We study the saturated state of an untapered free electron laser in the Compton regime, arising after exponential amplification of an initial low level of radiation by an initially monoenergetic, unbunched electron beam. The saturated state of the FEL is described by oscillations about an equilibrium state. Using the two invariants of the motion, and certain assumptions motivated by computer simulations, we provide approximate analytic descriptions of the radiation field and electron distribution in the saturation regime. We first consider a one-dimensional approximation, and later extend our approach to treat an electron beam of finite radial extent. Of note is a result on the radiated power in the case of an electron beam with small radius.

During the past few years the physics and technology of charged particle beams on which electron-positron linear colliders in the TeV region, storage rings from synchrotron radiation sources and Free Electron Lasers are based, has seen a remarkable development. The purpose of this series of schools is to address the physics and technology issues of this field, train young people and at the same time provide a forum for discussions on recent advances for scientists active in this field. The subjects chosen for this first course reflect the recent interest in TeV electron positron colliders, the possibility offered by Free Electron Lasers to power them and the developments in the production of high brightness electron beams.

Preliminary concepts -- Synchrotron radiation -- Basic FEL physics -- 1D FEL analysis -- 3D FEL analysis -- Harmonic generation in high-gain FELs -- FEL oscillators and coherent hard X-rays -- Practical considerations and experimental results for high-gain FELs

This dissertation focuses on the study of novel high-gain free-electron laser (FEL) operation schemes with external seed lasers. The technique of manipulating the phase space of the electron beam, which is widely used in novel seeded FEL schemes, is systematically studied. Several novel FEL schemes are proposed for the generation of intense coherent FEL pulses with short wavelength, sub-femtosecond pulse length or multiple carrier frequency properties, which meet the needs of FEL users. Results of experiments are described for the recently proposed FEL schemes such as echo-enabled harmonic generation and cascaded high-gain harmonic generation. New photon/electron beam diagnostic methods are also developed for these experiments and future high-gain FEL facilities.