Third-order transfer matrices calculated for an electrostatic toroidal sector condenser including fringing-field effects

The third-order transfer matrices are calculated for an electrostatic toroidal sector condenser using a rigorously conserved matrix method that implies the conservation of the beam phase volume at each step in the calculations. The transfer matrices (matrizants) obtained, include the fringing-field effect due to the stray fields. In the case of a rectangular distribution of the field components along the optical axis, the analytical expressions for all aberration coefficients, including the dispersion ones, are derived accurate to the third-order terms. In simulations of real fields with the stray field width other than zero, a smooth distribution of the field components is used for which similar aberration coefficients were calculated by means of the conserved numerical method . It has been found that for a smooth model, as the stray field width tends to zero, the aberration coefficients approach the corresponding aberration values in the rectangular model.