This paper presents a comprehensive study on the axial phase modulation of self-focused elliptical \(q\) -Gaussian laser beams propagating through cubic-quintic nonlinear media. The unique interplay between the beam’s elliptical geometry and its non-ideal Gaussian intensity profile, modulated by the \(q\) parameter, significantly influences both the beam’s self-focusing dynamics and its axial phase evolution. Using a variational approach, we investigate how the beam’s intensity-dependent refractive index variation leads to self-focusing and its axial phase modulation, with special emphasis on the effects of the cubic (Kerr) and quintic nonlinearities. The \(q\) parameter plays a critical role in shaping the beam’s profile and, consequently, the strength of axial phase modulation. Numerical simulations reveal how higher \(q\) values lead to weaker phase shifts and more stable propagation, while lower \(q\) values result in stronger modulation and more dynamic behavior. These findings have important implications for optical communication systems and nonlinear optical devices, where maintaining phase coherence is critical for long-distance signal transmission and beam stability.
