Self-focusing of the intense laser pulses as well as production of the well-collimated electron beams in the moderately underdense plasma of
n_{e}/n_{c}
≥ 10% 7 ×
10^{17}
W/cm
^{2}
. Progresses of a relativistic self-focusing in the plasma of a gaussian density profile were observed by an ultra-fast optical probing. Even if the large gas-target of the quasi-uniform density was used, the self-focusing was terminated after traveling approximately 1.2 mm by the abrupt transition of the propagation mode from the smooth channeling to the complex filamentation, which correlated with the emission of MeV-electron beams. High-energy electron beams have the power-law spectrum of
E^{-3.5}
reaching the maximum energy of 2 MeV. The maximum number of the electrons was (1.3 ± 0.6) ×
10^{11}
/sr/shot. Model calculations suggest that the most plausible mechanism of the production of high-energy electron beam in the moderately underdense plasma is that the phase of the electron oscillation in filaments of the laser beam randomly changes many times.
