Electron vortices: Beams with orbital angular momentum

The recent prediction and subsequent creation of electron vortex beams in a number of laboratories occurred after almost 20 years had elapsed since the recognition of the physical significance and potential for applications of the orbital angular momentum carried by optical vortex beams. A rapid growth in interest in electron vortex beams followed, with swift theoretical and experimental developments. Much of the rapid progress can be attributed in part to the clear similarities between electron optics and photonics arising from the functional equivalence between the Helmholtz equations governing the free-space propagation of optical beams and the time-independent Schrödinger equation governing freely propagating electron vortex beams. There are, however, key differences in the properties of the two kinds of vortex beams. This review is primarily concerned with the electron type, with specific emphasis on the distinguishing vortex features: notably the spin, electric charge, current and magnetic moment, the spatial distribution, and the associated electric and magnetic fields. The physical consequences and potential applications of such properties are pointed out and analyzed, including nanoparticle manipulation and the mechanisms of orbital angular momentum transfer in the electron vortex interaction with matter.

Figure 1

The phase character of $l=1$ and 3 vortex beams. (a) The single helix phase front of an $l=1$ vortex around the beam axis. (b) The corresponding continuous phase ramp in one of the transverse planes perpendicular to the beam axis. The total phase change on one rotation is exactly $2\pi$. For the $l=3$ vortex there are three helical surfaces of constant phase, each moving around the axis as shown in (c). This leads to a total phase change on one rotation of exactly $6\pi$. We used wrapped phase representation, with all phases mapped to values between 0 and $2\pi$ [represented between blue to red in the 2D phase maps in (b) and (d)], so there are three “artificial” phase jumps from 0 to $2\pi$ in the phase-wrap representation as shown in (d). In both cases, the only real phase discontinuity of significance is on the beam axis.

Figure 2

Schematic representation of the Gaussian profile showing the characteristic parameters, namely, the width $w(z)$, with $w_0$ the width at the narrowest part of the beam. $z_R$ is the Rayleigh range and $R(z)$ is the in-plane radius of curvature at axial position $z$

Figure 3

Intensity distribution patterns for the ${\mathrm{LG}}_{\text{pl}}$ modes, shown in the $z=0$ plane. Intensity is given by $|{\psi }_{p,l}^{\mathrm{LG}}{|}^2$. The concentric ring structure of the orbital angular momentum carrying modes is clear, with $p+1$ rings. The color scale shows the intensity variation within individual modes (not the relative intensity variation across all modes). The Laguerre-Gaussian modes with $l<0$ have the same intensity distributions as shown; however, the phase (not shown) has the opposite sign.

Figure 4

The intensity $|{\psi }_l^{\mathrm{B}}(\mathbf{r}){|}^2$ of a Bessel beam with (a) $l=0$, (b) $l=1$, and (c) $l=2$. The Bessel modes with $l<0$ have the same intensity distributions as shown; however, the phase (not shown) has the opposite sign.

Figure 5

The Fourier transform of the Bessel beam results in a set of waves of fixed $k_z$ and varying $k_x$ and $k_y$, such that $k_{\perp }=\sqrt{k_x^2+k_y^2}$. The vortex Bessel beam illustrated here has a phase factor $e^{il\phi }$, so that the phase changes by $2\pi l$ on rotation about the $k_z$ axis. This is illustrated for $l=1$. The relationship between $k_z$ and $k_{\perp }$ fixes the cone angle $\theta$.

Figure 6

The intensity and phase distribution of the transverse wave functions of FT-TBB at $z=0$. From [234] and are plotted for the same relative scale.

Figure 7

Computer-simulated fine structure of the diffraction of truncated Bessel beams (FT-TBB): the first row is for intensity and the second row for the corresponding phase distribution. From left to right, for the FT-TBB vortex beams with $l=1$ but with different radial modes ($p=0$, 1, and 2). Adapted from [234].

Figure 8

The electric fields of the Bessel beams of finite radial extent, for (a) $l=1$ and (b) $l=10$, respectively. Adapted from [162].

Figure 9

The magnetic fields of the Bessel beam of finite radial extent, for (a) $l=1$ and (b) 10, respectively. Note that the $z$ components of the magnetic fields are 2 orders of magnitude smaller than the $\varphi$ component. We assumed that the vortex beam is sufficiently long, as for a current carrying solenoid, allowing us to ignore beam ending effects. Adapted from [162].

Figure 10

The effect of the magnetic field on the circulating current of the nondiffracting Laguerre-Gaussian vortex modes for a magnetic field directed in the positive $z$ direction, such that the vector potential has the same sense of circulation as the $l>1$ vortex modes. The local character of the azimuthal current is indicated with arrows; it can be seen that the presence of the field induces circulation, even in the modes for which $l=0$. For those modes with $l>0$ the observable OAM (listed at the top of each figure) is increased from that of the bare mode, while those beams with $l\le 0$ have a fixed observable OAM, determined by the field strength and the radial quantum number $p$. For those beams with $l<0$ the azimuthal current changes direction at the point of maximum intensity. The top (bottom) panel is for the nondiffracting LG vortex modes with radial index $p=0$ ($p=1$). The scale bar length is $2w_B$. Adapted from [53].

