Unveiling the Orbital Angular Momentum and Acceleration of Electron Beams

Roy Shiloh,1,* Yuval Tsur,1 Roei Remez,1 Yossi Lereah,1 Boris A. Malomed,1 Vladlen Shvedov,2 Cyril Hnatovsky,2

Wieslaw Krolikowski,2 and Ady Arie11Department of Physical Electronics, Fleischman Faculty of Engineering, Tel Aviv University, Tel Aviv 6997801, Israel

2Laser Physics Centre, The Australian National University, Canberra ACT 0200, Australia(Received 27 November 2013; published 4 March 2015)

New forms of electron beams have been intensively investigated recently, including vortex beamscarrying orbital angular momentum, as well as Airy beams propagating along a parabolic trajectory. Theirtraits may be harnessed for applications in materials science, electron microscopy, and interferometry, andso it is important to measure their properties with ease. Here, we show how one may immediately quantifythese beams’ parameters without need for additional fabrication or nonstandard microscopic tools. Ourexperimental results are backed by numerical simulations and analytic derivation.

DOI: 10.1103/PhysRevLett.114.096102 PACS numbers: 68.37.Lp, 41.85.-p, 42.40.Jv

Introduction.—In the last few years, it became possibleto generate special shapes of electron beams. One of theseis the vortex beam [1–4] having a helical wave frontstructure and a phase singularity on axis. Its azimuthalphase dependence is exp ðilϕÞ, where integer l is thetopological charge and ϕ is the azimuthal angle. Thisbeam carries an orbital angular momentum (OAM) of ℏl[5]. Another interesting beam is the Airy beam, having atransverse amplitude dependence in the form of the Airyfunction, Aiðx=x0Þ, where x0 defines the transverse scale.It is shape preserving and moves along a parabolictrajectory in free space with a nodal trajectory coefficient,sometimes referred to as the “acceleration” coefficient,τ ¼ 1=ð2k2DBx30Þ, where kDB is the (de Broglie) wavenumber. These beams, generated and observed in a trans-mission electron microscope (TEM) are expected toopen new possibilities for interactions between electronsand matter, as well as for electron microscopy andinterferometry. For example, vortex beams were used inelectron energy loss spectroscopy in order to characterizethe magnetic state of a ferromagnetic material [2], whileAiry beams were proposed for realization of a new type ofelectron interferometer [6]. In order to utilize these newtypes of electron beams, it is required to develop methodsthat allow one to easily determine their defining properties—the OAM of a vortex beam or the nodal trajectorycoefficient of an Airy beam. Here, we first demonstratea straightforward method for single-mode OAM determi-nation. The method relies on the astigmatic Fourier trans-form of the beam; the concept and measuring procedure wedescribe are central to our main discussion: the measure-ment of the nodal trajectory of electron Airy beams.OAM determination.—In their 1991 article [7],

Abramochkin and Volostnikov discussed the mathematicsof beam transformations under astigmatic conditions.Since then, different authors proposed and demonstratedmode conversion in lasers [8], linear and nonlinear

optics [9,10], and free-electron beams in TEM [11,12].In the latter works, McMorran et al. and Schattschneideret al. have shown in theory and experiment howLaguerre-Gauss modes transform to Hermite-Gaussmodes. Recently, it was proposed, in light optics, that acylindrical lens acting as a mode converter [13] be used toquantify the OAM of optical vortices [14]. Thus, a vortexof integer topological charge converts into a correspond-ing Hermite-Gauss-like mode, where the number ofdark stripes precisely indicates the topological charge.The advantage of this method is in its simplicity andgenerality: the only addition to the setup is a cylindricallens. In the TEM, the lens is inherent—the condenserand objective stigmators may be used to impose a strongastigmatism along a desired axis, thereby implementingthe astigmatic transformation required to perform modeconversion.To observe this phenomena, in our first experiment we

designed a two-dimensional binary hologram—a forkgrating which is mathematically written

hðx; yÞ ¼ sgn

�sin

�2πxΛ

− l arctan 2ðy; xÞ þ δπ

2

�þ Δ

�;

