Up frequency conversion in a two resonant wave free electron laser

In a free electron laser there are two different resonant frequencies, with positive and negative slippage. In free space these frequencies are widely different, but they move together when a waveguide is used to control the slippage. We investigate the waveguide operation in the high gain regime and we describe the nonlinear interaction between the two resonant waves by a set of partial differential equations. We show that by injecting a signal at the lower resonant frequency, strong bunching and signal at the upper frequency can be obtained. This up frequency conversion process is maximum when the frequency ratio is an integer number, which can be in principle arbitrarily large.