Nondestructive Measurement of Orbital Angular Momentum for an Electron Beam

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by  Hugo Larocque

Non-destructivemeasurementofanelectronbeam’sorbitalangularmomentum
Hugo Larocque,1 Fre´de´ric Bouchard,1 Vincenzo Grillo,2 Alicia Sit,1 Stefano Frabboni,2,3
Rafal E. Dunin-Borkowski,4 Miles J. Padgett,5 Robert W. Boyd,1,6 and Ebrahim Karimi1,7,∗
1The Max Planck Centre for Extreme and Quantum Photonics, Department of Physics,
University of Ottawa, 25 Templeton St., Ottawa, Ontario, K1N 6N5 Canada
2CNR-Istituto Nanoscienze, Centro S3, Via G Campi 213/a, I-41125 Modena, Italy
3DipartimentoFIMUniversita´ diModenaeReggioEmilia,ViaGCampi213/a,I-41125Modena,Italy
4ErnstRuska-CentreforMicroscopyandSpectroscopywithElectrons,ForschungszentrumJu¨lich,Ju¨lich52425,Germany
5School of Physics and Astronomy, Glasgow University, Glasgow, G12 8QQ, Scotland, UK
6Institute of Optics, University of Rochester, Rochester, New York, 14627, USA
7DepartmentofPhysics, InstituteforAdvancedStudiesinBasicSciences, 45137-66731Zanjan, Iran
Freeelectronswithahelicalphasefront,referredtoas“twisted”electrons,possessanorbitalangularmomen-
7 tum(OAM)andhenceaquantizedmagneticdipolemomentalongtheirpropagationdirection. Thisintrinsic
1 magnetic moment can be used to probe material properties. Twisted electrons thus have numerous potential
0 applicationsinmaterialsscience. Measuringthisquantityoftenreliesonaseriesofprojectivemeasurements
2 thatsubsequentlychangetheOAMcarriedbytheelectrons.InthisLetter,weproposeanon-destructivewayof
measuringanelectronbeam’sOAMthroughtheinteractionofthisassociatedmagneticdipolewithaconduc-
n
tiveloop. Suchaninteractionresultsinthegenerationofinducedcurrentswithintheloop,whicharefoundto
a
J bedirectlyproportionaltotheelectron’sOAMvalue. Moreover,theelectronexperiencesnoOAMvariations
andonlyminimalenergylossesuponthemeasurement,andhencethenon-destructivenatureoftheproposed
2
technique.
1

]
h Electrons can possess net quantized orbital angular momen- theinvestigationofabeam’sOAMcomponentslong,tedious,
p tum(OAM)whileundergoingfree-spacepropagation[1].The and inefficient. Moreover, the beam’s OAM content, after
t- wavefunction ψ associated with such an electron includes an passingthroughaphasemask,willhaveavaluedifferentfrom
n
exp(i(cid:96)ϕ) term arising from its helical phase-fronts, where (cid:96) thatoftheinitialstate[16].
a
u and ϕ are an integer and the azimuthal coordinate, respec-
q tively. Beams consisting of these “twisted” electrons are re- InthisLetter,weproposeanalternativewayofmeasuring
[ ferredtoaselectronvortex-beams. Differenttechniques,such anelectronbeam’sOAMrelyingonelectricfieldsinducedby

as direct imprinting of a phase variation [2], amplitude [3] time-varyingmagneticfields. Theprincipleofourtechnique
1
v and phase [4] holograms, and magnetic needles [5] have ex- isrelatedtoonewhereamagnetisdroppedthroughaconduc-
9 perimentallybeenshowntogeneratesuchelectronbeams. In tivetube(orring).Thefallingmotionofthemagnetgenerates
9 turn,theseelectronbeamspossessquantizedOAMandcircu- currentswithinthetube,thatinturnproduceamagneticforce
1
latingcurrentdensities J inaplaneorthogonaltotheirprop- counteringthemagnet’sdescent[18–20]. Byusingasimilar
3 ϕ
agation direction. It thus follows that these current densities reasoning,inthenon-relativisticregime,onecancalculatethe
0
. causetwistedelectronbeamstocarryamagneticdipolemo- induced current inside a micro-scale conductive ring due to
1
ment (cid:96)µ in addition to their intrinsic spin magnetic dipole themotionofanOAM-carryingelectrontravelingthroughit.
