Symmetry and planar chirality measured with a log-polar transformation in a transmission electron microscope

Transmission electron microscopy can be described in terms of the measurement, in an appropriate quantum basis, of the quantum state of an electron after scattering through a sample. Recently developed methods for measuring the orbital angular momentum (OAM) of an electron beam using an OAM sorter can be used to reveal the n-fold symmetry of a sample. Here, we introduce and demonstrate a new composite planar chirality operator, which is based on rotation and scaling operators. By using synthetic electron holograms, we show that an electron OAM sorter can be used to determine the scaling symmetries and planar chiralities of relevant objects such as biomolecules, for which the technique could provide faster identification or help to achieve more detailed three-dimensional reconstructions by solving upside-down orientation ambiguities. A statistical analysis shows that, even for very low doses, this can be a solid indicator, even without the need for full imaging of the particle.