The high-gain free-electron laser equation: exact, perturbative, and approximated solutions

Abstract

We use the Laplace-transform technique for a general analysis of the integral equation that describes the Free-Electron Laser (FEL) small-signal evolution. We include both pulse propagation effects and inhomogeneous broadening contributions. The solutions are obtained in exact form and by using perturbative expansions. It is also shown that these solutions can be approximated with elementary functions and that most of the FEL high-gain evolution can be handled with almost negligible computational effort. The analysis is extended to includee-beam pre-bunching effects, and a generalization of the Madey theorem is proved.