0 B
7 moment±µB,whereµBistheBohrmagneton[6]. Hence,un- Becausetheelectron’sOAMandmagneticmomentarequan-
1 likeitsintrinsicspin,themagneticmomentassociatedwithits tized,themagneticfieldemanatingfromtheelectronwillalso
: twistedwavefrontisinprincipleunbounded,allowingvalues bequantizedandwillproducediscreteinducedcurrentsinside
v
i ashighas200µB tobeachievedexperimentally[7,8]. Such theringthatcanberelateddirectlytotheOAMcarriedbythe
X
alargeunboundedmagneticmomentmayfindapplicationsin electron.
r materialsscience[9],overcomingthefactthatthegeneration
a
ofspin-polarizedelectronbeamshashistoricallybeenaffected We use a semi-classical approach to describe the interaction
byempiricalandfundamentaldifficulties[10]. Amongfuture between a propagating electron vortex beam and a conduc-
potential applications are investigations related to magnetic tivematerial. Letusconsideranelectronwitharestmassme
dichroisminmaterials[11], thefundamentalnatureofradia- propagatingalongaspecificaxis,e.g. thez-axis,andpossess-
tion [12], exotic physics such as virtual forces [13], and the ingawell-definedcentralkineticenergyEandmomentump0,
interaction of twisted electrons with light beams [14]. Many wherecisthevelocityoflightinvacuum. Undertheslowly-
oftheseexamplesrequiretheanalysisoftheelectronbeam’s varyingamplitudeapproximation, thewavepacketassociated
OAMcontent,aprocessadoptedfromitsopticalcounterparts withthiselectronmustsatisfytheparaxialSchro¨dingerequa-
andthatisusuallycarriedoutbymakingthebeamgothrough tion. Thecorrespondingwavefunctionisquantized,andholds
phase-flattening projective measurements by means of phase aspecificshapebasedonitsinitialprobabilityandphasedis-
holograms[15–17].However,theanalysisofeachOAMcom- tributionconditions. Forinstance,itmaybequantizedinthe
ponentrequirestheuseofadistincthologram,thatcanmake transverse plane as well as in the longitudinal direction [1],
whichyieldsthefollowingwavepacketincylindricalcoordi-
2
natesr,ϕ,z y
y
x 0
ψp,(cid:96),n(r,ϕ,z;t)=uLp,G(cid:96)(r,ϕ;t)uHnG(ζ)ei(p0z−Et)/(cid:126), (1)
x
where uLG and uHG are Laguerre-Gauss and Hermite-Gauss Orest � h 0
modes [21], respectively, in which p and n are positive inte-
p
0
gersdefiningtheelectron’sdistributioninthetransverseplane dh
w
and the longitudinal direction. (cid:96) is an integer number that is O a
lab
associatedwiththeOAMcarriedbythebeamandalsodefines
its transverse distribution. The electron wavepacket’s centre
z
ofmassisdenotedbyζ = z− p t/m while(cid:126)isthereduced
0 e
Planck constant. On account of the electron’s OAM, its rest L
framefour-currentdensityconsistsonlyofascalarandanaz-
imuthalcomponent,accordingtotheexpression
FIG.1. Systeminwhichanelectronvortexbeamwithacentralen-
jαrest =(cid:16)cρ,Jr,Jϕ,Jz(cid:17)=(cid:32)−ceP,0,m(cid:126)e(cid:96)r P,0(cid:33), (2) e(urgLp,Gy(cid:96)(Er,ϕa;ntd)umH0oGm(ζe)netiu(pm0z−pE0t),/(cid:126)w),hipcrhopisaginateitssltohwroeurglhonagictyuldiinndaelrmwoidthe
conductivityσandpermeabilityµ.Therelativemotionofbothenti-
where ρ, J , J , and J correspond to charge density, and tiesresultsinthegenerationofacurrentintheinfinitessimalloopof
r ϕ z
thicknessdh.
