Orbital-Angular-Momentum Mode Selection by Rotationally Symmetric Superposition of Chiral States with Application to Electron Vortex Beams

A general orbital-angular-momentum (OAM) mode selection principle is put forward involving the rotationally symmetric superposition of chiral states. This principle is not only capable of explaining the operation of vortex generating elements such as spiral zone plate holograms, but more importantly, it enables the systematic and flexible generation of structured OAM waves in general. This is demonstrated both experimentally and theoretically in the context of electron vortex beams using rotationally symmetric binary amplitude chiral sieve masks.

Figure 1

Management of the OAM modes using rotationally symmetric masks constructed from different rotationally symmetric motifs: (a) the five-pinhole mask, (b) the simulated intensity pattern and (c) phase pattern at defocus $\mathrm{\Delta }f=-33.6\mu \mathrm{m}$, and (d) the corresponding power spectra in the OAM mode $\ell$. (e)–(h) are based on five short pinhole curves (motifs). (i)–(l) are based on logarithmic spirals. (m)–(p) are based on Archimedean spirals. (q)–(t) are based on the Fermat spirals. Here, the expansion area of the complex fields that were decomposed into the LG basis set was $10\times 10\mathrm{nm}$.

Figure 2

Generation of concentric electron vortex beams and the measurement of their topological charges. (a) Simulation of a structured electron sieve. [(b) and (c)] The corresponding simulations of the intensity (b) and phase (c) at defocus $\mathrm{\Delta }f=-74.6\mu \mathrm{m}$. (d) The scanning electron microscope image of the electron sieve. (e) The experimental results of the intensity patterns. (f) A zoomed-in view of the center part of (c). The white curves in (f) indicate increasing direction of phase. [(g) and (h)] The experimental results (g) and the corresponding simulation (h) of the intensity pattern after astigmatic transformation.

Figure 3

The propagation of vortex beams generated by electron sieves. (a) Experimental results of the intensity pattern in the $y\text{−}z$ plane. (b) The corresponding result of simulation. The experimental result in (a) was reconstructed from 140 slices of the $x\text{−}y$ plane intensity pattern recorded in the experiment. $P1–P4$ represent four transverse planes perpendicular to the propagation axis. $P2$ corresponds to the intensity patterns shown in Fig. 2, and $P4$ denotes the focal plane of the condenser lens.