Coupling between two resonant waves in a waveguide free-electron laser

The coupling between two resonant branches of a waveguide free-electron laser is studied. It is shown that the induced electron energy spread, which is produced along with the bunching by the radiation, strongly influences this coupling. In a low-gain oscillator, the growth of a lower frequency branch results in large induced energy spread, which stops the development of the upper frequency branch. On the other hand, in a high-gain amplifier, a strong signal can be obtained at high frequency by injecting a signal at low frequency that bunches the electron beam without large induced energy spread. In this way, the contradiction about the nonlinear interaction of two resonant waves in a waveguide free-electron laser is explained.