Large X-ray free-electron laser (FEL) facilities have been built around the world. In this talk, I first review the three-dimensional (3D) high-gain FEL theory which was successfully developed in the 1980s and 1990s. Van Kampen’s normal modes, originally used to describe plasma oscillations, are employed here to solve for the 3D Maxwell-Vlasov equations. This theory is then extended to include slowly varying beam and undulator parameters via the WKB approximation. Finally, the theoretical results are applied to both tapered and reverse-tapered undulators.
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