Spin-to-Orbital Angular Momentum Conversion and Spin-Polarization Filtering in Electron Beams

1 MB·English

by  Ebrahim Karimi

Spin-to-orbital angular momentum conversion and spin-polarization filtering in
electron beams
Ebrahim Karimi,1 Lorenzo Marrucci,1,2,∗ Vincenzo Grillo,3,4 and Enrico Santamato1
1Dipartimento di Scienze Fisiche, Universita` di Napoli “Federico II”,
Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy
2CNR-SPIN, Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy
3CNR-Istituto Nanoscienze, Centro S3, Via G Campi 213/a, I-41125 Modena, Italy
4CNR-IMEM, Parco delle Scienze 37a, I-43100 Parma, Italy
2
We propose the design of a space-variant Wien filter for electron beams that induces a spin
1
half-turn and converts the corresponding spin angular momentum variation into orbital angular
0
momentum (OAM) of the beam itself, by exploiting a geometrical phase arising in the spin ma-
2
nipulation. When applied to a spatially-coherent input spin-polarized electron beam, such device

n cangenerateanelectronvortexbeam,carryingOAM.Whenappliedtoanunpolarizedinputbeam,
a the proposed device in combination with a suitable diffraction element can act as a very effective
J spin-polarization filter. The same approach can be also applied to neutron or atom beams.

9
2
Phase vortices in electronic quantum states have been and this leads us to conceiving the proposed apparatus

] widely investigated in condensed matter, for example in essentially as a space-variant Wien filter. Such appara-
h
p connection with superconductivity, the Hall effect, etc. tuscanbeexploitedforgeneratingvortexelectronbeams
- Only very recently, however, free-space electron beams whenaspin-polarizedbeamisusedasinput. Conversely,
m
exhibiting controlled phase vortices have been experi- ifapurevortexbeamisusedininput,bymeanse.g.ofa
o mentally generated in transmission electron microscope holographic method, one can use the STOC process for
t
a (TEM) systems, using either a spiral phase plate ob- filtering a single spin-polarized component of the input
. tainedfromastackofgraphitethinfilms[1],ora“pitch- beam, as we will show further below.
s
c fork” hologram manufactured by ion beam lithography Let us consider an electron beam propagating in vac-
i
s [2, 3]. In a cylindrical coordinate system r,φ,z with uumalongthez-axisandcrossingaregionofspacelying
y the z axis along the beam axis, a vortex electron beam betweenz =0andz =Linwhichitissubjecttoelectric
h
is described by a wavefunction having the general form and magnetic fields E = Φ and B = A, where
p −∇ ∇×
ψ(r,φ,z,t) = u(r,z,t)exp(iℓφ), where ℓ is a (nonzero) Φ and A are the scalar and vector potentials, respec-
[
integerandu vanishesatr =0. As inthe caseofatomic tively. Inthenon-relativisticapproximationandneglect-

1 orbitals, ℓ is the eigenvalue of the z-component orbital ingallCoulombself-interactioneffects(smallchargeden-
3v angularmomentum(OAM)operatorLˆz =−i∂φ(inunits sity limit), the electron beam quantum propagation and
6 ofthe reducedPlanckconstant¯h),andthereforeanelec- spinevolutionaregenerallydescribedbyPauli’sequation
0 tronbeamofthisformcarriesℓ¯hofOAMperelectron[4].
6 Therecentexperimentsonelectronvortexbeamswerein- i¯h∂ ψ˜= 1 ( i¯h∇ eA)2+eΦ B µˆ ψ˜ (1)
1. spiredbythesingularopticsfield, inwhichsimilarphase t 2m − − − ·
(cid:20) (cid:21)
0 or holographic tools have been used in the last twenty
2 years (see, e.g., [5] and references therein). In optics, a where ψ˜ is the spinorial two-component wave-function
1
recently introduced alternative approach to the genera- of the electron beam, e = e and m are the electron
: −| |
v tion of vortex beams is based on the “conversion”of the charge and mass, ∂ is the derivative with respect to the
t
i angularmomentumvariationoccurringinaspin-flippro- time variable t, µˆ = 1gµ σˆ is the electron magnetic
X −2 B
cess into orbital angular momentum of the light beam, moment, with µ = ¯he/2m the Bohr’s magneton, g
r B | | ≃
a when the latter is propagating through a suitable spa- 2 the electron g-factor, and σˆ = (σˆ ,σˆ ,σˆ ) the Pauli
x y z
tially variant birefringent plate [6, 7]. In this paper, we matrix vector.
