Electron vortex studies have proliferated in the last few years, with many examples of their
production, measurement and some examples of application.
However, these studies focus on the more standard cylindrically-symmetric ideal electron vortex
beam. We have recently noted an abundance of exotic vortex behaviours, outwith cylindrical
symmetry [1]. Vortex-antivortex loops were formed and annihilated at many different positions
within the beam, with varying sizes. Other studies have shown how a vortex core can wend its way
through a crystal lattice, with the vortex always remaining conserved [2].
In the sister field of optical vortices, it has recently been discovered that it is possible, with
simultaneous manipulation of holography and careful limiting apertures, to produce ‘knots’ with the
vortex cores [3]. These knots are not knots as known to the layman, but specific forms of multiply
interlinked loops. These knots are unusual, and robust topological structures, and take the form of
knotted dark threads within the beam [4].
Here, we investigate the feasibility of producing these exotic structures in the rather more confined
space of a modern TEM, with finite aperture positioning and strict paraxial limitations. Adjusting
the phase hologram technique of Leach et al [3], we have produced a design for a TEM phase mask
(see figure 1 and 2). Such a design is more detailed and technologically challenging than those
produced so far in the electron vortex research field [5], but may just be within reach to enable the
study of these unusual topological electron structures for the first time.

References
[1] Clark, L., et al. “Symmetry-constrained electron vortex propagation.” Under review, arXiv preprint
arXiv:1603.00687 (2016).
[2] Lubk, Axel, et al. “Topological analysis of paraxially scattered electron vortex beams.” Physical Review A 87.3
(2013): 033834.
[3] Leach, J., et al. “Vortex knots in light.” New Journal of Physics 7.1 (2005): 55.
[4] Leach, Jonathan, et al. “Laser beams: knotted threads of darkness.” Nature 432.7014 (2004): 165.
[5] Shiloh, Roy, et al. “Sculpturing the electron wave function using nanoscale phase masks.” Ultramicroscopy 144
(2014): 26-31.

Acknowledgements
LC and JV acknowledge funding from the European Research Council under the 7th Framework Program (FP7), ERC
Starting Grant No. 278510-VORTEX. JV acknowledges financial support from the European Union under the 7th
Framework Program (FP7) under a contract for an Integrated Infrastructure Initiative (Reference No. 312483
ESTEEM2).

Figures:

Figure 1: Modelled focal plane beam amplitude.

Figure 2: Modelled focal plane beam phase.

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