Electric Charging of Thin Films Measured Using the Contrast Transfer Function

The phase shift of electron waves due to charging of thin films is investigated using the contrast transfer properties of the microscope. We take two photos, one with film at the back focal plane and the other one without film. The phase difference between the contrast transfer functions of the two photos is evaluated using our theoretical predictions. The theoretical model is based on an analytical solution of the Laplace equation with appropriate boundary conditions. From the resulting electrostatic potential function the phase shift of electron waves is derived in a weak lens approximation. With this method, information about the radius of the electron beam and the magnitude of the electrostatic potential at the thin film is obtained. The excellent agreement between the theoretical model and experimental results is observed.