propose that a beam of electrons traveling in free space As a first step, we consider the simpler case in which
undergoes a similar “spin-to-orbital angular momentum the electric and magnetic fields are taken to be uniform,
conversion”(STOC)processinthepresenceofasuitable lyinginthetransverseplanexy,andarrangedasinstan-
space-variant magnetic field. The same approach may dard Wien filters [8, 9], i.e. perpendicular to each other
work also for neutrons, or any other particle endowed and balanced so as to cancel the average Lorentz force,
with a spin magnetic moment (e.g., atoms, ions). Of i.e. E = B p /m where E and B are the electric and
0 0 c 0 0
course,in the case of electrons,as for other chargedpar- magnetic field moduli, andp the averagebeam momen-
c
ticles, the magnetic field, besides acting onthe spin, will tum. The magneticfieldB isalsotakentoformanarbi-
also induce forces that must be compensated in order to traryangleαwiththeaxisxwithinthexyplane. Forthis
avoid strong beam distortions or deflections. Such com- case, we solved the full Pauli’s equation in the paraxial
pensation may be obtained by a suitable electric field, slow-varying-envelope approximation for an input beam
2
havinga gaussianprofile and anarbitraryuniform input
spinstate ψ =a +a ,where and denote
in 1 2
| i |↑i |↓i |↑i |↓i
astateforwhichthespinisparallelorantiparalleltothe
z axis, respectively. The complete expression of the re-
sultingspinorialwave-functionisgiveninthesupplemen-
HaL HbL HcL HdL
talmaterial(SM)[10],whileherewesummarizethemain
findings. Thebeampropagationbehaviorcorrespondsto
the well known astigmatic lensing in the plane perpen-
dicular to the magnetic field. More precisely, the beam
undergoes periodic width oscillations, with a spatial pe-
riod Λ = πR , where R = p /(eB ) is the cyclotron
2 c c c 0
| |
radius. This lensing phenomenon is also predicted by a
FIG. 1. (color online) Electric (upper panels) and magnetic
classical ray theory, when properly taking into account (lowerpanels)fieldq-filtergeometriesfordifferenttopological
the effect of the input fringe fields [10]. The output spin charges: (a) q =−2, (b) q =−1, (c) q =1, and (d) q =+2;
state is instead givenby the following generalexpression in all cases β=π/2.
(Eq. 4 in SM [10])
ψ =a cos(δ/2) +sin(δ/2)ieiα
| iout 1 |↑i |↓i ometries, which do not require to have a field source at
+a(cid:2)2 cos(δ/2) +sin(δ/2)ie−iα(cid:3) , (2) r = 0. For example, the q = 1 case corresponds to the
|↓i |↑i −
standard quadrupole geometry of electron optics, while
whereδ =4πL/Λ (cid:2)andΛ =4πR /g 2Λ . This(cid:3)spino-
1 1 c ≃ 2 q = 2 corresponds to the hexapole one. Wien filters
rialevolutioncorrespondstotheclassicalLarmorpreces- −
withsuchgeometrieshavebeenalreadydevelopedinthe
sion of the spin with spatial period Λ /2, δ being the
1 past for the purpose of correcting chromatic aberrations
total precession angle. However, in addition to the spin
[12,13]. Moreover,inhomogeneousWienfiltersincluding
precession, Eq. (2) predicts the occurrence of wavefunc-
severalmultipolartermshavebeenalsoconsideredforthe
tion phase shifts. In particular, for a or input
|↑i |↓i purpose of spin manipulation, with the added advantage
state and a total spin precession of exactly half a turn,
ofobtainingastigmaticlensingbehavior[14]. Apossible
i.e. δ = π or L = Λ /4, the wavefunction acquires a
1 design of the q = 1 filter with quadrupolargeometry is
phase shift given by α+π/2, where α is the magnetic −
± shown in Fig. 2. In such non-uniform field geometry we
field orientation angle mentioned above and the sign
±
is fixed by the input spin orientation (+ for and
|↑i −
for ). These phase shifts can be interpreted as a spe-
|↓i
cialcase of geometric Berry phases arisingfromthe spin
manipulation [11].
Let us now move on to the case of a spatially vari-
ant magnetic field. We consider multipolar transverse
field geometries with cylindrical symmetry, described by
the following expression for the magnetic field (with
the vector given in cartesian components): B(r,φ,z) =
B (r)(cosα(φ),sinα(φ),0),wherethe angleαisnowthe
0
following function of the azimuthal angle:
α(r,φ,z)=qφ+β (3)
FIG. 2. (color online) Electrodes and magnetic poles geom-
etry of a q-filter with q = −1 (quadrupole), seen in cross-
where q is an integer and β a constant. Clearly, such
section (a)andinthree-dimensionalrendering(b). Thefilter
a field pattern must have a singularity of topological
length is set to 50 cm. In panel (a), the calculated vector
chargeq atr =0. Inparticular,by imposing the vanish- potential Az (in false colors), which is also roughly propor-
ing of the field divergence, we find that the radial factor tional to the electric potential, and the projection on the xy
B (r) r−q,i.e.,the fieldvanishesontheaxisforq <0, plane of the simulated electron ray trajectories for 100 keV
0
while i∼t divergesfor q >0. Inthe latter case,there must energy are also shown (see SM for details about the simula-
tions[10]),foraring-shaped inputbeam withradiusr=100
beafieldsourceontheaxis. Wecall“q-filter”abalanced
µm (ray color grows darker for increasing z). In panel (c)
Wienfilterwhosemagneticfielddistributioninthebeam
a zoomed-in view of the central region. The magnetic field
transverseplaneobeysEq.(14). Theelectricfieldwillbe at r needed to obtain the tuning condition δ = π is 3.5 mT,
takentohaveanidenticalpattern,exceptforalocalπ/2 with a corresponding electric field of 575 kV/m. These are
rotation,soastobalancethe Lorentzforce. Someexam- obtainedwithanelectrodepotentialdifferenceof≈9kVand
ples of such q-filter field distributions are shown in Fig. a magnetization of 135 A/mm. The fields need to be set to
1. We are particularly interested in the negative q ge- thedesign-values with a precision of 1 part in 104.
3
cannot solve analytically the full Pauli’s equation. How- the introduction, and it occurs for q = 1 because this
ever, the beam propagation is already well described by geometryisrotationallyinvariantandthereforenoangu-
classicaldynamicsandcanbederivedeitheranalytically, lar momentum can be exchanged with the field sources
using a power-expansion in r [14], or by numerical ray in the filter. In the q = 1 case, the input spin still con-
6
tracing. Intheformercase,wefindthattofirstorderthe trols the sign of the OAM variation, but the total beam
q-filterforq =0isalreadystigmatic,i.e.,itpreservesthe angular momentum is not conserved and some angular
6
beam circular symmetry. Only second-order corrections momentum is exchangedwith the field sources. We note
introduce aberration effects [14]. This behavior is con- that this OAM variation can be also explained as the
firmed by our numericalray-tracingsimulations (see SM effect of the spin-relatedmagnetic-dipole force acting on
for details[10]), whichshowrelativelyweakhigher-order theelectronswithinthemagneticfieldgradients,asmore
aberrations (see Fig. 2). It should be noted that these fully discussed in the SM [10].