ð1Þ

where 2πx=Λ is a linear grating along x, with period Λ,upon which the four-quadrant inverse-tangent function [15]modulates the spiral phase of the vortex, l being thetopological charge. δ reduces fabrication errors in thecenter by preserving continuity: its value is chosen equalto zero (one) if l is odd (even). Δ ¼ 0.25, related to dutycycle, is chosen so that both even and odd diffraction ordersare visible. A Raith IonLine focused ion beam was used tomill into a 100 nm SiN membrane coated by 50 nm Au,from the gold side. This layer prevents charging of thesample and assures that the membrane behaves as anamplitude (binary) grating.

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A fork grating with l ¼ 3 topological charge wasfabricated [Fig. 1(e)], based on a Λ ¼ 0.750 μm period,along with a linear (Bragg) grating of periodΛ ¼ 0.445 μmwhich was fabricated for calibration purposes [Fig. 1(f)].The sample was mounted onto a Tecnai F-20 FEG-TEMwith a single tilt specimen holder and, subsequently,observed in the microscope in low angle diffraction modethrough a 10 μm aperture.In order to form the far field diffraction pattern, the

condenser lens was set to measure a focused spot. The forkgrating was then aligned under the beam and the hologramimposed a spiral phase such that in each diffraction orderm,a vortex of OAM lm emerged [Fig. 1(a)]. The intenseon-axis (zero-order) beam was blocked to prevent damageto the CCD.Since the electron beam impinging on the hologram does

not have a Gaussian distribution, the emerging vortices arenot pure Laguerre-Gauss modes. However, the resultingdiffraction pattern is undeniably dominated by solutions ofthe Schrödinger equation

�∇2⊥ þ 2ikDB

∂∂z

�ψ ¼ 0; ð2Þ

where kDB ¼ 1=λDB is the de Broglie wave number.This equation governs the paraxial dynamics of theslowly varying envelope of the free electron beam in theTEM [16], a statement verified by the results of ournext step.The electron beam is made sufficiently astigmatic along

the desired axis so it becomes elliptic; in the present case,this axis must be parallel to the lateral direction of thefork. As a reference, the linear grating was first measuredunder these conditions. As will be explained below,we purposely measured slightly out of the focal plane[Fig. 1(c)]. This reference serves to assess the efficacy ofthe astigmatic transformation. When the beam is suffi-ciently astigmatic, the vortices in the different diffractionorders are converted into Hermite-Gauss-like modes[Fig. 1(b)], and the resulting dark stripes indicate theOAM carried by each order. In addition, the angle ofthe transformed orders signifies the rotation direction ofthe vortex: clockwise or counterclockwise. The lineargrating’s diffraction spots become lines, as expected froma cylindrical lens [Fig. 1(d)]. Note that, if the lineargrating’s diffraction pattern was observed in focus ratherthan out of it, the spots would be so small that theequipped Gatan 694 CCD would not be able to distinguishthe ellipticity of the beam; also, the additional diffractionlines visible in the negative orders [Fig. 1(d)] are aresult of asymmetry of the sample relative to the beam,introduced by using a single-tilt rather than a double-tiltholder. In the Supplemental Material [17], we show asimilar, additional measurement of at least 10ℏ. Apotential application of this method would be the

characterization of localized defects in materials: previ-ously, it was shown that spontaneously stacked graphitefilms can be modeled as spiral phase plates for electronsand, therefore, add OAM to the electron beam [3], but, inorder to characterize this OAM, an interference measure-ment was needed. The astigmatic transformation methodpresented here provides a simpler and more direct meas-urement of the direction of helicity and the thicknessvariation of the defect. Moreover, the electron energy canbe tuned to achieve a 2π phase step for the spiralphase plate.

FIG. 1. Experimental results: diffraction patterns of (a) vorticesgenerated by a charge 3 fork grating, using a focused stigmaticbeam (zero-order blocked); (b) corresponding vortices afterastigmatic transformation; (c) diffraction pattern generated bya linear calibration grating using a stigmatic beam, out of focus;(d) corresponding lines resulting from the astigmatic transforma-tion, out of focus. Below: TEM images of (e) charge 3 fork(0.750 μm period); (f) linear grating (0.445 μm period). Note:brightness levels in the third order parts, as marked by the dashedlines in (a) and (b), have been modified for visibility.

