General theory of the magnetic einzel lens based on the axial field distribution model H(z)=H0C∣ z/a ∣m-1/(1+∣ z/a ∣2m)

Abstract

Investigations are described which show that the magnetic einzel lens given by the axial field distribution H(z)=H0C mod z/a mod m-1/(1+ mod z/a mod 2m), where m is the parameter depending on the double-gap pole-piece geometry and the magnetic flux concentration, is the simple magnetic einzel lens whose solutions of paraxial ray paths can be obtained in the form of hypergeometric functions. Based on these solutions, the important focal quantities and third-order aberrations are expressed analytically in terms of the maximum width amax of the field and the lens strength k, mostly depending on the ampere-turns for various magnetic einzel lenses specified by m. The important parameters m and amax and their dependence upon double-gap pole-piece geometry are derived approximately in accordance with the potential theory for a cylindrical magnetic einzel lens. The optimum lenses with the minimum focal length and aberrations, and satisfying the anastigmatic conditions, are described and discussed in terms of practical design.