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.

Hardly any other discovery of the nineteenth century did have such an impact on science and technology as Wilhelm Conrad Röntgen’s seminal find of the X-rays. X-ray tubes soon made their way as excellent instruments for numerous applications in medicine, biology, materials science and testing, chemistry and public security. Developing new radiation sources with higher brilliance and much extended spectral range resulted in stunning developments like the electron synchrotron and electron storage ring and the freeelectron laser. This handbook highlights these developments in fifty chapters. The reader is given not only an inside view of exciting science areas but also of design concepts for the most advanced light sources. The theory of synchrotron radiation and of the freeelectron laser, design examples and the technology basis are presented. The handbook presents advanced concepts like seeding and harmonic generation, the booming field of Terahertz radiation sources and upcoming brilliant light sources driven by laser-plasma accelerators. The applications of the most advanced light sources and the advent of nanobeams and fully coherent x-rays allow experiments from which scientists in the past could not even dream. Examples are the diffraction with nanometer resolution, imaging with a full 3D reconstruction of the object from a diffraction pattern, measuring the disorder in liquids with high spatial and temporal resolution. The 20th century was dedicated to the development and improvement of synchrotron light sources with an ever ongoing increase of brilliance. With ultrahigh brilliance sources, the 21st century will be the century of x-ray lasers and their applications. Thus, we are already close to the dream of condensed matter and biophysics: imaging single (macro)molecules and measuring their dynamics on the femtosecond timescale to produce movies with atomic resolution.

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