Quantum theory of the free-electron laser: Large gain, saturation, and photon statistics

Abstract

Starting from the general QED Hamiltonian for the free-electron laser (FEL), a quantum-mechanical, many-particle theory of the FEL in the laboratory frame is presented. When suitable variables are introduced, the Hamiltonian is seen to a good approximation to be formally identical to the nonrelativistic Hamiltonian in the resonant (Bambini-Renieri) frame used in other treatments. The derivation is given for a general multimode laser field, although only the simpler case of single-mode operation is discussed in detail. It is shown how the large-gain evolution equations for the field, in the small-signal regime, may be obtained from the quantum theory. Then, from fourth-order perturbation theory, the change in the first two moments of the photon-number distribution in a single pass through the FEL is computed. Large-gain and saturation terms are obtained, for arbitrary values of the quantum recoil (i.e., both the classical and quantum-mechanical regimes are included). The evolution of the photon statistics over many cavity round trips is discussed. In the small-signal regime, the variance of the photon-number distribution is shown to correspond, to a good approximation, to that of thermal, or ‘‘chaotic,’’ radiation. At saturation, a significant reduction of the fluctuations is expected, but no conclusions can be drawn from the perturbation-theory approach. A numerical calculation of the buildup of the field from vacuum (through spontaneous emission) is presented. For simplicity, a uniform, circularly polarized, static wiggler is assumed in the text; however, the theory may be generalized (along lines shown) to deal with nonuniform, linearly polarized and/or traveling electromagnetic-wave wigglers.