As espoused by Montgomery¹ and later by Durnin et al.², the Helmholtz equation admits solutions which are invariant with respect to free space propagation, known as non-diffractive waves. Perhaps the simplest such solution is a scalar wave function where the amplitude of the wave function is in the form of a Bessel function of the first kind and is independent of the propagation distance z. Thus the intensity of the beam will not disperse laterally during propagation. In addition to this propagation-invariant property, Bessel beams can also reform after scattering from an object; the so-called “self-healing” property.  With such unique features, electron Bessel beams has potential applications in transmission electron microscopy (TEM). For example, in electron tomography, which requires electron probe intensity distributions to be invariant with sample depth; or in particle trapping, in analogy with optical tweezers.

How can one efficiently generate an electron Bessel beam for TEM? A simple method is to use a thin annular aperture to obtain a Bessel beam in the far-field diffraction plane³. This is based on the fact that the Hankel transform of a one dimensional delta function is a Bessel function. However, this method has extremely low efficiency since most of the beam is blocked by the aperture. A different approach has been taken by Grillo et al.⁴, who used a nanoscale-manufactured kinoform as a binary electron phase grating to create substantial depth of focus, utilising the intrinsic diffraction free property to demonstrate the suitability for electron tomography.

In this work, we present a simple, alternative approach to generate an electron Bessel beam, which is analogous to the axicon lens used in light optics.

In light optics, an axicon lens has circular symmetry and a thickness which varies linearly along the radial direction, so that a linear phase shift is imparted to the incident light along the radial direction (see fig.1), (in contrast to the usual quadratic phase shift from a thin lens). The incident beam is tilted due to the linear phase ramp, thereby forming a cone structure which generates the Bessel beam in the near field.

Can an axicon lens be fabricated for electrons? One could potentially use a conical nanostructure. The electron phase shift would be linearly modified by the varied thickness for a homogenous material of constant mean inner potential, for a sufficiently thin nanostructure. Alternatively, to reduce the size, the phase plate could be modulated with periodically varying thickness but this would then be even more challenging to fabricate.

In the present work, we introduce a natural and generic approach to efficiently create electron Bessel beams using magnetic vortex structures. For these ubiquitous dipole moment configurations, the magnetic vector potential imparts a linear phase shift upon incident electron waves, for specimens of constant thickness, such as thin films, resulting in a conical wavefront deformation centred about the vortex core. Thus magnetic vortex structures naturally behave as effective axicon lenses in the absence of electrostatic potential variations.

We prove this experimentally here, in a TEM using thin films of FeCo based nanocrystalline alloys (Fig 2). An electron Bessel beam was observed in the near field using Lorentz microscopy. The propagation-invariant property was verified using a through focal series. The coherent exit wave was recorded using off-axis electron holography, and the maximal non-diffractive distance was measured.  Utilising reciprocity to provide a further cross-check, a narrow annulus donut beam was also observed in the far field diffraction plane.

1. W. D. Montgomery, J. Opt. Soc. Am. 58(8), 1112–1124 (1968).

2. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).

3. K. Saitoh, K. Hirakawa, H. Nambu, N. Tanaka, and M. Uchida, J. Phys. Soc. Jpn. 85, 043501 (2016).

4. V. Grillo, E. Karimi, G.C. Gazzadi, S. Frabboni, M.R. Dennis and R.W. Boyd, Phys. Rev X, 4(1), p.011013 (2014).

Figures:

Fig.1: Comparison between a convex lens (a) and an axicon lens (c). Both lenses have circular symmetry but the convex lens introduces a quadratic phase shift (b) while the axicon lens introduces a linear phase shift along the radial direction (d). A Bessel beam will be formed in the near field.

Fig.2: Generation of electron Bessel beam using a generic magnetic vortex structure. (a) The electron holography phase image shows magnetic vortex structure in a FeCo based soft magnetic alloy (core indicated by the white arrow). (b) Electron Bessel beam generated at different propagation distances. (c) Simulated electron Bessel beam formed with a magnetic vortex structure.

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