Simplified computation of third-rank image aberrations of electron-optical systems with curved axis

Using the classical perturbation theory only the image aberrations of second rank of an electron-optical system with a curved axis are easy to obtain. Analytical integral expressions of these second-rank aberrations were already published in German by Plies and Typke (Z. Naturforsch. 33a (1978) 1361, 35a (1980) 566). They employed the eikonal method in the case of arbitrary electrostatic and stationary magnetic fields without space charge and currents in the beam region. Their results are repeated briefly in English and a method is discussed of how a magnetic round lens field can be considered in the case of a system with a single-section symmetry, i.e., an orthogonal system with no torsion of the curved optic axis. Then the important second-rank fundamental trajectories are introduced which are directly related to the integral expressions already given in Plies and Typke (1978). Then the secondary aberrations (obtained by the second step of the successive approximation method) and the complexity of the third-rank aberrations are discussed. After that two new methods are proposed for a simplified calculation of the third-rank image aberrations. Both proposals use the second-rank fundamental trajectories and the second proposed method additionally uses the spatial potential and field achieved by using a three-dimensional field solver. Since these two simplified computation methods are still rather complicated an additional method based on ray tracing is discussed. For this purpose a trajectory equation with respect to the curved optic axis and well suited for numerical integration with high accuracy is derived. This offers the possibility of exact numerical ray tracing of selected and representative axial and field rays. After subtracting from such a ray the known first- and second-rank ray with the same trajectory parameters (aperture, off-axial coordinate, energy deviation) one obtains the complete residual aberration.