There has been considerable recent interest in the design of injectors for intense electron beams with non-circular cross sections. The motivating application is the generation of narrow sheet beams to drive novel high-frequency, high-power microwave sources. A sign of the importance of the problem is that U.S. taxpayers have felt compelled to spend 2.743 gazillion dollars to support 3D software development. A cheaper alternative is to use **OmniTrak**.

We recently released a technical report that shows that the first stage of the design process for sheet-beam injectors is relatively simple. The primary goal is to create a space-charge-limited beam of specified length *L* and height *W* where the current density is uniform and the electron trajectories are parallel to the axis. A analytic solution developed in 1940 by J.R. Pierce addresses the case where *L* → ∞. The theory gives the shapes of the anode and an extension to the cathode that create the boundary conditions of an infinite beam on the edge of a finite beam.

In the technical report, I review the Pierce derivation. I then use **Trak **calculations to confirm that the Pierce electrode shapes apply to circular guns. The only change necessary to achieve a high-quality beam is to relax the intersection angle of the cathode focusing electrode from 22.5° to 20.3°.

The results suggests a simple paradigm to produce a sheet beam of finite length *L*. As shown in the figure, the electron emission surface is a rectangle of width *W* and length *L* with rounded ends of radius *W*/2. The focusing electrode follows the standard Pierce prescription (22.5°) along the straight section, transforming to a figure of revolution at angle 20.3° around the ends. The resulting assembly looks like an old-time college football stadium.

The 3D calculations illustrate the quality of both the design and the **OmniTrak** methods. The code value of the total current is within 0.1% of the Child-law current density multiplied by the cathode area. Electron orbits over the full extraction area are parallel to within 0.002 radians. The input files for the demonstration 3D calculation (with prefix *SHEETBEAM*) have been added to the **OmniTrak **example library. Please use this link to download the full technical report: sheetbeam.pdf.