Abstract
A Free Electron Laser (FEL) operating in the quantum regime can provide a compact and monochromatic Xโray source. Here we review the basic principles of a highโgain quantum FEL starting from noise, with special emphasis on the selfโamplified spontaneous emission (SASE) mode of operation. In the first part of the paper, a condition for the neglect of the fermionic character of the electrons is derived and the full quantum theory of the Nโparticle and singleโradiationโmode FEL Hamiltonian is presented. Quantum effects such as cooperative gain, discrete spectrum and line narrowing are described, both in the multiโparticle and in the second quantization formalism. In the second part, propagation effects (i.e. slippage) are described and the main features of the quantum SASE regime are discussed. The broad and spiky radiation spectrum observed in classical SASE reduces in the quantum regime to a series of narrow lines, associated with sequential transitions between adjacent momentum states. A simple interpretation of the discrete nature of the spectrum and of the linewidth of the single spike observed in the quantum regime is presented.