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.