Generation of a spin-polarized electron beam by multipoles magnetic fields
Ebrahim Karimi, Vincenzo Grillo, Robert W. Boyd, Enrico Santamato
GGeneration of a spin-polarized electron beam by multipoles magnetic fields
Ebrahim Karimi, ∗ Vincenzo Grillo, Robert W. Boyd,
1, 3 and Enrico Santamato
4, 5 Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, K1N 6N5 Canada CNR-Istituto Nanoscienze, Centro S3, Via G Campi 213/a, I-41125 Modena, Italy Institute of Optics, University of Rochester, Rochester, New York, 14627, USA Dipartimento di Scienze Fisiche, Universit`a di Napoli “Federico II”,Compl. Univ. di Monte S. Angelo, 80126 Napoli, Italy Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, Napoli
The propagation of an electron beam in the presence of transverse magnetic fields possessinginteger topological charges is presented. The spin–magnetic interaction introduces a nonuniformspin precession of the electrons that gains a space-variant geometrical phase in the transverse planeproportional to the field’s topological charge, whose handedness depends on the input electron’s spinstate. A combination of our proposed device with an electron orbital angular momentum sorter canbe utilized as a spin-filter of electron beams in a mid-energy range. We examine these two differentconfigurations of a partial spin-filter generator numerically. The results of these analysis could proveuseful in the design of improved electron microscope.
I. INTRODUCTION
A few years ago, the existence of orbital angular mo-mentum (OAM) for an electron beam was predictedtheoretically [1]. A couple of years later, two differenttechniques, based on the holography and random phasechanges in a graphite sheet, were used to generate elec-tron beams carrying OAM experimentally [2–4]. Suchan intriguing topic is of particular interest to materialsscientists as it opens up new opportunities for theircommunity [5]. The OAM as a “ rotational-like ” degreeof freedom of an electron beam induces a magneticmoment in addition to the spin magnetic moment,even up to few hundred Bohr magneton per electron,which gives a possibility to interact with an externalmagnetic field [3, 5]. The interaction of OAM magneticmoments with a uniform longitudinal magnetic fieldsor a fluxes have been recently examined theoreticallyand experimentally [6–8]. Indeed, this interaction wouldenhance or diminish the beam’s kinetic OAM, whicheventuates in an additional opportunity to measureor sort electron’s OAM spatially. However, beside itsinteresting and fascinated applications, this novel degreeof freedom of electrons would be utilized to rock somefundamental quantum concepts such as the
Bohr-Pauliimpossibility of generating a spin-polarized free electronbeam [9, 10].The spin-orbit coupling in a non-uniform balancedelectric-magnetic field, named a “ q -filter”, was proposedby some of the authors as a novel tool to generate anelectron vortex beam from a pure spin-polarized electronbeam. In that configuration, the spin of an electronfollows the Larmor precession up and acquires a geo-metrical phase, which depends on both spin–magneticfield direction and the time of interaction as well. Anon-uniform magnetic field introduces a non-uniform ∗ Corresponding author: [email protected] phase structure the same as the topological structure ofthe magnetic field. Several different topological chargeconfigurations, proposed in the previous article, canbe manufactured practically [10]. A local orthogonalelectric field was proposed to compensate the “net”magnetic force. Furthermore, the reverse process wassuggested to filter the spin component of an elec-tron beam spatially, where two different longitudinalelectron’s spin components suffer opposite precessiondirections, thus possess opposite OAM values.In this work, we suggest a scheme based on non-uniformmagnetic fields , rather than a balanced space-variant