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Determining the Airy beam’s nodal trajectory.—Opticalgeneration and manipulation of Airy beams [18,19] hasbeen recently under the spot-light; here, we present, to thebest of our knowledge, the first example of Airy beams withspecial transformations, specifically, the astigmatic trans-formation. It is of interest to measure the nodal trajectorycoefficient of 2D Airy beams; in order to do so from astigmatic pattern, one must either record the propagation ofthe Airy beam in different planes and measure the actualtrajectory, or directly measure the density of the Airy lobesand relate it to the length scale x0 (for more details, see [6]).These two methods are, unfortunately, time consuming andarduous; the latter case, for example, is generally dangerousto perform experimentally, since such a measurementrequires the pattern to be in focus, to exhibit high contrastand low signal-to-noise ratio so as to make the lobesdiscernible and measurable with precision, all the whileprotecting the CCD camera from damage. Using astigmatictransformation, we show that the energy of the beam isdistributed over a large area and the beam’s accelerationcoefficient is deduced from the curvature of the generatedpattern.For the purpose of generality, we express the trans-

formation of a beam, fðx; yÞ, under general astigmaticconditions by [[7], Eq. (8)]

Fðθx; θy; aÞ ¼Z

∞

−∞

Z∞

−∞eiðθxxþθyyÞþiψðx;y;a;αÞfðx; yÞdxdy:

ð3ÞHere, we define the Cartesian coordinates ðx; yÞ in the

plane of the holographic mask, and similarly, the geometricangles ðθx; θyÞ in the diffraction plane. ψðx; y; a; αÞ ¼a½ðx2 − y2Þ cos 2αþ 2xy sin 2α�, a is the measure of astig-matism and α is a rotation angle with respect to the curvey ¼ x. In Ref. [7], it is shown that, in the special case ofψðx; y; a; α ¼ π=4Þ ¼ 2axy, Hermite-Gauss and Laguerre-Gauss beams interchange; this mathematical paradigmexplains our previous results.Applying the transformation to Airy beams, we write the

Fourier transform

Fðθx; θy; aÞ ¼Z

∞

−∞

Z∞

−∞eiðθxxþθyyÞþi2axyeið3x

30Þ−1ðx3þy3Þdxdy;

ð4Þ

which yields an Airy beam in the diffraction plane.Note that fðx; yÞ ¼ exp ½ið3x30Þ−1ðx3 þ y3Þ� is a cubicphase term and x0 is the Airy-related transverse scale.Figure 2 depicts numerical simulations for constantastigmatism a ¼ 4 μm−2 and two different values of x0,with the main (high-intensity) lobes marked; the curvatureof these lobes is related to a and x0, and they, in turn,are related to the nodal trajectory; thus, a method offinding these parameters in situ could be devised. While

calculation of the Fourier integral in Eq. (4) in thestigmatic (a ¼ 0) case is easy and results in the two-dimensional Airy pattern, obtaining a closed-form analyticsolution to the astigmatic (a ≠ 0) case is difficult. However,one can describe the formation of the diffraction patternand particularly the main lobes using geometric optics.In this approach, we perform ray-tracing analysis of

the cubic phase imposed by the holographic mask andthe phase of the impinging, astigmatic beam, and seek ananalytic expression for the main lobes. To reduce thecomplexity of the derivation, we utilize the symmetry ofthe astigmatic Airy and the invariance of the pattern toadditional focus and find that the four high-intensity lobesof the astigmatic Airy are described geometrically by

ða2x30Þ−1θxð∓1Þ ¼ ∓ð3þ coshuÞ3=2sinh u2

Outer lobes;

ða2x30Þ−1θxð�2Þ ¼ �ð−3þ coshuÞ3=2cosh u2

Inner lobes;

