Superoscillating electron wave functions with subdiffraction spots

Almost one and a half centuries ago, Abbe [Arch. Mikrosk. Anat. 9, 413 (1873)] and shortly after Lord Rayleigh [Philos. Mag. Ser. 5 8, 261 (1879)] showed that, when an optical lens is illuminated by a plane wave, a diffraction-limited spot with radius $0.61\lambda /sin\alpha$ is obtained, where $\lambda$ is the wavelength and $\alpha$ is the semiangle of the beam's convergence cone. However, spots with much smaller features can be obtained at the focal plane when the lens is illuminated by an appropriately structured beam. Whereas this concept is known for light beams, here, we show how to realize it for a massive-particle wave function, namely, a free electron. We experimentally demonstrate an electron central spot of radius 106 pm, which is more than two times smaller than the diffraction limit of the experimental setup used. In addition, we demonstrate that this central spot can be structured by adding orbital angular momentum to it. The resulting superoscillating vortex beam has a smaller dark core with respect to a regular vortex beam. This family of electron beams having hot spots with arbitrarily small features and tailored structures could be useful for studying electron-matter interactions with subatomic resolution.

Figure 1

Schematic description of superoscillating electron wave-function generation. The desired wave function is created in the $+1$ and $-1$ diffraction orders.

Figure 2

Scanning electron micrographs of binary amplitude masks [(a), (d), (g), (j)], shown alongside simulations [(b), (e), (h), (k)] and experimental results [(c), (f), (i), (l)] for a circular aperture and superoscillation masks of diameter 10 µm for $\alpha =3.7\mathrm{mrad}$ (rows 1 and 2), and for diameter 20 µm and $\alpha =5.2\mathrm{mrad}$ (rows 3 and 4). The measurements of superoscillating electron beams (f) and (l) exhibit central hot spots of radius 114 and 106 pm, respectively, compared to diffraction-limit's Airy disk radii of 334 and 235 pm, respectively. (m) Three central orders of the diffraction pattern of the mask in (d).

Figure 3

(a) Simulation of a superoscillating electron beam for an aberration-corrected microscope in both the probe and the imaging parts with $\alpha =25\mathrm{mrad}$ and $r_{\pi }/r_{\max }=0.64$. The radius of the central hot spot, 20 pm, is only ten times the electron wavelength. The dashed black line marks the diffraction-limited Airy disk. (b) Central lobe radius plotted as a function of $r_{\pi }/r_{\max }$. Theoretically, an arbitrarily small electron hot spot can be achieved, as long as factors such as spatial coherence and signal-to-noise ratio are addressed.

Figure 4

The effect of spatial and temporal incoherence on a 10λ superoscillating hot spot for $\alpha =25\mathrm{mrad}$. (a) The superoscillating probe assuming perfect coherence. (b) Gun shape after demagnification, having a full width at half maximum of 7 pm. (c) The convolution of (a) and (b). (d) Cross sections of (a) (solid) and (c) (dashed). (e) The effect of chromatic aberration for $\alpha =25\mathrm{mrad}, \mathrm{Cc}=1.35\mathrm{mm},E=300\mathrm{keV}$, for $\mathrm{\Delta }E=0\mathrm{eV}$ (solid blue) and $\mathrm{\Delta }E=0.5\mathrm{eV}$ (dashed red). Incoherent summation was carried out over an ensemble of electrons having a Gaussian energy distribution with ${\sigma }_E=\mathrm{\Delta }E/\sqrt{8ln\left(2\right)}=0.21\mathrm{eV}$.

Figure 5

Hot-spot peak intensity divided by sidelobe peak intensity, shown as a function of hot-spot radius. The calculations were performed for a convergence semiangle of 25 mrad and $\lambda =2\mathrm{pm}$. Each point in this plot corresponds to a different value of $r_{\pi }/r_{\max }$.

Figure 6

Superoscillating electron vortex beams. Vortex and superoscillating $r_{\pi }/r_{\max }=0.68$ vortex beams with $\mathrm{OAM}=1$ (rows 1 and 2, respectively) and $\mathrm{OAM}=3$ (rows 3 and 4). (m) Schematic description of equiphase surfaces of $\mathrm{OAM}=3$ superoscillatory electron vortex (row 4), with π-phase-shifted inner and outer rings.