  Flat-beam transforms (FBTs) provide a technique for controlling the emittance
partitioning between the beam's two transverse dimensions. To date, nearly all
FBT studies have been in regimes where the beam's own space-charge effects can
be ignored, such as in applications with high-brightness electron linacs where
the transform occurs at high, relativistic, energies. Additionally, FBTs may
provide a revolutionary path to high power generation at high frequencies in
vacuum electron devices where the beam emittance is currently becoming a
limiting factor, which is the focus of this paper. Electron beams in vacuum
electron devices operate both at a much lower energy and a much higher current
than in accelerators and the beam's space charge forces can no longer be
ignored. Here we analyze the effects of space-charge in FBTs and show there are
both linear and nonlinear forces and effects. The linear effects can be
compensated by retuning the FBT and by adding additional quadrupole elements.
The nonlinear effects lead to an ultimate dilution of the lower recovered
emittance and will lead to an eventual power limitation for high-frequency
traveling-wave tubes and other vacuum electron devices.
