Measuring the orbital angular momentum spectrum of an electron beam

Abstract

Electron waves that carry orbital angular momentum (OAM) are characterized by a quantized and unbounded magnetic dipole moment parallel to their propagation direction. When interacting with magnetic materials, the wavefunctions of such electrons are inherently modified. Such variations therefore motivate the need to analyse electron wavefunctions, especially their wavefronts, to obtain information regarding the material's structure. Here, we propose, design and demonstrate the performance of a device based on nanoscale holograms for measuring an electron's OAM components by spatially separating them. We sort pure and superposed OAM states of electrons with OAM values of between -10 and 10. We employ the device to analyse the OAM spectrum of electrons that have been affected by a micron-scale magnetic dipole, thus establishing that our sorter can be an instrument for nanoscale magnetic spectroscopy.

Figure 1. Schematics of the electron sorter.

These schematics also show an electron beam's experimental transverse intensity profile recorded at various planes in the sorting apparatus. A hologram in the sorter's generator plane that corresponds to the electron microscope's condensor, produces an electron beam carrying orbital angular momentum (OAM). In this particular case, the beam consists of a superposition of ±5 OAM states. The beam then goes through a hologram in the apparatus' sorter plane, positioned at the microscope's sample holder, that performs the required conformal mapping . Once the beam is unwrapped, it passes through a hologram in the sorter's corrector plane corresponding to the microscope's selected area diffraction (SAD) aperture. This hologram brings corrections to any phase defects to the beam to stabilize its propagation through the rest of the sorter. At the sorter's output, the original beam's OAM content is spatially resolved on a screen and captured by a CCD camera. A more detailed schematic of our sorter's implementation is included in Supplementary Fig. 1 and Supplementary Note 1 where we provide details concerning the electron microscope lenses required to perform the mapping. Scanning electron microscopy (SEM) images of the depicted holograms, the ones in the generator, sorter and corrector planes, are shown in a–c, respectively.

Figure 2. Experimental OAM spectra of electron beams.

Spectrum of a beam consisting of electrons defined by: (a) OAM of +1, ψ₊₁, produced with a spiral phase plate; (b) a superposition of ±4 OAM states, , generated by a phase mask; (c) a superposition of ±5 OAM states, , generated by a phase mask; and (d) OAM of +10, ψ₊₁₀, produced by a spiral phase plate. Scanning electron microscopy (SEM) images of the devices used to generate the analysed electron beams are provided in the insets of their respective spectra.

Figure 3. OAM spectrum of a beam affected by a magnetic dipole.

(a) Scanning electron microscopy (SEM) image of the analysed magnetic bar configurated as a dipole. The bar is defined by dimensions of 100 nm (thickness) by 200 nm (width) by 2.8 μm (length). (b) Magnetic field lines of the magnetic bar measured using electron holography. (c) Expected (curve) and obtained (bars) OAM spectrum acquired by the electron beam upon interacting with the magnetic bar. The expected curve was calculated assuming that its magnetic field is saturated. Unlike the measurement performed in (b), the experimental data were obtained while the dipole was exposed to a field in the condenser plane of the electron microscope.