It is well known that the wavefunction for a free electron can be a plane wave. In 2007, a vortex wavefunction for the electron was predicted^[1] and was demonstrated soon.^[2–5] An electron vortex beam (EVB) carries orbital angular momentum and has magnetic moment. These features have been actively explored as chiral-dependent energy-loss spectroscopy in ultramicroscopy for detecting magnetic properties in condensed matter and spintronics^[6–8] at nano scale or even down to atomic size. Using the electron’s orbital angular momentum to generate spin-polarized electron beams has been proposed.^[9] Moreover, the mechanical properties of an EVB^[10] can be used for nanoparticle manipulation.^[11,12] When EVBs are investigated for probing materials,^[13] it is found that transition radiation induced by giant magnetic moment could be detected,^[14–16] especially quantum correction to Cherenkov radiation could be detected^[17,18] and could be further developed as a macroscopic diagnostic tool for electron vortex beams^[19] or Cherenkov concentrator.^[20,21] Moreover, optical vortex^[22–25] can be generated by undulation of twisted electrons.^[26,27]

In electron energy-loss spectroscopy (EEL),^[3,28–32] a relativistic electron beam of 100–200 keV is typically used, calling for relativistic description using the Dirac equation.^[33] Recently, Bliokh et al.^[34] have found a relativistic Bessel-type solution to the Dirac equation. It was shown that compared with its nonrelativistic counterpart, the relativistic electron vortex beam (REVB) has an intrinsic spin angular momentum and spin–orbit coupling.^[34] How to demonstrate the relativistic quantum effect of REVB is yet to be investigated. Further, to study the interaction of an electron beam with materials, it is essential to study the feature of the electromagnetic field generated by an EVB. The electromagnetic field generated by the relativistic EVB and its mechanical properties is pending for investigation.

In this paper, we investigate the electric and magnetic fields generated by a relativistic EVB.^[34] It is shown that the electric field and the magnetic field can be separated into two parts, one is only related to the orbital quantum number as a demonstration of the nonrelativistic quantum effect, and the other is related to the spin–orbit coupling effect intrinsic in the REVB. Thus we propose to measure the relativistic quantum effect through the electromagnetic field of the REVB. We find that the electromagnetic field of the REVB is a vortex field. We further study mechanical properties (the linear and angular momentums) of the electromagnetic vortex field.

The paper is organized as follows. We develop the theory of electromagnetic field of a relativistic Bessel beam in Section 2, give the numerical results in Section 3, and finally present the inclusions.

Our analysis starts from a Bessel-type relativistic EVB of the topological charge l with the radial and axial wave vectors k_⊥ and k_z, given by^[34]

Bliokh et al.^[34] have obtained the probability density of an REVB as follows:

In this paper, we note that the probability density and current density can be written as a sum of two parts. One is related to nonrelativistic quantum effect and the other corresponds to the relativistic quantum effect. For the probability density, (hereafter, superscripts (nr) and (r) are used to represent nonrelativistic quantum effect and relativistic quantum effect, respectively) with , which is irrelevant to the spin state and thus comes from the nonrelativistic quantum effect, and , in which the quantum number l and s cannot be separated, thus coming from the relativistic quantum effect characterized by the spin–orbit coupling. For the current density, we note that 2s = ±1. Using the recurrence relation of the Bessel function,^[37] equation (4) can be rewritten as , in which

Using Gauss’s law ∇·E_l,s (r) = –eρ_l,s(r)/ε₀ with the electric charge e and vacuum dielectric constant ε₀ for electric field E_l,s(r), we obtain the electric field of the REVB, with

Further using Ampere’s circuit law ∇ × B_l,s (r) = –μ₀ej_l,s(r) with vacuum magnetic permeability μ₀ for the magnetic field B_l,s(r), we obtain the magnetic field B_l,s(r) generated by the EVB, B_l,s(r) = B_l,φe_φ + B_l,s,z e_z with

We also note that the electric field and the magnetic field in the transverse plane are zero at the vortex center ρ = 0(E_l,s(0), B_l,φ(0) = 0). Thus the transverse electric field and magnetic field are vortex fields.

