Holographic Generation of Highly Twisted Electron Beams

Free electrons can possess an intrinsic orbital angular momentum, similar to those in an electron cloud, upon free-space propagation. The wave front corresponding to the electron’s wave function forms a helical structure with a number of twists given by the angular speed. Beams with a high number of twists are of particular interest because they carry a high magnetic moment about the propagation axis. Among several different techniques, electron holography seems to be a promising approach to shape a conventional electron beam into a helical form with large values of angular momentum. Here, we propose and manufacture a nanofabricated phase hologram for generating a beam of this kind with an orbital angular momentum up to $200\hslash$. Based on a novel technique the value of orbital angular momentum of the generated beam is measured and then compared with simulations. Our work, apart from the technological achievements, may lead to a way of generating electron beams with a high quanta of magnetic moment along the propagation direction and, thus, may be used in the study of the magnetic properties of materials and for manipulating nanoparticles.

Figure 1

(a) Designed CGH for generating an EVB with 200 twists. This CGH imprints a specific phase delay between [$-\pi$, $\pi$] onto the impinging beam. (b) SEM image of the nanofabricated phase hologram shown in (a). The hologram forms a pitchfork shape with 200 dislocations at the center. Because of high oscillation of the optical phase around the origin, shown in pink circles, the carrier changes more rapidly in a specific region where the phase alteration goes beyond the spatial resolution. (c) Thickness map of the shaded region of the fabricated hologram (see Supplemental Material for more details [25]).

Figure 2

(a) Experimentally (exp) observed and simulated (sim) Fraunhofer diffraction of a conventional EB from the CGH of Fig. 1. The upper and lower bright doughnuts are the $\pm 1$ orders of diffraction. In order to increase the visibility of the first order of diffraction, the central beam is blocked by a beam stop. The hologram is symmetric, and thus both first orders of diffraction possess the same OAM value, nonetheless having an opposite sign. The distribution of the electrons that possess a phase singularity at the upper side (a) is calculated and shown in (b). The singularity tends to concentrate in a ring at the periphery of the dark region. This is achieved by measuring phase variation about an infinitesimal region close to each pixel at the upper diffraction area. As can be seen, the residual of the second order of diffraction with an OAM value of $400\hslash$ (upper doughnut) is partially superimposed with the beam of first order. This lies in the fact that the diffraction angle of the electrons leaving the hologram is smaller than the divergence angle of the EB. Similar results are achieved for the hologram that its central region, shown in Fig. 1, is blocked.

Figure 3

(a) Calculated phase distribution of the electron wave function around the first order of diffraction. This is calculated by Fourier transforming the experimentally determined transfer function of the hologram from a measured map of the hologram thickness [see Fig. 1]. The phase distribution is shown in polar coordinates, where the phase value is represented in a hue color. The OAM spectrum of the generated beam at the first order of diffraction from the hologram with and without the central dead region are shown in (b) and (c), respectively. This is measured by performing a one-dimensional Fourier transform of the azimuth coordinate. Because of imperfections at the origin and low spatial frequency for the case of (b), the beam is not pure and the value of OAM is delocalized centered around an OAM of $(210.95\pm 0.030)\hslash$, while for the hologram with the excluded central dead region (c) the OAM spectrum central peak is about $(202.75\pm 0.032)\hslash$. Indeed, the OAM spectrum, for both cases, moves to higher values mainly due to a small contribution from the second order of diffraction which carries an OAM of $400\hslash$ [see Fig. 2]. Nevertheless, the hologram with the excluded central dead region (c) gives much better results.

Figure 4

Radial index distribution of the generated highly twisted EB with $m=200$ from the pitchfork hologram shown in Fig. 1. For simplicity, we assumed a hologram with unlimited resolution, and the size of hologram is reasonably larger than that of the impinging beam. The generated beam is delocalized in $p$ space and has a peak around $p=4900$.