First, I think you are running up against an essential conservation law in beam physics, so you will not be able to generate a dramatically better focus by designing a "better lens."

Essentially, the following product is a constant (unless you chop off some part of the beam)

A * omega * sqrt(KE),

where A = the area of the beam, omega = the solid angle of the 2d divergence/convergence of the beam, and KE is the kinetic energy

I can't say much about your system, because i need to know something about the geometry, but I can "solve" for the geometrical factor at a specific energy. At 5 keV, the ratios of the solid angles should be

omega_src/omega_dst = (50 um / 15 um)^2 * sqrt( .5 eV / 5000 eV ) = 9

It looks like you are collecting the beam close to the source and trying to focus it at a distant target using a single lens? The shrinking of the focal size is 100% due to the acceleration of the beam, which is fighting against the geometrical factor.

The better design would be to design the system using a "zoom lens" rather than an Einzel, and to move the target much closer to the lens. This would greatly improve the geometric factor.

In case you *must* use einzel lenses, try a two lens relay. where the first einzel lens to collect the beam near the source, and the einzel second to focus the beam near the target. To meet the "normal" periodic focusing conditions, the beam should come to a focus somewhere between the two einzel lenses. The complication here is that it takes much less voltage difference to focus the electrons at lower energy than at higher energy.

Electron microscopes often use magnetic focusing, which has some properties that are better suited to manipulating high-ish energy beams.