Aberration integrals for the low-voltage foil corrector

Foil aberration correctors make use of a transparent foil to correct for the spherical and chromatic aberration of electron lenses. The low-voltage foil corrector is a novel type, in which the electrons are retarded to almost 0 eV at the foil. For designing a system with this corrector, analytical expressions for the aberrations are required. Such aberration integrals allow combinations of the corrector with other lenses to be calculated, even if the fields overlap. Also many different configurations can be calculated within a reasonable time. Most existing aberration integrals for foil correctors are not suitable because of an integration by parts that had been applied to them, such to obtain expressions without the fourth derivative of the axial potential. This simplification is not allowed when the electron energy approaches zero and is not necessary nowadays because the fourth derivative can be calculated accurately. Additionally, for a characterization of the corrector independent of the probe forming system, the aberration coefficients are less suitable and the geometrical and chromatic slope aberrations must be derived. In this paper, integrals have been obtained that describe correctly the spherical and chromatic aberration correction for the low voltage foil corrector. A program for the numerical calculation of the electrostatic potential can use these integrals to calculate the Cs, Cc and the slope aberrations. The results show that the low-voltage foil corrector can correct for both the spherical and chromatic aberration simultaneously.