ð5Þ

whereffiffiffi2

pθy ¼ ða2x30Þsinh2u, and θx,θy are the coordinates

in Fourier space. Here, each of the top and bottom rowsrepresents the two outer and inner curves, respectively,as depicted in Fig. 2. It is worth noting that, since ourderivation is based on geometric optics, there is notheoretical limitation to our results as long as the Airybeam is generated through an aperture large enough withrespect to x0, such that enough oscillations of the Airyenvelope are transmitted. The full mathematical details arein the Supplemental Material [17].To see this in experiment, we designed binary holo-

graphic masks of varying diameters (20–80 μm) using theexpression

hðx; yÞ ¼ signfcos ½ðx3 þ y3Þ=3x30�g: ð6Þ

FIG. 2 (color online). Examples of astigmatic Airy beams withdifferent x0 values and astigmatic work point a ¼ 4 μm−2. In theleft panel, low x0 manifests in a shape very similar to the familiarAiry, while in the right panel the high acceleration (in conjunctionwith the value of a) yields an astigmatic Airy. The magentaand blue lines mark the θxð∓1Þ and θxð�2Þ solutions, respectively.(a) x0 ¼ 0.21 μm and (b) x0 ¼ 0.63 μm.

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The holographic masks were fabricated on a membranesimilar to that of the vortex masks, with x0 rangingð0.33 ≤ x0 ≤ 1.6Þ × 10−6 m, some of which are depictedin Fig. 3; our considerations in selecting these values werepurely technical and are based on our FIB’s fabricationcapability, but the method itself is in no way limited bythese values. The cubic phase modulation imposed on thebeam yields an Airy pattern in the diffraction plane. Usingthe TEM’s stigmators, we set the astigmatism to a constantvalue and measure the curved lobes of the resultingastigmatic Airy patterns in proximity to the diffractionplane, as seen in Figs. 3(a)–3(d). Note that, contrary to thesimulation, in experiment, we find a mirror-symmetricastigmatic Airy pattern around the astigmatic axis; thetwo patterns are the real and virtual image of our inlineholographic mask, and appear so, in part, due to the binarymask fabrication.The value of the astigmatism is related to a, which can be

referred to as the “astigmatic working point”, can be chosenappropriately to make the measurement of a range of x0easier: as we have shown earlier, upon normalizationthe astigmatic Airy is shape preserving and dependslinearly on the scaling a2x30; thus, if a is too small, onewould need a very large value of x0 to observe theastigmatic Airy, and if a is too large, moderate values ofx0 might yield a pattern whose details are impossible todiscern. Different methods of determining a could bedevised; here, we will deduce it from our series ofmeasurements: as analyzed in Figs. 3(e)–3(f), throughlinear fitting, we deduce a ¼ 3.46� 0.09 μm−2. Thisvalue, calibrated by our measurement of known x0 values,may now be used for the measurement of unknownx0 for other Airy beams and, subsequently, the nodaltrajectory τ ¼ 1=ð2k2DBx30Þ, which in our case rangesð7.7 ≤ τ ≤ 876Þ × 10−7 m−1.The astigmatic work point gives us a range of measur-

able x0 which spans at least 1 order of magnitude (in ourexperiment), a range that may be extended with an addi-tional working point or simply a camera with higherresolution. One may, otherwise, devise a relation betweenthe stigmator settings and a, and be free to measure Airybeams with nearly arbitrary x0.In this Letter, we first examined a method of measuring

the OAM of electron vortex beams by means of astigmatictransformation, previously employed in light optics usingcylindrical lenses. The transformation is applied, literally,by the turn of a knob in any standard TEM by correctlymanipulating the lens stigmators. We extended the scope ofthis transformation and applied it to Airy beams insimulation and experiment, and found an analytic,closed-form expression [Eq. (5)] without any approxima-tions, out of which the curvature of the resulting astigmaticshape and the Airy beam’s nodal trajectory may be found.In effect, following calibration, this is a viable method offinding the nodal trajectory, or acceleration coefficient, τ,

from one simple measurement. Our results warrant furthertheoretical investigation into the astigmatic transform ofother beams to which it may be relevant, such as Besselbeams [20], parabolic beams [21], and beams with arbitrarycaustic curves [22–24], unveiling their underlying propa-gation parameters.

