Quantum free-electron laser: A fluid model

Abstract

Free-Electron Lasers (FELs) are today a very important area of research. Such devices can generate short pulses of high-power coherent radiation in wavelengths that are unreachable to conventional molecular lasers, such as X-Rays, and are based on the radiation emitted by a relativistic electron beam that performs a waving movement induced by an alternating electromagnetic field. They can be described by classical or quantum models. Classical models are simpler than the last, but they are valid only if the one-photon momentum recoil is not greater than the beam momentum spread, where the FEL operates in a new Quantum regime, and quantum models should be used. It is the case for high-energy fotons and low-energy electron beams. In this work we present a hydrodynamical model which incorporates quantum effects. Starting from Poisson equation and a Schrödinger-like equation deduced from the total relativistic energy of the electron under the action of a ponderomotive potential V associated with the combined wiggle and radiation fields, we obtain, by performing a Madelung transformation to the electron wave function, a set of fluid equations (continuity and momentum) to the beam dynamics, where a Bohm potential accumulates the quantum information. By coupling the wave equation, under the SVEA hypothesis, we get a set of nonlinear PDE system which describes the quantum-FEL as a three-wave interaction phenomena, where the amplified radiation is seen as Compton or Raman backscattered radiation. Our model is simpler than previous quantum models, and can be used in laser amplification theory as well.