The research leading to these results has received funding from the European Union’s Horizon 2020 research and innovation programme under project/grant agreement Q-SORT No. 766970.

The present invention relates in general to devices and methods for manipulating and measuring orbital angular momentum of particles.

The Orbital Angular Momentum (hereinafter, also 0AM) is an important quantity in quantum physics. It is as fundamental as linear momentum. For a paraxial beam (either light or matter waves) the most important component is the one along the average beam propagation direction and differently from linear momentum it is characterized by a discrete spectrum.

Among the methods to measure the angular momentum of a particle, mention is made to the 0AM sorter. In such device, two phase elements introduce a conformal mapping of the wave function from polar to Cartesian coordinates. In particular, a log-polar transformation is introduced: arctan (y, x) . (1)

Thanks to this mapping the 0AM is transformed into a linear momentum and can be simply analysed by a lens system.

The overall phase introduced by the two elements can be written as [1] where a, b are free parameters, f is the focal length of the Fourier lens between the two phase elements, x, y are the Cartesian coordinates in an input plane orthogonal to the beam axis and u, v are the Cartesian coordinates in an output plane orthogonal to the beam axis. The Fourier lens transfers onto the output plane the Fourier (or Fraunhofer diffraction) image of the input plane. The parameter A is the wavelength of the beam.

This idea has also been transported to electron microscopy. Presently, there are three main approaches to the experimental realization of the phase elements composing the OAM sorter: 1) using holograms made of electron transparent materials such as Silicon Nitride [2], 2) using correctly shaped magnetic currents [3], 3) using electrostatic elements (electrodes).

As to electrostatic OMA sorters, McMorran et al [4] have proposed a theoretical solution based on electrostatic phase elements. The first phase element (phase unwrapper) is based on a needle-shaped electrode and a flat counter-electrode acting as an electrostatic mirror. The second phase element (phase corrector) is based on an array of oppositely charged elongated electrodes forming a periodic array. The arrangement proposed by McMorran et al requires that the length of the needle is much larger than the cross-sectional size of the electron beam (infinite length approximation). When a needle of finite length is taken into consideration, aberrations are produced that require additional electrostatic elements in order to be fixed.

A solution to this problem has been proposed by Pozzi et al [5]. This solution entails using two more needle-shaped electrodes, orthogonal to the main one but not directly illuminated by the electron beam. Their effect is to introduce and electrostatic field that compensate for the aberration due to the finite length of the main needle.

A further drawback is that it is unpractical to integrate the electrostatic mirror using MEMS devices. Such mirror would be indefinitely long in the vertical direction but this is incompatible with the small spaces available within an electron microscope.

The present invention aims at overcoming the above mentioned limitations on the practical realization of the first phase changing element. In view of the above, there is proposed a device for transforming between an azimuthal phase and a linear phase in a charged particle beam, the device comprising a first phase changing element and a second phase changing element arranged along a beam axis at conjugate Fourier planes of a lens system, said first phase changing element being configured to transform between orbital angular momentum states and planar waves in the charged particle beam, and said second phase changing element being configured to correct a phase distortion introduced by the first phase changing element, wherein the first phase changing element comprises a planar electrostatic arrangement of needle-shaped electrodes arranged orthogonally to the beam axis, said electrostatic arrangement comprising a first needle-shaped electrode having a distal tip arranged at the beam axis and a pair of second needle-shaped electrodes for correcting aberrations introduced by the first needle-shaped electrode, said second needle-shaped electrodes being arranged on opposite sides of, and oriented orthogonally to, the first needle-shaped electrode, wherein the electrostatic arrangement further comprises a third needle-shaped electrode and a pair of fourth needle-shaped electrodes arranged on opposite sides of, and oriented orthogonally to, the third needle-shaped electrode, said third needle-shaped electrode and fourth needle-shaped electrodes being arranged mirror-wise with respect to the first needle-shaped electrode and second needle-shaped electrodes, about a symmetry plane arranged orthogonally to the first needle-shaped electrode, wherein the third needle-shaped electrode and fourth needle-shaped electrodes are reversely charged with respect to the first needle- shaped electrode and second needle-shaped electrodes.

The arrangement comprising needle-shaped electrodes and an electrostatic mirror is replaced by a fully planar arrangement of needle-shaped electrodes. This allows for a very compact architecture for the first phase changing element, which can be realized using MEMS devices and installed in small spaces, for instance within an electron microscope.

While the present invention has been conceived, in particular, to overcome some of the limitations of known electron OAM sorters, the skilled person will easily recognize that it can be used potentially in any charged beam-based analyser.

Furthermore, the device according to the invention can be configured for generating vortex beams and coherent superpositions of vortex beams from plane wave beams according to the reciprocity principle. In this case the device comprises, in sequence along the beam axis, the second phase changing element and the first phase changing element.

