This article investigates the relativistic self-focusing of circularly polarized Laguerre-Gaussian (LG) laser beam with the mode LG \(^0_1\) under the influence of exponential density ramp in magnetized cold quantum plasma. A static axial magnetic field has been applied externally to the cold quantum plasma, influencing the self-focusing effect. Additionally, a preformed exponential density ramp has also been included to study the impact of its density gradient on the self-focusing effect. From the analysis of the influence of the axial magnetic field and exponential density ramp on the nonlinear propagation of the circularly polarized LG beam in cold quantum plasma, it has been found that an exponentially increasing density ramp induces a fast self-focusing effect on the laser beam. Moreover, the application of an axial magnetic field significantly improves self-focusing, leading to even stronger and faster self-focusing, although this effect slows down at very high field strengths. Similarly, high beam intensities produce fast self-focusing. However, excessively high intensities are observed to hinder self-focusing due to an excessively violent electron motion. The results of this analysis have been illustrated with the help of graphs.
