One-dimensional theory of harmonically seeded free-electron laser

We revisit free-electron laser (FEL) equations in the presence of a planar undulator to investigate effects on longitudinal FEL dynamics upon seeding by a harmonic. For this study, we adopt slowly varying phase approximation, collective variables, and Klimontovich description of particle beams to explore the effects of harmonic seeding on low-gain saturation, low-gain amplification, and exponential growth for on-axis radiation emission. We also investigate the effects of frequency detuning and nonzero energy spread on power gain and growth rate for low-gain and high-gain FELs, respectively.