Wien q -filter, to manipulate electron OAM. As the q -filter the proposed scheme imprints onto the incomingelectron beam the magnetic field topological charge, witha handedness depending on the longitudinal componentof electron spin, positive for spin up | ↑(cid:105) and negative forspin down | ↓(cid:105) with the advantage that no compensatingelectric field is needed. Unlike in the q -filter, however, thebeam structure is now strongly affected by the magneticfield. A TEM Gaussian beam after passing throughthe nonuniform magnetic field of the device splits outinto multi Gaussian-like beams, each beam oscillatingalong the vector-potential minima. In particular, the in-cident Gaussian electron beam splits into two and threesemi-Gaussian beams in quadrupole and hexapole mag-netic fields, respectively. It is worth noting, however,that the multi Gaussian-like beam does recover its origi-nal Gaussian shape at certain free-space propagation dis-tance, provided its phase distribution does not acquiresudden changes in the transverse plane. In this work, weintroduce and numerically simulate two realistic config-urations of spin-filter for electron beams, based on thenew proposed device. Our numerical simulations con-firms that a portion of electrons, usually small, remainspolarized after passing through the device and can beeasily separated form the rest of the beam by suitableapertures. a r X i v : . [ phy s i c s . op ti c s ] J un II. PROPAGATION OF ELECTRON BEAMS INAN ORTHOGONAL UNIFORM MAGNETICFIELD
Let us assume that the electron beam moves along the z -direction perpendicular to a uniform magnetic field B = B (cos θ, sin θ, x, y ) transverse planeat angle θ with respect to the x -axis. As associated vectorpotential we may take is A = B (0 , , y cos θ − x sin θ ).We assume a non relativistic electron beam, so that we can use Pauli’s equation i (cid:126) ∂ t ˜ ψ = (cid:26) m ( − i (cid:126) ∇ − e A ) − B · ˆ µ (cid:27) ˜ ψ, (1)where ˜ ψ is a two-component spinor and ˆ µ = gµ B ˆ σ is the electron magnetic moment – µ B = (cid:126) e/ m is Bohr’s magneton, g is the electron g -factor, andˆ σ = (ˆ σ x , ˆ σ y , ˆ σ z ) is Pauli’s vector, respectively. Weassume a paraxial beam with average linear mo-mentum p c and average energy E c = p c / m , sothat ˜ ψ ( x, y, z, t ) = exp[ i (cid:126) − ( p c z − E c t )]˜ u ( x, y, z ), with˜ u ( x, y, z ) slow-envelope spinor field [1]. Inserting this ansatz into Eq. (1) and neglecting the second derivativesof ˜ u with respect to z , we obtain the paraxial Pauli equa-tion (cid:26) ik c ∂ z + ∇ ⊥ + 2 k c e (cid:126) A − e (cid:126) A + 2 m (cid:126) B · ˆ µ (cid:27) ˜ u ( x, y, z ) = 0 , (2)where ⊥ stands for the transverse coordinate, and k c = p c / (cid:126) is a central de Broglie wave-vector.Equation (2) is solved with initial Cauchy data at z = 0,˜ u ( r, φ,
0) = ˜ a exp( − r /w ) corresponding to a Gaus-sian beam of width w in the cylindrical coordinates of( r, φ, z ). The constant spinor ˜ a = ( a , a ) describes thepolarization state | ψ (cid:105) = a | ↑(cid:105) + a | ↓(cid:105) of the input beamin the | ↑(cid:105) , | ↓(cid:105) basis where the spin is aligned parallel orantiparallel to the beam propagation direction, respec-tively. We assume the normalization | a | + | a | = 1. Astraightforward calculation shows that the required solu-tion of the paraxial Pauli equation is given by˜ u ( r, φ, z ) = G ( r, φ, z ) ˆ M ( z )˜ a, (3)where ˆ M ( z ) is a matrix given byˆ M ( z ) = (cid:18) cos πz Λ ie − iθ sin πz Λ ie iθ sin πz Λ cos πz Λ (cid:19) , (4) with Λ = π (cid:126) k c mgµ B B . The matrix ˆ M ( z ) accounts for theaction of the magnetic field on the particle spin. Theaction on the electron motion is described in Eq. (3) bythe Gaussian-Coherent factor G ( r, φ, z ) given by G ( r, φ, z ) = (cid:115) − k c z R π q (cid:107) ( z ) q ⊥ ( z ) e ik c (cid:18) fg ( r,φ )2 q (cid:107) ( z ) + fc ( r,φ,z )2 q ⊥ ( z ) + z (cid:19) (5)with f g ( r, φ ) = r cos ( θ − φ ) , (6) f c ( r, φ, z ) = i (cid:16) π Λ (cid:17) (cid:18) Λ π + r sin ( θ − φ ) (cid:19) (cid:32) i (cid:18) Λ π (cid:19) + (cid:16) π Λ (cid:17) z R (cid:16) q ⊥ ( z ) + iz R cos (cid:16) πz Λ (cid:17)(cid:17) (cid:18) Λ π + r sin ( θ − φ ) (cid:19)(cid:33) + cos (cid:16) πz Λ (cid:17) (cid:32) (cid:18) Λ π (cid:19) + r sin ( θ − φ ) (cid:18) (cid:18) Λ π (cid:19) + r sin ( θ − φ ) (cid:19)(cid:33) . The complex curvature radii of the Gaussian-Coherentfactor G ( r, φ, z ) are given by q (cid:107) ( z ) = z − iz R (7) q ⊥ ( z ) = (cid:18) Λ π (cid:19) sin (cid:16) πz Λ (cid:17) − iz R cos (cid:16) πz Λ (cid:17) , (8) FIG. 1. (a) Cross section of an electron gaussian