Generation of very high-frequency waves by up-conversion in a plasma-loaded free-electron laser

A free-electron laser loaded with a plasma is able to resonate at two different frequencies. The two waves are copropagating, one with positive slippage while the other has negative slippage. We deduce the nonlinear partial differential equations describing the interaction between the two waves in the slowly-varying-envelope approximation. By injecting a signal at the low frequency, a strong signal is produced at the harmonically related high frequency, with a lethargy time much smaller than that of the spontaneous vacuum emission. This effect could be applied in the generation of very short wavelength radiation, up to the range of hard x rays.