Oscillator evolution in free-electron lasers

In a free-electron laser oscillator, both the amplitude and frequency of the optical field are free to evolve. We use a Lagrangian formalism to describe the dynamics of the interacting electron beam and optical wave in a single pass. The character of the evolution over many passes is controlled by the design of the undulating magnetic field in the free-electron laser interaction region. Several laser magnet designs are presented under the same formalism and compared: the undulator, the tapered undulator, the optical klystron, and transverse gradient (''gain-expansion'') undulator. For each of these designs we numerically calculate the gain as a function of optical amplitude and frequency. These ''gain surfaces'' are used to infer a variety of properties of oscillator evolution and clearly demonstrate the relative merits of each magnet design.