Coherent Harmonic Radiation

The electron beam energy required for operation at a given frequency varies inversely as the square root of the harmonic number. For this reason, the mechanisms for the generation of harmonic radiation in free-electron lasers are of interest for many applications in which it is desired to reduce the beam energy requirement in order to generate a given wavelength. Harmonic generation in free-electron lasers occurs by both linear and nonlinear mechanisms. The principal difficulty with linear harmonic generation, however, is that the beam quality requirement associated with the growth rate of the linear instability increases with the harmonic number. In general, the free-electron laser resonance condition at the hth harmonic is ω ≈ (k + hkw)υ||; hence, thermal effects have a significant and deleterious impact on the interaction whenever Δυ||/υ|| ≈ Im k/(Re k + hkw). Nonlinear harmonic generation is driven parasitically off of a high power fundamental and is less sensitive to the electron beam quality than the linear instability. Harmonics are not commonly seeded in free-electron lasers. The growth of harmonic radiation starts by a combination of shot noise on the electron beam and the initiation of harmonic bunching due to the growth of power at the fundamental. This gives rise to a rapid initial growth of the harmonic power which quickly rolls over to the slower growth associated with the linear instability. The linear instability drives harmonic growth until the fundamental reaches a power level necessary for the nonlinear mechanism to take over, after which the harmonic power grows exponentially with a growth rate that scales as the product of the harmonic number and the growth rate of the fundamental. In this regime, the harmonic grows substantially faster than the fundamental until the harmonic saturates. Saturation of the harmonic typically occurs slightly prior to that of the fundamental due to over-bunching of the electrons at the harmonic wavelengths. In this chapter, both linear and nonlinear harmonic generation is discussed in both helical and planar wiggler configurations.