radial, azimuthal, and longitudinal current densities, re-
spectively; while P = P(r(cid:48),ϕ(cid:48);z(cid:48)) = |ψ (r,ϕ,z;t)|2 is
p,(cid:96),n
the probability density function of the electron’s position
in its rest frame defined by the coordinates r(cid:48),ϕ(cid:48),z(cid:48); and andsecondkind,respectively[22]. AsdepictedinFig.1,we
−e is the electron charge. The four-current densities in consider such electrons passing through a tube of thickness
the laboratory frame that the electron perceives as traveling w, radius a, conductivity σ and length L. The tube radius is
along the z-direction can then be calculated via an inverse largeenoughtoensurethattheelectron’swavefunctionnearly
Lorentz transformation jα = (Λα)−1jβ , yielding jα = vanishesatitsinnerradius. Inparticular,for p=0modedis-
(cid:16) lab(cid:17) β rest lab
−ceγP,0, (cid:126)(cid:96) P,−γβceP , whereΛα istheLorentztrans- tributionsdefinedbyanarbitrary(cid:96)index,thetuberadiusais
formation mmaetrrix, β = p /(m c) and γβ = (1−β2)−1/2 [22]. chosentobemuchgreaterthantheradiusr(cid:96),i.e. a(cid:29)r(cid:96). The
0 e
conductivetubecanbeconsideredasasequenceofinfinites-
Likewise, a Lorentz boost along the z-axis must also be ap-
imalcircle-loopspositionedatalongitudinaldistancehfrom
plied to the electron’s rest-frame coordinates to express its
thetube’scentre. AspredictedbyFaraday’slawofinduction,
current densities with respect to the laboratory frame coor-
dinates, i.e., xα = (Λα)−1xβ , where xα = (ct,r). One when the twisted electron travels through the tube, its longi-
lab β rest tudinalmagneticfieldinducesaneddycurrentineachofthe
mayassociatethefirst,third,andlasttermsofthefour-vector
tube’sinfinitesimalloops. AccordingtoLenz’slaw,thedirec-
current density with an electrostatic potential V, and the az-
tion of these currents must generate a magnetic field that is
imuthalandlongitudinalvectorpotentialsA andA ,respec-
ϕ (cid:107)
tively. Theazimuthalcurrentdensity J =((cid:126)(cid:96)/m r)Pe gen- opposed to the motion of the electron beam. Its value, how-
eratesamagneticfieldB=∇×Aorientϕedalongteheelecϕtron’s ever,willdependonthetimevariationofthemagneticfluxΦB
througheachloop,i.e.−∂Φ .Neithertheelectrostaticpoten-
propagationdirection,i.e. thez-axis,where∇isthegradient t B
tial V nor the longitudinal vector potential A contributes to
operator, and e is the azimuthal unit vector. The vector po- (cid:107)
ϕ themagneticfluxΦ . OnlytheazimuthalvectorpotentialA
tentialA atagivenpositionrcanthenbeexpresseddirectly B ϕ
ϕ
as a solution to one of Poisson’s equations, namely A (r) = isrelevanttotheanalysis. Duetothecylindricalsymmetryof
µ /(4π)(cid:82) d3r(cid:48)G(r,r(cid:48))J (r(cid:48)), where G(r,r(cid:48)) = |r − rϕ(cid:48)|−1 is theelectron-tubesystem,thevectorpotentialisalsoindepen-
0 ϕ
dentofϕ. Theinducedelectricfieldonthecircle-looplocated
thecorrespondingGreen’sfunction. Theelectron’stransverse
at position h, and hence the induced current, is therefore az-
motionforanyvalueof(cid:96) isthenconsideredasa“localized”
currentloopdefinedbyI = e(cid:126)/(πm w2)e ,asprescribedby imuthal and expressed as Eϕ = −∂tAϕ(a,z − h) where z is
the relation (cid:96)µ = I (πr2e), where r e=0w ϕ√(cid:96)/2 is the radius theelectron’srelativelongitudinalposition. Onecanshow,by
B e (cid:96) (cid:96) 0 means of Ohm’s law, dI = σE (wdh), that the total current
at which an electron is maximally distributed and w is the ϕ
0
withinthetubeinducedbyanelectronwithamagneticdipole
minimumradiusofitsGaussiandistribution.
