A multifrequency model is developed for a Compton free-electron laser and is solved to investigate the continuous-electron-beam limit. Both numerical simulations and perturbation expansions allow us to exhibit a mechanism leading to the broadening of the spectrum. Then, the nonlinear asymptotic behavior of the spectrum is analyzed.PACS numbers: 42.55.Tb, 52.35.Mw One of the most challenging questions in freeelectron-laser (FEL) physics is related to the nonlinear evolution of the spectrum, including the sideband instability 1_3 and mode competition (MC). 4 New nonlinear effects can alter the further evolution of a FEL spectrum, especially in the asymptotic regime which corresponds to a large number of round trips for an oscillator. Of course, optical elements can select a single frequency, 5 but we are concerned here with the natural physical evolution of the spectrum in the absence of frequency discrimination. Moreover, we focus on the limit of a continuous electron beam, which means that finite-pulse issues, such as the natural Fourier spread or temporal overlap effects, are disregarded.In the continuous-beam limit (CBL), our numerical simulations exhibit the following essential property: Any spectral pattern, for instance, two frequencies with the same amplitude, can duplicate itself. We confirm this numerical prediction by expanding the multifrequency model with two different perturbation theories. Then, the nonlinear evolution of the spectrum is investigated by means of computer simulations. We observe some expected results, such as the transition between narrow and broad spectra depending upon the electronic current or the cavity losses. A more challenging numerical prediction in the saturation regime is a scaling law between the spectral width and the extracted efficiency.To investigate the FEL spectral dynamics, we use a specific model for the CBL which is appropriate for electronic pulses much longer than the slippage distance. 6 To work out this model, we expand the laser field A L as the product of rapid phases and slowly varying envelopes S"iz):
S"(z)where k n =(\+n/N)kL is a wave number close to the central mode /cz, = 27rAz,. Each complex laser field S" J satisfies a paraxial equation:
n/N)k H w z G n -^Q--a w e 2Kwhere a w and k w are the normalized amplitude and the wave number of the magnetic field. The longitudinal electronic distribution g(z,y/,y) satisfies the associated Vlasov equations:[3 z + v8 vr +r(8 y -l/y)]g(z,^y)-0,
v-k w -(k L /2Y 2 Hl+ ja£) ,
F= J a w \m 2 &me(3)
dy/J dyg(z,y/,y)mc=p e ,
2TTNJOwhere p e is the electron number per unit of volume. The electron phase space is characterized x by the resonant phase y/^ikL + k^z -co L t and the kinetic energy y normalized to the electronic mass m. The variable y/ depends on the central frequency cot^ckL, but Eqs. (l)-(3) are invariant under the choice of co L when we consider the limit of the continuous Fourier expansion in , -i(\+n/N)y 2y(2)Eq. (1). Indeed, the phase (\+n/N)y/-(n/N)k w z, in Eqs. (2) and (3), is precisely the resonant ...