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A linear theory for a free electron laser with a one-dimensional helical
wiggler an axial magnetic field in the collective regime is presented. The
configuration consists of a cylindrical waveguide with arbitrary ratio of
electron beam radius $a$ to waveguide inner radius $R$. Parametric decay of
the wiggler pump wave, in the beam frame, into a space-charge wave and an
electric-magnetic (EH) waveguide mode is analyzed in three dimensions. A
nonlinear wave equation for the three-wave interaction is derived and employed
to obtain a formula for the spatial growth rate of the excited eigenmodes. It
was found that the relativistic treatment of the electron oscillations in the
wiggler field destroys the cyclotron resonance which appears in the
nonrelativistic case. Nevertheless, appreciable amplification was found.
Numerical analysis is conducted to study the growth rate, radiation
wavelength, and required relativistic factor as functions of axial magnetic
field $B_0$ and radius ratio $a/R$. The suitable value for $a/R$ was found to
be around 0.65 for our choice of parameters.
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