moment(cid:96)µ isgivenbytheexpression
Thevectorpotentialassociatedwithsuchanazimuthalcur- B
rentcanbeexpressedintheform 3 (cid:32)p (cid:33) (cid:90) L/2 γ2(z−h)dh
I = 0 (σµwa) ((cid:96)µ ) , (4)
µ I η (cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) 4π m B (a2+γ2(z−h)2)5/2
A (r,z)= 0 e uK 2η2 −(u+v)E 2η2 , (3) e −L/2
ϕ πv3/2
where µ is the tube’s permeability. The proportionality of
where u = r2 +r2 +z2, v = 2r r, η = v1/2(u+v)−1/2, and this relation describes the quantization of the induced cur-
(cid:96) (cid:96)
K(.) and E(.) are the complete elliptic integrals of the first rentwithinthetubeduetothediscretenatureoftheelectron’s
3
OAM.ByintegratingEq.(4),ananalyticalexpressionforthis electron can only realistically reach a value on the order of
current can be obtained and is plotted as a function of elec- 10−20 eV when an electron with an OAM (cid:96) = 100 is con-
tronpositionrelativetothecylinder’scentreinFig.2-(a)for sidered (we assume the parameters reported in Fig. 2). Such
various values of electron OAM. As a result, one can con- energy values are obtained when deducing the force applied
ceive a device for OAM measurement by detecting the cor- ontheelectronbythetube. Anotherwayofobtaininginsight
responding quantized current induced inside a tube or a thin on the electron’s energy loss is to calculate the total energy
loop circuit. As shown in Fig. 2-(a), currents of order of contained within the fields generated by the relative motion
of the electron. This energy can be calculated numerically
byfirstfindingthemagneticfieldgeneratedbythecylinder’s
loops of currents using the Biot-Savart law [26] and then in-
tegrating the total energy stored in these magnetic fields and
within the electric fields associated with the currents them-
selves. In particular, this method was employed to produce
theenergydensityplotsfoundinFig.2(b)-(c). Onecansee
that the act of measuring the electron’s OAM has nearly no
effect on the electron itself. Indeed, unlike projective mea-
surementtechniques,wheretheelectron’sphase-frontisflat-
tenedandprojectedonaGaussianmode,theelectron’sOAM
doesnotchangeduringthemeasurement. Forthesereasons,
FIG.2. (a)Theoreticallycalculatedtotalinducedcurrentinacon-
theelectron’smotionthroughthetubeleavesitlargelyunper-
ductivetubebyanelectronvortexbeam. Weassumedthattheelec-
turbed.Upuntilthispoint,weonlyconsideredthecasewhere
tronbeamcarriesOAMof(cid:96) = 1,5and10,andthattheconductive
an electron travels perfectly along the center of the tube. A
tube is made of platinum. Longitudinal cross section of the tube
depicting the relative magnetic energy density generated by its in- simple extension of this analysis reveals that the calculated
duced eddy currents when an electron consisting of a high OAM inducedcurrentsarenotsignificantlyaffectedbybreakingthe
quantum ((cid:96) = 100) is (b) entering the tube and (c) in the middle apparatus’cylindricalsymmetry. Wefurtherdiscusshowthe
ofthetube. Here,weassumedanelectronbeamwithcentralenergy currentsareaffectedbyasymmetriesinexperimentalapparati
E=100keV,andaplatinumtubewithlengthL=20µm,thickness
inthesupplementalmaterial[23].
w=1µmandradiusa=10µm.
ThisnondestructiveapproachtomeasuringOAMmaycre-
ateaconceptualparadox. Onemaymistakenlyarguethatbe-
tens of pA are induced in the loop and could be potentially
cause this measurement leaves the electron’s quantum state
readoutusinganamperemeter(e.g. Tektronix6485Picoam-
(OAM and energy) unchanged, it could challenge the valid-
meter). Therefore, this technique can potentially be used to
ity of a wave-particle duality experiment (quantum comple-
measure OAM values of twisted electron beams. The direc-
mentarity). Consider a double-slit experiment in which, due
tionoftheinducedcurrentadditionallyprovidesinformation
to an electrostatic interaction, the electron wavefunction is
on the sign of the OAM value. Moreover, since the gener- split into two parts |u(cid:105) and |d(cid:105). Both parts are then coher-
atedcurrentisdirectlyproportionaltothematerial’sconduc-
ently recombined and interfere at a screen, as illustrated in
tivity, it follows that by using a more conductive material
Fig. 3(a) [27]. The electron’s state can be described as a su-
one could increase the current by several orders of magni-
perpositionofbothpaths,asifitisinacoherentsuperposition
tude. Though these induced currents are rather short-lived, ofstates|u(cid:105)and|d(cid:105), resultingintheformationofaninterfer-
a combination of fast electronics, optimized cylinder dimen-
ence pattern on the screen. In the case when the electron is
sions, and secondary methods such as autocorrelation tech-
equally likely to take e√ach path, its state may be described
niques can be used to overcome experimental difficulties re- as |ψ(cid:105) = (|u(cid:105)+eiδ|d(cid:105))/ 2 where δ is the relative phase be-
lated to the short interaction between the electron and the
tween the states. The corresponding density matrix is pure
cylinder [23]. Our proposed technique has no influence on ρ:=|ψ(cid:105)(cid:104)ψ|=(|u(cid:105)(cid:104)u|+e−iδ|u(cid:105)(cid:104)d|+eiδ|d(cid:105)(cid:104)u|+|d(cid:105)(cid:104)d|)/2. Inthis
theOAMoftheelectronbeamsincetheelectron’scanonical expression,thetermse−iδ|u(cid:105)(cid:104)d|andeiδ|d(cid:105)(cid:104)u|carrytheinterfer-
OAMisconservedinthepresenceofanexternallongitudinal
ence pattern’s phase information and can therefore be asso-
magneticfield[24]. Theonlypropertythatthemeasurement
ciated with the fringe visibility, which is unity for this ideal
affectsistheenergycarriedbytheelectron[25].Thisisdueto case. The terms |u(cid:105)(cid:104)u| and |d(cid:105)(cid:104)d|, respectively, describe the
thefactthattheinducedcurrentswillcounterthemotionofthe probabilityoffindinganelectroninthe|u(cid:105)or|d(cid:105)path,bothof
electron. Theenergylossduetotheelectron-tubeinteraction
whichareequiprobableeventsforthiscase[28].
