Electron mirrors

Abstract

The conventional theory of electron motion in static fields considers the z-coordinate of a charged particle as the independent variable measured along the optic axis. Usually, the optic axis is chosen to coincide with the central trajectory of a bundle of rays regardless of whether this trajectory is straight or curved. We have replaced derivatives with respect to the time in the path equation and in the Lagrange function by derivatives with respect to the z-coordinate utilizing the conservation of energy. As a result, the lateral position coordinates x = x(z) and y = y(z) of the electron are functions of the z-coordinate instead of the time t. This approach is valid as long as the axial velocity component does not reverse its direction of flight. If, in addition, the motion is confined to the region near the optic axis, the slope components x ′(z) and y′(z) remain sufficiently small. However, large ray gradients do occur in the vicinity of turning points at which the axial direction of flight of the particle changes its sign or near the emitting surfaces of cathodes. Examples for systems with turning points are electron mirrors, ion traps, and the magnetic bottle. Because the components x′(z) and y′(z) of the ray gradient diverge at the turning point, we must describe in this case the position coordinates x, y, and z as functions of an appropriate independent variable T, which must not necessarily be the time t. Large ray gradients also occur in cathode lenses. Although electron mirrors have been studied at the very beginning of electron optics, they were not considered as promising elements for correcting aberrations. The reason for this belief may stem from early work by Ram-berg who found that the dimension of a mirror must be unrealistically small for correcting aberrations. Later studies showed that this pessimistic view does not hold true. An electrostatic mirror had been utilized in electron microscopy only as a reflection element for an imaging energy filter. The long-time negligence of studying thoroughly the correction properties of mirrors may be attributable to the difficulties associated with the violation of the standard conditions for the paraxial trajectories in the region of the turning point where the ray gradient diverges. Revived interest in electron mirrors originated from the work of Rempfer and Mauck in the context to correct the spherical and chromatic aberration of a low-voltage photoemission electron microscope (PEEM) by means of a hyperbolic two-electrode mirror. However, for adjusting the chromatic and spherical aberration of the mirror for a fixed focal length, we must increase the number of electrodes from two to four. Unlike a light-optical mirror, where the reflection occurs at the physical surface, the electron mirror represents a "soft" mirror, which allows the electrons to penetrate into the inhomogeneous refracting medium formed by the electrostatic potential. The depth of penetration depends on the energy and the direction of the electron in front of the mirror. We can conceive the total reflection as the sum of consecutive refractions on a continuous set of electrostatic equipotentials. © 2009 Springer-Verlag Berlin Heidelberg.