simulationshavebeenperformedforrealisticvaluesofthe The “tuning” condition L = Λ /4 or δ = π can be
1
electric and magnetic fields, as required to obtain a spin achievedinprincipleforagivenradiusrbyadjustingthe
precessionofhalfaturnacrossapropagationdistanceof strength of the magnetic and electric fields or the device
50 cm at a beam radius of 100 µm. These calculations length L. Since the precession angle δ is r-dependent,
are expected to reproduce very well the electron density however,this tuning condition can be applied to the en-
behavior(andspinprecession)aswouldbeobtainedfrom tire beam only if it is shaped as a ring, i.e. with all elec-
Pauli’s equation. However, Pauli’s equation predicts an tron density peaked at a given radius r. Vortex beams
additionalpurelyquantumphenomenon,namelythegeo- with OAM ℓ = 0 typically have a doughnut shape, so
6
metricphasealreadydiscussedabove. Inasemi-classical theyapproximatearingfairlywell. Ontheotherhanda
approximation,the geometricphasewillstillbe givenby gaussian input beam (with ℓ=0) cannot be fully trans-
α+π/2, where α is however now position-dependent formed, as δ = 0 at r = 0, where the beam has the
±
and given by Eq. (14). More specifically, neglecting the maximumdensity. Insuchcases,only afractionf ofthe
aberrations, each possible electron trajectory within a electrons would be converted.
beam is straight and parallel to the z axis. Therefore, So far we have assumed a spin-polarized input beam.
the electrons travelling in a given trajectory will expe- However, high brightness (i.e., spatially coherent) spin-
rience a constant magnetic field of modulus B0(r) and polarized electron beams, suitable for high-resolution
orientation α(φ). The set of all electrons travelling at TEM applications, are not so easily available. State-of-
a given radius r will then undergo a uniform spin pre- the-art spin-polarized sources may achieve a brightness
cession by angle δ(r) and, if δ(r) = π (i.e., for a spin of 107 A cm−2 sr−1 and a polarization purity of up to
half-turnrotation),they willalsoacquireaspace-variant 90% [15] (and the source decays with time due to laser-
geometric phase givenby α(φ)+π/2= qφ β+π/2, induced damage). It is interesting then to analyze the
± ± ±
with the sign determined by the input spin orienta- effect of the q-filter on a initially unpolarized electron
±
tion. Inotherwords,the outgoingwavefunctionacquires beam, having arbitrary initial OAM ℓ. Such input can
a phase factorexp(iℓφ), with ℓ= q,correspondingto a be simply viewed as a statistical mixture in which 50%
±
vortex beam with OAM q¯h. In a quantum mechanical of the electrons are in the state ,ℓ and 50% in the
notation,spin-polarizedi±nputelectronswithgiveninitial state ,ℓ . After passing throug|h↑a tiuned q-filter, the
OAM ℓ passing through a q-filter undergo the following beam|b↓ecoimes a 50-50 mixture of states ,ℓ+q and
transformations: ,ℓ q , for which spin and OAM are cor|re↓lated(iif the
|↑ − i
,ℓ cos(δ/2) ,ℓ +ieiβsin(δ/2) ,ℓ+q q-filter is not tuned, the fraction of converted electrons
|↑ i→ |↑ i |↓ i decreasestof/2ineachspinorbitstate,andtherewillbe
,ℓ cos(δ/2) ,ℓ +ie−iβsin(δ/2) ,ℓ q (4)
|↓ i→ |↓ i |↑ − i a residual 1 f fraction of electrons in states ,ℓ and
− |↑ i
where the ket indices now specify both the spin state ,ℓ ). Aswediscussnow,thisspin-OAMcorrelationcan
|↓ i
(arrows) and the OAM eigenvalue. be exploited for making an effective electron beam spin-
Equations (4) show that, in passing through the q- polarizationfilter. Suchfilterrequiresfourbasicelements
filter, a fraction f = sin2(δ/2) of the electrons in the in sequence (as shown in Fig. 2 of SM [10]): (i) anOAM
beamwillfliptheirspinandacquireanOAM q¯h,while manipulation device, such as a fork hologram [2, 3], to
±
theremainingfraction1 f =cos2(δ/2)willpassthrough set ℓ =0; (ii) a q-filter with q =ℓ, generating a mixture
− 6
the filter with no change. When L = Λ /4, then δ = π of electrons in states ,2ℓ and ,0 ; (iii) a free propa-
1
|↓ i |↑ i
and all electrons are spin-flipped and acquire the cor- gation(or imaging) stage that allows these two states to
responding OAM. In the specific case q = 1, the spin developdifferentradialprofilesby diffraction,becauseof
angular momentum variation for the electrons undergo- their different OAM values; (iv) a circular aperture for
ing the spin inversion is exactly balanced by the OAM finally separating the two states. In particular, in stage
variation, so that the total electron angular momentum (iii) state ,2ℓ will acquire a radial doughnut distribu-
|↓ i
remainsunchangedincrossingthefilter. Thisisthepure tion,asinLaguerre-GaussianmodeswithOAM2ℓ,which
“spin-to-orbitalconversion”STOC process mentionedin vanishes close to the beam axis as r2ℓ, while state ,0
|↑ i
4
will become approximately gaussian, with maximum in- effectoffringefieldscanbeneglectedifthelength-to-gap
tensity at the beam axis, as shown in Figs. 3a,b. There- ratio of the filter is large enough [10, 14].
