Three-dimensional analysis of harmonic generation in self-amplified spontaneous emission

In a high-gain free-electron laser, strong bunching at the fundamental wavelength can drive substantial harmonic bunching and sizable power levels at the harmonic frequencies. In this paper, we investigate the three-dimensional evolution of the harmonic fields based on the coupled Maxwell-Vlasov equations that take into account the nonlinear harmonic interaction. Each harmonic field is the sum of a self-amplified term and a term driven by the nonlinear harmonic interaction. In the exponential gain regime, the growth rate of the dominant nonlinear term is much faster than that of the self-amplified harmonic field. As a result, the gain length and the transverse profile of the first few harmonics are completely determined by those of the fundamental. A percentage of the fundamental power level is found at the third harmonic frequency right before saturation for the current self-amplified spontaneous emission projects