Nonlinear harmonic generation of coherent amplification and self-amplified spontaneous emission

Nonlinear harmonic generation in a uniform planar undulator is analyzed using the three-dimensional Maxwell–Klimontovich equations that include both even and odd harmonic emissions. After a certain stage of exponential growth, the dominant nonlinear harmonic interaction is caused by strong bunching at the fundamental. As a result, gain length, transverse profile, and temporal structure of these harmonic radiations are eventually determined by those of the fundamental. Transversely coherent third-harmonic radiation power is found to approach one percent of the fundamental power level for current high-gain FEL projects, while the power of the second-harmonic radiation is less but still significant for relatively low-energy FEL experiments.