In summary, we have developed a nanoscale platform for generating and manipulating structured EVBs in free-space. By designing the generalized spiral phases and imparting them onto the incident free electrons, we experimentally demonstrate that EVBs can be engineered with different intensity patterns. Unlike classic vortices, the orbital motions of structured EVBs are not attributed to the collective behavior of many electrons, and no external field or force are necessary for the generation. Thanks to its additional controllable degree of freedom, the structured EVB as a quantum electron probe holds great potential in electron microscopy³² and can further promote various in-situ applications, such as electron manipulation of nanoparticles along designed trajectories³³, pattern-dependent interaction of electron OAM with matter³⁴ and selectively exciting and probing surface plasmon modes^{35, 36}. The structured EVBs also can be directly used in lithography to produce shaped nanostructures without the need to scan the beam. Moreover, such concept and generation approach are convenient to generalize to other particle systems, such as neutron³⁷, proton³⁸, atom and molecule^{39, 40}.

where $\phi$ is the phase, $s$ is the unit vector tangential to the closed path C, and $l$ is the topological charge. Ordinarily, the phase of EVB uniformly varies following a circular path around the vortex center and can be simply expressed as $l\theta$ ($\theta$ is azimuthal angle). Here, for generalizing the definition of EVB, we rewrite the wave function of EVB as $\Psi =A{\text{e}}^{\text{i}lf\left(\theta \right)}$, where A is the amplitude, and $f\left(\theta \right)$ is parameterized azimuthal angle under constraint condition $f\left(2\pi \right)-f\left(0\right)=2\pi$. The phase gradient $\partial \phi /\partial \theta =lf\text{'}\left(\theta \right)$ is no longer a constant but a function varying along the azimuthal direction. According to the ray theory, the local divergence angle of the electron beam is proportional to the phase gradient $lf\text{'}\left(\theta \right)$ (see Supplementary information Section 1 for more details). Therefore, for a given topological charge l, the modulated electron beam would evolve into a structured vortex in the far-field with the geometry same as the contour of the parameterized azimuth gradient $f\text{'}\left(\theta \right)$. In other words, through deliberately designing the trajectory of $f\text{'}\left(\theta \right)$ in polar coordinates, the structured EVB is expected to exhibit desired intensity distribution. As proof-of-concept, herein we employ computer-generated hologram and design binary phase masks to demonstrate the creation of three types of structured EVBs, including a clover EVB with three-fold rotational symmetry, a non-closed spiral EVB and a custom arrowhead EVB, as illustrated in Fig. 1.

In principle, a free-space EVB is formed by continuous phase distributions around the center singularity that can be evaluated as the loop integral of the phase gradient in the transverse plane²⁸,

Thus far, all the experimentally generated EVBs with integer topological charge exhibit isotropic doughnut intensity patterns. One approach to break the isotropy is to introduce a fractional topological charge^{24, 25}, which would consistently generate C-shaped EVBs. Actually, a few of early studies in optics suggest that the intensity pattern of vortex structure can be engineered on-demand by introducing nonuniformity into the phase gradient around the central singular point^{26, 27}. Here, we generalize the concept of EVBs and present an experimental generation of structured EVBs in free-space. By designing and fabricating nanoscale holographic masks to shape the incident free electrons in the transmission electron microscope (TEM), three structured EVBs with identical OAM are engineered to exhibit customizable and completely different intensity patterns. Furthermore, based on the modal decomposition, we quantitatively investigate the OAM spectral distributions of structured EVBs and reveal that, although macroscopically the structured EVBs can be quantified and characterized by a single integer that describes the global topological invariance, which is similar to the ordinary ones, microscopically they present a superposition of a series of different eigenstates induced by the locally varied geometries. As the structured EVBs simultaneously have two independent controllable degree of freedom for electron beam manipulation including topological charge and intensity pattern, we envision this work may open new frontiers in related fundamental research and technical application of particle physics.

