Measuring an electron beam's orbital angular momentum spectrum

Quantum complementarity states that particles, e.g. electrons, can exhibit wave-like properties such as diffraction and interference upon propagation. \textit{Electron waves} defined by a helical wavefront are referred to as twisted electrons~\cite{uchida:10,verbeeck:10,mcmorran:11}. These electrons are also characterised by a quantized and unbounded magnetic dipole moment parallel to their propagation direction, as they possess a net charge of $-|e|$~\cite{bliokh:07}. When interacting with magnetic materials, the wavefunctions of twisted electrons are inherently modified~\cite{lloyd:12b,schattschneider:14a,asenjo:14}. Such variations therefore motivate the need to analyze electron wavefunctions, especially their wavefronts, in order to obtain information regarding the material's structure~\cite{harris:15}. Here, we propose, design, and demonstrate the performance of a device for measuring an electron's azimuthal wavefunction, i.e. its orbital angular momentum (OAM) content. Our device consists of nanoscale holograms designed to introduce astigmatism onto the electron wavefunctions and spatially separate its phase components. We sort pure and superposition OAM states of electrons ranging within OAM values of $-10$ and $10$. We employ the device to analyze the OAM spectrum of electrons having been affected by a micron-scale magnetic dipole, thus establishing that, with a midfield optical configuration, our sorter can be an instrument for nano-scale magnetic spectroscopy.

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