Instabilities for a relativistic electron beam interacting with a laser irradiated plasma

The effects of a radiation field (RF) on the unstable modes developed in relativistic electron beam--plasma interaction are investigated assuming that $\omega_{0} >\omega_{p}$, where $\omega_{0}$ is the frequency of the RF and $\omega_{p}$ is the plasma frequency. These unstable modes are parametrically coupled to each other due to the RF and are a mix between two--stream and parametric instabilities. The dispersion equations are derived by the linearization of the kinetic equations for a beam--plasma system as well as the Maxwell equations. In order to highlight the effect of the radiation field we present a comparison of our analytical and numerical results obtained for nonzero RF with those for vanishing RF. Assuming that the drift velocity $\mathbf{u}_{b}$ of the beam is parallel to the wave vector $\mathbf{k}$ of the excitations two particular transversal and parallel configurations of the polarization vector $\mathbf{E}_{0}$ of the RF with respect to $\mathbf{k}$ are considered in detail. It is shown that in both geometries resonant and nonresonant couplings between different modes are possible. The largest growth rates are expected at the transversal configuration when $\mathbf{E}_{0}$ is perpendicular to $\mathbf{k}$. In this case it is demonstrated that in general the spectrum of the unstable modes in $\omega$--$k$ plane is split into two distinct domains with long and short wavelengths, where the unstable modes are mainly sensitive to the beam or the RF parameters, respectively. In parallel configuration, $\mathbf{E}_{0} \parallel \mathbf{k}$, and at short wavelengths the growth rates of the unstable modes are sensitive to both beam and RF parameters remaining insensitive to the RF at long wavelengths.