beam prop-agating in a uniform magnetic field along the x -axis. Duringpropagation the beam starts to move up in the positive y -direction, orthogonal to the magnetic field. The figure showsthe beam position at z = α Λ with α = 4 . × − . The side-bar shows the strength of vector potential in false-color. (b) y -displacement of the beam center as function of the z coordi-nate. The oscillation is sinusoidal and recovers its transverseposition and shape at planes z = n Λ ( n integer). The sim-ulation was performed for an electron beam having energy E c = 100 KeV, waist w = 10 µ m, in a magnetic field ofstrength B = 3 . where z R = k c w , Λ = π (cid:126) k c eB . From Eq. (4) we see that,the beam spin state oscillates during propagation withspatial period Λ . Moreover, unlike in the Wien-filter,which does not affect the beam mean direction, now thebeam oscillates perpendicularly to the magnetic field B towards the minima of the vector potential with spatialperiod 2Λ = ( g/ (cid:39) Λ / ∝ B − . Figure 1-(a) showsthe beam intensity profiles at two different z -planes; atthe entrance plane z = 0 (central spot) and at z = α Λwith α = 0 .
44% (upper spot). The electron trajectory in yz -plane, orthogonal to magnetic field, is shown in Figure1-(b): the electron beam follows a sinusoidal oscillationwith spatial period 2Λ and amplitude 2Λ /π . III. PROPAGATION OF ELECTRON BEAMS INAN ORTHOGONAL NONUNIFORM MAGNETICFIELD POSSESSING A SPECIFICTOPOLOGICAL CHARGE
Equation (3) represents an exact solution of the beamparaxial equation, with an explicit boundary condition,for a uniform constant magnetic field at angle θ withrespect to the x -axis. If the angle θ = θ ( x, y ) changesslowly in the transverse plane, we may assume that thesolution (3) is still approximately valid. This GeometricOptics Approximation (GOA) is quite accurate in thepresent case, since the electron beam wavelength in atypical Transmission Electron Microscope (TEM) is inthe range of tens of picometers, while θ changes overlength of several microns. Within this slowly varyingapproximation, the effect of a nonuniform magnetic fieldis obtained simply by replacing θ with θ ( x, y ) in Eqs. (4),(5), and (6). We assume singular space distribution of themagnetic field where θ ( x, y ) = θ ( r, φ ) is given by θ ( φ ) = qφ + β, (9)where φ = arctan ( y/x ) is the azimuthal angle in thebeam transverse plane and β is a constant angle, whichdefines the inclination on the x -axis. Finally q is an inte-ger which fixes the topological charge of the singular mag-netic field distribution. Such magnetic structures can begenerated in practice by multipolar lenses (for negativecharges q ) or by a set of appropriate longitudinal currentsat origin (for positive charge q ). Inserting Eq. (9) intoEqs. (4), (6), yieldsˆ M ( z ) = (cid:18) cos πz Λ ie − iqφ e − iβ sin πz Λ ie iqφ e iβ sin πz Λ cos πz Λ (cid:19) (10)and f g ( r, φ ) = r cos (( q − φ + β ) , (11) f c ( r, φ, z ) = i (cid:16) π Λ (cid:17) (cid:18) Λ π + r sin (( q − φ + β ) (cid:19) (cid:32) i (cid:18) Λ π (cid:19) + (cid:16) π Λ (cid:17) z R (cid:16) q ⊥ ( z ) + iz R cos (cid:16) πz Λ (cid:17)(cid:17) (cid:18) Λ π + r sin (( q − φ + β ) (cid:19)(cid:33) + cos (cid:16) πz Λ (cid:17) (cid:32) (cid:18) Λ π (cid:19) + r sin (( q − φ + β ) (cid:18) (cid:18) Λ π (cid:19) + r sin (( q − φ + β ) (cid:19)(cid:33) . Equation (10) shows that in passing through a multipolesmagnetic field of length L , a fraction | η | = (cid:12)(cid:12)(cid:12) sin πL Λ (cid:12)(cid:12)(cid:12) of the electrons in the beam flip their spin and acquirea phase factor exp( ± iqφ ) accordingly if the initial spinwere up | ↑(cid:105) or down | ↓(cid:105) , respectively. The rest of theelectrons, i.e., 1 − | η | , pass through without changingtheir initial spin state. When L = Λ / π ), allelectrons of the beam emerge with their spin reversedand acquire the above mentioned phase factor, which means that an amount of ± (cid:126) q is added to their initialOAM value. In this case the Spin-To-OAM Conversion(STOC) process is complete and we say that the deviceis “ tuned ” [10]. Tuning can be made by acting on thestrength of the magnetic field or changing the devicelength. The STOC process is governed by the exp( ± iqφ )factors in Eq. (10), which is of geometrical origin [11].As a consequence, the STOC process occurs even if thefield amplitude B is (slowly) dependent on the radialcoordinate r in the beam transverse plane. The capa- FIG. 2. Propagation of an electron gaussian beam througha quadrupole ( q = − β = − π/