Figure 11

Phase and intensity of the electron wave packet around the circular cyclotron trajectory. The wave packet can be seen to rotate at ${\omega }_L={\omega }_c/2$, such that after one rotation the OAM vector points opposite to the original direction. The length and width of the wave packet also oscillates on rotation with the cyclotron frequency. Note that the scale of the plots of $\Re ({\mathrm{\Psi }}_1)$ is roughly 5 orders of magnitude smaller than that of the $|{\mathrm{\Psi }}_1{|}^2$ plots, so that the phase can be seen. From [92].

Figure 12

The typical forked mask pattern produced by binarization of the interference pattern between a Bessel vortex wave and a plane reference wave for (a) $l=1$ beam, and (b) $l=3$ beam.

Figure 13

The far-field diffraction pattern and phase distribution of the binary CGH amplitude mask to produce ($l=1$, $p=1$) FT-TBB beams shown in Fig. 7. (a) The diffracted beams, with several diffraction orders present; high intensity is indicated by white, zero by black. The phase of the beams is shown in (b), the opposite phase of the two sets of sidebands can be seen, with the $n$th order beams displaying a phase change of $2\pi n$. The rainbow scale indicates phase change from 0 (red) to $2\pi$ (purple). (c) The intensity of the various vortex beams. The central minima of all the diffracting beams are clear in the intensity patterns.

Figure 14

Comparison of the theory and the simulation of the nonzero order approximate Bessel-type electron vortex beam. The topological order of the beams is denoted by $n$, which is identical to the $l$ defined in this review. Adapted from [101].

Figure 15

The blazed phase grating. (a) An SEM image of the mask, with electron-energy-loss measured thickness profile in (b). (c) Typical heights of the grating maxima and minima, with an ideal blazed profile in (d) for comparison. Adapted from [98].

Figure 16

Spiral interference patterns and masks of vortices interfering with spherical waves. (a) The in-plane intensity pattern for a nonvortex beam interfering with an outward propagating spherical wave; binarization of this intensity pattern, forming a Fresnel zone plate, is shown in (d). (b), (e) The continuous and binarized interference patterns, respectively, for an $l=1$ vortex beam interfering with the spherical wave. (c), (f) The same for the $l=3$ vortex beam.

Figure 17

Vortex probes at focus of the condenser lens of a JEOL 2200FS aberration-corrected tranmission electron microscope operating at 200 kV. The central spot is the image of the nonvortex beam, while the other spots are images of the diffracted vortex beams produced by a forked diffractive hologram with a binary pattern similar to that given in Fig. 12. Sizes of the first and second vortex cores at FWHM are 3.4 and 2.6 nm, respectively. Because of the incoherent effect, the intensity dip is only partially visible for the second-order vortex beam.

Figure 18

The atomic model system interacting with a vortex beam whose cylindrical axis is along the $z$ direction of the laboratory frame of reference. The vectors ${\mathbf{r}}_e,{\mathbf{r}}_p,\mathbf{R}$, $\mathbf{q}$, and ${\mathbf{r}}_v$ refer, respectively, to the position vectors of the atomic electron, the nucleus, the center of mass, the internal position relative to the center of mass, and the position variable of the vortex electron. The corresponding $\varphi$’s represent the azimuthal angles and these are important for the description of the phase factors.

Figure 19

Illustration of the various interaction channels available in (a) the case in which the atom is situated on axis, and (b) the off-axis case. In (a), any change in OAM of the atomic electron is immediately apparent as a change in OAM of the transmitted beam $\mathrm{\Delta }l=l^{\prime }-l$. The case in (b) is more complicated due to the expansion modes having various OAM $p$, which may each transfer any number of units of OAM to the atomic electron. The resulting changes $\mathrm{\Delta }l$ values may correspond to a variety of combinations of expansion modes and multipolar transitions, as shown. However, due to the relative strengths of the expansion modes and multipolar atomic transitions, the dominant interaction channels arise from the dipole interaction with the $p=l$ mode. Note that $p$ is used here only to indicate the OAM modes involved in an off-axis vortex beam, not as a radial index as in the rest of the text. From [255].

Figure 20

Schematic of suggested experimental setup for OAM based spectroscopy using electron vortices. A vortex beam produced by a holographic mask or other suitable method (not shown) is incident onto a thin sample in the specimen plane. After interaction with the sample, the transmitted beam is then passed through a forked mask in order to separate the various vortex components. A pinhole placed in the diffraction plane allows isolation of those modes that have an OAM of 0 after passing though the mask; these can then be detected to obtain EELS.

Figure 21

Spatial maps of electron-energy-loss probability for (a) magnetic resonance at 0.863 eV, with an $l=1$ beam, and (b) electric resonance at 0.8 eV with a plane wave beam, both at 100 keV beam energy. The two plasmon resonance maps show markedly different spatial profiles, as well as the magnetic resonance response being approximately an order of magnitude less than the electric response. From [181].

Figure 22

Four snapshots of Au nanoparticles rotated by second-order vortex beams selected from a video at 1.2 s intervals. The center dark core surrounded by the bright ring of the first-order vortex beams is partially visible at the bottom right corner. The angles of lattice fringes correspond to 99.5°, 99.0°, 87.0°, and 84.5°, respectively. The insets show the corresponding fast Fourier transform.