Comments on Aberration Correction in Symmetric Imaging Energy Filters

Imaging energy filters are becoming an essential part of high-quality electron microscopes. They enhance the contrast, allow element specific imaging, and elemental decomposition by spectroscopy of inelastically scattered electrons. All filters which are commercially available are not completely corrected to second order; however, some use symmetric arrangements for the cancellation of the most destructive second order effects. However, completely corrected symmetric arrangements have been tested already. For the construction of these systems it is important to know what consequences the symmetry of the optical arrangement has: which aberrations cancel due to symmetry, which aberrations are interrelated and vanish simultaneously when implementing multipole correctors, which higher-order aberrations remain after cancellation of the leading-order aberrations. These questions can in principle be answered by the Eikonal method as well as by the transfer map method. Here we demonstrate that a combination of both methods answers these question in a very simplified fashion. This simplicity allows to draw some novel conclusions.