4. Conclusion

Generally, we aimed to review our results about self-focusing of high-intense laser beams through various plasmas such as cold and warm quantum plasmas. We, in addition, have introduced a ramp density profile for plasma as an effective factor in improving self-focusing and even ion acceleration. The propagation of cosh-Gaussian intensity profile for high-power and short-wavelength laser was also investigated by means of higher-order paraxial approximation. From these investigations, the effects of ramp density profile and off-axial contribution on enhancing the further focusing of laser beam can be drawn. The results show that inclusion of the quadratic r<sup>4</sup> term in the eikonal function modifies radial profile of the pulse. It can have a direct impact on producing ultra-high laser irradiance over distances much greater than the Rayleigh length which can be used for various laser-plasma applications.
