Analytical analysis and numerical calculation of third- and fifth-order aberrations in Glaser's bell-shaped magnetic lens

Using Hamiltonian representation, the author (J. Ximen) has developed canonical aberration theory, and has performed a generalized integration transformation on Hamiltonian-representation eikonals, which are position-dependent and momentum-independent. Meanwhile, electron optical aberrations have been derived for up to the ninth-order approximation in rotationally symmetric magnetic and electrostatic lenses. In the present article, this powerful technique of integration transformation on eikonals has been applied to Glaser's bell shaped magnetic lens. Consequently, third- and fifth-order geometric aberrations have been analytically derived and numerically calculated in detail. It has been shown that the fifth-order spherical aberration coefficient changes its sign in a range of the lens strength parameter for Glaser's bell-shaped magnetic lens. It would lead to a new break through in theoretical electron optics, and would be a new challenge to O. Scherzer's theorem. It is to be expected that this theoretical research may be useful for estimating variations and effects of higher-order aberrations in relationally symmetric magnetic lenses.