Theory of double resonance parametric excitation in plasmas

Parametric instabilities in a plasma driven by a long wavelength electric field with two “pump” frequencies ω1 and ω2 which lie near the resonant frequency for Langmuir oscillations, their difference Δ = ω1 − ω2 being chosen close to a low frequency resonance, linear or nonlinear, at Ω − i Γ are studied. A general dispersion relation in terms of linear susceptibilities, χ, is derived by retaining, on a selective basis, terms of fourth order in the pump amplitudes. Illustrative calculations are carried out using resonant fluid approximations for the χ. The most interesting cases occur when Δ = Ω or Δ = 2Ω. A lowering of the net power threshold for instability is found in both cases, when the linear damping rate of the electronic wave is large compared with Ω. In addition, a coupling between the “decay” and “oscillating two-stream” instabilities occurs when Δ = Ω, the threshold for exciting the latter with the ω2 pump being arbitrarily small when the ω1 pump amplitude is near the usual decay instability threshold.