Properties of the odd harmonics of the radiation from SASE FEL with a planar undulator

Recent theoretical and experimental studies have shown that Self-Amplified Spontaneous Emission Free Electron Laser (SASE FEL) with a planar undulator holds a potential for generation of relatively strong coherent radiation at the third harmonic of the fundamental frequency. Here we present detailed study of the nonlinear harmonic generation in the SASE FEL obtained with time-dependent FEL simulation code. Using similarity techniques we present universal dependencies for temporal, spectral, and statistical properties of the odd harmonics of the radiation from SASE FEL. In particular, we derived universal formulae for radiation power of the odd harmonics at saturation. It was also found that coherence time at saturation falls inversely proportional to harmonic number, and relative spectrum bandwidth remains constant with harmonic number.