This paper explores the self-modulation of the axial phase of \(q\) -Gaussian laser beams propagating through relativistic plasma channels. We utilize moment theory to examine the nonlinear interaction between the laser beam and the plasma, concentrating on the evolution of the axial phase. Our findings demonstrate a intricate relationship among the laser’s initial parameters, the plasma density profile, and relativistic effects, culminating in the axial phase modulation of the laser beam. This modulation has significant implications for applications such as laser-driven particle accelerators and high-harmonic generation.
