Aberrations

The main task of electron optics concerns the design of systems, which possess distinct imaging or beam-guiding properties. Therefore, we must solve an inverse problem by finding the geometry of the electrodes and pole pieces and the strengths of the currents and voltages, which produce the electromagnetic fields required for refracting the electrons appropriately. Owing to this difficulty, entire numerical methods are not suited for finding optimum systems composed of numerous different elements, such as solenoids and mul-tipoles. However, numerical methods are indispensable for the final design of the system after its outlay has been roughly determined by means of the analytical calculations and symmetry considerations employing the paraxial approximation for the trajectories and aberration integrals. Computer programs are nowadays available for calculating numerically field distributions throughout a given system very accurately by means of high-order finite element or finite difference procedures. Computing the particle trajectories by direct ray tracing [123, 124] yields directly the overall aberrations. The main disadvantages of this method are that the individual Seidel-order aberration terms cannot be determined with reliable accuracy and that it does not provide information how to suppress or eliminate appropriately the performance-limiting aberrations. To find such means, we must calculate analytically the integral expressions for the aberration integrals and investigate the structure of these integrals giving information how to nullify them. To determine the performance of systems corrected for the primary aberrations, one must calculate the next higher aberrations. Unfortunately, the number of aberrations and the complexity of the aberration coefficients increase drastically with increasing order. To avoid errors in the time-consuming analytical calculations, special algebraic computer programs have been developed for automatically deriving analytical expressions for the aberration coefficients. These programs are particularly useful for calculating the higher-order aberration coefficients of multielement systems, such as the SMART microscope [125].