An Analysis of Nonlinear Harmonic Generation in High Gain Free-Electron Laser

The odd harmonic evolution in the free-electron laser (FEL) employing planar undulator is studied analytically. The evolution equation of the harmonic is deduced, the dependence relation of the harmonics interactions is revealed, and both the linear and the nonlinear harmonic evolutions can be described. The analytical formula of the nonlinear harmonic power in self-amplified spontaneous emission FELs is given. For the saturation power of the harmonic, an estimation formula is given. The harmonic evolution equation for a high-gain harmonic generation (HGHG) FEL is also deduced, and in the fundamental case, it gives the optical field equation of HGHG, which describes the optical field evolution from the coherent enhancement to the exponential gain until the saturation occurrence.