The relationship between optical guiding and the relative phase in free-electron lasers

The relative phase in this case is defined as the shift in the wavenumber from the vacuum value integrated over the interaction length. The relative phase is studied from the standpoint of the linear stability analysis in both the high- and low-gain regimes, and the qualitative implications in each of these regimes of the relative phase on the refractive guiding of the signal are identical. Specifically, the relative phase is found to be negative at the low-frequency end of the gain band. The relative phase increases with increasing frequency over this band until it turns positive at a frequency approximately 10% below the frequency of peak gain. Thus optical guiding is indicated over a large portion, but not all, of the gain band. The specific example of interest involves the low-gain regime prior to saturation. In this case, it is shown that the analytic result is in substantial agreement with the calculation of the relative phase