is ∆E = −(2πσaw) (cid:16)mp0e(cid:17) (cid:82)−+∞∞ dz(cid:82)−LL//22(cid:16)∂zAϕ(a,z+h)(cid:17)2 dh, Now,considertwoconductivecircuitsintroducedintoeach
whichslightlydeceleratestheelectron. Thisdecelerationcan of the possible paths, as shown in Fig. 3(b). As mentioned
potentially reach a relatively high value, resulting in a large above, these circuits have the capacity to measure an elec-
radiatedelectromagneticpoweremittedfromtheelectron,as tron’sOAMwithminimalenergyloss,allowingforthedetec-
indicated by the Larmor formula. However, due to its very tion of whether an OAM-carrying electron has taken a given
shorttimeofinteractionwiththetube,theenergylostbythe path. Whennoelectrontravelsthroughthecircuit,thecircuit
4
(cid:68) (cid:12) (cid:69)
is in a state |0(cid:105)c. When an electron with an OAM number (cid:96) 0(cid:12)(cid:12)i(cid:96) ,willvarywiththecouplingbetweenthefreeelectron
c
travelsthroughthecircuit, itwillinduceaquantizedcurrent, and the circuit’s state which is determined by various exper-
changing the circuit to a state defined by |i(cid:96)(cid:105) which can be imental parameters. Such parameters, which include the cir-
c
expressedasasuperpositionoftheloop’scurrenteigenstates cuit’sradiusforinstance,canbemodifiedtoprovideavarying
c|no(cid:105)ecffi,ic.ei.en|it(cid:96)(cid:105)dcep=en(cid:80)dnincgn|onn(cid:105)c,vawrhioeuresecxnp=er(cid:68)imn(cid:12)(cid:12)(cid:12)ein(cid:96)(cid:69)taclispaarnamexeptearnssidoen- αcIonecffiocnicelnutsiaoffne,cwtinegptrheesefnrtinagenovnis-dibeisltirtyu.ctivetechniquethat
scribingtheinteractionbetweenthefreeelectronandtheloop can be used to measure the OAM of an electron beam. The
itself. Hence, the system consisting of both circuits can be technique is based on the interaction of the quantized mag-
describedbythetensorproduct|i (cid:105) |i (cid:105) where|i (cid:105) and|i (cid:105) , neticdipolemomentofthetwistedelectronandaconductive
u c d c u c d c
respectively, represent the state associated with the electrical tube. Thebeam’sOAMcomponentsaremeasuredbydetect-
currentgoingthroughcircuitsinthe|u(cid:105)and|d(cid:105)paths.Because ing the quantized induced eddy currents in the tube. These
electronssufferminimalenergylossesandthemethodisnon-
BP destructive.Toillustratethelimitationsofthemethod,wealso
u
(a) | i BP describethepossibilityofusingsuchadeviceinagedanken
quantumexperiment,inwhichtheknowledgeofanelectron’s
presenceisneeded.Doingsowouldresultinreducingthevis-
BP u d ibilityofobservedinterferenceasprescribedbycomplemen-
| i | i
(b)
BP tarity. Aprospectiveextensiontothemethodcouldbeusing
thetubetogenerateradiationwithanapproachsimilartothat
of [30] through the formation of plasmons by introducing a
d
| i discontinuity in the tube, such as the absence of conductive
materialatagivenazimuthalangle. However,thiswouldre-
sult in larger energies being lost by passing electrons. This
method’s minimal electron energy loss is an essential aspect
FIG.3. Proposedexperimentinwhichtheeffectofconductivecir- to its non-destructive nature which, along with the preserva-
tionoftheelectron’soriginalOAM,presentsthistechniqueas
cuitsinanOAM-carryingelectrondoubleslitexperimentisconsid-
ered.(a)Theelectrondoubleslitexperimentinwhichnocircuitsare aviablealternativetomodernprojectivemeasurements.
