Abstract
A set of one-dimensional equations has been deduced in the time domain from the Maxwell-Lorentz system with the aim of describing the free-electron laser radiation without using the slowly varying envelope approximation (SVEA). These equations are valid even in the case of arbitrarily short electron bunches and of current distributions with ripples on the scale of or shorter than the wavelength. Numerical examples are presented, showing that for long homogeneous bunches the new set of equations gives results in agreement with the SVEA free-electron laser theory and that the use of short or prebunched electron beams leads to a decrease of the emission lethargy. Furthermore, we demonstrate that in all cases in which the backward low frequency wave has negligible effects, these equations can be reduced to a form similar to the usual 1D SVEA equations but with a different definition of the bunching term.
Highlights
The usual equations describing the free-electron laser radiation process were deduced in the framework of the slowly varying envelope approximation (SVEA) [1,2,3,4,5,6]
We demonstrate that, under the assumption that the backward low frequency wave is negligible, these new equations can be reduced to the usual SVEA 1D equations, apart from the fact that the bunching term is computed on an average length smaller than the wavelength, thereby retrieving a model very close to those of Refs. [19,21]
Vs z=LG (Lpulse being the length of the radiation wave packet and LG the gain length), solutions of Eqs. (9)–(12), while the blue curves are the usual SVEA results, obtained by integration of Eq (15), the computational steps being very much shorter in the NOSVEA case, because the spatial and temporal oscillations on the wavelength have to be resolved
Summary
The usual equations describing the free-electron laser radiation process were deduced in the framework of the slowly varying envelope approximation (SVEA) [1,2,3,4,5,6] This procedure requires that all the characteristic lengths L (for instance: the pulse modulation width, the length of the gradients, the dimension of the electron beam Lb, the gain length LG, the cooperation length LC) are much longer than the wavelength of the radiation, L ). Few works [14,15,16,17] reintroduced the backward wave in the model, associating to the particle equations not one, but a couple of radiation equations, written for two wave packets centered, respectively, on two different single resonances and correlated only via the electron dynamics We demonstrate that, under the assumption that the backward low frequency wave is negligible, these new equations can be reduced to the usual SVEA 1D equations, apart from the fact that the bunching term is computed on an average length smaller than the wavelength, thereby retrieving a model very close to those of Refs. [19,21]
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