Test and verification of temporal aberration theory for electron optical imaging systems by an electrostatic concentric spherical system model

On theory of temporal aberration of dynamic electron optical imaging systems, we have put forward two methods: 'τ-variation method' and 'direct integral method' to calculate the temporal aberration coefficients. The difference between them is that the 'τ-variation method' needs to solve the differential equations for geometrical temporal aberration coefficients of second order, while the 'direct integral method' needs only to carry out the integral calculations. The aim of the present paper is to test and verify the correctness of the two methods using an electrostatic concentric spherical system model. Results show that these two methods can obtain identical solutions and the solutions of temporal aberration coefficients of dynamic electron optical imaging systems are the same with the analytical solutions. Thus it can be concluded that these two methods are equivalent and correct, but the 'direct integral method' is more convenient for computation and it could be suggested to use in the practical design.