  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.
