Grating mirror for diffraction of electrons

The ability to imprint a phase pattern onto a coherent electron wave would find many applications in electron optics, in analogy to what is already possible with photons in light optics. Spatially dependent phase manipulation is achieved in transmission electron microscopy by passing the beam through a phase plate. However, in transmission mode this technique suffers from crystal imperfections and electron-matter interaction. If instead the electron wave is reflected of a spatially modulated potential, these difficulties can be circumvented. To demonstrate this principle, we consider here a periodic topological mirror structure that results in a sinusoidal plane of reflection for the incident electron. The reflection of the electron then takes place just above the physical mirror surface. Such “electron grating mirror” is expected to diffract the incident wave upon reflection by the introduced path length difference. The mirror can then be used as an electron beam splitter and coupler, analogous to semitransparent mirrors used in light optics. This enables for instance a lossless Mach-Zehnder interferometer for electrons. A numerical model that solves the Schrödinger equation for this system is obtained to enable a quantitative description of the grating mirror. The results show that the obtained diffraction order intensities behave like squared Bessel function of their respective order, and thus for instance the results show how an increase in grating pitch reduces the sensitivity to energy spread in the incident electron beam. Additionally, we show how the use of the WKB approximation enables faster calculations in the case of general patterns.

Figure 1

(a) Parameters describing the grating mirror and coordinate system. (b) Details of region I, defining the pitch and amplitude of the pattern. Equipotential lines are shown schematically. (c) Axial and side potential corresponding to the details of region I.

Figure 2

(a) WKB approximation with integral paths parallel to the optical axis of the system. The arrows indicate (schematically) the starting point for the integration. (b) Obtained phase accumulation as a function of pattern potential (legend items with respect to $-2000$ V beam energy) and transverse coordinate. The offset is subtracted such that only the modulated part of the phase is shown. (c) The effect of grating pitch on the relative phase accumulation.

Figure 3

(a) Intensity of the four most dominant orders as function of pattern potential. Solid lines (direct solution) and dashed lines (WKB approximation) of the Schrödinger equation are shown. (b) The amplitude $\left(\delta x_R\right)$ of the classical mirror plane $U_{\text{class.}}=\mathcal{E}/e$ [V] and corresponding diffraction order intensities are shown.

Figure 4

Diffraction order intensities for a fixed pitch of (a) 500 nm and (b) 1000 nm. An increase in pitch results in a wider bias potential window for complete attenuation of the central order beam.

Figure 5

(a) Airy function of first kind squared is plotted. The classical turning point coordinate is indicated. Notice that the reflected electron has finite possibility of reaching behind this point. (b) 1D potential ramp starting in free space. The gray line indicates the energy that corresponds to the classical turning point of the electron. (c) Comparison between analytical and numerically obtained phase angles for the potential bump at $x_0$, obtained at $x_s$.

Figure 6

Schematic representation of the grid, showing the single pitch, boundary conditions, and the labeling of index points.