title: Analytic solution for a quartic electron mirror.
abstract: A converging electron mirror can be used to compensate for spherical and chromatic aberrations in an electron microscope. This paper presents an analytical solution to a diode (two-electrode) electrostatic mirror including the next term beyond the known hyperbolic shape. The latter is a solution of the Laplace equation to second order in the variables perpendicular to and along the mirror's radius (z(2)-r(2)/2) to which we add a quartic term (kλz(4)). The analytical solution is found in terms of Jacobi cosine-amplitude functions. We find that a mirror less concave than the hyperbolic profile is more sensitive to changes in mirror voltages and the contrary holds for the mirror more concave than the hyperbolic profile.
authors.affiliation: Portland State University, Department of Physics, PO Box 751, Portland, OR 97207, United States. Electronic address: straton@pdx.edu.
authors.forename: Jack C
authors.initials: JC
authors.lastname: Straton
keywords: Aberration correction; Electron microscope; Electron mirror; Jacobi cosine-amplitude functions; Microscopy