2) magnetic field; (a) atthe quadrupole pupil (b) after a propagation distance whereis given by z = α Λ. As shown in (b), the beam splits out intotwo different astigmatic quasi-Gaussian beams along vector-potential minima inside the quadrupole, along x -axis. Thebar side shows the strength of vector potential in a false-color,and simulation was performed for α = 4 . × − . bility of changing the electron OAM and of creating acorrelation between OAM and spin are the main featuresof the non-uniform magnetic multipoles. Equation (11)shows that the device alters deeply the transverse profileof the beam, which acquires a multi quasi-Gaussianshapes as shown in Figs. 2–(b) and 3–(b) for quadrupoleand hexapole magnetic field, respectively. The Gaussianbeam at the entrance after propagation through themagnetic field breaks up respectively into two and threeastigmatic quasi-Gaussian shape beams for quadrupoleand hexapole. The cleft beams number depends on thenumber of vector potential minima. Both the STOC andnon-STOC part of the beam possess the same intensityprofile; but the STOC part only acquires a helical phasestructure according to the magnetic field topologicalcharge. For a small beam distortion, it can be shownthat in the far-field the multi gaussian-like beam withthe helical structure assumes a doughnut shape, whilethe vortex free non-STOC beam assumes a Gaussianshape. IV. FRINGE FIELDS AND ITS EFFECT ONTHE SPIN-FILTERING
In practice, it is impossible to generate a completelytransverse magnetic field. A further non-transverse mag-netic fields known as the fringe fields cannot be avoided.In this section, we examine the effect of the fringe fieldson the spatial pattern distribution of electron beam whenour proposed device is not tuned. A non-tuned device,based on its length and magnetic field strength, convertsonly a portion of the incoming beam flipping the electronlongitudinal–spin state and gaining OAM - the remainingpart of the beam left unchanged. When the non–tuneddevice is applied to an unpolarized electron beam, the beam exits in a mixture state of spin up and down withopposite OAM values, given by FIG. 3. Propagation of an electron gaussian beam througha hexapoles ( q = − β = − π/
2) magnetic field; (a)at the pupil of hexapoles (b) after a propagation distancewhere is given by z = α Λ. As shown, during propagation thebeam splits out into three different astigmatic quasi-Gaussianbeams along vector-potential minima. The bar side shows thestrength of vector potential in a false-color, and simulation isbeen performed for α = 4 . × − . | ψ (cid:105) STOC = η (cid:26) | ↑ , − (cid:96) (cid:105) for spin | ↓(cid:105) input | ↓ , + (cid:96) (cid:105) for spin | ↑(cid:105) input , (12)where | (cid:96) (cid:105) stands for the OAM state given by topolog-ical charge of device, i.e., (cid:96) = q , and | η | = | sin πL Λ | is the device STOC’s efficiency. The main fringe fieldsappear at the entrance and the exit face of the device asnonuniform longitudinal magnetic fields. The interactionof longitudinal magnetic field with both spin and orbitalangular momentum of electron beams has been recentlytheoretically investigated [7]. The longitudinal magneticfield introduces a phase rotation Φ( (cid:96), s ) ∝ B z ( (cid:96) + g s ) z onthe beam, where s = ± (cid:126) / up and down , respectively,and B z is in general a function of transverse radial co- FIG. 4. First proposed scheme to generate a spin-polarizedelectron beam based on space-variant magnetic fields. ordinate r . [7]. This phase rotation comes out from theZeeman interaction. The longitudinal fringe field intro-duces a different phase change in each term of Eq. (12)which is therefore changed into | ψ (cid:105) STOC = η (cid:26) e i Φ( − (cid:96), | ↑ , − (cid:96) (cid:105) for spin | ↓(cid:105) input e i Φ( (cid:96), − | ↓ , + (cid:96) (cid:105) for spin | ↑(cid:105) input . (13)Since the fringe fields are nonuniform, in general, thephases in Eq. (13) are coordinate dependent and