To better understand the physical picture of structured EVB, we further investigate the probability current density in quantum electron states, defined as $j=\hslash \text{Im}({\psi }^{*}\nabla \psi )/m$, where $m$ is mass of a nonrelativistic electron and wave function $\psi$ can be described by amplitude and phase²³. According to its definition, the probability current density can reflect the local phase gradient in an arbitrary electron field, and thus is very suitable to characterize the angular momentum property of a free-space electron wave function. Figure 3(p−r) present the in-plane components of the experimentally measured probability current density vectors of three structured EVBs. For each case, although the current density is nonuniformly distributed, it follows the gradient of the electron phase and produces an effective loop of charge current, and thus forms a vortex around the beam axis. These results agree well with the calculated ones shown in Supplementary information Fig. S5. The probability current density can be further used to calculate the average electron OAM via the following volume integration: $\langle L_z\rangle =m\left|{\displaystyle \int }\rho \times j\text{d}V\right|/\left({\displaystyle \int }{\psi }^{*}\psi \text{d}V\right)$, where $\rho$ is the radial coordinates. For these three EVBs, the corresponding average OAM per electron are obtained as 28.1$\hslash$, 28.3$\hslash$ and 28.6$\hslash$, respectively, which are all close to the expectation OAM corresponding to the imparted topological charge l = 30. Moreover, based on modal decomposition approach, we quantitatively analyze the OAM spectral distributions of three structured EVBs (see Supplementary information Section 5 for details). By projecting the wave function into the helical harmonics of different orders, the weight factor of each intrinsic OAM mode is obtained, as shown in Fig. 4. In contrast to ordinary EVBs, the OAM modes of structured EVBs span a large spectral range, which indicates that they present as a superposition of a series of eigenstates. Therefore, although these structured EVBs have almost identical average OAM, their locally varied geometries lead to completely different spectral distributions.

Vortices widely exist on all scales in nature, such as gravitational vortices around black holes in universe, typhoon vortices on earth and quantum vortices in super-fluids. Among them, quantum vortices are particularly interesting because they can be represented by free movement of a single particle, which was first proposed by Allen et al. in 1992¹. It was discovered that an optical vortex beam with integer topological charge l can carry a quantized orbital angular momentum (OAM) l$\hslash$ per photon (where $\hslash$ is the reduced Planck’s constant). This discovery unveiled a new understanding of the connection between macroscopic optical phenomenon and quantum mechanism^2-7. The quantized OAM of photon offers a new degree of freedom of light and has significantly broadened the avenues of application including optical trapping^{8, 9}, high-resolution microscopy^{10, 11} and information processing^12-14.

${{\displaystyle \oint }}_C\nabla \phi \cdot \text{d}s=2\pi l\mathrm{,}$(1)

As another important elementary particle, electrons have drawn considerable attention in the fields from electron microscopy to nanofabrication. Different from massless neutral photons, electrons are massive charged particles and obey Fermi-Dirac statistics. In spite of this, electrons can still be represented by waves and free-electron wave functions. As a result, similar to photons, electrons are also able to be shaped to produce electron vortex beam (EVB) carrying OAM. In 2010, EVB was first generated by using spiral phase plates consisting of spontaneously stacked graphite thin films to impart OAM onto the incident electron beam¹⁵. Thereafter, several approaches, such as employing holographic masks^16-19, magnetic lens aberrations²⁰ and magnetic needles²¹, are reported to generate EVBs. Because EVB can effectively interact with matter as well as with external electric and magnetic fields, it is now starting to find applications as nanoscale magnetic probes for characterizing materials^{22, 23}.

In addition to generation, we also investigate the coherent superposition of two structured EVBs with different topological charges of $l_1$ and $l_2$. The morphology of phase masks designed for creating superposed states of structured EVBs are determined by the product of two transmission phase functions respectively corresponding to $l_1$ and $l_2$. For each type of structured EVB, we fabricate phase mask simultaneously imparted with topological charge l₁ = 30 and l₂ = 33, as shown in Fig. 5(a–c). The experimentally recorded transverse intensity distributions of three states are shown in Fig. 5(d–f). In spite of different intensity distributions, all the superposed states clearly exhibit petal-like interference patterns with exactly 3 lobes, which are consistent with the calculated results (Fig. 5(g–i)). This demonstration verifies that although microscopically structured EVBs consist of a range of discrete OAM modes, the coherent superposition state of two structured EVBs still depends on their global topological invariances and thus yields interference pattern with $\left|l_1-l_2\right|$ petals, behave similarly as the unstructured ones.

