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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

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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.

For permissions, please e-mail: journals.permissions@oup.com

at Oak Ridge National Laboratory on November 8, 2011

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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

Vortex beams for atomic resolution dichroism

299

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