One-dimensional free-electron laser equations without the slowly varying envelope approximation

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.