fore, a suitable iris (Fig. 3c) will select preferentially the Wenotethatthespin-filterapplicationdiscussedabove
electrons in the fully polarized state ,0 . An opti- is a new counterexample of the old statement by Bohr
| ↑ i
cal OAM sorter exploiting a similar approach has been thatfreeelectronscannotbespin-polarizedbyexploiting
demonstratedrecently[16]. Aspecificcalculationforthe magnetic fields, due to quantum uncertainty effects [17–
19]. The reasonwhy we can overcome Bohr’s arguments
is essentially that we do not use the magnetic forces di-
rectly to obtain the separation, but take advantage of
quantum diffraction itself, as also proposed recently in
Ref. [20] (see SM for a fuller discussion [10]).
In conclusion, we believe that the q-filter device de-
scribedinthispapercanbemanufacturedrelativelysim-
ply, for applications in standard electron beam sources
suchasthoseusedinTEMsorotherkindsofelectronmi-
croscopes. Incombinationwithcurrentfield-effectunpo-
larizedelectronsources,suchfilter might provide a spin-
polarized source with a brightness 109 A cm−2 sr−1,
∼
about two orders of magnitude higher than the current
state ofthe art. This result, if it willbe provedpractical
enough, may open the way to a spin-sensitive atomic-
scale TEM, e.g. suitable for investigating complex mag-
netic order in matter or for spintronic applications.
We acknowledge the support of the FET-Open Pro-
gramwithinthe7th FrameworkProgrammeoftheEuro-
pean Commission under Grant No. 255914,Phorbitech.
FIG. 3. (color online) Electron beam profiles in the far field
of the q-filter for the ℓ = 0 component (panel a) and the
ℓ=2component(panelb),andapossible discriminatingiris APPENDIX: SUPPLEMENTARY MATERIAL
radius(panelc)r=wtobeusedtoseparatetheminorderto
makeaspin-polarization filter. Panel (d)shows theintensity
Gaussian electron beam in homogeneous Wien filter
profiles of the same components (ℓ = 0 - blue dot-dashed
line, ℓ = 2 - green dashed line) and of the possible residual
ℓ = 1 component for an untuned q-filter (red solid line). w Let us consider an electron beam propagating in vac-
is the gaussian beam waist radius in the far-field plane. A uum along the z-axis, subject to transverse and or-
realisticvaluefortheirisradiusisoftheorderofseveraltens thogonal electric and magnetic fields E and B lying
of microns (obtained by setting the aperture some distance in the xy plane. In this Section, we assume the two
after thefocal plane of thesecond condensor)
fields to be spatially uniform, as in standard Wien fil-
ters, but we will release this assumption later on. If
case q = ℓ = 1 and an iris radius equal to the beam the magnetic field direction makes an angle α with
| | | |
waist w in the “far field” yields a transmission efficiency the x-axis, we may write E = E (sinα, cosα,0) and
0
ofourdeviceof55.5%(notincludingthelossesarisingin B = B (cosα,sinα,0). As scalar and−vector poten-
0
theOAM-manipulationdevice)andapolarizationdegree tials, we may then take Φ = E (xsinα ycosα) and
0
(cIu↑rr−enIt↓s),/o(If↑+9I7↓.)5,%w.hHerieghIe↑r,↓daergeretehseotfwpooslpariniz-aptoiloanriczaend rAela=tiBvi0st(i0c,0a,pypcroosxαim−atxiosninaαn)−d, rnesepgeleccttivinegl−y.alIlnCthoeulnoomnb-
∼
be obtained at the expense of the efficiency by reducing self-interaction effects (small charge density limit), the
theirisdiameterorbyemployinghigherq values(orvice electron beam propagation is described by Pauli’s equa-
versa). It is worth noting that this apparatus works also tion
withapartiallytunedq-filter,asinthiscasetheunmod-
ified electron beam component is left in the initial OAM i¯h∂ ψ˜= 1 ( i¯h∇ eA)2+eΦ B µˆ ψ˜ (5)
t
state ℓ=q andthereforeis alsocut-awayby the iris. An 2m − − − ·
(cid:20) (cid:21)
untuned q-filter will however have an efficiency reduced
by the factor f =sin2(δ/2). The aberrations introduced where ψ˜ is the spinorial two-component wave-function
by the q-filter, even if left uncorrected, might also affect of the electron beam, e = e and m are the electron
−| |
its efficiency but not its main working principle, as this charge and mass, ∂ is the derivative with respect to the
t
is basedonthe vortexeffect, whichis protectedby topo- time variable t, µˆ = 1gµ σˆ is the electron magnetic
−2 B
logicalstability. Finally, the possible spin depolarization moment, with µ = ¯he/2m the Bohr’s magneton, g
B
| | ≃
5
2 the electron g-factor, and σˆ = (σˆ ,σˆ ,σˆ ) the Pauli In addition to the spin dynamics, from Eq. (12), we
x y z
matrix vector. find that the wavefront complex curvature radius q (z)
2
We seek a monochromatic paraxial-wave solution in the direction perpendicular to the magnetic field B
with average linear momentum p and average energy changesperiodically with spatialperiod Λ . These oscil-
c 2
E = p2/2m in the form ψ˜(x,y,z,t) = exp[i(p z lations correspond to the well known astigmatic lensing
c c c −
E t)/¯h]u˜(x,y,z), where u˜(x,y,z) is taken to be a slow- effect of the Wien filter.
