| Summary: | Approved for public release, distribution unlimited === The dynamics of the Free Electron Laser are governed by Maxwell's equations which causes many highly nonlinear regimes to exist in Free Electron Laser Physics. This thesis will examine two such areas and develop simple models to describe the highly dynamic and rich behavior two of these regimes. In the strong-field, high current regime, the Free Electron Laser driving current can be modeled by a single macroparticle representing the trapped electrons. When the trapped electrons act collectively as a macroparticle representing the trapped electrons. When the trapped electrons act collectively as a macroparticle, solutions which include synchrotron oscillations can be found for the self-consistent pendulum and wave equations. In an FEL oscillator with low single-pass gain, the evolution of the optical wave can lead to sideband development. This phenomenon is studied by applying Maxwell's equations to an oscillator with two optical modes and deriving a two-mode wave and pendulum equation. The two-mode wave and pendulum equations are implemented numerically on computers so that the onset of the sideband can be explored.
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