producedual converging and diverging astigmatic effects. How-ever, even in the presence of fringe fields, the two spinstates of the emerging beam are still labeled by the twovalues ± (cid:96) of OAM so that an OAM sorter can be used toseparate the electrons according to their spin value. Inthe next section, we calculated numerically the efficiencyof this way to obtain polarized electron beam by usinga pitch-fork hologram with topological charge (cid:96) as OAMsorter. The final electrons state after an OAM sorter,then, is | ψ (cid:105) final = η (cid:26) e i Φ( − (cid:96), | ↑ , (cid:105) for spin | ↓(cid:105) input e i Φ( (cid:96), − | ↓ , +2 (cid:96) (cid:105) for spin | ↑(cid:105) down . (14)The first term tends to recover the Gaussian shape, whilethe second term will have a doughnut shape in the far-field. Both of the non-STOC terms own OAM= (cid:96) sincethe hologram, i.e., OAM sorter, is spin independent. Aspatial selector can be use, e.g., a pinhole, to select thecentral part, which has a uniform coherent spin up state.The efficiency and purity of the spin filter depends onthe the pinhole radius, which for an optical field has beendiscussed in [12]. So, combination of this device with anOAM sorter yields an electron spin-filter that can be real-istic since the multipolar magnetic magnets are availablecommercially as an aberration corrector for TEM as wellas OAM sorters. V. NUMERICAL SIMULATION ANDTECHNICAL DISCUSSIONS
The GOA analytical solution presented in Section IImight be used to obtain preliminary information on theSTOC efficiency and beam intensity profile in the de-vice. However, more details on the interaction of a space-variant magnetic field with an electron beam can be ex-amined by implementing both a ray tracing techniqueand the spin–orbit interaction simultaneously. In orderto overcome this issue, we developed a research–didacticsoftware based on the multi-slice method used in the elec-tron microscopy, where Eq. (2) has been considered as afree particle motion under the force of a “ local potential ”of A , A and Pauli terms. Therefore, one can evolveboth temporal and z -dependence of the wave-functionbased on a well-know Dyson –like decomposition, sinceboth the operators of the scalar wave equation and localpotential do not commute [13, 14]. The potential wasdivided and projected into multi-slices where the beamwave-function was constructed by a free-space propaga-tion between each slices. After each iteration the wave-function turns into
FIG. 5. Simulated diffraction pattern of the beam generatedby a quadrupole after passing through a pitch-fork hologram(right spot assumed to be the first order of diffraction). Ina fraction of the beam the nonuniform magnetic field of thequadruple couples electron spin to OAM. Up and down sub-figures inside each row show the non–converted and convertedparts of an input 50-50 mixture of spin up and down , respec-tively. The electrons spin state in the first order of diffractionhas been indicated at the right side of images. u ( r ⊥ , z j +1 ) = K ⊗ (cid:16) e ( i (cid:126) v (cid:82) zj +1 zj V ( r ⊥ ,ζ ) dζ ) · u ( r ⊥ , z j ) (cid:17) , (15)where v is the electrons velocity, ⊗ is the convolution withthe wave-function inside parenthesis, z j and V ( r ⊥ , ζ )stand for position of j th slice and the local potential(third, fourth and fifth term of Eq. (2)), respectively(see Ref. [15] for more details). K is the Fresnel prop-agator between each adjoined two slices spaced by ∆ z ,which is given by K = − ik c π ∆ z e ikc z ( x + y ) . (16)Apart the simple concept, one may extend such apowerful algorithm to the relativistic case as well [16].However, since electron microscopes work at mid-range energy, our simulation was carried out in thenon-relativistic regime. We considered two possibleconfigurations to generate a spin-polarized electronbeam in an electron microscope. (i) A first scheme can be in principle adopted inmany microscopes using a condenser stigmator. In thisfirst scheme, a Gaussian beam is directed to a quadrupo-lar magnetic field, and then to a pitch-fork hologramas shown in Fig. (4). A lens condenser can be used to
