Electrostatic lenses have traditionally been designed by analyzing and combining different electrode configurations. Computational complexity typically limits such systems to a few geometrically simple elements, where the component interactions are neglected and not exploited to combat the various aberrations. Recently, Szilagyi and Szep have demonstrated that an axially symmetric column of circular plates, with the electrode potentials optimized for focusing, can surpass the typical conventional designs by many times in performance. Following the footsteps of pioneers like Burfoot and Hawkes, we partition the plates in order to transcend the limitations set by Scherzer's theorem on the chromatic and spherical aberrations of axially symmetric structures. Two algorithms, one based upon integral asymptotics and one upon the Levinson algorithm. for Toeplitz matrix inversion, are developed to complement the charge-density method in analyzing the new column structures. Various optimization schemes are combined to avoid shallow minima at a reasonable computational cost. With each plate partitioned into four sectors, we show that the interactions between the monopole and the quadrupole components can increase the output current density by more than 400% over the axially symmetric structure. By adjusting the sector potentials, we can realize systems capable of both focusing and deflecting the beam. In comparison to some existing designs, our systems excel in both performance and compactness, sometimes by many hundred percents. We then further partition the plates to generate the "octupole" deflectors and correctors. We show that the "octupole" deflectors can drastically slow down the beam degradation with deflection distance and that the correctors can further increase the output current density by more than 300%. Finally, we apply linear system theories to the study of the first-order properties of optical systems with different symmetries. We showed, without resorting to perturbational mathematics, that the higher multipole components, with more than 2 folds of rotational symmetry, can induce no first-order influences. We also find that there are systems with other symmetries that can replace an axially symmetric structure in the first-order approximation. This latter study is the beginning of our investigation for the optimum system geometry.
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