HIGHLIGHT
Autori
Ebrahim Karimi, Lorenzo Marrucci, Vincenzo Grillo and Enrico Santamato
Abstract
We propose the design of a space-variant Wien filter for electron beams that induces a spin half-turn and converts the
corresponding spin angular momentum variation into orbital angular momentum of the beam itself by exploiting a geometrical
phase arising in the spin manipulation. When applied to a spatially coherent input spin-polarized electron beam, such a device
can generate an electron vortex beam, carrying orbital angular momentum. When applied to an unpolarized input beam, the
proposed device, in combination with a suitable diffraction element, can act as a very effective spin-polarization filter. The
same approach can also be applied to neutron or atom beams.
TEXT
The recent introduction of vortexes on an electron beam (namely beams carrying an orbital angular momentum
OAM) has recently opened the road to many innovative studies in electron optics. In particular the use of
‘‘pitchfork’’ hologram manufactured by ion beam lithography [1,2] has made this feature available in an electron
microscope.
In a recent article we have found the way to turn this control of orbital angular momentum in intrinsic spin
polarization of the electron beam. This result is even more important if one considers that for long time the
polarization of a free electron beam was considered impossible for fundamental reasons [4].
The main optical element in this transformation is a space variant Wien Filter (hereafter q-filter) formed by
electric and magnetic fields that produce a compensation between the electric and the Lorentz forces.
A peculiar characteristic of this device in order to have an actual conversion is that the magnetic fields needs to
̅=
have topological charge, namely it should be possible to write in Cartesian coordinates B
B0 (cos(α) , sin(α), 0) with α = qϕ + β. Here q is an integer (the topological charge ), ϕ the azimuth angle and
φ0 an arbitrary constant angle. Such is the case for example of quadrupolar or in general multipolar lenses.
Quantum mechanics calculation demonstrate that such a lens, when applied to a polarized beam produces a
vortex with well defined OAM while when applied to an unpolarized state with OAM= ℓ produces a separation
of the beam into two spin-OAM states at the end of the filter which can be represented as |↓, ℓ + q⟩ and |↑,ℓ −
q⟩ . Here the spin states with quantization axis along the z axis of beam propagation are indicated by arrows, the
OAM is expressed by the quantum orbital number as a function of the initial OAM number ℓ and the above
mentioned topological charge of the lens q.
In a far field the different OAM states evolve in different spatial arrangement of the wave. This means that the
spin-OAM correlation can be exploited for making an effective electron beam spin-polarization filter. Such filter
requires four basic elements in sequence: (i) an OAM manipulation device, such as a fork hologram [1], to set ℓ≠
0; (ii) a q-filter with q = ℓ, generating a mixture of electrons in states |↓ ,2ℓ⟩ and |↑,0⟩ ; (iii) a free propagation
(or imaging) stage that allows these two states to develop different radial profiles by diffraction, because of their
different OAM values; (iv) a circular aperture for finally separating the two states that will have different final
shapes because of their different OAM.
Fig 1 shows a scheme of the optical configuration showing that a beam with mixed polarization at entrance is
divided into two beam with different angular distribution and different polarization.
[1] J. Verbeeck, H. Tian, and P. Schattschneider, Nature 467, 301 (2010).
[2] B. J. McMorran et al., Science 331, 192 (2011).
[3] E. Karimi, L. Marrucci, V. Grillo, E. Santamato, Phys. Rev. Lett. 108, (2012) 044801
[4] O. Darrigol, Historical studies in the physical sciences 15, 39 (1984).
b
)
Fig1 a) Scheme of a working device for the spin separation of an unpolarized beam. At the exit of the qfilter the two different polarizations can be separated by their different angular distribution.b) simulation
of the wavefunction at far field: the two polarized states are shown in green and red.