c
varying envelope spinor field. Inserting this ansatz in
Eq.(5)andneglectingsmalltermsinthe z derivativesof
Space-variant Wien filter and ray-tracing simulations
u˜, we obtain the paraxial Pauli equation
i e2 2m In this Section, we consider “q-filter” field geometries
∂ u˜= 2 A2+ B µˆ u˜ (6)
z 2kc (cid:18)∇⊥− ¯h2 ¯h2 · (cid:19) withcylindricalsymmetry,describedbythefollowingex-
pression for the magnetic field (with the vector given in
where 2 =∂2+∂2 is the Laplacianin the beam trans- cartesian components):
∇⊥ x y
verse plane and k = p /¯h is the De Broglie wavevec-
c c
B(r,φ,z)=B (r)(cosα(φ),sinα(φ),0) (13)
tor along the beam axis. In deriving Eq. (6) we set 0
E0 = pcB0/m, so as to cancel out the average Lorentz where the angle α is now the following function of the
force on the beam. azimuthal angle:
Equation (6) determines the evolution of the spinor
field u˜ along the axis z. This equation can be solved α(r,φ,z)=qφ+β (14)
analytically for the case of a gaussian input beam, as
where q is an integer and β a constant. As discussed in
given by u˜(r,φ,0) = a˜exp[ r2/w2] for z = 0, where
− 0 the main paper, such a field pattern must have a singu-
w is the beam waist and a˜ = (a ,a ) the input spinor,
0 1 2 larity of topological charge q at the beam axis r =0. In
corresponding to an arbitrary spin superposition ψ =
particular, by imposing the vanishing of the field diver-
| i
a1|↑i+a2|↓i. A straightforwardcalculation shows that gence,wefindthattheradialfactorB0(r) r−q,i.e.,the
the solution is given by ∼
field vanishes on the axis for negative values of q, while
it diverges for positive values of q. In the latter case,
u˜(r,φ,z)=G(r,φ,z)Mˆ(z)a˜ (7)
there must be a field source on the axis. The electric
where Mˆ(z) is the matrix field will be taken to have an identical pattern, except
for a local π/2 rotation, so as to balance everywhere the
cos2πz ie−iαsin2πz Lorentz force. We are particularly interested in the neg-
Mˆ(z)= ieiαsinΛ12πz cos2πzΛ1 (8) ative q geometries, which do not require to have a field
(cid:18) Λ1 Λ1 (cid:19) source at the beam axis. For example, the q = 1 case
−
whereΛ =(2π¯h2k )/(mgµ B )andthescalargaussian correspondsto the standard quadrupole geometry, while
1 c B 0
q = 2 corresponds to an hexapole one.
factor G(r,φ,z) is given by −
Theelectronpropagationprobleminsuchnon-uniform
fieldcanonly be solvedapproximatelyor numerically. A
G(r,φ,z)=g (z)eikcr2(cid:18)cos22q1(α(z−)φ)+sin22q(2α(z−)φ)(cid:19) (9) second-order geometrical optics solution is reported in
0
Ref. [? ], which includes also a detailed analysis of spin
with precession but without considering the geometric phase
effect (as well as the magnetic dipole forces). This anal-
−iarctan z −iarctan Λ2 tanπz
√πzRe 2 (cid:16)zR(cid:17) e 2 (cid:16)πzR Λ2(cid:17) ysis shows that, to first order, a balanced Wien filter
g (z)= (1,0)
0 1 having quadrupole or hexapole geometry but vanishing
(zR2 +z2)14 π2zR2 cos2 Λπz2 +Λ22sin2 Λπz2 4 dipole term (as in our q-filter) is already stigmatic, i.e.
q (z)=z iz , (cid:16) (cid:17) (11) it preserves the cylindrical symmetry without need for
1 R
−
further corrections. The only beam lensing or distortion
q (z)= Λ2 Λ2sin(Λπz2)−iπzRcos(πΛz2) , (12) effectsenterashigher-orderaberrations(i.e., the Gterm
2 π Λ2cos(Λπz2)+iπzRsin(πΛz2)! in Eq. (31) of Ref. [? ]).
Tofurtheranalyzethebehaviorofourq-filter,wehave
where zR = 21kcw02 and Λ2 = πeh¯Bk0c =gΛ1/4≃Λ1/2. performed ray-tracing simulations of the electron propa-
The matrix Mˆ(z) given in Eq. (8) corresponds to Eq. gation for a quadrupole geometry (q = 1). The mag-
−
2 in the main paper and describes the magnetic-field- neticandelectricfieldshavebeencalculatedusingCOM-
induced Larmor spin precession during propagation and SOL finite-elements simulation (www.comsol.com). The
the associated geometric phases discussed in the main ray-tracingroutines therein containedhave been used to
paper. The precession length Λ corresponds to two full assesstheshapeofthebeamattheexitofthefilterstart-
1
spin rotations. ing from an input beam having 100 keV of enery that is
6
shapedasaring,witharadiusrof100µm. Themagnetic Magnetic dipole forces and angular momentum
field at r needed to obtain the tuning condition δ =π is
3.5 mT, with a correspondingelectric field of 575kV/m. Wenowevaluatetheeffectsoftheforceassociatedwith
Theseareobtainedinourgeometrywithanelectrodepo- themagneticdipoleoftheelectronsinthe magneticfield
tential difference of 9 kV and a magnetization of 135 gradient of the Wien filter. This is the force used in
≈
A/mm. The fields need to be set to the design-values Stern-Gerlach experiments and the only spin-dependent
with a precision of 1 part in 104. The electron velocity force acting on the electrons, so it is important to assess
to be used as initial conditions in the simulations must its effects in detail. At any given point inside the filter,
be obtained from a model of the fringe fields. For the assumingthatthe magnetic dipole ofthe electronis µ=
simplest description of the input fringe fields, i.e. the gµ S, where S is the corresponding spin vector, the
B
so-calledsharpcut-offfringingfield(SCOFF)model,the a−ssociated magnetic force is
transverse velocity components remain constant and nil,
i.e. vx = vy = 0, while the z-component vz(x,y) “just F = (µ B) (16)
∇ ·
inside” the filter is given by the following expression:
where the gradient must be taken while keeping µ con-
eΦ(x,y) eA (x,y)
z
vz(x,y)=v0 =v0 (15) stant, i.e. acting only on the magnetic field coordinates.
