Novel Vortex Generator and Mode Converter for Electron Beams

A mode converter for electron vortex beams is described. Numerical simulations, confirmed by experiment, show that the converter transforms a vortex beam with a topological charge $m=\pm 1$ into beams closely resembling Hermite-Gaussian ${\mathrm{HG}}_{10}$ and ${\mathrm{HG}}_{01}$ modes. The converter can be used as a mode discriminator or filter for electron vortex beams. Combining the converter with a phase plate turns a plane wave into modes with topological charge $m=\pm 1$. This combination serves as a generator of electron vortex beams of high brilliance.

Figure 1

Electron optical arrangement: A vortex fills the aperture in the FFP of a magnetic astigmatic lens. The phase is color coded (rainbow wheel). The astigmatic line foci along the $x$ and $y$ axes are at a mutual distance of $2df$. The vortex is composed of two phase-shifted HG-like components [Eq. (3)], one of which is symbolized as it passes the lens. Also drawn are the Gouy phases for the ${\mathrm{HG}}_{01}$ and the ${\mathrm{HG}}_{10}$ components for the case where the astigmatism equals the Rayleigh range, $df=z_R$. The observation plane is in the BFP midway between the line foci; there, the phase difference between the two HG modes is $\pi /2$.

Figure 2

Experimental mode conversion. Upper row: vortex with topological charge $m=-1$ after passage of a tunable lens. Left: astigmatism $df=0$, yielding a focused vortex. Middle: astigmatism $df=220\mathrm{nm}$. Right: simulation for 200 kV. Bottom row: same for $m=1$. Squares have a side length of 5 nm. Intensities in all figures are in false color.

Figure 3

Vortex with topological charge $m=1$ after passage of a tunable lens with astigmatism $df=700\mathrm{nm}$. Left: experiment. Right: simulation for 200 kV. Scale as in Fig. 2.

Figure 4

Geometry for a vortex generator. (a) A Hilbert plate in the FFP imposes a phase shift of $\pi$ on one half of the incident plane wave. This makes a rotated ${\mathrm{HG}}_{10}$-like beam, here symbolized as it passes the lens. The Gouy phase shift of the astigmatic lens creates an ${\mathrm{LG}}_{10}$-line beam in the BFP. (b) Shadow image of the phase plate. (c) Experimental $p$-orbital-like intensity in the BFP obtained with the phase plate and no astigmatism.

Figure 5

Ideal output of the vortex generator. Left: intensity and cophasal lines increasing from 0 to $2\pi$ with the phase singularity in the center, characteristic of a topological charge $m=1$. Right: same with superimposed circular quantum mechanical current density. Scale as in Fig. 2.

Figure 6

Left: experimental vortex generator with nominal $df$ as in Fig. 5. Right: optimized simulation (20% absorption, $df=500\mathrm{nm}$, defocus 400 nm). Despite astigmatic distortions, the central phase singularity is well visible. Scale as in Fig. 2.