Finally, we explore the mechanical properties of REVB-induced electromagnetic field, such as the linear momentum and the angular momentum. The electromagnetic momentum density^[10] . The linear momentum of the electromagnetic fields is given by

The angular momentum of the electromagnetic field reads

In Section 2, we have shown that the differences of the electric fields and z-component of magnetic fields between spin-up and spin-down states, E_l,1/2(ρ) – E_{l, –1/2}(ρ) and B_l,1/2,z(ρ) – B_l,–1/2,z(ρ) can be used to demonstrate the relativistic quantum effect. Therefore we numerically investigate how they could be controlled with the topological charge l, the beam kinetic energy E_k and the opening angle α, using radial range nm, m in the calculation. The numerical results are presented in Fig. 1. Figure 1(a), plotted with α = 0.7 rad and kinetic energy E_k = 0.2 MeV, shows that with the increasing topological charge, the differences of both the electric and magnetic fields between two spin states can be reduced. Thus it is better to use the REVB with small topological charge for testing the relativistic quantum effect intrinsic in the REVB. Figure 1(b), plotted with α = 0.7 rad and l = 1 for kinetic energy E_k = 0.2 MeV, 0.6 MeV, and 1 MeV, shows that with the increasing kinetic energy, the difference of electric field between the two spin states can be increased. In contrast, the difference of the magnetic field strength between the two spin states is not changed, but the peak positions have been shifted. Figure 1(c), plotted with l = 1, kinetic energy E_k = 0.2 MeV for α = 0.2 rad, 0.7 rad, and 1.2 rad, shows that the increasing opening angle α can enhance the difference between the electric or magnetic field strengths in the two spin states. Equations (8) and (11) show that the strengths of both E_l,1/2(ρ) – E_l,–1/2(ρ) and B_l,1/2,z(ρ) – B_{l, – 1/2,z}(ρ) are proportional to sinα, while the dependence of the peak positions of both E_l,1/2(ρ) – E_{l, – 1/2}(ρ) and B_l,1/2,z(ρ) – B_{l, – 1/2,z}(ρ) on α arises from k_⊥ = ksinα.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. The differences of electric fields (left) and magnetic fields (right) of the REVB between the spin-up (s = 1/2) and spin-down (s = –1/2) states, El,1/2(ρ) – El, – 1/2(ρ) and Bl,1/2,z(ρ) – Bl, – 1/2,z(ρ), vs. ρ for three cases: (a) l = 1, 5, 10, α = 0.7 rad, Ek = 0.2 MeV; (b) Ek = 0.2 MeV, 0.6 MeV, 1 MeV, l = 1, α = 0.7 rad; (c) α = 0.2 rad, 0.7 rad, 1.2 rad, l = 1, Ek = 0.2 MeV.

We further study the relation of the linear and angular momentums of the electromagnetic vortex fields in Fig. 2 to topological charge l, opening angle α, kinetic energy E_k as well as the spin state. Figure 2 shows that the linear momentum of the vortex field is weakly dependent on spin state, while the difference between the angular momentums in spin-up and spin-down states is remarkable. With the increasing topological charge l, the linear momentum decreases. In contrast, the angular momentum increases, especially the difference of the angular momentums between the two spin states gets pronounced, as a result that the spin–orbit coupling effect increases.^[36] Equations (12)–(17) shows that both the linear momentum and the angular momentum decrease with k_⊥. Thus figures 2(b) and 2(c) show that both the linear momentum and angular momentum decrease with increasing the opening angle (k_⊥ = ksin a) and the incident kinetic energy E_k.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. Linear momentum and angular momentum of the electromagnetic field of the relativistic EVBs, respectively, in spin-up (s = 1/2) and spin-down (s = –1/2) states vs (a) topological charge l for opening angle α = 0.7 rad, kinetic energy Ek = 0.2 MeV, (b) opening angle α for l = 1 rad, Ek = 0.2 MeV, (c) kinetic energy Ek for α = 0.7 rad, and l = 1.

In summary, we have deduced the formula for electromagnetic field generated by a truncated relativistic electron vortex beam, and show that the electric and magnetic fields are vortex fields in the transverse plane. Moreover, they are dependent on spin-state and vortex topological charge. The electric field includes two parts, i.e., the relativistic effect and the nonrelativistic effect. The magnetic field has azimuth component related to the nonrelativistic quantum effect and part longitudinal component corresponding to a relativistic effect. We further study the mechanical properties (the linear momentum and angular momentum) of the electromagnetic field. We find that the linear momentum of the vortex electromagnetic field are weakly dependent on the spin state of the electron vortex beam, while its angular momentum is remarkably dependent on the spin state. The topological charge of the electron vortex beam has strong influence on the linear and angular momentums, i.e., the linear momentum is reduced with the increasing topological charge, while the angular momentum linearly increases with the topological charge. The electromagnetic field and its linear/angular momentum can be controlled by the opening angle and kinetic energy.

The electromagnetic field of the REVB can be used to explore the relativistic quantum effect, since the difference between electromagnetic fields in spin-up and spin-down states is purely related to spin–orbit coupling effect in the REVB. Moreover, the electromagnetic and mechanical properties of the REVB could be useful for studying the interaction between the REVB and materials.