present. (b) The electron double-slit experiment in which a circuit Acknowledgments: E.K.thanksProfs. GerdLeuchsandIs-
ispresentineachofthepossiblepaths,|u(cid:105)or|d(cid:105),takenbytheelec- raelDeLeonforfruitfuldiscussionsinvolvingthetopic.H.L.,
tronwherethecoefficentα = 0(i.e. theelectronandtheloopsare
F.B, A.S., R.W.B. and E.K. acknowledge the support of the
perfectlycoupled).BPannotationsrefertobiprisms.
Canada Research Chairs (CRC) and Canada Excellence Re-
searchChairs(CERC)Program. R.D.B.thankstheEuropean
ResearchCouncilforanAdvancedGrant.
thecircuithastheabilitytoprovideinformationaboutwhich
paththeelectronhastaken,itprovidesinformationaboutthe
particle nature of the electron, while the presence of fringes
givesinformationaboutitswavenature. Therefore,thepres-
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M.Uchida,PhysicalReviewLetters111,074801(2013). Ammeter
[18] W.Saslow,Am.J.Phys60,693(1992).
[19] C.S.MacLatchy,P.Backman, andL.Bogan,Am.J.Phys61,
1096(1993).
[20] G.Donoso,C.L.Ladera, andP.Mart´ın,Am.J.Phys79,193
(2011).
[21] A.E.Siegman,“Lasers.”MillValley (1986).
[22] J.D.Jackson,ClassicalElectrodynamics(Wiley,1999). FIG. 4. Proposed experimental schematic for the non-destructive
[23] Seesupplementalmaterialforfurtherdetails. measurementofanelectronbeam’sorbitalangularmomentum. An
[24] C. Greenshields, R. L. Stamps, S. Franke-Arnold, and S. M. OAM-carryingelectronbeam,whichpossessesquantizedmagnetic
Barnett Phys.Rev.Lett.113,240404(2014). dipoles,isbeingfocusedthroughaconductiveloop. Themagnetic
[25] Due to the interaction with the very weak induced magnetic dipoleinducesaquantizedcurrentintheloopthatisreadoutbyan
field,thetwistedelectronbeamgainsanegligibleglobalphase ammeter.
of 10−28 rad, e.g. see C. Greenshields, R. L. Stamps, and S.
Franke-Arnold, NewJ.Phys14,103040(2012).
[26] R.Good,EuropeanJournalofPhysics22,119(2001).
[27] Y.Hasegawa,K.Saitoh,N.Tanaka,S.Tanimura,M.Uchida,J.
idealpositionontheaxisofthecylinder.Placinganinteractingelementper-
Phys.Soc.Jpn.82,033002(2013).
pendicularlytoanelectronbeamisusuallynotapracticalissue.Asillustrated
[28] B.-G.Englert,PhysicalReviewLetters77,2154(1996).
inFig.5,whereweconsideredtheparametersemployedinFig.2ofthemain
[29] W.H.Zurek,ReviewsofModernPhysics75,715(2003).
text,asimpleextensionofouranalysisrevealsthattheorderofmagnitudeof
[30] Z. Mohammadi, C. P. Van Vlack, S. Hughes, J. Bornemann, thecurrentsrelatedtotheelectron’sOAMisnotaffectedbytheseoffsets.In
andR.Gordon,OpticsExpress20,15024(2012).
