Three waves theory in virtual cathode oscillator (VCO)

The linear and nonlinear dispersion relations for the microwave in VCO have been obtained when the complex motion of electron is simplified to the motion of a single charge sheet with time-dependent charge number. It is found in the linear dispersion relation that the field with only one mode in VCO can´t grow. Meanwhile, it is shown that there are two modes at least growing simultaneously in VCO. In the case of only two growing waves, the pumping wave which supplies energy is produced by the oscillation of the charge in virtual cathode (VC). This is similar to the matching condition of three waves interaction. The frequency of the pumping wave is the interval between the two growing waves.