Generation of Nondiffracting Electron Bessel Beams

Abstract

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.

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Plain Language 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.