Influence of the synchrotron radiation on particle dynamics in a rectangular undulator

The synchrotron radiation from an undulating electron beam with a rectangular waveguide is investigated. It is shown analytically and numerically that the radiated energy spectrum may differ significantly from the free space result when the undulator length divided by the Lorentz factor of the electron beam is larger than the transverse size of the waveguide. The undulator radiation is identified with the wakefield in beam instabilities. The concepts of wake function and impedance are introduced to formulate the problem in the same manner as the beam instability problem. It is shown that the obtained impedances satisfy the Panofsky-Wenzel theorem and other properties inevitable for wakefields. The longitudinal wake function includes necessary information about how particles will be accelerated or decelerated by the radiation fields. From this, it is possible to calculate the bunching of particles and eventually explain the stimulated emission process of free-electron lasers