Figure 2(a–c) show the calculated parameterized azimuth gradients $f\text{'}\left(\theta \right)$ for these three structured EVBs. For the clover EVB and spiral EVB, $f\text{'}\left(\theta \right)$ are given by $f_1\text{'}\left(\theta \right)=1+\text{cos}\left(3\theta \right)/2$ (Fig. 2(a)) and $f_2\text{'}\left(\theta \right)=1-\text{cos}\left(\theta /2\right)/2$ (Fig. 2(b)), respectively. For more complex structured EVB with unformulated pattern, such as the custom arrowhead vortex beam proposed here, $f_3\text{'}\left(\theta \right)$ can be presented by the Fourier series expansion (see Supplementary section 2 for more details). Through integration of $f\text{'}\left(\theta \right)$ and then multiplying topological charge l, we can obtain the desirable parameterized phase profile ${\phi }_1$, ${\phi }_2$ and ${\phi }_3$ (Fig. 2(d–f)), which will be used to shape the incident free electrons and generate the corresponding structured EVBs. Based on these results, Fig. 2(g–i) present the calculated wavefronts and intensity distributions of three structured EVBs with identical topological charge l = 30. Different from the ordinary helical wavefront of EVB with a constant phase gradient, the helical wavefronts of structured EVBs exhibit non-uniformly distributed phase gradients. As expected, the calculated intensity distributions in the far-field reveal three EVBs exactly exhibit clover, spiral and arrowhead intensity patterns.

In experiment, we fabricate binary holographic phase masks composed of nanoscale forked gratings to create the structured EVBs. The phase masks can be imprinted with desired phase shift by adjusting the thickness of electron-transparent material^29-31. Here, a transverse composite phase profile $\phi$, which is the superposition of a generalized spiral phase and a Bragg carrier phase, imparts to the phase masks (see Supplementary Section 3 for more details). The additional Bragg carrier phase term can separate different diffraction orders and avoid crosstalk between them when electron beams pass through the grating masks. Corresponding to three structured EVBs mentioned above, the fabricated phase masks on 100 nm-thick Si₃N₄ membranes by focused-ion beam (FIB) milling are shown in Fig. 3(a–c). Each mask is etched in a circular area with a diameter d = 15 $\mu \text{m}$, and the grating is designed with a periodicity p = 150 nm. The enlarged insets of the fabricated masks show clear feature of the fork dislocations which indicates the encoded topological charge l = 30.

When illuminated with free electrons, the phase masks can impart the predesigned phase front onto the electron wave packets at the first order of diffraction, which would form the structured EVBs in the far-field. These generated EVBs are captured at the focal plane of magnetic lens system in the TEM. Figure 3(d−f) show the experimentally recorded images of the diffracted electron waves through the phase masks (+1 diffraction order), which clearly show clover, spiral and arrowhead intensity patterns, respectively. The electron phase maps, acquired by using the defocus series reconstruction technique based on the Gerchberg-Saxton transmission iterative algorithm (see Supplementary information Section 4 for more details), are given in Fig. 3(g−i). The phase distributions show that the modulated electron beams have a 2$\pi$l azimuthal phase swirl in free-space, which confirms their vortex features. These experimental results match very well with the calculated ones those are obtained by Fourier transform on the corresponding binary phase distributions (Fig. 3(j−o)). Additionally, in order to quantitatively analyze the geometries of the structured EVBs, the diffraction angle distributions are also experimentally measured. Owing to a relatively constant focal length of the Fourier transform lens during the measurements, the diffraction angle can be conveniently used as a substitute for the radial distance in the intensity pattern of EVBs. As shown in Supplementary information Fig. S4, the measured diffraction angle distributions are consistent with the theoretical predictions.

In this work, the binary phase masks are fabricated on 100 nm-thick silicon nitride membranes by using focused-ion-beam (FIB) milling. The FIB tool utilizing Ga-ions is operated at acceleration voltage of 30 keV and beam current of 7.7 pA. The patterning is performed with a minimum step size of 4 nm and a dwell time of 0.42 ms.

The evolution of electron vortex is observed in a Tecnai TF20 field emission gun TEM, which produces a coherent monoenergetic electron beam with an energy of 200 keV, corresponding to the relativistic de Broglie wavelength of approximately ~2.51 pm. The phase masks are placed at the front focal plane of the magnetic lens. When illuminated with coherent electron beam, the phase mask can shape the electron wave packets to form structured EVBs in the back focal plane of magnetic lens system. The diffraction patterns are recorded using low angle diffraction (LAD) mode, for which the magnetic lens is operated at low current. The LAD mode enables long camera length, so the diffractive angles for electrons <1 μrad can be measured. A charge-coupled device (CCD) camera is used to capture the propagation dynamics of the generated EVBs.