The authors would like to acknowledge Dr. Yigal Lilachfor his support in the fabrication process. The workwas supported by the Israel Science Foundation,Grant No. 1310/13, by the German-Israeli ProjectCooperation (DIP), and by the Australian Research Council.

FIG. 3 (color online). Measurements of several astigmatic Airybeams with different x0 values and corresponding fitted curves ofthe high-intensity main lobes. The only fitting parameter here isa2=3x0. From these measurements, we deduce the value of a withour theoretical considerations. In (a), we may scarcely observean additional, much weaker diffraction order, which is a result ofthe binary nature of the grating. Our linear fit in (e), yieldinga ¼ 3.46� 0.09 μm−2, is reinforced by the distribution observedin the residuals (inset) and figure of merit R2 ¼ 0.9965. Theerrors were derived from a 95% confidence interval scheme.(f) Example of a fabricated Airy mask. The field of view in theseimages is 100 μm−1; the brightness and contrast were altered forvisibility. (a) x0 ¼ 1.2 μm, (b) x0 ¼ 0.96 μm, (c) x0 ¼ 0.74 μm,(d) x0 ¼ 0.33 μm, (e) Fit to a2=3x0 ½m−1=3�.

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*royshilo@post.tau.ac.il[1] K. Y. Bliokh, Y. P. Bliokh, S. Savel’ev, and F. Nori, Phys.

Rev. Lett. 99, 190404 (2007).[2] J. Verbeeck, H. Tian, and P. Schattschneider, Nature

(London) 467, 301 (2010).[3] M. Uchida and A. Tonomura, Nature (London) 464, 737

(2010).[4] B. J.McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing,

H. J. Lezec, J. J. McClelland, and J. Unguris, Science 331,192 (2011).

[5] L. Allen, M.W. Beijersbergen, R. J. C. Spreeuw, and J. P.Woerdman, Phys. Rev. A 45, 8185 (1992).

[6] N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, andA. Arie, Nature (London) 494, 331 (2013).

[7] E. Abramochkin and V. Volostnikov, Opt. Commun. 83, 123(1991).

[8] G. Machavariani, A. A. Ishaaya, L. Shimshi, N. Davidson,A. A. Friesem, and E. Hasman, Appl. Opt. 43, 2561 (2004).

[9] T. Ellenbogen, I. Dolev, and A. Arie, Opt. Lett. 33, 1207(2008).

[10] N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler,and A. Arie, Phys. Rev. Lett. 108, 233902 (2012).

[11] B. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing,H. Lezec, J. J. McClelland, and J. Unguris, in CLEO:2011 -Laser Applications to Photonic Applications (IEEE, Piscat-away, NJ, 2011), p. QMA1.

[12] P. Schattschneider, M. Stöger-Pollach, and J. Verbeeck,Phys. Rev. Lett. 109, 084801 (2012).

[13] V. G. Shvedov, C. Hnatovsky, W. Krolikowski, and A. V.Rode, Opt. Lett. 35, 2660 (2010).

[14] V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. N. Neshev,W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar,Opt. Express 17, 23374 (2009).

[15] arctan 2ðy; xÞ returns the angle in the interval ½−π; π� asopposed to arctan which is limited to ½−π=2; π=2�, i.e., onlythe two quadrants in the positive x half-space.

[16] L. Reimer and H. Kohl, Transmission ElectronMicroscopy - Physics of Image Formation 5th ed. (Springer,New York, 2008).

[17] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.114.096102 for math-ematical derivations and additional measurements.

[18] M. V. Berry, Am. J. Phys. 47, 264 (1979).[19] G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32,

979 (2007).[20] V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R.

Dennis, and R.W. Boyd, Phys. Rev. X 4, 011013(2014).

[21] M. A. Bandres, Opt. Lett. 33, 1678 (2008).[22] E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys.

Rev. Lett. 106, 213902 (2011).[23] L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L.

Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, Opt.Express 19, 16455 (2011).

[24] I. Epstein and A. Arie, Phys. Rev. Lett. 112, 023903(2014).

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