Three-dimensional Stacking of Phase Plates for Advanced Electron Beam Shaping

Tuneable phase plates for free electrons are a highly active area of research. However, their widespread implementation, similar to that of spatial light modulators in light optics, has been hindered by both conceptual and technical challenges. A specific technical challenge involves the need to minimize obstruction of the electron beam by supporting films and electrodes. Here, we describe numerical and analytical mathematical frameworks for three-dimensional stacks of phase plates that can be used to provide near-arbitrary electron beam shaping with minimal obstruction.