Analytic solution for a quartic electron mirror

• Published in: Ultramicroscopy Vol. 148; pp. 168 - 179
• Main Author: Straton, Jack C.
• Format: Journal Article
• Language: English
• Published: Netherlands Elsevier B.V 01.01.2015
• Subjects: Aberration; Aberration correction; Diodes; Electric potential; Electron microscope; Electron microscopes; Electron mirror; Exact solutions; Jacobi cosine-amplitude functions; Laplace equation; Mathematical analysis; Microscopy; Voltage; Electron microscope; Jacobi cosine-amplitude functions; Aberration correction; Microscopy; Electron mirror
• ISSN: 0304-3991; 1879-2723
• DOI: 10.1016/j.ultramic.2014.09.003

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 (z2−r2/2) to which we add a quartic term (kλz4). 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. •We find the analytical solution for electron mirrors whose curvature has z4 dependence added to the usual z2 – r2/2 terms.•The resulting Jacobi cosine-amplitude function reduces to the well-known cosh solution in the limit where the new term is 0.•This quartic term gives a mirror designer additional flexibility for eliminating spherical and chromatic aberrations.•The possibility of using these analytical results to approximately model spherical tetrode mirrors close to axis is noted.