FIG. 6. Second proposed scheme to generate a spin-polarizedelectron beam based on coupling with two magnetic quadru-ples (spherical aberration corrector). The second quadruplecorrects the aberration induced by the first one. Nevertheless,the spin-to-orbit coupling efficiency after the second quadru-ple can be completely neglected. form the far-field image of the hologram. However, theintensity required for the magnetic field might make theoperation difficult. It also turns out difficult to polarizeelectrons before the specimen, except in microscopeswith two or even three condensers aperture planes.The simulated electron beam shapes after passingthrough the quadrupole and the pitch-fork hologram forspin up and down input are shown in Fig (5).The first and second row in Fig. (5) show the far-fieldintensity pattern for input electrons with spin up and down state, respectively. The subrows are correspondto expected spin state for the first order of diffraction(left spots). The non-STOC part of both spin up and down , i.e., first and last rows of Fig. (5), formsa Hermite-Gaussian shape of the first order, which isaffected by the quadrupole’s astigmatism and splitsinto two parts. Conversely the STOC parts are shapeddifferently depending on the initial spin state; one asthe
Hermite-Gaussian of second order and the other oneforms Gaussian beam bearing some astigmatic distor-tion. It can be seen that an aperture with appropriatesize can be used to select the central spot only, whichpossesses opposite polarization. (ii)
In the second case, a simplified scheme of a sphericalaberration corrector has been considered, see Fig (6),where two quadrupoles with opposite polarization werecoupled through two cylindrical lenses (transport lenses)having magnetic field along the propagation direction.Differently from the real device, we will assume thata limiting aperture can be added inside the correctorin correspondence of the focal plane of the transportlenses. Figure (7) shows the simulated evolution of
FIG. 7. As the beam with OAM=+1 propagates inside theaberration corrector system (quadrupole - condenser - aper-ture and a rotated quadrupole), it sees an astigmatism effectintroduced by the quadrupoles and, based on the time inter-action, a portion of electron suffers spin-to-orbit conversion,second and fourth column. The first two rows show phaseand intensity distributions of electron beam after the firstquadrupole for a mixture of 50-50 spin up and down , respec-tively. The last row shows the output intensity distributionfor both converted and non-converted part of spin up anddown after interacting with whole system. The simulation wasmade for an input electron beam size of 1 µ m, a quadrupolemagnetic field of 0 . µ m. This relatively small thickness was cho-sen to simplify and speed up the numerical calculation and toavoid too large phase deformations. the wave-function in the system for both spin up and down . At the exit face of the first quadrupole thetwo polarizations are indistinguishable, but evolve toa different intensity distribution in the focal planewhere an aperture selects the central part of the beamcontaining mainly the | ↑ , (cid:105) electron state. The secondquadruple, indeed, implies aberration correction intothe selected | ↑ , (cid:105) beam. It is worth noticing that thespin-to-orbit coupling, due to strength of magnetic field,after the second quadruple can be neglected. VI. CONCLUSIONS
We presented two possible practical devices to generate aspin polarized electron beam via a spin-to-orbit conver-sion in the presence of a nonuniform transverse magneticfield. The heart of the devices is a multipolar magnetgenerating a singular transverse magnetic structure withnegative integer topological charge. The device action ona pure spin-polarized electron beam is spin-to-orbit con-version and the beam gains a nonuniform phase struc-ture, defined by the magnetic field topological charge,where the sign is given by the input electron spin value.When the device is combined with an OAM sorter, spi-ral phase plate, hologram, or even longitudinal magneticfield, it can be used as a spin-filter to polarize the elec-tron beam. However, because of the strong astigmatism,the efficiency of the devices proposed here are lower thanthe ones discussed in Ref [10].
VII. ACKNOWLEDGEMENT
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