Further characteristics and advantages of the proposed device will be presented in the following detailed description, which refers to the attached drawings, provided purely by way of non-limiting example, in which:

Figure 1 is a schematic representation of a device according to the invention;

Figures 2 is a plan-view image of a first phase changing element of the device of Figure 1;

Figures 3 is a schematic representation of a further device according to the invention;

Figure 4 is a schematic perspective representation of a variant of the device of Figure 3;

Figure 5 is a schematic plan view representation of an additional component which can be added to the device of Figure 3;

Figure 6 is a schematic plan view representation of a further additional component which can be added to the device of Figure 3.

A device for transforming an azimuthal phase of a charged particle beam into a linear phase is shown schematically in Figure 1. A charged particle beam is designated with 10. The following description will specifically refer to an electron beam; it is however intended that the invention can be potentially applied to any kind of charged particle. A beam axis representing the average direction of the charged particle beam 10 is designated with 20.

A first phase changing element S 1 (hereinafter also indicated as phase unwrapper element) in the input plane of a first Fourier lens system LI is followed by a second phase changing element S2 (hereinafter indicated as phase corrector element) in the Fourier plane of the first lens system LI. After the phase corrector element S2 there is a second, Fourier-transforming lens system L2. A detecting device such as a CCD camera can be positioned at a plane 30 conjugate to the plane of the phase corrector element S2, to capture the spatially resolved spectrum of the 0AM. Coming back to the embodiment of Figure 1, the above described device is essentially based on unwrapping the azimuthal phase variations associated with an electron’s OAM into variations over a Cartesian coordinate of the plane transverse to the electron’s propagation 20. This effectively causes the electron’s original helical wavefront to become planar and inclined with respect to the beam’s original direction of propagation 20. Namely, the degree to which these wavefronts are tilted will increase with the azimuthal variation of the electron’s original phase profile, that is, its OAM. As a result, focusing these unwrapped waves with a Fourier lens system L2 will cause electrons originally carrying different OAM values to focus at correspondingly separate lateral positions. By using this method, it is possible to decompose an electron beam’s OAM content by measuring the relative electron intensity at each of these possible focusing positions.

The unwrapping process, as detailed in [1], requires the beam’s transverse profile in polar coordinates (r, <p) to be physically mapped to Cartesian coordinates. Such a transformation can be achieved by means of a conformal mapping between the Cartesian coordinates of the initial beam’s profile (x, y) and those of its final profile (u, v . The coordinates (it, v are log-polar coordinates and can be related to the Cartesian coordinates (x, y) via the transformations x = exp (it) cos(v) and y = exp (it) sin(v) or equivalently and v = tan ^-1 (y/x).

Figure 2 is a plan- view image of the first phase changing element S 1. A scale bar which is not to be intended as limiting the scope of the invention, is sketched at the lower side of the image.

The first phase changing element S 1 comprises a planar electrostatic arrangement of needle- shaped electrodes arranged orthogonally to the beam axis 20. The beam axis is orthogonal to the plane of Figure 2 which is also the plane of the electrostatic arrangement of needle- shaped electrodes, and the dashed circle B represents the cross-section of the electron beam 10.

The electrostatic arrangement of needle-shaped electrodes comprises a first needle-shaped electrode 41 extending in the plane of the electrostatic arrangement and having a distal tip 41a arranged at the beam axis 20. The electrostatic arrangement of needle-shaped electrodes further comprises a pair of second needle-shaped electrodes 42 for correcting aberrations introduced by the first needle-shaped electrode 41. The second needle-shaped electrodes 32 extend in the plane of the electrostatic arrangement and are arranged on opposite sides of, and oriented orthogonally to, the first needle-shaped electrode 41. The distal tips 42a of the second needle-shaped electrodes 42 are arranged close to the distal tip 41a of the first needle- shaped electrode 41, but outside of the area B occupied by the electron beam 10.

The electrostatic arrangement of needle-shaped electrodes further comprises a third needle- shaped electrode 43 extending in the plane of the electrostatic arrangement, as well as a pair of fourth needle-shaped electrodes 44 extending in the plane of the electrostatic arrangement and being arranged on opposite sides of, and oriented orthogonally to, the third needle- shaped electrode 43. The distal tips 44a of the fourth needle-shaped electrodes 44 are arranged close to the distal tip 43a of the third needle-shaped electrode 43.

The third needle-shaped electrode 43 and fourth needle-shaped electrodes 44 are arranged mirror-wise with respect to the first needle-shaped electrode 41 and second needle-shaped electrodes 42, about a symmetry plane M arranged orthogonally to the first needle-shaped electrode 41. In Figure 2, the symmetry plane M about which the first needle-shaped electrode 41 and second needle-shaped electrodes 42 are reflected into the third needle-shaped electrode 43 and fourth needle-shaped electrodes 44 is represented by a dashed straight line.

The proximal ends of the needle-shaped electrodes 41 to 44 are carried by a support S while the needle-shaped electrodes 41 to 44 cantilever from an edge of a hole formed in the support S. The needle-shaped electrodes 41 to 44 are conductively connected to a control system (not shown). The third needle-shaped electrode 43 and fourth needle-shaped electrodes 44 are reversely charged with respect to the first needle-shaped electrode 41 and second needle- shaped electrodes 42 in order to ensure the neutrality of the electric charge of the electrostatic arrangement of needle-shaped electrodes.

