Generalization of the Matsumoto-Tonomura approximation for the phase shift within an open aperture

As shown by Matsumoto and Tonomura, the phase shift imposed on an electron beam by an electrostatic phase plate is constant for all (straight) electron trajectories passing through a circular aperture, provided that (1) the electric field goes to zero at distances far above and below the aperture and (2) the value of the phase shift at the boundary (i.e. perimeter of the aperture) is constant [5]. We now point out that the result can be valid for any shape of the hole in the aperture, and, furthermore, it requires only that the electric field is equal and opposite at large distances above and below the aperture, respectively. We also point out that the conditions of validity of the Matsumoto-Tonomura approximation constrain the phase shift across the open aperture to a quadratic algebraic form when the phase shift is not constant around the perimeter. Finally, it follows that the projection approximation for calculating the phase shift must fail for strong phase shifts of higher than quadratic form. These extensions of the original result of Matsumoto and Tonomura give further insight to the analysis of charging phenomena observed with apertures that are designed to produce contrast in in-focus images of weak phase objects.