Linear theory of resonance self-modulation of an intense laser pulse in homogeneous plasma and plasma channels

The analytical solutions describing the linear stage of the intense-laser-pulse self-modulation, which results in a strong plasma wakefield excitation, are studied in terms of the paraxial approximation. The attention is focused on phase relations that were ignored in the previous studies. It is shown that the value of the phase velocity of the plasma wake wave differs from the pulse group velocity so that under some specific conditions, the relativistic factor corresponding to the phase velocity can be substantially less than that for the group velocity. This may be important for the particle acceleration in the self-modulated laser wakefield accelerator.