼0
1
in½ð?1ÞnþlJnðjÞJnþlðhÞþilJnþlðjÞJnðhÞ?þ2p
n;m¼1
inð?1ÞmJnðjÞJmðhÞdl;mþn:
ðA.2Þ
Eq.(A.2) can be simplified even further by perform-
ing the following analysis. First, the amplitude of
the final solution must preserve an axial symmetry.
By setting ξ or η to zero, the only term that survives
in Eq. (A.2) is 2π(−1)lJl(η) or 2πilJl(ξ), respectively.
The previous result means that the amplitude and
phase of Eq. (A.2) can simply be expressed as
2πJl(κ) and eiν, respectively, where κ2= ξ2+η2and ν
is a linear function of l and the angle θ, i.e. ν= l(θ+
constant). The value of the constant is obtained by
noting that for θ=π/2 (ξ =0), eiν=(−1)land for θ=0
(η=0),eiν=il. Thismeans
Performing the change of variable of τ= κ=kr, Eq.
(A.1) can be reduced to
that
ν =lθ+lπ/2.
LlðkÞ ¼ eilu1
k2
ðkR
0
tJlðtÞdt:
ðA:3Þ
Note that in Eq. (A.3), the variable a does not
appear. Nevertheless, a can be recovered by defin-
ing k2=(kx±1/a)2+ky
on the sign of l. In the definition of Λl(k), we left
out the phase lπ/2 and the constant 2π. Λl(k) has
some interesting mathematical and physical proper-
ties which we will briefly discuss next for all values
of l, with the exception of l=0, since Λ0 (k)
becomes the well-known
Fraunhofer diffraction of a circular aperture.
(i) Λl (k =0)=0. When R!∞, Λl (k) is con-
strained around the vicinity of k=0 but it never
reaches k= 0.
(ii) Λl(k) by definition carries angular momen-
tum lh?. This statement can easily be verified
by applying the momentum operator Lz¼ ?ih?@uto
Λl(k),
(iii) kLlðkÞjLl0ðkÞl / dl;l0. Note as well that vortex
beams always come in pairs as illustrated in Eq. (1).
2, where the sign of a depends
solution forthe
J.C. Idrobo and S.J. Pennycook
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8 Krivanek O L (2011) Private communication, January.
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The former statement is another way of expressing
the law of conservation of angular momentum (i.e.
the total angular momentum before the plane wave
goes through the dislocation grid is zero).
(iv) Λl(k) vanishes when k!∞.
(v) Λl (k) is not normalized. The integral with
respect to r results in a generalized hypergeometric
function of the form0F0(;;z) that needs to be cal-
culated with the help of a computer for each value
of l and R. For the sake of simplicity and mathemat-
ical aesthetics, we decided to define Λl (k) as a
non-normalized wave function.
References
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