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An analysis of the parametric instability which forms the basis
for operation of a free electron laser with a longitudinal
electrostatic wiggler is presented. A theory is developed that
allows for an arbitrary axial magnetic field and arbitrary ratio
of the electron beam radius \emph{a} to the waveguide inner radius
\emph{R} without imposing the electrostatic approximation on the
space-charge wave. Dispersion characteristic of the eigenmodes of
a magnetized electron beam inside a waveguide specially
$\emph{EH}_{01}$ electromagneric and $\emph{SC}_{01}$ space-charge
mode have been studied. A formula for the spatial growth rate of
parametrically excited eigenmodes is derived. variations of the
growth rate, efficiency, radiation wavelength, and required pump
wavelength as functions of the ratio $\emph{a}/\emph{R}$ are
studied numerically. It is shown that excitation of the upper
branch or lower branch of an $\emph{EH}_{01}$ waveguide mode can
be achieved with suitable chosen values for the system parameters,
and a fully electromagnetic analysis can produce modest changes on
the operative quantities, as compared to the imposition of
electrostatic approximation on the space-charge wave.
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