Collective effects in the induced emission of an electron beam in an undulator with an axial magnetic field

A theory is derived for the collective (plasma) effects in a free-electron laser consisting of a combination of longitudinal and undulator magnetic fields for two different configurations. In the linear theory, the process can reduce to the stimulated scattering of an electromagnetic wave by the intense relativistic beam passing through the undulator. The limiting cases of Raman scattering by a plasma wave and Compton scattering by free electrons are analyzed. A general analytic expression is derived for the instability growth rate. This expression applies to both field configurations. In the case of an axisymmetric field, the growth rate is slightly lower than for a helical field. In the presence of space charge, the longitudinal magnetic field alters the nature of the undulator motion of the electrons and influences their longitudinal motion. If the transverse velocity of the electrons is sufficiently high, a space-charge wave becomes unstable. This instability imposes an upper limit on the extent to which the gain can be raised by raising the amplitude of the undulator field.