− mv − m
0
For our classicalestimate, the magnetic dipole evolution
where v is the velocity outside the filter. This condi-
0 can be described by the precession equation
tion is derived from the conservation of energy and also
ensures the conservation of the canonical momentum of dµ g e
= | |B µ (17)
the electrons entering the Wien filter and hence the con- dt 2m ×
servation of their wavefront orientation. The results of
Assuming a starting magnetic moment parallel to the
z axis, the magnetic dipole will rotate around the local
magnetic field direction. In the approximation in which
aberrations are neglected, each electron travels along a
parallel ray with constant transverse coordinates, and
hence sees a uniform magnetic field, given by Eq. (13).
Therefore, the magnetic dipole at any given point inside
the filter will be given by the following expression:
µ(r,φ,z)=µ [ sinδ(r,z)sinα(φ),sinδ(r,z)cosα(φ),
B
−
cosδ(r,z)] (18)
where δ(r,z) is the total spin precession angle at z, for
a trajectory at radius r. Taking the scalar product be-
tweenµandB andthenthegradientwithrespecttothe
coordinates of B only, we find that the magnetic dipole
force is only directed along the azimuthal direction and
FIG. 4. Projection on the xy plane of the simulated trajec-
toriesof theelectrons startingfrom acircleofradiusr=100 given by
µm. The trajectories are shown with a color growing darker
as z is increased. F =φˆqµBB0(r) sinδ(r,z) (19)
r
ray-tracingsimulationsarereportedinFig.2ofthemain
paper and in Fig. 4 of the present manuscript, which where φˆ denotes the unit vector along the azimuthal di-
show how a circle of electron positions representative rectionandwehaveusedEq.(14)fortakingtheφderiva-
of the input beam becomes deformed during the prop- tive.
agation into a quadrupole-lobed shape, due to second- Equation(19) can be used for two purposes. First, we
or higher-order aberrations. For optimizing the perfor- can verify that the ray deflection inside the Wien filter
mances of our proposeddevice, in particularin the spin- due to this force is negligible. Indeed, integrating the
polarization filter application, these aberrations should forceinthetimeT neededforafullπrotationofthespin
probably be compensated by additional electron optics (or spin flip), we obtain the overall momentum change
(e.g., octupole electrostatic lenses). Nevertheless, even
T π dδ −1
for no aberration compensation, the vortex effect giving
∆p = F (t)dt= F (δ) dδ
φ φ φ
rise to the radial separation of the two spin components Z0 Z0 (cid:18)dt(cid:19)
discussedinthespin-polarizationapplicationisexpected 2m π 2q¯h
= F (δ)dδ = (20)
to be preserved, owing to its topological stability. g eB φ gr
| | 0 Z0
7
We can see from this equation that the momentum vari- component of the electric field plays a role, by ensuring
ation is of the same order as the quantum uncertainty the validity of the condition (15). The z component of
in momentum of the electron wavefunction, taking into the magnetic fields does not affect the ray propagation,
account that the position uncertainty is of the same or- since it exerts a vanishing Lorentz force. However, the
der of the beam size r (as it is also expected based on latter may affect the spin precession. If we assume that
Bohr impossibility theorem, as discussed further below). theinputelectronshaveaspinparalleltothez axis,then
Thismomentumchangeproducesnegligibleeffectsinthe the z component of the fringe magnetic field does noth-
electronpropagationacrossthefilter,althoughitinstead ing. However, the spin may be somewhat rotated away
becomes relevant in the far field. Indeed, it is precisely from the z axis by the transverse fringe magnetic field,
this force that produces the orbital angular momentum and in this case the longitudinal component will induce
change at the basis of our spin selection. Indeed, we will somenon-uniformprecessionaroundthe z axisthatmay
now show that the torque associated with the magnetic partly depolarize the beam. This effect will however be
dipole force is that responsible for the OAM change of proportional to the time spent by the particles in the
the electrons. The z-component torque of this force is fringe field region, and hence will be negligible in the
given by large length-to-gap ratio limit.
