Parametric Coupling of Langmuir Waves in an Inhomogeneous Plasma

Parametric coupling of two Langmuir waves due to a combined action of a small density gradient and an externally applied, spatially homogeneous, high-frequency electric field is investigated. It is shown that the space-dependence of the pump field plays a decisive role in the parametric coupling, the space-dependence being produced due to the oscillation of the electrons relative to the ions by the external field. On the basis of a fluid description of the plasma, a coupled system of partial differential equations, which describe the relevant parametric coupling, are derived to the lowest significant order with respect to the pump field. For the case the perturbed velocity varies as \(exp[i∫ˣk(x')dx']\), a nonlinear dispersion relation is obtained, with the use of which various characteristics of the parametric coupling, such as the frequency shift, frequency locking and instability, are investigated.