The interaction between high-brilliance electron beams and counter-propagating
laser pulses produces X rays via Thomson back-scattering. If the laser
source is long and intense enough, the electrons of the beam can bunch
and a regime of collective effects can establish. In this case of dominating
collective effects, the FEL instability can develop and the system
behaves like a free-electron laser based on an optical undulator. Coherent
X-rays can be irradiated, with a bandwidth very much thinner than that
of the corresponding incoherent emission. The emittance of the electron
beam and the distribution of the laser energy are the principal mechanisms
that limit the growth of the X-ray signal. In this work we analyse
with a 3-D code the transverse effects in the emission produced by
a relativistic electron beam when it is under the action of an optical
laser pulse and the X-ray spectra obtained. The scalings typical of
the optical wiggler, characterized by very short gain lengths and overall
time durations of the process make possible considerable emission of
radiation.