We investigate the linear stability of the system formed by an electron beam and its return plasma current within a general framework, namely, for any orientation of the wave vector k with respect to the beam and without any a priori assumption on the orientation of the electric field with respect to k . We apply this formalism to three configurations: cold beam and cold plasma, cold beam and hot plasma, and cold relativistic beam and hot plasma. We proceed to the identification and systematic study of the two branches of the electromagnetic dispersion relation. One pertains to Weibel-like beam modes with transverse electric proper waves. The other one refers to electric proper waves belonging to the plane formed by k and the beam, it divides between Weibel-like beam modes and a branch sweeping from longitudinal two-stream modes to purely transverse filamentation modes. For this latter branch, we thoroughly investigate the intermediate regime between two-stream and filamentation instabilities for arbitrary wave vectors. When some plasma temperature is allowed for, the system exhibits a critical angle at which waves are unstable for every k . Besides, in the relativistic regime, the most unstable mode on this branch is reached for an oblique wave vector. This study is especially relevant to the fast ignition scenario as its generality could help clarify some confusing linear issues of present concern. This is a prerequisite towards more sophisticated nonlinear treatments.

The linear instability that induces a relativistic electron beam passing through a return plasma current to filament transversely is often related to some filamentation mode with wave vector normal to the beam or confused with Weibel modes. We show that these modes may not be relevant in this matter and identify the most unstable mode on the two-stream/filamentation branch as the main trigger for filamentation. This sets both the characteristic transverse and longitudinal filamentation scales in the non-resistive initial stage.