Time-dependent perturbation theory for electron mirrors

Time-dependent perturbation theory was originally studied for analyzing the aberration properties of electrostatic mirrors. The describing parameter of the theory is time and the perturbation is constructed based on the velocity of electrons, instead of slope of trajectories with respect to the optic axis, to avoid divergence at the reflection. This article gives a detailed review and further investigation on the time-dependent theory of the system of electron mirrors, composed of both rotationally symmetric electrostatic and magnetic fields. We review everything from the concept of the time-dependent theory to the procedure of perturbation method. Then, the second-rank aberration coefficients, the third-order geometrical aberration coefficients, and the aberration coefficients induced by variation in voltages and currents, including the off-axis aberrations, are derived. We give the time-dependent small-angle deflection aberration theory up to the third order, for a system composed not only of rationally symmetric fields but also of small angle deflection fields, such as a few degrees.