SUPPLEMENTALMATERIALFOR“NON-DESTRUCTIVE
MEASUREMENTOFANELECTRONBEAM’SORBITAL
ANGULARMOMENTUM”
ExperimentalSchematics
We propose the following electron microscope-based apparatus, whose
FIG.5. Maximuminducedcurrentinducedbytheelectronuponits
schematicisprovidedinFig.4,tonon-destructivelymeasureanelectron’s propagationasafunctionoftheelectron’sradialoffsetalongwiththe
orbitalangularmomentum(OAM).TheOAM-carryingelectronbeampasses
corresponding relative ratio between currents induced by electrons
throughanobjectivelensfocusingitthroughtheconductiveloop. Thelatter
definedbydifferentOAMvalues.Thesecurrentscorrespondtothose
isheldbythemicroscope’ssampleholderandalsopossessesathinslitin
whichleadsareplacedandconnectedtoanammeter. Astheelectronpasses intheregionofthecylinderthatwillexperiencethegreatestdropin
throughtheconductiveloop,itinducesquantizedEddycurrentsassociated electromotiveforceuponbreakingitssymmetry.
withitsOAM,givenbyEq. (4)ofthemaintext,whicharethereafterread
outbytheammeter.Shouldthediscontinuityintroducedbythethinslitpro-
videanyobstacleshinderingourproposedmeasurement,analternativeway
ofmeasuringapassingelectron’sOAMwouldbetowrapthecylinderwitha particular,thecurrentsshowninFig.5arefoundbasedontheazimuthalelec-
nanoscalesolenoid.Muchlikethecylinderitself,thesolenoidwouldpickup tricfieldinducedintheregionoftheconductiveloopthatwillexperiencethe
theelectromotiveforcecausedbythepassageoftheelectron,thusresulting lowestinduction. Theratiobetweensuchcurrentsdefinedbypassingelec-
inthegenerationofquantizedcurrentswithinitscoils. tronswithdifferentOAMvaluesremainsconstant. Therefore,breakingthe
apparatus’cylindricalsymmetrywillnotaffectthedetectablenatureofthe
electronsnorwillitchangetherelativeresponsesassociatedwithelectronsof
differentOAMvalues.
AsymmetricalConsiderations
Thoughtheelectron’selectricfieldmayalsoaddunwantedcontributions
tothesecurrents,theydonotvarywiththeelectron’sOAM.Theseunwanted
Wenowconsiderhowtheinducedcurrentsareaffectedbybreakingthe contributionsmayalsobesuppressedbyoptimizingcertainexperimentalpa-
apparatus’cylindricalsymmetry. Weassumethatthemostlikelyimperfec- rameterssuchasthedevice’stransversepositionwithrespecttotheaxisalong
tionisthattheelectrontrajectoryisoffsetinthetransversedirectionfromits whichtheelectronisdisplaced.
6
SuggestedMeasurementMethods (a)
50
ApotentialobstacletoourproposedOAMmeasurementconsistsofthe
rathershortinteractionbetweentheconductiveloopandtheelectron.Inpar-
0
ticular,fortheparametersusedinFig.2ofthemaintext,thedurationofthis
interactioniswithintheorderof0.1ps,thusmakingtherealizabilityofthis
measurementparticularlychallenging. Thoughthefrequenciesofsuchin- -50
ducedcurrentscanpotentiallyreachhighvalues,e.g.intheorderof1013Hz,
thefactthattheyaresignificantlybelowthoseofopticalfrequenciesrefrains -0.2 -0.1 0 0.1 0.2
the occurence of anomalous phenomena such as the generation of surface (b)2.5
)
plasmons. Therefore,theonlyobstaclethatthisshortinteractionintroduces A
irbseestcohaleuuscteihoatnhlle(eene.ggle.ecHtorfyodmnr’easahtsiamurprei)notghfaiptt.acsHasnaogwmeeatvkheerro,sutuhgcehhretahaetarelsoksoompmuseccahelleeesacswtireoirtn.hiMcthsoewrleiootohvpep’rss, urrent (p 21..05
length,alongercylindercanbeusedtomaketheinteractionmorenoticeable. d C 1.0
Likewise,lowerenergyelectronscanalsobeemployedtoachievethesame e
c
effect.Suchelementarymeasurescan,toacertainextent,overcomedifficul- du 0.5
tiesrelatedtothisshortinteraction.Wealsoproposetwoadditionalmethods In 0
toaddressissuesassociatedwiththeelectrons’stimeofpassagethroughthe 0 0.5 1.0 1.5 2.0
loop. (c) 6
Ononehand,theelectron’sOAMcanbemeasuredwithanautocorrelation 5
methodinvolvingtheuseoftwoconductiveloopsinwhichcurrentswillbe
4
inducedbythepassageofdistinctelectrons. Withthesecurrents, onecan
calculatetheelectronbeam’sautocorrelationtrace,C(τ),asafunctionofthe 3
timedelaybetweentheelectrons’passagethroughtheirrespectiveloops.This
2
autocorrelationfunctionisprovidedinEq.(5)
(cid:90) ∞ 1
C(τ)∝ i(cid:96)(t)i(cid:96)(t−τ)dt. (5) 0
−∞ 0 0.5 1.0 1.5 2.0
Thistracewillscalewiththesquareoftheelectron’sOAMasdepictedin Time(ps)
Fig.6whereweprovidetheautocorrelationfunctionscalculatedusingthe
parametersemployedinFig.2ofthemaintext. FIG.7. (a)Currentsinducedincircuitswithinductancesof0,0.05,
Ontheotherhand,methodsto“extend”thelifetimeoftheinducedcur- 0.1,and0.2pHbyanelectroncarryingoneunitofOAM.(b)Zoomed
rentswouldalsomakethemmorenoticeable.Forinstance,aninductorcould inregionof(a)overanextendedtimescale. (c)Currentsinducedin
beaddedtothecircuitconsistingoftheammeterandtheconductiveloop. a circuit with an inductance of 0.1 pH by electrons carrying OAM
AccordingtoLenz’slaw,therateatwhichthemeasuredcurrentincreasesas valuesof(cid:96)=1,5,and10.
