Up-frequency conversion in a two-resonant-wave high-gain free-electron-laser amplifier

A free-electron laser is able to resonate at two different frequencies, both in free space and in a waveguide. The two waves have positive and negative slippage. We describe the nonlinear interaction between the two waves by a set of partial differential equations which in free space do not require the slowly varying envelope approximation (SVEA). In a waveguide a less restrictive SVEA is applied to each wave. By injecting a small signal at the low frequency, a strong signal and bunching are produced at the high frequency. This effect suggests a new method of generating short wavelength radiation.