The phase constancy of electron waves traveling through Boersch's electrostatic phase plate

We draw attention to the observation that an electrostatic phase plate, originally proposed by Boersch in 1947, introduces a uniform phase shift. The device consists of a ring electrode, shielded by earthed electrodes. The phase shift of the electron wave traveling through the interior of the ring is uniform and constant independent of the radius of the ring for a very low applied voltage. This can be shown by considering the Laplace equation, which is satisfied by the electrostatic scalar potential subjected to the boundary conditions. The phase shift, which is proportional to the potential integrated along the optic axis, is constant inside the ring as if there were a thin film of uniform thickness. With such a device in the back focal plane of the objective lens, a phase contrast electron microscope could be constructed. The device could also be used as a constant phase shifter for electron interferometry.