Stability of a large amplitude plasma wave to oscillating two-stream instability

The stability of a long wavelength large amplitude plasma wave, generated in a laser wakefield accelerator at moderately relativistic laser intensity, to oscillating two-stream instability has been examined. In the limit when the oscillatory velocity of electrons due to the plasma wave, V^→ ₀, exceeds the electron thermal speed, the short wavelength plasma wave turns out to be ω² = ω_p ² + |k^→.V^→ ₀|² / 2. In the four wave parametric process, involving the pump plasma wave, a short wavelength low frequency quasimode and two short wavelength plasma wave sidebands, the pump and the sidebands exert a ponderomotive force on the electrons driving a low frequency quasimode. The electron density perturbation associated with this mode couples with the pump driven electron oscillatory velocity to produce nonlinear currents driving the sidebands. We find that this process has no growth when the ion motion is ignored. However, with the inclusion of ion motion the parametric instability is important on the time scale of an ion plasma period.