  I theoretically found eigenmodes and growth rates of relativistic current
filamentation instability in collisional regimes, deriving a generalized
dispersion relation from self-consistent beam-Maxwell equations. For
symmetrically counterstreaming, fully relativistic electron currents, the
collisional coupling between electrons and ions creates the unstable modes of
growing oscillation and wave, which stand out for long-wavelength
perturbations. In the stronger collisional regime, the growing oscillatory mode
tends to be dominant for all wavelengths. In the collisionless limit, those
modes vanish, while maintaining another purely growing mode that exactly
coincides with a standard relativistic Weibel mode. It is also shown that the
effects of electron-electron collisions and thermal spread lower the growth
rate of the relativistic Weibel instability. The present mechanisms of
filamentation dynamics are essential for transport of homogeneous electron beam
produced by the interaction of high power laser pulses with plasma.