We may also consider the idealized limit of an ex-
M =(r F) =qµ B (r)sinδ =q(µ B)
z × z B 0 × z tremely thin fringe field region in which the electric and
dS
z magnetic fields vary rapidly from the vanishing values
= q (21)
− dt they have away from the Wien filter to the values they
whereS isthez componentoftheelectronspinangular acquirewellinsidethefilter,whichthenremainperfectly
z
momentumS. Since this torque acts onthe OAMofthe constant. This is the SCOFF limit mentioned above. In
electrons L , we find the following angular momentum particular, in the SCOFF limit the vector potential A
z
coupling law: must go from zero to the value A = [0,0,Az(x,y)] de-
scribing the multipolar magnetic field inside the filter.
dL dS
z = q z (22) This rapid transition does not give rise to any magnetic
dt − dt field, because it is curl-free. Therefore,the SCOFF limit
Once integrated for the entire flight time, corresponding has vanishing magnetic fringe fields and therefore abso-
to a full spin flip, this coupling law returns the same lutely no effect on the spin, nor on the lensing behavior
resultreportedinthe mainpaper forthe OAMvariation of the filter. On the other hand, in the SCOFF limit the
induced by the spin flip. electricpotentialmust gofromzeroto the nonzerovalue
This correspondence proves that the effect, and the Φ(x,y) describing the multipolar electric field inside the
only effect, of the magnetic dipole force acting on the filter. This in turn implies the presence of a very strong
electrons is the change of OAM already considered by longitudinalelectricfringefieldthatisresponsibleforthe
including the geometric phase shift in the wavefunction changeof velocitygivenby Eq. (15). We stressthat this
emerging from the filter. Therefore, the only significant idealizedSCOFFdescriptionofthefringefieldsneedsnot
effect of this force appears in the far field and is exactly being realistic, but it is sufficient to identify the inelim-
the effectthatwe proposetoutilize toseparateelectrons inableeffectsthatthefringefieldimposetotheelectrons,
according to their spin. and therefore the possible fundamental limitations that
may arise from them.
Fringe field effects
Layout of a spin-polarized TEM microscope
The actual fringe fields depend on the electrodes and
magnetic poles geometry. It is however safe to assume We report in Fig. 5 the sketch of a possible TEM mi-
that the fringe fields extend only for a longitudinal dis- croscopeincorporatingtheproposedspin-polarizationfil-
tance along the z axis that is comparable to the trans- ter based on the q-filter. Such a device could be used to
verse aperture of the Wien filter, and therefore much performspin-sensitiveTEMmicroscopyonmagneticma-
smaller than the filter length, in the large length-to-gap terials or spintronic devices. It includes the four stages
ratio limit. Moreover, the transverse components of the discussed in the main paper, i.e. (i) the fork hologram
fringe fields can be taken to have exactly the same mul- to set the initial OAM, (ii) the q-filter to couple OAM
tipolargeometryasthefieldsinsidethe Wien,andhence to spin, (iii) an imaging stage to go in the Fourier plane,
will give rise to similar effects both in lensing and spin (iv) an aperture to select a single spinorbit component,
precession. Theiroveralleffectisthereforeonlytochange aswellasthe additionalelectronopticsneededforimag-
the effective length of the filter. However, in addition ingandcollimating the beam. The input apertureofthe
there will also be a longitudinal z component that must system is defined by the fork hologram mask and has a
be taken into account. In the lensing effect, only the z typicalradius of few microns [2]. The output iris needed
8
shown in Fig. 3 of the main paper, has been evaluated
usingafullyquantummechanicaltreatment. Thediffrac-
tion effects are obviously important in the propagation
from near to far field, and in particular in determining
thepresenceorabsenceofdestructiveinterferenceeffects
at the vortex position that are essential in our proposed
scheme.
One may wonder if we can legitimately ignore the
diffraction effects in the other stages of our proposed
setup. Forthepropagationthroughtheq-filter,wecould
assessdirectlythe roleofdiffractioninthe caseofhomo-
geneous field, for which we have both the exact solution
of Pauli’s equation (including diffraction) and the classi-
cal trajectory solutions. We find that diffraction is only
relevant if we look at the electron beam very close to a
focal point, where its width becomes of the order of few
nanometers. Away from the focus, the ensemble of clas-
sical trajectories reproduces perfectly well the quantum
beamevolution. Thisresultisnotsurprising,asintheq-
filter all length scales are 8–9 ordersof magnitude larger
FIG. 5. Sketch of a possible TEM microscope incorporating than the electron wavelength λ=2π/k , which is of few
c
theproposed spin-polarization filterbased on theq-filter.
picometersforenergiesoftheorderof100keV(typicalof
electron microscopes). Therefore, we may safely assume
that diffraction can be ignored also in inhomogeneous q-
to select a single spin polarization can be set to a radius
filter geometries,as long asno focusing occursinside the
offewtensofmicrons,bycarefullyselectingtheaperture
filter. Our classical ray-tracing simulations confirm that
plane position with respect to the focal plane. It will be
we avoid a focusing inside the q-filter. It should be also
presumably convenient to work slightly off focus to ob-
noted that in designing and analyzing electron-optics el-
tain larger beam waists without the need to introduce
ements as those commonly used in electron microscopy,
additional magnifying optics.
it is quite standard to ignore quantum effects and use
We note here in passing that in the new generation
classical ray tracing or geometrical optics.
of TEM microscopes with aberration-corrected probe a
The input and output apertures, being again about 7
larger number of lenses are available for beam control,
ordersofmagnitudelargerthantheelectronwavelength,
and in particular a pair of hexapoles. It could be worth
will also give negligible diffraction effects (only some ra-
then exploring the possibility of using directly these
dial fringes extremely close to the aperture edges are ex-
lenses as an effective q-filter, without introducing addi-
pected,whichshouldnotaffecttheproperworkingofthe
tional optical elements. In this configuration, the mag-
proposed setup).
netic hexapoles with opposite sign could be used to pro-
ducethepolarization,whilealsocompensatingtheintro-
duced aberration. However stability issues might arise
Violation of Bohr’s “theorem”
forlargefieldsandtheeffectoffringingfieldsonthespin
may also not be negligible, in this case, as the length-
As reported in Ref. [17], few years after the discov-
to-gap ratio would not be very large. Therefore, further
ery of electron spin, several experiments were proposed
analysisandsimulationswillbe necessaryinordertoas-
(andsometimesattempted)aimedatseparatingelectrons
sess this possibility.
according to their spin, mainly by exploiting inhomoge-
neousmagneticfieldsactingonthemagneticdipoleasso-
ciated with it (similar to the Stern-Gerlachexperiment).
