Generation of Nondiffracting Electron Bessel Beams

Almost 30 years ago, Durnin discovered that an optical beam with a transverse intensity profile in the form of a Bessel function of the first order is immune to the effects of diffraction. Unlike most laser beams, which spread upon propagation, the transverse distribution of these Bessel beams remains constant. Electrons also obey a wave equation (the Schrödinger equation), and therefore Bessel beams also exist for electron waves. We generate an electron Bessel beam by diffracting electrons from a nanoscale phase hologram. The hologram imposes a conical phase structure on the electron wave-packet spectrum, thus transforming it into a conical superposition of infinite plane waves, that is, a Bessel beam. We verify experimentally that these beams can propagate for 0.6 m without measurable spreading and can also reconstruct their intensity distributions after being partially obstructed by an obstacle. Finally, we show by numerical calculations that the performance of an electron microscope can be increased dramatically through use of these beams.

Popular Summary

Diffraction, the spreading of a wave upon propagation, is in the soul of waves. Having waves that are immune to diffraction would then seem impossible. Almost 30 years ago, however, a diffraction-free optical beam was discovered. The key was to have a transverse intensity profile in the form of a well-known mathematical function, the so-called Bessel function of the first order. As electrons have their wave nature, too, Bessel beams of electrons are expected to exist. In this experimental paper, we report the generation of such electron beams, for the first time, by putting electrons through a nanoscale pure phase hologram, known as a “kinoform” in optics.

The kinoform we use is a thin film of silicon nitride milled to have nanoscale grooves, in other words, different thicknesses across. Electrons going through this transparent thin film experience different paths of propagation and acquire different phases as a result. Certain phase modulations lead to Bessel beams. The key to generating electron Bessel beams of different orders, therefore, lies in mapping the phase modulation needed for each type of Bessel beam into the nanoscale patterns and then fabricating the kinoform. We have succeeded in generating Bessel beams whose shapes remain unchanged for more than 60 cm, using electrons from a standard transmission electron microscope.

Electron Bessel beams will certainly find their applications in electron microscopy. Our experiment should enable the development of novel electron microscopes with better resolution and image quality.

Figure 1

(a) Scanning-electron-microscope image (in a tilted direction) of the manufactured kinoform with a zoom-in image of the central region shown in (c). The kinoform is made of milled silicon nitride ($\mathrm{Si}{}_3\mathrm{N}{}_4$), which is transparent to the electron beam emitted in the transmission electron microscope. Different depths modify the local atomic potential; thus, electrons see different effective paths (analogous to “optical paths”) at grooves. (b) The thickness profile of the silicon nitride calculated by energy-loss mapping of the close-up region in (c). The thickness is expressed in units of $L$, the inelastic electron mean-free path in $\mathrm{Si}{}_3\mathrm{N}{}_4$. The estimated value of $L$ is about 130 nm. The profile is largely nonrectangular and closer to a sinusoidal form. This depth results in an effective phase change of $1.6\pi$ for electrons with a de Broglie wavelength of ${\lambda }_{\mathrm{dB}}=2.5\mathrm{pm}$.

Figure 2

(a) Far-field distribution of the diffracted electrons by the kinoform shown in Fig. 1. Only the zeroth and first orders are shown. In the first order of diffraction, the Bessel beam forms a ring in the far field. (b) Simulated transverse distribution of the electron Bessel beam of the zeroth kind at the first order of diffraction. (c) Measured transverse distribution of the electron Bessel beam of the zeroth order generated by the kinoform in Fig. 1. The number of symmetric multirings present defines the imprinted conical phase front to the electron wave packet. (d) Radial distribution of the detected electrons (solid blue curve) for the electron Bessel wave function. The plot legend shows the simulated Bessel beams with different convergences $\alpha$ on the kinoform plane.

Figure 3

Observed image of the probability-density distribution of electrons for a Bessel beam of the zeroth order in different propagation planes. Note that the beam diameter remains mainly constant as a function of propagation distance. The small variation in the radius (the distance from the central to the first ring) occurs because we are using an approximate Bessel beam. This variation has already been predicted and observed in the optical domain [9]. The right inset shows the raw images of electron distributions upon propagation at different plans.

Figure 4

Experimentally observed (left panels) and simulated (right panels) images of the higher-order electron Bessel beams with topological charges of $n=1$ and $n=2$.

Figure 5

(a) Intensity distribution of the electron Bessel beam of the zeroth order upon propagation after being obstructed by an asymmetrical obstacle. (b),(c) The distribution of the electrons after the obstacle and at the plane of 0.1 m. As seen in (c), the beam recovers its “Bessel-like" intensity distribution, revealing the self-healing feature.

Figure 6

Simulated HAADF-STEM image for two particles lying on a 60°-inclined graphene sheet. (a) The two spheres are displaced in depth by $17$ nm. Notice that graphene is not visible in the simulation because of its low contrast. (b) The simulated image of the particles with a conventional STEM probe with $\alpha =15\mathrm{mrad}$. (c) The simulated image of the particles with an electron Bessel probe in a HAADF-STEM. The upper and lower insets in (b) and (c) are Fourier transforms of the images (diffractograms) of the upper and lower particles, respectively. (d) Numerically simulated images (left column) and diffractograms (right column) of a particle located at different planes of a focused aperture-limited probe and a zeroth-order Bessel-beam probe. As is seen, the atomic resolution of the particle is clearly visible for a Bessel-beam probe at different positions upon propagation. A conventional STEM probe with convergence of $\alpha =15\mathrm{mrad}$ is assumed.