Fig. 1. Schematic of the generation of structured EVBs. The binary holographic phase masks can be engineered with the generalized spiral phase to shape the incident free electrons to generate structured EVBs with customizable intensity patterns. The phase mask is composed of nanoscale forked gratings fabricated on 100 nm-thick silicon nitride membranes.

Fig. 2. Theoretical construction of structured EVBs. (a–c) Three parameterized azimuth gradients ${\text{f}}_{\text{1}}\text{'}\left(\theta \right)$, ${\text{f}}_{\text{2}}\text{'}\left(\theta \right)$ and ${\text{f}}_{\text{3}}\text{'}\left(\theta \right)$, corresponding to the patterns of clover, spiral and arrowhead, respectively. (d–f) Desired generalized spiral phase of three EVBs ${\phi }_{\text{1}}$, ${\phi }_{\text{2}}$ and ${\phi }_{\text{3}}$ with identical topological charge l = 30. (g–i) Calculated wavefronts and intensity distributions of three structured EVBs.

Fig. 3. Experimental generation of structured EVBs using nanoscale holographic phase masks with fork dislocations. (a–c) The scanning electron microscope (SEM) images of phase masks for three EVBs. Scale bars: 4 μm. Insets are the enlarged middle portions of phase masks. Scale bars: 1 μm. (d–f) Experimentally recorded far-field intensity distributions and (g–i) phase profiles retrieved by the transmission iterative algorithm. (j–l) Calculated far-field intensity distributions and (m–o) phase profiles. Scale bars of (d–o): 3 μrad. (p–r) In-plane components of the probability current density vectors of three EVBs extracted from experimental measurements. The red arrows indicate the direction of the probability current of electron. The lengths of the red arrows are given in arbitrary unit normalized to the maximum value. Scale bars of (p–r): 3 μrad.

Fig. 4. Modal decomposition of EVBs. The OAM spectral distributions of three structured EVBs extracted from the experimental results. The blue, green and red histograms correspond to clover, spiral and arrowhead EVBs, respectively.

Fig. 5. Coherent superposition of structured EVBs. (a–c) The SEM images of fabricated phase masks to generate superposition states of topological charge l₁ = 30 and l₂ = 33, corresponding to three structured EVBs. Scale bars: 4 μm. Insets are the enlarged middle portions of phase masks. Scale bars: 1 μm. (d–f) Experimentally recorded interference patterns of the superposition states. (g–i) Simulated interference patterns of superposition states. Scale bars of (d–i): 3 μrad.