  An analysis of steady-state electron trajectories by simultaneous solution of
the equation of motion and the dispersion relation (DR) for electromagnetic
wave wiggler in free-electron laser (FEL) with axial magnetic field is
presented. The effects of the normalized axial magnetic field and the
normalized angular frequency of electromagnetic wave wiggler on axial and
transverse velocity for group I and II orbits are investigated. A fluid model
is used to obtain the DR for electrostatic wave and the right and the left
circularly polarized electromagnetic waves with all relativistic effects
included. This dispersion relation is solved numerically to investigation the
unstable coupling among all waves. When the transverse velocity is small, only
the FEL instability is found. In group II orbits, with large transverse
velocity, new coupling between the negative and positive energy space charge
waves as well as between the left circular wave and positive energy space
charge wave are found.