theelectronenterstheloopwillsignificantlybedecreased. However,asthe
electronexitstheloop,thecircuit’scurrentwillbemaintainedbytheinductor
inordertominimizetherateatwhichthemagneticfluxthroughitdecreases.
Therefore,thoughaninductorwilllowertheregisteredcurrentstosmalleryet
stilldetectablemagnitudes,itwillalsoextendtheirduration.
Weillustratethisconceptinthefollowing.First,weassumethattheelec- theresistanceisthatassociatedwiththedimensionsandthematerialofthe
tromotiveforceinducedbytheelectron’spassageinsidetheloopisnotaf- cylinderR,andtheinductanceisthatoftheinductorL. Theresultingdiffer-
fectedbytheadditionoftheinductor. Second,tosimplify,weassumethat entialequationdescribingtheinducedcurrent’sI(t)timeevolutionisthus
theEMFinducedinsidetheloopisuniformoveritslengthasopposedtowhat
wascalculatedinthemaintext. Last,weassumethatoursystemconsisting dI
oftheloopandtheinductorcanberepresentedbyaseriesRLcircuitinwhich E(t)−L =IR. (6)
dt
thedrivingvoltageistheelectromotiveforceinducedinsidethecylinderE(t),
WenumericallysolveEq.(6)fortheparametersusedinFig.2ofthemain
textforvariousOAMandinductancevalueswheret=0correspondstothe
momentwheretheelectronishalf-waythroughtheloop. Thesolutionsfor
)ps 1500 thecurrentsinducedbyanelectroncarryingoneunitofOAMandcircuits
.2pA 1000 wsaimtheisnodluuctitoanncsearvealduiessploafy0ed,0i.n0F5,ig0..17,(ba)ndov0e.r2apzHooamreesdhoinwrneginioFnigo.f7F(iag)..7T(hae)
(
n (highlightedinblue)correspondingtoalongertimeinterval.Wecanclearly
o
ati 500 observetheinfluenceoftheinductor’spresenceinthecircuit,i.e.,thecurrent
el varieslessabruptlyupontheelectron’sarrivalanddecaysatamuchslower
orr 0 rateaftertheelectronexitstheloop.Morespecifically,astheelectronenters
c
o theloop,theinductorconsiderablyreducesthecircuit’scurrentinsuchaway
Aut -500 tominimizeanyvariationsinitsmagneticflux.Unlikethecasewherethereis
noinductor,thecurrentkeepsondiminishingastheelectronarrivesmidway
-0.4 -0.2 0.0 0.2 0.4 throughtheloopinordertocounterthefactthattheinducedEMFgoesback
tozero. Thecurrentstartstocomebackuponcetheelectronstartsleaving
Time Delay (ps)
theloopandeventuallystartsdecaying.
FIG.6.Autocorrelationfunctionsassociatedwithelectronscarrying Fig7(c)displaystheinducedcurrentastheelectronleavestheloopfor
OAMvaluesof(cid:96)=1,5,10.Thefunctionswerecalculatedusingthe OAMvaluesof(cid:96)=1,5,and10andacircuitwithaninductanceof0.1pH.
Asinthecasewherethereisnoinductor,thecurrentsremainquantizedand
electronandloopparametersemployedinFig.2ofthemaintext.
proportionaltotheelectron’sOAMvalueinadditiontokeepingobservable
valuesoveralongerperiodoftime.