Calculations and measurements for a magnetic quadrupole lens with a large aperture and a bell-shaped field distribution

Abstract

A magnetic quadrupole lens of an aperture radius of 12.5 cm and a physical over-all length of 12 cm is described. The fringe field is linearized by means of rounded-off pole pieces and parallel end plates. The distribution of the transverse field components along the axis is nearly bell-shaped. The field strength necessary to focus parallel incoming electrons of an energy of 20 keV has been measured. The effect of the field distribution on the motion of electrons through the quadrupole lens has been calculated in first order theory using a hard-edge and a bell-shaped model. The bell-shaped model gives 4 and 3% larger fields than the hard-edge model, respectively, for two different focal lengths. There is a good agreement between measurement and bell-shaped model calculations for paraxial rays. For beams crossing the axis at large angles the fringe field linearization reduces the third order aberration by a factor of 4 compared to a pure bell-shaped quadrupole field. Moreover, the focusing fields were obtained by a stepwise numerical integration of the trajectory equations using the measured field distribution. The agreement with the measurements is satisfactory. Finally the stigmatic focusing of parallel beams by a quadrupole doublet consisting of two of these lenses has been considered. The bell-shaped model yields better results than the hard-edge model as is shown by numerically calculated trajectories using the measured field distributions and the field gradients obtained by the models.