The role of electron diffraction In1929,NielsBohrdemonstratedasortof“impossibility
theorem” [17], stating that no Stern-Gerlach-like exper-
In our treatment we fully included the diffraction ef- iment (including Knauer’s proposal, in which an elec-
fects only in the free propagation(througha lensingsys- tric fieldwasusedtobalancethe Lorentzforce)couldbe
tem) taking place after the q-filter device. Indeed, this usedtoseparateelectronsaccordingtotheirspinbymore
stage is where the correlation between OAM and radial than a small fraction of the beam transverse extension.
profile develops and is then exploited for separating the This was related to the uncertainty principle and to the
different spin components. This free propagation, end- unavoidableorthogonalmagneticfieldaccompanyingthe
ing in the far-field and giving rise to the beam profiles required magnetic field gradient. A simple explanation
9
of this result is that the spin magnetic moment vanishes destructive interference effects. Finally, we note that in
intheclassicallimit ¯h 0,soallattempts atseparating our geometry the unavoidable orthogonal magnetic field
→
particles according to the magnetic-momentforce occur- associated with the azimuthal field gradient generating
ring in a magnetic field gradient give rise to a vanishing the dipole forces, which is the main source of problems
separation in the classical limit, and hence a separation in the Stern-Gerlachapproachto spin-polarized electron
that is comparable to quantum uncertainty. Actually, separation,isalreadytakenintoaccountandcorresponds
this statement has been later proved not to be strictly totheradialmagneticfieldofthequadrupoleorhexapole
valid (see, e.g., Refs. [18, 19]), although it remains ex- configuration. Soits effecthasbeenalreadyfully consid-
tremely hard to overcome in practice. ered in our treatment.
Our approachis howeverfundamentally different from
past attempts. Indeed, we exploit the quantum nature
itself of the electrons to operate the spin separation (as
such, our proposal bears some similarities with a very
∗
[email protected]
recently proposed scheme for spin-polarization in which
[1] M. Uchidaand A. Tonomura, Nature 464, 737 (2010).
electron diffraction is also exploited [20]). As we have
[2] J. Verbeeck, H. Tian, and P. Schattschneider, Nature
seen,the magneticdipole force inourdevice acts onlyin
467, 301 (2010).
the azimuthal direction, which is not the direction along [3] B. J. McMorran et al., Science 331, 192 (2011).
which we separate the electrons. The torque associated [4] K. Y. Bliokh, Y. P. Bliokh, S. Savel’ev, and F. Nori,
with this force imparts the OAM variation correspond- Phys. Rev.Lett. 99, 190404 (2007).
ing to the azimuthal geometric phase. However, at the [5] S. Franke-Arnold, L. Allen, and M. J. Padgett, Laser
exit of our Wien filter the electrons show no significant Photonics Rev.2, 299 (2008).
[6] L. Marrucci et al., Phys. Rev.Lett. 96, 163905 (2006).
spatial separation according to the spin. It is then the
[7] L. Marrucci et al., J. Opt.13, 064001 (2011).
subsequent free propagation and diffraction that acts to
[8] V. Tioukine and K. Aulenbacher, Nucl. Instr. Meth. A
separate the electrons along the radial direction, which
568, 537 (2006).
is not the direction of the main field gradient. In par- [9] J.Gramesetal., Proc.of2011particleacceleratorconf.,
ticular, an efficient radial separation is ensured by the New York,NY,USA (2011).
presence of a vortex at the beam axis for one spin com- [10] See thesupplemental material in theAppendix.
ponentandnotfor the other. Inotherwords,we use the [11] A. Shapere and F. Wilczek, eds., Geometric Phases in
Physics (World Scientific,Singapore, 1988).
quantum wavy nature of the electrons to separate them
[12] H. Rose, Optik 32, 144 (1970).
according to the spin, and hence our method is not sub-
[13] J. Zach and M. Haider, Nucl. Instr. and Meth. A 363,
ject to Bohr’s impossibility theorem. As a further argu-
316 (1995).
ment,evenintheclassicalStern-GerlachapproachBohr’s [14] M. R.Scheinfein, Optik 82, 99 (1989).
theorem does not forbid the creation of spin-polarized [15] N. Yamamoto et al., J. Phys.: Conf. Ser. 298, 012017
components at the edges of the beam, so that by suit- (2011).
able aperturing it should be possible to generate spin- [16] E. Karimi et al., Appl. Phys.Lett. 94, 231124 (2009).
[17] O. Darrigol, Hist. Stud.Phys. Sci. 15, 39 (1984).
polarized beams, although with very low efficiency. Our
[18] H. Batelaan et al., Phys. Rev.Lett. 79, 4517 (1997).
method is also based on selecting only an edge of the
[19] G. A.Gallup et al., Phys.Rev.Lett. 86, 4508 (2001).
beam, although this is done in the radial direction and
[20] S. McGregor et al., New J. Phys.13, 065018 (2011).
the efficiency is strongly improved by the vortex-related