The distance h of the distal tip 41a of the first needle-shaped electrode 41 from the symmetry plane M has only a minor effect on the potential. The exact electrostatic formula of the phase an electron plane wave acquires as it propagates through the potential generated by the first phase changing element S 1 is [4] : where Q/L is the charge density of the first needle-shaped electrode 41, C _E is a constant that depends only on the energy of the beam, and L is the length of the first needle-shaped electrode 41.

However, it is usually assumed that h » x, y and in these conditions the effect of the distance h can be removed.

A simplified formula is disclosed by Pozzi et al [5] that assumes only a single charge compensation: where 6 is a scale factor with the dimension of length.

Even in this case if h is assumed to be very large its effect is just a constant phase. Nevertheless the mirror charge provided by the third needle-shaped electrode 43 and fourth needle- shaped electrodes 44 is necessary to ensure neutrality. If this is not ensured, a long range electrostatic field would exist and its compensating charge could be everywhere in the microscope column. This would introduce complex and uncontrollable aberrations.

To conclude, the distance h must be such that h » x, y (i.e. of the beam size) and smaller than any other electrostatic element, for example the grounding electrodes at the perimeter of the hole in the support S. If the device of the invention is applied in an electron microscope, the distance h could be comprised between 20 pm and 100 pm.

The specific construction of the second phase changing element S2 is not essential to the present invention. For instance, the second phase changing element S2 can be formed by a set of electrostatic elements, such as proposed e.g. by McMorran et al [4], or elements configured to support magnetic currents as disclosed in WO 2020/144630 Al.

Further embodiment of a device for transforming an azimuthal phase of a charged particle beam into a linear phase are shown schematically in Figures 3 and 4. Elements corresponding to those of the preceding embodiment are designated with the same reference numerals; these elements will be not discussed further.

The embodiments of Figure 3 and 4 further comprise a diffraction grating G1. In the embodiment of Figure 3, the diffraction grating G1 is arranged in an intermediate plane between the first phase changing element SI and the second phase changing element S2. According to an alternative embodiment, shown in Figure 4, the diffraction grating G1 could be arranged in the same optical plane as the first phase changing element SI.

The addition of this grating of only amplitude, appropriately calibrated, allows to obtain a lateral repetition (reference numerals 10’, 10”, 10”’) of the electron beam transformed by the phase changing device SI. Therefore, the diffraction grating G1 produces a plurality of repeated images of the charged particle beam transformed by the first phase changing element; as shown in Figures 4 and 5, these repeated images are arranged along a lateral direction orthogonal to the beam axis. This repetition improves the resolution of the device by a factor of 70% by making sure that the peaks corresponding to different quanta of angular momentum are completely separated. It should be noted that by using an intermediate plane and additional lenses XL, it is possible to precisely modulate the separation between repetitions.

The architecture with the grating is inspired by what is called "fan-out sorter" in optics. In fact, an analogous form exists in optics [6] but in a purely diffractive form while here a planar grating is combined with an electrostatic phase which can be adjusted independently of the grating.

It is to be noted that in Figures 4 and 5 the second phase changing element S2 comprises a plurality of oppositely charged elongated electrodes forming a periodic array as disclosed in [4]. As explained above, this construction is not essential to the invention. For the sake of simplicity, the first phase changing element SI is shown as having the first needle-shaped electrode 41 only.

Figure 6 shows a circular opening 51 which can be added to the above discussed device to limit the aperture angle thereof. The circular opening 51 can be formed through an aperture angle limiting element 52 of absorbent material arranged around the diffraction grating (G 1 ). Advantageously, this circular opening can be formed in the same support which holds the diffraction grating G1 (not shown in Figure 6). As used herein, the term “absorbent material” is intended to mean any material that can absorb the charged particles of the charged particle beam 10.

Alternatively, an aperture angle limiting element 61 can be arranged in the plane of the second phase changing element S2. This aperture angle limiting element 61 can consist in an absorbent lamina arranged under the array of electrodes of the second phase changing element S2, and protruding laterally beyond an edge of said array. In the new coordinates after the conformal transformation the lamina is equivalent to the circular opening 51 in the plane of the first phase changing element SI (the cross-sectional shape of the electron beam after the conformal transformation is represented by the closed polygonal line B’). Therefore, the aperture phase changing element 61 forms a ledge that intercepts a lateral portion of the repeated images of the charged particle beam (10) transformed by the first phase changing element (SI). Advantageously, by adjusting the potentials of the second phase changing element S2, the working distance and therefore the angular acceptance can be changed continuously.

Bibliographic references [1] Berkhout GC et al Phys. Rev. Lett. 105, 153601 (2010)

[2] Grillo V et al Nat Comm 8 (2017) 15536

[3] WO 2020/144630 Al

[4] McMorran B et al New Journal of Physics 19 (2017), 023053

[5] Pozzi G et al Ultramicroscopy 208 (2020), 112861 [6] Ruffato et al Sci Rep. 2018 Jul 6; 8 (1): 10248