Modelo hidrodinâmico para Quantum Free-electron lasers

Abstract

Free-Electron Laser is today a very important área of research. These devices can generate high power of coherent radiation by releasing the energy of a relativistic electron beam into electromagnetic field energy. It works by injecting the relativistic beam into a devide called undulator (or wiggler), that consists of a magnetistatic field or, alternatively, a counterpropagating laser pulse (electromagnetic, or optical, unduladors). The wiggler forces the beam, that moves in a straight line at the beginning og interaction, to execute a wave movement and consequently it emits electromagnetic radiation. FELs are able to operate in a wide range of the electromagnetic spectrum, in wave-lengths are unreachable to convencional lasers. Such frequency flexibility is due to the fact that the coherent radiation wavelength, [lambda]r, is mainly determined by the beam energy and by the wiggler period, wich satisfies the approximate resonance condition [lambda]r = [lambda]w/4[gama]z^2, to optical wigglers, or [lambda]r = [lambda]w/2[gama]z^2, to magnetostatic wigglers, where [gama]z is the normalized longitudinal energy of the electron beam and [lambda]w is the wiggler period. It is well known the FELs theory was originally conceived in the framework of Quantum Mechanics by Madey and co-workers. In their notable work they calculated the radiation gain by using the Weizsäcker-Williams methid. Subsequently, it was shown that classical models could describe FELs equally well, if the one foton momentum recoil in not greater than the beam momentum spread, hence quantum effects can be neglected. Otherwise, Quantum-Mechanics effects can not neglected, and quantum models are necessary. In the last years it has been happening a great interest in the experimental realization of FELs based on optical on wigglers. The fact of operating in wavelenghs of the order of visible light or less, gives to a laser pulse the advantage of working as a very-small-period wiggler. Such device also has the advantage of working with a much lower energetic-beam, in the range of a few MeVs, instead of tens of GeVs by using a magnetostatic undulator, to FELs working in the range of X-Raysm for example. Then, laser wigglers might to make feasible the construction of low-dimension coherent radiation emission devices in the X-ray and Gama ray band (table-top X-Ray Free-Electron Lasers) based on Stimulaed Compton Backscattering. But it was showed that quantum effects might not be neglected in this domain of low-energy electron beam and high foton energy, and the FEL get into to work in a Quantum Regime. In this work we will present a hydrodynamical model to study Free-Electron Lasers that includes quantum effects. Starting from the electron total relativistic energy equation that includes a ponderomotive potencial term, we will deduce a equation to the electron eave-function evolution under the Slow-Vary Envelope Approximation (SVEA) hypotesis, which has the same shape as the Schrodinger equation. By using the Madelung transformation, we will deduce a sustem of fluid equations to the electron beam dynamic. The coupling of the electron beam fluid equations to the field ones will allow us to deduce a cubic dispersion relation that includes space-charge effects to the FEL instability. Subsequently we will show a ser of nonlinear quantum plasma fluid equations of coupled-mode type to quantumplasmas, which is able to describe a relativistic electron beam interacting with a stimulated radiation inside an optical wiggler under the assumption of the Slow-Varying Envelope Approximation. Numerical results will be showed at the end.