Rigorous Treatment of the Electrostatic Immersion Lens Whose Axial Potential Distribution Is Given by: φ(z)=φ0e K arctan z

Abstract

Investigations have shown that the electrostatic immersion lens, given by the axial potential distribution φ(z)=φ0 exp(K arctan z), is the simplest electrostatic lens for which Newton's image equations are satisfied throughout the field. The general solution of the paraxial ray differential equation is derived. Based on this solution and the fact that Newton's image equations are satisfied, exact expressions are derived for all important optical quantities such as the focal lengths, the location of the four cardinal points, magnification, the object-image relation, and the spherical and chromatic aberration for any object position. The optimum positions of an object are determined so that the chromatic and spherical aberrations are reduced to a minimum. All quantities are represented graphically. A plot of the equipotential lines in space is given for a special value of the voltage ratio.