Beam shaping - the ability to engineer the phase and the amplitude of massive and massless particles - has long interested scientists working on communication, imaging and the foundations of quantum mechanics. In light optics, the shaping of electromagnetic waves (photons) can be achieved using techniques that include, but are not limited to, direct manipulation of the beam source (as in X-ray Free Electron Lasers (XFELs) and Synchrotrons), deformable mirrors, spatial light modulators, mode converters and holograms. The recent introduction of holographic masks for electrons provides new possibilities for electron beam shaping. Their fabrication has been made possible by advances in micrometric and nanometric device production using lithography and focused ion beam patterning. This article provides a tutorial on the generation, production and analysis of synthetic holograms for transmission electron microscopy. It begins with an introduction to synthetic holograms, outlining why they are useful for beam shaping to study material properties. It then focuses on the fabrication of the required devices from theoretical and experimental perspectives, with examples taken from both simulations and experimental results. Applications of synthetic electron holograms as aberration correctors, electron vortex generators and spatial mode sorters are then presented.

This project has received funding from the European Union's Horizon 2020 research and innovation programme (Grant No. 766970, project "Q-SORT"; Grant No. 856538, project "3D MAGiC"; Grant No. 823717, project "ESTEEM3"), from the Deutsche Forschungsgemeinschaft (Project-ID 405553726 – TRR 270), from the DARPA TEE program (Grant MIPR# HR0011831554) and from an Ontario Early Researcher Award (ERA) and a Canada Research Chair (CRC).